Design and Analysis of 15 MW SPM Vernier Generator for Direct-Drive Wind Turbine Applications

: This paper presents the design and an analysis of a surface PM vernier generator (SPMVG) for MW-scale direct-drive (DD) wind turbine application. An SPMVG has the advantage of higher torque density; however, especially at higher power ratings with increased electrical loadings, the power factor worsens and there are some serious concerns including magnetic saturation of cores and PM demagnetization. These issues are directly related to machine design parameters such as PM dimensions, applied electrical loading, slot geometry and the choice of slot–pole combination. It is determined that depending on the PM thickness and a few other design variables, each slot–pole combination has an optimal value of speciﬁc electrical loading. The use of the optimal value of speciﬁc electrical loading ensures that the machine is not saturated, the performance is optimum and the power factor is not unnecessarily degraded. Moreover, under certain design constraints, design criteria are developed that ensure the proper choice of various entailed design variables. By using the developed design criteria, the trends of various electromagnetic performances with variation in the slot–pole combination are discussed. The obtained trends clearly show that each slot–pole combination offers a certain torque density and power factor; thus, it serves as a guide for the selection of the slot–pole combination considering the required torque density and/or certain power factor limit. Finally, by using the developed design approach, an SPMVG for rated power of 15 MW is designed; the design objectives are to maximize torque per volume with a power factor limit of 0.4. Moreover, the various aspects of the performances of the designed SPMVG are comprehensively compared against a conventional PM DD 15 MW generator.


Introduction
Over the last decade, research and development related to the wind energy has grown very rapidly; in particular, research trends have been focused on multi-MW direct-drive (DD) offshore wind turbines [1][2][3].The main motivation behind the upscaling of wind turbines is to lower the levelized cost of energy (LCOE).According to the Global Wind Energy Council 2022 report, 21.1 GW was added in the offshore market in 2021, three times more than in 2020, showing year-over-year growth of 58% [1].Considering the reliability and energy yield, DD topology is preferred over the traditional geared wind turbine system [4].However, direct-drive generators, due to the high torque requirements, suffer from large volume and higher material consumption which results in higher cost.To overcome these issues, it is necessary to use higher torque/power density generators.
Flux modulation machines (FMMs) owing to the flux modulation effects have higher torque densities and thus FMMs are naturally suitable for direct-drive applications [5][6][7][8][9][10][11][12][13][14][15].Among the family of FMMs, the surface PM (SPM) vernier machine is famous due to its higher torque density, higher torque quality and simple mechanical structure [5,6].Aside from the advantages, the low power factor of the SPM vernier machine, getting significantly lower especially at higher power ratings (MW scale), remains a challenge and there are some serious concerns including magnetic saturation and PM demagnetization [11,13].
The SPM vernier machine is magnetically identical to a conventional PM machine with an integrated magnetic gear where the degree of the magnetic gearing effect is defined by the term gear ratio that is a function of the machine's slot-pole combination [7].Due to the unique operation principle of the SPM vernier machine, it has certain design constraints that are not that relevant for regular PM machines [5].From that point of view, the conventional design procedure based on the pre-set magnetic and electric loadings is not proper for flux modulation machines [7].
In the literature, the SPM vernier machine, due to its high torque density, has been discussed with regard to the application of direct-drive wind generators with power ratings up to a few kilowatts [7,9,12].On the other hand, in recent years, the SPM vernier machine with an MW-scale power range has been considered in several studies [11,[13][14][15].In [11], the impact of scaling (from 3 kW to 3 MW) on the electromagnetic performance of the SPM vernier machine was investigated; it was indicated that the magnetic saturation especially at low slot-pole combinations is serious and of course there is a problem with regard to a poor power factor; these issues fundamentally arise due to the use of higher electrical loading [11]; similar issues concerning the SPM vernier machine are reported in [13,14].Two SPM vernier models-with 1 and 3 MW-are optimally designed in [15] and the authors concluded that, considering the power factor and the total machine mass, an SPM vernier machine with lower gear ratios is more promising.
In previous studies related to the SPMVG for MW-scale wind turbine application, the design of the generator is largely based on numerical optimization, which greatly limits the extent of the insights into the nature of the design problem.Moreover, considering the numerous slot-pole combinations and the various entailed design parameters, it is not proper to utilize the numerical optimization methods at the pre-design stages.The fundamental issues of SPMVG that become more serious at higher power levels are related to the design parameters and the most important parameter is the specific electrical loading.It will be shown in this study that each slot-pole combination depending on certain design parameters has its own specific electrical loading value; unfortunately, the present literature has neglected addressing the matter from that point of view.In addition, with regard to the proper selection of various important design parameters, the general trends of the resulting electromagnetic performances have not yet been discussed and the scope of most studies is limited to a specific gear ratio.Therefore, in order to analyze the feasibility of SPMVG with regard to the MW scale, first it is important to establish a proper and systematic design procedure that clearly defines the ways to properly choose the various entailed design parameters.
With the aim of evaluating the feasibility of surface PM vernier generators (SPMVGs) for MW-scale wind turbine applications, this paper presents the design and an analysis of an outer rotor SPMVG.The fact that the SPMVG has higher torque density makes it suitable for DD applications; however, concerns such as its worse power factor and the magnetic saturation of cores that are more prominent at higher power ratings (MW scale) should be carefully considered.The severity of these issues depends on certain design parameters, including slot-pole combination, PM dimensions and the applied specific electric loading.In order to address these issues, the basic operation principle and the involved design variables are introduced, and the relations between important design aspects such as the selection of slot-pole combination, the PM dimensions and the choice of specific electrical loading are discussed.Based on certain design constraints, proper design criteria are developed ensuring the proper selection of various entailed design parameters.The various trends of key electromagnetic performance with variation of slot-pole combination are investigated.Finally, based on the proposed design criteria, an SPMVG for a rated power of 15 MW is designed under the design objectives of maximum torque density while having a power factor limit of 0.4.To clearly highlight the pros and cons of SPMVG, the Energies 2023, 16, 1094 3 of 14 various aspects of the electromagnetic performances are comprehensively investigated and compared against a 15 MW DD SPM generator developed by NREL [3].

Basic Operation Principle and Analytical Modeling
Like all flux modulation machines, a SPM vernier machine also works on the principle of magnetic gearing effect, alternatively called the flux modulation effect [5].In order to utilize the flux modulation effect, the stator windings should be configured to obtain a 3-phase balanced back EMF E ph from the modulation flux whose number of pole pairs p − mod is given as Z s -Z r , where Z r and Z s represent the number of PM pole pairs and stator slots, respectively.The ratio of Z r to p − mod is defined as gear ratio G r , and the choice of both p − mod and G r are crucial in order to effectively utilize the flux modulation effects.
The basic structure of SPMVG in outer rotor configuration along with the important design variables is shown in Figure 1.The stator of the structure shown in Figure 1 is installed with the most common fractional slot winding configuration having the ratio of 3 slots/2 poles, where the number of poles means the modulation poles, not the actual PM poles.Moreover, the stator of the machine has Q s main slots for the windings and each slot has auxiliary teeth n aux , and therefore the total number of slots is Z s = n aux Q s .
Energies 2023, 16, x FOR PEER REVIEW 3 of 15 SPMVG for a rated power of 15 MW is designed under the design objectives of maximum torque density while having a power factor limit of 0.4.To clearly highlight the pros and cons of SPMVG, the various aspects of the electromagnetic performances are comprehensively investigated and compared against a 15 MW DD SPM generator developed by NREL [3].

Basic Operation Principle and Analytical Modeling
Like all flux modulation machines, a SPM vernier machine also works on the principle of magnetic gearing effect, alternatively called the flux modulation effect [5].In order to utilize the flux modulation effect, the stator windings should be configured to obtain a 3-phase balanced back EMF Eph from the modulation flux whose number of pole pairs  is given as Zs -Zr, where Zr and Zs represent the number of PM pole pairs and stator slots, respectively.The ratio of Zr to  is defined as gear ratio Gr, and the choice of both  and Gr are crucial in order to effectively utilize the flux modulation effects.The basic structure of SPMVG in outer rotor configuration along with the important design variables is shown in Figure 1.The stator of the structure shown in Figure 1 is installed with the most common fractional slot winding configuration having the ratio of 3 slots/2 poles, where the number of poles means the modulation poles, not the actual PM poles.Moreover, the stator of the machine has Qs main slots for the windings and each slot has auxiliary teeth naux, and therefore the total number of slots is Zs = nauxQs.

Air-Gap Flux Density and Back EMF
For the structure shown in Figure 1, the air-gap magneto-motive force (MMF) developed by the PMs and the specific air-gap permeance can be expressed as ( 1) and (2), respectively, where Fm1 is the magnitude of the 1st component of the air-gap MMF [5] due to the PMs.In (2), the average permeance P0 and the first harmonic permeance P1 are expressed in terms of the air-gap geometries as (3) and (4), respectively, where geff is the effective air-gap length, c0 is the ratio of slot opening o to slot pitch td, and  is given in (5).

Air-Gap Flux Density and Back EMF
For the structure shown in Figure 1, the air-gap magneto-motive force (MMF) developed by the PMs and the specific air-gap permeance can be expressed as (1) and (2), respectively, where F m1 is the magnitude of the 1st component of the air-gap MMF [5] due to the PMs.In (2), the average permeance P 0 and the first harmonic permeance P 1 are expressed in terms of the air-gap geometries as (3) and (4), respectively, where g eff is the effective air-gap length, c 0 is the ratio of slot opening o to slot pitch t d , and β is given in (5).
By using the air-gap MMF of (1) and air-gap permeance of (2), the air-gap flux density due to the PM can be expressed as (6), where the main flux B 0 = F m1 P 0 and the modulation flux B 1 = 0.5F m1 P 1 , p ± mod = |Z s ± Z r |.The third term in (3) with p + mod has negligible contribution due to its very low speed; however, here, in this study, for the sake of accurate modeling, p + mod is not neglected.As the magnitude of the modulation flux B 1 is related with the F m1 and the harmonic permeance P 1 and both P 1 and F m1 are linked with the PM thickness and specifically as given by ( 4), P 1 also depends on the choice of stator slot opening ratio c 0 and the value of β.Therefore, to effectively utilize the modulation flux, the choice of PM thickness g m and c 0 is important.Considering the magnitude of P 1 the optimal value of stator slot opening ratio c 0 lies somewhere around 0.5~0.6 [5], in this study, the value of c 0 is set as 0.55.
Considering the winding span of the fractional slot winding configuration and by using the Faraday's law with (6), the back EMF of SPM vernier machine can be obtained as (7), where ω m is the mechanical angular speed of the rotor, N ph is the number of turns per phase, D g is the air-gap diameter, l stk is the stack length of the machine, k w1 is the fundamental winding factor, G ± r is the gear ratio G ± r = Z r /p ± mod and k leak is the leakage factor that is a function of the PM thickness and PM pole pitch.

Specfic Electric Loading and Torque Density
The specific electric loading, alternatively called surface current density, is commonly represented by the symbol K s , and it can be expressed as (8), where m is the number of phases and I ph is the RMS value of the phase current.The choice of K s has significant effect on the key electromagnetic performances of a machine, i.e., power factor, torque density, magnetic saturation.While designing an electric machine, it is a very common practice to decide the value of surface current density empirically [7], but it is not suitable especially for the flux modulation machines.It will be shown in the upcoming sections that depending on the design parameters the optimal value of K s varies with the slot-pole combination of the machine.
Under the condition that the applied current is in phase with the back EMF of (7), the machine produces the maximum torque, and thus the torque per rotor-volume (in case of outer rotor, air-gap volume), TRV is obtained as given by (9).From (9), it is clear that for a given winding configuration and magnet material having specific B r , the TRV of SPMVG is proportional to the available surface current density K s , the gear ratio of the machine and the choice of slot-pole combination which affects the value of g eff and β.

Flux Densities of Teeth and Back Yoke
To avoid the saturation problem, the flux density in the iron core parts should be kept constant.In order to estimate the flux density levels in the iron core parts of the machine in operation, it is important to consider both the MMF sources, i.e., the PMs and the stator windings which are configured to utilize the modulation flux having pole p − mod pairs.Under the condition of maximum torque per ampere, the PM MMF F m and the stator windings MMF F w will be orthogonal to each other; thus, the net air-gap MMF F g can be expressed as (10), where F w1 is expressed as (11) having the term of surface current density K s .
Energies 2023, 16, 1094 5 of 14 By using the net air-gap MMF of ( 10) with the air-gap permeance of (2), the net air-gap flux density B g.net can be obtained as where . By integrating B g.net of ( 12) over one winding pole area and a tooth area individually, both the back yoke flux and tooth flux are obtained as ( 13) and ( 14), respectively, where First, it should be noted that both the main and the modulation fluxes contribute to the flux density in the core parts.Depending on the required flux density levels in the stator/rotor yokes, the heights of stator and rotor yoke can be determined by using (13).
On the other hand, regarding the tooth flux density, as mentioned before, considering the flux modulation effects, the stator slot opening ratio c 0 of SPMVG is generally fixed with a value of 0.5~0.6 at that time the available tooth area is limited; therefore, the tooth saturation problem is tricky to avoid.Using (14), the tooth flux density B teeth can be derived and approximated as (15) since the difference of α 1 and α 2 is normally less than π /3.
From (15), it is obvious that for a given slot-pole combination (that is, Z s and Z r ) with a fixed c 0 , the tooth flux density is determined by B conv and B ver which depend on F m1 linked with the PM thickness and F w1 related with the current density K s .Because the PM thickness is determined to obtain the maximum back EMF, the teeth saturation can be avoided by careful selection of the surface current density K s .
In machine design, the maximum tooth flux density is commonly set as 1.8T, but it should be noted that the flux density B teeth of ( 15) is composed of two fluxes with different wavelengths.To demonstrate the peculiar nature of vernier machine, the conceptual flux density waveforms of regular PM machine with Z r pole pairs and vernier PM machine with 5Z r pole pairs is shown in Figure 2. If the saturation of iron core is assumed to begin from 1.5T and if both the machines are designed to obtain 1.8T of the tooth flux density, the saturated core area of vernier PM machine is comparatively much smaller than that of the conventional PM machine as depicted in Figure 2. Therefore, when using (15), the flux density can be higher than 1.8 for design of the vernier PM machine.

Inductance and Power Factor
The power factor of a PM machine can be expressed as ( 16), where Xs represents the synchronous reactance of the machine.As given by ( 16), the power factor is inversely proportional to the synchronous reactance Xs and the current Iph.It will be shown in the fol-

Inductance and Power Factor
The power factor of a PM machine can be expressed as ( 16), where X s represents the synchronous reactance of the machine.As given by ( 16), the power factor is inversely proportional to the synchronous reactance X s and the current I ph .It will be shown in the following section that the values of I ph , X s and E ph vary with the slot-pole combination, and at that time, each slot-pole combination offers a specific power factor as given by ( 16).
The synchronous reactance X s is related with the average permeance P 0 of (3) and the operating frequency that depends on Z r .Considering the slot leakage and end turn effects, synchronous reactance X s is calculated in [7] as given by ( 17).It can be noted that the X s is proportional to the PM pole pairs Z r and hence the gear ratio.Form ( 7) and ( 17), it is obvious that as the gear ratio increases, both the back EMF and the reactance increase, but the increase in X s is more than that of the back EMF; therefore, the power factor is inversely related with the gear ratio.

Nature of the Design Problem and Design Criteria
As discussed in detail, the choice of the specific electric loading K s has a significant influence on the key electromagnetic performances, and especially the magnetic saturation in the tooth region is also sensitive to the applied specific electric loading.Considering the unique operation principle of PM vernier machine, certain design constraints exist; for example, to keep the modulation flux higher, a slot opening ratio should be kept around 0.5~0.6.As a result of fixed slot opening ratio, the available tooth area is limited and it leads to higher tendency of tooth saturation in PM vernier machine.Furthermore, it is important to keep a specific PM thickness called the optimal PM thickness in order to obtain the maximum back EMF, and obviously higher back EMF is advantageous for both the torque density and the power factor.
In the following section, various performances of SPMVG will be discussed, and it will be shown that the performances are sensitive to certain design parameters.Thus, to effectively utilize the advantages of SPMVG, the proper selection of various design parameters is important.First, in order to demonstrate an important aspect of the design problem, the graph of the gear ratio with variation of the PM pole pairs Z r and the total stator slots Z s is shown in Figure 3.Under the condition of balanced 3-phase stator winding configuration, it is possible to operate everywhere on the graph except the infeasible region shown in Figure 3. Now, it is to be decided which slot-pole combination should be selected among the various slot-pole combinations with different gear ratios and modulation pole pairs.
To demonstrate the various trends of the design parameters and electromagnetic performances of SPMVG, three gear ratios (G r = 5, 8, 11) with a range of modulation pole pairs of up to 25 are chosen.As mentioned, the 15 MW DD conventional PM generator designed by NREL [3] is taken as a reference.Initially, in order to comparatively analyze the various aspects of the design parameters and the performances, the basic geometry of the 15 MW reference model (see Table 1) will be used for the SPMVG.To demonstrate the various trends of the design parameters and electromagnetic performances of SPMVG, three gear ratios ( = 5, 8, 11) with a range of modulation pole pairs of up to 25 are chosen.As mentioned, the 15 MW DD conventional PM generator designed by NREL [3] is taken as a reference.Initially, in order to comparatively analyze the various aspects of the design parameters and the performances, the basic geometry of the 15 MW reference model (see Table 1) will be used for the SPMVG.

Back EMF Perofrmance and Slection of Optimal PM Thickness
The back EMF performance of SPMVG is very sensitive to certain design parameters such as the slot opening ratio c0, slot-pole combination, and the PM thickness gm.To show the back EMF trends, a gear ratio of 11, rated speed of 7.56 rpm, slot opening ratio c0 of 0.55, air-gap diameter and stack length of the reference model (see Table 1) is used.By using those basic design parameters, the back EMF of SPMVG with variation of PM thickness and modulation pole pairs  is shown in Figure 4. First, it can be noted that the magnitude of the back EMF is higher when  is lower, and back EMF starts decreasing as  increases.Moreover, for each  there exists a specific PM thickness called the optimal PM thickness that produces the maximum back EMF as shown by the black dotted line in Figure 4, and it should be noted that as the  increases, the optimal PM thickness starts decreasing.The back EMF performance of SPMVG is very sensitive to certain design parameters such as the slot opening ratio c 0 , slot-pole combination, and the PM thickness g m .To show the back EMF trends, a gear ratio of 11, rated speed of 7.56 rpm, slot opening ratio c 0 of 0.55, air-gap diameter and stack length of the reference model (see Table 1) is used.By using those basic design parameters, the back EMF of SPMVG with variation of PM thickness and modulation pole pairs p − mod is shown in Figure 4. First, it can be noted that the magnitude of the back EMF is higher when p − mod is lower, and back EMF starts decreasing as p − mod increases.Moreover, for each p − mod there exists a specific PM thickness called the optimal PM thickness that produces the maximum back EMF as shown by the black dotted line in Figure 4, and it should be noted that as the p − mod increases, the optimal PM thickness starts decreasing.At rated speed of 7.56 rpm, by using ( 7), the back EMF performance of SPMVG at the optimal PM thickness for the three gear ratios ( = 5, 8, 11) and their  = 1~25 is obtained as shown in Figure 5; for comparison, the back EMF of the reference model is also depicted in the graph.First it should be noted that with the same air-gap geometry, At rated speed of 7.56 rpm, by using (7), the back EMF performance of SPMVG at the optimal PM thickness for the three gear ratios (G r = 5, 8, 11) and their p − mod = 1 ∼ 25 is obtained as shown in Figure 5; for comparison, the back EMF of the reference model is also depicted in the graph.First it should be noted that with the same air-gap geometry, the back EMF of SPMVG with each gear ratio and its p − mod is significantly higher than that of the reference model; obviously, it is due to the additional advantages of the flux modulation effects.Moreover, the back EMF trends suggest that higher gear ratio with lower p − mod offers the higher back EMF.For a given gear ratio, as the p − mod increases, the optimal PM thickness thins; as a result, the magnitude of F m1 starts decreasing, and the leak factor k leak starts increasing, due to which the back EMF decreases as it is given by (7).At rated speed of 7.56 rpm, by using (7), the back EMF performance of SPMVG at the optimal PM thickness for the three gear ratios ( = 5, 8, 11) and their  = 1~25 is obtained as shown in Figure 5; for comparison, the back EMF of the reference model is also depicted in the graph.First it should be noted that with the same air-gap geometry, the back EMF of SPMVG with each gear ratio and its  is significantly higher than that of the reference model; obviously, it is due to the additional advantages of the flux modulation effects.Moreover, the back EMF trends suggest that higher gear ratio with lower  offers the higher back EMF.For a given gear ratio, as the  increases, the optimal PM thickness thins; as a result, the magnitude of Fm1 starts decreasing, and the leak factor kleak starts increasing, due to which the back EMF decreases as it is given by (7).

Available Surface Current Density
Unlike the problem of magnetic saturation in the back yoke, the flux density in the tooth region is tricky to control because the available tooth region is limited due to the fixed stator slot opening ratio c0.When the optimal magnet thickness obtained in the previous section is used for a given slot-pole combination, the only adjustable variable for Bteeth of ( 15) is the surface flux density Ks.Therefore, it is necessary to determine the proper Ks to keep the value of Bteeth in order to avoid the saturation problem.Considering that Bteeth of a vernier machine can be higher than that of a conventional one, it is set to 1.8/0.95= 1.92 T in this study.Thus, the available surface current density Ks that ensures the constraint of Bteeth = 1.92T for each gear ratio and its  are obtained and shown in Figure 6.It can be noted that the available Ks is somewhat inversely proportional to Gr and is almost proportional to  .In addition, the results in Figure 6 clearly explain why it is not proper to take commonly used Ks value of the reference model, and notes that Ks for SPMVG can be much smaller depending on its slot-pole combination.

Available Surface Current Density
Unlike the problem of magnetic saturation in the back yoke, the flux density in the tooth region is tricky to control because the available tooth region is limited due to the fixed stator slot opening ratio c 0 .When the optimal magnet thickness obtained in the previous section is used for a given slot-pole combination, the only adjustable variable for B teeth of (15) is the surface flux density K s .Therefore, it is necessary to determine the proper K s to keep the value of B teeth in order to avoid the saturation problem.Considering that B teeth of a vernier machine can be higher than that of a conventional one, it is set to 1.8/0.95= 1.92 T in this study.Thus, the available surface current density K s that ensures the constraint of B teeth = 1.92 T for each gear ratio and its p − mod are obtained and shown in Figure 6.It can be noted that the available K s is somewhat inversely proportional to G r and is almost proportional to p − mod .In addition, the results in Figure 6 clearly explain why it is not proper to take commonly used K s value of the reference model, and notes that K s for SPMVG can be much smaller depending on its slot-pole combination.

Torque Density and Power Factor
As the SPMVG has to be designed for a rated power of 15 MW, the rated speed, airgap diameter, air-gap length, number of turns and volume current density of SPMVG are set the same as those of the reference model (see Table 1).Under those design constraints, by using the calculated available surface current density for SPMVG, the required stack length lstk for rated power of 15 MW is calculated as shown in Figure 7.It can be noted that for the same rated power, as the number of modulation pole pairs  increases, the

Torque Density and Power Factor
As the SPMVG has to be designed for a rated power of 15 MW, the rated speed, air-gap diameter, air-gap length, number of turns and volume current density of SPMVG are set Energies 2023, 16, 1094 9 of 14 the same as those of the reference model (see Table 1).Under those design constraints, by using the calculated available surface current density for SPMVG, the required stack length l stk for rated power of 15 MW is calculated as shown in Figure 7.It can be noted that for the same rated power, as the number of modulation pole pairs p − mod increases, the required stack length of SPMVG first decreases, and at higher number of p − mod it starts increasing, regardless of the higher K s .Moreover, it can be seen that the lower gear ratio offers more advantage in terms of lower l stk due to the fact that the available K s increases as the gear ratios decreases.The stack length of the reference model is also depicted in Figure 7.It can be noted that for the same rated power, SPMVG, depending on the gear ratio and number of modulation pole pairs, mostly offers much lower stack length that that of the reference model.

Torque Density and Power Factor
As the SPMVG has to be designed for a rated power of 15 MW, the rated speed, airgap diameter, air-gap length, number of turns and volume current density of SPMVG are set the same as those of the reference model (see Table 1).Under those design constraints, by using the calculated available surface current density for SPMVG, the required stack length lstk for rated power of 15 MW is calculated as shown in Figure 7.It can be noted that for the same rated power, as the number of modulation pole pairs  increases, the required stack length of SPMVG first decreases, and at higher number of  it starts increasing, regardless of the higher Ks.Moreover, it can be seen that the lower gear ratio offers more advantage in terms of lower lstk due to the fact that the available Ks increases as the gear ratios decreases.The stack length of the reference model is also depicted in Figure 7.It can be noted that for the same rated power, SPMVG, depending on the gear ratio and number of modulation pole pairs, mostly offers much lower stack length that that of the reference model.The torque per air-gap volume commonly known as TRV is calculated by using (9) as shown in Figure 8; the TRV of the reference model is also depicted on the graph.It can be noted that each gear ratio and its modulation pole pairs  , depending on the available surface current density Ks, offer a certain TRV, and as the modulation pole pairs  change, the available TRV changes significantly.As compared to the TRV of the reference model, that is, 110.38 kPa, the SPMVG with Gr = 5 and  = 21 offers up to 80% higher TRV that is quite significant.As shown in Figure 8, it should be noted that at lower number of  , higher gear ratios, despite their lower Ks, provide comparatively higher TRVs, but as  starts increasing, lower gear ratio has a clear advantage of having higher TRV.Furthermore, as shown in Figure 8, it can be noted that at higher number of modulation pole pairs, especially for higher gear ratios, the TRV starts decreasing despite having higher available Ks.In fact, it is due to the fact that at higher number of  , the PM-PM leakage flux is significant, and so is the decaying back EMF due to lower magnitude of Fm1 and P1.The torque per air-gap volume commonly known as TRV is calculated by using ( 9) as shown in Figure 8; the TRV of the reference model is also depicted on the graph.It can be noted that each gear ratio and its modulation pole pairs p − mod , depending on the available surface current density K s , offer a certain TRV, and as the modulation pole pairs p − mod change, the available TRV changes significantly.As compared to the TRV of the reference model, that is, 110.38 kPa, the SPMVG with G r = 5 and p − mod = 21 offers up to 80% higher TRV that is quite significant.As shown in Figure 8, it should be noted that at lower number of p − mod , higher gear ratios, despite their lower K s , provide comparatively higher TRVs, but as p − mod starts increasing, lower gear ratio has a clear advantage of having higher TRV.Furthermore, as shown in Figure 8, it can be noted that at higher number of modulation pole pairs, especially for higher gear ratios, the TRV starts decreasing despite having higher available K s .In fact, it is due to the fact that at higher number of p − mod , the PM-PM leakage flux is significant, and so is the decaying back EMF due to lower magnitude of F m1 and P 1 .As discussed before, the available surface current density has direct and the back EMF has inverse relation with the modulation pole pairs  ; thus, from ( 16) it is obvious that the power factor will decreases as the  will increase.The calculated power factor of SPMVG for the three gear ratios and their  is shown in Figure 9; it can be seen that the power factor starts decreasing as the  increases, and of course the power factor is inversely related with the gear ratio.Moreover, it can be noted that the power factor of the SPMVG is much lower in comparison with the reference model.It is interesting to note that, considering the power factor, it is better to use lower  that is As discussed before, the available surface current density has direct and the back EMF has inverse relation with the modulation pole pairs p − mod ; thus, from (16) it is obvious that the power factor will decreases as the p − mod will increase.The calculated power factor of SPMVG for the three gear ratios and their p − mod is shown in Figure 9; it can be seen that the power factor starts decreasing as the p − mod increases, and of course the power factor is inversely related with the gear ratio.Moreover, it can be noted that the power factor of the SPMVG is much lower in comparison with the reference model.It is interesting to note that, considering the power factor, it is better to use lower p − mod that is a conflicting problem with the TRV.As discussed before, the available surface current density has direct and the back EMF has inverse relation with the modulation pole pairs  ; thus, from (16) it is obvious that the power factor will decreases as the  will increase.The calculated power factor of SPMVG for the three gear ratios and their  is shown in Figure 9; it can be seen that the power factor starts decreasing as the  increases, and of course the power factor is inversely related with the gear ratio.Moreover, it can be noted that the power factor of the SPMVG is much lower in comparison with the reference model.It is interesting to note that, considering the power factor, it is better to use lower  that is a conflicting problem with the TRV.Considering the TRV, it is better to operate at higher number of modulation pole pairs ; however, at higher  , apart from the worse power factor, there exists few more limitations including the PM-PM leakage flux and the PM demagnetization issues.As the PM-PM leakage flux is directly proportional to the number of PM pole pairs, at higher number of  , due to the increased leakage flux, the machine performance starts decreasing as can be observed in Figure 8; at higher number of  , despite having higher available Ks, the TRV starts decreasing.Moreover, as mentioned before, as  increases, the optimal PM thickness decreases, and due to the higher surface current density, the winding MMF increases.Both these factors, i.e., thin PM thickness and higher demagnetization force clearly indicate the concerns of the PM demagnetization at higher number of modulation pole pairs.Therefore, considering the practical limitations of having the low power factor and the PM demagnetization, it is feasible to design the machine with a lower gear ratio and moderate number of  depending on the required TRV or power factor.
The main focus of the study is to propose a systematic design procedure with the aim to properly choose the various entailed design parameters.In that regard, first, the key analytical expressions were obtained, and it was shown that the main electromagnetic performances such as the TRV, power factor and flux density in the core parts are directly linked with the choice of specific electric loading.Therefore, it is very important to Considering the TRV, it is better to operate at higher number of modulation pole pairs p − mod ; however, at higher p − mod , apart from the worse power factor, there exists few more limitations including the PM-PM leakage flux and the PM demagnetization issues.As the PM-PM leakage flux is directly proportional to the number of PM pole pairs, at higher number of p − mod , due to the increased leakage flux, the machine performance starts decreasing as can be observed in Figure 8; at higher number of p − mod , despite having higher available K s , the TRV starts decreasing.Moreover, as mentioned before, as p − mod increases, the optimal PM thickness decreases, and due to the higher surface current density, the winding MMF increases.Both these factors, i.e., thin PM thickness and higher demagnetization force clearly indicate the concerns of the PM demagnetization at higher number of modulation pole pairs.Therefore, considering the practical limitations of having the low power factor and the PM demagnetization, it is feasible to design the machine with a lower gear ratio and moderate number of p − mod depending on the required TRV or power factor.
The main focus of the study is to propose a systematic design procedure with the aim to properly choose the various entailed design parameters.In that regard, first, the key analytical expressions were obtained, and it was shown that the main electromagnetic performances such as the TRV, power factor and flux density in the core parts are directly linked with the choice of specific electric loading.Therefore, it is very important to properly choose the specific electric loading to effectively utilize the advantages of SPMVG while avoiding problems such as core saturation, PM demagnetization and excessively lower power factor.To deal with those concerns, a design criterion under some design conditions is developed, which clearly shows that depending on certain design parameters, each slot-pole combination corresponds to a certain value of specific electric loading.Thus, it is obvious that each slot-pole combination and the resulting gear ratio of SPMVG offers a certain TRV and power factor.By using the developed design criteria, with variation of the slot-pole combination, the key electromagnetic characteristics and design parameters such as the back EMF characteristic at optimal PM dimensions, available surface current density, TRV and power factor trends are thoroughly discussed.The obtained results, especially the TRV and the power factor variation trends, provide a clear guide for the selection of the gear ratio as well as the number of modulation pole pairs.It was clearly shown that considering the TRV and the power factor, it is better to use lower gear ratio; however, the selection of the number of modulation pole pairs p − mod is a compromising problem between the TRV and the power factor.
In this study, with the aim to achieve maximum TRV while having a power factor limit around 40%, the SPMVG with a gear ratio of 5 and p − mod = 13 is selected as depicted in Figure 8.The detailed design parameters of the selected SPMVG model in comparison with the reference model are given in Table 1.Form the comparison of the stack length and the surface current density of SPMVG with that of the reference model, it is important to note that the SPMVG with much lower air-gap volume and reduced value of surface current density is providing the same rated power of 15 MW.In fact, it highlights the specific nature of the SPMVG and of course the additional advantages of the flux modulation effects.

FEM Modeling and Results
The detailed geometry of the reference model is specified in [3], the main design parameters of both the reference model and the SPMVG are given in Table 2, and by using those parameters, both the generators are modeled in 2D-FEM.Firstly, for both the prototype models, the stator winding configuration and the no-load flux lines with the specified PM thickness are shown in Figure 10, where the letters A, B and C represent the three phases and the signs show the direction of the winding conductors.At the same rated speed of 7.56 rpm, the back EMF waveforms of both the prototype models are calculated and compared as shown in Figure 11.The magnitude of the fundamental components of the back EMF waveforms are given in Table 2.It can be noted that at the same rated speed, despite much smaller dimensions of SPMVG, its back EMF is almost similar to that of the reference model.It is due to the additional advantages of the flux modulation effects.properly choose the specific electric loading to effectively utilize the advantages of SPMVG while avoiding problems such as core saturation, PM demagnetization and excessively lower power factor.To deal with those concerns, a design criterion under some design conditions is developed, which clearly shows that depending on certain design parameters, each slot-pole combination corresponds to a certain value of specific electric loading.Thus, it is obvious that each slot-pole combination and the resulting gear ratio of SPMVG offers a certain TRV and power factor.By using the developed design criteria, with variation of the slot-pole combination, the key electromagnetic characteristics and design parameters such as the back EMF characteristic at optimal PM dimensions, available surface current density, TRV and power factor trends are thoroughly discussed.The obtained results, especially the TRV and the power factor variation trends, provide a clear guide for the selection of the gear ratio as well as the number of modulation pole pairs.It was clearly shown that considering the TRV and the power factor, it is better to use lower gear ratio; however, the selection of the number of modulation pole pairs  is a compromising problem between the TRV and the power factor.
In this study, with the aim to achieve maximum TRV while having a power factor limit around 40%, the SPMVG with a gear ratio of 5 and  = 13 is selected as depicted in Figure 8.The detailed design parameters of the selected SPMVG model in comparison with the reference model are given in Table 1.Form the comparison of the stack length and the surface current density of SPMVG with that of the reference model, it is important to note that the SPMVG with much lower air-gap volume and reduced value of surface current density is providing the same rated power of 15 MW.In fact, it highlights the specific nature of the SPMVG and of course the additional advantages of the flux modulation effects.

FEM Modeling and Results
The detailed geometry of the reference model is specified in [3], the main design parameters of both the reference model and the SPMVG are given in Table 2, and by using those parameters, both the generators are modeled in 2D-FEM.Firstly, for both the prototype models, the stator winding configuration and the no-load flux lines with the specified PM thickness are shown in Figure 10, where the letters A, B and C represent the three phases and the signs show the direction of the winding conductors.At the same rated speed of 7.56 rpm, the back EMF waveforms of both the prototype models are calculated and compared as shown in Figure 11.The magnitude of the fundamental components of the back EMF waveforms are given in Table 2.It can be noted that at the same rated speed, despite much smaller dimensions of SPMVG, its back EMF is almost similar to that of the reference model.It is due to the additional advantages of the flux modulation effects.At the rated load conditions, the flux density distribution in the cores of the SPMVG and the reference model are shown in Figure 12.It shows that the SPMVGs do not have severe saturation spots, which means the machine is properly designed.Furthermore, for both the prototypes, considering their surface current densities, the corresponding rated current is applied in phase with the back EMF, the resulting torque waveforms are shown in Figure 13, and the percent torque ripple that represents the ratio of peak-peak torque to average torque is calculated for both the prototype models, and the values are given in Table 2.At the rated load conditions, the flux density distribution in the cores of the SPMVG and the reference model are shown in Figure 12.It shows that the SPMVGs do not have severe saturation spots, which means the machine is properly designed.Furthermore, for both the prototypes, considering their surface current densities, the corresponding rated current is applied in phase with the back EMF, the resulting torque waveforms are shown in Figure 13, and the percent torque ripple that represents the ratio of peak-peak torque to average torque is calculated for both the prototype models, and the values are given in Table 2.At the rated load conditions, the flux density distribution in the cores of the SPMVG and the reference model are shown in Figure 12.It shows that the SPMVGs do not have severe saturation spots, which means the machine is properly designed.Furthermore, for both the prototypes, considering their surface current densities, the corresponding rated current is applied in phase with the back EMF, the resulting torque waveforms are shown in Figure 13, and the percent torque ripple that represents the ratio of peak-peak torque to average torque is calculated for both the prototype models, and the values are given in Table 2.For both the prototype models, the summary of the electromagnetic performance along with the power factor, TRV and the efficiency is given in Table 2.It can be noted that despite the fact that the Ks of the SPMVG is 7.5% less than that of the reference model, the TRV of the SPMVG is almost 1.67 times higher compared to the reference model.The torque quality of the reference model is comparatively higher than that of the SPMVG but still the torque ripple of the SPMVG is acceptable as the machine is used without any modification in the structure.Moreover, the compromising power factor, as is well known that the power factor of flux modulation machines remains a challenge, the designed SPMVG has a power factor of 0.37.As discussed, it is possible to improve the power factor by compromising the TRV; however, the development of more effective ways of increasing the power factor requires further research.Furthermore, the efficiency of the SPMVG is calculated by considering the copper and the iron losses, as given in Table 2.The SPMVG offers efficiency that is almost similar to that of the reference model.
Finally, by using the geometrical information of both the generators, the active material consumption for the various generator parts such as PM, copper winding, stator and For both the prototype models, the summary of the electromagnetic performance along with the power factor, TRV and the efficiency is given in Table 2.It can be noted that despite the fact that the K s of the SPMVG is 7.5% less than that of the reference model, the TRV of the SPMVG is almost 1.67 times higher compared to the reference model.The torque quality of the reference model is comparatively higher than that of the SPMVG but still the torque ripple of the SPMVG is acceptable as the machine is used without any modification in the structure.Moreover, the compromising power factor, as is well known that the power factor of flux modulation machines remains a challenge, the designed SPMVG has a power factor of 0.37.As discussed, it is possible to improve the power factor by compromising the TRV; however, the development of more effective ways of increasing the power factor requires further research.Furthermore, the efficiency of the SPMVG is calculated by considering the copper and the iron losses, as given in Table 2.The SPMVG offers efficiency that is almost similar to that of the reference model.
Finally, by using the geometrical information of both the generators, the active material consumption for the various generator parts such as PM, copper winding, stator and rotor is obtained as given in Table 3.It can be noted that the designed SPMVG has a clear advantage in terms of the PM material and copper consumption, both of which are comparatively expensive materials.Moreover, the designed SPMVG also has much lower stator steel mass than the reference model; however, the rotor mass of the SPMVG is comparatively higher.From the perspective of the control stability, the higher rotor mass/inertia of the SPMVG is actually advantageous.As given in Table 3, it can be noted that the net generator mass of the SPMVG is 8.5% lower than that of the reference model, which is quite beneficial.It should be noted that the SPMVG structure is used in its most fundamental form, thus there still exists a margin to further lower the material consumption.

Conclusions
In this paper, a surface PM vernier machine (SPMVG) for the application of MW-scale DD wind turbine is designed and analyzed against a conventional SPM DD generator.The various aspects of the design problem are discussed, and it is shown that for each slot-pole combination, the proper choice of certain design parameters such as specific electric loading and PM dimensions are necessary to effectively utilize the advantages of PM vernier machine.By considering the interrelationship between various design parameters, a design criterion under certain design constraints is developed.The developed design criterion ensures the proper selection of various entailed design parameters.By using the developed designed criterion, various electromagnetic performances of SPMVG with variation in the slot-pole combination are presented.It is shown that the surface current density has substantial impact on the various key electromagnetic performances, and it is determined that for each slot-pole combination of SPMVG, there exists a certain value of surface current density.Therefore, each slot-pole combination of SPMVG due to its unique available surface current density offers a certain torque density and power factor which ultimately serves as a guide for the selection of the gear ratio and the number of modulation pole pairs.
Based on the developed design procedure, an SPMVG for a rated power of 15 MW is designed, and its electromagnetic performance is comprehensively analyzed against a counterpart conventional DD generator.It was clearly shown that compared to the reference model, the designed SPMVG has almost 67% less air-gap volume despite its lower surface current density.Moreover, the designed SPMVG has comparatively less active material consumption; however, the only compromise is the 44% lower power factor of SPMVG.

Energies 2023 ,of 15 Figure 2 .
Figure 2. Conceptual distribution of the flux density.

Figure 2 .
Figure 2. Conceptual distribution of the flux density.

Energies 2023 ,
16, 1094 7 of 14 Energies 2023, 16, x FOR PEER REVIEW 7 of 15selected among the various slot-pole combinations with different gear ratios and modulation pole pairs.

Figure 3 .
Figure 3. Gear ratio Gr as a function of Zs and Zr.

Figure 3 .
Figure 3. Gear ratio G r as a function of Z s and Z r .

Figure 12 .
Figure 12.Flux density distribution at rated conditions (a) reference model, (b) designed SPMVG.Figure 12. Flux density distribution at rated conditions (a) reference model, (b) designed SPMVG.