Elsevier

Ocean Engineering

Volume 213, 1 October 2020, 107816
Ocean Engineering

Performance enhancement of submerged wave energy device using bistability

https://doi.org/10.1016/j.oceaneng.2020.107816Get rights and content

Highlights

  • An additional nonlinear stiffness can give near optimal power performance.

  • Bistability can improve the robustness of submerged wave energy converters.

  • Consideration of nonlinearities are important in bistable wave energy systems.

  • Benefits are due to phase matching and exposure to a range of natural frequencies.

  • Demonstrated potential for passive or semi-active controllers for wave energy devices.

Abstract

The performance of a submerged cylindrical point absorbing wave energy converter was explored under the addition of different nonlinear stiffness (bistable) conditions. The limitations of previous studies were addressed by incorporating higher-fidelity modelling. Devices employing bistability in other energy harvesting applications, have improved the amount of power generated. For wave energy converters, most theoretical models with bistability were limited to one-degree-of-freedom, neglect nonlinearities such as viscous drag, and are excited by unrealistic sinusoidal waves. Such simplifications lead to neglecting features such as modal interactions. The presented model investigated a three-degree-of-freedom submerged point absorber with bistability subjected to regular and irregular waves. The bistable mechanism was an adjustable magnetic model such that a range of potential profiles were examined and parameterised, for generality, by features common between mechanisms. For this device, bistability may be used to obtain near optimal results and was suitably robust for changing ocean conditions. Regions of improvement were identified in terms of the changing natural frequency due to a nonlinear stiffness, and a phase matching property. In varying sea-states, a selected bistable condition demonstrated a 10–20% improvement in power production. The consistency implies that semi-active elements may be able to adjust the bistability to enhance power production.

Introduction

Ocean wave energy generation has been the subject of over two centuries of research (Cruz, 2008). Typical wave energy converters (WEC) can be broadly classed as one of three types: an attenuator; a point absorber (PA); or a terminator (Drew et al., 2009). There are other significant devices which are difficult to categorise into these classes such as oscillating water columns or overtopping devices. However, this paper focuses on a PA type WEC, which are systems in which the buoy is small relative to the wavelengths of incident waves, and, for a single PA, is subsequently relatively insensitive to wave direction. A representative diagram of the simplified CETO PA (Carnegie Clean Energy, 2018) used in this study is shown in Fig. 1. Whilst smaller than typical wavelengths, the CETO is one of the large PAs, and therefore is more accurately described as a quasi-PA. For the purposes of this paper, the CETO device will be referred to as a PA. While the simple operation of a generic PA WEC is well known, there remains many challenges for wave energy (Greaves and Iglesias, 2018). Due to the complexity of these unique challenges in the context of wave energy, the technology as a whole remains at a low technology readiness level.

One of the areas which remains an actively studies topic is the methodology behind controlling PA devices. Hybrid control, in which passively applied dynamics supplement an active controller, may provide some advantages over active control algorithms (Wu et al., 2018). Bistable mechanisms using nonlinear stiffness have received notable attention over the past few years. A generic bistable system is given in Fig. 2. For further clarity specific to wave energy, a characteristic time history of the WEC operating at low, medium, and high levels of bistability in terms of the central potential barrier is shown in Fig. 3, and the corresponding frequency domain displacement and generated power amplitudes are provided in Fig. 4. These figures simply show how increasing levels of bistability changes the resulting motion and power generation. That is, higher levels of bistability tends to force the WEC to oscillate about two different stable regions and leads to an increase in the bandwidth of power generated, while decreasing the peak performance. This finding is, in general, consistent with many bistable or nonlinear dynamics, that nonlinear systems may be very sensitive to small changes in system parameters — particularly parameters responsible for transitions between linear and nonlinear regimes.

Substantial benefits have been observed when using a bistable mechanism within the power take off unit (PTO) with simple single degree of freedom (DOF) simulation models (Zhang et al., 2016, Wu et al., 2018, Younesian and Alam, 2017, Xiao et al., 2017, Zhang et al., 2018). There have been some attempts to optimise a nonlinear reactive force for power production (Abdelkhalik and Darani, 2018), however no method of implementing this force profile was provided. Methods using an asymmetric mass distribution to emulate a nonlinear effective stiffness have also been proposed (Meng et al., 2019b, Meng et al., 2019a). A model using a bistable mechanism composed of magnets (Xiao et al., 2017) found that a bistable WEC can harvest more energy than a linear WEC when the excitation frequency is less than the natural frequency of the linear system. Another magnetic design composed of coaxial cylinders concluded that bistability can enhance the efficiency of a floating WEC (Zhang et al., 2019b). However, this study was restricted to a single DOF and used a linear stiffness only valid for small oscillations. Though the damping in the system was optimised, the linear stiffness was not. The addition of bistability in this case did show significant improvement on the suboptimised system.

Experimental investigations using different devices showed that the realistic benefit of bistability is significant and in some cases may be as much as a three times power increase due to phase matching (Têtu et al., 2018, Todalshaug et al., 2016). Since the geometries and general operation of WEC devices and mechanisms differ significantly, bistability is employed in various ways to either excite super harmonics to convert low frequency oscillations into high frequency oscillations (Harne and Wang, 2013), or to provide passive phase matching for a floating PA WEC (Todalshaug et al., 2016).

Conventional PAs are designed to be floating rather than submerged. Accordingly, there has been more research into floating systems than submerged systems. While both systems may seem similar, they undergo fundamentally different excitation and forces. As a result, submerged PAs typically generate lower levels of power at longer wavelengths, have reduced bandwidths, but more readily capture power from multiple DOF when compared to floating systems (Sergiienko et al., 2017). In particular, there is limited work in taking into account nonlinearity in stiffness in the theoretical modelling of a submerged WEC (Wang et al., 2018), and such work is related to a device that does not operate in the vertical heave direction. In addition to restricted DOFs, many of the models previously employed do not include drag. A study of a submerged WEC compares the effect of bistability in 1-DOF and 3-DOF which includes drag (Schubert et al., 2020). The study concluded that for regular waves, an improvement is only seen under certain circumstances when the device is non-optimally tuned. The study was limited to regular wave scenarios, and was primarily directed at observing if there was potential for any benefit but did not attempt to explore the cause of the benefit.

There have been numerous ideas proposed for the mechanism to provide a bistable force including mechanical springs (Zhang et al., 2019a), magnetic systems (Xiao et al., 2017), and pneumatic systems (Todalshaug et al., 2016). Each of these suggestions provide unique benefits. Generally, electromagnetic reactive mechanisms in the form of direct drive systems can be expected to have smaller reactive power losses compared to alternatives such as hydraulic systems (Pecher and Kofoed, 2017). How a submerged PA WEC subject to irregular waves behaves when bistability is included, and what bistable attributes are beneficial within the context of submerged WEC, remains a gap within current literature.

In this paper, the limitation of previous studies will be addressed by incorporating higher-fidelity system modelling including nonlinear coupling between the dynamic degrees of freedom. In particular, the following features differentiate the current work and contribute to the enhanced fidelity:

  • (1)

    The 1-DOF model used by prior authors with relatively simple dynamics has been extended to 3 DOF to allow for coupling between DOF.

  • (2)

    A drag coefficient has been included to improve the estimation of velocity dependent forces acting on the buoy and consequently preventing unrealistically large amplitudes of motion.

  • (3)

    The regular wave excitation has been extended to irregular waves to better estimate the buoy performance under a more realistic broadband excitation.

  • (4)

    Systems with bistability are often compared to poorly tuned monostable counterparts in the aforementioned studies. Such an assumption can lead to systems with nonlinear stiffnesses showing better performance, but could be further improved by careful selection of a linear stiffness. This study analyses both regular and irregular results with respect to the well tuned monostable counterparts to understand the real potential for improvement.

The combination of both the improved dynamic model and the broader range of simulation conditions furthers the current understanding of the applicability of bistability to submerged point absorbing WECs.

The impact of bistable control on a submerged 3-DOF PA WEC was explored to quantify how simple hydrodynamics and the passive control system interact. A 3-DOF theoretical model of the CETO device being developed by Carnegie Clean Energy (2018) was constructed. The model was subjected to a set of regular waves and three different irregular waves (or sea states). A nonlinear stiffness mechanism was included and varied to understand what characteristics are important for power production capability. The mathematical models are described in Section 2, with the representation of the magnetic bistable mechanism given in Section 2.1.4, as well as a dimensionless parameter pertinent to any bistable system as a means to generalise results. The considerations and limitations of the used modelling method are outlined in Section 3, and the results of these simulations are presented and discussed in Section 4, with a summary of the findings in Section 5.

Section snippets

Mathematical models

The governing equation for both regular and irregular scenarios can be derived from the contributing forces represented in the equation Mẍ(t)=Fe(x,t)+Fr(ẍ,ẋ,t)+Fh+FPTO(ẋ,x,t)+FD(ẋ,x,t)+Fbi(x,t),where the inertia of each DOF of the system is represented by the matrix M, and x is the position vector containing the surge (x), heave (z), and pitch (θ) coordinates. Also, F represents a 3-DOF force vector and the subscripts e, r, h, PTO, D, and bi indicate the excitation, radiation, hydrostatic,

Simulation considerations

For each simulation, there were an number of considerations and parameters specific to each scenario. These considerations and parameters will be provided in this section. In the regular wave simulations, the WEC system was excited by a series of monochromatic waves and the resulting dynamic behaviour was analysed for each frequency to quantify motion and power performance. For each frequency, the optimal stiffness found through a local optimisation search was used in initial regular wave

Results and discussion

The results for the simulation scenarios described in Section 3 are presented and discussed in the following sections.

Conclusion

The impact of a nonlinear stiffness or a bistable force acting on a submerged point absorbing wave energy converter was investigated. A model of a 3-DOF CETO-shaped device was constructed and subjected to regular waves. The degree of bistability was varied. The resultant motion and time averaged power showed that for an optimally tuned system, bistability provided no benefit, whereas for a non-optimally tuned system, certain levels of bistability were able to improve the performance for

CRediT authorship contribution statement

Benjamin W. Schubert: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Visualization. William S.P. Robertson: Conceptualization, Methodology, Formal analysis, Writing - review & editing, Visualization, Supervision. Benjamin S. Cazzolato: Conceptualization, Methodology, Formal analysis, Writing - review & editing, Supervision. Mergen H. Ghayesh: Methodology, Formal analysis, Writing - review & editing, Supervision.

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.

Acknowledgement

This research has been supported by the Australian Government Research Training Program Scholarship .

References (37)

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