Numerical and experimental modelling for wave energy arrays optimization

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UNIVERSITATISACTA UPSALIENSIS

UPPSALA

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1968

Numerical and experimental

modelling for wave energy arrays optimization

MARIANNA GIASSI

ISSN 1651-6214 ISBN 978-91-513-1012-1

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Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 20 November 2020 at 09:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Gregorio Iglesias (University College Cork).

Abstract

Giassi, M. 2020. Numerical and experimental modelling for wave energy arrays optimization.

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1968. 85 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-1012-1.

Many wave power conversion devices, especially point-absorbers, do not provide alone the necessary amount of converted electricity to be cost effective, instead they are designed to be deployed in arrays of many units. Such arrays, or parks, can satisfy a large-scale energy demand, reduce the costs of the produced electricity and improve the reliability of the system.

The performance of a wave energy park is affected by multiple and mutually interacting parameters, and the complex problem that arises during its design is called array optimization.

The scope of the present thesis is to study such systems and their design, by the development of an optimization routine able to predict the best layout of a wave energy park under fixed constraints. The wave energy converter considered is the point-absorber developed at Uppsala University, which consists of a linear electric generator located on the seabed and a floating buoy at the surface.

An optimization routine based on a genetic algorithm was created, which allows simultaneous optimization of the geometry of the buoys, the damping coefficient of the linear generators and the geometrical layout of the park.

Finally, an experimental campaign with a single device and three arrays of six devices was conducted in order to compare the theoretical results with experimentally acquired data.

The results identify optimal configurations of wave energy arrays, and highlight the effect of optimizing upon different objective functions, including economical ones. In the experiments, standard models and common assumptions used for wave energy park optimizations were tested against realistic conditions.

Keywords: Wave energy, Point absorber, Arrays, Parks, Farms, Optimization, Array layout, Genetic algorithm, Cost function, Wave tank experiment, Hydrodynamics, Economical model Marianna Giassi, Department of Electrical Engineering, Electricity, Box 534, Uppsala University, SE-751 21 Uppsala, Sweden.

urn:nbn:se:uu:diva-420773 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-420773)

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To my grandparents Ai miei nonni

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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 Giassi M, Göteman M, Parameter optimization in wave energy design by a genetic algorithm;

Proceedings of the 32nd International Workshop on Water Waves and Floating Bodies, Dalian, China, April 23–26, IWWWFB (2017) II Giassi M, Göteman M, Layout design of wave energy parks by a

genetic algorithm;

Ocean Engineering, 154 (252–261), (2018) doi:10.1016/j.oceaneng.2018.01.096

III Giassi M, Göteman M, Thomas S, Engström J, Eriksson M, Isberg J, Multi-parameter optimization of hybrid arrays of point absorber wave energy converters;

Proceedings of the 12th European Wave and Tidal Energy Conference, Cork, Ireland, August 27–31, EWTEC (2017)

IV Giassi M, Castellucci V, Engström J, Göteman M, An economical cost function for the optimization of wave energy converter arrays;

Proceedings of the 29th International Ocean and Polar Engineering Conference, Honolulu, Hawaii (USA), June 16–21, ISOPE (2019) V Giassi M, Castellucci V, Göteman M, Economical layout optimization

of wave energy parks clustered in electrical subsystems;

Applied Ocean Research, 101, 102274, (2020) doi:10.1016/j.apor.2020.102274

VI Giassi M, Thomas S, Tosdevin T, Engström J, Hann M, Isberg J, Göteman M, Capturing the experimental behaviour of a

point-absorber WEC by simpliﬁed numerical models;

Accepted for publication in Journal of Fluids and Structures, (2020) VII Giassi M, Thomas S, Shahroozi Z, Engström J, Isberg J, Tosdevin T,

Hann M, Göteman M, Preliminary results from a scaled test of arrays of point-absorbers with 6 DOF;

Proceedings of the 13th European Wave and Tidal Energy Conference, Naples, Italy, September 1–6, EWTEC (2019)

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VIII Giassi M, Engström J, Isberg J, Göteman M, Comparison of wave energy park layouts by experimental and numerical methods;

Journal of Marine Science and Engineering, 8(10), 750 (2020) doi:10.3390/jmse8100750

IX Göteman M, Giassi M, Engström J, Isberg J, Advances and challenges in wave energy park optimization - A review;

Frontiers in Energy Research 8:26, (2020) doi:10.3389/fenrg.2020.00026

Reprints were made with permission from the publishers.

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

1. Bozzi S, Giassi M, Moreno Miquel A, Bizzozero F, Gruosso G, Archetti R, Passoni G, Wave farm design in real wave climate: the Italian offs- hore; Energy, 122 (378–389), (2017)

2. Thomas S, Giassi M, Göteman M, Eriksson M, Isberg J, Engström J;

Optimal constant damping control of a point absorber with linear ge- nerator in different sea states: comparison of simulation and scale test;

Proceedings of the 12th European Wave and Tidal Energy Conference, Cork, Ireland, August 27–31, EWTEC (2017)

3. Göteman M, McNatt C, Giassi M, Engström J, Isberg J, Arrays of point- absorbing wave energy converters in short-crested irregular waves; Ener- gies, 11(4), 964, (2018)

4. Göteman M, Mathew J, Engström J, Castellucci V, Giassi M, Waters R, Wave energy farm performance and availability as functions of weather windows; Proceedings of the 3rd International Conference on Renewa- ble Energies Offshore, Lisbon, Portugal, October 8–10, RENEW (2018) 5. Göteman M, Giassi M, McNatt C, Wave energy park interactions in short-crested waves; Proceedings of the 33nd International Workshop on Water Waves and Floating Bodies, Guidel-Plagues, France, April 4–

7, IWWWFB (2018)

6. Thomas S, Giassi M, Eriksson M, Göteman M, Isberg J, Ransley E, Hann M, Engström J, A model free control based on machine learning for energy converters in an array; Big Data and Cognitive Computing, 2(4), 36, (2018)

7. Thomas S, Giassi M, Göteman M, Hann M, Ransley E, Isberg J, Eng- ström J, Performance of a direct-driven wave energy point absorber with high inertia rotatory power take-off ; Energies, 11(9), 2332, (2018) 8. Tosdevin T, Giassi M, Thomas S, Engström J, Hann M, Isberg J, Gö-

teman M, Ransley E, Musiedlak PH, Simmonds D, Greaves D, On the calibration of a WEC-Sim model for heaving point absorbers; Procee- dings of the 13th European Wave and Tidal Energy Conference, Naples, Italy, September 1–6, EWTEC (2019)

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Nomenclature and abbreviations

Symbol¹ SI Unit Quantity

a m Wave amplitude

AEPy kWh/y Annual energy production

B kNs/m Radiation damping

c m/s Wave celerity

CapEx EUR Capital expenditure

CF_y EUR/y Cash ﬂow at year y

d m Buoy draft

F N Total force

fcost variable Cost/objective function

fexc kN/m Freq. dom. amplitude of the excitation force

FIT EUR/MWh Feed-in tariff

g m/s² Gravitational acceleration

h m Water depth

H m Wave height

H_rms m RMS wave height

Hs m Signiﬁcant wave height

It_max - Maximum GA iterations allowed

k 1/m Wave number

y Lifetime

m kg Total mass

madd kg Added mass

m_b kg Buoy mass

mbig kg Total mass of arrays consisting of ﬂoats with geometry #2

msmall kg Total mass of arrays consisting of ﬂoats with

geometry #1

m_t kg Translator mass

n - Number of iteration without improvements

N_b - Number of buoys/WECs in the park

nss - Number of electrical substations in the park OpEx_y EUR/y Operational expenditure per year

P kW WEC power

P¯ kW Time averaged WEC power

1The symbols used in the comprehensive summary might differ from notation used in the published papers.

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Symbol¹ SI Unit Quantity

P¯_big kW Time averaged power of arrays consisting of ﬂoats with geometry #2

P¯_small kW Time averaged power of arrays consisting of

ﬂoats with geometry #1

Ptot kW Total array power

P¯tot kW Time averaged array power

P^SW kW Power of WEC in isolation

P^PTO W Experimental power from PTO

P^ΔL W Experimental power from buoy motion

q - Interaction factor

¯r= (r,θ,z) m, deg Cylindrical coordinates

R m Buoy radius

R1 m Buoy radius R1

R2 m Buoy radius R2

r_d m Distance from IMU to disc center

s % Discount rate

S m² Wetted buoy surface

T s Period

T_e s Energy period

¯v= (u,v,w) m²/s Fluid velocity

¯x= (x,y,z) m Cartesian coordinates

y - Year

z m Vertical position of the ﬂoat

Γ kNs/m Generator damping coefﬁcient

η m Free surface elevation

λ m Wavelength

λ0 m Wavelength (deep water approx.)

ρ kg/m³ Density

φ m²/s Velocity potential

φD m²/s Diffraction potential

φI m²/s Incident potential

φR m²/s Radiation potential

φS m²/s Scattering potential

ω rad/s Angular frequency

ωd rad/s PTO angular velocity

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Symbol¹ Quantity

R(r) Radial eigenfunctions

Θ(θ) Angular eigenfunctions

Z(z) Vertical eigenfunctions

Amn Unknown coefﬁcients

B_mn Unknown coefﬁcients

Hn,Jn Bessel function

K_n,In Modiﬁed Bessel function

L Distance buoy bottom-sea ﬂoor

A0,n Incident wave coefficient α˜mn Combined coefficients inφ_Dî(II) amn Coefficients inφ_Dî(I)

V Velocity of the buoys in heave

Tmln Expression for Graf’s addition theorem

k_m Wave numbers

λm Variables inφ_D^i(I)

B_D Single-body diffraction matrix BR Single-body radiation matrix Abbreviations Description

A, B, C, D, E, F ID of experimental buoys A1, A2, A3 Array layout 1, 2 and 3

CapEx Capital expenditure

CMA Covariance matrix adaptation

D Damping ID in the experiment

DoF Degree of freedom

DS Damping sweep ID in the experiment

G Gauges

GA Genetic algorithm

IMU Inertia measurement unit

IW Irregular wave

LCOE Levelized cost of energy

NPV Net present value

OpEx Operational expenditure

PA Point-absorber

PS Parameter sweep

PTO Power take-off

RE Renewable energies

RW Regular wave

UU Uppsala University

WEC Wave energy converter

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1. Introduction

We do not inherit the earth from our ancestors, we borrow it from our children.

Native American Proverb

In recent years, many governments and authorities around the world are put- ting effort into the transition from fossil fuels to more sustainable sources of energy, in order to reduce the carbon emissions and to mitigate climate change.

In addition, changes in the prices of fossil fuels, an increase of energy demand and the declining cost of many renewable technologies have encouraged the pursuit of sustainable forms of energy supplies.

Renewable energies (RE) represent a form of energy that comes from natural resources and that can renew itself over cycles related to natural pheno- mena (e.g. sun, tides, waves, winds, river ﬂows, earth’s heat). Among various RE technologies, ocean energy, i.e. the energy stored in seawater, represents a source of enormous potential, if efﬁciently exploited. Tidal currents and tidal ranges, ocean currents, waves, ocean thermal energy and salinity gradient are different variations of ocean energy resources. Wave energy, i.e. the energy stored in the ocean waves, is the main subject of the present thesis.

Over the past 40 years, wave energy conversion has been widely studied, but some simple applications of it were already designed and implemented starting from the end of the 18th century. However, the research and development of wave energy technologies has gained particular attention and funding after the oil crisis in the 1970s.

Since then, many wave energy converter (WEC) technologies have been designed, patented and tested all over the world. They differ from each other in their shapes, dimensions, operational principles, locations in the sea, power take-off (PTO) and directional characteristics, for example. However, to produce electricity in the MW range, many technologies will have to be deployed in commercial parks of many units, called "arrays" or "farms". This is particu- larly true for small point-absorber (PA) devices. The full performance of such energy systems is affected by the interaction among the devices, which can be hydrodynamical (scattered and radiated waves), electrical (grid connection, sea cables, substations), mechanical (shared mooring and foundations) and economical (shared capital and operational costs).

Today, there is no device that has yet reached the necessary maturity for full scale commercialization. This is mostly due, among other aspects, to the following issues: it is extremely difﬁcult to design and build devices that are reliable and able to survive the harsh ocean environment, which is regu- larly subjected to storms and extreme weather conditions; secondly, the cost of the technology is still too high, which implies that only large-scale wave

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energy projects would be economically viable, due to costs sharing in the in- frastructures and a large-scale production. However, it is challenging to secure investments in the ﬁeld, also due to a long history of failures.

For these reasons, the reduction of the costs would help the technology to reach the commercial stage and therefore it is one of the main goals of the Eu- ropean Union and the Swedish Energy Agency action plans [1, 2]. One way to reduce the costs is to deploy large arrays consisting of many units. The improvement of the technology reliability and durability through development of models of predictions is one of the goals, among others, that needs addressing;

increased knowledge is also needed when it comes to up-scaling conceptual individual units to parks. On the other hand, as the system increases in size, so does its complexity.

1.1 Aim of the thesis

The goal of the thesis is to find a way to systematically optimize a park of point-absorber wave energy converters, in order to design economically feasi- ble wave energy arrays. The tool should be reliable and useful to be used in the design process, in order to minimize the costs and maximize the electricity production or the revenue. The work aims to include as many influencing parameters as possible and to develop a simulation routine able to predict the best layout of a wave energy parks under fixed constraints.

1.2 Thesis outline

In order to achieve the aforementioned goals, two complementary approaches have been pursued: a numerical and an experimental method combined. An optimization routine has been created from scratch based on a genetic algo- rithm (GA), which allows the optimization of the geometrical layout of parks of many devices. Subsequently, an economical model of the wave energy converter developed at Uppsala University (UU) has been built and nested into the optimization routine, allowing the economical optimization of wave energy parks. Finally, an experimental campaign has been conducted in order to compare the theoretical results with experimentally acquired data.

The thesis is divided into two main parts:

Part I:

Describes the background knowledge upon which this work is based. More speciﬁcally, chapter 2 presents an introduction to wave energy, the description of the WEC developed at Uppsala University, wave energy parks and a brief literature review of the research ﬁeld. Chapter 3 outlines the theoretical

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fundamentals of the modelling work, including linear potential ﬂow theory, wave-structure interaction and the genetic algorithm optimization routine.

Part II:

Describes the main work of this doctoral project. Chapter 4 outlines the optimization model created, including the economical model and the simulation work. The experimental campaign is presented in chapter 5 while results are displayed in chapter 6, both for the numerical and experimental modelling, and further discussed in chapter 7. Conclusions are drawn in chapter 8. A suggestion of improvement and future work that could follow this thesis is expressed in chapter 9.

A brief summary of the papers included in this thesis is outlined in chapter 10.

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Part I:

Background

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2. Wave energy

The shore is safer, but I love to buffet the sea.

Emily Dickinson

This chapter provides an overview on wave energy and the role that wave energy parks play in the development of this form of renewable energy. The wave energy converter developed at Uppsala University, upon which this the- sis work is based, is presented. The ﬁnal section outlines advantages and challenges of wave energy parks and introduce the wave energy park optimi- zation problem. Finally, a short survey of the current state of the art regarding optimization of arrays of wave energy converters is made.

2.1 General aspects

Wave energy originates from the irradiation of the sun, which asymmetrically illuminates the Earth’s surface at different latitudes. From the resulting tempe- rature variation follows an atmospheric pressure change, which induces motion of air masses from areas with high to low pressure, creating winds. Winds transfer their energy into the waves by blowing over the ocean surface, and waves store this energy as potential energy (in the mass of water displaced from the mean sea level) and kinetic energy (in the motion of water particles). The height and the period of the resulting waves are inﬂuenced by the wind speed, the duration the wind blows and the length of the generation area (also called

Figure 2.1. Annual mean wave power density (colors) and annual mean direction of the power density vectors (→) (Source: [3]).

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fetch) [4]. The distribution of wave energy over the world is uneven and varies seasonally, which is a common feature of RE sources. Thanks to the ﬂux of regular westerly winds, the world coastal areas on the west side of continents, mostly between 40^◦ – 60^◦ latitude, have the highest incident wave potential (Fig. 2.1). Along the Swedish west coast the energy ﬂux is estimated to be around 5 kW/m [5], which is considered as a "mild" wave climate compared to open Atlantic coasts like UK or Portugal.

Compared to other renewable energy sources, such as wind and solar, wave energy has some advantages including a high density, persistence and good predictability. In fact, waves transport energy efﬁciently, since they travel long distances without much energy loss. The theoretical potential of wave energy in the world has been estimated to be around 3 TW [6].

2.2 Wave energy converters

Technologies to extract wave energy are called wave energy converters. The basic wave energy conversion process can be stated in very general terms as follows: the force produced in a system by an incident wave causes relative motion between an absorber and a reaction point, which acts directly on, or drives a working ﬂuid through, a generator prime mover [4]. A huge number of WECs have been developed over the last decades [7], and they can be classiﬁed according to [8]:

1. operational principle (oscillating water columns [9], overtopping devices [10], oscillating bodies or wave-activated bodies [11]);

2. wave interaction with respective motions (heaving, surging, pitching);

3. location or distance from shore (shoreline, near-shore, offshore);

4. water depth (deep, intermediate, shallow waters);

5. power take-off system, which is the main mechanisms that convert the energy of the wave into mechanical and/or electrical energy (air turbines, linear generators, and hydraulic systems);

6. directional characteristics (point-absorber [12], attenuator [13], termina- tor [14]).

Different coastal areas with different wave climates characteristics will require different types of technologies. Therefore, even though until now no device has reached the required level of reliability for full scale commercialization, it will be more likely that a small variety of technologies will possibly conquer the future wave energy market.

2.2.1 Uppsala University WEC

Uppsala University has been developing a heaving point-absorber wave energy converter since 2006, which consists of a linear generator located on the seabed, connected via a rope to a buoy on the surface (Fig. 2.2). The linear

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Figure 2.2. Uppsala University WEC: (a) sketch and principle of operation (from Paper II); (b) photograph of a UU prototype deployment at the Lysekil research site.

generator consists of a ﬁxed part, called stator, and a moveable part, called translator. Permanent magnets are mounted on the surface of the translator, while the stator contains coil windings. When waves lift the buoy, the relative movement of the magnets with respect to the coils produce a variation in the magnetic ﬂux inducing electricity, according to Faraday’s law. Stator and translator are placed in a hull anchored to the seabed by a concrete founda- tion. Several UU-WECs are connected to an offshore marine substation [15], which operates as a hub for power transmissions and measurements. The marine substation collect the generated electricity and transmits it to an onshore measurement cabin [16].

2.3 Wave energy parks

Many wave power conversion devices do not provide sufﬁcient converted electricity to motivate for the costly offshore infrastructure and deployment, so that they are designed to be deployed in arrays of many units. Examples of pre-commercial visualization of wave energy parks are shown in Fig. 2.3, for different type of converters. Among the beneﬁts of such systems it can be mentioned:

• utility scale power generation;

• the modular construction;

• sharing of the electrical cables;

• redundancy and higher reliability to failures, including ability to perform maintenance without shutting down the production;

• quality or smoothness of the power output;

• cost-effective deployment.

These characteristics are likely to make the wave energy park a system with high power production, reduced costs/kW and high reliability.

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Figure 2.3. Examples of arrays of different wave energy converters: cSeabased AB (top left), cAquamarine Power Oyster (top right), cCarnegie CETO (bottom left),

Langlee Wave Power (bottom right).c

2.3.1 The wave energy park optimization problem

The park optimization problem arises from the complexity of the system, which is affected by external variables and by the internal interactions among devices. The external variables are, for example, the number of devices and their minimum separation distance, the park layout, mooring conﬁgurations, electrical and power take-off systems, rated power of individual devices and constraints of subsystems. The interaction among devices can be of different types and can be divided in:

• hydrodynamical (scattered and radiated waves),

• mechanical (shared mooring and foundations),

• electrical (sea cables, substations, grid connection),

• economical (shared capital and operational costs).

Costs, performance and reliability of the park are affected by such interactions.

Moreover, additional parameters that influence the wave energy array design are the size of the permitted area, the wave climate, the wave direction and va- riability, the water depth, currents, distance of the deployment site from shore and effects on marine life and coastal processes. Therefore, different optimal solutions are required for different initial constraints and external parameters (i.e. for different locations). All the aforementioned aspects have a direct or indirect effect on the power production and costs of the power plant. Fig. 2.4 shows a block diagram with some of the influencing variables of the array design, where arrows represent influence on the variable they are pointing at.

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The complexity of the problem is understood by observing the numerous mutual and multi-directional relations among the parameters. The sketch is still a simpliﬁed representation of the reality, and each "box" of Fig. 2.4 represents an engineering problem in itself.

In this context, researchers all over the world have tried to solve the problem by utilizing optimization routines, which are a systematic identification of the best solution from a set of alternatives, given some constraints. In other words, optimization can be seen as the process of finding the best value of an objective function in a defined parameter space. Considering a wave energy park and the variables shown in Fig. 2.4, the ideal optimization process should consider all aspects of the park, including all costs and total revenue over the lifetime, reliability, constraints regarding available ocean area, deployment and maintenance, allowed power fluctuations, water depth, hydrodynamic interaction, etc. However, performing full optimization of a wave energy park system is extremely challenging.

The optimization routine implemented in this thesis work is mainly, but not only, a layout optimization, i.e. the layout of the park is the optimized vari- able, while other aspects are considered constraints or variable inﬂuenced in cascade by the layout. The inﬂuenced variables are the hydrodynamic inte-

Figure 2.4. Variables that play a role in the optimization of a wave energy farm.

Arrows represent mutual inﬂuence.

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ractions, the length of electrical cables, economical aspects (capital and operational costs (CapEx and OpEx) and levelized cost of energy (LCOE)), while the constraints are the available ocean area, the number/size/rated power of the WECs, the wave climate and the lifetime. The optimization routine is outlined in chapter 3, while the objective functions that have been used are explained in details in chapter 4.

2.3.2 Literature review

Given the importance of solving the array optimization problem, in the recent years the ﬁeld has been widely studied with many different approaches and assumptions.

This section is a brief outline of the state of the art on the topic and is taken from the comprehensive review of optimization of wave energy arrays that was carried out in Paper IX. Here, only works that performed systematic optimizations with global metaheuristic optimization algorithms are reported.

Metaheuristic optimization methods [17] are useful tools when the solution space of the optimization problem is too large and multi-peaked [18]. They search the solution space until a sufﬁciently good solutions is found, given speciﬁed constraints and convergence criteria. Several different algorithms have been developed and applied to wave energy park optimization such as, among others, genetic algorithms, covariance matrix adaptation (CMA) and particle swarm algorithms.

Child and Venugopal [18] were the ﬁrst to apply the MATLAB GA toolbox for the optimization of the layout of 5 wave energy converters and compared the results to a parabolic intersection method. In Child et al. [19], the work was further developed to obtain optimal layouts or PTO settings for point- absorber arrays. Subsequently, GAs were utilized by many other researchers.

Sharp and DuPont [20] used a binary genetic algorithm to optimize the layout of 5 devices with constraints of different separating distances between the WECs, including a simpliﬁed economic cost function. In Sharp et al. [21]

the method was coupled with an active device control. Fang et al. [22] presented an adaptive mutation operator, in order to speed up the convergence.

Faraggiana et al. [23] compared a particle swarm optimization method with GA to optimize the separation distance between devices, the PTO tuning and the rated power of three WaveSub WECs. One other important research project in the ﬁeld of array optimization is the so called DTOcean (optimal design tools for ocean energy arrays) [24], where a CMA evolution optimization algorithm is utilized. The method was compared to both a GA and a particle swarm optimization algorithm by Ruiz et al. [25]. It was found that the CMA method required less computational cost, but the GA and the particle swarm algorithm performed slightly better. Ferri [26] compared the DTOcean optimization tool to a meta-model optimization method, and it was concluded

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that the former was more accurate, but computationally heavier. Sarkar et al.

[27] compared the GA with a Monte-Carlo method and it was found that the GA was able to ﬁnd layouts with higher q–factor than the Monte-Carlo simu- lations, using the same number of evaluations. Many different metaheuristic optimization algorithms were applied and compared in the studies by Neshat et al. [28, 29, 30, 31, 32], for arrays of submerged CETO WECs. Neshat et al.

[28, 29, 30] optimized the coordinates of the WECs to achieve maximal absor- bed power under the constraints of a given ocean area and minimal separation distance between the units. In Neshat et al. [31, 32], the optimization of the individual PTO parameters was included as well. Several algorithms such as the CMA strategy, differential evolution, grey wolf optimization, or a particle swarm optimization were compared and used to optimize the parameters. Me- taheuristic evolutionary algorithms with multi-objective optimizations were applied by Arbonès et al. [33, 34].

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3. Theory

Nature is written in mathematical language.

Galileo Galilei

This chapter outlines the mathematical fundamentals of the thesis, from the linear potential ﬂow theory used in the description of the ocean waves, to the wave-structure interaction model and the genetic algorithm optimization routine. All these models are combined in the array optimization routine pre- sented in this thesis, which will be fully explained in chapter 4.

3.1 Wave energy park numerical model

The in house built hydro-mechanical model of the wave energy park [35] as- sumes the validity of linear potential flow theory. It includes a semi-analytical model, based on the multiple scattering method [35], which is used to calcu- late the hydrodynamic coefficients such as added mass, radiation damping and excitation amplitude of the floats. The equations of motion of the converters are solved in the frequency domain and, from the motion of the devices, the power production is finally calculated. A similar version of this chapter was published in the author’s licentiate thesis [36].

3.2 Linear potential ﬂow theory

Linear potential flow theory, or Airy wave theory [37], is used to describe the physics of water waves in a three dimensional space, and is based on the following assumptions: inviscid and incompressible fluid, irrotational flow, small wave steepness, small ratio between amplitude of waves and water depth, small body motions and only the force of gravity is acting on the fluid.

The hypothesis of incompressibility results in the following continuity equation:

∇ · ¯v = ∂u

∂x+∂v

∂y+∂w

∂z = 0, (3.1)

where ¯v= (u,v,w) is the ﬂuid velocity in the Cartesian coordinates ¯x = (x,y,z).

The origin is set at the mean water line, with the z axis pointing upwards. The assumption of incompressible fluid and irrotational motion presuppose that the fluid velocity is a conservative vector field, implying the existence of a velocity potential. The velocity potentialΦ is defined by

¯v= −∇Φ, (3.2)

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and thus satisﬁes

u=dx

dt = −∂Φ

∂x = −Φx (3.3a)

v=dy

dt = −∂Φ

∂y = −Φy (3.3b)

w=dz

dt = −∂Φ

∂z = −Φz. (3.3c)

From Eq. (3.1) and Eq. (3.2) it can be concluded that the potential satisﬁes the Laplace equation:

ΔΦ =∂²Φ

∂x² +∂²Φ

∂y² +∂²Φ

∂z² = 0. (3.4)

To solve the Laplace equation it is necessary to apply speciﬁc boundary conditions:

• Bottom boundary condition: the sea bottom is assumed to be ﬁxed, horizon- tal and impermeable, then a no ﬂow condition should be applied:

w= −∂Φ

∂z = 0 at z= −h (3.5)

where h is the water depth.

• Kinematic free surface boundary condition: at the free surface the vertical velocity ˙z(t) of the ﬂuid particles should coincide with the wave velocity η (x,y,t), and it can be decomposed as˙

Φz= −ηt+ ηx· Φx+ ηy· Φy at z= η (3.6)

• Dynamic free surface boundary condition: at the free surface, the pressure should coincide with the atmospheric one, assumed constant, so we derive the condition from Bernoulli’s energy conservation balance as

η (x,y,t) + 1 2g

Φ²x+ Φ²y+ Φ²z

=1

g· Φt at z= η (3.7)

• Periodic boundary condition (here constrained for a wave travelling in the x-direction):

Φ(x,t) = Φ(x + λ,t) for every t (3.8a)

Φ(x,t) = Φ(x,t + T) for every x (3.8b)

whereλ 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 termηx· Φx+ ηy· Φy

andΦ²x+ Φ²y+ Φ²z. Moreover,η is unknown, being itself part of the problem solution. It is therefore not possible to ﬁnd an analytical exact solution. By

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analysing the order of magnitude of the non-linear terms, on the assumption that the wave length is higher than the wave height, so H/λ 1, we can neglect the second order terms. In this way, Φ and η are calculated at z = 0, instead of at z= η, ignoring the non-linearity laying in the fact the free boun- dary shape is unknown.

The linearized problem becomes:

Φxx+ Φyy+ Φzz= 0 at 0< x < λ and − h < z < 0 (3.9)

Φz= 0 at z= −h (3.10)

Φz+ ηt= 0 at z= 0 (3.11)

η (x,y,t) =1

g· Φt at z= 0. (3.12)

Eq. 3.11 and 3.12 can be joined as:

Φz= −1

g· Φtt at z= 0. (3.13)

From the Laplace equation combined with the new linear boundary conditions we obtain the dispersion relation, which describes how a ﬁeld of propagating waves consisting of many frequencies would separate or disperse due to the different phase velocities of the various components,

ω²= gk · tanh(kh) (3.14)

whereω = ²_T^π is the wave angular frequency and k=²_λ^π is the wave number and c is deﬁned as the phase velocity, also called celerity, of the waves:

c²=ω k

2

=g

k· tanh(kh). (3.15)

With some algebraic manipulation of the last equation, we will ﬁnd the relati- onship for the wave length:

λ = gT² 2π · tanh

2π λ · h

. (3.16)

In deep water h λ, so the wave length can be simply written as λ0=gT²

2π . (3.17)

Both the wave length and celerity depend only on wave period T . Finally, by separation of variables, a solution of the linearized problem, valid for waves travelling in the x-direction, can be found as:

η (x,t) = H

2 · cos(ωt − kx) (3.18)

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Φ(x,z,t) = ω · a

k ·cosh(k(h + z))

sinh(kh) · sin(ωt − kx) (3.19) where a= H/2 is the wave amplitude. The extension of the solution for waves travelling also in the y-direction is straightforward.

3.3 Wave-structure interaction

3.3.1 Multiple scattering and body radiation problem

Consider Nbpoint-absorber floats which, in the case of the present study, will be cylindrical buoys. With the assumption of linear potential flow theory presented in section 3.2, the fluid diffracted velocity potential φD of the buoy i can be decomposed as the sum of three components: the incident waves φI, the scattered wavesφSand radiated wavesφR.

φDⁱ = φIⁱ+ φSⁱ+ φRⁱ. (3.20) By Fourier transform the ﬂuid potential in the frequency domain results in

φˆⁱ(¯x,ω) = ^∞

−∞φⁱ(¯x,t)eⁱ^ωtdt. (3.21) The velocity potential needs to satisfy the Laplace equation and the linear boundary conditions. Consider now to transfer the problem in local cylindrical coordinates ¯r= (ri,θi,z) with origin at the center of each buoy (xi,yi,0), and that the ﬂuid domain is divided into two regions, one beneath the buoys (I) and one outside the buoys (II) (see Fig. 3.1).

Figure 3.1. Sketch of the regions in which the ﬂuid domain has been divided (Adapted from [38]).

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By separation of variables, is it possible to ﬁnd a solution to the Laplace equation with the linear boundary constraints in cylindrical coordinates, in terms of vertical functions, radial functions and angular functions. Hence, in each ﬂuid domain the general expression of the velocity potential can be written as an ansatz in terms of eigenfunction expansions:

Φ(ri,θi,z) = R(ri)Θ(θi)Z(z)

=_n_=−∞

∑

^∞

∑

^∞

m=0[AmnK_n(kmr_i) + BmnI_n(kmr_i)]eⁱⁿ^θⁱZ_m(z) (3.22) where A_mnand B_mn are unknown coefﬁcients, K_n and I_n are modiﬁed Bessel functions, Zm(z) are the vertical eigenfunctions, and eⁱⁿ^θⁱis the angular eigenfunction.

If we consider multiple bodies, the diffracted potential in the exterior domain will be the sum of the incident potential, the scattered and radiated waves of the body i, plus the all contributions from the other bodies ( j= i) in terms of scattered and radiated waves (Eq. 3.23).

In the external domain:

φ_Dî(II)= φIⁱ+ φ_Rî(II)+ φ_Sî(II)+

∑

j=i

φ_R^j(II)+ φ_S^j(II)

i

=_n_=−∞

∑

^∞

Z0(z)

α˜_0nⁱ H_n(kr)

Hn(kRⁱ)+ Jn(kr)(Aⁱ_0n+

∑

j=i

∑

∞ l=−∞

T_0ln^{i j}α˜_0l^j)

+

∑

^∞

m=1

Zm(z) α˜mnⁱ

Kn(kmr)

K_n(kmRⁱ)+ In(kmr) I_n(kmRⁱ)

∑

j=i

∑

∞ l=−∞

T_mln^{i j} α˜_ml^j eⁱⁿ^θⁱ.

(3.23) In the domain underneath the buoys:

φ_D^i(I)= Vⁱ 2Lⁱ

(z + h)²−r² 2

+_n_=−∞

∑

^∞

aⁱ_0n

r Rⁱ

_|n|

+ 2

∑

^∞

m=1

aⁱ_mncos(λ_mⁱ(z + h))I_n(λmⁱr) In(λmⁱRⁱ)

eⁱⁿ^θⁱ (3.24)

where Lⁱ= h − dⁱ, Aⁱ₀_,n is the incident wave coefficient, ˜αmnⁱ is the combined coefficient that include contributions from scattered and radiated waves, Vⁱ is the heave velocity of the buoys, Z_m(z) are the vertical eigenfunctions, Hnand Jn are Bessel functions, Knand Inare modified Bessel functions, T_mln^{i j} are the expression needed for Graf’s addition theorem (to write outgoing waves from one cylinder as incoming waves in the local coordinates of one other cylinder), λm= mπ/Lⁱ and k_m are the wave numbers that solve the dispersion relation (Eq. 3.14).

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The scattering problem (i.e. all the buoys are ﬁxed and there is only the pro- pagation of an incident wave) and the radiation problem (i.e. all buoys oscillate independently in heave and there is no incident wave) are solved separately.

In absence of incident waves Aⁱ₀_,n= 0 for all i, while in absence of radiated waves Vⁱ= 0 for all i.

Continuity between equation (3.23) and (3.24) and their radial derivatives at the boundaries r= Rⁱ, combined with truncation of the sum of the vertical and radial eigenfunctions, results in an expression of the coefﬁcients aⁱ_mn in the internal solution in terms of ˜αⁱand a ﬁnite system of linear equations:

⎡

⎢⎢⎢⎣

1 −B¹_DT¹² ··· −B¹_DT^1N^b

−B²_DT²¹ 1 ··· −B²_DT^2N^b ... ... . .. ...

−B^N_D^bT^N^b¹ −B^N_D^bT^N^b² ··· 1

⎤

⎥⎥⎥⎦

⎡

⎢⎢⎢⎣

α˜¹ α˜² ... α˜^N^b

⎤

⎥⎥⎥⎦=

⎡

⎢⎢⎢⎣

V¹B¹_R V²B²_R

... V^N^bB^N_R^b

⎤

⎥⎥⎥⎦+

⎡

⎢⎢⎢⎣

B¹_DA¹ B²_DA²

... B^N_D^bA^N^b

⎤

⎥⎥⎥⎦

(3.25) where B_Dand B_R are the single-body diffraction and radiation matrices, respectively. Once we have solved for ˜αⁱand calculated the coefficients aⁱ_mn, we can compute the potential in the region under the floater (φ_Dî(I)).

3.3.2 Equations of motion

In this work we consider only long-crested waves propagating in the posi- tive x-direction, i.e. unidirectional, and the equation of motion is restricted to 1 DoF (vertical displacement, or heave). Moreover, the connection between translator and the buoy is assumed stiff, so they have the same displacement, and thus a single equation of motion can be written for the buoy-translator system. These assumptions are implicit in the equations derived henceforth.

With the velocity potentialφ_D^i(I), the dynamical forces acting on the bodies can be calculated as a surface integral on the wetted surface S of the ﬂoat:

Fⁱ= iωρ

Sφ_D^i(I)dS. (3.26)

The excitation force is computed from the scattering problem and the radiation force from the radiation problem. The equation of motion of the i^th WEC in the frequency domain then takes the form

−ω²

mⁱ+ mⁱadd(ω)

− iω

Bⁱ(ω) + Γⁱ

− ρgπRⁱ² zⁱ(ω)

= ηⁱ(ω) f_excⁱ (ω), (3.27) where mⁱ= mⁱ_b+ mⁱt is the total mass of the moving system (buoy and translator),Γⁱis the constant power take-off coefﬁcient, and zⁱis the vertical position of the buoy; the radiation force has been divided into added mass (mⁱ_add) and

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radiation damping (Bⁱ) as F_radⁱ (ω) = [ω²mⁱ_add(ω) + iωBⁱ(ω)]zⁱ(ω); fexcⁱ (ω) andηⁱ(ω) are the frequency domain amplitudes of the excitation force and of the incident waves, respectively. The position of the buoy in the time domain can be obtained by Fourier transform zⁱ(t) = zⁱ(ω). With zⁱ(t), the instanta- neous power absorption of the WEC i is given as P_i(t) = Γⁱ· ˙zi(t)², while the power output of the full park will be the sum of all NbWECs:

Ptot(t) =

∑

^N^b

i=1

Pi(t), (3.28)

and the time averaged value:

P¯tot=

∑

^N^b

i=1

P¯i. (3.29)

3.4 Optimization

Optimization is the procedure of identifying the best solution from some set of available alternatives, under given constraints. Optimization of wave energy converter arrays involve a large parameter space, is multi-objective and the shape of the objective function is not known. For such problems, evolutionary algorithms are suitable, since they rely on intelligent search over a large but ﬁnite solution space using statistical methods [39] and they are less likely to get stuck in local minima. Evolutionary algorithms are based on the theory of biological evolution and perform optimization by implementing "computational evolution". Subsection 3.4.1 summarizes the procedure of a genetic algorithm optimization process.

3.4.1 Genetic algorithm

To describe the genetic algorithm, we can refer to Fig. 3.2, where the general procedure is represented as a sequence of diagram boxes. The genetic algorithm routine [40] is based on the arbitrary choice of some parameters which can influence the achievement of convergence and the output results. The opti- mization process starts with the random creation of the first population, which is a set of a fixed number of chromosomes. Each chromosome contains a cer- tain number of genes. Each gene represents a variable that will be optimized during the process.

Initial population

In the beginning of the optimization routine, a ﬁrst population (or ﬁrst set of chromosomes) is generated by uniform random sampling over a pool of

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possible values. It represents the ﬁrst set of solutions from which the algorithm will start its optimization routine. The ﬁrst population can contain different genes according to the optimization goals.

Evaluation

By means of an objective function, or cost function, every chromosome of the first population is associated with a fitness value. Different objective functions have been defined and compared in this thesis work.

Ranking

After 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).

Convergence

The number of iterations in the evolution depends on whether an acceptable solution is found or a preset number of iterations is exceeded [39]. Therefore, in order to stop the search of the genetic algorithm, some convergence criteria have been implemented:

1) a maximum number of iterations (Itmax) is reached;

2) all the chromosomes in the actual population are the same;

3) the solution ceases to improve after a certain number n of iterations.

If one of these conditions is fulfilled, the algorithm stops and the first chromosome of the ranked population is taken as the final optimal solution.

Reproduction

A reproduction step is carried out whenever convergence is not reached. It consists of four parts: natural selection, pairing, mating – crossover, elitism and mutation.

a. Natural selection: this operator selects the upper percentage of indivi- duals (or chromosomes) in the ranked population that will survive and continue to the next generation.

Figure 3.2. Genetic algorithm procedure.

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b. Pairing: the selected part of the individuals represents the new parents;

odd and even rows are paired from top to bottom to generate a new part of the population called offspring.

c. Mating – crossover: the crossover operator exchanges genetic material between two parent chromosomes during reproduction, so that potentially positive distinctive genes from both individuals will be inherited by every child. This procedure is performed in a slightly different way according to the optimization problem, but always with a single-point crossover.

d. Elitism and Mutation: mutation introduces new genetic material in the population by randomly changing a chosen percentage of genes. This ensures that other regions in the solution space will be explored, preven- ting that the algorithm gets stuck in a local minima. Elitism protects the ﬁrst upper set of the ranked individuals from potentially negative muta- tions, in order to preserve the best solutions unaltered in the following generation.

New population

The combination of the ﬁrst selected population (parents) and the new generated offsprings, after mutation, represents the new population that will be evaluated in the next generation or iteration and so forth until the algorithm reaches convergence.

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Part II:

Optimization of wave energy parks

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4. Simulations

To know that we know what we know, and to know that we do not know what we do not know, that is true knowledge.

Nicolaus Copernicus

This chapter describes the optimization routine for wave energy arrays that was built and the simulation work performed with the models. The work pre- sented in this chapter was published in Papers I, II, III, IV, V.

As mentioned in chapter 2, the work in the present thesis focuses mostly on the layout optimization of wave energy park, meaning that the spatial location of the devices within the park is the variable that is systematically optimized, while all the other parameters are treated as constraints or dependent variables.

The model was built with the goal to solve the array optimization problem described in section 2.3.1, and is based on the genetic algorithm routine described in section 3.4. In its most extended version, the model included all the parameters of the diagram in Fig. 2.4, except electrical interaction, advanced PTO settings and control strategies. A further modification of the model has allowed the optimization of different parameters, such as the dimension of the floats or the damping coefficient, given some fixed constraints.

The modelling work has evolved as follow:

1. Optimization of a single device;

2. Array layout optimization based on power output, which was conducted with both a discrete and a continuous real-coded GA;

3. Multi-parameter optimization of an array, which include the optimization of additional parameters such as ﬂoat dimensions and PTO damping coefﬁcient;

4. Array layout optimization by means of an economical objective function, conducted with a discrete real-coded GA.

A scheme of the model is shown in Fig. 4.1 and consists of different and connected sub-models: an optimization model, a wave energy park model (including a hydrodynamic model) and an objective function model. The genetic algorithm optimization is the outer shell and main part of the routine. In- side the optimization model a WEC park dynamic and hydrodynamic model is nested, which computes the hydrodynamic coefﬁcients, the motions and the power output of the WECs in the park. Finally, an objective function calcu- lates the ﬁtness value, which determines the goodness of the solution and the optimization convergence rate. The objective function can consist of, or uti- lize, a full model itself, such as an economical model of the wave energy park system.

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Figure 4.1. Schematic overview of the nested models in the optimization routine.

4.1 Optimization of a single device

The optimization of a single PA WEC of the type developed at Uppsala Uni- versity consists in ﬁnding the optimal value of buoy radius (R), draft (d) and generator damping coefﬁcient (Γ). The values are chosen from a set of possible numbers: R ∈ {1 : 0.5 : 5} m, d ∈ {0.2 : 0.05 : 0.4} m and Γ ∈ {15 : 1 : 2000} kNs/m. The objective function minimizes the negative value of the average hourly produced power output:

f_cost^A = min(− ¯Ptot). (4.1) Twenty simulations with different initial random seeds have been performed in order to evaluate the goodness of the model, which was also compared to results obtained with parameter sweep (PS) (i.e. computation of the power output for every possible combination of radius, draft and damping coefﬁcient) for validation.

The work of the single device optimization and validation is published in Paper I and Paper II, and the results are presented in section 6.1.1.

4.2 Array layout optimization based on power output

The first array layout optimization was conducted with the objective to max- imise the power output of the devices, or park, and the problem consists in placing identical devices according to mutual hydrodynamic interactions. In fact, wave energy converters, differently from wind turbines, can influence not only devices located in their shadow, but also converters located in the front by scattered and radiated waves propagating in all directions [41]. This leads to the important fact that the interaction in wave energy parks can be con- structive, and not only destructive as in wind energy farms. The goal is to find a layout which improves the power production of the park, under the given

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constraints. The objective function maximises the array power output:

f_cost^B = min(− ¯Ptot). (4.2)

The input waves are unidirectional and long-crested, travelling along the po- sitive direction of the x-axis (from left to right). To solve this problem two different GAs codes have been created:

• Discrete code: it performs optimization of the spatial coordinates [xi,yi] where the solution space is discrete, i.e. the ocean area is gridded every 10 m in both x and y directions and i number of WECs can take random positions only on the knots (Fig. 4.2a). The GA chromosome and cros- sover method utilized are shown in Fig. 4.3, where N_bgenes (i.e. couple of coordinates) are contained; during crossover a gene is randomly selected as separation frontier between two parts that will be swapped.

• Continuous code: it performs optimization of the spatial coordinates [xi,yi] where the solution space is continuous, i.e. WECs allowed positions are continuous numbers (Fig. 4.2b). A minimum separation distance constraint between the center of the devices is set to 10 m, in order to avoid overlapping. The chromosome and crossover method is similar to the discrete code procedure, however the crossover gene is also suppo- sed to blend, as shown in Fig. 4.4.

(a) (b)

Figure 4.2. (a) Sketch of the layout optimization problem over a gridded area; (b) Sketch of the layout optimization problem over a continuous area; minimum edge-to- edge distance (red dotted line) and minimum center-to-center distance (orange).

The work of the array layout optimization based on power output is published in Paper II, and the results are presented in section 6.1.2.

Numerical and experimental modelling for wave energy arrays optimization

Numerical and experimental

modelling for wave energy arrays optimization

List of papers

Contents

Nomenclature and abbreviations

1. Introduction

1.1 Aim of the thesis

1.2 Thesis outline

Part I:

Background

2. Wave energy

2.1 General aspects

2.2 Wave energy converters

2.2.1 Uppsala University WEC

2.3 Wave energy parks

2.3.1 The wave energy park optimization problem

2.3.2 Literature review

3. Theory

3.1 Wave energy park numerical model

3.2 Linear potential ﬂow theory

3.3 Wave-structure interaction

3.3.1 Multiple scattering and body radiation problem

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

3.3.2 Equations of motion

∑

∑

3.4 Optimization

3.4.1 Genetic algorithm

Part II:

Optimization of wave energy parks

4. Simulations

4.1 Optimization of a single device

4.2 Array layout optimization based on power output