Elsevier

Applied Ocean Research

Volume 31, Issue 4, October 2009, Pages 251-259
Applied Ocean Research

On the capture width of wave energy converters

https://doi.org/10.1016/j.apor.2010.04.001Get rights and content

Abstract

This paper extends the theory on capture width, a commonly used performance indicator for a wave energy converter (WEC). The capture width of a linear WEC is shown to depend on two properties: the spectral power fraction (a property introduced in this paper), which depends entirely on the sea state, and the monochromatic capture width, which is determined by the geometry of the WEC and the chosen power take off (PTO) coefficients. Each of these properties is examined in detail. Capture width is shown to be a measure of how well these two properties coincide. A study of the effects of PTO control on the capture width suggests that geometry control, a form of control that has not been the focus of much academic research, despite its use in the wave energy industry, deserves more attention. The distinction between geometry control and PTO control is outlined. While capture width is a valuable design tool, its limitations must be recognised. The assumptions made in the formulation of capture width are listed, and its limitations as a tool for estimating annual power capture of a WEC are discussed.

Introduction

Capture width is a parameter that characterises the performance of a wave energy converter (WEC). It is the width of the wavefront (assuming uni-directional waves) that contains the same amount of power as that absorbed by the WEC. In the literature, the term capture width is used to describe the performance of a linearised WEC model in a sinusoidal sea, as well as in a sea state containing many frequencies. Here the terms monochromatic and polychromatic are used to distinguish between these cases. This paper presents a new formulation for capture width that clarifies the relationship between the polychromatic and monochromatic capture widths.

Existing work on capture width appears in two forms: either capture width is described algebraically as a function of undefined coefficients, or the function is plotted for specific values of these coefficients. Thomas [1] presents an equation for polychromatic capture width in terms of the hydrodynamic coefficients, power take off (PTO) coefficients, and wave elevation. It is clear that an equation of this nature must have been in use earlier than this because it is required to plot the function referred to by Thomas and Gallagher as power absorption density [2]. It is possible that an equation of this nature was in use as early as 1982, as there is a graph in [3] that appears to be the power absorption density.

The new formulation presented in this paper agrees with Thomas’ [1] and is different in two respects. Firstly, it assumes the use of the discrete Fourier transform (DFT) rather than the continuous Fourier transform, which means it can be used directly with the discrete hydrodynamic data that are the standard outputs of experimental and numerical methods. Secondly, and more significantly, the polychromatic capture width is now expressed as a function of just two parameters: one describing the WEC performance, namely the monochromatic capture width; and the other describing the sea state, namely the newly introduced spectral power fraction.

This concept of the interplay between device performance and a sea state parameter is such a useful tool that several representations of this type can be seen in the literature. For instance, Falnes plotted the performance together with the spectrum [4], while Weber and Thomas plotted the performance together with the spectrum normalised by the peak spectral component [5], [1]. However, when plotting the WEC performance (monochromatic capture width) and a representation of the sea state on the same axes, it is more meaningful to choose the spectral power fraction than the spectrum or normalised spectrum: the product of the spectral power fraction and WEC performance gives the power absorption density, which can be integrated to give the polychromatic capture width. As the use of monochromatic and polychromatic capture widths is so well established, it is conceptually appealing to have a single sea state parameter linking them. This link is confirmed by theoretical consideration of monochromatic behaviour (36). This new way of presenting the capture width gives insights into the control problem, for instance, the difference between PTO control and geometry control.

A brief outline of the paper now follows. Section 2 begins with the definition of the unrestrained response of a generalised linear model of a WEC. A specific geometry with well-known hydrodynamic parameters is considered. Two control conditions are then described: one that results in complete absorption of the waves (ideal), and another that is realisable in real time. Section 3 defines the power equations from first principles, and highlights the modelling assumptions inherent in these formulations. Section 4 combines the power equations to give a formulation for capture width in terms of parameters describing WEC performance and the sea state. Section 5 uses this new formulation to give a fresh perspective on concepts such as reactive PTO control, geometry control, and amplitude constraints.

Note that the equations in Section 2 are for continuous functions of frequency, while the equations from Section 3 onwards are for discrete frequencies. In the literature [4], [1] the convention is for continuous functions of frequency, which are related to their time domain counterparts via the continuous Fourier transform. In this paper, discrete functions of frequency were used because they are related to their time domain counterparts via the DFT. When calculating hydrodynamic coefficients using experimental results, or using a numerical method such as the commercial code WAMIT [6], time and frequency must be discretised; hence the DFT is required.

Section snippets

Frequency domain modelling

This section presents a generalised version of the well-known frequency domain model of a WEC. Non-linear systems cannot be modelled in the frequency domain, so this approach only models the sub-set of linear behaviour. Non-linear terms are either linearised, or if the linear component is sufficiently small, not included.

Equations are presented such that the forces, motions and transformation coefficients can be interpreted as either scalars (for a single degree of freedom) or vectors and

Average power

Discrete frequency notation (ωj) is used for the remaining part of this paper as most experimental and numerical results appear in a discrete form. Plancherel’s theorem, a generalised version of the better known Parseval’s theorem, can be used to describe the product of two time domain functions, integrated over time, as the product of their Fourier transforms, integrated over frequency. For discrete quantities, the product of two time domain functions, summed over time, is related to the

Definition of capture width

Capture width is a widely used measure of the performance of a WEC. This section presents a new formulation that can be used to explain the properties of capture width. The theory presented here is also relevant to the relative capture width, sometimes referred to as efficiency, which is the capture width normalised with respect to geometry.

Capture width is defined as the ratio of the power absorbed (PA) to the wave energy transport (PIW): CW=PAPIW.

For small periodic waves, wave energy

Matching spectral and device properties with PTO

Fig. 3 demonstrates why monochromatic capture width is often optimised at the frequency for which the spectral power fraction is the highest. Thus, by matching the impedance at the frequency where the waves have the most power, there is a good chance of getting a high value of capture width. However, this does not necessarily mean that the best choice of operating frequency is the spectral peak frequency. This can be understood intuitively by considering the case of a spectrum with several

Conclusions

Capture width is a powerful performance indicator for wave energy converters (WECs). For the first time, an equation for capture width was explicitly presented in terms of WEC performance and a sea state parameter. This equation was used to emphasise the distinction between capture width in monochromatic and polychromatic conditions. The monochromatic capture width describes the performance of a WEC independently of the sea state, as a function of frequency, while the polychromatic capture

Acknowledgements

The authors thank Jørgen Hals, Julien Cretel and David Forehand for advice and proof-reading. The first author was supported by the EPSRC funded SuperGen-Marine program. The second author was supported by the EPSRC funded SuperGen-Amperes program. This work was carried out in the Joint Research Institute with the Heriot-Watt University, a part of the Edinburgh Research Partnership, which is supported by the Scottish Funding Council.

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