Optimization of Point Absorber Wave Energy Parks

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(1)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.

(2) Abstract Renewable energies are believed to play the key role in assuring a future of sustainable energy supply and low carbon emissions. Particularly, this thesis focus on wave energy, which is created by extracting the power stored in the waves of the oceans. In order for wave energy to become a commercialized form of energy, modular deployment of many wave energy converters (WECs) together will be required in the upcoming future. This design will thus allow to benefit, among others, from the modular construction, the shared electrical cables connections and moorings, the reduction in the power fluctuations and reduction of deployment and maintenance costs. When it comes to arrays, the complexity of the design process increase enormously compared 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 deployment held in 2006. The device is classified as a point absorber and consists in a linear electric generator located on the seabed, driven in the vertical direction by the motion of a floating buoy at the surface. Nowadays, one of the difficulties of the sector is that the cost of electricity is still too high and 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 tools for parks of many units. In this thesis, a first attempt of systematic optimization for arrays of the Uppsala University WEC has been proposed. A genetic algorithm (GA) has been used to optimize the geometry of the floater and the damping coefficient of the generator of a single device. Afterwards, the optimal layout of parks up to 14 devices has been studied using two different codes, a continuous and a discrete variables real coded GA. Moreover, the method has been extended to study arrays with devices of different dimensions. A deterministic evaluation of small array layouts in real wave climate has also been carried out. Finally, a physical scale test has been initiated which will allow the validation of the results. A multi–parameter optimization of wave power arrays of the Uppsala University WEC has been shown to be possible and represents a tool that could help to reduce the total cost of electricity, enhance the performance of wave power plants and improve the reliability..

(3) List of papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I. II. 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, 23-26 April, IWWWFB 2017. Giassi M., Göteman M.; Layout design of wave energy parks by a genetic algorithm; Under revision for Ocean Engineering, 2017.. 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, 27-31 August, EWTEC 2017.. IV. Thomas S., Giassi M., Göteman M., Eriksson M., Isberg J., Engström J.; Optimal constant damping control of a point absorber with linear generator in different sea states: comparison of simulation and scale test; Proceedings of the 12th European Wave and Tidal Energy Conference, 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: the Italian offshore; Energy, 122 (378-389), January 2017. DOI: 10.1016/j.energy.2017.01.094. Reprints were made with permission from the publishers..

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(5) Contents. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 Wave energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Uppsala University concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Wave energy sector challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Multi-units arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Research question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Structure of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Wave energy farm model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Linear wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Waves-structures interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Dynamic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Optimization theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 13 13 16 18 18 19. 3. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Single device optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Array layout optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Multi-parameters array optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Deterministic array evaluation - A case study . . . . . . . . . . . . . . . 3.2 Wave tank experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22 22 22 22 24 27 29. 4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Single device optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 First GA validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Array layout optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Multi-parameters array optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Second GA validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Deterministic array evaluation - A case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Wave tank experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31 31 31 32 35 36 37 39. 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Computational time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 GA parameters sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41 41 50 51. 6. Conclusions. 52. .................................................................................................

(6) 7. Future work. 8. Summary of papers. 9. Svensk sammanfattning. ................................................................................................. 55. .............................................................................. 57. .................................................................................... 58. ......................................................................................................... 59. 10 Acknowledgements References. ..................................................................................... 54.

(7) 1. Introduction. The EU energy and climate action goal is to reduce greenhouse gas emissions by 40% by 2030, and that the proportion of renewable energy (RE) will have to cover at least 27% of the total energy use [1]. According to the Swedish energy policy, the share of renewable energy shall be at least 50% of total energy use by 2020 and 100% by 2040 [2].. 1.1 Wave energy Renewable energies represent the solutions to a future of sustainable energy supply and no carbon emissions. Ocean energy, i.e. the energy that can be harvested from seawater, is the general term which includes the following energy resources: tidal currents, ocean currents, tidal range, waves, ocean thermal energy and salinity gradient. The overall potential of all these resources is enormous. The origin of wave energy is the unbalanced irradiation of the sun at different latitudes. Due to this temperature variation on the Earth’s surface, the atmospheric pressure varies and induces motion of air masses from high to low pressure areas, creating winds. As the wind blows over water, some of the energy is transfered to the ocean, forming waves, which 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 wind speed, the length of time the wind blows and the length of the generation area will influence the height and the period of the resulting waves [3]. Waves usually travel long distances without much energy loss and therefore are really efficient in the energy transport. The theoretical potential of wave energy in the world has been estimated to be around 3 TW [4]. Like other renewable energy sources, wave energy is available with seasonal and geographical variability. The areas with 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 different concepts, and they can be classified according to operational principle, location, power take off (PTO) and directional characteristics, for example. Over the last decades, a huge number of wave energy converters (WECs) have been developed, patented and tested. However, until now, there is no device that has reached the required level of reliability for full scale commercialization. 7.

(8) Figure 1.1. Annual mean wave power density (colors) and annual mean direction of the power density vectors (→) [5]. Nevertheless, different coastal areas with different wave climates will require different type of technologies, making it more likely that a small number of devices will be conquering the market. The average Swedish energy flux along the west Atlantic coast is estimated to be around 5 kW/m [6]. Such wave climates are considered "mild" and they require small rated power WECs compared to open Atlantic coast like UK or Portugal, for example. Large scale electricity production will benefit from the deployment of the devices in multi-units arrays or parks: cost reductions, modularity, redundancy, power quality, sharing of the electrical cables and utility scale power generation are just some examples of the advantages provided by these systems.. 1.2 Uppsala University concept Uppsala University has been developing a point absorber wave energy converter 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 has permanent magnets mounted on his surface, while the stator contains coil windings. When waves lift the buoy, the relative movement of the magnets with respect to the coils induce electricity according to Faraday’s law.. 1.3 Wave energy sector challenges The research during the last decades has resulted in many important achievements. However, to become cost competitive with other energy sources and to get the support and interest of investors, the wave energy sector has still to face and solve many challenges. The European Union and the Swedish Energy 8.

(9) Figure 1.2. Uppsala University WEC and principle of operation (from Paper II).. Agency have developed and funded specific regulations and action plans to help the delivery of ocean energy [1],[7], according to which some of the most significant 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 cost reduction progress. • The EU’s transmission grid infrastructures expansions onshore and offshore to deliver the new generated power; in addition, other infrastructures improvement such as port facilities and specialized vessels for deployment operation and maintenance. • More knowledge about the environmental impact to mitigate the negative effect on the marine environment, as well as the social acceptability. • Development of systems, subsystems and components related to power transmission 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 upscaling conceptual individual units to parks.. 1.4 Multi-units arrays There are many ways to reduce the cost of the technology. One option is to deploy large arrays of many units (examples sketch in Fig. 1.3). Having a park of wave energy converters instead of one or few bigger units has a lot of advantages: the modular construction, sharing of the electrical cables connections and moorings, quality and smoothness of the power output, redundancy, maintenance can be done without shutting down the entire production, higher reliability to failures, higher power production and cost-effective deployment. 9.

(10) Figure 1.3. Outline examples of arrays. Seabased AB (top left), Aquamarine Power Oyster (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 actual physical 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 the power production of the plant. Normally, problems with multi objective goals are solved by optimization routines. However, optimization of an array of wave energy converter is not an easy task, although very important and crucial at this stage of the wave energy development. The complexity of the problem can be understood by looking at Fig. 1.4, where some of the most important variables 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 multidirectional. Note that this diagram includes many simplification and that the problem, in reality, can be much more complex than that. The ideal optimization routine would optimize all these variables simultaneously, taking in account that, if one variable is changed, automatically all the variables that the box is pointing at will be modified. 10.

(11) Figure 1.4. Variables that play role in the optimization of a wave energy farm. Arrows represent mutual influence.. 1.5 Research question At this stage, the wave energy sector is in urgent need of reliable simulation tools able to guide the decision making process before huge investments are done. The aim of this thesis is to help answering the question: how can we at best optimize a wave energy park of Uppsala WECs during the design process and would that be able to help reducing the cost of energy?. 1.6 Structure of the work The 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. 11.

(12) • Results: summary of the results obtained with different models and results from the scale experiment. • Discussion: considerations about the results. • Conclusions: summary of the main findings. • Ongoing and future work: description of the necessary future development of the research.. 12.

(13) 2. Theory. 2.1 Wave energy farm model The model used to simulate the hydro-mechanical behaviour of the wave energy park consists of two main parts. An analytical model based on the multiple scattering method [8] is used to calculate the hydrodynamic coefficients such as added mass, radiation damping and excitation amplitude. The method assumes the validity of linear potential flow theory. The coefficients are then used to solve the equation of motion in the frequency domain. The motion of the buoys are then directly linked to the power production.. 2.1.1 Linear wave theory Linear potential flow theory, or Airy wave theory, is used to describe the physics of water waves in a two dimensional space, given 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 gravity force acting on the fluid. The hypothesis of incompressibility results in the following continuity equation: ∂u ∂w ∂v + + = 0, (2.1) ·v = ∂x ∂y ∂z where v = (u, w, v) is the fluid velocity in Cartesian coordinates. In this work we consider only waves propagating in one direction. Without loss of generality, the direction of wave propagation is defined as the positive x-direction. The assumption of incompressible fluid and irrotational motion imply that the fluid velocity is a conservative vector field, implying the existence of a velocity potential and a stream function for waves. The velocity potential Φ satisfies ∂Φ dx =− = −Φx dt ∂x ∂Φ dz =− = −Φz v= dt ∂z. u=. v = −Φ.. (2.2a) (2.2b) (2.3). From Eq. 2.1 and Eq. 2.3 it can be concluded that the potential satisfies the Laplace’s equation: Φ =. ∂ 2Φ ∂ 2Φ ∂ 2Φ + 2 + 2 = 0. ∂ x2 ∂y ∂z. (2.4) 13.

(14) To solve the Laplace equation it is necessary to apply specific boundary conditions: • Bottom boundary condition: The sea bottom is assumed to be fixed, horizontal and impermeable, then a no flow condition should be applied: v=−. ∂Φ =0 ∂z. 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 coincide 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 atmospheric one, assumed constant, so we derive the condition from Bernoulli’s energy conservation balance as η (x,t) +. 1 1 2 Φx + Φ2z = · Φt 2g g. at z = η. (2.7). • Periodic boundary condition: Φ (x,t) = Φ (x + L,t) Φ (x,t) = Φ (x,t + T ). for every t for every x.. (2.8a) (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 and Φ2x + Φ2z . Moreover, η is unknown, being itself part of the problem solution. It is therefore not possible to find an analytical exact solution. By 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/L 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 boundary shape is unknown. The linearized problem becomes: Φxx + Φzz = 0. at 0 < x < L and − d < z < 0 Φz = 0. 14. at z = −d. (2.9). (2.10).

(15) Φz + ηt = 0. at z = 0. 1 · Φt at z = 0 g Eq. 2.11 and 2.12 can be joined in a single condition: η (x,t) =. 1 Φz = − · Φtt g. at z = 0. (2.11) (2.12). (2.13). From the Laplace equations combined with the new linear boundary conditions we obtain the dispersion equation, that describes the manner in which a field of propagating waves consisting of many frequencies would separate or disperse due to the different celerities of the various components. ω 2 = gk · tanh (kd). (2.14). 2π where ω = 2π T is the wave angular frequency and k = L is the wave number and c is defined as the phase velocity, also called celerity, of the waves: ω 2 g 2 (2.15) = · tanh (kd) . c = k k. With some algebraic manipulation of the last equation, we will find the relationship for the wave length: 2π gT 2 · tanh ·d . (2.16) L= 2π L 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 wave period T, not on wave height. Finally, by separation of variables, the solution of the linearized problem is: η (x,t) = Φ (x, z,t) = Or. H · cos (ωt − kx) 2. ω · a cosh (k (d + z)) · · sin (ωt − kx) k sinh (kd). H · cos (kx − ωt) 2 g · a cosh (k (d + z)) Φ (x, z,t) = − · · sin (kx − ωt) ω cosh (kd) η (x,t) =. (2.18) (2.19). (2.20) (2.21) 15.

(16) 2.1.2 Waves-structures interaction Multiple scattering and body radiation problem We now consider Nb point absorber floating devices. In this study we will only consider cylindrical buoys. With the assumption of linear potential flow theory, the fluid diffracted velocity potential φD of the buoy i can be decomposed in the sum of three components: the incident waves φI , the scattered waves φS and radiated waves φR . φDi = φIi + φSi + φRi .. (2.22). By Fourier transform the fluid potential in the frequency domain results in ¯ ω) = φˆ i (x,. ∞ −∞. iωt φ i (x,t)e ¯ dt. (2.23). The velocity potential needs to satisfy the Laplace equation and the linear boundary conditions. Consider now to transfer the problem in local cylindrical coordinates (ri , θi , z) with origin at the center of each buoy (xi , yi , 0) and the the fluid domain is divided into two regions, one beneath the buoys (II) and one 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 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 term of eigenfunction expansions: Φ(ri , θi , z) = R(ri )Θ(θi )Z(z) =. ∞. ∞. ∑ ∑ [Amn Kn (km ri ) + Bmn In (km ri )]einθi Zm (z). n=−∞ m=0. 16. (2.24).

(17) where Amn and Bmn are unknown coefficients, Kn and In are modified Bessel functions, Zm (z) are the vertical eigenfunctions, einθi is the radial 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. 2.25). In the external domain:. φDi(III) = φIi + φRi(III) + φSi(III) + ∑ φRj(III) + φSj(III). j=i. i. ∞ ij j i Hn (kr) i ˜ ˜ α Z (z) α (kr)(A + T ) + J n 0 ∑ ∑ ∑ 0ln 0l 0n 0n Hn (kRi ) n=−∞ j=i l=−∞

(18) ∞ ∞ Kn (km r) In (km r) ij j i einθi . T α˜ + + ∑ Zm (z) α˜ mn i) i ) ∑ ∑ mln ml K (k R I (k R n m n m m=1 j=i l=−∞. =. ∞. (2.25) In the domain underneath the buoys: Vi r2 2 (z + h) − 2Li 2

(19) r |n| ∞ ∞ In (λm r) inθi i i i e . (2.26) + 2 ∑ amn cos(λm (z + h)) + ∑ a0n i R In (λm Ri ) n=−∞ m=1. φDi(II) =. i = α i +V i bi are the combiwhere Ai0,n is the incident wave coefficient, α˜ mn mn mn 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 and ij are the Jn are Bessel functions, Kn and In are modified Bessel functions, Tmln 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), 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 the propagation of an incident wave) and the radiation problem (i.e. all buoys oscillates 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 V i = 0 for all i. Continuity between equation (2.25) and (2.26) and their radial derivatives at the boundaries r = Ri , combined with truncation of the vertical and radial eigenfunctions, results in an expression of the coefficients aisn in terms of α˜ i. 17.

(20) and a finite system of linear equations: ⎡ 1 −B1M T 12 · · · −B1M T 1N ⎢ −B2 T 21 1 · · · −B2M T 2N M ⎢ ⎢ .. .. .. .. ⎣ . . . .. ⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣. 1. −BNM T N1 −BNM T N2 · · ·. α˜ 1 α˜ 2 .. . α˜ N. ⎤. ⎡. ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎦ ⎣. V 1 B1rad V 2 B2rad .. .. ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥+⎢ ⎦ ⎣. B1diff B2diff .. .. V N BNrad. BNdiff (2.27) Once we have solved for α˜ i and calculated the coefficients aisn , we can compute the potential in the region under the floater (φDi(II) ).. 2.1.3 Dynamic equation With the velocity potential φDi(II) , the dynamical forces acting on the bodies can be calculated as a surface integral on the wetted surface S of the float: F i = iωρ. . S. i(II). φD dS. (2.28). In the following, only heave forces are considered. The excitation force is computed from the scattering problem and the radiation force from the radiation problem. The equation of motion in the frequency domain then takes the form. 2 2 −ω 2 mi + miadd (ω) − iω Bi (ω) + Γi − ρgπ (Ri ) − (Riin ) zi (ω) = i (ω) = Ai (ω) fexc. (2.29). where mi = mib + mti is the total mass of the moving system (buoy and translator), Γi is the constant power take-off coefficient, and zi is the vertical position of the buoy. The connection between translator and the buoy is assumed stiff, so they have the same displacement. The equations of motion can equivalently be expressed as zi (ω) = Ai (ω)H i (ω) where H i (ω) is the transfer function. The position of the buoy in the time-domain can be obtained by Fourier transi (ω). With zi (t), the instantaneous power absorption of the WEC form zi (t) = z i is given as Pi (t) = Γi · z˙i (t)2 , while the power output of the full park will be the sum of all Nb WECs: Nb. Ptot (t) = ∑ Pi (t).. (2.30). i=1. 2.2 Optimization theory The problem we have to face during optimization of wave energy converter arrays involve a large parameter space, is multi-objective and the shape of 18. ⎤ ⎥ ⎥ ⎥ ⎦.

(21) the objective function is not known. For such kind of problems, evolutionary algorithms are suitable, since they rely on intelligent search over a large but finite solution space using statistical methods [10] and they are less likely to get stuck in local minima. The procedure and the nomenclature of evolutionary algorithms, and among these genetic algorithm, are based on the biological evolution of natural creatures. The GA procedure is fully explained in paper II and specific modifications are explained in each paper theory section. Here, some of the most important general aspects are summarized.. 2.2.1 Genetic algorithm To describe the genetic algorithm, we can refer to Fig. 2.2, where the general procedure 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 parameters which can influence the achievement of convergence and the output results. The optimization process starts with the random creation of the first population, which is a set of a fixed number of chromosomes. Each chromosome contains a certain number of genes. Each gene represents a variable that will be optimized in 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 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 code. For example it can be a set of possible combinations of R (radius), d (draft) and Γ (damping coefficient), each set representing a parameter configuration of a WEC, or a set of coordinates [xi , yi ] of the WECs that forms a specific array layout. 19.

(22) Evaluation By means of an evaluation function, every chromosome of the first population is associated with a fitness value. Different objective functions have been defined and compared. The evaluation functions required the computation of the power output of the arrays; therefore the evaluation step contains the semi-analytical multiple scattering model explained in section 2.1.2 for the calculation of the hydrodynamic parameters of the fully coupled WECs in the array. The input to this model consists of 20 min time series of irregular waves measured 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 . Ranking After evaluation, the population is ranked in descending order, from the "best" chromosome (higher fitness value or best solution), to the "worst" (lower fitness value or worst solution). Convergence The number of iterations that evolve depends on whether an acceptable solution is reached or a set number of iterations is exceeded [10]. Therefore, in order to stop the search of the genetic algorithm, some convergence criteria have 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 chromosome of the ranked population is taken as 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 individuals in the ranked population that will survive and continue to the next generation. b. Pairing: The selected part of the population chromosomes 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. 20.

(23) d. Elitism and Mutation: Mutation introduces new genetic material in the population by randomly changing a chosen percentage of variables (i.e. genes). This ensures that other regions in the solution space will be explored, preventing that the algorithm gets stuck in a local minima. Elitism protects the first upper set of the ranked individuals from potentially negative mutations, 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 is stopped.. 21.

(24) 3. Methods. 3.1 Simulations The simulation work can be divided into different parts explained in the next few paragraphs. One part regards the systematic optimization of wave energy parks. In this work the wave energy farm model explained in section 2.1 and the optimization method explained in section 2.2 have been combined together in a unique routine, as shown in Fig. 2.2. One other simulation part regards the deterministic study of small arrays configurations and orientations in real wave climate, through a case study.. 3.1.1 Single device optimization A genetic algorithm optimization for a single device parameters has been developed (Fig. 3.1). The goal of the single WEC GA is to find the optimal value of the radius (R) and draft (d) of the buoy and the damping coefficient (Γ) of the generator, upon calculation of hourly average power output ( fcost = −Ptot ). In fact, radius, draft and Γ of the generator influence the resulting output power of a device firstly in terms of hydrodynamics and secondly according to Eq. 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. Each combination of parameters would give a different power output of the device. In total 20 simulations have been performed. Crossover method is single variable, uniformly randomly chosen and then swapped. The other parameter settings of the GA are outlined in Table 3.1. Results from this part of the work will be used to validate the capability and accuracy of the custom built genetic algorithm code.. 3.1.2 Array layout optimization The first problem in wave energy converter array design is the spatial distribution of the units within a certain area, for which the deploying company/organization has received the legal permits. The relative position of the WECs 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.

(25) Figure 3.1. Sketch of the parameters of the WEC (From paper III).. Figure 3.2. Chromosome of the single device problem (From paper II).. differently from wind turbine parks, can influence not only devices located in the shadow, but also in the front by scattered and radiated waves propagating in all directions of the park [11]. The nature of the interactions is complex and not well understood or predictable. Hence, the method has been extended to perform layout optimization of parks. Two different codes have been implemented: discrete variables and continuous variable based. At this stage all the WECs are identical (R = 3 m, d = 0.45 m and Γ = 140 kNs/m) and the aim is to improve the power production ( fcost = −Ptot ). In all different cases presented, the waves considered are long crested and travelling along the x-axis (from left to right). Discrete code This model 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 WECs can take random position only on the knots (Fig. 3.3). The total available area is of 2500 m2 (or 6400 m2 for some simulations). Fig. 3.4 represents the chromosomes and crossover method utilized. Each of them contains Nb genes (i.e. couple of coordinates); during crossover a gene is randomly selected as separation frontier Figure 3.3. Sketch of the layout optibetween two parts that will be swapped. mization problem over a gridded area.. 23.

(26) Figure 3.4. Chromosome of the discrete GA code (From paper II). Continuous code This model performs optimization of the spatial coordinates [xi , yi ] where the solution space is continuous, i.e. WECs positions are continuous numbers (Fig. 3.5). To avoid overlapping, a minimum distance constraint between the devices center is set to 10 m. The solution space is much bigger than previously, and different configuration from gridded ones are allowed. Fig. 3.6 represents the chromosomes and crossover method utilized. The procedure is the same as in the previous example, but the crossover gene in this case is also blending. A summary of the parameter settings of the GA is reported in Table 3.1.. Figure 3.5. Sketch of the layout optimization problem over a continuous area. Minimum bord-to-bord distance (red dotted line) and minimum centerto-center distance (orange).. Figure 3.6. Chromosome of the continuous GA code (From paper II).. 3.1.3 Multi-parameters array optimization Until now, the problem we have tried to solve consisted in locating identical devices according to hydrodynamic mutual influence, and the optimization routine has been driven by the maximization of the power output of the single device or of the park. But what happens if we deploy devices of different sizes 24.

(27) Table 3.1. Genetic algorithm parameters settings. Parameter. Description. 1 WEC. Array. Nb nPop nGene Nvar MaxIt selection rate mutation rate elitism rate cut off. Number of WEC Initial population size (number of chromosomes) Number of different genes Number of genes in every chromosome Maximum number of iteration/generation Fraction of nPop that is selected for mating % of population to be mutated Best solutions unaltered in the next generation Iteration to converge if solution doesn’t improve. 1 16 3 3 100 0.5 0.2 1 25. 4,5,7,9,14 12 1 Nb·nGene 250, 500 0.5 0.2 3 −. 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 different sizes. • 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-units array (i.e. optimizing 3 x Nb parameters). The investigation has been divided into three different cases; moreover, the effect of having different cost functions has also been considered. This is motivated by earlier studies that have shown an improved total performance 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 gridded regular layout of 9 WECs at 15 m distance (Fig. 3.7). As previously mentioned, the method allows simultaneous optimization of radius, draft and Γ of the generator, but here it performs the optimization only on two different values of radii. A value of the draft and PTO gamma is assigned accordingly to Table 3.2. It is clear that there are 512 possibilities of combinations. Hence, the problem is small enough for a parameter sweep optimization to be viable, and A is the non–dimensionalised can be used for validation. The cost function fcost power to mass ratio expression in Eq. (3.1), where the mass acts as a crude estimation for the installation cost of a wave device. The chromosome contains a value of the radius for every floater. A =− fcost. (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 power of the park when all WECs are of the largest allowed dimension, whereas Psmall 25.

(28) Figure 3.7. Sketch of the radii optimization for 9 WECs in a fixed layout (From paper 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 two different sizes: 6 WECs of geometry #1 and 6 WECs of geometry #2, as specified in Table 3.2. The possible positions allowed are placed on a 6 x 6 grid with 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 to be noted that, in this situation, the total mass of the park will be equal for all cases; hence, the fitness function is taken as the negative value of the power production of the park, as in the Eq. (3.2).. Figure 3.8. Sketch of the layout optimization for 12 WECs of 2 different size (From paper III).. 26.

(29) B fcost = −Ptot .. (3.2). Case 3 In the third studied application, the parameters radius, draft and PTO coefficients have been simultaneously optimized in an array of 4 WECs. The coordinates 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 to take values within the ranges R ∈ [2 : 0.5 : 3.5] m, d ∈ [0.3 : 0.05 : 0.6] m and Γ ∈ [15 : 1 : 250] kNs/m. The alternative cost function (Eq. 3.3) is used for comparison with the non-dimensional cost function explained before. The chromosomes contains 12 genes as well, i.e. one value of Ri , di and Γi for each device.. Figure 3.9. Sketch of the parameter optimization for 4 WECs in a fixed layout (From paper III).. C fcost =−. Ptot . mtot. (3.3). 3.1.4 Deterministic array evaluation - A case study Despite the fact that wave directionality on array power production is believed to be a key parameter to choose the best orientation of wave energy farms and achieve a maximum in production, the optimization method described above does not include this variability yet. A first attempt to study the influence of the wave climate and directions for a 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 the results compared. Differently from the optimization method, arrays of four devices are simulated in the time domain by a hydrodynamic-electromagnetic model, and a boundary element code is used for the estimation of the hydrodynamic parameters. The arrays consist of four heaving point absorbers with diameter equal to 4 m, height equal to 0.8 m and draft equal to 0.4 m. The nominal power of the linear generator was set equal to 10 kW. Fig. 3.10 shows the different 27.

(30) Figure 3.10. Simulated array layouts: (a) linear, (b) square and (c) rhombus. (From paper V).. array geometries simulated. For each layout, four distances among the units were investigated: 20, 40, 80 and 120 m, i.e. 5, 10, 20 and 30 diameters. The wave direction rose was discretized into 12 direction bins, every 30◦ . For each incident wave direction, 90 sea states were simulated, considering 9 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 each sea states the power output of the farms was calculated and the power matrix obtained. Hence, the power matrix has been used for a site specific study, in order to evaluate 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 with different absolute orientations (with respect to north). According to the array’s simulations computed with different incident wave angles, six geographical deployment orientations are possible for linear and rhombus layouts and three for the square one. Each oriented layout was also simulated considering four different distances between units, thus leading to 60 wave farm designs for each site. By assuming 12 directions of wave propagation, the probability of occurrence WC(i, j) of the sea state (i, j) was multiplied by the corresponding power matrix PM(i, j) taking into account the direction of wave propagation 28.

(31) with respect to the oriented layouts. Then, the annual energy production of the farm was obtained by summing the energy production over all the twelve wave directions.. 3.2 Wave tank experiments In order to verify the results from the array behaviour simulations and optimizations, a scale experiment project has been started, in conjunction with the study of different control strategies. As a first step, an electrical linear motor has been used to build a 1:10 model 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 connection line between the buoy and the translator (i.e. a rope), the PTO and a control computer. The setup allows the use of constant damping and other control strategies like latching and adjustable damping. The float was made out of Styrofoam and coated with glass fiber. A pulley system (2 on the bottom of the tank and 2 on the gantry) ensured the transmission of the forces from the buoy to the translator. Like the real existing PTOs, the model was equipped with two end stop springs on both ends of the rod. Characteristic dimensions of 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 perform by controlling the damping with a constant optimum value for each one of 41 different sea states. The wave height ranged from 0.75 : 0.5 : 3.25 m, while the wave periods ranged between 3.5 : 1 : 9.5 s. The optimal values have been obtained in advance with a time domain two-body simulation by a parameter sweep over the damping values. Afterwards, the result is applied to the scaled model and the power output measured. In addition, we wanted to verify simulations 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 of 3 m. Table 3.3. Full scale and scaled geometrical characteristics.. Translator mass. 1:1. 1:10. 6000 kg. 6 kg. Buoy mass. 5300 kg. 5.3 kg. Stroke length. 3m. 0.3 m. Buoy diameter. 5m. 0.5 m. Buoy height. 2.5 m. 0.25 m. Sea states duration. 14.22 min. 4.5 min. 29.

(32) Figure 3.12. Overview of the test setup at the COAST lab.. 30.

(33) 4. Results. 4.1 Single device optimization Results of the single device optimization allow to find the best setup for a wave energy converter; however, in this specific case, there are "only" 89370 different possible combinations with the values of R, d and Γ set. For these reasons, the results of the genetic algorithm have been validated against parameter sweep, i.e power output calculation from all the combinations of radius, drafts and PTO constants.. 4.1.1 First GA validation Comparison of the results from GA simulations and parameters sweep is shown in Fig. 4.1. Here only the graph with a fixed radius is shown (for the complete representation see paper I). The black cross represents the absolute maximum value. All the solution from GA are located in the region of maximum power output 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.

(34) The GA code was able to find the exact optimal radius and draft 100% of the times. The diamonds in Fig. 4.1 are very close to the black cross (real optimum) but not exactly overlapped to that because the value of Γ found by the optimization routine differs from the real optimum Γ by at most 8%. One explanation for this result can be related to the number of iterations that the tool has performed. A larger number of iterations would most likely lead to a perfect solution. However, the difference is neglectable for the results, in the sense that the final average power outputs calculated by the two methods have anyhow less than 0.2% difference in all 20 simulations. This is a simple case of our application; in fact, we know the shape of the cost function since we can calculate all the results from all the different possible combinations of variables by parameter sweep.. 4.2 Array layout optimization Fig. 4.2 shows an example of results from paper II. In the left column the best layout obtained for 4 and 9 WECs park with the gridded code is shown, while on the right column the results from the continuous code for the same array. In general, results have shown a tendency for the devices to align in a few number of lines as possible perpendicular to wave direction. The fact that WECs do not spread as far as possible from each other is a result of some positive interaction due to scattering and radiation. The continuous code produces results that tend to be similar as the gridded one, but Figure 4.2. Example of best solution found by GA for 4 and 9 WECs. Gridded code (left) and conti- requires a higher number of iterations, since the solution nuos code (right). space is much extended. The interaction factor (q-factor) represent the effect of hydrodynamic interactions between WECs in a park and is defined as: q=. Ptot Nb IS ∑i=1 Pi. (4.1). Ptot represents the total power of the array, while PiIS is the power of the i-th device in isolation. Fig. 4.3 shows that for parks up to 9 WECs, a configura32.

(35) tion with positive interaction gain (i.e. power production > power production of the same amount of WECs in isolation) can be found by the tool in 250 iterations. The q-factor is therefore larger than 1. However, the results are specifically dependent on the size of the ocean area in relation to the number of WECs. In paper II it was also shown that both codes have advantages and disadvantages. The continuous code search within a smaller solution space, reaching convergence faster, but the results are dependent on the choice of the grid. On the other hand the continuous GA finds the best separation distance between devices but it needs a very large number of iterations for parks with over 5 devices to get the same solution as the gridded code. Since the gridded code has in most of the cases produced a solution with higher power production compared to the continuous code, it was suggested as more suitable for the application. Regularity of the layout is also a design advantage related to the utilities. The results also show that destructive interactions can influence the power absorption to a larger extent than positive interactions.. Figure 4.3. q-factor of park of different sizes. Best and worst solution from gridded GA code (code A) and continuous GA code (code B) (From paper II).. 33.

(36) To characterize the evolution of the layouts through the GA iterations, the following coefficients have been calculated for every layouts and iterations: ∑ ShadowDistance + 1 Nb dtot D= Nb T = S·D S=. (4.2a) (4.2b) (4.2c). S is a measurement of the shadowing effects within the park, and is given by 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 between the 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 park is spread out in the available marine area. T is the overall coefficient given by the product of S and D. The mean values over the best three layouts for every iteration is shown in Fig. 4.4. We can conclude that as the algorithm proceeds, the shadowing coefficient in average decreases until the minimum shadowing distance feasible according to the number of WECs and the area size is reached. Distance coefficient D has also a descending trend, meaning that the WECs don’t need to be spread apart to get the maximum of the power output and that it is possible to reduce the 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 alignment perpendicular to the incident wave with a small offset angle, and the WECs are staggered in maximum two rows along the wave front direction. The general trends in the results and the q-factors values are consistent. However, different models and parameter spaces have been used in the different studies, so that a straightforward comparison is difficult.. 34.

(37) Figure 4.4. Coefficients for different park sizes: a) S, b) D, c) T (From paper II).. 4.3 Multi-parameters array optimization Fig. 4.5 represents the values of the non-dimensionalized power ratio (numerator 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 power production, according to the internal location of big and small buoys. The highest ratio between the relative produced power and the relative mass (with 35.

(38) respect to a park with all buoys of the smallest and largest geometry) is shown with a red dot and consists of a park of 6 small and 3 big devices (shown in Fig. 4.5 upper left part). In this work, we denote parks with devices of different dimensions hybrid parks.. Figure 4.5. Value of the non-dimensionalized power ratio (numerator of equation (3.1)) as a function of the non-dimensionalized mass ratio (denominator of equation (3.1)). Upper left the optimal solution represented by the red dot.. Regarding the second application the best hybrid park solution is shown in Fig. 4.6. The power production of this configuration was then compared with the 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 deployment 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 devices. It means that, theoretically, we can achieve larger power production if we 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 validation To verify the reliability of the genetic algorithm optimization routine, a second verification has been carried out by means of parameter sweep of all the possible combinations in case 1. The results, shown in Fig. 4.8 and 4.9, have been confirmed. 36.

(39) Figure 4.6. Optimal solution found for case 2 (From paper III). Figure 4.7. Optimal solution found for case 1 (From paper III). Figure 4.8. Optimal solution found for case 1 with genetic algorithm.. Figure 4.9. Optimal solution found for case 1 with parameter sweep.. 4.4 Deterministic array evaluation - A case study First, the results from the JONSWAP spectrum simulations over all the sea states and directions have been used to understand the sensitivity of the array performance to the design parameters, i.e. geometrical layout, separating distance and wave direction. It can be concluded that, although the effect of the separating distance depends on the array layout, all farm geometries have some common features: • the behaviour of the q-factor with respect to distance is typically not monotonic, because hydrodynamic interactions depend on the ratio between WEC distance and wavelength [9,11], • there is no optimal spacing between the units, but rather the best WEC distance should be selected as a function of the incident wave direction [9] • the effect of wave interactions is inversely proportional to the distance between the units and it is a few percent once the separating distance is larger than 30D [13]. Regarding the specific real locations, results in term of best and worst design is shown in Fig. 4.10 for all the four geometries. 37.

(40) Figure 4.10. Wave farm designs maximizing/minimizing the annual energy production at the study sites and associated energy gain (green) and loss (red). Geographical North pointing towards the top of the page (From paper V).. The highest power output is obtained with a rhombus layout with WECs distance equal to 20 diameters at Alghero and a linear layout with 5 diameters spacing at the other locations. In all the sites, the layouts should be oriented by aligning the most productive wave farm orientation with the prevailing wave direction. The optimum wave farm designs lead to power gains from 1.5% (at Alghero) to 3.4% (at Mazara del Vallo). However, the difference in the annual energy 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 possible both in theoretical unidirectional wave fields and in real multi-directional wave climates. It was possible to design a four WECs farm off the Italian coasts performing better than four isolated devices, taking advantage of constructive wave interferences. However, as long as the devices are separated by at least 10 buoy diameters and the layouts are oriented to achieve the maximum energy absorption for the prevailing wave direction, the effect of wave interactions on absorbed power is quite low, as already observed in previous studies on small wave farms [13]. 38.

(41) 4.5 Wave tank experiments.

(42) & ' '. Fig. 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 can be seen that the damping values increase over the wave height and decrease over the wave period. Fig. 4.13 shows the power matrix obtained in the scale test and Fig. 4.14 shows a comparison with the theoretical one. It can be noticed that the real absorbed power is much less than the simulated one. The reasons for this huge difference lies in the high non linear and non uniformly distributed friction, which couldn’t be completely compensated during the experiments. Furthermore the data were very sensitive to small changes between the sea states due to the high scaling factor. The theoretical obtained power matrix shows that the absorbed energy increases with increasing wave height and has a maximum point for a certain wave period. By looking at the measured power matrix (Fig. 4.13) the trend is 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 (From paper IV).. . . . . '"() *! . . . Figure 4.12. Simulated power matrix [kW](From paper IV).. 39.

(43) ! " # # . . . . . $ %& ' . . . &

(44) ' &' . 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).. 40.

(45) 5. Discussion. As for any modelling technique, the optimization procedure presented in this thesis has some strengths and limitations. It has been proven to be reliable through validation with parameter sweep and faster in finding the optimum for problems that depend on many variables. It can optimize device coordinates together with buoy’s and generator’s parameters simultaneously. The number of 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 interacting 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 and therefore undefined rated power of the park). In the next few sections some of the most significant points aforementioned will be further discussed.. 5.1 Cost function As previously mentioned, the choice of the cost function in an optimization problem is crucial. In paper III three different functions have been used and compared. As in most engineering problems, there are usually more than one objective to achieve which are conflicting with each other. In our case, if we just look at the simplified problem, we want to maximise the power output and minimize the costs (represented here by the mass). This multi-objective optimization problem has been transformed for simplicity into a single-objective optimization problem, by including both objectives in the same function as a ratio. To investigate the influence of the choice of cost function on the results some comparisons have been carried out for Case 1 outlined in section 3.1.3. 41.

(46) In fact, it was possible to evaluate all the possible solutions by parameter sweep given the limited variable space. The following cost functions have been computed: (P −P. )/(P −P. big small A = − tot small fcost (mtot −m )/(m −m small. (P −P. big. )/(P −P. big small A = − tot small fcost (mtot −m )/(m −m small. big. ). (5.1). small ). )+1. (5.2). small )+1. B = −P fcost tot Ptot C fcost = − mtot D = −q = fcost. (5.3) (5.4). Ptot Nbig Nsmall IS IS (∑m=1 Pbig +∑n=1 Psmall ). (5.5). A , f B and f C have been previously introduced, while f A is a mofcost cost cost cost A and f D is the q-factor calculation for arrays with WECs dification of fcost cost of different sizes, where the power output of the park is compared with the power 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 a single-objective one. Another possible single-objective optimization approach would be to calculate the minimum distance from an ideal point (which often doesn’t exist) that optimizes all the objective functions, called utopia. A Cost function fcost Fig. 5.1 shows the results in the objective space when the nondimensionalized cost function in Eq. (5.1) is used. On the axis are shown the two objectives separately, which both range from 0 to 1. Maximizing this ratio (or, equivalently, minimizing its negative value) means that the best solution will be the point belonging to the line with the highest slope. Conversely, the worst solution is the one belonging to the line with the lowest slope value. The nondimensionalization of the terms leads to the same relative weight between the increment 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 be classified, since, graphically, it belongs to all the lines with different slopes and, numerically, results in an indeterminate form. Best and worst solutions according the minimum distance from the utopia point are also plotted in Fig. 5.1.. A Cost function fcost Fig. 5.2 shows the results in the objective space when the undimensionalized cost function in Eq. (5.2) is used. This time, the numerator and denominator. 42.

(47) Figure 5.1. Undimensionalized power output versus undimensionalized mass of a park of 9 WECs in a fixed grid. The grey lines represent the highest and lowest slope of the solutions points.. of the cost function range both from 1 to 2. As before, optimizing the ratio in Eq. (5.2) leads to the solution point belonging to the line with the highest slope. A and This cost function solves the problem of the point {0, 0} arising with fcost give similar results in the ranked solution, but not identical. To be noted also the fact that the points {1, 1} and {2, 2} (i.e. park with only small devices and park with only big devices) have the same fitness value. B Cost function fcost Optimization only upon power output is a reasonable choice only for hydrodynamic considerations among identical devices (as in paper I and II) or for a qualitative first approach. With parks of WECs where the dimension of the floater is free to change, optimizing over power output will clearly lead to the biggest possible devices, which is misleading.. 43.

(48) Figure 5.2. Undimensionalized power output versus undimensionalized mass of a park of 9 WECs in a fixed grid. The grey lines represent the highest and lowest slope of the solutions points.. C Cost function fcost Fig. 5.3 shows the scatter plot of all the possible solutions in the objective space when the objective function is the dimensional ratio between power output and mass (as crude estimation of capital costs), i.e. Eq. (5.4). On the axis are shown the two objectives separately. In this case the solutions of our problem 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 array with lowest mass (9 small devices) and the worst solution results in the array with highest mass (9 big devices). It can be seen that the relative weight between the power output and mass is not the same, i.e. the increment in the mass (ΔM) when substituting a small device with a big device is higher than the respective increment in power production (ΔP), as already discussed in section 4.3. It is to be noted that in case ΔP > ΔM, one would obtain the opposite results in term of best/worst solutions.. 44.

(49) Figure 5.3. Power output versus mass of a park of 9 WECs in a fixed grid. The grey lines represent the highest and lowest slope of the solutions points. D Cost function fcost Optimizing over the q-factor when the park is composed of different sized WECs leads to the same optimal solution as in the other two undimensionalized cost functions. Since the devices in the array are not all identical, optimizing 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 according to the number of big devices. It can be seen that some certain array layouts lead to a more advantageous hydrodynamic interactions.. 45.

(50) Figure 5.4. q-factor values for all the 512 layouts (top) and q-factor values vs number of 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 number of big devices in the array is plotted. One other way to see the same results is highlighted in Fig. 5.6, where the percentage number of big devices in the array is plotted against the all different park configurations, ordered from the worst to the best. Results in term of best and worst layout, according to the five 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. In other words, given a certain array formation (for example 6 small devices and 3 big devices), what is important is the gain in the power output according to the different internal location inside the array, which leads to different hydrodynamic interactions and thus different power output. However, assuming the same relative weight between power output and masses is not straightforward and is a choice that has to be carefully evaluated a priori. • if ΔM = ΔP, decisive is the horizontal spread of the solutions (seen as the slope of the interpolant line). In other words we are comparing different formations only according to the number of small and big devices included, but their arrangement inside the group is not influential. B and f C ) the ranking of the • when using dimensional cost functions ( fcost cost solutions is determined by the number of big WECs, and the hydrodynamic interactions resulting from the internal locations of big and small devices are not influential. 46.

(51) Figure 5.5. Comparison of all the fcost as a function of the number of big WECs in the layouts.. 47.

(52) Figure 5.6. Comparison of all the fcost by plotting the percentage number of big devices for each park configurations (ordered from worst to best on the x axis).. 48.

(53) Figure 5.7. Best layout (left) and worst layout (right) obtained with the different cost functions.. 49.

(54) . A , f A and f D ), the dif• when using nondimensional cost functions ( fcost cost cost ferent internal locations rather than the actual number of big and small WECs determine the results of the optimization. Hence, we see that the choice of the cost function is very critical for the outcome of the optimization. In paper III we then suggested the use of a objective non-biased economical cost function for the optimization process with a single-objective optimization approach. This function would be defined as the ratio between the total income of the produced electricity (function of the 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 evaluated.. 5.2 Computational time The computational time between the PS and GA methods has been evaluated with the single WEC optimization and with 9 devices on a fixed grid (case 2 of the multi–parameter optimization). In single GA optimization (results showed in Fig. 5.8), the GA was able to save more than 88% of the computational time.. Figure 5.8. Simulation time (left) and number of iterations (right) performed with GA and PS for 20 identical simulations of a single device.. In the array size optimization the parameter sweep evaluation of the 512 different layout possibilities took around 145 minutes, while the GA converged after 350 min. For the GA to be faster it should have converged within 32 iterations. In this case solution was found at iteration 73, because the routine evaluates the same solution several times. A feature to avoid redundancy is important for the future to be implemented in the code. Anyhow, when the 50.

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