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Optimization of Point Absorber Wave Energy Parks MARIANNA GIASSI UURIE 353-18L ISSN 0349-8352 Division of Electricity Department of Engineering Sciences Licentiate Thesis Uppsala, 2018
Optimization of Point Absorber Wave Energy Parks
MARIANNA GIASSI
UURIE 353-18LISSN 0349-8352
Division of ElectricityDepartment of Engineering SciencesLicentiate Thesis
Uppsala, 2018
Abstract
Renewable energies are believed to play the key role in assuring a future of sustainable energysupply and low carbon emissions. Particularly, this thesis focus on wave energy, which is crea-ted by extracting the power stored in the waves of the oceans.
In order for wave energy to become a commercialized form of energy, modular deploymentof many wave energy converters (WECs) together will be required in the upcoming future. Thisdesign will thus allow to benefit, among others, from the modular construction, the shared elec-trical cables connections and moorings, the reduction in the power fluctuations and reduction ofdeployment and maintenance costs.
When it comes to arrays, the complexity of the design process increase enormously com-pared with the single WEC, given the mutual influence of most of the design parameters (i.e.hydrodynamic and electrical interactions, dimensions, geometrical layout, wave climate etc.).
Uppsala University has developed and tested WECs since 2001, with the first offshore de-ployment held in 2006. The device is classified as a point absorber and consists in a linearelectric generator located on the seabed, driven in the vertical direction by the motion of afloating buoy at the surface.
Nowadays, one of the difficulties of the sector is that the cost of electricity is still too highand not competitive, due to high capital and operational costs and low survivability. Therefore,one step to try to reduce these costs is the development of reliable and fast optimization toolsfor parks of many units.
In this thesis, a first attempt of systematic optimization for arrays of the Uppsala UniversityWEC has been proposed. A genetic algorithm (GA) has been used to optimize the geometryof the floater and the damping coefficient of the generator of a single device. Afterwards, theoptimal layout of parks up to 14 devices has been studied using two different codes, a continuousand a discrete variables real coded GA. Moreover, the method has been extended to study arrayswith devices of different dimensions. A deterministic evaluation of small array layouts in realwave climate has also been carried out. Finally, a physical scale test has been initiated whichwill allow the validation of the results.
A multi–parameter optimization of wave power arrays of the Uppsala University WEC hasbeen shown to be possible and represents a tool that could help to reduce the total cost ofelectricity, enhance the performance of wave power plants and improve the reliability.
List of papers
This thesis is based on the following papers, which are referred to in the textby their Roman numerals.
I Giassi M., Göteman M.; Parameter optimization in wave energydesign by a genetic algorithm; Proceedings of the 32nd InternationalWorkshop on Water Waves and Floating Bodies, Dalian, China, 23-26April, IWWWFB 2017.
II Giassi M., Göteman M.; Layout design of wave energy parks by agenetic algorithm; Under revision for Ocean Engineering, 2017.
III Giassi M., Göteman M., Thomas S., Engström J., Eriksson M., IsbergJ.; Multi-parameter optimization of hybrid arrays of point absorberWave Energy Converters; Proceedings of the 12th European Wave andTidal Energy Conference, Cork, Ireland, 27-31 August, EWTEC 2017.
IV Thomas S., Giassi M., Göteman M., Eriksson M., Isberg J., EngströmJ.; Optimal constant damping control of a point absorber with lineargenerator in different sea states: comparison of simulation and scaletest; Proceedings of the 12th European Wave and Tidal EnergyConference, Cork, Ireland, 27-31 August, EWTEC 2017.
V Bozzi S., Giassi M., Moreno Miquel A., Bizzozero F., Gruosso G.,Archetti R., Passoni G.; Wave farm design in real wave climate: theItalian offshore; Energy, 122 (378-389), January 2017.DOI: 10.1016/j.energy.2017.01.094
Reprints were made with permission from the publishers.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1 Wave energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Uppsala University concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Wave energy sector challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Multi-units arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Research question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Structure of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Wave energy farm model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Linear wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Waves-structures interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.3 Dynamic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Optimization theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1 Single device optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Array layout optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.3 Multi-parameters array optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.4 Deterministic array evaluation - A case study . . . . . . . . . . . . . . . 27
3.2 Wave tank experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1 Single device optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.1 First GA validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Array layout optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Multi-parameters array optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3.1 Second GA validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Deterministic array evaluation - A case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 Wave tank experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.1 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Computational time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 GA parameters sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9 Svensk sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
10 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1. Introduction
The EU energy and climate action goal is to reduce greenhouse gas emissionsby 40% by 2030, and that the proportion of renewable energy (RE) will haveto cover at least 27% of the total energy use [1].
According to the Swedish energy policy, the share of renewable energy shallbe at least 50% of total energy use by 2020 and 100% by 2040 [2].
1.1 Wave energyRenewable energies represent the solutions to a future of sustainable energysupply and no carbon emissions. Ocean energy, i.e. the energy that can be har-vested from seawater, is the general term which includes the following energyresources: tidal currents, ocean currents, tidal range, waves, ocean thermalenergy and salinity gradient. The overall potential of all these resources isenormous.
The origin of wave energy is the unbalanced irradiation of the sun at dif-ferent latitudes. Due to this temperature variation on the Earth’s surface, theatmospheric pressure varies and induces motion of air masses from high tolow pressure areas, creating winds. As the wind blows over water, some of theenergy is transfered to the ocean, forming waves, which store this energy aspotential energy (in the mass of water displaced from the mean sea level) andkinetic energy (in the motion of water particles). The wind speed, the lengthof time the wind blows and the length of the generation area will influence theheight and the period of the resulting waves [3]. Waves usually travel longdistances without much energy loss and therefore are really efficient in theenergy transport. The theoretical potential of wave energy in the world hasbeen estimated to be around 3 TW [4]. Like other renewable energy sources,wave energy is available with seasonal and geographical variability. The areaswith the highest incident wave potential are the western coasts of continents,between 40◦−60◦ latitude, due to the flux of regular westerly winds (Fig. 1.1).
Technology to extract wave energy consists nowadays of many differentconcepts, and they can be classified according to operational principle, loca-tion, power take off (PTO) and directional characteristics, for example. Overthe last decades, a huge number of wave energy converters (WECs) have beendeveloped, patented and tested. However, until now, there is no device thathas reached the required level of reliability for full scale commercialization.
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Figure 1.1. Annual mean wave power density (colors) and annual mean direction ofthe power density vectors (→) [5]
Nevertheless, different coastal areas with different wave climates will requiredifferent type of technologies, making it more likely that a small number ofdevices will be conquering the market.
The average Swedish energy flux along the west Atlantic coast is estimatedto be around 5 kW/m [6]. Such wave climates are considered "mild" and theyrequire small rated power WECs compared to open Atlantic coast like UK orPortugal, for example. Large scale electricity production will benefit from thedeployment of the devices in multi-units arrays or parks: cost reductions, mo-dularity, redundancy, power quality, sharing of the electrical cables and utilityscale power generation are just some examples of the advantages provided bythese systems.
1.2 Uppsala University conceptUppsala University has been developing a point absorber wave energy con-verter since 2006, which consists of a linear generator located on the seabed,connected via a rope to a floater on the surface (Fig. 1.2). The generator haspermanent magnets mounted on his surface, while the stator contains coil win-dings. When waves lift the buoy, the relative movement of the magnets withrespect to the coils induce electricity according to Faraday’s law.
1.3 Wave energy sector challengesThe research during the last decades has resulted in many important achie-vements. However, to become cost competitive with other energy sources andto get the support and interest of investors, the wave energy sector has still toface and solve many challenges. The European Union and the Swedish Energy
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Figure 1.2. Uppsala University WEC and principle of operation (from Paper II).
Agency have developed and funded specific regulations and action plans tohelp the delivery of ocean energy [1],[7], according to which some of the mostsignificant aspects that have to be addressed are:• The reduction of the cost of the technology; prototype demonstration is
difficult and expensive, due to the harsh marine environment. Moreover,the number of technologies under development decreases the capital costreduction progress.
• The EU’s transmission grid infrastructures expansions onshore and offs-hore to deliver the new generated power; in addition, other infrastruc-tures improvement such as port facilities and specialized vessels for de-ployment operation and maintenance.
• More knowledge about the environmental impact to mitigate the nega-tive effect on the marine environment, as well as the social acceptability.
• Development of systems, subsystems and components related to powertransmission quality, control and monitoring.
• Device performance development.• Improve installation, operation and maintenance strategies.• Improving reliability and durability through development of models of
predictions; increased knowledge is also needed when it comes to up-scaling conceptual individual units to parks.
1.4 Multi-units arraysThere are many ways to reduce the cost of the technology. One option is todeploy large arrays of many units (examples sketch in Fig. 1.3). Having a parkof wave energy converters instead of one or few bigger units has a lot of ad-vantages: the modular construction, sharing of the electrical cables connecti-ons and moorings, quality and smoothness of the power output, redundancy,maintenance can be done without shutting down the entire production, higherreliability to failures, higher power production and cost-effective deployment.
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Figure 1.3. Outline examples of arrays. Seabased AB (top left), Aquamarine PowerOyster (top right), Carnegie CETO (bottom left), Langlee Wave Power (bottom right).
However, since the system becomes much more complex than a single WEC,there are many aspects that need to be studied and understood before the actualphysical realization of the power plant, such as:• Multi-device interaction analysis (hydrodynamical and electrical).• Layout geometry of the power plant.• Effect of the wave climate on the layout.• Optimal utilization of the available ocean area.• Power take off characteristics and control strategies• Effects on marine life and coastal processes.• Economical analysis (CapEX and OpEX costs).
All the aforementioned aspects will have a direct or indirect influence on thepower production of the plant. Normally, problems with multi objective goalsare solved by optimization routines. However, optimization of an array ofwave energy converter is not an easy task, although very important and crucialat this stage of the wave energy development. The complexity of the problemcan be understood by looking at Fig. 1.4, where some of the most importantvariables of an array design are represented.
Arrows represent influence on the "box" or variable they are pointing at.It can be seen that the mutual relations among variables are many and multi-directional. Note that this diagram includes many simplification and that theproblem, in reality, can be much more complex than that.
The ideal optimization routine would optimize all these variables simulta-neously, taking in account that, if one variable is changed, automatically allthe variables that the box is pointing at will be modified.
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Figure 1.4. Variables that play role in the optimization of a wave energy farm. Arrowsrepresent mutual influence.
1.5 Research questionAt this stage, the wave energy sector is in urgent need of reliable simulationtools able to guide the decision making process before huge investments aredone.
The aim of this thesis is to help answering the question: how can we at bestoptimize a wave energy park of Uppsala WECs during the design process andwould that be able to help reducing the cost of energy?
1.6 Structure of the workThe thesis is divided in the following chapters:• Introduction: wave energy overview, challenges of the sector, wave energy
parks introduction, aim of the thesis.• Theory: description of the theoretical fundamentals of the model (linear
wave theory, hydrodynamics and optimization)• Methods: description of the different models that have been built, of the
simulations performed and of the scale experiment.
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• Results: summary of the results obtained with different models and re-sults from the scale experiment.
• Discussion: considerations about the results.• Conclusions: summary of the main findings.• Ongoing and future work: description of the necessary future develop-
ment of the research.
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2. Theory
2.1 Wave energy farm modelThe model used to simulate the hydro-mechanical behaviour of the wave energypark consists of two main parts. An analytical model based on the multiplescattering method [8] is used to calculate the hydrodynamic coefficients suchas added mass, radiation damping and excitation amplitude. The method assu-mes the validity of linear potential flow theory. The coefficients are then usedto solve the equation of motion in the frequency domain. The motion of thebuoys are then directly linked to the power production.
2.1.1 Linear wave theoryLinear potential flow theory, or Airy wave theory, is used to describe the phy-sics of water waves in a two dimensional space, given the following assumpti-ons: inviscid and incompressible fluid, irrotational flow, small wave steepness,small ratio between amplitude of waves and water depth, small body motionsand only gravity force acting on the fluid.
The hypothesis of incompressibility results in the following continuity equa-tion:
� ·v = ∂u∂x
+∂w∂y
+∂v∂ z
= 0, (2.1)
where v = (u,w,v) is the fluid velocity in Cartesian coordinates. In this workwe consider only waves propagating in one direction. Without loss of gene-rality, the direction of wave propagation is defined as the positive x-direction.The assumption of incompressible fluid and irrotational motion imply that thefluid velocity is a conservative vector field, implying the existence of a velocitypotential and a stream function for waves. The velocity potential Φ satisfies
u =dxdt
=−∂Φ∂x
=−Φx (2.2a)
v =dzdt=−∂Φ
∂ z=−Φz (2.2b)
v =−�Φ. (2.3)
From Eq. 2.1 and Eq. 2.3 it can be concluded that the potential satisfies theLaplace’s equation:
�Φ =∂ 2Φ∂x2 +
∂ 2Φ∂y2 +
∂ 2Φ∂ z2 = 0. (2.4)
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To solve the Laplace equation it is necessary to apply specific boundary con-ditions:• Bottom boundary condition:
The sea bottom is assumed to be fixed, horizontal and impermeable, thena no flow condition should be applied:
v =−∂Φ∂ z
= 0 at z =−d (2.5)
where d is the water depth.• Kinematic free surface boundary condition:
At the free surface the vertical velocity z(t) of the particle should coin-cide with the wave velocity η (x, t), and it can be decomposed as
Φz =−ηt +ηx ·Φx at z = η (2.6)
• Dynamic free surface boundary condition:At the free surface, the pressure should coincide with the atmosphericone, assumed constant, so we derive the condition from Bernoulli’s energyconservation balance as
η (x, t)+12g
(Φ2
x +Φ2z)=
1g·Φt at z = η (2.7)
• Periodic boundary condition:
Φ(x, t) = Φ(x+L, t) for every t (2.8a)Φ(x, t) = Φ(x, t+T ) for every x. (2.8b)
where L is the wavelength and T is the wave period.While the Laplace equation and the bottom condition are linear, the two
conditions at the free surface are non linear, because of the term ηx ·Φx andΦ2
x +Φ2z . Moreover, η is unknown, being itself part of the problem solution.
It is therefore not possible to find an analytical exact solution. By analysingthe order of magnitude of the non linear terms, on the assumption that thewave length is higher than the wave height, so H/L� 1, we can neglect thesecond order terms. In this way Φ and η are calculated at z = 0, instead of atz = η , ignoring the non-linearity laying in the fact the free boundary shape isunknown.
The linearized problem becomes:
Φxx+Φzz = 0 at 0 < x < L and −d < z < 0 (2.9)
Φz = 0 at z =−d (2.10)
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Φz+ηt = 0 at z = 0 (2.11)
η (x, t) =1g·Φt at z = 0 (2.12)
Eq. 2.11 and 2.12 can be joined in a single condition:
Φz =−1g·Φtt at z = 0 (2.13)
From the Laplace equations combined with the new linear boundary condi-tions we obtain the dispersion equation, that describes the manner in which afield of propagating waves consisting of many frequencies would separate ordisperse due to the different celerities of the various components.
ω2 = gk · tanh(kd) (2.14)
where ω = 2πT is the wave angular frequency and k = 2π
L is the wave numberand c is defined as the phase velocity, also called celerity, of the waves:
c2 =(ω
k
)2=
gk· tanh(kd) . (2.15)
With some algebraic manipulation of the last equation, we will find the relati-onship for the wave length:
L =gT 2
2π· tanh
(2πL·d
). (2.16)
In deep water d � L, so the wave length can be simply written as
L0 =gT 2
2π(2.17)
It is important to notice that the wave length and celerity depend only on waveperiod T, not on wave height.
Finally, by separation of variables, the solution of the linearized problem is:
η (x, t) =H2· cos(ωt− kx) (2.18)
Φ(x,z, t) =ω ·a
k· cosh(k (d+ z))
sinh(kd)· sin(ωt− kx) (2.19)
Orη (x, t) =
H2· cos(kx−ωt) (2.20)
Φ(x,z, t) =−g ·aω· cosh(k (d+ z))
cosh(kd)· sin(kx−ωt) (2.21)
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2.1.2 Waves-structures interactionMultiple scattering and body radiation problem
We now consider Nb point absorber floating devices. In this study we willonly consider cylindrical buoys. With the assumption of linear potential flowtheory, the fluid diffracted velocity potential φD of the buoy i can be decompo-sed in the sum of three components: the incident waves φI , the scattered wavesφS and radiated waves φR.
φ iD = φ i
I +φ iS+φ i
R. (2.22)
By Fourier transform the fluid potential in the frequency domain results in
φ i(x,ω) =∫ ∞
−∞φ i(x, t)eiωtdt (2.23)
The velocity potential needs to satisfy the Laplace equation and the linearboundary conditions. Consider now to transfer the problem in local cylindricalcoordinates (ri,θi,z) with origin at the center of each buoy (xi,yi,0) and thethe fluid domain is divided into two regions, one beneath the buoys (II) andone outside the buoys (III) (Fig. 2.1).
Figure 2.1. Sketch of the regions in which the fluid domain has been divided. Here,only cylidrical buoys without moonpool have been considered (figure from [9])
By separation of variables, is it possible to find a solution to the Laplaceequation with the linear boundary constraints in cylindrical coordinates, interms of vertical functions, radial functions and angular functions. Hence,in each fluid domain the general expression of the velocity potential can bewritten as an ansatz in term of eigenfunction expansions:
Φ(ri,θi,z) = R(ri)Θ(θi)Z(z)
=∞
∑n=−∞
∞
∑m=0
[AmnKn(kmri)+BmnIn(kmri)]einθiZm(z) (2.24)
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where Amn and Bmn are unknown coefficients, Kn and In are modified Bes-sel functions, Zm(z) are the vertical eigenfunctions, einθi is the radial eigen-function.
If we consider multiple bodies, the diffracted potential in the exterior dom-ain will be the sum of the incident potential, the scattered and radiated wavesof the body i, plus the all contributions from the other bodies ( j �= i) in termsof scattered and radiated waves (Eq. 2.25).
In the external domain:
φ i(III)D = φ i
I +φ i(III)R +φ i(III)
S +∑j �=i
[φ j(III)
R +φ j(III)S
]i
=∞
∑n=−∞
[Z0(z)
(α i
0nHn(kr)Hn(kRi)
+ Jn(kr)(Ai0n+∑
j �=i
∞
∑l=−∞
T i j0lnα j
0l))
+∞
∑m=1
Zm(z)(
α imn
Kn(kmr)Kn(kmRi)
+In(kmr)In(kmRi) ∑
j �=i
∞
∑l=−∞
T i jmlnα j
ml
)]einθi .
(2.25)
In the domain underneath the buoys:
φ i(II)D =
V i
2Li
((z+h)2− r2
2
)
+∞
∑n=−∞
[ai
0n
( rRi
)|n|+2
∞
∑m=1
aimn cos(λ i
m(z+h))In(λmr)In(λmRi)
]einθi . (2.26)
where Ai0,n is the incident wave coefficient, α i
mn = α imn+V ibi
mn are the combi-ned coefficients that include contributions from scattered and radiated waves,V i is the velocity of the buoys, Zm(z) are the vertical eigenfunctions, Hn andJn are Bessel functions, Kn and In are modified Bessel functions, T i j
mln are theexpression needed for Graf’s addition theorem (to write outgoing waves fromone cylinder as incoming waves in the local coordinates of one other cylinder),km are the wave numbers that solve the dispersion relation (Eq. 2.14).
The scattering problem (i.e. all the buoys are fixed and there is only thepropagation of an incident wave) and the radiation problem (i.e. all buoysoscillates independently in heave and there is no incident wave) are solvedseparately. In absence of incident waves Ai
0,n = 0 for all i, while in absence ofradiated waves V i = 0 for all i.
Continuity between equation (2.25) and (2.26) and their radial derivativesat the boundaries r = Ri, combined with truncation of the vertical and radialeigenfunctions, results in an expression of the coefficients ai
sn in terms of α i
17
and a finite system of linear equations:⎡⎢⎢⎢⎣
� −B1MT 12 · · · −B1
MT 1N
−B2MT 21
� · · · −B2MT 2N
......
. . ....
−BNMT N1 −BN
MT N2 · · · �
⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣
α1
α2
...αN
⎤⎥⎥⎥⎦=
⎡⎢⎢⎢⎣
V 1B1rad
V 2B2rad
...V NBN
rad
⎤⎥⎥⎥⎦+
⎡⎢⎢⎢⎣
B1diff
B2diff...
BNdiff
⎤⎥⎥⎥⎦
(2.27)Once we have solved for α i and calculated the coefficients ai
sn, we can com-pute the potential in the region under the floater (φ i(II)
D ).
2.1.3 Dynamic equationWith the velocity potential φ i(II)
D , the dynamical forces acting on the bodies canbe calculated as a surface integral on the wetted surface S of the float:
Fi = iωρ∫∫
Sφ i(II)
D dS (2.28)
In the following, only heave forces are considered. The excitation force iscomputed from the scattering problem and the radiation force from the radia-tion problem. The equation of motion in the frequency domain then takes theform[−ω2(mi+mi
add(ω))− iω
(Bi(ω)+Γi)−ρgπ
((Ri)
2− (Riin)
2)]zi(ω) =
= Ai(ω) f iexc(ω) (2.29)
where mi = mib+mi
t is the total mass of the moving system (buoy and transla-tor), Γi is the constant power take-off coefficient, and zi is the vertical positionof the buoy. The connection between translator and the buoy is assumed stiff,so they have the same displacement. The equations of motion can equivalentlybe expressed as zi(ω) = Ai(ω)Hi(ω) where Hi(ω) is the transfer function.The position of the buoy in the time-domain can be obtained by Fourier trans-form zi(t) = zi(ω). With zi(t), the instantaneous power absorption of the WECi is given as Pi(t) = Γi · zi(t)
2, while the power output of the full park will bethe sum of all Nb WECs:
Ptot(t) =Nb
∑i=1
Pi(t). (2.30)
2.2 Optimization theoryThe problem we have to face during optimization of wave energy converterarrays involve a large parameter space, is multi-objective and the shape of
18
the objective function is not known. For such kind of problems, evolutionaryalgorithms are suitable, since they rely on intelligent search over a large but fi-nite solution space using statistical methods [10] and they are less likely to getstuck in local minima. The procedure and the nomenclature of evolutionaryalgorithms, and among these genetic algorithm, are based on the biologicalevolution of natural creatures.
The GA procedure is fully explained in paper II and specific modificationsare explained in each paper theory section. Here, some of the most importantgeneral aspects are summarized.
2.2.1 Genetic algorithmTo describe the genetic algorithm, we can refer to Fig. 2.2, where the generalprocedure is represented as a sequence of diagram boxes.
Figure 2.2. Schematic procedure of a GA.
The genetic algorithm routine is based on the arbitrary choice of some pa-rameters which can influence the achievement of convergence and the outputresults. The optimization process starts with the random creation of the firstpopulation, which is a set of a fixed number of chromosomes. Each chromo-some contains a certain number of genes. Each gene represents a variable thatwill be optimized in the process.
Initial populationIn the beginning of the optimization routine, a first population (or first set ofchromosomes) is generated by uniform random sampling over a pool of pos-sible values. It represents the first set of solutions from which the algorithmwill start its optimization routine. The first population can contain differentgenes according to the optimization code. For example it can be a set of pos-sible combinations of R (radius), d (draft) and Γ (damping coefficient), eachset representing a parameter configuration of a WEC, or a set of coordinates[xi,yi] of the WECs that forms a specific array layout.
19
EvaluationBy means of an evaluation function, every chromosome of the first populationis associated with a fitness value. Different objective functions have beendefined and compared. The evaluation functions required the computationof the power output of the arrays; therefore the evaluation step contains thesemi-analytical multiple scattering model explained in section 2.1.2 for thecalculation of the hydrodynamic parameters of the fully coupled WECs in thearray. The input to this model consists of 20 min time series of irregular wavesmeasured off-shore at the Lysekil research test site at the west coast of Sweden,characterized by a specific significant wave height Hs and energy period Te.
RankingAfter evaluation, the population is ranked in descending order, from the "best"chromosome (higher fitness value or best solution), to the "worst" (lower fit-ness value or worst solution).
ConvergenceThe number of iterations that evolve depends on whether an acceptable solu-tion is reached or a set number of iterations is exceeded [10]. Therefore, inorder to stop the search of the genetic algorithm, some convergence criteriahave been implemented:
1) a maximum number of iterations (MaxIt) is reached;2) all the chromosomes in the actual population are the same;3) the solution ceases to improve after a certain number (I) of iterations.
If one of this conditions is fulfilled, the algorithm stops and the first chromo-some of the ranked population is taken as final optimal solution.
ReproductionA reproduction step is carried out whenever convergence is not reached. Itconsists of four parts: natural selection, pairing, mating – crossover, elitismand mutation.
a. Natural selection: This operator selects the upper percentage of indivi-duals in the ranked population that will survive and continue to the nextgeneration.
b. Pairing: The selected part of the population chromosomes representsthe new parents; odd and even rows are paired from top to bottom togenerate a new part of the population called offspring.
c. Mating – crossover: The crossover operator exchanges genetic materialbetween two parent chromosomes during reproduction, so that poten-tially positive distinctive genes from both individuals will be inheritedby every child. This procedure is performed in a slightly different wayaccording to the optimization problem, but always with a single pointcrossover.
20
d. Elitism and Mutation: Mutation introduces new genetic material in thepopulation by randomly changing a chosen percentage of variables (i.e. ge-nes). This ensures that other regions in the solution space will be explo-red, preventing that the algorithm gets stuck in a local minima. Elitismprotects the first upper set of the ranked individuals from potentially ne-gative mutations, in order to preserve the best solutions unaltered in thefollowing generation.
New PopulationThe combination of the first selected population (parents) and the new ge-nerated offsprings, after mutation, represents the new population that will beevaluated in the next generation or iteration and so forth until the algorithm isstopped.
21
3. Methods
3.1 SimulationsThe simulation work can be divided into different parts explained in the nextfew paragraphs.
One part regards the systematic optimization of wave energy parks. In thiswork the wave energy farm model explained in section 2.1 and the optimiza-tion method explained in section 2.2 have been combined together in a uniqueroutine, as shown in Fig. 2.2.
One other simulation part regards the deterministic study of small arraysconfigurations and orientations in real wave climate, through a case study.
3.1.1 Single device optimizationA genetic algorithm optimization for a single device parameters has been de-veloped (Fig. 3.1). The goal of the single WEC GA is to find the optimal valueof the radius (R) and draft (d) of the buoy and the damping coefficient (Γ) ofthe generator, upon calculation of hourly average power output ( fcost =−Ptot).In fact, radius, draft and Γ of the generator influence the resulting output po-wer of a device firstly in terms of hydrodynamics and secondly according toEq. 2.30. The number of different genes in each chromosome is three (R,d,Γ)(Fig. 3.2). These parameters are sampled from a "pool" of possible values:R = 1 : 0.5 : 5 m, d = 0.2 : 0.05 : 0.4 m and Γ = 15 : 1 : 2000 kNs/m. Eachcombination of parameters would give a different power output of the device.In total 20 simulations have been performed. Crossover method is single va-riable, uniformly randomly chosen and then swapped. The other parametersettings of the GA are outlined in Table 3.1.
Results from this part of the work will be used to validate the capability andaccuracy of the custom built genetic algorithm code.
3.1.2 Array layout optimizationThe first problem in wave energy converter array design is the spatial dis-tribution of the units within a certain area, for which the deploying com-pany/organization has received the legal permits. The relative position of theWECs within a park influence the hydrodynamic and electrical interaction,thus modifying the power output. It is in fact well known that arrays of WECs,
22
Figure 3.1. Sketch of theparameters of the WEC(From paper III).
Figure 3.2. Chromosome of the single device pro-blem (From paper II).
differently from wind turbine parks, can influence not only devices located inthe shadow, but also in the front by scattered and radiated waves propagatingin all directions of the park [11]. The nature of the interactions is complex andnot well understood or predictable. Hence, the method has been extended toperform layout optimization of parks. Two different codes have been imple-mented: discrete variables and continuous variable based. At this stage all theWECs are identical (R = 3 m, d = 0.45 m and Γ = 140 kNs/m) and the aimis to improve the power production ( fcost = −Ptot). In all different cases pre-sented, the waves considered are long crested and travelling along the x-axis(from left to right).
Discrete code
Figure 3.3. Sketch of the layout opti-mization problem over a gridded area.
This model performs optimization of thespatial coordinates [xi,yi] where the so-lution space is discrete, i.e. the oceanarea is gridded every 10 m in both x andy directions and WECs can take randomposition only on the knots (Fig. 3.3). Thetotal available area is of 2500 m2 (or6400 m2 for some simulations).
Fig. 3.4 represents the chromosomesand crossover method utilized. Each ofthem contains Nb genes (i.e. couple ofcoordinates); during crossover a gene israndomly selected as separation frontierbetween two parts that will be swapped.
23
Figure 3.4. Chromosome of the discrete GA code (From paper II).
Continuous code
Figure 3.5. Sketch of the layout op-timization problem over a continuousarea. Minimum bord-to-bord distance(red dotted line) and minimum center-to-center distance (orange).
This model performs optimization ofthe spatial coordinates [xi,yi] wherethe solution space is continuous, i.e.WECs positions are continuous num-bers (Fig. 3.5). To avoid overlap-ping, a minimum distance constraint be-tween the devices center is set to 10m. The solution space is much big-ger than previously, and different con-figuration from gridded ones are allo-wed.
Fig. 3.6 represents the chromosomesand crossover method utilized. The pro-cedure is the same as in the previous ex-ample, but the crossover gene in this caseis also blending.
A summary of the parameter settingsof the GA is reported in Table 3.1.
Figure 3.6. Chromosome of the continuous GA code (From paper II).
3.1.3 Multi-parameters array optimizationUntil now, the problem we have tried to solve consisted in locating identicaldevices according to hydrodynamic mutual influence, and the optimizationroutine has been driven by the maximization of the power output of the singledevice or of the park. But what happens if we deploy devices of different sizes
24
Table 3.1. Genetic algorithm parameters settings.
Parameter Description 1 WEC Array
Nb Number of WEC 1 4,5,7,9,14nPop Initial population size (number of chromosomes) 16 12nGene Number of different genes 3 1Nvar Number of genes in every chromosome 3 Nb·nGeneMaxIt Maximum number of iteration/generation 100 250, 500selection rate Fraction of nPop that is selected for mating 0.5 0.5mutation rate % of population to be mutated 0.2 0.2elitism rate Best solutions unaltered in the next generation 1 3cut off Iteration to converge if solution doesn’t improve 25 −
within the same park or cluster?To help answer this question the tool has been extended by:• the inclusion of an improved hydrodynamics calculations model [9],
which give the possibility to optimize layout with WECs of differentsizes.
• the possibility to optimize all the three parameters of the Uppsala WEC(R,d,Γ), not any more as a single device, but as part of a multi-unitsarray (i.e. optimizing 3 x Nb parameters).
The investigation has been divided into three different cases; moreover, theeffect of having different cost functions has also been considered.
This is motivated by earlier studies that have shown an improved total per-formance when devices of different dimensions are deployed together [12],[9].
Case 1
In the first case study the optimal buoy geometry is sought for a fixed griddedregular layout of 9 WECs at 15 m distance (Fig. 3.7). As previously mentio-ned, the method allows simultaneous optimization of radius, draft and Γ of thegenerator, but here it performs the optimization only on two different valuesof radii. A value of the draft and PTO gamma is assigned accordingly to Ta-ble 3.2. It is clear that there are 512 possibilities of combinations. Hence, theproblem is small enough for a parameter sweep optimization to be viable, andcan be used for validation. The cost function f A
cost is the non–dimensionalisedpower to mass ratio expression in Eq. (3.1), where the mass acts as a crude es-timation for the installation cost of a wave device. The chromosome containsa value of the radius for every floater.
f Acost =−
(Ptot−Psmall)/(Pbig−Psmall)
(mtot−msmall)/(mbig−msmall). (3.1)
Here, Ptot is the total mean power of the considered array, Pbig is the total powerof the park when all WECs are of the largest allowed dimension, whereas Psmall
25
Figure 3.7. Sketch of the radii optimization for 9 WECs in a fixed layout (Frompaper III).
Table 3.2. WECs buoys geometry and PTO constants.
Geometry Radius Draft Mass PTO Γ
#1 2.0 m 0.5 m 6440 kg 70 kNs/m
#2 3.5 m 0.6 m 23668 kg 200 kNs/m
is the total power when all WECs are of the smallest dimension. Analogously,the subscripts of the mass m refer to the same conditions.
Case 2
The second case implemented is a layout optimization of 12 WECs with twodifferent sizes: 6 WECs of geometry #1 and 6 WECs of geometry #2, asspecified in Table 3.2. The possible positions allowed are placed on a 6 x 6 gridwith a separating distance of 15 m. There are 12 genes in each chromosome,so that every device is represented by a couple of coordinates [xi,yi]. It is tobe noted that, in this situation, the total mass of the park will be equal for allcases; hence, the fitness function is taken as the negative value of the powerproduction of the park, as in the Eq. (3.2).
Figure 3.8. Sketch of the layout optimization for 12 WECs of 2 different size (Frompaper III).
26
f Bcost =−Ptot. (3.2)
Case 3
In the third studied application, the parameters radius, draft and PTO coef-ficients have been simultaneously optimized in an array of 4 WECs. Thecoordinates of the WECs are fixed on a grid with separation distance 15 m(Fig. 3.9). The radius, draft and PTO coefficient of each WEC are free totake values within the ranges R ∈ [2 : 0.5 : 3.5] m, d ∈ [0.3 : 0.05 : 0.6] mand Γ ∈ [15 : 1 : 250] kNs/m. The alternative cost function (Eq. 3.3) is usedfor comparison with the non-dimensional cost function explained before. Thechromosomes contains 12 genes as well, i.e. one value of Ri , di and Γi foreach device.
Figure 3.9. Sketch of the parameter optimization for 4 WECs in a fixed layout (Frompaper III).
f Ccost =−
Ptot
mtot. (3.3)
3.1.4 Deterministic array evaluation - A case studyDespite the fact that wave directionality on array power production is believedto be a key parameter to choose the best orientation of wave energy farms andachieve a maximum in production, the optimization method described abovedoes not include this variability yet.
A first attempt to study the influence of the wave climate and directions fora specific study site has been carried out in paper V in a deterministic way,i.e. a fix number of array configurations have been modelled a priori and theresults compared.
Differently from the optimization method, arrays of four devices are simu-lated in the time domain by a hydrodynamic-electromagnetic model, and aboundary element code is used for the estimation of the hydrodynamic para-meters. The arrays consist of four heaving point absorbers with diameter equalto 4 m, height equal to 0.8 m and draft equal to 0.4 m. The nominal powerof the linear generator was set equal to 10 kW. Fig. 3.10 shows the different
27
Figure 3.10. Simulated array layouts: (a) linear, (b) square and (c) rhombus. (Frompaper V).
array geometries simulated. For each layout, four distances among the unitswere investigated: 20, 40, 80 and 120 m, i.e. 5, 10, 20 and 30 diameters. Thewave direction rose was discretized into 12 direction bins, every 30◦.
For each incident wave direction, 90 sea states were simulated, considering9 significant wave height Hs (0.25 : 0.5 : 4.25 m) and 10 peak periods Tp (2.5 :1 : 11 s). The monodirectional JONSWAP spectrum was used. Thus, for eachsea states the power output of the farms was calculated and the power matrixobtained.
Hence, the power matrix has been used for a site specific study, in order toevaluate the optimal wave farm designs for four locations off the Italian coasts:Alghero, Mazara del Vallo, Ponza and La Spezia (Fig. 3.11).
Figure 3.11. Location of the study sites (From paper V).
The aim was to find out the most productive wave farm configuration, i.e.the one providing the highest annual energy output (AEO), at each study site.For this purpose, it was considered that each array layout can be deployed withdifferent absolute orientations (with respect to north). According to the array’ssimulations computed with different incident wave angles, six geographicaldeployment orientations are possible for linear and rhombus layouts and threefor the square one. Each oriented layout was also simulated considering fourdifferent distances between units, thus leading to 60 wave farm designs foreach site. By assuming 12 directions of wave propagation, the probability ofoccurrence WC(i, j) of the sea state (i, j) was multiplied by the correspondingpower matrix PM(i, j) taking into account the direction of wave propagation
28
with respect to the oriented layouts. Then, the annual energy production ofthe farm was obtained by summing the energy production over all the twelvewave directions.
3.2 Wave tank experimentsIn order to verify the results from the array behaviour simulations and optimi-zations, a scale experiment project has been started, in conjunction with thestudy of different control strategies.
As a first step, an electrical linear motor has been used to build a 1:10model of the PTO, driven by a cylindrical buoy in the wave tank (Fig. 3.12).The experimental setup is divided into four parts: the floater, a connectionline between the buoy and the translator (i.e. a rope), the PTO and a controlcomputer. The setup allows the use of constant damping and other controlstrategies like latching and adjustable damping. The float was made out ofStyrofoam and coated with glass fiber. A pulley system (2 on the bottom ofthe tank and 2 on the gantry) ensured the transmission of the forces from thebuoy to the translator. Like the real existing PTOs, the model was equippedwith two end stop springs on both ends of the rod. Characteristic dimensionsof the full scale and scaled parts are reported in Table 3.3.
The goal of this first experiment was to assess how well the device performby controlling the damping with a constant optimum value for each one of 41different sea states. The wave height ranged from 0.75 : 0.5 : 3.25 m, whilethe wave periods ranged between 3.5 : 1 : 9.5 s. The optimal values have beenobtained in advance with a time domain two-body simulation by a parametersweep over the damping values. Afterwards, the result is applied to the sca-led model and the power output measured. In addition, we wanted to verifysimulations output results with real data.
The scale tests were carried out at the COAST lab of Plymouth University(UK). The wave basin has a width of 15.5 m, a length of 35 m and a depth of3 m.
Table 3.3. Full scale and scaled geometrical characteristics.
1:1 1:10
Translator mass 6000 kg 6 kg
Buoy mass 5300 kg 5.3 kg
Stroke length 3 m 0.3 m
Buoy diameter 5 m 0.5 m
Buoy height 2.5 m 0.25 m
Sea states duration 14.22 min 4.5 min
29
Figure 3.12. Overview of the test setup at the COAST lab.
30
4. Results
4.1 Single device optimizationResults of the single device optimization allow to find the best setup for awave energy converter; however, in this specific case, there are "only" 89370different possible combinations with the values of R, d and Γ set. For thesereasons, the results of the genetic algorithm have been validated against para-meter sweep, i.e power output calculation from all the combinations of radius,drafts and PTO constants.
4.1.1 First GA validationComparison of the results from GA simulations and parameters sweep is shownin Fig. 4.1. Here only the graph with a fixed radius is shown (for the completerepresentation see paper I). The black cross represents the absolute maximumvalue. All the solution from GA are located in the region of maximum poweroutput of the WEC (i.e. our current objective function).
Figure 4.1. Optimal power solution obtained with the genetic algorithm (diamonds)vs parameter sweep (surface) for a fixed radius value (R = 5 m).
31
The GA code was able to find the exact optimal radius and draft 100% ofthe times. The diamonds in Fig. 4.1 are very close to the black cross (realoptimum) but not exactly overlapped to that because the value of Γ found bythe optimization routine differs from the real optimum Γ by at most 8%. Oneexplanation for this result can be related to the number of iterations that thetool has performed. A larger number of iterations would most likely lead to aperfect solution. However, the difference is neglectable for the results, in thesense that the final average power outputs calculated by the two methods haveanyhow less than 0.2% difference in all 20 simulations.
This is a simple case of our application; in fact, we know the shape ofthe cost function since we can calculate all the results from all the differentpossible combinations of variables by parameter sweep.
4.2 Array layout optimizationFig. 4.2 shows an example of results from paper II. In the left column thebest layout obtained for 4 and 9 WECs park with the gridded code is shown,
Figure 4.2. Example of best solution found by GAfor 4 and 9 WECs. Gridded code (left) and conti-nuos code (right).
while on the right column theresults from the continuouscode for the same array. Ingeneral, results have showna tendency for the devices toalign in a few number of li-nes as possible perpendicu-lar to wave direction. Thefact that WECs do not spreadas far as possible from eachother is a result of some posi-tive interaction due to scatte-ring and radiation. The con-tinuous code produces re-sults that tend to be simi-lar as the gridded one, butrequires a higher number ofiterations, since the solutionspace is much extended.
The interaction factor (q-factor) represent the effect of hydrodynamic inte-ractions between WECs in a park and is defined as:
q =Ptot
∑Nbi=1 PIS
i(4.1)
Ptot represents the total power of the array, while PISi is the power of the i-th
device in isolation. Fig. 4.3 shows that for parks up to 9 WECs, a configura-
32
tion with positive interaction gain (i.e. power production > power productionof the same amount of WECs in isolation) can be found by the tool in 250iterations. The q-factor is therefore larger than 1. However, the results arespecifically dependent on the size of the ocean area in relation to the numberof WECs.
In paper II it was also shown that both codes have advantages and disadvan-tages. The continuous code search within a smaller solution space, reachingconvergence faster, but the results are dependent on the choice of the grid. Onthe other hand the continuous GA finds the best separation distance betweendevices but it needs a very large number of iterations for parks with over 5devices to get the same solution as the gridded code. Since the gridded codehas in most of the cases produced a solution with higher power productioncompared to the continuous code, it was suggested as more suitable for theapplication. Regularity of the layout is also a design advantage related to theutilities.
The results also show that destructive interactions can influence the powerabsorption to a larger extent than positive interactions.
Figure 4.3. q-factor of park of different sizes. Best and worst solution from griddedGA code (code A) and continuous GA code (code B) (From paper II).
33
To characterize the evolution of the layouts through the GA iterations, thefollowing coefficients have been calculated for every layouts and iterations:
S =∑ShadowDistance+1
Nb(4.2a)
D =dtot
Nb(4.2b)
T = S ·D (4.2c)
S is a measurement of the shadowing effects within the park, and is givenby the sum of all the distances between shadowed buoys (Shadow Distance)divided by the total number of WECs in the park (Nb). D is the ratio betweenthe distance from all the WECs to a point that minimizes the total distance(dtot), and the number of buoys. It gives an indication of how much the parkis spread out in the available marine area. T is the overall coefficient given bythe product of S and D. The mean values over the best three layouts for everyiteration is shown in Fig. 4.4.
We can conclude that as the algorithm proceeds, the shadowing coefficientin average decreases until the minimum shadowing distance feasible accordingto the number of WECs and the area size is reached. Distance coefficient Dhas also a descending trend, meaning that the WECs don’t need to be spreadapart to get the maximum of the power output and that it is possible to reducethe ocean area occupied by the park without loosing power output.
Results from this study can be compared with some previous literature [13]-[14]-[15], where the optimal layouts show the same tendency towards align-ment perpendicular to the incident wave with a small offset angle, and theWECs are staggered in maximum two rows along the wave front direction.The general trends in the results and the q-factors values are consistent. Ho-wever, different models and parameter spaces have been used in the differentstudies, so that a straightforward comparison is difficult.
34
Figure 4.4. Coefficients for different park sizes: a) S, b) D, c) T (From paper II).
4.3 Multi-parameters array optimizationFig. 4.5 represents the values of the non-dimensionalized power ratio (nume-rator of equation (3.1)) as a function of the non-dimensionalized mass ratio(denominator of equation (3.1)). It shows that given a certain capital cost or,in other words, a fixed value of the mass, is it possible to get different powerproduction, according to the internal location of big and small buoys. Thehighest ratio between the relative produced power and the relative mass (with
35
respect to a park with all buoys of the smallest and largest geometry) is shownwith a red dot and consists of a park of 6 small and 3 big devices (shown inFig. 4.5 upper left part). In this work, we denote parks with devices of differentdimensions hybrid parks.
Figure 4.5. Value of the non-dimensionalized power ratio (numerator of equa-tion (3.1)) as a function of the non-dimensionalized mass ratio (denominator of equa-tion (3.1)). Upper left the optimal solution represented by the red dot.
Regarding the second application the best hybrid park solution is shown inFig. 4.6. The power production of this configuration was then compared withthe deployment of two distinctive parks of big and small WECs separately(located on two adjacent line facing the wave front). It was shown that the de-ployment of two of hybrid parks would give a total power output around 2.7%higher than having two distinguished homogeneous parks of big and small de-vices. It means that, theoretically, we can achieve larger power production ifwe deploy arrays of mixed sizes, for a given number of big and small devices.
Fig. 4.7 shows the optimal solution for the third case after 5000 iterations.
4.3.1 Second GA validationTo verify the reliability of the genetic algorithm optimization routine, a secondverification has been carried out by means of parameter sweep of all the possi-ble combinations in case 1. The results, shown in Fig. 4.8 and 4.9, have beenconfirmed.
36
Figure 4.6. Optimal solution found forcase 2 (From paper III)
Figure 4.7. Optimal solution found forcase 1 (From paper III)
Figure 4.8. Optimal solution found forcase 1 with genetic algorithm.
Figure 4.9. Optimal solution found forcase 1 with parameter sweep.
4.4 Deterministic array evaluation - A case studyFirst, the results from the JONSWAP spectrum simulations over all the seastates and directions have been used to understand the sensitivity of the ar-ray performance to the design parameters, i.e. geometrical layout, separatingdistance and wave direction. It can be concluded that, although the effect ofthe separating distance depends on the array layout, all farm geometries havesome common features:• the behaviour of the q-factor with respect to distance is typically not mo-
notonic, because hydrodynamic interactions depend on the ratio betweenWEC distance and wavelength [9,11],
• there is no optimal spacing between the units, but rather the best WECdistance should be selected as a function of the incident wave direction[9]
• the effect of wave interactions is inversely proportional to the distancebetween the units and it is a few percent once the separating distance islarger than 30D [13].
Regarding the specific real locations, results in term of best and worst designis shown in Fig. 4.10 for all the four geometries.
37
Figure 4.10. Wave farm designs maximizing/minimizing the annual energy productionat the study sites and associated energy gain (green) and loss (red). GeographicalNorth pointing towards the top of the page (From paper V).
The highest power output is obtained with a rhombus layout with WECsdistance equal to 20 diameters at Alghero and a linear layout with 5 diametersspacing at the other locations. In all the sites, the layouts should be oriented byaligning the most productive wave farm orientation with the prevailing wavedirection. The optimum wave farm designs lead to power gains from 1.5% (atAlghero) to 3.4% (at Mazara del Vallo). However, the difference in the annualenergy output between the best and worst array configuration is rather small,between 7% and 9%, depending on the deployment location.
The results showed that an increase in the energy production is possibleboth in theoretical unidirectional wave fields and in real multi-directional waveclimates. It was possible to design a four WECs farm off the Italian coastsperforming better than four isolated devices, taking advantage of constructivewave interferences. However, as long as the devices are separated by at least10 buoy diameters and the layouts are oriented to achieve the maximum energyabsorption for the prevailing wave direction, the effect of wave interactions onabsorbed power is quite low, as already observed in previous studies on smallwave farms [13].
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4.5 Wave tank experimentsFig. 4.11 and 4.12 show the optimal damping values and the output power,respectively, obtained with the simulations for the selected sea states. It canbe seen that the damping values increase over the wave height and decreaseover the wave period. Fig. 4.13 shows the power matrix obtained in the scaletest and Fig. 4.14 shows a comparison with the theoretical one. It can be no-ticed that the real absorbed power is much less than the simulated one. Thereasons for this huge difference lies in the high non linear and non uniformlydistributed friction, which couldn’t be completely compensated during the ex-periments. Furthermore the data were very sensitive to small changes betweenthe sea states due to the high scaling factor.
The theoretical obtained power matrix shows that the absorbed energy in-creases with increasing wave height and has a maximum point for a certainwave period. By looking at the measured power matrix (Fig. 4.13) the trendis not as clear, due to the mechanical problems of the PTO and, subsequently,the bad repeatability of the tests.
Figure 4.11. Optimal damping values [kNs/m] resulted from parameter sweep (Frompaper IV).
Figure 4.12. Simulated power matrix [kW](From paper IV).
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Figure 4.13. Experimental power matrix [kW] (From paper IV).
Figure 4.14. Absorbed power in the wave tank / Absorbed power in the simulations[%] (From paper IV).
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5. Discussion
As for any modelling technique, the optimization procedure presented in thisthesis has some strengths and limitations. It has been proven to be reliablethrough validation with parameter sweep and faster in finding the optimum forproblems that depend on many variables. It can optimize device coordinatestogether with buoy’s and generator’s parameters simultaneously. The numberof WECs in the array is theoretically unlimited.
However, there are some limitations related to:• Limited numbers of variables that are optimized at this stage;• Unidirectionality of the incoming waves;• Difficult choice of the cost function for arrays of mixed sized WECs,
which determine the outcome of the optimization;• High computational time for big parks of fully hydro-dynamically inte-
racting devices.• The variability of the wave climate, which is not yet included in the
model.• Genetic algorithm parameter choice.• Limitation of the full optimization problem by either the choice of the
total power rated of the park (with a fixed number and size of devices)or the choice of the number of devices (with undefined dimensions andtherefore undefined rated power of the park).
In the next few sections some of the most significant points aforementionedwill be further discussed.
5.1 Cost functionAs previously mentioned, the choice of the cost function in an optimizationproblem is crucial. In paper III three different functions have been used andcompared.
As in most engineering problems, there are usually more than one objectiveto achieve which are conflicting with each other. In our case, if we just look atthe simplified problem, we want to maximise the power output and minimizethe costs (represented here by the mass). This multi-objective optimizationproblem has been transformed for simplicity into a single-objective optimiza-tion problem, by including both objectives in the same function as a ratio.
To investigate the influence of the choice of cost function on the resultssome comparisons have been carried out for Case 1 outlined in section 3.1.3.
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In fact, it was possible to evaluate all the possible solutions by parametersweep given the limited variable space. The following cost functions havebeen computed:
f Acost =− (Ptot−Psmall)/(Pbig−Psmall)
(mtot−msmall)/(mbig−msmall)(5.1)
f A′cost =− (Ptot−Psmall)/(Pbig−Psmall)+1
(mtot−msmall)/(mbig−msmall)+1 (5.2)
f Bcost =−Ptot (5.3)
f Ccost =− Ptot
mtot(5.4)
f Dcost =−q = Ptot
(∑Nbigm=1 PIS
big+∑Nsmalln=1 PIS
small)(5.5)
f Acost, f B
cost and fCcost have been previously introduced, while f A′
cost is a mo-dification of f A
cost and f Dcost is the q-factor calculation for arrays with WECs
of different sizes, where the power output of the park is compared with thepower of a corresponding number and size of devices in isolation. Eqs. (5.3)and (5.5) represent pure single objective optimization, while Eqs. (5.1), (5.2)and (5.4) represent a multi-objective optimization problem simplified into asingle-objective one.
Another possible single-objective optimization approach would be to cal-culate the minimum distance from an ideal point (which often doesn’t exist)that optimizes all the objective functions, called utopia.
Cost function f Acost
Fig. 5.1 shows the results in the objective space when the nondimensionalizedcost function in Eq. (5.1) is used. On the axis are shown the two objectivesseparately, which both range from 0 to 1. Maximizing this ratio (or, equi-valently, minimizing its negative value) means that the best solution will bethe point belonging to the line with the highest slope. Conversely, the worstsolution is the one belonging to the line with the lowest slope value. The non-dimensionalization of the terms leads to the same relative weight between theincrement in the mass (ΔM) and the respective increment in power production(ΔP). Solutions are now aligned along the bisector of the first quadrant.
This evaluation function has the problem that the point {0,0} cannot beclassified, since, graphically, it belongs to all the lines with different slopesand, numerically, results in an indeterminate form.
Best and worst solutions according the minimum distance from the utopiapoint are also plotted in Fig. 5.1.
Cost function f A′cost
Fig. 5.2 shows the results in the objective space when the undimensionalizedcost function in Eq. (5.2) is used. This time, the numerator and denominator
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Figure 5.1. Undimensionalized power output versus undimensionalized mass of a parkof 9 WECs in a fixed grid. The grey lines represent the highest and lowest slope of thesolutions points.
of the cost function range both from 1 to 2. As before, optimizing the ratioin Eq. (5.2) leads to the solution point belonging to the line with the highestslope.This cost function solves the problem of the point {0,0} arising with f A
cost andgive similar results in the ranked solution, but not identical. To be noted alsothe fact that the points {1,1} and {2,2} (i.e. park with only small devices andpark with only big devices) have the same fitness value.
Cost function f Bcost
Optimization only upon power output is a reasonable choice only for hydro-dynamic considerations among identical devices (as in paper I and II) or fora qualitative first approach. With parks of WECs where the dimension of thefloater is free to change, optimizing over power output will clearly lead to thebiggest possible devices, which is misleading.
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Figure 5.2. Undimensionalized power output versus undimensionalized mass of a parkof 9 WECs in a fixed grid. The grey lines represent the highest and lowest slope of thesolutions points.
Cost function f Ccost
Fig. 5.3 shows the scatter plot of all the possible solutions in the objectivespace when the objective function is the dimensional ratio between power out-put and mass (as crude estimation of capital costs), i.e. Eq. (5.4). On the axisare shown the two objectives separately. In this case the solutions of our pro-blem in the objective space align themselves along a straight line with slope< 1 not passing through the origin, so that the best solution results in the ar-ray with lowest mass (9 small devices) and the worst solution results in thearray with highest mass (9 big devices). It can be seen that the relative weightbetween the power output and mass is not the same, i.e. the increment inthe mass (ΔM) when substituting a small device with a big device is higherthan the respective increment in power production (ΔP), as already discussedin section 4.3. It is to be noted that in case ΔP > ΔM, one would obtain theopposite results in term of best/worst solutions.
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Figure 5.3. Power output versus mass of a park of 9 WECs in a fixed grid. The greylines represent the highest and lowest slope of the solutions points.
Cost function f Dcost
Optimizing over the q-factor when the park is composed of different sizedWECs leads to the same optimal solution as in the other two undimensiona-lized cost functions. Since the devices in the array are not all identical, opti-mizing over the q-factor is not the same as optimizing over the power output.Fig. 5.4 shows the numerical values for every park and its variation accordingto the number of big devices. It can be seen that some certain array layoutslead to a more advantageous hydrodynamic interactions.
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Figure 5.4. q-factor values for all the 512 layouts (top) and q-factor values vs numberof big devices in the layout (bottom). Full size of array is 9 WECs.
In Fig. 5.5 the variation of all the cost functions as a function of the numberof big devices in the array is plotted. One other way to see the same resultsis highlighted in Fig. 5.6, where the percentage number of big devices in thearray is plotted against the all different park configurations, ordered from theworst to the best. Results in term of best and worst layout, according to thefive different cost functions, are reported in Fig. 5.7.
To summarize, it can be concluded that:• if ΔM ΔP, decisive is the vertical spread of the solutions from the
bisector of the first quadrant, related to the total mass of the array. Inother words, given a certain array formation (for example 6 small devicesand 3 big devices), what is important is the gain in the power outputaccording to the different internal location inside the array, which leadsto different hydrodynamic interactions and thus different power output.However, assuming the same relative weight between power output andmasses is not straightforward and is a choice that has to be carefullyevaluated a priori.
• if ΔM �= ΔP, decisive is the horizontal spread of the solutions (seen asthe slope of the interpolant line). In other words we are comparing dif-ferent formations only according to the number of small and big devicesincluded, but their arrangement inside the group is not influential.
• when using dimensional cost functions ( f Bcost and fC
cost) the ranking of thesolutions is determined by the number of big WECs, and the hydrody-namic interactions resulting from the internal locations of big and smalldevices are not influential.
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Figure 5.5. Comparison of all the fcost as a function of the number of big WECs in thelayouts.
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Figure 5.6. Comparison of all the fcost by plotting the percentage number of bigdevices for each park configurations (ordered from worst to best on the x axis).
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Figure 5.7. Best layout (left) and worst layout (right) obtained with the different costfunctions.
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• when using nondimensional cost functions ( f Acost, f A′
cost and f Dcost), the dif-
ferent internal locations rather than the actual number of big and smallWECs determine the results of the optimization.
Hence, we see that the choice of the cost function is very critical for theoutcome of the optimization. In paper III we then suggested the use of aobjective non-biased economical cost function for the optimization processwith a single-objective optimization approach. This function would be definedas the ratio between the total income of the produced electricity (function ofthe output power) and the capital (CAPEX) and operational (OPEX) costs(function of the total mass).
fcost =−g(Pout)
h(mtot). (5.6)
Suitability of a pure multi-objective optimization instead is still to be evalua-ted.
5.2 Computational timeThe computational time between the PS and GA methods has been evaluatedwith the single WEC optimization and with 9 devices on a fixed grid (case 2 ofthe multi–parameter optimization). In single GA optimization (results showedin Fig. 5.8), the GA was able to save more than 88% of the computationaltime.
Figure 5.8. Simulation time (left) and number of iterations (right) performed with GAand PS for 20 identical simulations of a single device.
In the array size optimization the parameter sweep evaluation of the 512different layout possibilities took around 145 minutes, while the GA conver-ged after 350 min. For the GA to be faster it should have converged within 32iterations. In this case solution was found at iteration 73, because the routineevaluates the same solution several times. A feature to avoid redundancy isimportant for the future to be implemented in the code. Anyhow, when the
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parameter space is big, the CPU time of the parameter sweep will increase ata point that it will become infeasible.
In general, the CPU time is related to the inversion of the diffraction matrixin the hydrodynamic calculations, which increases exponentially by addingWECs in the array. Some approaches to speed up calculations with an accep-table level of accuracy have been shown by [8] with the introduction of aninteraction distance cut off or the grouping of devices in different clusters.
5.3 GA parameters sensitivityThe internal parameters of the GA routine have been selected by trial and errorto ensure fast convergence of the results. The trade-off between members ofthe population and number of iteration needs to be balanced. Thus, in thiswork we have chosen the maximum iterations as a compromise between fastCPU (small MaxIt) so that it is possible to study a large set of arrays and cases,and high reliability and repeatability in the results (high MaxIt).
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6. Conclusions
The research question stated in the introduction was: how can we optimize awave energy park of Uppsala WECs during the design process? would that beable to help reducing the cost of energy?
To help answer this questions a custom optimization model based on ge-netic algorithm has been built. The combination with a fast semi–analyticalmethod for the computation of the hydrodynamics has made possible the eva-luation of thousands of wave energy parks configurations within the procedure,which would have not been possible with a BEM methods software.
The optimization method applied to a single WEC’s parameters gave accu-rate results in little computational time, compared to parameter sweep of thesame variables.
Two layout optimization routines for arrays of identical point–absorber waveenergy converters were presented and their applicability and efficiency tes-ted. The solutions are characterized as spatial configurations which avoid de-structive hydrodynamical interactions. The solutions were independent fromthe number of WECs and consist of a regular geometrical pattern. This couldbe useful for extrapolation of general rules for layout deployment in relationto the size of the WEC and the ocean area in a pre–evaluation stage.
From an engineering point of view, the real gain from collective behaviourof WECs in a park is anyhow not likely to happen (or at least is yet to be provenby real data), due to all simplifications included in modelling approaches andsecondary effects related to real working conditions. With that in mind, andconsidered the complexity of the problem, the tool developed in this study canbe a useful and practical help in designing a configuration with the target valueof the q-factor 1. Given a specific wave climate, the wave energy convertercan first be optimized internally (parameters); subsequently, an array layoutoptimization can be performed avoiding a negative park effect and setting theq-factor close to 1.
From that we have moved on to optimization of parks with different WECsgeometry and generators, where a decisional support tool for effective deploy-ment is needed, given the even higher complexity and unpredictability of thehydrodynamic interactions. Even though the method allows simultaneous op-timization of radius, draft, PTO coefficient and coordinates of the WECs, thefirst three applications were done by fixing some of these parameters. In thisway it has been possible to validate the model against parameter sweep.
Different cost functions have been used and compared, highlighting the factthat the choice of cost function determines the outcome of the optimization. It
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has been concluded that an ideal objective and non–biased cost function wouldbe an economical one.
The effects of wave interactions in small arrays of point absorbers, hypot-hetically deployed at some Italian locations, has been investigated in a de-terministic way. The study was a first attempt of study small arrays in realmulti–directional wave climates. A site–specific optimization has been car-ried out for the selected study sites and new insights have been provided forwave farm design in real wave climates.
Regarding the physical wave tank experiment it can be concluded that someimprovements need to be implemented, mostly in the mechanical system of thepower take-off, in order to overcome the problems faced and get more reliableresults.
Challenges still to be overcome have been highlighted, such as:• The complete optimization of a big park or clusters (more than 25 WECs)
is still very computationally demanding; some solution need to be foundto reduce the computational time.
• The generator is still simplified as a linear constant damper;• The combination of the systematic optimization routine (i.e. the GA)
with a real multi–directional wave climate, applicable to real study sites;• The cost function is to be related to some economical assessment of the
wave power park.• A reliable and more robust mechanical system for the scale test is requi-
red.• Validation of the simulation results with physical data.
Nevertheless, optimization is just an instrument and, as any other modellingand engineering technique, requires hypotheses and simplifications. However,although the model leaves room for changes and improvements, this work hasshown that multi–parameter optimization of wave power arrays of the UppsalaUniversity WEC is possible and could provide an effective tool in reducingthe total cost of electricity. This will help enhancing the performance of wavepower plants and reducing the price of electricity.
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7. Future work
Many aspects of the present work still need to be further investigated. Cur-rently, we are working on the development of an optimization model whichtake into consideration the variability of the wave climate, as well as the dif-ferent incoming directions of the wave field.
Other interesting aspects that are being evaluated or will be studied in thefuture are the following:• A model suitable for case specific studies;• Validation of the genetic algorithm results in a tank test with more devi-
ces in array;• An economical model for a more robust and reliable cost function;• The inclusion of a more complex and realistic model to simulate the
generator behaviour.• The full optimization problem, meant as the simultaneous optimization
of all the variables, including wave climate variability, wave direction,number of WECs, layout.
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8. Summary of papers
Paper IParameter optimization in wave energy design by a genetic algorithm
The paper introduces a tool for optimization of a single wave energy converterradius, draft and PTO coefficient and a spatial layout of an array up to 9 devi-ces. The method is based on a real-coded discrete-variable genetic algorithm.Validation of the results against parameter sweep of the optimization variablesis carried out.The author built the model, performed the simulations and wrote the paper.Presented by the author at the 32nd International Workshop on Water Wavesand Floating Bodies in Dalian, China, 2017.
Paper IILayout design of wave energy parks by a genetic algorithm
In this paper results are shown when the tool is used to find the optimal spatiallayout of parks of identical devices up to 14 devices, both in a gridded anda continuous solution space (discrete- and continuous- variables genetic algo-rithm). Results from the two approach are then compared.The author built the models, performed the simulations and wrote the paper.Under revision for Ocean Engineering.
Paper IIIMulti-parameter optimization of hybrid arrays of point absorber Wave
Energy Converters
This paper investigates the presence of different sized WECs in the park viaoptimization process. Three different cases are investigated, together withthree different cost functions in the optimization scheme.The author built the models, performed the simulations and wrote the paper.Presented by the author at the 12th European Wave and Tidal Energy Confe-rence in Cork, Ireland, 2017.
Paper IVOptimal constant damping control of a point absorber with linear gene-
rator in different sea states: comparison of simulation and scale test
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In this paper a comparison of a numerical two-body simulation in time-domainand a 1:10 scale test performed at the COAST lab at Plymouth University wascarried out for a single floating point absorber. The PTO consists of a directdriven generator with a optimal constant damping coefficient.The author participated in the preparation and execution of the physical expe-riment at Plymouth University COAST Lab.
Paper VWave farm design in real wave climate: the Italian offshore
This paper investigates the effect of hydrodynamic interactions among diffe-rent parks of four devices. Different layouts, WEC separation distances andincident wave directions were considered to assess the effect of design para-meters on the power production. Then, a site-specific design optimization iscarried out for four Italian locations.The author contributed in building the model and performed the simulations.
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9. Svensk sammanfattning
För att säkerställa en framtid för hållbar energiförsörjning och låga koldioxid-utsläpp kommer förnybar energi spela en nyckelroll. Särskilt fokuserar dennalicentiatavhandling påvågenergi, som skapas genom att extrahera kraften somlagras i havets vågor.
För att vågenergi ska bli en kommersialiserad energiform, kommer det för-modligen krävas att vågkraftverk placeras ut i parker. Detta är fördelaktigtbland annat för de gemensamma elektriska kabelförbindelserna och förtöj-ningarna, minskningen av effektfluktuationerna och minskningen av installa-tions - och underhållskostnader. När det gäller parker ökar komplexiteten idesignprocessen enormt jämfört med den enskilda vågkraftverk, eftersom deolika designparametrarna (exempelvis hydrodynamisk och elektrisk interak-tion, dimensioner, geometrisk layout, vågklimat) påverkar varandra.
Uppsala universitet har utvecklat och testat vågkraftverk sedan 2001, ochdet första storskaliga vågkraftverket sjösattes 2006. Enheten klassificeras somen punktabsorbator och består av en linjär elgenerator påhavsbotten, som drivsi vertikal riktning genom en flytande böja vid ytan.
En av branschens svårigheter är idag att elkostnaden fortfarande är för högoch inte konkurrenskraftig; kapitalkostnaderna och driftskostnaderna är förhöga och överlevnaden till havs inte demonstrerad. Därför är ett steg för attförsöka minska dessa kostnader utvecklingen av pålitliga och snabba optime-ringsverktyg.
I denna licentiatavhandling har ett första försök om systematisk optimeringför parker av vågkraftverk föreslagits. En genetisk algoritm (GA) har använtsför att optimera bojens geometri och dämpningskoefficienten hos generatornför enskilda vågkraftverk. Därefter har den optimala uppläggningen av parkermed upp till 14 enheter studerats genom att använda tvåolika koder: en kon-tinuerlig och en diskret kodad GA. Vidare har metoden utvecklats till studierav parker bestående av vågkraftverk med olika dimensioner. En determinis-tisk utvärdering av parker i verkligt vågklimat har ocksåutförts. Slutligen harfysiska skalexperiment initierats vilket möjliggör validering av resultaten.
En multiparameteroptimering av vågkraftverk från Uppsala universitet harvisat sig vara möjlig och representerar ett verktyg som kan bidra till att minskaden totala kostnaden för el, förbättra vågkraftverkens prestanda och förbättratillförlitligheten.
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10. Acknowledgements
I would like to express my sincere gratitude to my main supervisor MalinGöteman for her invaluable help, advice and support.
The SUPERFARMS group members: Simon, Jens, Jan I., Mikael, thankyou for the time spent preparing and performing the experimental work inPlymouth.
I would like to thank Mats Leijon for the wonderful opportunity to be hereat the Division of Electricity doing my PhD in such an interesting field.
Many thanks also to Deborah, Edward, Martin, Peter, Kiran, Tara, Liz andOliver from Plymouth University. Your help was precious.
Thanks to all the members of the group in Politecnico di Milano, in Bolognaand in Plymouth.
Thank to Johan Blåbäck for useful discussion and advice regarding the ge-netic algorithm.
A very special thanks to Valeria, which inspired me into wave energy someyears ago.
Anke, Juan, Valeria, Johan, Saman, Arianna, Markus, Katja, Milena, Ulf,Jessica, José, Nattakarn, Joel, thank you for the wonderful moments duringfree time; from Norreda to Valborg to Sunday afternoon fika. I know I’m notalone in Uppsala.
To all my colleagues and friends here at the Division of Electricity, thankyou for all the time spent together, at conferences, fika time and lunches. I’mreally happy to have met and work with such wonderful people.
Thomas Götschl, Emma Holmberg, Lena Gamova and Ingrid Ringård, thankyou for the technical and administrative support. Skatteverket is not a myste-rious word any more.
Cristina, thank you for being always there for me from distance.Last but not least, a huge thanks to my always supporting family: my pa-
rents Tiziana and Giampiero, my sister Giulia and my brother Davide togetherwith Caroline.
I would finally like to thank the Swedish Research Council, the SwedishEnergy Agency, StandUp for Energy, the Lundström-Åman foundation, andMiljöfonden for supporting my research.
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