sustainability
Review A Comprehensive Review on Brushless Doubly-Fed Reluctance Machine
Omid Sadeghian 1 , Sajjad Tohidi 1, Behnam Mohammadi-Ivatloo 1,2,* and Fazel Mohammadi 3,*
1 Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 5166616471, Iran; [email protected] (O.S.); [email protected] (S.T.) 2 Department of Energy Technology, Aalborg University, 9220 Aalborg, Denmark 3 Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON N9B 1K3, Canada * Correspondence: [email protected] (B.M.-I.); [email protected] or [email protected] (F.M.)
Abstract: The Brushless Doubly-Fed Reluctance Machine (BDFRM) has been widely investigated in numerous research studies since it is brushless and cageless and there is no winding on the rotor of this emerging machine. This feature leads to several advantages for this machine in comparison with its induction counterpart, i.e., Brushless Doubly-Fed Induction Machine (BDFIM). Less maintenance, less power losses, and also more reliability are the major advantages of BDFRM compared to BDFIM. The design complexity of its reluctance rotor, as well as flux patterns for indirect connection between the two windings mounted on the stator including power winding and control winding, have restricted the development of this machine technology. In the literature, there is not a comprehensive review of the research studies related to BDFRM. In this paper, the previous research studies are reviewed from different points of view, such as operation, design, control, transient model, dynamic model, power factor, Maximum Power Point Tracking (MPPT), and losses. It is revealed that the BDFRM is still evolving since the theoretical results have shown that this machine operates efficiently if it is well-designed. Keywords: brushless doubly-fed machine (BDFM); brushless doubly-fed reluctance machine Citation: Sadeghian, O.; Tohidi, S.; (BDFRM); doubly-fed induction generator (DFIG); brushless machines; electrical machinery; Mohammadi-Ivatloo, B.; power electronics Mohammadi, F. A Comprehensive Review on Brushless Doubly-Fed Reluctance Machine. Sustainability 2021, 13, 842. https://doi.org/ 1. Introduction 10.3390/su13020842 In recent decades, Doubly-Fed Machines (DFMs) [1–3] have been extensively used Received: 17 October 2020 in different applications due to using partially-rated converters to control the machine, Accepted: 13 January 2021 leading to lower converter cost [4]. Such machines are used in applications where the speed Published: 16 January 2021 changes slightly with time, such as large wind turbines [5–7], as well as power drives [8] like pump drives [9] or even turbo-electric distributed propulsion systems [10]. The common Publisher’s Note: MDPI stays neu- type of DFMs is induction-rotor DFM so-called Doubly-Fed Induction Machine (DFIM). tral with regard to jurisdictional clai- In this machine, two separate windings are mounted on the stator and rotor, in which the ms in published maps and institutio- stator winding directly connects to power grids while the rotor winding indirectly connects nal affiliations. to power grids via two back-to-back converters [11]. The main advantage of such machines is using partially-rated converters, which reduce the cost of power electronics converters compared to fully-rated converters in Permanent Magnet Synchronous Machines (PMSMs). The presence of brush and slip-ring for connecting the rotor to the converters increases Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. the need for maintenance, probability of failure, and power losses [12]. To tackle this This article is an open access article issue, the brushless type of DFM, namely the Brushless Doubly-Fed Machine (BDFM), distributed under the terms and con- has been presented [13]. The main advantage of BDFM technology compared to DFM is ditions of the Creative Commons At- the absence of brush and slip-ring, which reduces the need for maintenance services and tribution (CC BY) license (https:// additionally, improves the machine robustness. Different types of BDFM, such as wound- creativecommons.org/licenses/by/ rotor, cage-rotor, dual-stator, and hybrid-rotor BDFMs, have been introduced in [13]. The 4.0/). widely-used type of BDFM is Brushless Doubly-Fed Induction Machine (BDFIM) type [14].
Sustainability 2021, 13, 842. https://doi.org/10.3390/su13020842 https://www.mdpi.com/journal/sustainability Sustainability 2021, 13, 842 2 of 39
In BDFIM, the rotor is a short-circuit wound-rotor or a cage-rotor, whereas the stator has two windings. One winding is directly connected to power grids, while the other winding is connected to power grids via two back-to-back converters. Thus, both the windings are on the stator, while they have indirect coupling via the rotor and there is no direct coupling between them. BDFM is attractive for being used in many applications, such as variable-frequency speed regulation systems and variable-speed constant frequency generation systems. Therefore, the design and control of BDFM have been attracted much interest [13]. The significant rotor losses and poor efficiency caused by the cage or wound rotor make the machine considerably parameter-dependent and controller operation compli- cated. Such drawbacks make the BDFIM difficult to implement [15]. Accordingly, another promising BDFM technology has been proposed that has a reluctance-type rotor called Brushless Doubly-Fed Reluctance Machine (BDFRM). BDFRM was firstly introduced in the early 1970s [16]. In this machine, there is no wound-or cage-rotor. Hence, it has higher reliability and lower power losses compared to its induction counterpart, i.e., BD- FIM [9]. Additionally, BDFRM is approximately maintenance-free due to the absence of rotor winding compared to wound-rotor BDFIM. Simpler structure, smaller size, higher power density, higher efficiency, and lower maintenance cost are the other advantages of BDFRM in comparison with traditional machines [17]. Furthermore, the BDFRM control is much easier due to its cageless rotor [18]. The BDFRM technology has a high potential to be used in large-power restricted-speed applications due to its manufacturing cost, as well as reliability and robustness criteria [19]. It also can be used in applications, such as turbomachinery, adjustable-speed constant- frequency wind, and hydro-power applications, air conditioning, commercial heating, and large pump drives [20,21]. The application of BDFRM in wind turbines especially offshore ones, has accelerated the research studies on BDFRM [22], due to the advantages of wind turbines in viewpoints of zero energy generation cost and environmental issues [23,24]. Due to the application and the importance of BDFRM, the requirements of BDFRM-based wind turbines are discussed in [6]. However, its complicated operation because of the coupling concept between the stator windings via the reluctance rotor is less investigated [15]. One implication of the unusual operating basics and improper design is the insignificant torque per volume for the constructed BDFRMs, and thus, a bigger and more expensive BDFRM is needed to achieve the torque of an equivalent synchronous reluctance or a cage induction machine [25]. However, recent advances have revealed that BDFRMs can be operated at high efficiency and torque density (torque per volume ratio) when properly designed [26]. Additionally, decreasing their mass per torque ratio has also been investigated in literature [27]. Furthermore, it is revealed that the BDFRM can achieve better torque density than a cage rotor induction machine using the same frame [28]. Due to the mentioned advantages of BDFRM compared to BDFIM, such as low manufacturing cost arising from its reluctance rotor, high reliability arising from the brushless structure, low losses, and lower maintenance needs, research studies have been conducted since the 1990s to analyze such machines [29]. For using low-capacity power converters, the BDFRM should operate in a narrow speed range around the synchronous speed, since it is a slip energy recovery machine [30]. For a typical speed range of 2:1 in wind turbines or pump drives, BDFRM allows to use of partially-rated converters with only 25% of the machine rating [31], which shows BDFRM is a slip power recovery machine. In recent years, some reports focusing on the performance of BDFRM have been released [32,33]. In [34], different aspects of BDFRM are discussed. The superiorities and deficiencies of the BDFRM technology compared with BDFIM are presented. Specific characteristics of BDFRM compared to BDFIM are addressed in the literature [35]. Simpler control and higher efficiency of BDFRM compared to BDFIM are achieved by the improvement of the reluctance rotor design. Such reasons made it necessary to conduct more research on the Brushless Doubly-Fed Reluctance Generator (BDFRG) [36]. In Table1, the structure features of DFMs are compared. Sustainability 2021, 13, x FOR PEER REVIEW 3 of 38
BDFRM compared to BDFIM are addressed in the literature [35]. Simpler control and higher efficiency of BDFRM compared to BDFIM are achieved by the improvement of the reluctance rotor design. Such reasons made it necessary to conduct more research on the Brushless Doubly-Fed Reluctance Generator (BDFRG) [36]. In Table 1, the structure fea- Sustainability 2021, 13, 842 tures of DFMs are compared. 3 of 39
Table 1. Structure features of DFMs. Table 1. Structure features of DFMs. Machine Technology Structure of DFM MachineDFM Technology Using partially-rated Structure ofconverters DFM DFMUsing Using partially-rated partially-rated converters converters BDFM Using partially-rated converters BDFM Brush and slip-ring removal Brush and slip-ring removal Using partially-rated converters Using partially-rated converters BDFRMBDFRM BrushBrush and slip-ring and slip-ring removal removal UsingUsing reluctance-rotor reluctance-rotor
Figure1 1 illustrates illustrates thethe mainmain keywordskeywords ofof previousprevious researchresearch studiesstudies inin thethe fieldfield ofof BDFRM associated with the temporaltemporal relationshiprelationship amongamong them.them. As observed,observed, in recentrecent years, such research studiesstudies havehave notablynotably increased.increased.
Figure 1. Main keywords of previous research studies about Brushless BDFRM associated with temporal relationship Figure 1. Main keywords of previous research studies about Brushless BDFRM associated with temporal relationship amongamong them.them. In this paper, a comprehensive literature survey on BDFRM is performed. The publi- In this paper, a comprehensive literature survey on BDFRM is performed. The pub- cations are categorized into different points of view. The conducted study covers all the lications are categorized into different points of view. The conducted study covers all the previous research studies on BDFRM design improvement, operation, dynamic modeling, previous research studies on BDFRM design improvement, operation, dynamic modeling, power factor control, Maximum Power Point Tracking (MPPT), and power losses. The power factor control, Maximum Power Point Tracking (MPPT), and power losses. The previous research studies have focused on one or more aspects of BDFRM, while this paper previous research studies have focused on one or more aspects of BDFRM, while this pa- deals with the overview of all the previous research studies in different fields of this topic. per deals with the overview of all the previous research studies in different fields of this The remarkable features of this paper are highlighted as: topic. The remarkable features of this paper are highlighted as: • A comprehensive review is accomplished on all the previous works about BDFRM. • • OperationA comprehensive principle review of BDFRM is accomplished and its components on all the areprevious discussed. works about BDFRM. • • DetailOperation characteristics principle of of BDFRM the previous and its research components studies are are discussed. presented. • • TheDetail research characteristics studies of are the compared previous inresearch different studies aspects, are suchpresented. as machine type and machine size, etc. • The conducted research studies in different fields, such as operation, design, and control of BDFRM, are reviewed.
2. BDFRM Principles This topology is composed of a reluctance rotor, two-winding stator, two back-to-back power electronics circuits, which are inverters, forming Voltage-Sourced Inverter (VSI), Sustainability 2021, 13, x FOR PEER REVIEW 4 of 38
• The research studies are compared in different aspects, such as machine type and machine size, etc. • The conducted research studies in different fields, such as operation, design, and con- trol of BDFRM, are reviewed.
2. BDFRM Principles
Sustainability 2021, 13, 842 This topology is composed of a reluctance rotor, two-winding stator, two back-to-4 of 39 back power electronics circuits, which are inverters, forming Voltage-Sourced Inverter (VSI), with a capacitor in the intermediate circuit, called DC-link capacitor located in par- withallel between a capacitor the in converters. the intermediate Indeed, circuit, the VSI- calledconverter DC-link acts capacitoras an AC/DC/AC located in converter parallel betweenunit. One theof the converters. converters, Indeed, namely the the VSI-converter Machine-Side actsConverter as an AC/DC/AC(MSC), plays the converter role of unit.controlling One of machine the converters, active and namely reactive the power. Machine-Side Active power Converter reference (MSC), can plays be determined the role of controllingfrom Maximum machine Power active Point and Tracking reactive power.(MPPT) Active in wind power energy reference conversion can be systems determined [37]. fromOn the Maximum other hand, Power the Pointsecond Tracking converter, (MPPT) i.e., Grid-Side in wind energy Converter conversion (GSC), systemscontrols [the37]. DC- On thelink other voltage. hand, Furthermore, the second converter,it may absorb/injec i.e., Grid-Sidet reactive Converter power (GSC),from/into controls power the grids DC-link [37], voltage.and thus, Furthermore, there is no need it may for absorb/inject reactive power reactive compensation power from/into [38]. power grids [37], and thus,In there some is research no need forstudies, reactive current power source compensation converters [are38]. used instead of voltage source ones In[19,39]. some researchThe BDFRM studies, structure current is source presented converters in Figure are used2, where instead 𝜔 , of 𝜔 voltage , and 𝜔 source are onesthe angular [19,39]. velocities The BDFRM of the structure power iswinding, presented control in Figure winding,2, where andω rotorp, ωc ,(rad./s), and ωr arerespec- the angulartively, 𝜔 velocities is the ofmechanical the power angular winding, velocity control of winding, the rotor and (mech.-rad./s), rotor (rad./s), and respectively, 𝑝 shows ω p therm rotoris the pole mechanical numbers. angular Additionally, velocity 𝑝 of theand rotor 𝑝 are (mech.-rad./s), the pole-pair andnumbersr shows of power the rotor and polecontrol numbers. windings, Additionally, respectively.pp Theseand p cparameteare the pole-pairrs are used numbers to analyze of power the operation and control of windings,BDFRM. respectively. These parameters are used to analyze the operation of BDFRM.
Grid Reluctance Rotor
Power winding Control winding
pole pairs pr pole pairs pp pc
ω ω p c ω rm
Magnetic Coupling
Control
AC/DC/AC Converter
DC Link GSC (AC/DC) MSC (DC/AC)
Figure 2.2. ConfigurationConfiguration ofof BDFRMBDFRM (Combining(Combining thethe configurationsconfigurations usedused inin [[25,40,41]).25,40,41]).
As previously mentioned, BDFRM has no winding on the rotor that is the superiority of this machine compared to BDFIM [40]. The operation of BDFRM is based on an indirect connection between the windings mounted on the stator. The windings are named as primary and secondary windings. The primary winding directly connects to power grids, whereas the control winding indirectly connects to power grids via a power electronics unit [42]. In BDFRM and other BDFM topologies, direct coupling between the windings must not be established. Because the power drained from power grids by the control winding can be reversed to power grids via the power winding. Hence, the machine cannot be controlled when there is a direct coupling between them. This is while the windings have Sustainability 2021, 13, x FOR PEER REVIEW 5 of 38
As previously mentioned, BDFRM has no winding on the rotor that is the superiority of this machine compared to BDFIM [40]. The operation of BDFRM is based on an indirect connection between the windings mounted on the stator. The windings are named as pri- mary and secondary windings. The primary winding directly connects to power grids, whereas the control winding indirectly connects to power grids via a power electronics unit [42]. In BDFRM and other BDFM topologies, direct coupling between the windings must not be established. Because the power drained from power grids by the control winding Sustainability 2021, 13, 842 5 of 39 can be reversed to power grids via the power winding. Hence, the machine cannot be controlled when there is a direct coupling between them. This is while the windings have indirectindirect coupling coupling via via the the rotor. rotor. In In other other words, words, both both the the windings windings have have direct direct coupling coupling withwith the the reluctance reluctance rotor, rotor, separately separately [43]. [43]. Figure Figure 33 shows shows the the mentioned mentioned coupling coupling concept concept inin BDFRM. BDFRM.
No direct coupling
Stator
Power Winding Control Winding
Rotor
FigureFigure 3. 3.ElectromagneticElectromagnetic coupling coupling in in BDFRM BDFRM [43]. [43 ].
Accordingly,Accordingly, the the pole pole numbers mountedmounted onon the the stator stator windings windings must must be be different different [44 ]. [44].Ideally, Ideally, no no coupling coupling is is created created between between the the windings windings withwith a different number number of of pole pole pairspairs when when the the rotor rotor has has a round a round structure structure [45]. [The45]. key The factor key in factor this machine in this machine is its salient is its rotor,salient which rotor, is whichrequired is requiredfor indirect for coupling indirect couplingbetween the between stator thewindings, stator windings,and thus, for and thethus, desired for the operation desired operationof this machine of this technology. machine technology. Formation Formation of appropriate of appropriate flux density flux harmonicsdensity harmonics for linking for the linking windings the windingsneeds a flux needs path a fluxwith path variable with reluctance variable reluctance in the ma- in chinethe machine essentially essentially for modulating for modulating the Magnetom the Magnetomotiveotive Force (MMF) Force waveforms (MMF) waveforms of the stator of the [45].stator [45]. Generally,Generally, the the stator stator has has two two sets sets ofof windings windings with with three-phase three-phase including including p±1p ± 1polepole pairs,pairs, while while the the rotor rotor has has ha halflf of of the the summation summation of of the the stator stator winding winding poles, poles, i.e., i.e., pp. .For For instance,instance, for for p=2p = ,2 the, the stator stator windings windings pole pole numbers numbers are are two two and and six, six, and and thus, thus, the the rotor rotor windingwinding pole pole number number is is four. four. It Itseems seems that that a ahigher higher number number of of pole pole numbers numbers cannot cannot result result inin effective effective operation operation [8]. [8]. In In BDFIM, BDFIM, the numbernumber ofof polespoles for for the the windings windings mounted mounted on on the therotor rotor is is simultaneously simultaneously equal equal to to both both the the stator stator windings windings pole pole number number by by using using a a special spe- cialrotor rotor structure, structure, while while in BDFRM,in BDFRM, the the air air gap gap permeance permeance density density is theis the key key to to modulate modulate the theMMFs MMFs for for generating generating flux flux waves, waves, and an thus,d thus, generating generating cross-coupling cross-coupling [8]. [8]. TheThe primary primary winding winding has has a fixed-frequencyfixed-frequency voltagevoltage with with roughly roughly constant-magnitude constant-magni- tude(due (due to its to connection its connection to power to power grids) grids) [25] [25] while, while, the controlthe control winding winding is supplied is supplied from froma converter a converter with with variable variable voltage voltage and frequency and frequency [46], in [46], which in thewhich frequency the frequency is changed is by the MSC [47]. The power converter unit can absorb energy based on the machine’s changed by the MSC [47]. The power converter unit can absorb energy based on the ma- condition. However, when BDFRM is used as pump drives, the power converter unit chine’s condition. However, when BDFRM is used as pump drives, the power converter absorbs energy only during start-up [48] since for most of the pump applications, the range unit absorbs energy only during start-up [48] since for most of the pump applications, the of the speed variation is roughly 70–100% of the nominal speed. range of the speed variation is roughly 70–100% of the nominal speed. However, the V/f ratio may retain fixed [20,35], i.e., rotor speed, changes based on the load [8]. A bidirectional converter with 20% power rated of the machine rating can enable a 40% speed control range with constant average torque output [49]. The control winding must satisfy a synchronizing requirement, and thus, it is limited as Equation (1) [8,50] or Equation (2) [18–20,39,51–53]. ωc = prωrm − ωp (1)
ωr = ωc + ωp = prωrm (2)
Therefore, the power and control windings are 2pp-pole and 2pc-pole, respectively, while the rotor is pp-pole. Firstly, some assumptions are described in order to make the analysis feasible [48]: Sustainability 2021, 13, x FOR PEER REVIEW 6 of 39
However, the V/f ratio may retain fixed [20,35], i.e., rotor speed, changes based on the load [8]. A bidirectional converter with 20% power rated of the machine rating can enable a 40% speed control range with constant average torque output [49]. The control winding must satisfy a synchronizing requirement, and thus, it is limited as Equation (1) [8,50] or Equation (2) [18–20,39,51–53].
ωc = prωrm − ωp (1)
휔푟 = 휔푐 + 휔푝 = 푝푟휔푟푚 (2)
Therefore, the power and control windings are 2푝푝-pole and 2푝푐-pole, respectively, Sustainability 2021, 13, 842 6 of 39 while the rotor is 푝푝-pole. Firstly, some assumptions are described in order to make the analysis feasible [48]: a.a. TheThe iron iron permeability permeability of of the the machine machine is isinfinite. infinite. b.b. TheThe stator stator windings windings of ofthe the machine machine are areassumed assumed to be to adequately be adequately modeled modeled as spa- as tiallyspatially sinusoidally sinusoidally distributed. distributed. c.c. TheThe power power and and control control windings windings are are three three phases. phases. d.d. TheThe three three-phase-phase temporal temporal current current of of the the stator stator windings windings is is ideal. ideal. e.e. TheThe air air gap gap of of the the machine machine is ismodeled modeled using using a asinusoidal sinusoidal air air gap gap function. function. MMFsMMFs for for a a single single phase phase of of the the power power and and control control windings windings of of a a stator stator are are illust illustratedrated inin Figure 4 4[ 54[54]].. These These MMFs MMFs are are for for a stator a stator with with two two and sixand poles, six poles, respectively, respectively, for power for powerand control and control windings. windings.
Fp( ) n p i p cos( ) nipp Fc( ) n c i c cos(3 )
nicc MMF / Airgap / MMF
FigureFigure 4 4.. MMFMMFss of of a a single single phase phase of of power power and and control control windings. windings.
TheThe air air gap gap function function is is as as follows follows [55] [55]::
−−11 g푔 ((θ휃, θ, 휃rm푟푚) )==m푚++n푛coscos(p(푝r(푟θ(휃−−θrm휃푟푚)))) ((3)3) 푚 푛 wherewherem and nandare the are fixed the and fixed variable and variable parts of theparts air of gap, the respectively, air gap, respectively so that m ,≥ son that> 0 . 푚 ≥ 푛 > 0. Additionally, 휃푟푚 and 휃 are the mechanical angle of the rotor and the me- Additionally, θrm and θ are the mechanical angle of the rotor and the mechanical angle chanicalaround theangle machine around with the referencemachine with to phase referencea, respectively. to phase 푎 Therefore,, respectively. the maximumTherefore, andthe maximum and minimum air gap (푔푚푎푥 and 푔푚푖푛) are as follows: minimum air gap (gmax and gmin) are as follows: 1 푔푚푎푥 = 1 (4) g = 푚 − 푛 (4) max m − n 1 푔 = (5) 푚푖푛 푚1+ 푛 g = (5) min + The inverse air gap function is diagrammaticallym n illustrated in Figure 5. The inverse air gap function is diagrammatically illustrated in Figure5. Sustainability 2021, 13, 842 7 of 39
Figure 5. The inverse air gap function with pr poles.
By assuming the sinusoidal three-phase current of the power and control windings as ideal, the current equations are as follows: iap = Ip cos ωpt (6)
iac = Ic cos(ωct − α) (7)
where iap and iac are current of phase a for power and control windings, respectively. Additionally, α is the angle difference of current for power and control windings, and Ip and Ic are the magnitudes of the mentioned current. It is worth mentioning that the current −2π 2π of phases b and c have 3 and 3 difference with the current of phase a. The resultant three-phase MMFs per air gap for the power and control windings, i.e., Fp(θ) and Fc(θ), become as follows: Fp(θ) = Fpm cos ωpt − ppθ (8)
Fc(θ) = Fcm cos(ωct − α − pcθ) (9)
where pp and pc are the pole pair numbers of the power and control windings, respectively, θ is the periphery angle around machine (mech.-rad.). Moreover, the MMFs’ magnitude of power and control windings for phase a, i.e., Fpm and Fcm, are as follows:
3 Fpm = 2 np Ip 3 (10) Fcm = 2 nc Ic
where np and nc are obtained as follows:
n0 n = p p pp 0 (11) n = nc c pc
0 0 where np and nc represent the peak conductor density of the power and control windings in (conductors/rad.), respectively. Sustainability 2021, 13, 842 8 of 39
Based on the MMFs of the windings and the air gap function, the flux density dis- tribution of power and control windings, i.e., Bp and Bc with respect to θ for each of the individual windings is obtained using the following expressions.
−1 Bp(θ, θrm) = µ0g (θ, θrm)Fp(θ) −1 (12) Bc(θ, θrm) = µ0g (θ, θrm)Fc(θ)
where µ0 represents the electromagnetism permeability. G In several research studies, it is assumed that m = n = 2 and based on Equations (3)–(5), the following equation is derived [48,54,56,57].
G g−1(θ, θ ) = (1 + cos(p (θ − θ )) (13) r 2 r rm Then, G B (θ, θ ) = µ (1 + cos(p (θ − θ ))F cos ω t − p θ (14) p rm 0 2 r rm pm p p G B (θ, θ ) = µ (1 + cos(p (θ − θ ))F cos(ω t − p θ) (15) c rm 0 2 r rm cm c c One of the pre-requisites mentioned above is that the component field of one power winding must be rotating in the same direction as the fundamental field of the other stator winding, and the spatial pole numbers must be the same. This means that the two values including cosine terms of the secondary fundamental component and the primary harmonic component must be equal [45,58,59]. cos (ω1 − prωrm)t − pp − pr θ = cos(ω2t − α − pcθ) (16) Hence, ±(ω1 − prωrm) = ω2 and ± pp − pr = pc. Accordingly, for the positive sign, Equations (17) and (18) results in Equation (19).
ω1 − ω2 +(ω1 − prωrm) = ω2 → ωrm = (17) pr + pp − pr = pc → pr = pp − pc (18)
ωp − ωc ωrm = (19) pp − pc In addition, for the positive sign (+), Equations (20) and (21) results in Equation (22).
ω1 + ω2 −(ω1 − prωrm) = ω2 → ωrm = (20) pr − pp − pr = pc → pr = pp + pc (21)
ωp + ωc ωrm = (22) pp + pc The negative case (−), i.e., Equations (20)–(22), can be disregarded. This note is consid- ered in problem formulation [2,7,48,50,60–68]. The electro-mechanical energy conversion occurs under the following conditions, and this shows the share of each winding [60]. ωp + ωc ωp ωc ωc ωp ωc ωrm = = + = 1 + = 1 + ωsyn (23) pp + pc pr pr ωp pr ωp
Teωp Teωc Pm = Teωrm = + = Pp + Pc (24) pr pr
where Pp and Pc are the power of the primary and secondary windings, i.e., the power and control windings, respectively, and Pm and Te are the mechanical power and electromag- Sustainability 2021, 13, 842 9 of 39
Sustainability 2021, 13, x FOR PEER REVIEW 9 of 38 netic torque of the machine, respectively. Additionally, the synchronous speed (ωsyn) is as follows [44]: ωp + ωc ωsyn = (25) pp ± pc 𝜔 −𝜔 It is worth mentioning that in BDFRM,𝑠= the slip is as follows [50]: (26) 𝜔
ωp − ωc s = (26) In some research studies, the pole numbersωp of rotor (𝑝 ) may be limited as follows [20,48,54,56,57,60,62]: In some research studies, the pole numbers of rotor (pr) may be limited as fol- lows [20,48,54,56,57,60,62]: 𝑝 = 𝑝 ±𝑝 (27) = ± As mentioned, both the positive pandr negative pp pc signs are valid, only the positive sign(27) (+) isAs considered mentioned, in bothpractice the [63,69]. positive Further, and negative the research signs are studies valid, have only revealed the positive that sign the negative(+) is considered sign (−) in in Equation practice [(27)63,69 does]. Further, not give the favorable research couplings studies havebetween revealed the windings that the fornegative some signpole (combinations−) in Equation [26], (27) and does thus, not this give case favorable may result couplings in impractical between thedesigns windings [45]. Therefore,for some pole unlike combinations the conventional [26], and machines, thus, this th casee number may result of poles in impractical on the rotor designs can be [45 an]. oddTherefore, number unlike [29,30,70]. the conventional For instance, machines, for four and the numbertwo pole of numbers poles on for the stator rotor windings, can be an theodd number number of [29 poles,30,70 on]. the For rotor instance, should for be four three and [18,30]. two pole The numbers unusual foroperating stator windings, principle ofthe the number BDFRM of polesleads onto this the rotorinteresting should design be three feature [18,30 [30].]. The unusual operating principle of theUnlike BDFRM BDFM, leads the to thisBDFRM interesting rotor has design no winding. feature [ 30However,]. the rotor design is more complicatedUnlike BDFM,in this machine the BDFRM technology rotor has [56,57,70]. no winding. The However, BDFRM rotor the rotor design design is based is more on increasingcomplicated its insaliency this machine to improve technology machine [56 operation,57,70]. The [47]. BDFRM The BDFRM’s rotor design rotor is designed based on inincreasing different its types saliency according to improve to its application. machine operation Figure [647 shows]. The different BDFRM’s types rotor of is BDFRM’s designed rotor.in different It should types be according noted that to the its rotor application. velocity Figure is independent6 shows different of the rotating types of field BDFRM’s speed [8].rotor. It Itis shouldrevealed be that noted the that axially-laminated the rotor velocity rotor is independent type results of thein high rotating iron field losses speed of the [8]. rotor,It is revealed which have that thenegative axially-laminated effects on the rotor effi typeciency, results as well in high as the iron thermal losses ofstability the rotor, of BDFRMwhich have [8]. negative effects on the efficiency, as well as the thermal stability of BDFRM [8].
(a) (b) (c)
Figure 6. Different rotor types used in BDFRM: ( a)) Salient-pole, Salient-pole, ( (b)) Axially-laminated, Axially-laminated, and and ( (cc)) Radially-laminated Radially-laminated ducted- ducted- rotor [[35,44,50,65,71].35,44,50,65,71].
The flux flux density with air gap permeance of different rotor types is discussed in [[72]72] while in [26], [26], the flux flux density harmonics ar aree calculated from the MMF and ideal coupling factors.factors. The The characteristics characteristics of of different different roto rotorr types types are are compared compared in in[13,56]. [13,56 Moreover,]. Moreover, in [26],in [26 the], the salient-rotor salient-rotor designs designs including including simp simplele salient-pole salient-pole and and duct ducted-rotored-rotor structures are studied. Different Different rotor rotor configurations configurations and their their characteristics characteristics are are discussed discussed in in [6,47]. [6,47]. Furthermore, designing twotwo differentdifferent rotorrotor structuresstructures for for wind-driven wind-driven BDFRM BDFRM including including a asalient-pole salient-pole radially-laminated radially-laminated and and axially-laminated axially-laminated rotor rotor is performed is performed in [ 73in]. [73]. In [ 74In], [74], the thecharacteristics characteristics of axially-laminated, of axially-laminated, as well as well as salient-pole as salient-pole radially-laminated radially-laminated reluctance reluc- tancerotors, rotors, are compared. are compared. A 2-D A form 2-D of form three of rotor three types rotor including types including the traditional the traditional salient-pole, sali- ent-pole,the segment, the andsegment, the axially-laminated and the axially-lamina rotors areted presentedrotors are inpresented [72]. A comparison in [71]. A compari- between sontwo between rotor structures two rotor of BDFRM structures including of BDFRM axially-laminated including axially-laminated and salient rotors and is performedsalient ro- torsin [70 is]. performed Ideal and practicalin [70]. Ideal ducted and rotors practi arecal discussedducted rotors in [ 63are]. discussed in [63].
As mentioned, in BDFRM, both windings are mounted on the stator. The circuit of stator windings is investigated in [43,45,75,76]. Additionally, the winding diagrams of the Sustainability 2021, 13, 842 10 of 39
As mentioned, in BDFRM, both windings are mounted on the stator. The circuit of stator windings is investigated in [43,45,75,76]. Additionally, the winding diagrams of the stator with six and two poles for power and control windings, respectively, are presented in [47]. Figure7 illustrates a typical stator diagram with eight-pole and four-pole windings for the power and control windings, respectively [45]. Figure8 depicts the winding connection diagrams of the stator for the one-pole pair (for power winding) and the three-pole pair (for control winding) on the inner and outer sides of it, respectively. In this figure, the number of slots is equal to 2 × 3 × pp = 24. It is worth noting that Figures8 and9 do not refer to an identical machine. The winding of the lower layer is wounded with the pitch 30, while the other winding in the upper layer is wounded with the pitch 10. BDFRM machines may operate in sub-synchronous or super-synchronous modes. The operation mode for BDFRG is shown in [60,62,63,77–79]. In [80], the operation mode for both the motor and generator in BDFMs is presented. The operation modes are illustrated in Figure9. Sustainability 2021, 13, x FOR PEER REVIEW 10 of 38 Sustainability 2021, 13, x FOR PEER REVIEW 10 of 38
stator with six and two poles for power and control windings, respectively, are presented statorin [47]. with six and two poles for power and control windings, respectively, are presented in [47].Figure 7 illustrates a typical stator diagram with eight-pole and four-pole windings for theFigure power 7 illustrates and control a typical windings, stator respectively diagram with [45]. eight-pole Figure 8 depictsand four-pole the winding windings con- fornection the power diagrams and controlof the stator windings, for the respectively one-pole pair [45]. (for Figure power 8 depicts winding) the andwinding the three- con- nectionpole pair diagrams (for control of the winding) stator for on thethe one-poleinner and pair outer (for sides power of it, winding) respectively. and theIn this three- fig- pole pair (for control winding) on the inner and outer sides of it, respectively. In this fig- ure, the number of slots is equal to 2×3×𝑝 =24. It is worth noting that Figures 8 and ure,9 do the not number refer to of an slots identical is equal machine. to 2×3×𝑝 The winding =24. It of is theworth lower noting layer that is woundedFigures 8 andwith 9the do pitchnot refer 30, whileto an identicalthe other machine. winding Thein the winding upper oflayer the islower wounded layer iswith wounded the pitch with 10. theBDFRM pitch machines30, while maythe other operate winding in sub-synchron in the upperous orlayer super-synchron is wounded ouswith modes. the pitch The 10.op- BDFRMeration modemachines for BDFRGmay operate is shown in sub-synchron in [60,62,63,77–79].ous or super-synchron In [80], the operationous modes. mode The for bothop- erationthe motor mode and for generator BDFRG is in shown BDFMs in [60,62,63,is presented.77–79]. The In operation [80], the operation modes are mode illustrated for both in theFigure motor 9. and generator in BDFMs is presented. The operation modes are illustrated in Sustainability 2021, 13, 842 Figure 9. 11 of 39
8-pole winding 4-pole winding 8-pole winding 4-pole winding
Phase a Phase a
Phase b Phase b
Phase c Phase c
FigureFigure 7. 7. DiagramDiagram of of a a BDFRM BDFRM stator stator with with two two 8-pole 8-pole and and 4-pole 4-pole wi windingsndings [45] [45] (The (The 8-pole 8-pole winding winding is is the the power power winding, winding, Figurewhich 7. is Diagram wounded of in a BDFRMthe inner stator side withof the two stator, 8-pole while and the 4-pole 4-pole windings winding [45] is (Thethe control 8-pole windingwinding, is which the power is wounded winding, in which is wounded in the inner side of the stator, while the 4-pole winding is the control winding, which is wounded in the whichthe outer is wounded side of it). in the inner side of the stator, while the 4-pole winding is the control winding, which is wounded in theouter outer side side of of it). it).
A 7 8 9 10 11 12 A1112 23 24 35 36 A 7 8 9 10 11 12 A1112 23 24 35 36 43 44 45 46 47 48 43 44 45 46 47 48 47 48 59 60 71 72 47 48 59 60 71 72
B 31 32 33 34 35 36 B1920 31 32 43 44 B 31 32 33 34 35 36 B1920 31 32 43 44 67 68 69 70 71 72 67 68 69 70 71 72 55 56 67 68 7 8 55 56 67 68 7 8
C 55 56 57 58 59 60 C2728 15 16 51 52 C 55 56 57 58 59 60 C2728 15 16 51 52 19 20 21 22 23 24 19 20 21 22 23 24 39 40 3 4 63 64 39 40 3 4 63 64
(a) (b) (a) (b) Figure 8. Winding connection diagrams of a typical stator: (a) Power winding, and (b) Control winding. Sustainability 2021, 13, x FOR PEER REVIEW 11 of 38
Sustainability 2021, 13, 842 12 of 39 Figure 8. Winding connection diagrams of a typical stator: (a) Power winding, and (b) Control winding.
Pp Pp Motoring mode
Pm Pc Pm Pc
Sub-Synchronous Super-Synchronous
Pp Pp Generating mode
Pm Pc Pm Pc
Sub-Synchronous Super-Synchronous
Figure 9. The reference active power directio directionsns for motoring and generating mo modesdes and super-synchronous and and sub- synchronous modes [[80].80].
Electrical EquivalentEquivalent Circuit Circuit (EEC) of BDFRM(EEC) isof presented BDFRM in [ 6,17is,34 ,48presented,50,54,56, 57,73in, [6,17,34,48,50,54,56,57,72,81–87].81–87]. In [56], the equivalent electric In [56], circuit the equivalent for both the electric case of circuit considering for both and the ignoring case of the rotor current is presented. A typical equivalent electric circuit of BDFRM ignoring rotor considering and ignoring the rotor current is presented. A typical equivalent electric→ cir-→ cuitcurrent of BDFRM is presented ignoring in Figure rotor 10 current. The power is presented and control in Figure induction 10. The voltage, power i.e., andEp controland Ec are as follows: 𝐸 ⃗ 𝐸 ⃗ induction voltage, i.e., and are→ as follows:→ Ep = jωp Lpcip]γ (28) ω → ω → Rp jLpp jL R Ec = jωc Lccpc ic ]γ c (29) where Lpc is the mutual inductance between the windings, γ is the initial rotor position and it is equal to prθrm0. This term is equivalent to the load angle in a conventional synchronous machine.ip θrm0 is the initial position of θrm and as mentioned,ic θrm is the mechanical angular velocity of the rotor (mech.-rad./s).+ DifferentE γ approaches are investigated to model the V BDFRM with the aim of analyzing– its operation.c Classification of the BDFMsV modeling p approaches is performed in Figure 11. c γ + In this study,Ep the previous– research works are compared from different points of view. Table2 lists the main characteristics of the previous research studies in the field of BDFRM.
Power Winding Control Winding Figure 10. EEC of BDFRM [34,54,57,82].
𝐸 ⃗ = 𝑗𝜔 𝐿 𝚤 ⃗∡𝛾 (28)