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Materials

# Constrained Optimal Control of a Heaving Buoy Wave-Energy Converter

[+] Author and Article Information
Jørgen Hals1

Centre for Ships and Ocean Structures (CeSOS), Norwegian University of Science and Technology (NTNU), Otto Nielsens v. 10, 7491 Trondheim, Norwayjorgen.hals@ntnu.no

Johannes Falnes, Torgeir Moan

Centre for Ships and Ocean Structures (CeSOS), Norwegian University of Science and Technology (NTNU), Otto Nielsens v. 10, 7491 Trondheim, Norway

Mathematically, what we define for this context is $a.

1

Corresponding author.

J. Offshore Mech. Arct. Eng 133(1), 011401 (Nov 03, 2010) (15 pages) doi:10.1115/1.4001431 History: Received February 22, 2009; Revised October 31, 2009; Published November 03, 2010; Online November 03, 2010

## Abstract

The question of optimal operation of wave-energy converters has been a key issue since modern research on the topic emerged in the early 1970s, and criteria for maximum wave-energy absorption soon emerged from frequency domain analysis. However, constraints on motions and forces give the need for time-domain modeling, where numerical optimization must be used to exploit the full absorption potential of an installed converter. A heaving, semisubmerged sphere is used to study optimal constrained motion of wave-energy converters. Based on a linear model of the wave-body interactions, a procedure for the optimization of the machinery force is developed and demonstrated. Moreover, a model-predictive controller is defined and tested for irregular sea. It repeatedly solves the optimization problem online in order to compute the optimal constrained machinery force on a receding horizon. The wave excitation force is predicted by use of an augmented Kalman filter based on a damped harmonic oscillator model of the wave process. It is shown how constraints influence the optimal motion of the heaving wave-energy converter, and also how close it is possible to approach previously published theoretical upper bounds. The model-predictive controller is found to perform close to optimum in irregular waves, depending on the quality of the wave force predictions. An absorbed power equal to or larger than 90% of the ideal constrained optimum is achieved for a chosen range of realistic sea states. Under certain circumstances, the optimal wave-energy absorption may be better in irregular waves than for a corresponding regular wave having the same energy period and wave-power level. An argument is presented to explain this observation.

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## Figures

Figure 1

A heaving sphere of radius 5.0 m is used as an example of a point absorber. It is semisubmerged at equilibrium, and connected to the bottom through a PTO machinery, which in its most general form can act instantaneously with any vertical force on the sphere. The heave excursion from the equilibrium position is given by η(t) and the heave velocity by v(t)=η̇(t).

Figure 2

Retardation function k(t) for the heaving sphere, sometimes also referred to as the memory kernel. Convoluted with the velocity it gives the radiation force Fr(t) (see, e.g., Ref. 34, Sec. 5.3).

Figure 3

Optimal velocity (blue, dash-dotted) and position (black, fully drawn) for the heaving sphere excited by a regular wave of period 9 s and varying wave height: 0.5 m (top), 1.0 m (middle), and 3.0 m (bottom). The excitation force is a sine function where the amplitude has been attenuated by a factor increasing linearly from 0 to 1 during the first half period (0–4.5 s). It is given by the dashed, red curve with scale on the right-hand axis. The heave amplitude was constrained to ±3 m. The corresponding average absorbed powers are 45.9 kW, 183 kW, and 851 kW for the three wave heights, respectively.

Figure 4

Optimal motion for an incident regular wave of period 9 s and height 3 m when the heave amplitude is restricted to 3 m while the machinery force is unconstrained. The upper diagram gives the position η (black, fully drawn) and velocity v (blue, dash-dotted), while the lower diagram gives the machinery force Fm (green, fully drawn). Both diagrams also contain the excitation force (red, dashed). The dotted curves, giving the optimal solution (position, velocity, and machinery force) under the additional constraint of sinusoidal motion, are included for comparison.

Figure 5

Convergence of the direct force optimization as the time step is reduced: Ts=0.25 s (fully drawn), 0.05 s (dashed), and 0.02 s (dash-dotted). The solution found by velocity optimization is included by the dotted curve. The excitation force is the same as given in the middle diagram of Fig. 3 and the heave amplitude has in this case been constrained to |η|≤1 m. Due to the rapidly increasing computational burden as the time step is reduced, only one wave period has been solved for, although the solution is still in its transient phase.

Figure 6

Optimal motion for an incident regular wave of period 9 s and height 3 m when the heave amplitude is restricted to ±3 m and the machinery force is restricted to ±1.5 MN. The upper diagram gives the position η (black, fully drawn) and velocity v (blue, dash-dotted), while the lower diagram gives the machinery force Fm (green, fully drawn). Both diagrams also contain the excitation force Fe (red, dashed). The results may be compared with the curves of Fig. 4 obtained without force constraints.

Figure 7

Optimal functions with an incident regular wave of period 9 s and height 2 m: for position (top), for velocity (middle), and for machinery force (bottom). The heave motion has been restricted to ±3 m, and the excitation has the same phase as in Fig. 3. The dash-dotted curves result from unbounded machinery-force input, whereas the dashed curves have been calculated by including a force constraint of 1.5 MN in the velocity optimization. Finally, the fully drawn curves stem from first optimizing the velocity without input constraints, and then finding the constrained input that minimizes the deviation from the found optimal velocity. One oscillation cycle at steady state is shown.

Figure 8

Absorbed useful power for the heaving sphere with excursions restricted to ±3 m and an incident wave height of 1 m. The thin dashed curves show the theoretical upper bounds: the limit for heaving axisymmetric bodies (Eq. 35, ascending right) and Budal’s upper bound (Eq. 36, descending right). A thin dotted curve is drawn at half the Budal bound, and its intersection with the ascending limit is indicated by a vertical dotted line and point A. The four remaining curves originate from different control strategies: passive (dash-dot), latching (dashed), reactive control restricted to sinusoidal oscillation (fully drawn), and constrained optimal control (with squares), of which the first three curves are taken from an earlier work (27).

Figure 9

The Budal diagram extended to three dimensions by including the wave amplitude as a variable for the incident regular wave. This surface gives theoretical limits to the possible power absorption for an axisymmetric heaving body of a chosen volume, in this case equal to 524 m3. The crossing point of the limiting curves shift toward lower periods as the wave amplitude increases, giving the skew shape of the surface.

Figure 10

Surface giving the upper bound for the power that can be absorbed by a heaving sphere of diameter 10 m when the vertical excursion is restricted to ±3 m. The surface would be covered by the surface in Fig. 9, coinciding for low wave periods in the same way as the curves in Fig. 8, where the amplitude restriction has not yet come into play.

Figure 11

The system model is broken into two complexity levels: the process plant model, representing the real dynamics of the system, and the control plant model, used for calculation of the controller signal to the machinery. Measurements and predictions of the wave excitation force are fed into the controller, which computes the machinery force set point using a QP optimization algorithm.

Figure 12

Example of time series for MPC control of the heaving sphere, with amplitude restricted to η≤3 m. The upper diagram gives the position η (black, fully drawn) and velocity v (blue, dash-dotted), while the lower diagram gives the machinery force Fm (green, fully drawn). The excitation force is represented by the red dashed curves and corresponds to a regular wave of period 9.0 s and height 1.0 m. The dotted curves give the corresponding position, velocity, and machinery force solutions found by the numerical optimization described in Sec. 3.

Figure 13

Performance of the MPC controller (fully drawn, black curve) compared with the optimal absorbed power (blue squared curve) under amplitude constraints |η|≤3 m for incident regular waves with height 1.0 m and varying period. The difference between the optimal and MPC results is given by the black, diamond-marked curve, which has been scaled up by a factor of 10. The optimization horizon is set to Th=8.8 s≈2T0 and the predictor time step is 0.15 s. For comparison, the dashed, red curve gives the frequency domain solution for reactive control under the constraint of sinusoidal motion, and the green, dotted curve gives the result for latching control (both also shown in Fig. 8). The dashed, black curves give the upper theoretical bounds.

Figure 14

The ratio between extreme values of instantaneous power Pm,max flowing through the machinery and average absorbed power Pm,avg found from simulations with incident regular waves and the MPC controller using ideal predictions of the excitation force. The filled red squares gives the minima (machinery working as a motor) and the filled blue circles give the maxima (machinery working as a generator) for incoming regular waves of height H=1.0 m and varying wave period (given on the horizontal axis). The corresponding open squares and circles show the same ratio for latching control, which has no reactive power flow.

Figure 15

Performance of the MPC controller for varying optimization horizons in irregular waves synthesized from a modified Pierson–Moskowitz spectrum, and with restrictions on only the heave amplitude. On the horizontal scale, the fraction of optimization horizon Th to eigenperiod T0≈4.4 s is given. The vertical scale is absorbed power PIR,ideal using an ideal predictor in irregular waves relative to the optimum absorbed power PR for a regular wave of the same energy period and same wave-power level. The three different diagrams correspond to different significant wave heights (given on top), and the plus, circle and cross marks refer to energy periods of 6 s, 9 s, and 12 s, respectively.

Figure 16

Performance achieved with a optimization horizon of Th=8.8 s, corresponding to twice the resonance period T0. This is the longest horizon studied here and from the convergence observed in Fig. 1, the achieved performance is expected to be very close to optimal. The absorbed power on the horizontal scale is given as relative absorption width da/d, with the buoy diameter d=10 m. The labels on the vertical scale indicates the energy period, significant wave height, and wave-power level of the incident irregular wave, which was synthesized from a modified Pierson–Moskowitz spectrum. The sea states on the vertical axis have been ordered according to increasing wave-power level.

Figure 17

Illustration of how we may think that the shape of the limiting curve affects the ratio between power absorbed in irregular waves PIR and regular waves PR with the same energy period and wave-energy transport. We may, as an approximation, think of an irregular wave as a series of half wave cycles having different wave periods and wave heights. If the ability to absorb power did not depend on the period of the incoming wave we would have case A. When there is a convex limit to the possible absorbed power, the absorbed power may increase when the incoming wave period is allowed to vary about Te (case B). A straight-line limit (as in case C) compares to case A, whereas a concave limit (case D) causes a reduction in the absorbed power for irregular waves relative to the corresponding regular wave.

Figure 18

The augmented Kalman filter run on an excitation force time series resulting from a sea state of significant wave height Hs=1.41 m and energy period Te=9 s (with a modified Pierson–Moskowitz wave spectrum). The force signal is predicted 2.2 s ahead (dashed, red curve), and this is compared to the true values (fully drawn, black curve).

Figure 19

Performance of MPC controller when the force prediction is made by an augmented Kalman filter based on a damped harmonic oscillator model for the wave force. The vertical axis gives absorbed power PIR,Kalman relative to the corresponding case with ideal prediction, PIR,ideal, as shown in Fig. 1. On the horizontal scale, the fraction of optimization horizon Th to resonance period T0≈4.4 s is given. The three different diagrams correspond to different significant wave heights (given on top), and the plus, circle, and cross marks refer to energy periods of 6 s, 9 s, and 12 s, respectively. The waves were synthesized from a modified Pierson–Moskowitz spectrum.

Figure 20

The same data and coding as for Fig. 1, but here absorbed power PIR,Kalman, are given relative to the absorbed power PIR,ideal achieved with an ideal predictor and horizon Th=2T0 (cf. Fig. 1), which is expected to be very close to optimal. The two points on the leftmost 6 s curve has been left out of the scale to keep a higher resolution. Their values are 0.53 for Th/T0=1 and 0.44 for Th/T0=2.

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