Linear Permanent Magnet Vernier Generators for Wave Energy Applications: Analysis, Challenges, and Opportunities

Abstract

Harvesting energy from waves as a substantial resource of renewable energy has attracted much attention in recent years. Linear permanent magnet vernier generators (LPMVGs) have been widely adopted in wave energy applications to extract clean energy from oceans. Linear PM vernier machines perform based on the magnetic gearing effect, allowing them to offer high power/force density at low speeds. The outstanding feature of providing high power capability makes linear vernier generators more advantageous compared to linear PM synchronous counterparts used in wave energy conversion systems. Nevertheless, they inherently suffer from a poor power factor arising from their considerable leakage flux. Various structures and methods have been introduced to enhance their performance and improve their low power factor. In this work, a comparative study of different structures, distinguishable concepts, and operation principles of linear PM vernier machines is presented. Furthermore, recent advancements and innovative improvements have been investigated. They are categorized and evaluated to provide a comprehensive insight into the exploitation of linear vernier generators in wave energy extracting systems. Finally, some significant structures of linear PM vernier generators are modeled using two-dimensional finite element analysis (2D-FEA) to compare their electromagnetic characteristics and survey their performance.

1. Introduction

Rotational electrical machines equipped with linear to rotary motion converters had been employed for linear applications before the introduction of linear structures [1]. Even though linear motion can be realized by using rotational machines and linear to rotary motion converters, some unfavorable concerns are inevitably arisen, including higher costs, lower accuracy of position sensing, higher inertia, higher maintenance, and shorter lifetime of the system [2]. Hence, devices to transform the linear to the rotary motion were eliminated and replaced by linear machines as more profitable alternatives. Linear structures have been broadly employed in different applications such as rail transit systems [3], ropeless elevators [4,5], automatic doors [6], electromagnetic suspension systems [7], free-piston engines [8,9], and wave energy applications [10]. The main scope of this research is to investigate the utilization of linear machines in harvesting wave energy, known as one of the most significant renewable energy resources.

The utilization of wave energy conversion systems has become extensively pervasive to generate electrical power [11,12]. Wave energy has the highest power density among other resources of renewable energy and is also more predictable. The capacity of ocean wave energy was estimated between 8000 to 80,000 TW$·$h/yr or 1 to 10 TW, indicating the importance of wave energy conversion technology [13]. However, this technology is moderately immature in comparison with other consequential energy resources, such as wind [14] and solar energy [15,16]. In order to harvest the environmentally friendly wave energy of oceans, wave energy converters (WECs) are employed. The devices to harness wave energy are categorized based on different criteria, such as the wave direction or their operation principles [17]. Two of the most widely used WECs that can be integrated with linear generators are Archimedes wave swing (AWS) and point absorber converters. Linear generators can produce electricity from the reciprocating motion of waves yielded by wave energy converters [18]. Different types of linear generators have been proposed, such as linear permanent magnet synchronous generators (LPMSGs) [19], linear flux-switching permanent magnet generators [20], and linear permanent magnet vernier generators (LPMVGs) [21].

Linear PM synchronous generators are outstanding options for wave energy harvesting due to offering high power capability and high efficiency [22,23]. On the other hand, the study on magnet materials and the improvement of their characteristics, specifically neodymium–iron–boron (NdFeB) magnets, have led to growing attention on PM machines [24]. However, a high number of poles are required for linear PM synchronous generators to be able to provide a high power density [25,26]. Linear PM vernier generators are regarded as excellent candidates for this aim. Linear vernier structures operate based on the same principles as magnetic gears, known as the magnetic gearing effect, which enables them to offer a high power/force density at low speeds [27]. A linear magnetic gear with the capability of adjustable gear ratio was surveyed in [28] used in wave energy conversion systems. The integration of a linear tubular magnetic gear and a tubular PM generator was proposed in [29] for extracting wave energy with a higher power density at low speeds. Even though magnetic gears deliver a high power density, the integration of magnetic gears and linear generators makes the system suffer from a bulky size and a complicated structure. Accordingly, linear PM vernier machines (LPMVMs) are preferable, in which they function based on the principles of magnetic gearing effect [30]. Different configurations and novel structures of linear PM vernier machines have been proposed in numerous pieces of research. Despite the merit of the high power capability of linear PM vernier structures, they inherently possess a low power factor. The poor power factor of vernier structures has resulted from their high leakage flux, stemming from the magnetic gearing effect. Consequently, high rating converters should be employed for low power factor structures, which is not an economically viable solution [31]. In recent years, different methods have been suggested to enhance the poor power factor of linear vernier machines. Furthermore, the longitudinal end effect of linear vernier machines is regarded as the other problem which uniquely exists for linear structures compared to rotational ones. Linear machines have two open ends contrary to the rotational structures, leading to the undesirable longitudinal end effect and causing a high force ripple [32].

In recent years, different types of linear PM vernier generators have been proposed to further enrich their high power/force capability and diminish their unwanted disadvantages. A comprehensive study is required to offer a comparative analysis and study of different linear vernier generators for more investigation in the future. This research focuses on surveys that have been accomplished to improve the performance of linear PM vernier machines, specifically for wave energy applications. In addition, different concepts of linear vernier generators are evaluated, and operation principles are investigated in this paper. In Section 2, the significance of wave energy harvesting and a succinct description of wave energy systems will be explained. The operation principles and analysis of the linear PM vernier machines will be surveyed in Section 3. Different types of LPMVMs will be categorized and studied in Section 4. The solutions to improve the performance of linear vernier structures will be investigated in Section 5 based on a generic classification. Moreover, four outstanding structures of linear generators will be thoroughly examined and compared in Section 6 by employing two-dimensional finite element analysis (2D-FEA). Finally, the conclusions and some suggestions for future research on this topic will be drawn in Section 7.

2. Wave Energy Applications

The motivation to handle the environmental issues and also the necessity to eradicate the dependence upon fossil fuels have encouraged many researchers to seek clean and free renewable resources of energy [33,34,35]. Among the significant renewable energy resources, including wind and solar energy, wave energy offers the highest power density. Approximately 15 to 20 times more energy per square meter can be harnessed by using wave energy of oceans in comparison with wind energy [36]. In addition, wave energy is more predictable and leaves fewer adverse impacts on the environment. Wave energy extraction systems can generate electricity for approximately 90% of the time, while the availability of other vital resources of renewable energy is 20% to 30% [37]. The distribution of mean power per year that can be harvested from waves is shown in Figure 1.

It can be inferred that there are outstanding opportunities to exploit the energy from oceans; principally, the southern hemisphere is more prosperous in terms of wave energy extracting capabilities. Moreover, the installation of wave energy systems is predicted to reach 100 $GW$ in Europe by 2050, which can supply the electricity required by 76 million households [39]. This shows the tremendous possibilities for many countries to benefit from a clean and free resource of energy. However, there are some regions in the world that are deprived of this kind of energy due to their inaccessibility to oceans.

Research on the prospects of extracting wave energy and the introduction of practical devices to transform the motion of waves into electrical power is increasingly required. Many researchers work on developing novel structures and improving the technology of wave energy conversion from oceans [40,41,42].

2.1. Wave Energy Converters (WECs)

In order to extract the energy from waves, wave energy converters are extensively employed, which can be classified into several categories based on different criteria. In this paper, wave energy conversion systems are classified based on their principles of operation [43,44,45], as follows:

• oscillating water column;
• overtopping converter;
• oscillating body system.

Among the mentioned classifications, four types of oscillating body converters perform based on linear power take-off systems. Oscillating water column and overtopping converters use rotational generators to produce electricity [46]. The study of wave energy converters is beyond the scope of this research. Therefore, two of the most commonly used WECs that can be integrated with linear generators are surveyed, i.e., Archimedes wave swing (AWS) and point absorber converters.

2.1.1. Archimedes Wave Swing WEC

Generally, Archimedes wave swing systems are compromised of a chamber filled with air and a floater attached to the translator of a linear generator. AWS converters function based on the difference in the air pressure of the chamber. The movement of waves provides the translator of a linear generator with reciprocal motion. As the waves pass over the AWS, the floater heaves downward, and the air inside the chamber is compressed. In addition, when the AWS is placed under a trough wave, the air pressure causes the floater to move upward in parallel to the movement of the linear generator. The concept of harvesting wave energy by using AWS converters is depicted in Figure 2.

The AWS is an off-shore converter and is entirely submerged. Hence, AWS converters possess the advantages of being protected from storms, not imposing obstacles for ships, and being rescued from human disturbance. However, the maintenance is complicated, and the installation is demanding [47].

2.1.2. Point Absorber WEC

Linear generators can be driven to produce electricity through the utilization of point absorber converters. Point absorber devices can harvest wave energy from all directions and convert the wave movements into electricity using a linear generator. The simplicity of prototyping and installing point absorber devices has made them promising candidates for wave energy applications [48,49]. Nevertheless, the weak performance of point absorbers in conditions that the frequency of the wave and the frequency of the converter differ leads to the requirement for more advanced controlling mechanisms. Otherwise, the efficiency of the system decreases considerably [50]. The necessity to tune the wave energy converter to be in resonance with the frequency of waves has led to the introduction of two-body point absorbers, in which the active control is not required anymore [51]. Two-body point absorbers extract wave energy from the relative motion between the buoy and the submerged body [52]. The straightforward configurations of single-body and two-body point absorber converters are illustrated in Figure 3. As can be seen, a two-body converter is compromised of a surface body and a submerged body, which are connected to the linear generator. The surface body performs as an energy harvester, and the submerged body creates inertia to assist the converter in having a more resonant frequency with waves.

2.2. Linear Generators

A power take-off (PTO) system is employed in capturing wave energy to transform the mechanical motion into electricity. A PTO system contains a linear generator or a rotational electrical generator connected to a mechanical interface, such as air turbines, hydraulic, hydro turbines, or mechanical direct-drive systems. Air turbines are used to convert the wave energy into the mechanical energy required by the rotational generator, which is primarily employed in oscillating water column systems and operates based on the concept of the oscillating water level. The hydraulic converters perform as mechanical interfaces to drive the rotational generators in wave energy conversion systems through the activation of the hydraulic arm or piston and the pressure increase of the hydraulic oil. Overtopping devices or hydraulic pump systems use hydro turbines to be used in wave energy extraction systems. The mechanical direct-drive systems are the other types of mechanical interfaces to connect the wave energy converter to the electrical generator. The utilization of rotational electrical generators connected to the mechanical interfaces can lead to higher maintenance, lower reliability, and decreased efficiency [53,54].

An efficient strategy to extract wave energy from oceans is adopting direct-drive linear generators, in which the oscillatory motion of waves can be converted into electrical power without any intermediate equipment. The importance of linear structures in extracting wave energy cannot be denied owing to being less complicated, less bulky, and more efficient compared to conventional rotational generators [55,56]. Various kinds of linear generators have been commercialized and widely used in wave energy conversion systems. Correspondingly, a point absorber wave energy converter based on the direct-drive linear PM generator was developed at Uppsala University [57] to provide a less complicated system. In addition, a wave energy harvesting device integrated with a linear PM generator was proposed at Oregon State University [58].

Despite the outstanding merits of linear generators used in wave energy harvesting, they have some weaknesses. The frequency and amplitude of the voltage supplied by the linear generator are not constant during their operation, which complicates the transmission of power [59]. Although linear PM generators offer high power density, they are costly, and PMs are at the risk of demagnetization. In addition, conventional PM synchronous generators possess a bulky size and need a high number of poles to offer a high power density at low speeds. Consequently, linear PM vernier generators can be practical solutions with improved performance; the rest of this research focuses on linear vernier structures.

3. Operation Principles and Analysis of Linear PM Vernier Machines

The concept of linear vernier structures is developed based on rotational machines, as shown in Figure 4.

Accordingly, the linear to motion converters can be eliminated, and the linear movement can be generated by using a linear structure and without additional equipment. Linear PM vernier machines can produce a high power/force density at low speeds, an outstanding feature, particularly for wave energy applications. Linear PM vernier machines function based on the magnetic gearing effect, which makes them distinguishable from linear PM synchronous counterparts. Accordingly, the operation principles of linear vernier structures must be surveyed initially.

3.1. Magnetic Gearing Effect

Mechanical gearbox systems have been used to connect the moving part of electrical machines to the load to transfer speed and torque/force. In recent years, magnetic gears have become more prevailing in several applications [60]. Therefore, mechanical gearbox systems can be superseded by magnetic gears, and the following advantages can be accomplished effectively [61,62,63]:

• no need for maintenance;
• isolation between the output and input shafts;
• inherent overload protection;
• no mechanical vibration and minimum acoustic noise;
• higher efficiency;
• higher reliability;
• no need for lubrication.

A rotational magnetic gear system was firstly introduced in [64], providing high torque capability. Until recently, different structures have been developed from magnetic gears and are used in different applications, including wave energy conversion [65]. A radial flux magnetically geared generator was proposed in [66], designed for wave energy applications. The proposed structure is depicted in Figure 5 and offers a high power capability. Nevertheless, the proposed structure utilizes a huge amount of magnets and has a complicated structure.

Owing to the notable benefits acquired by magnetic gear systems, diverse types of linear magnetic gears have been conceptualized and introduced. For the first time, a linear magnetic gear was proposed by K. Atallah et al. [67], based on the magnetic gearing effect. The proposed linear magnetic gear adopts the tubular structure, offering high thrust force capability. Similar to rotational magnetic gears, a conventional linear magnetic gear consists of a low-speed mover with a higher number of poles, a high-speed mover with a fewer number of poles, and a stationary part (ferromagnetic pole pieces). The ferromagnetic pole pieces modulate the magnetic field of the mover with a different number of poles. Thus, in order to take advantage of the magnetic gearing effect, the number of ferromagnetic pole pieces of the stationary part must be equal to the sum of the number of pole pairs of the moving parts.

A linear magnetic gear system comprises a linear magnetic gear and a conventional linear PM synchronous machine (LPMSM), in which the inner translator of the linear magnetic gear is connected to the translator of the PM machine. The inner translator of the linear magnetic gear can be replaced by the stator of the LPMSM if they have the same number of poles. In the next step, the stator and ferromagnetic pole pieces of the system can be integrated as stationary parts in order to result in the concept of linear vernier structures. Linear vernier machines are introduced based on the concept of the magnetic gearing effect. However, less permanent magnet volume is adopted compared to magnetic gear systems, and vernier structures possess smaller dimensions because of the integration of ferromagnetic pole pieces with the stator teeth. Accordingly, linear PM vernier machines are perceived as excellent options for harvesting wave energy. The stator teeth of a linear vernier machine perform as the modulators and conduct the modulation of the airgap magnetic field of the translator to adapt to the stator magnetic field with different poles. In order to exploit the magnetic gearing effect, the following relationship of slot/pole combination must be satisfied:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill Z_r=Z_s\pm p\end{array}\end{array}$$

(1)

where $Z_r$ is the number of translator pole pairs, p is the number of stator armature winding pole pairs, and $Z_s$ is the number of stator teeth. The electrical speed of the mover of a linear vernier machine equals the effective flux electrical speed; consequently, the following equation can be used to define the mechanical speeds.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill v_{eff}=GR\times v\end{array}\end{array}$$

(2)

The effective flux pitch is also expressed as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\tau }_{eff}=\frac{1}{2}GR\times {\tau }_m\end{array}\end{array}$$

(3)

where $v_{eff}$ is the mechanical speed of the effective flux, v is the mechanical speed of the moving part, ${\tau }_{eff}$ is the pitch of the effective flux, and $\tau$ is the tooth pitch of the mover. Based on the relationships, the mechanical speed of the effective flux is greater than the translator speed, which is resulted from the magnetic gearing effect of vernier machines.

Contrary to conventional PM synchronous machines, linear vernier structures possess two airgap components, i.e., the fundamental component ($Z_r$th order) and the modulated component ($Z_s\pm Z_r$th order). The back-EMF is produced as a result of the interaction between the fundamental component of magnetomotive force (MMF) produced by PMs ($Z_r$th order) and the airgap permeance of stator slots ($Z_s$th order). The reciprocal velocity of the modulated component of $|Z_s-Z_r|$th order is $Z_r/(Z_s-Z_r)$ times greater than the fundamental component. This ratio is known as the gear ratio of vernier machines. The higher the gear ratio, the higher velocity of the modulated component is expected; accordingly, a higher back-EMF can be obtained [68]. Indeed, linear PM vernier machines undergo a significant flux change in a short displacement of the translator owing to the utilization of the magnetic gearing effect, providing a higher thrust force density.

3.2. Analysis and Optimization of Linear PM Vernier Generators

The fundamental relationships and operation principles of linear PM vernier machines are required to be studied to analyze their characteristics. Different methods to investigate vernier structures had been carried out, which failed to explain their characteristics accurately. The design principles of vernier structures were investigated carefully by Kim et al. [69], and a reliable analysis was accomplished for the first time. The analytical analysis of a linear vernier machine was realized in [70] based on flux harmonic theory. In addition, two structures of surface-mounted and consequent-pole LPMVMs were analytically surveyed in [71]. Although the proposed procedure can be accomplished in a short time, the analysis did not consider the magnetic saturation and longitudinal end effect. A nonlinear equivalent magnetic network (EMN) was developed in [72] to survey the performance of a linear PM vernier machine by considering the cogging force and end effect. A novel mesh generation approach was studied in [73] based on equivalent magnetic network modeling to increase the accuracy. Linear vernier structures can be optimized to offer more beneficial performance in different applications. A multi-objective optimization method was carried out in [74] in order to improve the performance of a linear PM vernier machine. In addition, a linear vernier structure with the capability of high fault tolerance was proposed in [75] and optimized to meet the requirement of a multi-objective optimization process.

High leakage flux of linear PM vernier structures is a considerable problem that can deteriorate their ideal operation. Generally, the kinds of leakage flux in a linear PM vernier machine can be divided into tooth-tip leakage flux, air-gap leakage flux, and magnet-end leakage flux for a surface-mounted LPMVM. An analytical procedure was carried out in [76] to survey the tooth-tip leakage flux of the linear vernier structure based on the translator position. In addition, the fringing flux was considered into account. However, many approximations were regarded, decreasing the accuracy of the analysis, such as neglecting the saturation effect and ignoring the longitudinal end effect. Reducing the leakage flux of vernier machines is aimed by many researchers to improve the performance of vernier structures, which will be thoroughly discussed in this paper.

4. Linear Permanent Magnet Vernier Generators (LPMVGs)

Linear PM vernier generators are extensively employed in wave energy systems. In order to provide a comprehensive study and investigate the performance of linear PM vernier structures, they can be classified into the following major categories, as shown in Figure 6:

• linear flat and tubular;
• long/short stator and short/long translator;
• single-sided and double-sided;
• permanent magnet type;
• armature winding.

4.1. Flat and Tubular Structures

Linear PM vernier machines can be divided into two primary classifications of flat and tubular structures, which have been commercialized and used in wave energy applications. Some unconventional types of linear generators have also been proposed but are not efficient choices, such as octagonal linear PM generators [77] or four-sided linear generators [78]. In both linear flat and tubular structures, the stator is the stationary part, and the translator is driven by a wave energy converter to extract the reciprocal motion of waves [79,80].

A hybrid tubular PM vernier generator used in wave energy applications was proposed in [81], in which the armature windings and magnets are located on the stator. The hybrid tubular machine was developed from a linear PM vernier configuration depicted in Figure 7, offering higher flux density compared to the linear flat counterpart. The process of developing the tubular vernier generator from the flat structure is illustrated in Figure 8. Due to the higher ratio of force to weight for the tubular structure, the active volume can be decreased.

A high temperature superconducting dual stator tubular vernier machine (HTS-DSTVM) was proposed in [82], using HTS bulks to improve flux linkage. A two-dimensional cross-section of the tubular linear PM vernier generator is shown in Figure 9. The armature windings are only located on the outer stator, and HTS bulks are inserted inside the teeth of the inner stator. Although the HTS tubular generator offers a high thrust force density, the structure is relatively costly and complicated.

Tubular and flat vernier machines perform based on the same operation principles but have different topologies. Tubular structures are symmetrical around an axis; therefore, the forces are applied from all directions. In addition, their symmetrical structure provides a more uniform distribution of magnetic flux compared to flat counterparts [83,84,85]. However, the complexity of prototyping and analysis of tubular structures is the most deterrent element that has pushed researchers to pay more attention to linear flat machines. Correspondingly, this paper concentrates more on linear flat vernier configurations.

4.2. Long/Short Stator and Short/Long Translator

Another criterion to compartmentalize linear PM vernier machines is based on the structures of the stator and translator, i.e., long stator and short translator or short stator and long translator. The moving part of a linear vernier machine with a short stator and long translator is longer than the stator, in which the preponderance of linear PM vernier machines belongs to this sort. A part of the long stator of a linear vernier machine with a short moving part is not interacting with the translator. Hence, the copper losses increase if the armature windings are located on the stator. An approach is disconnecting the redundant part of the stator with the aid of power electronic converters. However, the system becomes complex, and the costs grow due to the employment of additional equipment. An alternative strategy to avoid these problems and reduce the costs is locating both armature windings and magnets on the short moving part and adopting a reluctance core for the long stationary part [86]. However, using a moving translator equipped with armature windings is also a challenging concern for designers.

4.3. Single-Sided and Double-Sided

Single-sided linear structures have been developed from the rotational concept of electrical machines, possessing only a stator and a moving part [87]. An attraction force exists in the vertical direction of the reciprocation of linear single-sided machines. This force is known as the normal force, which leads to a high thrust force ripple. In order to attenuate the normal force, double-sided structures are excellent choices. In addition, a higher power capability can be obtained by using a double-sided generator instead of the single-sided one [88].

4.4. Permanent Magnet Type

Different types of magnets can be adopted for linear PM vernier generators. As shown in Figure 10, magnet types can be classified into five major categories, and the other configurations are considered as subdivisions:

• surface-mounted type;
• interior type;
• spoke type;
• halbach/Quasi-Halbach arrays;
• V-type.

Figure 10. Different types of magnets: (a) Surface-mounted type, (b) Interior type (c) Spoke type, (d) Quasi-Halbach arrays, and (e) V-type.

Figure 10. Different types of magnets: (a) Surface-mounted type, (b) Interior type (c) Spoke type, (d) Quasi-Halbach arrays, and (e) V-type.

Permanent magnets can be located on the surface of the stator or translator of linear vernier machines, known as the surface-mounted type. The surface-mounted magnets with a simple structure have been widely adopted for different linear vernier machines [89]. Surface-mounted PMs with the sinusoidal shape can be helpful in decreasing the back-EMF distortion [90]. Nevertheless, this type of magnet has not been used in many prototyped structures due to the highly increased cost of using magnets with a unique configuration. Permanent magnets are regarded as one of the most vulnerable and expensive parts of a linear PM vernier machine. The consequent-pole magnet type is considered a practical option to downsize the costs by decreasing the amount of PMs. By exploiting the consequent pole instead of surface-mounted PMs, not only the volume of PMs is considerably reduced, but the leakage flux is also diminished. Figure 11 shows how the leakage flux in conventional surface-mounted and consequent-pole structures are produced.

As can be seen, due to the elimination of PM poles that do not participate in the production of the flux linkage, the leakage flux of the consequent-pole type can be reduced [91]. The leakage flux of the surface-mounted structure (${\varphi }_1$) and the consequent-pole structure (${\varphi }_2$) can be calculated as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\varphi }_1=\frac{F_m}{R_m+\frac{R_{ag1}+R_{ag2}}{2}}\end{array}\end{array}$$

(4)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\varphi }_2=\frac{F_m}{R_m+R_{ag1}+R_{ag2}}\end{array}\end{array}$$

(5)

where $R_m$ is the reluctance of the magnet, $R_{ag1}$ and $R_{ag2}$ are the air-gap reluctance of the slot and tooth, respectively. By comparing the leakage flux of both structures, the following relationship can be concluded, indicating a lower leakage flux for the consequent-pole type:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\varphi }_1>{\varphi }_2\end{array}\end{array}$$

(6)

Moreover, the leakage flux of a consequent-pole type can be further weakened by using the tapered ferromagnetic poles [92]. By optimizing the ratio of the ferromagnetic pole tip to the pole pitch, as shown in Figure 12, the machine performance can be improved.

A linear consequent pole stator permanent magnet vernier machine (LCPSPMVM) with improved force capability was compared with a surface-mounted counterpart in [93]. A 2D schematic of the consequent-pole linear machine can be seen in Figure 13.

The volume of magnets for the consequent-pole machine is decreased by 25% compared to the surface-mounted counterpart with the same dimensions and design characteristics. Despite using less PM volume, thrust force density is increased due to the reduction of the leakage flux. Nevertheless, due to the fact that the consequent-pole and surface-mounted structures have the same reluctance, a low power factor is expected for both types.

Interior magnet types are preferred over surface-mounted magnets to decrease the air gap length. There are different types of interior magnets [94,95,96]. Figure 10b shows a simple configuration of interior magnets with a linear shape. The usage of conventional interior magnets with a linear shape is restricted. In order to improve the flux linkage of linear vernier machines, other configurations are preferred, such as Halbach arrays or V-type magnets.

Spoke-type magnets have been exploited in different structures of linear vernier machines. Particularly, by adopting spoke-type magnets and the double-sided structure with a shift as half of the slot pitch, the flux focusing effect can be obtained. Thus, a higher power capability and power factor are anticipated [97].

In order to increase the flux linkage of linear PM vernier machines, Halbach arrays are excellent choices. A higher force density, self-shielding effect, low cogging force, and more sinusoidal distribution of airgap flux density are the advantages of Halbach magnets [98,99]. However, the construction process of Halbach magnets is complex, and the costs are increased. Accordingly, quasi-Halbach arrays are more profitable candidates. The configuration of quasi-Halbach magnets is shown in Figure 10d.

The V-type magnets can be regarded as a subclassification of interior types. However, V-type is an excellent solution to fulfill the flux focusing effect, which is preferred over the interior magnets with a linear shape. By employing V-type PMs for linear vernier machines, higher airgap flux density can be achieved, and the risk of irreversible demagnetization can be reduced [100]. Accordingly, higher thrust force density and higher power factor can be realized. The angle between two magnets and the dimensions can be optimized to improve the performance of V-type linear vernier structures. Furthermore, the iron bridges of the V-type magnets can be eliminated in order to decrease the leakage flux [101]. The bridged and bridgeless structures are compared in Figure 14.

4.5. Armature Winding

Different types of distributed, concentrated, and toroidal windings can be used for the armature winding of a linear PM vernier machine. The distributed winding offers a more sinusoidal and higher back-EMF. On the other hand, by utilizing armature windings with fractional poles, more variety of slot/pole combinations can be adopted for linear machines contrary to rotational structures. Indeed, 0 and 360 mechanical degree positions do not coincide for a linear configuration, in which this feature uniquely provides linear vernier machines with the possibility to select a non-integer number of poles. A fractional pole-pair linear PM vernier machine (FP-LPMVM) with 18 slots and 3.5 armature pole pairs was proposed in [102]. The topology of the FP-LPMVM and the configuration of the armature winding are demonstrated in Figure 15 and Figure 16, respectively. The proposed linear vernier machine offered a low thrust force ripple of 1.6%, owing to the unique configuration of winding and the combination of slots and poles.

However, the long end-windings cause the linear structure to suffer higher copper losses. Accordingly, concentrated winding for the armature of linear vernier machines is preferred in some cases due to a shorter end-winding and a higher capability of fault tolerance. In addition, the flux modulation poles (FMPs) have been utilized for the stator of a linear machine with concentrated windings in many cases. The modulation by FMPs instead of traditional teeth causes the fundamental flux components to have less interaction with the armature coils, leading to a lower force capability. Moreover, the toroidal winding type can be utilized for the armature winding of a linear PM vernier machine to decrease the length of end windings [103,104]. Not only the copper losses are decreased, but a higher fault-tolerant capability can also be obtained. However, the utilization of toroidal winding is restricted to some specific configurations of linear vernier structures.

5. Performance Improvement of Linear PM Vernier Generators

In order to increase the productivity of linear PM vernier machines in wave energy applications, their performance is required to be improved. As aforementioned, the most outstanding characteristic of vernier machines is their ability to offer high force/power at low speeds. Different configurations have been proposed to further develop the force density of linear PM vernier machines. On the other hand, many studies have concentrated on decreasing the negative sides of a linear PM vernier machine, including the reduction of leakage flux and the decrease of thrust force ripple. In addition, the cost and reliability of linear vernier generators are other concerns that must be taken into account. Overall, the methods to develop the operation of linear vernier generators are classified as:

• thrust force capability improvement;
• power factor development;
• thrust force ripple reduction;
• cost reduction.

5.1. Thrust Force Capability Improvement

The superiority of linear PM vernier machines is providing high thrust force density at low speeds, which is accounted outstanding merit and makes them different from linear PM synchronous machines. The capability of high thrust force of linear PM vernier structures originates from the principles of the magnetic gearing effect, which can be estimated by the following relationship [105]:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill F=\frac{P}{v}=\frac{\sqrt{2}}{2}\frac{k_w{B}_{gmax}{k}_{s}L{L}_{stk}}{k_d}GR\end{array}\end{array}$$

(7)

where P is the electromagnetic power of the linear PM vernier machine, and v is the speed. $k_w$ is the winding factor, $B_{gmax}$ is the maximum flux density, $k_s$ is the electric loading, $k_d$ is the flux leakage coefficient, L is the effective length, $L_{stk}$ is the stack length, and $GR$ is the gear ratio. Equation (7) defines the parameters that directly influence the thrust force of a linear vernier structure. The gear ratio is a decisive parameter in which a higher force density can be obtained by selecting a high value of the gear ratio. In addition, by exploiting different configurations to increase the flux linkage, the force capability of vernier structures can be enhanced effectively [106,107].

A single-sided linear PM vernier machine with increased thrust force performance was proposed in [108], employing consequent-pole and Halbach PMs (CPHPMs). A 2D schematic of the CPHPM linear vernier machine is depicted in Figure 17a. As can be seen, Halbach arrays and armature winding are inserted inside the translator slots, and the stator employs the consequent-pole structure. Halbach arrays are the excitation source, and the stator teeth perform as the modulation pole-pieces. In addition, two other structures similar to CPHPM linear machine were studied, i.e., conventional surface-mounted and consequent-pole LPMVMs, as shown in Figure 17b,c. The proposed CPHPM LPMVM offers an average thrust force of 1.65 kN, while the surface-mounted and consequent-pole LPMVMs have an average force of 1.13 kN and 1.24 kN, respectively. By employing the proposed CPHPM linear vernier machine, thrust force density is improved by 46% and 33% compared to the surface-mounted and consequent-pole linear machines, respectively. However, the power factor of the proposed linear machine is decreased compared to the consequent-pole structure due to higher leakage flux. On the other hand, there are some considerations related to the reliability and the risk of demagnetization of the PMs of the CPHPM linear machine.

5.2. Power Factor Development

The most critical drawback of linear PM vernier machines is their low power factor, which necessitates more consideration for vernier structures. Generally, the power factor of an electrical machine is defined as the ratio of the real power to the reactive power. Owing to the inherent low power factor of linear vernier structures, they require high rating converters to deliver the same real power. Indeed, the low power factor of vernier machines becomes vital when the system is considered from the economic point of view. In recent years, many researchers have proposed different methods and configurations to enhance the performance of LPMVMs by developing their power factor. The power factor of vernier machines can be defined based on a phasor diagram, depicted in Figure 18.

The power factor can be calculated by using the following realtionship [109,110], in which the resistance of the stator is neglected and the d-axis current equals zero:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill cos\theta =\frac{1}{\sqrt{1+\frac{L_{s}I}{{\mathrm{\Psi }}_{PM}}}}\end{array}\end{array}$$

(8)

where $L_s$ is the synchronous inductance, I is the current, and ${\mathrm{\Psi }}_{PM}$ is the flux linkage produced by PMs. Based on the classical method of calculating the power factor, increasing the flux linkage and decreasing the synchronous inductance leads to a higher power factor. The majority of research in this field can be divided into adopting the following approaches:

• optimal magnetic gear ratio;
• appropriate configuration of PMs;
• high-temperature superconducting (HTS) bulks.

5.2.1. Optimal Magnetic Gear Ratio

A principal step in defining the electromagnetic characteristics of linear PM vernier machines is selecting an optimal combination of slots and poles [111]. Different slot/pole combinations can be adopted for vernier structures based on the following expression [112]:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{p\pm Z_r}{GCD(p\pm Z_r,p)}=mk\end{array}\end{array}$$

(9)

where m is the number of phases, k is an integer, and $GCD$ is the greatest common divisor. For a linear PM vernier machine with PMs on the translator, the gear ratio is defined as the ratio of the number of translator PM pole pairs to the number of stator pole pairs.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill GR=\frac{Z_r}{p}\end{array}\end{array}$$

(10)

Overall, higher values of gear ratio can lead to a higher force capability; nevertheless, due to the increment of the leakage flux, particularly the leakage flux resulting from PM fringing, the power factor decreases [113]. So, a practical way to improve the power factor is selecting a low gear ratio, while a considerable reduction of thrust force capability must be avoided. Certainly, a trade-off is required to be taken into account to offer a linear PM vernier machine with improved performance. In addition, a specific gear ratio can be obtained with different combinations when the ratio of the number of translator PM pole pairs and the number of stator pole pairs are constant.

5.2.2. Appropriate Configuration of PMs

An excellent approach to improving the power factor of linear vernier machines is adopting the appropriate configuration of magnets. Different types of magnets have been used in various structures to increase the flux linkage.

A dual-stator spoke-type linear vernier machine (DSSLVM) was proposed in [114]. In order to obtain the flux focusing effect, the stators are not aligned, and a position displacement as half of the slot pitch is considered. Hence, the flux linkage is concentrated to increase the power factor. The configuration of the spoke-type linear machine is depicted in Figure 19. However, the power factor equals 0.40, which is still a low value.

5.2.3. High-Temperature Superconducting (HTS) Bulks

The power factor and power density of linear PM vernier generators can be significantly developed by adopting High-temperature superconducting (HTS) bulks. The relative permeability of HTS bulks is approximately equal to zero for the case that operation temperature does not exceed the critical temperature based on the Meissner effect [115]. Hence, the leakage flux is shielded, which results in the increment of the flux linkage and improving the power factor and power density [116]. A direct current (DC) is applied to the HTS bulks, and the airgap flux density can be controlled by adjusting the current [117]. A linear PM vernier generator fed by an additional DC excitation undergoes two separate flux paths, namely the flux produced by PMs and flux generated by the additional DC field windings. The DC excitation can be regulated to raise the airgap flux density when the flux of PMs and the flux excited by the DC field are aligned in the same direction [118].

In order to further improve the operation of the dual-stator spoke-type linear vernier machine (DSSLVM), shown in Figure 19, a novel structure equipped with high temperature superconducting (HTS) bulks was proposed in [119]. The configuration of the HTS dual-stator spoke-type linear vernier machine (HTS-DSSLVM) is depicted in Figure 20, in which HTS bulks of yttrium boron copper oxide (YBCO) are located inside the flux modulation poles. The HTS-DSSLVM has a power factor of 0.60, which is improved compared to the DSSLVM. However, the equivalent airgap lengths of the HTS LPMVM are not the same owing to inserting HTS bulks inside the FMPs, leading to a higher detent force.

Despite the capability of increasing the power factor by adjusting a DC excitation, some concerns have inevitably arisen. The efficiency is influenced, and the temperature is increased [120]; thus, a cooling system is required [121,122]. Due to the movement of the translator used in wave energy applications, using the cooling system imposes more complexity and increases costs. A soft tube was exploited in [123] to connect the cooling system and pipes in a linear PM vernier generator, possessing the ability to be compressed by the movement of the translator. As a result, the weight increases, which is a negative point, particularly for wave energy extraction systems.

5.3. Thrust Force Ripple Reduction

In order to take advantage of a smooth operation and eradicate the mechanical vibrations and acoustic noise, linear vernier machines with low thrust force ripple are needed. The thrust force ripple can be calculated by using the following relationship:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \%F_{ripple}=\frac{F_{max}-F_{min}}{F_{avg}}\times 100\end{array}\end{array}$$

(11)

where $F_{max}$ is the maximum of thrust force, $F_{min}$ is the minimum of thrust force, and $F_{avg}$ is the average thrust force. The fundamental sources of thrust force ripple in linear PM vernier structures can be itemized as follows [124]:

• linear longitudinal end effect;
• cogging force;
• normal force in the perpendicular direction of reciprocation;
• non-sinusoidal back-EMF.

5.3.1. Linear Longitudinal End Effect

The longitudinal end effect of linear machines leads to a high thrust force ripple. As aforementioned, the linear concept is derived from the rotational structure (as shown in Figure 4), in which two open ends cause the flux passing through the ends not to have a closed magnetic circuit, unlike rotational counterparts. Certainly, the magnetic paths are not symmetric in linear vernier structures. The coils of the end slots generate the flux linkage with different amplitudes compared to other coils. Therefore, the linear structures inherently suffer from the longitudinal end effect, and they are expected to have a higher thrust force ripple [125].

5.3.2. Cogging Force

The undesired cogging force is caused by the slot effect of machines that employ permanent magnets, which can be defined as the tendency of translator PMs and stator teeth to be aligned in a position to have minimum reluctance. The cogging force and end force cause the detent force in linear PM vernier machines, which results in vibration and noise, decreasing the accuracy of positioning, and complicating the characteristics of the controlling system [126].

In order to diminish the adverse impact of detent force, the design characteristics of linear PM vernier generators can be optimized. Another effective method is selecting a proper combination of slots and poles for linear PM vernier generators. The cogging force period can be calculated for linear PM machines and is proportional to the inverse of its fundamental amplitude [127].

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T_{cogging}\propto \frac{1}{LCM(2Z_r,Z_s)}\end{array}\end{array}$$

(12)

where $T_{cogging}$ is the cogging force period, and $LCM$ is the lowest common multiple. Accordingly, it can be concluded that the higher $LCM(2Z_r,Z_s)$ results in a lower cogging force. Furthermore, the type of slots can be altered to offer a low cogging force. However, in linear PM vernier structures, the impact of slot type on the modulation must be considered into account to avoid the attenuation of the magnetic gearing effect.

5.3.3. Normal Force

The destructive normal (attraction) force of single-sided linear PM vernier machines is another substantial element that causes a high force ripple. An applicable method to reduce the normal force is adopting double-sided structures rather than single-sided counterparts. Hence, the force that exists in the vertical direction of reciprocating motion is diminished. In addition, the supporting system of the double-sided linear generator must be designed and assembled delicately to maintain a symmetrical configuration.

5.3.4. Non-Sinusoidal Back-EMF

The design procedure and analysis of electrical machines are carried out based on ideal back-EMF waveforms to obtain a smooth operation. Non-sinusoidal back-EMF is another element that generates a high thrust force ripple of linear vernier machines. Therefore, a control strategy is demanded to feed the linear vernier machines with sinusoidal back-EMF waveforms and avoid a high thrust force ripple. Furthermore, the harmonics of the current must be regarded into account from the viewpoint of the controller [124].

5.4. Cost Reduction

Linear PM machines are rather costly compared to linear structures without magnets. Reducing the cost of linear PM vernier generators used in wave energy applications is a significant issue, particularly for industrial projects on a large scale. Consequently, the cost reduction of linear vernier machines is an exciting subject to propose cost-effective structures.

The cost of energy ($CE$) can be defined as the ratio of the total cost to the total produced energy [128], as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill CE=\frac{TotalCost}{TotalProducedEnergy}\end{array}\end{array}$$

(13)

Total costs of electrical machines generally include initial capital costs and operational costs, and the relationship of the cost of energy ($CE$) can be rewritten:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill CE=\frac{InitialCapitalCosts+OperationalCosts}{Power\times \eta }\end{array}\end{array}$$

(14)

where $\eta$ is the efficiency of the electrical machine. Generally, the operational costs of brushless linear PM structures are less than their initial costs, mainly due to the consumption of PMs.

The cost of a power converter is linearly proportional to the kVA rating, which can be included in the following expression:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill CE=\frac{A\times PMVolume+B\times 1/powerfactor}{\eta }\end{array}\end{array}$$

(15)

where A is the coefficient defined by the volume of consumed permanent magnets, and B is the cost coefficient related to the power converter. Therefore, an effective approach to decrease the costs is consuming less volume of PMs [129]. By employing hybrid configurations of linear vernier machines, the costs can be decreased, in which the magnets and armature windings can be adopted for the shorter side. Moreover, the cost of energy is inversely proportional to the power factor and efficiency of the linear vernier machine. In other words, by improving the power factor and efficiency of the linear vernier structure, the costs can be reduced [128].

In order to present a comprehensive comparison among different linear vernier machines, the classifications and outstanding merits and demerits of the linear generators are compared in

Table 1. A general comparison of different linear PM vernier generators.

Reference	Characteristics	Advantages	Disadvantages
Xiao et al. [26]	Flat Long translator Double-sided Surface-mounted Distributed	High force capability Decreased flux leakage owing to utilization of HTS bulks Low thrust force ripple	Complicated structure Requirement of cooling system
Botha et al. [71]	Flat Long translator (mover) Single-sided Surface-mounted/ Consequent-pole Concentrated	Study of airgap flux density analytical model A quick analysis approach	Ignoring saturation effect Not modeling end effect
Baker et al. [81]	Tubular Long translator Single-sided Surface-mounted Concentrated	High force capability Improved power factor Mass saving	Difficulty of prototyping Still a low power factor
Baloch et al. [82]	Tubular Long translator Double-sided Spoke-type Distributed	High force capability Improved flux linkage Improved power factor	Still a low power factor Complicated structure Uneconomical structure
Almoraya et al. [92]	Flat Long translator Double-sided Consequent-pole Concentrated	High force capability Low thrust force ripple Cost effective	Estimated low power factor
Shi et al. [102]	Flat Long stator Single-sided Consequent-pole Fractional pole	High force capability Low thrust force ripple High power factor	Low fault tolerance Uneconomical structure due to PMs on the long side
Zhang et al. [104]	Flat Long stator Double-sided Surface-mounted Toroidal	High force capability	Uneconomical structure High thrust force ripple
Khaliq et al. [114]	Flat Long translator Double-sided Spoke-type Distributed	High force capability Improved flux linkage owing to adopting flux focusing effect Low thrust force ripple	Low power factor
Baloch et al. [119]	Flat Long translator Double-sided Spoke-type Distributed	High force capability Improved power factor	High thrust force ripple Uneconomical structure
Ching et al. [120]	Flat Long stator Single-sided Surface-mounted Concentrated	High force capability Improved power factor	High copper losses Complicated structure
Du et al. [123]	Flat Long translator Double-sided Surface-mounted Distributed	High power density Low cogging force Low thrust force ripple	Requirement of cooling system Uneconomical structure

. Overall, the most considerable advantage of linear vernier generators is their high force capability. The double-sided structure is preferred in some configurations to further increase the force capability. Moreover, using the structures with the flux focusing effect is also a practical approach to obtaining a high force density at low speeds. Nevertheless, the low power factor is a critical issue in vernier configurations, regarded as their most significant downside. Even though practical methods are proposed in [81,82,114] to increase their performance, the power factor values are reported as 0.44, 0.5, and 0.40, respectively. On the other hand, the power factor of a linear vernier machine was reported as high as 0.92 in [102], while the force density is also increased. However, there are still opportunities to improve the power factor and further increase the force density of vernier machines.

6. Analysis and Comparison of Different Linear PM Vernier Generators

In this section, some of the most functional structures of linear PM vernier generators used in wave energy conversion systems are compared to provide a comprehensive investigation. Two-dimensional finite element analysis (2D-FEA) is employed to analyze the electromagnetic behavior of four different linear PM vernier generators based on a system of Maxwell’s equations. The outstanding linear vernier machines are selected as case studies to examine their capabilities to be used in wave energy applications. In order to define the boundary conditions, the magnetic vector potential of the outer region is set to be zero based on the Dirichlet boundary conditions. In addition, the non-linear characteristics of the translator and stator cores, the longitudinal end effect, and the leakage flux are considered into account, and the eddy current of permanent magnets is neglected.

A direct-drive linear primary PM vernier machine (LPPMVM) was proposed in [130], designed for wave energy applications. The LPPMVM exploits the single-sided structure, and both armature windings and PMs are located on the stator; therefore, the proposed structure is more robust and thermally stable. The configuration of the single-sided generator is depicted in Figure 21.

The structure and operation principles of the single-sided generator have been studied in [130]. In addition, by considering the saturation effect, a magnetic equivalent circuit (MEC) method was proposed in [131].

The slot/pole combination of the single-sided LPPMVM is selected based on (17) to obtain the magnetic gearing effect. The single-sided linear machine possesses 6 primary teeth, and 5 PMs are located on each of the stator teeth. In addition, the armature winding is distributed to form 2 poles, and the proposed structure has a gear ratio of 17. As aforementioned, the performance of vernier structures is influenced by the gear ratio, in which a high force density and a low power factor can be estimated for the LPPMVM compared to similar structures with lower gear ratios. The most considerable advantage of the single-sided LPPMVM is offering a high thrust force density of 1.61 kN at the speed of 1 m/s. The thrust force density and the average thrust force per PM volume are 283 kN/m${}^3$ and 13.4 N/cm${}^3$, respectively.

Different structures have been developed based on the LPPMVM proposed in [130]. A linear stator spoke-type permanent magnet vernier machine (LSSPMVM) was proposed in [132] with similar topology and identical operation principles as LPPMVM. In order to decrease the leakage flux of the linear generator, surface-mounted PMs are replaced by spoke-type magnets, and a non-magnetic space is inserted between magnets and stator. A 2D schematic of the LSSPMVM with non-magnetic space is depicted in Figure 22.

The magnetic flux distributions of the LSSPMVM with and without non-magnetic space are illustrated in Figure 23. As can be seen, adopting the non-magnetic space is an effective approach to decreasing the leakage flux, and the LSSPMVM with non-magnetic space has a lower amount of the leakage flux.

Figure 24 compares the flux linkage waveforms of LPPMVM and LSSPMVM with non-magnetic space, showing higher values for the LSSPMVM. The maximum amplitude of the flux linkage is improved by approximately 58% for the case of using spoke-type magnets and adopting non-magnetic space.

Airgap flux density and harmonic spectra of the LPPMVM and LSSPMVM are illustrated in Figure 25. The amplitude of the 18th harmonic order is 0.79 T for the LSSPMVM and 0.68 T for the LPPMVM, which is improved by 16%. Adopting the non-magnetic space has resulted in the improvement of the airgap flux density. On the other hand, although the LSSPMVM has a higher volume of PMs than LPPMVM, the effectiveness of using the non-magnetic space was validated in [132] in comparison with a similar structure using the same volume of PMs.

The LSSPMVM with non-magnetic space offers an average thrust force of 2.18 kN, and the thrust force density equals 327 kN/m${}^3$. By exploiting spoke-type PMs and the non-magnetic space, thrust force density is improved by 15.5% compared to the surface-mounted LPPMVM. Furthermore, both structures have low values of detent force, lower than 2% of the average thrust force. However, the LPPMVM and LSSPMVM have low power factor values of 0.26 and 0.27, respectively.

Two topologies of linear PM vernier generators employing inset magnet consequent pole (IMCP) and V-shaped consequent pole (VCP) structures have been proposed in [133]. The configurations of linear vernier generators are depicted in Figure 26. As can be seen, the consequent pole structures are preferred for both linear machines with different magnet types to decrease the leakage flux.

The IMCP vernier generator possesses a double-sided E-core stator, and each stator tooth adopts three magnets. The proposed IMCP machine has 9 PM pole pairs, 1 armature winding pole pair, and 10 translator teeth. Accordingly, the magnetic gearing effect can be obtained, and a high force capability is estimated for the linear vernier structure. The flux distribution of the IMCP vernier generator at different positions of the translator is shown in Figure 27. As can be seen, a short displacement of the translator provides a considerable change in the flux distribution, resulting from the magnetic gearing effect. Hence, the linear vernier machine has a high thrust force density.

The V-type structure is another practical choice for obtaining the flux focusing effect and increasing the flux linkage. As shown in Figure 26b, the iron bridges of the V-type configuration are eliminated in order to further decrease the leakage flux. Average thrust force values of IMCP and V-type linear vernier generators are reported as 739 N and 812 N, respectively. In addition, the thrust force density is 354 kN/m${}^3$ for the IMCP machine and 389 kN/m${}^3$ for the VCP machine. The linear vernier generator with the VCP structure has 10% higher thrust force density than the IMCP counterpart. Furthermore, the linear generator with VCP magnets has a higher power factor of 0.65 compared to 0.51 for the IMCP counterpart. Figure 28 shows the flux linkage waveforms of the IMCP and VCP LPMVMs. The maximum flux linkage values of phase A are reported as 0.14 Wb and 0.18 Wb for the IMCP and VCP machines, respectively. The higher flux linkage of the VCP linear vernier structure results from the unique configuration of V-type magnets and the flux focusing effect.

The detent force of IMCP and VCP linear machines are compared in Figure 29. The peak to peak values are reported as 59 N and 48 N for the IMCP and VCP LPMVMs, respectively. The linear structure with V-type magnets has a lower detent force and excels the IMCP machine in terms of the low force ripple.

The temperature rise of different parts affects the performance and lifetime of the linear generators. Consequently, thermal analysis is regarded as one of the principal steps in the evaluation procedure of linear machines used in wave energy systems. The finite element analysis thermal modeling is exploited to thermally analyze the linear machines and provide a reliable analysis. The ambient temperature is assumed to be 23 °C (296.15 K) to provide an accurate operational condition. In addition, the thermal properties of the materials are presented in

Table 2. Thermal characteristics of the materials [134].

Material	Thermal Conductivity [W/(m$\mathit{·}$K)]	Specific Heat Capacity [J/(kg$\mathit{·}$K)]	Density [kg/m${}^3$]
Steel	30	460	7650
Magnet	7.6	460	7500
Copper	401	385	8933

.

Once the electromagnetic behavior analysis of the linear generators is realized, the thermal field analysis can be performed. The core and copper losses of linear PM vernier machines are regarded as heat sources. The copper loss model is obtained based on the resistive heating, and the equation of the model is assumed to be:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill P_{cu}=\frac{1}{T}{\int }_{T_{end}}^{T_{end}-T}J·Edt\end{array}\end{array}$$

(16)

where T is the electrical period, $T_{end}$ is the end time, J is the current density, and E is the electric field. Furthermore, the Bertotti loss model is exploited to accurately characterize the hysteresis, eddy current, and excess losses in linear vernier generators. The Bertotti loss model can be defined as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill P_{fe}=k_{h}f{B}_{gmax}^{\beta }+k_c{f}^2{B}_{gmax}^2+k_e{f}^{1.5}{B}_{gmax}^{1.5}\end{array}\end{array}$$

(17)

where $k_h$ is the hysteresis loss coefficient, $\beta$ is Steinmetz constant, $k_c$ is the eddy current loss coefficient, and $k_e$ is the excess loss coefficient [135,136,137]. The losses are calculated through the utilization of finite element analysis. The LSSPMVM possesses the maximum core loss, reported as 61.5 W. The core losses of LPPMVM, IMCP LPMVM, and VCP LPMVM are 44.4 W, 18.4 W, and 22.0 W, respectively. The volumetric loss density for the linear PM vernier generators is shown in Figure 30.

The heat sources lead to the temperature increment of linear machines; therefore, the losses calculated in the electromagnetic analysis are used to survey the thermal behavior. Defining heat transfer mechanisms is an important stage in thermal analysis. Conduction and convection mechanisms are applied for the thermal analysis of the linear machines in this section, while the radiation is neglected. Heat transfer coefficients (HTCs) have been calculated through the use of empirical expressions to accurately model the thermal behavior. The linear machines operate under natural cooling convection. Finally, a commercial software package based on finite element analysis is exploited to estimate the temperature rise of different parts by applying the heat sources and boundary conditions. The process ends when the simulation is converged. The temperature rise of the stators is shown in Figure 31 during 6 h of the simulation process. The LSSPMVM undergoes the maximum temperature rise of the stator among the case studies, in which the stator temperature rises to reach the maximum temperature of 347 K. The maximum temperature is below the Curie temperature of magnets, and the temperature rise can not significantly influence the performance of the linear machines. On the other hand, the IMCP LPMVM has the minimum temperature rise owing to the lower losses.

The IMCP and VCP LPMVMs are more advantageous than LPPMVM and LSSPMVM in terms of higher efficiencies. The efficiency values are reported as 84.8%, 87.8 %, 91.6%, and 92.0% for LPPMVM, LSSPMVM, IMCP LPMVM, and VCP LPMVM, respectively. The information on copper losses, core losses, and efficiencies of the linear generators are briefed in

Table 3. Characteristics of different linear PM vernier generators.

	LPPMVM [130]	LSSPMVM [132]	IMCP LPMVM [133]	VCP LPMVM [133]
Structure	Flat	Flat	Flat	Flat
Single/double-sided	Single-sided	Single-sided	Double-sided	Double-sided
Armature winding type	Distributed	Distributed	Concentrated	Concentrated
PM type	Surface-mounted	Spoke-type	Consequent pole	V-type
Location of PMs	Stator	Stator	Stator	Stator
No. of PM pole pairs	18	18	9	9
No. of winding pole pairs	1	1	1	1
No. of translator teeth	17	17	10	10
Gear ratio	17	17	10	10
No. of PMs per stator tooth	5	5	3	6
Frequency [Hz]	50	50	50	50
No. of phases	3	3	3	3
PM volume [cm${}^3$]	120	150	64.8	64.8
Magnet remanence [T]	1.2	1.2	1.24	1.24
Active length [mm]	360	360	232	232
Thickness of linear machine [mm]	158	185	180	180
Stack length [mm]	100	100	50	50
Translator tooth pitch [mm]	21.17	21.17	24	24
Stator tooth pitch [mm]	60	60	80	80
Airgap length [mm]	1	1	1	1
Rated speed [m/s]	1	1	1.2	1.2
No. of winding turns per phase	142	140	90	90
Current density [A/mm${}^2$]	4.3	4.3	3.5	3.5
No-load back-EMF [V]	60	87	45	57
Flux linkage [Wb]	0.19	0.30	0.14	0.18
Average thrust force [kN]	1.61	2.18	0.739	0.812
Thrust force density [kN/m${}^3$]	283	327	354	389
Force/PM volume [N/cm${}^3$]	13.4	14.5	11.1	12.5
Power factor	0.26	0.27	0.51	0.65
Copper loss [W]	243.5	240.1	62.6	62.6
Core loss [W]	44.4	61.5	18.4	22.0
Efficiency [%]	84.8	87.8	91.6	92.0
Detent force pk2pk [N]	31 (1.9%)	36 (1.65%)	59 (7.9%)	48 (5.9 %)
Thrust force pk2pk [N]	50	93	66	51
Thrust force ripple [%]	3.1	4.2	8.9	6.2

.

To conclude, the thrust force of four linear generators is illustrated in Figure 32 to make a comparison based on their force capabilities. However, due to the different dimensions and features of electrical machines, average force per volume is regarded as a relatively logical indicator to compare the force capabilities. Thrust force density of LPPMVM, LSSPMVM, IMCP LPMVM, and VCP LPMVM are reported 283 kN/m${}^3$, 327 kN/m${}^3$, 354 kN/m${}^3$, and 389 kN/m${}^3$, respectively. The VCP linear vernier machine offers the highest force density and also the highest power factor. The power factor of the VCP LPMVM is 0.65, which is improved compared to the IMCP counterpart owing to the flux focusing effect. Another significant criterion is the effective usage of PMs, in which the average force per PM volume of LSSPMVM equals 14.5 N/cm${}^3$, known as the highest value among the four machines. Moreover, the linear machines have low detent force and force ripple values. The LPPMVM has the minimum thrust force ripple, reported as 3.1%. A fair comparison can be carried out by considering different elements that influence the performance of linear machines. The characteristics of the LPPMVM, LSSPMVM with non-magnetic space, IMCP LPMVM, and VCP LPMVM are tabulated in Table 3. Generally, vernier structures are suitable candidates for wave energy applications due to their high force capability. Moreover, their low power factor can be enhanced by using practical strategies.

The results obtained from FEA show a negligible difference between the simulation results and experimental results obtained from the main references. The difference arises inevitably from using FEA software packages, applying operational conditions, the meshing, and other elements considered by the designers. The analysis in this paper validates the characteristics of the linear vernier machines and confirms the advantages of the proposed structures for wave energy harvesting systems.

7. Conclusions and Outlooks

Linear PM vernier machines are excellent choices to offer high power/force densities at low speeds. Consequently, linear vernier generators are practical candidates for wave energy applications. In this paper, a comprehensive study on different linear vernier structures has been carried out. Different linear vernier machines have been categorized, and their operation principles are evaluated and compared with other structures. Furthermore, different approaches to improve the performance of linear vernier structures are classified and surveyed. Finally, some practical linear vernier generators have been selected to analyze their performance using 2D-FEA. The results validated the outstanding merit of the high force density of vernier machines, making them beneficial options for harvesting wave energy.

Overall, linear vernier structures investigated as case studies offer a high force density. The LPPMVM is an excellent candidate for wave energy systems in terms of offering the high capability of thrust force. By adopting the non-magnetic space, the LSSPMVM is proposed to decrease the leakage flux. Accordingly, the thrust force density is improved by 15.5% compared to the LPPMVM. However, both structures have a low power factor, necessitating high rating converters. The IMCP and VCP linear vernier machines have been studied by using 2D-FEA, possessing higher values of power factor and thrust force density. The VCP LPMVM provides 10% and 19% higher force density in comparison with the IMCP machine and LSSPMVM, respectively. Even though the VCP machine has a higher force ripple than the LPPMVM and LSSPMVM, the advantages of high force density and higher power factor outweigh its relatively higher force ripple. The thrust force ripple values of LPPMVM and LSSPMVM are lower than the VCP linear generator; however, the difference cannot make a considerable change. Consequently, the VCP structure can be regarded as a functional configuration to be employed in wave energy systems.

Also, the following outlooks can be conducted in order to further improve the performance of linear PM vernier machines:

• The most considerable disadvantage of linear PM vernier machines is their poor power factor. Innovative techniques are required to further develop the low power factor of linear vernier structures.
• Due to the time-consuming process of analyzing linear PM vernier generators based on FEA, accurate analytical methods are desired to provide analysis models in a short time.
• One of the most important criteria that must be considered into account is the cost of linear PM vernier generators, which are required to be declined by offering more economically viable structures and the decrease of the volume of magnets.
• The reduction of the weight of linear PM vernier generators can be realized by reducing the active material. There are opportunities to employ more lightweight linear generators for wave energy harvesting systems.
• Unconventional topologies can be introduced to improve the performance of linear vernier generators and facilitate their utilization in wave energy applications.
• The unwanted longitudinal end effect of linear PM vernier generators imposes a disadvantageous impact on the machine performance, which is desired to be diminished.
• Thermal analysis of linear PM vernier generators is another interesting subject in which not enough research has been accomplished so far.
• The magnetic gear ratio significantly affects the performance of linear PM vernier generators used in wave energy applications. Analytical and numerical methods are required to investigate the optimal gear ratio values.
• The configuration and number of flux modulation poles can have an impact on the gear ratio and the machine performance, which can be surveyed for linear vernier structures.
• A system-level optimization process is needed to improve the efficiency and performance of a linear vernier generator used in wave energy applications.
• Linear PM vernier machines utilize more number of PMs compared to linear PM synchronous machines; thus, the study on the possibilities to avoid irreversible demagnetization can be very useful.