# Wind Turbines Optimal Operation at Time Variable Wind Speeds

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## Abstract

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## 1. Introduction

## 2. Wind Turbine’s Mathematical Model

## 3. PMSG’s Optimal Power Determination based on the Kinetic Motion Equation

- the first term, which is dependent on the moment of total inertia, ${P}_{inertial}$ is called the “inertial power”,

- the second term, ${P}_{WT}$, which is dependent on the PMSG’s angular speed, $\omega $, and on the wind speed, ${S}_{w}$, as depicted by Equation (1), represents the “wind turbine’s power”.
- the third term, which is dependent on the PMSG’s angular speed, $\omega $, and on the load resistance, represents the PMSG’s power, ${P}_{PMSG}$.

- the first term, ${E}_{0}$, represents the wind energy captured by the wind turbine during the time interval, $\Delta t$,$${E}_{0}={\int}_{{t}_{k-1}}^{{t}_{k}}{P}_{WT}dt=\frac{{P}_{WT\_avg}}{\Delta t}$$
- the second term, ${E}_{E}$, represents the energy delivered by the PMSG to the main grid,$${E}_{E}={\int}_{{t}_{k-1}}^{{t}_{k}}{P}_{PMSG}dt=\frac{{P}_{PMSG\_avg}}{\Delta t}$$
- the third term, ${E}_{kinetic}$, represents the kinetic energy,$${E}_{kinetic}=J{\scriptscriptstyle \frac{{\omega}_{k}^{2}-{\omega}_{k-1}^{2}}{2}}$$

## 4. PMSG’s Optimal Power Control Based on the PI-type Regulator

## 5. PMSG’s Optimal Power Control Based on the PID-type Regulator

## 6. Case Study

#### 6.1. PMSG’s Optimal Power Determination

#### 6.2. PI-Based Control of the PMSG’s Optimal Power

#### 6.3. PID-Based Control of the PMSG’s Optimal Power

- the maximum wind turbine’s power when operating at MPP, defined in Equation (47) (${P}_{WTmax}$)
- the PMSG’s optimal power calculated from the kinetic motion equation, defined in Equation (49) (${P}_{PMSGoptim}\left(t\right)$)
- the PMSG’s power obtained by using either the PI- or PID-type regulators, ${P}_{PMSG\_PI\_PID}\left(t\right)$, resulting from either Equation (62) or Equation (67).

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

## Appendix A

Type | Horizontal Axis Wind Turbine with Variable Rotor Speed |
---|---|

Rotor diameter | 100 m |

Power regulation | Independent electromechanical pitch system for each blade |

Rated power | 2500 kW |

Hub height | 100 m |

Rated rotational speed | 14.05 rpm |

Operating range rotational speed | 3.83–15.61 rpm |

Cut-in wind speed | 3 m/s |

Rated wind speed | 12 m/s |

Cut-out wind speed (10-min mean) | 25 m/s |

Extreme wind speed (50-year mean) | 37.5 m/s |

Annual average wind speed | 7.5 m/s |

Design life time | 20 years |

IEC 61400-1, class | IIIA |

Variable | Description | Units |
---|---|---|

${\mathrm{E}}_{0}$ | delivered energy | (J) |

${\mathrm{E}}_{\mathrm{E}}$ | captured wind energy | (J) |

${\mathrm{E}}_{\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}}$ | kinetic energy | (J) |

$\mathrm{J}$ | total moment of inertia | ($\mathrm{kg}\times {\mathrm{m}}^{2}$) |

${\mathrm{P}}_{\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$ | inertial power | (W) |

${\mathrm{P}}_{\mathrm{P}\mathrm{M}\mathrm{S}\mathrm{G}}$ | PMSG’s power | (W) |

${\mathrm{P}}_{\mathrm{P}\mathrm{M}\mathrm{S}\mathrm{G}-\mathrm{a}\mathrm{v}\mathrm{g}}$ | PMSG’s average power | (W) |

${\mathrm{P}}_{\mathrm{P}\mathrm{M}\mathrm{S}\mathrm{G}\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}}$ | PMSG’s optimal power | (W) |

${\mathrm{P}}_{\mathrm{P}\mathrm{M}\mathrm{S}\mathrm{G}\_\mathrm{P}\mathrm{I}\_\mathrm{P}\mathrm{I}\mathrm{D}}$ | PMSG’s power obtained by using either the PI- or PID-type regulators | (W) |

${\mathrm{P}}_{\mathrm{P}\mathrm{M}\mathrm{S}\mathrm{G}\_\mathrm{P}\mathrm{I}}$ | PMSG’s power in the case of PI control | (W) |

${\mathrm{P}}_{\mathrm{P}\mathrm{M}\mathrm{S}\mathrm{G}\_\mathrm{P}\mathrm{I}\mathrm{D}}$ | PMSG’s power in the case of PID control | (W) |

${\mathrm{P}}_{\mathrm{W}\mathrm{T}}$ | wind turbine’s power | (W) |

${\mathrm{P}}_{\mathrm{W}\mathrm{T}\_\mathrm{a}\mathrm{v}\mathrm{g}}$ | wind turbine’s average power | (W) |

${\mathrm{P}}_{\mathrm{W}\mathrm{T}\mathrm{max}}$ | wind turbine’s maximum power | (W) |

${\mathrm{T}}_{\mathrm{P}\mathrm{M}\mathrm{S}\mathrm{G}}$ | PMSG’s torque | (Nm) |

${\mathrm{T}}_{\mathrm{W}\mathrm{T}}$ | wind turbine’s torque | (Nm) |

${\mathrm{S}}_{\mathrm{w}}$ | wind speed | (m/s) |

${\mathrm{S}}_{\mathrm{w}\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$ | measured wind speed | (m/s) |

$\mathrm{\omega}$ | PMSG’s angular speed | (rad/s) |

${\mathrm{\omega}}_{\mathrm{max}}$ | PMSG’s maximum angular speed | (rad/s) |

${\mathsf{\omega}}_{\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}}$ | PMSG’s optimal angular speed | (rad/s) |

${\mathsf{\omega}}_{\mathrm{k}}$ | the PMSG’s angular speed at time instant $\mathrm{k}$ | (rad/s) |

${\mathsf{\omega}}_{\mathrm{k}-1}$ | the PMSG’s angular speed at time instant $\mathrm{k}-1$ | (rad/s) |

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**Figure 1.**Time variations of the approximated wind speed, ${S}_{w}\left(t\right)$, of its derivative, $\frac{d{S}_{w}\left(t\right)}{dt}$, and of the measured the wind speed, ${S}_{wmeas}\left(t\right)$.

**Figure 4.**Time variation of the optimal PMSG‘s power curve, ${P}_{PMSGoptim}\left(t\right)$ for: (

**a**). ${k}_{d}=10000$, (

**b**). ${k}_{d}=1000$, (

**c**). ${k}_{d}=100$, (

**d**). ${k}_{d}=10$, (

**e**). ${k}_{d}=1$, (

**f**). ${k}_{d}=0.1$, (

**g**). ${k}_{d}=0.001$, (

**h**). ${k}_{d}=0.0001$.

**Figure 5.**Time variation of the three power curves: ${P}_{PMSGoptim}\left(t\right)$, ${P}_{PMSG\_PI\_PID}\left(t\right)$ and ${P}_{WTmax}$.

**Figure 6.**Time variation of the PMSG’s angular speed $\omega \left(t\right)$ and of the wind speed derivative $\frac{d{S}_{w}}{dt}\left(t\right)$.

t (s) | S_{wmeas}(t) (m/s) |
---|---|

0 | 6.24 |

30 | 6.25 |

60 | 6.26 |

90 | 6.27 |

120 | 6.28 |

150 | 6.29 |

180 | 6.3 |

210 | 6.31 |

240 | 6.32 |

270 | 6.32 |

300 | 6.32 |

330 | 6.315 |

360 | 6.31 |

390 | 6.29 |

420 | 6.27 |

450 | 6.255 |

480 | 6.24 |

510 | 6.225 |

540 | 6.21 |

570 | 6.14 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ancuti, M.-C.; Musuroi, S.; Sorandaru, C.; Dordescu, M.; Erdodi, G.M. Wind Turbines Optimal Operation at Time Variable Wind Speeds. *Appl. Sci.* **2020**, *10*, 4232.
https://doi.org/10.3390/app10124232

**AMA Style**

Ancuti M-C, Musuroi S, Sorandaru C, Dordescu M, Erdodi GM. Wind Turbines Optimal Operation at Time Variable Wind Speeds. *Applied Sciences*. 2020; 10(12):4232.
https://doi.org/10.3390/app10124232

**Chicago/Turabian Style**

Ancuti, Mihaela-Codruta, Sorin Musuroi, Ciprian Sorandaru, Marian Dordescu, and Geza Mihai Erdodi. 2020. "Wind Turbines Optimal Operation at Time Variable Wind Speeds" *Applied Sciences* 10, no. 12: 4232.
https://doi.org/10.3390/app10124232