# Multi-Mode Wave Energy Converter Design Optimisation Using an Improved Moth Flame Optimisation Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Modelling

#### 3.1. Wave Energy Converter

#### 3.2. Site Location and Wave Resource

#### 3.3. Equations of Motion

- Step 1.
- Select the sea state of interest and calculate the incident wave spectrum ${S}_{\eta}(\omega )$ for the given ${H}_{s}$ and ${T}_{p}$. A Pierson–Moskowitz wave spectrum was used in this study [39].
- Step 2.
- Calculate the power spectral density (PSD) matrix of the excitation force:$${\mathbf{S}}_{\mathbf{F}}(\omega )={S}_{\eta}(\omega ){\widehat{\mathbf{f}}}_{exc}(\omega ){\widehat{\mathbf{f}}}_{exc}^{\ast}(\omega ),$$
- Step 3.
- Evaluate the WEC transfer function:$$\mathbf{H}(\omega )={\left[i\omega \left(\mathbf{M}+{\mathbf{A}}_{rad}(\omega )\right)+\left({\mathbf{B}}_{rad}(\omega )+{\mathbf{D}}_{pto}+{\mathbf{B}}_{visc}\right)-i\frac{{\mathbf{K}}_{pto}}{\omega}\right]}^{-1},$$
- Step 4.
- Calculate the PSD matrix of the buoy velocity:$${\mathbf{S}}_{\mathbf{u}}(\omega )=\mathbf{H}(\omega ){\mathbf{S}}_{\mathbf{F}}(\omega ){\mathbf{H}}^{\ast}(\omega ).$$
- Step 5.
- Compute the covariance matrix of the buoy velocity:$${\sigma}_{\mathbf{u}}^{2}=cov[\mathbf{u},\mathbf{u}]={\int}_{0}^{\infty}{\mathbf{S}}_{\mathbf{u}}(\omega )d\omega .$$
- Step 6.
- Approximate the viscous damping matrix ${\mathbf{B}}_{visc}$ as [35]:$$\begin{array}{c}\hfill {\mathbf{B}}_{visc}=-\u2329\frac{\partial {\mathbf{F}}_{visc}}{\partial \mathbf{u}}\u232a=\frac{1}{2}\sqrt{\frac{8}{\pi}}{\rho}_{w}{\mathbf{C}}_{d}{\mathbf{A}}_{d}{\sigma}_{\mathbf{u}}^{2}.\end{array}$$
- Step 7.
- Check the convergence:$$|{\mathbf{B}}_{visc}\left[n\right]-{\mathbf{B}}_{visc}[n-1]|<\delta ,$$

## 4. Optimisation Setup

## 5. Bio-Inspired Optimisation Algorithms

#### 5.1. Improved Moth–Flame Optimisation (IMFO)

#### 5.2. Diversification Strategy

## 6. Results and Discussions

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## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

WEC | Wave energy converters |

PTO | Power take-off |

PSD | Power spectral density |

MFO | Moth Flame Optimisation |

EAs | Evolutionary algorithms |

SI | Swarm intelligence |

CMA-ES | Covariance matrix adaptation evolution strategy |

PSO | Particle Swarm Optimisation |

GWO | Grey Wolf Optimiser |

WOA | Whale Optimisation Algorithm |

LSHADE-EpSin | Self-adaptive version of differential evolution |

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**Figure 3.**The Marettimo test site: (

**a**) wave scatter diagram and (

**b**) ten representative sea states identified using the k-means clustering method.

**Figure 4.**Statistical optimisation results of the improved MFO (IMFO) and the five other meta-heuristics. Each experiment runs ten times.

**Figure 5.**Convergence speed of the improved MFO (IMFO) and the five other meta-heuristics. The average optimisation results of ten experiments is plotted.

**Figure 6.**The historical exploration and exploitation trajectories of the best-found designs per generation in radius for (

**a**) standard MFO; (

**b**) improved MFO. Each line shows an independent run.

**Figure 7.**Trajectory of the best-found solution’s parameters of the (

**a**) MFO and the (

**b**) improved MFO algorithms for height of the converter.

**Figure 8.**The searchability of (

**a**) MFO and (

**b**) IMFO in order to find the best inclination angle of the tethers ($\alpha $).

**Figure 9.**A performance comparison between MFO and IMFO in terms of exploration and exploitation ability.

**Figure 10.**The ten best-found Kpto values per generation during the optimisation process using MFO (

**a**) and IMFO (

**b**).

**Figure 11.**The search pattern of (

**a**,

**b**) standard MFO, and (

**c**,

**d**) improved MFO for damping-spring settings.

Location | 37.96${}^{\circ}$ N, 12.04${}^{\circ}$ E |
---|---|

Type of data | Real sea measurement [32] |

Water depth | 10 m |

Mean wave power density | 6.38 kW/m |

WXSD resource class [33] | Class 1 |

Parameter | Unit | Min | Max | Length |
---|---|---|---|---|

a | m | 1 | 10 | 1 |

H | m | 1 | 10 | 1 |

$(H/a)$ | 0.4 | 2 | 1 | |

${\alpha}_{t}$ | deg | 10 | 80 | 1 |

${\alpha}_{ap}$ | deg | 10 | 80 | 1 |

${K}_{pto}$ | N/m | ${10}^{3}$ | ${10}^{8}$ | 10 |

${D}_{pto}$ | N/(m/s) | ${10}^{3}$ | ${10}^{8}$ | 10 |

**Table 3.**Details of the control parameters of the applied optimisation algorithms. All algorithms are limited to the same number as the fitness function evaluation.

Methods | Settings |
---|---|

CMA-ES [40] | $\lambda =25$ with the default settings recommended in reference [40]; |

PSO [41] | with $\lambda =25$, ${c}_{1}=1.5$, ${c}_{2}=2$, $\omega =1$ (reduced by a damping ratio ${w}_{f}=0.99$ exponentially); |

GWO [21] | $\lambda $= 25, $\alpha =2$ (linearly decreased to zero) |

WOA [22] | $\lambda =25$, ${\alpha}_{1}=2$ (declines linearly from 2 to 0), ${\alpha}_{2}=-1$ (linearly reduced from −1 to −2), $\beta =1$ |

MFO [23] | $\lambda =25$, $\alpha =-1$ (linearly drops from −1 to −2), $\beta =1$, $t=(\alpha -1)\times rand+1$; |

IMFO | $\lambda =25$, the same MFO’s control parameters |

**Table 4.**Optimisation performance of IMFO and the five other optimisation methods based on the best-found configuration per each experiment.

CMA-ES | PSO | GWO | WOA | MFO | IMFO | |
---|---|---|---|---|---|---|

Mean | 8.0621 × 10^{4} | 7.9192 × 10^{4} | 8.1757 × 10^{4} | 8.1134 × 10^{4} | 8.3430 × 10^{4} | 8.4448 × 10^{4} |

Min | 8.0003 × 10^{4} | 7.1264 × 10^{4} | 7.9016 × 10^{4} | 7.9956 × 10^{4} | 8.1406 × 10^{4} | 8.1534 × 10^{4} |

Max | 8.1032 × 10^{4} | 8.5783 × 10^{4} | 8.5985 × 10^{4} | 8.4410 × 10^{4} | 8.5933 × 10^{4} | 8.8274 × 10^{4} |

STD | 3.6621 × 10^{2} | 5.1481 × 10^{3} | 2.3205 × 10^{3} | 1.3638 × 10^{3} | 1.7451 × 10^{3} | 2.5527 × 10^{3} |

CMA-ES | PSO | GWO | WOA | MFO | IMFO | |
---|---|---|---|---|---|---|

a [m] | 10 | 10 | 9.97 | 9.99 | 10 | 10 |

H [m] | 10 | 10 | 10 | 10 | 10 | 10 |

${\alpha}_{t}$ [deg] | 80 | 36 | 10 | 79.99 | 80 | 79.85 |

${\alpha}_{ap}$ [deg] | 79 | 33 | 80 | 76.98 | 29.85 | 11 |

${\overline{)\sum}}_{i=1}^{{N}_{K}}$${K}_{pto}$$(\times {10}^{7})$ | 8.85 | 4.9415 | 4.4482 | 7.5518 | 5.8939 | 5.0907 |

${\overline{)\sum}}_{i=1}^{{N}_{B}}$${D}_{pto}$$(\times {10}^{7})$ | 9.575 | 6.3568 | 5.5574 | 9.4748 | 7.6913 | 6.7046 |

Power (Watt) | 8.10 × 10^{4} | 8.58 × 10^{4} | 8.60 × 10^{4} | 8.44 × 10^{4} | 8.59 × 10^{4} | 8.83 × 10^{4} |

**Table 6.**The percentage improvement of the WEC absorbed power using IMFO compared with the other optimisers applied in this study.

CMA-ES | PSO | GWO | WOA | MFO | |
---|---|---|---|---|---|

Mean | 4.75% | 6.64% | 3.29% | 4.08% | 1.22% |

Worst | 1.91% | 14.41% | 3.19% | 1.97% | 0.16% |

Best | 8.94% | 2.90% | 2.66% | 4.58% | 2.72 % |

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## Share and Cite

**MDPI and ACS Style**

Neshat, M.; Sergiienko, N.Y.; Mirjalili, S.; Majidi Nezhad, M.; Piras, G.; Astiaso Garcia, D.
Multi-Mode Wave Energy Converter Design Optimisation Using an Improved Moth Flame Optimisation Algorithm. *Energies* **2021**, *14*, 3737.
https://doi.org/10.3390/en14133737

**AMA Style**

Neshat M, Sergiienko NY, Mirjalili S, Majidi Nezhad M, Piras G, Astiaso Garcia D.
Multi-Mode Wave Energy Converter Design Optimisation Using an Improved Moth Flame Optimisation Algorithm. *Energies*. 2021; 14(13):3737.
https://doi.org/10.3390/en14133737

**Chicago/Turabian Style**

Neshat, Mehdi, Nataliia Y. Sergiienko, Seyedali Mirjalili, Meysam Majidi Nezhad, Giuseppe Piras, and Davide Astiaso Garcia.
2021. "Multi-Mode Wave Energy Converter Design Optimisation Using an Improved Moth Flame Optimisation Algorithm" *Energies* 14, no. 13: 3737.
https://doi.org/10.3390/en14133737