Hydrodynamic optimization of the geometry of a sloped-motion wave energy converter

Ocean Engineering 199 (2020) 107046

Contents lists available at ScienceDirect

Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Hydrodynamic optimization of the geometry of a sloped-motion wave energy converter Claudio A. Rodríguez a, *, Paulo Rosa-Santos b, c, Francisco Taveira-Pinto b, c a

Laboratory of Ocean Technology (LabOceano), Department of Naval Architecture and Ocean Engineering, Federal University of Rio de Janeiro – Parque Tecnol� ogico do Rio, Rua Paulo Emídio Barbosa 485 – Qda. 07A, Cidade Universit� aria, Rio de Janeiro, RJ, CEP. 21.941-907, Brazil b CIIMAR - Interdisciplinary Centre of Marine and Environmental Research of the University of Porto, Terminal de Cruzeiros do Porto de Leix~ oes, Av. General Norton de Matos, S/N, 4450-208, Matosinhos, Portugal c FEUP - Faculty of Engineering of the University of Porto, Department of Civil Engineering, Rua Dr. Roberto Frias, S/N, 4200-465, Porto, Portugal

A R T I C L E I N F O

A B S T R A C T

Keywords: CECO Exhaustive search method WEC tuning Experimental testing Frequency domain modelling Hydrodynamic efficiency

This paper presents the hydrodynamic optimization study of CECO, a point absorber wave energy converter (WEC) with sloped motion. To maintain the overall costs, the characteristic dimensions of the optimized solution were not allowed to change significantly. Instead, different geometrical shapes have been generated and numerically investigated based on the exhaustive search method with a heuristic approach. The assessment of the different geometries was based on a new index - the hydrodynamic capacity for wave energy conversion - which considers that the theoretical maximum WEC’s absorbed power from irregular waves is obtained assuming that for each individual wave component of the sea spectrum, the PTO system can operate with its optimum damping coefficient. The optimum geometry obtained is able to harvest twice as much wave energy than the original design of CECO. The numerical outcomes have been validated with the results of experimental tests with the new geometry. Unlike “pure” heaving WECs, a sloped-motion WEC can achieve natural oscillation periods within a broad range by controlling the inclination of the motion path, the submergence level or the shape of its floaters. Therefore, CECO can be tuned to any given sea state and avoid the need for active complex control strategies.

1. Introduction The global energy demand is expected to grow by more than 25% to 2040 (IEA, 2018). In addition, the share of renewables in energy gen­ eration should rise to over 40% by 2040 (from 25% today), which is essential in securing a sustainable future for the planet. Ocean wave energy, in spite of being on a pre-commercial stage, has the potential to provide utility scale power production in the future, due to the huge global offshore wave power resource, estimated at approximately 32, 000 TWh/yr (Reguero et al., 2015). That potential could be used to provide about 10% of the Europe electricity needs by 2050 (Rusu and Onea, 2018), suppling not only the onshore grids but also offshore ac­ tivities (Oliveira-Pinto et al., 2019). A large variety of technologies to convert wave energy has been developed and tested, but a dominant concept was not found yet (Taveira-Pinto et al., 2015). However, some promising designs are presently undergoing detailed optimization studies and demonstration testing at open sea, therefore approaching the commercial phase (e.g.,

Liang et al., 2017; Moretti et al., 2020; Sheng et al., 2017). A key variable in the optimization of a wave energy converter (WEC) is the amount of power absorbed, which shall take into account the wave climate at the installation site. In this process, the WEC is often tuned so that its natural period of oscillation matches the most probable sea-state period or that with the higher amount of energy on a yearly basis at the point of interest. However, the characteristics of the power matrix also have an important influence on the annual power absorption of the WEC (De Andres et al., 2015). This explains why in some studies the opti­ mization is done so as to harvest the maximum power over the largest range of periods possible considering the wave climate at site (Shadman et al., 2018). The two main alternatives to adjust and optimize the performance of a WEC are: (i) the modification of the physical characteristics of the device (e.g., size, mass or geometry), which has a direct impact on its natural oscillation periods; and (ii) the control of the power take off (PTO) system, which consists in the modification of the WEC absorption characteristics over time to maximize its power production.

* Corresponding author. E-mail address: [email protected] (C.A. Rodríguez). https://doi.org/10.1016/j.oceaneng.2020.107046 Received 19 October 2019; Received in revised form 31 December 2019; Accepted 30 January 2020 Available online 7 February 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

The tuning of WECs to the prevailing sea states can be done in different ways, depending on the type of device, namely by modifying the inertia of the system (e.g., Flocard and Finnigan, 2012), the mass distribution (e.g., Meng et al., 2019), the geometry of floating bodies (e. g., Goggins and Finnegan, 2014), the mooring system characteristics (e. �pez et al., g., Paredes et al., 2016), the PTO inclination angle (e.g. Lo 2018) or the PTO characteristics (e.g., Shi et al., 2019; Xiao et al., 2017). Moreover, different PTO control strategies are available to maximize power production (Ozkop and Altas, 2017) namely, the passive loading control, the latching control and the reactive loading control. However, in contrast to the characteristics of the PTO system (stiffness and/or damping), the physical characteristics of the WEC are often more diffi­ cult to vary according to the incoming waves and therefore are defined first. Different approaches of varying complexity can be applied in the optimization process of WECs. In a simplified form, the main goal is to maximize the mean absorbed power while the construction costs are minimized. To reach the tradeoff between multiple goals, multiobjective optimization techniques can be applied, namely those based on genetic (e.g., Alamian et al., 2019; McCabe, 2013) or differential evolutionary algorithms (e.g., Blanco et al., 2019). Other alternatives are the sequential optimization procedures, in which the optimal parame­ ters are set in a consecutive manner (Bouali and Larbi, 2017), or the optimization approach based on exhaustive search and driven by con­ strains, to reduce the number of variables and simplify the optimization process. In this last case, if constrains are incorrectly chosen, inappro­ priate results may be obtained, as Garcia-Rosa et al. (2015) demon­ strated when analyzed the impact of PTO restrictions on the shape optimization of a heaving buoy, since different control strategies resul­ ted in different optimal geometries. The geometry optimization is of relevance for all classes of WECs, since it determines the hydrodynamic characteristics of the fluidstructure interaction. Therefore, to ensure that a WEC works effi­ ciently at a given site, its geometry is often considered the first variable for the optimization of its design (Bozzi et al., 2018; Esmaeilzadeh and Alam, 2019). To maximize the absorbed power and the absorption bandwidth of a heaving point absorber, Shadman et al. (2018) optimized the draft and diameter of the floater while maintaining its natural period close to the predominant wave periods at site using a frequency domain method. The PTO was assumed to be a pure damper and its damping frequency independent and equal to the hydrodynamic damping of each floater shape at its resonant frequency, to maximize the power absorp­ tion at that frequency. Pastor and Liu (2014) optimized also a point absorber by testing different shapes, drafts and diameters of the buoy taking into account the wave conditions of the deployment site near the Gulf of Mexico. The PTO control parameters (i.e., spring and damping coefficients) were adjusted to tune the WEC to the incident wave con­ ditions and therefore to maximize its power absorption. Goggins and Finnegan (2014) presented a methodology for the geometry optimiza­ tion of a floating axisymmetric point absorber, which considers 6 basic geometric shapes derived from a vertical cylinder and have used the average annual wave energy spectrum for the case-study location, offshore the west coast of Ireland. An ideal linear damper was used where the PTO damping coefficient is maintained constant, since this approach facilitates the dynamic response calculation in the frequency-domain. The optimum damping was determined so as to maximize the mean power absorbed at the reference site. Even though the classical linear approach allows defining an opti­ mum PTO damping coefficient that maximizes the absorbed power for each particular wave period, the use of a single optimum damping co­ efficient for the whole set of waves that compose a given sea spectrum is questionable if the maximum power output is envisioned in the geo­ metric optimization work. The assessment of the power absorbed considering that the PTO system is operating with its optimum damping coefficient, for each individual period of the wave spectrum, would have the benefit of allowing to obtain the maximum power output, for each

tested geometry, avoiding the arbitrariness of choosing a particular damping coefficient. The advancement of computational resources facilitates the appli­ cation of optimization methodologies. The computational fluid dy­ namics models based on the Navier-Stokes equations allow considering the effect of all relevant nonlinearities on the hydrodynamic analysis (e. g., wave breaking, surface piercing, vortex shedding), but are still not a cost effective tool to be used in cases where a systematic analysis of a large number of variables needs to be done. Nonetheless, such models were already applied in some optimization studies of WECs (e.g., Shi et al., 2019). On the other hand, the use of frequency domain hydrodynamic methods (linear) based on potential flow theory in the optimization of WECs has two important advantages: a smaller computational cost and the possibility of carrying out systematic analyses considering a wide range of physical characteristics of the WEC. A large number of exam­ ples can be found in which those models were used in the optimization process (e.g., Goggins and Finnegan, 2014; Piscopo et al., 2016; Shad­ man et al., 2018; Tom et al., 2016). Time domain models are, however, recommended in the more advanced development phases, to validate the characteristics the optimized WECs. This paper presents the hydrodynamic optimization of the floaters of a sloped-motion WEC – CECO. Since the original geometry of the floaters was intuitively defined, an exhaustive search method with a heuristic approach is used to improve its ability to harvest wave energy. In order to maintain the costs of the different solutions in the same order of magnitude, the characteristic dimensions of CECO were not changed significantly. The different geometric shapes were assessed numerically and then compared using a new hydrodynamic efficiency parameter – the index of hydrodynamic capacity for wave energy conversion. The optimum geometry of the floaters has been validated experimentally. 2. CECO’s working principle The WEC concept for the present study (CECO) is a point absorber that harvests wave energy from the translational motions of its floaters along a straight-sloped path defined by the inclination of the guiding rods, Fig. 1. CECO consists of two structures: a moving part (composed of two lateral floaters attached to a central connecting frame) and a fixed-reference structure (that houses the power take-off system). When the moving structure is excited by the waves, its sloped oscillatory

Fig. 1. Scheme of the CECO concept. 2

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

motion is converted into rotations by a rack-pinion mechanism. The rack is attached to the connecting structure while the pinion is connected to a DC generator’s shaft. Finally, rotations are converted into electricity by the DC generator (which is fixed and housed in the reference structure). For the sake of simplicity, the axis of the central cylinder is installed perpendicular to the motion path of the floaters. The most outstanding features of CECO concept are its ability to harness energy from both surge and heave modes simultaneously and, as shown in previous studies (e.g., Rodríguez et al. (2019b); Rosa-Santos et al. (2019b)), the possibility of tuning the natural frequency of the WEC motion with the angle of its translation path. The nominal (design) inclination and submergence volume of the floaters was 45� and 60%, respectively. For this condition, the free-oscillation frequency of CECO’s floating part is around 8 s. Table 1 presents the main dimensions and nominal operational parameters of CECO.

assessed easily and independently of the other design variables. The behavior and effects of these “unessential” variables may be predicted in advance through preliminary analyses. 3.1. Optimization scope and criteria The results of previous studies with CECO highlighted the fact that its original design was not suitable for the wave conditions typical of the most energetic coastal stretches of the target sea site, for example, the northern Portuguese coast (Ramos et al., 2017; Rosa-Santos et al., 2019b). In this context, the first objective of the optimization task is to adapt the geometry of CECO so that its natural period of oscillation becomes resonant with the period of the target sea condition to improve its efficiency for wave energy harvesting. Since the focus is the hydrodynamic performance associated to wave energy extraction, the criteria for the assessment of the alternative ge­ ometries to be analyzed in the optimization process, in principle, can be expressed by:

3. Optimization strategy The standard form of the general optimization problem is typically represented by the objective (or goal) function and the constraints. Both are defined in terms of design variables which, in the optimization process, are modified to obtain the optimum. In the literature, several references can be found describing a large number of optimization techniques that exist for engineering optimization applications (e.g., Coello-Coello, 1999; Onwubolu and Babu, 2013; Rao, 2019; Venter, 2010). In general, depending on the complexity of the objective func­ tion, the associated constraints and the optimization technique, the optimization process requires substantial computational effort. In this study, the objective function is the absorbed power of the WEC, which is intrinsic to its hydrodynamic response. In other words, the complexity of the objective function and the associated constraints �pez et al. depends on the type of the adopted hydrodynamic model. Lo (2017a, 2017b) have performed time-domain simulations using a nonlinear numerical tool based on the potential flow theory. Although that approach is expected to be more accurate for the analysis of wave-body interaction, it still demands substantial computational effort when body dynamics in irregular wave conditions are required. More recently, Rodríguez et al. (2019a) used a frequency-domain linear nu­ merical model for the assessment of damping coefficients for regular and irregular wave conditions from model tests. Their results compared very well with those from the respective time-domain nonlinear approach with much less computational time. Therefore, the frequency-domain approach was considered adequate for the hydrodynamic optimization process. A significant amount of computational cost can be further saved if the design variables are considered as deterministic variables at their reference values, i.e., the number of analyses is reduced by systemati­ cally fixing “unessential” design variables throughout the optimization process (Kim et al., 2004). The term “unessential” may not necessarily refer to irrelevant design variables, but to variables whose effect can be

� the response amplitude operator (RAO) of the motions of the floating part considering only its inherent damping (i.e., without the PTO effect); � the efficiency in the wave energy absorption for the target sea condition. The first criterion assesses the capacity of the WEC for the conversion of wave excitation into motions. The greater the motions, the greater the available power that can be absorbed by the PTO system. Typically, the peak of the RAO curve is obtained when the wave excitation period matches the natural period of the system, i.e., at resonance. Therefore, the tuning between those periods is a key parameter in the optimization of WECs. The second criterion is related to the conditions of maximum power absorption specified in the traditional approach. However, there are two important aspects that should be highlighted. First, in the present approach, the second criterion refers to a given sea condition, i.e., irregular waves with random heights and periods, while the traditional approach is based in regular waves (single wave period and amplitude). In irregular waves, there is no uniform distri­ bution of wave energy (indeed, wave energy is unevenly distributed over a wide range of periods although concentrated around the peak period of the sea spectrum), so the WEC that maximizes wave power absorption is not necessarily the one whose natural oscillation period matches the sea spectrum’s peak. That WEC, in fact, can present a very high (sharp) peak of absorbed energy at its resonant period, but it may not absorb a significant amount of energy at the other periods of the sea spectrum. In contrast, a hypothetical WEC with its resonant period detuned from the sea’s peak period may absorb a larger amount of wave energy resultant from the overall contribution of a number of other components of the wave spectrum. To explain the latter case, the dy­ namic behavior of the WEC along the wave frequency domain should be considered. The aim is to keep the WEC0 motions significant in a broad range of wave periods, by putting, for instance, the sea’s peak period to excite oscillation periods at which the WEC does not display sufficiently large motions, and the periods of “smaller” wave energy to excite the resonant period of the WEC. The WEC’s motion will not be the largest in any case, but the system will be able, in principle, to absorb a larger amount of wave energy by adding the individual contributions from a broader range of periods. The second aspect is related to the PTO damping coefficient and the definition of the captured power efficiency for the WEC under irregular waves. As known, the amount of absorbed energy is directly dependent on the level of damping of the PTO system. For regular waves, the classical linear approach provides an optimum PTO damping coefficient that maximizes the WEC’s absorbed power for each single (regular) wave period. If this approach is applied to a (passive) WEC in irregular

Table 1 Main characteristics of CECO. Floaters Shape diameter width draught

semi-cylindrical 8.00 4.50 4.60

Connecting structure length width Inclination

20.00 5.68 45�

Reference structure shape diameter

Cylinder 4.00

m m m m m

m

3

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

pffiffiffiffiffiffi and the wave excitation force, respectively, i ¼ 1. This dynamic system has been numerically implemented in WAMIT® - a radiation/diffraction panel program developed for the linear analysis of the interaction of surface waves with various types of floating and submerged structures (Lee and Newman, 2013). For CECO, both the fixed cylinder and the oscillating floaters have been modelled to allow a multibody hydrodynamic analysis. It should be highlighted that different from traditional WEC applications, such as the heave point absorbers, CECO’s dynamics required the definition of a generalized (additional) mode of motion while the motions in the other 6-dof con­ ventional modes were restrained. The definition of the generalized mode allows the computation of hydrodynamic coefficients and wave excita­ tion force for the inclined degree of freedom of the floaters. Although eq. (1) refers only to the motion dynamics of CECO’s floating part, the effect of the central (fixed) cylinder is automatically accounted for by WAMIT® as a hydrodynamic interaction effect in the added mass co­ efficient (As) and in the hydrodynamic damping coefficient (Bs) of the floaters. When the amplitude of the response for any given wave excitation period is divided by the amplitude of the incident wave, the well-known RAO of motions for the floaters is obtained:

waves, a challenge arises because a single optimum PTO damping co­ efficient should be chosen for the whole set of waves that compose the given sea spectrum. The selection of that PTO damping coefficient can be arbitrary but is limited by the range of damping levels that the actual PTO system can provide, especially at the peak period of the wave spectrum and at the WEC’s resonant condition. However, intermediate values may not be disregarded since an overall optimum condition is sought for the full range of periods of the wave spectrum. For the assessment in the optimization study, this dependence on arbitrarily chosen values of the PTO damping coefficients is not desir­ able. To avoid that challenge, an alternative efficiency index for irreg­ ular sea conditions is proposed for the assessment of the WEC’s captured power. This index is denominated hydrodynamic capacity for wave energy conversion (ηhydro,max) and is defined as the ratio of the maximum WEC’s absorbed power under optimum PTO damping condition over the en­ ergy flux of the incident wave along the WEC’s width (see Eqs. (9)–(11)). For a given sea state, the optimum PTO damping condition assumes that for each individual wave period of the sea spectrum, the PTO system is operating with its optimum damping coefficient (obtained from the classical linear approach). Although, in practice, the optimum PTO condition in irregular waves may be impossible to be achieved, the hydrodynamic capacity index for wave energy conversion represents a theoretical condition in which the maximum absorbed power could be obtained from an irregular sea, if an ideal (theoretical) control system is applied to the WEC. This dimen­ sionless index concentrates in a single value the hydrodynamic perfor­ mance of the WEC in irregular waves. As such, it will be used as the objective function in the optimization problem, i.e., the optimum candidate will be the one that displays the highest index of hydrody­ namic performance.

RAOs ðωÞ ¼

1 PPTO; ​ reg ¼ BPTO ω2 s2a 2

� � BPTO; ​ opt ðωÞ ¼ B2s ðωÞ þ ωðm þ As ðωÞÞ

_ F ws ð

ωÞ ω2 ½m þ As ðωÞ� þ iω½Bs ðωÞ þ BPTO � þ Cs

(1)

ω

�2 �1=2

(5)

To describe the motions of the WEC in irregular waves, the spectral approach can be applied if the linear superposition principle is assumed for the description of wave elevation and the WEC’s response. According to this principle, the response of a floating body to an irregular sea can be approximated by superimposing the responses of the body to each one of the regular waves components of the irregular sea. This approach is typically applied for the assessment of motions of ships and offshore platforms in irregular waves (Faltinsen, 1993; Lewis, 1988) and can be expressed as: Ss ðωÞ ¼ RAO2s ðωÞ⋅Sζ ðωÞ

(7)

where Sζ(ω) and Ss(ω) denote the power spectral densities of the incident sea (wave spectrum) and the WEC’s response (response spectrum), respectively. For the computation of the WEC’s absorbed power in an incident irregular wave characterized by a wave spectrum Sζ(ω), the linear su­ perposition principle can also be applied, so that, for an arbitrary (constant) PTO damping coefficient, the average WEC’s absorbed power is expressed by: Z ∞ PPTO; ​ irr ¼ BPTO ω2 ½RAOs ðωÞ�2 Sζ ðωÞdω (8)

(2)

0

For the computation of the index of hydrodynamic capacity for wave energy conversion, the (theoretical) maximum absorbed power from an

_

where s and F ws are the complex amplitudes of the harmonic motion _

Cs

This (optimum) PTO damping coefficient allows the WEC to absorb maximum (average) power for that (regular) wave frequency: . _ 2 jF ws ðωÞj 4 Pmax; ​ opt ðωÞ ¼ (6) Bs ðωÞ þ BPTO; ​ opt ðωÞ

where m is the mass of CECO’s oscillating part, As, Bs and Cs are added mass, hydrodynamic damping and hydrostatic force coefficients, respectively. Fws and αs are the amplitude and phase of the wave exci­ tation force. ω is the wave-excitation frequency and t is the time. BPTO is the damping coefficient of the force exerted on CECO by the PTO system that is modelled as a linear damper system (proportional to the WEC’s velocity). Time derivatives are expressed by dots over the instantaneous position of CECO’s oscillating part along the inclined path (s). Subscripts s indicate that the coefficients of the forces refer to the inclined 1-dof. Since, typically, the purpose of the simulations in optimization studies is to compare different designs more qualitatively than quantitatively, for the sake of simplicity, the hydrodynamic damping coefficients consider only the potential (non-viscous) contribution. If the force coefficients in eq. (1) are assumed constant for a given regular wave condition, the equation of motion becomes linear and its time-dependence vanishes to give place to an algebraic (frequency domain) equation. The solution for this equation of motion can be expressed as: s ðωÞ ¼

(4)

According to Falnes (2002), the optimum PTO damping for a regular wave of frequency ω is given by (optimum amplitude condition):

Although CECO is composed of two bodies partially submerged (the oscillating part and the central fixed structure), in terms of dynamics, it can be modelled as a one-degree-of-freedom (1-dof) body. Notice that since the cylinder is fixed, it is not allowed to move in any degree of freedom. To describe the motions in regular waves of a 1-dof body (i.e., the floating oscillating part of CECO), the most common mathematical model is:

_

(3)

where ζa(ω) is the amplitude of the incident wave and sa the amplitude of the WEC’s response. For such a system, the time-average power absor­ bed by an arbitrary PTO damper in regular waves is expressed as:

3.2. Hydrodynamic model

ðm þ As Þ€ s þ ðBs þ BPTO Þs_ þ Cs s ¼ Fws cosðωt þ αs Þ

sa ðωÞ ζa ðωÞ

4

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

irregular sea is given by: Z ∞ Pmax; ​ irr ¼ BPTO;opt ðωÞω2 ½RAOs ðωÞ�2 Sζ ðωÞdω 0

The energy flux of the given sea state can be obtained by: Z ∞ Pζ ¼ ρgb cg ðωÞSζ ðωÞdω 0

null (corresponding to a pure surge oscillator) to 90� (pure heave oscillator). The floaters’ response is significantly affected not only in amplitude, but mainly in the location of its natural period – which, in turn, is dependent on hydrostatic restoring and hydromechanical iner­ tial characteristics. Both the restoring force and added mass coefficients are (strongly) affected by the inclination of the motion trajectory (Figs. 4 and 5). Moreover, their combined trends make the inclination effect even stronger. Starting from 90� inclination (pure heave) and moving towards smaller inclination angles, it is observed that added mass co­ efficients increase whereas restoring coefficients decrease. Both trends contribute to an increase of the natural oscillation period with the reduction of the path’s inclination. On the other side, the (unrealistic) large values of response amplitudes as the inclination is decreased from the vertical direction towards smaller angles, particularly for oscillation periods over 4 s, can be explained by the drastic decrease of the potential damping coefficient for all the inclination angles (Fig. 6) in spite of the decrease of the wave excitation force (Fig. 7). In practice, such (unre­ alistic) large response amplitudes are not expected to occur due to the presence of other sources of damping such as viscous drag and me­ chanical friction, besides the PTO damping. However, for the pre­ liminary analyses and subsequent optimization process, based on �pez et al., 2017a; previous validations against experimental results (Lo Rodríguez et al., 2019a, b), the potential model is believed to reproduce �pez well the behavior of CECO in regular and irregular waves. Both, Lo et al. (2017b) with a time-domain nonlinear tool and Rodríguez et al. (2019a, 2019b) with a frequency-domain linear model, have reported a satisfactory agreement in the prediction of RAO motions, particularly the CECO’s natural oscillation period. In the optimization strategy, the effect of those punctual unrealistic RAO values is strongly attenuated. First, in the computation of RAO velocities (when the RAOs of motion are multiplied by oscillation fre­ quencies) and then in the computation of the density spectrum of absorbed power, when the sea spectrum is introduced. It should be also noticed that the hydrodynamic index results from the integration of the spectrum of the absorbed power along the sea-spectrum frequencies. In summary, the smaller the inclination, the greater the natural oscillation period, e.g., by changing the inclination from 90� to 20� , the natural oscillation period can be shifted from 4.5 s to more than 16 s. Based on the above analyses and results, it may be concluded that the effect of inclination angle is known and can be readily predicted. Therefore, to reduce the number of design variables, hereafter, the inclination angle of the motion path will be set as constant in the opti­ mization process. Another parameter that was analyzed in advance is the level of

(9)

(10)

where cg represents the group velocity of the wave and b the WEC’s width, which is considered as twice the floater’s width. Therefore, the index of hydrodynamic capacity for wave energy conversion is given by:

ηhydro;max ¼

Pmax; ​ irr Pζ

(11)

3.3. Preliminary analyses The preliminary analyses involve the assessment of the hydrody­ namic interaction between the floating part and the central support structure as well as the impact on the response of CECO of the inclination of the trajectory of the floating part and the submerged volume of the floaters. In terms of wave energy absorption, the hydrodynamic per­ formance is directly dependent on the wave-induced motions of the WEC, so the preliminary analyses will be based on the dimensionless responses of the floating part of CECO in regular waves, i.e., the RAO. The assessment consists of comparing the RAOs of alternative configu­ rations of CECO with the reference design configuration (Marinheiro et al., 2015), i.e.: floaters with a semi cylindrical shape whose axis is horizontal and perpendicular to the direction of incident waves; about 60% of floaters’ submergence; 45-degree inclination for the path (translational trajectory) of the floating part; and a fixed support structure with a cylindrical cross-section in between the floaters. In the computation of the RAOs of CECO’s floating part, the damping of the system only included the potential radiation effects in eq. (2), i.e., viscous effects were neglected and the PTO damping was set to zero. The effect of the central fixed cylinder was investigated by comparing the motion response of the floaters with and without that structure for the inclinations of 45� and 30� . Fig. 2 shows that the effect of the central structure on performance can be considered negligible. So, the following numerical analyses can be further simplified by modelling only the two floaters as a single structure, instead of the actual multi­ body system (floaters þ fixed cylinder). Fig. 3 details the effect of the inclination of CECO’s trajectory by considering, besides the floaters’ response at the reference inclination (45� ), inclinations ranging from

Fig. 2. CECO’s RAO of motion with and without the fixed central cylinder. 5

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

Fig. 3. Effect of the inclination of motion path on the floaters’ response.

submergence of the floaters. For the reference inclination (45� ), different submergence levels have been simulated and the correspond­ ing wave-induced responses computed (see Fig. 8). Based on hydrostatic considerations, the increase of submergence increases the submerged volume (thus, the required mass of the system) and induces variations in the mean waterplane area. Due to the semi cylindrical shape of the floaters, the maximum waterplane area occurs for a submergence of 50%. Below and over that submergence level, a decrease in the water­ plane area is obtained towards the extreme draughts. Therefore, for submergences over 50%, the natural oscillation period increases with the increasing draught, due to the reduction in waterplane area and the increase in the submerged volume. For submergences below 50%, the reduction in draught decreases the waterplane area and the submerged volume together, so that, depending on the dominant effect, a decrease or increase in the natural oscillation period can be obtained. For the CECO floaters at 45� , a continuous increase in the natural period is observed as draught increases. Notice, however, that by controlling the submergence of the floaters, natural periods between 6 s and 9 s are obtained (Fig. 7), i.e., the effect of submergence is less intense than the inclination effect. Based on these results, the effect of submergence can be considered as known and predictable.

Fig. 4. Effect of inclination of motion path on the floaters’ hydro­ static coefficient.

Fig. 5. Effect of inclination of motion path on floaters’ added mass coefficient. 6

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

Fig. 6. Effect of inclination of motion path on floaters’ potential damping coefficient.

Fig. 7. Effect of inclination of motion path on floaters’ wave excitation force.

Fig. 8. Effect of the submergence level (in %) on the floaters’ response.

7

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

212.09 49.80 4.50 5.53 9.1 16.1 69.4 102.6 43.0 41.9% 245.16 41.91 4.50 4.66 10.7 7.5 51.8 102.6 41.2 40.2%

26

298.44 56.69 4.50 6.30 9.5 16.3 51.1 102.6 50.8 49.5%

25

258.10 51.33 4.00 6.42 9.0 14.3 79.5 91.2 40.1 44.0%

24

180.48 36.00 4.50 4.00 7.7 9.6 118.2 102.6 19.0 18.5%

23

273.77 37.37 4.50 4.15 10.0 5.7 6.4 102.6 40.5 39.5%

22

70.58 31.18 4.50 3.46 6.0 11.4 162.1 102.6 10.6 10.3% 126.45 47.32 4.50 5.26 6.8 21.4 204.8 102.6 18.6 18.1% 115.64 46.09 4.50 5.12 5.5 36.1 259.3 102.6 12.1 11.8% 57.93 35.99 4.50 4.00 4.7 30.4 224.3 102.6 9.0 8.8%

143.64 36.00 4.50 4.00 7.3 12.2 135.2 102.6 18.5 18.0%

17

73.80 50.91 4.50 5.66 4.8 50.8 313.0 102.6 11.6 11.3%

16

waterplane area [m2] waterline width [m] characteristic lenght [m] Natural period, Tn [s] BPTO,opt @ Tn [kN.s/m] BPTO,opt @ Tp [kN.s/m] Pζ [kW] Pmax,irr [kW] ηhydro, max [%]

mass [t]

73.80 36.00 4.50 4.00 6.7 13.9 162.7 102.6 15.2 14.8%

15 14 Shape sketch:

Shape Nº:

115.85 36.00 4.00 4.50 6.4 14.4 170.5 91.2 13.8 15.1% 115.85 36.00 4.50 4.00 6.3 17.5 176.9 102.6 13.3 13.0% 115.85 36.00 4.50 4.00 7.2 12.2 141.7 102.6 17.8 17.3% 139.04 35.54 4.50 3.95 7.8 10.7 114.5 102.6 21.7 21.2% waterplane area [m2] waterline width [m] characteristic lenght [m] Natural period, Tn [s] BPTO,opt @ Tn [kN.s/m] BPTO,opt @ Tp [kN.s/m] Pζ [kW] Pmax,irr [kW] ηhydro, max [%]

mass [t]

00

221.40 36.00 4.50 4.00 8.4 6.5 86.0 102.6 20.3 19.8%

115.85 36.00 4.00 4.50 6.5 15.0 169.6 91.2 13.4 14.7%

18

221.40 50.91 4.50 5.66 8.2 11.6 140.5 102.6 23.2 22.6%

115.85 50.22 4.00 6.28 5.5 32.8 280.2 91.2 12.3 13.5%

19

295.20 36.00 4.50 4.00 11.0 4.2 63.3 102.6 26.7 26.0%

90.55 62.35 4.50 6.93 4.8 72.3 386.6 102.6 12.9 12.6%

20

208.10 36.81 4.50 4.09 8.0 8.4 106.9 102.6 19.6 19.1%

141.16 62.35 4.50 6.93 5.7 41.9 335.9 102.6 14.8 14.4%

21

45.27 31.18 4.50 3.46 4.1 22.9 202.4 102.6 7.6 7.4%

13 12 11 10 09 08 07 06 05 04 03 02 01

8

Shape sketch:

The preliminary analyses and the adopted constraints have left the floaters’ shape as the unique parameter to be optimized in this study. The optimization of shapes would require, ideally, a numerical tool to automatize the systematic generation of a series of geometric shapes. However, the complexity in the specification of constraints to avoid

Shape Nº:

3.5. Shape optimization method

Table 2 Tested shapes in the optimization process.

Based on the preliminary analyses, the effect of some of the design variables on the hydrodynamic behavior of the WEC are already known and can be readily predicted. These design variables may be classified as “unessential” in the design process and will be set as constants at reference values, hereafter. However, there are still a significant number of parameters that, even with an efficient hydrodynamical numerical tool, make the optimization process still unfeasible. A typical practice to further reduce the number of design variables is to set some constraints in the configuration of the system. In the design of WECs for a particular sea site location, the first step is to determine the most frequent waves according to the sea site proba­ bility data (Sergiienko et al. (2017)) and to assess the energy content of each sea state in a yearly basis. In the present study, the choosen location was the northern Atlantic coast of the Iberian Peninsula, which has been extensively analyzed in Ramos et al. (2017) and Ramos et al. (2018). Those studies have shown that the main bulk of wave energy can be found in sea states whose peak periods are between 9 and 16 s, with significant wave heights ranging between 1.5 and 5.0 m. The maximum values of the mean annual wave energy resource are concentrated mainly around two peak periods, Tp (10 and 14 s), with significant wave heights, Hs, of 1.5 and 2.5 m, respectively. The average annual occur­ rence of the first maximum is approx. 19% (1737 h per year), while the second one is approx. 5% (442 h per year). As for this location, the level of wave energy for both sea states is quite similar (around 15–20 MWh/m), the criteria for choosing the target sea state was the wave occurrence probability, which resulted in the following sea state: Tp ¼ 10 s and Hs ¼ 1.5 m. Long-crested (unidirectional) waves are also assumed. Another imposed constraint is related to the main characteristic di­ mensions of CECO. In the design of heave oscillating WECs, typically the main dimensions of the oscillating body are systematically varied to achieve natural oscillating periods around the peak period of the target sea state. Variations of those main dimensions directly affect all the costs of the WEC (i.e., both CAPEX and OPEX), thus impacting the levelized cost of the produced energy. For this sloped-motion WEC, since only shape variations were allowed, the natural oscillation period should achieve the target peak period without significant changes in the main dimensions of CECO’s reference design. In addition, the angle of incli­ nation of the motion path will be set at the reference value of CECO’s original design, i.e. 45� . This inclination angle represents the bisectrix path relative to pure heave (vertical) - or pure surge (horizontal) - os­ cillations and allows equal wave energy absorption from heave and surge oscillation modes. Furthermore, for later adjustments or tuning of the WEC system, the 45� configuration also allows equal margin to both sides, facilitating the control of this parameter. For the sake of simplicity and to allow more versatility during operation, only the submergence level (or draught) corresponding to approx. 50% of the floaters’ height will be simulated. The water depth and the distance between the floaters will be considered constant parameters during the optimization process. Ac­ cording to Ramos et al. (2018), as the operating water depth increases, CECO’s power performance improves. However, this effect is almost negligible. Here, for the sake of comparison with the experimental test conditions of the reference design, the local water depth adopted for the optimization study was 16.0 m and the distance between the floaters 5.0 m.

73.80 36.00 4.50 4.00 5.5 19.5 202.8 102.6 10.2 9.9%

3.4. Constrained parameters

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

unrealistic geometries precludes the use of this kind of tools. Therefore, the WEC’s shape will be optimized for the target sea state using the technique of exhaustive search (also known as “brute-force type” or “generate and test”) method with a heuristic approach. The exhaustive search method is a problem-solving technique that consists of system­ atically enumerating all possible solution candidates and checking whether each candidate satisfies the problem’s statement. The bruteforce search is simple to implement, but its computational cost is pro­ portional to the number of candidate solutions – which in many practical problems tends to grow very quickly as the size of the problem increases. Therefore, the brute-force search is typically used when the problem size is limited, or when there are problem-specific heuristics that can be used to reduce the set of candidate solutions to a manageable size. The latter is the case of the present study, i.e., the heuristic approach (based on trial and error, rules of thumb, intuitive judgment supported by theory, experience from earlier experimental tests, etc.) was adopted for the generation of a reduced number of candidate geometric shapes.

characteristic length of the submerged body is small in comparison to the wavelength. From shape 14 to 22, non-cylindrical shapes within the overall di­ mensions of the original design were also simulated. Shape 16 (boxshaped) achieved a slightly better performance than shape 01, but without overcoming the original design (shape 00). These shapes also confirm that the best hydrodynamic performances were obtained for bodies whose natural periods were closer to 10 s (shape 19 and 21). Furthermore, shapes with protuberant (fore- and/or aft-) volumes exhibit greater performances. Indeed, shape 21 displayed a hydrody­ namic performance of 39.5%, i.e., close to twice the value of the original CECO’s design. After analyzing shapes 20 to 22, it was preliminary concluded that the fore-protuberant volume, which resembles a ship’s bulbous bow, has a significant beneficial effect in the WEC performance. So, to verify that hypotheses, shapes with only protuberant fore-volumes were generated (shapes 23 to 26). Indeed, all these shapes achieved a hydrodynamic index above 40%, being shape 26 the best (ηhydro, max ~ 50%), thus, evidencing a significant improvement in relation to CECO’s original design. Notice, that, although viscous effects are not considered in the potential hydrodynamic model, sharp edges were avoided in the final shapes to reduce losses associated with vortex-shedding. Fig. 9 presents the frequency response operator of absorbed power in regular waves while Fig. 10 presents the density spectrum of absorbed wave power for the target sea state, i.e., the integrand of eq. (9). Its integration along the wave frequency allows the computation of the maximum average absorbed power, i.e., under ideal PTO damping. Fig. 10 also highlights the better performance of shapes 24 and 25. However, despite the highest peak of shape 25 in the response in regular waves and in the density spectrum, the total amount of absorbed power of shape 24 is higher (thus, more hydrodynamically efficient) than shape 25 as evidenced by the index of hydrodynamic performance in Table 2. Therefore, shape 24 was selected as the shape to be adopted for the new CECO design. It should be noticed that the new CECO design can be further improved by fine adjustments in the angle of inclination and/or the submerged volume. Furthermore, the inclusion of the support cylinder and the additional damping in the WEC (from viscous effects, PTO damping, etc.) will lead to an increase in the natural oscillation period of the WEC and, consequently, the improvement in the hydrodynamic performance of the new design due to the better tuning with the peak period of the target sea state. Regarding the PTO damping, the optimum values corresponding to the WEC’s natural period and to the peak period of the target sea state have been also presented in Table 2. At the natural period, the values are, in general, low compared to those achieved during the model tests �pez et al., 2017a; Rodríguez et al., with CECO’s reference design (Lo 2019a). Fig. 11 displays the curve of optimum PTO damping that was used for the computation of the maximum absorbed power for regular and irregular waves shown in Figs. 9 and 10, respectively. Fig. 12 shows the response amplitude operator (RAO) of motion in regular waves considering only the radiation (potential) damping. It can be noticed that the difference between the amplitudes of the peaks of RAOs of motion between the new and the reference design is not as big as the difference between the peaks of frequency response of absorbed power (Fig. 9), but the shape of the motion response curve of the new design is broader than that of the reference design.

4. Numerical results and analyses In order to systematize and speed up the numerical optimization work, Eqs. (2)–(11) have been implemented in MATLAB®. The hydro­ dynamic coefficients and wave excitation forces required have been computed in WAMIT® for 60 regular-wave periods ranging between 2 and 50 s, for all shapes idealized. The characteristic dimension of the panels used in the numerical modelling (meshing) of the floaters and the central cylinder was around 0.25 m. For the description of the (irregular) wave spectrum and the WEC response spectrum, 5000 regular-wave components have been used. Table 2 shows the most relevant shapes of the floaters analyzed during the optimization search. For all of them, the submergence considered was around 50% of the total volume of the floaters and the width of each floater of approx. 4.5 m. The incident sea state was characterized by a JONSWAP spectrum (Hs ¼ 1.5 m, Tp ¼ 10 s, peak enhancement factor of 3.3) propagating in the direction perpendicular to the floaters and the angle of inclination of the trajectory path was fixed at 45� . The basis for shape generation was the shape of the floaters of CECO’s original design (shape 00). Shape 01 is similar to the reference design but, to allow more margin for later draught adjustments, the draught was set to approximately 50% of the total depth. Then, based on the fact that hydrodynamic performance is affected by the relative orientation between the submerged geometry and the incident flow, the first approach was to simulate different orthogonal orientations of the floaters while keeping the overall volume (and, in principle, the overall costs) constant (shapes 02 to 07). Shapes 02 to 04 were obtained by rotating the floaters of shape 01 around the vertical axis. Shape 05 kept the same mass of shape 01, but with a waterplane area greater than that of the reference design. Shapes 06 and 07 keep the waterplane area of shape 05, but, respectively, have a smaller and a greater volume than the shape 00. Since no improvements were observed in terms of hydrody­ namic efficiency (ηhydro, max), intermediate orientations were also analyzed. To speed-up the process of shape generation, this time changes in the displaced volume were allowed (shapes 08 to 10). So, shapes 08 and 09 have been rotated around the transversal axis 45� and shape 10 was rotated 135� also around the transversal axis. Again, none of these alternative shapes performed better than the original shape. The first ten alternative shapes allowed to conclude that, in general, deeper and fuller submerged shapes (such as shape 10) performed better than shallower shapes. To verify the above conclusion, shapes 11 to 13, with approximately half the overall dimensions of the original WEC, were also simulated. Indeed, shape 13 (box-shaped) displayed a significant improvement when compared to shape 12, but not relatively to shape 11 – suggesting that a smaller waterplane area is also an important parameter. In other words, the hydrodynamic performance is reduced when the

5. Validation of results After the optimization process, the new design of CECO, hereafter, referred to as CECOv2, is that corresponding to shape 24. To validate the conclusion of the optimization study regarding the new shape of the floaters, a new experimental campaign was carried out. The scale chosen for the model tests was 1:25 to allow the maximum expected motions of this WEC in the available water depth of the wave basin of the Hy­ draulics Laboratory of the Hydraulics, Water Resources and 9

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

Fig. 9. Frequency response operator of absorbed power in regular waves.

Fig. 10. Density spectrum of absorbed power for the target sea state (JONSWAP, Tp ¼ 10 s, Hs ¼ 1.5 m).

Environment Division of the Faculty of Engineering of the University of Porto (FEUP), Portugal. The available water depth in the basin is also associated to the operational limitations of the wave maker, and for CECOv2’s model tests corresponded to 20.0 m (full scale), i.e., slightly larger than the water depth used in the optimization studies (16.0 m). The effect of the water depth in the results has been investigated through numerical simulations and it was found to be negligible. To simplify the physical model test arrangements, the same set-up used in the experimental tests with CECO’s reference design (here­ �pez et al., 2017a, b; after, referred to as CECOv1) was adopted (Lo Rosa-Santos et al., 2019a). Thus, a central fixed cylinder, perpendicular to the motion path, was used to house the PTO system. Due to space limitation during the construction of the physical model, the separation distance between the two floaters needed to be increased to 6.25 m (full-scale value) in comparison to the distance adopted in the optimi­ zation simulations. The final separation distance resulted from the diameter of the central cylinder (5.0 m) plus an additional distance between the cylinder and each of the floaters to allow their fixation to

their connecting structure. Regarding the connecting structure, which is basically a metallic frame composed of tubular members, its effect on the hydrodynamic behavior has been considered negligible compared to that of the central cylinder. Thus, it was not included in the numerical model at this stage (Fig. 13). Still in the context of the numerical simulations, some previous calculations were necessary to validate the assumptions made in the preliminary analyses – which were based on CEVOv1. Those assump­ tions refer to the effect of the presence of the fixed support cylinder, the submergence of the floaters and the inclination of the motion path. Fig. 14 presents the response amplitudes of CECOv2’s reference design – characterized by a submergence (in terms of draught) of 52%, inclina­ tion of 45� and water depth of 20.0 m, together with alternative con­ figurations to assess the above effects. Fig. 14 confirms that neither the presence of the cylinder nor the water depth has a significant effect on CECOv2’s hydrodynamic per­ formance and that inclination and submergence have a strong impact on the natural oscillation period of this WEC. These conclusions are in 10

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

Fig. 11. Optimum PTO damping for maximum absorbed power in regular waves.

Fig. 12. Frequency response operator of floaters’ motion amplitude in regular waves.

considered, including tests without the PTO system. For the experi­ mental validation of the numerical conclusions of the optimization study, only a few physical model tests results are presented, namely those test conditions of CECOv2 without PTO damping. Fig. 15 com­ pares the experimental RAOs of CECOv1 and CECOv2, obtained from regular wave tests, for wave periods around the corresponding natural periods. These results confirm that CECOv2 is a better motion converter than CECOv1. For the period corresponding to the peak of the target sea state, the response of CECOv2 almost doubles the response of CECOv1. Fig. 16 shows the motion response spectra of CECOv2 for three irregularsea spectra, namely, the target sea state (Tp ¼ 10 s), a sea state with a lower peak period (Tp ¼ 7 s) and one with a larger peak period (Tp ¼ 13 s) – all with the same significant wave height (Hs ¼ 1.5 m). For the sake of comparison, the motion response spectra of CECOv1 for similar sea states are plotted together. Notice that the significant wave heights in the experimental tests with CECOv1 were larger (Hs ¼ 2.0 m). Despite the wave conditions for CECOv1 being more energetic, CECOv2 displayed larger excursions than CECOv1 for the target sea state, Fig. 16. Thus, confirming the numerically predicted resonant

Fig. 13. CECOv2’s layout: a) physical model and b) numerical model.

agreement with the tendencies observed for CECOv1. The overall experimental test campaign included pre-tests with the PTO system alone, decay tests and tests in regular and irregular waves. Also, systematic variations of the levels of PTO damping have been 11

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

Fig. 14. Effect of presence of the central cylinder, water depth, submergence of the floaters and inclination of the motion path on CECO’s motion amplitude operator in regular waves.

Fig. 15. Experimental RAO of CECOv1 versus CECOv2 without PTO damping.

Fig. 16. Experimental motion response spectra of CECOv1 versus CECOv2 without PTO damping.

12

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

behavior of CECOv2. For the sea state with lower peak-period, CECOv1 performs better than CECOv2. The reason is that the lower peak period for CECOv1 matches its resonant period (~ 8 s) while for CECOv2 this peak period (7 s) is quite far from its resonant period (~ 10s). For the sea state with the largest peak period, CECOv1 motions are slightly larger than those of CECOv2; however, it should be recalled that the waves for CECOv1 (Hs ¼ 2.0 m) are much more energetic than those for CECOv2 (Hs ¼ 1.5 m). The comparison of the experimental results obtained for CECOv1 and CECOv2 has allowed an indirect validation of the numerical outcomes of the optimization study. However, a more direct validation can be per­ formed by comparing numerical simulations of motions with experi­ mental results once damping has been calibrated using, for instance, the hybrid approach proposed by Rodríguez et al. (2019a). This approach relies on the numerical RAO of motion and the experimental incident wave time series to compute the (numerical) motion response spectra based on eq. (7). Starting with the radiation damping coefficient, an additional external damping is iteratively applied in the RAO of motion, until the numerical response spectra matches the experimental one to obtain the “calibrated” external damping coefficient. For the test conditions without PTO damping, the calibrated damp­ ing coefficient represents the hydrodynamic damping (potential damp­ ing plus losses including viscous effects). For the target sea state and CECOv2 at the 45� inclination, two additional PTO damping levels has been experimentally tested, too. The experimental response spectra and

the corresponding “calibrated” numerical spectra and numerical RAOs are shown in Fig. 17. Since the hybrid calibration procedure for irregular seas is based on the area under the response spectrum, it is inherently guaranteed that the significant response amplitude is the same for the experimental tests and the corresponding numerical simulation. For conditions BPTO_1 and BPTO_2, the shapes of the calibrated response spectra also resemble quite well the corresponding experimental spectra, thus, evidencing the suitability of the linear hydrodynamic model and the calibration pro­ cedure. The external damping coefficient (total damping excluding po­ tential effects) for the condition without PTO damping was 45 kN s/m, while for damping levels 1 and 2, the values were 110 kN s/m and 195 kN s/m, respectively. The absorbed power associated to those PTO damping coefficients were 18.3 kW and 18.6 kW with capture effi­ ciencies of 17.5% and 17.8%, respectively. Notice that these efficiencies correspond to arbitrary values of PTO damping tested during the experimental model tests. According to the results presented in Table 2, higher capture efficiencies may be achieved for lower PTO damping coefficients. Further analyses are necessary to determine the best PTO damping for a given WEC condition but are out of the scope of the present study. Currently, CECO is classified in the Technological Readiness Level (TRL) 3–4, where its performance is assessed through numerical studies and physical model tests. Higher stages of development of CECO will demand full-scale tests, where, in principle, the optimum geometry

Fig. 17. CECOv2’s response for different external damping levels (45, 110, 195 kN s/m) and the target sea state (Tp ¼ 10 s, Hs ¼ 1.5 m): a) Experimental and calibrated numerical response spectra b) Calibrated numerical RAO of motion. 13

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

found in the prior numerical and experimental tests is expected to be used. On the other hand, the PTO system will depend on the available technologies at the time of full-scale testing. Currently, most of the point-absorber WECs adopt direct mechanical drive PTO systems (i.e., the same type of PTO system used in the experimental tests carried out with CECO). However, more recently, direct electrical drive systems have become more popular and an attractive solution to reduce me­ chanical losses.

Fundaçaeo para a Ci^encia e a Tecnologia, I.P. References Alamian, R., Shafaghat, R., Safaei, M.R., 2019. Multi-objective optimization of a pitch point absorber wave energy converter. Water 11 (5), 969. Blanco, M., Lafoz, M., Ramirez, D., Navarro, G., Torres, J., Garcia-Tabares, L., 2019. Dimensioning of point absorbers for wave energy conversion by means of differential evolutionary algorithms. IEEE Trans. Sustain. Energy 10 (3), 1076–1085. https:// doi.org/10.1109/TSTE.2018.2860462. Bouali, B., Larbi, S., 2017. Sequential optimization and performance prediction of an oscillating water column wave energy converter. Ocean Eng. 131, 162–173. Bozzi, S., Besio, G., Passoni, G., 2018. Wave power technologies for the Mediterranean offshore: scaling and performance analysis. Coast Eng. 136, 130–146. Coello-Coello, C.A., 1999. A comprehensive survey of evolutionary-based multiobjective optimization techniques. Knowl. Inf. Syst. 1 (3), 269–308. De Andres, A., Guanche, R., Vidal, C., Losada, I., 2015. Adaptability of a generic wave energy converter to different climate conditions. Renew. Energy 78, 322–333. Esmaeilzadeh, S., Alam, M.-R., 2019. Shape optimization of wave energy converters for broadband directional incident waves. Ocean Eng. 174, 186–200. Falnes, J., 2002. Ocean Waves and Oscillating Systems: Linear Interactions Including Wave-Energy Extraction. Cambridge university press. Faltinsen, O., 1993. Sea Loads on Ships and Offshore Structures. Cambridge university press. Flocard, F., Finnigan, T., 2012. Increasing power capture of a wave energy device by inertia adjustment. Appl. Ocean Res. 34, 126–134. Garcia-Rosa, P., Bacelli, G., Ringwood, J., 2015. Control-informed geometric optimization of wave energy converters: the impact of device motion and force constraints. Energies 8 (12), 13672–13687. Goggins, J., Finnegan, W., 2014. Shape optimisation of floating wave energy converters for a specified wave energy spectrum. Renew. Energy 71, 208–220. IEA, 2018. World energy outlook 2018. https://doi.org/10.1787/20725302. Kim, N.H., Wang, H., Queipo, N., 2004. Adaptive reduction of design variables using global sensitivity in reliability-based optimization. In: 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, p. 4515. Lee, C., Newman, J., 2013. WAMIT User Manual, Version 7.0. WAMIT, Inc., Chestnut Hill, MA. Lewis, E.V., 1988. Principles of Naval Architecture, vol. 3. The Society of Naval Architects & Marine Engineers. Seakeeping and Controllability. Liang, C., Ai, J., Zuo, L., 2017. Design, fabrication, simulation and testing of an ocean wave energy converter with mechanical motion rectifier. Ocean Eng. 136, 190–200. L� opez, M., Ramos, V., Rosa-Santos, P., Taveira-Pinto, F., 2018. Effects of the PTO inclination on the performance of the CECO wave energy converter. Mar. Struct. 61, 452–466. L� opez, M., Taveira-Pinto, F., Rosa-Santos, P., 2017a. Influence of the power take-off characteristics on the performance of CECO wave energy converter. Energy 120, 686–697. L� opez, M., Taveira-Pinto, F., Rosa-Santos, P., 2017b. Numerical modelling of the CECO wave energy converter. Renew. Energy 113, 202–210. Marinheiro, J., Rosa-Santos, P., Taveira-Pinto, F., Ribeiro, J., 2015. In: Guedes Soares, C., Santos, T.A. (Eds.), Feasibility Study of the CECO Wave Energy Converter. Maritime Technology and Engineering. Taylor & Francis Group, London, pp. 1259–1267. McCabe, A., 2013. Constrained optimization of the shape of a wave energy collector by genetic algorithm. Renew. Energy 51, 274–284. Meng, F., Cazzolato, B., Li, Y., Ding, B., Sergiienko, N., Arjomandi, M., 2019. A sensitivity study on the effect of mass distribution of a single-tether spherical point absorber. Renew. Energy 141, 583–595. Moretti, G., Malara, G., Scial� o, A., Daniele, L., Romolo, A., Vertechy, R., Fontana, M., Arena, F., 2020. Modelling and field testing of a breakwater-integrated U-OWC wave energy converter with dielectric elastomer generator. Renew. Energy 146, 628–642. Oliveira-Pinto, S., Rosa-Santos, P., Taveira-Pinto, F., 2019. Electricity supply to offshore oil and gas platforms from renewable ocean wave energy: overview and case study analysis. Energy Convers. Manag. 186, 556–569. Onwubolu, G.C., Babu, B., 2013. New Optimization Techniques in Engineering. Springer. Ozkop, E., Altas, I.H., 2017. Control, power and electrical components in wave energy conversion systems: a review of the technologies. Renew. Sustain. Energy Rev. 67, 106–115. Paredes, G.M., Palm, J., Eskilsson, C., Bergdahl, L., Taveira-Pinto, F., 2016. Experimental investigation of mooring configurations for wave energy converters. Int. J. Mar. Energy 15, 56–67. Pastor, J., Liu, Y., 2014. Power absorption modeling and optimization of a point absorbing wave energy converter using numerical method. J. Energy Resour. Technol. 136 (2), 021207. Piscopo, V., Benassai, G., Cozzolino, L., Della Morte, R., Scamardella, A., 2016. A new optimization procedure of heaving point absorber hydrodynamic performances. Ocean Eng. 116, 242–259. Ramos, V., L� opez, M., Taveira-Pinto, F., Rosa-Santos, P., 2017. Influence of the wave climate seasonality on the performance of a wave energy converter: a case study. Energy 135, 303–316. Ramos, V., L� opez, M., Taveira-Pinto, F., Rosa-Santos, P., 2018. Performance assessment of the CECO wave energy converter: water depth influence. Renew. Energy 117, 341–356. Rao, S.S., 2019. Engineering Optimization: Theory and Practice. John Wiley & Sons. Reguero, B., Losada, I., M�endez, F., 2015. A global wave power resource and its seasonal, interannual and long-term variability. Appl. Energy 148, 366–380.

6. Conclusions The hydrodynamic optimization of the geometry of the floaters of a sloped-motion WEC, named CECO, has been performed based on a timeefficient frequency-domain numerical model. The optimization pro­ cedure relied on the use of the exhaustive search technique with heu­ ristic approach. Prior to the optimization study, some of the parameters that govern CECO’s hydrodynamics were numerically investigated to understand their effects in advance, so that the optimization procedure could only focus on the shape of the floaters. Those preliminary analyses allowed to conclude that the effects of the fixed central cylinder and of the water depth were negligible, while the submergence level of the floaters and the inclination of the motion path determine the natural period of CECO. The optimization study showed that the shape of the floaters controls also the natural period and depending on its tuning with the peak period of the target sea state, a more efficient wave energy converter can be developed. It was shown that the geometry obtained as the optimum solution from the numerical optimization study is able to harvest twice as much wave energy than the original CECO’ design (CECOv1). In order to validate the numerical outcomes, experimental model tests have been conducted with the optimized geometry. The experimental results with regular and irregular wave tests have confirmed that CECOv2 is, in fact, a better motion converter than CECOv1. In regular waves, the peak of the RAO of motion was almost twice the value measured for CECOv1. In irregular waves, the motion spectrum for the target sea state was significantly larger than the spectrum of CECOv1 (even considering a much more energetic sea state). Finally, it was demonstrated that unlike “pure” heave oscillators, where basically the main dimensions of the floaters control the natural frequency, for CECO concept, the inclination of the motion path, sub­ mergence or shape of the floaters can be adjusted to achieve any natural oscillating period within a relatively broad range. Thus, it is possible to adapt CECO parameters to tune it to any sea state so that high energy capture efficiencies can be obtained. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Claudio A. Rodríguez: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing, Supervi­ sion. Paulo Rosa-Santos: Conceptualization, Methodology, Validation, Writing - review & editing, Supervision. Francisco Taveira-Pinto: Conceptualization, Methodology, Validation, Writing - review & editing, Supervision. Acknowledgements This work was financially supported by: Project PTDC/MAR-TEC/ 6984/2014 - POCI-01-0145-FEDER-016882 - funded by FEDER funds through COMPETE2020 - Programa Operacional Competitividade e Internacionalizaç~ ao (POCI) and by national funds through FCT 14

C.A. Rodríguez et al.

Ocean Engineering 199 (2020) 107046

Rodríguez, C.A., Rosa-Santos, P., Taveira-Pinto, F., 2019a. Assessment of damping coefficients of power take-off systems of wave energy converters: a hybrid approach. Energy 1022–1038. https://doi.org/10.1016/j.energy.2018.12.081. Rodríguez, C.A., Rosa-Santos, P., Taveira-Pinto, F., 2019b. Experimental assessment of the performance of CECO wave energy converter in irregular waves. J. Offshore Mech. Arctic Eng. 141 (4), 041903. Rosa-Santos, P., Taveira-Pinto, F., Rodríguez, C.A., Coelho, G., Clemente, D., Mendonça, H., Moreira, A.P., 2019a. Development and Assessment of a New Geometry for CECO Wave Energy Converter, 13th European Wave and Tidal Energy Conference. Naples, Italy, pp. 1–10. Rosa-Santos, P., Taveira-Pinto, F., Rodríguez, C.A., Ramos, V., L� opez, M., 2019b. The CECO wave energy converter: recent developments. Renew. Energy 139, 368–384. Rusu, E., Onea, F., 2018. A review of the technologies for wave energy extraction. Clean Energy 2 (1), 10–19. Sergiienko, N.Y., Cazzolato, B.S., Ding, B., Hardy, P., Arjomandi, M., 2017. Performance comparison of the floating and fully submerged quasi-point absorber wave energy converters. Renew. Energy 108, 425–437.

Shadman, M., Estefen, S.F., Rodriguez, C.A., Nogueira, I.C., 2018. A geometrical optimization method applied to a heaving point absorber wave energy converter. Renew. Energy 115, 533–546. Sheng, S., Wang, K., Lin, H., Zhang, Y., You, Y., Wang, Z., Chen, A., Jiang, J., Wang, W., Ye, Y., 2017. Model research and open sea tests of 100 kW wave energy convertor Sharp Eagle Wanshan. Renew. Energy 113, 587–595. Shi, H., Han, Z., Zhao, C., 2019. Numerical study on the optimization design of the conical bottom heaving buoy convertor. Ocean Eng. 173, 235–243. Taveira-Pinto, F., Iglesias, G., Rosa-Santos, P., Deng, Z.D., 2015. Preface to Special Topic: Marine Renewable Energy. AIP Publishing. Tom, N., Lawson, M., Yu, Y.-H., Wright, A., 2016. Spectral modeling of an oscillating surge wave energy converter with control surfaces. Appl. Ocean Res. 56, 143–156. Venter, G., 2010. Review of Optimization Techniques. Encyclopedia of aerospace engineering. Xiao, X., Xiao, L., Peng, T., 2017. Comparative study on power capture performance of oscillating-body wave energy converters with three novel power take-off systems. Renew. Energy 103, 94–105.

15