Generalized Vectorial Formalism – Based multiphase series-connected motors control

SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 Generalized Vectorial Formalism – based multiphase series-connected motors control . Eric Semail . Ngac Ky Nguyen . Xavier Kestelyn . Tiago Dos Santos Moraes L2EP Laboratory, Arts et Métiers ParisTech, France (Manuscript Received on July 15, 2015, Manuscript Revised August 30, 2015) ABSTRACT Multiphase drives are more and more magnet synchronous motors (PMSM) fed by used in specific applications leading to a one voltage source inverter (VSI). Based on necessity of control strategy development. a decomposition of multiphase machine, a This paper presents the Generalized proposed control strategy has been Vectorial Formalism (GVF) theory to control achieved. Some experimental results are multiphase series-connected permanent given to illustrate this control method. Keywords: Multiphase drives, Generalized Vectorial Formalism, Multiphase series- connected machines. 1. INTRODUCTION Higher reliability of classical 3-phase drives The multiphase theory has been developed can be achieved by oversizing the converter- since 10 years ago [3]-[6] in objective to machine set but this solution increases the cost of understand deeply and allow using simple the whole system. Even if this oversizing is regulators for current and speed. Based on this chosen, in case of an open circuit fault appears in theory, a multiphase machine can be one or two phases of the drive, the system cannot decomposited to some equivalent fictitious ones. ensure a functioning even at reduced power. Classically, a three-phase machine in the Park Using multiphase drives instead of three- reference frame is the simplest case where we phase drives, makes possible to increase the have a d-q diphase machine and a homopolar one. power density, fault-tolerance capability and to If the machine is star connected, the latter one is reduce torque pulsations at low frequencies [1, 2]. not considered. Indeed, the Park (or Concordia) In fault mode, this kind of drives is able to work transformation is a linear mathematical at a reduced power with satisfactory application where machines will be modelled into performances. This aspect is very important in the eigen space and represented by the eigen systems which are designed for specific vectors of the matrix inductance. In a general case applications, such as offshore energy harvesting where a multiphase machine is considered, the or electrical vehicles. generalized Concordia (or Park) transformation is Trang 18 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 needed. After the transformation, the multiphase machine. Then we consider that the stator machine is modeled by some diphase machines inductance matrix is the characterization, in a and some homopolar machines according to the natural base, of a linear application also called an shape of the back-electromotive force (back- endomorphism. EMF). In a general way, to control a diphase In the next paragraphs, we give some of its machine, two independent currents are required. properties a n-phase machine and particularly a 5- That is why a 3-phase Voltage Source Inverter phase machine. (VSI) is needed to supply a 3-phase machine and 2.1 Endomorphism and stator inductance a n-phase VSI is required to supply a n-phase matrix machine. In the case where multiphase machines Let us consider the stator inductance matrix are series-connected, the model of each machine L n  of a multiphase machine. We consider it as is not changed but the control is more complex s  the matrix of an endomorphism  in an since phase currents across all machines. An s orthonormal base  classified as “natural”. This independent functionning (speed and torque) of n endomorphism  has properties independent of each machine requires a decoupled strategy of s control and imposes some contraints to machine the choice of the studying base: eigenvalues, design in term of the back-EMF. eigenvectors and eigenspace. To get them we have only to examine L n  . This paper presents firstly the theory of s  mutliphase machine based on the Generalized As mutual inductance between two windings Vectorial Formalism. A two series-connected 5- j and k are identical (Mjk=Mkj) then the matrix is phase machines supplied by one VSI structure is symmetrical. This symmetry implies the existence presented thereafter. Some experimental results of a base of eigenvectors. Moreover the will be given to confirm the feasibility of the eigenspaces of s are orthogonal each other and proposed structure. the dimension of an eigenspace Es is equal to the 2. GENERALIZED VECTORIAL multiplicity order of the associated eigenvalue  FORMALISM . For example, if the order of multiplicity is one In order to show that a polyphase machine is (respectively two), the eigenspace is a vectorial equivalent to a set of 1-phase and 2-phase line (respectively vectorial plan). To obtain an machines, we have to bring out vectorial orthonormal base of eigenvectors we have only to properties of the stator self inductance matrix. The choose in each eigenspace an orthonormal base. analysis of its properties enables the The classic transformation matrixes of Park or generalization of the transformation concept. Of Concordia are nothing else than tables that allow course, the matrices which generalize Park’s and us to find the coordinates of these eigenvectors. Concordia’s transformations for a n-phase If the order of multiplicity of all eigenvalues machine have already been defined and used in is one, then there is only one orthonormal base of particular cases. Our formalism defines a larger eigenvectors. Consequently, only one class of systems for which these transformations transformation that keeps the power invariant can can be used. This is possible because, in the then be elaborated. vectorial approach, transformations are only the On the other hand, if the order of multiplicity expression of vectorial properties linked to the is not one for one eigenvalue, then there is an stator inductance matrix. At first, an Euclidian infinity of orthonormal bases of eigenvectors. vector n-space is associated with an n-phase Trang 19 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015    d d d , the matrix of Consequently an infinity of transformations d  x1,,, x 2  x n  keeping invariant the power can be defined. This endomorphism  is given by: property explains the great number of s transformations that have been proposed in the 1 0 0  0  0  past. Ld   2  (6) s     0  2.2 Applying to a n-phase machine   0 0 0 n  In an orthogonal base  composed of the n Each eigenvalue is associated to an    vectors xn,,, x n x n it can be defined the  1 2 n  eigenspace whose the dimension depends on the voltage, current and stator flux linkage vectors as multiplicity of the eigenvalue. For example, if follows: there are two solutions equal to 1 , then the 1  n  n  n v vx1 1  v 2 x 2   vn x n (1) is belong to a 2-dimensional eigenspace.  n  n  n It can be noticed that all subspaces (the i ix1 1  i 2 x 2   in x n (2) eigenspaces) are orthogonal and it can be defined     n n n as a set of fictitious magnetically independent  1x 1   2 x 2   n x n (3) systems. These vectors are linked by:    For each subspace, the relationship between  d  d d v Ri   Ri ss  sf (4) the voltage and the current is given by: dt dt dt   d  di     v Ri j  Ri  j  e (7) where  depends on i and  is due to j jdt j j dt j ss sf  magnets on the rotor. where ej is the back-electromotive force. The relationship between the current vector Based on equation (7), we can consider that and the flux vector is given by the endormorphism in the new base, a multiphase can be decomposed  represented by the inductance matrix n  . In s L s  into several fictitious machines where each the natural frame, L n  is expressed as: machine is characterized by a resistor R, an s   inductance  and a vector of the back-EMF e LMMM11 12 13 1n  j j   LLMM21 22 23 2n . According to the dimension of the eigenspace Ln     (5) s          where the eigenvalue j is belong to, the LLLLn1 n 2 n 3  nn  fictitious machine can be monophase or diphase. where Lkk is the self-inductance of the phase Let us take an example in the case of a 3-phase k and Ljk is the mutual inductance between the machine. The new base is defined by the phases j and k. Concordia transformation where it exists a double As mentionned above, the symetrical root and a single root of eigenvalues by solving inductance matrix leads to a base of eigenvectors 3  the determination detLIs  3    0. The whose corresponding eigenvalues are given by double root corresponds to the d-q machine and detLIn    0. In this new base defined by s n   the single root is linked to the homopolar one. Trang 20 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 2.3 Applying to a 5-phase machine 0  ia    i i  m  b   T imachim    C5   i c  (9)     i s  id  i    s  ie  0  vaN    v v   m  bN  T (10) vvm    C  v  mach  5  cN  v s  vdN  Fig. 1. A 5-phase drive having star connection.     vs  veN  For a 5-phase PMSM, after a generalized Currents and voltages obtained using this Concordia transformation (given in equation (8)) transformation can be decomposed into two which brings electrical variables of machine to the subsystems associated with the main and eigenspace, we obtain thus: one single eigenvalue, secondary machines: two double eigenvalues. It means there are one 1- dimensional fictitious machine and two 2-   v v v t  v   v v  t dimensional ones. They are called respectively m m m    s  s  s   ; homopolar machine, main machine and secondary t   t (11) i i i   i   i i  machine. It is not always the case where these m m m    s  s  s   machines exist. Their presence depend on the As the machine is supplied by a VSI, shape of the back-EMF. Indeed, the homopolar vmach vvsi and consequently: machine is created by harmonics 5*k. The main     machine is issue from harmonics 1, 9, 11,, 5*k vm v mvsi ; v s  v svsi (12) ± 1 and the secondary is formed by harmonics 3, Equations (13) - (16) give the mathematical 7, 13,, 5*k ± 2 by considering only odd model of the two fictitious machines in the new harmonics [3]. Thus the Main Machine (MM) reference frame: (resp. the Secondary Machine (SM)) produces Tm  d   (resp. Ts) torque mainly thanks to the first vm R m i m  L m i m  e m (13) harmonic (resp. third harmonic) of the back-EMF. dt Relationships between actual phase variables  d   vs R s i s  L s i s  e s (14) (denoted with subscripts a, b, c, d and e) and dt values of fictitious machines are then defined by: TTTtot m  s (15) 1  1 0 1 0  2  where : 1 2 2  4  4     cos sin cos sin 5 5 5 5  2  em. i m T m  and e s . i s  T s  (16) 2 1 4 4  8  8   C   cos sin cos sin 5 5 5 5 5  Equation (15) means that the total torque of 5 2  1 6 6  12  12   the 5-phase machine is the sum of the torque cos sin cos sin 5 5 5 5  2  created by the two fictitious machine. We can 1 8 8  16  16   cos sin cos sin  consider that there is a fictitious mechanical 2 5 5 5 5  coupling between them. (8) Trang 21 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 Controlling a 5-phase machine leads to The special connection between the two finally control two equivalent fictitious diphase machines can be expressed by:  ones after the Concordia transformation. Each T i2  i 2a i 2 b i 2 c i 2 d i 2 e  ficitious machine is linked to some harmonics of T (17)  L5 i 1a i 1 b i 1 c i 1 d i 1 e  the back-EMF as discussed previously in section   L i 2.3. There are many works in the literature  5 1 presenting the control of 5-phase machines under 1 0 0 0 0    healthy and fault modes [6]-[12]. In both 0 0 0 1 0  with:     (18) operating modes, physical limits related to L5   0 1 0 0 0   voltages and currents of the drive have been also 0 0 0 0 1  studied [13]-[16]. This paper focuses to control 0 0 1 0 0  two 5-phase machine having a series connection   The vectors i1 and i 2 represent the current and they are fed by two isolated DC-buses. vectors of the 1st machine M1 and the second one 3. TWO SERIES-CONNECTED 5- M2 respectively. PHASE PMSM Using equations (9), (17) and the property 1 T The studied structure is shown in Fig. 2. The [][]CC5 5 , a relation between the current vectors objective is to control independently these two of the fictitious machines is given as follows: machines. Normally, to do this, only one VSI is required but a higher functional reliability can be 0  i2a   i 1 a  achieved by adding another VSI [19], specially in   i i   i   m 2  2b   1 b  short-circuit inverter switch fault.   TT imach2im 2  C 5  i 2 c    C 5  L 5  i 1 c        Controlling independently two 5-phase i s 2  i2d   i 1 d        machines leads to a necessity of 8 independently is 2  i2e   i 1 e  currents as degree of freedoms (DOF). In the 0  present structure, there are only 4 currents can be i  m1   TT  freely controlled. That is why a specific C5  L 5 C 5 im1   C 5  L 5 C 5 imach 1   connection between two machines is needed [20]. is1  i  3.1 Modelling s1  (19) Fig. 2. Structure of two 5-phase open-end winding machines under study Trang 22 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 where: main machine is almost linked to the 1st 1 0 0 0 0  (fundamental) harmonic while the second rd 0 0 0 1 0  machine is affected to the 3 harmonic.   rd T     Generally, the 3 harmonic can contribute to the CLC5  5  5   0 0 0 0 1 (20)   machine torque but when the machine is operated 0 1 0 0 0   at high speed (constraint of voltage has to be 0 0 1 0 0    applied), the current calculation becomes more Equations (19) and (20) lead to: complex, especially as in degraded mode where  TT some degrees of freedom are lost. That is why      is1  i s 1 i s  1    i m  2 i m  2  (21) until now, there are no analytical solutions for a  TT i i i    i  i  multiphase machine operating under limit of  m1 m 1 m  1   s  2 s  2  voltages and currents. For some specific or:   applications as electric vehicle where the drive is   is1 i m 2 almost operated at high speed thanks to the flux   (22) * weakening operation, a multiphase having im1 i s 2 sinusoidal back-EMF is preferred. Expression (22) lead to two significant In this study, two 5-phase PMSM are things: considered sinusoidal. As a consequence, the  Thanks to the swapping connection   back-EMF es1 and es 2 does not exist leading to a between two machines M1 and M2, the currents  very simple control strategy. The vector current im1 of the main machine MM1 (second SM1 controls the torque (and/or the speed) of the respectively) of the M1 are linked to the ones of  the secondary SM2 (main MM2 respectively) machine M1 and the vector current is1 is machine of the M2. This allows to control dedicated to the torque (and/or the speed) of the independently two machines M1 and M2 with machine M2. only 4 free currents. Let’s talk about the voltages limit that two  In point of view of electromagnetic machines can be fed.  r r r r torque, i will create a torque not only for the v= v - v + v (23) m1 å VSI1 VSI 2 N1 N 2 main machine of M1 but also for the second with: r r r machine of M2. In the same manner, is1 will v= v + v å MM1 2 (24) contribute to the torque created in the two fictitious machines SM1 and MM2. This leads to is the sum of the phase voltages of the two a more complex strategy of control to decouple machines M1 and M2 and  the previous mentioned interaction in order to T vVSI1  v 1111111111 aN v bN  v cN  v dN  v eN   obtain an independence of the two machines M1 (25) and M2.  T This second conclusion is obvious to take into vVSI2 v 2 aN 2 v 2'2 bN  v 2'2 cN  v 2' dN  2 v 2'2 eN   account for multiphase drives. As mentionned (26) above, a multiphase machine can bedecomposed  T into some fictitious ones representing by vNNNNNNNNNNNN12 v 12 v 12  v 12  v 12  v 12   corresponding harmonics of the back-EMF. The (27) Trang 23 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 Fig. 3. Speed control scheme. are the voltages of the two VSI and the voltage M2. A simplified strategy has been implemented between the two negative points of two DC buses, r r1 r in this work by using v= v = v . respectively. MM1 2 2 å Voltages delivered by two VSI can be 3.2 Control scheme obtained by some techniques. A Space Vector The control scheme is reported in Fig. 3. The Modulation has been proposed in [21] to exploit two machines M1 and M2 are required to rorate better the two DC-buses. In [22], some simple as Ω1ref and Ω2ref. PI controllers are used for speeds modulation strategies have been presented for and currents tracking. It can be noticed that only varying the machine voltages according to the the phase currents of M1 are measured since they rotor speed without flux weanking operation. In are the same ones acrossing the M2 by the L5 this paper, a bipolar modulation is chosen to transformation. maximize the voltage of machines. 4. EXPERIMENTAL RESULTS It’s interesting to discuss about the voltage In order to validate the proposed control sharing between the two machines. Indeed, by strategy, some experimental results have been observing the equation (24), voltages of M1 and carried out. Fig. 4. shows the platform for M2 can be shared in the optimal way while experimental verification. Tab. I. gives some respecting the voltage limit fed by two VSI based details of the drive parameters. on the bipolar modulation technique. As the M1 and M2 are independently controlled, an optimal A 5-phase VSI is communicated to a dSPACE strategy could be employed according to technical 1005 board through an I/O interface. A DC- specifications of the applications in term of programmable source is used to feed the drive torques and speeds of the two machines M1 and during motor mode and recover energy during braking. Trang 24 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 Fig. 4. Experimental test-bed. Table 1. Drive parameters Parameters 5-phase M1 5-phase M2 Phase resistance Rs = 2.24 [Ω] Rs = 9.1 [mΩ] Fig. 5. Experimental results in three studied Phase inductance Ls = 2.7 [mH] Ls = 0.09 [mH] cases: a) Ω2ref = 0; b) Ω2ref = 40 rad/s and Ω1ref is M = 0.02 Mutual induc. 1 M = 0.25 [mH] 1 1 [mH] varying; c) Both machine’s speeds are varying. Mutual induc. 2 M1 = -0.75 [mH] M2 = -0.01[mH] Fig. 5 gives the experimental verification. st Pole pair number p = 2 p = 7 Three study cases have been considered. The 1 one consists to keep the machine M2 at stand-still back-EMF constant En 0.51 En 0.1358 and the M1 is trained following a speed profile. The second test has been realized by keeping the Max. RMS current 15 [A] 147 [A] rotor speed of M2 at 40 rad/s and the M1 is tuned Maximum speed 1500 [rpm] 16000 [rpm] to track a speed profile. For the last case, both Maximum torque 20 [N.m] 50 [N.m] machines M1 and M2 are operated at two different Maximum power 3.1 [kW] 15 [kW] profiles of speed. Based on the results given in For experimentation tests, the two machines Fig. 5, we can conclude that the proposed control M1 and M2 are operated under speed control. strategy has been verified. It should be highlighted that there was no strategy for voltage sharing between M1 and M2. The speed profiles were chosen in the way that DC-buses are able to delivery enough voltages for two machines M1 and M2 Trang 25 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 5. CONCLUSION sinusoidal has been made to simplify to control In this paper, a generalized vectorial strategy. formalism has been presented and applied for This specific structure is suitable for multiphase drives. Based on this approach, two 5- applications where we have the constraint for phase PMSM having series connection (through compacity and weight of the whole systems. windings) can be independently controlled. However, one assumption that all machines are Điều khiển động cơ điện nhiều pha mắc nối tiếp dựa trên lý thuyết “Generalized Vectorial Formalism” . Eric Semail . Ngac Ky Nguyen . Xavier Kestelyn . Tiago Dos Santos Moraes L2EP Laboratory, Arts et Métiers ParisTech, France TÓM TẮT Bộ truyền động nhiều pha (hơn 3) đang pha tương ứng, bằng mô hình toán học, với dần được áp dụng trong nhiều ứng dụng đặc một vài máy điện ảo (hai pha và một pha). Số biệt dẫn đến sự cần thiết trong việc phát triển lượng máy điện ảo phụ thuộc số pha và cách các giải thuật điều khiển của các bộ truyền đấu dây giữa các pha của máy điện nhiều động này. Bài báo này trình bày lý thuyết pha. Dựa trên lý thuyết này, một giải thuật đã “Generalized Vectorial Formalism” để điều được đề ra để điều khiển một cách hoàn toàn khiển hai động cơ điện đồng bộ nhiều pha độc lập (vận tốc và moment xoắn) hai động mắc nối tiếp. Hai động cơ đồng bộ được cung cơ đồng bộ nhiều pha mắc nối tiếp với duy cấp bằng một bộ biến tần trong đó số nhánh nhất một bộ biến tần. Kết quả thực nghiệm của bộ biến tần này bằng với số pha của mỗi với hai máy điện 5 pha cho thấy sự đúng đắn động cơ. Theo lý thuyết “Generalized của giải pháp điều khiển này. Vectorial Formalism”, một máy điện nhiều Từ khóa: Bộ truyền động nhiều pha, Generalized Vectorial Formalism, Máy điện nhiều pha mắc nối tiếp. Trang 26 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 REFERENCES [1]. H. Jin, K. Min, Y. 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