# Wind Turbine Pitch Actuator Regulation for Efficient and Reliable Energy Conversion: A Fault-Tolerant Constrained Control Solution

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## Abstract

**:**

## 1. Introduction

## 2. HAWT Operational Model and Preliminaries

#### 2.1. HAWT Operational Model

**Assumption**

**1.**

#### 2.2. Technical Preliminaries

**Definition**

**1**

**.**Let’s assume that $V\left(x\left(t\right)\right)$ is positive definite continuous with respect to the solution of the system $\dot{x}\left(t\right)=f\left(x\left(t\right)\right)$ on an open region $\mathcal{D}$. If $V\left(x\left(t\right)\right)$ approaches to infinity, as $x\left(t\right)$ approaches to the boundary of the region $\mathcal{D}$, then $V\left(x\left(t\right)\right)$ is a BLF with continuous first order partial derivatives within all $\mathcal{D}$. Consequently, the inequality $V\left(x\left(t\right)\right)\le \mathcal{W}$, $\forall t\ge 0$ holds along with the solution of $\dot{x}\left(t\right)=f\left(x\left(t\right)\right)$ for $x\left(0\right)\in \mathcal{D}$, and some positive constant $\mathcal{W}$.

**Definition**

**2**

**.**$x\left(t\right)$ is Uniformly Ultimately Bounded (UUB) if there exists a number $T\left(K,x\left({t}_{0}\right)\right)$, and a $K>0$ such that for any compact set $\mathcal{S}$ and all $x\left({t}_{0}\right)\in \mathcal{S}$, $\Vert x\left(t\right)\Vert \le K$, for all $t\ge {t}_{0}+T$.

**Definition**

**3**

**.**Any continuous function $N\left(s\right)\in \mathbb{R}$ is a Nussbaum-type function of $s\in \mathbb{R}$, satisfying $\underset{s\to \infty}{lim}sup{{\displaystyle \int}}_{{s}_{0}}^{s}N\left(\tau \right)d\tau =+\infty $ and $\underset{s\to \infty}{lim}inf{{\displaystyle \int}}_{{s}_{0}}^{s}N\left(\tau \right)d\tau =-\infty $, for ${s}_{0}\in \mathbb{R}$.

**Lemma**

**1**

**.**Let’s assume that $V\left(t\right)>0$ and $\mathcal{F}\left(t\right)$ are smooth functions for any $t\in \left[0{t}_{f}\right)$. Then, if

**Lemma**

**2**

**.**For a real variable $\psi $ in $\left|\psi \right|<1$, the inequality $tan\left(\pi {\psi}^{2}/2\right)<\pi {\psi}^{2}se{c}^{2}\left(\pi {\psi}^{2}/2\right)$ holds true.

## 3. Baseline NACC Design

**Theorem**

**1.**

- P1. All the closed-loop system states are bounded;
- P2. For $i=1,2$, the constraint sets ${C}_{i}$ are not violated;
- P3. By the proper choice of the design parameters, the tracking error ${e}_{1}\left(t\right)$ can be made arbitrarily small.

**Proof.**

## 4. MNACC Design with Unknown Control Gain

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 5. MNACC Control with Arbitrary Initial Conditions

**Theorem**

**3.**

**Proof.**

**Remark**

**2.**

## 6. Fault-Tolerance Capability of MNACC

**Theorem**

**4.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

## 7. Feasibility Check and Design Algorithm

Algorithm 1 Proposed controller design procedure |

1. Offline computation: |

1.1. For the given ${\delta}_{{\omega}_{r}}$ and ${\dot{\omega}}_{r}$, solve OP to obtain ${\kappa}_{1}^{\ast}$. Select ${\kappa}_{2}$, ${\sigma}_{F1}$, ${\sigma}_{F2}$ and ${\epsilon}_{i}$, for i = 1,2, in accordance with Remark 2. |

1.2. For the given initial condition, compute ${\overline{a}}_{i}$ and ${\underset{\_}{a}}_{i}$, then select ${\overline{b}}_{i}$, ${\underset{\_}{b}}_{i}$ and ${\varphi}_{i}$, for i = 1,2. |

2. Online computation: |

2.1. Integrate the virtual controls Equations (41) and (42), the adaptive laws Equations (21) and (34). |

2.2. Compute the control gain Equation (33) and then the control signal Equation (18). |

## 8. Simulations and Discussion

#### 8.1. Control Parameters

#### 8.2. Fault Model

#### 8.3. Parameters of Measurement Errors

#### 8.4. Simulation Results and Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 4.**Tacking error ${e}_{1}$ (blue line) with constructed constraints (red dashed lines), under single (

**a**) and simultaneous faults (

**b**). The circle represents the initial value.

**Figure 5.**Tacking error ${e}_{2}$ (blue line) with constructed constraints (red dashed lines), under single (

**a**) and simultaneous faults (

**b**). The circle represents the initial value.

**Figure 6.**Rotor speed ${\omega}_{r}$ (blue line) with constructed constraints (red dashed lines) and nominal value (green dashed line), under single (

**a**) and simultaneous faults (

**b**).

**Figure 7.**Generator speed ${\omega}_{g}$ (blue line) with constructed constraints (red dashed lines) and nominal value (green dashed line), under single (

**a**) and simultaneous faults (

**b**).

**Figure 8.**Generated power ${P}_{g}$ (blue line) with constructed constraints (red dashed lines) and nominal value (green dashed line), under single (

**a**) and simultaneous faults (

**b**).

Pitch Actuator Condition | Parameters | Indicator |
---|---|---|

$\mathrm{Normal}\text{}\left(N\right)$) | ${\omega}_{n,N}=11.11$, ${\xi}_{N}=0.6$ | ${\alpha}_{{f}_{1}}={\alpha}_{{f}_{2}}=0$ |

$\mathrm{Pump}\text{}\mathrm{Wear}\text{}(PW$) | ${\omega}_{n,PW}=7.27$, ${\xi}_{PW}=0.75$ | ${\alpha}_{{f}_{1}}=0.63$, ${\alpha}_{{f}_{2}}=0.30$ |

Pitch Actuator Condition | Parameters | Indicator |
---|---|---|

Bias | $\mathsf{\Phi}\left(t\right)={10}^{\xb0}$ | $200\left(s\right)\le t\le 300\left(s\right)$ |

Effectiveness loss | $\rho \left(t\right)=0.7$ | $400\left(s\right)\le t\le 500\left(s\right)$ |

Pump wear | ${\alpha}_{{f}_{1}}=0.63$, ${\alpha}_{{f}_{2}}=0.30$ | $600\left(s\right)\le t\le 700\left(s\right)$ |

Hydraulic oil leak | ${\alpha}_{{f}_{1}}=1,{\alpha}_{{f}_{2}}=0.88$ | $800\left(s\right)\le t\le 900\left(s\right)$ |

High air content in oil | ${\alpha}_{{f}_{1}}=0.81$, ${\alpha}_{{f}_{2}}=1$ | $900\left(s\right)\le t\le 1000\left(s\right)$ |

Pitch Actuator Fault Type | Fault Effect | Fault Period |
---|---|---|

Bias | $\mathsf{\Phi}\left(t\right)={15}^{\xb0}$ | $100\left(s\right)\le t\le 400\left(s\right)$ |

Pump wear | ${\alpha}_{{f}_{1}}=0.63$, ${\alpha}_{{f}_{2}}=0.30$ | |

Effectiveness loss | $\rho \left(t\right)=0.5$ | $500\left(s\right)\le t\le 800\left(s\right)$ |

High air content in oil | ${\alpha}_{{f}_{1}}=0.81$, ${\alpha}_{{f}_{2}}=1$ | |

Hydraulic oil leak | ${\alpha}_{{f}_{1}}=1,{\alpha}_{{f}_{2}}=0.88$ | $900\left(s\right)\le t\le 1000\left(s\right)$ |

Sensor | Mean | Noise Standard Deviation | Error Compared to Nominal Values (%) | |
---|---|---|---|---|

$Set\text{}1$ | Rotor speed | $0$ | ${\mathsf{\sigma}}_{{\omega}_{r}}=0.002$ | $0.12$ |

Generator speed | $0$ | ${\mathsf{\sigma}}_{{\omega}_{g}}=0.5$ | $0.31$ | |

Generator torque | $0$ | ${\mathsf{\sigma}}_{{T}_{g}}=90$ | $0.28$ | |

Pitch angle | $0$ | ${\mathsf{\sigma}}_{\beta}=0.2$ | $1.16$ | |

$Set\text{}2$ | Rotor speed | $0$ | ${\mathsf{\sigma}}_{{\omega}_{r}}=0.004$ | $0.24$ |

Generator speed | $0$ | ${\mathsf{\sigma}}_{{\omega}_{g}}=1$ | $0.62$ | |

Generator torque | $0$ | ${\mathsf{\sigma}}_{{T}_{g}}=100$ | $0.31$ | |

Pitch angle | $0$ | ${\mathsf{\sigma}}_{\beta}=1$ | $5.8$ | |

$Set\text{}3$ | Rotor speed | $0$ | ${\mathsf{\sigma}}_{{\omega}_{r}}=0.008$ | $0.48$ |

Generator speed | $0$ | ${\mathsf{\sigma}}_{{\omega}_{g}}=3$ | $1.84$ | |

Generator torque | $0$ | ${\mathsf{\sigma}}_{{T}_{g}}=120$ | $0.37$ | |

Pitch angle | $0$ | ${\mathsf{\sigma}}_{\beta}=2$ | $11.6$ |

**Table 5.**Monte-Carlo simulation results in terms of PM% index. Letters B, A and W stand for best, average and worst values, respectively.

PM (%) | |||||||
---|---|---|---|---|---|---|---|

Maximum | Minimum | ||||||

B | A | W | B | A | W | ||

Set 1 | C1 | 1.038 | 1.151 | 1.422 | $6.14\times {10}^{-8}$ | $3.73\times {10}^{-4}$ | 0.016 |

C2 | 1.142 | 1.268 | 1.531 | $7.51\times {10}^{-8}$ | $6.14\times {10}^{-6}$ | $2.37\times {10}^{-4}$ | |

Set 2 | C1 | 1.033 | 1.230 | 1.641 | $1.13\times {10}^{-8}$ | $5.53\times {10}^{-4}$ | 0.014 |

C2 | 1.131 | 1.359 | 1.813 | $2.77\times {10}^{-10}$ | $1.14\times {10}^{-5}$ | $1.48\times {10}^{-4}$ | |

Set 3 | C1 | 1.042 | 1.419 | 2.428 | $7.65\times {10}^{-9}$ | 0.006 | 0.349 |

C2 | 1.136 | 1.446 | 2.482 | $0.21\times {10}^{-8}$ | $2.72\times {10}^{-5}$ | $5.39\times {10}^{-4}$ | |

Mean | Standard Deviation | ||||||

B | A | W | B | A | W | ||

Set 1 | C1 | 0.155 | 0.186 | 0.277 | 0.141 | 0.157 | 0.198 |

C2 | 0.169 | 0.197 | 0.334 | 0.158 | 0.168 | 0.207 | |

Set 2 | C1 | 0.155 | 0.249 | 0.576 | 0.144 | 0.167 | 0.209 |

C2 | 0.117 | 0.268 | 0.629 | 0.158 | 0.182 | 0.228 | |

Set 3 | C1 | 0.155 | 0.413 | 1.374 | 0.143 | 0.182 | 0.214 |

C2 | 0.171 | 0.353 | 1.285 | 0.158 | 0.191 | 0.234 |

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**MDPI and ACS Style**

Habibi, H.; Howard, I.; Simani, S.
Wind Turbine Pitch Actuator Regulation for Efficient and Reliable Energy Conversion: A Fault-Tolerant Constrained Control Solution. *Actuators* **2022**, *11*, 102.
https://doi.org/10.3390/act11040102

**AMA Style**

Habibi H, Howard I, Simani S.
Wind Turbine Pitch Actuator Regulation for Efficient and Reliable Energy Conversion: A Fault-Tolerant Constrained Control Solution. *Actuators*. 2022; 11(4):102.
https://doi.org/10.3390/act11040102

**Chicago/Turabian Style**

Habibi, Hamed, Ian Howard, and Silvio Simani.
2022. "Wind Turbine Pitch Actuator Regulation for Efficient and Reliable Energy Conversion: A Fault-Tolerant Constrained Control Solution" *Actuators* 11, no. 4: 102.
https://doi.org/10.3390/act11040102