An Enhanced Predictive Current Control Method for Asymmetrical Six ...

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 8, AUGUST 2011

An Enhanced Predictive Current Control Method for Asymmetrical Six-Phase Motor Drives Federico Barrero, Senior Member, IEEE, Joel Prieto, Student Member, IEEE, Emil Levi, Fellow, IEEE, Raúl Gregor, Member, IEEE, Sergio Toral, Senior Member, IEEE, Mario J. Durán, and Martin Jones, Member, IEEE

Abstract—The interest in predictive control approach and multiphase drives has been steadily growing during the last decade. Predictive control techniques have been recently introduced as a viable alternative to conventional PI controllers with carrier-based or space vector PWM techniques in the current regulation of multiphase power converters and drives. The developed schemes have demonstrated a good performance at the expense of a high computational cost, an unknown switching frequency, and the appearance of large undesirable stator current harmonic components. In this paper, an enhanced predictive current control technique with fixed switching frequency is introduced for an asymmetrical dual three-phase ac drive. Fast torque and current responses are achieved while favoring stator current harmonic suppression. The experimental results are provided to verify the benefits of the proposed control method when compared to the other existing predictive control methods. Index Terms—Current control, multiphase machines, predictive control.

I. I NTRODUCTION

P

REDICTIVE control algorithms are nowadays playing an important role in the development of modern highperformance power electronics and drive systems [1], [2]. A survey of the most important types of predictive control applied in these areas is presented in [2]. Predictive control encompasses a very wide class of controllers, but its main characteristic is the use of the model of the system for the prediction of the future behavior of the controlled variables. This information is then used by the controller to obtain the actuation according to a predefined optimization criterion. In power converters and electrical drives, it is possible to simplify the optimization problem, considering the discrete nature of the power converter.

Manuscript received May 20, 2010; revised August 27, 2010; accepted October 11, 2010. Date of publication October 28, 2010; date of current version July 13, 2011. This work was supported in part by the Spanish Government (reference DPI2009/07955) and in part by the Itaipu Binacional/Parque Tecnológico Itaipu-Py. F. Barrero, J. Prieto, and S. Toral are with the Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]; [email protected]; [email protected]). E. Levi and M. Jones are with the School of Engineering, Liverpool John Moores University, L3 3AF Liverpool, U.K. (e-mail: [email protected]; [email protected]). R. Gregor is with the FIUNA, National University of Asunción, Luque, 2060 Paraguay (e-mail: [email protected]). M. J. Durán is with the Electrical Engineering Department, University of Malaga, 29120 Malaga, Spain (e-mail: [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 10.1109/TIE.2010.2089943

Taking into account the finite set of possible switching states of the converter, which depends on the possible combinations of the ON/OFF switching states of the power switches, the optimization problem is reduced to the evaluation of all possible states and the selection of the one which minimizes the selected cost function. This control scheme, also called as “finite set model predictive control” (FS-MPC), had, in the past, very few applications in power converter control and drives due to the high amount of calculations needed in order to solve the optimization problem online, which is incompatible with the small sampling times used in converter control [3]. However, the increase in computing power of the modern microprocessors makes this strategy now plausible for controlling power electronic converters and variable speed drives [2], [3]. As a consequence, extensive work has been conducted in recent times to explore the advantages of this predictive control method in the aforementioned areas. The FS-MPC method has been recently applied to the current control of a two-level voltage source inverter (VSI) supplying three-phase static loads [4] and electrical drives [5], [6], current control of multilevel converters supplying static loads [7], [8] or in the operation on the grid [9], and current control of matrix converters feeding static loads [10] and induction motor drives [11], [12]. Further applications of the FS-MPC method can be found in the current and voltage controls of multilevel inverters [13], in the torque and flux controls of induction machines using two-level VSIs [5], [14], [15] and matrix converters [16], in the speed and torque controls of permanent magnet synchronous machines [17], [18], or in the power control using multilevel converters and induction motors for wind power conversion systems [19]. Other control schemes have also been modified, adopting the principles of predictive control, like the popular direct torque control [20], [21]. Last but not the least, predictive current control has been very recently examined in conjunction with multiphase drives in [22]–[30]. The predictive current control techniques presented in [22] and [23], termed as “model-based predictive control” (MBPC), show that predictive current control can provide a high dynamic performance in asymmetrical six-phase drives. The number of voltage vectors used to evaluate the predictive model has been reduced to 13 vectors while requiring the sinusoidal output voltage and considering the quasi-balanced operation of the drive. In this way, the optimizer has also been implemented using only 13 possible stator voltage vectors (12 active, corresponding to the largest vectors in the α–β subspace and the smallest ones in the x–y subspace plus a zero vector). The MBPC method using only 13 switching vectors required less computing

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time, and the real-time implementation has been shown to be viable. The variations of the MBPC method have been further analyzed to improve the harmonic content and to ensure the control method’s quality. Two research directions have been identified, with the view of improving the predictive current control performance in multiphase drives. The first one is related to the losses in the power converter legs, which are not equally distributed as the total number of commutations per fundamental period and leg is not the same. The idea is either to use the linear combination of active vectors plus a zero one during a sampling period, resulting in the predictive current control method termed as “one-step modulation predictive current control” (OSPC further on) [24], or to introduce pulsewidth modulation techniques to translate these desired voltages into switching patterns, resulting in a control technique termed as “predictive space vector PWM current control” (PSVPWM further on) [25]. The second direction relates to the computation time needed for the implementation, which should be reduced. Some predictive control techniques have been developed, which favor the speed of real-time implementation over other criteria [27]– [30]. The method called as “restrained search predictive control technique” favors online implementation at the expense of achieving suboptimal solutions. This paper improves the performance reported in the previous studies by searching the best distribution of the switching losses and by taking into account a new control scheme that considers the use of predictive and modulation schemes for the current control of an asymmetrical dual three-phase ac machine. The viability of the introduced method is studied, and its performance is analyzed in detail using real-time implementation in a DSP, thus providing the experimental results. A low ripple in the stator current response is achieved, while a fast dynamic behavior is maintained. The performance of the proposed control method is studied, and the obtained results are compared with other predictive control methods. This paper is organized as follows. An asymmetrical dual three-phase induction motor drive is described in Section II. Section III details the general principles of the proposed predictive current control method. Section IV compares the developed current control technique with other predictive current control methods, presenting a discussion of the obtained results. The conclusion is given in the last section. II. A SYMMETRICAL S IX -P HASE D RIVE An asymmetrical six-phase (dual three-phase) induction machine is considered. A schematic of the drive is shown in Fig. 1. The machine consists of two sets of three-phase stator windings that are spatially shifted by 30 electrical degrees, with two isolated neutral points. The power is supplied from a sixphase VSI, which has a total of 26 = 64 different switching states, defined by six switching functions, which correspond to the six inverter legs [Sa , Sb , Sc , Sd , Se , Sf ], where Si ∈ {0, 1}. Different switching states and the voltage of the dc link (Vdc ) define the phase voltages which can, in turn, be mapped into the α–β and x–y subspaces according to the vector

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Fig. 1. General scheme of an asymmetrical dual three-phase ac drive.

space decomposition (VSD) approach [31]. Consequently, the 64 different ON / OFF combinations of the six VSI legs lead to 64 voltage space vectors in the α–β and x–y subspaces. Fig. 2 shows the active voltage space vectors in the α–β and x–y subspaces. Each switching state is identified by two octal numbers which correspond to the binary numbers [Sa Sb Sc ] and [Sd Se Sf ], respectively. It must be noted that the 64 possible switching states imply only 49 different voltage space vectors in the α–β and x–y subspaces. Nevertheless, redundant states have to be considered because they have a different impact on the switching patterns. The six-phase machine is a continuous system which can be described by a set of differential equations. The predictive model applied here is based on the VSD approach under standard modeling assumptions, which directly maps the low-order current harmonics of the machine current spectrum into different 2-D subspaces of the multidimensional space. Consequently, the machine model consists of three pairs of windings in mutually orthogonal subspaces. One pair engages in electromechanical energy conversion (α–β subspace), while the others do not. The α–β subspace represents the fundamental supply component plus supply harmonics of the order 12n ± 1 (n = 1, 2, 3, . . .). The second component pair (x–y subspace) does not contribute to the air-gap flux and torque, and it includes all of the supply harmonics of the order 6n ± 1. The third pair (z1 −z2 subspace) corresponds to the zero sequence harmonic components. These harmonics cannot flow due to the isolated neutral points. The considered multiphase drive can be represented in the stationary reference frame using an amplitude invariant criterion in the transformation (constant power per phase) with the following mathematical model [22]: ⎡ ⎤ 2 ua ⎢ −1 ⎢ ub ⎥ ⎢ ⎥ V ⎢ ⎢ uc ⎥ dc ⎢ −1 ·⎢ ⎢ ⎥= ⎢ ud ⎥ 3 ⎢ 0 ⎣ ⎣ ⎦ 0 ue 0 uf ⎡

−1 −1 2 −1 −1 2 0 0 0 0 0 0

⎤⎡ ⎤ Sa 0 0 0 0 0 0 ⎥ ⎢ Sb ⎥ ⎥⎢ ⎥ 0 0 0 ⎥ ⎢ Sc ⎥ ⎥ · ⎢ ⎥ (1) 2 −1 −1 ⎥ ⎢ Sd ⎥ ⎦⎣ ⎦ −1 2 −1 Se −1 −1 2 Sf

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Fig. 2. Voltage space vectors and switching states in the α–β and x–y subspaces for a six-phase asymmetrical VSI. √ √ ⎡ ⎤ 3 ⎤ 1 − 21 − 12 − 23 0 ⎡ ua ⎤ uα 2 √ √ ⎢ ⎥ 3 1 1 − 23 −1 ⎥ ⎢ ub ⎥ ⎢0 ⎢ uβ ⎥ 2 2√ √2 ⎢ ⎥ 1 ⎢ ⎥⎢ ⎥ 3 1 ⎢u ⎥ ⎢ ⎢ ux ⎥ − 12 − 23 0 ⎥ 2 ⎢ ⎥ = · ⎢ 1 −√2 ⎥·⎢ c ⎥ √ ud ⎥ ⎢ uy ⎥ 3 ⎢ 0 − 3 ⎥ 3 1 1 −1 ⎥ ⎢ ⎢ ⎣ ⎦ ⎣ ⎦ 2 2 2 2 uz1 ue ⎣1 ⎦ 1 1 0 0 0 uz2 uf 0 0 0 1 1 1 ⎡ ⎤ ⎞ ⎛⎡ ⎤ ua τa ⎢ ub ⎥ ⎟ ⎜ ⎢ τb ⎥ ⎢ ⎥ ⎜⎢ ⎥ 1 ⎟ √ 2 ⎢ uc ⎥ ⎟ ⎜ ⎢ τc ⎥ = T · ⎢ ⎥ = T · · 2+ 3·Vdc · ⎜⎢ ⎥ − ⎟ (2) ⎢ ud ⎥ ⎜ ⎢ τd ⎥ 2 ⎟ 3 ⎣ ⎦ ⎠ ⎝⎣ ⎦ ue τe uf τf ⎡ ⎤ ⎡ ⎤ uαs 0 0 0 Rs 0 0 ⎥ 0 Rs ⎢ uβs ⎥ ⎢ ⎣ ⎦=⎣ ⎦ 0 Rr ωr · Lr 0 ωr · Lm −ωr · Lm 0 −ωr · Lr Rr 0 ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 Lm 0 iαs Ls iαs 0 Lm ⎥ ⎢ iβs ⎥ Ls ⎢i ⎥ ⎢ 0 · ⎣ βs ⎦ + ⎣ ⎦ ·p· ⎣ ⎦ (3) iαr Lm 0 Lr 0 iαr 0 Lm iβr 0 Lr iβr Rs 0 ixs Lls 0 ixs uxs = · + ·p· . (4) uys iys iys 0 Rs 0 Lls

⎡

Here, p is the time derivative operator, ωr is the rotor angular electrical speed, and Rs , Ls = Lls + Lm , Rr , Lr = Llr + Lm ,

⎡

and Lm are the electrical parameters of the machine, with index l denoting leakage inductances. A suitable model is required for the prediction of the future behavior of the drive on the basis of (1)–(4). The drive equations (1)–(4) are therefore written in the state space form, taking the stator currents in the α–β and x–y subspaces as state variables, and the machine model is discretized in order to be of use as a predictive model. A forward Euler method with a sampling time Tm is used in order to keep the computational burden at the minimum, producing equations in the required digital control form, with the predicted variables depending just on the past values and not on the present values of the variables X(k + 1) = A(k) · X(k) + B (U (k)) + C(k) (5) ⎤ ⎡ iαs ⎢i ⎥ X(k) = ⎣ βs ⎦ U = [Sa , Sb , Sc , Sd , Se , Sf ] . (6) ixs iys More details that are related to this model can be found in [22]. Matrix A depends on the electrical parameters of the machine and on the sampling time, matrix B additionally involves the VSI model that relates the switching states with the voltages, and matrix C arises from the immeasurable variables such as rotor current, shown in (7)–(9) at the bottom of the page. In (7)–(9), I is the 4 × 4 identity matrix, B t is the transpose matrix of B, ci = cos(i π/6), si = sin(i π/6), λ2 = Lr /(Ls · Lr − L2m ), λ3 = Lm /(Ls · Lr − L2m ), and λ5 = 1/Lls .

ωr (k) · Lm · λ3 −Rs · λ2 −Rs · λ2 ⎢ −ωr (k) · Lm · λ3 A(k) = I + Tm · ⎣ 0 0 0 0 ⎡ 2 −1 −1 0 0 0 ⎢ −1 2 −1 0 ⎢ Vdc 0 0 ⎢ −1 −1 2 · U (k) · ⎢ B t (U (k)) = Tm · 0 0 2 −1 ⎢ 0 9 ⎣ 0 0 0 −1 2 0 0 0 −1 −1

0 0

0 0 0

−Rs · λ5 0 −Rs · λ5 ⎤ ⎡ 1 0 1 0 0 ⎥ ⎢ c4 s4 c8 ⎥ ⎢ 0 ⎥ ⎢ c8 s8 c4 ⎥·⎢ −1 ⎥ ⎢ c1 s1 c5 ⎦ ⎣ −1 c5 s5 c1 2 c9 s9 c9

C(k) = X(k) − X(k − 1) − [A(k − 1)X(k − 1) + B (U (k − 1))]

⎤ ⎥ ⎦ ⎤ 0 ⎡ λ2 s8 ⎥ ⎥ s4 ⎥ ⎢ 0 ⎥·⎣ 0 s5 ⎥ ⎦ s1 0 s9

(7)

0 λ2 0 0

0 0 λ5 0

⎤ 0 0 ⎥ ⎦ 0 λ5

(8)

(9)


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Fig. 3. EFSMPC current controller for an asymmetrical dual three-phase induction machine. (a) Proposed scheme. (b) Pseudocode of the EFSMPC control algorithm. TABLE I PARAMETERS OF THE ASYMMETRICAL SIX-PHASE INDUCTION MACHINE1

III. E NHANCED FS-MPC T ECHNIQUE The principle of operation of the enhanced FS-MPC controller, developed here and abbreviated as EFSMPC further on, combines the methods reported separately in [24] and [25]. Fig. 3 shows a block diagram of the EFSMPC method, which can be summarized as follows. For a desired stator current vector i∗s , the proposed control scheme proceeds as in an OSPC method, using a predefined cost function g(x) to select the VSI switching states. The minimizer chooses the switching vector Sioptimum that provides the lowest value of g(x). The selected opt vector provides the optimum solution [uopt α , uβ ] in terms of the current errors in the α–β subspace. Then, the first submodulation problem is solved by computing the time τ that the active vector needs to be applied so that the desired stator current is achieved. The computation of the submodulation period τ is posed as an optimization problem aimed at minimizing the prediction error. A linearity assumption is made based on the time scales involved. In this way, the predicted error is obtained as a linear combination of the errors corresponding to the selected and zero voltages, allowing an analytical expression for τ to be derived. τ is obtained under the hypothesis that, for small periods of time, linearity holds with respect to the application time. In this way, the state after combining an active vector (Xu ) and a zero one (X0 ) would be Tm · X(k + 1) = τ · Xu (k + 1) + (Tm − τ ) · X0 (k + 1), while the predicted error would be Tm · e(k + 1) = τ · eu (k + 1) + (Tm − τ ) · e0 (k + 1). The latter expression allows obtaining the optimal value of τ by setting the derivative of the expected error to zero, which leads to the following equation [24]: τ=

|e0 |2 − |e0 | · |eu | · Tm . |e0 − eu |2

(10)

The second submodulation problem is solved next. Instead of applying the chosen voltage vector to the multiphase machine during the whole switching period, which is the procedure in

conventional predictive control schemes, the proposed method opt uses [τ · uopt α , τ · uβ , 0, 0, 0, 0] as the voltage references in the α–β–x–y–z1 –z2 subspaces to solve a new modulation problem. Since the x–y components are undesirable, the inputs for this second modulation problem are only the α–β components of the phase voltage, and the x–y inputs are set to zero. It must be highlighted here that the zeroing of the x–y components can only be achieved with a method like the one proposed here and not with the conventional predictive control. This is regarded as an interesting feature of this paper. The VSI duty cycles are then obtained from the mathematical expression of the phase voltages defined in [25], as shown in (11), where the duty cycles associated with the VSI legs τi have been introduced (τi takes values between zero and one) ⎡ ⎤ ⎡ ⎤ τa τ · uopt α opt ⎢ τb ⎥ ⎢ τ · uβ ⎥ ⎢ ⎥ 1 ⎢ ⎥ 3 ⎢ τc ⎥ ⎥ −1 ⎢ 0 ·T ·⎢ ⎢ ⎥= + ⎥ . (11) √ ⎢ τd ⎥ 2 2 · 2 + 3 · Vdc ⎢0 ⎥ ⎣ ⎦ ⎣ ⎦ τe 0 τf 0

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Fig. 4. Scheme of the experimental setup.

Fig. 6. M SEix−y performance parameter using the (left) MBPC and (right) OSPC methods.

Fig. 5. Summary of the experimental study using the EFSMPC method: performance parameters at the equilibrium points. The two upper figures depict the (left) M SEiα−β and (right) M SEix−y performance parameters, and the bottom figure shows the THD of a stator phase current (in percent).

IV. E XPERIMENTAL R ESULTS The experimental tests are performed in order to confirm the properties of the introduced current control technique. The test rig is based on the asymmetrical dual three-phase induction machine with six poles (the electrical and mechanical parameters are presented in Table I). A schematic of the test rig is shown in Fig. 4. The control system is based on the MSK28335 system and the TMS320LF28335 Texas Instruments DSP, offering a floating point unit and 12 independent PWM output signals that can be synchronized. The value of the dc bus voltage is around 200 V in the experiments. The same sampling frequency

Fig. 7. Experimental results using the PSVPWM and EFSMPC current control techniques. Stator current tracking in the α–β and x–y subspaces using the following operating point: 2 A @ 36-Hz stator current reference in the α–β subspace. (Upper plots) Temporal waveform for the α component. (Lower plots) Circle diagrams.


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(6.5 kHz) is established for comparison purposes. The cost function is defined as the distance between the reference and exy ) the predicted stator currents in the α–β (ˆ eαβ ) and x–y (ˆ subspaces, as it is shown in (12), where λxy is a weighing factor that allows putting more emphasis on the α–β or x–y subspaces. In this case and by following the recommendations obtained in [23], a value of 0.125 is used (λxy = 0.125). The delay compensation in the predictions for the experimental implementation of the predictive control methods is described in [22] exy . g(x) = ˆ eαβ + λxy ˆ

(12)

The induction machine is operated in different conditions, and Figs. 5–13 show the obtained results. First, an extensive study is performed to characterize the benefits of the introduced method (EFSMPC) with respect to the other already available predictive control techniques (MBPC, OSPC, and PSVPWM). Figs. 5 and 6 and Table II summarize the obtained results. A steady-state sinusoidal waveform reference signal for the currents is considered. The amplitude and the electrical frequency of the applied stator current waveform are identified with i (in amperes) and f (in hertz). At each point (i from 0 to 8 A and f from 0 to 50 Hz), the external mechanical load torque is made to coincide with the machine’s electromagnetic torque so that the system is in equilibrium. The following performance parameters are considered for the analysis: the mean square error of the α–β current components (M SEiα−β ), the mean square error of the x–y current components (M SEix−y ), and the total harmonic distortion (T HD). These performance parameters are evaluated, and the obtained results are summarized graphically in Figs. 5 and 6. The PSVPWM and EFSMPC techniques considerably offer a better performance in the obtained total T HD and the performance indicator in the x–y subspace, when compared to the MBPC and OSPC methods (see Table II). The overall performance of the PSVPWM and EFSMPC is mutually similar. However, there is a slightly better performance of the EFSMPC for all indicators and measurement points (Table II). For the sake of conciseness, only a sample of the results is included here in graphical form. Fig. 5 shows the obtained results for the EFSMPC technique, which are qualitatively similar to the results obtained using the PSVPWM method. Fig. 6 shows the mean square error of the x–y current components for the MBPC and OSPC methods. Higher values and different trends in the graphs are obtained when compared to the EFSMPC or PSVPWM methods. It is worth mentioning that the improvements using the EFSMPC and PSVPWM techniques also appear in the α–β subspace, and the M SEiα−β performance parameters are also quantitatively lower. However, the differences are less pronounced than in the x–y subspace. Furthermore, although the PSVPWM and EFSMPC techniques offer qualitatively similar performances in the α–β and x–y subspaces, a quantitatively better behavior is obtained using the EFSMPC (Table II), as discussed next. Three particular operating points are analyzed in more detail to quantify the improvements obtained using the proposed control method. Table II details the obtained experimental results



for three operating points (2 A @ 36 Hz, 5 A @ 24 Hz, and 8 A @ 12 Hz), where the numerical values of the performance parameters are given for the MBPC, OSPC, PSVPWM, and EFSMPC. It can be observed that all performance parameters are improved (reduced) using the introduced current control method. For instance, the obtained total harmonic distortion in the phase current components is seven and five times lower using the EFSMPC method than the MBPC and OSPC techniques, respectively, at the 2 A @ 36 Hz operating point. The obtained M SEiα−β and M SEix−y performance parameters are also reduced using the EFSMPC method, e.g., M SEiα−β is reduced from 0.702 (MBPC) and 0.409 (OSPC) to 0.235, and M SEix−y is reduced from 2.489 (MBPC) and 1.823 (OSPC)

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Fig. 10. Experimental results using the PSVPWM current control technique. Stator phase currents in different operating points. (Upper plots) Temporal waveforms for the one-phase stator current component. (Lower plots) Total harmonic distortion for the same-phase stator current component (up to 2 kHz).

Fig. 11. Experimental results using the EFSMPC current control technique. Stator phase currents in different operating points. (Upper plots) Temporal waveforms for the one-phase stator current component. (Lower plots) Total harmonic distortion for the same-phase stator current component (up to 2 kHz).

to 0.272 at the 2 A @ 36 Hz operating point. Consequently, the EFSMPC offers a superior behavior when compared to the MBPC and OSPC methods. The same three operating points have been further examined in detail by comparing the PSVPWM and EFSMPC methods (Figs. 7–11). The current tracking in the α–β–x–y subspaces is shown in Figs. 7–9. The EFSMPC technique offers a better tracking in the α–β and x–y subspaces than the PSVPWM method. For instance, the obtained M SEiα−β and M SEix−y performance parameters are 26% (reduction from 0.297 to 0.235) and 27% (reduction from 0.346 to 0.272) lower with the EFSMPC method than with the PSVPWM, respectively, at the 2 A @ 36 Hz operating point. Figs. 10 and 11 show the stator phase currents and the T HD obtained using the PSVPWM and EFSMPC techniques, respectively. The improvement obtained in the T HD perfor-

mance parameter is about 62% (a drop from 25.8% to 15.9%) using the EFSMPC method at the 2 A @ 36 Hz operating point. Notice that similar improvements are obtained in the other operating points as well. Consequently, the EFSMPC also offers a behavior that is superior to the PSVPWM. Next, the transient response using different predictive control methods (MBPC, OSPC, and PSVPWM) is compared. Figs. 12 and 13 show the obtained results. Fig. 12 shows the responses when the stator current reference at 10 Hz is stepped from 2.5 to 8 A. Fig. 13 shows the comparison of the responses when a ramp in the stator current reference at 20 Hz is generated again from 2.5 to 8 A. Again, a better stator current tracking is obtained in the α–β and x–y subspaces using the EFSMPC method when compared to the MBPC, OSPC, and PSVPWM. It is interesting to note that the PSVPWM and EFSMPC methods provide a significant improvement over the MBPC and OSPC


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Fig. 12. Experimental results using the MBPC, OSPC, PSVPWM, and EFSMPC current control techniques. Stator current tracking in the α–β and x–y subspaces during transients when a step in the stator current reference from 2.5 to 8 A @ 10 Hz is generated.

Fig. 13. Experimental results using the MBPC, OSPC, PSVPWM, and EFSMPC current control techniques. Stator current tracking in the α–β and x–y subspaces in transient states when a fast ramp in the stator current reference from 2.5 to 8 A @ 20 Hz is generated.

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TABLE II EXPERIMENTAL RESULTS. PERFORMANCE PARAMETERS OBTAINED IN THREE OPERATING POINTS: (A) 2 A @ 36 Hz, (B) 5 A @ 24 Hz, AND (C) 8 A @ 12 Hz

techniques, as already expected from the previous results. A better stator current tracking is also obtained in the α–β subspace, while the stator current x–y components significantly decrease with the EFSMPC and PSVPWM methods compared to the MBPC and OSPC techniques. To further examine the performance of the EFSMPC, a transient comprising the step change of the phase current reference from 2 to 8 A (peak) at 50 Hz is examined next. The dc voltage is, in this test, set to 300 V, and the control is expected to operate in the region of voltage saturation (it should be noted that a steady-state operation with 8-A current at 50 Hz is not possible due to the low maximum dc link voltage of 300 V; hence, the current is, after a few periods, stepped back to 2 A). The results, shown in Fig. 14, include the oscilloscope recording of the phase current, as well as the zoomed extracts of the stator current α–β reference and actual current components, and a duty cycle command in per unit. As can be seen from the results, an operation with the duty cycle command that is equal to unity does take place, and there is some distortion in the stator current and its components, especially during the transient (which is however very fast). Current reference tracking is still very good, indicating that the control is capable of dealing with the duty cycle command saturation. The last test, shown in Fig. 15, relates to the operation of the EFSMPC method at frequencies higher than the rated frequency. It serves the purpose of proving that the predictive current control and the forward Euler method, applied in the model discretization, are still capable of a satisfactory operation in the field-weakening region. In this test, the dc voltage is set to 300 V, and the phase current references are 2 A at 100 Hz. Fig. 15 shows the α–β reference and actual current components, as well as the x–y current components. The trajectories of the currents in two planes are also included. The current tracking in the α–β plane is, of course, worse than, for example, in Fig. 7, but it is still satisfactory. The current components in the x–y plane are maintained at near-zero values. The experimental results fully confirm the viability of the EFSMPC method and its advantages with respect to the previously introduced methods of predictive current control for multiphase induction machines. The main characteristics and advantages of the EFSMPC method, when compared with other

Fig. 14. EFSMPC current control technique. Stator current tracking when the stator current reference is stepped from 2 to 8 A @ 50 Hz. (Upper plot obtained from the scope) Phase current. (Lower left and middle plots) Reference and actual α–β current components. (Lower right plot) Duty cycle command.

Fig. 15. (Upper plot obtained from the scope) Stator current in a phase, (middle plots) α–β component tracking, (lower left) x–y current components, and (lower right) polar plots for operation, with a 2-A (peak) current reference at 100 Hz.


TABLE III QUALITATIVE COMPARISON BETWEEN PREDICTIVE CURRENT CONTROL METHODS APPLIED IN MULTIPHASE DRIVES

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dual three-phase ac drives. The proposed predictive method provides a better performance (a lower mean square error of the α–β and x–y stator current components and lower stator current harmonic components) for real-time applications than previously reported predictive current control techniques. The experimental results confirm the viability of the proposed current control method. R EFERENCES

TABLE IV QUANTITATIVE COMPARISON BETWEEN PREDICTIVE CURRENT CONTROL METHODS APPLIED IN MULTIPHASE DRIVES

predictive current control techniques for multiphase drives, are summarized in Tables III and IV, and they are as follows. 1) The obtained switching frequency is fixed, producing a well-defined discrete current and voltage spectra in contrast to the MBPC and OSPC. This is a desirable feature both for the selection of the proper switching of the VSI and for the tuning of the output filter in VSI gridconnected applications. The method can also achieve the zero average x–y current components during a sampling period, with this being in contrast with the MBPC and OSPC. 2) Since the EFSMPC includes a submodulation problem prior to the modulation algorithm, the voltage reference generation becomes more accurate than in the PSVPWM, which, in turn, leads to a higher current control accuracy. 3) From a computational cost point of view, there are no significant differences between predictive control techniques when the number of used vectors is the same (in contrast to the case when the comparison of methods is done using all available voltage vectors or using a subset of them). This is due to the fact that the computational cost principally depends on the predictions and not on the included submodulation problems. V. C ONCLUSION The area of multiphase induction motor drives has experienced a substantial growth in recent years. Research has been conducted worldwide, and numerous interesting developments have been reported in the literature, particularly in the current control of the VSI-driven asymmetrical dual threephase ac machine. PC techniques have been recently applied to power converters and drives due to their advantages and the appearance of fast microprocessors. In this paper, a variant of the predictive control strategy, called as EFSMPC, has been proposed for the current control of the VSI-driven asymmetrical

[1] A. Linder and R. Kennel, “Model predictive control for electrical drives,” in Proc. IEEE Power Electron. Spec. Conf., Recife, Brazil, 2005, pp. 1793–1799. [2] P. Cortés, M. P. Kazmierkowski, R. M. Kennel, D. E. Quevedo, and J. Rodríguez, “Predictive control in power electronics and drives,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4312–4324, Dec. 2008. [3] S. Kouro, P. Cortés, R. Vargas, U. Ammann, and J. Rodríguez, “Model predictive control—A simple and powerful method to control power converters,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1826–1838, Jun. 2009. [4] J. Rodríguez, C. A. Silva, P. Correa, P. Lezana, P. Cortés, and U. Ammann, “Predictive current control of a voltage source inverter,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 495–503, Feb. 2007. [5] K. Drobnic, M. Nemec, D. Nedeljkovic, and V. Ambrozic, “Predictive direct control applied to ac drives and active power filter,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1884–1893, Jun. 2009. [6] F. Morel, X. Lin-Shi, J. M. Rétif, B. Allard, and C. Buttay, “A comparative study of predictive current control schemes for a permanent-magnet synchronous machine drive,” IEEE Trans. Ind. Electron., vol. 56, no. 7, pp. 1838–2715, Jul. 2009. [7] R. Vargas, P. Cortés, U. Ammann, J. Rodríguez, and J. Pontt, “Predictive control of a three-phase neutral-point-clamped inverter,” IEEE Trans. Ind. Electron., vol. 54, no. 5, pp. 2697–2705, Oct. 2007. [8] M. A. Pérez, P. Cortés, and J. Rodríguez, “Predictive control algorithm technique for multilevel asymmetric cascaded H-bridge inverters,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4354–4361, Dec. 2008. [9] J. Castelló, J. M. Espí, R. García, and S. A. González, “A robust predictive current control for three-phase grid-connected inverters,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1993–2004, Jun. 2009. [10] P. Correa, J. Rodríguez, M. Rivera, J. R. Espinoza, and J. W. Kolar, “Predictive control of an indirect matrix converter,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1847–1853, Jun. 2009. [11] R. Vargas, J. Rodriguez, U. Ammann, and P. W. Wheeler, “Predictive current control of an induction machine fed by a matrix converter with reactive power control,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4362–4371, Dec. 2008. [12] R. Vargas, U. Ammann, J. Rodríguez, and J. Pontt, “Predictive strategy to control common-mode voltage in loads fed by matrix converters,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4372–4380, Dec. 2008. [13] P. Lezana, R. Aguilera, and D. E. Quevedo, “Model predictive control of an asymmetric flying capacitor converter,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1839–1846, Jun. 2009. [14] P. Correa, M. Pacas, and J. Rodriguez, “Predictive torque control for inverter-fed induction machines,” IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 1073–1079, Apr. 2007. [15] H. Miranda, P. Cortés, J. I. Yuz, and J. Rodríguez, “Predictive torque control of induction machines based on state-space models,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1916–1924, Jun. 2009. [16] R. Vargas, U. Ammann, B. Hudoffsky, J. Rodriguez, and P. Wheeler, “Predictive torque control of an induction machine fed by a matrix converter with reactive input power control,” IEEE Trans. Power Electron., vol. 25, no. 6, pp. 1426–1438, Jun. 2010. [17] S. Bolognani, L. Peretti, and M. Zigliotto, “Design and implementation of model predictive control for electrical motor drives,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1925–1936, Jun. 2009. [18] F. Morel, J. M. Rétif, X. Lin-Shi, and C. Valentin, “Permanent magnet synchronous machine hybrid torque control,” IEEE Trans. Ind. Electron., vol. 55, no. 2, pp. 501–511, Feb. 2008. [19] G. Abad, M. A. Rodríguez, and J. Poza, “Three-level NPC converterbased predictive direct power control of the doubly fed induction machine at low constant switching frequency,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4417–4429, Dec. 2008. [20] T. Geyer, G. Papafotiou, and M. Morari, “Model predictive direct torque control—Part I: Concept, algorithm, and analysis,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1894–1905, Jun. 2009.

3252


[21] G. Papafotiou, J. Kley, K. G. Papadopoulos, P. Bohren, and M. Morari, “Model predictive direct torque control—Part II: Implementation and experimental evaluation,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1906–1915, Jun. 2009. [22] M. R. Arahal, F. Barrero, S. Toral, M. J. Durán, and R. Gregor, “Multiphase current control using finite-state model-predictive control,” Control Eng. Pract., vol. 17, no. 5, pp. 579–587, May 2009. [23] F. Barrero, M. R. Arahal, R. Gregor, S. Toral, and M. J. Durán, “A proof of concept study of predictive current control for VSI-driven asymmetrical dual three-phase ac machines,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1937–1954, Jun. 2009. [24] F. Barrero, M. R. Arahal, R. Gregor, S. Toral, and M. J. Durán, “Onestep modulation predictive current control method for the asymmetrical dual three-phase induction machine,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1974–1983, Jun. 2009. [25] R. Gregor, F. Barrero, S. Toral, M. J. Durán, M. R. Arahal, J. Prieto, and J. L. Mora, “Predictive-SVPWM current control method for asymmetrical dual-three phase induction motor drives,” IET Elect. Power Appl., vol. 4, no. 1, pp. 26–34, Jan. 2010. [26] R. Gregor, F. Barrero, J. Prieto, M. R. Arahal, S. L. Toral, and M. J. Durán, “Enhanced predictive current control method for the asymmetrical dualthree phase induction machine,” in Proc. IEEE IEMDC, Miami, FL, 2009, pp. 265–272. [27] M. J. Durán, M. R. Arahal, F. Barrero, S. L. Toral, and R. Gregor, “Restrained search predictive control of dual three-phase induction motor drives,” in Proc. ICREPQ, Valencia, Spain, 2009. [28] M. J. Durán, F. Barrero, S. L. Toral, M. R. Arahal, and J. Prieto, “Improved techniques of restrained search predictive control for multiphase drives,” in Proc. EPE, Barcelona, Spain, 2009, pp. 1–9. [29] M. J. Durán, F. Barrero, S. L. Toral, M. R. Arahal, and J. Prieto, “Improved techniques of restrained search predictive control for multiphase drives,” in Proc. IEEE IEMDC, Miami, FL, 2009, pp. 239–244. [30] M. J. Durán, J. Prieto, F. Barrero, and S. Toral, “Predictive current control of dual three-phase drives using restrained search techniques,” IEEE Trans. Ind. Electron., DOI: 10.1109/TIE.2010.2087297. [31] 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.

Federico Barrero (M’04–SM’05) received the M.Sc. and Ph.D. degrees in electrical and electronic engineering from the University of Seville, Seville, Spain, in 1992 and 1998, respectively. In 1992, he joined the Electronic Engineering Department, University of Seville, where he is currently an Associate Professor. Dr. Barrero received the Best Paper Award from the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS in 2009.

Joel Prieto (S’10) received the B.Eng. degree in electronic engineering from the Universidad Católica Nuestra Señora de la Asunción, Asunción, Paraguay, in 2005 and the M.Sc. degree from the University of Seville, Seville, Spain, in 2009, where he is currently working toward the Ph.D. degree. In 2008, he joined the Electronic Engineering Department, University of Seville. He is a recipient of a scholarship from Itaipu Binacional/Parque Tecnológico Itaipu-Py for his Ph.D. studies.

Emil Levi (S’89–M’92–SM’99–F’09) received the M.Sc. and Ph.D. degrees from the University of Belgrade, Belgrade, Yugoslavia, in 1986 and 1990, respectively. From 1982 to 1992, he was with the Department of Electrical Engineering, University of Novi Sad, Novi Sad, Serbia. He joined Liverpool John Moores University, Liverpool, U.K., in May 1992, where he has been a Professor of electric machines and drives since September 2000. Dr. Levi serves as the Coeditor-in-Chief of the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS, as the Editor of the IEEE T RANSACTIONS ON E NERGY C ONVERSION, and as the Editor-in-Chief of the IET Electric Power Applications. He is the recipient of the Cyril Veinott Award of the IEEE Power and Energy Society in 2009.

Raúl Gregor (M’10) received the B.Eng. degree in electronic engineering from the Universidad Católica Nuestra Señora de la Asunción, Asunción, Paraguay, in 2005 and the M.Sc. and Ph.D. degrees from the University of Seville, Seville, Spain, in 2007 and 2010, respectively. In 2006, he joined the Electronic Engineering Department, University of Seville. In 2009, he joined the FIUNA, National University of Asunción, Luque, Paraguay, where he is currently an Associate Professor. Dr. Gregor received the Best Paper Award from the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS in 2009.

Sergio Toral (M’01–SM’06) received the M.Sc. and Ph.D. degrees in electrical and electronic engineering from the University of Seville, Seville, Spain, in 1995 and 1999, respectively. He is currently an Associate Professor with the Electronic Engineering Department, University of Seville. Dr. Toral received the Best Paper Award from the IEEE T RANSACTIONS ON I NDUSTRIAL E LEC TRONICS in 2009.

Mario J. Durán received the M.Sc. and Ph.D. degrees in electrical engineering from the University of Malaga, Malaga, Spain, in 1999 and 2003, respectively. He is currently an Associate Professor with the Electrical Engineering Department, University of Malaga. Dr. Durán received the Best Paper Award from the IEEE T RANSACTIONS ON I NDUSTRIAL E LEC TRONICS in 2009.

Martin Jones (M’07) received the B.Eng. (first class honors) and Ph.D. degrees from the Liverpool John Moores University, Liverpool, U.K., in 2001 and 2005, respectively. He was a research student at the Liverpool John Moores University from September 2001 to Spring 2005, where he is currently a Senior Lecturer. He was a recipient of the IEE Robinson Research Scholarship for his Ph.D. studies.