Design of a wave energy converter based on Halbach magnetic array

Abstract

With the continuous development of society and economy, people’s demand for electric energy is increasing. The low-carbon and energy-saving technologies of renewable energy especially wave energy have become the focus of current researches. Considering the increasingly serious energy problems, a wave energy converter (WEC) is proposed based on Halbach permanent magnetic array, which increases the output performance. The equivalent magnetic circuit model of the WEC is established. The static magnetic field modeling and structural parameters optimal permanent of the WEC are performed on this theoretical. Theoretical studies have found that the optimal permanent magnet thickness ratios for Halbach permanent magnet array structures is 0.6, and the ratio of permanent magnet to coil radial ratio is 0.7. The coil winding form and rectifying circuit of the WEC were designed. The WEC equivalent magnetic circuit model was verified by COMSOL Multiphysics finite element software, and the open circuit voltages of WEC was obtained. If the WEC moves at a speed of 0.1 m/s, the coil voltage can reach about 113 V after simulation. According to the testing requirements of the WEC, a test platform was built. The Halbach permanent magnet array structures greatly enhances the wave energy collection of WEC.

Keywords

WEC Halbach equivalent magnetic circuit model finite element analysis

1. Introduction

People consume more and more energy in production and life because the improvements of technology and living conditions. Traditional ore energy is inefficient, and it brings serious pollution. Long-term dependence on ore energy will lead to energy crisis [1]. The low-carbon renewable energy, especially wave energy, is inexhaustible and clean. These kinds of energy have become the focus of current researches. Wave energy is converted from wind energy, so it is a renewable energy [2]. Wave energy is a renewable energy. It has no pollution, wide distribution, high power density and large reserves. Compared to some other renewable energy sources, the spatial distribution of wave energy is more concentrated. Using wave energy converter (WEC) to harvest wave energy can provide electricity to islands or coastal areas [3].

There are many kinds of WECs, such as oscillating-body WEC (OWC) [4,5], float WEC [6], pendulum WEC [7], and etc. Various studies have achieved a lot of valuable results. Xia et al. [8] proposed a field-modulated tubular linear generator (FMTLG). The FMTLG, which is equipped with field-modulated technology and quasi-Halbach arrays for WEC, can overcome the disadvantages of heavy weight and inefficiency caused by low-speed ocean waves to some extent. Xuanlie et al. [9] proposed a system consisting of a front oscillating buoy type WEC and a rear fixed pontoon and it can improve the energy conversion performance of the original single pontoon breakwater-type WEC system. Zhong et al. [10] proposed an open-book-like triboelectric nanogenerator (TENG) for large-scale harvesting of water wave energy. The device integrates a large number of TENG units into an open-book-like structure in a limited space, greatly improving the volume density of the microstructured contact interface. Tan et al. [11] proposed a type of film-type generator made of carbon. A 15 cm²-sized generator can produce a voltage of 20 mV and a current of 10 microamp by optimizing carbon dosage, wave frequency and hoisting angle. Farrok et al. [12] proposed a new design of flux switching permanent magnet linear generator (FSPMLG). Under the same wave conditions, the efficiency of the proposed FSPMLG is 5.86% higher than the tradition FSPMLG.

However, these kinds of WEC have common problems, such as complicated structure, high cost, low energy conversion efficiency, poor adaptability, and short life. So it is lack of a simple structure which has ability to harvest wave energy. This study is based on a new type of WEC with higher efficiency and stronger applicability. The objective of this study is to design a Halbach permanent magnetoelectric transducer for a designed WEC. The dimension of this magnetoelectric transducer can be determined according to the size of the WEC. Then lumped parameter equivalent magnetic circuit model of this magnetoelectric transducer is proposed, and its parameters of each component can be determined. Its finite element model is established by the commercial software COMSOL. At last, the prototype of the magnetoelectric transducer is manufactured and fabricated, and test the prototype in the test system to evaluate the possibility of its application on the WEC.

2. WEC design and optimization

The WEC based on Halbach permanent magnetic array can convert wave energy into electrical energy directly. Some permanent magnet buoys and some coil buoys make up this new kind of WEC, and this two kinds of buoys alternately connected, as is shown in Fig. 1. The number of buoys can be determined according to the actual situation. The magnetoelectric transducer is located between a coil buoy and a permanent magnet buoy, and it includes a transmission mechanism, a permanent magnet array and coil windings. This magnetoelectric transducer is a kind of linear power generation structure, and it can directly convert its vibration energy into electrical energy without using other transmissions. The performance of magnetoelectric transducer is an important indicator for evaluating the performance of the WEC. The proposed magnetoelectric transducer with Halbach permanent magnet array is suitable for the WEC because Halbach permanent magnet array has the capability to enhance the single-side magnetic field strength.

Fig. 1.

Overall structure of WEC.

Fig. 2.

Magnetoelectric transducer.

The magnetoelectric transducer with Halbach permanent magnet array in this paper was designed based on linear power generation structure, and is shown in Fig. 2. The magnetoelectric transducer includes coil windings and permanent magnets. The directions of the arrows are the magnetization directions of the permanent magnets.

Before the design of the magnetic circuit, it is necessary to select a suitable permanent magnet material to ensure that it has better wave energy harvesting performance and it is also necessary to make sure that the size of the WEC is within a reasonable range. After making many comparisons, NdFeB 42UH is selected as the permanent magnet material for the WEC.

Establishing the equivalent magnetic circuit model based on the designed Halbach permanent magnet array transducer, as is shown in Fig. 3. In Fig. 3, τ is the length of the permanent magnet array, τ_ma is the thickness of the axial permanent magnet, and τ −τ_ma is the thickness of the radial permanent magnet. r_mi is the inner radius of the permanent magnet and r_mo is the outer radius of the permanent magnet. r_ci is the inner radius of the coil winding , and r_co is the outer radius of the coil winding.

Fig. 3.

Equivalent magnetic circuit model of the WEC.

In this model, select a pair of magnetic poles for analysis. The stator casing is made of high magnetic permeability material, so the magnetic resistance of the stator casing (R_c) can be ignored. It can be obtained by the Kirchhoff’s second law: $\begin{eqnarray}\displaystyle H_{cr}(r_{mo}-r_{mi})+H_{ca}{\tau}_{ma}=(R_{mr}+R_{ma}+2R_{g}){\Phi}_{g}. & & \displaystyle\end{eqnarray}$ (1)

H_c is the coercivity. B_r is the remanence. Φ_g is the magnetic flux of the air gap at the position of the coil. R_m and R_g represent the reluctance of the permanent magnet and the air gap, respectively.

In this structure, the cross-sectional area of the magnetic flux passing through the radial permanent magnet is: $\begin{eqnarray}\displaystyle A_{mr}={\displaystyle \frac{{\pi}(r_{mo}+r_{mi})({\tau}-{\tau}_{ma})}{2}}. & & \displaystyle\end{eqnarray}$ (2)

The cross-sectional area of the magnetic field lines passing through the axial permanent magnet is: $\begin{eqnarray}\displaystyle A_{ma}={\pi}(r_{mo}^{2}-r_{mi}^{2}). & & \displaystyle\end{eqnarray}$ (3)

The cylindrical surface area of the magnetic flux passing through the air gap: $\begin{eqnarray}\displaystyle A_{g}=\frac{{\pi}(r_{co}+r_{mo})({\tau}-{\tau}_{ma})}{2}. & & \displaystyle\end{eqnarray}$ (4)

Neglecting the magnetic flux leakage, the air gap flux density B_g can be approximated as: $\begin{eqnarray} \displaystyle B_{g}={\displaystyle \frac{B_{r}(r_{mo}-r_{mi}+{\tau}_{ma}){\mu}_{0}H_{c}}{2(r_{co}-r_{mo})B_{r}+(r_{mo}-r_{mi}+{\tau}_{ma}){\mu}_{0}H_{c}\left({\displaystyle \frac{A_{g}}{A_{mr}}}+{\displaystyle \frac{A_{g}}{A_{ma}}}\right)}}. & & \displaystyle \end{eqnarray}$ (5)

The H_c and B_r can be obtained from the BH characteristic curve of the NdFeB permanent magnet. μ₀ is the vacuum permeability of the permanent magnet.

The coil of WEC can reach the maximum damping coefficient (c) in the short-circuit state. The damping coefficient of the short-circuit state of the coil can be calculated. There are: $\begin{eqnarray}\displaystyle c={\displaystyle \frac{1}{{\rho}}}\displaystyle \int _{{\Gamma}}B_{g}^{2}d{\Gamma}={\displaystyle \frac{2A_{g}(r_{co}-r_{ci})B_{g}^{2}}{{\rho}}}. & & \displaystyle\end{eqnarray}$ (6)

𝛤 is the conductor volume, 𝜌 is the resistivity of the coil wire metal, and the resistivity 𝜌_Cu = 1.75 × 10⁻⁸Ω ⋅ m.

Assuming τ is 0.2 m, r_co is 0.15 m, and r_mi is 0.02 m. Define the permanent magnet thickness ratio of the Halbach permanent magnet array to 𝛼_h. Define the permanent magnet-coil radial size ratio in the Halbach permanent magnet array transducer to 𝛽_h: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{h}={\displaystyle \frac{{\tau}-{\tau}_{ma}}{{\tau}}}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\beta}_{h}={\displaystyle \frac{r_{mo}-r_{mi}}{r_{co}-r_{mi}}}.\end{eqnarray}$ (8)

The c of a single coil of the WEC can be calculated as shown in Fig. 4. It can be seen from Fig. 4 that both 𝛼_h and 𝛽_h are associated with the value of c. Within a certain range, c will increase with the increase of 𝛼_h and 𝛽_h, and then decrease. The inflection point occurs when 𝛼_h = 0.7 and 𝛽_h = 0.6, and then decreases rapidly as 𝛼_h and 𝛽_h value increases. At this time, the induced voltage of WEC reaches its maximum, and the increase or decrease of 𝛼_h or 𝛽_h will cause the decrease of its induced voltage Therefore, 𝛼_h = 0.6 and 𝛽_h = 0.7 are selected in the design of the magnetic circuit. After determining the external dimensions of the WEC, its internal parameters can be determined based on 𝛼_h and 𝛽_h.

Fig. 4.

Relationship between c, 𝛼_h and 𝛽_h under different size parameter combinations.

The size parameters of each part of the structure are calculated separately, in Table 1. N is the number of turns of the coil.

Table 1

Dimensions of the WEC

Parameter	r_mi∕m	r_mo ∕m	r_ci /m	r_co /m	τ_mr/m	τ_ma/m	N	R_{int, c} ∕Ω
Size	0.02	0.111	0.112	0.15	0.12	0.08	4000	71.74

Designed four phases winding configuration for the WEC, as in shown in Fig. 5. Designed the charging and rectifier circuit for the coil, as shown in Fig. 6. In the magnetic circuit, the coils of opposite phases are respectively connected in forward and reverse directions into one winding.

Fig. 5.

Four phase winding configuration of the WEC.

Fig. 6.

Charging and rectifier circuit.

The finite element model of the WEC is built in COMSOL Multiphysics 5.3. To simplify the model, the 3D model of the WEC is obtained by rotating in a coordinate system around the z-axis. Figure 8 shows the magnetic flux density at the center of the coil. The relative motion between the mover and the stator changes the magnetic flux of the coils to produce induced voltage. The velocity of relative motion is set to 0.1 m/s, and induced voltage of the coils and windings are shown in Fig. 9 and Fig. 10.

Fig. 7.

Magnetic field distribution of the WEC.

Fig. 8.

Radial flux density

Fig. 9.

Induced voltage of the coils.

Fig. 10.

Induced voltage of the windings.

Fig. 11.

Peak value and VRMS of induced voltage.

The peak value of the induced voltage of a winding is 226 V, and the voltage RMS (V_RMS) values is 152.6 V. Then calculate them at different velocities range of 0.01 to 0.1 m/s, in Fig. 11.

3. Experimental study

According to the conclusions obtained from the previous calculations, the size is appropriately reduced according to the actual situation, and the size of the prototype is determined. The prototype of the WEC is shown in Fig. 12. The test system for the WEC is shown in Fig. 13. The test system consists of a prototype of the WEC, a Hall current sensor, a NI CompactRIO embedded system, a vibration test bench (Su Shi DL-300-40), and an oscilloscope (Tectronic TDS 2012).

Fig. 12.

The prototype of the WEC.

Fig. 13.

Test system for the WEC.

Table 2

Dimensions of the prototype

Parameter	r_mi∕cm	r_mo ∕cm	r_ci ∕cm	r_co ∕cm	τ_mr∕cm	τ_ma∕cm	N	R_{int, c} ∕Ω
Size	0.5	2.95	3.05	4	2.4	1.6	2400	127.5

The peak values and the V_RMS values of induced voltage of a winding at velocities range of 0.01 to 0.1 m/s is shown in Fig. 14. It could be seen that the peak value of induced voltage of the Halbach permanent magnet array transducer at a velocity of 0.1 m/s is 47.54 V, and V_RMS value is 32.10 V. It can be seen that as the velocity of the WEC movement increase, the peak value and V_RMS value of induced voltage increase. The rising trend is as same as the calculation result of the finite element model.

Fig. 14.

Peak value and V_RMS of the prototype.

It can be seen from the figure that the induced voltage of the prototype differs greatly from the induced voltage of WEC. This is due to the size of the prototype being smaller than the size of WEC. Increasing the volume of WEC and increasing the number of turns of the coil can get lager induced voltage.

The WEC’s best matching load is related to coil internal resistance and coil inductance. Because the coil inductance is usually small, the best matching value of the load resistance is the value of the coil internal resistance. Therefore, the load resistance of a winding of the WEC in this experiment is 255 Ω. The experiment shows that the power of a winding harvesting wave energy is 1.010 W when the WEC at a relative velocity of 0.1 m/s. Because there are four windings in the prototype, the output power of the prototype is 4.041 W.

4. Conclusion

This paper proposed a magnetoelectric transducer with Halbach permanent magnet array to constitute the WEC that can harvest wave energy in ocean. The advantages of Halbach permanent magnet array can be used to increase the power of the WEC. The lumped parameter equivalent magnetic circuit model of magnetoelectric transducer with Halbach permanent magnet array is established. The size of the WEC is determined on the basis of this theory. After obtaining the optimal size ratio, the magnetoelectric transducer is analyzed by finite element modeling. Finally, using the conclusions that have been got, the prototype was built and tested. The prototype could harvest wave energy of 8.082W when WEC at a relative velocity of 0.1 m/s, and P_out is 4.041 W. This type of WEC can be used in ocean or lake, and we can charge its size according to the work scene. It provides new idea for the follow research of WEC.

Footnotes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant NO. 51675265), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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