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In this paper, an economic Model Predictive Control (MPC) is used to investigate the effects that arise from the model mismatch between the control and the system. It is shown that the average electrical power is affected by the modelling discrepancies, but that the performance is still acceptable. A move-blocking technique is incorporated into the...

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... and system. This shows that there needs to be a balance where the minimum control horizon is found while maintaining high performance. A cylindrical wave energy converter point absorber which is restricted to move in the heave direction is assumed in this paper. The model is based on linear wave theory. The hydrodynamic model (1), as shown in Fig. 1, consists of the hydrostatic force F h (t), the radiation force F r (t), the excitation force F e (t), controlled PTO force F P T O (t) and the non-linear viscous force F v (t). Initially in this paper, F v (t) is neglected, since it adds unnecessary complexity into the individual characteristic mismatch ...

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... is clearly shown from the waveforms in Fig. 10 that as the control horizon N c decreases, the control action across the prediction horizon becomes more disjointed and diverges from the control waveform when N c = N ; this results in the deterioration of average power. However, by incorporating a move-blocking system in the control horizon, less control variables need to be ...

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... the performance of the MPC with a perfect (matched) hydrodynamic model was investigated using a full control horizon, a reduced control horizon and a move-blocking control horizon. For all tests, a 1 m monochromatic excitation wave was used; linear constraints were assumed for WEC heave and velocity and for the PTO force. Fig. 11 shows the average electrical powers absorbed from the system when the different types of control horizons were used. When a standard reduced horizon is used with N c = 30, the average power has drastically diminished when compared to the average power results found when using a full control horizon with N c = N . However, when a ...

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... for N c < 10 show much improved power levels when compared with the standard control horizon reduction with a much greater control horizon N c . To show the serious advantages of using a move-blocking control horizon, the average optimisation solution times for the unconstrained and linearly constrained problems were recorded. As shown in Fig. 12, the difference between the solution times for constrained and unconstrained MPC is very clear. However, for both unconstrained and constrained cases Furthermore, this figure also shows the corresponding power ratio obtained for the various control horizons. The power ratio here is the ratio of the average power extracted using a ...

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... test the robustness of the system, the move-blocking technique was used on the mismatched hydrostatic system where a −20% hydrostatic stiffness coefficient was used; 1 m high monochromatic waves are used in this analysis. The resulting extracted average electrical power from the mismatched system are shown in Fig. 13. From Fig. 13 it is shown that to some degree, the inclusion of the move- blocking technique does not cause any significant difference in performance. It is only when the control horizon has been decreased to the point (N c = 10) where ∆u q (k + 2) is forced to become the same as ∆u q (k+1) that the average power starts to significantly deviate from ...

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... + 2) is forced to become the same as ∆u q (k+1) that the average power starts to significantly deviate from the average power extracted when using a full control horizon. On the other hand, when move-blocking results in a ∆u q (k +2) and ∆u q (k +1) which are equal to each other, the calculated PTO force ∆u q (k) becomes damped; this is shown in Fig. 14. This damped PTO force control action can lead to problems in satisfying the hard ...

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... Fig. 15 it is shown for a 1 m high monochromatic wave with a frequency of 0.419 rad.s −1 that the heave of the WEC and LPMG stay within the heave limitation of ±3.5 m when a control horizon of N c = 100 is used. However, when a control horizon of N c = 10 is implemented, the PTO force (Fig. 14) becomes more damped and the heave of the system ...

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... Fig. 15 it is shown for a 1 m high monochromatic wave with a frequency of 0.419 rad.s −1 that the heave of the WEC and LPMG stay within the heave limitation of ±3.5 m when a control horizon of N c = 100 is used. However, when a control horizon of N c = 10 is implemented, the PTO force (Fig. 14) becomes more damped and the heave of the system starts to exceed the heave limitation, which Time (s) could cause damage. If the control horizon N c is too low, then there is a higher chance of ∆u q (k + 2) = ∆u q (k + 1) which would degrade the systems performance. However, as stated in section. IV-A, if the control horizon N c is ...

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## Citations

... WEC control systems aim to keep the buoy velocity 410 in phase with the wave force. Modelling uncertainties and input/output disturbances may introduce errors in the reference and tracking loops leading to a significant degradation of the converter efficiency (O'Sullivan and Lightbody, 2017). For the controller shown in Figure 4, the main 415 sources of modelling errors and their effect on the buoy performance are listed in Table 2. Firstly, the current value of the excitation force, ˜ F exc (t), is estimated using the Extended Kalman filter which is based on the available model of the WEC. ...

Wave energy converters (WECs) capable of extracting power in multiple degrees of freedom require special attention from control engineers as the control problem becomes multivariable involving highly coupled dynamics of the plant. Taking the three-tether submerged buoy as an example of a multi-degree-of-freedom WEC, this paper presents three main steps that should be taken during control system development. Firstly, an understanding of the system dynamics, its rigid body modes of vibration and input/output controllability is built using the singular value decomposition approach. Then, a causal close-to-optimal controller developed for the single-tether heaving WEC is extended to the multivariable control problem, demonstrating a significant increase in the power output as compared to the simple spring-damper approach. At the final stage, technical requirements imposed by this controller on the power take-off (PTO) machinery are investigated showing that, in order to achieve a 15%-improvement in power absorption compared to a quasi-standard spring-damper control, the amount of reactive power should be increased by 50%, forcing one PTO unit to operate as an actuator all the time.

... Moreover, it can be seen that the steepness of the curve decreases as C d increases, confirming that overestimation of the drag coefficient is preferred. Such a result (better to overestimate than underestimate C d ) is consistent with sensitivity studies performed in [187,188], for example. Fig. 7.3, and summarized in Table 7.6. ...

One of the major challenges facing modern industrialized countries is the provision of energy: traditional sources, mainly based on fossil fuels, are not only growing scarcer and more expensive, but are also irremediably damaging the environment. Renewable and sustainable energy sources are attractive alternatives that can substantially diversify the energy mix, cut down pollution, and reduce the human footprint on the environment. Ocean energy, including energy generated from the motion of wave, is a tremendous untapped energy resource that could make a decisive contribution to the future supply of clean energy. However, numerous obstacles must be overcome for ocean energy to reach economic viability and compete with other energy sources. Energy can be generated from ocean waves by wave energy converters (WECs). The amount of energy extracted from ocean waves, and therefore the profitability of the extraction, can be increased by optimizing the geometry and the control strategy of the wave energy converter, both of which require mathematical hydrodynamic models that are able to correctly describe the WEC-fluid interaction. On the one hand, the accuracy and representativeness of such models have a major influence on the effectiveness of the WEC design. On the other hand, the computational time required by a model limits its applicability, since many iterations or real-time calculations may be required. Critically, computational time and accuracy are often mutually contrasting features of a mathematical model, so an appropriate compromise should be defined in accordance with the purpose of the model, the device type, and the operational conditions. Linear models, often chosen due to their computational convenience, are likely to be imprecise when a control strategy is implemented in a WEC: under controlled conditions, the motion of the device is exaggerated in order to maximize power absorption, which invalidates the assumption of linearity. The inclusion of nonlinearities in a model is likely to improve the model's accuracy, but increases the computational burden. Therefore, the objective is to define a parsimonious model, in which only relevant nonlinearities are modelled in order to obtain an appropriate compromise between accuracy and computational time. In addition to presenting a wider discussion of nonlinear hydrodynamic modelling for WECs, this thesis contributes the development of a computationally efficient nonlinear hydrodynamic model for axisymmetric WEC devices, from one to six degrees of freedom, based on a novel approach to the nonlinear computation of static and dynamic Froude-Krylov forces.

... An economic MPC controller typically uses a control horizon of the same size as the prediction horizon, N c = N . However, the computational burden can be lowered by utilising move-blocking, while maintaining the fidelity of an MPC with a full control horizon (N c = N ) [40].The move-blocking control horizon spreads the N c control variables appropriately across the prediction horizon N , where the control variables are concentrated over the early stages of the prediction horizon. The early control variables allow faster control action and constraint feasibility while the remainder of the control variables are used to estimate the power over the prediction horizon. ...

... In this section an upper instantaneous power limit P M AX is incorporated into the centralised NMPC; this is compared with the results obtained when applying a local power constraint P M AX /N w to each individual device. Also incorporated in both the global and local power constrained NMPC is a move-blocking technique [40], which reduces the computational effort of the QP algorithm while maintaining a similar performance to an economic MPC with a full horizon; in this case the selected move-blocking horizon is N c = 40. As shown in section IV-B, when an irregular excitation wave with sea state 3 (T p = 9.6 s and H s = 3 m) was implemented on the two arrays, the difference between the centralised control system and the decentralised control system was insignificant. ...

This work focuses on an array of point absorbers, with linear permanent magnet generators (LPMG) connected to the grid via back to back voltage source converters, controlled using economic model predictive control (MPC) that produces optimal electrical power generation. The main contribution of this paper is the comparison of the performance provided by using either a centralised or decentralised MPC scheme. In this study, it is shown how the inclusion of viscosity and system constraints limits the benefits to be obtained by the use of a centralised control scheme. Indeed, it was shown that a decentralised MPC scheme was sufficient for the provision of close to optimum electrical power extraction from the array when there was a reasonable separation between WEC devices. It was shown that the introduction of power constraints, either locally at each device or globally for the entire array, improved the quality of the power exported to the grid. Importantly, it was shown that from the viewpoint of power quality, that global predictive control of the wave energy array offered significant benefits over local decentralised control in increasing the average to peak power ratio of power exported to the grid.

Some active controls were proposed to maximize the power production for the point absorber type wave energy converter (WEC) with a linear generator. However, proposed conventional controls have some problems, e.g. constraint conditions cannot be considered. Model predictive control (MPC) has attracted attention as an alternative method to solve these problems. In this paper, the performance of the MPC for the WEC with the linear generator was evaluated by experiments and numerical simulations. In the experiments in regular waves, the MPC performance was comparable to the conventional control of the power production amount reasonably. Furthermore, the MPC was more effective than the conventional control in irregular waves which has broadband spectral distribution. The numerical simulations in the irregular waves analyzed three characteristics of the proposed MPC. Firstly, it was investigated how much the power production decreased due to wave predictive error. Secondly, the best predictive horizon of the proposed MPC was investigated. The simulation result showed the predictive horizon of the MPC needs to be longer than the averaged wave period in irregular waves. Finally, performance of the MPC was compared with that of the conventional control under displacement constraints. It was confirmed the MPC presented in this paper was more effective than the conventional control under the displacement constraints.