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Nonlinear model reduction for wave energy systems: a moment-matching-based approach

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Abstract

Wave energy converters (WECs) inherently require appropriate control system technology to ensure maximum energy absorption from ocean waves, consequently reducing the associated levelised cost of energy and facilitating their successful commercialisation. Regardless of the control strategy, the definition of the control problem itself depends upon the specification of a suitable WEC model. Not only is the structure of the model relevant for the definition of the control problem, but also its associated complexity: given that the control law must be computed in real-time, there is a limit to the computational complexity of the WEC model employed in the control design procedure, while there is also a limit to the (analytical) complexity of mathematical models for which a control solution can be efficiently found. This paper presents a systematic nonlinear model reduction by moment-matching framework for WEC systems, capable of providing control-oriented WEC models tailored for the control application, which inherently preserves steady-state response characteristics. Existence and uniqueness of the associated nonlinear moment for WECs are proved in this paper, for a general class of systems. Given that the definition of nonlinear moments depends upon the solution of a nonlinear partial differential equation, an approximation framework for the computation of the nonlinear moment is proposed, tailored for the WEC application. Finally, the use and capabilities of the framework are illustrated by means of case studies, using different WEC systems, under a variety of wave conditions.

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Notes

  1. Though rare due to the inherent complexity behind the energy-maximising control objective for WECs, we note that model-free control techniques also exists within the wave energy control literature. The interested reader is referred to, for instance, [15, 39].

  2. Linear conditions refers to so-called linear potential flow theory, see [13].

  3. Within the field of wave energy applications, internal stability is a fundamental requirement of a model representing the physical system not only for control/estimation, but also for motion simulation and power assessment purposes.

  4. The reader is referred to, for instance, [16, 27, 43] for comprehensive discussions on different sources of nonlinear effects (and associated modelling procedures) in WECs.

  5. Assuming that the nonparametric nature of \(\varSigma \) is only due to linear radiation dynamics. This is not always necessarily the case, since \(f_{\text {nl}}\) can be potentially nonparametric.

  6. Similar considerations can be drawn for multiple-input–multiple-output (MIMO) systems, by following the framework presented in [10, 28].

  7. From now on, the dependence on t is dropped when clear from the context.

  8. Let \(A\in \mathbb {R}^{n\times n}\). An eigenvalue \(a\in \lambda (A)\) is said to be simple if its algebraic multiplicity is equal to 1.

  9. All statements are local, although global versions can be straightforwardly derived.

  10. Throughout this manuscript, if a given system \(\varSigma \) is reduced by moment-matching to a system \({\tilde{\varSigma }}\), \(\varSigma \) and \({\tilde{\varSigma }}\) are referred to as the target and approximating systems, respectively.

  11. In the case of nonlinear restoring effects, the division between linear and nonlinear contributions is performed without any loss of generality, and to subsequently analyse the local properties of system (9) in Sect. 4.

  12. See [13] for the definition of \(A(\omega )\).

  13. The interested reader is referred to [42] for further detail on this class of integro-differential operators.

  14. Practical implications of both \({\mathbf {f}}\) and T (or, equivalently, \(\omega _0\)) in our model reduction framework, are discussed in detail in Sect. 8.1.

  15. This claim, which directly relates to Jacobian analysis, is considered standard in nonlinear dynamics. Further detail can be found in, for instance, [20, Chapter 8].

  16. The extension to multiple trajectories presented in this section is proposed in the spirit of the so-called \({\mathscr {U}}/{\mathscr {X}}\) variation [35].

  17. The selection of l depends upon the specific nature of the WEC input process. This is discussed in detailed in Sects. 7 and 8.

  18. The mapping \(f^{\text {nl}}_{\text {re}}\) is geometry dependent and, for the spherical heaving point absorber case, can be found in, for instance, [25].

  19. A small time-step is selected (with respect to the dominant system dynamics) to guarantee convergence in the benchmark response.

  20. The approach presented herein is simply one possibility: The user is free to select a finite set of initial conditions using different methods, according to specific application requirements.

  21. The use of the term ‘amplitude’ for \(A_p\) is justified in Remark 27.

  22. This is considered to simplify the case study, and focus on the performance of the nonlinear reduction technique.

  23. Computed randomly according to the SDF of Fig. 8.

  24. The methodology employed herein uses random amplitudes. The reader is referred to [24] for further detail.

  25. Ratio between the time required to compute the output of each corresponding model, and the length of the simulation itself. The computations are performed using Matlab®, running on a PC composed of an Intel Core i7-5550U processor with 8GB of RAM.

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Acknowledgements

This work was supported by the Science Foundation Ireland under Grant No. SFI/13/IA/1886. The authors are grateful with Prof. Alessandro Astolfi and Dr. Giordano Scarciotti from the Control and Power Group, Imperial College London, for useful discussions on nonlinear moment-based theory.

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Authors and Affiliations

  1. Centre for Ocean Energy Research (COER), Department of Electronic Engineering, Maynooth University, Kildare, Ireland

    Nicolás Faedo & John V. Ringwood

  2. Departamento de Ciencia y Tecnología, Universidad Nacional de Quilmes, Bernal, Argentina

    Francisco Javier Dores Piuma

  3. Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy

    Giuseppe Giorgi

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  1. Nicolás Faedo
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Correspondence to Nicolás Faedo.

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Faedo, N., Dores Piuma, F.J., Giorgi, G. et al. Nonlinear model reduction for wave energy systems: a moment-matching-based approach. Nonlinear Dyn 102, 1215–1237 (2020). https://doi.org/10.1007/s11071-020-06028-0

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