A harmonic balance framework for the numerical simulation of non-linear wave energy converter models in random seas

Numerical simulation is essential, to assist in the development of wave energy technology. In particular, tasks such as power assessment, optimisation and structural design require a large number of numerical simulations to calculate the wave energy converter (WEC) outputs of interest, over a variety of wave conditions or physical parameters. Such challenges involve a sound understanding of the statistical properties of ocean waves, which constitute the forcing inputs to the wave energy device, and computationally efficient numerical techniques for the speedy calculation of WEC outputs. This thesis studies the statistical characterisation, and numerical generation, of ocean waves, and proposes a novel technique for the numerical simulation of non-linear WEC models. The theoretical foundations, the range of validity, and the importance of the statistical representation of ocean waves are first examined. Under relatively mild assumptions, ocean waves can be best described as a stationary Gaussian process, which is entirely characterised by its spectral density function (SDF). Various wave superposition techniques are discussed and rigorously compared, for the numerical generation of Gaussian wave elevation time series from a given SDF. In particular, the harmonic random amplitude (HRA) approach can simulate the target statistical properties with perfect realism. In contrast, the harmonic deterministic amplitude (HDA) approach is statistically inconsistent (because the generated time-series are non-Gaussian, and under-represent the short-term statistical variability of real ocean waves), but can be advantageous in the context of WEC simulations since, if it can be verified that HDA results are unbiased, the HDA method requires a smaller number of random realisations than the HRA method, to obtain accurate WEC power estimates. When either HDA or HRA are used for the generation of wave inputs, the forcing terms of WEC mathematical models are periodic. Relying on a Fourier representation of the system inputs and variables, the harmonic balance (HB) method, which is a special case of spectral methods, is a suitable mathematical technique to numerically calculate the steady-state response of a non-linear system, under a periodic input. The applicability of the method to WEC simulation is demonstrated for those WEC models which are described by means of a non-linear integro-differential equation. In the proposed simulation framework, the WEC output, in a given sea state, is assessed by means of many, relatively short, simulations, each of which is efficiently solved using the HB method. A range of four case studies is considered, comprising a flap-type WEC, a spherical heaving point-absorber, an array of four cylindrical heaving point-absorbers, and a pitching device. For each case, it is shown how the HB settings (simulation duration and cut-off frequency) can be calibrated. The accuracy of the HB method is assessed through a comparison with a second-order Runge-Kutta (RK2) time-domain integration scheme, with various time steps. RK2 results converge to the HB solution, as the RK2 time step tends to zero. Furthermore, in a Matlab implementation, the HB method is between one and three orders of magnitude faster than the RK2 method, depending on the RK2 time step, and on the method chosen for the calculation of the radiation memory terms in RK2 simulations. The HB formalism also provides an interesting framework, for studying the sensitivity of the WEC dynamics to system parameter variations, which can be utilised within a gradient-based parametric optimisation algorithm. An example of WEC gradientbased parametric optimisation, carried out within the HB framework, is provided

Optimal trajectories, nonlinear models and constraints in wave energy device control

The optimal control problem for a generic, one-degree of freedom Wave Energy Converter (WEC) with dynamical nonlinearities is formulated in the frequency-domain. Previous research, concerning more specifically a heaving point-absorber with nonlinear restoring force, shows that the unconstrained optimal velocity trajectory is influenced neither by the linear inertial terms, nor by the linear or nonlinear static forces. Further to this result, in this paper, we examine the influence of velocity-dependent nonlinear forces on the optimal trajectory, as well as the effect of physical system constraints. In particular, we show that, under state constraints (e.g. position and velocity limitations), the optimal velocity trajectory remains uninfluenced by static forces; but this is no longer true for constraints involving the control force, such as force limitation and passivity constraints. In addition, unlike static terms and linear inertial terms, the velocity-dependent forces, such as viscous drag, significantly influence the optimal velocity trajectory, regardless of constraints, and must be carefully modelled at the control design stage. In any case, even when the optimal velocity trajectory is not affected by some of the forces considered, the optimal control force required to achieve it depends on all the model dynamics (inertial terms, velocity-dependent and static forces). Numerical simulations, in the specific case of a heaving point absorber, are used to validate and illustrate the theoretical results

A nonlinear extension for linear boundary element methods in wave energy device modelling

To date, mathematical models for wave energy devices typically follow Cummins equation, with hydrodynamic parameters determined using boundary element methods. The resulting models are, for the vast majority of cases, linear, which has advantages for ease of computation and a basis for control design to maximise energy capture. While these linear models have attractive properties, the assumptions under which linearity is valid are restrictive. In particular, the assumption of small movements about an equilibrium point, so that higher order terms are not significant, needs some scrutiny. While this assumption is reasonable in many applications, in wave energy the main objective is to exaggerate the movement of the device through resonance, so that energy capture can be maximised. This paper examines the value of adding specific nonlinear terms to hydrodynamic models for wave energy devices, to improve the validity of such models across the full operational spectrum

A nonlinear extension for linear boundary element methods in wave energy device modelling

To date, mathematical models for wave energy devices typically follow Cummins equation, with hydrodynamic parameters determined using boundary element methods. The resulting models are, for the vast majority of cases, linear, which has advantages for ease of computation and a basis for control design to maximise energy capture. While these linear models have attractive properties, the assumptions under which linearity is valid are restrictive. In particular, the assumption of small movements about an equilibrium point, so that higher order terms are not significant, needs some scrutiny. While this assumption is reasonable in many applications, in wave energy the main objective is to exaggerate the movement of the device through resonance, so that energy capture can be maximised. This paper examines the value of adding specific nonlinear terms to hydrodynamic models for wave energy devices, to improve the validity of such models across the full operational spectrum

Ocean forecasting for wave energy production

There are a variety of requirements for future forecasts in relation to optimizing the production of wave energy. Daily forecasts are required to plan maintenance activities and allow power producers to accurately bid on wholesale energy markets, hourly forecasts are needed to warn of impending inclement conditions, possibly placing devices in survival mode, while wave-by-wave forecasts are required to optimize the real-time loading of the device so that maximum power is extracted from the waves over all sea conditions. In addition, related hindcasts over a long time scale may be performed to assess the power production capability of a specific wave site. This paper addresses the full spectrum of the aforementioned wave modeling activities, covering the variety of time scales and detailing modeling methods appropriate to the various time scales, and the causal inputs, where appropriate, which drive these models. Some models are based on a physical description of the system, including bathymetry, for example (e.g., in assessing power production capability), while others simply use measured data to form time series models (e.g., in wave-to-wave forecasting). The paper describes each of the wave forecasting problem domains, details appropriate model structures and how those models are parameterized, and also offers a number of case studies to illustrate each modeling methodology

Wave Energy Control Systems: Robustness Issues

While traditional feedback control systems enjoy relatively good sensitivity properties, energy maximising wave energy converter (WEC) control systems have particular characteristics which challenge the application of traditional feedback and robust control methods. In particular, the relationship between plant and controller is largely defined by the need to maximise power transfer, and the controller contains a feedforward component which is difficult to robustify. Typically, WEC control systems are based on linear model descriptions, but this belies the true nonlinearity of WEC hydrodynamics (particularly under controlled conditions) and the associated power take-off (PTO) system. This paper examines two popular WEC control structures and examines the sensitivity of these structures to parameter variations, both in terms of closed-loop transfer functions and power absorbed. Some recommendations are also given on which WEC parameters need to be modelled with high accuracy

Finite-order hydrodynamic approximation by moment-matching (FOAMM) toolbox for wave energy applications

—Cummins’ equation is commonly used to describe the motion of Wave Energy Converters (WECs), where the radiation force is characterised by a convolution operation. The computational effort associated with the solution of the convolution term, often represents a drawback for e.g. optimisation or exhaustive-search studies. To overcome this disadvantage, and given that the convolution operator intrinsically defines a dynamical system, the convolution term is commonly approximated using suitable finite-order parametric models. To this end, the Centre for Ocean Energy Research has recently presented a moment-matching based identification method for the radiation force subsystem and the complete force-to-motion WEC dynamics (i.e. wave excitation force to device velocity). Motivated by the theory and the obtained results, already reported by the authors, the FOAMM MATLAB application has been developed, which systematically implements the moment-matching based identification strategy from raw frequency-domain data, provided by hydrodynamic solvers, in a user-friendly fashion. The aim of this paper is to describe the theoretical background behind the identification strategy, and the structure, organisation and characteristics of the developed application. Additionally, the relevant modes of operation, along with the different options of the toolbox are explained, and, at the end, a step-by-step example of how to use the FOAMM application is provided, along with recommendations from the author

A harmonic balance framework for the numerical simulation of non-linear wave energy converter models in random seas

Numerical simulation is essential, to assist in the development of wave energy technology. In particular, tasks such as power assessment, optimisation and structural design require a large number of numerical simulations to calculate the wave energy converter (WEC) outputs of interest, over a variety of wave conditions or physical parameters. Such challenges involve a sound understanding of the statistical properties of ocean waves, which constitute the forcing inputs to the wave energy device, and computationally efficient numerical techniques for the speedy calculation of WEC outputs. This thesis studies the statistical characterisation, and numerical generation, of ocean waves, and proposes a novel technique for the numerical simulation of non-linear WEC models. The theoretical foundations, the range of validity, and the importance of the statistical representation of ocean waves are first examined. Under relatively mild assumptions, ocean waves can be best described as a stationary Gaussian process, which is entirely characterised by its spectral density function (SDF). Various wave superposition techniques are discussed and rigorously compared, for the numerical generation of Gaussian wave elevation time series from a given SDF. In particular, the harmonic random amplitude (HRA) approach can simulate the target statistical properties with perfect realism. In contrast, the harmonic deterministic amplitude (HDA) approach is statistically inconsistent (because the generated time-series are non-Gaussian, and under-represent the short-term statistical variability of real ocean waves), but can be advantageous in the context of WEC simulations since, if it can be verified that HDA results are unbiased, the HDA method requires a smaller number of random realisations than the HRA method, to obtain accurate WEC power estimates. When either HDA or HRA are used for the generation of wave inputs, the forcing terms of WEC mathematical models are periodic. Relying on a Fourier representation of the system inputs and variables, the harmonic balance (HB) method, which is a special case of spectral methods, is a suitable mathematical technique to numerically calculate the steady-state response of a non-linear system, under a periodic input. The applicability of the method to WEC simulation is demonstrated for those WEC models which are described by means of a non-linear integro-differential equation. In the proposed simulation framework, the WEC output, in a given sea state, is assessed by means of many, relatively short, simulations, each of which is efficiently solved using the HB method. A range of four case studies is considered, comprising a flap-type WEC, a spherical heaving point-absorber, an array of four cylindrical heaving point-absorbers, and a pitching device. For each case, it is shown how the HB settings (simulation duration and cut-off frequency) can be calibrated. The accuracy of the HB method is assessed through a comparison with a second-order Runge-Kutta (RK2) time-domain integration scheme, with various time steps. RK2 results converge to the HB solution, as the RK2 time step tends to zero. Furthermore, in a Matlab implementation, the HB method is between one and three orders of magnitude faster than the RK2 method, depending on the RK2 time step, and on the method chosen for the calculation of the radiation memory terms in RK2 simulations. The HB formalism also provides an interesting framework, for studying the sensitivity of the WEC dynamics to system parameter variations, which can be utilised within a gradient-based parametric optimisation algorithm. An example of WEC gradientbased parametric optimisation, carried out within the HB framework, is provided

A harmonic balance framework for the numerical simulation of non-linear wave energy converter models in random seas

Numerical simulation is essential, to assist in the development of wave energy technology. In particular, tasks such as power assessment, optimisation and structural design require a large number of numerical simulations to calculate the wave energy converter (WEC) outputs of interest, over a variety of wave conditions or physical parameters. Such challenges involve a sound understanding of the statistical properties of ocean waves, which constitute the forcing inputs to the wave energy device, and computationally efficient numerical techniques for the speedy calculation of WEC outputs. This thesis studies the statistical characterisation, and numerical generation, of ocean waves, and proposes a novel technique for the numerical simulation of non-linear WEC models. The theoretical foundations, the range of validity, and the importance of the statistical representation of ocean waves are first examined. Under relatively mild assumptions, ocean waves can be best described as a stationary Gaussian process, which is entirely characterised by its spectral density function (SDF). Various wave superposition techniques are discussed and rigorously compared, for the numerical generation of Gaussian wave elevation time series from a given SDF. In particular, the harmonic random amplitude (HRA) approach can simulate the target statistical properties with perfect realism. In contrast, the harmonic deterministic amplitude (HDA) approach is statistically inconsistent (because the generated time-series are non-Gaussian, and under-represent the short-term statistical variability of real ocean waves), but can be advantageous in the context of WEC simulations since, if it can be verified that HDA results are unbiased, the HDA method requires a smaller number of random realisations than the HRA method, to obtain accurate WEC power estimates. When either HDA or HRA are used for the generation of wave inputs, the forcing terms of WEC mathematical models are periodic. Relying on a Fourier representation of the system inputs and variables, the harmonic balance (HB) method, which is a special case of spectral methods, is a suitable mathematical technique to numerically calculate the steady-state response of a non-linear system, under a periodic input. The applicability of the method to WEC simulation is demonstrated for those WEC models which are described by means of a non-linear integro-differential equation. In the proposed simulation framework, the WEC output, in a given sea state, is assessed by means of many, relatively short, simulations, each of which is efficiently solved using the HB method. A range of four case studies is considered, comprising a flap-type WEC, a spherical heaving point-absorber, an array of four cylindrical heaving point-absorbers, and a pitching device. For each case, it is shown how the HB settings (simulation duration and cut-off frequency) can be calibrated. The accuracy of the HB method is assessed through a comparison with a second-order Runge-Kutta (RK2) time-domain integration scheme, with various time steps. RK2 results converge to the HB solution, as the RK2 time step tends to zero. Furthermore, in a Matlab implementation, the HB method is between one and three orders of magnitude faster than the RK2 method, depending on the RK2 time step, and on the method chosen for the calculation of the radiation memory terms in RK2 simulations. The HB formalism also provides an interesting framework, for studying the sensitivity of the WEC dynamics to system parameter variations, which can be utilised within a gradient-based parametric optimisation algorithm. An example of WEC gradientbased parametric optimisation, carried out within the HB framework, is provided

Spectral Control of Wave Energy Converters with Non-Ideal Power Take-off Systems

Spectral control is an accurate and computationally efficient approach to power-maximising control of wave energy converters (WECs). This work investigates spectral control calculations with explicit derivative computation, applied to WECs with non-ideal power take-off (PTO) systems characterised by an efficiency factor smaller than unity. To ensure the computational efficiency of the spectral control approach, it is proposed in this work to approximate the discontinuous efficiency function by means of a smooth function. A non-ideal efficiency function implies that the cost function is non-quadratic, which requires a slight generalisation of the derivative-based spectral control approach, initially introduced for quadratic cost functions. This generalisation is derived here in some detail given its practical interest. Two application case studies are considered: the Wavestar scale model, employed for the WEC control competition (WECCCOMP), and the 3rd reference model (RM3) two-body heaving point absorber. In both cases, with the approximate efficiency function, the spectral approach calculates WEC trajectory and control force solutions, for which the mean electrical power is shown to lie within a few percent of the true optimal electrical power. Regarding the effect of a non-ideal PTO efficiency upon achievable power production, and concerning heaving point-absorbers, the results obtained are significantly less pessimistic than those of previous studies: the power achieved lies within 80–95% of that obtained by simply applying the efficiency factor to the optimal power with ideal PTO