**5. Simulation results**

12 Will-be-set-by-IN-TECH

The universe of discourse for the stator flux error input is defined in the closed interval [-0.5, 0.5]. The extreme MFs have trapezoidal shapes but the middle one takes triangular shape as is shown in Figure 13. However, the universe of discourse for electromagnetic torque error input is defined in the closed interval [-20, 20] but with the objective to see the shape of the MFs only is shown the interval [-5, 5] in Figure 14, the shapes of these MF are similar to the first input. For both inputs the linguistic labels N, Ze and P means Negative, Zero and Positive,

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−5 −4 −3 −2 −1 0 1 2 3 4 5

Where *FE* = *TE* = {*N*, *ZE*, *P*} are the fuzzy sets of the inputs and, **a** and **b** are coefficients of the first-order polynomial function typically present in the consequent part of the firs-order

**Electromagnetic Torque Error**

*ds* is determined by the rules of the form:

*ds* = *aEψ<sup>s</sup>* + *bE<sup>τ</sup>*

*qs* = −*bEψ<sup>s</sup>* + *aE<sup>τ</sup>*

*qs* is determined by the rules of the

**Stator Flux Error**

**N ZE P**

**N P ZE**

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

The direct component of the stator voltage *u*∗

**Figure 14.** Membership function for electromagnetic torque error input (*Eτ*)

However, the quadrature component of the stator voltage *u*∗

*Rx* : if *Eψ<sup>s</sup>* is *FE* and *E<sup>τ</sup>* is *TE* then *u*<sup>∗</sup>

*Ry* : if *Eψ<sup>s</sup>* is *FE* and *E<sup>τ</sup>* is *TE* then *u*<sup>∗</sup>

**Degree of membership**

**4.2. The fuzzy rule base**

Takagi-Sugeno fuzzy controllers.

form:

**Figure 13.** Membership function for stator flux error input (*Eψs*)

**Degree of membership**

respectively.

The simulations were performed using MATLAB simulation package which include Simulink block sets and fuzzy logic toolbox. The switching frequency of PWM inverter was set to be 10*kHz*, the stator reference flux considered was 0.47 Wb and the coefficients considered were *a* = 90 and *b* = 2. In order to investigate the effectiveness of the proposed control system and in order to check the closed-loop stability of the complete system, we performed several tests.

We used different dynamic operating conditions such as: step change in the motor load (from 0 to 1.0 pu) at fifty percent of rated speed, no-load sudden change in the speed reference (from 0.5 pu to -0.5 pu), and the application of an arbitrary load torque profile at fifty percent of rated speed. The motor parameters are given in Table 3.


**Table 3.** Induction Motor Parameters [16]

The Figure 15 illustrates the torque response of the DTC-SVM scheme with T-S fuzzy controller when the step change in the motor load is apply. The electromagnetic torque tracked the reference torque and in this test is obtained the following good performance measures: rise time *tr* = 1.1*ms*, settling time *ts* = 2.2*ms* and torque ripple *ripple* = 2.93%. Also is observed that the behavior of the stator current is sinusoidal.

**Figure 15.** Electromagnetic torque and stator current response for step change in the motor load at fifty percent of rated speed

−0.4 −0.2 0 0.2 0.4 0.6

1.5 2 2.5 3 3.5 4

1.5 2 2.5 3 3.5 4

1.5 2 2.5 3 3.5 4

W\* r Wr 355

T\* em Tem

> I as

ψds

The Figure 18 shows the behavior of the rotor angular speed *ωr*, the electromagnetic torque and the phase **a** stator current waveform when a step change in the reference speed from 0.5 pu to -0.5 pu is imposed, with no-load. The torque was limited in 1.5 times the rated torque how it was projected and the sinusoidal waveforms of the stator current shown that this control technique allowed also a good current control because it is inherent to the algorithm control

Rotor Angular Speed

Electromagnetic Torque

Stator Current

time (s)

**Figure 18.** Rotor angular speed, electromagnetic torque and phase **a** stator current when was apply the

no-load sudden change in the speed reference at fifty percent of rated speed

Stator Flux

The Takagi-Sugeno Fuzzy Controller Based Direct Torque Control with Space Vector Modulation for Three-Phase Induction Motor

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

**Figure 17.** Space of the stator flux quadrature components.

−100 0 100

> −20 0 20

> −20

0 20

T

I

s (A)

em (N.m)

Wr (rad/s)

ψqs

The Figure 16 presents the results when an arbitrary torque profile is imposed to DTC-SVM scheme with T-S fuzzy controller. In the first sub-figure the electromagnetic torque tracked the reference torque as expected, and in the next one the sinusoidal waveforms of the stator currents is shown. The Figure 17 shows space of the quadrature components of the stator flux and it shows the circular behavior of the stator flux when the torque profile is applied, and in consequence the proposed controller maintain the stator flux constant.

**Figure 16.** Electromagnetic torque and stator current response when is apply the load torque profile at fifty percent of rated speed

**Figure 17.** Space of the stator flux quadrature components.

14 Will-be-set-by-IN-TECH

Electromagnetic Torque

1.27 1.28 1.29 1.3 1.31 1.32 1.33 1.34

Stator Currents

1.27 1.28 1.29 1.3 1.31 1.32 1.33 1.34

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

Stator Currents

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

**Figure 16.** Electromagnetic torque and stator current response when is apply the load torque profile at

**Figure 15.** Electromagnetic torque and stator current response for step change in the motor load at fifty

The Figure 16 presents the results when an arbitrary torque profile is imposed to DTC-SVM scheme with T-S fuzzy controller. In the first sub-figure the electromagnetic torque tracked the reference torque as expected, and in the next one the sinusoidal waveforms of the stator currents is shown. The Figure 17 shows space of the quadrature components of the stator flux and it shows the circular behavior of the stator flux when the torque profile is applied, and in

Electromagnetic Torque

consequence the proposed controller maintain the stator flux constant.

T\* em Tem

> I as I bs I cs

T\* em Tem

> I as I bs I cs

−15 −10 −5 0 5 10 15

−15 −10 −5 0 5 10 15

I

fifty percent of rated speed

s (A)

T

em (N.m)

I

percent of rated speed

s (A)

T

em (N.m)

The Figure 18 shows the behavior of the rotor angular speed *ωr*, the electromagnetic torque and the phase **a** stator current waveform when a step change in the reference speed from 0.5 pu to -0.5 pu is imposed, with no-load. The torque was limited in 1.5 times the rated torque how it was projected and the sinusoidal waveforms of the stator current shown that this control technique allowed also a good current control because it is inherent to the algorithm control

**Figure 18.** Rotor angular speed, electromagnetic torque and phase **a** stator current when was apply the no-load sudden change in the speed reference at fifty percent of rated speed

proposed in this chapter. All the test results showed the good performance of the proposed DTC-SVM scheme with T-S fuzzy controller.

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