# Research on Maximum Penetration Ratio of Wind Power under the Voltage Stability Margin Constraint

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation and Background

#### 1.2. Related Work

## 2. Mathematical Model of Wind Generator Grid Connection

_{SG}and Z

_{WG}are the equivalent impedances of conventional units and wind generators to the grid-connected node, respectively; Z

_{ST}represents the Thevenin equivalent impedance of the system viewed from the grid-connected point; P

_{L}and Q

_{L}are the active and reactive loads of the grid; the calculation formula of Z

_{SG}and Z

_{WG}is as follows (1):

_{SG}and E

_{WG}of the power supply at the same grid-connected point is ignored, and the modulus value is assumed to be 1, the equivalent impedance of the power supply at the grid-connected node can be obtained as:

## 3. Impedance Mode Margin and Thevenin Model

#### 3.1. IMMI

_{L}is the load impedance, Z

_{th}is the system Thevenin equivalent impedance at the load point. When $\eta $ = 1, it indicates that the voltage stability of the load point is strong; when $\eta $ = 0, it indicates that it is at the critical point of voltage stability.

_{L}is the parallel connection of the constant impedance load, the constant current load, and the constant power load equivalent impedance, namely, ${Z}_{L}={Z}_{L,0}//{Z}_{L,1}//{Z}_{L,2}$. According to Z = U/I and Z = U

^{2}/S, it is easy to know that Z

_{L}

_{,1}and Z

_{L}

_{,2}are the primary and quadratic functions of the node voltage, respectively. If the load power factor is assumed to be the same, the calculation formula of Z

_{L}is:

_{LN}is the load impedance at the rated voltage, and a

_{0}, a

_{1}, and a

_{2}are the ratios of the constant impedance, constant current, and constant power loads under the rated voltage, respectively, satisfying a

_{0}+a

_{1}+a

_{2}= 1.

#### 3.2. Thevenin Equivalent Model

_{th}|=|Z

_{L}| under the condition of voltage critical stability; that is, the Thevenin impedance voltage drop, U

_{th}, is equal to the load operating voltage, U

_{op}, at this time. When the Thevenin equivalent impedance angle and the load power factor are kept constant, U

_{th}and U

_{op}can form a circular trajectory of the voltage phasors, and if the load power factor changes, a semicircle cluster can be formed, as shown in Figure 3.

_{th}, and the load impedance angle, θ

_{l}; namely, $\theta =\pi -({\theta}_{th}-{\theta}_{l})$. It is not difficult to calculate the corresponding critical stable voltage, U

_{c}, as follows (5):

## 4. Calculation of Maximum Wind Power Permeability under Voltage Stability Margin Constraints

#### 4.1. Thevenin Critical Impedance

_{u}≥ 10–15% according to the reactive power voltage criterion, and the calculation formula of k

_{u}is shown in (6).

_{op}is the voltage under normal operation, and U

_{c}is the critical value of voltage stability.

_{op,min}, under the constraint of the voltage stability margin k

_{u}can be obtained as (7):

_{th}satisfies (8):

_{u}, can be obtained as follows (9):

_{op,min}, and U

_{th,max}represents the maximum voltage drop of Thevenin impedance, the calculation formula can be described as follows (10):

_{th,c}, is affected by many factors. When the load is a pure constant impedance type, the relationship between Z

_{th,c}and the voltage stability margin, k

_{u}, is shown in Figure 3b.

#### 4.2. Maximum Penetration Ratio of Wind Power

_{eq}+Z

_{ST}, so it is necessary to satisfy Z

_{eq}+Z

_{ST}≤Z

_{th,c}under the constraint of the voltage stability margin so that the corresponding maximum penetration can be obtained. The formula for calculating the rate, λ

_{c}, is:

_{ST}represents the connection between the grid connection point and the system; and Z

_{th,c}mainly reflects the influence of load, including load type, size and power factor.

## 5. Case Analysis

#### 5.1. Maximum Wind Power Penetration Calculation

_{0}, a

_{1}, and a

_{2}of constant impedance, constant current, and constant power load, respectively, are each taken as one third, and the grid-connected line impedance multiple γ, after the wind generator replaces the conventional one, is set to four, and the voltage stability margin k

_{u}is taken as 10%, as required. Using Formula (12), the maximum penetration rate of wind power is calculated to be 74.7%.

#### 5.2. Influencing Factors Analysis of Maximum Wind Power Penetration Rate

#### 5.2.1. Load Power Factor

#### 5.2.2. Load Type Ratio

_{0}, a

_{1}, and a

_{2}, in the power grid. The results are shown in Table 1.

#### 5.2.3. Grid-Connected Node

#### 5.2.4. Grid-Connected Line Impedance

#### 5.2.5. Generator Terminal Voltage

#### 5.2.6. Reactive Power Compensation Capacity

#### 5.2.7. Receiver Power Grid Strength

#### 5.2.8. Voltage Stability Margin

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Load Type Proportion | Maximum Wind Power Penetration λ_{c} | ||
---|---|---|---|

a_{0} | a_{1} | a_{2} | |

0.33 | 0.33 | 0.33 | 74.7% |

0.4 | 0.3 | 0.3 | 79.2% |

0.3 | 0.4 | 0.3 | 75.3% |

0.3 | 0.3 | 0.4 | 68.9% |

0.6 | 0.2 | 0.2 | 90.8% |

0.2 | 0.6 | 0.2 | 77.2% |

0.2 | 0.2 | 0.6 | 47.1% |

Node | Z_{ST} | Maximum Wind Power Penetration λ_{c} |
---|---|---|

4 | 0.170j | 74.7% |

5 | 0.153j | 81.4% |

6 | 0.157j | 79.9% |

7 | 0.049j | 100% |

8 | 0.075j | 98.4% |

PV Node | Maximum Wind Power Penetration λ_{c} |
---|---|

2, 3 | 74.7% |

2, 3, 8 | 78.8% |

2, 3, 5, 8 | 96.03% |

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**MDPI and ACS Style**

Lei, C.; Wu, Y.; Huang, Y.; Liang, Y.; Nie, J.; Tang, M.; Yi, X.; Luo, Y.
Research on Maximum Penetration Ratio of Wind Power under the Voltage Stability Margin Constraint. *Sustainability* **2022**, *14*, 7217.
https://doi.org/10.3390/su14127217

**AMA Style**

Lei C, Wu Y, Huang Y, Liang Y, Nie J, Tang M, Yi X, Luo Y.
Research on Maximum Penetration Ratio of Wind Power under the Voltage Stability Margin Constraint. *Sustainability*. 2022; 14(12):7217.
https://doi.org/10.3390/su14127217

**Chicago/Turabian Style**

Lei, Cheng, Yaning Wu, Yu Huang, Yu Liang, Jinfeng Nie, Muyi Tang, Xuancheng Yi, and Yi Luo.
2022. "Research on Maximum Penetration Ratio of Wind Power under the Voltage Stability Margin Constraint" *Sustainability* 14, no. 12: 7217.
https://doi.org/10.3390/su14127217