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One-Step Modulation Predictive Current Control Method for the Asymmetrical Dual Three-Phase Induction Machine Federico Barrero, Senior Member, IEEE, Manuel R. Arahal, Member, IEEE, Raúl Gregor, Sergio Toral, Senior Member, IEEE, and Mario J. Durán
Abstract—Multiphase (more than three phases) drives exhibit interesting advantages over conventional three-phase drives. Over the last years, topics related to the extension of control schemes to these specific drives have been covered in depth in the literature. Direct torque control and predictive current control are normally used in conventional ac drives when fast electrical dynamic performance is required. In this paper, a one-step modulation predictive current control technique is proposed for asymmetrical dual threephase ac drives. Based on the use of a predictive model including the motor and the inverter, the control algorithm determines the switching state which minimizes errors between predicted and reference state variables. The period of application of the selected switching state is then obtained, resulting in a submodulation method. The proposed predictive current control algorithm uses a prediction horizon of one sampling period; however, two switching states are applied during the sampling period. The switching states are the selected optimum active vector and a null voltage combination. Simulation and experimental results are provided to examine the features of the control method. Performances, advantages, and limitations are also discussed. Index Terms—Asymmetrical dual three-phase ac machine, multiphase drives, predictive current control.
I. I NTRODUCTION
M
ULTIPHASE electrical drives have recently been proposed for applications where some specific advantages (lower torque pulsations, less dc-link current harmonics, higher overall system reliability, better power distribution per phase) can be better exploited [1]–[3]. Among multiphase drives, a very interesting option is the dual three-phase induction machine having two sets of three-phase windings spatially shifted by 30 electrical degrees with isolated neutral points (also called asymmetrical dual three-phase ac machine). The asymmetrical dual three-phase ac machine has been used in specific applications since the late 1920s [4]. For instance, in applications like Manuscript received October 27, 2008; revised December 29, 2008. First published March 16, 2009; current version published June 3, 2009. This work was supported in part by the Spanish Ministry of Education and Science within the I+D+I national project with reference DPI2005/04438. F. Barrero, R. Gregor, and S. Toral are with the Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: fbarrero@ esi.us.es). M. R. Arahal is with the Department of System Engineering and Automatic Control, University of Seville, 41092 Seville, Spain (e-mail:
[email protected]). M. J. Durán is with the Electrical Engineering Department, University of Málaga, 29013 Málaga, 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.2009.2016505
electrical vehicles, the low available dc-link voltage imposes high phase currents for a three-phase drive. In this case, the dual three-phase induction machine is an interesting alternative to the conventional three-phase counterpart [5]. High-performance applications are necessary in multiphase drives like in other conventional ac-drive systems. With this purpose, different control strategies have been applied during the last decade to the current control in voltage-source-inverter (VSI)-fed dual three-phase induction drives. The direct torque control (DTC) technique, developed in the mid-1980s, is the most widely used control technique when fast torque response is required. The basic principle of DTC is to select the appropriate stator voltage vectors from a table, according to the signs of the errors between the references of torque and stator flux and their estimated values, respectively [6], [7]. DTC advantages include low machine parameter dependence and fast dynamic torque response. A DTC strategy for dual three-phase induction motor drives has been recently discussed in [8]. However, the predictive control technique, a control theory developed at the end of the 1970s, also provides fast torque response being a more flexible control scheme. The predictive control scheme determines and applies during a sample time the optimal set of VSI switching states, based on a model of the real system [9], [10]. Its applicability is hindered due to the use of intensive computations, particularly in the multiphase case. The increase in computing power of microprocessors has recently made predictive current control plausible for controlling conventional three-phase power converter and electrical drives [11]–[19], and a conventional predictive current control (CPC) strategy for dual three-phase induction motor drives has been discussed in [20], [21]. Enhanced versions of the predictive current controller have been recently proposed for conventional drives [22], [23]. The idea uses the linear combination of active vectors plus a null one during a computation period, resulting in a modulated predictive current control method that is referred to as one-step modulation predictive controller (OSPC). This paper considers the use of OSPC for the current control of multiphase drives in high-performance applications. The main contribution of this paper is to prove the viability and effectiveness of OSPC in multiphase electrical drives. The asymmetrical dual three-phase ac machine is used as a case example due to its practical interest. The difficulties in the implementation of the proposed control method are discussed and analyzed. The performance of OSPC is studied, comparing the obtained results with a CPC strategy.
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BARRERO et al.: ONE-STEP MODULATION PREDICTIVE CURRENT CONTROL METHOD
Fig. 1.
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CPC method in electrical drives.
This paper is organized as follows. First, the general principles of the one-step modulation predictive current control method are shown in Section II, and its application to a dual three-phase induction motor drive is described in Section III. Then, Section IV presents simulation and experimental results obtained using OSPC and a CPC technique, comparing the performance, advantages, and limitations. Finally, the conclusions are detailed in the last section.
II. OSPC G ENERAL P RINCIPLES Systems controlled by an energy modulator that has a finite number of configurations has attracted considerable attention in recent years [23]. Traditionally, the output of a continuous controller is translated to a sequence of states of the converter using a switching algorithm such as a pulsewidth modulation. This is the case of ac motor drives, where the VSI is considered by the controller as a gain. When fast torque responses are required, high-performance current control is requested. Among these current control techniques, predictive current control techniques show very good performance compared to classical methods like DTC or vector control techniques [20], [21]. Predictive control uses a model of the set of the VSI and the machine. The predictive model is used at each sampling period to predict the machine state vector evolution for each possible VSI state; the control actions are obtained solving an optimization problem aimed at minimizing a cost function. Different cost functions can be used to express different control criteria. For instance, the distance between the reference and the predicted stator currents can be used, defining the cost function J = |i∗s − ˆis |, where i∗s is the reference stator current and ˆis is the predicted stator current which is computationally obtained using the predictive model. The direction of the stator current evolution can also be used to define a simple cost function. A block diagram of the CPC technique is shown in Fig. 1. The sequencer issues voltage vectors one at a time, while the minimizer chooses the one that provides the lower value of J. The selected voltage vector is applied to the physical system (the ac machine) during a sampling time. Each sampling period k, the control algorithm produces the gating signal
combination to be applied during the next sampling period k + 1, Sioptimum (k + 1), as follows. • Assign initial values Jo ← ∞, j ← 1. • While j ≤ N (being N the considered gating signal combinations). • Take Sij . • Compute stator voltages corresponding to gating signal combination Sij using the predictive model. • Use the predictive model to compute a prediction of the stator current for the next sampling period. • Compute the cost function J. • If J < Jo then Jo ← J, Sioptimum (k + 1) ← Sij . • Increment counter j. • End. OSPC is similar to predictive control but it allows combining two states of the VSI within one sample period. The principle of operation is as follows. For a desired stator current vector i∗s , OSPC proceeds as a CPC using the cost function to select a VSI configuration Sioptimum (k + 1). Then, a submodulation problem is solved, computing the time τ that the active vector is to be applied, being the rest of the sample time reserved for the null vector. The computation of the submodulation period τ is posed as an optimization problem aimed at minimizing the predicted 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 null voltage, allowing an analytical expression of τ to be derived. A block diagram of the OSPC technique is shown in Fig. 2, and a description of the proposed control method for the asymmetrical dual three-phase ac machine is detailed in the next section. III. OSPC FOR A SYMMETRICAL D UAL T HREE -P HASE AC M ACHINES The asymmetrical dual three-phase ac machine is a six-phase induction machine having two sets of three-phase windings spatially shifted by 30 electrical degrees with isolated neutral points. A detailed scheme of the drive is shown in Fig. 3. The analytical description of this machine follows two different paths: the d−q winding approach and the vector space
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Fig. 2. OSPC method in electrical drives.
The VSD approach under standard assumptions (negligible space harmonic and magnetic saturation) explains the presence of low-order current harmonics in the machine current spectrum, in contrast to the double d−q winding approach. Using this approach, the original 6-D space of the machine is decomposed into two orthogonal subspaces: α−β and x−y, and a practical model suitable for control is obtained as follows in the stationary reference frame, considering a normal squirrel cage induction motor: ⎤ ⎡ ⎤ 0 0 0 Rs uαs 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 ⎥ Ls ⎢i ⎥ ⎢ 0 ⎢ iβs ⎥ · ⎣ βs ⎦ + ⎣ ⎦·p·⎣ ⎦ iαr Lm 0 Lr 0 iαr iβr Lr iβr 0 Lm 0 ⎡
Fig. 3. General scheme of an asymmetrical dual three-phase ac drive.
decomposition (VSD) approach. According to the first approach [24], the machine can be decomposed into direct sum of a d−q subspace that produce rotating MMF, and a zero sequence subspace which is orthogonal to d−q (the isolated neutral points assumption applied in this case was not adopted). Notice that the proposed model is suitable for analysis of the machine behavior with an arbitrary angle of displacement. According to the second approach [25], VSD, the machine can be represented with three stator-rotor pairs of windings in orthogonal subspaces. One stator-rotor pair engages with electromechanical energy conversion (α−β subspace in what follows), while the others do not. The first stator-rotor pair represents the fundamental supply component plus supply harmonic of the order 12n ± 1 (n = 1, 2, 3, . . .). The other statorrotor pairs represent supply harmonic of the order 6n ± 1 (x−y subspace with n = 1, 3, 5, . . .) plus the zero sequence harmonic components which disappear if isolated neutral points are assumed.
uxs uys
=
Rs 0
i Lls 0 · xs + iys Rs 0
i 0 · p · xs iys Lls
(1) (2)
where p is the time derivative operator, ωr the rotor angular speed, and Rs , Ls = Lls + Lm , Rr , Lr = Llr + Lm and Lm the electrical parameters of the machine. Different aspects must be studied in detail to implement the OSPC algorithm in the asymmetrical dual three-phase ac drive. These aspects are related to the predictive model of the electrical machine, the considered switching voltage vectors, and the evaluation of the index modulation.
A. Predictive Model The machine equations (1) and (2) can be written in state space taking stator currents in α−β and x−y subspaces as state variables. The machine model must be discretized in order to be of use as a predictive model. A forward Euler method with a sampling time Tm can be used, producing equations in the
BARRERO et al.: ONE-STEP MODULATION PREDICTIVE CURRENT CONTROL METHOD
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Fig. 4. Voltage vectors applied in the α−β and x−y subspaces using a six-phase VSI. Notice that the same VSI switching state produces different voltage vector in α−β and x−y subspaces.
needed digital control form, with predicted variables depending just on past values and not on present values of variables ⎤ ⎡ iαs ⎢i ⎥ X(k) = ⎣ βs ⎦ , U = [Sa , Sb , Sc , Sd , Se , Sf ] (3) ixs iys X(k + 1) = A(k) · X(k) + B (U (k)) + C(k) (4) where Si is the switching state of the i-leg of the VSI, and the details can be found in [21]; matrix A depends on the electrical parameters of the machine and on the sampling time, matrix B also involves the VSI model relating switching states with voltages, and matrix C arises from unmeasured variables such as rotor current A(k) = I +Tm ⎤ ⎡ ωr (k)·Lm ·λ3 0 0 −Rs ·λ2 0 0 ⎥ ⎢−ω (k)·Lm ·λ3 −Rs ·λ2 ·⎣ r ⎦ 0 0 −Rs ·λ5 0 0 0 0 −Rs ·λ5 (5) V dc ·U (k) B t (U (k)) = Tm · ⎡ 9 ⎤ 2 −1 −1 0 0 0 ⎢ −1 2 −1 0 0 0 ⎥ ⎢ ⎥ ⎢ −1 −1 2 0 0 0 ⎥ ·⎢ ⎥ ⎢ 0 0 0 2 −1 −1 ⎥ ⎣ ⎦ 0 0 0 −1 2 −1 0 0 0 −1 −1 2 ⎡ ⎤ 1 0 1 0 ⎡ ⎤ ⎢ c4 s4 c8 s8 ⎥ λ2 0 0 0 ⎢ ⎥ ⎢ c s c s ⎥ ⎢ 0 λ2 0 0 ⎥ ·⎢ 8 8 4 4 ⎥·⎣ (6) ⎦ ⎢ c1 s1 c5 s5 ⎥ 0 0 λ5 0 ⎣ ⎦ c5 s5 c1 s1 0 0 0 λ5 c9 s9 c9 s9 C(k) = X(k)−X(k−1) − [A(k−1)X(k−1)+B (U (k−1))] (7) being I the 4 × 4 identity matrix, B t the transpose matrix of B, ci = cos(iπ/6), si = sin(iπ/6), λ2 = Lr /(Ls · Lr − L2m ), λ3 = Lm /(Ls · Lr − L2m ), and λ5 = 1/Lls .
B. Considered Voltage Vectors Computing time is a crucial factor for the predictive controller implementation. The total number of control signals combinations to be explored by the predictive model exponentially increases in multiphase electrical machines with the number of phases. Consequently, the number of voltage vectors to be explored by the predictive model needs careful consideration. The VSI is characterized with 26 = 64 vectors (60 active and 4 zero) which are mapped in the α−β and x−y subspaces. Fig. 4 shows the active vectors in the α−β and x−y subspaces, where each vector switching state is identified using the switching function by two octal numbers corresponding to the binary numbers [Sa Sb Sc ] and [Sd Se Sf ], respectively. The redundancy of the switching states results in only 49 different vectors (48 active and 1 zero). This reduces the time needed to obtain the optimal control action using exhaustive search. Nevertheless, the computation time needed increases in six-phase electrical drives compared to conventional three-phase machines. The number of voltage vectors to evaluate the predictive model can be further reduced if only the 12 outer vectors (the largest ones) are considered. This assumption is commonly used in current control of the asymmetrical dual three-phase ac machine [20], [25]–[27]. In this way, the optimizer can be implemented using only 13 possible stator voltage vectors (12 active and 1 zero vectors). The predictive current control with sinusoidal output voltage considering 13 switching vectors requires less computing time and favors the real-time implementation of the control algorithm. C. Modulation Index Evaluation Fig. 5 shows the evolution in the α−β subspace of the state vector for each possible switching state of the inverter. Point i(k) represents the α−β projection of the measured state, while point i∗ (k + 1) is the α−β projection of the desired state. Forty-nine directions of the state vector evolution can be obtained, corresponding to the 49 different voltage vectors. However, only 13 directions of the state vector evolutions have been represented for the sake of simplicity, corresponding to
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3) For the next sampling period, apply the selected vector u during time τ and the null vector during time (Tm − τ ). A consideration must be taken into account. Since the inverter needs some time to commute between states, the applications periods must comply with τ > τmin , and (Tm − τ ) < τmin , being τmin the time needed by the inverter to commute safely. IV. R ESULTS
Fig. 5. Snapshot of the state vector evolution in the α−β subspace, depending on the applied switching voltage vector.
the 12 outer voltage vectors plus the null one. The proposed algorithm takes into account these represented directions of the state vector. It has to be noted that the reference is not achievable in general using a CPC method, because only one among the possible switching states is applied during the whole sampling period. In the depicted example, the state vector evolves in the optimal control direction for each computation iteration; the ˆi4−4 point in the example. The OSPC method establishes a more appropriate control technique, corresponding to the application of the optimal voltage vector during a τ (0 < τ < Tm ) time. The time of application of the active voltage vector (τ ) is obtained under the hypothesis that for small periods of time linearity holds with respect to the application time (see Fig. 5). In this way, the state after combining an active vector (Xu ) and a null one (X0 ) would be Tm · X(k + 1) = τ · Xu (k + 1) + (Tm − τ ) · X0 (k + 1). (8) Similarly, the predicted error would be Tm · e(k + 1) = τ · eu (k + 1) + (Tm − τ ) · e0 (k + 1). (9) This latter expression allows one to obtain the optimal value of τ setting the derivative of the expected error to zero de =0 dτ
(10)
which leads to the following equation: τ=
|e0 |2 − |e0 | · |eu | · Tm . |e0 − eu |2
(11)
The control method can be summarized with the following pseudocode. 1) Compute the optimal control action (Sioptimum (k + 1)) according to cost function J. 2) Compute the application time according to (11).
A Matlab/Simulink simulation environment has been designed for the VSI-fed asynchronous dual three-phase induction machine, and simulations have been done to prove the effectiveness of the proposed control method. Moreover, an experimental test rig has been designed for obtaining experimental results [28]. A diagram and photos of the complete system are shown in Figs. 6 and 7. The test rig is based on a conventional 36 slots, two pairs of poles, 10-kW three-phase induction machine whose stator has been rewound to construct a 36 slots, three pairs of poles, dual three-phase induction machine. Two sets of stator three-phase windings spatially shifted by 30 electrical degrees have been included. Table I shows the parameters of the machine used to obtain the experimental results as well as the simulations. The control system is based on the TMS320LF2812 Texas Instruments digital signal processor and the MCK2812 system. The control code is written in C, performing closed loop current control, and using a sampling frequency of 5 kHz, obtained after using specialized floating-point mathematical libraries many sourcecode and compiler optimizations. A comparative study has been done, using the distance between the reference and the predicted stator currents as the cost function. A CPC technique and the proposed OSPC control technique have been implemented. A series of tests are performed in order to examine the OSPC properties. Notice that the same sampling period is used for comparison purposes. This is possible because the additional computation time introduced by the OSPC method represents only about 1% of the entire computation time required by the CPC technique. However, the OSPC method leads to a higher switching frequency in the VSI because submodulation requires more switching changes. The OSPC switching frequency would be double if no restrictions were placed on τ . Due to these restrictions, τ > τmin , and (Tm − τ ) < τmin , the resulting OSPC switching frequency is similar to the resulting CPC switching frequency when a high load torque is applied. In any case, the practitioner must be warned that the increase in switching frequency produces an increase of the stress of the power switches. First, an extensive simulation study is performed to characterize the benefits of the OSPC method against the CPC technique. The obtained results are summarized in Fig. 8. This figure shows the obtained results (average square current error in α and x components, and total harmonic distortion) for different steady states using the CPC technique and the OSPC method. Notice that the applied reference signal for currents is a steady-state sinusoidal waveform. The amplitude and the electrical frequency of the applied stator current waveform are denoted I ∗ (in amperes) and ωe (in radians per second), respectively. At each point of the level set, the external load is made to
BARRERO et al.: ONE-STEP MODULATION PREDICTIVE CURRENT CONTROL METHOD
Fig. 6.
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Scheme of the experimental setup.
Fig. 7. Photographs of the experimental setup including: 1) the power electronics, 2) the windings, 3) the stator connection grid, and 4) the machine test rig. TABLE I PARAMETERS OF THE ASYMMETRICAL DUAL THREE-PHASE INDUCTION MACHINE
Fig. 8. Summary of the simulation study using (left side) the CPC method proposed in [20] and [21], and (right side) the OSPC current control method. The two upper figures depict the average square current error in α component, the two figures in the middle represent the average square current error in x component, and the two lower figures show the total harmonic distortion in percent.
coincide with the electrical load, so the system is at equilibrium. The areas with less average square current error in α and x components and total harmonic distortion increase using the OSPC method, showing a qualitative improvement using the OSPC method versus the CPC method applied in [20] and [21].
When a high load torque is applied, the restrictions placed on τ make the OSPC method similar to the CPC technique, and minimum improvements are expected. Consequently, the study focuses on the obtained results far below the rated power.
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Fig. 9. Case study (1.5-A reference stator current at 33.4 Hz) using the CPC proposed in [20] and [21], and the OSPC current control method. Stator current tracking in the upper row, using (left side) the CPC technique and (right side) the OSPC method. Applied stator voltage in the lower row, using (green ink) the CPC technique and (black ink) the OSPC method.
The results are represented in level sets, where the average square current error in α and x components and the total harmonic distortion are shown. Each set depicts contour lines, identifying equal levels of the represented variable. Intermediate values correspond to zones between these isocontour lines. Please note that β components are very similar to α, and thus, they are not reported. The same holds for y components with respect to x ones. The improvement of the OSPC method can be quantified between 10% to 40% in the α−β average square current error, depending on the operation point. For instance, a case study (1.5-A reference stator current at 33.4 Hz) has been selected, and the comparative results are shown in Fig. 9. The average square current error improvement in α and x components can be quantified in a 17% (from 0.155 to 0.132 A) and a 160% (from 0.207 to 0.124 A), respectively, while total harmonic distortion improvement is about 30% (from 21.748% to 16.634%). A second case study (5-A reference stator current at 60 Hz) has been selected at a higher load torque level, and the comparative results are shown in Fig. 10. The average square current error improvement in α and x components can be quantified in a 12% (from 0.279 to 0.249 A) and a 3.5% (from 0.145 to 0.14 A), respectively, while total harmonic distortion improvement is about 1.5% (from 6.451% to 6.357%). The lower graphs in Figs. 9 and 10 compare the applied stator voltage using the OSPC and CPC techniques, showing the influence of the modulation process. The applied stator voltage using the OSPC method is lower than the one obtained using the CPC technique. However, the introduced restrictions in the modulation index evaluation diminish the differences between the control methods when the load torque increases (notice that the difference is higher in Fig. 9, where no load torque is applied, than in Fig. 10, where a load torque of 5 N · m is used). Then, a series of experimental tests are performed in order to verify the OSPC properties. Figs. 11–14 show the obtained
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Fig. 10. Second case study (5-A reference stator current at 60 Hz) using the CPC proposed in [20] and [21], and the OSPC current control method. Stator current tracking in the upper row, using (left side) the CPC technique and (right side) the OSPC method. Applied stator voltage in the lower row, using (green ink) the CPC technique and (black ink) the OSPC method.
Fig. 11. Experimental results for a 2.5-A reference stator current at 12 Hz, using (right side) the CPC technique proposed in [20] and [21], and (left side) the OSPC current control method. Stator current tracking in the α−β subspace, and submodulation index evolution in the OSPC method. The predicted stator current in the α component is shown in the upper side (zoom graphs, green curves). The lower figure shows the duty ratio term of the application time of the active voltage vector (τ ), evaluated as τ /Tm .
BARRERO et al.: ONE-STEP MODULATION PREDICTIVE CURRENT CONTROL METHOD
Fig. 12. Experimental results for a 2-A reference stator current at 18 Hz, using (right side) the CPC technique proposed in [20] and [21], and (left side) the OSPC current control method. Stator current tracking in the α−β subspace, and submodulation index evolution in the OSPC method. The predicted stator current in the α component is shown in the upper side (zoom graphs, green curves). The lower figure shows the duty ratio term τ /Tm .
results. Fig. 11 shows the current tracking in the α−β subspace using the OSPC method (left side) and the CPC technique (right side). A 2.5-A reference stator current at 12 Hz is established. Better stator current tracking is obtained using the OSPC method. The improvement can be quantified in the average square current error measurement which decreases from 0.3791 to 0.2925 (29.6% improvement using the OSPC method). Fig. 12 shows the current tracking in the α−β subspace using a 2-A reference stator current at 18 Hz. The obtained results using the OSPC method appears at the left side, while the obtained results using the CPC technique appears at the right side. Again, better stator current tracking is obtained using the OSPC method. In this case, the improvement can be quantified in a 20.8% improvement of the average square current error measurement using the OSPC method (from 0.4199 to 0.3475). Fig. 13 shows the current tracking in the α−β subspace using a 1.5-A reference stator current at 36 Hz. The improvement is quantified in this case in the reduction of the average square current error measurement from 0.6053 to 0.4889 (23.8% improvement using the OSPC method). Finally, Fig. 14 shows the experimental result obtained applying a step in the amplitude of the reference stator current, from 2.5 to 1.5 A. Better response is detected using the OSPC technique, as it is clearly shown in the current tracking.
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Fig. 13. Experimental results for a 1.5-A reference stator current at 36 Hz, using (right side) the CPC technique proposed in [20] and [21], and (left side) the OSPC current control method. Stator current tracking in the α−β subspace, and submodulation index evolution in the OSPC method. The predicted stator current in the α component is shown in the upper side (zoom graphs, green curves). The lower figure shows the duty ratio term τ /Tm .
V. C ONCLUSION The area of multiphase induction motor drives has experienced a substantial growth in the recent years. Research has been conducted worldwide and numerous interesting developments have been reported in the literature, particularly in the VSI-driven asymmetrical dual three-phase ac machine and in the implementation of fast torque control schemes. Although predictive control is a well-established control discipline, its applicability to fast processes like electromechanical drives is hindered due to the use of intensive computations. The increase in computing power of modern microprocessors has recently made this control strategy plausible for controlling multiphase drives. In this paper, a variant of the conventional predictive control strategy, called OSPC, is proposed for the fast current control of VSI-driven asymmetrical dual three-phase ac machines. It generates predictions from a model of the system like conventional predictive control methods, using a quality function which is evaluated based on those predictions over a finite receding horizon. However, OSPC allows performance optimization applying two voltage vectors in a sampling time, providing better performance for real-time applications than CPC techniques. Simulation and experimental results confirm
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Fig. 14. Experimental results for a change in the stator current reference, from 2.5 to 1.5 A at 12 Hz, using (right side) the CPC technique proposed in [20] and [21], and (left side) the OSPC current control method. Stator current tracking in the α−β subspace, and submodulation index evolution in the OSPC method. The lower figure shows the duty ratio term τ /Tm .
the viability and capability of the proposed current control method. R EFERENCES [1] E. Levi, R. Bojoi, F. Profumo, H. A. Toliyat, and S. Williamson, “Multiphase induction motor drives—A technology status review,” IET Elect. Power Appl., vol. 1, no. 4, pp. 489–516, Jul. 2007. [2] E. Levi, “Multiphase electric machines for variable-speed applications,” IEEE Trans. Ind. Electron., vol. 55, no. 5, pp. 1893–1909, May 2008. [3] D. Dujic, G. Grandi, M. Jones, and E. Levi, “A space vector PWM scheme for multifrequency output voltage generation with multiphase voltagesource inverters,” IEEE Trans. Ind. Electron., vol. 55, no. 5, pp. 1943– 1955, May 2008. [4] P. L. Alger, E. H. Freiburghouse, and D. D. Chase, “Double windings for turbine alternators,” AIEE Trans., vol. 49, pp. 226–244, 1930. [5] R. Bojoi, E. Levi, F. Farina, A. Tenconi, and F. Profumo, “Dual threephase induction motor drive with digital current control in the stationary reference frame,” Proc. Inst. Elect. Eng.—Elect. Power Appl., vol. 153, no. 1, pp. 129–139, Jan. 2006. [6] G. S. Buja and M. P. Kazmierkowski, “Direct torque control of a PWM inverter-fed AC motors—A survey,” IEEE Trans. Ind. Electron., vol. 51, no. 4, pp. 744–757, Aug. 2004. [7] G. Buja and R. Menis, “Steady-state performance degradation of a DTC IM drive under parameter and transduction errors,” IEEE Trans. Ind. Electron., vol. 55, no. 4, pp. 1749–1760, Apr. 2008. [8] R. Bojoi, F. Farina, G. Griva, F. Profumo, and A. Tenconi, “Direct torque control for dual three-phase induction motor drives,” IEEE Trans. Ind. Appl., vol. 41, no. 6, pp. 1627–1636, Nov./Dec. 2005. [9] R. Kennel and A. Linder, “Predictive control of inverter supplied electrical drives,” in Proc. 31st IEEE PESC, 2000, vol. 2, pp. 761–766.
[10] A. Linder and R. Kennel, “Model predictive control for electrical drives,” in Proc. 36th IEEE PESC, 2005, pp. 1793–1799. [11] J. Rodríguez, J. Pontt, C. 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. [12] X. Lin-Shi, F. Morel, A. M. Llor, B. Allard, and J. M. Rétif, “Implementation of hybrid control for motor drives,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 1946–1952, Aug. 2007. [13] M. Nemec, D. Nedeljkovic, and V. Ambrozic, “Predictive torque control of induction machines using immediate flux control,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 2009–2017, Aug. 2007. [14] 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. [15] M. A. Pérez, P. Cortés, and J. Rodríguez, “Predictive control algorithm technique for multilevel asymmetric cascade H-bridge inverters,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4354–4361, Dec. 2008. [16] R. Vargas, J. Rodríguez, 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. [17] R. Vargas, U. Ammann, J. Rodriguez, 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. [18] P. Antoniewicz and M. P. Kazmierkowski, “Virtual-flux-based predictive direct power control of AC/DC converters with online inductance estimation,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4381–4390, Dec. 2008. [19] E. S. de Santana, E. Bim, and W. C. do Amaral, “A predictive algorithm for controlling speed and rotor flux of induction motor,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4398–4407, Dec. 2008. [20] F. Barrero, S. Toral, M. R. Arahal, M. J. Duran, and R. Gregor, “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, Jun. 2009. to be published, DOI: 10.1109/TIE.2008.2011604. [21] 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, 2009. [22] F. Morel, X. Lin-Shi, J. M. Rétif, and B. Allard, “A predictive current control applied to a permanent magnet synchronous machine, comparison with a classical direct torque control,” Elect. Power Syst. Res., vol. 72, no. 8, pp. 1437–1447, Aug. 2008. [23] 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. [24] G. K. Singh, V. Pant, and Y. P. Singh, “Voltage source inverter driven multi-phase induction machine,” Comput. Elect. Eng., vol. 29, no. 8, pp. 813–834, Nov. 2003. [25] 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. [26] D. Hadiouche, L. Baghli, and A. Rezzoug, “Space-vector PWM techniques for dual three-phase AC machines: Analysis, performance evaluation, and DSP implementation,” IEEE Trans. Ind. Appl., vol. 42, no. 4, pp. 1112–1122, Jul./Aug. 2006. [27] R. Bojoi, A. Tenconi, G. Griva, and F. Profumo, “Vector control of a dual-three-phase induction-motor drive using two current sensors,” IEEE Trans. Ind. Appl., vol. 42, no. 5, pp. 1284–1292, Sep./Oct. 2006. [28] R. Gregor, F. Barrero, S. Toral, and M. J. Durán, “Realization of an asynchronous six-phase induction motor drive test-rig,” in Proc. ICREPQ, 2008, pp. 1–5. [Online]. Available: http://www.icrepq.com/icrepq-08/ 230-gregor.pdf
Federico Barrero (M’04–SM’05) was born in Seville, Spain, in 1967. He received the M.Sc. and Ph.D. degrees in electrical and electronic engineering from the University of Seville, Seville, in 1992 and 1998, respectively. Since 1992, he has been with the Electronic Engineering Department, University of Seville, where he is currently an Associate Professor. His recent interests include microprocessor and DSP device systems, control of electrical drives and power electronics, and information and communication technologies systems.
BARRERO et al.: ONE-STEP MODULATION PREDICTIVE CURRENT CONTROL METHOD
Manuel R. Arahal (M’06) was born in Seville, Spain, in 1966. He received the M.Sc. and Ph.D. degrees in electrical and electronic engineering from the University of Seville, Seville, in 1991 and 1996, respectively. From 1995 to 1998, he was an Assistant Professor with the Department of System Engineering and Automatic Control, University of Seville, where he is currently an Associate Professor. His current research interests include industrial applications of model predictive control, artificial intelligence, and
1983
Sergio Toral (M’01–SM’06) was born in Rabat, Morocco, in 1972. He 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. His main research interests include microprocessor and DSP devices, electrical drives, and real-time and distributed systems.
forecasting techniques.
Raúl Gregor was born in Asunción, Paraguay, in 1979. He received the M.Sc. degree in electronic engineering from the Universidad Católica Nuestra Señora de la Asunción, Asunción, in 2005. He is currently working toward the Ph.D. degree in electrical and electronic engineering in the Electronic Engineering Department, University of Seville, Seville, Spain. Since 2005, he has been with the Electronic Engineering Department, University of Seville. His recent interests include control of multiphase electrical drives.
Mario J. Durán was born in Málaga, Spain, in 1975. He received the M.Sc. and Ph.D. degrees in electrical engineering from the University of Málaga, Málaga, in 1999 and 2003, respectively. He is currently an Associate Professor with the Electrical Engineering Department, University of Málaga. His research interests include modeling and control of multiphase converters and machines.
