A geometrical optimization method applied to a heaving point absorber wave energy converter
Introduction
Wave energy conversion technology is in the pre-commercial stage. The cost of a wave energy converter (WEC) increases with its size. Therefore, the geometry optimization of the system has a significant role in the design process to have an economically feasible system. So far, one of the most promising WEC concepts under technical and economic evaluation is the point absorber (PA). It consists of a floater body, which has small dimensions relative to the incident wavelength and, a support system, which could be mooring cables connected to the seabed or bottom-mounted structures. A heaving PA is a wave energy device in which the heave motion due to the wave-body interaction is absorbed to produce power. The system that receives the mechanical power resulting from heave motion and delivers electricity, for instance, is called power take-off (PTO). Some examples of the PA WEC can be found in Refs. [1], [2], [3], [4]. Geometry optimization techniques applied on marine structures have been used for seakeeping of ships and minimizing the dynamic response of moored vessels [5], [6], [7], [8]. The objective of the geometry optimization of the offshore platforms, floating breakwaters etc. is to minimize the motion of the structure to improve the seakeeping ability. In contrast, in the case of point absorbers, the geometry of WEC must be optimized to make the system oscillates in the range of sea predominant wave frequencies allowing maximum possible motion amplitudes to produce more power. Until now, different works have been presented for geometry optimization of a PA. Vantorre et al. [9] presented an optimization study of a point absorber with controllable inertia through linear frequency domain analyses and experimental tests for the Belgian coastal area of the North Sea. The geometry of the heaving buoy, PTO damping and supplementary inertia are considered as variable parameters to maximize the power absorption. Soulard et al [10] presented a numerical modeling of a two-body heaving PA. they determined the optimum dimensions of the WEC applying a statistical analyses method called “design of experiments” (DOE). Sjokvist et al. [11] and Goggins and Finnegan [12] applied a frequency domain approach to optimize a heaving point absorber, with a pure damper as PTO system, for the Lysekil test site in Sweden and Atlantic marine energy test site (AMETS) respectively. The objective of the optimization was to determine the buoy dimensions that maximize its heaving velocity for that sea state and a certain PTO damping. To attain this, several runs with different diameters and drafts were performed for different PTO damping values. Several studies have also investigated the geometrical optimization process for other WEC concepts. For instance, Kramer and Frigaard [13] used the boundary element method to optimize wave reflectors by exploring their orientation and angle to increase the wave energy absorption for the wave energy converter Wave Dragon. Genetic algorithms have been used by Babarit and Clement [14] and McCabe [15] to the shape optimization of the SEAREV device, which is based on a pendulum that is placed in a closed buoy actuated by the swell through excitation forces [16], and a surging wave energy collector, respectively. The SEAREV optimization process goals were to maximize the absorbed power and minimize the costs. The objective of the shape optimization of the surging wave energy collector was to maximize the mean power, the ratio of the mean power to the characteristic length of the device shape, and the ratio of the mean power to the displaced volume of the collector. Kurniawan and Moan [17] used a multi-objective optimization algorithm to determine the optimal geometry of a wave absorber oscillating about a fixed submerged horizontal axis. They used two objective functions to be minimized, which were the ratios of the submerged surface to the maximum absorbed power and the maximum reaction force to the maximum absorbed power. Recently, Mahnamfar and Altunkaynak [18] and Son et al. [19] deployed experimental tests beside the numerical analyses for the shape optimization of a oscillating water column and a specific PA WEC respectively.
This paper presents a geometry optimization methodology, as a preliminary approach, applied on a simple one-body point absorber, considering the nearshore region of Rio de Janeiro, Brazil as the sea site. The WEC consists of a floating cylinder, which reacts against the seabed with an ideal pure damper as the PTO system (PTO stiffness is zero). The optimization process has been developed through linear hydrodynamic frequency domain analyses and the design of experiments (DOE) approach. Minitab [20] is used to apply the DOE and perform statistical analysis of the optimization process. The frequency domain analysis of the system hydrodynamics is performed using ANSYS-AQWA [21]. The diameter and draft of the floating cylinder (buoy) are considered as the geometrical parameters to be optimized. The objective of the optimization process is to determine the buoy that absorbs the maximum wave energy over the largest range of frequencies for the site's predominant waves. The presented methodology can be applied to geometrical optimization of different types of wave energy converters.
Section snippets
Methodology approach
The methodology to be used aims at obtaining a set of geometrical parameters (factors) to optimize the point absorber in relation to the following three primary requirements (responses):
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Buoy heave natural frequency, which should be within the prevailing sea wave frequencies;
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Resonance bandwidth that is defined as the frequency interval in which the absorbed power is more than half of its maximum value;
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Maximum mechanical power.
A simple wave energy converter is used as the WEC system for the
Sea characteristics
The information on the wave climate is obtained by either in situ measurements or numerical modeling. Third generation spectral wave models have emerged as a reliable tool for forecasting and hindcasting ocean conditions. The use of these models in hindcast mode allows for an assessment of the global wave climate [22] and the energy resources [23], [24], [25].
The wave power assessment along the Brazilian coast has been discussed in several studies [26], [27]. To describe the nearshore wave
Equations of motion
In the preliminary hydrodynamic modeling of the WECs, it is typically assumed that the hydrodynamic forces of the floating body in waves are those obtained from the linear diffraction theory, i.e., viscous effects are neglected and only potential forces are considered. Thus, the response of a single floating body in a wave is generally described using a mass-spring system. By assuming a linear system with 6° of freedom, the equations of motion for this analysis can be represented as follows:
Immature determination
As mentioned in previous sections, the optimization of the buoy starts by an immature determination, which includes the definition of the upper and lower bounds for its diameter and draft, D and L, respectively. These bounds should satisfy two design premises. The first one is associated with the maximum amount of power that the buoy can absorb from incident waves, and the second one is related to the maximum buoy response in heaving due to the incident waves. Evidently, both premises are
Design of experiments methodology
The design of experiments (DOE) is a systematic technique for studying any situation that involves a response that varies as a function of one or more independent variables. The DOE can address complex problems where more than one variable may affect a response (or a set of responses) and two or more variables may interact with each other. The technique provides answers to specific questions on the behavior of a system requiring an optimum number of experimental observations.
So far a range of
Results and discussion; mature determination
Thus far, the range of factors was determined based on the sea site characteristics and the principle conditions applied using the DOE method. In this step, the statistical analysis results are discussed, and the final buoys are determined based on the resultant contour and surface plots. The hydrodynamic analyses (experiments) are performed in the frequency domain using ANSYS/AQWA. Fig. 7 illustrates a typical geometrical model and its discretization as required by ANSYS/AQWA. An additional
Conclusion
A methodology for the geometric optimization of WECs based on a series of frequency domain analyses and a statistical analysis method known as Design of Experiments (DOE) was presented. The optimization process is applied to the preliminary design of a one-body point absorber with an axisymmetric floating cylinder for the nearshore region of Rio de Janeiro. An ideal pure damper is considered as power take-off (PTO) system, and the energy absorption is calculated for different wave frequencies.
Acknowledgements
The authors acknowledge CAPES, Ministry of Education/Brazil, for the D.Sc. Scholarship to the first author, and CNPq (305338/2013-7), Ministry of Science, Technology and Innovation/Brazil, for supporting research activities of the second author. Special thanks to FURNAS through ANEEL (contract number: 9000000692) (Brazilian Electrical Energy Agency) P&D Program for the financial support of the research in progress at Subsea Technology Laboratory (COPPE) on wave energy.
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