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176 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 3, NO. 1, JANUARY 2012 A Study of the Prediction Requirements in Real-Time Control of Wave Energy Converters Francesco Fusco and John V. Ringwood Abstract—It is widely acknowledged that real-time control of wave energy converters (WECs) can beneﬁt from prediction of the excitation force. The prediction requirements (how far ahead into the future do we need to predict?) and the achievable predictions (how far ahead can we predict?) are quantiﬁed when unconstrained reactive control is implemented. The fundamental properties of the ﬂoating system that inﬂuence the length of the required forecasting horizon, as well as the achievable prediction, are characterized. The possibility of manipulating the control, based on prior knowledge of the wave spectral distribution, is also proposed for the reduction of the prediction requirements, such that they are within the range of predictability offered by simple stochastic predictors. The proposed methodology is validated on real wave data and heaving buoys with different geometries. Index Terms—Optimal control, predictive control, wave energy, wave forecasting. NOMENCLATURE Fourier transform pair. Heaving position and velocity of ﬂoating body. Wave excitation force. Power takeoff force (control input). Intrinsic impedance of ﬂoating system. Impedance of power takeoff. Radiation resistance of ﬂoating system. Bandwidth of . Transfer function between excitation force and optimal velocity. Same as is removed. after singularity for Manipulation of for reduced causality. Same as is removed. after singularity for Lowest frequency pole of or . Manuscript received June 14, 2011; revised August 19, 2011; accepted September 24, 2011. Date of current version December 16, 2011. This work was supported by the Irish Research Council of Science, Engineering and Technologies (IRCSET) under the Embark Initiative. The authors are with the Centre for Ocean Energy Research, National University of Ireland Maynooth, Maynooth, County Kildare, Ireland (e-mail: francesco.fusco@eeng.nuim.ie; web: www.eeng.nuim.ie/coer). Color versions of one or more of the ﬁgures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identiﬁer 10.1109/TSTE.2011.2170226 RCW Main time-constant of or . Future time-horizon considered for implementation of noncausal . Relative capture width. I. INTRODUCTION O CEAN wave power can make an important contribution to the development of a more sustainable, and renewable, energy strategy in different countries worldwide [1]–[3]. In order to raise the economic performance of wave energy converters (WECs), still far from being competitive, a large scope exists for the improvement of their capacity factor through more intelligent control systems. In particular, the efﬁciency of WECs, of the oscillating type, can be signiﬁcantly increased through an automatic control that tunes its oscillations to the incident wave elevation, in such a way to improve the power transfer from the ocean to the system. Real-time control of WECs is based on the optimization of the system’s motion for maximum wave energy absorption. The unconstrained optimal solution, namely reactive control or complex-conjugate control, speciﬁes the conditions under which the system is always in resonance with the wave force but has some practical limitations [4], [5]. Alternative suboptimal control solutions, formulated as constrained optimization problems, have also been proposed, where the limitation imposed by the physics of the system (e.g., amplitude of motion or velocity, applicable forces) are also taken into account. In particular, some of these alternatives include latching [6], [7], where the oscillation in the system is delayed so to be in phase with the excitation from the waves, and model predictive control (MPC) [8]–[10], which handles the use of constraints. It is acknowledged that the effectiveness of the different realtime control strategies depends on, among others, the prediction of the future wave elevation or wave excitation force acting on the system [5], [6], [8], [11]–[14]. Note that simpler suboptimal, but at the same time effective, causal control solutions, that do not require predictions, have also been proposed in [15] and [16]. The problem of short-term wave forecasting has been studied by a number of researchers. Purely stochastic time series solutions have been proposed in [17], [18], and [13], where the future wave elevation (or force/motion) is extrapolated from a combination of its past values. Particularly in [18] it is shown, on real wave measurements, how autoregressive (AR) models allow the achievement of accurate forecasts for more than 1 wave period into the future. Alternative deterministic approaches have been also investigated, where the incident wave (or force/motion) is predicted from distant measurements [12], [19]–[22]. The latter solutions can produce more accurate 1949-3029/$26.00 © 2011 IEEE FUSCO AND RINGWOOD: STUDY OF THE PREDICTION REQUIREMENTS IN REAL-TIME CONTROL OF WECs 177 predictions for longer into the future, at the expense of more instrumentation and higher algorithm complexity. In [22], it is claimed that prediction, some 60 seconds in advance, of ship and platform motions, due to the incident wave elevation, is realized. The prerequisites for the implementation of the aforementioned control solutions, in terms of prediction, were never formally analyzed and quantiﬁed. In addition, it is of fundamental importance to understand how long into the future predictions are required for, and if there is any particular property of the WEC that may have an inﬂuence on this time horizon. Not only is such an analysis important to judge the adequateness of the forecasting solutions, but it is also important at the design stage of a WEC, when controllability may need to be considered. Throughout the paper, the focus is put on a WEC consisting of a single ﬂoating body constrained to move in one degree of freedom. While this assumption does not limit the generality of the results, it allows for a clearer understanding of the fundamental relations between the system and the prediction requirements. The quantitative analysis is conﬁned to unconstrained optimal control, but other control possibilities are also qualitatively discussed. After the ﬂoating system and the control solutions are introduced in Section II, the methodology proposed for the study of the prediction requirements is presented in Section III. Results obtained for some ideal systems over a variety of sea states are then discussed in Section IV. Section V, ﬁnally, points out the main conclusive remarks. A. The System II. CONTROL SYSTEM The WEC considered in this study consists of a generic ﬂoating body, oscillating in heave. The relative motion with respect to the sea bottom is converted into useful electricity by a power takeoff (PTO) mechanism, the detail of which is left unspeciﬁed at this stage. The motion of such a system, illustrated in Fig. 1, is described as follows: (1) where is the mass of the body, and its velocity and position, is the kernel function modeling the radiation damping, a constant modeling the friction, and the buoy- ancy coefﬁcient. The external forces acting on the system are the excitation force due to the incident waves, and a con- trollable load force produced by the PTO. Note that (1) is valid under the assumption of zero initial conditions, that is . Linearity is assumed for all the hydrodynamic forces, so that the model of the WEC can alternatively be expressed in the fre- quency domain (2) where all the properties of the system are incorporated in the intrinsic mechanical impedance deﬁned as Fig. 1. Floating cylinder oscillating in heave. In (3), the Fourier transform of the radiation kernel is expressed as , in terms of radiation resis- tance and added mass . Note that such a Fourier transform is only valid in a generalized sense, as does not, in general, vanish in the limit [4]. The excitation force is the effect that the incident wave elevation has on the system, determined by the noncausal excitation transfer function [4] (4) Note that in (4), wave elevation represents the Fourier transform of the . B. Unconstrained Optimal Control No speciﬁcations about the PTO mechanism which actually provides the load force is made at this stage. The only assumption is that it can produce any force calculated by a control law, assumed to be of the form (5) where the controller can be thought of as the impedance realized by the PTO. Without considering any constraints on the achievable forces (we did not make any assumptions about the PTO), and without any hypothesis about the disturbance , the only criterion for the design of the controller is that it allows maximum energy transfer from the excitation (disturbance) to the load. The average power , absorbed at the load over the time , is the time integral of the product of the load force and the system velocity (6) Such an expression for the average power can more conveniently be expressed in the frequency domain, by making use of Parseval’s theorem [4] (7) (3) The notation indicates the complex-conjugate operation. Note that Parseval’s theorem can only be applied if the force 178 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 3, NO. 1, JANUARY 2012 Fig. 2. General structure for a real-time controller of a WEC. and velocity signals are supposed to be zero outside the time in- terval . The integrand of (7) is proportional to the average power absorbed at each frequency (8) Using the dynamic equation in (2), and considering the control law (5), the velocity and the load force can be written as functions of the external disturbance only (9) (10) so that the average absorbed power, from (8), becomes The optimal controller in (12) is not realizable in practice because it is noncausal [4]. Predictions of the excitation force can be utilized, however, to estimate the optimal velocity from the following noncausal relation: (13) which is easily derived from (9) and (12). The system can then be forced to follow this reference velocity through a feedback control loop. The structure of the proposed real-time controller is shown in Fig. 2, where the noncausal condition (13) is implemented in a higher level control (HLC) that decides the optimal reference velocity to be imposed on the WEC. Estimation of the excitation force, which is not directly measurable, is required [8], along with its prediction. (11) Note that the minus sign in (6)–(11) is due to the fact that the excitation power entering the system is considered to be posi- tive, so that the power absorbed at the load (exiting the system) will, therefore, be negative. Maximization of (11), with respect to , yields the op- timal control law producing the load force and system’s velocity that allow maximum power transfer from the waves to the load [4] (12) and this is independent of the power spectral distribution of the disturbance . The optimal control law in (12) is termed, in the wave energy literature, complex-conjugate con- trol, as the load impedance required in order to produce the op- timal force is the complex-conjugate of the intrinsic impedance of the system. The name reactive control is also used, to highlight the pres- ence of a reactive power and the necessity to inject energy into the system during part of the cycle. This requirement of a re- versible power ﬂow has major implications on the practical im- plementation of reactive control. Although maximum mechan- ical energy is ensured, in fact, losses in the bidirectional ﬂow implemented by a nonideal PTO mechanism may in fact result in a poor overall efﬁciency of the system, if one considers the actual useful (e.g., electrical) energy produced. C. Other Control Approaches Other control architectures have been proposed, where the load force is analytically determined from the excitation force or from the velocity, using (10) and (12). The velocity of the ﬂoating system is then controlled, in a feed-forward fashion, by applying the calculated PTO force [14], [23], [24]. Such solutions, in the authors’ opinion, may lose their effectiveness in a real-world implementation, where inaccuracies in the model (nonlinearities, uncertainties in the parameters, or prediction errors) could lead to poor control actions. As mentioned in the introduction, practical constraints usually limit the applicability of reactive control. In particular, the condition in (12) may require infeasible amplitudes or velocities of the motion, as well as forces signiﬁcantly bigger than the PTO mechanism can provide, or that the system can bear. Another constraint could be the inability to provide reactive power from the load, if the PTO is purely passive. This is why alternative control strategies have been investigated, where the control problem is treated as a real-time constrained optimization problem and an optimal reference trajectory for the system’s variables (velocity and PTO force) is generated from (future) knowledge of the excitation force. Most common control strategies of this type are latching (passive) [6], [25] and model predictive control (MPC) [8]–[10]. While the optimal velocity given by reactive control, as in (13), is based on the minimization of the average power over an inﬁnite time horizon, MPC, on the other hand, optimizes the operation of the system (velocity and/or force) over a ﬁnite future FUSCO AND RINGWOOD: STUDY OF THE PREDICTION REQUIREMENTS IN REAL-TIME CONTROL OF WECs 179 horizon. In particular, at each time step , the absorbed power, over a future time horizon , is maximized (14) possibly subject to constraints on the system’s variables. Due to a number of reasons, however, the scope of the present study will be limited to reactive control, in Sections III and IV. In general, the solution of constrained optimization problems would converge towards reactive control when the system’s operating conditions are well within the constraints. In addition, when the limits of the constrained region are approached, it is not usually possible to derive an analytical solution and the prediction requirements can only be determined numerically and on a case by case basis. On the other hand, restraining the analysis to reactive control allows an understanding of the basic links that exist between the fundamental properties of a ﬂoating system (radiation and excitation) and its prediction requirements. III. MEASURES OF NONCAUSALITY AND PREDICTABILITY This section outlines a methodology for the quantiﬁcation of the prediction requirements concerning reactive control, structured as in Fig. 2. After a fundamental discussion of the noncausality of the control in Section III-A, quantiﬁcations of the prediction requirements and of the predictability are proposed in Sections III-B and III-C, respectively. The possibility of manipulating the control to achieve a reduction in the prediction requirements is also discussed in Section III-D. A. Analysis of Noncausality The noncausality of the control strategy described in Section II, and shown in Fig. 2, is encapsulated by the following relationship: (15) Fig. 3. Noncausal reference-generator transfer function between excitation force and optimal velocity. It is a real and even function of the frequency. , is now well deﬁned. Note, as far as noncausality is concerned, or have the same characteristics, as they only differ at time . The impulse response is real and even (the Fourier transform is real and even) and tends to zero for inﬁnite time (the Fourier transform tends to a constant value for ). The decay to zero is dominated, in general, by an exponential with a time constant which is the reciprocal of the lowest frequency pole of the corresponding transfer function. Due to the presence of the radiation resistance , is only known numerically for some frequencies, and an explicit ﬁnite-order representation is not immediately available. From the behavior shown in Fig. 3, valid for systems with a single- peaked radiation, it is, however, straightforward to identify the lowest frequency pole. The function can, in fact, be seen as a low-pass ﬁlter cascaded with a high-pass ﬁlter, where the attenuation band corresponds to the resonance band of the system’s radiation function. The lowest frequency pole, namely , can be estimated as the cutoff frequency (at 3 dB) of the low-pass ﬁlter. The time constant is the dominant parameter that inﬂuences the decay to zero of , and consequently the noncausality of the control action. As an example, Fig. 4 shows the relation between and for two different ﬂoating systems. which can be equivalently written in the time domain as B. Prediction Requirements (16) where is the inverse Fourier of (17) When , the radiation resistance goes to zero and, therefore, tends to the constant value . This means that the transform relationship in (17) is only valid in a generalized sense and, more correctly In Section III-A, a parameter strictly connected to the radiation of the ﬂoating system was identiﬁed as a measure of the noncausality of its control system. It would be interesting to see if a direct proportionality exists between and the actual forecasting horizon required for an effective approximation of the convolution in (16), for the generation of the optimal velocity reference. The required prediction horizon is determined from a range of suboptimal oscillation velocities, calculated from the integral in (16), truncated at different times (19) (18) where is the Dirac delta function. The inverse Fourier transform in the right-hand term of (18), yielding the function so that future values for the excitation force only up to (supposed to be perfectly known) are included. Note that the causal part of the integral is also truncated to maintain the 180 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 3, NO. 1, JANUARY 2012 where , are the coefﬁcients of the AR model, of order and, obviously, if (information acquired, no need for prediction). The performance of the prediction algorithm is measured in terms of the following index of goodness-of-ﬁt: Fig. 4. Relation between lowest frequency pole and noncausality of the im- pulse response. and calculated for two heaving buoys. evenness of the impulse response. In the extreme case of , the integral in (19) becomes causal and no future values of the excitation force are utilized. The velocity is then ideally imposed onto the system, using a load force that can be directly derived from (1). The relative capture width (RCW), obtained with the velocity thus calculated (and with the required PTO force), is (20) where is the absorbed power, calculated from (6), is the width of the device (the diameter in the case of a cylinder or sphere), and is the wave power per unit width [4] (21) In (21), is the acceleration due to gravity and is the water density. Also, deep water is assumed, that is [4], where is the wave number and is the water depth. Intuitively, it is expected that increases when larger values of are considered, but only up to a critical future horizon, beyond which the power gain from considering extra future information of the excitation force is negligible. Such a critical future horizon will be proportional to the time constant . C. Prediction of the Excitation Force The ability to predict the excitation force is tested with linear autoregressive (AR) models, whose identiﬁcation and estimation is operated accordingly to the methodology presented in [18]. In particular, at instant , the -step ahead prediction, , is calculated as (22) (23) In [18] it was shown how AR models allow for accurate pre- dictions of the wave elevation, at low frequencies, for more than one wave period into the future. From (4), the excitation force represents the wave elevation low-pass ﬁltered by the function , which encloses the excitation properties of the ﬂoating system. The stochastic nature of the AR model implies that the achievable prediction horizon cannot exceed the intrinsic cor- relation of the signal, as the physics of the phenomenon (wave propagation laws) is not exploited. As a consequence, and based on the fact that the correlation in a signal is in general inversely proportional to its bandwidth, it may be expected that the ability to predict the excitation force, by using a stochastic model is related to properties of the sea state [18] (peak frequency and bandwidth), but also to the system’s ﬁltering properties [26], speciﬁed by the function . For a given sea state, systems with a narrower excitation bandwidth will experience lower frequency waves and, there- fore, will allow for longer prediction horizons, in seconds. D. Reduction of Prediction Requirements In some situations, it may happen that the prediction require- ments, determined from Section III-B, are beyond the ability of the prediction model. One solution could be the adoption of complex forecasting solutions, making use of expensive instru- mentation and of more computationally intensive models [22]. A critical evaluation of the noncausality of , which also takes into account the typical frequency distribution of the ex- citation force, offers a much simpler alternative. For a number of reasons, the wave excitation force is always contained within the stopband of : 1) Waves do not appear at frequencies below 0.2–0.3 rad/s (30–40 s). 2) Waves at higher frequencies than the attenuation band of are ﬁltered out by the excitation transfer func- tion, . There exists, in fact, a correspondence be- tween the radiation and excitation bandwidth (based on the Haskind relation) [4], [26]. 3) If the system is well-suited to the location of deployment, its radiation response shall roughly match most of the sea states, and this corresponds to the attenuation band of , as discussed in Section III-B. Based on these considerations, it may be argued that the ref- erence-generator will mostly work at frequencies within the at- tenuation band of (0.4–1 rad/s in Fig. 3). As a conse- quence, there exists the possibility of manipulating the shape of in order to reduce its noncausality, while at the same time maintaining its effectiveness in most sea conditions. FUSCO AND RINGWOOD: STUDY OF THE PREDICTION REQUIREMENTS IN REAL-TIME CONTROL OF WECs 181 Fig. 5. Manipulation of , resulting in , such that the lowest frequency pole is moved rightward, thus decreasing the dominant time constant of the correspondent impulse response. Note that the impulse response instead of is shown, as from (18). In Section III-B, the noncausality was related to the time constant , which is the reciprocal of the pole at the lowest frequency of . A possible intuitive manipulation may, therefore, consist of lowering the gain of outside the attenuation band, where its input (the excitation force) will not usually appear. This would move the ﬁrst pole towards higher frequencies, thus reducing the time constant . As an example, consider Fig. 5(a), where the original is modiﬁed to a new function termed . As a result, the impulse response , which is still real and even, decays much more quickly to zero than , as shown in Fig. 5(b). Note that the modiﬁcation proposed here is not rigorous, in any sense, and is only based on qualitative considerations. It is not within the scope of the paper to ﬁnd the best reshaping of which minimizes the noncausality and at the same time maximizes the approximation of the optimality condition. A very important remark should also be mentioned. From Figs. 3 and 5(a), it may be noted that, over its attenuation band, the function is quite ﬂat. This observation may sug- gest that a constant approximation of could be close to optimal in most sea states. The possibility of approximating with a constant value is investigated in more detail in [16]. IV. RESULTS The wave data utilized for the analysis of the prediction requirements was provided by the Irish Marine Institute and comes from real measurements collected from a data buoy deployed off the West coast of Ireland, in the Belmullet wave energy test site, at approximately N, W. The data consists of a number of selected 30-min sets, sampled at 1.28 Hz, as in Fig. 6(b), covering a variety of sea condition in the year 2010. Fig. 6(a) shows the distribution of the peak Fig. 6. Selected wave data: (a) overall statistics of selected data sets; (b) sample data set. Location is Belmullet, off the West coast of Ireland. frequency, , and signiﬁcant wave height of the selected data sets. The wave energy conversion system consists of a bottom- referenced ﬂoating body constrained to move in the heave di- rection only, as modeled in Section II-A. Both cylinders and spheres are considered, with a range of values for the radius and the height. The detail of the geometries is speciﬁed in Table I, where : radius; : draught at rest; : submerged mass at rest; : resonance frequency; : bandwidth of the radiation resis- tance ; : main time constant of the control . As the radius of the sphere increases, the bandwidth and the reso- nance frequency get smaller, while the dominant time constant of the noncausal control increases. Regarding the cylinders, the time constant is mainly affected by the radius, while it only changes slightly for different heights. Based on the methodology proposed in Section III-B, reac- tive control is implemented with the (truncated) noncausal law in (19), where only future knowledge of the excitation force up to seconds into the future is included. The resulting perfor- mance is calculated for a range of values of the lead time . Fig. 7(a) shows the behavior of for the three spheres of Table I, in a particular sea state. As expected, the performance improves if more future information about the ex- citation force is included in the control. At some point, however, the optimum is approached and no further beneﬁt is gained from extra knowledge of the future. This critical future horizon de- pends on the geometry and increases with , as expected. Note that the optimal value of changes with the size of the device, in accordance with the general theory of axisymmetric heaving buoys [27]. In Fig. 7(b), the average lead time , for which , over all the considered sea states (about 200), is plotted against the time constant corresponding to the case of the three spheres. Interestingly, the relation between the future horizon required and is quite linear (the variance of the residuals of a linear ﬁt 0.78). Besides, the standard deviation 182 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 3, NO. 1, JANUARY 2012 Fig. 8. Future prediction horizon required for effective implementation of reactive control, applied to heaving cylinders. Fig. 7. Performance of reactive control, applied to heaving spheres, when only limited future information of the excitation force, up to seconds in the future, is considered. (a) Detail for a speciﬁc sea state. (b) Required horizon averaged over all sea states. TABLE I GEOMETRY OF HEAVING BUOYS. : CYLINDER; : SPHERE Fig. 9. Accuracy of AR prediction models applied to the excitation force produced on heaving cylinders. related to each mean value (dot) in Fig. 7(b) is very small (bar centered around each dot), which conﬁrms the independence of the prediction requirements from the speciﬁc wave conditions. In the case of the cylinders, there are two parameters to be considered: the height and the radius. The time constant is mainly inﬂuenced by the radius, so that the prediction require- ments are not expected to be signiﬁcantly affected by varia- tions in height. Fig. 8 shows the required prediction horizon , such that , averaged over all the considered sea states, against the time constant , for each cylinder. As for the spheres, the forecasting horizon required increases with (and with the radius). In addition, for a given height, the relation is linear (variance of the residuals of a linear ﬁt 0.04). To a lesser extent, changes in height also affect the prediction requirements and the trend is consistent: the cylinders are less demanding in terms of prediction when their height increases. The parameter remains, though, the most critical in relation to the prediction requirements. Having quantiﬁed the required forecasting horizon, the ability to predict the excitation force is discussed. As proposed in Section III-C, linear AR forecasting models are utilized, identiﬁed and estimated as in [18]. Focus is put on the cylinders of Table I, and predictions are calculated for all the selected wave data. The accuracy of the prediction, at each lead time , is measured with the index , from (23). In Fig. 9(a), the behavior of for the excitation force produced by a particular sea state, on three cylinders (with different band- widths), is shown. The forecasting horizon is expressed as a fraction of the peak period. The effectiveness of the prediction obviously decreases with the forecasting horizon, and the level of accuracy is higher for systems with narrower bandwidth, as expected [26]. In Fig. 9(b), the forecasting horizon , for which , averaged over all the sea states, is shown against the bandwidth of . The predictability clearly tends to decrease with increasing bandwidth. Note also that the standard deviation is relatively large, due to the fact that the properties of the waves also have a signiﬁcant effect on the predictions [18]. In general, however, less than 1 wave period in the future (about 6 to 15 s at the considered location) can be predicted with a . In comparison with the requirements, well above 10 s in most cases (Figs. 7 and 8), the performance of the predictor is quite poor. The possibility of manipulating the control, such that the noncausality is reduced, was proposed in Section III-D. In- FUSCO AND RINGWOOD: STUDY OF THE PREDICTION REQUIREMENTS IN REAL-TIME CONTROL OF WECs 183 qualitatively derived from and lower performance losses could be achieved through an optimization that takes into account the spectral distribution of the sea states, as also mentioned in Section III-D. By applying the procedure to all the cylinders, and calcu- lating the prediction requirements in all the sea states, the re- sults of Fig. 10(c) are obtained. The prediction requirements are signiﬁcantly reduced to an horizon shorter than 10 s, which is well within the ability of stochastic wave forecasting models, as shown in Fig. 9 and in [18]. Also, it is interesting to note that the differences between the cylinders are quite ﬂattened, so that no geometry seems to be particularly advantageous in terms of required forecasting horizon. Fig. 10. Reduction of prediction requirements through manipulation of the noncausal control law: (a) Detail on a speciﬁc sea state. (b) Max RCW, when using , as percentage of ideal RCW, obtained with . (c) Average over all sea states. tuitive modiﬁcations to the reference generator are applied, as shown in the example of Fig. 5(a). As a result of implementing reactive control based on the modiﬁed , Fig. 10(a) shows the behavior of for two different systems, cylinders and of Table I. A stark reduction of the prediction horizon required for an effective implementation of reactive control is achieved. In addition, the performance drop is minor in most cases. The deviation from the optimal performance, in particular, will be affected by the bandwidth of the system (ﬂatness of ) and by the excitation force spectral distribution, as noted in Section III-D. Fig. 10(b) shows the distribution, with the bandwidth of the system, of the maximum RCW achievable, when is adopted, as percentage of the RCW obtained from ideal reactive control. The average (dot) performance is usually well above 80%, and increases with the bandwidth of the system. The standard deviation (bar centered around dot) is quite large only for very narrow-banded systems. Note that was only V. CONCLUSION A study of the prediction requirements in real-time control of WECs was presented. Focus was put on unconstrained reactive control, which allowed the characterization of the relation between such requirements and some fundamental properties of the ﬂoating system. A time constant, denoted , related to the shape of the radiation resistance of a ﬂoating system, was identiﬁed as the most critical parameter inﬂuencing the noncausality of the control law. From time-domain simulations of different heaving buoys (spheres and cylinders), over a wide variety of real wave conditions, it was veriﬁed that the relation between and the future horizon required for an effective implementation of reactive control is practically linear. The predictability of the excitation force was also determined using stochastic AR models that were already successfully adopted for the prediction of the wave elevation [18]. Floating systems with narrower bandwidth were found to allow better prediction of the excitation force, as already found in [26]. However, accurate predictions were obtained for lead times between 0.5 and 1 wave period, which is well below the requirements for most systems (well above 20 s). Simple manipulation of the control, based on prior knowledge of the excitation force spectral distribution, was also proposed for the reduction of the prediction requirements. Such reshaping of the noncausal control law allows a signiﬁcant decrease in the prediction requirements, so that they fall within the range of predictability achievable with simple stochastic predictors. Minor performance losses, in terms of energy capture, are veriﬁed in most sea states. The possibility of making the control law causal was also proposed, and this is investigated in more detail in [16]. In order to maintain the focus of the paper on the prediction requirements, effects of the prediction errors on the control performance were not investigated. Future work will, therefore, involve the quantiﬁcation of the mapping of the excitation force prediction error on the reference velocity, and on the design of a lower level control loop that could possibly reject unwanted (erroneous) components in the reference velocity. Preliminary work on such a topic can be found in [28]. ACKNOWLEDGMENT The authors would like to thank the Irish Marine Institute for providing the wave data from the Belmullet test site. 184 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 3, NO. 1, JANUARY 2012 REFERENCES [1] D. Halamay, T. Brekken, A. Simmons, and S. McArthur, “Reserve requirement impacts of large-scale integration of wind, solar and ocean wave power generation,” IEEE Trans. Sustain. Energy, vol. 2, no. 3, pp. 321–328, Jul. 2011. [2] F. Fusco and J. Ringwood, “Variability reduction through combination of wind and waves: A irish case study,” Energy, vol. 35, pp. 314–325, 2010. [3] X. Yang, Y. Song, G. Wang, and M. 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Thomas, “An application of model predictive control to a wave energy point absorber,” in Proc. IFAC Conf. Control Methodologies and Technology for Energy Efﬁciency (CMTEE 2010), Portugal, 2010. [10] J. Hals, J. Falnes, and T. Moan, “Constrained optimal control of a heaving buoy wave-energy converter,” J. Offshore Mech. Arctic Eng., vol. 133, pp. 011401-1–011401-15, 2011. [11] A. Price, D. Forehand, and A. Wallace, “Time-span of future information necessary for theoretical acausal optimal control of wave energy converters,” in Proc. Eur. Control Conf. (ECC), Budapest, Hungary, 2009, pp. 3761–3766. [12] M. R. Belmont, “Increases in the average power output of wave energy converters using quiescent period predictive control,” Renew. Energy, vol. 35, no. 12, pp. 2812–2820, 2010. [13] M. P. Shoen, J. Hals, and T. Moan, “Wave prediction and robust control of heaving wave energy devices for irregular waves,” IEEE Trans. Energy Convers., vol. 26, no. 2, pp. 627–638, Jun. 2011. [14] J. Hals, J. Falnes, and T. Moan, “A comparison of selected strategies for adaptive control of wave energy converters,” J. Offshore Mech. Arctic Eng., vol. 133, no. 3, pp. 031101-1–031101-12, 2011. [15] M. Lopes, J. Hals, R. Gomes, T. Moan, L. Gato, and A. Falcao, “Experimental and numerical investigation of non-predictive phase-control strategies for a point-absorbing wave energy converter,” Ocean Eng., vol. 36, no. 5, pp. 386–402, 2009. [16] F. Fusco and J. Ringwood, “Suboptimal causal reactive control of wave energy converters using a second order system model,” in Proc. Int. Society of Offshore and Polar Eng. (ISOPE), Maui, HI, 2011. [17] U. A. Korde, “Efﬁcient primary energy conversion in irregular waves,” Ocean Eng., vol. 26, pp. 625–651, 1999. [18] F. Fusco and J. Ringwood, “Short-term wave forecasting for real-time control of wave energy converters,” IEEE Trans. Sustain. Energy, vol. 1, no. 2, pp. 99–106, Jul. 2010. [19] K. Budal and J. Falnes, “The Norwegian wave-power buoy project,” in Second Int. Symp. Wave Energy Utillization, Trondheim, Jun. 1982, pp. 323–344. [20] J. Tedd and P. Frigaard, “Short term wave forecasting, using digital ﬁlters, for improved control of wave energy converters,” in Proc. Int. Offshore and Polar Eng. Conf. (ISOPE), 2007, pp. 388–394. [21] M. Belmont, J. Horwood, R. Thurley, and J. Baker, “Filters for linear sea-wave prediction,” Ocean Eng., vol. 33, pp. 2332–2351, 2006. [22] H. Van Den Boom, OWME Project Makes Technological Breakthrough 2009, Marin Report. [23] U. A. Korde, “On control approaches for efﬁcient primary energy conversion in irregular waves,” in Proc. IEEE Oceanic Eng. Society (OCEANS’98), 1998, vol. 3, pp. 1427–1431. [24] U. A. Korde, “Control system applications in wave energy conversion,” in Proc. OCEANS 2000 MTS/IEEE Conf. Exhibition, 2000, vol. 3, pp. 1817–1824. [25] A. Babarit, G. Duclos, and A. Clement, “Comparison of latching control strategies for a heaving wave energy device in random sea,” Appl. Ocean Res., vol. 26, pp. 227–238, 2004. [26] F. Fusco, J.-C. Gilloteaux, and J. Ringwood, “A study on prediction requirements in time-domain control of wave energy converters,” in Proc. Control Appl. Marine Syst. (CAMS), Germany, 2010, pp. 405–410. [27] J. Cruz, Ocean Wave Energy. Current Status and Future Perspectives. New York: Springer-Verlag, 2008. [28] F. Fusco and J. Ringwood, “A model for the sensitivity of non-causal control of wave energy converters to wave excitation force prediction errors,” in Proc. 9th Eur. Wave and Tidal Energy Conf. (EWTEC), Southampton, U.K., 2011. Francesco Fusco received the Master’s degree in industrial automation engineering, in 2008, from Universita’ Politecnica delle Marche (UNIVPM), Ancona, Italy, with his ﬁnal project carried out at the National University of Ireland Maynooth (NUIM), Ireland. Since January 2009, he has been working toward the Ph.D. degree at the Center for Ocean Energy Research (COER), NUIM. His research project is focused on the optimization of wave energy converters control systems through prediction of the future incident wave elevation or wave excitation force. In 2010 and 2011, he also worked as an occasional lecturer in the Electronic Engineering Department at NUIM. Mr. Fusco is a Chartered Engineer in Italy. John V. Ringwood received the Diploma in electrical engineering from Dublin Institute of Technology and the Ph.D. degree (in control systems) from Strathclyde University, Scotland, in 1981 and 1985, respectively. He is currently Professor of Electronic Engineering with the National University of Ireland (NUI), Maynooth and Associate Dean for Engineering in the Faculty of Science and Engineering. He was Head of the Electronic Engineering Department at NUI Maynooth from 2000 until 2005, developing the Department from a greenﬁeld site. His research interests cover time series modeling, wave energy, control of plasma processes, and biomedical engineering. Dr. Ringwood is a Chartered Engineer and a Fellow of the Institution of Engineers of Ireland.

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