Electrical Systems for Wave Energy Conversion

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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 Leijon, M., Boström, C., Danielsson, O., Gustafsson, S., Haikonen, K., Langhamer, O., Strömstedt, E., Stålberg, M., Sundberg, J., Svensson, O., Tyrberg, S., and Waters, R., “Wave energy from the North Sea: expe-riences from the Lysekil research site” Surveys in Geophysics, Springer, 29(3):221–240, 2008.

II Leijon, M., Waters, R., Rahm, M., Svensson, O., Boström, C., Strömst-edt, E., Engström, J., Tyrberg, S., Savin, A., Gravråkmo, H., Bernhoff, H., Sundberg, J., Isberg, J., Ågren, O., Danielsson, O., Eriksson, M., Lejerskog, E., Bolund, B., Gustafsson, S., and Thorburn, K., “Catch the wave to electricity: the conversion of wave motions to electricity using a grid-oriented approach” IEEE Power and Energy Magazine, 7(1):50– 54, 2009.

III Waters, R., Rahm, M., Eriksson, M., Svensson, O., Strömstedt, E., Boström, C., Sundberg, J., and Leijon, M., “Ocean wave energy absorption in response to wave frequency and amplitude - offshore experiments on a wave energy converter” Conditionally accepted for publication in IET Renewable Power Generation. Revision submitted in December 2010.

IV Boström, C., Lejerskog, E., Stålberg, M., Thorburn, K., and Leijon, M., “Experimental results of rectification and filtration from an offshore wave energy system” Renewable Energy, 34(5):1381–1387, 2009. V Boström C., Waters, R., Lejerskog, E., Svensson, O., Stålberg, M., and

Leijon, M., “Study of a wave energy converter connected to a nonlinear load” IEEE Journal of Oceanic Engineering, 34(2):123–127, 2009. VI Boström, C., Lejerskog, E., Tyrberg, S., Svensson, O., Waters, R.,

Savin, A., Bolund, B., Eriksson, M., and Leijon, M., “Experimental results from an offshore wave energy converter” Journal of Offshore Mechanics and Arctic Engineering, 132(4):041103, 2010.

VII Boström, C., and Leijon, M., “Operation analysis of a wave energy converter under different load conditions” Accepted for publication in IET Renewable Power Generation, December 2010.

VIII Leijon, M., Boström, C., Lejerskog, E., Rahm, M. and Svensson, O. “A wave power unit”, International patent WO 2010/085188, published 2010-07-29.

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IX Boström, C., Svensson, O., Rahm, M., Lejerskog, E., Savin, A., Ström-stedt, E., Engström, J., Gravråkmo, H., Haikonen, K., Waters, R., Björk-löf, D., Johansson, T., Sundberg, J., and Leijon, M., “Design proposal of electrical system for linear generator wave power plants” Proceed-ings of the IEEE Industrial Electronics, IECON2009, Porto, Portugal, PD-027448:4429–4434, 2009.

X Rahm, M., Boström, C., Svensson, O., Grabbe, M., Bülow, F., and Lei-jon, M., “Offshore underwater substation for wave energy converter ar-rays” IET Renewable Power Generation 4(6):602–612, 2010.

XI Svensson, O., Boström, C., Rahm, M., and Leijon, M. “Description of the control and measurement system used in the low voltage marine substation at the Lysekil research site” Proceedings of the 8th European Wave and Tidal Energy Conference, EWTEC2009, Uppsala, Sweden, pp. 44–50, 2009.

XII Boström, C., Rahm, M., Svensson, O., Strömstedt, E., Savin, A., Wa-ters, R., and Leijon, M. “Temperature measurements in a linear gen-erator and marine substation for wave power” Submitted to Journal of Offshore Mechanics and Arctic Engineering, June 2010.

XIII Rahm, M., Svensson, O., Boström, C., Waters, R., and Leijon, M. “Ex-perimental results from the operation of aggregated WECs” Submitted to IET Renewable Power Generation, December 2010.

XIV Boström, C., Ekergård, B., Waters, R., Eriksson, M., and Leijon M. “Linear generator connected to a resonance circuit” Submitted to Re-newable Energy, January 2011.

Patents pending, not official.

XV Leijon, M., Boström, C., and Eriksson, M., “Resonance circuit” Patent submitted to PCT/EPO, PCT/SE2010/051356, 2010-12-09.

Other contributions of the author, not included in the thesis.

XVI Lundin, J., Goncalves, J., Boström, C., Yuen, K., Kjellin, J., Rahm, M., Bernhoff, H., and Leijon, M., “Dynamic stability of a generation system based on renewable energy” Submitted to the 21st International Confer-ence and Exhibition on Electricity Distribution, CIRED2011, Frankfurt, Germany, 6–9 June 2011, January 2011.

XVII Savin, A., Svensson, O., Strömstedt, E., Bolund, B., Boström, C., and Leijon, M. “Determining the service life of a steel wire under a work-ing load in the wave energy converter (WEC)” Proceedwork-ings of the ASME 28th International Conference on Ocean, Offshore and Arctic Engineer-ing, OMAE2009, Honolulu, Hawaii, OMAE2009-79164, 2009.

XVIII Tyrberg, S., Stålberg, M., Boström, C., Waters, R., Svensson, O., Strömstedt, E., Savin, A., Engström, J., Langhamer, O., Gravråkmo, H., Haikonen, K., Tedelid, J., Sundberg, J., and Leijon, M., “The Lysekil wave power project: Status update” Proceedings of the 10th

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World Renewable Energy Congress (WRECX), Glasgow, UK, pp. 1061–1066, 2008.

XIX Rahm, M., Boström, C., Svensson, O., Grabbe, M., Bülow, F., and Lei-jon, M., “Laboratory experimental verification of a marine substation” Proceedings of the 8th European Wave and Tidal Energy Conference, EWTEC2009, Uppsala, Sweden, pp. 51–58, 2009.

XX Boström, C., Lejerskog, E., Tyrberg, S., Svensson, O., Waters, R., Savin, A., Bolund, B., Eriksson, M., and Leijon, M., “Experimental results from an offshore wave energy converter” Proceedings of the ASME 27th International Conference on Ocean, Offshore and Arctic Engineering, OMAE2008, Estoril, Portugal, OMAE2008-57415, 2008. XXI Boström, C., Rahm, M., Svensson, O., Strömstedt, E., Savin, A.,

Wa-ters, R., and Leijon, M., “Temperature measurements in a linear gener-ator and marine substation for wave power” Proceedings of the ASME 29th International Conference on Ocean, Offshore and Arctic Engineer-ing, OMAE2010, Shanghai, China, OMAE2010-20881, 2010.

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Nomenclature

A [m2] Cross-sectional area

A_abs [%] Absorbed power

Af ac Active stator area

B [T] Magnetic flux density

C, CDC [F] Capacitance

CC [F] Sea cable capacitance

cg [m/s] Group velocity

D [C/m2] Electric displacement field

D_b [m] Diameter of buoy

E [V/m] Electric field

Ei, ei, ea, eb, ec [V] internal EMF or no-load voltage

f [Hz] Frequency

F_b [N] Buoy force

Fem [N] Electromagnetic force

F_es [N] End stop force

Fs [N] Spring force

g [m/s2] Acceleration due to gravity

H [m] Wave height

h [m] Water depth

H [A/m] Magnetizing field

Hm0, HSand H1/3 [m] Significant wave height

i [A] Current

iain, ibin, icin [A] Incoming current

iDC [A] Current in DC-load

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j [A/m2] Free current density or electric current density

l [m] Length

LC [H] Sea cable inductance

LS [H] Generator synchronous inductance

m [kg] Mass

mn varies Spectral moment

N Number of turns

P [W] Power

P_abs, PG_res, PG_re f [W] Absorbed power

PDC [W] Power in DC-load

P_in [W] Incoming power or power in resistive load

PL_res, PL_re f [W] Power in load

Ploss [W] Resistive power losses in sea cable and generator

Pmax [W] Maximum power

P_out [W] Power out from generator

PRC [W] Resistive power losses in sea cable

PRG [W] Resistive power losses in generator

Q Quality factor

r [m] Radius of buoy

R0 [Ω] DC-resistance

R_C [Ω] Sea cable resistance

R_i [Ω] Internal resistance of switching device

RG [Ω] Generator resistance

RDC, RL [Ω] Load resistance

R_PE [Ω] Power electronic resistance

S( f , θ ) [m2/Hz] Directional wave spectra

t [s] Time

t₀, ts [s] Time when diodes stop and start conducting

TE, Tm0−1 [s] Mean energy period

TZ, T02 [s] Zero-upcrossing period

V_d [V] Line-to-line voltage rms

v_D [V] Voltage drop over diode

vDC, VDC, vL [V] Voltage over load

vin [V] Segments of three phase voltages

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V_{W EC} [V] Controlled DC-voltage w_p [m] Pole width of magnets

x [m] Translator position

Zeq [Ω] Equivalent impedance

α [Np/s] Damping ratio

γ [Ns/m] Damping function

λ [m] Wave length

ω [rad/s] Angular frequency ω0 [rad/s] Resonant frequency

Φ [Wb] Magnetic flux ρ [kg/m3] Density ρc [C/m3] Charge density ρr [Ωm] Resistivity θ Degrees Direction ξ Damping factor

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Abbreviations

AC Alternating Current

D Diode

DC Direct Current

EMF Electromotive force FEM Finite Element Method

FPGA Field Programmable Gate Array

HF High Frequency

IGBT Insulated Gate Bipolar Transistor KCL Kirchhoff’s Current Law

KVL Kirchhoff’s Voltage Law

L1–L9 Refers to WEC 1–9 installed at Lysekil

MOSFET Metal Oxide Semiconductor Field Effect Transistor OWC Oscillating Water Column

PAC Programmable Automation Controller PWM Pulse Width Modulation

WEC Wave Energy Converter

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Preface

Converting the energy in ocean waves into usable electric power is not trivial. At the Division for Electricity, Uppsala University, a team of scientists are working together in the wave power research field. Today, there are 15 PhD students working in the team. The main goals with the research are to optimize and develop the wave energy system further and to study the wave energy project from an overall perspective, from the waves to the grid connection point. Beside the main goals, all the PhD students have their own individual goals with the research.

The aim of the work performed by the author has been to study what im-pact the electrical system has on the WEC performance and how the electrical system could be designed. An electrical conversion system is needed between the WEC and the grid. Depending on how the first conversion step is carried out, converting a fluctuating AC into DC, the generator will be subjected to different loads which affect the power production and absorption of the WEC. Based on the results presented in this thesis, conclusions about system oper-ation characteristics can be drawn and they can act as a foundoper-ation for future system and WEC designs.

The aim of the following chapters is to give the reader a deeper understand-ing of the research presented in the appended papers and to present some of the preparatory work made to achieve the results. The author also wishes to give the reader an overall picture of the project and the work carried out by the research group.

Previous work by the author was presented in a Licentiate thesis in 2009 [1]. A detailed study of the WEC connected to a diode rectifier and filter was made in the thesis and the experimental set-up was described in detail.

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1. Introduction

Since the late 19th century, the Western world’s society has been developed into an infrastructure highly dependent on electric generation [2, 3]. To ensure the social welfare in the future, a central issue will be to secure a long-term energy supply and energy production. Therefore, it has been important to find efficient and environmentally friendly ways to utilize electric energy. This is one of the reasons for the increasing use of sustainable energy resources for electricity production [4]. One energy resource that can be seen as a sustain-able energy source is the energy from the ocean waves.

1.1 Ocean waves – a source of energy

Ocean waves can be created in several ways. The source creating the waves could for example be an earthquake, a ship or the tide. However, when study-ing wave energy, the ocean waves created by the winds, wind waves, are the most important [5]. When the wind sweeps over the ocean surface some of its kinetic energy is transferred to the ocean due to the friction between the wind and the ocean surface. There are also pressure variations in the wind which force the water particles into motion resulting in waves [6].

The potential of ocean waves for energy utilization is large, it has both a high utilization factor and a high power density [7]. Fig. 1.1 shows a map of how the wave-power level is distributed over the globe in kW/m crest length. On an yearly average, the wave climate is estimated to be better on the southern hemisphere because of its smaller variations in sea state, but on a monthly basis, the northern and southern hemisphere can have similar sea states [8]. The global power potential represented by waves is estimated to be 1–15 TW [9–11].

Technically, the potential available in Sweden is estimated to 5–10 TWh annually [13] which can be compared to Sweden’s total electric energy con-sumption during one year, 144 TWh (2008) [14]. Additional reports about the wave climate in Swedish waters can be found in [15, 16].

1.2 Wave energy technologies

Scientists have struggled for centuries to find a way to utilize the energy in the oceans, and a series of different concepts have come up over the years. In con-nection with the oil crisis in the middle of the seventies, the research started

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Figure 1.1: Estimated average annual power density of ocean waves expressed in kW/m [12].

to emerge in Europe. However, in the middle of the eighties, several gov-ernments, the UK government amongst others, decided to stake only power utilization projects with constructions rated 2 GW or more [17]. The conse-quence was that many smaller projects got funding problems and were forced to phase out their research [8]. About 15 years later, research in the area started to grow again, and today there are a number of different ongoing wave power projects with representatives from all over the world [18]. In 2007, there were approximately 75 different projects and around 16 of them had come to a full scale prototype that had been tested in real sea conditions [19].

There are different proposals on how a WEC could be designed and which kind of generator should be used [20]. A WEC could either operate offshore or be a shoreline device. Today, the offshore market is predicted to have the largest opportunities for expansion since shoreline devices are limited by the low number of available sites and high installation costs [6]. Moreover, a WEC can be grouped in different ways, see for example [11, 21–23]. One classifi-cation used is based on how the device utilizes its energy and is as follows: OWC, overtopping devices and wave activated bodies, see Fig. 1.2.

An OWC consists of a chamber with an open bottom. The incident waves will force the water in the chamber to rise and fall which compresses and expands the air in the chamber, Fig. 1.2a. The energy is extracted from the air-flow using a unidirectional air turbine. Two devices based on this technology are further described in [24, 25].

The operation of an overtopping device is similar to the technique used in a hydropower plant. The devices use a reservoir at a higher level than the surrounding sea level. The reservoir is filled with water by the waves and the water is led back to the ocean via a turbine. The principle is illustrated in Fig.

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Air

Turbine

Reservoir

Heaving Pitching Surging

a) b)

c)

Figure 1.2:Schematic of different operating principles divided into a) OWC, b) over-topping devices and c) wave activated bodies.

1.2b. Wave dragon [26] and the Sea-wave Slot-cone Generator (SSG) [27] are two examples of overtopping devices.

The last group, wave activated bodies or oscillating bodies, use one or sev-eral of the oscillating motions in the waves to run a generator. Alternatively, the energy can be extracted first by using a hydraulic system. The system can have two moving parts moving relative to each other, or one moving body that moves relative to a fixed reference. Wave activated bodies are often clas-sified according to how their power take-off moves in the water, i.e. if they have a pitching, surging or heaving motion, see Fig. 1.2c. One example of a pitching system is Pelamis1. The device consists of cylindrical bodies that are connected by hinged joints. The wave motion will get the joints to pump high pressure fluid through a hydraulic motor, which in turn drives an electric gen-erator. WaveRoller2 is a device using the surging motion of the oceans. The power take-off device is a vertical plate placed on the seabed. The plate will get a surging motion by the waves and the kinetic energy is transferred to a hydraulic system by a pump.

The WEC treated in this thesis is one example of a device that mainly uti-lizes its energy from the heaving and surging motion of the waves. Two other examples of wave activated bodies using the heaving motion are Wave Bob3 and AquaBuOY [28].

Even if two concepts belong to the same group, their systems to convert the wave energy into electricity can be completely different, see Fig. 1.3 and [29].

1_{Pelamis Wave Power, http://www.pelamiswave.com/ Accessed 2010-12-13} 2_{Waver Roller, http://www.aw-energy.com/ Accessed 2010-12-13}

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Fluid power Mechanical power Electric power

Wave PM linear generator PE conversion

system Grid

Wave PE conversion

system Grid

Hose

pump Accum-ulator Impulseturbine generatorPM a)

b)

Figure 1.3:Different conversion schemes for wave activated bodies using the heaving motion of the waves. a) The conversion principle of the Uppsala University concept. b) The conversion principle of AquaBuOY.

1.3 Wave

energy

system

developed

at

Uppsala

University

The technology treated in this thesis is based on a direct driven synchronous permanent magnet linear generator. Instead of having one large WEC with a high power rating, several smaller units are interconnected to get the desired power out of the system, see Fig. 1.4 for an illustration.

Figure 1.4:An illustration of how a wave power system could look in the future. ©Seabased Industry AB

The design should be based on having as many full load hours per year as possible [7]. Therefore, the generator will have different power ratings de-pending on the sea state it will operate in. For example, a generator on the

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Swedish west coast could be designed to have a nominal power of 5–20 kW, whereas on the coast of Norway, a more appropriate rating could be 50–100 kW.

A direct drive approach is used to eliminate mechanical parts, like gear boxes, turbines etc., that otherwise are required to connect a conventional high speed generator to the low speed motion of the waves. Thereby, the mainte-nance work is believed to decrease. However, a direct drive system tends to have a somewhat more complicated electrical system since a conversion is needed before the connection to the grid. Furthermore, a generator moving with a low speed is larger compared to a 50 Hz generator with the same power rating [30, 31].

There are several researchers working with linear generators for wave en-ergy conversion, see for example [32–37]. However, to the best of the author’s knowledge, no results have been published for offshore operation of WECs using the same electrical system as presented in this thesis.

1.3.1 Wave energy converter

A schematic of the WEC is illustrated in Fig. 1.5. The generator is placed on the seabed and connected to a point absorbing buoy on the ocean surface. The WEC acts as a point absorber if the diameter of the buoy is approximately less than half of the wave length [8].

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The buoy, and thereby the translator with permanent magnets mounted on it, moves upward with wave crests. At wave troughs, the mass of the system and possible springs mounted at the bottom of the generator pulls the translator downwards. When the translator moves in relation to the fixed stator, a voltage is induced in the stator windings. The voltage will drive a current in the cable windings if the generator is connected to a load.

A linear generator can be seen as an unrolled rotary generator. Many of the magnetic and electric properties are similar to a rotary generator, but there are some essential differences [30]. First, a linear generator has in almost all applications a varying speed and cannot be connected directly to the grid. Sec-ond, a linear generator has open magnetic circuits at both ends of the generator which influence the magnetic flux in the generator [38, 39].

Simulations, calculations and laboratory experiments on a four-sided lin-ear generator based on Uppsala University’s concept can be found in [40–43] and an octagonal linear generator based on Uppsala University’s concept is investigated in [44, 45].

1.3.2 Conversion and transmission system

Since the translator moves with a varying speed, the generator will produce an irregular voltage and current which needs to be converted before grid con-nection. The transmission system can be designed in several ways depending on the size of the wave power plant, the distance to the shore, the need to reduce losses and costs etc. Transmission schemes for different power plant cases are presented in [20, 46–48]. Moreover, there exist several studies on electrical conversion and control schemes for direct driven generators, see for example [49–55]. An overview diagram indicating how a conversion system could look is shown in Fig. 1.6.

The conversion strategy shown in Fig. 1.6, is to first rectify the varying AC into a DC with diode rectifiers [56, 57]. The generators are then intercon-nected in parallel on the DC-side to achieve power smoothening [58]. After the interconnection, the DC-voltage is stabilized and inverted back to an AC with a constant frequency and amplitude. Finally, the AC is transformed one or two times, depending on distance to shore and the number of generators, to decrease the transmission losses.

Both generators and substations are placed on the seabed, see Fig. 1.4. In this way, the cable installation becomes easier and the most expensive parts in the system are protected against the harsh climate at the ocean surface. There are also good cooling possibilities for power electronics since the cold sur-rounding water can be used as cooling element. However, the maintenance work is more difficult, especially in a research stage. In the future, with a line production of WECs and marine substations, the strategy is to change a de-fected part (WEC or substation) with a new one and perform the maintenance work onshore.

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Grid Sea Cable Sea Cable Sea Cable Sea Cable WEC Sea Cable Sea Cable Sea Cable Sea Cable Sea Cable

Medium Voltage Marine Substaion Low Voltage Marine Substaion

Low Voltage Marine Substaion WEC

WEC

Figure 1.6:Overview diagram of a conversion and transmission system.

1.3.3 Generator connected to different loads

The conversion system required to connect the generators to the grid has an impact on the damping of the generator and depending on how the system is designed, the generator is subjected to different loads. In turn, the damping has an impact on the power absorption and the power production. The effect of the damping is shown and discussed in Paper VII and is a key issue for the thesis. Four different load cases are studied in the appended papers. The studied cases are a generator connected to:

• a resistive load (Paper III)

• a diode rectifier and filter (Paper IV–VI)

• an actively controlled DC-voltage (Paper VII and Paper X) • a resonance circuit (Paper XIV)

A resistive load has been used when designing the generator but it will not represent a load case for a grid connected generator. To improve the design of the WEC, it has been important to show the difference between the resis-tive load case and the case when the generator is connected to an electrical conversion system.

The study with the generator connected to the diode rectifier and filter was made to see to which extent the power could be smoothened out from one WEC and to investigate the non-linear load behaviour which occurs when the generator is connected to a stable DC-voltage.

By connecting the generators to a stable DC-voltage, a number of genera-tors are controlled by one converter and a power smoothening is achieved at an early stage in the system. Moreover, the system between the converter and the grid can be performed in a straightforward way.

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Initial studies have been done on the WEC connected to a novel resonance circuit. The purpose with the circuit is to increase the damping of the generator and at the same time increase the amounts of power produced and increase the voltage.

As mentioned before, there exist a wide variety of load cases for a grid connected generator and the studied cases in this thesis are just a few of them. Other common existing control strategies are to use an active rectifier [32, 53] or control the motion of the buoy by a latching control, discrete control, or a reactive control, continuous control [59, 60].

1.3.4 Theses published by the wave power group

So far, the project has resulted in four Licentiate theses and six Doctoral the-ses. The first PhD thesis [61] was written in 2006. The author of the thesis, Dr. Thorburn, has done simulations, calculations and suggestions of different de-signs of the wave energy conversion and transmission systems. An extensive theoretical study of the electromagnetic properties of the linear synchronous generator can be found in the PhD thesis written by Dr. Danielsson [62]. A model including the interaction between wave, buoy and generator was devel-oped and verified with experimental data by Dr. Eriksson [63]. The mechani-cal design, construction of the first full smechani-cale generator and offshore results are presented by Dr. Waters in [64]. Dr. Waters has also done a study of the wave climate off the Swedish West coast and made a program that can calculate the incident power in the waves on the research site using data from the wave measuring buoy. Dr. Rahm presented in his thesis a detailed description of the mechanical design of the first marine substation [65]. Dr. Rahm focused on studying array operation of WECs. The effects of a wave power plant on the marine environment are studied by Dr. Langhamer in [66].

The Licentiate thesis by Mrs. Ivanova covers a theoretical study of an oc-tagonal permanent magnet generator for wave energy conversion [67]. Mr. Tyrberg presented an observation system that can be used to study how the buoys will move in the water and what impact the sea level will have on the power production in his Licentiate thesis [68]. Mr. Engström has continued with Dr. Erikssons work and studies the hydrodynamic wave/device interac-tion [69].

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2. Theory

This chapter presents basic theory about wave energy, generator modelling and circuit theory. For the reader who would like more information about the different areas, there are references in each section that expand on the subject.

2.1 Waves and wave energy spectrum

Physically, a wave can be described by its length, λ , and height, H, see Fig. 2.1. To describe the behaviour of a wave more accurately, different wave the-ories can be applied depending on the water depth and the steepness of the waves. For deep water, λ < 0.5h [70], 95% of the energy in the waves is available between the surface and a depth h = λ /4 [17]. The water particles are moving in an oscillating pattern and the amplitude of the circles decrease exponentially with water depth. For waves in deep water, the linear wave the-ory developed by Airy (1845) can be applied [71]. If the waves become steep, e.g. the ratio between the wavelength and wave height is small, the linear wave theory is not valid, and instead, Stokes second order theory is used [72]. The particles in waves at shallow waters will move in an elliptical pattern and the theory for solitary waves is preferable to use [72].

l

h

H Crest

Trough mean water level

Behaviour of particles beneath the surface (deep water).

wave direction

Sea bed

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To describe the wave energy resource at a site, a spectrum analysis is often performed [73]. One commonly used spectrum is the directional wave spec-trum, S ( f , θ ), [8]. The term describes the distribution of energy density in frequencies, f , and directions, θ . Some parameters typically used to charac-terize a sea state are expressed in terms of the spectral moment, mn,

mn=

Z ∞ 0

fnS( f ) d f (2.1)

where m0is the total energy (variance) of the wave system.

The significant wave height, HS, can be expressed in two different ways; H_1/3 or Hm0 [8]. When the term H1/3 is used, the significant wave height is defined as the average crest-to-trough height of one third of the highest waves in the spectrum. The other term, Hm0, represents the spectral expression, see Eq. 2.2. However, the two terms are approximately equal, 0.9Hm0 < H1/3< Hm0 [74].

H_m₀= 4√m₀ (2.2)

The mean energy period, Tm0−1, is defined as

Tm0−1= m−1/m0. (2.3)

Tm0−1depends on low frequency components and is often denoted as TE. Another term that is used to express the time period is the zero-upcrossing period, T02or TZ

T02= p

m0/m2, (2.4)

where TZdepends on high frequency components. TZis not used as commonly as TE in ocean wave theory [75].

According to linear wave theory and the deep water condition, the wave-power level or wave-power density [70, 76], can be expressed as

J= ρg Z ∞ 0 cg( f , h) S ( f , θ ) d f = ρ g2 64πTm0−1H 2 m0 (2.5)

where ρ is the density, g is the acceleration due to gravity and cgis the group velocity.

2.2 Generator theory

Depending on the purpose of the study, a generator can be modelled differ-ently. In the design process of the generator, it is important to have a model where the physical dimensions and the electric and magnetic behaviour can

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be set and studied. A field based model of the generator is often chosen in this case.

In a power system, the generator is usually represented by its equivalent cir-cuit. In this case, the no-load voltage, power rating and internal inductance and resistance must be known before the study. With an equivalent circuit, short circuit analysis as well as system analysis (generator connected to different loads) can be performed.

2.2.1 Electromagnetic field theory

Field theory is used to explain electric and magnetic phenomena in generators. Mathematically, there are two different kinds of fields of interest when study-ing generators, scalar fields and vector fields [77]. A scalar field associates to a scalar value given for each point in a space and could for example be the field of temperature or pressure. A vector field has a vector linked to each point in space and could be an electric field or a magnetic field. The relation between scalar and vector fields underlies the fundamental laws of electromagnetism and can be represented by Maxwell’s equations Eq. 2.6–2.9 [78],

∇ × E = −∂ B ∂ t , (2.6) ∇ × H = j +∂ D ∂ t , (2.7) ∇ · D =ρc, (2.8) ∇ · B = 0, (2.9)

where E is the electric field, B is the magnetic flux density, H is the magnetiz-ing field, j represents the electric current density, D is the electric displacement field and ρcis the charge density.

The first equation, Eq. 2.6, can be seen as a vector formula of Faraday’s law of induction and states: “An electromotive force is induced in a closed circuit when the magnetic flux, Φ, linking the circuit changes” [30]. If the circuit consists of a tightly-wound N-turn coil of wires, the induced voltage, e, is given by

e= −NdΦ

dt . (2.10)

Where Φ can be expressed in terms of the electric and the magnetic field as in Eq. 2.11 and Eq. 2.12.

Φ =

Z

S

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dΦ dt = −

I

c

Eds (2.12)

A field based model of the generator can be done by using Maxwell’s equa-tions and inserting boundary condiequa-tions. To solve the model numerically, a finite element method (FEM) could be used. The method is described and used in literature; see for example [79–81].

2.2.2 Equivalent circuit model

Once the electrical characteristics of the generator are well-defined, an equiv-alent circuit model can be created. The generator can be represented by resis-tances, inductances and voltage sources.

It should be mentioned that an equivalent circuit is a rough model of a linear generator and several of the non-linear effects are neglected. An example of an equivalent circuit representation of a generator can be seen in Fig. 2.2.

ei

RG LS

Generator

i

Figure 2.2:Equivalent circuit of a generator.

In Fig. 2.2, ei is the internal EMF of the generator, RGis the resistance in the armature winding and LS is the synchronous inductance. In reality, both the resistance, RG, and inductance, LS, will vary. The resistance is mainly tem-perature dependent, but also dependent of the frequency. Since the frequency is low for the generator considered in this thesis, the DC-resistance, R0, can be used to express the resistance in the generator

RG≈ R0= ρrl

A (2.13)

where ρris the resistivity of the conductor, l is the length of the conductor and Ais the cross-sectional area of the conductor.

The inductance, Ls, is proportional to the magnetic flux, Φ, and the current, i, flowing in the circuit.

Ls∝Φ

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Since the magnetic flux is not symmetrical as in a round machine, i.e. the magnetic flux will change with translator position, the synchronous tance will vary for a linear generator. Other parameters affecting the induc-tance are saturation, air gap variations and other anomalies.

2.3 Forces acting on the generator and the damping

function

There are a number of different forces acting on the generator. Some of the forces damp the translator motion and some of them contribute with energy to the system. The forces acting on the generator can be expressed with Newton’s second law

mx¨= Fb+ Fs+ Fes+ Fem+ mg. (2.15)

Where m is the mass of the moving parts in the system, i.e. the mass of the buoy and the translator if the rope does not slacken and ¨xis the acceleration of the translator. The first force in the equation, Fb, is the lifting force from the buoy. This is the driving force in the system. Fsis the force from possible springs mounted at the bottom of the generator. The force pulls the translator downwards. Fes is the force of the end stop at the top of the generator. The force Fem is a consequence of the damping from the electrical system. The term describes how the electrical system affects the translator motion, e.g. how the generator is electrically damped. Eq. 2.15 is further described and used in [56, 82].

The electromagnetic force, Fem, has an influence on the WEC’s ability to absorb energy. Femis proportional to the damping function, γ, and the transla-tor speed ˙x[82];

Fem= Af ac· γ · ˙x, (2.16)

where Af ac is the active area of the stator, i.e. the part of the stator that is covered by the translator.

The damping function can be derived from the statement that the power is equal to the force multiplied with the velocity;

P_abs= Femx.˙ (2.17)

Eq. 2.16 and Eq. 2.17 give the following expression for γ, assuming Af ac equal to one;

γ =Pabs ˙

x2 , (2.18)

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2.4 Generator connected to different loads

The theory behind the four different studied load cases are presented in this section.

2.4.1 Resistive load

The simplest load to connect the generator to is a purely resistive load, as in Fig. 2.3. ei RG LS Generator RL Load vL i

Figure 2.3:Generator connected to a resistive load.

The voltage, vL, over the load, RL, can according to KVL be expressed as;

v_L(t) = ei(t) − i(t)RG− LS di

dt. (2.19)

The expression of the current, i(t), is derived from Ohm’s law;

i(t) =vL(t) RL

. (2.20)

2.4.2 Diode rectifier, filter and actively controlled DC-voltage

loads

If the generator is connected to a diode rectifier and capacitor as in Fig. 2.4, the diodes make the circuit non-linear [83, 84]. There are two different cases: when the diodes are conducting and when they are blocking.

Knowing that only two diodes, one of the positive diodes, D1, D3or D5, and one of the negative diodes, D2, D4or D6, are conducting at the same time, the circuit in Fig. 2.4 can be described with the equivalent circuit shown in Fig. 2.5. Where vin(t) is made up of segments of the line-to-line voltages, ea(t), eb(t) and ec(t) [83].

When the diodes are conducting, the voltage and the current in Fig. 2.5 can according to KVL and KCL be described with Eq. 2.21 and Eq. 2.22 respectively.

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ea RG LS Generator RDC Load D1 CDC eb RG LS ec RG LS n D4 D3 D6 D5 D2 ia vDC

Figure 2.4:Circuit diagram of a generator connected to a diode bridge rectifier and filter. CDC vin RG LS Generator RDC Load vDC i DP DN RG LS

Figure 2.5:Equivalent circuit of the system shown in Fig. 2.4.

vDC(t) = vin(t) − 2i(t)RG− 2LS di

dt − 2vD (2.21)

where vDis the voltage drop across the diode.

i(t) = CDC dvDC

dt + vDC(t)

R_DC (2.22)

When the diodes are blocking, there is no current flowing in the circuit, i(t) = 0, and the voltage, vDC(t), over the load is expressed as

dvDC dt = −

1

C_DCR_DCvDC(t). (2.23)

The solution to Eq. 2.23 is

v_DC(ts) = vDC(t0)e − ts−t0

CDCRDC_{, t}₀_{< t < t}_s _(2.24)

where t0 is the time when the diodes stop to conduct and tsis the time when the diodes start to conduct again. The capacitance needed to maintain a stable DC-voltage can be calculated from Eq. 2.24.

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If CDCRDCis much larger than ts− t0, the voltage over the load in Eq. 2.24 can be written as [83]

vDC(ts) = vDC(t0) = VDC (2.25)

and the capacitance, CDC, and resistance, RDC, can be approximated to a con-stant voltage source, VDC, and the circuit in Fig. 2.5 can be expressed as in Fig. 2.6. VDC vin RG LS Generator Load i DP DN RG LS

Figure 2.6:Equivalent circuit of the system shown in Fig. 2.4, if CDC and RDC are

assumed to be larger than ts− t0.

Another way to achieve a constant DC-voltage, without using capacitors with a high capacitance is to control the current on the DC-side of the recti-fier, and thereby controlling the power. By controlling the power on the DC-side, the DC-voltage can be set to different values [85–87]. The active device controlling the power could for example be an IGBT or a MOSFET in a buck-boost converter or in an inverter. To be able to maintain a constant DC-voltage, the load after the active device should have a small impedance value.

A simplified model of the control is shown in Fig. 2.7.

vin RG LS Generator RDC Load VDC i DP DN RPE VWEC RG LS

Figure 2.7:Generator connected to a load with active power regulation. The resis-tance, RPE, represents the active device in the circuit.

The active device is represented by a variable resistor in series with the load. The resistor could be switched to have an average resistance, RPE, between Ri and infinity. Where Ri is the internal resistance of the switching component. By decreasing the value of RPE, the current delivered to the load RDC will in-crease and the controlled DC-voltage, VW EC, decrease, and vice versa, when

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R_PEis increased, VW ECis increased. However, there will be a limit on the max-imum power, Pmax, the system can supply to the DC-side without changing the DC-voltage, VW EC. The limit is given by Eq. 2.26.

Pmax=

V_{W EC}2 Ri+ RDC

. (2.26)

Fig. 2.7 is a rough model of a circuit with an active control and would in reality also include other components such as capacitors and inductors for filtering. However, the aim with the figure is to show the principle of the op-eration.

2.4.3 Resonance circuit

Resonance is a phenomenon that occurs in electrical circuits when the capac-itive reactance is equal to the inductive reactance.

The simplest way to achieve resonance in an inductive circuit is to add a capacitor to the system as in Fig. 2.8. The circuit in Fig. 2.8 is used to define some basic expressions used in general to describe resonance circuits.

ei

RG LS

C

Generator

i

Figure 2.8:Generator connected to a capacitor representing a simple resonance cir-cuit.

Fig. 2.8 can be described by the second order equation

de_i dt = RG di dt+ LS d2i dt2+ 1 Ci(t). (2.27)

The characteristic expression for Eq. 2.27 can be written as

s2+RG L_Ss+

1

L_SC = 0. (2.28)

The solution to Eq. 2.28 is given in terms of resonant frequency, ω0, and damping ratio, α:

s1,2= −α ± q

α2− ω₀2 (2.29)

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ω0= 1 √

LSC

(2.30)

and the damping attenuation is

α = RG 2LS

. (2.31)

The resistance, RG, will give rise to a damping of the oscillations in the circuit and depending on the relative magnitude of α and ω0, the system will behave differently. To describe the behaviour of the system, the damping factor, ξ = α/ω0, and the quality factor, Q = 1/(2ξ ), are used [88]. There are three different cases:

1. If ξ > 1 or Q < 0.5, the system is over-damped. 2. If ξ = 1 or Q = 0.5, the system is critically damped. 3. If ξ < 1 or Q > 0.5, the system is under-damped.

At steady-state, the equivalent impedance of the circuit in Fig. 2.8 can be written as

Z_eq= RG+ jωLS+ 1

jωC. (2.32)

Inserting ω0in Eq. 2.32, Zeqbecomes purely resistive. The small impedance occurring at resonance will result in high currents and voltages especially if the resistance in the circuit is small. Therefore, resonance is often something to avoid in power circuits. However, in some circuits resonance is preferable as in resonant converters [83,88] and in wireless energy transmission [89]. It is also common to use a capacitor for decreasing the phase angle if the generator has a high inductive reactance, but never to reach the resonance mode [90].

If a load is added to the circuit in Fig. 2.8, as in Fig. 2.9.

ei RG LS C Generator i RL vL Load

Figure 2.9:A load, RLis added to the resonance circuit.

The power delivered to the load, PL_res, in Fig. 2.9 will not be significantly increased compared to a system without the capacitor, see Fig. 2.10. In the

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example shown in Fig. 2.10, the resistance RG= 1 Ω, the inductance LS= 20 mH, the voltage ei = 400 V, the load RL= 10 Ω and the capacitance C = 12.665 mF in the resonance case and C is not present in the reference case.

When the frequency is smaller than the resonant frequency, ω/ω0< 1, the capacitive reactance is larger compared to the inductive reactance and the volt-age over and power delivered to the resistive part of the load, RLis larger. For frequencies higher than the resonant frequency, ω/ω0> 1, the inductive reac-tance is dominating and the voltage drop over the inductor is increasing. The power delivered and voltage over the resistive part of the load decreases since the current through the capacitor increases.

w/w0 P/P L_ref PG_res PL_res PG_ref PL_ref 1 0.5 0 1.5 2 0 10 20 30

Figure 2.10:Power for a generator equivalent circuit with a constant voltage source, ei, connected to a capacitor and resistive load, the resonance case, and a generator

connected to a purely resistive load, the reference case. PG_resand PG_re f refer to the

power in the generator, the absorbed power. PL_res and PL_re f refer to the power in

load. The average value of PL_re f is used as a reference on the y-axis.

The significant difference in the resonance circuit compared to the circuit without a capacitor can be observed by studying the generated power in the generator, PG_res and PG_re f in Fig. 2.10 where PG_res is much larger than PG_re f resulting in a higher damping function (Eq. 2.18).

Novel resonance circuit

A novel resonance circuit is presented in this thesis and a patent has been submitted, Paper XV. The basic concept of the circuit is shown in Fig. 2.11. A single-phase generator is used to describe the theory behind the circuit. For a

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three-phase generator, the resonance circuit will consist of three single-phase circuits connected in parallel on the DC-side.

The purpose with the circuit is to have more power delivered to the load at resonance. The power generated in the generator will not be as large as in the pure resonance case in Fig. 2.10, but the power supplied to the load can be much larger compared to a circuit with a single-phase full bridge rectifier, see Fig. 2.12. ei RG LS Generator i RDC vDC Load D1 D2 D3 D4 C1 C2

Figure 2.11:The generator connected to the novel resonance circuit.

PG_res PL_res PG_ref PL_ref w/w0 0 1 2 3 P/P L_ref 0 5 10 15

Figure 2.12:The power in the load, PLand the power generated by the generator, PG,

in two different cases. In one case the generator is connected to the circuit in Fig. 2.11 and in the other case, the reference case, it is connected to a single-phase full bridge rectifier.

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In, Fig. 2.12, the same values for the parameters as in Fig. 2.10 are used. For frequencies larger than ω0, PG_resis not decreasing as much as for the pure resonance case in Fig. 2.10.

By assuming that the diodes are ideal, the circuit shown in Fig. 2.11 can be drawn as in Fig. 2.13. e_i RG LS _i RDC C2 C1

Figure 2.13:A circuit diagram of the circuit shown in Fig. 2.11 assuming that the diodes are ideal and conducting.

The circuit in Fig. 2.13 can then be expressed as

dei dt = RG di dt+ LS d2i dt2+ 1 R_DC di dt+ 1 2Ci(t) (2.33) with C = C1= C2

The characteristic expression for Eq. 2.33 can be written as

s2+RG+ 1 RDC LS s+ 1 2LSC = 0. (2.34)

The resonant frequency for the circuit will then be

ω0= 1 √

2LSC

(2.35)

and the damping attenuation is

α =

RG+_R1 DC 2LS

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3. The Lysekil research site

A review is done by Ref. [18] of the present research sites and test centres for testing wave power devices in Europe, see Fig. 3.1. The authors to [18] have divided the test sites into three different categories: “pre-prototype stage gate requirements test sites”, “pre-production stage gate requirements test sites” and “pre-commercial stage gate requirements test sites”.

Figure 3.1: Different test sites in Europe. Test sites marked in green are in the prototype stage gate requirements stage, those marked in purple are in the production stage gate requirements stage and test sites marked in blue are in the pre-commercial stage gate requirements stage. [18]

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To be in the third category, it should be possible to test the following at the site [18]:

• multiple units performance • device array interactions • power supply interaction • environmental impacts issues

• full technical and economic due diligence

There are four test sites in Europe belonging to the third category and one of them is the Lysekil research site. The other three sites are the Agucadoura test site in Portugal where Pelamis Wave Power has tested their device1, the Santona test site where the Ocean Power Technologies has tested their device “the Power Buoy”2 and the Mutriku test site in the north of Spain where an OWC has been tested [91].

The work with the Lysekil research site started during 2003 and in 2004 permissions were granted by the County Administration to set up a research park consisting of 10 generators and approximately 30 buoys for environmen-tal studies. A site on the Swedish west coast near the town Lysekil was chosen because of its proximity to field stations and harbours. The location of the site is shown in Fig. 3.2.

Figure 3.2:A sea chart over the test area with the research site, sea cable and measur-ing station label.

1_{Pelamis Wave Power, http://www.pelamiswave.com/our-projects/agucadoura/ Accessed} 2010-12-13

2_{Ocean Wave Technologies, http://www.oceanpowertechnologies.com/spain.htm/ Accessed} 2010-12-13

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The site has a fairly good sea state (especially during the winter months) and the distance to shore is about 2 km. The sea state at the site is investigated in [92], and annually, the wave climate at the site, based on eight years of satellite data, is estimated to be 2.6±0.3 kW/m.

The seabed conditions at the test site were investigated in May 2006. The investigation showed that the bottom area consists mainly of a 1 m thick sandy silt material. The area is fairly level and the water depth varies between 24–25 m. More of the results from the study are presented in [93]. The water level variations at the site are investigated in [94].

Some of the main components installed at the site can be seen in Fig. 3.3 and are further described in the following subsections.

25 m 2.9 km 200 m Wave power plant Marine Substation Wave measuring buoy Observation tower 12 m 150 m Biology buoy Measuring station & dump load 2 km

Figure 3.3:Overview of the different components installed at the research site.

3.1 Biology buoys

The purpose of having a number of biology buoys at the site is to investigate what impact a wave power plant will have on the surrounding eco system and vice versa. One of the purposes is to see what the environmental impact will be if one places large structures on an otherwise rather empty sandy bottom and what will be the consequences of having large structures on the ocean surface. To get prior and after results, the measurements started in 2004 before the biology buoys were placed in the area. The first buoys were launched in 2005 and some of them can be seen in Fig. 3.4d.

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Results from measurements can be found in [95,96] where it is investigated if a wave power plant can act as an artificial reef.

Another important issue is to investigate what effect the environment has on the WEC. It is especially important to study if the amount of biofouling that occurs on the generator and buoy has an impact on the energy absorption and the material of the WEC. This was studied in [97].

Figure 3.4:Photos of installations at the research site. a) Observation tower. b) Wave measuring buoy. c) Buoy in ice formation. d) Biology buoys.

3.2 Observation tower

An observation tower equipped with a Sony SNC-RX550 remotely control-lable network camera is located approximately 150 meters from the WECs. A photo of the tower is shown in Fig. 3.4a. Images from the network camera will make it possible to correlate voltage data and line force data with the move-ment of the buoy in the water. The construction of the tower and some results are presented in [68].

The tower will be equipped with a weather station to measure tempera-ture, precipitation and barometric pressure. In connection to the tower, a hy-drophone system will be installed to measure the underwater noise from the generators and the substations.

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3.3 Wave measuring buoy

The first wave measuring buoy was installed at the site in April 2004. The buoy is a Datawell Waverider buoy3 and is shown in Fig. 3.4b. The sampling frequency of the wave data is 2.56 Hz and the data are transmitted to shore via an HF radio link and sampled in 20 minute files. A description of its operation can be found in [98]. The winter of 2009/2010 was quite cold and the harsh climate resulted in an ice formation at the site, see Fig. 3.4c. The Datawell Waverider buoy was not built for such conditions which resulted in a line breaking of the elastic part of the line and the buoy was drifting away. In May 2010, a new Datawell Waverider buoy was installed at the site with the same properties as the first buoy.

3.4 Wave energy converters

So far, eight WECs have been installed at the research site. The work with the first WEC, L1, started in 2005 and the WEC was launched in March 2006. Some of the design characteristics and physical dimensions of the WEC are presented in Table 3.1 and a picture of the WEC can be seen in Fig. 3.5a. Results from L1 are presented in [99, 82, 100, 101] and Paper I–VI.

Main parameters of L1 Nominal power at 0.7 m/s 10 kW Voltage, line-to-line, rms at 0.7 m/s, Vd 200 V Generator resistance, RG 0.44±1.5% Ω Generator inductance, LS 11.7 mH Air gap 3 mm

Size of magnet block 6.5x35x100 mm3

Pole width, wp 50 mm

Number of stator sides 4

Vertical stator length 1264 mm

Vertical translator length 1867 mm

Translator resp. stator width 400 mm

Translator weight 1000 kg

Table 3.1: Mechanical parameters and electrical design parameters for L1.

The next generators to be installed were L2 and L3, see Fig. 3.5b and c. They are electrically similar to L1, but some changes in the mechanical con-struction were done. They were launched in February 2009 and brought to shore in October the same year since the stator windings were connected

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Figure 3.5:The WECs that have been installed at the research site, with exception for L5. In a) L1, b) L2, c) L3, d) L4, L7 and L8 and e) L9.

correct, resulting in a phase shift of one of the phases. Results from L2 and L3 are presented in Paper VI and Paper VII.

L9 was installed in October 2009 and a photo is shown in Fig. 3.5e. L9 is a second generation WEC with both electrical and mechanical changes. Electrically, the no-load voltage has been increased and instead of mounting the stators on an inner frame, they are mounted directly on the hull walls. The weight has been reduced with approximately 50% compared to L1. Data for L9 are presented in Table 3.2.

During 2010, four more WECs were installed, L4–L8. Three of the WECs can be seen in Fig. 3.5d. The WECs L4–L8 have been built by Seabased In-dustry AB. L4–L6 are three phase generators electrically similar to L1–L3.

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Main parameters of L9 Nominal power at 0.7 m/s 20 kW Voltage, line-to-line, rms at 0.7 m/s, Vd 450 V Generator resistance RG 1±1.5% Ω Generator inductance LS 20 mH Air gap 3 mm

Size of magnet block 6.5x47x230 mm3

Pole width, wp 55 mm

Number of stator sides 4

Vertical stator length 2000 mm

Vertical translator length 2000 mm

Translator resp. stator width 230x2 mm

Translator weight 2700 kg

Table 3.2: Mechanical parameters and electrical design parameters for L9.

L7 and L8 are one phase generators but are otherwise similar to L1–L3 with exception of the size of the magnets.

The magnets used to magnetise the generators are of the N40 NE-Fe-B type of magnets and have a remanence induction of 1.3 T and a maximum energy production of 320 kJ/m4.

The generators have been connected to buoys with different shapes and weights. Three of the buoys can be seen in Fig. 3.6. The purpose with using different kinds of buoys is to see what impact the shape and size of the buoy have on the power absorption.

Figure 3.6:Three different kind of buoys that the generators have been connected to.

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3.5 Marine substations

To be able to interconnect the WECs and transmit the aggregated power to shore, a substation is needed. A first marine substation was installed in 2009 to interconnect three of the WECs, L1–L3. A circuit diagram of the system can be seen in Fig. 3.7. Since the generators do not have any “on/off”-switch, the WECs are connected to dummy loads when they are disconnected from the system, Paper VIII. The dummy loads are made of delta-connected resistive loads consisting of immersion heaters.

L1 L2 L3 Marine Substation Measuring Station Aux

Figure 3.7:Circuit diagram of the system consisting of three generators, one substa-tion and a measuring stasubsta-tion.

The marine substation is placed on the seabed in a similar way to the WECs. The substation is a 3 bar pressure vessel with a volume of 3 m3and it was filled with nitrogen before deployment. The generators are connected to the substa-tion by underwater connectors in the lower dished end, see Fig. 3.8a. Heat sensitive components, such as power electronics and capacitors, are mounted on the vessel wall of the substation to take advantage of the cold water on the outside of the vessel.

The voltage from each WEC is rectified in a six-pulse diode rectifier. After the rectification, the WECs are interconnected in parallel on a common DC-busbar. The rectified voltage is filtered by electrolytic capacitors connected in parallel to the DC-busbar to even out the DC-voltage further before it is inverted to a 50 Hz AC. The capacitors are shown in Fig. 3.8b. A three phase IGBT bridge with an IR2130 driver circuit is used as an inverter. No filter is placed after the inverter.

The PWM pulses controlling the inverter are created in LabVIEW and im-plemented in the control system. In the process of time, the control algo-rithm can be modified and the operation characteristics of the inverter can be changed. Finally, a tap-changed Y-Y transformer is placed after the inverter, Fig. 3.8c. The transformer has five different taps (80-100-125-180-250/1000 V) and is controlled by contactors to maintain the voltage fairly constant out from the substation.

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Figure 3.8:The first installed substation. a) The substation casing. b) Picture taken from the inside of the substation where sets of DC-link capacitors and auxiliary bat-teries can be seen. c) The transformer with the contactors that are used to switch between different winding taps.

The substation has an auxiliary system which distributes power to the break-ers and the control system. The system is supplied from the main circuit after the transformer. Thereby, the auxiliary system can be supplied from the gen-erators, and if necessary, from shore.

A control system has been developed in a National Instruments FPGA-based PAC computer for measurement acquisition and control.

Another marine substation is currently under construction. The new substa-tion can handle the power from seven WECs. The substasubsta-tion is larger than the first one, 5 m3, and some changes in the mechanical construction have been made, see Fig. 3.9.

3.6 Measuring station and grid connection point

In connection with the test site, a measuring station has been built on the nearby island of Hermanö (frequently called Gullholmen). The measuring sta-tion is located in a nature reserve and is built to fit in the surrounding environ-ment as much as possible, see Fig. 3.10a.

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Figure 3.9:a) Substation casing of the new marine substation. b) Capacitors mounted on a curved disk to achieve better contact with the casing. c) Assembly of contactors.

A sea cable transmits the power from the offshore research site to the mea-suring station. Data for the sea cable are presented in Table 3.3.

Main parameters of the sea cable

Sea cable resistance RC 0.54±1.5% Ω Sea cable inductance LC <0.01 mH

Sea cable capacitance CC 145 µF

Table 3.3: Parameters for the sea cable.

Before the launching of L1, the measuring station was equipped with resis-tive loads and a control and measurement system that enables data acquisition and the possibility to control the system remotely from Uppsala.

During the winter of 2006/2007, the system was complemented with a six-pulse diode rectifier and a large power storage packet consisting of ultra capac-itors. New resistive loads, DC-loads, were constructed in several steps during the year. The measurement system was supplemented with current measure-ments. Hall transducers and resistive shunts were used for the current mea-surement.

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In 2010, IGBTs where installed to actively control the DC-voltage and a resonance circuit was installed. A simplified overview diagram of the measur-ing station is shown together with the rest of the system in Fig. 3.7.

Figure 3.10:a) The measuring station on Hermanö. b) The transformer station and grid connection point.

In future, the WEP will be connected to the 11 kV grid on Hermanö and preparations for the grid connection have been made. A cable and a trans-former station transforming 1 kV to 11 kV have been installed, see Fig. 3.10b.

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4. Simulations

There exist several simulation programs for simulating electrical circuits. Some widely used programs are presented and described in [102]. The tools that have been used for simulating electrical systems in the present work are the commercially available programs PSpice (used in Paper IV) and MATLAB Simulink (used in Paper VII, X and XIV).

Since circuit oriented simulation tools are used, the modelling of the elec-trical components as capacitors and diodes can be well defined in the sim-ulations. The critical part of the simulations is to model the generator in an accurate way.

4.1 Modelling of generator

The modelling of the generator can be done in different ways depending on the study, i.e. how to choose ea, eband ecin Fig. 4.1.

ea RGa LSa Generator eb RGb LSb ec RGc LSc

Figure 4.1:Circuit diagram of the model of the generator in the simulations.

When the size of the filter was investigated (Paper IV), it was important to include the real movement pattern of the translator. To get the shape of the voltage that was expected during operation in real sea conditions, the varia-tions in frequency and amplitude were applied to the model. A simple way of doing this in the simulations is to implement a set of experimentally measured voltage data as voltage sources, ea, eband ec.

In Paper X, with the substation connected to two experimental generators running with a constant speed, a good representation of ea, eband eccould be

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a constant voltage source corresponding to the no-load voltage of the experi-mental generator at the operating speed. The inductance and the resistance in the cable windings are then added to the circuit.

If the no-load voltage and frequency are known for a certain translator speed, a rough estimation of the produced power and losses can be elicited from the simulations. The results are approximate since the model is linear and does not take into account any non-linear effects. However, it can be a good starting point when evaluating different load cases of the generator.

In Paper VII and XIV, the voltage is assumed to increase linearly with the translator speed. The assumptions made to simplify the models are:

• The induced voltage is assumed to be proportional to the translator speed. Iron losses, mechanical and non-linear effects are ignored.

• The stator is assumed to be 100% covered by the translator, Af ac= 1. • The value of LSwill be constant for all x.

The frequency of the voltage from the generator, f , can be expressed as

f= x˙ 2 · wp

(4.1)

where ˙xis the speed of the translator and wpis the pole width of the magnets. The no-load voltage is, with the assumptions given above, increasing lin-early with translator speed

Ei= ˙ x ˙

x_d ·Vd (4.2)

where ˙x_dand Vdare the design speed and the no-load voltage at design speed respectively. The different values for Eiand f and the electrical data in Table 3.1–3.3 are used as input parameters for the generator and sea cable modelling in the simulations.

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5. Experiments

A summary of the offshore experiments is presented in this chapter. The first two sections describe the voltage and current measurements and the calcula-tion of power. The following seccalcula-tions describe the experimental set-up for the different load cases.

5.1 Voltage and current measurements

The voltage and current are measured at two different places in the measur-ing station, see Fig. 5.1. The current denoted iinis measured with hall current transducers on the three incoming phases, iain, ibin and icin. The voltage de-noted vin in Fig. 5.1, is measured between the three incoming phases and the neutral conductor and are vain, vbinand vcin. Before the voltage is sampled in the data acquisition system, it is decreased by voltage division to ±10 V. The sampling frequency of the measurements carried out on the AC-side is either 50 Hz or 256 Hz.

The DC-current, iDC= iDC+= −iDC−, is measured with shunt resistors and hall current transducers. The DC-voltage, denoted vDC= vDC+− vDC−, where vDC+ is measured between the positive conductor and neutral conductor and v_DC−is measured between the negative conductor and neutral conductor and divided in a similar way as the AC-voltage. The sampling frequency on the DC-side will vary depending on whether the study is carried out over a long time (hours) or over a shorter time period (seconds).

Measuring Station Sea cable vin iin RL vDC iDC Resonance circuit RDC CDC Rectifier

Figure 5.1:The electrical system in the measuring station with the measuring points marked. RLrepresents the Y-equivalent value of the delta-connected load.

The voltage and current measurements carried out in the substation are per-formed in a similar way as in the measuring station. The measuring points in the substation are the same as in the measuring station, at the incoming AC

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from each WEC, on the DC-side and on the AC-side before and after the trans-former. The voltage and current are sampled with 256 Hz before the inverter and the sampling frequency is 500 Hz after the inverter.

The accuracy of the voltage measurements is after calibration estimated to ±2.2% and the accuracy of the current measurements with hall sensors in the measuring station is after calibration estimated to ±3.1%. In the substation, the accuracy of the voltage measurements is estimated to ±1.2% and the ac-curacy of the current measurements is estimated to ±1.7%, see Paper XII.

5.2 Calculation of power

Mainly three different power calculations are discussed in the papers. First, the power dissipated in the resistive load, Pin in the purely resistive case, Eq. 5.1, and PDC in the non-linear cases, Eq. 5.2. Second, the power produced by the generator, Pout, which is calculated as Pin added to the resistive losses in the sea cable, PRC. Third, the absorbed power by the generator, Pabs, Eq. 5.5. In Paper I, III, V, VI, the absorbed power, Pabs, is calculated as the power delivered to the measuring station, Pin, with the resistive losses added. In Paper XIV, the absorbed power is calculated by adding the resistive losses, Plossto the power in DC-load, PDC.

Pin= vainiain+ vbinibin+ vcinicin (5.1)

PDC= vDCiDC= v2_DC RDC

(5.2)

P_loss= (RC+ RG) i2ain+ ibin2 + i2cin = PRC+ PRG (5.3)

P_out= Pin+ PRC (5.4)

Pabs= Pin+ Ploss (5.5)

The calculated absorbed power, Pabs, is an approximate value since there exist losses that are not included in the calculations. The losses in the genera-tor are discussed in [62, 64, 65] and can be divided into three different groups: mechanical losses, iron losses and conductor losses. The mechanical losses will occur between moving parts in the generator due to friction. Iron losses are a consequence of the magnetic field in the stator and can be divided into hysteresis losses, eddy currents and excess losses. Finally, there will be con-ductor losses. The concon-ductor losses due to the resistance in the generator and the sea cable can be calculated with Eq. 5.3. In Paper XIV there will also exist

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losses in the resonance circuit which are not included in the calculations of Pabs.

An expression for the absorbed power by the buoy in percent of the incident power can be written as:

A_abs= 100 Pabs Db· J

(5.6)

where Dbis the diameter of the buoy. In the case of a torus shaped buoy, the diameter of the buoy is calculated as in Eq. 5.7.

D_b= 2 q

r2₂− r2

1 (5.7)

where r1is the inner radius and r2is the outer radius of the buoy.

The incident power, J, measured in W/m wave crest is calculated with the help of data collected from the Datawell Waverider buoy described in Section 4.3.

5.3 Generator connected to resistive load

Two different kinds of loads have been used in the experiments; the generator has either been connected to resistive loads at the measuring station consisting of a number of aluminium housed resistors, Fig. 5.2a, or it has been connected to its dump load, Fig. 5.2b.

Figure 5.2:a) Resistive loads mounted on the roof and wall of the measuring station. The resistive loads are both used on the AC-side, RL, and on the DC-side, RDC. b)

Dump load placed on the seabed.

A number of different values of the resistive load have been tested in the ex-periments. The resistive values were changed by connecting/disconnecting a number of resistors. The resistance was measured after installation and the different values of the delta-connected load were 2.2 Ω, 4.9 Ω and 10 Ω

Electrical Systems for Wave Energy Conversion

List of Papers

Contents

Nomenclature

Abbreviations

Preface

1. Introduction

1.1

Ocean waves – a source of energy

1.2

Wave energy technologies

1.3

Wave

energy

system

developed

at

Uppsala

University

1.3.1

Wave energy converter

1.3.2

Conversion and transmission system

1.3.3

Generator connected to different loads

1.3.4

Theses published by the wave power group

2. Theory

2.1

Waves and wave energy spectrum

2.2

Generator theory

2.2.1

Electromagnetic field theory

2.2.2

Equivalent circuit model

2.3

Forces acting on the generator and the damping

function

2.4

Generator connected to different loads

2.4.1

Resistive load

2.4.2

Diode rectifier, filter and actively controlled DC-voltage

loads

2.4.3

Resonance circuit

3. The Lysekil research site

3.1

Biology buoys

3.2

Observation tower

3.3

Wave measuring buoy

3.4

Wave energy converters

3.5

Marine substations

3.6

Measuring station and grid connection point

4. Simulations

4.1

Modelling of generator

5. Experiments

5.1

Voltage and current measurements

5.2

Calculation of power

5.3

Generator connected to resistive load