Introduction
Ocean wave energy is renewable, they have a high energy density, and they could be more predictable than solar and wind energy, which are affected by the weather and climate [1], [2], [3], [4], [5]. Generating power from ocean waves has become an important area of research in the wake of the increasing energy demand. For this reason, different wave energy converter (WEC) technologies have been proposed [6], [7], [8], [9], [10], [11]. The kinetic energy in the up and down motion of ocean waves is harnessed by using WEC and converted into electric power. Thus, many linear generators have been proposed to generate electric power from ocean wave energy. Example of that, among others, are the work conducted by Refs. [12], [13], [14], [15], [16], [17], [18], [19].
Seawater has a density that is 840 times than that of air, and this determines the strength of the force that ocean waves carry. An important characteristic of ocean waves that perform Brunt-Väisälä oscillation is long period due to the high density. The oscillation of ocean waves slowly drives the power generation mechanism of the WEC. Such a slow-changing magnetic field has a low electromotive force (EMF) in the conductor because general WEC obtains power by electromagnetic induction. The generated power should be greatly improved without increasing the weight or size of power generation devices because the high output is necessary for the spread of WEC. The ocean wave amplitudes are larger with faster offshore wind currents, but installing and operating power generation plants in rough weather is problematic in terms of maintenance. Thus, WEC is difficult to use in large wave height waters. On the other hand, the stroke length obtained by wave power generation systems from near-shore with small wave heights is short, and together with their prolonged period, the power output is small. Therefore, a WEC device with improved output power despite its short stroke length is necessary. In addition, the high density of seawater generates a large external force on the WEC. Hence, a particular attention should be paid to the survivability of a linear power generation module for WEC because the large force places a significant load on the power generating mechanism of a linear generator, which could cause damage to the power generating device. For this reason, there is a necessity to damp the large loads from ocean waves in a WEC.
In this paper, we propose a compact linear power generator with high-temperature superconducting bulks and a concept design system we call a dual translator power structure for a direct drive WEC. The proposed linear generator is aimed to solve key issues associated with ocean power generating devices, which are, the reduction of output power due to the lengthy period of waves and short stroke length obtained in near-shore sea areas. To improve the output power, the size and weight of the power generating device increase, which may make it difficult to install generators and inhibit power generation operation. These are damaging to the practical application of WECs. Another issue to be resolved is the structural damage caused by large loads of ocean waves. It is effective to install dampers on the WEC in parallel with the power generation mechanism to protect it from damage and keep it in operation. However, since the added dampers do not contribute to the increase in power generated, but may often suppress it, it is therefore not ideal to add mass to a linear generator which does not improve output power.
Concept design structure
Toward the practical application of ocean energy power generation, we studied a WEC design that amplifies short stroke length and increases output power under a long period of ocean wave without requiring a buffer mechanism that protects the structure but adds weight and reduces power generation efficiency. Fig. 1 shows a simple schematic of a single WEC with the lateral view of the proposed HTS dual linear power generator structure. We use a heaving buoy as our primary part of our power take off system. The buoy captures energy from the horizontal and vertical motions of ocean waves, where its vertical motion is slow, but it exerts excessively large loads onto the buoy as we mentioned earlier. This slow motion and large loads/forces of ocean waves make it difficult for wave power generating devices to obtain the output power of practical application and survive structural damage. To avoid the reduction of the output power and structural damage, we came up with the idea of increasing power density in time and per volume, that is, the strength and rate of change of the magnetic field through the armature coils are enhanced. In this sense, the proposed generator holds a concept structure of a direct-drive linear generator (linear type of power generation structure) in which the non-fixed armature copper coils and field pole magnets come and go in opposite directions to each other to achieve enhanced output power and electromagnetic force. Since the induced EMF is greater when the magnetic flux density is higher, a strong magnet should constitute the field pole. By employing HTS bulks for the field magnets, we will improve the output of the WEC. In this concept design, we plan to pre-magnetize the HTS bulk magnets to a strong magnetic field before incorporating it into the generator, but for practical use, an efficient method of magnetization must be separately investigated in future. The strong magnetic field by HTS magnets suggests an increase in WEC output, but at the same time, the strong electromagnetic force may suppress the drive of the WEC, but suitable for marine power applications such as WEC to act as a damper. Unlike typical single translator linear generator, where the armature coil is stationary, the design concept which we have named a dual translator power generation system moves the armature copper coils and HTS bulk field poles in opposite directions as a function of time. The device is equivalent to increasing the short strokes obtained from small wave force amplitudes in non-open seas by a factor of two to generate electric power. The dual translator power generation system allows for increased output power and damping ability without increasing the size or weight of the linear power generator.
Fig. 1. Single WEC with a lateral view of the proposed HTS dual translator linear power generator. (a) During crest ocean wave condition. (b) Trough ocean wave condition. |
We illustrate the working principle of the proposed dual translator concept design in Fig. 1 and compare it to a single translator system shown in Ref. [20]. The primary translator is made up of the armature copper coils, whereas the secondary translator is comprised of HTS bulks field poles. The HTS bulks will be kept under cryogenic temperature by the cooling system composed of cold head, condenser, adiabatic tube, as well as a cryostat. The heaving buoy exerts a linear force onto the primary translator. In response, to this force, the primary translator moves upward during wave crest as shown by the blue arrow in Fig. 1(a). The motion transfers kinetic energy into the hydraulic oil shown by the green-colored arrows, which moves through the hydraulic tube until its kinetic energy is transferred into the secondary translator as shown in Fig. 1(a). This results into the secondary translator which move at the same time with the primary translator but in the downward direction until it reaches z = 0 as shown by the red arrows in Fig. 1(a). During wave trough, the motions reverse, that is, the primary translator moves downwards, which forces the hydraulic oil through the hydraulic tube to move the secondary translator in the upwards direction as indicated by the red arrows in Fig. 1(b).
Method of analysis
Our linear generator was modeled using magnetic field interface whereby a quadratic discretization is used to solve for the magnetic vector potential. The Dirichlet or magnetic insulation boundary condition with a normal of the magnetic vector potential, $n \times A=0$ was set on all outer boundary sides. In addition, we imposed continuity across pair boundaries by using a continuity boundary condition, $\boldsymbol{A}_{\text {destination }}=\boldsymbol{A}_{\text {source }} \text {. }$ COMSOL Multiphysics® software automatically generated meshes to delimit the 2-dimensional cross section of the model of our linear generator by elements. A progressively finer mesh size was selected, with a maximum of 43.7 mm and a minimum of 0.148 mm respectively. The mesh type was structured quadrilateral mesh, and unstructured triangular mesh type was used for the remaining regions of the dual translator linear generator. The result of the analysis of the meshed model using the finite element method (FEM) is shown in Fig. 2. The model was solved in stationary and time-dependent approach, where we used Multifrontal Massively Parallel sparse direct Solver (MUMPS), with a direct non-linear newton method.
Fig. 2. Results of the FEM analysis. (a) Example of mesh set up for armature and field magnet. (b) 2-dimensional magnetic flux density distribution in the internal cross section of the linear generator. |
We determined the main parameters of our dual translator linear generator, such as field element pole pitch and the number of field poles by using an iterative procedure considering the efficiency of the linear generator and the total cross-sectional surface area of the generator internal parts. This was done to achieve a lightweight linear power generating machine. Table 1 shows its basic specifications. The finite element method of electromagnetic field analysis used for the linear power generator is illustrated in the flowchart in Fig. 3. From the field analysis implemented in COMSOL, we obtained EMF under loads conditions using (1), (2):
$e=-N \frac{d \Phi_{B}}{d t}$
$\Phi_{B}=\int \boldsymbol{B} \cdot d \boldsymbol{A}$
where $e, \Phi_{B}, N, B$ and $A$ are induced EMF, flux passing through a single turn coil, number of turns in an armature copper coil, magnetic flux density, and cross-sectional area of the armature coil wire. In addition, we computed the electromagnetic forces by integrating maxwell’s stress tensor over the armature coil domain as expressed in (3)
$\boldsymbol{F}=\int_{S} \overleftrightarrow{\boldsymbol{T}} \cdot \widehat{\boldsymbol{n}} d A$
Where $\overleftrightarrow{T} F, \widehat{n}$ and A are force per unit area acting on the surface of the armature coil, the total electromagnetic force on the charges in the armature coil domain, unit vector, and cross-sectional area of the armature coil domain. The total cross-sectional area of the linear generator’s internal parts is used as a proxy to estimate or infer the weight of the linear generator in a 2-dimensional analysis. We minimize the total cross-sectional area of the generator’s internal parts to optimize the parameters of the field elements. The total cross-sectional area of the linear generator’s internal parts is calculated using (4):
$S=l_{C u} h_{C u}+l_{F e} h_{F e}+l_{H T S, \text { bulk }} h_{H T S, b u l k}$
where S, $l_{C u}, h_{C u}, l_{F e}, h_{F e}, l_{H T S, b u l k}$ and $h_{H T S, b u l k}$ are the total cross-sectional area of armature copper coil, back iron, and field pole magnets (linear generator internal parts), armature copper coil, back iron, HTS bulk length and thickness, respectively.
Table 1. Basic specifications of the proposed dual translator linear generator machine. |
| Items | Value |
|---|---|
| Maximum stroke length [mm] | 1160 |
| Number of field poles | 27-70 |
| Pole pitch of field pole [mm] | 66-82 |
| Field pole magnet type | REBaCuO |
| Diameter of HTS bulk [mm] | 40-70 |
| Thickness of HTS bulk [mm] | 14-26 |
| Armature coil type | Copper |
| Diameter of armature coil [mm] | 54-84 |
| Thickness of armature coil [mm] | 16-28 |
| Diameter of copper wire [mm] | 2 |
| Armature coils self-inductance [μH] | 650 |
| Number of armatures coils | 6 |
| Airgap length [mm] | 2 |
| Speed of generator [m/s] | 0.3 |
Fig. 3. Design flowchart of the proposed dual translator linear generator machine. |
Results and discussion
During the simulation, the armature copper coils move vertically at the velocity of 0.3 m/s with respect to its HTS bulks maintaining a mechanical air gap of 2 mm. We developed a WEC that can generate improved output power despite the constraints that are imposed on the stroke of the translator by the geometric relationship between the ocean wave amplitude and WEC as presented in Section ‘Concept design structure’. We attempted to determine the field pole pitch based on a total surface area of the generator internal parts and generator efficiency. As shown in Fig. 4, an optimum field pole pitch was obtained at around 73 mm from the analysis, where the cross-sectional area was small with a relatively high efficiency. This corresponds to the lowest weight of our linear generator with high efficiency. The efficiency of the linear power generation module is calculated by:
$\boldsymbol{\eta}=\left(\frac{P_{\text {out }}}{P_{\text {out }}+P_{\text {Out }}+P_{F e}}\right)$
where $P_{C u}$ and $P_{F e}$ represent armature copper coil losses and iron losses. These losses are referred to as electromagnetic losses. Armature copper coil losses are mainly influenced by ohmic resistance in the armature coil and the current through the coil, whereas iron losses are mainly created by the changing magnetic flux density in the back iron, and they are due to the hysteresis properties and eddy currents. Since the copper coil losses are a lot higher that the iron losses, we therefore only show how the armature copper coil losses were calculated using (5), (6).
$P_{C u}=3 R I^{2}$
where R is the phase resistance of armature coil. $I^2$ is the RMS value of current in the coil. The resistance is determined by the geometry of the armature coil. Therefore, it can be calculated as:
$R=\frac{N l}{A}$
where N is the number of turns. l is the length of the copper wire. A is the cross-sectional area of the copper wire.
Fig. 4. Estimation of optimal field pole pitch. |
Fig. 5 shows the results of the analysis for the stroke length suitable for a linear generator optimized for a pitch of 73 mm. As the number of field poles decreases, the dimensional ratio of armature coils and field poles results in a small ratio of effective armature coil diameter, which generates induced EMF. This leads to an increase in armature resistance loss, which makes it difficult to achieve our target output power we mentioned in the design flowchart without a rapid increase in the total surface area of the generator internal parts as it can be observed below 40 poles in Fig. 5. Above 45 poles, the ratio of effective armature coil diameter is compromised but less. This leads to a less severe increase in the total surface area of generator internal parts above 45 optimum field poles. The cross-sectional area inside the generator at 45 poles was then estimated to be around 0.09 m2. The stroke length at that time was 896 mm.
Fig. 5. Estimation of optimal number of field poles at 73 mm pole pitch. |
The HTS bulk used in this study was validated with experimental data of an HTS bulk used in Ref. [21]. This HTS bulk trapped around 2.17 T of magnetic flux density measured by hall sensor at 2 mm above the HTS bulk at 60 K of operating temperature. The loses shown in Table 2 includes the AC losses in the armature copper, which was determined to increase with the increase in relative speed of the dual translator linear generator. The increase in relative speed increases the AC losses of the linear generator, which has a negative impact on the overall efficiency of the linear generator. For this reason, the efficiency of the generator is relatively low given that we did not consider the mechanical losses and losses of the cryogenic system. The increase in the changing magnetic field caused by the dual translator system increases flux jump in the HTS bulks, which could increase AC loses in the bulk material. However, an analysis of AC loses in the HTS bulk material caused by the increase in relative speed will be investigated in future study.
Table 2. Analysis results of the dual translator linear generator machine. |
| Items | Value |
|---|---|
| Output power at rated operation [kW] | 13 |
| Operating Temperature [K] | 60 |
| Airgap flux density [T] | 2.18 |
| Output voltage at rated operation [V] | 347.9 |
| Trapped magnetic flux density of HTS bulk [T] | 2.17 |
| Number of field poles | 45 |
| Pole pitch of field pole [mm] | 73 |
| Diameter of HTS bulk [mm] | 66 |
| Thickness of HTS bulk [mm] | 19 |
| Diameter of armature coil [mm] | 74 |
| Thickness of armature coil [mm] | 20 |
| Stroke length [mm] | 896 |
| Total surface area of generator internal parts [m2] | 0.09 |
| Losses [kW] | 1.5 |
We did not include losses of the cryogenic system of the superconducting magnet system. However, the high relative speed of our dual translator linear generator reduces the efficiency due to the high AC losses associated with high relative speed. For this reason, the efficiency as shown in Fig. 4 is slightly lower than typical given than mechanical loss and loss of cryogenic system is not yet considered.
We repeated the analysis of connecting load resistors of various values to our dual translator linear generator model and finally determined 14 Ω to be the optimal load in our WEC for maximum output power. The load condition was performed when the magnetic interface was connected to an electrical circuit interface. The armature copper coils (magnetizing coil) used during load conditions have been modeled using the coil subdomain in COMSOL Multiphysics. The results that follow show the case where this optimal load is connected to generate electricity. In this section, we introduce typical output power and electromagnetic force at optimum load. The HTS linear generator with a dual translator can obtain higher output power despite the decrease in its stroke length in comparison to its HTS single translator counterpart with its design specifications conforming to that of the dual translator. Fig. 6, Fig. 7 show that the maximum electromagnetic force and the average output power of the dual translator system are about 5 % and 11 % higher than that of a single translator system. The increase of electromagnetic force and the output power of the dual translator is coming from the increase in the rate of change of magnetic field in time and space through the armature copper coils, which is caused by the increase in the synthetic speed between the armature copper coils and HTS bulks. A dual translator increases the relative speed of our linear generator, which increases the rate of change of the magnetic field through the armature copper windings. An increase in the rate of change of the magnetic field is one way to increase the electromagnetic force and output power of a generator. This caused the deviation between the HTS dual and HTS single translator linear generators observed in Fig. 6, Fig. 7.
Fig. 6. Electromagnetic force of different translator structures, HTS dual translator and HTS single translator. |
Fig. 7. The output power of both HTS dual and single translator structures. |
The results of the analysis show that the proposed dual translator linear machine concept has the potential to improve the electromagnetic force and output power of linear generators for WEC despite the constraints of stroke length. By doing so, we aim to achieve practical output power by achieving miniaturization. The stroke length of the linear generator could be reduced during this process because the length and diameter of the linear generator are reduced. From this perspective, our proposed dual translator linear generator has the potential to help us achieve practical applications of ocean power generation. Our proposed dual translator power generation system has the potential to help us achieve practical applications of ocean power generation systems from a second perspective, which is, its application is suitable in near-shore seas. Where the wave period is long, and the amplitude of ocean waves is smaller, which leads to a constraint stroke length for WEC. Installing WEC away from the open seas makes the installation logistics easier, which can help in promoting the practical applications of ocean power generation systems.
Conclusion
In this study, prior to the improvement of power output and electromagnetic force, we proposed a uniquely design concept of a linear power generator for direct drive WEC that is suitable for nearshore seawater power generation applications where long wave periods and small ocean wave amplitudes limit the stroke length. We attempted to optimize the concept through FEM analysis. The concept of a dual translator power generation system with HTS bulk field poles were used to enhance the strength and rate of change of the magnetic field through armature windings. The electromagnetic force and output power generated by the dual translator is superior to that of a single translator. The HTS dual translator generates electromagnetic force and output power with an increasing difference relative to that of the HTS single translator when the stroke length is decreasing. A combination of the dual translator power generation system and HTS bulks keeps a potential to provide us to achieve high output power despite low ocean wave speed, which is important to achieve practical application of ocean power generation systems.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers 21H01541 (2021-2024) and SECOM Science and Technology Foundation (2021-2024).
