Elsevier

Ocean Engineering

Volume 65, 1 June 2013, Pages 10-18
Ocean Engineering

A geometric tool for the analysis of position and force constraints in wave energy converters

https://doi.org/10.1016/j.oceaneng.2013.03.011Get rights and content

Highlights

  • A discretization method is applied to express the system model as a linear system of equations.

  • Satisfaction of force and position constraints is done purely from the hydrodynamic model.

  • No simulation is required to evaluate constraints across a set of specific wave conditions.

  • The method is graphical and intuitively allows constraints to be adjusted to optimism power capture.

  • Multi-body devices are considered, with results for 1- and 2-body cases presented.

Abstract

Most wave energy devices are subject to finite constraints on both the power take-off (PTO) stroke length and the maximum force that the PTO can tolerate. It is also often the case that greater stroke lengths can reduce the maximum force in the PTO and vice versa. Ultimately, some informed choice of PTO constraints must be made in order to ensure that PTO constraints are not violated and that the trade-off between position and force constraints is made in such as way that maximum energy is captured by the converter. This paper presents a tool to allow device developers to check the satisfaction of constraints for a given hydrodynamic model and set of sea conditions and, where constraints are not satisfied, shows how to relax the constraints to maximize energy capture. The tool is algebraic, requiring no simulation and the results are presented through intuitive geometrical constructs. Sample application results are presented for single- and two-body wave energy systems.

Introduction

Wave energy conversion technology is at a relatively early stage of maturity, with few commercial devices in the water. However, even at this stage, it is largely accepted that most devices will have two modes of operation. The ‘normal’ mode of operation is that where power is converted from the waves into a useable form (water pressure, electricity, etc.) and a further ‘survivability’ mode, which the device will need to enter when the wave excitation forces become too great for the constraints of the normal mode to bear. How each device survives is not the focus of this paper and, indeed, there have been a number of methods proposed, many of which are particular to specific device types. Rather, we concern ourselves with the interplay between the wave excitation force and the physical constraints of the wave energy device, namely the force and position constraints and will assume that, where system constraints cannot be observed, the wave energy device must enter survival mode. We note that there may also be velocity constraints related to limitations of rate of movement of various system components. While velocity constraints can also be included in our formulation, we focus on the main constraints of force and position, for brevity.

The suggested interplay between the force and position (amplitude) constraints, and the wave excitation force, is illustrated by Fig. 1. A situation which satisfies the constraints for a given wave excitation force, Fe, is indicated by an intersection of the force and position sets. In general, larger position and force constraints are indicated by circles of greater radii, while larger excitation forces are represented by a greater distance between circle centers. Ideally, we would like to identify the functions f1(), f2() and f3() so that the construct in Fig. 1 can be generated for a specific wave energy device. In addition, it would be beneficial if some indication of the energy landscape could be provided so that, where constraints are not immediately satisfied, some guidance is available to allow captured energy to be maximized by virtue of a constraint relaxation.

The feasibility problem, as identified in the preceding paragraph, has, to date, not been addressed. A number of researchers address the issue of position constraints and design a controller which actuates PTO force in order to satisfy constraints, usually focussing position constraints alone. However, such studies pre-suppose a particular control design method. For example, amplitude constraints, based on a frequency-domain approach, are addressed in Evans (1981), Pizer (1993), and Falnes (2000), while a number of authors have utilized the model predictive control (MPC) framework to maximise energy capture subject to constraints; for example Cretel et al. (2011). The work reported in Hals et al. (2011) is interesting in that it uses MPC to solve for the optimal velocity profile of the wave energy device, subject to amplitude constraints, while then examining the PTO force required to achieve this velocity profile, where the PTO force is also subject to constraints.

Our development is based on the discretization of the equations of motion of the wave energy converters (WECs) by means of the approximation of the forces and of the velocities with a linear combination of basis functions. A special case is considered, in which truncated Fourier series are used for the approximation. This case is particularly interesting because the results are in harmony with the frequency domain theory of wave energy conversion, as in Falnes (2002). The discretization was initially motivated by the study of an optimal controller for the maximization of the energy converted by the device, subject to constraints. For brevity, only the theoretical details crucial to the development of the constraint framework are described here. The full details of the underlying control framework are given in Bacelli et al. (2011).

While it is (theoretically) possible to determine the satisfaction of system constraints for a given wave climate via extensive simulation, such an approach would be both time consuming and may not capture the complete set of circumstances. In contrast, the constraint analysis tool developed in this paper is analytical, requiring a hydrodynamic model of the wave energy device of interest together with a specification of the force and position constraints and the excitation force in order to give an immediate conclusion regarding the satisfaction of the constraints. Furthermore, the graphical nature of the answer provides some insight into the interplay between cf, cp and Fe, placed against the background of absorbed energy and therefore provides the basis for a design tool.

In the paper we present the theoretical development for a multi(two)-body device, where single body analysis can be achieved as a special case. The paper proceeds as follows: Section 2 presents the mathematical model of a two-body self-reacting point absorber and the discretization performed by means of the Galerkin method. The force and oscillation amplitude constraints are defined in Section 3; the procedure for approximating the constraints by means of the 2-norm is also presented, in conjunction with the geometrical interpretation of the approximated constraints. In Section 4, the case of a single-body point absorber is considered as a special case of the two-body device, while sample results for both single-body and two-body WECs are described in Section 5.

Section snippets

Self-reacting point absorber model

The general form of device considered is a two-body self-reacting point absorber restricted to heave motion only, as depicted in Fig. 2, and described by the frequency domain model (Falnes, 1999)iωmA+BA+SAiωVA=FeA+FrAFptoiωmB+BB+SBiωVB=FeB+FrB+Fptowhere VA and VB are the vertical velocities of body A and body B, respectively. The radiation force Fr is FrAFrB=ZVAVBwithZ=ZAAZABZBAZBBwhere Z=Z(ω) is the radiation impedance matrix.

FeB and FeB denote the excitation forces, mA and mB are the

Specification and approximation of constraints

The force constraint is defined asf^ptoFmax,while the constraint on the relative amplitude isΔu^ΔUmax,where the infinity norm · is defined in the appendix (Eq. (A.2)) and Δu^(t)=u^A(t)u^B(t), withu^A(t)=u0A+j=1NbnAnω0(1cos(nω0t))+anAsin(nω0t),u^B(t)=u0B+j=1NbnBnω0(1cos(nω0t))+anBsin(nω0t).

Special case of single body

A single body heaving buoy is now considered as a special case of the self-reacting point absorber described in Section 2, given by the frequency domain model(iωm+B+Z(ω)+S/iω)V=FeFpto.

The corresponding time domain model is(m+m)v̇(t)+k(t)(⁎)v(t)+Bv(t)+Su(t)=fe(t)fpto(t),and the energy absorbed by the PTO, neglecting losses, isJ(T)=0Tfpto(t)v(t)dt.

The energy maximization problem is then discretized by following the same steps performed for the case of a self-reacting device (Section 2.1).

Sample results

Initially, in Section 5.1, we will take a single body device and perform a constraint analysis for a monochromatic sea. This will allow us to develop some insight into the use of the geometrical tool and to consider it from a design perspective. While monochromatic analysis might seem restrictive, appropriate choice of the frequency, such as the resonant frequency of the device, or the peak energy frequency of the incident sea, could be sufficient to determine the peak loading condition.

Conclusions

In this paper, we have developed a procedure to examine if physical constraints will be satisfied for a particular wave energy device, under a particular set of sea conditions. The method does not require any simulation and does not depend on the control method used to calculate the PTO force. However, the method does require hydrodynamic model parameters which are used to evaluate the matrix G in (14).

The method relies on a discretization of the equation of motion in frequency, using Fourier

References (14)

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