Nine-Phase Six-Terminal Induction Machine ...

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Nine-Phase Six-Terminal Induction Machine Modelling using Vector Space Decomposition Ayman S. Abdel-Khalik, Senior Member, IEEE, Ahmed M. Massoud, Senior Member, IEEE, and, Shehab Ahmed, Senior Member, IEEE

Abstract— The nine-phase six-terminal Induction Machine (IM) has been recently proposed as a promising contender to the conventional asymmetrical six-phase type in terms of torque density, stator winding simplicity, and fault tolerant capability. The stator is composed of nine phases, which are connected in a fashion to only provide six stator terminals. Therefore, this connection combines the high performance of a nine-phase winding with the terminal behaviour of a six-phase machine. This paper introduces the machine mathematical model based on the Vector Space Decomposition (VSD) modelling approach. The required current and voltage sequence transformation matrices are derived such that the machine is mathematically regarded as an equivalent six-phase IM with only three decoupled subspaces. This way, the same VSDbased controller structures conventionally applied to sixphase based systems can be preserved. A 1.5Hp prototype IM is used to experimentally validate the machine model under both healthy and open-phase conditions. Index Terms— Asymmetrical six-phase winding, ninephase machine, vector space decomposition, high-power machines, fault-tolerant operation, dynamic modelling.

I. INTRODUCTION

T

HE growing demand for high fault-tolerance capability in modern drive systems has in turn provoked the upward proliferation of multiphase machines in many industrial applications. Considerable effort has therefore been devoted in the literature to investigate different types of multiphase machines [1]. Multiphase machines with multiple three-phase winding sets have demonstrated their attractiveness, especially in high-power drives and multi-MW wind energy conversion systems [2]. Standard three-phase converters can still be used to supply these machines. A well-known example of multiple winding three-phase machines is the asymmetrical (or split-phase) six-phase (A6P) machine. The A6P machines have inevitably been prevailing in most practical applications due to different system cost/size trade-offs. Therefore, the control of this machine topology is still the focus of many researchers in various innovative Manuscript received January 10, 2018; revised March 20, 2018 and April 16, 2018; accepted April 23, 2018. A. S. Abdel-Khalik is with Electrical Engineering Department, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt (e-mail: [email protected]). A. M. Massoud is with the Department of Electrical Engineering, Qatar University, Doha, Qatar (e-mail: [email protected]). S. Ahmed is with Texas A&M University at Qatar, Doha, Qatar (email: [email protected]).

applications [3]-[7]. Moreover, some operational problems associated with this specific winding topology still require more investigation. This includes the intolerable circulating stator currents due to the relatively small secondary subspace impedance [8] and the impact of mutual coupling between different winding sets [9]. In [9], a new nine-phase six-terminal (9P6T) Induction Machine (IM) with a concentrated single layer winding layout has been proposed as a promising alternative to the conventional A6P machines in medium-voltage high-power applications. This winding layout is capable of providing a unity winding factor by employing a single-layer concentrated winding layout, which highly simplifies the winding construction and insulation, while increasing the achievable slot fill factor. Additionally, the induced line current ripple was highly suppressed, owing to the higher secondary subspace impedance. This 9P6T winding layout employs the asymmetrical nine-phase stator winding shown in Fig. 1(a). However, the different phases are connected using the connection shown in Fig. 1(b) in order to create six-line terminals. Therefore, under balanced operation, the current phase shift between different winding sets will be -200, while the phase shift between the terminal line currents will be -400 to emulate the standard asymmetrical nine-phase case [9]. It has also been concluded that the required voltage phase shift between the two applied three line-voltage groups (abc1 and abc2) should be -26.160. These anomalous voltage and current phase shift angles stand out as a substantial difference from A6P machines, where both the voltage and current phase shifts are typically -300. As far as the fault tolerant capability is concerned, the postfault control of the 9P6T IM was presented in [10] and compared with the conventional A6P case. The comparison showed that the improved performance of this promising connection is not only limited to the healthy case but was also extended to the postfault operation. The maximum achievable torque using the well-known postfault control strategies was notably improved when compared with the conventional A6P with a single neutral (1N) configuration; the latter has recently been shown optimum among different standard six-phase winding layouts [5]. Due to the lack of a clear established mathematical model, the machine controller provided in [10] was therefore based on the well-known double dq modelling approach that is characterized by its heavy cross-coupling between voltage equations of each winding set. Besides, it has been proven in the literature that the two-individual dq current

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control may incorporate some instability problems if there is strong magnetic coupling between different winding sets [11]. Another modelling approach, which has been predominantly preferred in controlling multiphase machines, is the Vector Space Decomposition (VSD) modelling approach [12]. In VSD modelling, the vector space of an n-phase machine is decomposed into a single flux/torque producing (α-β) subspace and multiple orthogonal non-flux/torque secondary (x-y) subspaces, where the different low order current/voltage harmonics will be mapped to [2]. When compared with twoindividual dq current control, the VSD-based current control theoretically provides a complete decoupling between different subspaces. This offers more flexibility to individually optimize the PI gains of current controllers of the (α-β) and (x-y) subspaces [11]. Unfortunately, the VSD transformations for different layouts of conventional six-phase windings [5] cannot directly be applied to the 9P6T winding due to the irregular voltage and current phase shift relations. Nevertheless, and from a mathematical perspective, although the machine originally has nine phases, it should eventually be equivalent to a six-phase machine. The analysis given in [10] proved that the number of independent states of a 9P6T winding is only five, which is similar to an A6P with 1N configuration. This paper introduces the mathematical modelling of a 9P6T machine based on VSD modelling approach and is organized as follows. First, Section II compares the harmonic mapping of different subspaces for both conventional A6T and 9P6T stators. Starting from the conventional dq model of the ninephase IM, the obtained coupling between different subspaces due to the 9P6T connection is then clarified in Section III. Thus, the required transformation matrices that decompose the machine terminal values into their equivalent and independent six sequence components are therefore derived. This way, the machine model will only comprise three decoupled subspaces, namely, (α-β), (x-y), and zero subspaces. With this given fact, Section IV then proves that VSD based controller structure of conventional A6P induction machines can still be kept up under both healthy and postfault operation. The developed model is then experimentally verified in Section V using a 1.5kW prototype IM under different operating conditions. Finally, the main conclusions are presented in Section VI. s1

200

s1

s2

a1

s

200 3

b1 c1

s9

s4

Nc s2

s7

s5 s3

a2

s8

s7

s5

s4

b2 c2

s8

s6 s9

Nc

Nc/1.88

s6 (a) (b) Fig. 1. (a) Conventional asymmetrical nine-phase winding. (b) Sixterminal connection.

II. EFFECT OF STATOR EXCITATION ON MMF DISTRIBUTION In this section, the Magneto Motive Force (MMF) distributions of both 9P6T and A6P windings with 1N are

compared under different subspace excitations. A 36-slot/4pole stator frame will be used as an illustrative example. A. MMF Spectrum of A6P Winding It is well known that the MMF distribution of a multiphase winding can be modified based on the excited subspace. In A6P machines, the MMF distribution will therefore depend on the current phase shift angle between the two three-phase winding sets [13]. For example, a fundamental flux can be produced by exciting the fundamental (α-β) subspace using a phase shift angle between the two current groups of -300. On the other hand, the fundamental component can completely be eliminated by exciting the secondary (x-y) subspace, which corresponds to a current phase shift of -2100 [13]. This can be proved as follows. The MMF space phasor of any harmonic component (ℎ = 1, −5,7, … . ) of a three-phase winding fed from balanced three-phase currents can be written as; 3 4 𝑁𝑐 𝐼𝑚 𝐾𝑤ℎ 𝑗(𝑝ℎ𝜃−𝜔𝑡) (1) 𝐹̅𝑠ℎ = ∙ 𝑒 2 𝜋 2 ℎ where, 𝑁𝑐 is the number of turns per phase, 𝐼𝑚 is the phase current magnitude, 𝑝 is the number of pole pairs, 𝜔 is the supply frequency, 𝜃 is the stator peripheral angle, and 𝐾𝑤ℎ is the winding factor of the harmonic order ℎ (the sign of the harmonic order indicates the rotation direction of the corresponding flux component). For an A6P winding with two three-phase winding sets having a 300 spatial phase shift, the total MMF space phasor of the harmonic component ℎ is given by; 𝜋 3𝑁𝑐 𝐼𝑚 𝐾𝑤ℎ 𝑗(𝑝ℎ𝜃−𝜔𝑡) ) ℎ (𝑒 𝐹̅𝑠(A6P) = + 𝑒 𝑗ℎ(𝑝𝜃−6 ∙ 𝑒 −𝑗(𝜔𝑡−𝛾) ) 𝜋 ℎ (2) where 𝛾 is the current phase shift angle between the two threephase winding sets. This can be reduced to; 𝜋ℎ 3𝑁𝑐 𝐼𝑚 𝐾𝑤ℎ 𝑗(𝑝ℎ𝜃−𝜔𝑡) ) ℎ (3) 𝐹̅𝑠(A6P) = 𝑒 ∙ (1 + 𝑒 𝑗(𝛾− 6 ) 𝜋 ℎ 0 It is clear that for 𝛾 = 30 , the MMF exhibits a 2p-pole distribution, while the MMF harmonic components corresponding to ℎ = −5 or 7 add to zero. On the other hand, for 𝛾 = 2100 , the MMF fundamental component (ℎ = 1) is cancelled, and the first dominant MMF harmonics will be the 5th and 7th order harmonics. Using a 36-slot/4-pole stator frame with an A6P double layer distributed winding [9], the MMF spectra under these two different excitation cases are shown in per unit in Figs. 2(a) and (b), respectively. Clearly, the 5th and 7th order harmonics are dominating under (x-y) subspace excitation, while having small magnitudes compared to the fundamental component. This is why, this subspace is characterized by its low input inductance [14]. Under 1N connection, a zero-sequence circulating component will also take place, which corresponds to a pulsating third order harmonic induction with the maximum magnitude shown in Fig. 2(c) [14]. Needless to say, this component is avoided using an isolated neutral configuration. B. MMF Spectrum of 9P6T Winding The 9P6T winding shown in Fig. 1(a) comprises three threephase winding sets shifted in space by 200 electrical degrees. With the given connection shown in Fig. 1(b), the phase

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currents of the winding group (s2, s5, s8) represent the summation of the two three-phase current groups (s1, s4, s7) and (s3, s6, s9). Assuming a general current phase shift angle 𝛾 between the two winding sets (s1, s4, s7) and (s3, s6, s9), the winding current relation is therefore given by; 𝑖𝑠258 = 𝑖𝑠147 + 𝑖𝑠369 = 𝐼𝑚 𝑒 −𝑗𝜔𝑡 + 𝐼𝑚 𝑒−𝑗(𝜔𝑡−𝛾) (4) = 2cos(𝛾⁄2) 𝐼𝑚 ∙ 𝑒 −𝑗(𝜔𝑡−𝛾⁄2) To emulate a conventional asymmetrical 9-phase winding, the phase shift between the current groups of the winding sets (s1, s4, s7) and (s3, s6, s9) should be -400. The magnitude of the current group of the winding set (s2, s5, s8) will therefore be 1.88 times the line current with a phase shift of -200, as given by (4). Thus, the number of turns of the winding group (s2, s5, s8) should be 0.532 times the number of turns of other phases to maintain a balanced MMF production from all phases, while the conductor cross sectional area is increased by a factor of 1.88 to maintain the same copper volume for all stator phases [9]. This case corresponds to a fundamental α-β subspace excitation. This can be proved by summing up the MMF space phasors of the three three-phase winding sets. Those for the winding groups (s1, s4, s7) and (s3, s6, s9) are given by; 3𝑁𝑐 𝐼𝑚 𝐾𝑤ℎ 𝑗(𝑝ℎ𝜃−𝜔𝑡) ℎ (5) 𝐹̅𝑠(147) = 𝑒 𝜋 ℎ 2𝜋 3𝑁𝑐 𝐼𝑚 𝐾𝑤ℎ 𝑗ℎ(𝑝𝜃− ) −𝑗(𝜔𝑡−𝛾) ℎ 9 ∙𝑒 (6) (𝑒 ) 𝐹̅𝑠(369) = 𝜋 ℎ It can therefore be shown that the MMF space phasor of the winding group (s2, s5, s8) will be given by; ℎ 𝐹̅𝑠(258) = 𝛾 (7) 6(0.532𝑁𝑐 ) cos 2 𝐼𝑚 ∙ 𝐾𝑤ℎ 𝑗ℎ(𝑝𝜃−𝜋) −𝑗(𝜔𝑡−𝛾) 9 ∙𝑒 2 ∙𝑒 𝜋 ℎ The total MMF space phasor is therefore given by; ℎ ℎ ℎ ℎ 𝐹̅𝑠(9P6T) = 𝐹̅𝑠(147) + 𝐹̅𝑠(369) + 𝐹̅𝑠(258) (8) which can be put is the following simple form; 3𝑁𝑐 𝐼𝑚 𝐾𝑤ℎ 𝑗(𝑝ℎ𝜃−𝜔𝑡) ℎ 𝐹̅𝑠(9P6T) = 𝑒 𝜋 ℎ (9) 2𝜋 𝛾 𝜋 𝛾 ) ) ∙ [1 + 𝑒 𝑗(𝛾−ℎ 9 + 1.064 cos ∙ 𝑒 𝑗(2−ℎ 9 ] 2 From (9) and with 𝛾 = 400 , the 5th and 7th order harmonics are completely eliminated while the MMF distribution exhibits a fundamental flux distribution, which clearly represents the α-β subspace excitation case for this winding layout. On the other hand, the fundamental MMF component will completely diminish for a phase shift angle of 206.160, while the dominant air gap harmonics will be the 5th and 7th order harmonics, similar to the x-y excitation in A6P windings. The MMF distribution for the same 36-slot stator equipped with an equivalent 9P6T winding is shown in Figs. 3(a) and (b) for the current phase shift angles 𝛾 = 400 and 206.160, respectively. The corresponding current and MMF phasor diagrams are shown in Figs. 4(a) and (b), respectively. As for the zero-sequence component, and similar to the A6P winding, this current component will give rise to a third harmonic pulsating flux component, as is clear from Fig. 3(c), and is slightly higher in the 9P6T winding. One of the main problems of conventional A6P stators with double layer windings is their relatively low inductance of the (x-y) subspace because of the effect of the mutual leakage coupling between different coil sides sharing same slots. This

yields intolerable circulating current, which highly distorts the phase current waveform with a significant ripple current component if not properly controlled [14]. By comparing the magnitudes of different MMF harmonic components in relation to the fundamental components in both A6P and 9P6T winding layouts, it is evident that the magnitudes of the 5th and 7th MMF harmonics, which are both mapped to the (x-y) subspace, are relatively higher in the 9P6T winding thanks to its higher winding factor and its single layer layout. Yet, the input inductance of this subspace will therefore be higher, and hence, the induced current ripples in the (x-y) subspace will notably be suppressed in 9P6T winding, which has been proved in [10]. III. PROPOSED MACHINE DYNAMIC MODEL In this section, the equivalent six-phase VSD modelling of the 9P6T IM is inferred from the conventional dq model of a nine-phase machine [15]. The first step is to derive the suitable VSD transformation that transforms the line values to their equivalent sequence components. This transformation will then be used to obtain the machine equivalent six-phase dq model. The following assumptions will be made in the given derivation: 1) Linear magnetic circuit is assumed, and the effects of hysteresis and eddy currents are neglected. 2) Dominant harmonic for each subspace is only considered, while higher order harmonics are neglected. 3) All sources of asymmetries are neglected. A. Nine-phase DQ Model For the original asymmetrical 9-phase winding, the generalized Clarke’s transformation, [𝑇9 ], for a nine-phase system [15] decomposes the nine-phase quantities [𝑥 𝑠1→𝑠9 ], where, 𝑥 ∈ {𝑣, 𝑖, 𝜆}, to four orthogonal planes and one zero 9 ] = [𝑇9 ][𝑥 𝑠1→𝑠9 ]. These decoupled sequence component, [𝑥𝛼𝛽 subspaces represent the effect of the fundamental, third, fifth and seventh space harmonics. The machine voltage equations are then given in matrix from as; 𝑑 9 9 [𝑣𝑠𝛼𝛽 ] = [𝑅𝑠9 ][𝑖𝑠𝛼𝛽 ] + [𝜆9𝑠𝛼𝛽 ] (10) 𝑑𝑡 𝑑

9 [0] = [𝑅𝑟9 ][𝑖𝑟𝛼𝛽 ] + [𝜆9𝑟𝛼𝛽 ] + [𝐺 9 ]𝜔𝑟 [𝜆9𝑟𝛼𝛽 ] (11) 𝑑𝑡 The flux linkage equations are defined by (12) and (13). 9 9 (12) [𝜆9𝑠𝛼𝛽 ] = [𝐿9𝑠 ][𝑖𝑠𝛼𝛽 ] + [𝐿9𝑚 ][𝑖𝑟𝛼𝛽 ] 9 9 9 9 9 (13) [𝜆𝑟𝛼𝛽 ] = [𝐿𝑟 ][𝑖𝑟𝛼𝛽 ] + [𝐿𝑚 ][𝑖𝑠𝛼𝛽 ] 9 The sequence vector [𝑥𝛼𝛽 ], which may represent voltage, current, or flux linkage, is defined as 9 [𝑥𝛼𝛽 ] = [𝑥𝛼1 𝑥𝛽1 𝑥𝛼3 𝑥𝛽3 𝑥𝛼5 𝑥𝛽5 𝑥𝛼7 𝑥𝛽7 𝑥0 ]𝑡 The subscripts 𝑠 and 𝑟 denote referencing to stator and rotor, respectively. The impedance matrices are given as functions of different subspace parameters as [𝑅𝑠9 ] = 𝑑𝑖𝑎𝑔([𝑟𝑠 𝑟𝑠 𝑟𝑠 𝑟𝑠 𝑟𝑠 𝑟𝑠 𝑟𝑠 𝑟𝑠 𝑟𝑠 ]), [𝑅𝑟9 ] = 𝑑𝑖𝑎𝑔([𝑟𝑟1 𝑟𝑟1 𝑟𝑟3 𝑟𝑟3 𝑟𝑟5 𝑟𝑟5 𝑟𝑟7 𝑟𝑟7 0]), [𝐿9𝑠 ] = 𝑑𝑖𝑎𝑔([𝐿𝑠1 𝐿𝑠1 𝐿𝑠3 𝐿𝑠3 𝐿𝑠5 𝐿𝑠5 𝐿𝑠7 𝐿𝑠7 𝑙𝑠0 ]), [𝐿9𝑟 ] = 𝑑𝑖𝑎𝑔 ([𝐿𝑟1 𝐿𝑟1 𝐿𝑟3 𝐿𝑟3 𝐿𝑟5 𝐿𝑟5 𝐿𝑟7 𝐿𝑟7 0]), [𝐿9𝑚 ] = 𝑑𝑖𝑎𝑔([𝐿𝑚1 𝐿𝑚1 𝐿𝑚3 𝐿𝑚3 𝐿𝑚5 𝐿𝑚5 𝐿𝑚7 𝐿𝑚7 0]),

0.5 0

0

5

10

15

MMF har., pu

0.04

1

MMF har., pu

MMF har., pu

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0.02

0

20

0

Harmonic order

5

10 15 Harmonic order

(a)

Current phasors

0.4 0.2 0

20

200 200

0

5

10

15

Is369 =1pu

0

5

10

15

20

Harmonic order

0.2

MMF har., pu

MMF har., pu

MMF har., pu

0

0.1 0

0

5

10

15

20

Fs369 =1pu 0.2

0

5

10

15

and [𝐺 9 ] = 0 1 0 3 0 5 0 7 ] [ ] [ ] [ ] 0]). 𝑑𝑖𝑎𝑔 ([[ −1 0 −3 0 −7 0 −5 0 Since a simple single-layer concentrated winding is employed, the stator impedance (𝑟𝑠 and 𝑙𝑠 ) of different subspaces can be assumed the same [2]. This is different from a conventional A6P machine with a double layer winding, where the stator inductance is affected by the mutual leakage inductance between different layers that yields different stator leakage inductance among different subspaces [14], [16]. The average torque, 𝑇𝑒 , expression can be calculated from; 9

𝑡

(14)

B. Reducing Winding to the Supply Side As explained in [10], some of the phases have different numbers of turns, namely phases s2, s5, s8, which entails winding transformation to reduce all phase variables. Voltage and current are referred to the stator terminal side using the winding turns ratio 𝑛 = 0.532. The current connection matrix [𝐶𝑖 ] that correlates the referred phase currents with the input 𝑠 ] = [𝐶𝑖 ][𝑖𝑙𝑠 ] is derived in [10] and given by line currents [𝑖𝑝ℎ (15), where, [𝑖𝑙𝑠 ] = [𝑖𝑎1 𝑖𝑏1 𝑖𝑐1 𝑖𝑎2 𝑖𝑏2 𝑖𝑐2 ]𝑡 and ′ 𝑠 ′ ′ [𝑖𝑝ℎ ] = [𝑖𝑠1 𝑖𝑠2 𝑖𝑠3 𝑖𝑠4 𝑖𝑠5 𝑖𝑠6 𝑖𝑠7 𝑖𝑠8 𝑖𝑠9 ]𝑡 are the line and phase current vectors, respectively. 1 𝑛 0 0 0 0 0 0 0𝑡 0 0 0 1 𝑛 0 0 0 0 [𝐶𝑖 ] = 0 0 0 0 0 0 1 𝑛 0 (15) 0 𝑛 1 0 0 0 0 0 0 0 0 0 0 𝑛 1 0 0 0 [0 0 0 0 0 0 0 𝑛 1] The inverter phase voltage vector is obtained from the phase 𝑠 ] 𝑠 ] using the voltage connection voltage vector [𝑣𝑖𝑛𝑣 = [𝐶𝑣 ][𝑣𝑝ℎ 𝑡 matrix [𝐶𝑣 ] = [𝐶𝑖 ] [10], where ′ 𝑠 ′ ′ [𝑣𝑝ℎ ] = [𝑣𝑠1 𝑣𝑠2 𝑣𝑠3 𝑣𝑠4 𝑣𝑠5 𝑣𝑠6 𝑣𝑠7 𝑣𝑠8 𝑣𝑠9 ]𝑡 𝑠 𝑡 and [𝑣𝑖𝑛𝑣 ] = [𝑣𝑎1 𝑣𝑏1 𝑣𝑐1 𝑣𝑎2 𝑣𝑏2 𝑣𝑐2] . C. Vector Space Decomposition From the mathematical point of view, although there are nine phase variables, the number of independent currents will be only five, with the fact that the summation of all line currents

76.920

Fs258 =0.241pu 76.920

Fs258 =1pu Fs369 =1pu

76.920

Current phasors Is147 =1pu

MMF phasors Fs147 =1pu

20

Harmonic order

Harmonic order

Is258 =0.453pu 76.920

0.4

0

Fs147 =1pu

(a) Is369 =1pu

(a) (b) (c) Fig. 3. MMF spectra of 9P6T winding for different excitations. (a) α-β (𝛾 = 400 ). (b) x-y (𝛾 = 206.160 ). (c) 0+0- .

9 ] [𝐺 9 ][𝜆9𝑟𝛼𝛽 ] 𝑇𝑒 = 2 𝑝[𝑖𝑟𝛼𝛽 where, 𝑝 is the number of machine pole-pairs.

Is258 =1.88pu

(c)

Fig. 2. MMF spectra of A6P winding for different excitations. (a) α-β (𝛾 = 300 ). (b) x-y (𝛾 = 2100 ). (c) 0+0- .

0.5

200 200

20

Harmonic order

(b)

1

MMF phasors

Is147 =1pu

(b) Fig. 4. Current and MMF phasor diagrams for different excitations. (a) α-β. (b) x-y.

should always be zero. Therefore, the zero-sequence component will always add to zero [10]. There will also be some sort of state dependency between different 𝛼𝛽𝑗 sequence components. As the number of independent states should equal those of a conventional A6P winding, the terminal values of a 9P6T machine can similarly be decomposed to only three equivalent subspaces, namely, 𝛼𝛽6 , 𝑥𝑦 6 , 0+ 06− . The main torque producing 𝛼𝛽1 components of the nine-phase winding will be mapped to the fundamental subspace 𝛼𝛽6 of this equivalent six-phase system. Both 5th and 7th harmonic components (𝛼𝛽5 and 𝛼𝛽7 ) are mapped to the equivalent 𝑥𝑦 6 subspace; keeping in mind that the 5th and 7th air gap flux components should be travelling in opposite directions in this equivalent six-phase system. Finally, the third harmonic components 𝛼𝛽3 and the single zero-sequence component will belong to the zero-sequence subspace 0+ 06−. This sequence current component dependency can mathematically be proven by expressing the nine-phase sequence current components using the independent line current vector [𝑖𝑧 ] = [𝑖𝑎1 𝑖𝑏1 𝑖𝑐1 𝑖𝑎2 𝑖𝑏2 ]𝑡 using (16). 𝑠9 𝑠 [𝑖𝛼𝛽 ] = [𝑇9 ][𝑖𝑝ℎ ] = [𝑇9 ][𝐶𝑖 ][𝐵][𝑖𝑧 ] (16) where, 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 [𝐵] = 0 0 0 1 0 0 0 0 0 1 [−1 −1 −1 −1 −1]

(17)

The rank of the product [𝑇9 ][𝐶𝑖 ][𝐵] is only five, which entails three dependent sequence current components, after discarding the null zero-sequence current component 𝑖𝑠0 . The following relations can then be derived from (16); 1 1 1 0 0 𝑖 1 [𝑖𝑠𝛼3 ] = [− 1 − 1 − 1 0 0] [𝑖𝑧 ] (18) 3 𝑠𝛽3 √3 √3 √3 𝑖 𝑖 [𝑖𝑠𝛼7 ] = 0.3264 [ 1 √3 ] [𝑖𝑠𝛼5 ] (19) 𝑠𝛽7 √3 −1 𝑠𝛽5 1

Clearly, 𝑖𝑠𝛽3 = − 𝑖𝑠𝛼3 , which emulates the relation between √3 zero sequence current components in A6P winding with 1N 6 6 arrangement, where 𝑖0+ = −𝑖0− . From (19), the sequence components 𝑖𝛼7 , 𝑖𝛽7 are shifted by 60 degrees with respect to 𝑖𝛼5 , 𝑖𝛽5 and rotate in opposite

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direction, which complies with the well-known relation between the 5th and 7th harmonic components in either three- or six-phase systems. Now, it is required to suggest a suitable connection matrix 𝑠9 ] to the that correlates the original sequence current vector [𝑖𝛼𝛽 𝑠6 equivalent sequence vector [𝑖𝛼𝛽 ]. The suggested current invariant connection matrix is therefore given by (20); 𝑠6 𝑠9 [𝑖𝛼𝛽 ] = [𝐷𝑖69 ][𝑖𝛼𝛽 ] (20) where [𝐷𝑖69 ] is defined as; 𝛼6 1 0 0 0 0 0 0 0 0 𝛽6 0 1 0 0 0 0 0 0 0 𝑥6 0 0 0 0 1 0 1 0 0 [𝐷𝑖69 ] = 6 (21) 𝑦 0 0 0 0 0 1 0 −1 0 06+ 0 0 1 0 0 0 0 0 0 06− [0 0 0 √3 0 0 0 0 0] The transformation [𝐷𝑖69 ] is defined such that the equivalent 𝛼𝛽6 components will be equal to those of the fundamental 𝛼𝛽1 subspace. For the 𝑥𝑦 6 subspace, the equivalent 𝑥 component represents the summation of the 𝛼′𝑠 components of the 5th and 7th subspaces, while the equivalent 𝑦 component is defined as the difference between the 𝛽′𝑠 components to account for the inherent opposite sequence relation between these two harmonic components. Finally, the equivalent zero-sequence components 0+ 06− will be opposite and equal, similar to an A6P winding with 1N configuration. The relation between the line currents [𝑖𝑙𝑠 ] and the sequence 𝑠6 ] can then be expressed using (22). currents [𝑖𝛼𝛽 𝑠6 [𝑖𝛼𝛽 ] = [𝐷𝑖69 ][𝑇9 ][𝐶𝑖 ][𝑖𝑙𝑠 ] = [𝑇6𝑖 ][𝑖𝑙𝑠 ] (22) where the required current transformation matrix, [𝑇6𝑖 ], is calculated as given by (23), assuming [𝑇9 ] is a current-invariant Clarke’s transformation. 1 [𝑇6𝑖 ] = × 3 1 −0.605 −0.395 0.844 −0.898 0.054 0.121 0.805 −0.927 0.550 0.456 −1.006 1 −0.395 −0.605 −0.844 0.898 −0.054 0.121 −0.927 0.805 0.550 0.456 −1.006 0.844 0.844 0.844 −0.156 −0.156 −0.156 [0.532 0.532 0.532 1.532 1.532 1.532 ] (23) From the inverse of (23), balanced 𝛼𝛽6 current components will produce two three phase current sets (abc1 and abc2) with a phase shift angle of -400. The phase shift angle will however be -206.160 for an 𝑥𝑦 6 excitation, which complies with the discussion made in section II. Since [𝐷𝑖69 ] is not a square matrix, the inverse transformation, [𝐷𝑖96 ], is found as follows 𝑠9 𝑠6 𝑠6 [𝑖𝛼𝛽 ] = [𝑇9 ][𝐶𝑖 ][𝑇6𝑖 ]−1 [𝑖𝛼𝛽 ] = [𝐷𝑖96 ][𝑖𝛼𝛽 ] (24) Then [𝐷𝑖96 ] is calculated as shown in (25). 1 0 0 0 [𝐷𝑖96 ] = 0 0 0 0 [0

0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0.577 0.638 −0.272 0 0 0.272 0.638 0 0 0.362 0.272 0 0 0.272 −0.362 0 0 0 0 0.113 0.113]

(25)

𝑠9 𝑠6 ] = [𝐷𝑣96 ][𝑣𝛼𝛽 ] The sequence voltage transformation [𝑣𝛼𝛽 𝑡 can be obtained by defining [𝐷𝑣96 ] = 𝐾[𝐷𝑖69 ] . Since current invariant transformation is adopted in (21), the required value for the constant 𝐾 to maintain same total power is found to be 1/1.51. Obviously, the 1.51 factor represents the relation between the inverter and machine winding phase voltages under 𝛼𝛽6 excitation, as proved in [9]. Interestingly enough, this calculated factor approximately equals the phase voltage relation between two equivalent nine- and six-phase machines having same phase currents, which equals 6/9 [17]. The relation between the machine terminal voltages and the equivalent sequence voltage components can therefore be written as, 𝑠 ] 𝑠9 𝑠6 [𝑣𝑖𝑛𝑣 ] = [𝐶𝑣 ][𝑇9 ]−1 [𝐷𝑣96 ][𝑣𝛼𝛽 ] (26) = [𝐶𝑣 ][𝑇9 ]−1 [𝑣𝛼𝛽 𝑠6 [𝑣𝛼𝛽 ]= The equivalent voltage transformation 𝑠 ] [𝑇6𝑣 ][𝑣𝑖𝑛𝑣 [𝑇 ], matrix, 6𝑣 is then calculated as; [𝑇6𝑣 ] = 0.336 −0.168 −0.168 0.257 −0.316 0.058 0 0.291 −0.291 0.216 0.115 −0.331 0.336 −0.168 −0.168 −0.301 0.280 0.023 0 −0.291 0.291 0.148 0.187 −0.335 0.374 0.374 0.374 −0.130 −0.130 −0.130 [0.038 0.038 0.038 0.206 0.206 0.206 ] (27) Unlike a conventional A6P system, the voltage and current transformations, [𝑇6𝑣 ] and [𝑇6𝑖 ] are clearly different. It can also be shown that the inverse of (27) will give two balanced threephase voltage groups (abc1 and abc2) with voltage phase shift angles of −26.160 and 1400 under balanced 𝛼𝛽6 and 𝑥𝑦 6 excitations, respectively. Similarly, the inverse of [𝐷𝑣96 ] is found from 𝑠6 𝑠9 𝑠9 [𝑣𝛼𝛽 ] = [𝑇6𝑣 ][𝐶𝑣 ][𝑇9 ]−1 [𝑣𝛼𝛽 ] = [𝐷𝑣69 ][𝑣𝛼𝛽 ] (28) where [𝐷𝑣69 ] = 1.51 ×

1 0 0 0 0 [0

0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.638 0.272 0.362 0.272 0 0 0 −0.272 0.638 0.272 −0.362 0 1 0 0 0 0 0 0.227 0 0.577 0 0 0 0 0.227]

(29) D. Equivalent Six-Phase DQ model The derived transformation and connection matrices are used along with the original dq model of the nine-phase machine to obtain the equivalent six-phase dq model. The voltage and current connection matrices [𝐷𝑣69 ] and [𝐷𝑖69 ] are applied to voltage equations given by (10) and (11) and the equivalent sixphase dq model will then be given by (30) and (31). 𝑑 6 6 (30) [𝑣𝑠𝛼𝛽 ] = [𝑅𝑠6 ][𝑖𝑠𝛼𝛽 ] + [𝜆6𝑠𝛼𝛽 ] 𝑑𝑡 𝑑 6 (31) [0] = [𝑅𝑟6 ][𝑖𝑟𝛼𝛽 ] + [𝜆6𝑟𝛼𝛽 ] + [𝐺 6 ]𝜔𝑟 [𝜆6𝑟𝛼𝛽 ] 𝑑𝑡 where the equivalent impedance matrices are given by, [𝑅𝑠6 ] = [𝐷𝑣69 ][𝑅𝑠9 ][𝐷𝑖96 ], [𝑅𝑟6 ] = [𝐷𝑣69 ][𝑅𝑟9 ][𝐷𝑖96 ], [𝐿6𝑠 ] = [𝐷𝑣69 ][𝐿9𝑠 ][𝐷𝑖96], [𝐿6𝑟 ] = [𝐷𝑣69 ][𝐿9𝑟 ][𝐷𝑖96 ], (32) [𝐿6𝑚 ] = [𝐷𝑣69 ][𝐿9𝑚 ][𝐷𝑖96 ], and [𝐺 6 ] = [𝐷𝑣69 ][𝐺 9 ][𝐷𝑣96 ]

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TABLE I. IMPEDANCE MATRICES OF DIFFERENT SUBSPACES OF THE EQUIVALENT SIX-PHASE MACHINE MODEL. 6 [𝑅𝑠𝛼𝛽 ]

1 1.51𝑟𝑠 [ 0 6 ] [𝑅𝑠𝑥𝑦

6 [𝑅𝑟𝛼𝛽 ]= 1 1.51𝑟𝑟1 [ 0

= 0 ] 1

6 [𝑅𝑠0+0− ]

1.55 = 𝑟𝑠 [ 0.04

0 ] 1

0 ] 1

0.04 ] 0.54

[𝐿6𝑟𝛼𝛽 ] =

1 1.51𝐿𝑠1 ∙ [ 0

6 ] [𝑅𝑟𝑥𝑦 = 𝑟𝑟𝑥𝑦 0 [ ], 0 𝑟𝑟𝑥𝑦 𝑟𝑟𝑥𝑦 = 0.73𝑟𝑟5 + 0.31𝑟𝑟7 6 [𝑅𝑟0+0− ] 1.51 0 = 𝑟𝑟3 [ ] 0 0.5

=

1 1.037𝑟𝑠 [ 0

[𝐿6𝑠𝛼𝛽 ] =

[𝐿6𝑠𝑥𝑦 ] = [

0 ] 1

𝐿𝑠𝑥𝑦 0

1 1.51𝐿𝑟1 ∙ [ 0 0

𝐿𝑠𝑥𝑦

],

𝐿𝑠𝑥𝑦 = 0.73𝐿𝑠5 + 0.31𝐿𝑠7 [𝐿6𝑠0+0− ]

= 1.51𝐿𝑠3 + 0.04𝑙𝑠0 ∙[ 0.04𝑙𝑠0

0.04𝑙𝑠0 ] 0.5𝐿𝑠3 + 0.04𝑙𝑠0

TABLE II. OPTIMAL REFERENCE CURRENTS UNDER DIFFERENT MODES. Variable

Healthy

MT

ML

̅ 𝐼𝑎1 𝐼𝑏1 𝐼𝑐1 = 𝐼𝑎2 𝐼𝑏2 [𝐼𝑐2 ]

1∠00 1∠−1200 1∠1200 1∠−400 1∠−1600 [ 1∠800 ]

0 1.472∠−88.90 1.472∠143.20 1.472∠−18.40 1.472∠171.50 [ 1.472∠34.70 ]

0 1.005∠−120.60 1.005∠120.40 1.817∠−20.70 1.197∠−163.40 [ 1.091∠64.60 ]

0 0 0 [ 0

[𝐾𝑥𝑦 ]

0 ] 0 0 ] 0

[

[𝐾0 ]

−0.617 −0.696 −0.384 [ 0.384 [

id* , iq*

id*

𝜔*

iα , iβ

[P]-1

ix* , iy* ix , iy i0+* , i0-* i0+ , i0-

[P]

PI PI

[P]-1

i0+* , i0-* [K0]

vα , vβ va1 vb1 vc1

vx , vy

[P]

[P]-1

PI PI PI

[P]-1

abc1

v0+, v0-

[P]

Current Controller

abc2

va2 vb2 vc2

[P]

9P6T IM

𝜔

VSC 2 iα , iβ ix , iy i0+ , i0-

ia1 , ib1 , ic1 [T6i]

0 ] 1

1 1.51𝐿𝑚1 [ 0

0 ] 1

6 [𝐺𝛼𝛽 ]= 0 1 [ ] −1 0

[𝐿6𝑟𝑥𝑦 ] = 1 0 𝐿𝑟𝑥𝑦 ∙ [ ] , 0 1 𝐿𝑟𝑥𝑦 = 0.73𝐿𝑟5 + 0.31𝐿𝑟7

[𝐿6𝑚𝑥𝑦] = 1 0 𝐿𝑚𝑥𝑦 [ ], 0 1 𝐿𝑚𝑥𝑦 = 0.73𝐿𝑚5 + 0.31𝐿𝑚7

6 ] [𝐺𝑥𝑦 = 0.544 × 6 −1.21 [ ] 1.21 6

[𝐿6𝑟0+0− ] = 1.51 0 [ ]𝐿 0 0.504 𝑟3

[𝐿6𝑚0+0− ] = 𝐿𝑚3 1.51 0 [ ] 0 0.5

6 [𝐺0+0− ]= 0 3√3 [ ] −√3 0

As is clear from Table I, the equivalent machine impedances of the 𝛼𝛽6 and 𝑥𝑦 6 subspaces are 1.51 and 1.037 times the phase impedance of the original nine-phase machine. The parameters of 𝛼𝛽6 subspace only depend on those of the fundamental 𝛼𝛽1 subspace. On the other hand, the equivalent 𝑥𝑦 6 subspace impedance depends on the parameters of both 5th and 7th harmonic planes. Finally, the equivalent 0+ 06− subspace impedance will depend on the impedances of the 𝛼𝛽3 subspace of the original nine-phase winding. For this latter subspace, a cross coupling between the two zero sequence components, although neglected, appears in both stator resistance and self6 ] and [𝐿6𝑠0+0− ], respectively. This inductance matrices [𝑅𝑠0+0− cross coupling has nothing to do with the rotor circuit since this is mainly due to the effect of the single zero-sequence component of the original nine-phase system. IV. VSD-BASED CURRENT CONTROL

PWM

[T6v]-1 [P]-1

ia1 , ib1 , ic1 VSC 1

[P]

[P]-1

0 ] 0 0 ] 0

ix* , iy*

θs* PI

−0.66 [ 0 −0.34 [ 0.34

[Kxy]

IRFOC

𝜔 iα* , iβ*

iα* , iβ* [D]-1

PI

[P]

−0.197 ] −0.358 0.197 ] −0.197

[𝐿6𝑚𝛼𝛽 ] =

ia2 , ib2 , ic2

ia2 , ib2 , ic2

Fig. 5. VSD based control scheme.

The transformed flux linkage equations will be given as; 6 6 [𝜆6𝑠𝛼𝛽 ] = [𝐿6𝑠 ][𝑖𝑠𝛼𝛽 ] + [𝐿6𝑚 ][𝑖𝑟𝛼𝛽 ] (33) 6 6 6 6 6 [𝜆𝑟𝛼𝛽 ] = [𝐿𝑟 ][𝑖𝑟𝛼𝛽 ] + [𝐿𝑚 ][𝑖𝑠𝛼𝛽 ] (34) The final torque equation will be 𝑡 1 9 6 ] [𝐺 6 ][𝜆6𝑟𝛼𝛽 ] 𝑇𝑒 = 1.51 2 𝑝[𝑖𝑟𝛼𝛽 (35) The impedance matrices given by (32) can be rewritten as (36) and are detailed in Table I. 6 6 ], [𝑅 6 ], [𝑅𝑠𝑥𝑦 [𝑅𝑠6 ] = 𝑑𝑖𝑎𝑔([[𝑅𝑠𝛼𝛽 𝑠0+0− ]]), 6 6 ], [𝑅 6 [𝑅𝑟6 ] = 𝑑𝑖𝑎𝑔([[𝑅𝑟𝛼𝛽 ], [𝑅𝑟𝑥𝑦 𝑟0+0− ]]), 6 6 6 6 ]]), [𝐺 ] = 𝑑𝑖𝑎𝑔([ [𝐺𝛼𝛽 ], [𝐺𝑥𝑦 ] , [𝐺0+0− 6 6 ], 6 6 [𝐿 ] [𝐿 ], [𝐿 [𝐿𝑠 ] = 𝑑𝑖𝑎𝑔([ 𝑠𝛼𝛽 𝑠𝑥𝑦 𝑠0+0− ]) , (36) [𝐿6𝑟 ] = 𝑑𝑖𝑎𝑔([[𝐿6𝑟𝛼𝛽 ], [𝐿6𝑟𝑥𝑦 ], [𝐿6𝑟0+0− ]]), and [𝐿6𝑚 ] = 𝑑𝑖𝑎𝑔([𝐿6𝑚𝛼𝛽 ] [𝐿6𝑚𝑥𝑦 ] [𝐿6𝑚0+0− ])

In this section, the required VSD based controller is derived, which is based on that of conventional A6P induction machines [5]. This controller has the same structure for both healthy and open-line cases. In [10], the postfault optimal reference currents are derived for one-line open under the two possible postfault scenarios, namely, Maximum Torque (MT) and Minimum Loss (ML) modes. These reference line currents are shown in Table II assuming balanced 𝛼𝛽6 current components with 1pu magnitude. Similar to the concept presented in [5] and based on VSD of A6P stator, the 𝑥𝑦 6 and 0+ 06− current components are expressed in terms of the reference torque producing 𝛼𝛽6 components as given by (37). The calculated values for these matrix gains under different modes are also given in Table II. Needless to say, the reference secondary current components are set to zero under balanced healthy case. 𝑖𝛼6 𝑖𝛼6 𝑖6 𝑖6 [𝐾 ] [ 𝑥6 ] = [𝐾𝑥𝑦 ] [ 6 ] and [ 0+ ] [ ] (37) = 0 6 𝑖𝑦 𝑖𝛽 𝑖𝛽6 𝑖0− The standard Indirect Rotor Field Oriented Control (IRFOC) can be used to generate the reference torque producing 𝛼𝛽6 current components. The 𝑥𝑦 6 and 0+ 06− components can then be calculated using (37) based on the suggested postfault scenario. As is clear from Table II, the optimal reference currents will correspond to unbalanced secondary current components and Proportional Resonant (PR) controllers should therefore be used. Alternatively, dual-PI regulators in synchronous and anti-synchronous reference frames can be employed [5], as depicted by the full controller block diagram shown in Fig. 5. The dual PI-controllers are used to control the

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sequence current components obtained from the measured line currents using the matrix [𝑇6𝑖 ]. The sequence voltage components are then transformed back to their phase values using the inverse of the voltage transformation [𝑇6𝑣 ]. Under healthy case, the negative sequence PI controllers of the dq subspace will simply be deactivated to simplify the controller. V. MODEL VERIFICATION A. Experimental IM The proposed VSD modelling is verified using the prototype machine with the specifications given in Table III. The machine is fed from two three-phase PWM modulated inverters connected to the same dc link. The switching frequency is 5kHz. The IM is mechanically loaded using a coupled PM generator connected to load bank resistors. The machine model is verified under open loop free running mode and IRFOC. The controller is implemented using a digital signal processor board eZdspTM based on Texas Instruments F28335 DSP while a CAN bus is used for online measurements. B. Parameter Identification The machine parameters of the fundamental subspace (𝛼𝛽1 = 𝛼𝛽6 ) can be estimated based on the original nine-phase winding as explained in [10] using the terminal measurements of conventional no-load and blocked rotor tests. Unfortunately, the parameters of other subspaces cannot easily be estimated based on experimental tests owing to their small magnetizing inductances. Alternatively, they can roughly be estimated based on the derived formulas given in [17], as shown in Table IV. The equivalent six-phase machine parameters can then be calculated from the estimated parameters of the nine-phase machine based on the derived formulas given in Table I. These calculated parameters are given in Table V. Otherwise, these equivalent parameters can experimentally be obtained by directly applying the identification technique given in [14] for A6P machines, which gives the readings in Table VI. Clearly, the measured blocked rotor impedance under 𝛼𝛽 excitation, using [14], represents the summation of the stator and rotor impedances, while it equals 1.51 times the per phase impedance of the original nine-phase machine. However, for 𝑥𝑦 excitation, the input impedance is approximately 1.037 the per phase impedance. Therefore, conventional tests of A6P can simply be employed with this winding layout. In order to clearly include the effect of all space harmonics, the parameters given by Tables IV and V will however be used to build the simulation model. C. Dynamic Model Verification under Open Loop Control In this subsection, the machine open loop dynamic response under healthy and one-line open cases are presented based on the mathematical model given in section III. In a conventional A6P IM, the primitive harmonic-free models are usually assumed for the 𝑥 − 𝑦 and zero subspaces, while the machine dynamic response is mainly decided from the 𝛼 − 𝛽 subspace. Although these assumptions have usually been employed for A6P, a clear evidence to include or discard the effect of low order harmonics under different neutral configurations has not been confirmed yet [14]. For the 9P6T proposed model given

in section III.D, the effect of dominant low space harmonics is however included, and their effect on the dynamic response will be evident. The expected secondary subspace effect on the machine dynamics will mainly be limited to the transient periods under machine acceleration/deceleration and unbalanced operation. These harmonic components, namely 3 rd, 5th, and 7th, will expectedly give rise to additional parasitic torque components. The severity of different harmonics will of course depend on their relative magnitudes. As evident from Fig. 3, the induced 3rd harmonic, due to zero-sequence excitation while having a relatively high magnitude with respect to the fundamental flux component, exhibits a pulsating nature. Hence, its effect will sound more than the other harmonics (5th and 7th). TABLE III. PROTOTYPE IM SPECIFICATIONS Rated phase voltage (V) Rated Power (Hp) Rated phase current (A)

110 1.5 2.7

Rated frequency (Hz) Rated speed (rpm) Pole number

60 1700 4

TABLE IV. MACHINE PARAMETERS OF THE ORIGINAL 9-PHASE MACHINE 𝑘 𝑘 𝑘 𝑘

=1 =3 =5 =7

𝑟𝑠𝑘 (Ω) 1.98 1.98 1.98 1.98

𝑙𝑠𝑘 (mH) 7.7 7.7 7.7 7.7

𝑟𝑟𝑘 (Ω) 1.68 1.55 1.81 2.35

𝑙𝑟𝑘 (mH) 9 3.36 3.4 4.3

𝐿𝑚𝑘 (mH) 135.3 34.6 19.1 11.95

TABLE V. CALCULATED EQUIVALENT SIX-PHASE MACHINE PARAMETERS USING TABLE I 𝛼𝛽 6 𝑥𝑦 6 0+ 06−

𝑟𝑠𝑘 (Ω) 3 2.05 3

𝑙𝑠𝑘 (mH) 11.6 8 10.6

𝑟𝑟𝑘 (Ω) 2.54 2.04 2.34

𝑙𝑟𝑘 (mH) 21 3.8 5.06

𝐿𝑚𝑘 (mH) 204.3 17.6 52.3

TABLE VI. BLOCKED-ROTOR AND NO-LOAD MEASURED IMPEDANCES USING THE IDENTIFICATION TECHNQIUE OF CONVENTIONAL A6P [14] 𝛼𝛽 6 𝑥𝑦 6 0+ 06−

6𝑏𝑙 𝑅𝛼𝛽 = 5.55Ω 6 𝑅𝑥𝑦 = 2.38Ω 6 𝑅0+0− = 3.2Ω

𝐿6𝑏𝑙 𝛼𝛽 = 23.3𝑚𝐻 𝐿6𝑥𝑦 = 9.4𝑚𝐻 𝐿60+0− = 15𝑚𝐻

𝐿6𝑛𝑙 𝛼𝛽 = 224𝑚𝐻

To illustrate this expected effect, the machine is started under free-running mode at a rated supply frequency of 60Hz. To avoid saturating the current measurement boards in the experimental validation due to inrush starting currents, especially under open phase conditions, the voltage magnitude is reduced to half its rated value under this initial simulation case. Although the achievable load torque will highly be reduced to approximately one quarter the torque at rated voltage case, this test is the best approach to depict the effect of different space harmonics on the machine torque-speed curve assuming that core saturation is neglected. The simulation as well as experimental results are shown for both healthy and with line a1 disconnected cases. The machine is started mechanically unloaded then step-loaded with a load torque of 1.4Nm (¼ rated torque). Figs. 6(a) and (b) show the machine speed and RMS line current profiles under healthy operation. Under open-line case, the machine speed response is given in Fig. 6(c), while the simulated and measured line currents are shown in Figs. 6(d) and (e), respectively. It is evident that the simulation results coincide with an acceptable degree to the experimental results, which verify the fidelity of the proposed model. Due to the reduction in the achievable machine torque under open-line case, the machine took a longer time to reach its final steady-state speed when compared with the healthy

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case. In addition, for the same rated torque, the corresponding steady-state speed will be less. The starting profiles of different line currents under open-line clearly show a relative magnitude diversion due to this unbalance case. The machine instantaneous torque profiles under both healthy and open-phase cases are compared in Fig. 6(f), while the average torque variation and its sequence components versus speed are plotted in Figs. 6(g) and 6(h), respectively for the same two cases. Under healthy conditions, the torque production is mainly due to the main (α-β) subspace, while the torque components due to other subspaces add to zero, as evident From Fig. 6(g). On the other hand, non-negligible torque components will be produced due to the effect of (x-y) and zero sequence current components under unbalanced operation, as clear from Fig. 6(h). The torque component of the zero-subspace due to the third harmonic airgap flux component sounds more effective than the 5th and 7th order harmonics. A clear sub synchronous speed point appears at 1800⁄3 = 600 rpm as a direct consequence of the presence of a notable zero-

(a)

sequence current component. This can be confirmed by plotting the variation of the RMS line currents with the speed as given by Fig. 6(i), which indicates the same effect around the same speed point. These parasitic torque components will cause a torque reduction and eventually increase the starting time. To compare the effect of discarding the different air gap harmonics on the machine dynamic response, the machinestarting period, which mainly depends on the area under the torque-speed curve, is simulated under rated voltage and rated frequency, while the simulation results are given in Table VII. Clearly, the harmonic-free model, usually employed in conventional A6P machines, represents a rather theoretical case, while reductions by 14% and 12.5% in the starting and maximum torques, respectively, are obtained when employing the full machine model. Besides, the starting time increases by approximately 16%. Finally, neglecting the 3rd harmonic sounds more significant than the 5th and 7th harmonics.

(b)

(c)

(d)

(e)

(f)

(g)

(h) (i) Fig. 6. Open loop free running mode results. (a) Speed healthy case. (b) Line currents healthy case. (c) Speed for line a1 open. (d) Simulated line currents for line a1 open. (e) Experimental line currents for line a1 open. (f) Torque profiles with time. (b) Simulated average torque components versus speed under healthy conditions. (h) Simulated average torque components versus speed for line a1 open. (i) Simulated line currents versus speed for line a1 open.

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Ia1 Ib1 Ic1 Ia2 Ib2 Ic2

Iαβ Ixy I0

(a) (b) (c) Fig. 7. Simulated steady-state current waveforms and their sequence components in per unit at 1000 rpm and full load torque. (a) Healthy. (b) MT. (c) ML. Ia1 Ib1 Ic1 Ia2 Ib2 Ic2

Iαβ Ixy I0

(a) (b) (c) Fig. 8. Measured steady-state current waveforms and their sequence components in per unit at 1000 rpm and full load torque. (a) Healthy. (b) MT. (c) ML TABLE VII. EFFECT OF DIFFERENT SPACE HARMONICS ON MACHINE STARTING (TORQUE IN NM AND TIME IN SECONDS) Fault Parameter Healthy Harmonic Without Without Full free 3rd 5th + 7th model Starting 7 5.7 5.3 5.4 4.9 torque Max. torque 12.2 10.69 10.23 10.03 9.35 Starting time 0.72 0.86 0.91 0.94 1 TABLE VIII. CONTROLLER PARAMETERS Current controllers Speed controller

Kp = 10 and Ki = 200 Kp = 0.1 and Ki = 0.1

D. IRFOC with Optimal Current Control It has been shown in pervious subsection that the induced current components in the secondary subspaces under unbalanced operation will affect the machine dynamic response if not properly controlled, which is the case under open loop case. Under VSD-based controller, the 𝑥 − 𝑦 and the zero sequence components are controlled under fault conditions in accordance with the reference 𝛼 − 𝛽 current components using (37), while the fundamental subspace components are always kept balanced to provide a pre-fault operation [10].

First, the machine current waveforms are investigated under steady-state operation. The reference direct current component is set to 1.5A while the reference speed is set to 1000 rpm. All PI controllers are tuned via trial and error [18] and given in Table VIII. The simulated line current waveforms as well as their corresponding sequence current components are shown in Fig. 7 under both healthy and open-line cases, while the corresponding experimental results are shown in Fig. 8. Both MT and ML modes are presented for open line operation. Clearly, the VSD-based controller ensures balanced 𝛼 − 𝛽 current components under all cases and the line currents are perfectly matching their optimal references given in Table II. As far as the effect of the non-fundamental subspaces on the dynamic response is concerned, the machine is tested under both step reference speed and step mechanical loading conditions. The machine reference speed is set to 500 rpm. At 1s, the speed reference is step increased to 1000 rpm then brought back to 500 rpm. This test is carried out while the machine is mechanically unloaded. The simulated as well as the measured speed responses are shown in Fig. 9(a). While, the experimentally measured reference and actual dq current components of the fundamental subspace are compared and

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higher fundamental torque component 𝑇𝛼𝛽 is required to maintain same speed profile under fault conditions, if for no other reason than to compensate for theses parasitic torque components. This is also translated to a higher quadrature current component than the required current component under healthy conditions, as clear from Fig. 9(d). In the declaration period, these secondary torque components can be considered as braking torque components. Thus, the fundamental torque component 𝑇𝛼𝛽 , and the corresponding quadrature current component, will be less than the healthy case. The machine is also tested under step mechanical loading and the comparison between simulation and experimental results for the machine speed and the fundamental quadrature current component are shown in Fig. 12. Under fault conditions, the quadrature current component is slightly higher than the healthy case (MT is also higher than the ML) to compensate for the small negative torque components of the secondary subspaces as explained previously.

shown in Fig. 9(b) for the healthy case. The simulated and experimentally measured direct current components under healthy and different postfault scenarios are shown in Fig. 9(c). While, those for the quadrature current components are given in Fig. 9(d). The good matching between experimental and simulation results does verify the proposed harmonic model. The simulated developed torque and its sequence components during both the acceleration and deceleration periods are shown in Figs. 10 and 11, respectively. Since the fundamental subspace is controlled to ensure balanced α-β current components, the torque component 𝑇𝛼𝛽 is shown to be ripple free. However, the torque components corresponding to non- fundamental subspaces give rise to a notable torque ripple component, which is higher under MT postfault scenario. Meanwhile, these subspaces also produce a negative average torque component since the machine slip corresponding to these space harmonics will be higher than 1. This negative torque component is also higher in the MT scenario. Therefore, a

(a)

(b)

(c)

(d) Fig. 9. Comparison between simulation and experimental step speed dynamic response under IRFOC and different postfault scenarios. (a) Speed. (b) Measured reference ad actual fundamental dq current components. (c) I d. (d) Iq.

(a) (b) (c) Fig. 10. Developed torque during accelerating period from 500 rpm to 1000 rpm. (a) Total torque. (b) T αβ. (c) Txy. (d) T0+0-.

(d)

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(a) (b) (c) Fig. 11. Developed torque during decelerating period from 1000 rpm to 500 rpm. (a) Total torque. (b) T αβ. (c) Txy. (d) T0+0-.

(a) Fig. 12. Dynamic machine performance under step loading and IRFOC. (a) Speed. (b) 𝑖𝑞 .

In conclusion, with the employed VSD-based controller, the machine response under fault case will be quite similar to the healthy case provided that the remaining healthy phases can withstand the corresponding current overloading. Since the secondary as well as the zero subspace are now fully controlled, the sluggish machine response obtained under open-loop control will also be avoided.

(b)

and open-line cases, while the same controller structure is preserved. It was also proven that the pre-fault dynamic response can perfectly be preserved under VSD-based controller provided that the remaining healthy phases can withstand the corresponding current overloading. REFERENCES [1]

VI. CONCLUSION This paper introduced the dynamic mathematical model of the 9P6T IM based on a VSD modelling approach. The required transformation matrices that decompose the machine terminal variables into their sequence components were derived such that the machine will be equivalent to a conventional six-phase system. Unlike conventional six-phase systems, the sequence voltage and current transformation matrices were found to be different. The equivalent machine parameters of different subspaces have been also calculated and experimentally identified. It has been shown that the conventional tests usually used with the A6P machines can be applied in this case. The machine mathematical model was used to simulate the machine under open-loop control, and the simulated results were experimentally verified. It was proven that the harmonic-free model usually used in the A6P case will not adequately represent the actual machine dynamic response due to the notable effect of the non-fundamental subspaces since a single layer concentrated winding is employed in a 9P6T IM. The derived VSD approach also allowed for the employment of conventional six-phase VSD-based controllers using standard IRFOC to fully control the 9P6T machine under both healthy

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[2]

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M.J. Duran, E. Levi, and F. Barrero, “Multiphase electric drives: Introduction,” Wiley Encyclopedia of Electrical and Electronics Engineering, 2017. E. Levi, R. Bojoi, F. Profumo, H. A. Toliyat, and S. Williamson, “Multiphase induction motor drives – a technology status review,” IET Electr. Power Appl., vol. 1, no. 4, pp. 489-516, Jul. 2007. E. Ariff, O. Dordevic, and M. Jones, “A space vector PWM technique for a three-level symmetrical six phase drive,” IEEE Trans. Ind. Electron., vol. 64, no. 11, pp. 8396-8405, Nov. 2017. M. Diab, A. Elserougi, A. S. Abdel-Khalik, A. Massoud, and S. Ahmed, “A nine-switch-converter-based integrated motor drive and battery charger system for EVs using symmetrical six-phase machines,” IEEE Trans. Ind. Electron., vol. 63, no. 9, pp. 5236-5335, Sep. 2016. W. Munim, M. Duran, H. S. Che, M. Bermudez, I. Gonzalez-Prieto, and N. Abd Rahim “A unified analysis of the fault tolerance capability in sixphase induction motor drive,” IEEE Trans. Power Electron., vol. 32, no. 10, pp. 7824-7836, Oct. 2017. F. Baneira, J. Doval-Gandoy, A. Yepes, O. Lopez, and D. Perez-Estevez, “Control strategy for multiphase drives with minimum losses in the full torque operation range under single open-phase fault,” IEEE Trans. Power Electron., vol. 32, no. 8, pp. 6275-6285, Aug. 2017. A. Yepes, J. Doval-Gandoy, F. Baneira, D. Perez-Estevez, O. Lopez, “Current harmonic compensation for n-phase machines with asymmetrical winding arrangement and different neutral configurations,” IEEE Trans. Ind. Appl., vol. 53, no. 6, pp. 5426-5439, Nov./Dec. 2017. I. Gonzalez-Prieto, M. Duran, J. Aciego, C. Martin, and F. Barrero, “Model predictive control of six-phase induction motor drives using virtual voltage vectors,” IEEE Trans. Ind. Electron., vol. 65, no. 1, pp. 2737, Jan. 2018.

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A. Abdel-Khalik, S. Ahmed, and A. Massoud, “A nine-phase six-terminal concentrated single layer winding layout for high-power medium-voltage induction machines,” IEEE Trans. Ind. Electron., vol. 64, no. 3, pp. 17961806, Mar. 2017. A. Abdel-Khalik, M. Hamad, A. Massoud, and S. Ahmed, “Postfault operation of a nine-phase six-terminal induction machine under single open-line fault,” IEEE Trans. Ind. Electron., vol. 65, no. 2, pp. 10841096, Feb. 2018. Y. Hu, Z. Q. Zhu, and M. Odavic, “Comparison of two-individual current control and vector space decomposition control for dual three-phase PMSM,” IEEE Trans. Ind. Appl., vol. 53, no. 5, pp. 4483-4492, Sep./Oct. 2017. Y. Zhao and T. A. Lipo, "Space vector PWM control of dual three-phase induction machine using vector space decomposition," IEEE Trans. Ind. Appl., vol. 31, no. 5, pp. 1100-1109, Sep./Oct. 1995. Ivan Subotic, N. Bodo, E. Levi, M. Jones, and V. Levi, “Isolated chargers for EVs incorporating six-phase machines,” IEEE Trans. on Ind. Electron., vol. 63, no. 1, pp. 653-664, Jan. 2016. HS. Che, A. Abdel-Khalik, O. Dordevic, and E. Levi, “Parameter estimation of asymmetrical six-phase induction machines using modified standard tests,” IEEE Trans. Ind. Electron., vol. 64, no. 8, pp. 6075-6085, Aug. 2017. A. Rockhill and T.A. Lipo, “A Simplified model of a nine-phase synchronous machine using vector space decomposition,” in Conf. PEMWA 2009, pp. 1-5. D. Hadiouche, H. Razik, and A. Rezzoug, “On the modeling and design of dual-stator windings to minimize circulating harmonic currents for VSI fed AC machines,” IEEE Trans. Ind. Appl., vol. 40, no. 2, pp. 506–515, 2004. A. Abdel-Khalik, M. Masoud, S. Ahmed and A. Massoud, “Effect of current harmonic injection on constant rotor volume multiphase induction machine stators: A comparative study,” IEEE Trans. Ind. Appl., vol. 48, no. 6, pp. 2002-2013, Nov./Dec. 2012. H. S. Che, E. Levi, M. Jones, W. P. Hew, and N. A. Rahim, “Current control methods for an asymmetrical six-phase induction motor drive,” IEEE Trans. on Power Electron., vol. 29, pp. 407–417, Jan 2014.

Ayman S. Abdel-Khalik (SM’12) received the B.Sc. and M.Sc. degrees in electrical engineering from Alexandria University, Alexandria, Egypt, in 2001 and 2004, respectively, and the Ph.D. degree in electrical engineering from Alexandria University, and Strathclyde University, Glasgow, U.K., in 2009, under a dual channel program. He is currently an Associate Professor with the Electrical Engineering Department, Faculty of Engineering, Alexandria University, Alexandria, Egypt. He serves as an Associate Editor of IET Electric Power Applications Journal and an Associate Editor of Alexandria Engineering Journal. His current research interests include electrical machine design and modelling, electric drives, energy conversion, and renewable energy.

Ahmed M. Massoud (SM’11) received the B.Sc. (first class honors) and M.Sc. degrees in Electrical Engineering from Alexandria University, Egypt, in 1997 and 2000, respectively, and the Ph.D. degree in Electrical Engineering from Heriot-Watt University, Edinburgh, U.K., in 2004. He is currently an Associate Professor at the Department of Electrical Engineering, College of Engineering, Qatar University. His research interests include power electronics, energy conversion, renewable energy and power quality. He holds five U.S. patents. He published more than 100 journal papers in the fields of power electronics, energy conversion and power quality.

Shehab Ahmed (SM'12) received the B.Sc. degree in electrical engineering from Alexandria University, Alexandria, Egypt, in 1999, and the M.Sc. and Ph.D. degrees from the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA, in 2000 and 2007, respectively. He was with Schlumberger Technology Corporation, Houston, TX, USA, from 2001 to 2007, developing downhole mechatronic systems for oilfield service products. He is currently an Associate Professor with Texas A&M University at Qatar, Doha, Qatar. His research interests include mechatronics, solid-state power conversion, electric machines, and drives.