Optimal control of wave energy converters

Wave Energy Converters (WECs) are devices designed to absorb energy from ocean waves. The particular type of Wave Energy Converter (WEC) considered in this thesis is an oscillating body; energy conversion is carried out by means of a structure immersed in water which oscillates under forces exerted by waves. This thesis addresses the control of oscillating body WECs and the objective of the control system is to optimise the motion of the devices that maximises the energy absorption. In particular, this thesis presents the formulation of the optimal control problem for WECs in the framework of direct transcription methods, known as spectral and pseudospectral optimal control. Direct transcription methods transform continuous time optimal control problems into Non Linear Programming (NLP) problems, for which the literature (and the market) offer a large number of standard algorithms (and software packages). It is shown, in this thesis, that direct transcription gives the possibility of formulating complex control problems where realistic scenarios can be taken into account, such as physical limitations and nonlinearities in the behaviour of the devices. Additionally, by means of spectral and pseudospectral methods, it is possible to find an approximation of the optimal solution directly from sampled frequency and impulse response models of the radiation forces, obviating the need for finite order approximate models. By implementing a spectral method, convexity of the NLP problem, associated with the optimal control problem for a single body WEC described by a linear model, is demonstrated analytically. The solution to a nonlinear optimal control problem is approximated by means of pseudospectral optimal control. In the nonlinear case, simulation results show a significant difference in the optimal behaviour of the device, both in the motion and in the energy absorption, when the quadratic term describing the viscous forces are dominant, compared to the linear case. This thesis also considers the comparison of two control strategies for arrays of WECs. A Global Control strategy computes the optimal motion by taking into account the complete model of the array and it provides the global optimum for the absorbed energy. In contrast, an Independent Control strategy implements a control system on each device which is independent from all the other devices. The final part of the thesis illustrates an approach for the study of the effects of constraints on the total absorbed energy. The procedure allows the feasibility of the constrained energy maximisation problem to be studied, and it provides an intuitive framework for the design of WECs relating to the power take-off operating envelope, thanks to the geometrical interpretation of the functions describing both the total absorbed energy and the constraints

Optimal control of wave energy converters

Wave Energy Converters (WECs) are devices designed to absorb energy from ocean waves. The particular type of Wave Energy Converter (WEC) considered in this thesis is an oscillating body; energy conversion is carried out by means of a structure immersed in water which oscillates under forces exerted by waves. This thesis addresses the control of oscillating body WECs and the objective of the control system is to optimise the motion of the devices that maximises the energy absorption. In particular, this thesis presents the formulation of the optimal control problem for WECs in the framework of direct transcription methods, known as spectral and pseudospectral optimal control. Direct transcription methods transform continuous time optimal control problems into Non Linear Programming (NLP) problems, for which the literature (and the market) offer a large number of standard algorithms (and software packages). It is shown, in this thesis, that direct transcription gives the possibility of formulating complex control problems where realistic scenarios can be taken into account, such as physical limitations and nonlinearities in the behaviour of the devices. Additionally, by means of spectral and pseudospectral methods, it is possible to find an approximation of the optimal solution directly from sampled frequency and impulse response models of the radiation forces, obviating the need for finite order approximate models. By implementing a spectral method, convexity of the NLP problem, associated with the optimal control problem for a single body WEC described by a linear model, is demonstrated analytically. The solution to a nonlinear optimal control problem is approximated by means of pseudospectral optimal control. In the nonlinear case, simulation results show a significant difference in the optimal behaviour of the device, both in the motion and in the energy absorption, when the quadratic term describing the viscous forces are dominant, compared to the linear case. This thesis also considers the comparison of two control strategies for arrays of WECs. A Global Control strategy computes the optimal motion by taking into account the complete model of the array and it provides the global optimum for the absorbed energy. In contrast, an Independent Control strategy implements a control system on each device which is independent from all the other devices. The final part of the thesis illustrates an approach for the study of the effects of constraints on the total absorbed energy. The procedure allows the feasibility of the constrained energy maximisation problem to be studied, and it provides an intuitive framework for the design of WECs relating to the power take-off operating envelope, thanks to the geometrical interpretation of the functions describing both the total absorbed energy and the constraints

Nonlinear optimal wave energy converter control with application to a ap-type device

Wave energy converters (WECs) require active control to maximise energy capture over a wide range of sea conditions, which is generally achieved by making the device resonate. The exaggerated device motion arising at resonance, however, may result in nonlinear effects that are ignored by the linear models that are typically employed. In particular, nonlinear viscous forces are significant for particular device types, such as hinged aps, which we take as a case study in this paper. The paper develops a general nonlinear WEC control methodology based on pseudospectral methods. The continuous time energy maximization problem is fully discretised (both state and control), and the optimal solution is obtained by solving the resulting finite dimensional optimization problem. By way of example, the nonlinear viscous damping for a hinged ap WEC is incorporate into the control model. It is shown that the ratio of energy captured to energy dissipated is significantly increased with the nonlinear controller, compared to the linear case

Control-Influenced Layout Optimization of Arrays of Wave Energy Converters

In this paper we compare the optimal configurations for an array of WECs given two control schemes, a real-time global control and a passive sea-state based tuning scheme. In a particular wave climate and array orientation with its axis normal to the prevailing wave direction, closely-spaced symmetrical arrays of 2, 3, 4, 5, and 6 cylinders of different radiative properties are simulated for varying inter-device separation distances. For each device and control type, we focus on the factors that influence the optimal layout, including number of devices, separating distance and angular spreading. The average annual power output is calculated for each optimal configuration

Impedance matching controller for an inductively coupled plasma chamber: L-type Matching Network Automatic Controller

Plasma processing is used in a variety of industrial systems, including semiconductor manufacture (deposition and etching) and accurate control of the impedance matching network is vital if repeatable quality is to be achieved at the manufacturing process output. Typically, impedance matching networks employ series (tune) and parallel (load) capacitors to drive the reflection coefficient on the load side of the network to zero. The reflection coefficient is normally represented by real and imaginary parts, giving two variables to be controlled using the load and tune capacitors. The resulting problem is therefore a nonlinear, multivariable control problem. Current industrial impedance matching units employ simple single-loop proportional controllers, which take no account of interaction between individual channels and, in many cases, may fail to tune altogether, if the starting point is far away from the matching point. A hierarchical feedback controller is developed which, at the upper level, performs a single-loop tuning, but with the important addition of a variable sign feedback gain. When convergence to a region in the neighbourhood of the matching point is achieved, a dual single-loop controller takes over, which gives fine tuning of the matching network

Control, forecasting and optimisation for wave energy conversion

This paper presents an overview of the motivation, background to and state-of- the-art in energy maximising control of wave energy devices. The underpinning mathematical modelling is described and the control fundamentals established. Two example control schemes are presented, along with some algorithms for wave forecasting, which can be a necessary requirement, due to the non-causal nature of some optimal control strategies. One of the control schemes is extended to show how cooperative control of devices in a wave farm can be beneficial. The paper also includes perspectives on the interaction between control and the broader objectives of optimal wave energy device geometry and full techno-economic optimisation of wave energy converters

State space model of a hydraulic power take off unit for wave energy conversion employing bondgraphs

In this work, the modeling of a Power Take- Off (PTO) unit for a point absorber wave energy converter is described. The PTO influences the energy conversion performance by its efficiency and by the damping force exerted, which affects the motion of the body. The state space model presented gives a description of the damping force and of the internal dynamics of the PTO. The aim of this work is to develop a model for the PTO as a part of a complete wave-to-wire model of a wave energy converter as in Figure 1, used for the design control techniques. Figure 1: Wave-to-wire model structure A bondgraph is employed to model the physical system that provides transparent and methodical means of formulating state space equations and of visualizing energy transfer throughout the system. Bondgraphs have already been shown to be a very useful tool for the modeling of PTO for wave energy converters (2). The dynamic of the mathematical model is then analyzed respect to the variation of parameters; in particular, the non-linear system obtained is linearized and its eigenvalues are calculated as function of the accumulator size and pre-charge pressur

State space model of a hydraulic power take off unit for wave energy conversion employing bondgraphs

In this work, the modeling of a Power Take- Off (PTO) unit for a point absorber wave energy converter is described. The PTO influences the energy conversion performance by its efficiency and by the damping force exerted, which affects the motion of the body. The state space model presented gives a description of the damping force and of the internal dynamics of the PTO. The aim of this work is to develop a model for the PTO as a part of a complete wave-to-wire model of a wave energy converter as in Figure 1, used for the design control techniques. Figure 1: Wave-to-wire model structure A bondgraph is employed to model the physical system that provides transparent and methodical means of formulating state space equations and of visualizing energy transfer throughout the system. Bondgraphs have already been shown to be a very useful tool for the modeling of PTO for wave energy converters (2). The dynamic of the mathematical model is then analyzed respect to the variation of parameters; in particular, the non-linear system obtained is linearized and its eigenvalues are calculated as function of the accumulator size and pre-charge pressur

A control system for a self-reacting point absorber wave energy converter subject to constraints

The problem of the maximization of the energy produced by a self reacting point absorber subject to motion restriction is addressed. The main objective is to design a control system suitable for real-time implementation. The method presented for the solution of the optimization problem is based on the approximation of the motion of the device and of the force exerted by the power take off unit by means of a linear combination of basis functions. The result is that the optimal control problem is reformulated as a non linear program where the properties of the cost function and of the constraint are affected by the choice of the basis functions. An example is described where the motion and the force are approximated using Fourier series; an optimization algorithm for the solution of the non linear program is also presented. The control system is implemented and simulated using a real sea profile measured by a waverider buoy

On the solution of multi-body wave energy converter motions using pseudo-spectral methods

Multi-body wave energy converters are composed of several bodies interconnected by joints. Two different formulations are adopted to describe the dynamics of multi-body systems: the Differential and Algebraic Equations (DAEs) formulation and the Ordinary Differential Equations (ODEs) formulation. While the number of variables required for the description of the dynamics of a multi-body system is greater in the DAE formulation than in the ODE formulation, the ODE formulation involves an extra computational effort in order to describe the dynamics of the system with a smaller number of variables. In this paper, pseudo-spectral methods are applied in order to solve the dynamics of multi-body wave energy converters using both DAE and ODE formulations. Apart from providing a solution to the dynamics of multi-body systems, pseudo-spectral methods provide an accurate and efficient formulation for the control of multi-body wave energy converters. As an application example, this paper focuses on the dynamic modeling of a twobody hinge-barge device. Wave-tank tests were carried out on the device in order to validate the DAE and ODE formulation against experimental data. The comparison between pseudospectral methods and a method based on the integration of the equations of motion, e.g. the Runge-Kutta method, showed that pseudo-spectral methods are computationally more stable and they require a less computational effort for short time steps