Optimizing efficiency of a ship’s two-magnet energy harvesting device using the Cuckoo search algorithm

Abstract

Establishing an energy harvesting device to supply sustainable energy to deep sea sensors is a challenging task due to the complex subsea marine phenomenon. To overcome this, we propose a two-magnet KMC energy harvester hanging on the cabin of the ship with a spherical bearing. The magnets connected with springs move vertically up and down when the ship is periodically actuated by surface waves, generating electricity to power the deep sea sensors through a cable. We present a case study using wave speed (U_ow = 2.0 m/s), wave amplitude (Ho = 0.4 m), and wave length (λ = 4.0 m) to demonstrate the efficacy of the two-magnet energy harvester. Before optimizing the energy harvester, a sensitivity analysis is conducted on the electrical power with respect to marine parameters and the energy harvester’s geometric parameters. To maximize the electrical power, we adopt a Cuckoo Search Algorithm as an optimizer. By using the objective function of root-mean-square electrical power in conjunction with the Cuckoo Search, we obtain optimal electrical power. Simulations reveal that the energy harvester produces 0.1497 Watts of power.

Introduction

In the past decade, offshore exploration for oil and natural gas has significantly increased, leading to the discovery of more resources. However, the potential for oil and gas pipeline leaks or ruptures during natural disasters, such as earthquakes and storms, has made it necessary to establish a monitoring system that can detect such incidents. This monitoring system requires a large number of environmental sensors, which need a sustainable source of electricity to function continuously. Therefore, it is essential to find an easy and sustainable way of generating electrical energy for the sensors. To address this issue, the design of a self-generated energy harvester that can extract energy from sea currents is a viable approach.

Common methods of power generation include piezoelectric power generation, which is suitable only for high-frequency vibration systems. However, wave reciprocating motion falls within the low-frequency domain, making piezoelectric power generation unsuitable. To resolve the power supply issue, self-generated energy harvesters that extract energy from sea currents have been developed. Two types of electricity generation systems, namely, cantilever systems^1,2 and spring-mass systems,^3,4 have been developed. While cantilever systems are suitable for smaller vibrations, they are mainly limited to higher frequencies and smaller systems such as MEMS. ⁵ In contrast, spring-mass systems are more suitable for extracting energy from low-frequency and large-displacement vibrations, such as those present in sea wave environments.

Several researchers have developed energy harvesting systems using ambient air vibrating motion, ⁶ vibrating electromagnetic energy extractors,^7–10 and linear generators using wave motion, ¹¹ but these systems are primarily limited to onshore applications. Others have suggested piezoelectric power generation systems using offshore buoys and waves,^12–14 but the electrical power output is usually small due to the low-frequency of ocean currents. An et al. ¹⁵ in 2022 utilized electromagnetic energy harvesting to generate electrical power for condition monitoring sensors. Later, research by Muscat et al. ¹⁶ suggested that vibrational energy harvesters offer a promising solution for powering sensors, especially in areas with prevalent vibrations. Among different types of vibrational harvesters, electromagnetic ones stand out for their capacity to capture kinetic energy within a low-frequency range. Kurt et al. ¹⁷ introduced an elementary electromagnetic energy harvester featuring two magnets in 2023. Their study focused on harnessing electricity from a broad spectrum of vibrational sources. The findings demonstrated that this electromagnetic device can efficiently harvest power across a wide frequency range, spanning from 2 Hz to 1.5 KHz. Its cost-effective and straightforward design makes it suitable for various applications, enabling continuous energy harvesting from diverse ambient vibrations. In addition, Digregorio and Redout, ¹⁸ in 2024, introduced a compact electromagnetic energy harvester (EMEH) comprising two coils within a head-to-head permanent magnet tower. Their results showed that this inertial EMEH, operating at 10 Hz and 1.32 g RMS acceleration, consistently generates an output power of 1696 µW, making it suitable for biomedical and wearable applications.

In previous studies, Chiu et al. ¹⁹ proposed a sustainable energy harvester installed on the sea bed, but the small thrust of the ocean current resulted in a low induced electricity output. Later, Chiu ²⁰ developed an electrical energy harvesting system using a long rod connected to a buoy, but the study did not address the issue of unforeseen fatigue and shearing problems of the rod caused by strong deep-sea currents. To overcome this issue, Chiu et al. ²¹ developed a subsea generator, replacing the long rods with cables. The induced electricity from the subsea generator can be intermittently generated through unidirectional rotation when the buoy moves up and down with the wave to pull and release the cable. However, the lateral forces of ocean currents on the cable can decrease the induced electricity output from the submarine generator. Furthermore, Chiu and Cheng ²² devised a wave energy harvesting device designed for use in a lake, aimed at enhancing the extraction of wave energy.

In the past, installations were directly exposed to water waves underwater, which posed risks due to unpredictable sea conditions. Cross waves and wind waves underwater could cause damage and corrosion to the equipment. Although underwater power generation equipment ²¹ could offer higher electrical power, the primary drawback is the risk associated with operating in hazardous underwater conditions.

To address the drawbacks mentioned above, the study we present here is to place the energy harvesting system on the water surface to avoid hardware damage issues. This study utilizes a mechanical vibration system with parameters K (spring stiffness), M (system mass-magnet), and C (damping effect), combined with coil-wound induction coils, to perform magnetic induction power generation. This magnetic-electric induction system is designed for low-frequency vibration sources. The design of this energy harvesting system (as shown in Figure 1) is portable, so to increase its power generation capacity, multiple independent compartments can be arranged within ship-related tools. The energy harvesting device is designed to avoid unforeseen risks of the subsea, such as lateral forces and harsh underwater conditions. To keep the energy harvester in a vertical position when the ship tilts, a spherical bearing and gravity effect are employed. In this position, the energy harvester can generate sustainable electricity offshore by utilizing the base-excitation of the cabin, which is periodically actuated by the sea current. The two-magnet energy harvester, surrounded by two sets of coils, generates induced electrical power when magnetic flux variations occur due to the relative motion of the magnets and the energy harvester base excited by the sea current.

Figure 1.

A two-magnet energy harvester installed on the deck of a ship which has been excited up and down periodically by the surface wave.

To maximize the electrical power output, an optimizer is necessary. Traditional gradient methods ²³ have a limitation in optimization as they rely on a gradient-guided approach that can potentially lead to local optima. In contrast, the Cuckoo Search algorithm, a more recent metaheuristic optimization algorithm, has the ability to avoid local optima and move towards the global optimum. Swarm intelligence algorithms have proven successful in solving complex optimization problems in engineering, including genetic algorithms (GA), ²⁴ particle swarm optimization (PSO), ²⁵ differential evolution (DE), ²⁶ ant colony optimization (ACO), ²⁷ artificial immune methods (AIM), ²⁸ and simulated annealing (SA). ²⁹ Research by Al-Abaji ³⁰ indicates that cuckoo search achieves better results than PSO and GA. The Cuckoo Search Algorithm is a nature-inspired metaheuristic algorithm that draws its principles from the brood parasitism observed in certain cuckoo species, combined with Levy flights random walks. To efficiently search a global solution, The Cuckoo Search Algorithm is then adopted in optimizing our energy harvesting system.

Therefore, this study employs the cuckoo search algorithm to optimize the ship-based energy harvesting device.

Mathematical background

The system includes (1) spherical bearing unit; (2) energy harvesting unit; (3) energy storage unit. The energy harvesting system utilizes a spherical bearing unit, supporting a dual-mass point energy harvester connected to an electromagnetic induction system, and coupled with a charging unit. The upper part of the spherical bearing unit can be hung with hooks to suspend the energy harvester in any compartment of a ship. Due to the combined effect of the energy harvester’s weight and the freely rotating spherical bearing, the energy harvester’s shell remains vertical. Therefore, as the ship rocks in all directions due to waves, the energy harvester’s shell maintains its vertical position, while the dual-mass point magnets and springs inside the shell undergo vertical oscillations, performing base-excitation mechanical vibrations. Using the Lagrange equation energy method, the governing equation for base-excitation mechanical vibrations is derived. The base-excitation source is the reciprocating vibration of waves, with damping in the system coming from mechanical damping and magnetic-electric damping generated by coil-wound induction. For problem solving, we use the real and imaginary parts of the complex method to represent external forces acting on the system, to solve for the displacement and velocity of the two magnets in the mechanical reciprocating vibration. We also calculate the induced electromotive force and corresponding induced electrical power due to changes in magnetic flux using Faraday’s law.

Base-excitation motion

In Figure 1, the energy harvester is shown hanging on top of the ship’s cabin. To ensure that the energy harvester remains in a vertical position regardless of the ship’s tilt in any direction, a spherical bearing and gravity center effect are utilized, as illustrated in Figure 2. Consequently, the two-magnet energy harvester will be excited vertically by the wave, resulting in periodic upward and downward motion. This movement, as depicted in Figure 3, is how the energy harvester will be actuated by the wave.

Figure 2.

A hanging type energy harvester keeping in vertical position.

Figure 3.

A two-magnet energy harvester.

As depicted in Figures 1 and 3, the base-excitation motion at the top of the cabin is induced by the surface wave as

$z_0\left(t\right)=H_0\hspace{0.17em}\cos \hspace{0.17em}\left(\omega t\right)$

(1a)

$\omega =2\pi \frac{U_{sea}}{\lambda }$

(1b)

where $U_{sea}$ is the wave speed and $\lambda$ is the wave length.

Dynamic system model

A mathematical model of the two-magnet hanging type energy harvester is analyzed as a KMC (spring + mass + damper) system and displayed in Figure 4.

Figure 4.

A scheme of the two-magnet hanging energy harvester.

Using the energy equations yields

$\mathrm{T}=\frac{1}{2}{M}_1{\dot{zz}}_1^2+\frac{1}{2}{M}_2{\dot{zz}}_2^2$

(2a)

$\mathrm{V}=\frac{1}{2}{k}_1{\left(L_2-{zz}_1\right)}^2+\frac{1}{2}{k}_2{\left({zz}_1-{zz}_2\right)}^2$

(2b)

$\mathrm{D}=\frac{1}{2}{C}_1{\left({\dot{zz}}_1\right)}^2+\frac{1}{2}{C}_2{\left({\dot{zz}}_1-{\dot{z}z}_2\right)}^2$

(2c)

where T, V, and D represent the kinetic, potential, and damping energies.

Using the Lagrange function $\mathrm{L}=\mathrm{T}-\mathrm{V}$ and the Lagrange equations, it has

$\mathrm{L}=\mathrm{T}-\mathrm{V}=\frac{1}{2}{M}_1{\dot{zz}}_1^2+\frac{1}{2}{M}_2{\dot{zz}}_2^2-\frac{1}{2}{k}_1{\left({zz}_1\right)}^2-\frac{1}{2}{k}_2{\left({zz}_1-{zz}_2\right)}^2$

(3a)

$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{zz}}_i}\right)-\frac{\partial L}{\partial z_i}+\frac{\partial D}{\partial {\dot{z}}_i}=Q_i\left(t\right)$

(3b)

where $Q_i\left(t\right)$ represents the external force applied at the i-th degree of freedom. Substituting equations (2a)–(2c) into equations (3a) and (3b) and doing an rearrangement in a matrix form yields

$\left[\begin{array}{cc}{M}_1& 0\\ 0& M_2\end{array}\right]\left[\begin{array}{c}{\ddot{zz}}_1\\ {\ddot{zz}}_2\end{array}\right]+\left[\begin{array}{cc}{C}_1+C_2& -C_2\\ -C_2& C_2\end{array}\right]\left[\begin{array}{c}{\dot{zz}}_1\\ {\dot{zz}}_2\end{array}\right]+\left[\begin{array}{cc}\left(k_1+k_2\right)& {-k}_2\\ -k_2& k_2\end{array}\right]\left[\begin{array}{c}{zz}_1\\ {zz}_2\end{array}\right]=\left[\begin{array}{c}{ff}_1\left(t\right)+{ff}_2\left(t\right)\\ 0\end{array}\right]$

(4)

As deduced in Appendix A, the displacements of zz₁(t) and zz₂(t) are

${zz}_1\left(t\right)=\text{Real}\hspace{0.17em}\left({zz}_{1p\left(i=1\right)}\left(t\right)\right)+\text{Im}\hspace{0.17em}\left({zz}_{1p\left(i=2\right)}\left(t\right)\right)$

(5a)

${zz}_2\left(t\right)=\text{Real}\hspace{0.17em}\left({zz}_{2p\left(i=1\right)}\left(t\right)\right)+\text{Im}\hspace{0.17em}\left({zz}_{2p\left(i=2\right)}\left(t\right)\right)$

(5b)

The relative displacement of magnet #1 (M₁) relating to the top of the cabin is

$\begin{array}{l}{zz}_{r1}\left(t\right)\\ ={zz}_1\left(t\right)-z_0\left(t\right)\\ ={\gamma }_{1\left(i=1\right)}\hspace{0.17em}\cos \left(\omega t+{\varphi }_{1\left(i=1\right)}\right)+{\gamma }_{1\left(i=2\right)}\mathrm{s}in\left(\omega t+{\varphi }_{1\left(i=2\right)}\right)-H_0\hspace{0.17em}\cos \left(\omega t\right)\\ ={\mathrm{\Gamma }}_1\hspace{0.17em}\sin \hspace{0.17em}\left(\omega \mathrm{t}+{\mathrm{\phi }}_1\right)\end{array}$

(6)

Similarly, the relative displacement of magnet #2 (M2) with respect to the top of the energy harvester can be expressed as

$\begin{array}{l}{zz}_{r2}\left(t\right)\\ ={zz}_2\left(t\right)-z_0\left(t\right)\\ ={\gamma }_{2\left(i=1\right)}\hspace{0.17em}\cos \left(\omega t+{\varphi }_{2\left(i=1\right)}\right)+{\gamma }_{2\left(i=2\right)}\mathrm{s}in\left(\omega t+{\varphi }_{2\left(i=2\right)}\right)-H_0\hspace{0.17em}\cos \left(\omega t\right)\\ ={\mathrm{\Gamma }}_2\hspace{0.17em}\sin \hspace{0.17em}\left(\omega \mathrm{t}+{\mathrm{\phi }}_2\right)\end{array}$

(7)

Review of electromagnetic-based electricity^22,23

The jth set of coils, which surround the jth magnet and consist of NNc(j) turns, is depicted in Figure 5. Based on the derivation in Appendix B and considering the number of coil layers (NNL1), the electrical voltage induced by the kth turn of the coil is given by

$\begin{array}{l}\mathrm{\epsilon }{\mathrm{\epsilon }}_{\mathrm{k}\left(\mathrm{j}\right)}\left(\mathrm{t}\right)\\ =-\frac{\mathrm{d}{\Phi \Phi }_{\mathrm{k}\left(\mathrm{j}\right)}}{\text{dt}}\\ =\frac{\mathrm{d}}{\text{dt}}\left({\text{AA}}_{\mathrm{c}\left(\mathrm{k}\right)}\left(\mathrm{j}\right)·{\text{BB}}_{{\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)}\right)\\ =-\frac{\pi \mathrm{D}{\mathrm{D}}_{\mathrm{C}\left(\mathrm{j}\right)}^2}{4}·{\text{BB}}_{\mathrm{r}\left(\mathrm{j}\right)}·\left[\begin{array}{l}\frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)+\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}-\\ \frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)-\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}\end{array}\right]·{\dot{\text{zz}}}_{\mathrm{r}\left(\mathrm{j}\right)}\end{array}$

(8)

Figure 5.

The allocation of coil surrounding magnet with NN_c(j) turns.

To calculate the total electrical voltage εε_T(j) induced by the jth coil set, the electrical voltages (εε_{T(j, k)}) induced in all coil turns (k) of the jth set needs to be summed up as follows

${\epsilon \epsilon }_{\mathrm{T}\left(\mathrm{j}\right)}={\displaystyle \sum }_{\mathrm{k}=1}^{{\text{NN}}_{\mathrm{C}\left(\mathrm{j}\right)}}\left(-\frac{\mathrm{\pi }{\text{DD}}_{\mathrm{C}\left(\mathrm{j}\right)}^2}{4}\right)·{\text{BB}}_{\mathrm{r}\left(\mathrm{j}\right)}·\left[\begin{array}{l}\frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)+\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+\mathrm{D}{\mathrm{D}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}-\\ \frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)-\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+\mathrm{D}{\mathrm{D}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}\end{array}\right]·{\dot{\text{zz}}}_{\mathrm{r}\left(\mathrm{j}\right)}$

(9)

Taking into account the number of coil layers NN_L(j) of the jth coil set, the total electrical voltage εε_T(j) and electrical current I_T(j) of the jth coil set can be expressed as

${\epsilon \epsilon }_{\mathrm{T}\left(\mathrm{j}\right)}={\displaystyle \sum }_{\mathrm{k}=1}^{{\text{NN}}_{\mathrm{C}\left(\mathrm{j}\right)}}\left(-\frac{\mathrm{\pi }{\text{DD}}_{\mathrm{C}\left(\mathrm{j}\right)}^2}{4}\right)·\left({\text{NN}}_{\mathrm{L}\left(\mathrm{j}\right)}\right)·{\text{BB}}_{\mathrm{r}\left(\mathrm{j}\right)}·\left[\begin{array}{l}\frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)+\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+\mathrm{D}{\mathrm{D}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}-\\ \frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)-\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}\end{array}\right]·{\dot{\text{zz}}}_{\mathrm{r}\left(\mathrm{j}\right)}$

(10a)

${\mathrm{I}}_{\mathrm{T}\left(\mathrm{j}\right)}={\displaystyle \sum }_{\mathrm{k}=1}^{{\text{NN}}_{\mathrm{C}\left(\mathrm{j}\right)}}\frac{1}{{\text{RR}}_{\text{load}\left(\mathrm{i}\right)}}\left(-\frac{\mathrm{\pi }{\text{DD}}_{\mathrm{C}\left(\mathrm{j}\right)}^2}{4}\right)·\left({\text{NN}}_{\mathrm{L}\left(\mathrm{j}\right)}\right)·{\text{BB}}_{\mathrm{r}\left(\mathrm{j}\right)}·\left[\begin{array}{l}\frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)+\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}-\\ \frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)-\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}\end{array}\right]·{\dot{\text{zz}}}_{\mathrm{r}\left(\mathrm{j}\right)}$

(10b)

Regarding the root-mean-square voltage induced by the jth coil set, it can be expressed as follows

$\begin{array}{l}{\overline{\epsilon \epsilon }}_{\mathrm{T}\left(\mathrm{j}\right)}={\displaystyle \sum }_{\mathrm{k}=1}^{{\text{NN}}_{\mathrm{C}\left(\mathrm{J}\right)}}{\overline{\epsilon \epsilon }}_{\mathrm{k}\left(\mathrm{j}\right)}={\displaystyle \sum }_{\mathrm{k}=1}^{{\text{NN}}_{\mathrm{C}\left(\mathrm{j}\right)}}{\int }_0^{\mathrm{T}/2}\frac{{\epsilon \epsilon }_{\mathrm{k}\left(\mathrm{j}\right)}^2\left(\mathrm{t}\right)}{\mathrm{T}/2}\text{dt}\\ ={\displaystyle \sum }_{\mathrm{k}=1}^{{\text{NN}}_{\mathrm{C}\left(\mathrm{j}\right)}}\left(-\frac{\mathrm{\pi }{\text{DD}}_{\mathrm{C}\left(\mathrm{j}\right)}^2}{4}\right)·\left({\text{NN}}_{\mathrm{L}\left(\mathrm{j}\right)}\right)·{\text{BB}}_{\mathrm{r}\left(\mathrm{j}\right)}·\left[\begin{array}{l}\frac{\mathrm{D}{\mathrm{D}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)+\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}-\\ \frac{{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2}{\sqrt{{\left[4{\left({\mathrm{z}}_{\mathrm{k}}\left(\mathrm{j}\right)-\frac{{\text{HH}}_{\mathrm{m}\left(\mathrm{j}\right)}}{2}\right)}^2+{\text{DD}}_{\mathrm{m}\left(\mathrm{j}\right)}^2\right]}^3}}\end{array}\right]·\left(\frac{{\mathrm{\Gamma }}_{\mathrm{j}}·\mathrm{\omega }}{\sqrt{2}}\right)\end{array}$

(11)

The total root-mean-square voltage ( ${\overline{\epsilon \epsilon }}_{\text{TT}}$ ) can be calculated as follows

${\overline{\epsilon \epsilon }}_{\text{TT}}={\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\overline{\epsilon \epsilon }}_{\mathrm{T}\left(\mathrm{j}\right)}$

(12)

where m is the number of the magnets.

The related electrical power is given as

${\text{WW}}_{\mathrm{T}}\left(\mathrm{t}\right)={\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}\frac{{\left({\epsilon \epsilon }_{\mathrm{T}\left(\mathrm{j}\right)}\left(\mathrm{t}\right)\right)}^2}{{\text{RR}}_{\text{load}\left(\mathrm{j}\right)}}$

(13)

Moreover, the related electrical power in root-mean-square term can be express as

${\overline{\text{WW}}}_{\mathrm{T}\left(\text{RMS}\right)}={\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}\frac{{\left({\mathrm{\epsilon }\overline{\mathrm{\epsilon }}}_{\mathrm{T}\left(\mathrm{j}\right)}\right)}^2}{{\text{RR}}_{\text{load}\left(\mathrm{j}\right)}}$

(14)

Ship-base energy harvester model verification

In Figure 6, the experimental setup involves an electrical power generating system. A ceiling attached to the spherical bearing of the energy harvester is periodically moved up and down. To capture the vibratory energy of the system, a Volt Ohm Milliammeter (VOM) connected to the energy harvester is used. The energy harvester, as depicted in Figure 6, consists of two sets of magnets surrounded by three sets of coils. Each coil has 42 turns distributed across seven layers, while each magnet measures 0.02 m in diameter and 0.02 m in height. The stiffness of the spring is 98 N/m. One scenario with a H_o value of 0.15 m and an angular frequency of 1 Hz is considered. Using equations (10a)–(10b) to calculate the theoretical electrical voltage, the results are plotted in Figure 7. The figure shows that the peak values of the electrical voltage at H_o = 0.15 are 0.028 volts. Simultaneously, experimental data obtained through the VOM indicates a peak power value of 0.03 volts. The theoretical total peak values of 0.028 volts closely align with the experimental data of 0.03 volts. The deviation between the theoretical values and experimental data falls within the range of 0.002 volts. These discrepancies may be attributed to potential measuring errors. However, given the small magnitude of the error, the theoretical model for the energy harvesting system appears acceptable for subsequent calculations.

Figure 6.

An experimental platform used to simulate the ship-based energy harvesting system.

Figure 7.

The theoretical electrical voltage to time at H_o = 0.15 m.

Sensitivity assessment

Before optimizing the two-magnet hanging energy harvesting device, we conducted a sensitivity analysis of the electrical power with respect to marine parameters and energy harvester’s geometric parameters. In accordance with the criteria of ocean waves with a ratio of wave amplitude to wave length of 1/7, ³⁰ we selected a slope of 1/10 for wave amplitude (H_o) to wave length (λ). Using equation (13), we calculated the induced electrical power ${\text{WW}}_{\mathrm{T}}\left(\mathrm{t}\right)$ and plotted the profile of induced electrical power over time at various wave parameters (H_o, λ), as shown in Figures 8 and 9. From Figure 8, it is evident that the induced electrical power is positively proportional to the wave amplitude. Conversely, Figure 9 indicates that the electrical power is inversely proportional to the wave length. Additionally, we conducted a sensitivity analysis of the electrical power with respect to the energy harvester’s geometric parameters (DD_m1, HH_m1, k₁, NN_C1, NN_L1, DD_m2, HH_m2, k₂, NN_C2, NN_L2), as demonstrated in Figures 10–19. It can be seen from Figures 10 and 11 that the induced electrical power is positively proportional to DD_m1 and DD_m2.

Figure 8.

The influence of induced electrical power relating to H₀. λ

Figure 9.

The influence of induced electrical power relating to λ.

Figure 10.

The influence of induced electrical power relating to DD_m1.

Figure 11.

The influence of induced electrical power relating to DD_m2.

Figure 12.

The influence of induced electrical power relating to HH_m1.

Figure 13.

The influence of induced electrical power relating to HH_m2.

Figure 14.

The influence of induced electrical power relating to k₁.

Figure 15.

The influence of induced electrical power relating to k₂.

Figure 16.

The influence of induced electrical power relating to NN_C1.

Figure 17.

The influence of induced electrical power relating to NN_C2.

Figure 18.

The influence of induced electrical power relating to NN_L1.

Figure 19.

The influence of induced electrical power relating to NN_L2.

Moreover, it is observed from Figures 12 and 13 that the induced electrical power is strongly correlated with HH_m1 and HH_m2. Figures 14 and 15 show that k₁ and k₂ have a significant impact on the induced electrical power. Additionally, Figures 16 and 17 demonstrate that the number of coil turns NN_C1 and NN_C2 has a noticeable effect on the electrical power, although the trend of electrical power with different coil layers is fluctuating.

Finally, as shown in Figures 18–19, it is clear that the electrical power is positively proportional to the number of coil layers NN_L1 and NN_L2.

All parameters, including wave parameters (H_o, λ), and geometric parameters of the energy harvester (DD_m1, HH_m1, k₁, NN_C1, NN_L1, DD_m2, HH_m2, k₂, NN_C2, NN_L2), were analyzed in a sensitivity analysis to investigate their influence on the electrical power generated by the energy harvester. The results showed that all parameters play an essential role in determining the induced electrical power.

Case study

As discussed in Section of sensitivity assessment, both marine parameters (Y_o, λ) and the geometric parameters of the energy harvester (DD_m1, HH_m1, k₁, NN_C1, NN_L1, DD_m2, HH_m2, k₂, NN_C2, NN_L2) have a significant impact on the induced electricity. To optimize the induced electrical power by adjusting the geometric parameters of the energy harvester, equation (14) is used, with the marine situation of U_sea, Y_o, and λ set to 2 m/s, 0.4 m, and 4.0 m, respectively. The ranges of the design parameters for DD_m1, HH_m1, k₁, NN_C1, NN_L1, DD_m2, HH_m2, k₂, NN_C2, and NN_L2 are preset and shown in Table 1. We utilize the Cuckoo Search (CS) optimizer to maximize the electricity production of the two-magnet hanging energy harvester.

Table 1.

The ranges of the parameters for the two-magnet hanging energy harvester.

	Design parameter
Ranges of the parameters	DDm1	HHm1	k1	NNc1	NNL1	DDm2	HHm2	k2	NNc2	NNL2
	[0.01, 0.08]	[0.01, 0.08]	[1000, 1000000]	[20, 400]	[1,50]	[0.01, 0.08]	[0.01, 0.08]	[1000, 1000000]	[20,400]	[1,50]

Cuckoo searching optimization

The CS (Cuckoo Search) algorithm is a well-known optimization algorithm used to solve optimization problems. ²⁶ It was first introduced by Yang and Deb in 2009 ²⁷ and draws inspiration from the brooding parasitism behavior of cuckoo species in nature. Due to the aggressive breeding behavior of cuckoos and the characteristics of Lévy flights, the cuckoo search algorithm has been successfully applied in the numerical optimization field. Cuckoos have a reproductive strategy of obligate brood parasitism, which involves laying their eggs in the nests of other host birds, often of other species. Some species, such as the ani and Guira cuckoos, lay their eggs in public nests, sometimes removing other eggs to increase the hatching probability of their own eggs. ²⁸ There are three types of brood parasitism: intraspecific brood parasitism, mutual breeding, and nest occupation. When the host bird discovers the alien eggs, it may either throw them away or abandon the nest and build a new one. Interestingly, the timing of egg-laying for some species of cuckoo birds is astonishing, as the cuckoo eggs often hatch a little earlier than the host’s eggs. When the first cuckoo chick hatches, its first instinct is to throw out the host eggs by blindly propelling them out of the nest. Furthermore, the cuckoo chick may also mimic the call of host chicks to gain more feeding opportunities.

In the simple Cuckoo Search algorithm, three idealized rules are adopted:

1. Each cuckoo lays one egg at a time, which is dumped in a randomly chosen nest.

2. The best nests with high-quality eggs are preserved for the next generations.

3. The number of host nests is preset, and the cuckoo egg will be discovered by the host bird with a probability p_a ∈ [0, 1]. The host bird can either throw the egg away or abandon the nest and build a new one.

For simplicity, the fraction p_a of the n nests is defined as the nests that are replaced by new nests with new random solutions. Assuming that each egg in a nest represents a solution and a cuckoo egg represents a new solution, the goal is to use the new solutions (cuckoos) that are potentially better to replace not-so-good solutions in the nests. In more complicated cases, each nest may contain multiple eggs, representing a set of solutions.

For a cuckoo i, a new generated solution x(t + 1) is obtained using a Lévy flight

${\mathrm{x}}_{\mathrm{i}}^{\mathrm{t}+1}={\mathrm{x}}_{\mathrm{i}}^{\mathrm{t}}+\mathrm{\alpha }\oplus \acute{\mathrm{L}}\text{evy}\left(\mathrm{\lambda }\right)$

(15)

Here, α (> 0) is the step size, typically set to 1.

Equation (15) is a stochastic equation for a random walk. A random walk is a Markov chain whose next status/location only depends on the current location ${\mathrm{x}}_{\mathrm{i}}^{\mathrm{t}}$ and the transition probability α ⊕ Lévy(λ). The product ⊕ denotes entry-wise multiplication. The Lévy flight provides a random walk, where the random step length is drawn from a Lévy distribution

$\acute{\mathrm{L}}\text{evy}\sim \mathrm{u}={\mathrm{t}}^{-\mathrm{\lambda }},\left(1\hspace{0.17em}<\hspace{0.17em}\mathrm{\lambda }\le 3\right),$

(16)

The steps in the Cuckoo Search algorithm follow a random walk process with a heavy-tailed power-law step-length distribution. To speed up local search, some of the new solutions should be generated by performing a L´evy walk around the best solution obtained. However, a significant proportion of the new solutions should also be generated by randomization in the far field, with locations that are far enough from the current best solution to avoid getting trapped in a local optimum. Cuckoo Search is now widely used in engineering optimization, image processing, scheduling, planning, feature selection, forecasting, and more. ²⁹ Figure 20 shows the block diagram of the Cuckoo Search method used in the energy harvesting optimization discussed in this study. The optimization process terminates when the maximal iteration reaches the maximum value of iter_max.

Figure 20.

A flow chart of the Cuckoo Search Algorithm.

The optimal performance of the Cuckoo Search (CS) algorithm is influenced by its control parameters, namely, N_p (population size), p_a (discovery rate of alien eggs/solutions), and iter_max (maximal iteration number). Therefore, selecting an appropriate set of control parameters is necessary to achieve the best possible outcome during the energy harvesting device optimization. In this study, the CS’s control parameters are varied step by step during the optimization process to reach better parameters. Section of sensitivity assessment showed that the induced electrical power of the energy harvester is affected by its geometric parameters, including DD_m1, HH_m1, k₁, NN_C1, NN_L1, DD_m2, HH_m2, k₂, NN_C2, and NN_L2. To optimize the ten design parameters using the CS algorithm, the wave characteristics are chosen as U_sea = 2 (m/s), λ = 2.0 (m), and H_o = 0.2 (m), while the electrical resistances of the load attached to the two coil sets of the harvester, RR_load1 and RR_load2, are set to be 200 Ω. The objective function (OBJ) for optimizing the energy harvesting device using Cuckoo Search (CS) is defined as the root-mean-square of the induced electrical power output, which is a function of the ten design parameters

$\text{OBJ}={\overline{\text{WW}}}_{\mathrm{T}\left(\text{RMS}\right)}\left({\text{DD}}_{\mathrm{m}1},{\text{HH}}_{\mathrm{m}1},{\mathrm{k}}_1,{\text{NN}}_{\mathrm{C}1},{\text{NN}}_{\mathrm{L}1},{\text{DD}}_{\mathrm{m}2},{\text{HH}}_{\mathrm{m}2},{\mathrm{k}}_2,{\text{NN}}_{\mathrm{C}2},{\text{NN}}_{\mathrm{L}2}\right)$

(17)

Setting α (the step size) to 1, in order to obtain the optimal design data, the CS control parameters (N_p, p_a, $\mathrm{i}{\text{ter}}_{\max }$ ) need to be varied step by step during the optimization process. The values of these parameters used in this study are:

N_p = (10, 20, 30, 40, 50), p_a = (0.15, 0.20, 0.25, 0.30, 0.35), $\mathrm{i}{\text{ter}}_{\max }$ = (100, 500, 1000, 2000, 4000, 8000).

Results and discussion

Results

The optimal simulated results were obtained using equation (17) and Table 1 with the CS optimizer, and are displayed in Table 2. It can be seen from Table 2 that the maximum root-mean-square electrical power of 0.1497 Watt was achieved at the 14th set of solutions. The corresponding optimal values of the design parameters, including DD_m1, HH_m1, k₁, NN_C1, NN_L1, DD_m2, HH_m2, k₂, NN_C2, and NN_L2, are 0.080 m, 0.080 m, 1,000,000 N/m, 84 turns, 50 layers, 0.080 m, 0.080 m, 1,000,000 N/m, 86 turns, and 50 layers, respectively. It should be noted that the optimal value of the induced root-mean-square electrical power ( ${\overline{\text{WW}}}_{\mathrm{T}\left(\text{RMS}\right)}$ ) is 0.1497 Watt, and this was achieved when the CS control parameter set ( ${\mathrm{N}}_{\mathrm{p}}$ , p_a, ${\text{iter}}_{\max }$ ) was (20, 0.15, 8000).

Table 2.

Optimization results with respect to various FPA control parameters for the two-magnet hanging energy harvester.

Item	FPA Control Parameter			Design Parameter										OBJ
	Np	p a	itermax	DDm1 (m)	HHm1 (m)	k1 (N/m)	NNC1 (turns)	NNL1 (layer)	DDm2 (m)	HHm2 (m)	k2 (N/m)	NNC2 (turns)	NNL2 (layer)	PW (Watt)
1	10	0.15	100	0.072	0.035	1000000	63	14	0.080	0.080	1000000	77	50	0.0738
2	20	0.15	100	0.079	0.080	533402	84	50	0.080	0.072	188956	76	50	0.1343
3	30	0.15	100	0.080	0.078	939661	80	36	0.080	0.080	292592	77	50	0.0867
4	40	0.15	100	0.080	0.080	104834	72	50	0.080	0.080	630290	88	42	0.1049
5	50	0.15	100	0.059	0.073	634355	83	50	0.080	0.060	325738	59	49	0.1002
6	20	0.20	100	0.080	0.060	1000000	61	50	0.064	0.066	298022	59	50	0.0980
7	20	0.25	100	0.073	0.059	1000000	65	48	0.080	0.051	185786	62	50	0.1072
8	20	0.30	100	0.073	0.059	1000000	65	48	0.080	0.051	185786	62	50	0.075
9	20	0.35	100	0.080	0.076	854104	61	50	0.075	0.034	711267	46	50	0.0875
10	20	0.15	500	0.076	0.059	875294	58	50	0.080	0.059	886585	64	50	0.1105
11	20	0.15	1000	0.076	0.059	875294	58	50	0.080	0.059	886585	64	50	0.1486
12	20	0.15	2000	0.080	0.080	1000000	84	50	0.080	0.080	1000000	84	50	0.1493
13	20	0.15	4000	0.080	0.080	1000000	85	50	0.080	0.080	779743	85	50	0.1496
14	20	0.15	8000	0.080	0.080	1000000	84	50	0.080	0.080	1000000	86	50	0.1497

After obtaining the optimal design parameters using the CS optimizer, the theoretical electrical power with respect to time is calculated and plotted in Figure 21 by inputting these parameters into theoretical formulas equation (13). Additionally, the optimal values of the design parameters are utilized in equations (10a) and (10b) to obtain the corresponding response curves of induced electrical voltage and electrical current with respect to time, which are displayed in Figures 22 and 23, respectively. To visualize the relative mechanical motion of the magnets in relation to the top of the energy harvesting device, the relative displacement curves of magnet #1 and magnet #2 are calculated using equations (6) and (7) and shown in Figure 24. By differentiating equations (6) and (7), the relative velocity curves of magnet #1 and magnet #2 are obtained and plotted in Figure 25.

Figure 21.

The response of electrical power to time (after optimization of two-magnet energy harvester).

Figure 22.

The response of electrical voltage to time (after optimization of two-magnet energy harvester).

Figure 23.

The response of electrical current to time (after optimization of two-magnet energy harvester).

Figure 24.

The response of magnets’ relative displacement to time (after optimization of two-magnet energy harvester).

Figure 25.

The response of magnets’ relative velocity to time (after optimization of two-magnet energy harvester).

Discussion

Wave energy is a vast natural resource, and harnessing electrical energy from waves is a green and inexhaustible sustainable energy source. This study aims to design a safer and portable energy harvesting generator system that can be installed inside general ships to increase their electrical endurance. In Section 4, it was found that the induced electrical power of the KMC type two-magnet energy harvester is positively correlated with wave amplitude and inversely proportional to wave length, as shown in Figures 8 and 9. Figures 10 and 11 also demonstrated that the induced electrical power of the energy harvester increases with increasing DD_m1 or DD_m2. The magnets’ heights (HH_m1, HH_m2) were found to have a strong correlation with induced electrical power, as shown in Figures 12 and 13, and the spring constants (k₁, k₂) also significantly impact electrical power, as demonstrated in Figures 14 and 15. The coils’ turns (NN_C1, NN_C2) were found to affect electrical power in Figures 16 and 17, and the induced electrical power was found to be positively proportional to the coils’ layers (NN_L1, NN_L2) in Figures 18 and 19. Therefore, the geometric parameters (DD_m1, HH_m1, k₁, NN_C1, NN_L1, DD_m2, HH_m2, k₂, NN_C2, and NN_L2) of the hanging energy harvester are highly sensitive to induced electrical power and are considered design parameters for optimizing the energy harvester. The maximal root-mean-square electrical power occurred at the 14th set of optimal results, as shown in Table 2, when the CS controller ( ${\mathrm{N}}_{\mathrm{p}}$ , p_a, ${\text{iter}}_{\max }$ ) was adjusted to be (20, 0.15, 8000). The maximal root-mean-square electrical power reached 0.1497 watt, as shown in Table 2. By plugging the optimal design parameters into the theoretical formulas (Equation (13)), the time response of electrical power was calculated and plotted in Figure 21, with a peak value of 0.3 watt. The peak values of electrical voltages with respect to magnet#1 and magnet#2 were found to be 5.5 V, as shown in Figure 22. The peak values of electrical currents with respect to magnet#1 and magnet#2 were 0.028 A, as shown in Figure 23. The related displacement amplitudes of magnet#1 and magnet#2 were found to be 0.4 m, as shown in Figure 24, and the related velocity of magnet#1 and magnet#2 was found to be 1.25 m/s, as shown in Figure 25.

Even though the induced electrical power is low, placing multiple sets of energy harvesting generator systems in these compartments can effectively increase the power generation output. Furthermore, there is a prevalent issue with the sustainable power supply for sensors deployed on the seabed, which is currently the most pressing concern. The purpose of the aforementioned seabed power generation facilities is also aimed at addressing this issue. To tackle this problem, we can establish a dedicated ship for wave power generation near the shore, with the energy harvesting generator system fully installed in the ship’s compartments. This setup can provide a large amount of electricity, which can then be transmitted through cables to power the seabed sensors.

Conclusion

To generate sustainable energy for deep sea sensors, a two-magnet energy harvester with two sets of coils was proposed and mounted on top of a ship’s cabin. By utilizing a spherical bearing and gravity effect, the energy harvester can remain in a vertical position despite the ship’s tilt caused by waves. Through the KMC mechanism and periodic wave motion, the magnetic flux variation induces electrical power. To optimize the energy harvester, a sensitivity analysis of marine parameters and geometric parameters was performed, and the objective function was set to maximize the root-mean-square electrical power. Using the CS optimizer, the optimal design resulted in a root-mean-square electrical power of 0.1497 (W), with electrical voltages induced by the two magnets of 5.5 (V) and electrical currents of the two coil sets of 0.028 (A). This study provides a straightforward approach to harvesting energy from the sea and optimizing energy harvesting with the CS optimizer.