**4. Simulation results**

A Matlab/Simulink simulation environment has been designed for the VSI-fed asynchronous asymmetrical dual three-phase induction machine, and simulations have been done to prove the efficiency of the scheme proposed. Numerical integration using fourth order Runge-Kutta

<sup>3</sup> 12 active, corresponding to the largest vectors in the (*<sup>α</sup>* <sup>−</sup> *<sup>β</sup>*) subspace and the smallest ones in the (*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>*) subspace plus a zero vector.

algorithm has been applied to compute the evolution of the state variables step by step in the time domain. Table 3 shows the electrical and mechanical parameters for the asymmetrical dual three-phase induction machine.


**Table 3.** Parameters of the asymmetrical dual three-phase induction machine

Computer simulations allow valuing the effectiveness of the proposed control system under unload and full-load conditions, with respect to the mean squared error (MSE) of the speed and stator current tracking. In all cases is considered a sampling frequency of 6.5 kHz, and that the initial conditions of the covariance matrix (*ϕ*(0)), and the process and measurement noise, are known. The Kalman Filter has been started with the following initial conditions; *ϕ*(0) = *diag* 1111 , in order to indicate that the initial uncertainty (rms) of the state variables is 1 A. Because *ϕ*(*k*) is time varying, the KF gain is sensitive to this initial condition estimate during the initial transient, but the steady final values are not affected [11]. The magnitudes of the process noise (*R�*) and measurement noise (*Rν*) are known and are generate using a Random Source block of the Simulink Signal Processing Blockset, assuming the following values, *<sup>R</sup>�* <sup>=</sup> <sup>15</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> and *<sup>R</sup><sup>ν</sup>* <sup>=</sup> <sup>25</sup> <sup>×</sup> <sup>10</sup>−3, respectively.

### **4.1. Efficiency of current control loop**

10 Induction Motor

The predictive model should be used 64 times to consider all possible voltage vectors. However, the redundancy of the switching states results in only 49 different vectors (48 active and 1 null) as shown on Figure 2. This consideration is commonly known as the optimal solution. 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 if sinusoidal output voltage is required and it is not necessary to synthesize (*x* − *y*) components. In this way, the optimizer can be implemented using only 13 possible stator voltage vectors3. This way of proceeding increases the speed at which the optimizer can be run, allowing decreasing the sampling time at the cost of losing optimality. A detailed study of the implications of considering the optimal solution can be found at [6]. For a generic multi-phase machine, where *f* is the number of phase and *ε* the search space (49 or 13 vectors), the control algorithm proposed produces the optimum gating signal combination

*3.2.2. Optimizer*

*Sopt* as follows:

**Algorithm 1** Proposed algorithm

**comment:** Optimization algorithm

**<sup>K</sup>***e*(*k*) = **<sup>Γ</sup>**(*k*) · **<sup>C</sup>***TR<sup>ν</sup>*

**S***<sup>i</sup>* ← **S***i*,*<sup>j</sup>* ∀ *j* = 1, ..., *f*

*Jo* <sup>←</sup> *<sup>J</sup>*, **<sup>S</sup>***opt* <sup>←</sup> **<sup>S</sup>***<sup>i</sup>*

*Jo* := ∞, *i* := 1 **while** *i* ≤ *ε* **do**

*Uαβxys* =

**if** *J* < *Jo* **then**

**4. Simulation results**

plus a zero vector.

**end if** *i* := *i* + 1 **end while**

**comment:** Compute the covariance matrix. Equation 23 **<sup>Γ</sup>**(*k*) = *<sup>ϕ</sup>*(*k*) <sup>−</sup> *<sup>ϕ</sup>*(*k*) · **<sup>C</sup>***T*(**<sup>C</sup>** · *<sup>ϕ</sup>*(*k*) · **<sup>C</sup>***<sup>T</sup>* <sup>+</sup> *<sup>R</sup>ν*)−<sup>1</sup> · **<sup>C</sup>** · *<sup>ϕ</sup>*(*k*) **comment:** Compute the KF gain matrix. Equation 22

**comment:** Compute stator voltages. Equation 12

**comment:** Compute the cost function. Equation 25

*T*

**comment:** Compute the correction of the covariance matrix. Equation 24

**comment:** Compute a prediction of the state. Equation 15

= *Vdc* · **T** · **M**

A Matlab/Simulink simulation environment has been designed for the VSI-fed asynchronous asymmetrical dual three-phase induction machine, and simulations have been done to prove the efficiency of the scheme proposed. Numerical integration using fourth order Runge-Kutta

<sup>3</sup> 12 active, corresponding to the largest vectors in the (*<sup>α</sup>* <sup>−</sup> *<sup>β</sup>*) subspace and the smallest ones in the (*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>*) subspace

*uαs*, *uβs*, *uxs*, *uys*, 0, 0

**<sup>X</sup>**ˆ(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*) = **AX**(*k*) + **BU**(*k*) + **<sup>H</sup>***�*(*k*)

*<sup>J</sup>* <sup>=</sup>� *<sup>e</sup>*ˆ*iαs*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*) + *<sup>e</sup>*ˆ*iβs*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*) �<sup>2</sup>

*<sup>ϕ</sup>*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) = **<sup>A</sup>** · **<sup>Γ</sup>**(*k*) · **<sup>A</sup>***<sup>T</sup>* <sup>+</sup> **<sup>H</sup>** · *<sup>R</sup>�* · **<sup>H</sup>***<sup>T</sup>*

A series of simulation tests are performed in order to verify the efficiency of current control loop in three points of operation of the machine. Figure 5 shows the current tracking in stationary reference frame (*α* − *β*) and (*x* − *y*) subspaces considering sub-optimal solution in the optimization process (12 active and 1 null vectors). The predicted stator current in the *α* component is shown in the upper side (zoom graphs and green curves). For all cases of analysis efficiency is measured with respect to the MSE of the currents tracking in (*α* − *β*)-(*x* − *y*) subspaces and the total harmonic distortion (THD), defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency, obtained from the Powergui-Continuous Simulink block. A 2.5 A reference stator current at 12 Hz is established for the case of Figure 5 (a). Figure 5 (b) shows the current tracking in the (*α* − *β*) and (*x* − *y*) subspaces using a 2 A reference stator current at 18 Hz and Figure 5 (c) shows the current tracking in stationary reference frame using a 1.5 A reference stator current at 36 Hz. Table 4 summarizes the results of the three previous trials where are considered different amplitudes and angular frequencies for the reference current.

From the obtained results can be emphasized as follows:


c. As increases the frequency of the reference currents the switching frequency decreases, consequently there is a degradation in the THD of the stator currents as can be seen in

The Asymmetrical Dual Three-Phase Induction Machine and the MBPC in the Speed Control 397

The structure of the proposed speed control for the asymmetrical dual three-phase induction machine based on a KF is shown in Figure 6. The process of calculation of the slip frequency (*ωsl*) is performed in the same manner as the Indirect Field Orientation methods, from the

machine (*Rr*, *Lr*) [13, 14]. The inner loop of the current control, based on the MBPC selects control actions solving an optimization problem at each sampling period using a real system model to predict the outputs. As the rotor current can not be measured directly, it should be estimated using a reduced order estimator based on an optimal recursive estimation algorithm

**Figure 6.** Proposed speed control technique based on KF for the asymmetrical dual three-phase

Different cost functions (*J*) can be used, to express different control criteria. The absolute current error, in stationary reference frame (*α* − *β*) for the next sampling instant is normally used for computational simplicity. In this case, the cost function is defined as Equation

which is computationally obtained using the predictive model. However, other cost functions can be established, including harmonics minimization, switching stress or VSI losses [6]. Proportional integral (PI) controller is used in the speed control loop, based on the indirect vector control schema because of its simplicity. In the indirect vector control scheme, PI speed

reference used by the predictive model is obtained from the calculation of the electric angle used to convert the current reference, originally in dynamic reference frame (*d* − *q*), to static

*<sup>s</sup>* is the stator reference current and *iαβ*(*k* + 1|*k*) is the predicted stator current

∗

∗ *ds*, *i* ∗

*qs*) and the electrical parameters of the

*ds* in dynamic reference frame. The current

Table 4

from the Equations 21-24.

induction machine

∗

controller is used to generate the reference current *i*

reference frame (*α* − *β*) as shown in Figure 6.

25, where *i*

**4.2. Proposed speed control method**

reference currents in dynamic reference frame (*i*

**Table 4.** Simulation results obtained from Figure 5

**Figure 5.** Stator current in (*α* − *β*) component tracking and (*x* − *y*) current components. (a) 2.5 A (peak) current reference at 12 Hz. (b) 2 A current reference at 18 Hz and (c) 1.5 A current reference at 36 Hz


c. As increases the frequency of the reference currents the switching frequency decreases, consequently there is a degradation in the THD of the stator currents as can be seen in Table 4

### **4.2. Proposed speed control method**

12 Induction Motor

Method Test MSE*α*, MSE*<sup>β</sup>* MSE*x*, MSE*<sup>y</sup>* THD*α*, THD*<sup>β</sup>*

(a) (b) (c)

<sup>4</sup> 0.005 0.015 0.025 1.5 2 2.5

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

**Figure 5.** Stator current in (*α* − *β*) component tracking and (*x* − *y*) current components. (a) 2.5 A (peak) current reference at 12 Hz. (b) 2 A current reference at 18 Hz and (c) 1.5 A current reference at 36 Hz

a. The MBPC is a flexible approach that, opposite to PWM based control methods, allows a straightforward generalization to different requirements only changing the cost function b. The MBPC method is discontinuous technique, so the switching frequency is unknown. This feature reduces the switching losses (compared to continuous techniques) at expense

**Table 4.** Simulation results obtained from Figure 5

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

*ia, i\*a* [A]



*iß, i\*ß* [A]

0

2

4

0

2

<sup>4</sup> 0.01 0.02 0.03 <sup>2</sup> 2.5 3

*ix, iy* [A]


0

2

4

*i* \* *i*

 *i* \* *i i*

*yi xi*

of an increase in the harmonics of the stator current

*iß, i\*ß* [A]


0

2

4

*ia, i\*a* [A]


0

2

*ix, iy* [A]


0

2

4

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

(a) 0.2105, 0.2322 0.9298, 0.9304 7.1330,7.5969 MBPC (b) 0.1989, 0.2141 1.0957, 1.0885 10.3610, 11.8192

(c) 0.2287, 0.2348 1.2266, 1.3102 15.8951, 17.4362

 *i* \* *i i*

*i* \* *i*

*ia, i\*a* [A]


0

2

<sup>4</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup>

1 1.5 2

*iß, i\*ß* [A]


0

2

4

*ix, iy* [A]


0

2

4

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

x 10-3

*i* \* *i*

 *i* \* *i i*

*y i xi*

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

Time [s]

<sup>0</sup> 0.05 0.1 0.15 -4

*y i xi* The structure of the proposed speed control for the asymmetrical dual three-phase induction machine based on a KF is shown in Figure 6. The process of calculation of the slip frequency (*ωsl*) is performed in the same manner as the Indirect Field Orientation methods, from the reference currents in dynamic reference frame (*i* ∗ *ds*, *i* ∗ *qs*) and the electrical parameters of the machine (*Rr*, *Lr*) [13, 14]. The inner loop of the current control, based on the MBPC selects control actions solving an optimization problem at each sampling period using a real system model to predict the outputs. As the rotor current can not be measured directly, it should be estimated using a reduced order estimator based on an optimal recursive estimation algorithm from the Equations 21-24.

**Figure 6.** Proposed speed control technique based on KF for the asymmetrical dual three-phase induction machine

Different cost functions (*J*) can be used, to express different control criteria. The absolute current error, in stationary reference frame (*α* − *β*) for the next sampling instant is normally used for computational simplicity. In this case, the cost function is defined as Equation 25, where *i* ∗ *<sup>s</sup>* is the stator reference current and *iαβ*(*k* + 1|*k*) is the predicted stator current which is computationally obtained using the predictive model. However, other cost functions can be established, including harmonics minimization, switching stress or VSI losses [6]. Proportional integral (PI) controller is used in the speed control loop, based on the indirect vector control schema because of its simplicity. In the indirect vector control scheme, PI speed controller is used to generate the reference current *i* ∗ *ds* in dynamic reference frame. The current reference used by the predictive model is obtained from the calculation of the electric angle used to convert the current reference, originally in dynamic reference frame (*d* − *q*), to static reference frame (*α* − *β*) as shown in Figure 6.

performance of the proposed algorithm, based on a Kalman Filter. The estimated rotor current converges to real values for these test conditions as shown in figures, proving that the observer

The Asymmetrical Dual Three-Phase Induction Machine and the MBPC in the Speed Control 399

In this chapter a new approach for the speed control of the asymmetrical dual three-phase induction machine has been proposed and evaluated. The speed control scheme uses an inner loop predictive current control based on the model, where the main advantage is the absence of modulation techniques. The MBPC is described using a state-space representation, where the rotor and stator current are the states variables. The proposed algorithm provides an optimal estimation of the rotor current in each sampling time in a recursive manner, even when internal parameters of the drive are not precisely known, and the measurements of the state variables are perturbed by gaussian noise. The theoretical development based on a Kalman Filter has been validated by simulations results. The method has proven to be efficient

The author gratefully the Paraguay Government for the economical support provided by means of a Conacyt Grant (project 10INV01). Also, wishes to express his gratitude to the

[1] Levi, E. (2008). Multiphase electric machines for variable-speed applications. *IEEE Transactions on Industrial Electronics*, Vol. 55, No. 5, (May 2008) page numbers

[2] Bucknal R. & Ciaramella, K. (2010). On the Conceptual Design and Performance of a Matrix Converter for Marine Electric Propulsion. *IEEE Transactions on Power Electronics*,

[3] Zhao Y. & Lipo, T. (1995). Space vector PWM control of dual three-phase induction machine using vector space decomposition. *IEEE Transactions on Industry Applications*,

[4] Boglietti, A.; Bojoi, R.; Cavagnino, A.& Tenconi, A. (2008). Efficiency Analysis of PWM Inverter Fed Three-Phase and Dual Three-Phase High Frequency Induction Machines for Low/Medium Power Applications. *IEEE Transactions on Industrial Electronics*, Vol.

[5] Arahal, M.; Barrero, F.; Toral, S.; Durán, M.; & Gregor, R. (2008). Multi-phase current control using finite-state model-predictive control. *Control Engineering Practice*, Vol. 17,

Vol. 25, No. 6, (June 2010) page numbers (1497-1508), ISSN 0885-8993

55, No. 5, (May 2008) page numbers (2015-2023), ISSN 0278-0046

No. 5, (October 2008) page numbers (579-587), ISSN 0967-0661

Vol. 31, No. 5, (October 1995) page numbers (1100-1109), ISSN 0093-9994

even when considering that the machine is operating under varying load regimes.

anonymous reviewers for their helpful comments and suggestions.

*Engineering Faculty of the National University of Asunción Department of Power and Control Systems, Asunción-Paraguay*

performance is satisfactory.

**5. Conclusions**

**Acknowledgments**

**Author details**

**6. References**

Raúl Igmar Gregor Recalde

(1893-1909), ISSN 0278-0046

(a) Simulation results for a ±320 rpm step wave speed comand tracking (b) Simulation results for a ±2.5 A step in the reference current comand tracking

(c) Simulation results for a 50 Nm trapezoidal load

**Figure 7.** Simulation results for a proposed speed control. The predicted stator current in the *α* component is shown in the upper side (zoom graphs, red curves)

Figure 7 (a) shows simulation results for a 200 revolutions per minute (rpm) trapezoidal speed reference, if we consider a fixed current reference (*i* ∗ *ds* = 1 A). The subscripts (*α* − *β*) represent quantities in the stationary frame reference of the stator currents. The measurement speed is fed back in the closed loop for speed regulation and a PI controller is used in the speed regulation loop as shown in Figure 6. The predicted stator current in the *α* component is shown in the upper side (zoom graph, red curve). Under these test conditions, the MSE in the speed and current tracking are 0.75 rpm and 0.15 A, respectively. Figure 7 (b) shows the step response for the induction machine to a change of ±2.5 A in the current reference (*i* ∗ *ds* see Figure 6), if we consider a fixed speed reference (*ω*<sup>∗</sup> *<sup>r</sup>* = 200 rpm). In these simulation results, the subscripts (*α* − *β*) represent the stator current in stationary reference frame. Under these test conditions, the MSE in the stator current tracking are 0.1 A for the reference current (*i* ∗ *ds*) and 0.18 A considering stationary reference frame. Finally, Figure 7 (c) shows a trapezoidal load application response, and the rotor current evolution (measured and observed) in stationary reference frame. These simulation results substantiate the expected performance of the proposed algorithm, based on a Kalman Filter. The estimated rotor current converges to real values for these test conditions as shown in figures, proving that the observer performance is satisfactory.
