**1. Introduction**

In the field of renewable energy sources, wind energy is gaining much importance. The increase in the level of generation has two main restrictions, one is due to the limitations of the switching frequency of the power devices with respect to the power drive of the Doubly Fed Induction Machine (DFIM) and second one is the requirement of good dynamic torque performance, these restrictions are addressed by the proposed new Direct Torque Control (DTC) approach.

The foremost torque control methods proposed were classified into Field Oriented Control (FOC) techniques, Blaschke [1] and the direct control techniques. Then after, the concept of DTC, Takahashi and Ohmori [2] and Direct Self Control (DSC), Depenbrock [3] were introduced to achieve good steady-state and transient torque control conditions. Moreover, direct control techniques do not require current regulators, nor coordinate transformations or specific modulations like Pulse Width Modulation (PWM) or Space Vector Modulation (SVM) for pulse generation. The disadvantages are the lack of direct current control, torque control difficulties at very low speeds and especially variable switching frequency behavior.

The last important drawback put forward several authors in recent years, thus, methods like Direct Mean Torque Control (DMTC), [4] and Direct Torque Control based on Discrete Space Vector Modulation (DTC–DSVM), [5], have already achieved constant switching frequency. Furthermore, the methods proposed in Kang and Sul [6, 7] have extended the solution to reduce torque ripple at the same time imposing the switching frequency, as well as for different voltage-source multilevel topologies.

• Implementation of a new predictive Direct Torque Control (DTC) strategy of the Doubly Fed Induction Machine (DFIM) is presented which is designed to

*Predictive Direct Torque Control Strategy for Doubly Fed Induction Machine for Torque…*

• The proposed DTC method effectively reduces the torque and flux ripples at low switching frequency, even under variable speed operation conditions.

The novelty of this Chapter is that the predictive DTC have been developed to control the different parameters of DFIM, to improve its performance during transient, steady state, tracking behavior and operation near synchronous speed. All

The DFIG model adopted is the qd0 rotating reference frame. It is because the model of DFIM is quite suitable with this frame of reference during transients. The transient solution of the DFIM model is possible because of the transformation from abc to qd0 by which the differential equations with time-varying inductances is

Similarly, the q and d-axis rotor voltages referred to the stator are given by,

The stator and rotor fluxes can be calculated by using (5) and (6) can be used

ð1Þ

ð2Þ

ð3Þ

ð4Þ

• Results are presented to validate the proposed control strategy.

converted into differential equations with constant inductances,

operate at a low constant switching frequency.

these cases are thoroughly investigated.

*DOI: http://dx.doi.org/10.5772/intechopen.89979*

(or) simply (1) can be written as

(or) simply (3) can be written as

to calculate the magnitudes.

**103**

**3. Modeling of DFIM**

Further the concept of DTC have been applied to the brushless doubly fed induction machine, [8], or the equivalent Direct Power Control (DPC) strategy for several grid connected converter applications, [9].

In this paper, the analysis on the Doubly Fed Induction Machine (DFIM), which is a common solution for variable speed wind turbines, is discussed. The control methods like FOC have been performed by many authors, for example [10]. DTC and DPC methods without switching frequency imposition have also been carried out in Gomez & Amenedo [11], Datta & Ranganathan [12], while the DPC at constant switching frequency has also been developed in [13].

These mentioned direct control techniques that achieve constant switching frequency behavior are based on predictive control with a prediction horizon equal to one sample period. In this paper, predictive DTC technique for the DFIM will be employed based on [13] and the performance is compared with the DTC strategy, at constant switching frequency and with reduced torque and flux ripples criteria. This control technique is based on a prediction of the torque and the flux evolution of the DFIM. Hence, the new Predictive DTC (PDTC) strategy presented in this paper is based on a direct control of the electromagnetic torque and the rotor flux of the machine. Simulation results are presented and discussed, and at last, the results show that the predictive DTC technique presents good dynamic response compared to classical DTC concept.

In this Chapter, predictive DTC technique for the DFIM has been implemented and its performance is compared with the classical DTC of DFIM based on certain parameters like constant switching frequency, torque ripple and flux ripple. This control technique is based on a prediction of the torque and the flux evolution of the DFIM. To validate the proposed control scheme, results are presented. From these results, it is observed that the predictive DTC technique gives good dynamic response compared to classical DTC concept.

The control strategy even reduces the switching losses of the converter and reduces the electromagnetic torque and flux ripples at low switching frequency even under variable speed operating conditions.

In Section 1, the introduction of the Chapter is given.

In Section 2, Contributions and Novelty of this Chapter is explained.

In Section 3, Modeling of the DFIM is given.

In Section 4, the basic control principle of predictive DTC is explained and also implementation of the proposed predictive DTC strategy of DFIM along with selection of rotor voltage vectors for constant switching frequency and reduction of switching power losses are described.

In Section 5, Results are presented to validate the proposed control strategy.

In Section 6, the Conclusions of the Chapter are described.

In Section 7, the summary of the Chapter is given.

#### **2. Contributions and novelty**

The main contributions of this Chapter are as follows:

*Predictive Direct Torque Control Strategy for Doubly Fed Induction Machine for Torque… DOI: http://dx.doi.org/10.5772/intechopen.89979*


The novelty of this Chapter is that the predictive DTC have been developed to control the different parameters of DFIM, to improve its performance during transient, steady state, tracking behavior and operation near synchronous speed. All these cases are thoroughly investigated.

## **3. Modeling of DFIM**

The DFIG model adopted is the qd0 rotating reference frame. It is because the model of DFIM is quite suitable with this frame of reference during transients. The transient solution of the DFIM model is possible because of the transformation from abc to qd0 by which the differential equations with time-varying inductances is converted into differential equations with constant inductances,

$$\begin{aligned} \mathbf{v\_{qs}} &= \mathbf{R\_s}\mathbf{i\_{qs}} + \alpha\_\mathbf{s}\boldsymbol{\upmu\_{ds}} + \frac{d\boldsymbol{\upmu\_{qs}}}{dt} \\\\ \mathbf{v\_{ds}} &= \mathbf{R\_s}\mathbf{i\_{ds}} - \alpha\_\mathbf{s}\boldsymbol{\upmu\_{qs}} + \frac{d\boldsymbol{\upmu\_{ds}}}{dt} \end{aligned} \tag{1}$$

(or) simply (1) can be written as

$$
\overline{\mathbf{v}}\_{\mathbf{s}}^{\mathbf{s}} = \mathcal{R}\_{\mathbf{s}} \overline{\mathbf{r}}\_{\mathbf{s}}^{\mathbf{s}} + \frac{\mathbf{d} \overline{\mathbf{p}}\_{\mathbf{s}}^{\mathbf{s}}}{\mathbf{d} \mathbf{t}} \tag{2}
$$

Similarly, the q and d-axis rotor voltages referred to the stator are given by,

$$\begin{aligned} \mathbf{v}'\_{\mathbf{q}\mathbf{r}} &= \mathbf{R}'\_{\mathbf{r}} \mathbf{i}'\_{\mathbf{q}\mathbf{r}} + (\boldsymbol{\alpha}\_{\mathbf{s}} - \boldsymbol{\alpha}\_{\mathbf{r}}) \boldsymbol{\upmu}'\_{\mathbf{d}\mathbf{r}} + \frac{\mathbf{d} \boldsymbol{\upmu}'\_{\mathbf{q}\mathbf{r}}}{\mathbf{d}\mathbf{t}} \\\\ \mathbf{v}'\_{\mathbf{d}\mathbf{r}} &= \mathbf{R}'\_{\mathbf{r}} \mathbf{i}'\_{\mathbf{d}\mathbf{r}} - (\boldsymbol{\alpha}\_{\mathbf{s}} - \boldsymbol{\alpha}\_{\mathbf{r}}) \boldsymbol{\upmu}'\_{\mathbf{q}\mathbf{r}} + \frac{\mathbf{d} \boldsymbol{\upmu}'\_{\mathbf{d}\mathbf{r}}}{\mathbf{d}\mathbf{t}} \end{aligned} \tag{3}$$

(or) simply (3) can be written as

$$\overline{\mathbf{v}}\_{\mathbf{r}}^{\mathbf{s}} = \mathbf{R}\_{\mathbf{r}} \overline{\mathbf{r}}\_{\mathbf{r}}^{\mathbf{s}} + \frac{\mathbf{d} \overline{\mathbf{p}}\_{\mathbf{r}}^{\mathbf{s}}}{\mathbf{d}\mathbf{t}} - \mathbf{j} \boldsymbol{\alpha}\_{\mathbf{m}} \overline{\mathbf{p}}\_{\mathbf{r}}^{\mathbf{s}} \tag{4}$$

The stator and rotor fluxes can be calculated by using (5) and (6) can be used to calculate the magnitudes.

$$\begin{aligned} \vert \Psi \rangle\_{\rm s} &= \mathcal{L}\_{\rm s} \overline{\mathcal{I}}\_{\rm s} + \mathcal{L}\_{\rm m} \overline{\mathcal{I}}\_{\rm r} \\ \vert \Psi \rangle\_{\rm r} &= \mathcal{L}\_{\rm m} \overline{\mathcal{I}}\_{\rm s} + \mathcal{L}\_{\rm r} \overline{\mathcal{I}}\_{\rm r} \\ \vert \psi \vert\_{\rm s} &= \sqrt{\Psi\_{\rm ds}^{2}} + \Psi\_{\rm q,s}^{2} \end{aligned} \tag{5}$$
 
$$\begin{aligned} \vert \psi \vert\_{\rm r} &= \sqrt{\Psi\_{\rm dr}^{2}} + \Psi\_{\rm q,r}^{2} \end{aligned} \tag{6}$$

The electromagnetic torque of DFIM is given in (7).

$$\begin{aligned} \mathbf{T\_{em}} &= \frac{3}{2} \text{pIm} \{ \overline{\mathbf{\upmu}}\_{\texttt{g}}^{\texttt{\text{\textdegree}}}, \overline{\mathbf{i\_{s}}} \} \text{ (or)} \frac{3}{2} \text{p} \left( \mathbf{\upmu}\_{\texttt{ea}} \mathbf{i\_{s}} - \mathbf{\upmu}\_{\texttt{g}\texttt{\textdegree}} \mathbf{i\_{sa}} \right) \\\\ \text{(or)} &\frac{3}{2} \text{p} \left( \mathbf{\upmu}\_{\texttt{ds}} \mathbf{i\_{qs}} - \mathbf{\upmu}\_{\texttt{qs}} \mathbf{i\_{ds}} \right) \text{(or)} \approx \frac{3}{2} \frac{\text{p}}{2} \text{L}\_{\texttt{m}} (\mathbf{i\_{dr}} \mathbf{i\_{qs}} - \mathbf{i\_{qr}} \mathbf{i\_{ds}}) \end{aligned} \tag{7}$$

The active and reactive powers are given by:

$$\begin{aligned} \mathbf{P\_s} &= \frac{3}{2} \operatorname{Re} \{ \overline{\mathbf{v}}, \overline{\mathbf{i}}^\star \} \begin{pmatrix} \operatorname{or} \end{pmatrix} \frac{3}{2} \{ \mathbf{v\_a} \mathbf{i\_a} + \mathbf{v\_b} \mathbf{i\_b} \} \begin{pmatrix} \operatorname{or} \end{pmatrix} \frac{3}{2} \{ \mathbf{v\_d} \mathbf{i\_d} + \mathbf{v\_q} \mathbf{i\_q} \} \overline{\mathbf{w\_r}} = \mathbf{L\_m I\_s} + \mathbf{L\_r I\_r} \\\\ \mathbf{Q\_s} &= \frac{3}{2} \operatorname{Im} \{ \overline{\mathbf{v}}, \overline{\mathbf{i}}^\star \} \{ \operatorname{or} \} \frac{3}{2} \{ \mathbf{v\_\beta} \mathbf{i\_a} - \mathbf{v\_a} \mathbf{i\_\beta} \} \operatorname{or} \} \frac{3}{2} \{ \mathbf{v\_q} \mathbf{i\_d} - \mathbf{v\_d} \mathbf{i\_q} \} \end{aligned} \tag{8}$$

## **4. Description of proposed predictive direct torque control strategy for torque and flux ripple minimization**

The main drawback of classical DTC scheme is nonconstant switching behavior; it is avoided by the proposed predictive Direct Torque Control strategy. The constant switching behavior is achieved by increasing slightly the complexity of control strategy.

**4.1 Effect of voltage vector on the DFIM**

*with zero vectors at constant torque and rotor flux.*

*DOI: http://dx.doi.org/10.5772/intechopen.89979*

by [2, 13]

**Figure 1.**

sions (9) and (10).

**105**

Using (2), (4)–(7), the predictive expressions for torque and rotor flux are given

*(a) Simplified prediction scheme of torque and rotor flux of DFIM. (b) Variation of electromagnetic torque*

*Predictive Direct Torque Control Strategy for Doubly Fed Induction Machine for Torque…*

The below space vector representations are used in order to analyze the expres-

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ð11Þ

ð12Þ

ð13Þ

A sequence of three voltage vectors will be introduced at a constant switching period of which two are active vectors always followed by a zero vector, in order to reduce the ripples of both directly controlled variables compared to selection of four rotor voltage vectors depending on the position of rotor flux in classical DTC scheme. For that purpose, ripple reduction criteria based on a prediction of the electromagnetic torque and rotor flux evolution over time is implemented which is derived from (7) and (10) [13] and illustrated in **Figure 1(a)**.

The basic principles of the predictive DTC scheme are firstly, constant switching period 'ts' is defined. In predictive DTC, the control procedure is discretized unlike in DTC scheme, which is based on time domain. Secondly, at steady state condition, by taking the electromagnetic torque and flux amplitude errors to be minimized, the three different rotor voltage vectors are injected during ts. Note that according to the chosen three vector sequence together with the specified time intervals for each vector, the electromagnetic torque and flux evolutions within the switching period can be different. Thirdly, this procedure is repeated at constant period ts.

The Doubly Fed Induction Machine is modeled using (2), (4)–(6). The torque is calculated by using (7) in terms of complex conjugate of rotor flux and stator flux.

*Predictive Direct Torque Control Strategy for Doubly Fed Induction Machine for Torque… DOI: http://dx.doi.org/10.5772/intechopen.89979*

**Figure 1.**

*(a) Simplified prediction scheme of torque and rotor flux of DFIM. (b) Variation of electromagnetic torque with zero vectors at constant torque and rotor flux.*

#### **4.1 Effect of voltage vector on the DFIM**

Using (2), (4)–(7), the predictive expressions for torque and rotor flux are given by [2, 13]

$$\frac{\mathbf{d}\left|\boldsymbol{\upPsi}\_{\mathrm{r}}\right|}{\mathbf{d}\mathbf{t}} = \frac{1}{\left|\boldsymbol{\upPsi}\_{\mathrm{r}}\right|} \left[ \left(\frac{\mathbf{R}\_{\mathrm{r}}\mathbf{L}\_{\mathrm{m}}}{\sigma\mathbf{L}\_{\mathrm{s}}\mathbf{L}\_{\mathrm{r}}}\right) \mathrm{Re}\{\boldsymbol{\upPsi}\_{\mathrm{r}} \cdot \boldsymbol{\upPsi}\_{\mathrm{s}}\} - \left(\frac{\mathbf{R}\_{\mathrm{r}}}{\sigma\mathbf{L}\_{\mathrm{r}}}\right) \left|\boldsymbol{\upPsi}\_{\mathrm{r}}\right|^{2} + \mathrm{Re}\{\boldsymbol{\upPsi}\_{\mathrm{s}} \cdot \bar{\mathbf{v}}\_{\mathrm{r}}\} \right] \tag{9}$$

$$\frac{\mathrm{d}\mathbf{T}\_{\mathrm{om}}}{\mathrm{d}\mathbf{t}} = \frac{3}{2}\mathrm{p}\frac{\mathrm{L}\_{\mathrm{m}}}{\mathrm{e}\mathrm{L}\_{\mathrm{a}}\mathrm{L}\_{\mathrm{r}}} \left[ \left( \frac{\mathrm{R}\_{\mathrm{a}}}{\mathrm{e}\mathrm{L}\_{\mathrm{a}}} + \frac{\mathrm{R}\_{\mathrm{r}}}{\mathrm{e}\mathrm{L}\_{\mathrm{r}}} \right) \mathrm{Im} \{ \overline{\boldsymbol{\upmu}}\_{\mathrm{r}}, \boldsymbol{\upupup}\_{\mathrm{a}}^{\circ} \} - \mathrm{o}\mathrm{m}\_{\mathrm{m}} \mathrm{Re} \{ \overline{\boldsymbol{\upmu}}\_{\mathrm{r}}, \boldsymbol{\upupup}\_{\mathrm{a}}^{\circ} \} + \mathrm{Im} \{ \overline{\boldsymbol{\upmu}}\_{\mathrm{s}}, \boldsymbol{\upupup}\_{\mathrm{r}}^{\circ} \} + \mathrm{Im} \{ \overline{\boldsymbol{\upmu}}\_{\mathrm{s}}, \boldsymbol{\upupup}\_{\mathrm{r}}^{\circ} \} \right] \tag{10}$$

The below space vector representations are used in order to analyze the expressions (9) and (10).

$$\left|\upPsi\_{\mathbf{r}} = \left|\upPsi\_{\mathbf{r}}\right|\mathbf{e}^{j\alpha\_{\mathbf{s}}\mathbf{t}}\tag{11}$$

$$\left|\upPsi\_{\mathfrak{s}} = \left|\upPsi\_{\mathfrak{s}}\right|\mathbf{e}^{\mathbf{j}\left\{\alpha\_{\mathfrak{s}}\mathbf{t} + \mathfrak{s}\right\}}\tag{12}$$

$$\left|\bar{\mathbf{v}}\_{\mathfrak{s}}\right| = \left|\bar{\mathbf{v}}\_{\mathfrak{s}}\right| \mathbf{e}^{\mathbf{j}\left(\omega\_{\mathfrak{s}}\mathbf{t} + \left(\frac{\mathbf{r}}{2}\right) + \mathbf{0}\right)}\tag{13}$$

*Direct Torque Control Strategies of Electrical Machines*

$$\overline{\mathbf{v}}\_{\mathbf{r}} = \frac{2}{3} \mathbf{V\_{DC}} \mathbf{e^{j(\alpha\_m \mathbf{t} + \pi/3 \cdot (n - 1))}} \tag{14}$$

torque variation. This fact implies that near synchronous speed the amplitude of rotor voltage vector is small demanding the condition of zero vectors. As shown in **Figure 1(b)**, this transition from positive to negative torque slope is different to the

*Predictive Direct Torque Control Strategy for Doubly Fed Induction Machine for Torque…*

The first vector is selected depending on the errors of torque and flux and the sector where the rotor flux lies, the look up table for vector selection is shown in **Table 1**. From the **Table 1**, it is noticed that the required rotor voltage vector should produce either positive or negative slope variation depending on the output of the

The first vector is actually selected based on the classical DTC; the second vector

is selected such that it is always followed by zero vectors in order to reduce the

rotor voltages are chosen in such a way that it reduces the switching losses.

From **Figure 2**, it can be seen that from the derivative calculations block the required values of torque and flux evolutions depending on the each rotor voltage selected are fed to the ripple reduction criteria block, in which based on the slopes calculations by (15) and (17), the required rotor voltage vectors which are selected are active for the time period of this constant slopes. This constant time period of the constant slopes is fed to the switching table, where in the switching operation of

From (15) and (17), it can be inferred that the rotor flux is constant for zero vector and it produces opposite sign for torque variation for first two active vectors. These two active vectors along with zero vectors are useful to control torque and rotor flux. By the two active vectors the rotor flux as one vector produce positive slope, the other vectors produce negative slope because of this the flux ripple is not eliminated completely as compared to torque ripple and further it affects the rotor and stator currents. **Table 2** shows the selection of second active vector, after the first active vector is selected. It clearly shows that, one vector cannot be selected, as the flux would have a very big or small variation, which leads to poor quality of flux

The right choice of zero vectors V0 and V7 implies that there is reduction in switching power loss of the converter. Two different switching sequences exist for each pair of required active vectors, which allows the commutations of the converter to be reduced. **Table 3** shows the correct sequence of vectors, which allows only four commutations per switching period ts. The candidate sequences are such

Error value of rotor flux 1 V(n 1) V(n + 1)

**Error of electromagnetic torque**

1 V(n 2) V(n + 2)

**1 1**

torque or flux hysteresis comparators that is either 1 or 1, respectively.

**4.3 Selection of second and third vector rotor voltage vector**

torque and flux ripples based on the predictive DTC strategy.

synchronous speed.

output.

*n = sector.*

**Table 1.**

**107**

*Selection of voltage vectors [13].*

**4.2 Selection of first rotor voltage vector**

*DOI: http://dx.doi.org/10.5772/intechopen.89979*

**4.4 Reduction of switching power losses**

where *n* is representation of sectors from 0 to 7. In (9), if Eqs. (11)–(14) are substituted, then

$$\frac{d\left|\overline{\boldsymbol{\upmu}}\_{\rm r}\right|}{d\mathbf{t}} = \left[ \left( \frac{\mathbf{R}\_{\rm r}\mathbf{L}\_{\rm m}}{\sigma\mathbf{L}\_{\rm s}\mathbf{L}\_{\rm r}} \right) \left| \overline{\boldsymbol{\upmu}}\_{\rm s} \right| \cos\Theta - \left( \frac{\mathbf{R}\_{\rm r}}{\sigma\mathbf{L}\_{\rm r}} \right) \left| \overline{\boldsymbol{\upmu}}\_{\rm r} \right| + \frac{2}{3} \mathbf{V}\_{\rm DC} \cos\left( \mathbf{o}\_{\rm alip}\,\mathbf{t} - \frac{\mathbf{n}}{3} \text{(\"n--1\)} \right) \right] \tag{15}$$

where

$$
\boldsymbol{\mathfrak{op}}\_{\text{align}} = \boldsymbol{\mathfrak{op}}\_{\mathbf{s}} - \boldsymbol{\mathfrak{op}}\_{\mathbf{m}} \tag{16}
$$

Eq. (15) comprises of a cosine term and two constant terms, the cosine term with depends on DC bus voltage and it indicates that the cosine term of rotor flux variation is constant for zero vectors and only depends on active vectors.

Eq. (10) can be simplified considering the stator flux vector module which is nearly constant.

$$\frac{\mathrm{d}T\_{\mathrm{em}}}{\mathrm{d}t} = \mathrm{T\_{\mathrm{em}}} \left( \frac{\mathrm{\sigma\_{\mathrm{alip}}}}{\tan \, \mathrm{\delta}} - \left( \frac{\mathrm{R\_{\mathrm{s}}}}{\sigma \mathrm{L\_{\mathrm{s}}}} + \frac{\mathrm{R\_{\mathrm{r}}}}{\sigma \mathrm{L\_{\mathrm{r}}}} \right) \right) + \mathrm{p} \frac{\mathrm{L\_{\mathrm{m}}}}{\mathrm{\sigma\_{\mathrm{b}}} \mathrm{L\_{\mathrm{r}}}} \mathrm{V\_{\mathrm{DC}}} \left| \left. \overline{\mathrm{\sigma}\_{\mathrm{s}}} \right| \sin \left( \left. \mathrm{\sigma\_{\mathrm{alip}}} \mathrm{t} + \mathrm{\delta} - \frac{\mathrm{m}}{3} \mathrm{\mathrm{(n-1)}} \mathrm{t} \right) \right) \tag{17}$$

Eq. (17) is similar to rotor flux derivation with only one constant and one sine term.

Equations (15) and (17) are used practically as shown in **Figure 1(a)**, instead of (9) and (10). As shown in **Figure 1(a)**, the torque and flux derivatives depends on only four terms when considering the DC bus voltage, stator voltage and flux as constant magnitudes. In the expression (17), the constant term depends on Tem and, that means it depends on machine operating condition, which can be positive or negative as shown in **Figure 1(b)**. Because of this reason only, the phase shift order of each active vector varies, whereas, it is not like that for rotor flux derivative. The key point is knowing the slopes or the derivatives of torque and flux for each of the rotor voltage vector based on **Figure 1(a)** and **Table 1** and considering these slopes to be constants within the specific time in the given switching period, ts, the torque and flux ripples can be maintained within the limits.

With constant values

$$\mathbf{K\_1 = \frac{3}{2} p \frac{\mathbf{L\_m}}{\sigma \mathbf{L\_s} \mathbf{L\_r}} \left| \psi\_s \right|} \left| \psi\_s \right| \tag{18}$$

$$\mathbf{K\_2} = \frac{\mathbf{2}}{\mathbf{3}} \mathbf{v\_{DC}} \tag{19}$$

$$\mathbf{K\_3 = \frac{\mathbf{R\_s}}{\sigma \mathbf{L\_s}} + \frac{\mathbf{R\_r}}{\sigma \mathbf{L\_r}}} \tag{20}$$

From the **Figure 1(b)**, it can be seen that the slope of torque derivative varies proportionally to the slip speed provided at fixed torque and rotor flux operating conditions and also near synchronous speed, the slope of the zero vector becomes smaller obeying the fact that from the expression (17) the zero vector produce small

### *Predictive Direct Torque Control Strategy for Doubly Fed Induction Machine for Torque… DOI: http://dx.doi.org/10.5772/intechopen.89979*

torque variation. This fact implies that near synchronous speed the amplitude of rotor voltage vector is small demanding the condition of zero vectors. As shown in **Figure 1(b)**, this transition from positive to negative torque slope is different to the synchronous speed.
