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American Journal of  
Geospatial Technology (AJGT)

Steady-State Harmonic and Time Domain Point Absorber Modelling
Saad Al-Sahlawi1*, Garvey Seamus1, Kathy Johnson1

Volume 3 Issue 1, Year 2024
ISSN: 2833-8006 (Online)

DOI: https://doi.org/10.54536/ajgt.v3i1.2882
https://journals.e-palli.com/home/index.php/ajgt

Article Information ABSTRACT

Received: May 01, 2024

Accepted: May 27, 2024

Published: June 01, 2024

This paper analysed the development of  a steady-state harmonic model which, along the 
vertical axis, can simulate a single buoy’s motion with one degree of  freedom (heave). The 
model can optimise a buoy’s geometric and control parameters to maximise power absorp-
tion from incident waves. The steady-state model revealed that at resonance, maximum pow-
er absorption of  the buoy occurred in two region values with either low or high range of  
values of  radiation damping coefficient (c2). In practice, achieving operation in the low c2 
region is difficult. Therefore, it is recommended that the devices be designed to operate in 
the high c2 region to increase power capture. The model also revealed the best value for c1 
(PTO damping coefficient) when the buoy with the peak frequency of  the sea state is at 
resonance, and its mass is optimum. The PTO device size can therefore be manufactured 
accordingly to maximise power absorption. Out of  all the tested buoy shapes (spike, bullet, 
and bi-cone), the bi-cone (60o/120o) buoy performed the best, and its response was most 
similar to the optimum mass response predicted by the model.

Keywords
Heaving Point Absorber, Damping, 
Steady-State Harmonic Model, 
Time Domain Model, Power 
Take-Off  (PTO), Buoy Shape 
Optimisation, Resonance

1 Department of  Mechanical Engineering, University of  Nottingham, Nottingham NG7 2RD, United Kingdom
* Corresponding author’s e-mail: SaadAl-Sahlawi@outlook.com

INTRODUCTION
Increasing the energy absorption of  heaving point 
absorbers under regular waves is focused on tuning the 
Wave Energy Conversion (WEC) system to oscillate 
in resonance with the incoming waves (Al Shami et al., 
2018; Pastor & Liu, 2014). If  a point absorber buoy is 
in resonance with incident waves, it will see increased 
displacement amplitudes and velocities and absorb more 
energy than when it is not (Falnes & Kurniawan, 2020; 
Kara, 2020). To achieve the resonance, the phase and 
amplitude of  the point absorber oscillation have to be 
chosen carefully to ensure optimal performance. Phase 
and amplitude can be varied independently through 
phase and amplitude control factors (Cruz, 2007; Sinha 
et al., 2016). When maximum power absorption is 
achieved, these control factors are considered optimal for 
a desired point absorber motion. This can be calculated 
for a single buoy in multiple degrees of  freedom motion 
or a single degree of  freedom motion (Ahmed et al., 
2022). An optimal result can be obtained by forcing the 
amplitude and phase to take particular values (Haider 
et al., 2021). The various stages required to convert the 
power absorbed from a wave to the final useful energy 
are as follows:

• Stage One is the flow of  power between wave (0) and 
primary interface (1), which gives the intercepted power.

• Stage Two is the power flow between the primary 
interface and the (PTO) (2), giving the captured power.

• Stage Three is the power flow between the PTO and the 
final power conversion stage (3), giving the delivered power.
In a point absorber system, the buoy is connected to 
a PTO-driven generator to capture the wave-induced 
motion and turn it into electrical power. A point absorber 

is a mechanical system with two parameter sets, including 
control and geometrical parameters (Bubbar & Buckham, 
2020). The structure of  the buoy is defined by geometrical 
parameters, which are unchanged once the buoy is built 
(Sun et al., 2021). The control parameters are related to 
the PTO damping mechanism, also known as variable 
parameters that can be tuned to match a given sea state. The 
focal point of  the WEC devices is motion control, which 
aims to increase the performance and competitiveness 
of  particular devices in the energy market. Therefore, 
numerous techniques are used, including latching and 
unclutching or phase control (Salter et al., 2002; Tona 
et al., 2019). They have been developed and applied to 
wave devices to improve efficiency and minimise energy 
production costs. These will be discussed in detail below.
With regular waves, the maximum power absorption for 
a heaving point absorber WEC can be achieved when:

1. The excitation force of  the waves is in phase with the 
velocity profile of  the device (Rahmati & Aggidis, 2016). 

2. The velocity excursion of  the wave cycle maximized 
when energy is supplied (Rahmati & Aggidis, 2016). 
The second condition requires very complex PTO 
mechanisms; it is rarely considered. However, a few types 
of  solutions have been proposed for condition one. 
These include:

Linear Damping
Applying a constant linear damping coefficient (Nolan et 
al., 2005).

Freewheeling or Declutching
By allowing the device to freewheel (unloaded) from the 
extrema (i.e., causing velocity to increase and later applying 



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load only after a specific threshold of  velocity is reached) 
(Garcia-Rosa & Ringwood, 2015; Salter et al., 2002).

Latching
Locking the buoy at the instant in a position when its 
velocity reaches zero and releasing it after a specific 
period (Budar & Falnes, 1975; Saupe et al., 2014). 

LITERATURE REVIEW
Early theoretical works showed that a single-point absorber 
could capture power from a wave crest width greater 
than the width of  the buoy itself, giving potential greater 
power extraction when collecting power from a given 
length of  the beach. The heave amplitudes required for 
maximum power extraction are available only in resonant 
conditions for real buoys. Outside resonant frequencies, 
the power absorbed decreases markedly (Kara, 2016). 
Additionally, the power absorption of  a semi-submerged 
sphere was plotted as a function of  wave frequency. At 
resonant frequencies with no frictional damping, a linear 
PTO damper (generator) ‘s power recovery reached the 
theoretical maximum (Kurniawan et al., 2014). However, 
on either side of  the resonant frequency, the dynamic 
heaving response of  the buoy was insufficient to reach the 
levels required for maximum theoretical power recovery 
(Duncan & Brown, 1982). Another study reviewed the 
physical aspects related to this behaviour. It noted that 
to broaden a buoy’s frequency response, the external 
PTO damping coefficient should be set higher than the 
optimum at resonance (Sakr et al., 2020). 
Studies have also been undertaken to optimise the 
hydrodynamic performance of  various WEC devices 
to improve their energy extraction efficiency from 
waves. Nonetheless, PTO damping, buoy geometry, and 
supplementary inertia are significant for phase control 
(Aderinto & Li, 2019). They maximise efficiency and 
increase the capture width of  a heaving point absorber 
device deployed in the North Sea off  Belgian coastlines 
(Piscopo et al., 2016). A point absorber’s performance was 
assessed in multiple studies where the resonance was tuned 
by changing the PTO characteristics. Results depicted an 
increased power capture in the regular waves by 50% of  
the rated power (Al Shami et al., 2018). In addition, a past 
study examined irregular waves, analyzing that maxmised 
power capture can be achieved by continuously tuning the 
natural frequency to the incoming wave frequency (Burgaç 
& Yavuz, 2020). Falcao (2007) performed a time domain 
analysis to examine the hydrodynamic performance of  a 
coupled hydraulic PTO unit and heaving floating device. An 
algorithm was developed for the performance optimisation 
of  the device, showing performance on the wave period to 
be dependent weakly irrespective of  wave height simulated 
in conditions of  the real sea (Falcão, 2007). He further 
examined and included a phase control strategy of  latching 
in the frequency domain to increase power absorption 
(Falcão, 2008). In his next paper, a two-body heaving point 
absorber’s geometrical configuration was optimised using 

phase control in the frequency domain (Falcao, 2010). 
Al Shami et al. reviewed multiple studies analysing the 
effects of  geometry, mass distribution, and mooring 
system on a tethered WEC in irregular waves. Results 
revealed that tuning the system for a specific wave 
climate is critical for avoiding potential WEC failure 
and maximum energy capture (Al Shami et al., 2018). 
Reportedly, Sjokvist et al. (2014) utilised a velocity ratio 
to study the impact of  radius and buoy draft on WEC 
energy and motion absorption. Findings showed that 
an optimal buoy geometry can be identified for a given 
generator damping. When designing the point absorber, 
the velocity ratio was more informative than the capture 
width ratio. For example, the optimum buoy radius for 
7.4 kNs/m, 20 kNs/m, and 30 kNs/m of  damping is 
around 1.75 m, 2.5 m, and 3 m, respectively. For damping 
of  20 kNs/m and 30 kNs/m, however, there was no 
difference in the results for buoy radius values of  2.5 m, 
3 m, or 3.5 m. While larger buoys were more favourable 
with increasing damping, smaller buoys were favourable 
with lower damping (Sjökvist et al., 2014). Another study 
showed the modeling of  a PTO system, demonstrating 
that oversimplification during the simulation phase of  
WEC development can lead to incorrect design decisions 
and additional delays and costs (Cargo et al., 2016).
Using phase control to shift the natural frequency of  a 
point absorber near the resonance condition by adding 
a supplementary mass or a negative mechanical spring 
has also been studied (Piscopo et al., 2016). They also 
showed that phase control tunes the natural frequency of  
the device suitably into the fully submerged body based 
on PTO damping. Passive control was also studied by 
Piscopo et al. (2016), combined with the development of  
a new optimisation process for heaving point absorber 
hydrodynamic performance to maximize energy 
production yearly. Their optimisation procedure allows 
for modifying the PTO damping and fully submerged 
added mass until the optimum configuration is detected. 
It was concluded that the deployment site is essential 
in the WEC assessment for optimum configuration 
(Piscopo et al., 2016).
A new approach for optimising the geometry and 
performance of  a point absorber was proposed in this 
paper, to increase wave device efficiency while minimising 
energy production costs. A steady-state harmonic model 
was designed to simulate motion with one degree of  
freedom (heave) along the vertical axis of  a single buoy. It 
is used to optimise its geometrical and control parameters, 
and its power absorption is maximised from incident 
waves. The steady-state harmonic model was verified 
against a time domain model by comparing the models’ 
predictions for steady state response. The geometrical and 
control parameters of  the heaving buoy were optimised to 
achieve a cost-effective buoy design that maximised power 
absorption from incident waves. The buoy behaviour was 
simulated for the case of  regular waves. The approach 
used was numerical, and the dynamic equations were 



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formulated, and the results were calculated in the time 
domain using the software MATLAB. The PTO damper 
c1  was assumed to be a constant damping coefficient for 
simplicity, and the force of  force was decomposed into a 
linear radiation-damping term c2 which reflected the system 
geometry. Further analyses were carried out to determine 
the damping profiles of  varied buoy shapes: bullet, spike, 
and bi-cone (60o/120o).

Generalised Heaving Point Absorber (Buoy) Model
Point Absorber System 
The basic design and functional principles of  a point 
absorber system are illustrated in Figure 2, which shows 
a rigid floating body (buoy) attached to the sea bed. The 
purpose of  the floating buoy is to act as the energy-
absorbing body of  the system and hold the system 
upright. These two functions result from the buoyancy 
of  the buoy and the induced pressure forces due to 
water particle movement in the wavefront. The generator 
consists of  a spring-damper system where the PTO 
provides the damping. The generator’s primary function 
is to take the energy captured from the wave-induced 
motion of  the float by the PTO and convert it into 
electricity. The PTO device is placed somewhere between 
the sea bed and the buoy to determine the direction of  
the buoy’s motion, which is crucial for power production. 
The generator also contains a spring, which acts as a pre-
tensioner to keep the line straight. The point absorber 
buoy system, which has a mass, a damping coefficient, 
and a stiffness coefficient in classical mechanics, can be 
treated as a mass-spring-damper system. 
The buoy system is modelled as rigidly attached to the 
generator to move the entire system in phase. This is one 
of  the main assumptions used in the analysis presented 
in this paper.
When the point absorber buoy system is displaced 

force is proportional to the velocity, the damped oscillator 
is the harmonic oscillator (Fitzpatrick, 2018). Using the 
damping coefficient, the system can:

• Oscillate with a frequency lower than in the non-
damped case, and the displacement amplitude decreases 
with time (underdamped system).

• Decay to the equilibrium position without any 
oscillation (overdamped system).
The boundary between overdamped and underdamped 
behaviour occurs at a specific value of  damping 
coefficient where the system is called “critically damped.” 
In this case, the damping coefficient is called the critical 
damping coefficient, cc, is given by:
cc=2√km (Ns/m)                  (1)
The ratio of  the actual damping coefficient to the critical 
damping coefficient is called the damping ratio,ξ, and is 
given by:
ξ=c/cc                   (2)

Mass-Spring-Damper Representation
In this model, a point absorber system’s simple motion is 
compared to a mechanical oscillator containing a mass-
spring-damper system with a single degree of  freedom 
of  motion working in the direction of  the degree of  
freedom subjected to an external force. Figure 3 presents 
a schematic representation of  the system.

Figure 1: Basic design of  a point absorber system

from equilibrium, a restoring force proportional to its 
displacement is experienced. Notably, the system is called 
a simple harmonic oscillator if  the only force acting on 
the system is the restoring force (Korde & Ringwood, 
2016). This means that, with a constant amplitude and 
frequency, it undergoes simple harmonic motion. If  the 

Figure 2: Schematic representation of  mass-spring-
damper system

The system’s damping is linear with a damping coefficient, 
c (Ns/m). A harmonic force is applied with angular 
frequency, ω (rad/s) and amplitude, A (m). The spring 
constant, k (N/m) to the system, is a restoring mechanical 
force proportional to the system’s vertical displacement, 
y, from its original position.

Time Domain Model
The Equation of  motion of  this buoy can thus be found 
from Newton’s second law:
mÿ+(c1+c2)ẏ +ky=A [kcos(ωt)- c2 ωsin(ωt)]               (3)
Equation (3) was linearised by expressing each term on 
the left-hand side as a velocity function. The buoy relative 
velocity v(t) was then calculated using the Runge-Kutta 
4th order method in MATLAB (suitable for non-stiff  
cases) by setting.(y) (t)= v̇(t) and ẏ(t)=v(t)  and applying 



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the initial conditions y ̇(0)= v(0) = 0. Based on the results, 
the rate of  energy dissipation in the damper over time t1  
was computed and is given as:
Ed (t1)=∫t1

0 P1 dt= ∫t1
0 c1 ẏ

2 dt                  (4)
In theory, this energy is maximised when Equation (4) is 
maximised over one period of  the excitation wave force. 
It results in the condition:
ωn = c1 √(k/m)= ω               (5)
When this condition is met, the device’s velocity profile 
is in phase with the excitation wave force, consistent with 
condition one for maximising the power absorption.

Equation of  Motion
Due to the vertical wave velocity u(t), the buoy is displaced 
from its equilibrium position to an actual position of  
y(t). The differential Equation of  motion of  the point 
absorber can be found in Newton’s second law:
mÿ+ c1 ẏ = k (u-y)+ c2  (u̇- ẏ )               (6) 
The spring force is expressed in Equation (6) as k (u-y). The 
damping force exerted by the PTO system is expressed as 
c1 ẏ. The damper c1 is the main power extraction unit, and 
the PTO system is assumed to be linear for simplicity. The 
radiation force is expressed as c2  (u̇- ẏ ).

Power Absorption
The velocity,v, in the power extraction damper c1 is 
expressed as:
v(t)=(y(t)) = Re (V× ejωt )                 (7) 
The mean power extracted by damper c1 is given by:
P=  1/2 c1 |jωY|2 =  1/2  c1 ω

2  |Y|2               (8)
Substituting Equation (7) into Equation (8) gives:
P=  1/2 ω2 c1 × ((k2 + ω2 c2

2 ))/(((k- ω2 m)2+ ω2  (c1+ c2 )
2 

))  × U2                   (9) 
It can be seen that although the wave magnitude,U, does 
not affect the optimal values of  m, c1 and c2, it can affect 
the value of  the extracted power.

Power Absorption Optimisation
To optimise the buoy power extraction rate, the term H 
defined below needs to be differentiated concerning m, 
c1 and c2. The partial derivatives ∂H/∂m, ∂H/(∂c1) and 
∂H/(∂c2) are then set equal to zero, and their stationary 
values are obtained.
H= (c1 (k

2 + ω2 c2
2 ))/(((k- ω2 m)2+ ω2 (c1+ c2 )

2 ))      (10)
Setting the numerator in the partial derivative (∂H/∂m ) 
equal to zero gives the stationary values for m as:
m=  k/ω2               (11) 
Setting the numerator in the partial derivative (∂H/(∂c1 )) 
equal to zero, substituting for m= k/ω2   and rearranging 
gives: 
c1 = c2                 (12)
Setting the numerator in the partial derivative (∂H/(∂c2 )) 
equal to zero, and rearranging gives: 
c2 =  k2/(ω2 c1)               (13)
By substituting for m from Equation (11) and c2 from 
Equation (13) into Equation (9), the expression for the 
optimum power extraction rate was obtained as follows:
P=  1/2  ×  (ω2 c1 k

2)/((ω2 c2
1+ k2 ) )  × U2            (14)

The system has 5 dimensions: k,m,c1,c2 and  ω. To reduce 
the number of  dimensions when calculating the buoy 
optimum power response,   m was fixed at its optimum 
value and c1 was fixed at its reference value. The unknown 
parameter c2 was varied for a wave magnitude U = 1 
m. The buoyancy-stiffness and wave natural frequency 
were kept constant at k =10006 N/m and  ω=2π rad/s. 
The natural frequency of  the buoy was assumed to be 
equal to the peak frequency of  the sea state, i.e., it was 
assumed to be in resonance. Optimisation of  the buoy 
power response was carried out in MATLAB using the 
following:
PTO external damping coefficient c1= k⁄ω  (Ns/m). 
Buoy mass m=  k⁄ω2 (kg).
Radiation damping coefficient c2 = cc [10(-8) to 108]  
(Ns/m). The c2 value was varied using 500 time steps.

RESULTS
The first set of  simulations was performed by fixing m 
and c1 at their optimum values and varying c2 to determine 
the optimum power extraction rate over the range of  
regular wave conditions. The mean power extracted was 
determined using Equation (9), and the results are plotted 
in Figure 3.

Figure 3: Power capture at c1= 1592.50 Ns/m and  m= 
(a) 100 kg (b) 253.4549 kg, and (c) 500 kg



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The results indicate that there are two power extraction 
rate regions: the left region is driven by the low value of  
c2  and the right region is driven by the high value of  c2  
as seen in Figure 3. With this model, the optimum mass 
and value of  c1 were found to be 253.4549 kg and 1592.50 
Ns/m, respectively (Figure 3b).
It was clear from the simulations that using the optimum 
mass gave the highest power extraction rates in the right 
and left regions of  the graph (Figure 3a). The power 
extraction rate was constant for c2<102 and c2>104 
(Ns/m), and the lowest power extraction rates occurred 
for c2  values between 102 and 104 (Ns/m). These results 
are similar to the results from the time domain model.
Further analyses were conducted with mass values above 
and below the optimum mass. Mass values of  100 kg 
(Figure 3a) and 500 kg (Figure 3c) resulted in only lower 
power capture in the left region. This showed that there 
is no visible effect of  increasing the mass to 500 kg 
when the c2 value is high. Mass was only found to play a 
significant role in low values of  c2. The maximum power 
extraction rate of  a buoy with optimum mass     and     was 
1585 W/s. 

Validation
The time for time domain and steady-state harmonic 
models to reach steady state was 60s, and the maximum 
power absorption achieved for both models was 36.53 
kW. The results from both analyses were almost identical. 
This indicated that the steady-state harmonic model 
developed for analysing the steady-state conditions of  the 
WEC system gave good predictions for buoy behaviour. 
Therefore, the steady-state harmonic model was utilised 
for further optimisation.

Steady-State Harmonic Model
In this section, the steady-state model is presented, and 
oscillation of  a point absorber in a harmonic wave is 
discussed to a fixed reference. The buoy is restricted to 
heave motion only. It is assumed that one end of  the 
system is fixed to an inertial frame similar to the time 
domain model discussed in the previous section. A 
schematic of  a point absorber system under vertical wave 
velocity u(t) is shown in Figure 4.

Steady-State Harmonic Model Optimisation
The objective of  this section is to examine the optimal 
damping profile for a point absorber (buoy) using 
the values for c1 and c2 which gave the greatest power 
absorption. The shape is assumed to be independent of  
the specific device parameters for the optimal damping 
profile.
From Equation (12), the optimum performance is 
achieved when c1= c2. The power response was thus 
maximised by using the best value for c2 from the region 
driven by high c2 for determining the best value of  c1 for 
a mass equal to the optimum mass.

Figure 4: Schematic representation of  a heaving point 
absorber

Figure 5: Power capture at c1= 6.54 × 105 Ns/m and c2 
= 6.54 × 105 Ns/m

Figure 6: Contour plot for the relationship between c1 
and c2 at 

RESULTS
For the resonance condition, the best value of  the PTO 
damping coefficient c1 for the optimal radiation damping 
coefficient c2 = 6.54 × 105 Ns/m is illustrated in Figure 
5. The results of  the simulation confirm the validity of  



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the linear relationship derived in Equation (12) (see the 
contour plot of  c1= c2 in Figure 6).
From Figure 7, it can be seen that the highest power 
occurs at c1= c2 with the optimum mass of  253.4549 
kg. The value of  c2 is defined by the structure of  the 
buoy and cannot be changed once the buoy has been 

constructed. The term c1 is used to model the PTO 
damping mechanism, a variable parameter that can be 
tuned to match the sea state. It is therefore recommended 
that buoys are constructed with large c2 values such that 
the optimum c1 value will equal c2 and will not change 
when the system is not at resonance.

Figure 7: Log power vs c1 and c2  at      = 253.4549 Kg

Figure 8: The relationship between c1 and c2 with m

At optimum mass, the system is in resonance, and to 
maximise the power absorbed, the condition of  c1= 
c2 must be met. Further analyses were conducted with 
mass values of  99.99% and 100.01% of     . The tests 
revealed that for a system that is not in resonance, the 
linear relationship between c1 and c2 no longer holds. This 
was true even when mass deviated by only 0.01% from its 

optimum value, as seen in Figures 8a and 8c.
It was concluded that the relationship c1= c2 is only valid if  
the point absorber is at its optimum mass (Figure 8b) and 
is in resonance with the peak frequency of  the sea state.

Buoy Shape Optimisation
Validating the steady-state harmonic model against 



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experimental data was conducted. The steady-state 
harmonic model determined the optimal damping 
profiles for different point absorber (buoy) shapes. As 
already shown, wave energy extraction is a complicated 
procedure, and every change in the geometrical or 
control parameters has a major impact on the system’s 
power extraction. It was therefore decided to carry out 
the optimisation process by keeping the diameter of  the 
buoys fixed and comparing the buoys purely based on 
their shape and draft.
The damping characteristics of  the three different point 
absorber (buoy) shapes were determined experimentally, 
and the results were compared against those of  the 
steady-state harmonic model to determine the optimum 
operating region for each buoy shape. Three buoy shapes 
were chosen, namely (a) bullet, (b) spike, and (c) bi-cone 
(60o/120o). The dimensions of  the buoy prototypes are 
illustrated in Figure 9.

All the shapes have the same cross-sectional area (diameter 
= 0.3125 m) but have different mass and draft values, as 
shown in Table 1 in the next section. The bullet and spike 
buoys had relatively longer drafts (0.755 m) than the bi-
cone buoy (0.22 m).

Figure 9: Shapes and dimensions of  prototype buoys

Table 1: Bullet, spike, and bi-cone decay tests and statistical results
Buoy Shape Mean 

Damping 
Ratio

Buoy 
Mass (kg)

Mean 
Draft (m)

Mean Damped 
Frequency 
(Hz)

Mean  Radiation 
Damping 
Coefficient,
C2 (Ns/m)

Sample 
Variance 
s2 (Ns/m)2

Spike 0.128 27.72 0.755 0.729 32.51 3.6074
Bullet 0.124 33.44 0.755 0.667 34.84 9.2916
Bi-cone 0.094 11.63 0.220 1.164 16.0 7.8795
Damped Spike 0.114 27.30 0.755 0.727 28.51 0.8511
Damped Bullet 0.111 31.24 0.755 0.685 29.54 10.5423
Extra Damped Spike 0.099 27.58 0.755 0.715 24.55 5.3399
Extra Damped Bullet 0.095 31.01 0.755 0.693 25.55 2.37

Decay tests were performed at the Kuwait Institute 
for Scientific Research (KISR) utilising the Coastal 
Management Program’s wave flume tank facility. The 
approach for this work consisted of  two parts: (a) 
conducting decay tests to determine the radiation-
damping coefficient, c2, for each buoy shape; and (b) 
examining the behaviour of  each buoy under different 
damping conditions. Only the bullet and spike buoys 
were used for part (b). The different damping conditions 
were achieved by adding one (damped) or two (extra-
damped) round disks to the lower section of  the buoys. 
When two disks were used, they were attached 2 cm and 
3 cm apart for the bullet and spike buoys, respectively. 
After testing, the values of  c2 obtained from the decay 
tests were verified by checking whether they were close 
to the worst c2 value obtained from the optimised steady-
state harmonic model or approached the best c2 values.

Buoy Power Optimisation
To optimise the power absorbed by each buoy at 
resonance and under zero damped, damped, and extra 
damped conditions, the steady-state harmonic model 
was used to determine the optimal. c2 damping profiles 
for each shape. The values of  c2 obtained during decay 

testing were verified to ensure they were not near the 
worst c2 value obtained from the model. All the damping 
coefficient values were scaled down to match the physical 
models. The diameter used for the steady-state harmonic 
analyses was d = 1 m, and the physical models had d = 
0.3125 m.
The performance of  the bullet, spike, and bi-cone 
buoys was optimised using the following hydrodynamic 
parameters in MATLAB:
Buoyancy-stiffness corresponding to an occupied area of  
0.08 m2 in seawater, k = (πd2/4) × 9.81 × 1000 N/m.
Wave frequency, ω=2π = 6.28 rad/s. 
Resonance frequency, ωn= 2π/0.9 rad/s. This was 
calculated based on the experimental results, which gave 
resonance at T= 0.9 s (see Chapter 4). 
Mass = mass of  buoys used in the decay tests (see Table 
3.1).
Radiation damping coefficient c2=damping ratio × cc 
(Ns/m). The damping ratio of  each buoy shape was 
taken from the decay tests and used to determine c2 the 
value associated with it.
Radiation damping coefficient c2= cc [(10-8) to (108)]  
(Ns/m). The c2 value in the model was varied using 500-
time steps.



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RESULTS
During the experimental testing, the opportunity to vary 
the performance of  the buoys at very little cost was utilised. 
Based on the steady-state harmonic model predictions, it 
was concluded that operating in the high-damping region 

Figure 10: Mean c2 for the bullet, spike, and bi-cone buoys

is recommended to maximise the power extraction rate, 
which suggests that adding some form of  damping 
is beneficial. This was tested during the experimental 
investigation, and extra damping was provided in the 
form of  round discs mounted on the lower parts of  the 
bullet and spike buoys. A comparison between the mean 
c2 values for all the shapes were collected; the results are 
illustrated in Figure 10.
The extra-damped bullet and spike buoys had the lowest 
radiation-damping values compared to the zero-damped 
and damped buoys. This behaviour was confirmed by the 
statistical t-test, where probabilities above 95% provided 
very high confidence in the results.
The variation in the power absorbed by the zero-damped 
buoys as a function of  the radiation-damping values is 
illustrated in Figures 11a, 11b, and 11c. In these figures, 
the c2 values derived from the decay tests for each shape 
are marked in blue and the worst c2 values from the model 
are marked in red. 

Figure 11: Maximum power capture at     for zero damped buoys



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Am. J. Geo Spat. Technol. 3(1) 35-45, 2024

Figures 11a and 11b, for the bullet and spike buoys, 
respectively, showed lower power capture in the left-low 
c2 region only. This means that their mass is far from 
optimum, as indicated in Figure 3, which shows results 
for masses above and below optimum. The c2 values for 
the bullet and spike buoy shapes are close to the worst 
values for c2  predicted by the model. In Figure 11a, for 
the bullet buoy, the power recorded in the highest and 
lowest c2 regions were 107.78 W and 45.67 W, respectively. 
The worst power region for the bullet buoy occurred for 
c2 between 2.41 and 1.51 × 104 Ns/m. For this buoy to 
achieve a power absorption of  107.78 W, its c2 the value 
should be higher than 1.51 × 104 Ns/m. 

CONCLUSION
A new approach for optimising the geometry and 
performance of  a point absorber was proposed to 
increase wave device efficiency while minimizing energy 
production costs. The simulation of  the motion of  a 
single buoy for a steady-state harmonic oscillator model 
was attempted with one degree of  freedom (heave) to 
optimise control and geometric parameters along the 
vertical axis. It is vital for maximisation of  the power 
absorption from incident waves. The steady-state 
harmonic model was verified against a time domain 
model by comparing the models’ predictions for steady-
state response. The geometrical and control parameters 
of  the heaving buoy were optimised to achieve a cost-
effective buoy design which maximised power absorption 
from incident waves. The buoy behaviour was simulated 
for the case of  regular waves. The approach used was 
numerical, and the dynamic equations were formulated, 
and the results were calculated in the time domain 
using the software MATLAB. The PTO damper c1 
was assumed to be a constant damping coefficient for 
simplicity, and the radiation force was decomposed into 
a linear radiation-damping term c2 which reflected the 
system geometry. Further analyses were carried out to 
determine the damping profiles of  different buoy shapes: 
bullet, spike, and bi-cone (60o/120o).
The steady-state model revealed that at resonance, 
maximum power absorption of  the buoy occurred in 
two region values with either low or high range of  values 
of  radiation damping coefficient. In practice, it is critical 
to achieve operation in the low c2 region; therefore, 
it is advised to use the devices designed with high c2 
region to maximise power capture. This suggests that 
external damping should be added to the buoy structure. 
The results also indicated that there was an optimum 
operating range for each buoy shape for the PTO-
driven generator where wave energy capture is greatest 
and thus the electrical power. The model revealed that 
the best value for c1 is c1= c2=k/ω, when the buoy is in 
resonance with the peak frequency of  the sea state and its 
mass, is optimum. The PTO device size can therefore be 
manufactured accordingly to maximise power absorption. 
Out of  all the tested buoy shapes (spike, bullet, and bi-
cone), the bi-cone (60o/120o) buoy’s response was most 

similar to the optimum mass response predicted by the 
model, which indicated that its mass of  11.63 kg was 
closest to optimum. It was also noted that the c2 damping 
associated with the tested bi-cone shape was quite 
different from the c2 value predicted by the model to 
give the worst performance. The optimisation approach 
developed could be used to realise major economic design 
benefits for any point absorber buoy shape. The model 
presented in this work provides an analytical framework 
for an improved understanding of  point absorber PTO 
devices.

Future Developments
The PTO system has been modelled as a linear damper. 
This might be considered the models’ main limitation 
because approaches that consider irregular waves tend 
to be more accurate as they better replicate real sea 
conditions. In reality, the linear approach is invalid, and 
power losses due to the PTO will always be present. The 
PTO system thus needs to be modelled more accurately. 
An extensive study on modelling a PTO system can be 
conducted where existing literature could be used further 
to improve the steady-state harmonic model in this work. 
Cargo (2012) demonstrated that over-simplification 
of  the PTO during the phase of  simulation of  WEC 
development could lead to subsequent delays and costs 
and incorrect design decisions. Therefore, future studies 
need to consider economic considerations during the 
design process. The radiation damping drives optimal 
design parameters such as buoy shape, dimensions, and 
structure c2 which in turn is dependent on the costs 
involved.

Nomenclature
A Amplitude of  excitation force [m]
Aw Area occupied by the buoy in seawater [m2]
c Damping coefficient [Ns/m]
c1 PTO damping coefficient [Ns/m]
    Optimum PTO damping coefficient [Ns/m]
c2 Radiation damping coefficient [Ns/m] 
    Optimum radiation damping coefficient [Ns/m]
ca Actual damping coefficient [Ns/m]
cc Critical damping coefficient [Ns/m]
K Spring constant, used to represent buoyancy-stiffness              
[N/m]
M Mass [kg]
    Optimum mass [kg]
P Power absorbed by the PTO [W]
S Standard deviation in results
T Time [s]
u(t) Wave vertical velocity [m/s]
V Velocity across damper [m/s]
x(t) Wave displacement [m]
Y Vertical displacement [m]
ẏ  & v Vertical velocity [m/s]
ÿ& v̇ Vertical acceleration [m/s2]
ξ Damping ratio
λ Frequency of  displacement for calculating time constant



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∅ Amplitude of  displacement for calculating time constant
τ Time constant representing time to settle to steady state
η Overall power absorption efficiency [%]
ωα Angular frequency of  excitation force [rad/s]

Acknowledgements 
The authors wish to express their gratitude and 
thanks for the University of  Nottingham and Kuwait 
Scientific Research Centre (KISR) for providing all the 
infrastructure facilities to carry out this work. 

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