# A New Bi-Level Optimisation Framework for Optimising a Multi-Mode Wave Energy Converter Design: A Case Study for the Marettimo Island, Mediterranean Sea

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## Abstract

**:**

## 1. Introduction

## 2. Modelling

#### 2.1. Wave Energy Converter

#### 2.2. Wave Climate

#### 2.3. Equations of Motion

_{t,k}$={F}_{t0}+{K}_{pto}\Delta {\ell}_{k}+{B}_{pto}\Delta {\dot{\ell}}_{k}\phantom{\rule{0.166667em}{0ex}}(k=1\dots 3)$ being proportional to the tether extension $\Delta \ell $, the rate of change of the tether length $\Delta \dot{\ell}$ and includes the initial tension ${F}_{t0}$. The PTO stiffness ${K}_{pto}$ and damping ${B}_{pto}$ coefficients take the same values for all three tethers. The transformation between the buoy velocity $\dot{\mathbf{x}}$ and the tether velocity vector $\dot{\mathbf{q}}={\left[\Delta {\dot{\ell}}_{1}\phantom{\rule{1.em}{0ex}}\Delta {\dot{\ell}}_{2}\phantom{\rule{1.em}{0ex}}\Delta {\dot{\ell}}_{3}\right]}^{\mathsf{T}}$ has a form of $\dot{\mathbf{q}}\left(t\right)={\mathbf{J}}^{-1}\left(\mathbf{x}\right)\dot{\mathbf{x}}\left(t\right)$, where ${\mathbf{J}}^{-1}\left(\mathbf{x}\right)\in {\mathbb{R}}^{3\times 6}$ is the inverse kinematic Jacobian that depends on the buoy position at each time instance [34]. So the tether force vector can be converted to the Cartesian space according to ${\mathbf{F}}_{tens}=-{\mathbf{J}}^{-\mathsf{T}}{\mathbf{F}}_{t}$.

**F**

_{visc}and the generalised tether tension force

**F**

_{tens}. Due to the fact that geometric nonlinearity contained within Ftens is much weaker than the quadratic nonlinearity in

**F**

_{visc},

**F**

_{tens}can be linearised around the zero position without loss of accuracy for the proposed configuration. If nonlinear effects from tethers become relevant, the equivalent terms can be derived as shown in [38,41,42]. Moreover, it should be noted that other nonlinear forces can be included in the model but omitted in this study, e.g., nonlinear Froude–Krylov force that becomes relevant when the buoy experiences large motion amplitudes [43]. As a result, a nonlinear dynamic Equation (1) is replaced by the equivalent frequency domain model:

**A**(ω) and radiation damping matrix $\mathbf{B}\left(\omega \right)$, ${\widehat{\mathbf{F}}}_{rad}(\omega )=-(-{\omega}^{2}\mathbf{A}(\omega )+i\omega \mathbf{B}(\omega ))\widehat{\mathbf{x}}(\omega )$, the tether tension force is linearised as ${\widehat{\mathbf{F}}}_{tens}\left(\omega \right)=-(i\omega {\mathbf{B}}_{pto}+{\mathbf{K}}_{pto})\widehat{\mathbf{x}}\left(\omega \right)$ (see [44] for more details), and the viscous drag force is replaced by ${\widehat{\mathbf{F}}}_{visc}\left(\omega \right)=-i\omega {\mathbf{B}}_{eq}\widehat{\mathbf{x}}\left(\omega \right)$. The equivalent damping term

**B**

_{eq}is unknown and determined iteratively (for each wave condition separately) using the procedure explained in [38]:

**C**

_{d}and

**A**

_{d}are the matrices of the drag coefficients and the cross-section areas of the buoy perpendicular to the direction of motion respectively, and ⊙ represents the Hadamard product (element-wise multiplication). Note that only the body velocity (not the relative fluid/body velocity) has been considered in the drag force formulation. A detailed methodology of how to incorporate the wave-particle velocity into the spectral-domain model is demonstrated in [45].

- Step 1.
- Define the sea state and corresponding incident wave spectrum ${S}_{\eta}\left(\omega \right)$.
- Step 2.
- Compute the power spectral density (PSD) matrix of the excitation force:$${\mathbf{S}}_{\mathbf{F}}\left(\omega \right)={S}_{\eta}\left(\omega \right){\widehat{\mathbf{f}}}_{exc}\left(\omega \right){\widehat{\mathbf{f}}}_{exc}^{*}\left(\omega \right),$$
- Step 3.
- Calculate the WEC response matrix assuming ${\mathbf{B}}_{eq}={\mathbf{0}}_{6\times 6}$ in the first iteration:$$\mathbf{H}\left(\omega \right)=[-{\omega}^{2}(\mathbf{M}+\mathbf{A}\left(\omega \right))+i\omega (\mathbf{B}\left(\omega \right)+{\mathbf{B}}_{pto}+{\mathbf{B}}_{eq})+{\mathbf{K}}_{pto}]{}^{-1}.$$
- Step 4.
- Establish the power spectral density matrix of the buoy motion:$${\mathbf{S}}_{\mathbf{x}}\left(\omega \right)=\mathbf{H}\left(\omega \right){\mathbf{S}}_{\mathbf{F}}\left(\omega \right){\mathbf{H}}^{*}\left(\omega \right).$$
- Step 5.
- Calculate the covariance matrix of the WEC velocity:$${\sigma}_{\dot{\mathbf{x}}}^{2}=cov[\dot{\mathbf{x}},\dot{\mathbf{x}}]={\int}_{0}^{\infty}{\omega}^{2}{\mathbf{S}}_{\mathbf{x}}\left(\omega \right)d\omega .$$
- Step 6.
- Estimate the equivalent damping matrix ${\mathbf{B}}_{eq}$ using the analytical expression from [38]:$$\begin{array}{c}\hfill {\mathbf{B}}_{eq}=-{\displaystyle \langle}\frac{\partial {\mathbf{F}}_{visc}}{\partial \dot{\mathbf{x}}}{\displaystyle \rangle}=\frac{1}{2}\sqrt{\frac{8}{\pi}}{\rho}_{w}{\mathbf{C}}_{d}{\mathbf{A}}_{d}{\sigma}_{\dot{\mathbf{x}}}^{2}.\end{array}$$
- Step 7.
- Check the convergence criteria:$$|{\mathbf{B}}_{eq}\left[n\right]-{\mathbf{B}}_{eq}[n-1]|<\delta .$$

**B**

_{eq}and the WEC response in irregular waves. Once calculated, the average power absorbed by each PTO unit $k=1\dots 3$ is calculated as [38]:

**B**

_{eq}= 0, the spectral-domain model is specified in Equation (2) where

**B**

_{eq}is estimated iteratively for each sea state, and the time-domain model is represented by Equation (1). Good agreement is achieved between the spectral-domain and time-domain models, while the frequency domain model significantly overestimates power generation potential of the WEC.

#### 2.4. Economic Model

- -
- The mass of the buoy is calculated based on a given geometry as ${m}_{b}=0.5{\rho}_{w}\pi {a}^{2}H$;
- -
- The needed mass of the anchoring system (three piles) relays on the tether tension associated with buoyancy and the wave force, and can be approximated by ${m}_{as}\approx 0.116{F}_{t}^{peak}$ using case presented in [47] as a reference. The tether peak force ($99\%=2.57{\sigma}_{{F}_{t}}$) is estimated from the spectral-domain model.

#### 2.5. Implementation

**M**= diag(m

_{b}, m

_{b}, m

_{b}, I

_{xx}, I

_{yy}, I

_{zz}) with moments of inertia calculated for the cylindrical body. Hydrodynamic parameters of the WEC, including the added mass

**A**(ω), hydrodynamic damping

**B**(ω), and excitation force vector ${\widehat{\mathbf{F}}}_{exc}\left(\omega \right)$are estimated using a semi-analytical model [48,49].

**B**

_{eq}is calculated based on the iterative procedure explained in Section 2.3.

**B**

_{eq}, are highly dependent of the ratio between the cylinder height to its diameter, especially for the heave mode. Therefore, it order to develop an optimisation procedure that can accommodate WEC geometries with various aspect ratios ($H/a$), the drag coefficient in heave is expressed as a function ${C}_{{d}_{3}}=-0.12(H/a)+1.2$ based on published data [50] shown in Figure 4. Drag coefficients in other directions are not sensitive to the cylinder aspect ratio and are kept fixed ${C}_{{d}_{1}}={C}_{{d}_{2}}=1$ for surge and sway, and ${C}_{{d}_{4}}={C}_{{d}_{5}}=0.2$ for roll and pitch. The irregular waves from Table 1 are modelled using the Bretschneider (modified Pierson–Moskowitz) spectrum according to [51].

## 3. Optimisation Configuration Models

- (i)
- The average annual produce power output computed utilising Equation (14), that is maximised as$${f}_{O1}=\underset{\mathbf{z}}{argmax}\phantom{\rule{0.277778em}{0ex}}{P}_{AAP}\left(\mathbf{z}\right),\phantom{\rule{0.166667em}{0ex}}\mathrm{subject}\mathrm{to}:{\mathbf{z}}_{\mathbf{1}}\in [{\mathbf{z}}_{min},{\mathbf{z}}_{max}]$$
- (ii)
- The LCoE is minimised using the below equation that is specified in Equation (16):$${f}_{O2}=\underset{\mathbf{z}}{argmin}\phantom{\rule{0.277778em}{0ex}}\mathrm{LCOE}\left(\mathbf{z}\right),\phantom{\rule{0.166667em}{0ex}}\mathrm{subject}\mathrm{to}:{\mathbf{z}}_{\mathbf{2}}\in [{\mathbf{z}}_{min},{\mathbf{z}}_{max}]$$

## 4. Optimisation Algorithms

#### 4.1. All-at-Once Optimisation

#### 4.1.1. L-SHADE with an Ensemble Pool of Sinusoidal Parameter Adaptation (LSHADE-EpSin)

#### Mutation Strategy with External Archive

#### Ensemble of Parameter Adaptation

_{CR}and variance at 0.1. The successful crossover probabilities S

_{CR}are recorded and updated at each generation. The μ

_{CR}is initialised by 0.5 and in the next generation it is updated by Equation (24).

_{A}is a simple arithmetic mean. Likewise, the mutation factor F

_{i}of each x

_{i}is separately generated at each generation, as stated in a Cauchy distribution with the mean μ

_{F}and scale parameter 0.1. (Equation (25))

_{F}at the end of each generation. The value of μ

_{F}is updated using Equation (26).

_{L}is the Lehmer mean [65] and computed as follows:

_{min}is the minimum population size, and initialised at 4 that is required to make the current-to-pbest mutation strategy. The four required solutions are ${x}_{i},\phantom{\rule{3.33333pt}{0ex}}{x}_{best}^{p},\phantom{\rule{3.33333pt}{0ex}}{x}_{{r}_{1}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{x}_{{r}_{2}}.$ The mutant vector of this strategy is generated using Equation (29).

#### Local Search

#### 4.2. Bi-Level Optimisation

#### Tuning the Local Search

## 5. Optimisation Results and Discussions

#### 5.1. Multi-Modality of Search Space

#### 5.2. Power Landscape Analysis

Algorithm 1 Bi-level Optimisation method (LSHADE-EpSin+NM)
| |

procedureBi-level Optimisation method | |

Initialization$\mathit{P}=\{\langle {a}_{1},{H}_{1},{\alpha}_{{t}_{1}},{\alpha}_{{ap}_{1}},{K}_{1}^{1},...,{K}_{1}^{10},{B}_{1}^{1},...,{B}_{1}^{10}\rangle ,\dots $ | |

$\dots ,\langle {a}_{N},{H}_{N},{\alpha}_{{t}_{N}},{\alpha}_{{ap}_{N}},{K}_{N}^{1},...,{K}_{N}^{10},{B}_{N}^{1},...,{B}_{N}^{10}\rangle \}$ | ▹ initial population |

M:$\mu $F =$\mu $CR=0.5 | ▹ initialise memory of first control settings |

M${}_{freq}$:$\mu $freq = 0.5,$Imp-rat{e}_{d}=Imp-rat{e}_{\alpha}=1$ | ▹ initialise memory of second control settings |

Upper-Level (Global search method) | |

for $iter$ in $ite{r}_{max}$do | ▹ termination criteria |

if $iter>\frac{ite{r}_{max}}{2}$ then | |

Call second control parameter settings | |

${S}_{F}={S}_{CR}=\varnothing $ | ▹ Reset successful mean vectors |

${r}_{i}=rand(1,H)$ | ▹ Generate a random index, H is memory size |

${F}_{i}=randc(\mu {F}_{{r}_{i}},0.1)$,$C{R}_{i}=randn(\mu C{R}_{{r}_{i}},0.1)$ | |

end if | |

if $iter\le \frac{ite{r}_{max}}{2}$ then | |

Call first control parameter settings | |

$c=rand(0,1)$ | |

if $c<0.5$ then | |

${F}_{i}=\frac{1}{2}\times (sin(2\pi \times freq\times iter+\pi )\times \frac{ite{r}_{max}-iter}{ite{r}_{max}}+1)$ | |

else | |

${F}_{i}=\frac{1}{2}\times (sin(2\pi \times freq\times iter)\times \frac{iter}{ite{r}_{max}}+1)$ | |

end if | |

Generate $C{R}_{i}$ same as first control parameters (Equation 23) | |

end if | |

for $i=1$ to N do | |

Generate $p=rand(0,1)\times n$, $n=0.1\times N$ | |

${v}_{i}={x}_{i}+{F}_{i}\times ({x}_{pbest}-{x}_{i})+{F}_{i}\times ({x}_{{r}_{1}}-{x}_{{r}_{2}})$ | ▹ Mutation $current$-$to$-$pbest/1$ |

${\mathit{u}}_{\mathit{i},\mathit{iter}}^{\mathit{j}}=\{\begin{array}{cc}{\mathit{v}}_{\mathit{i},\mathit{iter}}^{\mathit{j}},\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}(\mathit{rand}<{\mathit{CR}}_{\mathit{i}})\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}(j=={j}_{rand})\hfill \\ {P}_{i,iter}^{j},\hfill & \mathrm{Otherwise}\hfill \end{array}$ | ▹ Binomial Crossover |

${\mathit{P}}_{\mathit{i},\mathit{iter}+\mathit{1}}=\{\begin{array}{cc}{\mathit{u}}_{\mathit{i},\mathit{iter}},\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}(\mathit{f}\left({\mathit{u}}_{\mathit{i},\mathit{iter}}\right)>\mathit{f}\left({\mathit{P}}_{\mathit{i},\mathit{iter}}\right))\phantom{\rule{4.pt}{0ex}}Maximisation\hfill \\ {P}_{i,iter},\hfill & \mathrm{Otherwise}\hfill \end{array}$ | ▹ Selection |

Store successful ${F}_{i}$ and $C{R}_{i}$ | |

end forUpdate the memory according to used settings Update the population size by Equation (28) ${N}_{diff}={N}_{g}-{N}_{g+1}$ Sort ${P}_{iter}$ based on the fitness function Remove worst solutions ${N}_{diff}$ from ${P}_{iter}$ AND Select the best solution ${P}_{best}$ | |

Lower-Level (Local search method) | |

if $Imp-rat{e}_{d}$> 0.001% then | ▹ Optimise Cylinder dimension |

${P}_{best}(a,H)=Nelder-Mead({P}_{best}(a,H),Ma{x}_{eval})$ Compute improvement rate $Imp-rat{e}_{d}$ end if | |

if $Imp-rat{e}_{\alpha}$> 0.001% then | ▹ Optimise tether angles |

${P}_{best}({\alpha}_{t},{\alpha}_{ap})=Nelder-Mead({P}_{best}({\alpha}_{t},{\alpha}_{ap}),Ma{x}_{eval})$ | |

Compute improvement rate $Imp-rat{e}_{\alpha}$ | |

end if | |

Update ${P}_{iter}^{best}$ by the best-found NM configurations | |

end for | |

end procedure |

#### 5.3. The Annual Average Power Output Maximisation

#### 5.4. LCoE Minimisation

_{t}< 35 °. Another important finding is that the power production of WECs optimised for LCoE is relatively low leading to 28.3 kW.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

WEC | Wave Energy Converter |

PTO | Power Take-off system |

PSO | Particle Swarm Optimisation |

DE | Differential Evolution |

SaDE | Self adaptive Differential Evolution |

CMA-ES | Covariance Matrix Adaptation Evolution Strategy |

LSHADE | Local Success-history Adaptive Differential Evolution |

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**Figure 2.**The wave climate at the Marettimo deployment site, Italy (12.04°E, 37.96°N, 6.38 kW/m mean annual wave power resource) [37]: (

**a**) wave scatter diagram, and (

**b**) clustering of the wave data where crosses correspond to ten representative states.

**Figure 3.**Power production of a three-tether WEC in irregular waves estimated using three different models: frequency-, spectral-, and time-domain. Parameters of the WEC are $a=5.5$ m, $H=5.5$ m, ${\alpha}_{ap}={\alpha}_{t}=45deg$, ${K}_{pto}=200$ kN/m, ${B}_{pto}=150$ kN/(m/s)), irregular waves have the significant wave height of ${H}_{s}=3$ m and modeled using the Pierson–Moskowitz spectrum.

**Figure 4.**Drag coefficient of the cylindrical body in axial flow as a function of its aspect ratio $H/a$.

**Figure 5.**A general sketch of the bi-level optimisation applied in order to maximise the produced power.

**Figure 6.**The effect of computational budget on tuning the local search iterations. (

**a**) dimension optimisation ($a,H$), (

**b**) Tether angles optimisation (${\alpha}_{t},{\alpha}_{ap}$).

**Figure 7.**Twenty independent NM runs with the random initial solutions. (

**a**) The NM’s trajectory in the cylinder’s dimension (radius and height) optimisation, (

**b**) 3D NM’s trajectory in the cylinder’s dimension and the absorbed power. (

**c**) NM’s trajectory in the initial value of the damping (${B}_{pto}$) and spring (${K}_{pso}$) array. (

**d**,

**e**) two examples of 3D NM’s trajectory in ${B}_{pto}$ and ${K}_{pso}$.

**Figure 8.**A power landscape of the cylinder with the fixed angles ${\alpha}_{t},{\alpha}_{ap}=45$ and various dimensions and PTO parameters.

**Figure 9.**Each method runs 10 times. (

**a**) Average annual produced power, (

**b**) Levelised cost of energy (LCoE).

**Figure 10.**The average convergence rate comparison of the absorbed power and LCoE of the cylinder. Each method runs 10 times. (

**a**) Average annual produced power, (

**b**) Levelised cost of energy (LCoE).

**Figure 11.**Search history and trajectory of the best solution per each population in all decision variables. (

**a**) the optimisation process (power maximisation) of DE, (

**b**) Bi-level-2.

Sea State | ${\mathit{T}}_{\mathit{p}}$, s | ${\mathit{H}}_{\mathit{s}}$, m | Probability O, % |
---|---|---|---|

1 | 3.82 | 0.24 | 8.06 |

2 | 5.13 | 0.44 | 14.62 |

3 | 6.20 | 0.61 | 17.80 |

4 | 7.18 | 0.90 | 18.01 |

5 | 8.30 | 0.73 | 12.10 |

6 | 8.43 | 1.92 | 9.58 |

7 | 9.68 | 1.08 | 8.68 |

8 | 10.24 | 2.76 | 5.78 |

9 | 11.56 | 1.46 | 3.30 |

10 | 12.99 | 3.69 | 2.07 |

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

radius, a | m | 1 | 20 | 1 |

height, H | m | 1 | 30 | 1 |

aspect ratio, $(H/a)$ | 0.4 | 2 | 1 | |

Tether inclination angle, ${\alpha}_{t}$ | deg | 10 | 80 | 1 |

Tether attachment angle, ${\alpha}_{ap}$ | deg | 10 | 80 | 1 |

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

PTO damping, ${B}_{pto}$ | N/(m/s) | ${10}^{3}$ | ${10}^{8}$ | 10 |

**Table 3.**The details of the optimisation methods settings. All approaches are restricted to the same evaluation number.

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

Nelder–Mead [53] | Nelder–Mead simplex direct search (NM) |

1+1EA [54] | mutation step sizes are ${\sigma}_{a}={\xi}_{1}\times ({U}_{a}-{L}_{a})$, ${\sigma}_{H}={\xi}_{1}\times ({U}_{H}-{L}_{H})$,${\sigma}_{{\alpha}_{t}}={\sigma}_{{\alpha}_{ap}}={\xi}_{1}\times ({U}_{{\alpha}_{t}}-{L}_{{\alpha}_{t}})$, ${\sigma}_{{K}_{pto}}={\sigma}_{{B}_{pto}}={\xi}_{2}\times ({U}_{{K}_{pto}}-{L}_{{K}_{pto}})$, and Probability mutation rate=$\frac{1}{N}$, ${\xi}_{1}=0.3,{\xi}_{2}=0.01$ |

CMA-ES [55] | with the default settings and $\lambda =13$; |

PSO [56] | with $\lambda =25$, ${c}_{1}=1.5$, ${c}_{2}=2$, $\omega =1$ ( decreased with a damping ratio ${w}_{f}=0.99$ exponentially); |

GWO [35] | with $\lambda $= 25, $\alpha =2$ (linearly decreased to zero) |

DE [57] | with $\lambda =25$, $F=0.5$, ${P}_{cr}=0.8$ |

SaDE [58] | with $\lambda =25$, $LP=50$, $NumSt=4$ |

LSHADE-EpSin [36] | $\lambda =25$, historical memory size $H=5$, $Nu{m}_{LS}=10$ |

Bi-level-1 | SaDE +NM, WEC’s dimensions and tether angles are optimised in the lower-level, default settings of SaDE |

Bi-level-2 | LSHADE-EpSin + NM, WEC’s dimensions and tether angles are optimised in the lower-level, default settings of LSHADE-EpSin |

Parameter | 1+1EA | CMA-ES | PSO | GWO | DE | SaDE | LSHADE-EpSin | Bi-Level-1 | Bi-Level-2 |
---|---|---|---|---|---|---|---|---|---|

a [m] | 16.62 | 16.10 | 19.99 | 16.68 | 15.46 | 15.50 | 15.49 | 15.61 | 14.51 |

H [m] | 30 | 30 | 14.80 | 30 | 30 | 30 | 30 | 30 | 30 |

${\alpha}_{t}$ [deg] | 70 | 26 | 60 | 14 | 48 | 26 | 39 | 50 | 10 |

${\alpha}_{ap}$ [deg] | 10 | 13 | 63 | 28 | 10 | 11 | 29 | 40 | 67 |

${}_{i=1}^{{N}_{k}}{K}_{pto}(\times {10}^{7})$ | 0.665 | 0.863 | 3.796 | 1.51 | 1.894 | 2.883 | 0.882 | 0.665 | 0.514 |

${}_{i=1}^{{N}_{B}}{B}_{pto}(\times {10}^{7})$ | 2.765 | 9.928 | 4.676 | 1.51 | 3.775 | 4.036 | 2.479 | 2.095 | 1.129 |

${P}_{AAP}\left[\mathrm{K}\mathrm{W}\right]$ | 259 | 248 | 239 | 261 | 259 | 261 | 262 | 265 | 279 |

**Table 5.**Performance comparison of various optimisation methods based on the maximum, minimum and average power output and LCoE of the best-found design per each experiment.

Power [MW] | ||||||||

1+1EA | CMA-ES | PSO | GWO DE | SaDE | LSHADE-EpSin | Bi-Level-1 | Bi-Level-2 | |

Mean | 0.2325 | 0.2329 | 0.2208 | 0.2537 0.2501 | 0.2537 | 0.2541 | 0.2551 | 0.2612 |

Min | 0.1941 | 0.2121 | 0.1934 | 0.2467 0.2327 | 0.2498 | 0.2473 | 0.2526 | 0.2544 |

Max | 0.2590 | 0.2476 | 0.2392 | 0.2615 0.2589 | 0.261 | 0.2621 | 0.261 | 0.2792 |

STD | 0.0234 | 0.0117 | 0.0181 | 0.0049 0.0087 | 0.0036 | 0.0046 | 0.0032 | 0.0088 |

LCoE | ||||||||

1+1EA | CMA-ES | PSO | GWO DE | SaDE | LSHADE-EpSin | Bi-Level-1 | Bi-Level-2 | |

Mean | 0.0443 | 0.0303 | 0.0678 | 0.0315 0.0334 | 0.0309 | 0.028 | 0.0295 | 0.0268 |

Min | 0.0316 | 0.0284 | 0.0556 | 0.0297 0.0282 | 0.0277 | 0.0248 | 0.0267 | 0.0243 |

Max | 0.0599 | 0.0382 | 0.0794 | 0.0335 0.0514 | 0.0329 | 0.0361 | 0.0324 | 0.0285 |

STD | 0.0109 | 0.0036 | 0.0071 | 0.0014 0.0079 | 0.0019 | 0.0041 | 0.0019 | 0.0012 |

Parameter | 1+1EA | CMA-ES | PSO | GWO | DE | SaDE | LSHADE-EpSin | Bi-Level-1 | Bi-Level-2 |
---|---|---|---|---|---|---|---|---|---|

a [m] | 7.31 | 6.40 | 14.32 | 7.00 | 7.38 | 6.57 | 5.00 | 6.15 | 5.00 |

H/a | 0.40 | 0.40 | 0.40 | 0.4 | 0.40 | 0.40 | 0.40 | 0.40 | 0.40 |

αt [deg] | 28 | 29 | 10 | 10 | 31 | 25 | 35 | 31 | 34 |

αap [deg] | 10 | 11 | 10 | 31 | 14 | 11 | 10 | 12 | 10 |

${}_{i=1}^{{N}_{k}}{K}_{pto}(\times {10}^{7})$ | 0.647 | 0.919 | 3.90 | 0.651 | 3.50 | 0.383 | 2.094 | 0.77 | 2.071 |

${}_{i=1}^{{N}_{B}}{B}_{pto}(\times {10}^{7})$ | 0.577 | 0.332 | 3.52 | 0.847 | 1.15 | 1.481 | 1.350 | 0.256 | 1.914 |

LCoE | 0.0316 | 0.0284 | 0.0556 | 0.0297 | 0.0287 | 0.0277 | 0.0248 | 0.0267 | 0.0243 |

PAAP [kW] | 53.1 | 43.6 | 131 | 51.4 | 64.8 | 50.6 | 27.1 | 43.5 | 28.3 |

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**MDPI and ACS Style**

Neshat, M.; Sergiienko, N.Y.; Amini, E.; Majidi Nezhad, M.; Astiaso Garcia, D.; Alexander, B.; Wagner, M.
A New Bi-Level Optimisation Framework for Optimising a Multi-Mode Wave Energy Converter Design: A Case Study for the Marettimo Island, Mediterranean Sea. *Energies* **2020**, *13*, 5498.
https://doi.org/10.3390/en13205498

**AMA Style**

Neshat M, Sergiienko NY, Amini E, Majidi Nezhad M, Astiaso Garcia D, Alexander B, Wagner M.
A New Bi-Level Optimisation Framework for Optimising a Multi-Mode Wave Energy Converter Design: A Case Study for the Marettimo Island, Mediterranean Sea. *Energies*. 2020; 13(20):5498.
https://doi.org/10.3390/en13205498

**Chicago/Turabian Style**

Neshat, Mehdi, Nataliia Y. Sergiienko, Erfan Amini, Meysam Majidi Nezhad, Davide Astiaso Garcia, Bradley Alexander, and Markus Wagner.
2020. "A New Bi-Level Optimisation Framework for Optimising a Multi-Mode Wave Energy Converter Design: A Case Study for the Marettimo Island, Mediterranean Sea" *Energies* 13, no. 20: 5498.
https://doi.org/10.3390/en13205498