The Scientific World Journal

Volume 2015 (2015), Article ID 980613, 14 pages

http://dx.doi.org/10.1155/2015/980613

## Stochastic Control of Inertial Sea Wave Energy Converter

^{1}Department of Mechanics and Aerospace Engineering, Polytechnic University of Turin, Corso Duca degli Abruzzi 24, 10129 Turin, Italy^{2}Environmental Hydraulics Institute “IH Cantabria”, C/Isabel Torres 15, Santander, 39011 Cantabria, Spain

Received 27 June 2014; Revised 19 August 2014; Accepted 21 August 2014

Academic Editor: Linni Jian

Copyright © 2015 Mattia Raffero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The ISWEC (inertial sea wave energy converter) is presented, its control problems are stated, and an optimal control strategy is introduced. As the aim of the device is energy conversion, the mean absorbed power by ISWEC is calculated for a plane 2D irregular sea state. The response of the WEC (wave energy converter) is driven by the sea-surface elevation, which is modeled by a stationary and homogeneous zero mean Gaussian stochastic process. System equations are linearized thus simplifying the numerical model of the device. The resulting response is obtained as the output of the coupled mechanic-hydrodynamic model of the device. A stochastic suboptimal controller, derived from optimal control theory, is defined and applied to ISWEC. Results of this approach have been compared with the ones obtained with a linear spring-damper controller, highlighting the capability to obtain a higher value of mean extracted power despite higher power peaks.

#### 1. Introduction

Oceans represent a wide, distributed reservoir of energy and, in them, waves are by far the most conspicuous form of energy. The global power potential represented by waves in open oceans, where energy is not dissipated due to seabed friction or wave breaking, has been estimated to be in the order of 10 TW (1 terawatt = 10^{12} W), a quantity comparable with the present world power demand [1].

For more than two centuries, many devices have been proposed for harvesting such a huge power source: the earliest patent was filed in 1799 in France [2]. Traditionally, the father of modern wave energy exploitation is considered Masuda, who started his studies in the 1940s and developed a navigation buoy powered by an air turbine which has been later commercialized [3]. Since then a lot of devices have been conceived and developed while a few of them arrived to the precommercial stage [4].

Among these machines, a considerable role is played by gyroscopic converters. Gyroscopes have been widely used on ships with the task of roll stabilization [5, 6]. The first use of gyroscopes for wave energy extraction is due to Salter, who invented the Duck device at the University of Edinburgh in the 1970s [7, 8]; the ISWEC belongs to the last generation of this kind of energy converters. Many problems have still to be solved in order to develop a reliable and economically sustainable wave energy converter (WEC). A proof of that is given by the little number of surviving WEC concepts [9, 10]. The main issue is the “reaction problem”: in order to generate an action on the power take-off (PTO: the component aimed at the power conversion, e.g., the electric generator) to generate energy, a reaction is needed and has to be given by either the seabed, the water, inertia, or other structures [11]. Moreover, sea waves involve low-frequency and alternating high forces, making it necessary to use strong structures and heavy conversion systems and therefore increasing the technology costs. Other problems to be faced are related to corrosion of components in contact with the sea water, possible leakage of oil (if hydraulic conversion systems are used), survivability in case of extreme events, maintenance, and environmental and visual impacts [12–15].

In this paper, one of the most important issues for the power optimization of a WEC is faced: the control problem. Developing a good control scheme is challenging and many solutions have been proposed in the recent years [16]. In Section 2, the main existing WECs control algorithms are described. The reviewed algorithms are as follows: the linear proportional-derivative (PD) controller, the latching and declutching controller, the optimal controller, and finally the stochastic suboptimal controller. Afterwards the ISWEC is presented and the equations describing its working principle are discussed. Given the physical characteristics of the full scale prototype, which will be installed in 2014 in real sea, a performance analysis is carried out, comparing the results obtained with the PD controller and the stochastic control algorithm, for some representative wave conditions registered at the installation site. Moreover, the effect of the maximum PTO torque constraint is analyzed in order to take into account the real machine limits.

#### 2. WECs Control System Outlook

In this section, a review of the existing control algorithms for wave energy converters is given, so that the reader can have an overview of the state of the art in this field. In most cases, when analyzing the power extraction capabilities of a WEC, a one degree of freedom system is analyzed. As described in Section 3 of the paper, in the simplest case the hydrodynamic model of the device may be approximated by a 2nd order linear differential equation whose coefficients are frequency dependent. In the following considerations such a simple model may be a good reference for a reader that does not have a deep knowledge of this field.

Often, the first step is to develop a control strategy able to maximize the power output under plane (2D problem) monochromatic waves. Of course this means that the wave profile is composed of a single frequency contribution and this is not what happens in real sea. Afterwards, the case of plane polychromatic wave is analyzed generating a wave time series based on the spectrum of a specific sea state or using acquired wave data. In the most recent studies a 3D sea state is analyzed taking into account wave contributions coming from different directions.

Many control strategies have been proposed with varying levels of complexity. The main ones are described here.

##### 2.1. PD Controller

One of the simplest ways to control a WEC is to apply on the floater an action proportional to its velocity. This kind of controller can be called “proportional controller (P)” and the ratio of force to velocity is the damping coefficient. In this case, the power output is related to the square of the wave height; moreover, if the wave is monochromatic and its frequency matches the natural frequency of the device, the velocity and the force are in phase and the power absorbed by the WEC is maximum [16]. The natural frequency of a floating body is dependent on its physical features and could be varied acting on its mass, for example, in order to match the incident wave frequency, thus maximizing its response amplitude. Another way to obtain such a result, without acting on the physical quantities of the device, is to use a reactive controller. This kind of controller can also be called “proportional-derivative controller (PD)” since the torque acting on the floater is composed of two contributions: the first one, proportional to the speed such as in the P controller, and the second one, proportional to the displacement of the body (with respect to the hydrostatic equilibrium condition). The ratio between the last force term and the displacement is the stiffness coefficient. As shown in Section 4.1 in this case it is possible to tune the response of the device in order to make the device resonant with the incoming wave [17]. A problem often arises with this kind of controller: the PTO can provide an action up to a maximum value, thus limiting the capacity of the system to adapt itself to the incoming wave. Moreover, the PD controller implies reactive power thus increasing the power losses due to the action generated by the PTO on the floater. After these considerations, it is clear that the floater has to be designed properly in order to reduce the control reactive component for most of the incoming waves.

##### 2.2. Latching/Declutching Controller

The latching control technique has been firstly proposed for a heaving body, independently, by Falnes and Budal [18], French [19], and Guenther et al. [20]. This strategy is particularly suitable for waves longer than the WEC natural period; it basically consists in locking the floating body when its velocity approaches the zero value, by means of a clamping mechanism, and then releasing it at some point so that its velocity will be at its highest point simultaneously with the wave force; at this point the PTO force is set to its maximum value. The action on the system can thus be regarded as binary; that is, either the body is locked, or it is moving under maximum PTO action—thus resulting in a highly nonlinear control force. The declutching controller is similar to the previous one, but it is applied for waves shorter than the WEC natural period [21]. Different from before, the floater is normally free to move and when its velocity reaches some desired value, the maximum PTO force is applied.

The use of genetic algorithms indicated that, if applicable, the latching and declutching control is among the best control techniques for a wave energy converter; see Nolan et al. [22]. A drawback of these strategies is that they need some kind of prediction of the incoming wave force, in order to actuate the device at the right time (autoregressive models and Kalman filter have been widely used in this context); however, as an advantage with respect to the previously mentioned “PD controller,” any reactive power flow is eliminated from the power take-off. The result is a suboptimal control strategy that is best suitable using hydraulic power take-off systems. Experimental tests have been carried out during time, including wave prediction, which proved the reasonable goodness of these control strategies especially if compared to applying linear damping; see Budal et al. [23], Hals et al. [24], Falnes and Bjarte-Larsson [25], and Lopes et al. [26].

However, these considerations apply to devices for which the control force is directly applied on the floater main degree of freedom, so that this could be locked or released at the desired time instant. The wave energy converter considered in this paper is not suitable for the implementation of this strategy, since in such a device it is not possible to lock/release the relative motion between floater and gyro at a desired time instant.

##### 2.3. Optimal Controller

Optimal control theory, as described in [27, 28], has already been applied on a wave energy converter model by Nielsen et al. [29]. The objective of this control strategy is to maximize the power transfer from waves to the floater in a wide range of sea states.

Here, the idea is to make the controller compensate for the dynamics of the floater and then damp its oscillation, so that its motion is in phase with the wave excitation force and thus the power flow is unidirectional, from the waves to the WEC. In this controller, an infinite time horizon is needed thus resulting in a noncausal control law. In order to overcome such noncausality, an approximation is introduced. The convolution integral is split into two parts: the causal part remains as it is, whilst the noncausal part is replaced by a damping term, whose value is obtained by means of a stochastic analysis of the wave-structure interaction aimed at maximizing the expected value of the power output. A more detailed explanation of this approach can be found in Section 4.3 of this paper after the hydrodynamic model description.

#### 3. The ISWEC

In this section the ISWEC device is introduced. After a brief description of the device, the hydrodynamic model of the floater and the mechanical model of the gyroscope are described. Finally, the features of the ISWEC first full scale prototype, analyzed in this paper and to be deployed in autumn 2014, are reported.

##### 3.1. Description of the System

ISWEC (inertial sea wave energy converter) is a device designed to exploit wave energy through the gyroscopic effect of a flywheel [30–33]. A lot of studies and experimental tests have been carried out on this device proving the concept feasibility [34, 35] and estimating its annual energy production [36].

Figure 1 shows the four main components of the gyroscopic system: the floater, the flywheel, the gyro structure, and the PTO. To describe the system dynamics, two reference frames have to be introduced: a hull-fixed coordinate system , , and a gyroscope structure-fixed coordinate system , , . Both have their origins coincident with the centre of gravity of the system. The -axis is oriented towards the bow and coincides with the sea wave direction. The hull rotates about the -axis with the induced pitching motion due to the wave-floater-gyro interaction. Due to the angular momentum conservation of the flywheel, the combination of the pitch speed with the flywheel speed about the -axis generates a gyroscopic torque around the -axis, which can be exploited by the PTO to generate electrical power. The device involves two main phenomena: the hull hydrodynamics and gyroscope mechanics. There is a strong coupling between them due to torques and energy interactions as shown in the following paragraphs.