Wave-body interactions among energy absorbers in a wave farm

doi:10.1016/j.apenergy.2018.09.131

Applied Energy

Volumes 233–234, 1 January 2019, Pages 1051-1064

https://doi.org/10.1016/j.apenergy.2018.09.131 Get rights and content

Highlights

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An efficient semi-analytical method is used to model hydrodynamics of cylinder arrays.
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The Haskind relation is generalized for wave-exciting forces on an array or a member.
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Cylinders need not be identical and can oscillate independently with 6 DOFs.
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Effects of cylinder spacing and layout symmetry on power extraction are examined.
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Optimal power of a large farm with multiple arrays is studied by a combined method.

Abstract

A semi-analytical method is developed to investigate surface-wave interactions among an array of wave-energy converters, each modeled as a truncated cylinder, and the interaction effects on power absorption from the array is studied. Each cylinder can have independent movements with six degrees of freedom. The method of matched eigen-function expansions is applied to solve the wave radiation problem. To achieve fast computation, effects of evanescent modes of local scattering waves from one cylinder is neglected in the near fields of neighboring cylinders, but the far-field radiated waves are retained. Wave-exciting forces and moments on an individual cylinder or a group of cylinders, situated among an array, are evaluated by a new, generalized form of the “Haskind relation” applicable to an array of arbitrary configuration, which only requires the solution to the radiation problem. Hydrodynamic properties and wave-exciting loads are presented for arrays of different configurations. This efficient computation facilitates investigating wave-interaction effects on the optimal power output of a cylinder array. Effects of the cylinder numbers, their spacing, and the layout geometry on power extraction are discussed. The interaction factor for a large wave farm consisting of multiple small arrays was evaluated by the current method combined with the point-absorber approximation.

Introduction

Comparing to solar and wind, wave energy exhibits better predictability with more uniform power output throughout the day. In the U.S., with favorable wave conditions for power generation along the coastline, the Electric Power Research Institute (EPRI) in 2011 reported that the estimated recoverable wave energy resource could achieve 1170 TWh/yr, which would cover almost one third of the 4000 TWh of electricity used in the U.S. that year and power about 102.7 million homes [1]. Certainly, wave energy deserves serious consideration in the diversification of renewable energy portfolio. However, wave energy technology is still in its early stage of development, desiring cost-reduction innovations.

Efforts are made in various aspects to increase the cost competitiveness of the wave energy. Power Take-Off (PTO) systems with real-time control strategies were designed to enhance the energy-absorption efficiency for various types of WECs, for example, heaving or surging point absorbers [2], [3] and the attenuator Pelamis [4]. Numerical wave-to-wire model coupling CFD and PTO system [5] was established to increase the accuracy in modeling a WEC, as well as model tests for validating the performance of designs[6], [7]. In addition, the concept of a wave farm, consisting of arrays of small units of wave-energy converters are pursued to gain commercial viability [8].

For a wave farm, wave interactions among the units can have significant effects on the power production of the array, compared to the one produced by multiple isolated devices, when the devices are closely spaced. Avoiding those effects requires the WECs placed farther apart, which can increase the cost of production and operation. Thus, for optimizing the layout design of a WEC array, wave-interaction effects need to be considered to accurately estimate the generated power from an array. In 1980s, expressions for evaluating optimal power generated by an array of oscillating bodies were obtained by Evans [9] and Falnes [10] respectively, where an interaction factor q was defined as the ratio of the power per device in the array to the power produced by a single device. The q factor is later widely used as an indicator for the power-capture capability of a WEC array in the layout-optimization study [11], [12]. However, solving hydrodynamic properties for each device in an array, which are required to compute the optimal power, can be a complex problem because of scattering waves among the array.

Review on existing methods for solving such hydrodynamic problems of wave interactions among a cylinder array can be found in [13], [14]. Numerical methods [15], [16] can be directly applied, but the computational cost may become prohibitive with the increase of the size of the array. Semi-analytical method based on linear theory is hence preferred for faster computation, and it also provides physical insights into the problem. In this category, point-absorber approximation used in [9], [10] and plane-wave approximation applied in [17], [18] can estimate hydrodynamic properties of an array of cylinders in a substantially simplified way. Yet, errors can be significant. As a result, these methods may not be applicable to study detailed performance of the array of absorbers. Multiple scattering method [19], [20], based on an multi-level iterative scheme, is able to achieve any order of accuracy in principle, by increasing the order of scattering waves taken into account. Indeed, Ohkusu’s [20] approach is well known. However, even for low order of accuracy, the amount of computations associated with iterations will become infeasible with increase of the number of cylinders. Kagemoto & Yue proposed an interaction theory [21] to study hydrodynamics of an arbitrary array based on the diffraction properties of its individual elements. Such an interaction theory fully takes into account the effect of multiple scattering waves, including the interaction of evanescent modes, which is considered “exact” in the context of linear theory and adopted in other studies [22], [23], [24]. In this paper, we show that the interaction of evanescent modes of scattering waves is negligible for the majority of practical cases; most importantly, neglecting evanescent modes can reduce computational efforts in solving this problem, which is essential in efficiently evaluating hydrodynamic properties and estimating power production for an arbitrary WEC arrays. This approach was pursued by Yeung & Sphaier [25] in studying tank-wall interferences.

For computing the optimal power for a WEC array, both damping coefficients from solving the radiation problem and wave-exciting forces from solving the diffraction problem are required, as explained in [9], [10]. In existing studies of WEC arrays, the two problems were treated separately. A method mentioned above will be applied twice to solve the two problems and obtained the required coefficients, which can be computationally costly for the array problems. While for a single body, it is well-known that Haskind relation [26] can be used to obtain wave-exciting forces and moments, which does not require knowledge of diffraction properties of the structure, but rather, depends on radiation potential of the body in the far field, (see Newman [27] and Wehausen [28]). Applications can be found in Yeung [29] for a truncated vertical cylinder and Chau & Yeung [30] for dual coaxial cylinders. Here, we generalize Haskind relation and apply it to an array of cylinders, so that all of the first-order hydrodynamic coefficients can be obtained by only solving the radiation problem, which are then used to find the optimal power of the array.

In the present study, a cohesive semi-analytical method is developed and used to investigate wave interaction among multiple truncated circular cylinders in arbitrary configurations. Each cylinder is considered dynamically independent with six degrees of freedom. Following Yeung [29] and others, we let the velocity potential of the flow field will be obtained by matching eigen-function expansions for separated fluid domains. To achieve fast computation, we will indeed assume that evanescent modes of scattering waves from one cylinder will not significantly affect the pressure field around the other cylinders. These effects of neglecting the evanescent modes in the interaction will be assessed. Then generalized Haskind relations will be derived for an individual, or a group of cylinders, situated among an array of cylinders. Results from this formula will be compared with those from directly solving the diffraction problem, a much more lengthy procedure. Added mass, damping coefficients and wave-exciting loads for arrays of different configurations will be presented as results to demonstrate the importance of interference effects. Most importantly, the wave-interaction effects on the optimal power captured by an array, represented by the “q” factor, was computed for arrays of up to 24 heaving point absorbers in various configurations with different wave conditions. Discussion is made regarding achieving configurations of constructive wave-interaction effects.

Section snippets

Modeling analysis of a system of bodies

Within the context of linearized potential flow theory, consider N floating vertical cylinders of finite draft d, oscillating harmonically in water of depth h. We define N local cylindrical coordinates $(r_{j}, θ_{j}, z)$ ( $j = 1, 2, \dots, N$ ) fixed in the undisturbed free surface with the origin $O_{j}$ on the axis symmetry of the body and z-axis pointing upwards, as shown in Fig. 1. The motion of j-th cylinder in q-th mode can be described by: $ζ_{q}^{j} (t) = Re [{\bar{ζ}}_{q}^{j} e^{- i σ t}], q = 1, 2, \dots, 6$ where $\bar{ζ}$ , being time-independent, is the

Wave-exciting forces and moments

With potential theory, wave-exciting forces and moments on a number of cylinders can be obtained by $(\begin{matrix} F_{ex} \\ M_{ex} \end{matrix}) = Re \{i σ ρ e^{- i σ t} A \iint_{S} (ϕ_{0} + ϕ_{7}) (\begin{matrix} n \\ r \times n \end{matrix}) d S\}$ where A is the amplitude of incident waves, $ϕ_{0}$ and $ϕ_{7}$ are unit-amplitude potentials for incident waves and diffracted waves respectively, and S denotes the surface of the cylinders. The governing Eqs. (3a), (3b), (3c) and the radiation condition (5) should be satified for $ϕ_{7}$ ; and on the surface of body j, denoted by $S_{j}$ , ${\frac{\partial (ϕ_{0} + ϕ_{7})}{\partial n^{j}}|}_{S_{j}} = 0 j = 1, 2, \dots, N$

Given $ϕ_{0}$ , solving for $ϕ$

Results of hydrodynamic study

To validate the present method and investigate wave interference effects on multi-cylinder structures, a computational solver was developed based on the theory in Sections 2 Modeling analysis of a system of bodies, 3 Wave-exciting forces and moments, and applied to a number of configurations shown in Fig. 4, where waves progress in a direction that makes an angle $β$ with the x-axis. We compare results with boundary integral method by Matsui & Tamaki [32] and interaction theory by Kagamoto & Yue

Power extraction from a WEC array

The proposed method takes advantage of the cylindrical shape of a body and efficiently computes hydrodynamic properties for a system of independently oscillating bodies with high-order accuracy. It well fits the need of a quick estimation of wave-interaction effects for a preliminary design of a WEC array. It has been derived in [9], [10] that the optimal power extracted by an array of oscillating bodies from regular waves can be computed as $P_{opt} = \frac{1}{8} F_{exc}^{*} B^{- 1} F_{exc}$ which is achieved when the velocity

Conclusions

The computational problem of waves interacting with an array of WECs, each modeled as a truncated cylinder, in arbitrary configurations is solved by the matched eigen-function expansion method. Graf’s addition theorem was applied to transform coordinates and express the presence of a multitude of scattering waves. To improve computational efficiency, we argued that evanescent modes generated by one cylinder are negligible in the near field of its neighboring cylinders, whereas the far-field

Acknowledgements

The first author gratefully acknowledges the American Bureau of Shipping (ABS) for an Ocean-Technology Fellowship. Partial support of the research via an ABS Endowed Chair in Ocean Engineering held by the second author at UC Berkeley is appreciated.

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Wave-body interactions among energy absorbers in a wave farm

Highlights

Abstract

Introduction

Section snippets

Modeling analysis of a system of bodies

Wave-exciting forces and moments

Results of hydrodynamic study

Power extraction from a WEC array

Conclusions

Acknowledgements

Appl Energy

Appl Energy

Renewable Energy

Appl Energy

Appl Energy

Appl Energy

Appl Ocean Res

Ocean Eng.

Ocean Eng

Appl Ocean Res

Appl Ocean Res

Appl Ocean Res

Ocean Eng

Ocean Eng

Ocean Eng

Appl Ocean Res

Appl Ocean Res

Ocean Eng

Ocean Eng