# Incorporating Mooring Dynamics into the Control Design of a Two-Body Wave Energy Converter

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mooring Models

#### 2.1.1. Catenary Mooring

#### 2.1.2. Heavy Catenary

#### 2.1.3. Taut-Leg

#### 2.1.4. Lazy-S

#### 2.2. System Modeling and the SRPA Canonical Form

#### 2.2.1. Mooring as a Circuit Element

#### 2.2.2. Equivalent Single-Body Circuit (The Canonical Form)

#### 2.2.3. Control Strategies

#### 2.3. Mooring Characterization

#### 2.3.1. Mooring Linearization

#### 2.3.2. System Identification

#### 2.3.3. SID Results

## 3. Results

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#### 3.1. Influence on Useful Power

#### 3.2. Influence on WEC Design

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Acronyms | ${F}_{{k}_{i}}$ | Hydrostatic stiffness force | |

DOF | Degrees of freedom | ${F}_{{m}_{i}}$ | Inertia force |

FFT | Fast Fourier transform algorithm | ${F}_{moor,mag}$ | Magnitude of mooring force on spar |

PTO | Power take-off | ${F}_{P}{{}_{T}}_{O}$ | Force due to power take-off |

SID | System identification | ${F}_{Th}$ | Thevenin equivalent single-body force |

SRPA | Self-reacting point absorber | ${k}_{1}$ | Hydrostatic stiffness of float |

WEC | Wave energy converter | ${k}_{2}$ | Hydrostatic stiffness of spar |

C1 | Case 1: power capture and control without mooring dynamics | ${m}_{1}$ | Mass of float |

C2 | Case 2: power capture with mooring dynamics | ${m}_{2}$ | Mass of spar |

C3 | Case 3: power capture and control with mooring dynamics | $\begin{array}{l}N\left(\omega \right),\\ D\left(\omega \right)\end{array}$ | Polynomial numerator of transfer function, polynomial denominator of transfer function |

Parameters | |||

${\varphi}_{moor}$ | Phase of mooring force with respect to spar velocity | $t$ | Time (seconds) |

$\omega $ | Frequency of wave exciting the system (Hz) | ${u}_{1}$ | Heave velocity of float |

${\omega}_{n}$ | Discrete frequency system oscillates at (rad/s) | ${u}_{2}$ | Heave velocity of the spar |

${A}_{1}$ | Added mass of the float | ${u}_{Th}$ | Relative velocity between float and spar |

${A}_{2}$ | Added mass of the spar | ${x}_{2}$ | Heave position of spar |

${b}_{1}$ | Radiation damping of float | ${Z}_{e{q}_{1}}$ | Total impedance of the float |

${b}_{2}$ | Radiation damping of spar | ${Z}_{e{q}_{2}}$ | Total impedance of the spar |

${F}_{e{x}_{1}}$ | Excitation force per unit wave amplitude acting on the float | ${Z}_{m}{{}_{o}}_{o}{}_{r}$ | Impedance of the mooring |

${F}_{e{x}_{2}}$ | Excitation force per unit wave amplitude acting on the spar | ${Z}_{P}{{}_{T}}_{O}$ | Impedance of the power take-off device |

${F}_{{A}_{i}}$ | Added mass force | $\begin{array}{l}{Z}_{PT{O}_{AC}},\\ {Z}_{PT{O}_{AC}}\end{array}$ | PTO impedance governed by amplitude (AC) and complex conjugate (CC) control |

${F}_{{b}_{i}}$ | Radiation damping force | ${Z}_{i}$ | Thevenin equivalent single-body intrinsic impedance |

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**Figure 1.**Generic SRPA WEC modeled after a design studied by [27].

**Figure 2.**Mooring structure: (

**a**) catenary mooring, (

**b**) heavy catenary, (

**c**) taut-leg, and (

**d**) lazy-S.

**Figure 3.**Mechanical circuit of self-reacting point absorber with a mooring. Adapted from [20].

**Figure 5.**FFT of mooring force. Vertical markers bound the content assigned to the excitation frequency.

**Figure 6.**Difference in mooring motion upwards versus downwards showcasing non-linear motion of mooring. Snapshots shown are taken from the heavy catenary test at 0.13 Hz.

**Figure 8.**Percentage of total frequency content of mooring response retained by linearization for each mooring.

**Figure 10.**Frequency response of best-fit transfer function and state-space models expressed as magnitude and phase for (

**a**) catenary, (

**b**) heavy catenary, (

**c**) taut-leg, and (

**d**) lazy-S moorings.

**Figure 13.**Cumulative useful power produced with amplitude control and varying moorings normalized against the no mooring case.

**Figure 14.**Cumulative useful power produced with complex conjugate control and varying moorings normalized against the no mooring case.

**Table 1.**Characterized parameters of full-scale self-reacting point absorber scaled up from [5].

Parameter | Symbol | Value |
---|---|---|

Float excitation force per unit wave amplitude | ${F}_{e{x}_{1}}\left(\omega \right)$ | 18,500,000 to 28,000,000 N/m |

Spar excitation force per unit wave amplitude | ${F}_{e{x}_{2}}\left(\omega \right)$ | 1,000,000 to 6,000,000 N/m |

Mass of float | ${m}_{1}$ | 187,500 kg |

Mass of spar | ${m}_{2}$ | 1,797,000 kg |

Added mass of float | ${A}_{1}\left(\omega \right)$ | 422,500 to 517,000 kg |

Added mass of spar | ${A}_{2}\left(\omega \right)$ | 233,500 to 235,500 kg |

Hydrostatic stiffness of float | ${k}_{1}$ | 1,251,000 N/m |

Hydrostatic stiffness of spar | ${k}_{2}$ | 318,400 N/m |

Radiation damping of float | ${b}_{1}\left(\omega \right)$ | 27,630 to 168,300 Ns/m |

Radiation damping of spar | ${b}_{2}\left(\omega \right)$ | 47,980 to 70,400 Ns/m |

Mooring Design | Wire Rope 8 cm Ø, 40 kg/m | Wire Rope 8 cm Ø, 580 kg/m | Dyneema™ Rope 10 cm Ø, 7 kg/m | Studlink Chain 8 cm Ø, 40 kg/m | Trawl Float 2 m Ø, 1680 kg |
---|---|---|---|---|---|

Catenary | 65 m | - | - | 105 m | - |

Heavy Catenary | - | 65 m | - | 105 m | - |

Taut leg | 40 m | - | - | - | - |

Lazy-S | - | - | 65 m | 50 m | 34 m (on rope) |

${\mathit{Z}}_{\mathit{m}\mathit{o}\mathit{o}\mathit{r}}$ Included in the Calculation of: | ||
---|---|---|

Case | System Plant | Pto Controller |

C1 | X | X |

C2 | ✓ | X |

C3 | ✓ | ✓ |

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**MDPI and ACS Style**

Funk, S.; Haider, A.S.; Bubbar, K.; Buckham, B.
Incorporating Mooring Dynamics into the Control Design of a Two-Body Wave Energy Converter. *J. Mar. Sci. Eng.* **2023**, *11*, 2347.
https://doi.org/10.3390/jmse11122347

**AMA Style**

Funk S, Haider AS, Bubbar K, Buckham B.
Incorporating Mooring Dynamics into the Control Design of a Two-Body Wave Energy Converter. *Journal of Marine Science and Engineering*. 2023; 11(12):2347.
https://doi.org/10.3390/jmse11122347

**Chicago/Turabian Style**

Funk, Spencer, Ali Shahbaz Haider, Kush Bubbar, and Brad Buckham.
2023. "Incorporating Mooring Dynamics into the Control Design of a Two-Body Wave Energy Converter" *Journal of Marine Science and Engineering* 11, no. 12: 2347.
https://doi.org/10.3390/jmse11122347