# Fault Tolerant Control of DFIG-Based Wind Energy Conversion System Using Augmented Observer

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## Abstract

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## 1. Introduction

_{∞}is used to guarantee the robustness and estimate the output noise. An extended sliding mode observer is designed to estimate external disturbances and system states simultaneously, which widens the application scope of fault diagnosis observer in [16]. Reference [17] reduced the influence of process disturbance by constructing augmented state vector composed of system states and related faults, and estimating system states and related faults. In [18], the discrete linear model is used to design the sliding mode controller. The stability and robustness of the nonlinear system are improved by adding discrete operators to improve the discrete sliding mode controller. A new design method of augmented fault diagnosis observer is proposed in [19], which separates the observer from the output feedback fault-tolerant device and simplifies the design process. In [20], the augmented system, unknown input fuzzy observer and linear matrix inequality are combined to design robust fault estimation and fault tolerance control approach for T-S fuzzy systems, which are applied to 4.8-MW wind turbines system.

## 2. Mathematical Modeling of Double-fed WECS

_{wt}is the mechanical power captured by the wind turbine, ρ is the air density, ν is the wind speed, R is the fan blade length; C

_{p}(λ,β) is the wind energy conversion coefficient, which is a function of tip velocity ratio λ and pitch angle β. Λ is the ratio of tip speed to wind speed, that is λ = Ω

_{l}(R/ν), where Ω

_{l}is the angular velocity of the wind turbine rotor, that is, the low speed axis.

_{p}(λ,β). If the pitch angle β of the wind turbine remains unchanged, the wind energy conversion coefficient C

_{p}is only related to the tip speed ratio λ. For different types of wind turbines, there is an unique optimal tip speed ratio λ to ensure the best wind energy conversion coefficient C

_{p}and achieve maximum wind energy capture.

_{Γ}(λ,β) = C

_{p}(λ,β)/λ is the torque coefficient.

_{h}is the rotor speed (high speed shaft) of the generator, that is Ω

_{l}= i

_{o}× Ω, i

_{o}is the gear transmission speed ratio. Γ

_{G}is the electromagnetic torque of the generator, η is the transmission efficiency, J

_{h}is the inertia of the high-speed axis, J

_{t}is the inertia of the low-speed axis.

_{G}is the electromagnetic time constant. Taking Ω

_{h}and Γ

_{G}as state vectors, the state equation of the WECS is obtained as shown in (6):

_{h}Γ

_{G}]

^{T}, u(t) = ${\mathsf{\Gamma}}_{ref}^{\ast}$, ${A}^{\prime}=\left[\begin{array}{cc}\frac{{\mathsf{\Gamma}}_{wt}({i}_{0}{\mathsf{\Omega}}_{h},\nu}{{i}_{o}{J}_{t}{\mathsf{\Omega}}_{h}}& -\frac{1}{{J}_{t}}\\ 0& -\frac{1}{{T}_{G}}\end{array}\right]$, $B=\left[\begin{array}{c}0\\ \frac{1}{{T}_{G}}\end{array}\right]$, $C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector.

## 3. Actuator Fault Model of the WECS

^{n}is the state variable, u ∈ R

^{m}is the input vector and y ∈ R

^{p}is the measurable output vector, f

_{a}∈ R

^{q}represents an unknown but bounded actuator fault of the system, A′ ∈ R

^{m}

^{×n}, B ∈ R

^{n}

^{×m}, C ∈ R

^{p}

^{×n}, D ∈ R

^{n}

^{×q}. The system matrix A′ is split into the form of the sum of an uncertain matrix and a constant matrix, that is A′ = ΔA, $A=\left[\begin{array}{cc}0& -\frac{1}{{J}_{t}}\\ 0& -\frac{1}{{T}_{G}}\end{array}\right]$, $\Delta A=\left[\begin{array}{cc}\frac{{\mathsf{\Gamma}}_{wt}\left({i}_{0}{\mathsf{\Omega}}_{h},\nu \right)}{{i}_{o}{J}_{ty}{\mathsf{\Omega}}_{h}}& 0\\ 0& 0\end{array}\right]$ where $\Delta Ax=Md\left(x,u,t\right)={\left[\frac{{\mathsf{\Gamma}}_{wt}\left({i}_{o}{\mathsf{\Omega}}_{h},\nu \right)}{{i}_{o}{J}_{t}}0\right]}^{T}$, where d(x,u,t) ∈ R

^{h}, M ∈ R

^{n}

^{×}

^{h}. d(x,u,t) is regarded as an unknown input disturbance of the system, and Equation (7) can be converted into (8):

_{a}satisfy $\Vert d\left(x,u,t\right)\Vert \le {d}_{0}$ and $\Vert {f}_{a}\Vert \le {\alpha}_{0}$, where d

_{0}> 0, α

_{0}> 0 is a known constant.

## 4. Design of the Augmented Sliding Mode Observer

_{p}∈ R

^{(n+q)}

^{×p}is the undetermined gain matrix of the sliding mode observer, and L

_{s}∈ R

^{(n+q)}

^{×(q+h)}is the sliding mode gain matrix of the sliding mode observer. u

_{s}∈ R

^{q}

^{+h}is a non-continuous sliding mode input term, which eliminates the effect of system actuator faults and uncertainties. It is defined as Equation (12):

^{(q+h)}

^{×p}is a parameter matrix determined by the Lyapunov matrix P.

^{1}satisfies the Lyapunov equation $-{\left(\mu I+{\overline{E}}^{-1}\overline{A}\right)}^{T}X-X\left(\mu I+{\overline{E}}^{-1}\overline{A}\right)=-{\overline{C}}^{T}\overline{C}$ and μ > 0, which satisfies $\mathrm{R}e\left[{\lambda}_{i}\left({\overline{E}}^{-1}\overline{A}\right)\right]>-\mu $.

_{p}and μ

_{s}are decomposed into the following Equation (21):

_{s}can be approximated by Equation (23) with arbitrary precision, as follows:

_{p}and the estimated value ${\widehat{f}}_{a}$ of actuator fault f

_{a}can be obtained.

## 5. Design of Active Fault Tolerant Controller for WECS

_{sm}> 0. a

_{2}depends on the steady-state objective of the system, that is, ${\lambda}_{opt}:\dot{\mathsf{\Omega}={a}_{1}{\mathsf{\Omega}}_{opt}+{a}_{2}{\mathsf{\Gamma}}_{opt}=0}$, λ

_{opt}is the optimum tip speed ratio, Ω

_{hopt}not in any equation is the optimum value of high speed shaft speed, Γ

_{hopt}not in any equation is the optimum value of generator electromagnetic torque, then ${a}_{2}=-{a}_{1}\frac{{\mathsf{\Omega}}_{opt}}{{\mathsf{\Gamma}}_{opt}}$.

_{eq}is the equivalent control input and un is the switching part, as shown in Equation (26):

^{2}, ${{C}^{\prime}}_{p}\left(\lambda \right)$ is the differential of power coefficient λ and sgn

_{h}(σ) is the hysteresis function with bandwidth h.

_{h}and Γ

_{G}of the controller are changed, which leads to the abnormal control signal ${\mathsf{\Gamma}}_{ref}^{\ast}$ fed back to the system, and then it affects the maximum wind energy capture of the WECS. Through active fault-tolerant control, the fault output is compensated, the output signal of the actuator is corrected, and the active fault-tolerant control target of the actuator fault is realized, so that the performance of the fault system can be restored to the same level as that of the fault-free system.

## 6. Simulation Analysis

_{p}is determined by the following Equation (29):

_{p}can reach the ideal maximum, which can be kept at about 0.476 in 5~100 s, as shown in Figure 6.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**Reference value of electromagnetic torque ${\mathsf{\Gamma}}_{ref}^{\ast}$ when actuator fails.

**Figure 10.**Reference value of electromagnetic torque ${\mathsf{\Gamma}}_{ref}^{\ast}$ after fault-tolerant control.

**Figure 11.**Wind Energy Conversion Coefficient Value C

_{p}without Fault after Fault Tolerant Control.

Parameter Names | Parameter Values |
---|---|

Rated voltage V_{S} | 220 V |

Rated speed w_{S} | 100 πrad/s |

Rated electromagnetic torque Γ_{G}_{max} | 40 N·m |

Electromagnetic time constant T_{G} | 0.02 |

Air density ρ | 1.25 kg/m^{3} |

Transmission efficiency η | 95% |

Transmission speed ratio i_{0} | 6.25 |

Blade length R | 2.5 m |

Moment of interia of high speed axis J_{t} | 0.0092 kg·m^{2} |

Moment of interia of low speed axis J_{wt} | 3.6 kg·m^{2} |

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**MDPI and ACS Style**

Wang, X.; Shen, Y.
Fault Tolerant Control of DFIG-Based Wind Energy Conversion System Using Augmented Observer. *Energies* **2019**, *12*, 580.
https://doi.org/10.3390/en12040580

**AMA Style**

Wang X, Shen Y.
Fault Tolerant Control of DFIG-Based Wind Energy Conversion System Using Augmented Observer. *Energies*. 2019; 12(4):580.
https://doi.org/10.3390/en12040580

**Chicago/Turabian Style**

Wang, Xu, and Yanxia Shen.
2019. "Fault Tolerant Control of DFIG-Based Wind Energy Conversion System Using Augmented Observer" *Energies* 12, no. 4: 580.
https://doi.org/10.3390/en12040580