**1. Introduction**

26 Will-be-set-by-IN-TECH

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Recent research has focused on exploring the advantages of multiphase<sup>1</sup> machines over conventional three-phase systems, including lower torque pulsations, less DC-link current harmonics, higher overall system reliability, and better power distribution per phase [1]. Among these multiphase drives, the asymmetrical dual three-phase machines with two sets of three-phase stator windings spatially shifted by 30 electrical degrees and isolated neutral points is one of the most widely discussed topologies and found industrial application in more-electric aircraft, electrical and hybrid vehicles, ship propulsion, and wind power systems [2]. This asymmetrical dual three-phase machines is a continuous system which can be described by a set of differential equations. A methodology that simplifies the modeling is based on the vector space decomposition (VSD) theory introduced in [3] to transform the original six-dimensional space of the machine into three two-dimensional orthogonal subspaces in stationary reference frame (*α* − *β*), (*x* − *y*) and (*z*<sup>1</sup> − *z*2). From the VSD approach, can be emphasized that the electromechanical energy conversion variables are mapped in the (*α* − *β*) subspace, meanwhile the current components in the (*x* − *y*) subspace represent supply harmonics of the order 6*n* ± 1 (*n* = 1, 3, 5, ...) and only produce losses, so consequently should be controlled to be as small as possible. The voltage vectors in the (*z*<sup>1</sup> − *z*2) are zero due to the separated neutral configuration of the machine, therefore this subspace has no influence on the control [4].

Model-based predictive control (MBPC) and multiphase drives have been explored together in [5, 6], showing that predictive control can provide enhanced performance for multiphase drives. In [7, 8], different variations of the predictive current control techniques are proposed to minimize the error between predicted and reference state variables, at the expense of increased switching frequency of the insulated-gate bipolar transistor (IGBTs). On the other hand are proposed control strategies based on sub-optimal solutions restricted the available voltage vectors for multiphase drive applications aiming at reducing the computing cost and

<sup>1</sup> The multiphase term, regards more than three phase windings placed in the same stator of the electric machine.

©2012 Gregor Recalde, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Gregor Recalde, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

improving the drive performance [9]. This chapter wide the concept of the MBPC techniques to the speed control of a dual three-phase induction machine, by using an Kalman Filter (KF) to improve the estimation of states through an optimal estimation of the rotor current. The KF is an efficient recursive filter that estimates the internal state of a dynamic system from a series of noisy measurements. Its purpose is to use measurements that are observed over time that contain noise (random variations) and other inaccuracies (including modeling errors), and produce values that tend to be closer to the true values of the measurements and their associated calculated values. This feature is an attractive solution in the predictive control of induction machines based on the model, mainly if not precisely known internal parameters of the drive, and the measurement of the state variables are perturbed by gaussian noise.

two-dimensional orthogonal subspaces in stationary reference frame (*α* − *β*), (*x* − *y*) and (*z*<sup>1</sup> − *z*2), by means of a 6 × 6 transformation matrix using an amplitude invariant criterion:

> √3 2

> > √3 2

The VSI has a discrete nature and has a total number of 26 = 64 different switching states defined by six switching functions corresponding to the six inverter legs [*Sa*, *Sd*, *Sb*, *Se*, *Sc*, *Sf* ], where **S***<sup>i</sup>* ∈ {0, 1}. The different switching states and the voltage of the DC link (Vdc) define the phase voltages which can in turn be mapped to the (*α* − *β*) − (*x* − *y*) space according to the VSD approach. Consequently, the 64 different on/off combinations of the six VSI legs lead to 64 space vectors in the (*α* − *β*) and (*x* − *y*) subspaces. Figure 2 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 [*SaSbSc*] and [*SdSeSf* ], respectively. For the sake of conciseness, the 64 VSI switching vectors will be usually referred as voltage vectors, or just vectors, in what follows. It must be noted that the 64 possibilities imply only 49 different vectors in the (*α* − *β*) − (*x* − *y*) space. Nevertheless, redundant vectors should be considered as different vectors because they have a different impact on the switching frequency even though they generate identical torque and losses in

2 − √3 <sup>2</sup> <sup>−</sup><sup>1</sup>

10 1 0 1 0 01 0 1 0 1

1 2 −

√3 <sup>2</sup> <sup>−</sup><sup>1</sup>

1 2 <sup>2</sup> 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(1)

<sup>2</sup> 0

0-1 2-0

7-1

6-5

3-3

0-2

4-6

5-2

5-3 7-2

4-2

*x*

(2)

4-3 6-2

1-2

5-6

5-0

5-7 7-6

*y*

1-3

0-3 2-2

7-3

4-7 6-6

4-1 6-0

6-3

2-3

6-7

5-1

6-1

2-1

�

0-4

7-4

1-6

1-0

1-7 3-6

5-5

0-6

5-4

3-2

3-7 4-0

4-4

4-5

2-4

1-4

1-5 3-4

3-5

2-5

0-5

7-5

2-7

3-1

3-0 1-1

2-6

6-4

√3 <sup>2</sup> −1

The Asymmetrical Dual Three-Phase Induction Machine and the MBPC in the Speed Control 387

√3 <sup>2</sup> −1

**<sup>T</sup>** <sup>=</sup> <sup>1</sup> 3

0-1 1-0

7-1

5-3

3-5

0-4

4-6

4-4

4-5

**Figure 2.** Voltage vectors and switching states in the (*α* − *β*) and (*x* − *y*) subspaces for a 6-phase

To represent the stationary reference frame (*α* − *β*) in dynamic reference (*d* − *q*), a rotation

*cos*(*δr*) −*sin* (*δr*) *sin* (*δr*) *cos*(*δr*)

2-4

6-6

6-0

6-7 7-6

2-5

0-5 1-4

7-5

4-7 5-6

5-4

6-5 7-4

6-4

5-5

5-0 4-1

1-5

5-7

transformation must be used. This transformation is given by:

**T***dq* =

�

where *δ<sup>r</sup>* is the rotor angular position referred to the stator as shown in Figure 1.

6-1

5-1

1-1

0-2

7-2

2-0

2-7 3-6

6-3

2-6

1-6

5-2

0-6

6-2

3-4

3-7 4-0

4-2

4-3

1-2

2-3 3-2

3-3

asymmetrical VSI

2-2

1-3

0-3

7-3

1-7

3-1

3-0 2-1

the six-phase machine.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 √3 <sup>2</sup> <sup>−</sup><sup>1</sup>

0 <sup>1</sup> 2

1 − √3 <sup>2</sup> <sup>−</sup><sup>1</sup> 2

0 <sup>1</sup> 2 −

The chapter includes simulation results of the current control based on a predictive model of the asymmetrical dual three-phase induction machine and proposes a new approach to speed control based on MBPC technique. The results provided confirm the feasibility of the speed control scheme for multi-phase machines. The rest of the chapter is organized as follows. Section 2 introduces an asymmetrical dual three-phase AC drive used for simulations. Section 3 details the general principles of the predictive current control method for AC drives. Section 4 shows the simulation results obtained from the inner loop of predictive current control and proposed a new approach to speed control for the dual three-phase induction machine, on the other hand presents a discussion of the obtained results from the proposed approach. The chapter ends with Section 5 where the conclusions are presented.
