# Maximum Power Extraction from Wind Turbines Using a Fault-Tolerant Fractional-Order Nonsingular Terminal Sliding Mode Controller

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## Abstract

**:**

## 1. Introduction

_{cut-in}and above the cut-out speed V

_{cut-out}, respectively. Accordingly, the WT is either unable to provide the expected power (region I) or would suffer from high mechanical loads leading to damage (region IV); hence, the WT is shut down in both regions. When the wind speed is between the cut-in and the rated speed ${V}_{rated}$ (region II or partial-load region), where the rated speed is the speed at which the WT is able to provide the power with its maximum capacity, torque control approaches are used to control the generator torque and maximize the power capture. When the wind speed exceeds the rated value and is below the cut-out speed, the WT enters its third operating region (region III or full-load region), where pitch control and power regulation approaches keep the captured power at the rated value (${P}_{g,rated}$) and prevent the turbine from damage. In order to maximize the wind energy extraction in a variable speed WT operating in region II, the rotor speed is regulated to track its continuously fluctuating optimum value, and hence maintain the optimum tip-speed ratio [5].

- A design that integrates the fractional calculus into NTSMC to effectively enhance the finite-time convergence speed and simultaneously alleviate the chattering phenomenon. Therefore, the optimum rotor speed tracking is achieved with little error, resulting in more power extracted from the wind;
- Validation and performance assessment of the fault-tolerant capability of proposed design using partial loss on the generator torque;
- Comparative performance analysis of the developed control strategy with conventional SMC [38] and second-order fast terminal SMC [39]. Accordingly, taking advantage of the proposed control law, a desirable optimum rotor speed tracking performance with fewer fluctuations and faster transient response is achieved.

## 2. Problem Formulation

#### 2.1. Wind Turbine Model

**Remark**

**1.**

#### 2.2. Problem Statement

#### 2.3. Actuator Faults

## 3. Controller Design

#### 3.1. Preliminaries on Fractional Calculus

#### 3.2. Proposed FNTSMC Controller

**Remark**

**2.**

**Remark**

**3.**

**Theorem**

**1.**

**Proof of Theorem 1.**

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CART | Controls Advanced Research Turbine |

DFIG | Doubly-Fed Induction Generator |

FNTS | Fractional Nonsingular Terminal Sliding |

FNTSMC | Fractional Nonsingular Terminal Sliding Mode Control |

FTC | Fault Tolerant Control |

MPC | Model Predictive Control |

NN | Neural Network |

NTSMC | Nonsingular Terminal Sliding Mode Control |

PI | Proportional Integral |

PID | Proportional Integral Derivative |

PMSG | Permanent-Magnet Synchronous Generator |

PSO | Particle Swarm Optimization |

RL | Riemann–Liouville |

SMC | Sliding Mode Control |

SOFTSMC | Second-order Fast Terminal Sliding Mode Control |

TSMC | Terminal Sliding Mode Control |

WECS | Wind Energy Conversion System |

WT | Wind Turbine |

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**Figure 2.**Schematic of the two-mass model. (Note: the blades are not included as a separate mass, and are shown as an illustration).

**Figure 3.**Illustration of the actuator health indicator. $\zeta \left(t\right)=1$ denotes a healthy actuation and $\zeta \left(t\right)<1$ represents a partial loss of actuation power.

**Figure 6.**Rotor speed tracking; a comparison between SMC, SOFTSMC, and FNTSMC approaches. The insets show the detail of the regions highlighted by dashed black lines.

**Figure 8.**Generator speed; a comparison between SMC, SOFTSMC, and FNTSMC approaches. The inset show the detail of the region highlighted by dashed black lines.

**Figure 9.**Generator torque; a comparison between SMC, SOFTSMC, and FNTSMC approaches. The inset show the detail of the region highlighted by dashed black lines.

**Figure 11.**Electric power; a comparison between SMC, SOFTSMC, and FNTSMC approaches. The inset show the detail of the region highlighted by dashed black lines.

Parameter | Value | Unit | Parameter | Value | Unit |
---|---|---|---|---|---|

R | 21.65 | $\mathrm{m}$ | $\rho $ | 1.308 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${J}_{r}$ | 3.25$\times {10}^{5}$ | $\mathrm{kg}.{\mathrm{m}}^{2}$ | ${J}_{g}$ | 34.4 | $\mathrm{kg}.{\mathrm{m}}^{2}$ |

${D}_{r}$ | 27.36 | $\mathrm{N}.\mathrm{m}.\mathrm{s}/\mathrm{rad}$ | ${D}_{g}$ | 0.2 | $\mathrm{N}.\mathrm{m}.\mathrm{s}/\mathrm{rad}$ |

${D}_{ls}$ | 2.691$\times {10}^{5}$ | $\mathrm{N}.\mathrm{m}.\mathrm{s}/\mathrm{rad}$ | ${k}_{ls}$ | 9500 | $\mathrm{N}.\mathrm{m}/\mathrm{rad}$ |

${n}_{g}$ | 43.165 | − | ${P}_{e,nom}$ | 600$\times {10}^{5}$ | $\mathrm{W}$ |

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**MDPI and ACS Style**

Mousavi, Y.; Bevan, G.; Küçükdemiral, I.B.; Fekih, A.
Maximum Power Extraction from Wind Turbines Using a Fault-Tolerant Fractional-Order Nonsingular Terminal Sliding Mode Controller. *Energies* **2021**, *14*, 5887.
https://doi.org/10.3390/en14185887

**AMA Style**

Mousavi Y, Bevan G, Küçükdemiral IB, Fekih A.
Maximum Power Extraction from Wind Turbines Using a Fault-Tolerant Fractional-Order Nonsingular Terminal Sliding Mode Controller. *Energies*. 2021; 14(18):5887.
https://doi.org/10.3390/en14185887

**Chicago/Turabian Style**

Mousavi, Yashar, Geraint Bevan, Ibrahim Beklan Küçükdemiral, and Afef Fekih.
2021. "Maximum Power Extraction from Wind Turbines Using a Fault-Tolerant Fractional-Order Nonsingular Terminal Sliding Mode Controller" *Energies* 14, no. 18: 5887.
https://doi.org/10.3390/en14185887