# Model Predictive Control of a Wave Energy Converter with Discrete Fluid Power Power Take-Off System

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## Abstract

**:**

## 1. Introduction

## 2. Discrete Power Take-Off System

## 3. Dynamic Modeling

#### 3.1. Wave Model

#### 3.2. Float Model

#### 3.3. Discrete PTO System Model

## 4. Model Predictive Control of Discrete PTO System

- Measure (or partially estimate) the full system state $x\left(k\right)$ at the current sampling time $t\left(k\right)$.
- Find the N optimal future system inputs$$\begin{array}{c}\hfill {\mathcal{U}}_{k}=[u\left(k\right),\phantom{\rule{4pt}{0ex}}u(k+1),\phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}u(k+N-1)]\end{array}$$$$\begin{array}{c}\hfill {\mathcal{X}}_{k}=[x(k+1),x(k+2),\phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}x(k+N)]\end{array}$$
- Apply only the first optimal controller output $u\left(k\right)$ to the system and loop back to step 1 for the next sampling instant.

#### 4.1. Prediction Model

#### 4.2. MPC Objective Functions

#### 4.3. MPC Objective Functions—Included Losses

#### 4.4. Loss Models

## 5. Results

#### 5.1. Discrete Reactive Control

#### 5.2. Methodology

#### 5.3. MPC Time Step and Prediction Horizon Analysis

#### 5.4. MPC and Reactive Control Performance Analysis

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Sea state model: (

**a**) Pierson-Moskowitz spectrum for three sea states and (

**b**) example of wave height and excitation torque in time domain for ${T}_{\mathrm{wp}}=4.62$ s and ${\mathrm{H}}_{\mathrm{m}}=1$ m.

**Figure 4.**Bode diagram of the radiation damping term (5) normalized to 1 MNm/(rad/s).

**Figure 5.**Bode diagram of the float dynamics (6) normalized to 1 rad/MNm.

**Figure 9.**Average absorbed power versus prediction horizon for a 25 ms time step for three sea states.

**Figure 11.**Examples of PTO force and chamber pressures for MPC${}_{1}$ and MPC${}_{2}$ using ${T}_{\mathrm{s}}=200$ ms.

**Figure 12.**Average absorbed and harvested power versus sea states using different controllers. (

**a**) ${T}_{\mathrm{s}}$ = 400 ms; (

**b**) ${T}_{\mathrm{s}}$ = 300 ms; (

**c**) ${T}_{\mathrm{s}}$ = 200 ms; (

**d**) Reactive controller versus MPC${}_{3}$.

${J}_{\mathrm{arm}}$ | 2.45 $\times {10}^{6}$ | $\left[{\mathrm{kgm}}^{2}\right]$ |

${J}_{\mathrm{add},\infty}$ | 1.32 $\times {10}^{6}$ | $\left[{\mathrm{kgm}}^{2}\right]$ |

${K}_{\mathrm{res}}$ | 14 $\times {10}^{6}$ | $[\mathrm{Nm}/\mathrm{rad}]$ |

$[{b}_{5},\cdots ,{b}_{0}]$ | $[0.01,1.44,62.4,816,1310,144]\times {10}^{4}$ | |

$[{a}_{5},\cdots ,{a}_{0}]$ | $[0.0010,0.0906,1.67,6.31,13.3,9.18]$ |

Chamber Areas | Valve Flow Coef. | Pressure Levels |
---|---|---|

${A}_{1}=-235$ cm${}^{2}$ | ${k}_{\mathrm{v},1}=705\frac{\mathrm{l}/\mathrm{min}}{\sqrt{5\phantom{\rule{3.33333pt}{0ex}}\mathrm{bar}}}$ | ${p}_{\mathrm{L},1}=20$ bar |

${A}_{2}=+122$ cm${}^{2}$ | ${k}_{\mathrm{v},2}=366\frac{\mathrm{l}/\mathrm{min}}{\sqrt{5\phantom{\rule{3.33333pt}{0ex}}\mathrm{bar}}}$ | ${p}_{\mathrm{L},2}=100$ bar |

${A}_{3}=+87$ cm${}^{2}$ | ${k}_{\mathrm{v},3}=261\frac{\mathrm{l}/\mathrm{min}}{\sqrt{5\phantom{\rule{3.33333pt}{0ex}}\mathrm{bar}}}$ | ${p}_{\mathrm{L},3}=180$ bar |

**Table 3.**Average absorbed power, harvested power and efficiency for three different sea states for different WPEAs.

Absorbed Power (kW) | Harvested Power (kW) | Efficiency (-) | |||||||
---|---|---|---|---|---|---|---|---|---|

Sea State | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

MPC${}_{1}$ | 13.14 | 29.63 | 42.02 | 6.11 | 22.33 | 34.71 | 0.47 | 0.75 | 0.86 |

MPC${}_{2}$ | 12.48 | 29.53 | 42.00 | 9.10 | 25.39 | 37.43 | 0.73 | 0.86 | 0.89 |

MPC${}_{3}$ | 12.32 | 29.37 | 41.65 | 9.15 | 25.48 | 37.43 | 0.74 | 0.87 | 0.90 |

Reactive | 11.23 | 25.41 | 36.36 | 8.68 | 21.86 | 32.52 | 0.77 | 0.86 | 0.89 |

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**MDPI and ACS Style**

Hedegaard Hansen, A.; F. Asmussen, M.; Bech, M.M. Model Predictive Control of a Wave Energy Converter with Discrete Fluid Power Power Take-Off System. *Energies* **2018**, *11*, 635.
https://doi.org/10.3390/en11030635

**AMA Style**

Hedegaard Hansen A, F. Asmussen M, Bech MM. Model Predictive Control of a Wave Energy Converter with Discrete Fluid Power Power Take-Off System. *Energies*. 2018; 11(3):635.
https://doi.org/10.3390/en11030635

**Chicago/Turabian Style**

Hedegaard Hansen, Anders, Magnus F. Asmussen, and Michael M. Bech. 2018. "Model Predictive Control of a Wave Energy Converter with Discrete Fluid Power Power Take-Off System" *Energies* 11, no. 3: 635.
https://doi.org/10.3390/en11030635