# Hybrid Model Predictive Control of a Two-Body Wave Energy Converter with Mechanically Driven Power Take-Off

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## Abstract

**:**

## 1. Introduction

## 2. Wave Energy Converter Model

#### 2.1. Structure

_{exc}represents the wave excitation force; F

_{radij}is the radiation force imposed on the body i by the oscillation of the body j; F

_{b}is the net buoyancy restoring force; F

_{pto}is the PTO force; and F

_{u}is the control force. The wave excitation force F

_{exc}, radiation force F

_{rad}, and net buoyancy restoring force F

_{b}are hydrodynamic forces, which can be calculated using hydrodynamic coefficients provided by frequency-domain boundary element method (BEM) solvers, such as WAMIT, AQWA, and NEMOH.

#### 2.2. Hydrodynamic Force

_{j}is the wave amplitude at angular frequency ${\omega}_{j}$, which can be obtained from the wave spectrum; ${\varphi}_{j}$ is the randomized phase angle at angular frequency ${\omega}_{j}$, and N is the number of frequency bands selected to discretize the wave spectrum. The Pierson–Moskowitz (PM) spectrum was chosen to describe incident waves in this paper.

_{s}is the significant wave height; f

_{p}is the peak frequency; and dω

_{j}is the frequency interval.

_{ij}is the added mass at infinite frequency and K

_{rij}is the radiation impulse response function. To improve the computing efficiency, a state-space approximation of the convolution term in Equation (6) is needed. It is constructed as follows:

**A**is the system matrix, the dimension of which depends on the accuracy of approximation representation;

_{r}**B**is the input matrix;

_{r}**C**is the output matrix;

_{r}**D**is the feedthrough matrix, which is assumed to be zero to maintain the causality of the system and will be omitted in the following equations; and

_{r}**q**and $\dot{q}$ are the system state and its derivative, respectively. The system input is the heave speed of the corresponding body. Figure 2 shows the impulse response functions and their state-space realizations. The approximate accuracies of all state-space realizations are above 95%. The dimensions of all system matrices are $2\times 2$.

_{H}is the hydrostatic stiffness coefficient.

#### 2.3. Power Take-Off Force

_{i}is the input moment of the gearbox, and the transmission ratio of the gearbox is denoted as n

_{g}. The angular speed and input moment of the generator are calculated as follows:

_{g}is the angular speed of the generator, and M

_{g}is the input shaft moment of the generator, which is the sum of the electric torque load and the torque induced by the inertia of the generator.

_{r}is the electric moment induced by the current in the energy harvesting circuit, and I

_{g}is the moment of inertia of the generator.

_{t}is the torque constant of the generator; i is the output current of the generator; V is the output voltage of the generator; R

_{i}is the internal resistance (motor armature resistance) in the circuit; and R

_{o}is the external resistance (passive load) in the circuit [24,25,26].

_{pto}is the mass of the PTO system; c

_{pto}is the damping of the PTO system; and k

_{pto}is the stiffness of the PTO system. By putting together Equations (14) and (15), it can be deduced as follows.

#### 2.4. Control Force

_{d}is opposite to that of the control force ${F}_{u}$, i.e., F

_{d}= −F

_{u}.

_{u}and relative velocity ${\dot{x}}_{1}-{\dot{x}}_{2}$ at each time, which can be described as a non-linear inverse function of the damper force, as follows:

#### 2.5. State-Space Model

_{s}(T

_{s}= 0.25 s in this study) to obtain the discrete state-space representation for the WEC system.

## 3. Hybrid MPC Problem Formulation

#### 3.1. Hybrid Dynamical Model

_{v}.

_{v}.

_{l}and v

_{u}are lower and upper bounds to the function ${\dot{x}}_{1}-{\dot{x}}_{2}$, respectively; $\epsilon >0$ is a small tolerance, typically the machine precision.

_{1}, z

_{2}, and z

_{3}$\in \mathbb{R}$ are continuous auxiliary variables, which are imposed constraints.

_{1}, a

_{2}, b

_{1}, b

_{2}, f

_{1}, and f

_{2}are constants with approximate dimensions, which can be translated into integer linear inequalities of the form.

_{i}and F

_{i}are lower and upper bounds to ${a}_{i}x+{b}_{i}u+{c}_{i}$, $i\in \left\{1,2\right\}$. Each if–then–else equation in Equation (29) can be translated into four linear inequality constraints.

_{min}and x

_{max}are the maximum and minimum relative displacement that the system can achieve. ${\dot{x}}_{min}$ and ${\dot{x}}_{max}$ are the maximum and minimum relative velocity. The relative displacement constraint represents a finite stroke limit of the WEC device, which prevents the PTO hardware and the surrounding structure from impulsive loads [29].

**A**,

_{d}**B**, and

_{du}**B**are the discrete system matrices from the original state-space Equation (25);

_{de}**B**and

_{dδ}**B**are the zero matrices in our case; and

_{dz}**E**,

_{2}**E**,

_{3}**E**,

_{1}**E**, and

_{4}**E**are matrices of suitable dimensions.

_{5}#### 3.2. Hybrid MPC Problem

_{p}is the optimization horizon length.

## 4. Numerical Investigation and Result

#### 4.1. Case Study

^{3}and the gravity acceleration takes the value of 9.81 m/s

^{2}.

_{p}= 8 s and H

_{s}= 2.5 m wave input. The spectrum and corresponding wave elevation are shown in Figure 5. The prediction horizon N

_{p}= 32 (8 s). The entire simulation time is 120 s.

#### 4.2. Control Force Constraint

_{3}and β

_{4}cannot be zero. To investigate the effect of parameters β

_{3}and β

_{4}on the performance of hybrid MPC, we assumed that β

_{3}is equal to β

_{4}and the wave excitation force prediction is exact. The average power $\overline{P}$ over the horizon length N

_{s}in terms of the number of samples can be calculated as follows:

_{3}and β

_{4}are listed in Table 5. The simulated result is under the wave condition of T

_{p}= 8 s and H

_{s}= 2.5 m, and the prediction horizon N

_{p}is 32 (8 s). The average generated power of uncontrolled WEC is 76.808 kW.

_{3}and β

_{4}is maximum. The power generation gradually decreases as the coefficients β

_{3}and β

_{4}increase, and there is a critical value β

_{c}(10

^{4}< β

_{c}< 10

^{5}); the power generation will be less than the case of no control when the boundary coefficients are greater than this value. The variable damper is placed in parallel with the PTO so that the captured energy by WEC is converted into PTO energy and hydraulic energy, with the former being subsequently converted into available electricity and the latter being converted into thermal energy dissipation. The energy converted via internal damping can be improved by adjusting the damper in a small interval of damping coefficient when less energy is dissipated by the variable damper, and most energy is converted into electrical energy. However, a too-large damping coefficient will result in too much energy being dissipated by the damper, reducing the improved performance of variable damping. In reality, a minor boundary coefficient of the electrohydraulic damper is challenging to obtain due to the limitations of the openings, and so a larger boundary coefficient (β

_{3}= β

_{4}= 1 × 10

^{4}Ns/m) was applied in subsequent studies.

#### 4.3. Prediction Horizon and Wave Condition

_{s}and T

_{p}of irregular waves range from 1~5 m to 6~12 s, respectively, which cover most of the possible sea conditions.

_{p}≤ 16 (4 s). With the increase in prediction horizon N

_{p}≥ 32 (8 s), the power improvement percentage tends to be steady, and the effect of increasing the prediction horizon on power stops being apparent. This indicates that the proposed hybrid MPC has a high requirement for the prediction capability of wave excitation force. However, the simulations here assume that the future wave excitation force is precisely known, which is different from the actual situation. The prediction of the wave excitation force is subject to some error as the prediction horizon increases, which also affects the performance of the hybrid MPC. The effects of the prediction error of wave excitation force will be discussed in the following section.

_{p}= 48, T

_{p}= 11 s, H

_{s}= 1 m). It should be noted that the simulations all assume a large boundary coefficient (β

_{3}= β

_{4}= 1 × 104 Ns/m). If the smaller coefficient can be selected, the power generation by the hybrid MPC system will improve.

#### 4.4. Prediction Error

_{s}= 2.5 m, T

_{p}= 8 s). The GoF

_{1}and GoF

_{2}are the prediction accuracy for the float and reaction section, respectively. It can be seen that the GoF is large within a short prediction horizon, and the prediction error gradually increases as the prediction horizon increases. The AR model has sufficient capacity to predict future wave excitation force when the prediction horizon is less than 6 s, with over 94% GoF for the float and reaction section. However, the prediction accuracy decays rapidly when the prediction horizon exceeds 8 s. Another noteworthy phenomenon is that the prediction accuracy for the reaction section is greater than that of the float for the same prediction horizon. The possible reason is that the motion response of the reaction plate is smooth and tends to be more of a stationary time series.

_{1}and Power

_{2}are the generation power under the exact prediction and the generation power under the AR prediction, respectively. It can be seen that the power generation under the inaccurate prediction is better than that under exact prediction within a short horizon (N

_{p}≤ 16). This indicates that the solution of the hybrid MPC is suboptimal, and a more accurate solution method needs to be investigated. As the prediction error increases, the degree of power degradation for non-accurate prediction tends to be extended. In summary, the determination of the prediction horizon is essential, which has a systemic impact on MPC.

## 5. Conclusions

_{p}and small H

_{s}wave conditions. The prediction horizon has a systemic impact on hybrid MPC performance. A long prediction horizon can improve generation power compared to no-control devices, although a large prediction error accompanies this.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Average output power ratio of different prediction horizons under irregular wave conditions.

## References

- Ringwood, J.V.; Bacelli, G.; Fusco, F. Energy-Maximizing Control of Wave-Energy Converters: The Development of Control System Technology to Optimize Their Operation. IEEE Control Syst. Mag.
**2014**, 34, 30–55. [Google Scholar] [CrossRef] - Lehmann, M.; Karimpour, F.; Goudey, C.A.; Jacobson, P.T.; Alam, M.-R. Ocean Wave Energy in the United States: Current Status and Future Perspectives. Renew. Sustain. Energy Rev.
**2017**, 74, 1300–1313. [Google Scholar] [CrossRef] - Montoya Andrade, D.-E.A.; de la Villa Jaen, A.; Garcia Santana, A. Improvements in the Reactive Control and Latching Control Strategies Under Maximum Excursion Constraints Using Short-Time Forecast. IEEE Trans. Sustain. Energy
**2016**, 7, 427–435. [Google Scholar] [CrossRef] - Falnes, J.; Kurniawan, A. Ocean Waves and Oscillating Systems: Linear Interactions Including Wave-Energy Extraction, 2nd ed.; Cambridge University Press: Cambridge, UK, 2020; ISBN 978-1-108-67481-2. [Google Scholar]
- Budal, K.; Falnes, J. Interacting Point Absorbers with Controlled Motion. In Power from Sea Waves; Academic Press: Cambridge, MA, USA, 1980; pp. 381–399. [Google Scholar]
- Teillant, B.; Gilloteaux, J.C.; Ringwood, J.V. Optimal Damping Profile for a Heaving Buoy Wave Energy Converter. IFAC Proc. Vol.
**2010**, 43, 360–365. [Google Scholar] [CrossRef] - Feng, Z.; Kerrigan, E.C. Declutching Control of Wave Energy Converters Using Derivative-Free Optimization. IFAC Proc. Vol.
**2014**, 47, 7647–7652. [Google Scholar] [CrossRef] - Feng, Z.; Kerrigan, E.C. Latching-Declutching Control of Wave Energy Converters Using Derivative-Free Optimization. IEEE Trans. Sustain. Energy
**2015**, 6, 773–780. [Google Scholar] [CrossRef] - Wang, Z.; Luan, F.; Wang, N. An Improved Model Predictive Control Method for Wave Energy Converter with Sliding Mode Control. Ocean Eng.
**2021**, 240, 109881. [Google Scholar] [CrossRef] - Li, L.; Gao, Y.; Ning, D.Z.; Yuan, Z.M. Development of a Constraint Non-Causal Wave Energy Control Algorithm Based on Artificial Intelligence. Renew. Sustain. Energy Rev.
**2021**, 138, 110519. [Google Scholar] [CrossRef] - Lin, Z.; Huang, X.; Xiao, X. A Novel Model Predictive Control Formulation for Wave Energy Converters Based on the Reactive Rollout Method. IEEE Trans. Sustain. Energy
**2022**, 13, 491–500. [Google Scholar] [CrossRef] - Sergiienko, N.Y.; Cocho, M.; Cazzolato, B.S.; Pichard, A. Effect of a Model Predictive Control on the Design of a Power Take-off System for Wave Energy Converters. Appl. Ocean Res.
**2021**, 115, 102836. [Google Scholar] [CrossRef] - Faedo, N.; Olaya, S.; Ringwood, J.V. Optimal Control, MPC and MPC-like Algorithms for Wave Energy Systems: An Overview. IFAC J. Syst. Control
**2017**, 1, 37–56. [Google Scholar] [CrossRef] - Li, X.; Chen, C.; Li, Q.; Xu, L.; Liang, C.; Ngo, K.; Parker, R.G.; Zuo, L. A Compact Mechanical Power Take-off for Wave Energy Converters: Design, Analysis, and Test Verification. Appl. Energy
**2020**, 278, 115459. [Google Scholar] [CrossRef] - Jusoh, M.A.; Ibrahim, M.Z.; Daud, M.Z.; Albani, A.; Mohd Yusop, Z. Hydraulic Power Take-Off Concepts for Wave Energy Conversion System: A Review. Energies
**2019**, 12, 4510. [Google Scholar] [CrossRef] - Falcão, A.F.O.; Henriques, J.C.C.; Gato, L.M.C. Self-Rectifying Air Turbines for Wave Energy Conversion: A Comparative Analysis. Renew. Sustain. Energy Rev.
**2018**, 91, 1231–1241. [Google Scholar] [CrossRef] - Penalba, M.; Ringwood, J.V. A Review of Wave-to-Wire Models for Wave Energy Converters. Energies
**2016**, 9, 506. [Google Scholar] [CrossRef] - Li, X.; Liang, C.; Chen, C.-A.; Xiong, Q.; Parker, R.G.; Zuo, L. Optimum Power Analysis of a Self-Reactive Wave Energy Point Absorber with Mechanically-Driven Power Take-Offs. Energy
**2020**, 195, 116927. [Google Scholar] [CrossRef] - Martin, D.; Li, X.; Chen, C.-A.; Thiagarajan, K.; Ngo, K.; Parker, R.; Zuo, L. Numerical Analysis and Wave Tank Validation on the Optimal Design of a Two-Body Wave Energy Converter. Renew. Energy
**2020**, 145, 632–641. [Google Scholar] [CrossRef] - Yang, L.; Huang, J.; Congpuong, N.; Chen, S.; Mi, J.; Bacelli, G.; Zuo, L. Control Co-Design and Characterization of a Power Takeoff for Wave Energy Conversion Based on Active Mechanical Motion Rectification. IFAC-Pap.
**2021**, 54, 198–203. [Google Scholar] [CrossRef] - Li, Q.; Li, X.; Mi, J.; Jiang, B.; Chen, S.; Zuo, L. Tunable Wave Energy Converter Using Variable Inertia Flywheel. IEEE Trans. Sustain. Energy
**2021**, 12, 1265–1274. [Google Scholar] [CrossRef] - Yang, Y.; Chen, P.; Liu, Q. A Wave Energy Harvester Based on Coaxial Mechanical Motion Rectifier and Variable Inertia Flywheel. Appl. Energy
**2021**, 302, 117528. [Google Scholar] [CrossRef] - Zhong, Q.; Yeung, R.W. Model-Predictive Control Strategy for an Array of Wave-Energy Converters. J. Mar. Sci. Appl.
**2019**, 18, 26–37. [Google Scholar] [CrossRef] - Liu, Y.; Xu, L.; Zuo, L. Design, Modeling, Lab, and Field Tests of a Mechanical-Motion-Rectifier-Based Energy Harvester Using a Ball-Screw Mechanism. IEEE/ASME Trans. Mechatron.
**2017**, 22, 1933–1943. [Google Scholar] [CrossRef] - Lin, T.; Pan, Y.; Chen, S.; Zuo, L. Modeling and Field Testing of an Electromagnetic Energy Harvester for Rail Tracks with Anchorless Mounting. Appl. Energy
**2018**, 213, 219–226. [Google Scholar] [CrossRef] - Li, X.; Chen, C.-A.; Chen, S.; Xiong, Q.; Huang, J.; Lambert, S.; Keller, J.; Parker, R.G.; Zuo, L. Dynamic Characterization and Performance Evaluation of a 10-kW Power Take-off with Mechanical Motion Rectifier for Wave Energy Conversion. Ocean Eng.
**2022**, 250, 110983. [Google Scholar] [CrossRef] - Qin, Y.; Zhao, F.; Wang, Z.; Gu, L.; Dong, M. Comprehensive Analysis for Influence of Controllable Damper Time Delay on Semi-Active Suspension Control Strategies. J. Vib. Acoust.
**2017**, 139, 031006. [Google Scholar] [CrossRef] - Borrelli, F.; Bemporad, A.; Morari, M. Predictive Control for Linear and Hybrid Systems; Cambridge University Press: Cambridge, UK, 2017; ISBN 1-108-15829-3. [Google Scholar]
- Lao, Y.; Scruggs, J.T.; Karthikeyan, A.; Previsic, M. Discrete-Time Causal Control of a Wave Energy Converter With Finite Stroke in Stochastic Waves. IEEE Trans. Control Syst. Technol.
**2022**, 30, 1198–1214. [Google Scholar] [CrossRef] - Torrisi, F.D.; Bemporad, A. HYSDEL—A Tool for Generating Computational Hybrid Models for Analysis and Synthesis Problems. IEEE Trans. Contr. Syst. Technol.
**2004**, 12, 235–249. [Google Scholar] [CrossRef] - Yu, Y.H.; Lawson, M.; Li, Y.; Previsic, M.; Epler, J.; Lou, J. Experimental Wave Tank Test for Reference Model 3 Floating-Point Absorber Wave Energy Converter Project; National Renewable Energy Laboratory (NREL): Golden, CO, USA, 2015.

**Figure 6.**The wave excitation force, radiation force, and buoyancy restoring force under no control.

**Figure 7.**The wave excitation force, radiation force, and buoyancy restoring force under hybrid MPC.

Body | Diameter (m) | Draft (m) | Mass (t) | CoG (m) | |
---|---|---|---|---|---|

Float | 20 | 3 | 727.01 | −0.72 | |

Reaction section | Spar | 6 | 30 | 878.30 | −21.29 |

Plate | 30 |

Symbol | Quantity | Parameter Explanation |
---|---|---|

n_{g} | 10 | gearbox ratio |

I_{g} | 0.54 kg∙m^{2} | generator moment of inertia |

L | 0.1 m | ballscrew lead |

k_{t} | 2.5 Nm/A | generator torque constant |

k_{e} | 120 Vs/rad | generator speed constant |

R_{i} | 2.1 Ω | internal resistance |

R_{o} | 100 Ω | external resistance |

Parameter | β1 | β2 | β3 | β4 | β5 | α1 | α5 |
---|---|---|---|---|---|---|---|

Value | 1.2 × 10^{6} | 2.4 × 10^{6} | 1.0 × 10^{4} | 1.0 × 10^{4} | 1.8 × 10^{6} | 6.0 × 10^{5} | −4.2 × 10^{5} |

Symbol | Quantity | Parameter Explanation |
---|---|---|

x_{max} | 8 m | maximum relative displacement |

x_{min} | −8 m | minimum relative displacement |

v_{min} | 5 m/s | maximum relative velocity |

v_{max} | −5 m/s | minimum relative velocity |

β_{3}, β_{4} (10^{4}) | Average Power (kW) | Power Improvement Percentage (%) |
---|---|---|

0 | 82.214 | 7.4454 |

0.01 | 82.206 | 7.4349 |

0.1 | 82.139 | 7.3474 |

1 | 81.469 | 6.4718 |

10 | 75.406 | −1.4520 |

50 | 55.483 | −27.4893 |

100 | 41.453 | −45.8251 |

N_{p} | GoF_{1} (%) | GoF_{2} (%) | Power_{1} (kW) | Power_{2} (kW) | Power Improvement Percentage (%) |
---|---|---|---|---|---|

8 | 99.99 | 99.99 | 75.73 | 77.02 | 1.70% |

16 | 99.56 | 99.79 | 79.48 | 79.56 | 1.00% |

24 | 94.31 | 96.67 | 81.31 | 81.06 | −0.31% |

32 | 81.62 | 91.93 | 81.47 | 81.44 | −0.04% |

40 | 67.28 | 78.06 | 81.34 | 80.57 | −0.95% |

48 | 39.58 | 60.09 | 81.28 | 77.50 | −4.65% |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Z.; Qin, J.; Wang, D.; Huang, S.; Liu, Y.; Xue, G.
Hybrid Model Predictive Control of a Two-Body Wave Energy Converter with Mechanically Driven Power Take-Off. *J. Mar. Sci. Eng.* **2023**, *11*, 1618.
https://doi.org/10.3390/jmse11081618

**AMA Style**

Zhang Z, Qin J, Wang D, Huang S, Liu Y, Xue G.
Hybrid Model Predictive Control of a Two-Body Wave Energy Converter with Mechanically Driven Power Take-Off. *Journal of Marine Science and Engineering*. 2023; 11(8):1618.
https://doi.org/10.3390/jmse11081618

**Chicago/Turabian Style**

Zhang, Zhenquan, Jian Qin, Dengshuai Wang, Shuting Huang, Yanjun Liu, and Gang Xue.
2023. "Hybrid Model Predictive Control of a Two-Body Wave Energy Converter with Mechanically Driven Power Take-Off" *Journal of Marine Science and Engineering* 11, no. 8: 1618.
https://doi.org/10.3390/jmse11081618