# Extending Complex Conjugate Control to Nonlinear Wave Energy Converters

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## Abstract

**:**

## 1. Introduction

## 2. CCC and PDC3

## 3. Electrical Power Networks, Mechanical Oscillators, and Linear Limit Cycles

#### 3.1. Electrical Power Networks

#### 3.2. Mechanical Systems

#### 3.3. Linear Limit Cycles

## 4. Nonlinear Feedback Linearization and PDC3

## 5. HSSPFC and Nonlinear Limit Cycles

#### 5.1. Hour-Glass (HG) WEC Design Model Development

#### 5.2. RCC WEC Design Model Development

## 6. Case Study Simulation Results

#### 6.1. Nonlinear Resonator Results

#### 6.2. Single Frequency Results

#### 6.3. Bretschneider Multi-Spectrum Results

`Bretschneider`and corresponding time domain data by

`spec2dat`Matlab functions from the toolbox in [32]. The varying sea state parameters are given in Table 3 with the corresponding Bretschneider spectrum in the frequency domain shown in Figure 18.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description | SI Units |

$\widehat{}$ | Estimate of parameter | - |

A | Cross-sectional area | m^{2} |

$\alpha $ | HG buoy geometry cone steepness angle | deg |

b | Linear damping coefficient | N s/m |

c | Linear damping coefficient | N s/m |

${c}_{NL1}$ | Nonlinear damping coefficient | N |

${c}_{NL2}$ | Nonlinear damping coefficient | N/(m/s)^{2} |

C | Electrical capacitance | F |

${E}_{max}$ | Maximum harvested energy | MJ |

$\eta $ | Wave elevation | m |

${F}_{ex}$ | External force | N |

${F}_{u}$ | Control force | N |

${F}_{{u}_{NL}}$ | Nonlinear control force | N |

${F}_{{u}_{PDC3}}$ | PDC3 control force | N |

${F}_{reac}$ | Reactive force | N |

${F}_{real}$ | Real force | N |

${F}_{h}$ | Hydrostatic force | N |

${F}_{g}$ | Gravitation force | N |

${F}_{buoy}$ | Buoy force | N |

${F}_{{p}_{i}}$ | Bretschneider ith sea state spectral peak frequency | Hz |

${F}_{0}$ | External force magnitude | N |

g | Gravitational constant | m/s^{2} |

${H}_{s}$ | Bretschneider significant wave height parameter | m |

h | Buoy height, HG, RCC | m |

${h}_{half}$ | Height of buoy and one-half of total draft | m |

${\mathcal{H}}_{cyclic}$ | Hamiltonian over a cycle | J |

$\mathcal{H}$ | Hamiltonian, total energy with subscripts; electrical (e), mechanical (m) | J |

$\dot{\mathcal{H}}$ | Hamiltonian rate, power flow with subscripts; electrical (e), mechanical (m) | W |

j | Sum index for number of force components | - |

$k,K$ | Stiffness coefficient | N/m |

${k}_{LIN2}$ | Linear stiffness coefficient | N/m |

${k}_{NL}$ | Nonlinear stiffness coefficient | N/m^{3} |

${k}_{NL2}$ | Nonlinear stiffness coefficient | N/m^{3} |

${K}_{p}$ | Proportional control gain | kg/s^{2} |

${K}_{d}$ | Derivative control gain | kg/s |

L | Electrical inductance | H |

$m,M$ | Mass of system | kg |

N | Maximum number of force components | - |

$\omega $ | Mechanical system natural frequency | rad/s |

$\mathsf{\Omega}$ | Extern force excitation frequency | rad/s |

$\overline{\omega}$ | Electrical system natural frequency | rad/s |

${P}_{reac}$ | Reactive power | MW |

${P}_{real}$ | Real power | MW |

q | Electrical charge | C |

$\dot{q}$ | Electrical charge rate (equal to current) | C/s |

$\ddot{q}$ | Electrical charge acceleration | C/s^{2} |

r | Buoy radius, HG, RCC | m |

R | Electrical resistance | Ohms |

${R}_{opt}$ | Optimal damping coefficient | N s/m |

${\overline{R}}_{opt}$ | Effective optimal damping coefficient $={R}_{opt}+b$ | N s/m |

$\rho $ | Buoy material density | kg/m^{3} |

${S}_{w}$ | Non-uniform water plane area | m^{2} |

$S\left(w\right)$ | Bretschneider spectral density | m^{2}s/rad |

$\tau $ | Cycle time | sec |

t | time | sec |

$\mathcal{T}$ | Kinetic energy with subscripts; electrical (e), mechanical (m) | J |

${T}_{p}$ | Bretschneider spectral peak period parameter | sec |

V | Volume with subscripts; cone, buoy, RCC, HG | m^{3} |

$V\left(z\right)$ | Volume as a function of heave displacement | m^{3} |

${v}_{0}$ | External voltage magnitude | V |

$\mathcal{V}$ | Potential energy with subscripts; electrical (e), mechanical (m) | J |

x | Displacement | m |

$\dot{x}$ | Velocity | m/s |

$\ddot{x}$ | Acceleration | m/s^{2} |

$\widehat{x}$ | x-coordinate | - |

$\widehat{y}$ | y-coordinate | - |

$\widehat{z}$ | z-coordinate | - |

z | Heave displacement | m |

$\dot{z}$ | Heave velocity | m/s |

$\ddot{z}$ | Heave acceleration | m/s^{2} |

$\zeta $ | Vertical position of the center of volume of buoy | m |

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**Figure 4.**For a single frequency the resonance $\omega =\mathsf{\Omega}$, (

**left**) and off-resonance $\omega \ne \mathsf{\Omega}$, (

**right**) plots.

**Figure 5.**For a single frequency the resonance $\omega =\mathsf{\Omega}$ and off-limit cycle $\omega \ne \mathsf{\Omega}$ comparisons, 3d (

**left**) and phase-plane (

**right**) plots.

**Figure 8.**For a single frequency utilizes RCC with nonlinear (NL) cubic spring (

**left**) and NL HG (

**right**), respectively.

**Figure 20.**SS4 external Bretschneider wave input (

**left**) and external wave force (

**right**), respectively.

Parameter | Symbol | HG Range | RCC Value | Unit |
---|---|---|---|---|

Radius | r | 5.72–10.0 | 4.47 | m |

Height | h | 8.18–2.68 | 4.47 | m |

Angle | $\alpha $ | 50–70 | 0.00 | deg |

Parameter | Unit | RCC_{NL} | RCC | HG | RCC_{10} | HG_{10} | RCC_{20} | HG_{20} | RCC_{30} | HG_{30} |
---|---|---|---|---|---|---|---|---|---|---|

$\alpha $ | deg | N/A | N/A | 59.5 | N/A | 56.5 | N/A | 53.5 | N/A | 50.9 |

${\overline{R}}_{opt}$ | $\left(\mathrm{N}\frac{\mathrm{s}}{\mathrm{m}}\right)\xb7{10}^{5}$ | 3.844 | 4.456 | N/A | 4.848 | N/A | 5.242 | N/A | 5.746 | N/A |

${h}_{limit}$ | m | 4.47 | 4.47 | 4.53 | 4.47 | 4.896 | 4.47 | 5.274 | 4.47 | 5.614 |

${E}_{max}$ | MJ | 129 | 146 | 146 | 160 | 171 | 173 | 197 | 183 | 226 |

Sea State | ${\mathit{H}}_{\mathit{s}}$ (m) | ${\mathit{T}}_{\mathit{p}}$ (sec) | Duration (sec) |
---|---|---|---|

1 | 5.7 | 8.0 | 300.0 |

2 | 6.6 | 6.6 | 300.0 |

3 | 7.8 | 7.8 | 300.0 |

4 | 6.9 | 11.0 | 300.0 |

Angle | Draft | Sea State 1 | Sea State 2 | Sea State 3 | Sea State 4 |
---|---|---|---|---|---|

$\mathbf{\alpha}$ | h_{half} | E_{max} | E_{max} | E_{max} | E_{max} |

deg | m | MJ | MJ | MJ | MJ |

55 | 5.084 | 26.485 | 23.935 | 174.63 | 32.230 |

60 | 4.470 | 43.240 | 39.235 | SAT | 48.564 |

65 | 3.8767 | 67.170 | 61.550 | – | 69.790 |

70 | 3.2864 | SAT | 92.752 | – | SAT |

75 | 2.680 | – | SAT | – | – |

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## Share and Cite

**MDPI and ACS Style**

Wilson, D.G.; Robinett, R.D., III; Bacelli, G.; Abdelkhalik, O.; Coe, R.G.
Extending Complex Conjugate Control to Nonlinear Wave Energy Converters. *J. Mar. Sci. Eng.* **2020**, *8*, 84.
https://doi.org/10.3390/jmse8020084

**AMA Style**

Wilson DG, Robinett RD III, Bacelli G, Abdelkhalik O, Coe RG.
Extending Complex Conjugate Control to Nonlinear Wave Energy Converters. *Journal of Marine Science and Engineering*. 2020; 8(2):84.
https://doi.org/10.3390/jmse8020084

**Chicago/Turabian Style**

Wilson, David G., Rush D. Robinett, III, Giorgio Bacelli, Ossama Abdelkhalik, and Ryan G. Coe.
2020. "Extending Complex Conjugate Control to Nonlinear Wave Energy Converters" *Journal of Marine Science and Engineering* 8, no. 2: 84.
https://doi.org/10.3390/jmse8020084