# Optimization Study of Hydraulic Power Take-off System for an Ocean Wave
Energy Converter

Erfan Amini  School of Civil Engineering, College of Engineering, University
of Tehran, Tehran. Iran<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Hossein Mehdipour  School of Civil Engineering, College of
Engineering, University of Tehran, Tehran. Iran<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Emilio Faraggiana  Department of
Mechanical and Aerospace Engineering, Polytechnic University of Turin, Torino,
Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>Danial Golbaz
School of Civil Engineering, College of Engineering, University of Tehran,
Tehran. Iran<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Sevda Mozaffari  Department of Water Engineering, Urmia
University, Urmia. Iran<EMAIL_ADDRESS>Giovanni Bracco  Department of
Mechanical and Aerospace Engineering, Polytechnic University of Turin, Torino,
Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>Mehdi Neshat∗
Center for Artificial Intelligence Research and Optimization, Torrens
University Australia, Brisbane, Australia<EMAIL_ADDRESS>
(*Corresponding author)

###### Abstract

Ocean wave renewable energy is fast becoming a key part of renewable energy
industries over the recent decades. By developing wave energy converters as
the main converter technology in this process, their power take-off (PTO)
systems have been investigated. Adjusting PTO parameters is a challenging
optimization problem because there is a complex and nonlinear relationship
between these parameters and the absorbed power output. In this regard, this
study aims to optimize the PTO system parameters of a point absorber wave
energy converter in the wave scenario in Perth, on Western Australian coasts.
The converter is numerically designed to oscillate against irregular and
multi-dimensional waves and a sensitivity analysis for PTO settings is
performed. Then, to find the optimal PTO system parameters which lead to the
highest power output, ten optimization algorithms are incorporated to solve
the non-linear problem, Including Nelder-Mead search method, Active-set
method, Sequential quadratic Programming method (SQP), Multi-Verse Optimizer
(MVO), and six modified combination of Genetic, Surrogate and fminsearch
algorithms. After a feasibility landscape analysis, the optimization outcome
is carried out and gives us the best answer in terms of PTO system settings.
Finally, the investigation shows that the modified combinations of Genetic,
Surrogate, and fminsearch algorithms can outperform the others in the studied
wave scenario, as well as the interaction between PTO system variables.

Keywords: Wave Energy Conversion; Wave Models; Power Take-off; Optimization
Algorithms

##  1 Introduction

Harvesting ocean waves energy has become more prominent since the oil crisis
in 1973. Ocean energy has huge potential in providing energy especially for
coastal communities, which made the governments, industries and engineers very
attracted to this matter and resulted in more than 1000 patents worldwide [1,
2]. Wave energy has immense reserves, high power potential, higher power
density than solar and wind energy, is geographically diverse and makes small
interference in the environment [3, 4].

One of the key parts of a WEC is the power take-off system which converts the
mechanical energy absorbed by the device into electrical energy [5, 3]. Since
the wave energy is fluctuating by nature, regular PTOs are not very effective
at absorbing wave power [i31]. The development of hydraulic PTO systems has
been a huge breakthrough in the field of ocean energy applications, due to the
fast-frequency response, easy and high controllability, high efficiency,
hydraulic overload protection, and adaptability to high power and low-
frequency of the HPTO systems [5, 3, 6]. These characteristics made HPTO the
best PTO type for the point absorber wave energy devices, which can reach up
to 90% efficiency [6].
There has been an increasing number of studies about the HPTO application in
different WECs in recent years [7, 8, 9, 10, 11], but the number of studies is
still lower than the studies WECs with Linear PTO, and there are a lot of
unexplored areas that need to be addressed in the future. Most of these
studies investigated the performance and efficiency of the HPTO systems, but
they did not consider the effect of HPTO model parameters. These parameters
are really important because they affect the unit’s efficiency and power
output. Only a small number of studies addressed this problem, especially the
effect of simultaneous change in the HPTO parameters has been investigated in
very few studies, for example in [12, 6]. That is why in this paper we
investigate the effect of HPTO parameters, namely piston area, volume and pre-
charged pressure of the low-pressure gas accumulator (LPA) and the volume of
the high-pressure gas accumulator (HPA) simultaneously on the power output of
the WEC.
Vantorre et al. [13] studied the influence of a few factors such as the WEC
hull geometry, the external damping and additional inertia on a point absorber
wave-energy device performance. In [14], the functioning of the WindWaveFloat
concept, which includes a spherical shape WEC, the SWEDE, was analyzed. Yu et
al. [15] studied the performance of a floating point absorber utilizing a
Computational Fluid Dynamics (CFD) method which is based on Reynolds-averaged
Navier–Stokes (RANS). They found that the non-linear parameters, particularly
viscous damping and wave overtopping, could reduce the absorbed energy of the
floating point absorber notably, especially in bigger waves. In [16], the
effects of buoy’s radius and draft on the performance of a point absorber
device have been studied utilizing a velocity ratio. The authors reported that
there are optimal values for these parameters according to the generator
damping of the device.
Clark et al. [17] used the damage equivalent loads (DELs) on the power take-
off system to investigate the influence of wave energy device shape and PTO
reliability. They recommend not to take into account PTO-stroke constraint
when evaluating the force-time series, because of inaccuracies of this method.
Li et al. [18] developed and investigated, a two-body self-react point
absorber with 2 types of power take-off system, the mechanical motion
rectifier (MMR) and non-MMR based. They found that the PTO inerter can
increase the power output or move the peak frequency. Also, the device’s drag
damping decreases the power capture in both scenarios.
In [19] a two-body point absorber with an innovative electromechanical power
take-off system was designed and analyzed. In [20], the performance of a point
absorber was investigated using Smoothed Particle Hydrodynamics (SPH) method.
They reported that keeping the device at water level puts the device under
maximum loads, and in order to reduce it significantly, the buoy should be
immersed at a specific depth. In [1], an innovative wave energy device with a
tuned inertial mass (TIM) is designed, in which a supplementary tuning spring
and a rotational inertial mass are affixed to the original system, which
causes the oscillation of the damper to increase, which in turn can improve
the WEC’s performance considerably.
In [21], a WEC emulator was developed, which can be used to evaluate a WEC’s
performance, saving money and time. In [22], in order to decrease the cost of
a wave energy converter PTO, the peak-to-average power (PTAP) ratio should be
decreased while increasing the average absorbed energy by the device. Results
showed that the most influential parameters on the PTAP ratio are gear ratio,
crank radius, and generator properties. Zhang et al. [4] investigated the
performance of the bistable energy absorption mechanism in long-term ocean
power capture. And found that, generally around truncated cone-shaped buoy
with a bigger size and lower non-linear parameter can improve long-term energy
absorption.
Numerical and metaheuristic algorithms have been utilized in WEC optimization
in a lot of studies, here we briefly review a few of them. In [23], the
authors tried to minimize the LCOE of a multi-float WaveSub device using GA &
PSO. They considered a few decision variables, and found that the float-
reactor separation is the key parameter to change. Also, both algorithms had
equally good performances. In [24] the performance of 5 metaheuristics
algorithms was compared in optimizing the power take-off system of a point
absorber. Results showed that MVO had the best performance.
Noad et al. [25] investigated the performance of an OSWEC array. They
determined the optimal parameters for a single OSWEC device using a numerical
optimization method, and suggested that the gap length is the most important
factor influencing the power output. They also studied the optimal
arrangements of a WEC array.
Babarit et al. [26] did a shape optimization study on a wave energy device,
the objective was to maximize the power output and to minimize the cost
simultaneously. They utilized a 2 layer process to achieve this purpose, each
layer using a different optimization algorithm, one of them was GA, and the
other one was a gradient-based method. In [6] researchers compared the
performance of GA and NLPQL in optimizing the parameters of a hydraulic PTO
system for a WEC. Both algorithms did improve the device’s performance
significantly.
Plummer et al. [27] developed an HPTO unit for a point absorber, and assessed
the performance of the control strategies in both cases of with and without
considering losses in the HPTO. Also, they showed that using a smaller motor
and pump in case of smaller wave conditions can increase the device’s
efficiency. In several recent studies [24, 28, 29], researchers studied
different aspects of PTO systems utilizing optimization methods. For instance,
A comparison has been done between two different WECs PTO systems, the
Carnegie CETO 6 and the WaveRoller with a linear PTO and a hydraulic one. They
compared the performance of three optimization strategies. Additionally, the
effect of piston area on the power output of both devices was investigated.
Further, [30] developed a cutting edge swarm cooperative optimization
framework for optimize power output of WEC farms.
Yu et al. [31] developed an HPTO model for a two-body floating point absorber
(FPA) to investigate the trade-off between the mean captured power and the
power fluctuation using three power smoothing methods. These methods were able
to lower the power fluctuation of the device notably. They also analyzed the
effect of HPA Volume on the Power Output and its fluctuations.
In another study [3], the effects of the HPA volume and pre-charge pressure,
flow control valve and oil tank on the operational stability of an HPTO
connected to a pendulum WEC were investigated. According to the findings, by
increasing the accumulator pre-charge pressure or reducing its volume, the
response speed for stabilizing the system state is decreased, but the stable
state becomes more volatile in return. Finally, modifying these two parameters
can improve the quality of the response speed, overshoot and operational
stability of the system.
Cargo et al. [32] modeled a point absorber WEC using an HPTO with and without
losses, to find whether the losses affect the parameters that maximize the
power output. They could have altered 3 different parameters, but they only
changed the motor displacement, because in practice this option is easier to
manipulate, and found that changing this parameter can improve the performance
in both cases. They continued their research in another study [33] where they
examined a practical active tuning method for a heaving point absorber. They
showed that over-simplification of the PTO dynamics in the design stage can
lead to non-optimal designs. In this study, the influence of motor capacity
and generator load on the absorbed power was analyzed.
Liu et al. [9] investigated the effect of 7 HPTO system parameters, namely
piston area, motor capacity, effective damping of the generator, mounting
position of cylinder, rod-to-piston area ratio, volume and pre-charge pressure
of the HPA on the power output of a two-raft-type WEC. They reported that the
best values for the first 4 parameters are mostly dependent on the wave
period. Also the rod area should be minimized while maintaining the necessary
strength. And finally, higher values of HPA volume and pre-charge pressure
don’t affect the capture width ratio, but lower values of these two parameters
can indeed be consequential.
Zhang et al. [5] investigated a state-dependent model of a hydraulic PTO for
an inverse pendulum WEC. Depending on the design criteria, HPA volume and pre-
charge pressure have different trends. In order to maintain constant pressure
in the HPTO system, these values should preferably be higher, but lower values
of HPA volume and pre-charge pressure are better for controlling the damping
force applied to the hydraulic motor. In another study, Zhang et al. [34]
showed that the performance of the WEC can be improved by altering the
resistance load, the accumulator pre-charge pressure and the size of the
opening of the regulating throttle valve.
Jusoh et al. [12] optimized the power output of a multi-point WEC using GA
with 5 decision variables: the diameters of piston and rod, accumulator’s
volume and pre-charge pressure and motor displacement. They were able to
reduce the size of all of the 5 parameters up to 38%, which in turn reduced
the operational cost. They also reduced the PTO force applied to device, which
makes it possible for the WECs to be employed in smaller wave conditions. The
authors continued their work on the same WEC and added two more parameters to
their power optimization study [6] to make a total of seven decision
variables, namely LPA volume capacity and pre-charge pressure. They also used
NLPQL (Non-Linear Programming by Quadratic Lagrangian) alongside GA. The
latter algorithm had slightly better results with performance improvements up
to 97%.
According to the spotted gap in the literature, this paper attempts to find
the optimal parameters of an HPTO unit of a point absorber in a case study of
western Australian coasts in order to maximize the output power of the WEC.
Different optimization algorithms are used for this purpose, and their
performances are compared at the end of the simulations. Moreover, a modified
combination of Genetic, Surrogate and fminsearch algorithms is developed.
Finally, the simultaneous interaction between different system settings has
been investigated. Finally, the simultaneous interaction between different
system settings has been investigated.
In this regard, section 2 explains the research method considering wave
resources, wave energy converter simulation and governing equations. It is
followed by the PTO dynamic model and optimization approaches including
numerical and metaheuristics methods. Then, section 3 represents the
feasibility study together with the sensitivity analysis. Next, the
optimization convergence for different algorithms is presented in section 4.
Afterward, section 5 provides a comparative assessment of the optimization
algorithms. Finally, the conclusion is given in section 6.

##  2 Research Method

In this section, we cover the methodology of the research. It begins by
describing the wave resource, including the wave scatter data and power
matrix. It follows by wave energy conversion simulation details including the
hydrodynamic governing equations. Then the PTO dynamic model details are
explained, and its mechanical setup is presented. After that, we go through
the optimization algorithms’ specifications that we used in this study,
followed by a detailed representation of our modified combined optimization
algorithm.

###  2.1 Wave Resource

Looking back to the recent literature on potential wave energy in the
Australian coasts, Perth has been identified as one of the most promising
areas for the installation of such systems[35]. Accordingly, a wave data-set
including the time history of waves’ significant heights, periods, and
directions is incorporated from the Australian Wave Atlas [36], and according
to the literature, in this study, the Pierson- Moskowitz (PM) spectrum with a
Significant Wave Height of 4.06 m and a Peak Wave Period of 13.65 s has been
used to model the waves of this area. Figure 1 reveals the directional wave
rose diagram and scatter data in Perth. For instance, the highest potential
probability of occurrence is related to waves with a significant height of 2
meters, and a period of roughly 13 seconds. The power matrix of this area is
presented in Figure 2.

Figure 1: Wave rose of travelled waves in Perth (left), and wave scatter data
(right)  Figure 2: Power Matrix of waves in Perth

###  2.2 WEC Simulation

WEC-Sim is a code utilized in this study and created by Sandia National
Laboratories (SNL) and the National Renewable Energy Laboratory (NREL).
Following that, time-domain modeling was complemented in MATLAB software,
which is based on a multibody dynamic solver [37]. The wave energy converter
requires hydrodynamic input to describe power output results, wec-sim imports
are calculated by a frequency domain and pre-prepared by the BEMIO function.
In terms of the boundary element method’s source program, NEMOH is utilized.
The BEM is obtained by linear free surface potential flow theory and it
generated hydrostatic and hydrodynamic coefficients. There are some
contributors to relevant coefficients including mass, wave excitation term,
hydrodynamic damping, and hydrostatic stiffness, given that those will be
utilized by WEC-Sim. Considering the effect of these coefficients, relevant
input data to waves, oscillating body, constraints, and the type of PTO are
required by WEC-Sim. Providing a brief overview of how NEMOH determines the
mentioned coefficients. The BEMIO functions take the data from NEMOH as input
and calculate output non-dimensionalized hydrodynamic coefficients, as well as
excitation and radiation impulse response functions [38]. By using NEMOH, we
made the following assumptions [39, 40]. (i) The fluid is incompressible, (ii)
The fluid is inviscid, (iii) The fluid is irrotational, (iv) Both the waves
and the oscillating body have small amplitudes, (v) The seabed is flat, so the
depth of water $d_{w}$ is supposed to be constant. By discretizing the wetted
surface in flat panels on which the velocity potential is intended to remain
constant, the open-source BEM solver NEMOH calculates a solution for the
velocity potential. More information on how NEMOH solves this boundary value
problem may be found in [39]. As indicated in Equation 1, WEC-Sim solves WEC’s
governing equations of motion using the Cummins in 6 degrees of freedom (DOF).

$F_{e}(t)-\int_{-\infty}^{t}\mathfrak{f}_{r}(t-\mathcal{T})\dot{X}(\mathcal{T})d\mathcal{T}+F_{hs}(X)+F_{m}(X,\dot{X})+F_{PTO}(\dot{X})-F_{D}(\dot{X})=(m+A(\infty))\ddot{X}$
(1)

Where $F_{e}$ is the wave excitation force, the convolution integral is the
wave radiation force, $F_{hs}$ is the hydrostatic restoring force, $F_{m}$ is
the mooring force, $F_{PTO}$ is the PTO force, $F_{D}$ is the drag force, m is
the WEC mass and $A_{\infty}$ is the added mass at infinite wave frequency
[41]. Reference Model 3 (RM3) is one of the WEC models developed by the U.S.
Department of Energy[42]. And the design load elevation framework was
demonstrated with this reference model[43].RM3 is a heavy two-body point
absorber made up of a float and spar, whose full-scale dimensions and mass
properties are illustrated in figure (2) and table(1) RM3 model. In reaction
to incident waves, the WEC can move freely in all 6 DOF. In terms of the heave
direction, power is predominantly captured. As a result of previous DOE-funded
projects, there is a strong correlation between numerically design
characterized and experimentally. Additionally, it has basic operating
principles and is reflective of the wave energy industry’s WECs. The total
simulation time for cases simulated in this study is 400 s and the ramp time
is 100 s.

Table 1: RM3 mass properties Spar-plate Full scale Properties |  |
---|---|---
Axis |  Moment of Inertia [kg-$m^{2}$] | Mass [tonne] | CG [m]
X | 9.44E+07 | 0.00E+00 | 0.00E+00 | 878.30 | 0.00
Y | 0.00E+00 | 9.44E+07 | 2.18E+05 | 878.30 | 0.00
Z | 0.00E+00 | 2.18E+05 | 2.85E+07 | 878.30 | -21.29
Float Full scale Properties |  |
X | 2.09E+07 | 0.00E+00 | 0.00E+00 | 727.01 | 0.00
Y | 0.00E+00 | 2.13E+07 | 4.30E+03 | 727.01 | 0.00
Z | 0.00E+00 | 4.30E+03 | 3.71E+07 | 727.01 | -0.72
Figure 3: An illustrative look of the studied wave energy converter geometric
dimensions, with a hydraulic power take-off system.

###  2.3 PTO Dynamic Model

Most point absorbers capture energy based on body oscillations caused by wave
interaction. In such cases, conventional rotary electrical machines are not
directly compatible. In most cases, Hydraulic Power Take-Off (HPTO) systems
are chosen to deal with this problem by linking the WEC to an electrical
generator, since they are well suited to capture energy when dealing with
immense forces at low frequencies [44].
The hydraulic cylinder’s rod is forced up and down by the floating heaving
buoy, with respect to an actuator [44]. The first HPTO element is a double-
acting hydraulic piston pump, which directly turns the linear kinetic energy
of the heaving buoy into a pressurized, bi-directional fluid flow [45].
The piston chamber is linked to a series of rectifying valves (4 valves) via
upper and lower terminals. This mechanism rectifies the flow, meaning it
converts the bi-directional fluid flow into a unidirectional flow and passes
the fluid on to the rest of the HPTO system [45]. One of the valve’s roles is
to deliver fluid to the high-pressure side of the system where it is stored in
the High-Pressure gas Accumulator [45]. If the incident waves are almost
sinusoidal, then the flow from one part of the cylinder can be represented by
Figure4(a). Rectification valves transform the fluid flow into a
unidirectional flow that is depicted in Figure4(b). Then the HPA would smooth
this flow and send it to the hydraulic motor [46].

Figure 4: Flow-time representations for hydraulic WEC [46]

The role of HPA is to smoothen the flow across the motor by either providing
or accumulating hydraulic energy when needed, in other words, it reduces the
hydraulic fluctuation in the system [47]. The smoothing effect increases with
the accumulator volume and working pressure [48].
The hydraulic fluid power is converted into rotational energy by a variable
displacement motor. The motor’s shaft is connected directly (no gearboxes) to
the shaft of the generator, causing it to revolve and generate electricity.
The hydraulic motor was selected to meet the torque and speed required
circumstances [45]. Since waves generate large loads, low velocity, oftentimes
radial piston motor is used as it is well suited for these circumstances [44].

The generator provides a gear speed ratio that translates the slow movement of
the heaving buoy (WEC) to higher rotational speeds in the generator [31].
Electrical power is less than mechanical power because the generator block in
the Simulink is modeled such that the rotational speed and torque of the
electrical generator determine the efficiency of output electrical power,
meaning there are some losses in the generator [45]. The generator in the
Simulink is modeled as simple rotational inertia with a lookup table
containing speed, torque, and efficiency, based on a typical large industrial
asynchronous generator [49].

Next, the fluid enters the low-pressure side where the LPA provides a
pressurized reservoir. Whenever the hydraulic piston pump needs, it can draw
fluid from the reservoir (LPA), to complete the circuit [45]. The LPA is used
to re-energize the hydraulic fluid [31], it also supplies a small boost in the
pressure to reduce the risk of cavitation on the low-pressure side [46].

It is assumed that the hydraulic piston pump is directly coupled to the
heaving buoy, therefore the velocity of the cylinder is the same as the buoy
($v$) [47], which is the relative velocity between the float and the spar
[45], and the output force can be determined using the pressure values of the
upper and lower parts of the piston [28]:

$F_{PTO}=(-sign(v)).(P_{high}-P_{low})A_{p}$ (2)

with the $A_{p}$ the piston area, $P_{high}$ and $P_{low}$ the pressure in the
upper section and lower section of the piston. Also, the total mean absorbed
power of the HPTO is calculated by

$P_{abs}=F_{PTO}.v=(sign(v).\Delta p.A_{p}).v$ (3)

The next parameter is the volume flow $(Q_{piston})$ caused by the piston
movement that can be calculated [28]:

$Q_{piston}=A_{p}v$ (4)

the volume flow into the nitrogen-charged high-pressure accumulator (HPA) and
low-pressure accumulator (LPA) can be determined this way. They have the same
absolute value, but opposite signs. The rectifying valves guarantee that
further in the hydraulic PTO, flow is unidirectional: fluid flows from the
valves towards the HPA and from the LPA towards the valves [28].
The accumulator’s job is to smoothen the fluid flow to the hydraulic motor, to
dump the undesired power output peaks [28].
The incoming volume flow into the HPA ($Q_{in}$) is the sum of $Q_{piston}$
and $Q_{motor}$ \- see further. Origin of $Q_{motor}$ is the hydraulic motor.
In order to calculate the volume of fluid that flows into the HPA in time
($dt$), this volume is integrated [28]:

$V_{in}=Q_{in}.dt$ (5)

Next, the pressure inside the HPA ($p_{high}$) is determined according to an
isentropic process and is calculated per time step [28]:

$p_{high}=\frac{p_{pre-charge}}{(1-\frac{V_{in}}{V_{0}})^{1.4}}$ (6)

The low pressure in the LPA can be calculated in a similar way [28].
The angular velocity of the hydraulic motor ($\dot{\omega_{m}}$) can be
determined by integrating the following equation [28]:

$\dot{\omega_{m}}=\frac{((P_{h}-P_{l}).\alpha D-T_{g}-T_{f})}{I_{mg}}$ (7)

where $T_{g}$ is the generator torque, $T_{f}$ the torque due to the friction,
$\alpha$ the swashplate angle, $D$ the nominal motor displacement, and
$I_{mg}$ the total moment of inertia of the motor/generator. Finally, the
volume flow originating from the hydraulic motor ($Q_{motor}$) is given by
[28]:

$Q_{motor}=\omega_{m}\alpha D$ (8)

The generator damping can be calculated by:

$C_{gen}=(\Delta P.\alpha
D).(\frac{\omega_{M}}{\omega_{desired}}).(\frac{1}{1.05})$ (9)

where $C_{gen}$ represents the generator damping, $\Delta$P the pressure
difference between the HPA and LPA, $\alpha$D the hydraulic motor volume,
$\omega_{desired}$ the desired speed of the rotary generator (150 rad/s) and
1.05 accounts for the mechanical efficiency. Also the term
$(\frac{\omega_{M}}{\omega_{desired}})$ represents the volumetric efficiency
[50].

Finally, it is notable that our case uses a variable hydraulic motor
displacement, and its value can be determined by:

$\begin{cases}if~{}~{}4MPa<\Delta P<15MPa\quad\rightarrow\quad\quad\alpha
D=2.67e10^{-11}.\Delta P-8.52e10^{-5}(m^{3})\\\
else\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\rightarrow\quad\quad\alpha
D=2e10^{-5}(m^{3})\end{cases}$ (10)

###  2.4 Optimization Approaches

In this paper, we used six optimization methods for the described problem.
Three of them are taken from the numerical methods and the other three are
from metaheuristic methods. A brief description is given for each of the
methods.

####  2.4.1 Nelder-Mead

One of the most popular numerical methods is Nelder-Mead which is based on the
concept of the simplex method. To solve a problem with $n$ dimensions, Nelder-
Mead keeps $n+1$ test points to approximate a local optimum of a problem.
Noting that the maximum number of evaluations in each iteration is not more
than two. The main part of the algorithm is extrapolating the value of the
objective function in each test point to find a new test point. Then the new
test point is replaced with the older test points. This procedure continues to
achieve the solution. A simple way of extrapolating is to replace the worst
point in terms of objective value with a reflected point from the center of
the remaining n points. Two approaches are proceeding when extrapolating is
done. if the reflected point has the better answer, then the expanded point is
computed, however, if the reflected point is not the better answer, then then
the contracted point is computed [51, 52].

$\displaystyle X_{reflected}=X_{0}+\alpha(X_{0}-X_{n+1})\ ,\ \
\quad\quad\quad\alpha>o$ (11) $\displaystyle
X_{expanded}=X_{0}+\gamma(X_{r}-X_{0})\ ,\quad\quad\quad\quad\gamma>1$
$\displaystyle X_{contracted}=X_{0}+\rho(X_{n+1}-X_{0})\ ,\quad 0<\rho<=0.5$

####  2.4.2 Interior-Point Method

Another algorithm selected in this study is the active-set method that is one
of the iterative optimization methods. Active-set methods have two iterative
sections. The first is including a feasible region and feasible initial guess.
The latter is the phase that the objective is optimized for nonempty feasible
region [53]. The sequence of feasible iterates $X_{k}$ can be computed
according to the equation below.

$X_{k+1}=X_{k}+\alpha_{k}p_{k}~{}~{}~{}~{}~{}where:F_{X_{k+1}}>F_{X_{k}}$ (12)

where $\alpha_{k}$ is positive step length, $p_{k}$ is nonzero search
direction and $F_{X_{k+1}}$ is the cost function value [54].

####  2.4.3 Sequential Quadratic Programming

To solve a constrained nonlinear optimization problem, an iterative method
called Sequential Quadratic Programming (SQP) can be used in which the
objective function together with the constraints are smooth nonlinear function
which means can be at least twice continuously differentiable. The main idea
is to optimize the quadratic model of the objective function while considering
the linearized constraints. Generally, this algorithm is the enhanced version
of Newton’s method. To begin with, the Lagrangian of the problem and the
hessian of the Lagrangian must calculate, then identify the jacobian matrix of
the constraints. After that, the given problem is a nonlinear problem that
solves within the algorithm [55].

####  2.4.4 Multi-Verse Optimizer

An interesting algorithm inspired by cosmology and consists of a white hole,
black hole, and wormhole called Multi-Verse Optimizer (MVO). The concept of
the algorithm is transferring objects from the white hole (the universe with a
higher inflation rate has more probability to have a white hole) to the black
holes. The inflation rate refers to the solution of the fitness function. The
Roulette wheel was used to designate a white and black whole to the universes.
Because of transferring objects from higher to lower inflation rate universes,
the average inflation rate improves as the algorithm runs. It is noted that a
wormhole that transfers objects from the best universe to another universe is
used for enhancing the exploitation part [56].

$\begin{cases}X_{m}^{n}=X_{n}+TDR(Ur_{n}-Lr_{n})&Ran3<0.5\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad Ran2<WEP\\\
X_{n}-TDR(Ur_{n}-Lr_{n})Ran4+Lr_{n})&Ran3>0.5\\\
X_{m}^{n}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad Ran2>WEP\\\
\end{cases}$ (13)

Where $X_{n}$ is the nth parameter of the Best Universe so far, $TDR$ is the
traveling distance rate, $WEP$ is the wormhole existence probability, and
$Lr_{n}$, $Ur_{n}$ are the lower and upper bounds of nth variable, $x_{m}^{n}$
is the nth parameter of the mth Universe and $ran2$, $ran3$ and $ran4$ are
random numbers between 0 and 1.

####  2.4.5 Modified combination of Genetic, Surrogate and fminsearch
algorithms

The genetic algorithm (GA) is one of the most famous evolutionary algorithms
that has been used largely for wave energy applications [57]. It is a
metaheuristic algorithm that is inspired by natural selection. This algorithm
can solve global optimization problems and it is particularly effective when
the gradient of the objective function is not available. Furthermore, it can
be used in a parallel computation reducing the computational time required.
The simulation of each individual of the population is independent and could
be run in parallel. The population at each generation is improved using three
main operators that are the selection, cross-over, and mutation. The
Stochastic Universal Sampling has been used for the selection operator while
the cross-over and the mutation operators have been chosen respectively as the
uniform and the real-coded mutation. An open-source MATLAB code has been used
as a starting code for this purpose[58]. Then, several improvements have been
made to this algorithm as explained in [23, 59]. This code has been modified
to avoid repetitions and to enable new evaluations in the optimal region
found. It is also improved to allow the tuning of the mutation, cross-over
probability and the population as described in [59]. Then, an elitism factor
has been introduced in the genetic algorithm to insert new individuals in the
population from a surrogate model which increases the efficiency of the
algorithm. The huge advantage of the surrogate model is that enables the user
to estimate the results at a very low computational time compared to the real
simulation. The Kriging surrogate model is chosen because it is suitable for
highly nonlinear responses. It belongs to the family of linear least square
algorithms and the estimation of the response of a point x is based on a
linear combination of the results of the simulations and it is expressed as

$f(x)=\sum_{i=1}^{N}\lambda_{i}(x)f(x_{i})\\\ $ (14)

where $\lambda_{i}$ are the weights obtained from a regression analysis. Then,
a Gaussian process is built through the residuals. The open-source Oodace
toolbox has been used for the surrogate model [60] while a parallel
computation of the GA+surrogate has been implemented using the HPC HACTAR of
Politecnico of Torino [61]. Finally, the MATLAB "fminsearch" function has been
added after the GA+surrogate optimization to allow the final convergence of
the optimization. The function "fminsearch" uses the Nelder-Mead simplex
method that is very efficient to find the local optimum. So, this final
algorithm is suitable to refine the optimum found after that the design space
has been explored by the previous global optimization algorithm.

##  3 Feasibility Study and Sensitivity Analysis

In order to obtain the applicable range of the decision variables for the
optimization problem, a few relevant studies were used, in particular two
studies that had the identical point absorber as this study. In [31]
efficiency of an HPTO for wave energy converters and the trade-off between
generated power and oscillation between different power smoothing techniques
was assessed. In [28] a numerical model of a point absorber using HPTO with
three different control strategies was investigated. These two studies were
the main reference for the selected range of the 4 HPTO parameters. Also, in
[47] another point absorber was studied which was considerably smaller than
our model, but had a higher value for the High-Pressure Gas Accumulator
volume, which was taken into consideration for finalizing the ranges. These
ranges can be seen in Table 2.

In the next step, we did a sensitivity analysis using the obtained ranges from
the literature review, in which we ran the simulation with different values of
each decision variable. We did a total of 266 runs in this section. A summary
of these runs are presented in Table 2. In a number of cases, the values of
most outputs were not applicable in practice, especially the values for piston
pressures, the PTO force, absorbed power, and displacement of the device.
There were a total of 41 cases with these described properties. After further
investigations, a pattern was found in these cases, the values of Piston Area
and HPA Volume in these cases had something in common, the volume of HPA
Volume was lower than a certain threshold. The minimum acceptable value of HPA
Volume was found for different Piston Areas and the acceptable values were
found.

Table 2:  Metrics of Sensitivity Analysis Runs Before Data Reduction.
(consisting of 266 cases)

5 Parameters |  Piston Area (m2) |  HPA Volume (m3) |  LPA Volume (m3) |  LPA Pre-Charge Pressure (MPa) |  Max Piston Pressure (MPa) |  Min Piston Pressure (MPa) |  Max PTO Force (MN) |  Mean Absorbed Power (kW) |  Mean Generated Power (kW) |  Mean Electrical Power (kW) |  Max Float Displacement (m) |  Max Spar Displacement (m)
---|---|---|---|---|---|---|---|---|---|---|---|---
2 Max |  0.18 |  10 |  8 |  9.6 |  50428.55 |  17.90 |  204883.44 |  39323.99 |  343.10 |  275.75 |  435.55 |  780.67
Min |  0.045 |  0.5 |  0.5 |  3.5 |  12.71 |  $-370852.02$ |  0.39 |  84.62 |  68.73 |  59.62 |  3.23 |  1.25
Average |  0.103 |  5.03 |  4.80 |  6.27 |  655.80 |  $-1485.99$ |  1209.25 |  452.55 |  166.69 |  138.57 |  7.70 |  9.53
Standard Deviation |  0.032 |  2.62 |  1.87 |  1.49 |  4374.61 |  22774.38 |  13175.43 |  2438.14 |  41.39 |  32.65 |  29.03 |  51.00
5 |  |  |  |  |  |  |  |  |  |  |  |

After removing these 41 cases from our database, the same statistics were
gathered for the remaining 225 cases in Table 3 .

Table 3:  Metrics of Sensitivity Analysis Runs After Data Reduction.
(consisting of 225 cases)

5 Parameters |  Piston Area (m2) |  HPA Volume (m3) |  LPA Volume (m3) |  LPA Pre-Charge Pressure (MPa) |  Max Piston Pressure (MPa) |  Min Piston Pressure (MPa) |  Max PTO Force (MN) |  Mean Absorbed Power (kW) |  Mean Generated Power (kW) |  Mean Electrical Power (kW) |  Max Float Displacement (m) |  Max Spar Displacement (m)
---|---|---|---|---|---|---|---|---|---|---|---|---
2 Max |  0.18 |  10 |  8 |  9.6 |  201.51 |  15.76 |  19.92 |  385.53 |  229.19 |  188.11 |  8.28 |  8.64
Min |  0.045 |  1.5 |  0.5 |  3.5 |  12.71 |  0.37 |  0.39 |  84.62 |  68.73 |  59.62 |  3.29 |  1.25
Average |  0.103 |  5.79 |  4.74 |  6.29 |  53.63 |  5.97 |  5.29 |  246.78 |  161.90 |  135.02 |  4.04 |  3.61
Standard Deviation |  0.033 |  2.08 |  1.92 |  1.54 |  51.60 |  2.87 |  5.50 |  73.54 |  37.79 |  29.98 |  1.36 |  1.84
5 |  |  |  |  |  |  |  |  |  |  |  |

By comparing Tables 2 and 3, it can be seen that there is not much change in
the column corresponding with the four decision variables, namely piston area,
HP- and LP accumulators volumes, and LPA pre-charge pressure. But for
previously problematic outputs, there is a significant change. The value for
the maximum of max piston pressure has been decreased from 50428 MPa to 201
MPa and the minimum of min piston pressure has been increased from -370852 MPa
to 0.37 MPa. Also, the values for these two parameters’ average and standard
deviation are smaller and more reasonable. Data regarding the max PTO force,
average absorbed power and device’s displacement has been significantly
improved mainly in case of their standard deviation and some in case of their
average value. Also, the maximum value for most variables in the table has
been decreased by at least 2 orders of magnitude. Overall we can see that the
data reduction was necessary, and it is considered a success.

After investigating the effect of data reduction, and making sure that it was
helpful, we finalized the penalty area of our search space. This has been
illustrated in Fig. 5, in which the green section represents the feasible area
in this study.

Figure 5: Feasibility of Search Space of Piston Area and HPA Volume of the
HPTO

In the next step, we investigated the rough influence of HPTO parameters
values in groups of 2 on the mean output power. This resulted in 6 plots which
are presented in Fig. 6.

Figure 6: Sensitivity Analysis Plots for the Effects of Changes in the 4
Decision Variables Values on the Output Power

As it is apparent in the first 3 plots ((a), (b) and (c)) in Fig. 6 that
include piston area, a value below 0.1 m2 for this parameter results in a
significant drop in the power output, this is consistent with [62]. Also,
according to Fig. 6-a, the best results happen when the piston area is
relatively large and the HPA volume is relatively small, and if both reach the
extremum value in the feasible area the power output drops. Additionally,
based on Fig. 6-b, for the same values for the low-pressure gas accumulator
volume, the increase in the piston area generally improves the power output,
except in the cases with LPA volume below 2 m3.

According to Fig. 6-d, when the LPA pre-charge pressure is below 6.5 MPa, the
objective function is generally better, and it has especially good values when
LPA volume is low and LPA pre-charge pressure has a moderate value with
respect to our feasible range. As can be seen in Fig. 6-e, if the HPA volume
is constant, the LPA volume does not affect the absorbed energy notably,
except when the former is large, and the latter has a small value, which
results in exceptionally good answers. Finally, according to Fig. 6-f, for the
low values of high-pressure gas accumulator volume, LPA pre-charge pressure is
not as influential as HPA volume in the power output.

The fluctuating nature of the ocean waves is one of the main challenges in
utilizing wave energy devices. Consequently, WEC components have to be
designed to handle loads bigger than the average load, which subsequently
decreases mechanical and volumetric losses and increases the Levelised Cost Of
Energy (LCOE) of the WEC. WEC’s maximum power output can reach one order of
magnitude greater than the mean power output [31]. Another disadvantage of
these fluctuations in an energy type is that it makes that energy an unfitting
option for sensitive equipment that needs a stable voltage. For these reasons,
the peak-to-average power ratio should be decreased while maintaining a high
average of absorbed energy. This can be done by using accumulators and
advanced control methods. However, it should be noted that the cost of the PTO
is not directly proportional to the maximum absorbed energy, especially for
hydraulic systems similar to those investigated here [31]. In order to assess
fluctuations of the HPTO power output, we determine the power fluctuation
ratio [31]:

$R_{PF}=\frac{\Delta P_{E}}{P_{avg}}=\frac{P_{max}-P_{min}}{P_{avg}}$ (15)

where the $P_{max}$ and $P_{min}$ values are calculated using 99.9 and 0.1
percentiles of power peaks during the simulation time [31].

To investigate the effects of changing the HPTO parameters on the power
fluctuation, a similar sensitivity analysis to Fig. 6 are done in Fig. 7 but
this time for the $R_{PF}$.

Figure 7: Sensitivity analysis plots for the effects of changes in the 4
decision variables values on the power fluctuation ratio ($R_{PF}$)

According to Fig. 7-(a), low volumes of HPA intensify the $R_{PF}$
drastically, especially when its value is below 4 m3. If you take a closer
look at the plots that investigate LPA volume (plots (b), (d), and (e)), you
can see that the LPA volume does not seem to influence the $R_{PF}$
significantly, except in a few cases, for example, when LPA is very large or
very small, and LPA pre-charge pressure is close to its maximum feasible
value. Fig. 7-(b) and (c) show that low values for the piston area can
actually be better for the power fluctuation ratio. Fig. 7-(c) and (f) also
show that LPA pre-charge pressure increase can have opposite effects on the
$R_{PF}$ depending on the piston area value. Based on Fig. 7-(d), the low
values for the LPA pre-charge pressure result in lower $R_{PF}$s, which is
more advantageous. And as noted before, small volume and large pre-charge
pressure for the LPA can be undesirable. By analyzing the last two plots of
Fig. 7 (plots (e) and (f)), one can see that moderate amounts for the HPA
volume result in large $R_{PF}$s, which is not preferable.

##  4 Optimization Convergence

###  4.1 Numerical Methods

Three numerical optimization algorithms, namely Nelder-Mead, SQP, and Active-
set, are used to solve our optimization problem. The properties of these 3
runs are presented in table 4. According to this table, Nelder-Mead had the
most number of iterations before reaching the termination criteria, which is
almost 5 times bigger than the iterations Active-set needed to reach the said
criteria. Since we had an unfeasible area for our inputs, some of the
iterations in these 3 runs are not acceptable. Active-set had the most number
of unfeasible iterations despite having the least number of total iterations,
with a 9.8 % of evaluations being unfeasible. Next is SQP which had a 1.5 % of
unfeasible iterations, and lastly Nelder-Mead had 0 unfeasible iterations
which is very good.

The optimization convergence results are presented in table 6 and Fig. 8. The
HPTO parameters and power output does not show a very good convergence
overall. The only exception is the LPA pre-charge pressure, which range of
optimal parameters is 2.6 % of the total feasible design range for that
parameter. Also, they have converged next to the lowest value for the feasible
range. The LPA volume has the least convergence, its range of optimal
parameters is 99 % of the total feasible design range for LPA volume. Finally,
If Nelder-Mead is excluded, the other two algorithms have pretty good
convergence in piston area and the HPA volume, the range of optimal values
according to these algorithms are 3 % and 0.1 % of the total possible design
range, respectively.

Table 4:  Numerical Runs Details.

5 Numerical Algorithm Name |  Total Iterations |  Unfeasible Iterations
---|---|---
2 Nelder-Mead |  242 |  0
SQP |  195 |  3
Active-set |  51 |  5
5 |  |

Figure 8: Convergence curve of the mean output power for the 3 numerical
optimization algorithms (Nelder-Mead, SQP, and Active-set)

###  4.2 Modified Combined Method

The optimization convergence has been checked for six optimization runs and
their optimization settings are defined in Table 5. The population and the
surrogate elitism factor have been investigated with three and two different
values respectively. The number of generations of the GA and the number of
maximum evaluations of the fminsearch function have been instead fixed to 50
and 100 respectively. The total number of simulations is then determined by
the sum of the product between the population and the number of generations
and the number of evaluations of fminsearch. The optimisation convergence
results are shown in Table 6 and in Fig. 9. The design parameters and the
electrical power demonstrate a good convergence from these results. In fact,
the design parameters of each optimization run have converged to similar
design parameters. The HPA and LPA volumes design parameters have converged
really well, especially the LPA volume. The range of the optimal parameters of
these variables are respectively 2.7% and 0.3% of the full design space range
for the HPA and LPA volumes. They have converged also next to the highest and
lowest values of the range considered respectively. The least converged design
parameters are the piston area and the LPA pre-charge pressure. In fact, the
range of these optimal parameters is around 12% and 13% respectively. However,
this is mostly due to GSF1 and GSF4 that are the least converged optimization
runs, probably due to their lowest number of total simulations. The converged
range of these variables decreases to only around 3% if these two
optimizations runs are not accounted for. Table 6 shows also the minimum and
maximum piston pressure, the maximum PTO force, the power and the maximum
displacement of the float and the spar. The maximum and the minimum piston
pressure are respectively the most and the least converged results among them.
The optimal absorbed, generated and electrical powers give an optimal
electrical efficiency of around 72-75%. Fig. 9 shows the convergence plot of
the electrical power for the six optimization runs. Each optimization run has
a different number of simulations with GSF3 and GSF6 that have the largest
ones. The best results are given from GSF2 and GSF6 in which the optimal
electrical power converges around 230 kW.

Figure 9: Convergence curve of the mean output power for the 6 GSF
(GA+surrogate+fminsearch) optimization algorithms. A convergence step of 10
simulations is used for visualization purposes. Table 5:  Convergence set-up
for six optimisation runs.

5 Run |  npop |  surr. elit. factor |  ngen (GA+surrogate) |  nfminsearch |  Total simulations
---|---|---|---|---|---
2 GSF1 |  40 |  0.1 |  50 |  100 |  2100
GSF2 |  60 |  0.1 |  50 |  100 |  3100
GSF3 |  80 |  0.1 |  50 |  100 |  4100
GSF4 |  40 |  0.2 |  50 |  100 |  2100
GSF5 |  60 |  0.2 |  50 |  100 |  3100
GSF6 |  80 |  0.2 |  50 |  100 |  4100
5 |  |  |  |  |

###  4.3 Multi-Verse Optimizer (MVO)

Lastly, the Multi-Verse Optimization algorithm is used in order to find the
optimum parameters of an HPTO unit. The results are shown in Table 6 and Fig.
10. The number total of iterations was 2000 and the number of search agents
was 10, so in this run, a total number of 20000 evaluations was performed.
This algorithm by far had the most number of iterations but did not result in
the best possible solution.

Figure 10: Convergence curve of the mean output power for the MVO (Multi-Verse
Optimization) algorithm

##  5 Comparative Assessment

In this section, the performance of the 10 optimization algorithms in this
paper was compared. This was done by plotting the chronological variation of
the objective function and the 4 variables of the 10 algorithms in a figure.
Since the Multi-Verse Optimization algorithms had the most iterations between
our algorithms, 2000 iterations, and this was about 4 times more iterations
than the GSF3 and GSF6 which had the next more iterations, we decided to cut
the curves corresponding to the MVO ‘s performance and only show it until the
410th iteration (which is the same iterations as GSF3 and GSF6). As for the
other algorithms, we extended their curves horizontally to meet the 410th
iteration mark. This was done only for illustration and comparison purposes.

Here we present the best parameters obtained by each optimization algorithm,
and the non-optimized case, and other performance parameters in Table 6

Table 6:  Properties of the non-optimized case and the 10 optimized cases
using 10 different optimization algorithms

5 Run |  Number of Iterations |  Piston Area (m2) |  HPA Volume (m3) |  LPA Volume (m3) |  LPA Pre-Charge Pressure (MPa) |  Mean Absorbed Power (kW) |  Mean Generated Power (kW) |  Mean Electrical Power (kW) |  Power Fluctuation Ratio
---|---|---|---|---|---|---|---|---|---
2 Non-Optimized Case |  - |  0.038 |  8.50 |  6.00 |  9.60 |  74 |  65 |  57 |  2.63
Nelder-Mead |  242 |  0.146 |  9.49 |  0.58 |  3.66 |  320 |  263 |  213 |  2.47
SQP |  195 |  0.091 |  1.51 |  4.05 |  3.50 |  285 |  221 |  179 |  2.91
Active-set |  51 |  0.087 |  1.5 |  8 |  3.50 |  288 |  223 |  181 |  2.84
GSF1 |  210 |  0.132 |  9.97 |  0.51 |  5.29 |  310 |  281 |  228 |  2.31
GSF2 |  310 |  0.140 |  9.99 |  0.52 |  4.85 |  320 |  284 |  230 |  2.29
GSF3 |  410 |  0.136 |  9.76 |  0.50 |  5.03 |  314 |  282 |  229 |  2.30
GSF4 |  210 |  0.124 |  9.75 |  0.50 |  5.64 |  297 |  273 |  222 |  2.38
GSF5 |  310 |  0.137 |  9.91 |  0.50 |  5.06 |  315 |  283 |  230 |  2.30
GSF6 |  410 |  0.140 |  9.96 |  0.52 |  4.85 |  320 |  283 |  230 |  2.30
MVO |  2000 |  0.121 |  6.30 |  0.50 |  4.05 |  283 |  251 |  205 |  2.57
5 |  |  |  |  |  |  |  |  |

From the sensitivity analysis section, we know that lower values of $A_{p}$s
are not really desirable, which is consistent with all the solutions in Table
6. The two worst performers, i.e., the SQP and Active-set cases, have an
$A_{p}$ below 1 $m^{2}$. Also, the LPA pre-charge pressure value is not high
for the optimized cases, which conforms to Fig. 6.

The first plot for this purpose is the mean power output chronological
variation for the 10 algorithms which is displayed in Fig. 11.

Figure 11: Convergence curve of the mean output power for all the 10
optimization algorithms (6 GSF algorithms, 3 numerical algorithms, and MVO)

As can be seen, the 6 GSFs (Modified combination of Genetic, Surrogate and
fminsearch algorithms) had the best performance and were able to reach a mean
output power of 230 kW, the worst-performing member was the GSF4 with a 222 kW
power output. Since this algorithm had the least number of iterations in GSF
algorithms, with only 210 iterations, it is understandable that it did not
reach the 230 kW mark. Next in line is the Nelder-Mead algorithm with a power
output of 213 kW. From this figure, it seems that the HPTO output power
approaches the optimum value starting from iteration number 110. Next, we have
MVO, with an output power of 205, which is worse than Nelder-Mead’s
performance but better than the other 2 numerical algorithms. Also, MVO
reached this value after around 75 iterations. Finally, we have SQP and
Active-set, which had the worst performance with a similar output power of
around 180 kW. Overall GSFs really outshined the other algorithms in this
comparison.

Fig. 12 shows the chronological variation of the first decision variable, the
piston area, for the 10 investigated algorithms.

Figure 12: Convergence curve of the first decision variable, the piston area,
for all the 10 optimization algorithms (6 GSF algorithms, 3 numerical
algorithms, and MVO)

The cylinder area did not have the best convergence between the HPTO
parameters. All the 6 GSFs converged on a value between 0.124 and 0.14 m2, the
Nelder-Mead which is the next best performer had a piston area above this
range, 0.146 m2, and the other 3 algorithms reached a piston area below 0.124
m2, the MVO had a relatively similar piston area, but SQP and Active-set had a
value around 0.9 m2.

The next decision variable we plotted the chronological variation of is the
high-pressure gas accumulator volume in Fig. 13, which had a very diverse
range for the final value of this HPTO parameter.

Figure 13: Convergence curve of the second decision variable, the HPA volume,
for all the 10 optimization algorithms (6 GSF algorithms, 3 numerical
algorithms, and MVO)

Similar to the 2 previous plots, the optimum values found by the GSF
algorithms were very close, and they ranged from 9.9 to 10 m3, which is near
the maximum feasible value for this parameter. Nelder-Mead had an optimum
value of 9.5 m3 which was very close. However, the SQP and Active-set
algorithms found the optimum HPA volume value to be almost the minimum
possible value in the feasible search space with a value of 1.5 m3. And
Lastly, the MVO found a 7 m3 volume to be the best for the HPA.

The next studied parameter is the low-pressure gas accumulator, which its
chronological variation for the 10 optimization algorithms is illustrated in
Fig. 14.

Figure 14: Convergence curve of the second decision variable, the LPA volume,
for all the 10 optimization algorithms (6 GSF algorithms, 3 numerical
algorithms, and MVO)

This had the best converge out of all the 4 decision variables. As it is
apparent in Fig. 14 eight out of 10 algorithms found the optimum value to be
around 0.5 m3 which is the minimum possible value for this parameter. The
remaining algorithms are SQP and Active-set. This was the only decision
variable that these 2 algorithms did not find a close optimum value for. SQP
and Active-set converged on a value of 4 and 7 m3 respectively, which cover
over half of the feasible area.

The last figure in this section is Fig. 15 which illustrates the convergence
curve of all 10 algorithms for the low-pressure gas accumulator pre-charge
pressure.

Figure 15: Convergence curve of the second decision variable, the LPA pre-
charge pressure, for all the 10 optimization algorithms (6 GSF algorithms, 3
numerical algorithms, and MVO)

Based on Fig. 15, the GSF algorithms were able to agree on a relatively close
value for this parameter, which is around 4.8 to 5.6 MPa. The other 4
algorithms including the MVO and the numerical algorithms found the optimum
value near the minimum range of the search space, almost a 3.5 MPa pre-charge
pressure as the optimum value.

In this section, we aim to compare the performance of an RM3 heaving device
with an HPTO unit with two sets of parameters. One uses the parameters of the
Non-Optimized Case, which was extracted from WEC-Sim’s default case; the other
uses the parameters of the best solution found by the 10 optimization
algorithms, which belonged to the GSF2, with a 410 total number of iterations.
The comparison is made by investigating the performance of different parts of
the HPTO unit and plotting parameters like the pressures of the bottom and top
chambers of the hydraulic cylinders, the PTO force applied to the WEC device,
the hydraulic motor speed, the pressure differential between the HP- and LP-
accumulators, hydraulic motor volume, the generator damping, and power
throughout the simulation.

The first part of the hydraulic PTO is the hydraulic cylinder. We will look
into the pressure of both pump chambers, which is illustrated in Fig. 16.

Figure 16: Hydraulic Cylinder Chambers Pressures during the simulation for the
Non-Optimized Case

The value of top and bottom piston pressures vary from 16.1 to 24.6 MPa. It
also has multiple fast fluctuation periods during the simulation time. Piston
pressures reach their maximum and minimum value around the 293 s mark.

According to equation 2, the PTO force value is dependant on the pressure
difference between the upper and lower sections of the hydraulic piston; we
are plotting the chronological change of these two parameters in Figs 17 and
18.

Figure 17: Pressure Difference Between the Hydraulic Cylinder Chambers during
the simulation for the Non-Optimized Case Figure 18: PTO Force during the
simulation for the Non-Optimized Case

Since the $A_{p}$ is constant through a simulation, the value of piston
chambers pressure difference and the PTO force have a linear correlation.
These two plots have similar overall shapes as expected. The maximum piston
pressure difference value is 8.6 MPa which corresponds to the trough PTO force
(-0.324 MN) at 293.1 s, and the minimum piston pressure difference value is
-8.5 MPa which results in the peak PTO force (0.321), at 293.82 s.

Next, the HPA and LPA pressures and the hydraulic motor angular velocity are
shown in Fig. 19. The accumulators’ job is to maintain an approximate constant
$\Delta P$ across the hydraulic motor, resulting in constant motor angular
speed and steady energy [32].

Figure 19: HP- and LP-accumulators Pressures and Hydraulic Motor Speed during
the simulation for the Non-Optimized Case

According to Fig. 19, in the early stage of simulation, after around the 120 s
mark, the HPA and LPA pressures become relatively stable. The maximum HPA
pressure is 24.62 MPa, and the minimum LPA pressure is 16.04 MPa, which
happens simultaneously at 293.1 s. Also, in the same figure, we can see that
up until the 138 s, the motor speed is ramping up to reach the desired value,
which is 150 rad/s. At this stage, the volumetric efficiency becomes 100 %.

The following analyzed parameters are the accumulators’ pressure difference,
the motor’s volume, and the generator damping, illustrated in Fig. 20.

Figure 20: Pressure Differential Between the HP- and LP- Accumulators, the
Hydraulic Motor Volume, and the Generator Damping during the simulation for
the Non-Optimized Case

The relation between the parameters shown in Fig. 20, plus the motor speed in
Fig. 19, can be determined from Equation 9. All three plots in Fig. 20 have
similar trends, but it can be seen that the standard deviation for the first
parameter is lower than the other two. For example, although there is a mild
descending trend in accumulators’ pressure difference between 215 s and 236 s,
the more steep trend in this interval for the hydraulic motor volume resulted
in a larger generator damping drop during these 21 seconds. Notably, the
maximum value for all these three parameters occurs at 293.1 s, and they are
equal to 8.6 MPa, 143.8 cm3, and 7.8 kg.m2/s.

Although $\Delta$P reaches its average value after 75 s and oscillate around
that, because motor displacements ascending trend did not start until around
the 100 s mark, the 100 s value is considered as the model’s ramp time to
calculate the parameters, including the generator’s damping, after this point
in time.

Now, we will take a closer look at the power generation parameters. The
absorbed, mechanical and electrical powers are depicted in Fig. 21.

Figure 21: Absorbed Power, Mechanical Power, and Electrical Power during the
simulation for the Non-Optimized Case

As you can see, the absorbed power has multiple fluctuations at small time
intervals, but the other two powers have a more smooth profile, which is the
result of using accumulators. The maximum absorbed power is 407 kW which
happens at 286.5 s. Also, the electrical power is lower than the mechanical
power throughout the simulation, which is the result of losses in the
generator.

Now we will study the second case, i.e., the best solution found by the GSF2
algorithm. This will be done by plotting the same plots and comparing them
with their correspondents from the non-optimized case.

Fig. 22 shows the chronological alteration of piston chambers pressure, the
pressure difference between the two piston chambers, and the PTO force; these
values are connected by Equation 2.

Figure 22: Hydraulic Cylinder Top and Bottom Chambers Pressures, and Chambers
Pressure Difference, and PTO Force during the simulation for the Best Case
Found by the Optimization Algorithms (GSF2)

According to Fig. 22, the piston upper and lower section pressures have
multiple intervals with high-frequency fluctuations. The maximum and minimum
values for these parameters are 15.5 MPa and 0.5 MPa, which happen at the 304
s mark. Compared to the non-optimized case, the piston pressures have the same
range but are significantly lower and have no shared values.

Also, the piston pressure difference values have the same high-frequency
fluctuations intervals between -14.975 MPa and 14.995 MPa; these extreme
values are almost the ones for the non-optimized case, and they happen 304
seconds after the simulation started.

In the last plot of Fig. 22, one can see the PTO force throughout the
simulation, which according to Equation 2, is directly related to the piston
pressure difference. At 304 s, we can see the ultimate values -2.103 and 2.100
MN, almost 7 times the minimum and maximum PTO force values for the non-
optimized case. The increase of PTO force is by 556 % prevents the converter
from operating during the smaller wave condition [12].

Figure 23: HP- and LP-accumulators Pressures and Hydraulic Motor Speed during
the simulation for the Best Case Found by the Optimization Algorithms (GSF2)

According to Fig. 23, which presents the HPA and LPA pressure and the
hydraulic motor angular velocity for the GSF2 case, we can see that the HPA
pressure is steady at the first half of the simulation and the LPA pressure is
stable in the other half. Also, the range of HPA pressure variation is not the
same as the LPA pressure’s one, but for the non-optimized case, they were
almost the same, and also, their profile was virtually symmetrical. Also, the
motor speed reaches its desired value at 77.8 s, which is nearly half the
required time for the non-optimized case.

Fig. 24 shows the accumulators’ pressure difference, the motor volume, and the
generator damping for the best solution found by the optimization algorithms.
Again, these parameters are related to each other based on Equation 9.

Figure 24: Pressure Differential Between the HP- and LP- Accumulators, the
Hydraulic Motor Volume, and the Generator Damping during the simulation for
the Best Case Found by the Optimization Algorithms (GSF2)

The maximum values for these three parameters occur at 303.84 s when the
accumulator’s pressure difference is 14.995 MPa, and the hydraulic motor
volume is 315.17 cm3; finally, the generator damping is 30 km.m2/s. These
values are almost 2, 2, and 4 times the non-optimized case values.

Figure 25: Absorbed Power, Mechanical Power, and Electrical Power during the
simulation for the Best Case Found by the Optimization Algorithms (GSF2)

And finally, for the power generation of the HPTO, we plotted the absorbed,
generated, and electrical powers for the best-optimized case in Fig. 25.
Similar to Fig. 21, we can see that the absorbed power has high-frequency
fluctuations and generated and electrical power has more smooth profiles,
which is more smooth than the Fig. 21 plots. The reason can be using an HPA
with a higher volume in this case, despite the increase in the LPA volume [6].
Decreasing the HPA and LPA volumes nullifies the smoothing effect and makes
the electrical and mechanical powers behave more like the absorbed power [28].
The maximum absorbed power is 3293 kW which is 7 times more than the non-
optimized case value, and it happens at 286.92 s.

It is notable that since the capacity of the HPA in our optimized case has
increased, it results in an increase in size, weight and cost of the HPTO
unit, since the hydraulic accumulator and the hydraulic motor are the most
expensive parts of the HPTO system [6].

##  6 Conclusions

In the study, the configuration parameters of a hydraulic PTO connected to an
RM3 point absorber have been optimized to reach the maximum power output. Ten
optimization has been used for this purpose, including three numerical
algorithms, MVO, and six Modified Combined Methods. The
simulation–optimization of HPTO unit parameters has been carried out using
WEC-Sim in MATLAB software. We did a sensitivity analysis to find out the
effect configuration parameters on the power output and the power fluctuation
ratio ($R_{PF}$), and it showed that:

  * •

Low values of $A_{p}$ result in low power output and low $R_{PF}$.

  * •

Low values of $LPA$ pre-charge pressure result in good average power output
and low $R_{PF}$.

  * •

Small and moderate initial volumes of the HPA can increase the

  * •

No distinct pattern was found for the influence of LPA volume change on the
power output and the $R_{PF}$.

The modified Genetic, Surrogate, and fminsearch algorithm combinations had the
most promising results and were able to increase the mean power output by 304
% and decrease the $R_{PF}$ by 13 % by adjusting the HPTO parameters to their
optimal values. Good convergence of the results of this algorithm is obtained
by comparing six different optimisations, which generally achieve a parameter
convergence range of less than 3% of the initial range. The most convergent
parameters are the high and low pressure gas accumulator volumes, which
converge to the maximum and minimum values of the considered range
respectively. GSF2 and GSF6 produced the best results and an optimal power
around 230kW.
Future work could also consider the effect of the cost of the HPTO on the
optimal design of the system. For example, a larger HPA volume means an
increase in the size, weight, and cost of the HPTO system. Furthermore, the
simultaneous interaction of hydraulic transmission system specifications over
the change of incident waves’ frequency would be promising.

CRediT authorship contribution statement:
Erfan Amini: Conceptualization, Simulation, Methodology, Original draft
preparation, writing. Hossein Mehdiour: Simulation, Writing Original draft,
Methodology, Visualisation. Emilio Faraggiana: Simulation, Visualisation,
Writing original draft. Danial Golbaz: writing original draft. Sevda
Mozaffari: Edit-ing original draft. Giovanni Bracco: Supervision, Editing
original draft. Mehdi Neshat: Conceptualization, Supervision, Editing original
draft.

## Acknowledgement

The project’s results have carried out using High Performance Computing
Laboratory (HPCL) clusters of School of Civil Engineering at University of
Tehran , and the HACTAR High Performance Computing (HPC) of Politecnico of
Turin.

Funding: This research received no external funding.

Conflicts of interest: The authors declare no conflict of interest.

abbreviations: The following abbreviations are used in this manuscript:

WEC | Wave Energy Converter
---|---
WEC-Sim | Wave Energy Converter Simulator
PTO | Power Take-off
RM3 | Reference Model 3
OpenFOAM | Open Source Field Operation and Manipulation
OWC | Oscillating Water Column
OSWEC | Oscillating Surge Wave Energy Converter

## References

  * [1] Ruriko Haraguchi and Takehiko Asai.  Enhanced power absorption of a point absorber wave energy converter using a tuned inertial mass.  Energy, 202:117740, 2020.
  * [2] Silvia Bozzi, Adrià Moreno Miquel, Alessandro Antonini, Giuseppe Passoni, and Renata Archetti.  Modeling of a point absorber for energy conversion in italian seas.  Energies, 6(6):3033–3051, 2013.
  * [3] Xuhui Yue, Geng Dazhou, and Wang Weiyu.  Dynamic performance of key components for oil-hydraulic power take-off of wave energy converters.  2019\.
  * [4] Nianfan Zhang, Xiantao Zhang, Longfei Xiao, Handi Wei, and Weixing Chen.  Evaluation of long-term power capture performance of a bistable point absorber wave energy converter in south china sea.  Ocean Engineering, 237:109338, 2021.
  * [5] Dahai Zhang, Haocai Huang, Ying Chen, Haitao Zhao, and Wei Li.  State-dependent model of a hydraulic power takeoff for an inverse pendulum wave energy converter.  Marine Technology Society Journal, 49(5):38–48, 2015.
  * [6] Mohd Afifi Jusoh, Mohd Zamri Ibrahim, Muhamad Zalani Daud, Zulkifli Mohd Yusop, and Aliashim Albani.  An estimation of hydraulic power take-off unit parameters for wave energy converter device using non-evolutionary nlpql and evolutionary ga approaches.  Energies, 14(1):79, 2021.
  * [7] DE Galván-Pozos and FJ Ocampo-Torres.  Dynamic analysis of a six-degree of freedom wave energy converter based on the concept of the stewart-gough platform.  Renewable Energy, 146:1051–1061, 2020.
  * [8] Markel Penalba, Josh Davidson, Christian Windt, and John V Ringwood.  A high-fidelity wave-to-wire simulation platform for wave energy converters: Coupled numerical wave tank and power take-off models.  Applied energy, 226:655–669, 2018.
  * [9] Changhai Liu, Qingjun Yang, and Gang Bao.  Influence of hydraulic power take-off unit parameters on power capture ability of a two-raft-type wave energy converter.  Ocean engineering, 150:69–80, 2018.
  * [10] CJ Cargo, AJ Hillis, and AR Plummer.  Optimisation and control of a hydraulic power take-off unit for a wave energy converter in irregular waves.  Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 228(4):462–479, 2014.
  * [11] Moisés Brito, Rui ML Ferreira, Luis Teixeira, Maria G Neves, and Ricardo B Canelas.  Experimental investigation on the power capture of an oscillating wave surge converter in unidirectional waves.  Renewable Energy, 151:975–992, 2020.
  * [12] MA Jusoh, MZ Ibrahim, MZ Daud, ZM Yusop, A Albani, SJ Rahman, and S Mohad.  Parameters estimation of hydraulic power take-off system for wave energy conversion system using genetic algorithm.  In IOP Conference Series: Earth and Environmental Science, volume 463, page 012129. IOP Publishing, 2020.
  * [13] Marc Vantorre, Robert Banasiak, and Ronny Verhoeven.  Modelling of hydraulic performance and wave energy extraction by a point absorber in heave.  Applied ocean research, 26(1-2):61–72, 2004.
  * [14] Antoine Peiffer, Dominique Roddier, and Alexia Aubault.  Design of a point absorber inside the windfloat structure.  In International Conference on Offshore Mechanics and Arctic Engineering, volume 44373, pages 247–255, 2011.
  * [15] Yi-Hsiang Yu and Ye Li.  Reynolds-averaged navier–stokes simulation of the heave performance of a two-body floating-point absorber wave energy system.  Computers & Fluids, 73:104–114, 2013.
  * [16] Linnea Sjökvist, Remya Krishna, Magnus Rahm, Valeria Castellucci, Hagnestål Anders, and Mats Leijon.  On the optimization of point absorber buoys.  Journal of Marine Science and Engineering, 2(2):477–492, 2014.
  * [17] Caitlyn E Clark, Anna Garcia-Teruel, Bryony DuPont, and David Forehand.  Towards reliability-based geometry optimization of a point-absorber with pto reliability objectives.  In Proceedings of the European Wave and Tidal Energy Conference, Naples, Italy, pages 1–6, 2019.
  * [18] Xiaofan Li, Changwei Liang, Chien-An Chen, Qiuchi Xiong, Robert G Parker, and Lei Zuo.  Optimum power analysis of a self-reactive wave energy point absorber with mechanically-driven power take-offs.  Energy, 195:116927, 2020.
  * [19] Felipe A Castro and Luciano E Chiang.  Design optimization and experimental validation of a two-body wave energy converter with adjustable power take-off parameters.  Energy for Sustainable Development, 56:19–32, 2020.
  * [20] Pablo Ropero-Giralda, Alejandro JC Crespo, Bonaventura Tagliafierro, Corrado Altomare, José M Domínguez, Moncho Gómez-Gesteira, and Giacomo Viccione.  Efficiency and survivability analysis of a point-absorber wave energy converter using dualsphysics.  Renewable Energy, 162:1763–1776, 2020.
  * [21] Ephraim Bonah Agyekum, Seepana PraveenKumar, Aleksei Eliseev, and Vladimir Ivanovich Velkin.  Design and construction of a novel simple and low-cost test bench point-absorber wave energy converter emulator system.  Inventions, 6(1):20, 2021.
  * [22] Hayrettin Bora Karayaka, Yi-Hsiang Yu, and Eduard Muljadi.  Investigations into balancing peak-to-average power ratio and mean power extraction for a two-body point-absorber wave energy converter.  Energies, 14(12):3489, 2021.
  * [23] E Faraggiana, I Masters, and J Chapman.  Design of an optimization scheme for the wavesub array.  In Advances in Renewable Energies Offshore, Proceedings of the 3rd International Conference on Renewable Energies Offshore (RENEW), Lisbon, Portugal, pages 8–10, 2018.
  * [24] Erfan Amini, Danial Golbaz, Rojin Asadi, Mahdieh Nasiri, Oğuzhan Ceylan, Meysam Majidi Nezhad, and Mehdi Neshat.  A comparative study of metaheuristic algorithms for wave energy converter power take-off optimisation: A case study for eastern australia.  Journal of Marine Science and Engineering, 9(5):490, 2021.
  * [25] IF Noad and R Porter.  Optimisation of arrays of flap-type oscillating wave surge converters.  Applied Ocean Research, 50:237–253, 2015.
  * [26] A Babarit and AH Clement.  Shape otimisation of the searev wave energy converter.  In World Renewable Energy Conference, 2006.
  * [27] AR Plummer and M Schlotter.  Investigating the performance of a hydraulic power take-off.  In Proceedings of the eight European wave and tidal energy conference, Uppsala, 2009.
  * [28] Nicolas Quartier.  Numerical implementation of the power-take-off (pto) of two types of generic wave energy converters using wec-sim.  2018\.
  * [29] Erfan Amini, Rojin Asadi, Danial Golbaz, Mahdieh Nasiri, Seyed Taghi Omid Naeeni, Meysam Majidi Nezhad, Giuseppe Piras, and Mehdi Neshat.  Comparative study of oscillating surge wave energy converter performance: A case study for southern coasts of the caspian sea.  Sustainability, 13(19):10932, 2021.
  * [30] Mehdi Neshat, Seyedali Mirjalili, Nataliia Y Sergiienko, Soheil Esmaeilzadeh, Erfan Amini, Azim Heydari, and Davide Astiaso Garcia.  Layout optimisation of offshore wave energy converters using a novel multi-swarm cooperative algorithm with backtracking strategy: A case study from coasts of australia.  Energy, page 122463, 2021.
  * [31] Yi-Hsiang Yu, Nathan Tom, and Dale Jenne.  Numerical analysis on hydraulic power take-off for wave energy converter and power smoothing methods.  In ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering, pages 1–6. American Society of Mechanical Engineers Digital Collection, 2018.
  * [32] Christopher J Cargo, Andrew R Plummer, Andrew J Hillis, and Michael Schlotter.  Determination of optimal parameters for a hydraulic power take-off unit of a wave energy converter in regular waves.  Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 226(1):98–111, 2012.
  * [33] CJ Cargo, AJ Hillis, and AR Plummer.  Strategies for active tuning of wave energy converter hydraulic power take-off mechanisms.  Renewable Energy, 94:32–47, 2016.
  * [34] Wei Zhang.  Parameter adjustment strategy and experimental development of hydraulic system for wave energy power generation.  Symmetry, 12(5):711, 2020.
  * [35] Mehdi Neshat, Nataliia Y Sergiienko, Erfan Amini, Meysam Majidi Nezhad, Davide Astiaso Garcia, Bradley Alexander, and Markus Wagner.  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(20):5498, 2020.
  * [36] Australian wave energy atlas - australian renewable energy agency (arena), Feb 2018\.
  * [37] The Mathworks, Inc., Natick, Massachusetts.  MATLAB version 9.10 (R2021a), 2021.
  * [38] Wec-sim (wave energy converter simulator).  http://wec-sim.github.io/WEC-Sim/dev/index.html.
  * [39] Aurélien Babarit and Gérard Delhommeau.  Theoretical and numerical aspects of the open source bem solver nemoh.  In 11th European wave and tidal energy conference (EWTEC2015), 2015\.
  * [40] Griet De Backer.  Hydrodynamic design optimization of wave energy converters consisting of heaving point absorbers.  Department of Civil Engineering, Ghent University: Ghent, Belgium, 2009.
  * [41] Kelley Ruehl, Carlos Michelen, Samuel Kanner, Michael Lawson, and Yi-Hsiang Yu.  Preliminary verification and validation of wec-sim, an open-source wave energy converter design tool.  In International conference on offshore mechanics and arctic engineering, volume 45547, page V09BT09A040. American Society of Mechanical Engineers, 2014.
  * [42] Sandia energy.  https://energy.sandia.gov/programs/renewable-energy/water-power/projects/reference-model-project-rmp/, Feb 2021.
  * [43] YH Yu, M Lawson, Y Li, M Previsic, J Epler, and J Lou.  Experimental wave tank test for reference model 3 floating-point absorber wave energy converter project.  Technical report, National Renewable Energy Lab.(NREL), Golden, CO (United States), 2015.
  * [44] Arthur Pecher and Jens Peter Kofoed.  Handbook of ocean wave energy.  Springer Nature, 2017.
  * [45] Ratanak So, Asher Simmons, Ted Brekken, Kelley Ruehl, and Carlos Michelen.  Development of pto-sim: A power performance module for the open-source wave energy converter code wec-sim.  In ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers Digital Collection, 2015.
  * [46] Benjamin Drew, Andrew R Plummer, and M Necip Sahinkaya.  A review of wave energy converter technology.  Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 223(8):887–902, 2009.
  * [47] Sean Sean Michael Casey.  Modeling, simulation, and analysis of two hydraulic power take-off systems for wave energy conversion.  2013\.
  * [48] F de O António.  Phase control through load control of oscillating-body wave energy converters with hydraulic pto system.  Ocean Engineering, 35(3-4):358–366, 2008.
  * [49] Ratanak So, Sean Casey, Sam Kanner, Asher Simmons, and Ted KA Brekken.  Pto-sim: Development of a power take off modeling tool for ocean wave energy conversion.  In 2015 IEEE Power & Energy Society General Meeting, pages 1–5. Ieee, 2015.
  * [50] Jorge Leon.  Motor characteristics on pto-sim 550.  https://github.com/WEC-Sim/WEC-Sim/issues/550issuecomment-812215757_, March 2021.
  * [51] Mordecai Avriel.  Nonlinear programming: analysis and methods.  Courier Corporation, 2003.
  * [52] Jeffrey C Lagarias, James A Reeds, Margaret H Wright, and Paul E Wright.  Convergence properties of the nelder–mead simplex method in low dimensions.  SIAM Journal on optimization, 9(1):112–147, 1998.
  * [53] Msncuso Vaccari, GM Mancuso, J Riccardi, M Cantù, and G Pannocchia.  A sequential linear programming algorithm for economic optimization of hybrid renewable energy systems.  Journal of process control, 74:189–201, 2019.
  * [54] Elizabeth Wong.  Active-set methods for quadratic programming.  University of California, San Diego, 2011.
  * [55] Paul T Boggs and Jon W Tolle.  Sequential quadratic programming.  Acta numerica, 4:1–51, 1995.
  * [56] Seyedali Mirjalili, Seyed Mohammad Mirjalili, and Abdolreza Hatamlou.  Multi-verse optimizer: a nature-inspired algorithm for global optimization.  Neural Computing and Applications, 27(2):495–513, 2016.
  * [57] Anna Garcia-Teruel, Bryony DuPont, and David IM Forehand.  Hull geometry optimisation of wave energy converters: On the choice of the optimisation algorithm and the geometry definition.  Applied Energy, 280:115952, 2020.
  * [58] Xavier Blasco.  Basic Genetic Algorithm.  https://uk.mathworks.com/matlabcentral/fileexchange/39021-basic-geneticalgorithm, 2021\.
  * [59] E Faraggiana, JC Chapman, AJ Williams, and I Masters.  Genetic based optimisation of the design parameters for an array-on-device orbital motion wave energy converter.  Ocean Engineering, 218:108251, 2020.
  * [60] Ivo Couckuyt, Tom Dhaene, and Piet Demeester.  oodace toolbox: a flexible object-oriented kriging implementation.  Journal of Machine Learning Research, 15:3183–3186, 2014.
  * [61] Politecnico di Torino.  HPC HACTAR.  https://www.hpc.polito.it/hactar_cluster.php, 2021.
  * [62] José F Gaspar, Miguel Calvário, Mojtaba Kamarlouei, and C Guedes Soares.  Design tradeoffs of an oil-hydraulic power take-off for wave energy converters.  Renewable energy, 129:245–259, 2018.