TPEL REG 2018 05 1071.R

IEEE TRANSACTIONS ON POWER ELECTRONICS Elimination of Harmonic Currents Using a Reference Voltage Vector Based-Model Predictive Control for a Six-Phase PMSM Motor Yixiao Luo, Student Member, IEEE and Chunhua Liu *, Senior Member, IEEE  Abstract—This paper proposes a novel dead-beat current control (DBCC) based model predictive control for an asymmetrical six-phase permanent magnet synchronous machine (PMSM). First, the solution of DBCC is adopted to obtain the expected reference voltage vector (RVV). Then, two groups of virtual vectors, in the total number of 24 with different magnitudes, are defined for the sake of current harmonics suppression. Subsequently, two in-phase virtual vectors which are closest to the RVV are selected as the prediction vectors. The next step is to define a cost function which is composed of the error between the RVV and the available prediction vectors. Then, the selected two virtual vectors are evaluated and the one that minimizes the cost function will be applied in the next instant. In this way, only two prediction vectors need to be evaluated and the computation burden is highly alleviated. In the meantime, the weighting factor involved in predictive torque control is avoided. In addition, to achieve the readily implementation with standard PWM switching sequence, 18 virtual vectors are artfully replaced by their corresponding equivalent virtual vectors. Finally, the proposed method is comparatively studied and compared with other benchmark methods. Simulation and experimental results are offered to confirm the effectiveness of the proposed method. Index Terms—Deadbeat current control, reference voltage vector, model predictive control, PMSM motor, six-phase machine, multiphase machine, current harmonics. I. INTRODUCTION V ARIABLE speed drives of permanent magnet synchronous machine (PMSM) have been widely used in industrial applications. In particular, multiphase machines are Manuscript received May 21, 2018; revised August 07, 2018; accepted September 30, 2018. This work was supported by a grant (Project No. 51677159) from the Natural Science Foundation of China (NSFC), China. Also, it was supported by a grant (Project No. CityU 21201216) from the Research Grants Council of HKSAR, China. ( *Corresponding author: Chunhua Liu; [email protected]) Y. Luo and C. Liu are with the School of Energy and Environment, City University of Hong Kong, Hong Kong, China. Also, they are with the Shenzhen Research Institute, City University of Hong Kong, Shenzhen, 518057, China. (e-mail: [email protected]; [email protected]) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org Digital Object Identifier ******* receiving more and more attention due to their high torque density, fault tolerant capability and lower power rating per phase [1], [2]. In addition, the model predictive control (MPC) has been recently introduced to the application for high performance dynamic control of electric machines [3]-[9]. Generally, there are two types of MPC methods according to their constrained objectives, namely the model predictive current control (MPCC) [3]-[5] and the model predictive torque control (MPTC) [6]-[8]. In the MPTC method, the cost function composed of the errors of the predictive variables, namely torque and stator flux, is predefined to evaluate the available voltage vectors. However, in conventional MPTC, a weighting factor is always involved to control the torque and stator flux simultaneously due to their different magnitude and unit [9]. Unfortunately, the tuning of the weighting factor is still an uphill task due to the lack of theoretical procedures. There are generally two strategies to deal with this problem, either design a proper value of the weighting factor based on some certain principles, for instance torque ripple minimization [10]-[11], or eliminate the weighting factor directly [12]-[13]. However, large amount of calculations are involved in the methods proposed in [10]-[11]. Additionally, a MPC method for a three-phase surface PMSM presented in [14] adopts the deadbeat direct torque and flux control (DB-DTFC) to obtain a reference voltage vector (RVV), then the error between the RVV and available actual vectors is defined as the constraint of the cost function. In this way, the weighting factor in the cost function is avoided. However, it is complex to derive the RVV for a six-phase interior PMSM motor using DB-DTFC. Besides, directly applying the available actual vectors for six-phase machine will introduce large current harmonics. In MPCC method, the weighting factor is intrinsically absent [15]. So, it is a natural option to include the stator current as control variables in the cost function. Actually, the MPCC for six-phase machine have been investigated in [16]-[20]. In the existing MPCC methods for six-phase machine, the cost function is composed of the stator current in α-β subspace and x-y subspace to suppress the current harmonics [16]-[18]. Then the available voltage vectors, usually the largest 12 actual vectors [19] or the 12 virtual vectors synthesized by the largest and the second largest actual vectors [20], are evaluated by the cost function. Nevertheless, in [19] and [20], the number of voltage vectors to be evaluated are still 13 (12 largest active vectors and 1 null vector), which is a large number compared IEEE TRANSACTIONS ON POWER ELECTRONICS with 7 (6 active vectors and 1 null vector) prediction vectors in three-phase machine drives. Meanwhile the large computation time has always been a defect of the MPC compared with vector control and direct torque control. Many attempts have been made to reduce the computation burden, such as the binary search tree method [21], the sphere decoding algorithm [22], [23], or excluding the inapplicable voltage vectors in advance [24]. Unfortunately, the methods proposed in [21], [22] are difficult to be implemented. Meanwhile, the look-up table developed in [24] is complicated with two flux position observers involved. In the aforementioned MPC method for six-phase machine, either large computation time is required [18]-[20], or complicated predictive model is involved [24]. In addition, [25] fails to implement the virtual vectors using standard PWM switching sequence. The MPC method combined with DBCC [15] or DB-DTFC [14] are limited to three-phase machine and only considers the energy conversion related subspace, which is not applicable for the two-degree freedom six-phase machine. This paper proposes a RVV based model predictive control to reduce the computation time and harmonic currents by using a very simple and effective predictive model. The RVV is obtained based on the principle of dead-beat current control (DBCC). Instead of directly applying the 12 largest actual vectors, 24 virtual vectors are synthesized by the largest actual vectors and the second largest actual vectors to suppress the current harmonics. Then, with the position of the RVV determined, two in-phase virtual vectors closest to the RVV are selected as the prediction vectors. In this way, the number of voltage vector candidates is reduced from 12 to 2. Subsequently, a novel cost function composed of the error between these two virtual vectors and the RVV is defined, where the weighting factor involved in MPTC is absent and predictive model is simplified. Finally, the conventional MPCC method, the direct DBCC method and the proposed method are all implemented and compared. Also, experimental results are given to verify the validity of the proposed method. II. CONVENTIONAL MPCC CONTROL SCHEME In model predictive control, the first step is to predict the machine behavior at each sampling period using the discretized model of the machine. Then a cost function is defined based on the predicted machine state according to the control criteria. For instance, the error between the reference and the predicted stator currents in the α-β plane can be included in the cost function, which is expressed as g = |i*α-iα| + |i*β-iβ|. Subsequently, each available voltage source inverter (VSI) state is evaluated through solving an optimization problem to minimize the predefined cost function. The optimal voltage vector will be applied in the next instant. A. Predictive Model The asymmetrical six-phase PMSM motor studied in this paper has two sets of three-phase windings spatially shifted by 30 electrical degrees with two isolated neutral points. The six-phase PMSM motor is supplied by a six-phase two-level VSI, as shown in Fig. 1. The widely used vector space decomposition (VSD) [26] is adopted to model the asymmetrical six-phase PMSM machine in a decoupled 6p PMSM Sa1 c Vdc a d e f 30° b Sa2 Fig. 1. Scheme of the six-phase PMSM drive. manner. Based on the amplitude invariant constraint, the VSD transformation matrix is expressed as   1 1 3 3 0   1  2 2 2 2     3 3 1 1  1 0 2 2 2 2    1 1 1 3 3 (1) T  1    0 3 2 2 2 2    3 3 1 1 0  1 2 2 2 2   1 1 1 0 0 0   0 0 1 1 1  0 T v v vx v y vo1 vo 2    (2) T  T va vb vc vd ve v f  Using the first two rows in (1), the fundamental components and the harmonics in the order of 12n ± 1 (n = 1, 2, 3…) are mapped into the α-β subspace, which governs the energy conversion. The middle two rows in (1) maps the harmonics of the order 6n ± 1 (n = 1, 3, 5…) into the x-y subspace, which does not produce torque but causes additional losses. The last two rows in (1) represent the zero sequence harmonic components in the order of 3n (n = 1, 3, 5…), which are ignored due to the isolated neutral point connection. To obtain the synchronous frame model, the Park transformation is applied to the machine variables in the α-β plane,  cos  sin   Tdq   (3)    sin  cos   T T     (4) vd vq   Tdq v v  By using this approach, the practical model of the machine in the synchronous reference frame and the x-y subspace can be obtained as: r Lq  id   Ld  vd   R id  0         * p*     (5) R  iq   Lq  vq  r Ld iq  r f   vx   R   v y   0 ix  0  i x     Ll * p *    R  i y  i y  (6) IEEE TRANSACTIONS ON POWER ELECTRONICS β 2-6 6-6 L4 2-2 L2 3-6 2-0 0-2 3-2 2-3 7-2 3-7 3-0 1-2 3-3 2-4 6-2 L3 7-3 0-3 2-7 0-6 7-6 L1 3-4 6-3 4-2 6-0 4-6 0-4 7-4 2-5 2-1 1-71-0 3-1 1-3 6-5 4-4 1-5 1-4 3-5 4-3 0-1 7-1 5-7 5-0 4-0 4-7 5-4 0-5 7-5 1-1 7-4 3-7 3-0 3-5 2-4 7-5 0-5 1-0 0-6 7-6 0-2 7-2 1-3 5-3 4-2 6-6 1-1 6-4 2-2 3-3 4-5 0-1 7-1 6-0 6-7 2-3 6-5 2-1 5-1 4-6 5-7 5-1 2-5 (a) 5-0 2-6 3-1 55 5-2 4-4 L4 3-2 5-5 2-7 2-0 4-1 1-5 5-3 α 4-5 1-7 0-4 3-4 5-6 6-1 L2 3-6 5-6 1-2 5-4 L3 6-7 1-6 5-2 1-4 6-4 y 1-6 L1 4-0 4-7 0-3 7-3 x 6-2 4-3 4-1 6-3 6-1 (b) Fig. 2. Actual voltage vectors in the α-β and x-y subspace. where vd, vq are the stator voltage components in the d- and qaxis; id, iq are the stator current components in the d- and q- axis; R is the stator resistance; Ld, Lq are the stator inductance in the d- and q-axis; ωr is the rotor angular speed; p is the time derivative operator; ψf is the permanent magnet flux linkage; vx, vy are the stator voltage components in the x-y subspace; ix, iy are the stator current components in the x-y subspace; Ll is the leakage self-inductance. The forward Euler method is used to derive the discretized model based on (5) ~ (6), thus obtaining the predictive model of the machine. The details of how the stator currents at instant k+1 are predicted can be found in [16]. Since the energy conversion is only involved in the α-β subspace, the cost function can be defined as * * g  i  i (k  1)  i  i (k  1) (7) Or, it is an alternative to take into account of the x-y stator current components in the cost function to suppress current harmonics, which is expressed as g  i*  i (k  1)  i*  i (k  1)  ix*  ix (k  1)  i*y  i y (k  1) (8) The current components in the x-y subspace at instant k+1, ix(k+1) and iy(k+1) can be predicted in the similar way, using the forward Euler method based on (6). Subsequently, the available voltage vectors can be evaluated by (7) or (8). B. Prediction Voltage Vectors The six-phase two-level VSI is characterized with 26 = 64 switching states [SaSbScSdSeSf]. The stator phase voltage can be expressed in the form of the switching states and the dc voltage as below, va   2 1 1 0 0 0   Sa     1 2 1 0 0 0   S  vb    b  vc  V  1 1 2 0 0 0   Sc     dc  (9)   3  0 0 0 2 1 1  Sd  vd  v   0 0 0 1 2 1  Se   e     0 0 0 1 1 2   S f  v f  where Si = 1 (i = a, b, c, d, e, f) when the upper switching is ON; Si = 0 (i = a, b, c, d, e, f) when the upper switching is OFF. Then, the phase voltages are transformed into the stationary frame using VSD transformation matrix Tαβ, vαβxyo1o2  T * vabcdef (10) In this way, the voltage vectors are mapped into two orthogonal subspaces, namely the α-β and x-y subspaces, as shown in Fig. 2. Each voltage vector in Fig. 2 is identified using the decimal number equivalent to the binary number of [SaSbSc]-[SdSeSf]. It can be seen from Fig. 2 that there are totally 48 active voltage vectors. Evaluating all the 48 voltage vectors will be redundant since the computing time is a crucial factor for MPC implementation. A common practice is only evaluating the largest 12 voltage vectors [16]-[19]. Then the voltage vector that minimizes the cost function will be applied at next instant. The above content in this section describes the basic principle of the conventional MPCC method for six-phase PMSM motor. This method can present fast dynamic response and it is easy to be implemented practically. Unfortunately, evaluating 13 voltage vectors will still cost a large amount of time, compared with the three-phase machine drives, where only 7 vectors need to be evaluated. In addition, using cost function (8) can suppress the current harmonics at some extent but the inclusion of the x-y components will complicate the predictive model. While adopting (7) in the conventional MPCC will result in large current harmonics. Besides, the potential of the other active vectors are not fully exploited for the sake of harmonics reduction. IEEE TRANSACTIONS ON POWER ELECTRONICS includes four parts: the RVV calculation based on DBCC, the To overcome the aforementioned problems introduced in the new cost function design, the prediction vectors synthesization, conventional MPCC method, a DBCC based MPC method the switching pulse generation of the prediction vectors. The using virtual vectors is proposed here. The control diagram of details of the proposed method are elaborated in the following the proposed MPC method is illustrated in Fig. 3, which mainly text. Standard PWM Cost Prediction switching sequence function ref vector selection ωref iq β ref T +PI S v ' vdref Reference Equivalent vvopt vvm S vv Inverse opt voltage vector S α g(min) virtual vectors ω idref S Matrix calculation ref vvm+12 S replacement  vqref S III. PRINCIPLE OF PROPOSED METHOD s a b c d e f idk  2 iqk  2 Predictive model idk iqk Park transformation ik ik VSD transformation Pulse generation iabcdef Six-phase VSI Fig. 3. Control diagram of proposed MPC method. A. RVV Calculation The solution of DBCC is adopted to calculate the desired RVV. The obtained RVV will be included as a reference vector in the cost function to evaluate the feasible prediction vectors. Using the Euler method, namely di/dt = (i(k+1) – i(k))/Ts, (5) can be expressed as Ld  vd (k )  Rs id (k )  T (id (k  1)  id (k ))  ed s  (11)  L v (k )  R i (k )  q (i (k  1)  i (k ))  e s q q q q  q Ts  Then the phase currents in the d- and q-axis at instant k+1 can be predicted as Lq  RsTs T r Ts iq (k )  s vd (k ) )id (k )  id (k  1)  (1  Ld Ld Ld   R T L T i (k  1)  (1  s s )i (k )  d  T i (k )  s v (k ) q r s d q q Lq Lq Lq  (12) In digital implementation, the computation delay will be involved caused by the large computation time, which can deteriorate the control performance [14]. A valid solution is using a two-step prediction to compensate the computation delay [14], where the current at instant k+2 are predicted as Lq  RsTs T )id (k  1)  rTsiq (k  1)  s vd ( k  1) id (k  2)  (1  L L L  d d d  R T L T i (k  2)  (1  s s )i (k  1)  d  T i (k  1)  s v (k  1) q r sd q q Lq Lq Lq  (13) According to the DBCC principle, the following constraints should be satisfied, ref  id (k  2)  id (14)  ref  iq (k  2)  iq Substitute (14) into (13), the d- and q-axis components of the reference voltage vector can be expressed as Ld ref  ref vd  Rs id (k  1)  T (id  id (k  1))  ed s  (15)  L v ref  R i (k  1)  q (i ref  i (k  1))  e s q q q q  q Ts  In this way, the expected RVV is obtained, which can be expressed in the form of a complex as (16) v ref  vdref  jvqref where j is the imaginary unit. By transforming the RVV in (16) into the α-β plane using the inverse Park transformation, which is given by  vref   cos   sin    vdref     (17)   ref  sin  cos    vqref   v     vref  vref  jvref (18) Subsequently, the position of the RVV in the α-β plane can be obtained by  ref  arctan( vref vref ) (19) B. Prediction Vectors Selection With the position of the RVV determined, the feasible voltage vectors can be evaluated to select the optimal ones closest to the RVV. The prediction vectors can be the largest 12 vectors from group L4 in Fig. 2, as commonly adopted in [16]-[19]. However, the current harmonics fail to be regulated using these 12 actual vectors. An effective approach to suppress IEEE TRANSACTIONS ON POWER ELECTRONICS the current harmonics is using the virtual vectors which are synthesized by two actual vectors [20], [25]. It can be seen from Fig. 2(a) that voltage vector v5-6, v6-5 and v4-4 are aligned in phase in the α-β subspace, while in the x-y subspace, v6-5 is in opposite direction with v5-6 and v4-4. Therefore, the effect of vector v6-5 on the flux component in the x-y subspace is opposite with v5-6 and v4-4. Meanwhile, the flux components in the x-y subspace can be expressed as  x   Ll 0  ix  (20)      y   0 Ll  i y  According to (20), the flux amplitude in the x-y subspace is proportional to its harmonic current components. Therefore, weakening the flux amplitude in the x-y subspace can suppress the harmonic currents. Therefore, the vectors from group L3 can be adopted to synthesize a virtual vector to suppress the harmonic currents. There are two options to obtain the virtual vectors, using the actual vectors from group L4 and L3, or L3 and L1 as reported in [25], for the sake of torque ripple reduction. The virtual vectors are synthesized based on the constraint that the sum of the two vectors in the x-y subspace is zero, for instance, v4-4 and v6-5, or v5-6 and v6-5. Then the acting time of each actual vector, as well as the amplitude of the virtual vectors can be calculated. The details of the calculation can be found in [20]. There are totally 24 virtual vectors and specifically, 12 of them with larger amplitude are synthesized by the actual vectors from group L4 and L3, termed as group G1 in Fig. 4, another 12 with smaller magnitude are synthesized by the actual vectors from group L3 and L1, termed as group G2. The amplitude of the virtual vectors from group G1 is 0.597Vdc and G2 is 0.345Vdc. For the virtual vectors from group G1, the acting time of the actual vectors from group L4 and L3 are 0.731Ts and 0.269Ts, respectively. While for the virtual vectors from group G2, the acting time of the actual vectors from group L1 and L3 are 0.422Ts and 0.578Ts, respectively. β V G1 IV vv12 vv11 vv24 III vv1 vv13 vv23 VI VII vv10 G2 vv22 II vv2 vv14 vv15 I α vv16 vv9 vv21 vv20 vv17 vv19 vv18 vv8 VIII vv3 vv7 IX vv6 vv5 vv4 XII XI X Fig. 4. Diagram of the synthesized virtual vectors and the divided sectors. With the adopted prediction vectors determined, the optimal ones can be selected based on the position of above calculated RVV. The α-β plane is divided into 12 sectors by the middle lines of two adjacent virtual vectors, as shown in Fig. 4. Then the optimal prediction vectors should be located in the same sector with the RVV. For instance, if the calculated RVV lies in sector I, the virtual vectors vv3 and vv15 should be selected. The prediction vectors can be determined in the same manner when the RVV lies in other sectors. It is noted that there are always only two virtual vectors to be selected, vvm and vvm+12 (m = 1, 2, 3, …12) no matter where the RVV locates. C. Cost Function Design In section B, there are two prediction vectors, vvm and vvm+12 closest to the RVV selected. These two vectors are aligned in phase but with different amplitudes. A novel cost function is designed to evaluate the error between the amplitude of the RVV and these two selected virtual vectors, which is expressed as g  v ref  vvi (21) where vvi represents the selected two virtual vector candidates. These two candidates, along with a null vector are evaluated by (21) and the one that minimizes (21) will be selected and applied at next instant. It can be seen from (21) that the weighting factor involved in the MPTC method is avoided here. Besides, there are only three candidate vectors to be evaluated and therefore the computing time needed is much smaller. In the meantime, the x-y harmonics are regulated by the virtual vectors. D. Switching Pulse Generation With the optimal virtual vector selected in section C, the switching pulses corresponding to the selected virtual vector should be applied. However, the switching pulses for some of the virtual vectors are not standard PWM pulses in one period, which makes the implementation difficult. According to their different switching pulse generation features, the 24 virtual vectors can be divided into four groups, namely S1 (vv1, vv3, vv5, vv7, vv9, vv11), S2 (vv2, vv4, vv6, vv8, vv10, vv12), S3 (vv13, vv15, vv17, vv19, vv21, vv23), S4 (vv14, vv16, vv18, vv20, vv22, vv24). The switching pulse generation for virtual vectors vv1, vv2, vv13 and vv14 are illustrated in Fig. 5 as examples. It can be seen from Fig. 5(a) that for virtual vector vv1, the switching sequence is standard in one PWM period, indicating that it is easy to be implemented. While for virtual vector vv2, the switching sequence is non-standard since it can be seen from Fig. 5(b) that the level of Sb and Se are opposite at the middle of the PWM period. In addition, non-standard switching sequences can also be observed from Fig. 5(c)(d) for virtual vectors vv13 and vv14. The switching pulses generation for other virtual vectors can be analyzed in the same manner and only vv1, vv3, vv5, vv7, vv9, vv11 among all 24 virtual vectors can present standard PWM switching sequence. Other virtual vectors from S2, S3, S4 all present non-standard PWM switching sequences. To achieve easy implementation, a valid solution is adopting the actual vectors from group L2 in Fig. 2 to obtain equivalent virtual vectors to replace the virtual vectors with non-standard PWM switching sequence. Due to the different features of the virtual vectors from group S2 and S3, S4, the solution of their switching pulses generation will be discussed separately. First, for virtual vectors vv2, vv4, vv6, vv8, vv10 and vv12 from group S2, the two vectors from group L2 adjacent to the actual vector from IEEE TRANSACTIONS ON POWER ELECTRONICS group L4 in the α-β subspace will be used to form the equivalent vector to replace it. For instance, for virtual vector vv2, which is synthesized by v4-6 and v6-4, the actual vector v6-4 can be replaced by two actual vectors v0-4 and v6-7, since the sum of v0-4 and v6-7 is equivalent to v6-4 both in α-β and x-y subspaces. Therefore, the virtual vector synthesized by v4-6, v0-4 and v6-7 (termed as vv’2) is equivalent to the virtual vector vv2. Nevertheless, the virtual vector vv’2 can present standard PWM switching sequence and is easy to be implemented, as shown in Fig. 6(a). Similarly, other virtual vectors vv4, vv6, vv8, vv10 and vv12 can be replaced by virtual vectors vv’4 (v4-0, v5-4, v7-5), vv’6 (v0-1, v1-5, v5-7), vv’8 (v1-0, v3-1, v7-3), vv’10 (v0-2, v2-3, v3-7) and vv’12 (v2-0, v6-2, v7-6). vv14(v4-6, v2-5), S4 Ts v4-6 v2-5 v2-5 v4-6 vv13(v2-4, v4-2), S3 Ts v2-4 v4-2 v4-2 v2-4 vv2(v4-6, v6-4), S2 Ts v4-6 v6-4 v6-4 v4-6 vv1(v2-4, v6-6), S1 Ts v2-4 v6-6 v6-6 v2-4 Sa Sa Sa Sa Sb Sb Sb Sb Sc Sc Sc Sc Sd Sd Sd Sd Se Se Se Se Sf Sf Sf Tm 2 Tn 2 Tn 2 Tm 2 Tm 2 Tn 2 Tn 2 Sf Tm 2 Tm 2 Tn 2 (c) (a) (b) Fig. 5. Switching pulses generation for virtual vectors. (a) vv1. (b) vv2. (c) vv13. (d) vv14. Tn 2 Tm 2 Tm 2 Tn 2 (d) Tn 2 vv’2(v0-4, v4-6, v6-7) vv’13(v2-4, v6-6, v0-0, v7-7) vv’14(v0-4, v4-6, v6-7, v0-0, v7-7) Ts v6-7 Ts v0-0 v2-4 v6-6 v7-7 v7-7 v6-6 v2-4 v0-0 Ts v0-0 v0-4 v4-6 v6-7 v7-7 v7-7 v6-7 v4-6 v0-4 v0-0 v0-4 v4-6 v4-6 v0-4 Sa Sa Sa Sb Sb Sb Sc Sc Sc Sd Sd Sd Se Se Se Sf Sf Sf Tm Tn 2 2 Tm (a) Tn 2 Tm 2 Tz Tm' Tn' Tz Tz 4 2 2 4 4 Tn' Tm' Tz 2 2 4 Tz 4 (b) Fig. 6. The switching pulse generation for virtual vectors. (a) vv’2. (b) vv’13. (c) vv’14. For the virtual vectors vv13 ~ vv24, their PWM switching sequence are also non-standard. Unfortunately, the solutions proposed for virtual vectors in S2 are not applicable here. Even with the actual vectors from group L2 employed to obtain equivalent vectors to replace the vectors from group L1 or L3, their PWM switching sequence are still non-standard. As it can be seen that there is always one vector from group G2 aligning in phase with one virtual vector from group G1, and the amplitude of G2 is 57.8% of G1. Since the standard switching Tm' 2 Tn' 2 Tm' 2 Tz 4 Tz 4 (c) Tm' 2 Tn' 2 Tm' 2 Tm 2 Tz 4 sequence generation for virtual vector vv1 ~ vv12 have been solved above, inserting a null vector to adjust the duty ratio of vv1, vv’2, vv3, vv’4, vv5, vv’6, vv7, vv’8, vv9, vv’10, vv11 and vv’12 can obtain virtual vectors equivalent to vv13 ~ vv24 while present standard switching sequence simultaneously. For instance, the virtual vector vv13 is replaced by an equivalent virtual vector vv’13, which is synthesized by v2-4, v6-6, v0-0, v7-7, as shown in Fig. 6(b). Moreover, the virtual vector vv’14 synthesized by v0-4, v4-6, v6-7, v0-0, v7-7 is adopted to replace vv14. IEEE TRANSACTIONS ON POWER ELECTRONICS Current[A] The next step is to calculate the duration of the null vectors. The amplitude of the virtual vectors from group G1 and G2 are: |vvG1| = 0.597Vdc, |vvG2| = 0.345Vdc. To obtain the vectors equivalent to the virtual vectors in group G2, the duration of the null vectors inserted into vectors in G1 can be calculated as 0.345Vdc (22) Tz  Ts  Ts  0.422Ts 0.597Vdc In the meantime, as mentioned above, to achieve harmonic currents suppression, the acting time ratio of the actual vector from group L4 and L3 is 0.731Ts (23)  2.717 0.269Ts For the virtual vectors vv’13 ~ vv’24, the acting time of the actual vectors from group L4 (or L2) is T’m and that of the actual vectors from group L3 is T’n. The following constraints should be satisfied,  Tm'  '  2.717 (24)  Tn  ' ' Tm  Tn  0.578Ts Solving (24), T’m and T’n can be obtained as T’m = 0.422Ts and T’n = 0.156Ts. According to the above analysis, the 18 virtual vectors from S2, S3 and S4 with non-standard switching sequence are replaced by another 18 equivalent virtual vectors with standard PWM switching sequence. Now all 24 virtual vectors can be easily implemented. Generally, the core idea of the proposed method is consistent with the conventional MPCC method, namely looking for the optimal voltage vector through a cost function. However, they are realized in different manners. The calculation of the RVV is not involved in the conventional MPCC method, where the optimal voltage vector is obtained indirectly based on the cost function with the constraint of current errors. While the proposed MPCC calculates the RVV first and then defines a cost function to compare the RVV with the candidate voltage vectors directly. Compared with the conventional MPCC method, the number of prediction vectors is significantly reduced and the predictive model is simplified in the proposed 15 ia 0 -15 -15 Torque[Nm] Current[A] ix In this section, the simulations are carried out in the environment of Matlab/Simulink to verify the effectiveness of the proposed method. The conventional MPCC method and the direct DBCC method are both implemented as benchmark methods. The former one is characterized as directly evaluating the largest 12 voltage vectors using the cost function (8), and the latter one is defined as using two active adjacent vectors and one zero vector from group L4 to synthesize the obtained vector in (18). A 11 kW asymmetrical six-phase motor is used in the simulation. The parameters of the machine are listed in Table I. The sampling frequency is set as 10 kHz for all methods in the simulation. First, the steady-state performance is investigated when the machine is running at 800 rpm with 70 Nm load. It can be seen from Fig. 7 that the amplitude of the phase current reaches 15 A at 70 Nm. The current quality of the direct DBCC and conventional MPCC method are much poorer than the proposed method due to the large amount of harmonic currents in the x-y subspace. While it is observed that the harmonic currents of the proposed method are almost zero and sinusoidal phase currents are presented. In the meantime, it can be seen that the magnitude of the harmonic currents of the direct DBCC is larger than the conventional MPCC. This can be explained by the fact that the x-y components are included in the conventional MPCC and they are slightly suppressed at some extent. While the direct DBCC method fails to regulate the x-y subspace harmonics. Secondly, the dynamic response with sudden load change of the machine are investigated, as shown in Fig. 8. The load is changed from 35 Nm to 70 Nm at 0.5 s. It can be seen that the torque command is tracked smooth and fast. The phase current quality is also consistent with the performance in Fig. 7. In addition, the current quality is better at 70 Nm than 35 Nm for the direct DBCC and conventional MPCC method. iy ia 15 id 0.52 0.56 0.54 Time[s] (a) 0.58 4 2 0 -2 -4 72 68 0.6 0.5 ia id ix iy 0 ix iy -15 4 2 0 -2 -4 72 70 70 68 0.5 IV. SIMULATION PERFORMANCES id 15 0 4 2 0 -2 -4 72 method. In the meantime, the harmonic currents are effectively suppressed using the virtual vectors. 70 0.52 0.56 0.54 Time[s] (b) 0.58 0.6 68 0.5 0.52 0.56 0.54 Time[s] 0.58 0.6 (c) Fig. 7. Steady state performance of the machine under 800 rpm with 70 Nm load. (a) Direct DBCC method. (b) Conventional MPCC method. (c) Proposed method. Current[A] Torque[Nm] IEEE TRANSACTIONS ON POWER ELECTRONICS 75 75 75 55 55 55 35 35 35 15 ia 0 0 -15 0.4 -15 0.5 Time[s] 0.6 0.4 15 ia 15 ia 0 0.5 Time[s] (a) -15 0.6 0.4 0.5 Time[s] (b) 0.6 (c) Fig. 8. Dynamic response with sudden load change under 800 rpm. (a) Direct DBCC method. (b) Conventional MPCC method. (c) Proposed method. V. EXPERIMENTAL PERFORMANCES In this section, the experiments have been conducted to demonstrate the effectiveness of the proposed method. The experiment platform is shown in Fig. 9. The scaled-down 1.1 kW asymmetrical six-phase PMSM motor is supplied by a conventional six-phase two-level VSI with a single dc power supply. The control actions are performed using the DSP (TMS320F28335) from Texas Instruments. The parameters of the PMSM motor drives and its rated values are listed in Table. I. It is worth to mention that to better show the superiority of the proposed method, the method presented in [20] is also conducted through experimentation, where the virtual vectors from group G1 are evaluated through the cost function (7). The switching frequency of the direct DBCC method is constant inherently. However, in the MPC methods including the conventional MPCC and the proposed method, the switching frequency are variable. In the meantime, the average switching frequency of the proposed MPC is somewhat higher than the conventional MPCC under the same sampling frequency. This is because there are always two active vectors to be applied in each sampling period of the proposed MPC and the switching state changes twice in some periods. Therefore, the proposed MPC is implemented under 10 kHz and 5 kHz, while the conventional MPCC is implemented 10 kHz to ensure that the switching frequency of the conventional MPCC is in between the switching frequency of the proposed MPC tests. The method presented in [20] and the method in [24] are also implemented under 10 kHz sampling frequency. Fig. 9. Experimental setup. TABLE I. KEY PARAMETERS OF MACHINE AND CONTROL SYSTEM Value Specification Simulation Experimentation Rated motor power 11 kW 1.1 kW Rated speed 1500 rpm 1500 rpm 7 Nm Rated torque 70 Nm Number of pole pairs 3 3 Stator resistance 4.5 Ω 4.5 Ω d-axis inductance 0.035 H 0.035 H q-axis inductance 0.055 H 0.055 H Permanent magnet flux 0.55 Wb 0.225 Wb Rotary inertia Rotary inertia of the load machine 0.021 kg*m2 0.0011 kg*m2 --- 0.0037 kg*m2 A. Steady-State Performance First, the steady-state performances of the direct DBCC, conventional MPCC, the method in [20] and the proposed MPC method are investigated. The steady-state responses under 800 rpm with a rated load are shown in Fig. 10. From top to bottom, the waveforms given in Fig. 10 are stator phase currents, harmonic currents in the x-y subspace, electromagnetic torque and motor speed. A significant difference can be observed in terms of the x-y harmonic currents between the direct DBCC method, the conventional MPCC and the proposed MPC method. Large x-y harmonic currents are generated in the direct DBCC method and the conventional MPCC method as shown in Fig. 10(a)(b), while the amount of x-y currents are clearly limited due to the use of the virtual vectors as shown in Fig. 10(c). This can be explained by the fact that the x-y harmonic currents are ignored in the direct DBCC method, where only the variables in the α-β subspace are regulated. While in the conventional MPCC, though the x-y components are included in the cost function, they are not well eliminated since the voltage vector is not zero in the x-y subspace. This, in turn, results in a low power quality of the stator currents of the direct DBCC method and the conventional MPCC as shown in Fig. 10(a)(b). In contrast, the very sinusoidal phase currents are exhibited in the proposed MPC method shown in Fig. 10(d). In the meantime, even with the sampling frequency reduced half, the harmonic currents presented by the proposed MPC method are still much smaller compared with the conventional MPCC method, as shown in Fig. 10(e), though larger sampling step IEEE TRANSACTIONS ON POWER ELECTRONICS of the x-y harmonic components by the synthesized virtual vectors in the proposed MPC method, it is possible to obtain a significant improvement in the phase current power quality with a lower switching frequency. Though the method in [20] can also well suppress the harmonic currents in the x-y subspace (THD of ia = 8.36%), 13 prediction vectors are involved, which will introduce large computation time. Another benefit of the proposed MPC method is its reduced number of iterations since the number of prediction vectors to be evaluated is reduced from 13 to 3. The total execution time of direct DBCC, conventional MPCC, the method in [20] and the proposed method are measured as 31.2 µs, 55.6 µs, 56. 9 µs, 42.7 µs, respectively. The execution time of the MPC methods is much larger than the direct DBCC method and this is the inherent characteristic of the MPC control that large computation time is usually required. However, compared with the conventional MPCC method, the total execution time of the proposed method is reduced by 23%. The torque performance is also presented in Fig. 10. It can be seen that the torque ripple of the proposed MPC at 10 kHz (Fig. 10(d)) is slightly smaller than that of the direct DBCC and the conventional MPCC. This can be explained by the fact that there are two groups of vectors with different magnitudes in the proposed method instead of 12 vectors of the same magnitude in the conventional MPCC method. results in this slightly increased current ripple. Then the steady-state test under 800 rpm without load is investigated as shown in Fig. 11. Similar current performances can be observed that the phase currents are distorted in the direct DBCC method and the conventional MPCC method. In the meantime, it can be seen that the current quality is inferior to that in the full load condition for each method. The frequency spectrum of the phase currents are given in Fig. 12. The total harmonic distortion (THD) of phase a current for these four methods are obtained as 23.58%, 20.53%, 7.97% and 13.87%, respectively. The zoom-in plot of the harmonics in the low order is provided for better visualization. The fundamental frequency of current waveform is 40 Hz (800rpm) and large amount of 5th and 7th harmonics can be observed in Fig. 12(a)(b). While in Fig. 12(c)(d), it can be noticed that the 5th and 7th harmonics as well as the harmonics in the higher order are reduced. In addition, the average switching frequencies of the conventional MPCC, the proposed MPC at 10 kHz and 5 kHz are measured as 3520 Hz, 6210 Hz and 3352 Hz, respectively. Thus, it is confirmed that the proposed MPC method can present much better steady-state performance with a lower switching frequency. Though the switching frequency of the direct DBCC method is fixed, its current performance is poor. This is because the direct DBCC and the conventional MPCC always generate voltage in the x-y subspace, thus resulting in high harmonic currents. Thanks to the suppression ia 2A/div id 10ms/div ix 1A/div iy 10ms/div n 800 rpm 10ms/div id ix 1A/div iy id ix 1A/div 10ms/div Te 7 Nm 800 rpm 10ms/div (b) iy n 800 rpm 10ms/div id 2A/div ix 1A/div iy 10ms/div n ia ix 1A/div iy 10ms/div Te 7 Nm id 10ms/div 10ms/div Te 7 Nm (c) ia 2A/div 10ms/div 10ms/div n ia 2A/div 10ms/div Te 7 Nm (a) ia 2A/div n Te 7 Nm 800 rpm 10ms/div 800 rpm 10ms/div (d) (e) Fig. 10. Test 1 for all these three methods in steady state with full load. From top to bottom: the measured phase currents, the current in the x-y subspace, the electromagnetic torque and the motor speed. (a) Direct DBCC method. (b) Conventional MPCC method. (c) Method in [20]. (d) Proposed method with 10 kHz sampling frequency. (e) Proposed method with 5 kHz sampling frequency. To further investigate the average switching frequency of the conventional MPCC method and the proposed method, they are measured under different speed conditions with half load and rated load, respectively, as shown in Fig. 13. It can be seen that the proposed method with 10 kHz sampling frequency always introduces the highest average switching frequency in all occasions. While with the sampling frequency lowered to 5 kHz, its average switching frequency is still always slightly lower than that of the conventional MPCC. In the meantime, the phase current THD of the proposed method, the conventional MPCC, the method presented in [24] and the direct DBCC with different load conditions under 500 rpm and 1000 rpm are illustrated in Fig. 14(a)(b). It can be seen that the proposed method can significantly reduce the current THD compared with the conventional MPCC and direct DBCC under same sampling frequency. Even with the sampling IEEE TRANSACTIONS ON POWER ELECTRONICS suppressed to zero theoretically or practically. In the meantime, the predictive model is significantly simplified using the proposed method. This is because the prediction of the torque and flux in each sampling period using the method in [24] is avoided in the proposed method. Moreover, there is no need to tune the weighting factor. frequency lowered half, the proposed method still presents less phase current THD than the direct DBCC and conventional MPCC thanks to the regulation of the x-y harmonic components by virtual vectors. Though the harmonic currents are already effectively suppressed in [24], it is still slightly larger than the proposed method as shown in Fig. 14. It can be justified that in [24], the voltage components in the x-y subspace cannot be id ia ia id 1A/div 1A/div 10ms/div 10ms/div n 2Nm/div Te ia 1A/div n 2Nm/div 800rpm Te n 2Nm/div Te n 2Nm/div Te 800rpm 10ms/div 10ms/div (b) id 10ms/div 800rpm 10ms/div (a) ia 1A/div 10ms/div 800rpm 10ms/div id (d) (c) 25 20 15 th 5 10 7th 5 25 20 15 10 5 00 THD= 20.53% 4 8 12 16 Harmonic order 25 20 15 10 5 5th 7th 25 THD= 7.97% 20 15 10 5 0 0 4 8 12 16 Harmonic order 25 20 15 th 10 5 th 7 5 Content(%) 25 THD= 23.58% 20 15 10 5 0 0 4 8 12 16 Harmonic order Content(%) 25 20 15 5th 10 7th 5 Content(%) Content(%) Fig. 11. Test 2 for all these three methods in steady state without load. From top to bottom: the measured phase currents, the motor speed and the electromagnetic torque, (a) Direct DBCC method. (b) Conventional MPCC method. (c) Proposed method with 10 kHz sampling frequency. (d) Proposed method with 5 kHz sampling frequency. 25 20 15 10 5 00 THD= 13.87% 4 8 12 16 Harmonic order 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) (c) (d) (a) (b) Fig. 12. The THD analysis of phase current ia for all three methods. (a) Direct DBCC method. (b) Conventional MPCC method. (c) Proposed method with 10 kHz sampling frequency. (d) Proposed method with 5 kHz sampling frequency. Conventional MPCC, fs = 10 kHz Proposed MPC, fs = 10 kHz Proposed MPC, fs = 5 kHz 6 5 4 3 2 1 7 Average switching frequency[kHz] Average switching frequency[kHz] 7 Conventional MPCC, fs = 10 kHz Proposed MPC, fs = 10 kHz Proposed MPC, fs = 5 kHz 6 5 4 3 2 0 200 400 600 800 1000 1200 1400 Speed [rpm] (a) 1 0 200 400 600 800 1000 1200 1400 Speed [rpm] (b) Fig. 13. Measured average switching frequency for all methods under different speed conditions with (a) half load and (b) rated load. IEEE TRANSACTIONS ON POWER ELECTRONICS THD content [%] 35 30 25 20 15 35 30 25 20 15 10 10 5 5 0 0 0.6 0.4 Load (p.u.) (a) 0.2 0.8 1 Direct DBCC Conventional MPCC Proposed MPC Proposed MPC, fs = 5 kHz Method in [24] 40 THD content [%] Direct DBCC Conventional MPCC Proposed MPC Proposed MPC, fs = 5 kHz Method in [24] 40 0 0 0.6 0.4 Load (p.u.) (b) 0.2 0.8 1 Fig. 14. Phase current THD analysis for all methods in different load conditions under (a) 500 rpm and (b) 1000 rpm. B. Dynamic Responses with Change in Load Second, to further evaluate the control performance of the proposed method, the dynamic test with a step load change is carried out. At the start, the motor is running at 800 rpm and then a rated load is added suddenly. The transient state waveforms are given in Fig. 15, including the speed, n Te 800 rpm n Te 800 rpm n Te 800 rpm n Te 800 rpm 7 Nm 7 Nm 7 Nm 7 Nm 10 ms/div ia 2A/div 10 ms/div (a) electromagnetic torque, phase current. Fast dynamic response and good disturbance rejection performance can be observed for the conventional MPCC, the proposed MPC at 10 kHz and 5 kHz. Additionally, when the motor reaches the steady state, similar phase current performance with Fig. 10 can be observed. 10 ms/div ia 2A/div ia 2A/div 10 ms/div (b) 10 ms/div 10 ms/div ia 2A/div 10 ms/div 10 ms/div (c) (d) Fig. 15. Test 3 for all methods with sudden load change. From top to bottom: the measure motor speed, the electromagnetic torque and the phase current. (a) Direct DBCC method. (b) Conventional MPCC method. (c) Proposed method with 10 kHz sampling frequency. (d) Proposed method with 5 kHz sampling frequency. C. Acceleration Test Third, the acceleration responses for these three methods are investigated, where the machine accelerates from standstill to 1500 rpm with and without load, respectively. The machine performance under the acceleration without load is illustrated in Fig. 16. It can be observed that the transient state performance of the direct DBCC, conventional MPCC, the proposed MPC at 10 kHz and 5 kHz are similar. The speed waveforms are smooth without overshoot. However, when the machine speed reaches 1500 rpm, much better current power quality can be observed in Fig. 16(c). This is consistent with the steady-state performance in the first test that the proposed MPC method can present much better power quality phase currents than the direct DBCC and the conventional MPCC method. When a rated load is added to the machine as the acceleration starts, more time is needed to reach the speed command, as shown in Fig. 17. It can be seen that the acceleration time with full load is almost twice of that in the no-load condition. In the meantime, the speed waveforms are still smooth during the transient state. To sum up, the effectiveness of the x-y harmonic currents regulation of the proposed MPC method in different scenarios is proved by the experimental results. In the meantime, the computation burden is reduced, thus making the proposed MPC method more practical. Moreover, the predictive model of the proposed method is simplified than conventional MPCC method. Also, the capability of the proposed MPC method to regulate the speed and torque in dynamic conditions is confirmed. IEEE TRANSACTIONS ON POWER ELECTRONICS n Te 1500rpm 2Nm/div n Te 1500rpm 2Nm/div 0rpm 0rpm 0rpm 50ms/div ia 2A/div n Te 1500rpm 2Nm/div 0rpm 50ms/div 50ms/div ia 2A/div ia 2A/div 50 ms/div 50 ms/div 50ms/div ia 2A/div 50 ms/div 50 ms/div (c) (b) (a) n Te 1500rpm 2Nm/div (d) Fig. 16. Test 4 for all methods at acceleration state without load. From top to bottom: the measured rotor speed, the electromagnetic torque and the phase current. (a) Direct DBCC method. (b) Conventional MPCC method. (c) Proposed method with 10 kHz sampling frequency. (d) Proposed method with 5 kHz sampling frequency. n 2Nm/div Te 1500rpm 0rpm 50ms/div 8 6 0rpm 4 2 0 ia 2A/div Te n Te Te n n 2Nm/div 2Nm/div 1500rpm 1500rpm 1500rpm 8 8 8 6 0rpm 6 0rpm 6 4 4 4 2 2 2 50ms/div 50ms/div 50ms/div 0 0 0 2Nm/div ia 2A/div 50 ms/div 50 ms/div (a) ia 2A/div ia 2A/div 50 ms/div 50 ms/div (d) (c) (b) Fig. 17. Test 5 for all methods at acceleration state with full load. From top to bottom: the measured rotor speed, the electromagnetic torque and the phase current. (a) Direct DBCC method. (b) Conventional MPCC method. (c) Proposed method with 10 kHz sampling frequency. (d) Proposed method with 5 kHz sampling frequency. VI. CONCLUSION In this paper, a novel RVV based MPC with harmonic currents suppressed and computation burden reduced is proposed for an asymmetrical six-phase PMSM motor. The major contributions of this work include: 1) two groups of virtual vectors are synthesized aiming at harmonic currents reduction; 2) the DBCC solution is adopted to calculate the RVV and select the appropriate voltage vector candidates, thus reducing the computing time significantly; 3) a new cost function is defined to directly evaluate the error between the RVV and the selected virtual vectors, which significantly simplifies the predictive model; and 4) the 18 virtual vectors with non-standard PWM switching sequences are replaced by the equivalent virtual vectors with the standard PWM switching sequences, where all the virtual vectors can be easily implemented practically. Accordingly, the simulation and experimental results are offered to confirm the validity of the proposed MPC method. 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He is currently working toward the Ph.D. degree in electrical engineering in City University of Hong Kong. His research interests include power electronics and multiphase machine drives. Chunhua Liu (M’10–SM’14) received the B.Eng., M.Eng. and Ph.D. degrees in Automatic Control, Beijing Institute of Technology, China, and in Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, in 2002, 2005 and 2009, respectively. Currently, he serves as Assistant Professor with the School of Energy and Environment, City University of Hong Kong, Hong Kong, China. His research interests are in electrical energy and power technology, including electric machines and drives, electric vehicles, electric robotics and ships, renewables and microgrid, and wireless power transfer. In these areas, he has published over 160 refereed papers. Dr. Liu is currently an Associate Editor of IEEE Transaction on Industrial Electronics, Editor of IEEE Transactions on Vehicular Technology, and Guest Editor-in-Chief of IEEE Transactions on Energy Conversion. Also, he is an Editor of Energies, Subject Editor of IET – Renewable Power Generation, Associate Editor of Cambridge University – Wireless Power Transfer, Associate Editor of IEEE Chinese Journal of Electrical Engineering, Editor of IEEE Transactions on Magnetics – Conference, respectively. In addition, he is Chair & Founder, HK Chapter, IEEE Vehicular Technology Society.