Abstract
Aiming at the limitations of direct drive wave energy conversion (DD-WEC), especially the poor power density, low energy conversion efficiency, and a large volume of linear generators (LGs), a novel magnetic helix hybrid excitation rotary generator (MH-HERG) with the higher power density and higher energy conversion efficiency is proposed. The proposed MH-HERG can convert linear motion into rotary motion without contact with a hybrid excitation magnetic screw (HEMS) unit, so it has high energy conversion efficiency. Furthermore, a new quasi-Halbach magnetization array is used in the proposed MH-HERG to increase its power density and allows a hybrid excitation method to be used to make the thrust adjustable to further improve power density. The analytical solution model is established to derive the calculation equations of air gap flux density, which are validated through the finite element simulation. Arc-shaped permanent magnets (PMs), instead of tile-type PMs, are designed to weaken cogging torque and harmonic content in proposed MH-HERG’s no-load back electromotive force (back-EMF), thereby improving output power quality. Finally, the prototype is built and an experiment is conducted to ascertain the effectiveness and superiority of the proposed MH-HERG which has increased power density by 4.4 times and energy conversion efficiency by 3 times compared to existing LGs.
Keywords: Magnetic helix hybrid excitation rotary generator, Wave energy conversion, Power density, Energy conversion efficiency, Analytical solution model
Subject terms: Ocean sciences, Energy science and technology, Engineering
Introduction
Faced with the sharp reduction of traditional non-renewable and polluting fossil energy, wave energy, as a new kind of renewable energy with huge reserves, has attracted widespread attention worldwide. Wave energy can provide a significant portion of existing electricity demand, especially for coastal cities and island residents1.
Various wave energy conversion (WEC) devices have been deep into research, which linear generators (LGs) are appealing for directly converting the linear motion of buoys into electrical energy output. Existing LGs can also save costs and improve energy conversion efficiency by eliminating the need for intermediary mechanical converters2,3, but their large volume, low energy conversion efficiency, only about 20%, and low power density, only about 0.1 MW/m4–6. A permanent magnet (PM) Vernier LG for direct drive wave energy conversion (DD-WEC) systems is described which is the traditional topology of LG7. Another PMLG is suggested to lessen the combined mass and volume of the systems8, but its power density is only 0.12 MW/m3. The internal configuration of the tubular PMLG has been explored, with a saltwater air gap bearing surface being integrated into the buoy components9, but its volume is larger and power density is lower. A different PMLG design featuring an adjustable air gap length has been suggested as a solution to prevent the frequent occurrence of irreversible demagnetization seen in many PMLGs10,11, but its efficiency is very low. A high-power density LG is proposed12,13, where the armature windings are positioned within the translator. Although its power density is slightly increased, the built-in armature windings bring about an increase in temperature rise. Furthermore, the energy conversion efficiency remains very low.
Furthermore, in addition to the traditional LG structures mentioned above, various specialized types of the LGs have also been proposed. For instance, magnetic field modulation LGs are proposed to enhance generator output by increasing linear speed14–17, but this may result in an augmentation of generator volume and a reduction in power density. The transverse flux type LGs are introduced due to their high energy conversion efficiency18–20, however, actual testing has shown it only accounts for less than 40%. Although a linear-rotary generator is capable of both linear and rotational movements, its power quality, power density, and energy conversion efficiency are still low21–28. Therefore, these specialized LGs are essentially still linear generators with similar drawbacks like not high energy conversion efficiency, poor power density, and large volume.
In addition to the idea of using an LG for the WEC, experts, and scholars have been considering the combination of rotary generators and converters used in the WEC in recent years. The initial design involves a combination of a rotary generator (RG) and magnetic screw (MS)29–33, which enhances the output of the entire WEC device. However, this design results in a large volume, complex structure, and high maintenance cost. Subsequently, an improved model is proposed to reduce volume and enhance power generation but suffers from incomplete decoupling of internal and external magnetic circuits leading to poor quality electrical energy output by the generator, its efficiency is only about 55% and its power density is only about 0.2 MW/m34–36. Furthermore, due to the use of only PM excitation, its non-adjustable thrust limits its adaptability to various applications.
A novel magnetic helix hybrid excitation rotary generator (MH-HERG) with remarkable power density and impressive energy conversion efficiency is proposed to address the issues of poor energy conversion efficiency, not high power density, and large volumes of existing LGs used for DD-WEC. It consists of an external stator, an intermediate rotor, and an internal mover. The non-contact energy transfer by a hybrid excitation magnetic screw (HEMS) unit is capable of transforming the float’s linear movement into rotational motion of the proposed MH-HERG and output electrical energy. Furthermore, magnetic barriers are used to physically decouple the coupling between internal and external magnetic circuits, which are described in the second part. In the same way, the analytical solution model and simulation verification of cogging torque and air gap flux density are presented at second part. Arc-shaped PMs, instead of tile-type PMs, are proposed to be used in the rotor to optimize the suppression of cogging torque and total harmonic distortion (THD) of no-load back electromotive force (back-EMF), thereby improving the output power quality of the proposed MH-HERG. The detailed optimization process is mainly introduced in the third part. Finally, comparative experiments are conducted between the proposed MH-HERG and existing LGs to confirm the efficacy and superior performance of the proposed MH-HERG.
Topology and working principle
Topology
The proposed MH-HERG are mainly divided into three parts: an intermediate rotor, an outer stator, and an inner mover, which are shown in Fig. 1. Firstly, the inner mover is constituted by a backiron and PMs. The PMs are helical-shaped and adopt the Halbach magnetization method, which are depicted in the upper right-hand section of Figs. 1 and 2. Ideal helical PMs, limited by materials and processing technology, are generally replaced by equivalent skewed tile-shaped PM blocks, such as using 6 skewed tile-shaped PM blocks to replace a complete circumference.
Fig. 1.
The topology of the proposed MH-HERG used in the WEC system.
Fig. 2.
The schematic diagram of PM magnetization directions of the HEMS unit.
Then, as the most important energy transfer and motion form conversion mechanism, the intermediate rotor is composed of external PMs, rotor backiron, magnetic barrier, internal PMs, and excitation coil, as shown in the middle of Fig. 1. The external PMs of the rotor are arc-shaped and adopt radial magnetization method, while the internal PMs of the rotor is helical-shaped. The use of arc-shaped PMs instead of tile-type PMs is to suppress the cogging effect and weaken the cogging torque and harmonic content in back-EMF. A new quasi-Halbach magnetization array is first proposed and used on the rotor internal PMs, as shown in Figs. 1 and 2. Meanwhile, the hybrid excitation method composed of rotor internal PMs and the internal magnetic field of the proposed MH-HERG is generated by the excitation windings located within the rotor. The magnetic barrier is embedded in the backiron of the rotor and is used to physically decouple the internal and external magnetic fields of the proposed MH-HERG.
At the same time, the hybrid excitation and inner mover jointly form a hybrid excitation magnetic screw (HEMS) unit, which can non-contact convert linear motion into rotary motion, avoiding the disadvantages of difficult maintenance and high losses of mechanical structures, and also has the function of overload and automatic recovery. An outer stator, which are shown in the middle of Fig. 2, is divided into a 36-slot stator core and 3-phase armature winding, in which armature winding adopts a double-layer lap winding arrangement, and the advantages of the used double-layer lap winding are large output thrust density and power density, low input current and high efficiency, which is beneficial for improving output and reducing energy consumption.
Working principle
The proposed MH-HERG applicable to the WEC system is shown in Fig. 3a. The inner mover is connected to the outer float floating on the sea surface. Both the middle rotor and the outer stator are fixed in the inner float submerged under the sea surface. When the float is at the peak or trough of the wave in Fig. 3a, the relative displacement between the rotor inner PM and the mover PM is the greatest, and the proposed MH-HERG reaches the output peak. The outer float is connected to the inner mover of the MH-HERG and oscillates vertically along with the undulating movement of the ocean waves, thereby converting wave energy into mechanical energy.
Fig. 3.
Working principle diagram of the proposed MH-HERG in the WEC system. (a) Schematic diagram. (b) Energy transfer diagram.
As shown in the energy transfer block diagram of Fig. 3b, the wave energy collected by the float is converted into mechanical energy by the mover connected to the outer float, and then through the non-contact magnetic transmission of the HEMS unit, the mechanical energy of the linear motion of the mover is converted into the mechanical energy of the rotational motion of the rotor. Finally, the mechanical energy of the rotor’s rotation is converted into the output electrical energy of the stator through the law of electromagnetic induction. The transfer of mechanical energy is mostly based on the hybrid excitation magnetic field as the medium, which is jointly established by the rotor inner PMs, mover PMs and the induced magnetic field induced by excitation winding of rotor. The excitation winding can adjust the dimensions of the air gap coupling magnetic field by changing the input current, which is used to adjust the output thrust. It cannot be ignored that due to the existence of the cogging structure, reluctance still exists, so there is still a portion of mechanical energy transmitted through the reluctance.
Assuming that the wave motion follows a sinusoidal pattern and aligns with the traits of regular waves, the motion equation of WEC system is able to be illustrated by frequency range. Under the action of waves, the part of the buoy that moves in a straight line plays a dominant role in power generation. Currently, the forces acting on the entire WEC system are mainly composed of wave vertical excitation force 
, hydrostatic buoyancy force 
, wave radiation force 
, generator electromagnetic force 
, and mechanical friction force 
.
 |
1 |
Where 
is the velocity of the buoy, 
is mass of the whole WEC system, including devices such as buoys and the proposed MH-HERG. 
is unit imaginary number, 
is vertical deviation of the buoy, and acceleration speed of the buoy is:
 |
2 |
, 
, and 
are expressed as follows:
 |
3 |
 |
4 |
 |
5 |
Where 
is water density, 
is gravitational acceleration, 
is the damping of the buoy in the water, 
is the friction resistance of the proposed MH-HERG, 
is water plane area of the buoy when the water surface is not stationary.
Because the 
is different under no-load and load conditions, it is mainly composed of detent force under no-load conditions, while under load conditions, in addition to detent force, there is also load electromagnetic force. Due to the main purpose of calculating power generation, to obtain an output power of 1 kW, the average force of 
should be 3800 N, at an average linear speed of 0.35 m/s. At this time, the 
is calculated as:
 |
6 |
To simplify the analysis of the 
on the buoy, it is assumed that the vertical position of the buoy on the sea surface is fixed and unchanged. At this point, the wave serves as the sole source of power for the buoy, and the 
can be simplified as:
 |
7 |
Where 
is the unit normal vector, 
is static cross-sectional area, and for tubular buoy, it is the bottom area of the buoy. 
is the water depth at which the buoy is deployed into the sea area, 
is the wave height, 
, 
is the length of the wave.
Taking the derivative of Eq. (1), the linear velocity 
of the buoy can be obtained.
 |
8 |
After obtaining the linear velocity 
of the buoy, which can also be the motion velocity of the mover, the rotational speed 
, and the output power 
can be calculated based on the transmission ratio 
of HEMS.
 |
9 |
 |
10 |
Where 
, 
is the pole pitch of the PMs, 
and 
respectively represent the efficiency of converting the mover’s linear motion into the rotor’s rotational motion, and the efficiency of generating electricity by cutting magnetic field lines by the rotor.
Air gap flux density
From structure of proposed MH-HERG, it is inferred that it has two layers of air gaps. The external air gap is within an outer stator and an intermediate rotor, and internal air gap is within an intermediate rotor and an inner mover. The external air gap magnetic field is mainly used for energy conversion, Then, conversion of mechanical energy to electrical energy output. Internal air gap magnetic field is mainly used as a conversion of motion form, that is, the linear motion of the rotor is converted into the rotational motion of the mover. So, total air gap flux density 
of proposed MH-HERG mainly consists of two parts: the external air gap 
and the internal air gap 
, and its calculation equation is as follows.
 |
11 |
For the external air gap, ignoring cogging influence, air gap flux density of a PM of equal thickness on the armature surface is able to be approximately set as:
 |
12 |
Where 
is the remanence density distribution of the rotor outer PM, 
is the rotor outer PM thickness function, 
is effective air gap length, 
is rotor position angle.
Because arc-shaped PMs of proposed MH-HERG are rotor outer PMs of unequal thickness, thickness of rotor outer PM 
and effective length of the air gap 
change with the position Angle of rotor. The external air gap flux density 
is able to be obtained.
 |
13 |
Where 
is the effective remanence flux density distribution of PM.
 |
14 |
Where 
is the thickness of the rotor outer PM.
Arc-shaped PM has a certain edge thickness 
shown in Fig. 2.
 |
15 |
Where 
is the number of the rotor outer PM poles.
The equivalent remanence flux density and its Fourier decomposition coefficient are as follows.
 |
16 |
 |
17 |
Where 
is pole arc coefficient of rotor outer PM, 
is the maximum thickness of rotor outer PM.
External air gap flux density is able to derived from substituting all above equations into Eq. (13) for calculation. It can be seen from Fig. 4a that there is still some error between the analytical solution obtained by energy method and finite element solution obtained by finite element simulation. This is because the energy method ignores the existence of some magnetic flux leakage, which leads to errors. However, the cogging torque component of the proposed MH-HERG using the arc-shaped PM is very small, and air gap flux density has better sine quality. This can also be verified from Fig. 4b, where the amplitudes of each harmonic can be obtained through Fourier decomposition. As can be seen from the Fig. 4b, the 3rd and 5th harmonics, the 9th and 11th harmonics are relatively high. Since the generator is a 36-slot 6-pole one, this also conforms to the harmonic analysis rules of permanent magnet synchronous motors.
Fig. 4.
Comparison between analytical Solution and finite element Solution for air gap flux density. (a) Curve. (b) FFT analysis.
For internal air gap, internal air gap flux density 
mainly consists of two parts: air gap flux density resulted from rotor internal PM 
and air gap flux density generated by excitation winding 
.
 |
18 |
Moreover, due to the presence of a cogging structure inside the rotor, there is no cogging structure on the mover. The rotor-slotted model can be mapped to the rotor-slotless model by using the conformal transformation method. Air gap flux field solved in rotor-slotless model is then inversely transformed into the solutions in the rotor-slotted model. The accuracy of the solutions is high and the universality is strong, so the conformal transformation method is used to calculate the proposed MH-HERG’s internal air gap flux density 
.
A significant portion of the flux lines that penetrate the rotor slot are focused in its vicinity, for the convenience of the conformal transformation method, the schematic diagram of the infinite depth slot model under a single slot pitch is shown in Fig. 5. Among them, 
is the slot pitch angle of the proposed MH-HERG, 
and 
are the mechanical angles from the lower edge and upper edge of the notch to the reference axis, respectively. 
and 
are outer radius of mover and inner radius of the rotor.
Fig. 5.
The schematic diagram of the infinite depth slot model under a single slot pitch of the HEMS unit of the proposed MH-HERG.
The plane where the infinitely deep slot model resides is the complex plane S. Through four groups of the conformal transformation method, the model with slots in the S-plane can be converted to a slotless representation on the K-plane. The detailed process is shown in Fig. 6.
Fig. 6.
The flowchart of four groups of the conformal transformation method.
Firstly, as shown in Fig. 7, by applying a logarithmic transformation, the model in S-plane is transformed to a linear representation in Z-plane. Secondly, utilizing the Schwarz-Christoffel transformation, the linear model in the Z-plane is associated with the upper section of W-plane. After transforming upper portion of W-plane into a banded region on the T-plane, it becomes apparent that the transition from the W-plane to the T-plane involves a combination of logarithmic and shifting transformations. Finally, the banded domain on the T-plane is transformed into a circular ring domain on the K-plane by using a power transformation, thus equivalent to a rotor-slotless model.
Fig. 7.
The four planes of the conformal transformation method.
Therefore, an analytical expression for the air gap flux density generated through rotor internal PM 
is obtained through conformal transformation method.
 |
19 |
Where 
is air gap flux density in K-plane when it is slotless, 
is radial air gap flux density, 
is axial air gap flux density, 
and 
are imaginary and real parts of complex number 
, respectively.
 |
20 |
 |
21 |
Where 
 |
22 |
Where 
and 
are vacuum permeability and relative permeability, respectively. 
is thickness of the rotor inner PM.
 |
23 |
 |
24 |
Where 
and 
are the variable magnetization angle and variable width of the rotor inner PM, respectively. 
is the arc length of a single magnetic pole, 
, 
, 
, and 
are as follows.
 |
25 |
 |
26 |
 |
27 |
 |
28 |
By iteratively solving in Fig. 7, the coordinates of point w, point k, point s, point a, and point b can be obtained, respectively. By substituting the following equation, a complex number 
can be obtained.
 |
29 |
As shown below, air gap flux density produced by the excitation winding 
is solved by magnetic circuit method.
 |
30 |
Where 
is sum of the magneto motive force generated through the excitation winding, 
, 
, 
, and 
are the air gap reluctance, PM reluctance, rotor reluctance, and mover reluctance, respectively. Finally, by substituting Eq. (19) ~ Eq. (30) into Eq. (18), analytical solution for internal air gap flux density is able to be got. Comparing it with finite element solution, which are shown in Fig. 8, the error is relatively small, which verifies the calculation equations of the analytical solution. Compared with the finite element solution shown in Fig. 8a, the error is smaller, verifying the calculation equation of the analytical solution. This can also be verified from Fig. 8b. As can be seen from Fig. 8a and b, the air gap flux density waveform is approximately sinusoidal, but the peaks and troughs are relatively flat. Therefore, after Fourier analysis, the amplitudes of even harmonics are relatively high.
Fig. 8.
The Comparison waveform of internal air gap flux density of the proposed MH-HERG. (a) Curve. (b) FFT analysis.
In order to further verify the accuracy of the calculation of the magnetic density of the inner and outer air gap, a three-dimensional finite element model is established. To accurately display the analysis results, a high-density mesh is set, as shown in Fig. 9 (a). The magnetic flux density map of the proposed MH-HERG is shown in Fig. 9b, which the maximum magnetic density is 1.4 T. In order to show the independence of the magnetic field more clearly, the maximum scale of the magnetic density is set to 2.0 T. It is evident that in generator unit and HEMS unit, the direction of the magnetic field lines is clear and the magnetic flux density is not saturated shown in Fig. 9c and d.
Fig. 9.
The magnetic flux distribution of the proposed MH-HERG. (a) Mesh. (b) Magnetic flux density map. (c) Generator unit. (d) HEMS unit.
Optimization
As the proposed MH-HERG serves the WEC system, amplitude of fundamental wave and harmonic components of no-load back-EMF directly influence the MH-HERG’s output performance. Furthermore, cogging torque poses as a significant adverse factor impacting the output. To optimize and enhance the MH-HERG’s performance, we introduce the use of arc-shaped PMs. Additionally, we propose a chaos-based adaptive differential evolution algorithm (DEA) as a comprehensive optimization approach, aiming to maximize the MH-HERG’s performance. Traditional DEAs face challenges in solving mixed-variable optimization problems (MVOPs) and often get stuck in local optima. To overcome this, we incorporate a chaotic map to enhance population diversity and prevent local optimal solutions. Furthermore, we introduce an adaptive dynamic inertia weight to boost the algorithm’s search efficiency and convergence rate. Our primary optimization targets include minimizing cogging torque, maximizing power density, and optimizing fundamental amplitude and total harmonic distortion (THD) of no-load back-EMF. Table 1 outlines preliminary key parameters of proposed MH-HERG.
Table 1.
The preliminary main parameters of the proposed MH-HERG.
| Item | Unit | Value |
|---|---|---|
| The axial effective length of the proposed MH-HERG | mm | 100 |
| The number of stator slots | - | 36 |
| The thickness of the rotor outer PM | mm | 5 |
| The pole arc coefficient of the rotor outer PM | - | 0.8 |
| The eccentric distance | mm | 0 |
| The thickness of the rotor inner main PM | mm | 10 |
| The width of the rotor inner main PM | mm | 5.25 |
| The thickness of the rotor inner auxiliary PM | mm | 5 |
| The width of the rotor inner auxiliary PM | mm | 2.625 |
| The number of turns of the excitation winding | - | 20 |
| The thickness of the mover PM | mm | 10 |
| The pole pitch of the mover PM | mm | 10.5 |
| The rated linear speed | m/s | 0.35 |
Arc-shaped PMs
The quality of the output electrical energy of the proposed MH-HERG during no-load operation is mainly determined by the external air gap magnetic field, which is generated separately by rotor outer PM. Different PM structures have different effects on output performance of proposed MH-HERG. By optimizing the shape of the rotor outer PM, fundamental wave of no-load back-EMF can be increased and THD of no-load back-EMF is able to be reduced to improve output power quality of the proposed MH-HERG. Moreover, the cogging torque is also reduced. The arc-shaped PM optimizes the thickness of the rotor outer PM to a sine function in the direction of magnetization to improve the electromagnetic performance of the proposed MH-HERG. Moreover, the actual air gap length also becomes a sine function.
Inner pole arc contour and outer pole arc contour of the rotor outer PM are regarded as functions of the rotor angular position 
, denoted as 
and 
, respectively. The information depicted in Fig. 10 reveals that the function of thickness 
of the magnetization direction of a radially magnetized arc-shaped PM is equal to the difference between the inner pole arc contour 
and outer pole arc contour 
of the rotor outer PM. In the case of radial magnetization, the inner pole arc contour 
of the rotor outer PM is constant C and does not change with the rotor angular position 
. Therefore, the key to achieving an arc-shaped PM is to sine the thickness of the rotor outer PM, that is, to sine the contour function of the outer pole arc of the rotor outer PM.
 |
31 |
Fig. 10.
The radial magnetized arc-shaped PM structure.
Where the range of values for 
is 
, 
is pole arc coefficient of rotor outer PM. Expression for cogging torque of the proposed MH-HERG is obtained through the energy method in the second part, as shown below.
 |
32 |
Where 
is width of the stator slot, 
is effective length of stator, 
and 
are inner radius of stator and outer radius of rotor, respectively. 
and 
are greatest common divisor and least common multiple of number of stator slots and number of rotor outer PM poles, respectively. In the actual optimization process, eccentric distance h and pole arc coefficient 
of the proposed arc-shaped PM are generally selected as optimization factors.
 |
33 |
where 
and 
are maximum and minimum thicknesses of arc-shaped PM, respectively.
Optimization by improved DEA
Adjustments to pole arc coefficient in proposed MH-HERG alter PM shape and air gap length, subsequently influencing air gap magnetic field, consequently, no-load back-EMF and cogging torque. To analyse the impact of pole arc coefficient on these factors using control variable method, it’s crucial to maintain another parameters constant. Studying the effects at excessively low pole arc coefficients offers limited insight into no-load back-EMF, cogging torque, and power density. Therefore, we focus on a pole arc coefficient range of [0.7, 0.98]. We input relevant parameters into the analytical model and cross-verify them with finite element analysis (FEA) results for corresponding pole arc coefficients.
By comparing the analytical and finite element solutions, we derive the influence trends and comparisons of fundamental amplitude and total harmonic distortion (THD) of the no-load back-EMF as pole arc coefficient varies, as illustrated in Fig. 11a. Notably, with constant maximum thickness and eccentric distance of the arc-shaped PM, fundamental amplitude of no-load back-EMF increases with pole arc coefficient. The analytical solutions align closely with the FEA results, exhibiting minimal error.
Fig. 11.
The variation law and comparison of the output performance of the proposed MH-HERG with the pole arc coefficient. (a) The fundamental amplitude of the no-load back-EMF. (b) The THD of the no-load back-EMF. (c) Cogging torque. (d) Power density.
The variation and comparison of THD of no-load back-EMF with pole arc coefficient in analytical solutions and finite element solutions are shown in Fig. 11b. The information reveals that THD of no-load back-EMF decreases with improve of pole arc coefficient. The main reason for the error between analytical solutions and finite element solutions is that the stator-slotted effect is considered in analytical solution model, while effect of magnetic flux leakage is not taken into consideration, resulting in its solution result being slightly smaller than the finite element solutions.
By analysing the analytical solution model and the finite element solution model, when the pole arc coefficient is equal to 0.98, peak-to-peak value of cogging torque is minimized. According to Fig. 11c, it can be seen that peak-to-peak value of cogging torque increases between pole arc coefficient of 0.7 and 0.86. When pole arc coefficient is 0.86, peak-to-peak value of cogging torque is the highest, with a value of 3.15 N·m. The peak-to-peak value of cogging torque decreases when pole arc coefficient is between 0.86 and 0.98. When pole arc coefficient is 0.98, peak-to-peak value of cogging torque is the lowest, with a value of 1.11 N·m. Similarly, power density increases first and then decreases as pole arc coefficient increases by comparing results of analytical solution model and FEA solution model, as shown in Fig. 11d.
After using arc-shaped PMs, the thickness on both sides of the rotor outer PM changes. The larger the eccentric distance, the less PM is used, and the corresponding change in air gap length will occur, resulting in different air gap flux densities. The information depicted in Eq. (33) reveals that eccentric distance is related to the minimum thickness of arc-shaped PM. By changing the eccentric distance, analytical solution of no-load back-EMF is able to be quickly obtained, and FEA solution models with different eccentric distances can be established for comparison. The information depicted in Fig. 12a reveals that while keeping pole arc coefficient constant, no-load back-EMF’s fundamental value decreases with eccentric distance increase.
Fig. 12.
The variation law and comparison of the output performance of the proposed MH-HERG with the eccentric distance. (a) The fundamental amplitude of the no-load back-EMF. (b) The THD of the no-load back-EMF. (c) Cogging torque. (d) Power density.
The variation of THD of no-load back-EMF with eccentric distance in the analytical solution model and finite element model is shown in Fig. 12b. It can be seen that while keeping the pole arc coefficient constant, the THD of no-load back-EMF first decreases and then slightly increases with increase of eccentric distance, and a minimum point appears at the eccentric distance of about − 110 mm.
As shown in Fig. 12c, with increase of eccentric distance, length of air gap increases, and peak-to-peak value of cogging torque first decreases and then slightly increases. Arc-shaped PMs can effectively reduce the cogging torque, but in general, to achieve a certain performance index of the proposed MH-HERG, the eccentric distance should not be too large and should be comprehensively considered. Moreover, it can be seen from Fig. 12d that while keeping the pole arc coefficient constant, the power density is first unchanged and then significantly decreases with the increase of eccentric distance. The analytical solution model results are consistent with the finite element solution model results, and the error is small.
After that, based on the analysis of the optimization factors mentioned above, the proposed MH-HERG is optimized by the improved DEA based on chaos adaptation. A multi-objective optimization model based on the analytical solution model is established, with eccentric distance, thickness of the arc-shaped PM, and pole arc coefficient as optimization factors and cogging torque, power density, the fundamental amplitude of no-load back-EMF, and THD of no-load back-EMF as optimization objectives. The detailed flowchart is displayed in Fig. 13.
Fig. 13.
The Multi-objective optimization flowchart based on the improved DEA based on chaos adaptation.
At the same time, the information depicted in Fig. 14 reveals that response of the two optimization factors to the four optimization objectives has a high gradient within the optimization interval, and the optimal interval of the two optimization factors is different. Moreover, the premise of using multi-objective optimization algorithms has been proven to be valid.
Fig. 14.
The impact trend of optimization factors on optimization objectives. (a) The fundamental amplitude of the no-load back-EMF. (b) The THD of the no-load back-EMF. (c) Cogging torque. (d) Power density.
The design parameters of the proposed MH-HERG that have been optimized and changed are shown in Table 2.
Table 2.
The optimized design parameters of the proposed MH-HERG.
| Item | Unit | Value |
|---|---|---|
| The thickness of the rotor outer PM | mm | 3 |
| The pole arc coefficient of the rotor outer PM | - | 0.98 |
| The eccentric distance | mm | −87 |
Finally, after undergoing the above optimization process, the performance comparison of the proposed MH-HERG is shown in Fig. 15. The information depicted in Fig. 15a reveals that although amplitude of optimized air gap flux density waveform slightly decreases, waveform is more sinusoidal, and its THD has decreased from 53% to 9.5%. Fundamental amplitude of no-load back-EMF of proposed MH-HERG before and after optimization, as shown in Fig. 15b, is 329 V and 343 V respectively. Its THD decreased from 14.6% to 1.7%, a decrease of 88.4%. In addition, optimized power density and energy conversion efficiency is 0.32 MW/m3 and 93.86%, respectively, as shown in Fig. 15c, an increase of 18.5% and 16.6%. Finally, the cogging torque, torque mean value, and torque ripple, as shown in Fig. 15d, is 52.2 N·m and 0.3 N·m, 23.3 N·m and 22.1 N·m, 222.5%, and 2.6% before and after optimization, respectively. Compared before and after optimization, cogging torque, torque ripple, and THD of proposed MH-HERG are significantly weakened while improving performance.
Fig. 15.
The performance comparison of the proposed MH-HERG before and after optimization. (a) Air gap flux density. (b) No-load back-EMF. (c) Power density and energy conversion efficiency. (d) Cogging torque and torque.
Experiment
The proposed MH-HERG prototype is shown in Fig. 16d, mainly consisting of three parts: the external stator, the middle rotor and the internal mover. The axial stacking of the stator core, as shown in Fig. 16a, consists of 36 slots, which is beneficial for reducing eddy current losses. The external PMs of the rotor, as shown in Fig. 16c, adopt arc-shaped PMs instead of the traditional tile-shaped PMs, which not only reduces the amount of PMs used but also lowers the cogging torque and harmonic content of the prototype, which is conducive to improving the quality of the output electrical energy of the prototype. In actual manufacturing and assembly, the PMs inside the rotor and the mover PMs are the most difficult. Firstly, in actual manufacturing, the existing technology is basically impossible to produce a complete helical PM, as shown in Fig. 16b. Therefore, for the sake of nearsighted equivalence, six equivalent skewed tile-shaped PM blocks are used to replace the ideal helical PM required for one circle on the mover. Meanwhile, if the diameter of the prototype is large, it can be considered to use more equivalent skewed tile-shaped PM blocks within one circle, which is also applicable to the PM inside the rotor. Secondly, in the actual assembly, grooves are pre-made on the backiron of th rotor and mover, and the PMs are installed in the grooves. This not only facilitates the installation of the PMs but also helps them to be fixed. At the same time, it prevents the PMs from being misaligned, thus avoiding displacement differences that could damage the performance of the prototype.
Fig. 16.
The prototype of the proposed MH-HERG. (a) Stator core. (b) Mover helical PM. (c) Rotor outer arc-shaped PM. (d) Proposed MH-HERG.
Since the prototype is made for the subsequent ocean experiments, the wave height is generally between 0.5 m and 1 m. Therefore, the length of the mover in Fig. 17a is close to 2.5 m. During laboratory experiments and tests, due to the limitations of the prototype’s size, volume and weight, the prototype cannot be tested using the traditional motor testing platform. Therefore, the experimental test platform as shown in Fig. 17a is used for the test here. Among them, a hydraulic crane is adopted instead of the drive motor of the traditional motor test platform to drive the proposed MH-HERG. The no-load back-EMF of the prototype is tested by directly connecting the three-phase outgoing lines of the prototype to the oscilloscope, and the load back-EMF, power and energy conversion efficiency are tested by connecting the three-phase outgoing lines to the three-phase symmetrical resistive load. Among them, power and energy conversion efficiency are tested by a power analyzer. During the experimental test, the experimental method adopted is to change the frequency of the hydraulic crane, thereby altering the linear speed of the mover of the prototype, and finally obtaining the output of the proposed MH-HERG.
Fig. 17.
The test platform of the prototype of the proposed MH-HERG. (a) Laboratory testing. (b) Ocean testing.
During the ocean test, the experimental test platform is shown in Fig. 17b, and its main components are the prototype, the electronic cabin, the outer float, and the inner float. The proposed MH-HERG is installed in the float for ocean testing. The stator and rotor are fixed inside the inner float, and the mover is connected to the outer float, moving up and down in a straight line along with the undulation of the waves. The linear motion is converted into high-speed rotational motion through the HEMS unit, which then drives the prototype to generate electrical energy output. The electronic cabin is installed in the inner float along with the prototype and includes: rectifier filter circuit, boost converter, overvoltage protection circuit, torque sensor, voltage waveform sampling module, General Packet Radio Service (GPRS) data transmission module and energy storage battery. The electrical energy output by the prototype is rectified, filtered and boosted, and then stored in the battery. Meanwhile, the voltage output by the prototype is collected through the voltage waveform sampling module. The real-time torque of the prototype is obtained by installing a torque sensor on the rotor of the prototype, and the collected output data of the prototype is transmitted through a GPRS data transmission module with satellite signals.
Combined with the actual wave speed of Lianyungang City, where the ocean test is conducted, and the results of ocean experiments, the frequency of the hydraulic crane used in the laboratory experiments is set, and the range of linear speed of the mover is converted from 0.022 m/s to 0.35 m/s. Fifteen sets of experimental data of no-load and load are tested respectively. The no-load back-EMF and the load back-EMF are shown in Fig. 18a and Fig. 18c respectively. As the linear speed increases, the amplitude of the back-EMF also gradually rises, showing a linear change, and the sine degree of the waveform is very high. Since the main body of the power generation part of the prototype is a rotary generator, the cogging torque under no-load conditions is additionally tested, as shown in Fig. 18b. The cogging torque does not change with the increase of the linear speed and remains basically constant. Moreover, the experimental test data is basically consistent with the finite element solution, while the analytical solution is slightly lower. This is because the cogging torque obtained from the experiment also includes mechanical friction torque.
Fig. 18.
The performance of the prototype of the proposed MH-HERG. (a) No-load back-EMF. (b) Cogging torque. (c) Load back-EMF. (d) Power density. (e) Energy conversion efficiency. (f) Generator loss.
The analytical solution, finite element solution and experimental data of power density at different linear speed are basically consistent, as shown in Fig. 18d. This also verifies the correctness and validity of the analysis. The energy conversion efficiency gradually increases with the increase of the linear speed, as shown in Fig. 18e. This is because as the linear speed increases, the output electrical energy gradually increases, gradually exceeding the mechanical loss and electromagnetic loss of the prototype itself, and thus the efficiency is constantly improving. Finally, based on the data obtained from the experimental tests, the losses of the prototype are also calculated, mainly electromagnetic losses. Thus, as shown in Fig. 18f, the losses of the prototype increase with the increase of the linear speed.
On the other hand, with higher linear speeds, there is a shift in power density. This is because as the linear speed increases, the output of the generator increases, after reaching the rated value, the generator reaches magnetic saturation, and the output remains unchanged. Furthermore, the energy conversion efficiency is also changing according to the trend of power density change. Finally, the power density and energy conversion efficiency of the proposed model are up to 0.32 MW/m3 and 92%, which can be seen in Fig. 18. Moreover, compared with the existing LGs used in DD-WEC, the power density and energy conversion efficiency of the proposed MH-HERG have increased by 4.4 times and 3 times compared with Huang’s, respectively. After all, the power density and energy conversion efficiency of the existing LGs are only 0.06 MW/m3 and 23% respectively. The specific performance comparison of the two is shown in Fig. 19.
Fig. 19.
The Performance comparison of the proposed MH-HERG and the existing LGs.
Finally, the generator loss also changes with the increase of linear speed, because with the increase of linear speed, the generator current also increases, and then the copper loss gradually occupies the main part of the generator loss, so the generator loss is parabolic rise.
Conclusion
This paper presents a novel magnetic helix hybrid excitation rotary generator (MH-HERG) for wave energy conversion with higher power density and higher energy conversion efficiency. Analytical solution model of air gap flux density is first proposed for calculating the performance of the proposed MH-HERG and subsequent optimization, including two parts: internal air gap flux density and external air gap flux density. The comparison between FEA solution and the analytical solution calculated by the energy method and the magnetic circuit method can verify the correctness and effectiveness of the proposed analytical solution model.
Second, a hybrid excitation method containing a new quasi-Halbach magnetization array is proposed in the proposed MH-HERG to increase its power density and allows a hybrid excitation method to be used to make the thrust adjustable to further increase the power density of the proposed MH-HERG.
Thirdly, Arc-shaped PMs, instead of tile-type PMs, are proposed to weaken cogging torque and THD of back-EMF of the proposed MH-HERG, thereby improving the output power quality. The improved DEA based on chaos adaptation is proposed to maximize the performance of the proposed MH-HERG. The fundamental amplitudes of no-load back-EMF increased by 4.3%. The THD had decreased from 14.6% to 1.7%. In addition, cogging torque is significantly reduced, with peak to peak value of only 0.31 N·m, accounting for only 1.4% of the mean torque. Moreover, the power density is 0.32 MW/m3, an increase of 18.5%. Finally, the increased power density by 4.4 times and energy conversion efficiency by 3 times compared to the existing LGs used in the DD-WEC34–36.
Finally, The prototype test results have verified the effectiveness of the performance of the proposed MH-HERG, and compared with traditional linear generators, the power density and energy conversion efficiency are greatly improved.
In this study, only the size parameters of the rotor outer PM are considered to optimize the performance of the proposed MH-HERG. There are relatively few studies on the actual manufacturing and assembly of the prototype. In future research, the equivalent of the ideal helical PM is worthy of further study to facilitate the manufacturing and installation of PMs. Secondly, it is also very necessary to optimize the overall structure of the actual prototype. For instance, to optimize the structure of the rotor, only one set of helical PMs should be used instead of having PMs both inside and outside the rotor. Then, the overall installation of the prototype requires multiple bearings, including both linear and rotary bearings, which increases mechanical friction and is not conducive to performance improvement. It is necessary to further optimize the overall structure of the prototype to reduce mechanical friction. Finally, multi-variable and multi-objective optimization problems will also be the focus of future research, including increasing the temperature rise and moment of inertia of generators.
Author contributions
Y. L. made the main contribution to this article, while H. Y. and Q. Z. made significant contributions to proofreading.
Funding
This work is supported by the National Natural Science Foundation of China (Grant No. 42176211).
Data availability
Due to the requirements of the funding project and to protect the privacy of the study participants, the data in this article cannot be shared publicly, but any individual with a reasonable academic research need can contact the corresponding author to obtain the research data.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
- 1.Hong, Y. et al. Review on electrical control strategies for wave energy converting systems. Renew. Sustain. Energy Rev.31, 329–342 (2014). [Google Scholar]
- 2.Vermaak, R. & Kamper, M. J. Design aspects of a novel topology air cored permanent magnet linear generator for direct drive wave energy converters. IEEE Trans. Ind. Electron.59(5), 2104–2115 (2012). [Google Scholar]
- 3.Xiao, X., Huang, X. & Kang, Q. A hill-climbing-method-based maximum-power-point-tracking strategy for direct-drive wave energy converters. IEEE Trans. Ind. Electron.63(1), 257–267 (2016). [Google Scholar]
- 4.Baker, N. J., Raihan, M. A. H. & Almoraya, A. A. A Cylindrical Linear Permanent Magnet Vernier Hybrid Machine for Wave Energy. IEEE Trans. Energy Convers.34(2), 691–700 (2019). [Google Scholar]
- 5.Chen, M. et al. A Stator-PM transverse flux permanent magnet linear generator for direct drive wave energy converter. IEEE Access.9, 9949–9957 (2021). [Google Scholar]
- 6.Chen, M., Huang, L., Hu, M., Hu, B. & Ahmad, G. A Spiral Translator Permanent Magnet Transverse Flux Linear Generator Used in Direct-Drive Wave Energy Converter. IEEE Trans. Magn.57(7), 1–5 (2021). [Google Scholar]
- 7.Du, Y., Cheng, M. & Chau, K. T. Simulation of the linear primary permanent magnet Vernier machine system for wave energy conversion, in Proc. 20th Int. Conf. Electr. Mach. Syst., pp. 262–266 (2013).
- 8.Keysan, O., Mueller, M., McDonald, A., Hodgins, N. & Shek, J. Designing the C-GEN lightweight direct drive generator for wave and tidal energy. IET Renew. Power Gener6(3), 161–170 (2012). [Google Scholar]
- 9.Prudell, J., Stoddard, M., Amon, E., Brekken, T. K. A. & Von Jouanne, A. A permanent-magnet tubular linear generator for ocean wave energy conversion. IEEE Trans. Ind. Appl.46(6), 2392–2400 (2010). [Google Scholar]
- 10.Farrok, O. et al. Oceanic wave energy conversion by a novel permanent magnet linear generator capable of preventing demagnetization. IEEE Trans. Ind. Appl.54(6), 6005–6014 (2018). [Google Scholar]
- 11.Zhu, S., Cheng, M., Hua, W., Cai, X. & Tong, M. Finite element analysis of flux-switching PM machine considering oversaturation and irreversible demagnetization. IEEE Trans. Magn.51(11), 1–4 (2015).26203196 [Google Scholar]
- 12.Farrok, O. & Islam, M. R. Advanced Electrical Machines for Oceanic Wave Energy Conversionpp. 115–141 (Springer-, Jan. 2018).
- 13.Farrok, O., Islam, M. R., Muttaqi, K. M., Sutanto, D. & Zhu, J. Design and Optimization of a Novel Dual-Port Linear Generator for Oceanic Wave Energy Conversion. IEEE Trans. Ind. Electron.67(5), 3409–3418 (2020). [Google Scholar]
- 14.Xia, T., Yu, H., Guo, R. & Liu, X. Research on the Field-Modulated Tubular Linear Generator with Quasi-Halbach Magnetization for Ocean Wave Energy Conversion. IEEE Trans. Appl. Superconduct28(3), 1–5 (2018). [Google Scholar]
- 15.Chen, M. et al. Analysis of Magnetic Gearing Effect in Field-Modulated Transverse Flux Linear Generator for Direct Drive Wave Energy Conversion. IEEE Trans. Magn.58(2), 1–5 (2022). [Google Scholar]
- 16.Huang, L. et al. Analysis of a Hybrid Field-Modulated Linear Generator for Wave Energy Conversion. IEEE Trans. Appl. Superconduct28(3), 1–5 (2018). [Google Scholar]
- 17.Feng, N., Yu, H., Zhao, M., Zhang, P. & Hou, D. Magnetic field modulated linear permanent-magnet generator for direct-drive wave energy conversion. IET Electr. Power Appl.14(5), 742–750 (2020). [Google Scholar]
- 18.Sui, Y., Yin, Z., Wang, M., Yu, B. & Zheng, P. A tubular staggered-teeth transverse-flux PMLM with circumferentially distributed three-phase windings. IEEE Trans. Ind. Electron.66(6), 4837–4848 (2019). [Google Scholar]
- 19.Qiu, S., Zhao, W., Zhang, C., Shek, J. K. H. & Wang, H. A novel structure of tubular staggered transverse-flux permanent-magnet linear generator for wave energy conversion. IEEE Trans. Energy Convers.37(1), 24–35 (2022). [Google Scholar]
- 20.Li, Z. & Niu, S. Design of a novel hybrid-excited transverse flux tubular linear machine with complementary structure. IEEE Trans. Magn.58(8), 1–6 (2022). [Google Scholar]
- 21.Guo, K., Guo, Y. & Fang, S. Flux Leakage Analytical Calculation in the E-Shape Stator of Linear Rotary Motor with Interlaced Permanent Magnet Poles. IEEE Trans. Magn.58(8), 1–6 (2022). [Google Scholar]
- 22.Xinghe, F., Jianhao, W., Mingyao, L. & Systems Design and analysis of linear rotary permanent magnet generator for wind-wave combined power generation system, 2017 20th International Conference on Electrical Machines and (ICEMS), Sydney, NSW, pp. 1–6, Australia, (2017).
- 23.Nie, R. et al. Loss and Efficiency Comparison of Mover PM and Stator PM Linear-Rotary Generators for Wind-Wave Combined Energy Conversion System, 2023 26th International Conference on Electrical Machines and Systems (ICEMS), pp. 3470–3474, Zhuhai, China, (2023).
- 24.Szabó, L. On the use of rotary-linear generators in floating hybrid wind and wave energy conversion systems, IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR), Cluj-Napoca, pp. 1–6, Romania, (2018).
- 25.Zhang, G., Nie, R., Si, J., Gan, C. & Hu, Y. Optimal design of a novel double-stator linear-rotary flux-switching permanent-magnet generator for offshore wind-wave energy conversion, 2021 IEEE Energy Conversion Congress and Exposition (ECCE), pp. 4300–4305, Vancouver, BC, Canada, (2021).
- 26.Nie, R. et al. Optimization design of a novel permanent magnet linear-rotary generator for offshore wind-wave combined energy conversion based on multi-attribute decision making. IEEE Trans. Energy Convers.39(3), 1583–1600 (2024). [Google Scholar]
- 27.Xu, L., Zhu, X., Zhang, C., Zhang, L. & Quan, L. Power oriented design and optimization of dual stator linear-rotary generator with Halbach PM array for ocean energy conversion. IEEE Trans. Energy Convers.36(4), 3414–3426 (2021). [Google Scholar]
- 28.Guo, K. & Guo, Y. Design Optimization of Linear-Rotary Motion Permanent Magnet Generator With E-Shaped Stator. IEEE Trans. Appl. Superconduct31(8), 1–5 (2021). [Google Scholar]
- 29.Holm, R. K., Berg, N. I., Walkusch, M., Rasmussen, P. O. & Hansen, R. H. Design of a magnetic lead screw for wave energy conversion. IEEE Trans. Ind. Appl.49 (6), 2699–2708 (2013). Nov.-Dec. [Google Scholar]
- 30.Lu, K., Xia, Y., Wu, W. & Zhang, L. New helical-shape magnetic pole design for magnetic lead screw enabling structure simplification. IEEE Trans. Magn.51(11), 1–4 (2015).26203196 [Google Scholar]
- 31.Zhu, L. et al. A novel consequent-pole magnetic lead screw and its 3-D analytical model with experimental verification for wave energy conversion. IEEE Trans. Energy Convers.39(2), 1202–1215 (2024). [Google Scholar]
- 32.Pakdelian, S., Frank, N. W. & Toliyat, H. A. Magnetic Design Aspects of the Trans-Rotary Magnetic Gear. IEEE Trans. Energy Convers.30(1), 41–50 (2015). [Google Scholar]
- 33.Zhu, L. et al. A novel hybrid excitation magnetic lead screw and its transient sub-domain analytical model for wave energy conversion. IEEE Trans. Energy Convers.39(3), 1726–1737 (2024). [Google Scholar]
- 34.Huang, L., Li, Y., Wei, L. & Chen, M. Comparison Research on Axial Flux Modulation Generators for Linear-rotary Wave Energy Conversion, 2022 IEEE 20th Biennial Conference on Electromagnetic Field Computation (CEFC), Denver, CO, pp. 1–2, USA, (2022).
- 35.Ling, Z. et al. Design and manufacture of a linear actuator based on magnetic screw transmission. IEEE Trans. Ind. Electron.68(2), 1095–1107 (2021). [Google Scholar]
- 36.Ling, Z., Ji, J., Wang, J. & Zhao, W. Design optimization and test of a radially magnetized magnetic screw with discretized PMs. IEEE Trans. Ind. Electron.65(9), 7536–7547 (2018). [Google Scholar]
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
Due to the requirements of the funding project and to protect the privacy of the study participants, the data in this article cannot be shared publicly, but any individual with a reasonable academic research need can contact the corresponding author to obtain the research data.