An Application of Model Predictive Control to a Wave Energy Point Absorber

An Application of Model Predictive Control to a Wave Energy Point Absorber J. Cretel*. A. W. Lewis* G. Lightbody**. G. P. Thomas*** 

*Hydraulics and Maritime Research Centre, University College Cork, Cork, Ireland (Tel: (+353)(0)21 4250031; e-mail: [email protected]). **Department of Electrical and Electronic Engineering, University College Cork, Cork, Ireland. ***Department of Applied Mathematics, University College Cork, Cork, Ireland. Abstract: Optimal performance of wave energy converters requires appropriate control strategies. This is especially true of wave energy point absorbers, which are relatively small oscillators excited by waves. The two control methods for point absorbers which are most studied in the literature are reactive control and latching, which have major deficiencies. This article outlines a time-domain control method based on Model Predictive Control, which can be applied to any wave energy point absorber whose behaviour can be described by a linear state-space model. The control method is applied here to a semi-immersed vertical cylinder in deep water, excited by regular or irregular waves and oscillating in heave only; the motion may be either free or subject to amplitude constraints. Preliminary results from numerical simulations are presented and discussed. This control approach aims to obviate some of the limitations of already existing control strategies and to pave the way towards better control methods for point absorbers. Keywords: wave energy, point absorber, latching, reactive control, model predictive control, oscillator, convex optimization, quadratic programming. 

1. INTRODUCTION Optimal control of wave energy converters is fundamental if wave energy is to achieve its full potential and significantly contribute to the energy mix on a global scale. This is especially important for point absorbers (Falnes, 2002a), which can generally be described as oscillators, excited by waves, whose horizontal dimensions are much smaller than the prevailing wavelength. Such devices are intended to be deployed in arrays of several units, a few kilometres off the coast. Advanced control methods are expected to markedly improve the performance over passive control, without generating significant additional cost. Control of point absorbers has been extensively studied since the early days of wave energy, with two complementary reviews provided by Salter, Taylor & Caldwell (2002) and Falnes (2002b). Prominent in the literature are reactive control and latching control, which are briefly reviewed here. Reactive control (Budal & Falnes, 1977) is generally formulated in the frequency domain: the load impedance is chosen so as to verify some optimum amplitude and phase conditions (Falnes, 2002a) whereby the damping is optimal and the intrinsic reactance of the system is cancelled. Despite being acausal, reactive control has been theoretically extended to the time domain, in which causal approximations can be formulated (Naito & Nakamura, 1985). However, the

combination of large amounts of energy flowing in and out of the system and losses in the conversion chain generally prohibits the use of reactive control for wave energy harvest. Latching control, as proposed by Budal & Falnes (1980), does not involve any bidirectional flow of energy. Point absorbers generally have a natural frequency larger than wave excitation frequencies and therefore tend to lead in phase. By latching the oscillator, i.e. locking it into position during part of the wave cycle, it is possible to approximately achieve resonance, thus improving the performance over passive control (fixed damping). Originally developed for the case of sinusoidal excitation, latching control was extended to irregular waves by Hoskin (1988) by employing Pontryagin’s Maximum Principle. The same technique was applied and improved by Babarit & Clément (2006), who treated the radiation in a more rigorous manner. Eidsmoen (1998) modified the basic principle of latching with the intent of handling amplitude constraints. Falcão (2007) proposed a causal version of latching applicable to point absorbers equipped with a conventional hydraulic PTO machinery. Despite its popularity, latching still suffers from two major drawbacks: it may not be suitable for point absorbers equipped with direct-drive linear generators and it is likely to prove inadequate for arrays of WECs, as the optimum phase condition on which latching is founded is different if more than one oscillator are present, as stated by Falnes (1980) and Thomas & Evans (1981).

The development of new control methods for point absorbers is desirable, if wave energy is to succeed commercially (Ringwood, 2006). With the advent of more controllable, more efficient PTO systems for WECs, such as linear generators (Baker, 2003) and smart variable-displacement hydraulics (Payne et al., 2005), it is becoming possible to explore control strategies that optimise the control force in real time, as encouraged by Salter, Taylor & Caldwell (2002) and Molinas et al. (2007). Model Predictive Control (MPC) (Rossiter, 2003) appears as a promising candidate for this type of approach. MPC was first applied to a wave energy point absorber by Gieske (2007) and Hals (2009) recently applied MPC to a heaving point absorber, subject to amplitude constraints, with a more rigorous treatment of the radiation. The MPC-based method outlined in this article remedies some shortcomings of Hals’ approach: -

Observability of the proposed state-space model, a practical requirement of MPC, is assessed.

-

The objective function is chosen in such a way that no energy is allowed to be stored in the system at the end of the horizon, either in potential or kinetic form.

The control method also lays the foundations for a strategy intended to estimate and forecast the wave force over a few seconds, which is the subject of ongoing research. It is assumed in this study that the future wave excitation force is known over the time horizon considered. The control method outlined here is applicable to any wave energy point absorber whose behaviour can be described by a linear state-space model. Furthermore, it is potentially extendable to arrays of such devices. The method is here applied to the classic example of a heaving, semi-immersed vertical cylinder in deep water, excited by regular or irregular waves and either free or subject to amplitude constraints. Preliminary results from numerical simulations are presented and discussed.

expressed in terms of the product of the hydrostatic stiffness 𝑘𝑕 and the vertical displacement, i.e. 𝑓𝑕 (𝑡) = −𝑘𝑕 𝑧(𝑡) .

(2)

Cummins (1962) expresses the radiation force as 𝑓𝑟 (𝑡) = −𝜇𝑧(𝑡) −

𝑡 0

𝑕𝑟 𝜏 𝑧 𝑡 − 𝜏 𝑑𝜏 ,

(3)

where 𝜇 is the added mass at infinity (representing the inertia of the surrounding fluid) and 𝑕𝑟 is the radiation kernel (also called retardation function). The convolution integral reflects the fluid memory effect, whereby the motion of the float at time 𝑡 affects the motion of the surrounding fluid, not only at time 𝑡 but also at subsequent times. The surrounding fluid effectively retains in memory the past trajectory of the float and in turn affects the motion of the float via the radiation force. The added mass at infinity and the radiation kernel are connected to the frequency-domain hydrodynamic coefficients and were obtained with WAMIT® for this study. Introducing the substitutions 𝑢(𝑡) =

𝑓 𝑃𝑇𝑂 (𝑡) 𝑚 +𝜇

,

𝑢𝑤 (𝑡) =

𝑓𝑤 (𝑡) 𝑚 +𝜇

,

(4)

and combining (1) - (4) gives the equation of motion in the form 𝑚 + 𝜇 𝑧(𝑡) +

𝑡 0

𝑕𝑟 𝜏 𝑧 𝑡 − 𝜏 𝑑𝜏 + 𝑘𝑕 𝑧(𝑡) = 𝑚 + 𝜇 (𝑢(𝑡) + 𝑢𝑤 (𝑡)) .

(5)

3. FORMULATION OF THE CONTROL A discrete-time state-space representation of the system must be obtained before MPC can be applied. As the convolution integral present in (5) is not suitable for this application, the Prony method described by Duclos, Clément & Chatry (2001) was used to approximate the convolution integral by an 𝑛𝑡𝑕 order linear model. This is represented by the controllable canonical form of a continuous-time SISO statespace model, with state vector 𝒙𝒓 ∈ 𝑅𝑛 , giving the approximate form of the equations as 𝒙𝒓 𝑡 = 𝐴𝑟 𝒙𝒓 𝑡 + 𝐵𝑟 𝑧 𝑡 ,

2. MATHEMATICAL MODEL In linear wave theory (Newman, 1977), which is assumed throughout this paper, the total force acting in the vertical direction on a floating body can be expressed as a sum of 𝑓𝑕 (the hydrostatic restoring force), 𝑓𝑤 (the wave exciting force, corresponding to the interactions of incident waves with the body fixed in position) and 𝑓𝑟 (the radiation force, due to waves radiated by the oscillating body in an otherwise calm sea). The floating body of structural mass 𝑚 is constrained to oscillate in heave only and, if 𝑓𝑃𝑇𝑂 denotes the force applied by the PTO machinery, the time-domain equation of motion is 𝑚𝑧(𝑡) = 𝑓𝑕 𝑡 + 𝑓𝑤 (𝑡) + 𝑓𝑟 (𝑡) + 𝑓𝑃𝑇𝑂 (𝑡) ,

(1)

where 𝑧(𝑡) is the vertical displacement of the float around the equilibrium position. The hydrostatic restoring force can be

𝑡 0

𝑕𝑟 𝜏 𝑧 𝑡 − 𝜏 𝑑𝜏 ≈ 𝐶𝑟 𝒙𝒓 (𝑡) .

Although the components of 𝒙𝒓 do not have any physical meaning, the radiation state vector 𝒙𝒓 holds information about the state of the surrounding fluid. The state and output vectors 𝒙 ∈ 𝑅𝑛+2 and 𝒚 ∈ 𝑅2 corresponding to the whole system (oscillator and surrounding fluid) are respectively chosen as 𝑧 𝒙= 𝑧 , 𝒙𝒓

𝒚=

𝑧 . 𝑧

A linear time-invariant state-space representation of the system is thus given by

𝒙 𝑡 = 𝐴𝑐 𝒙 𝑡 + 𝐵𝑐 𝑢 𝑡 + 𝐹𝑐 𝑢𝑤 𝑡 , 𝒚 𝑡 = 𝐶𝑐 𝒙(𝑡) , 0

𝐴𝑐 =

1 0

−𝑘 𝑕 (𝑚 +𝜇 )

𝟎

𝟎

−1

(𝑚 +𝜇 )

𝐵𝑟

𝐴𝑟

0 𝐵𝑐 = 𝐹𝑐 = 1 ∈ 𝑅 𝟎

𝐶𝑐 =

1 0

0 1

𝐶𝑟 ∈ 𝑅 (𝑛+2)×(𝑛+2) ,

𝑛+2 ×1

,

By inspection of the observability matrix of this state-space model, the system is fully observable. This means that although the radiation states have no physical meaning and cannot be measured, they can be inferred from the measured outputs. Another important consequence of observability is that the wave input can be estimated, although it is not measured. Furthermore, using this estimation, forecasting the wave excitation force some seconds into the future may be possible. This possibility will be explored in a subsequent article. It is assumed herein that the state of the system at the current time is known, as is the future wave exciting force. Introduce the notation … 𝑘 to denote the vertical concatenation of the considered column vector at times 𝑘 + 1 through to 𝑘 + 𝑁, where 𝑁 is the length of the horizon. The open-loop prediction vector 𝒚(𝑘) can be expressed as a function of the current state and the future input increments, i.e.

𝟎 ∈ 𝑅2×(𝑛+2) . 𝟎

A discrete-time approximation of this continuous system is obtained by using a zero-order hold with sample time ∆𝑡 𝒙 𝑘 + 1 = 𝐴𝑑 𝒙 𝑘 + 𝐵𝑑 𝑢 𝑘 + 𝐹𝑑 𝑢𝑤 𝑘 , 𝒚 𝑘 = 𝐶𝑑 𝒙(𝑘) . The state and output vectors are then augmented, the input increments thus playing the role of inputs to the system, as follows

𝒚 𝑘 = 𝑃𝒙 𝑘 + 𝐻𝜟𝒖 𝑘 +𝐻𝑤 𝜟𝒖𝒘 (𝑘) .

The form of the matrices 𝑃, 𝐻, and 𝐻𝑤 follow from Rossiter (2003). The quantity sought to be maximised here is the mechanical energy 𝐸𝑎𝑏𝑠 absorbed by the PTO system over the time horizon 𝑇 and, using (5), corresponds to 𝐸𝑎𝑏𝑠 = − 𝑚 + 𝜇

𝒚 𝑘 =

𝒚 𝑘 𝑢(𝑘 − 1)

𝑡+𝑇 𝑡

𝑢 𝜏 𝑧 𝜏 𝑑𝜏 .

Thus, the discrete-time objective function, to be minimised, is chosen as 𝐽 𝑘 =

𝒙(𝑘) 𝒙 𝑘 = 𝑢(𝑘 − 1) ∈ 𝑅𝑛 +4 , 𝑢𝑤 (𝑘 − 1)

𝑁 𝑖=1 𝑢

𝑘+𝑖−1 𝑧 𝑘+𝑖 .

∈ 𝑅3 ,

1 2

𝒙 𝑘 + 1 = 𝐴𝒙 𝑘 + 𝐵∆𝑢 𝑘 + 𝐹∆𝑢𝑤 (𝑘) , 𝒚 𝑘 = 𝐶𝒙(𝑘) , 𝐴𝑑 𝟎 𝟎

𝐵𝑑 1 0

𝐹𝑑 0 ∈𝑅 1

𝐵𝑑 𝐵= 1 ∈𝑅 0 𝐹𝑑 𝐹= 0 1 1 𝐶= 0 0

∈𝑅 0 1 0

𝑛+4 ×1

𝑛+4 × 𝑛+4

(7)

It should be noted that the form of the objective function departs from conventional MPC: since the future reference is not known in advance, the optimisation cannot be expressed as a least-squares problem. The trajectory to be followed by the system is instead only accessible after minimisation of (7). This expression can be formulated as a quadratic function of 𝒚, 𝐽(𝑘) = 𝒚𝑇 𝑘 𝑄𝒚 𝑘 ,

𝐴=

(6)

,

,

(8)

where 𝑄 ∈ 𝑅3𝑁×3𝑁 is a block diagonal matrix, whose building block 𝑀 is repeated 𝑁 times along the diagonal, 0 𝑀= 0 0

0 0 1

0 1 . 0

Substituting (6) into (8) yields an expression of 𝐽 as a quadratic function of 𝜟𝒖. After removing the terms that have no dependence in 𝜟𝒖, the following objective function is obtained as 1

𝐽 = 𝜟𝒖𝑇 𝐻𝑇 𝑄𝐻𝜟𝒖 + 𝜟𝒖𝑇 𝐻𝑇 𝑄 𝑃𝒙+𝐻𝑤 𝜟𝒖𝒘 . 2

𝑛+4 ×1

0 0 0

⋯ ⋯ ⋯

, 0 0 0

0 0 1

0 0 ∈ 𝑅3×(𝑛+4) . 0

By inspection, the matrix 𝐻𝑇 𝑄𝐻 is positive-definite, so minimising 𝐽 amounts to solving a convex quadratic programming problem (Boyd & Vandenberghe, 2004), which is constrained if the case being considered is that of a point absorber subject to amplitude constraints. Only the first component of 𝜟𝒖∗ (𝑘) = 𝑎𝑟𝑔𝑚𝑖𝑛𝜟𝒖 {𝐽 𝑘 }, i.e. the optimal control move at time 𝑘, is used. The same

optimisation procedure is then run again for time 𝑘 + 1 and repeated for subsequent times.

significant performance degradation for low frequencies and more so for shorter horizons, as would be expected from the limited length of the horizon.

4. RESULTS A semi-immersed vertical cylinder of radius 5m and draft 8m is considered in deep water; a sample time of 0.05 s was used for the simulations. 4.1 Open-loop system The order 𝑛 of the system used for approximating the radiation convolution integral was tested for a wide range of values and 𝑛=6 was found to give the best compromise between approximation accuracy and complexity of the system (figure 1). Figure 2 indicates a good agreement between displacement amplitudes obtained from time-domain simulations of the open-loop system undergoing monochromatic excitation and response amplitude operators (RAOs) obtained from WAMIT®. It should be noted that in the case of an unconstrained system, linear wave theory is considered to be invalid when the amplitude of motion becomes large compared to the wave amplitude; a factor of 3 or more is often regarded as a practical limit. This occurs around resonance (figure 2) and is non-physical but figure 2 is nonetheless useful to assess the agreement between the frequency- and the time-domain models.

Figure 2. Open-loop system: comparison of the response amplitude operators (RAOs) obtained from WAMIT® and from the timedomain simulation, using a 6th order system.

Figure 3. Control applied to the case of excitation by monochromatic waves: mechanical power absorbed against angular frequency, for three different horizon lengths.

4.3 Irregular waves with amplitude constraints Figure 1. Approximation of the radiation impulse response function by a 6th order system (𝑛=6).

4.2 Monochromatic waves without amplitude constraints Figure 3 confirms that, for an unconstrained system under regular wave excitation, the control yields performance results in accordance with those obtained by reactive control. The power absorbed by passive control (with optimal damping at the frequency considered) has been plotted for comparison. The figure shows that the predictions from the longest horizon tested (6s) provides the best agreement with theoretical optimum and figure 4 shows that the wave exciting force and velocity are in phase, as is the case when reactive control is applied. However, figure 3 indicates a

The more realistic case of a system with an amplitude constraint of 5m and subject to irregular waves, defined by a time series corresponding to a Bretschneider spectrum (significant wave height: 3m; zero-crossing period: 8s), is reported in figures 5-8. As the control tends to become too aggressive in the constrained case, a penalty on the control effort over the horizon was added to the objective function (with 𝜆=1) 1

𝐽 = 𝒚𝑻 𝑄𝒚 + 𝜆𝜟𝒖𝑇 𝜟𝒖 . 2

Figure 4. Control applied to the case of excitation by monochromatic waves (amplitude: 1m; period: 7s) – wave exciting force and device velocity.

Figure 6. Control applied to the case of excitation by irregular waves – wave exciting force and device velocity.

Figure 5 indicates that the control is effective in keeping the device within amplitude constraints. Figure 6 shows that the wave exciting force and velocity are approximately in phase as in the case of monochromatic excitation (figure 4).

Figure 7. Control applied to the case of excitation by irregular waves – PTO force and device velocity.

Figure 5. Control applied to the case of excitation by irregular waves – wave elevation and device displacement.

Figure 7 indicates that the PTO force is not in antiphase with the velocity and a comparison of figures 6 and 7 shows that the PTO force is much larger than the wave exciting force. Consequently, a prohibitively large amount of energy must flow back and forth through the conversion chain, as is apparent on figure 8, which suggest that large excursions of the absorbed power both in the negative and the positive directions are occurring. This would require a high power capacity for the PTO system and would be very detrimental to performance, because a large fraction of the energy would effectively be lost in the conversion process, as pointed out by Hals, Bjarte-Larsson & Falnes (2002). It is therefore desirable to limit the amount of mechanical reactive power involved in the control.

Figure 8. Control applied to the case of excitation by irregular waves – instantaneous and mean mechanical power absorbed.

5. CONCLUSION The results presented in this paper confirm that Model Predictive Control is a method of interest for the control of a particular wave energy converter, the heaving buoy, which is classified as a point absorber and for which effective control is considered essential if such devices are to operate efficiently and profitably. The classical results are recovered in the case of regular waves and unconstrained body motions, under the condition that the prediction horizon is sufficiently long. For the case of irregular waves with constrained motion, which is close to practical application, the method ensures that the constraint is satisfied and highlights an issue with the PTO system that requires further consideration. Subsequent work will focus on the inclusion of a state estimator (which opens the possibility of forecasting the exciting wave force over the horizon) as well as on limiting the amount of energy flowing back and forth through the PTO system. These improvements would bring the control method a step closer to practical application. Future research is also expected to enhance the robustness of the method and to cover the extension of the control strategy to nonlinear point absorbers and arrays of point absorbers. ACKNOWLEDGMENTS The first author is a research fellow funded by the Marie Curie WaveTrain 2 training network and a member of the International Network on Offshore Renewable Energy (INORE). REFERENCES Babarit A. & Clément A.H. 2006. Optimal latching control of a wave energy device in regular and irregular waves. Applied Ocean Research, 28, 77-91. Baker N.J. 2003. Linear generators for direct drive marine renewable energy converters. Ph.D. thesis, University of Durham, UK. Boyd S. & Vandenberghe L. 2004. Convex Optimization. Cambridge University Press. Budal K. & Falnes J. 1977. Optimum operation of improved wave-power converter. Marine Science Communications, 3, 133-159. Budal K. & Falnes, J. 1980. Interacting point absorbers with controlled motion. In Power from sea waves, ed. By B.M. Count, 381-399. Academic Press, London. Cummins W.E. 1962. The impulse response function and ship motions. Schiffstechnik, 9, 101-109. Duclos G., Clément A.H. & Chatry G. 2001. Absorption of outgoing waves in a numerical wave tank using a selfadaptive boundary condition. International Journal of Offshore and Polar Engineering, 11(3), 168-175. Eidsmoen H. 1998. Tight-moored amplitude-limited heaving buoy wave energy converter with phase control. Applied Ocean Research, 20, 157-161. Falcão A.F. de O. 2007. Phase control through load control of oscillating-body wave energy converters with hydraulic

PTO system. Proc. 7th European Wave and Tidal Energy Conference, Porto, Portugal. Falnes J. 1980. Radiation impedance matrix and optimum power absorption for interacting oscillators in surface waves. Applied Ocean Research, 2, 75-80. Falnes J. 2002a. Ocean waves and oscillating systems: linear interaction including wave-energy extraction. Cambridge University Press. Falnes J. 2002b. Optimum control of oscillation of waveenergy converters. International Journal of Offshore and Polar Engineering, 12(2), 147-155. Gieske P. 2007. Model Predictive Control of a Wave Energy Converter: Archimedes Wave Swing. M.Sc. thesis, Delft University of Technology, the Netherlands. Hals J. 2009. Constrained optimal control of a heaving buoy wave energy converter. Norwegian University of Science and Technology, Trondheim, Norway. Accepted for publication in Journal of OMAE. Hals J., Bjarte-Larsson T. & Falnes J. 2002. Optimum reactive control and control by latching of a waveabsorbing semisubmerged heaving sphere. Proc. of OMAE'02, 21st International Conference on Offshore Mechanics and Arctic Engineering, Oslo, Norway. Hoskin R.E. 1988. Optimal control techniques for water power generation. Ph.D. thesis, University of Reading, UK. Molinas M. et al. 2007. Power electronics as grid interface for actively controlled wave energy converters. International Conference on Clean Electrical Power, Capri, Italy. Naito S. and Nakamura S. 1986. Wave energy absorption in irregular waves by feed-forward control system. In Hydrodynamics of ocean wave-energy utilization, ed. by D.V.Evans & A.F. de O. Falcão, 269-280. IUTAM Symposium, Lisbon. Springer-Verlag, Berlin. Newman J.N. 1977. Marine Hydrodynamics. MIT Press. Payne G.S. et al. 2005. Potential of digital displacement™ hydraulics for wave energy conversion. Proc. 6th European Wave and Tidal Energy Conference, University of Strathclyde, Glasgow, UK. Ringwood J. 2006. The dynamics of wave energy. Irish Signal and Systems Conference. Dublin, Ireland. Rossiter J.A. 2003. Model-Based Predictive Control: A Practical Approach. CRC Press. Ch.3. Salter S.H., Taylor J.R.M. & Caldwell N.J. 2002. Power conversion mechanisms for wave energy. Proc. of the Institution of Mechanical Engineers, Part M—Journal of Engineering for the Maritime Environment, 216, 1-27. Thomas G.P. & Evans D.V. 1981. Arrays of threedimensional wave-energy absorbers. J. Fluid Mech., 108, 67-88.