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Ocean Engineering

A Comparison of Selected Strategies for Adaptive Control of Wave Energy Converters

[+] 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

1

Corresponding author.

J. Offshore Mech. Arct. Eng 133(3), 031101 (Mar 29, 2011) (12 pages) doi:10.1115/1.4002735 History: Received December 07, 2009; Revised May 03, 2010; Published March 29, 2011; Online March 29, 2011

Wave-energy converters of the point-absorbing type (i.e., having small extension compared with the wavelength) are promising for achieving cost reductions and design improvements because of a high power-to-volume ratio and better possibilities for mass production of components and devices as compared with larger converter units. However, their frequency response tends to be narrow banded, which means that the performance in real seas (irregular waves) will be poor unless their motion is actively controlled. Only then the invested equipment can be fully exploited, bringing down the overall energy cost. In this work various control methods for point-absorbing devices are reviewed, and a representative selection of methods is investigated by numerical simulation in irregular waves, based on an idealized example of a heaving semisubmerged sphere. Methods include velocity-proportional control, approximate complex conjugated control, approximate optimal velocity tracking, phase control by latching and clutching, and model-predictive control, all assuming a wave pressure measurement as the only external input to the controller. The methods are applied for a single-degree-of-freedom heaving buoy. Suggestions are given on how to implement the controllers, including how to tune control parameters and handle amplitude constraints. Based on simulation results, comparisons are made on absorbed power, reactive power flow, peak-to-average power ratios, and implementation complexity. Identified strengths and weaknesses of each method are highlighted and explored. It is found that overall improvements in average absorbed power of about 100–330% are achieved for the investigated controllers as compared with a control strategy with velocity-proportional machinery force. One interesting finding is the low peak-to-average ratios resulting from clutching control for wave periods about 1.5 times the resonance period and above.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Block diagram showing alternatives paths for optimal control of the wave energy converter represented by Hu(s). The choice between Fm,1 and Fm,2 corresponds to the choice between complex conjugate control, and phase and amplitude control, respectively (3) (Fig.  67). We have assumed that the hydrodynamic parameters can be approximated by rational Laplace transform functions, R(ω)≈R(s)∣s=iω and mr(ω)≈mr(s)∣s=iω.

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Figure 2

An illustration of the heaving sphere used as a wave energy absorber in this work. Its radius r is 5 m and it is semisubmerged at its equilibrium position, i.e., the mass m=ρ2πr3/3. A power take-off system (PTO) is connected between the buoy and a fixed reference, giving a force Fm on the buoy.

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Figure 3

Hydrodynamic parameters for a semi-submerged heaving sphere of radius 5 m. The radiation resistance Rr(ω) is given by the black curve, the red dashed curve gives the sum of physical mass and added mass m+mr(ω), and the hydrostatic stiffness S, which is approximated by a constant, is shown by the blue dash-dotted curve.

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Figure 4

Intrinsic impedance terms for the heaving sphere. Also shown is the optimal load resistance Rmopt=Ri(ω)2+Xi(ω)2 for nonreactive control.

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Figure 5

Block diagram for the RL strategy. It gives a machinery force directly proportional to the buoy velocity v(s). The dynamics of the heaving buoy is represented by the transfer function Hu(s).

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Figure 6

Regular wave (T=9 s and H=1 m) time series example for the RL strategy. The red dashed curve gives the excitation force Fe, the blue dash-dotted curve gives the velocity v=η̇, and the fully drawn black curve gives the heave position η.

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Figure 7

Block diagram for the ACC control. The machinery force is the sum of terms proportional to position, velocity and acceleration.

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Figure 8

Regular wave (T=9 s and H=1 m) time series example for the ACC strategy. Line patterns as in Fig. 6.

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Figure 9

Block diagram for the approximate optimal velocity tracking (AVT) strategy. The velocity reference signal is tracked by the use of a PI controller ZPI(s).

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Figure 10

Regular wave (T=9 s and H=1 m) time series example for the AVT strategy. Line patterns as in Fig. 6.

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Figure 11

The moving prediction horizon principle of model-predictive control (MPC). The control force is changed in discrete time steps that are short compared with the system dynamics (i.e., in practice shorter than shown here). At each discrete time step a new optimization of the predicted response is calculated using the predicted excitation force as input, and the first element of the machinery force vector is applied to the system (indicated by the thick line).

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Figure 12

Block diagram for the MPC strategy. A QP optimization problem is solved at each discrete time step of the controller. The prediction horizon is set to span an interval over which the force prediction is reasonably accurate. Here Th=2.2 s have been used.

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Figure 13

Regular wave (T=9 s and H=1 m) time series example for the MPC strategy. Line patterns as in Fig. 6.

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Figure 14

Block diagram for the PML and TUL control strategies. The latching mechanism is used to hold the buoy in a fixed position. During motion the machinery force is proportional to the buoy velocity.

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Figure 15

Regular wave (T=9 s and H=1 m) time series example for the PML and TUL strategies. Line patterns as in Fig. 6.

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Figure 16

Block diagram for the PMC and TUC control strategies. The clutching mechanism is used to engage and disengage the machinery force. When engaged, the machinery force is proportional to the buoy velocity.

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Figure 17

Regular wave (T=9 s and H=1 m) time series example for the PMC and TUC strategies. Line patterns as in Fig. 6.

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Figure 18

Block diagram for the extremum-seeking control algorithm. A modulation signal a sin(ωmt) is added to the parameter setting θ̂. The objective function J is chosen as the output from the process, and it is high-pass filtered (HHP) to retain only value changes. The multiplication serves as a demodulation, and the result is low-pass filtered by the −γ/s block (38,30).

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Figure 19

Phase plot showing the amplitude constraints principle for velocity tracking control. The red dashed line gives a saturation limit for the combined state of position η and velocity η̇.

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Figure 20

Budal diagram showing the average absorbed power for the different control strategies with varying wave period of regular waves with height H=1 m

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Figure 21

Budal diagram showing the average absorbed power for the different control strategies with varying wave period. The incident waves were regular with height H=3 m.

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Figure 22

Peak-to-average ratio for the absorbed power in regular waves with height 1 m for the controllers giving reactive power flow

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Figure 23

Peak-to-average power ratio in regular waves with height 1 m for the controllers without reactive power flow

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Figure 24

Absorption width da=P¯u/J (m) (based on absorbed useful energy) for the simulation in irregular waves and different control strategies. Horizontal dashed lines are drawn at da=5 m and da=10 m. The sea states were synthesized from a Bretschneider spectrum, and significant wave height and energy period are given to the left and on the bottom of the graph, respectively. The wave power level for each sea state may be read from Table 1.

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Figure 25

Peak-to-average ratio for the power flowing through the machinery for incident irregular waves. The wave excitation corresponds to the results presented in Fig. 2. Horizontal dashed lines are drawn at the values −25, 25, and 50. Some of the bars go outside the scale. Their sizes are then given by the white numbers.

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Figure 26

Control parameter values for the three of the control strategies with automatic tuning. The red circle-marked curves result from parameter tuning by gain scheduling, and the blue square-marked curves are due to extremum-seeking control. The sea state is a weighted sum (see Eq. 19) of two sea states, one with Te=6 s and Hs=1.41 m and the other with Te=12 s and Hs=2.83 m.

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Figure 27

Accumulated useful energy corresponding to the cases presented in Fig. 2. The black triangle-marked curve gives the energy absorbed without parameter tuning (i.e., keeping the initial setting throughout the simulation).

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