# Data-Driven Model Predictive Control for Wave Energy Converters Using Gaussian Process

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## Abstract

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## 1. Introduction

- A novel data-driven WEC model using machine learning techniques and targeting the control perspective is proposed, promising to advance state-of-the-art WEC modelling. The PAWEC system dynamics are learned by the Gaussian Process model, which aims to capture the nonlinear system characteristics with mean value and uncertainties.
- Developing a new data-driven MPC scheme based on the GP model for efficient and real-time implementation in the actual operation of WEC. The cross-entropy technique is introduced to deal with the trajectory optimization for fast, sample-efficient and high performance.
- The investigation of the performance of the data-driven MPC compared with the classical complex-conjugate controller is expected to fill the gap in the literature.
- The developed GP-based MPC scheme is validated in a small-sized and single-type WEC in this study, which can generally be applied to any WECs across different deployment prototypes (e.g., sizes, shapes) and other energy-maximizing control problems.

## 2. Classical WaveStar PAWEC Modelling and Gaussian Process Regression

#### 2.1. Classical WaveStar PAWEC Modelling

^{2}means the equivalent moment of inertia and ${\mathbf{0}}_{m\times n}$ is a zero matrix with m rows and n columns.

#### 2.2. Gaussian-Process-Based Modeling Method

## 3. Control for Optimal Power Extraction from WEC

#### 3.1. Complex-Conjugate Control

#### 3.2. Data-Driven MPC Design with Cross-Entropy Optimization

#### 3.2.1. Cross-Entropy Optimization

Algorithm 1: The CEM optimization algorithm. |

#### 3.2.2. Data-Driven MPC Formulation

## 4. Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The scaled WaveStar device in the WEC-Sim [22].

**Figure 5.**Time–domain comparison of regression results for the WaveStar by different models. (

**a**) The WaveStar angular position. (

**b**) The angular velocity.

**Figure 6.**The irregular wave scenario used in the simulations. (

**a**) Wave amplitude. (

**b**) Spectral energy distribution.

**Figure 8.**The control action ${F}_{pto}$, the corresponding system buoy position X and velocity $\dot{X}$ under the proposed data-driven MPC strategy.

**Figure 10.**Comparison of results for the instantaneous and accumulated power under the two control strategies. (

**a**) The instantaneous power. (

**b**) The accumulated power.

Comparison Items | GP Regression Model | State-Space Model |
---|---|---|

NMSE of angular position ($\theta $) | 0.2971 | 0.9951 |

NMSE of angular velocity ($\dot{\theta}$) | 0.3609 | 0.9607 |

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**MDPI and ACS Style**

Liu, Y.; Shi, S.; Zhang, Z.; Di, Z.; Babayomi, O.
Data-Driven Model Predictive Control for Wave Energy Converters Using Gaussian Process. *Symmetry* **2022**, *14*, 1284.
https://doi.org/10.3390/sym14071284

**AMA Style**

Liu Y, Shi S, Zhang Z, Di Z, Babayomi O.
Data-Driven Model Predictive Control for Wave Energy Converters Using Gaussian Process. *Symmetry*. 2022; 14(7):1284.
https://doi.org/10.3390/sym14071284

**Chicago/Turabian Style**

Liu, Yanhua, Shuo Shi, Zhenbin Zhang, Zhenfeng Di, and Oluleke Babayomi.
2022. "Data-Driven Model Predictive Control for Wave Energy Converters Using Gaussian Process" *Symmetry* 14, no. 7: 1284.
https://doi.org/10.3390/sym14071284