# Direct Torque Control of an Induction Motor Using Fractional-Order Sliding Mode Control Technique for Quick Response and Reduced Torque Ripple

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## Abstract

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## 1. Introduction

- Control algorithm of an FOSMC for SVM-based DTC for induction motor drives (IMDs);
- Derivation of the electromagnetic torque of an IMD using an FOSMC-DTC scheme;
- Minimization of torque ripples during steady state;
- Improved system response times during load-changing conditions;
- Reduction of high-frequency chattering phenomenon;
- Pure sinusoidal stator current waveform with fewer distortions in a 3-Φ IMD;
- Comparative analysis of the FOSMC-DTC, classical SMC, and PI controller methods.

## 2. Fractional-Order Sliding Mode Controller for Induction Motors

#### 2.1. SVM-Based DTC Using Single PI Controller

#### 2.2. SVM-Based DTC with Conventional Sliding Mode Controller

#### 2.3. SVM-Based FOSMC-DTC of Induction Motors

## 3. Mathematical Model of FOSMC-DTC Scheme of IMD

_{P}, K

_{I}, and K

_{D}are the proportional, integral, and derivative constants, respectively.

_{ref}is the desired speed and the actual speed of the three-phase induction motor is ω

_{act}. The first-order derivative for the sliding surface given in Equation (8) is given in Equation (10).

## 4. Simulation Results and Discussions

_{s}. For three different slip angles, torque response plots are obtained. Figure 16 shows that torque increases at a rate proportional to the step change in dθs/dt until reaching maximum torque, and it attains a maximum torque value at dθ

_{s}/dt = 2 × π × 22.7. For lower values of speed, torque increases at a slower rate when compared with high-speed values. Therefore, the DTC-FOSMC exhibits a robust behavior when compared to DTC-SVM with the single PI controller and conventional SMC.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BLDC | Brushless DC motor |

DTC | Direct torque control |

EV | Electric vehicle |

SPWM | Space vector pulse width modulation |

SVM | Space vector modulation |

PI | Proportional and integral |

SMC | Sliding mode controller |

HOSMC | Higher-order sliding mode controller |

FOSMC | Fractional-order sliding mode controller |

IMD | Induction motor drive |

PMSM | permanent magnet synchronous motor drive |

SRM | Switched reluctance motor |

PID | Proportional-integral controller |

S | Sliding surface |

${s}^{g}$ | Surface gradient |

u | Fractional-order derivative gain |

λ | Fractional-order integral gain. |

${\mathsf{\omega}}_{ref}$ | Desired speed |

${\mathsf{\omega}}_{act}$ | Actual speed |

${S}_{T}^{g}$ | Surface gradient of torque control |

${S}_{\psi}^{g}$ | Surface gradient of flux control |

${T}_{em}$ | Electromechanical torque |

${T}_{L}$ | Load torque |

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**Figure 7.**Simulink diagram of the FOSMC-DTC scheme. (

**a**) Complete block diagram of the FOSMC-DTC scheme; (

**b**) flux sliding surfaces for FOSMC-DTC scheme; (

**c**) torque sliding surfaces for FOSMC-DTC scheme.

**Figure 8.**Steady-state speed response and its ripple content at rated torque of 30 Nm. (

**a**) DTC-SVM with single PI controller; (

**b**) conventional SMC–based DTC-SVM scheme; (

**c**) DTC-SVM with FOSMC.

**Figure 9.**Steady-state torque response and its ripple content at rated torque of 30 Nm. (

**a**) DTC-SVM with single PI controller; (

**b**) conventional SMC–based DTC-SVM scheme; (

**c**) DTC-SVM with FOSMC.

**Figure 10.**Transient response at rated torque reversal of 30 Nm and its settling times. (

**a**) DTC-SVM with single PI controller; (

**b**) conventional SMC–based DTC-SVM scheme; (

**c**) DTC-SVM with FOSMC.

**Figure 11.**Dynamic response for step change of 7.5 Nm for every 0.4 s from no load to full load. (

**a**) DTC-SVM with single PI controller; (

**b**) conventional SMC–based DTC-SVM scheme; (

**c**) DTC-SVM with FOSMC.

**Figure 12.**Three-phase stator currents of an induction motor and its ripple content in (

**a**) DTC-SVM with single PI controller, (

**b**) conventional SMC–based DTC-SVM scheme, and (

**c**) DTC-SVM with FOSMC.

**Figure 13.**Flux trajectory and its utilization at rated torque of 30 Nm. (

**a**) DTC-SVM with single PI controller; (

**b**) conventional SMC–based DTC-SVM scheme; (

**c**) DTC-SVM with FOSMC.

**Figure 14.**Harmonic analysis of stator current and its THD in (

**a**) DTC-SVM with single PI controller, (

**b**) conventional SMC–based DTC-SVM scheme, and (

**c**) DTC-SVM with FOSMC.

**Figure 15.**Flux linkage and its corresponding ripple content at rated torque of 30 Nm. (

**a**) DTC-SVM with single PI controller; (

**b**) conventional SMC–based DTC-SVM scheme; (

**c**) DTC-SVM with FOSMC.

**Table 1.**Squirrel cage induction motor data [25].

Rating of the Induction Motor, P | 5.4 HP |
---|---|

Voltage (L-L), V_{rms} | 440 V |

Power frequency, f_{s} | 50 Hz |

Rated torque, T | 30 N-m |

Rated current (Peak) | 16 A |

Nominal speed, N_{nom} | 1430 rpm |

Controller Parameters | Flux Sliding Surface | Torque Sliding Surface |
---|---|---|

λ | 0.5 | 0.56 |

u | 0.45 | 0.5 |

k_{P} | 1.5 | 5.0 |

k_{I} | 0.1 | 0.1 |

k_{D} | 1.0 | 1.0 |

**Table 3.**Steady-state torque ripples with PI controller, SMC, and proposed DTC-SVM–based FOSMC at full and one-fourth load conditions.

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**MDPI and ACS Style**

Gudey, S.K.; Malla, M.; Jasthi, K.; Gampa, S.R.
Direct Torque Control of an Induction Motor Using Fractional-Order Sliding Mode Control Technique for Quick Response and Reduced Torque Ripple. *World Electr. Veh. J.* **2023**, *14*, 137.
https://doi.org/10.3390/wevj14060137

**AMA Style**

Gudey SK, Malla M, Jasthi K, Gampa SR.
Direct Torque Control of an Induction Motor Using Fractional-Order Sliding Mode Control Technique for Quick Response and Reduced Torque Ripple. *World Electric Vehicle Journal*. 2023; 14(6):137.
https://doi.org/10.3390/wevj14060137

**Chicago/Turabian Style**

Gudey, Satish Kumar, Mohan Malla, Kiran Jasthi, and Srinivasa Rao Gampa.
2023. "Direct Torque Control of an Induction Motor Using Fractional-Order Sliding Mode Control Technique for Quick Response and Reduced Torque Ripple" *World Electric Vehicle Journal* 14, no. 6: 137.
https://doi.org/10.3390/wevj14060137