On the Importance of High–Fidelity Numerical Modelling of Ocean Wave Energy Converters under Controlled Conditions

Abstract. Numerical modelling tools are commonly applied during the development and optimisation of ocean wave energy converters (WECs). Models are available for the hydrodynamic wave structure interaction, as well as the WEC sub–systems, such as the power take–off (PTO) model. Based on the implemented equations, different levels of fidelity are available for the numerical models. Specifically under controlled conditions, with enhance WEC motion, it is assumed that non-linearities are more prominent, re- quiring the use of high–fidelity modelling tools. Based on two different test cases for two different WECs, this paper highlights the importance of high–fidelity numerical modelling of WECs under controlled conditions.


Introduction
The growing recognition of human induced global warming has fuelled the research and development (R&D) of novel technologies to harness renewable energy resources. Amongst these resources, marine renewable energy, and specifically ocean wave energy, shows significant potential to contribute to the global energy supply [1]. To increase the efficiency and, thereby, the economical feasibility of WECs, devices should be equipped with energy maximising control systems (EMCSs) [2]. Since the objective of EMCSs is to drive the system towards resonance with the incoming wave field, WEC motion of a controlled device is enhanced (see Figure 1), and the power conversion is increased.
During the design and optimisation of WECs, researchers and engineers rely on physical wave tank (PWT), as well as numerical wave tank (NWT) tests. Generally, by testing in a real physical environment, PWTs allow all the relevant details of the wave-structure interaction (WSI) to be captured. However, although still cheaper compared to open ocean trials, PWT experiments are associated with higher costs compared to NWT experiments [3]. The main cost drivers for PWT tests are instrumentation, construction of the prototype, test facilities, and staff. Additionally, the accuracy of PWT experiments potentially suffer from peculiarities of the test facility, such as reflections from the tank walls, friction in mechanical device components, measurement noise, and scaling effects.
Overcoming the drawbacks of high costs, measurement noise, mechanical friction, and, to a great extent, scaling effects, NWTs provide powerful tools for the analysis of WECs. Depending on the implemented equations for the solution of the WSI problem, different levels of fidelity, at different levels of computational cost, can be achieved [4]. Hydrodynamic, lower-fidelity models, such as Boundary Element Method (BEM) -based NWTs, neglecting nonlinear effects, such as viscosity, are associated with minimal computational cost, and are valuable tools for parametric studies or exhaustive-search optimisation algorithms. However, due to the required linearisation of the hydrodynamic equations, lower-fidelity models are only valid when considering linear conditions, i.e. small amplitude waves and device motions. Contrary, higher-fidelity NWTs, such as Computational Fluid Dynamics (CFD)-based numerical wave tanks (CNWTs), are able to capture all relevant hydrodynamic non-linearities by numerically solving the Navier-Stokes equations. Thus, CNWTs are valid over a wider range of test conditions, compared to lower-fidelity models.
Equally, when considering the sub-systems of a WEC device, such as the power-take off (PTO) or the mooring system, a range of numerical models with varying degree of fidelity are available and can be coupled with the hydrodynamic model [5]. Generally, the implementation of highfidelity models of the WEC sub-systems is desired when employing CNWTs, to prevent the lower-fidelity sub-system models from undermining the accuracy of the high-fidelity hydrodynamic model.
The importance of high-fidelity modelling of the WSI, as well as the WEC sub-system, specifically under controlled conditions, will be investigated in the present paper. To that end, two different case studies are considered, analysing two different WECs: (1) the Wavestar device; (2) a generic heaving point absorber (HPA) type WEC (see Figure 2). In the first case study, considering the Wavestar device, the influence of different design and evaluation frameworks for EMCSs will be investigated, employing three different EMCSs of varying aggressiveness. In the second case study, considering the HPA-type WEC, the influence of the fidelity of the hydrodynamic model and the coupled PTO model is investigated.
The remainder of the paper is organised as follows: Section 2 details the low-and high-fidelity numerical wave tanks employed in the two case studies. Furthermore Section 2 presents a description of the employed PTO models. The results of the first case study, assessing the evaluation framework of EMCSs, will be presented and discussed in Section 3.
The second case study, investigating the influence of the fidelity of the hydrodynamic and PTO model, is discussed in Section 4. Finally, conclusions are drawn in Section 5.
where x, ̇ and ̈ are the displacement, velocity and acceleration of the WEC, respectively. M is the mass of the WEC, ℎ the hydrostatic stiffness, ℱ ℯ ( ) the excitation force, ∞ the addedmass at infinite frequency, ζ(t) the radiation impulse response function (IRF), and ( ) the control law (PTO force).
The linear hydrodynamic model can be extended to include non-linear effects, such as nonlinear Froude-Krylov (FK) forces or viscous effects. Non-linear FK forces can be included using, e.g. the computationally efficient algebraic solution as presented in [7]. Viscous effects can be incorporated by using a Morison-like equation [8], where ρ is the density of water, the drag coefficient, the instantaneous cross-sectional area of the device, and ̇ the velocity of the undisturbed water particles.
CNWT. The CNWT simulations are performed using the open source CFD toolbox Open-FOAM. In OpenFOAM, the incompressible Reynolds Averaged Navier-Stokes (RANS) equations (3) and (4) are solved using the finite volume method.
Equation (3) is the continuity equation, describing the conservation of mass, and equation (4) is the momentum equation. describing the conservation of momentum. In equations (3) and (4) denotes the fluid velocity, the fluid pressure, ρ the fluid density, T the stress tensor, and external forces such as gravity or PTO forces.
To account for the two phase flow, the volume of fluid method, proposed in [9], is used, following: where α denotes the volume fraction of water, (t) is the relative velocity between the liquid and gaseous phases [10], and Φ is a specific fluid quantity, such as density. The free surface elevation is monitored by extracting the iso-surface of the volume fraction α = 0.5.
PTO. As for the hydrodynamic model, different levels of fidelity are available for the modelling of the PTO system of a WEC. One of the simplest models describes the PTO as a spring damper system, following: where is a damping coefficient and ̇( ) the linear velocity of the hydraulic PTO cylinder, is a spring stiffness, and ( ) the linear motion of the PTO. The damping and stiffness coefficient either represent the mechanical characteristics of, say, a hydraulic cylinder, or and are representing the EMCSs and are optimised for maximum energy absorption. Higher-fidelity PTO models are available, including e.g. a hydraulic transmission system and an electrical generator [11]. The mathematical model for the hydraulic cylinder may include end-stop constraints, friction losses, and compressibility and inertia effects, providing the final PTO force, following: where is the piston area, Δ the pressure difference between the different cylinder chambers, the friction force and the inertia force. For a detailed description of the individual effects, influencing the PTO force, the interested reader is referred to [11].

Case study 1: Assessment of the evaluation framework for EMCSs
In classical control applications, the mathematical models, used for the controller design, are often linearised around a desired operational point, according to the process under analysis. The controller is subsequently synthesised to drive the system towards this point and, thus, in the neighbourhood of this operational point, the linearising assumption is obeyed. The large amplitude motions, induced by a reactive WEC controller, may result in viscous drag and other non-linear hydrodynamic effects. Thus, contrary to the aforementioned classical control applications, the energy-maximising operating conditions do not comply with the linear assumption in the control design model. This contradiction between the control objective and the underlying mathematical model raises the question if the common practice of designing a controller in a linear design environment can deliver optimal reactive controllers for the application in physical, non-linear operational conditions.
In this case study, a CNWT and linear BNWT model 1 , as described in Section 2, are employed to investigate the influence of different numerical evaluation frameworks on the performance evaluation of EMCSs for the Wavestar device (see Figure 2a). The performance of the EMCSs will be evaluated by comparing the dynamics of the WEC subject to three different EMCSs: (1) momentbased energy-maximising control [12]; (2) reactive output feedback control; (3) resistive output feedback control. The three EMCSs will drive the WEC away from the linear assumption in the hydrodynamic model, dependent on the aggressiveness of the controller, with the resistive controller being the least aggressive and the moment-based control the most aggressive.
EMCSs. The main objective of a wave energy device is to harvest energy from the incoming wave field. Therefore, the optimal control objective is to maximise the absorbed energy over a time interval [t, t + T ], while respecting the physical limitations of the device/PTO on excursion ( ), velocity ̇( ), and PTO force ( ). Consequently, the optimal control objective can be formulated as The optimal control is a moment-based WEC formulation [12] which allows an efficient computation of the optimal control law u max in real-time based on the solution of the following inequality constrained quadratic program: The reader is referred to [12] for the formal definition (and corresponding proofs) of the matrices involved in the QP problem of (10).
Additionally to the moment-based controller, less aggressive EMCSs, i.e. reactive and resistive controllers, are considered herein. For the reactive control case, the PTO force follows where opt is the optimal damping coefficient and opt is the optimal spring stiffness. The optimal PTO coefficients have been determined through exhaustive search optimisation using the BNWT model. For the resistive control case, only the second term, opṫ( ), is considered. As in the case of reactive control, the optimal damping coefficient has been determined through exhaustive search optimisation using the BNWT model.
Results. For each EMCSs, simulations were performed in the BNWT and CNWT, resulting in a total of six simulations. Extracting the PTO displacement, velocity, and the PTO force, the normalised root mean squared deviation (nRMSD), as defined in equation (12), can be calculated.
BNWT denotes the results from the BNWT experiment, CNWT results from the CNWT experiment, and N is the number of samples.
For a qualitative comparison between the three different EMCSs, Figure 3 shows the PTO displacement, extracted from the BNWT experiment. In Figure 3, a clear trend can be observed. The moment-based and the reactive controller controller increases the PTO displacement, compared to the resistive controller.

Figure 3: Time traces of the PTO displacement x(t) from the BNWT experiment
The resulting nRMSD for the PTO displacement, velocity, and force are listed in Table 1. Overall smallest values of the nRMSD can be observed for the case of a resistively controlled device. For this control strategy, smallest deviations are indeed expected, since the device effectively acts as a wave follower, reducing the influence of non-linear hydrodynamic effects, such as non-linear Froude-Krylov forces or viscous drag effects.
For the cases with reactive and moment-based control, the values for the nRMSD in all PTO quantities increase. In Table 1, it can be seen that the differences between the PTO quantities for the two different controllers are relatively small. This can be attributed to the fact that both state (displacement) and input (PTO force) constraints are inactive for this particular input wave, i.e. both controllers effectively reflect the unconstrained optimal energy-absorption conditions. That said, it is expected that also the differences between the BNWT and CNWT are similar. Results. The time-average absorbed and generated power values from the numerical models (2)-(4) are compared against the values from the CNWT+W2W. Absorbed power refers to the mechanical power directly absorbed from ocean waves, while generated power refers to the final electric power output. Since the CNWT+W2W accounts for all relevant hydrodynamic non-| linearities, as well as the occurring non-linearities in the PTO drivetrain, this model is considered as the benchmark in this case study.
For a quantitative comparison, Table 2 shows that the relative deviation, , between the CNWT+W2W model and models (2)-(4) in absorbed power. Considering only the time-average, absorber power, relatively small differences ( < 8%) can be observed between models (1), (2), and (4), indicting a sufficient accuracy of the non-linear BNWT with a drag coefficient of C d = 1. Larger deviations can be observed between the linear BNWT, model (3), and the benchmark, model (1). This indicates that a linear hydrodynamic model is not able to capture all relevant hydrodynamic effects.
Considering the generated power for the comparison between the numerical models, the influence of the non-linear W2W model becomes visible. For the two numerical models including an idealised, linear, spring-damper type PTO model (i.e. model (2) and (3)), relative differences to the benchmark model (1) of up to 97.52% are calculated. For the two W2W models only a difference of < 3% can be observed, which is consistent with the findings for the absorbed power.

Conclusions
In this paper, two different case studies are presented, highlighting the importance of (consistent) highfidelity modelling of WECs under controlled conditions. From the presented results, two main conclusions can be drawn: 1. Considering aggressive EMCSs for WEC control, driving the system further away from the assumptions in the linear hydrodynamic model, high-fidelity hydrodynamic models have to be employed for the assessment of the performance of the EMCSs. Omitting high-fidelity hydrodynamic modelling in the evaluation stage of the control design can lead to an overprediction of the WEC performance. 2. The holistic performance of a WEC can only be evaluated in high-fidelity by means of a comprehensive W2W simulation platform, where both high-fidelity hydrodynamic and PTO models are coupled. Minor inaccuracies in either of these major stages of the W2W model can result in significant inaccuracy in generated power estimation.