Wave Farm Design: Preliminary Studies on the Influences of Wave Climate, Array Layout and Farm Control

1 2 3 3 J Cruz , R Sykes , P Siddorn and R Eatock Taylor

1 Garrad Hassan Ibérica S.L.U, Rua Nova do Almada, n° 59, 2° Andar, 1200-288 Lisboa, Portugal 2 Garrad Hassan and Partners Ltd., St Vincent’s Works, Silverthorne Lane, Bristol BS2 0QD, UK 3 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK

Abstract Nomenclature In this paper preliminary results regarding the a radius of the cylinder assessment of the energy absorption characteristics of A wave amplitude an array of wave energy converters (also referred to as A added-mass matrix a wave farm) are presented. Regular and irregular ij waves are used as input in a frequency domain Bij radiation damping matrix hydrodynamic model which allows iterations in the c clearance array layout and farm control strategy. Under such an Cij hydrostatic stiffness matrix approach each array element can be controlled independently while keeping the design objective Dext applied (external) damping coefficient (maximisation of the wave farm energy yield). f wave frequency The approach is initially verified by comparing the g modulus of the acceleration of gravity solution of the radiation and diffraction problems for k wavenumber h water depth the array with the analogous results from a semi- (h-c) draft of the cylinder analytic method developed at the University of Oxford. H significant wave height Additional regular wave simulations identify the most m0 promising areas by quantifying the interaction factor as M Number of bodies per WEC N Number of WECs in the array a function of the incident wave frequency and wave P Absorbed power (kW/m) heading. Irregular waves which describe representative P average absorbed power per WEC (irregular waves) frequency spectra are then used as input to quantify the n power absorption characteristics from an isolated wave P average absorbed power by the array (irregular energy converter. Finally, the same representative seas waves) are used as input when evaluating the energy q interaction coefficient absorption by an array of wave energy converters, S(f) frequency spectrum t time while iterating on the array layout and control settings associated with each array element. T−10 wave energy period The overall objective of the study is to quantify the Tpeak peak (wave) period influence of the array layout and farm control in the x, y Cartesian co-ordinate system performance of a wave farm under the action of Xi wave exciting force irregular waves. The results show that the energy yield Z radiation impedance matrix is affected by such factors, hence these can be seen as key design drivers when considering the potential for reduction in the uncertainty and thus the cost of energy β wave heading associated with a wave farm. Further studies may Φ velocity potential address additional constraints, either technical or φ complex velocity potential economical. This study is expected to contribute to the ξ Response Amplitude Operator (WEC motions) development of specific modules of GH WaveFarmer, a tool that aims to optimise the design of wave farms. ω wave angular frequency

Keywords: floating bodies; hydrodynamic interactions; wave Acronyms farm. BEM Boundary Element Method PTO Power Take-Off © Proceedings of the 8th European Wave and Tidal Energy WEC Wave Energy Converter Conference, Uppsala, Sweden, 2009

7361 1 Introduction addressed under the action of regular waves are control [9] and layout [e.g. 10], which in practice will be The commercial development of wave energy in conditioned by device access routes, mooring layout (if utility-scale projects will rely on the deployment of applicable) and overall footprint of the wave farm multiple units in an array, creating multi-MW wave (linked with the number of devices and thus the power farms. Several engineering challenges are associated rating of the farm). The previously mentioned tool will with the definition and optimisation of a wave farm, need to be able to quantify the influence of all of these, namely the geometrical layout, electrical connections plus additional factors such as real seas, device between array elements (and the link to shore), geometry and modes of motion linked with the wave mooring configurations (if applicable), control aspects power absorption (PTO modes). and hydrodynamic interactions between devices. In this paper a preliminary study which addresses the All of the above mentioned factors will have an influence of control and layout in the wave farm energy impact on the energy yield. It is therefore essential that yield under the action of irregular waves is presented. project developers have access to tools which are able Irregular waves should be seen as another critical to accurately assess and optimise the key factors which design factor which influences the design. Firstly define the wave farm, maximising the energy output. comparisons with results from a semi-analytical Such tools should be seen as vehicles which allow method for an uncontrolled array of truncated cylinders project developers and / or investors to assess the previously presented in [11] are made, in order to suitability and potential of a given technology to a verify the solution of the radiation and diffraction specific project, in a close parallel with the project problems, with the latter allowing also a discussion development standard procedures which are available related to the excitation forces associated with each in wind energy. array element. The optimal q for the array outlined in Array interactions have been a recurrent R&D topic [12] is then presented as a function of ka and β , in offshore engineering over the last decades. providing a benchmark and a starting point when Pioneering work has also been done in the wave energy extending the study to irregular waves. The findings of field [e.g. 1-6], focusing primarily on linear arrays of this paper are expected to contribute to the creation of a point absorbers, equally and unequally spaced. In the tool (GH WaveFarmer) that will assess, iterate and following paragraphs of this section a short review of optimise the design of wave farms of N elements, for some selected publications is presented. devices of arbitrary shapes and with an arbitrary In [7] an overview of the key results related to the number of bodies. performance of wave energy converters when placed in arrays are presented, along with the fundamental principles behind some hydrodynamic interaction theories used in the field, such as the point absorber 2 Motivation and Approach and the plane wave approximations. The latter are also The potential for constructive interference (i.e. the approached, in more detail, in [8]. In summary the possibility of achieving an energy yield with a farm of point absorber assumption neglects the scattered wave N units which exceeds the output of N units working in field, which is equivalent to assuming that ka  1. The isolation) has long been identified. Equally identified is plane wave approximation can be summarised as a the fact that it is not realistic to assume that such wide spacing assumption, resulting in an incident wave constructive behaviour will be present for the full range field made of plane waves for each array element. In of incident wave frequencies, thus there will be [7] regular waves results for the interaction coefficient destructive interference effects. The objective function q for small arrays of heaving or surging buoys are when designing a wave farm is therefore twofold: 1) presented as a function of the nondimensional spacing maximise the effects of constructive interference; 2) kd between array elements, for several wave headings minimise the effects of destructive interference. β . The results illustrate the potential for constructive The motivation for this preliminary study is to assess and destructive interference, i.e. the existence of wave the influence of several critical variables (wave climate, frequencies for which the array arrangement is layout and control strategy) in the design of a wave beneficial ()q > 1 or detrimental ()q < 1 with regard to farm. To verify the approach, which is based in the frequency domain and uses the solutions of the farm energy yield. Although early work shows that a radiation (hydrodynamic coefficients) and diffraction net gain may still be possible, in [6] it is mentioned that (exciting forces) problems provided by a Boundary a practical strategy for the design of wave farms may Element Method (BEM) solver (WAMIT), involve the minimisation of the destructive interference comparisons with semi-analytic methods [11] and [12] effects. Due to these effects the necessity of a tool to are firstly made for an array of uncontrolled devices. accurately quantify the energy output of a wave farm The same approach is then extended to a controlled becomes clear, as a means to reduce the uncertainty array, in which each individual element can be associated with the project development stage if independently controlled via an external damping term commercial wave energy projects are to be considered. associated with the relative heave motion. The energy The critical factors that condition the energy output absorption results are a function of the incoming sea and thus the q factor need to be identified and their state. influence quantified. Two that have been previously

7372 The array configuration under study is schematised 6MN (3) in Figure 1. It consists of a square array of four φRjj= ∑ ξφ , cylinders with radius a and draft 2a, with centres j=1 equally spaced by 4a. The water depth is also equal to and 4a. The regular wave results use a =1m, as in [11] and [12], whereas the irregular wave results use a = 10m, φ = φφ+ , (4) providing results which are more meaningful for wave SD0 energy conversion. where φ0 is the velocity potential associated with the

incident (incoming) waves and φD the diffraction

potential. In Eq. (3) ξ j are the complex amplitudes of oscillation in all the available degrees-of-freedom

(6MN) and φ j the corresponding unit-amplitude radiation potentials (those resulting from the body motion in the absence of an incident wave). Note that the summation is limited to 6MN, where M is the β number of bodies per WEC and N is the number of WECs in the array. In the present case each array element has just one body and N=4, thus 6MN = 24. Even in cases where M > 1 it is unlikely that all 6M degrees-of-freedom are available, i.e. a multi-body WEC will necessary have locked modes, with power Figure 1: Initial array configuration (from [11]). being absorbed in one or two relative modes (and thus a total of 7 or 8 degrees-of-freedom). It is important that Linear wave theory is applied throughout this paper. As the tools developed are able to cope with locked modes a result there are some well defined simplifying to reduce the computational effort while being flexible assumptions such as: to allow all possible combination of locked modes.

The complex amplitudes of oscillation ξ j are given 1. The free-surface and the body boundary by conditions are linearised;

2. The fluid is incompressible and the flow is 6MN 2 irrotational (potential flow): ∇ Φ=0 , where Φ is the ⎡−+++ω2 MME A ∑ ⎣ ()ij ij ij velocity potential; j=1 , (5) 3. Viscous effects like shear stresses and flow ++++=iBωξ BEE C C⎤ X separation are not considered; ()()ij ij ij ij j⎦ i 4. The bottom is assumed to be flat (and uniform); 5. Under these above mentioned assumptions all E E E variables can be expressed as a complex amplitude where Mij is the mass matrix, Mij , Bij and Cij the times eiωt (regular waves, sinusoidal motions). externally applied mass, damping and stiffness matrices

(respectively), Aij is the added-mass matrix, Bij is the Under these assumptions, the velocity potential can be given by radiation damping matrix, Cij is the hydrostatic

stiffness matrix and Xi the wave exciting force. Note Φ=Re φeitω , (1) E E {} that in this preliminary study Mij and Cij are everywhere null (i.e. inertia tuning by changing the where φ is the complex velocity potential, Re denotes mass of systems with e.g. ballast tanks is not the real part, ω is the angular frequency of the incident considered and there is no externally applied stiffness wave and t is time. By assuming the linear component via e.g. a mooring line). Aij , Bij and Xi decomposition of the problem, the velocity potential are calculated in WAMIT. can be obtained as the sum of the radiation and the The absorbed power and associated control of the wave exciting components, array and its elements are therefore limited to the E influence of Bij . In this preliminary approach the only φ =+φφRS, (2) E elements of Bij which are not null are the diagonal where φR is the radiation potential and φS the terms associated with the heave motion of each of the EEE E scattered potential, respectively given by four array elements ( BBB33,, 99 15 15 , B 21 21 ). The farm

control strategy therefore involves four independent

7383 external damping terms. By normalising the complex (squared layout described in Figure 1) are assessed. In amplitudes of oscillation ξ j with the wave amplitude the latter each of the four cylinders is controlled via an independent damping coefficient (thus a total of four A, leading to ξξ= / A , it is possible to derive an jj terms), and the average power absorbed by the array P auxiliary absorber power Paux which is given by is monitored. In other terms, and for each wave climate, the coupled influences of array layout and different farm control strategies consisting of a layer of four 1 E 2 2 PBaux= ijω ξ j , (6) independent variables (damping coefficients) are 2 assessed.

2 which has dimensions of W/m . The usefulness of Paux is particularly clear when evaluating the average 3 Results absorbed power per farm element, Pn , under the action of irregular waves (using the superposition principle). Comparison with analytical solutions By definition the frequency spectrum S(f) can be expressed by The first set of results is presented in Figures 2 and 3, where the solutions of the radiation and diffraction problems derived in [11] and [12] are compared with 1 A2 ()f Sf()= , (7) new BEM predictions. The configuration under 2 df analysis is the array described in Figure 1. The comparisons are limited to the hydrodynamic and thus Pn can be given by coefficients ( Aij and Bij ) and the exciting force ( Xi ). PPSfdf= 2 () . (8) With regard to the former, the comparisons made in naux∫ Figure 3 use representative terms of the real and the imaginary part of radiation impedance matrix Z,

Finally, the average absorbed power by an array can be respectively equal to Bij and ω Aij . As Equation (5) quantified by illustrates A , B and X are the only variables ij ij i N directly obtained from the BEM solution (the equation (9) PP= ∑ n . of motion is solved in a separate code, to ensure that n controlled multi-body WECs can be accurately modelled). Note that in the irregular wave case BE is The objective of these initial comparisons is to ij verify the approach, ensuring its applicability to the unchanged throughout the frequency range for each S(f) subsequent steps, namely when solving the equation of – ensuring that the superposition principle is valid as motion and adding the farm control layer(s). Figure 2 the system remains unchanged from frequency to shows comparisons with regard to the wave exciting frequency. In practice this corresponds to a force (x, y, z components) for β = 0 . The four (conservative) control strategy that aims to tune the cylinders are identified by the numbering listed in response to a particular sea state, described by S(f). The Figure 1, and the BEM results are marked with the objective function is the maximisation of P , under all ‘WAMIT’ legend. Overall the correlation between the the previously mentioned constraints. In this paper the analytical and the numerical results is very high, with assessment is limited to the influence of the sea state, most of the discrepancies occurring for low values of array geometry and control strategy. ka, and thus are more significant for longer waves and The next section (Results) is divided into the several small WECs (i.e. small values of a). Results for other stages that lead to the preliminary assessment of incident wave directions allow similar conclusions. relative influence of the several parameters listed above The radiation impedance matrix is [24 x 24], but in in the design of a generic wave farm of four cylindrical Figure 3 comparisons are limited to a reduced number heaving WECs. Firstly the results associated with the of terms. Minor discrepancies between the analytical verification of the approach with regard to the solution and numerical results are once again seen for low of the radiation and diffraction problems are presented, values of ka. These findings provide additional ensuring that the BEM method applied is consistent confidence in the approach and confirm its suitability to with the semi-analytical models. Secondly, optimal solve the radiation and diffraction problems in an array results for regular waves of various wave headings are (which is particularly critical for more complex presented for heaving and surging cylinders; such shapes). Recall that these are the only quantities results allow comparisons with the subsequent results derived from the BEM code and that the equations of which are related to irregular waves (final two motion are solved in a separate code; this approach is subsections of ‘Results’): firstly a sensitivity analysis well suited to multi-body WECs. related to the damping coefficient as a function of the incident frequency spectrum for an isolated cylinder is presented, and secondly arrays of four cylinders

7394 -3 x 10 7 1-3 Heave-Heave 6 1-3 Heave-Surge 1-3 Heave-Sway 5 1-3 Heave-Heave WAMIT

0.5 4 1-3 Heave Surge WAMIT 1-3 Heave-Sway WAMIT 3 (h-c)(g/a)

1 2 2 a

Cylinders 1,2 π g 1 0.9 Cylinders 3,4 ρ Cylinders 1,2 WAMIT 0.8 0

Cylinders 3,4 WAMIT Re(Z)/ 0.7 -1

0.6 -2 a(h-c)

π 0.5 -3 g

ρ 0 1 2 3 4 5

|/A ka x 0.4 |F -3 0.3 x 10 7 0.2 1-3 Heave-Heave 6 1-3 Heave-Surge 0.1 1-3 Heave-Sway 5 1-3 Heave-Heave WAMIT 0 1-3 Heave-Surge WAMIT 0 1 2 3 4 5 0.5 4 ka 1-3 Heave-Sway WAMIT 3 (h-c)(g/a)

2 2 a

Cylinders 1,2 π g 1 Cylinders 3,4 ρ 0.25 Cylinders 1,2 WAMIT 0

Cylinders 3,4 WAMIT Im(Z)/ -1 0.2 -2 a(h-c)

π 0.15 -3 g

ρ 0 1 2 3 4 5

|/A ka y |F 0.1 0.05 1-3 Surge-Surge 0.05 0.04 1-3 Surge-Sway 1-3 Sway-Sway 0.03 1-3 Surge-Surge WAMIT 0

0.5 1-3 Surge-Sway WAMIT 0 1 2 3 4 5 0.02 1-3 Sway-Sway WAMIT ka 0.01 (h-c)(g/a)

0.5 2 a

Cylinders 1,2 π 0 g

0.45 Cylinders 3,4 ρ Cylinders 1,2 WAMIT -0.01 0.4

Cylinders 3,4 WAMIT Re(Z)/ -0.02 0.35

0.3 -0.03 a(h-c)

π 0.25 -0.04 g

ρ 0 1 2 3 4 5

|/A ka z 0.2 |F 0.15 0.05 0.1 1-3 Surge-Surge 0.04 1-3 Surge-Sway 0.05 1-3 Sway-Sway 0.03 1-3 Surge-Surge WAMIT 0 1-3 Surge-Sway WAMIT 0 1 2 3 4 5 0.5 0.02 1-3 Sway-Sway WAMIT ka Figure 2: Comparisons with the wave exciting force 0.01 (h-c)(g/a) 2

results derived in [11] for β = 0 a

π 0 g ρ -0.01 Im(Z)/ -0.02

-0.03

-0.04 0 1 2 3 4 5 ka Figure 3: Comparisons with the radiation impedance results derived in [11] for β = 0

7405 Results for complex geometries and multi-body and 5: such assessment of q leads to the identification WECs are more difficult to compare, as analytical of areas and scenarios which may be starting points for solutions for such configurations are harder to derive. further, more refined studies which address e.g. the The usefulness of the BEM solution is clear in such effects of irregular waves. A preliminary study (limited cases, albeit it is important to ensure that adequate to absorption in heave) on the influence of comparisons with experimental results (validation) is representative seas in the energy yield from the array is conducted, particularly if comparison with analytical presented in the following subsections. Particular results (verification) is not possible. attention is given to the β = ±π /4 case, which can be physically associated with either a change in the Interaction factor for regular waves dominant direction of the incoming sea or a change in The assessment of the performance of an array of the layout of the array. WECs has been approached to date mostly under the action of regular waves, in the vast majority of the studies available in the literature [e.g. 1-10]. Even with 2.2 significant limitations (quite often the peak values of 2 1.8 the interaction factor q are obtained under unrealistic 2 amplitudes of motion) such an approach is still useful 1.6 1.5 to assess the main constructive and destructive 1.4 q interference areas and the wave headings that most 1 1.2 contribute to constructive effects. Nevertheless it is 1 important to emphasise that conclusions from regular 0.5 waves tests will always be limited, and the following 0.8 1.5 subsections of the present paper address more realistic 0.6 1 5 4 0.4 conditions (namely the influence of irregular waves and 3 0.5 2 suboptimal control settings). 1 0.2 0 In this subsection the interaction factor q, which β ka corresponds to the ratio between the power absorbed by Figure 4: Interaction factor q – cylinders absorbing in heave the array and N times the power absorbed by a single isolated device (recall that N is the number of WECs in the array) was calculated [13] as a function of the nondimensional wave frequency ka and the wave 1.4 heading β , for two scenarios: in Figure 4 the cylinders 1.2 can absorb energy in heave, whereas in Figure 5 the 1.4 1.2 energy absorption mode is surge. 1 In Figure 4 the row of peaks along 1 0.8 0.8 β = π /4corresponds to the array being oriented q 0.6 diagonally, in which near trapping was observed at 0.4 0.6 certain wavenumbers. The central peak corresponds to 0.2 q = 2.3, i.e. the array of four cylinders is absorbing 0.4 1.5 about nine times what a single cylinder of the same 5 0.2 1 4 dimensions could at this wavenumber. 3 0.5 2 If the same optimal conditions were applied to 1 0 β ka absorption in surge, the results would show very high values of q near β = π /2, as an isolated surging Figure 5: Interaction factor q – cylinders absorbing in surge cylinder can not absorb any power at this wave heading. The array can still absorb some small power at this heading, so the q factor tends to infinity. Figure 5 shows a suboptimal interaction factor, Suboptimal damping settings for irregular waves reflecting a more practical configuration for absorption in surge. The q factor was redefined to assume that the (single WEC) isolated surging cylinder always surges in the ideal In order to design a wave farm a realistic direction. As expected, when the cylinders can only representation of the sea state needs to be used as input surge perpendicular to the wave heading ( β = π /2), to the simulation. A preliminary assessment of the very little power can be absorbed. Conversely, influence of this additional factor is done in the performance is best when they oscillate with the wave following subsection, but first it is relevant to quantify heading. A significant amount of power can, however, the power absorption characteristics of a single WEC still be absorbed even when the cylinders oscillate at an (acting in isolation) under the same irregular waves in angle to the incoming wave. order to compare its response with that of each farm Although the regular wave investigation only allows element. limited conclusions, its usefulness is clear in Figures 4

7416 In this study the influence of different Furthermore, it is interesting to record not only the omnidirectional spectra (and different dominant wave value of the maximum absorbed power, averaged over directions, in the array layout) was assessed. Two the frequency spectra specified in Table 1 (calculated

spectral parameters, Hm0 and T−10 , plus the shape of from Equation 9), but also the damping coefficient the frequency spectrum, influence the results and thus Dext associated with such a situation, for comparison these parameters were used to define the irregular wave with the individual damping coefficients associated input. Table 1 summarises the frequency spectra that with each array element when deriving the maximum were assessed. In the two last cases the peak period power absorbed by the farm.

Tpeak is specified, and its value set equal to the period Table 2 summarises the maximum power and the associated damping coefficient for the four cases leading to the peak capture width (a regular wave identified in Table 1. Figures 6 and 7 illustrate the result). Such assessment tries to evaluate the effect of power absorption characteristics of the heaving the incoming sea in the device response, which given cylinder, as a function of the wave period and external the simplicity of the design is narrow banded with damping coefficient for selected sea states. Note that regard to wave frequency. the absorbed power is presented in Figures 6 and 7 with

dimensions of MW/Hz, corresponding to the absorbed T Formula / Case Hm0 T−10 peak β power associated with each monochromatic shape Id. [m] [s] [s] [rad] component, which is then integrated over the entire 1 2 7 period range allowing the quantification of the averaged absorbed power per damping coefficient (and 2 4 7 −π /2, thus the identification of the maximum absorbed power Bretschneider 3 2 10.205 −π /4 per sea state). As Table 2 shows the narrow frequency response of 4 4 10.205 the heaving cylinder is clear: the maximum absorbed power in cases 3 and 4 is about 2.64 times higher than Table 1: Definition of the frequency spectra tested that associated with cases 1 and 2, which illustrates the suitability of the WEC to the longer peak period as Note that the different dominant directions β are expected from its capture width characteristics. The wave period also dictates the external damping irrelevant in the single body case, as the cylinder is axisymmetric. Table 1 applies also for the array coefficients Dext associated with the maximum simulations under irregular waves (see the following absorbed power for each sea state. As a result, cases 1- subsection), hence the inclusion of β . Further studies 2 and 3-4 achieve the maximum Pn for the same Dext . can approach different spectral shapes (e.g. Pierson- Finally, it should also be noted that the peak power (in Moskowitz, JONSWAP) or real (measured) seas. MW/Hz) is not necessarily associated with the Rather than searching for the absolute optimum with maximum absorbed power (in kW) over the entire sea, regard to energy capture under the influence of the sea and thus the two events can (and in for the cases listed states described in Table 1 (recall that the optimal in Table 1 do) occur for different damping settings. solution is typically associated with large and These results are particularly relevant for unrealistic amplitudes of motion) suboptimal solutions comparisons between the array output and the output of following a conservative control approach were single cylinder times N (number of WECs). Also, the evaluated. It is recognised from inception that such a external damping values that were obtained can be seen constrained optimisation exercise will not result in the as a starting point for the farm control strategy absolute maximum in terms of energy absorption, yet it iteration, which involves four independent damping will result in an achievable estimate. The cylinder is terms, associated with each array element. controlled via a single control term (external damping,

Dext ), allowing an assessment of the sensitivity of the D solution to variations in the control setting under the Case Max Pn ext action of realistic seas. In the case of the array layout, Id. [kW] [kNs/m] each cylinder is controlled independently (hence the 1 39.2 farm control layer has four terms). To obtain more 590 representative dimensional results the radius of the 2 157.0 cylinder a was altered to 10m, and the other 3 103.5 geometrical properties listed in Figure 1 apply unless 640 otherwise stated. The single cylinder iterations provide 4 414.1 the benchmark for the calculation of the interaction factor q under the influence of irregular waves. Table 2: Maximum absorbed power (per sea state) by an isolated cylinder absorbing in heave with an external damping

coefficient Dext

7427 as for the isolated cylinder case the radius a was altered to 10m. The study was focused on the square array of four elements, and the influences of the sea sates 4 described in Table 1, the farm control strategy and two

3 dominant wave directions ( β =−π /2 and β = −π /4). The latter case corresponds physically to 2 the array being oriented diagonally (hence a layout 1 Power [MW/Hz] Power change). Table 3 summarises the key results: the maximum averaged absorbed power P and the 0 1000 interaction factor q. The suboptimal farm control 15 strategy iterates on each of the four external damping 500 10 5 coefficients which control the four array elements with 0 0 step changes of 10kNs/m. PTO Damping [kNs/m] Period [s]

Figure 6: Power absorption characteristics of a single Case β Max P heaving cylinder – Case Id. 1 q Id. [rad] [kW] 1 −π /2 146.6 0.93 1 −π /4 153.4 0.97 40 2 −π /2 586.2 0.93 30 2 −π /4 613.7 0.98 20 3 −π /2 380.1 0.92 10 Power [MW/Hz] Power 3 −π /4 392.9 0.95 0 1000 4 −π /2 1520.5 0.92 15 500 10 4 −π /4 1571.8 0.95 5 0 0 PTO Damping [kNs/m] Period [s] Table 3: Wave farm absorbed power and interaction factor (irregular waves) under a suboptimal control strategy Figure 7: Power absorption characteristics of a single heaving cylinder – Case Id. 4 It should be emphasised that, as in the isolated cylinder case, the control strategy that was Suboptimal damping settings for irregular waves implemented is suboptimal, as a single external damping value is implemented throughout the sea state (arrays of WECs) on each WEC (passive control per sea). A more This subsection presents the first preliminary complex control approach would involve changing this assessment of the combined influences of the incoming control term for each incoming wave period or for each sea state, array layout and control strategy in the wave representative wave group (active control). Further farm energy yield. Ultimately, it is expected that such complexity is introduced by controlling more than just fundamental research can benefit the creation of the external damping (e.g. stiffness) or by absorbing in components in software package which are able to more than one degree-of-freedom. More complex quantify the interaction effects, iterate and optimise the WECs are likely to involve more than one body, and wave farm design, taking these and other operational thus the potential to explore other control and layout constraints into account (e.g. electrical cable length, configurations is also higher. mooring strategy, minimum / maximum distance This methodology deliberately tries to obtain more between each WEC, etc.). achievable estimates than those linked with optimal In this preliminary assessment three aspects are control, although it is recognised that improvements particularly critical: 1) the influence of the layout of the can be obtained via the implementation of less array in the energy yield under the action of irregular simplistic approaches. The main value of this waves; 2) the potential to further improve the energy preliminary assessment is in the quantification of the yield by adjusting the control of each farm element combined effects of layout and control changes under (farm control strategy); 3) the quantification of the irregular waves in the farm energy yield, even in such interaction factor q for irregular waves in the above a simplistic case. As Table 3 shows there is an average mentioned scenarios. As in the previous subsection a increase of 4% due to the (coupled) layout and control suboptimal solution (one term control strategy per iterations (when changing β from −π /2 to −π /4), a WEC – external damping) was implemented, to obtain figure which would have a significant impact in the more conservative estimates. All the geometrical farm’s energy yield and thus on the revenue of a wave properties of the array are listed in Figure 1, although farm. It is further encouraging to acknowledge that

7438 such result was derived in a simplified scenario (simple applied in the array case). Similar variations are geometry, single body per WEC, one term controller obtained for the case listed in Table 3. This range may per WEC). induce changes in the engineering specifications of the It is also possible to compare the average power P PTO, when considering the implementation of the full with the power that would be absorbed if all the array range of damping coefficients in a real application. elements were controlled with the same external Under these simplified assumptions the interaction damping coefficient, set equal to the value derived in factor q is always below 1 for all the sea states tested, the previous subsection when analysing an isolated which places the farm design objective as the cylinder. The average drop for all scenarios is around minimisation of the effects of destructive interference. 1%, in this case exclusively due to the absence of fine Further iterations which can include other variables tuning of the farm control strategy. such as the array spacing (regular or irregular), may Figures 8 and 9 show the type of quantitative mitigate such aspects, and even alter the design assessment that can be made when evaluating different objective (if q > 1 the goal is to maximise the effects control approaches for each case. When a wide range of of constructive interference). damping settings is analysed, zooming out in such figures allows the assessment of the implications of under and over-damped scenarios, while zooming in 4 Final Remarks allows the fine tuning of the four term farm control layer to become more evident. A preliminary assessment of a simplified WEC array was conducted, quantifying the effects of layout and 700 800 600 650 control iterations under the action of irregular waves. 650 800 600 650 600 800 600 650 To increase confidence in the numerical predictions, 550 800 600 650 800 750 600 650 the fundamental hydrodynamic properties were firstly 750 750 600 650 benchmarked against analytical results, and a regular 700 750 600 650 650 750 600 650 wave investigation was also carried out to outline the 600 750 600 650 550 750 600 650 areas of interest. The ultimate objective of this 800 700 600 650 approach is the creation of a software tool that assesses 750 700 600 650 700 700 600 650 and quantifies the interaction between the several array 650 700 600 650 600 700 600 650 elements, optimising the output. Particular attention is

PTO (kNs/m) appliedcylinders 4 PTO (kNs/m) to to 1 550 700 600 650 given to forces (wave induced and externally applied) 800 650 600 650 750 650 600 650 and the wave farm’s power absorption characteristics, 611 611.5 612 612.5 613 613.5 614 allowing comparisons with N times the output of an Farm Power [kW] isolated WEC (N being the number of WECs in the Figure 8: Wave farm power absorption characteristics array). (Case 2: H = 4m , T = 7s , β =−π /4) m0 −10 For simplicity the study was focused on a simple 750 700 850 750 geometry (circular cylinder), a limited (and fixed) 700 700 850 750 650 700 850 750 number of WECs (four), and a suboptimal control 600 700 850 750 approach which allowed only one term to be controlled 550 700 850 750 per WEC (external applied damping in heave). Even in 900 650 850 750 850 650 850 750 this simplified scenario, layout and control iterations 800 650 850 750 led to an average increase of 4% in the farm’s energy 750 650 850 750 absorption properties, a figure that would have a 700 650 850 750 650 650 850 750 significant impact in the energy yield. Furthermore, the 600 650 850 750 suboptimal nature of the control strategy (deliberately 550 650 850 750 900 600 850 750 tested to ensure that conservative estimates were 850 600 850 750 obtained) limited the potential of the approach. For PTO (kNs/m) applied to cylinders 1 to 4 1 to cylinders applied to (kNs/m) PTO 800 600 850 750 example the individual damping coefficients are kept 750 600 850 750 1555 1560 1565 1570 1575 fixed for each sea state (passive strategy, suited to tune Farm Power [kW] a WEC in a statistical sense, i.e. for the duration over Figure 9: Wave farm power absorption characteristics which the sea state is characterised); a more complex methodology would involve changing these settings for (Case 4: Hm0 = 4m , Tpeak =10.205s , β =−π / 4 ) the next set of incoming waves (active strategy). Finally, it is also interesting to compare the values of Further studies can address (among other variables) the damping coefficients associated with each of the variations in the number of WECs (farm installed four cylinders and compare such values with the capacity), WEC spacing (which is limited for analogous coefficient linked to the isolated cylinder operational and non-technical reasons), and other externally applied forces (e.g. mooring loads). The case. For example, for case 2 ( Hm0 = 4m , T−10 = 7s ) implementation of alternative algorithms such as those with β =−π /4 the suboptimal damping setting for described in [8] and [14] may also provide a means to cylinders 1 to 4 are 650, 750, 600 and 650 kNs/m (see reduce the computational effort associated with the Figure 8), which can be compared to the 590 kNs/m design of a wave farm. (thus up to 27% smaller than the maximum value

7449 References [8] Mavrakos, S. and McIver, P. Comparison of Methods for Computing Hydrodynamic Characteristics of Arrays of [1] Budal, K. Theory for absorption of wave power by a Wave Power Devices. Applied Ocean Research, Vol. 19, system of interacting bodies. Journal of Ship Research, 283-291, 1998. Vol. 21, 248-253, 1977. [9] Justino, P. and Clément, A. Hydrodynamic Performance [2] Falnes, J. and Budal, K. Wave-power conversion by of Small Arrays of Submerged Spheres. Proc. 5th point absorbers. Norwegian Maritime Research, Vol. 6, European Wave Energy Conference, pp. 266-273, 2003. 2-11, 1978. [10] Fitzgerald, C. and Thomas, G. A preliminary study on [3] Evans, D. Some theoretical aspects of three-dimensional the optimal formation of an array of wave power devices. wave-energy absorbers. Proc. First Symposium on Wave Proc. 7th European Wave Energy Conference, 2007. Energy Utilization, Gothenburg, Sweden, pp. 77-113, 1979. [11] Siddorn, P. and Eatock Taylor, R. Diffraction and independent radiation by an array of floating cylinders. [4] Falnes, J. Radiation impedance matrix and optimum Ocean Engineering, Vol. 35, 1289-1303, 2008. power absorption for interacting oscillators in surface waves. Applied Ocean Research, Vol. 2, 75-80, 1980. [12] Yilmaz, O. and Incecik, A. Analytical solutions of the diffraction problem of a group of truncated cylinders. [5] McIver, P. Some Hydrodynamic Aspects of Arrays of Ocean Engineering, Vol. 25, 385-394, 1998. Wave-Energy Devices. Applied Ocean Research, Vol. 16, 61-69, 1984. [13] Siddorn, P. Wave energy absorption by arrays of oscillating bodies. MEng Final Report, University of [6] Thomas, G. and Evans, D. Arrays of three-dimensional Oxford, 2007, wave-energy absorbers. Journal of Fluid Mechanics, Vol. 108, 67-88, 1981. [14] Li, Y. and Mei C. C. Multiple resonant scattering of water waves by a two-dimensional array of vertical [7] McIver, P. The Hydrodynamics of Arrays of Wave- cylinders: Linear aspects. Physical Review, Vol. 76, Energy Devices. Wave Energy Converters – Generic Article Number 016302, Part 2, 2007. Technical Evaluation Study (Annex B1), Report to the Commission of European Communities, 1993.

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