Umesh A. Korde

Multiresonant Feedback Control of a Three-Degree-of-Freedom Wave Energy Converter

For a three-degree-of-freedom wave energy converter (heave, pitch, and surge), the equations of motion could be coupled depending on the buoy shape. This paper presents a multiresonant feedback control, in a general framework, for this type of a wave energy converter that is modeled by linear time invariant dynamic systems. The proposed control strategy finds the optimal control in the sense that it computes the control based on the complex conjugate criteria. This control strategy is relatively easy to implement since it is a feedback control in the time domain that requires only measurements of the buoy motion. Numerical tests are presented for two different buoy shapes: a sphere and a cylinder. Regular, Bretschnieder, and Ochi–Hubble waves are tested. Simulation results show that the proposed controller harvests energy in the pitch-surge-heave modes that is about three times the energy that can be harvested using a heave-only device. This multiresonant control can also be used to shift the energy harvesting between the coupled modes, which can be exploited to eliminate one of the actuators while maintaining about the same level of energy harvesting.

Velocity Control Using a Hydrodynamic Model

In this part, we examine the control problem based on the fundamentals of the hydrodynamic model. In particular, Chapter 7 uses Cummins’ equation as a starting point and develops the classic reactive control strategy leading to a velocity response that approaches the hydrodynamic velocity optimum. Chapter 7 also addresses the issue of the requirement for future wave elevation information and discusses two possible ways to utilize up-wave measurements for this purpose. One uses instantaneous up-wave measurements at a distance related to the prediction time ahead and assumes nondispersive propagation. The second accounts for dispersive propagation and uses up-wave measurements made over a duration of time and at a distance, both of which are directly related to the practical range of group velocities commonly found in wave spectra and the prediction time ahead. Chapter 8 uses a simplified hydrodynamic model, motivated by the idea that, while offering some compromise in performance, less detail in the model may offer improved robustness properties. The method proposed in Chapter 8 addresses the issue of future information by modeling the wave spectrum as a narrow-banded process, where an extended Kalman filter is used to make instantaneous measurements of wave period and amplitude. The control method can cater to device amplitude constraints, and a possible lower-level power take-off controller is presented.

Control by Optimizing a Performance Index

In this part, optimal controllers, based on the maximization of an energy performance index, are presented. The optimization problem can be solved either algebraically or numerically. Chapter 9 examines the special case of switching control, where the control force is a braking force to be switched on and off, and has proved popular in the WEC control systems literature. The approach seeks an optimum switching sequence by maximizing a performance index (absorbed energy) under a basic variational formulation, and explicitly uses the Pontryagin principle to derive that sequence. The need for information from the future is discussed, even though the control applied is nonreactive. The optimum switching sequence here represents an optimum latching sequence, where velocity is alternately locked and released to maximize the performance index. The use of a numerical optimization framework allows system amplitude, force, and velocity constraints to be explicitly considered in the optimized design, by virtue of a constrained optimization solution. In order to achieve an efficient numerical solution (considering real-time control requirements), a pseudo spectral representation is chosen for the system variables, which parameterizes the variables as a set of (Fourier) basis functions. Chapter 10 presents the basic constrained optimal control formulation for a single device (which can be multibody), while Chapter 11 examines the opportunity for coordinated control of arrays of WECs, examining both independent and global control strategies. As in Part IV, future wave elevation is required, and Chapter 12 examines wave forecasting strategies based on both autoregressive (using only historical information) and up-wave philosophies.

Bodies Oscillating in Air

Excellent discussions and examples pertaining to oscillating systems in air can be found in texts such as [58, 59]. As we have seen in Chapter 1 many wave energy conversion concepts essentially consist of moored bodies floating on the ocean surface and oscillating in response to waves. Many devices are designed (or otherwise constrained) such that motion in a particular direction dominates. We first consider such single degree of freedom devices here, as they enable us to focus just on the aspects fundamental to their dynamics and control. In calm water (absent waves, currents, or winds), such a body is in stable static equilibrium with its weight balanced by buoyancy. Any initial displacement off the equilibrium will disturb this balance, with the excess or deficit in buoyancy opposing the displacement and trying to restore equilibrium as a spring force would. Other forces would also try to oppose the bodys motion; for instance, the force due to viscosity would act to damp the motion by dissipating some of its energy. Dynamic equilibrium would then require that the inertial force of the body, determined by its mass and acceleration, should balance the sum of the restoring (spring) and damping forces on the body. Power Absorption from an Oscillatory Force In the following, we first seek to understand the need and place for control in a hypothetical situation where the mass, spring, damping combination above are in vacuum or, for our purposes, in air. We thus ignore any forces that the air itself may apply on the mass. In the next chapter we shall consider the real situation where the body performs its motion on the water surface and begin the process of understanding of how this complicates its dynamics and any attempt to control it. Because we have a mass and a spring, we have the basic elements necessary to set off oscillations. We suppose next that an arbitrary oscillatory force is now applied to the mass. To help develop a firm understanding, we proceed slowly and incrementally at first, and start by reviewing the dynamics underlying the transfer of power from this oscillatory force to our oscillator.