# Research on Frequency Response Modeling and Frequency Modulation Parameters of the Power System Highly Penetrated by Wind Power

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background and Literature Review

#### 1.1.1. Background

#### 1.1.2. The Impact of Strategies of DFIG on Frequency Response

#### 1.1.3. The Research on System Frequency Response Model

_{max}), the lowest point of system dynamic frequency, and the maximum deviation of system steady-state frequency [21,22].

#### 1.2. Research Gap and Motivation

#### 1.3. Contribution and Organization

#### 1.3.1. Contribution of This Paper

- The frequency response model of the wind power highly penetrated system proposed this paper can fit the actual system better than TSFR.
- The calculation formulas of three indexes of system frequency stability are derived based on this model.
- The different effects of virtual inertia and virtual droop control strategies on the frequency response of the system are analyzed.
- The parameter setting of DFIG participating in frequency modulation is calculated according to the frequency stability requirements of the actual system.

#### 1.3.2. Organization of This Paper

## 2. Frequency Response Model of the Wind Power Highly Penetrated System

#### 2.1. Frequency Response Model of Traditional Power System

_{1}(s) of the power system and the frequency response function G

_{2}(s) of the engine load:

#### 2.2. Wind Power System Modeling

_{p}(λ,β) determines the efficiency of wind energy captured by turbines [13]. Affected by the control mode and ambient wind speed, C

_{p}(λ,β) is in dynamic change:

_{m}is the power captured by wind turbines; ρ is the air density; V is the upwind speed of the wind turbine rotor; S is the area of wind turbine blades; λ is the tip speed ratio; β is the pitch angle.

_{m}caused by the change of wind speed ΔV and rotor frequency Δw

_{r}, the function fitting method can be used to reduce the amount of calculation:

_{m}due to the evolution of rotor frequency Δw and wind speed ΔV is as follows:

_{e}, and the rotor outer loop control is diverse. The multi-loop control strategy leads to the decoupling of the rotor frequency and grid frequency. However, the variation of DFIG input and output power is provided by the interpretation of rotor kinetic energy:

_{DFIG}is the rotor inertia of DFIG.

_{r}occurs, the calculation method of the DFIG inertia time constant can be obtained:

_{r}, so the change of DFIG root speed will affect ΔP

_{M}

_{PPT}:

_{opt}is the maximum power tracking coefficient of MPPT.

_{1}is the gain of virtual droop control and A

_{2}is the gain of virtual inertia control.

_{wind}can reflects the change of system frequency by the virtual control strategies and wind speed by the MPPT strategy. The wind power permeability is adjusted through K. The calculation of system parameters is the same as that of TSFR [18], and the whole wind farm is equivalent to a DFIG.

#### 2.3. Model Frequency Response Analysis

_{MPPT}due to Δw

_{r}and ΔV is ignored, the equivalent powertrain speed regulation function ${G}_{1}{}^{\prime}(s)$ and the equivalent engine load frequency response function ${G}_{2}{}^{\prime}(s)$ are defined according to Figure 2 as follows:

_{max}), the lowest point of dynamic frequency and the maximum deviation of steady-state frequency can be effectively improved by reasonably setting the frequency response parameters of wind power. Thus, the frequency deterioration caused by the increase of wind power permeability can be improved. At the same time, it shows that the virtual inertial control can effectively improve the frequency change rate, and the virtual droop control can effectively improve the dynamic frequency lowest point and steady-state frequency deviation of the system.

## 3. Frequency Stability Analysis of the Wind Power Highly Penetrated System

#### 3.1. Steady-State Performance Analysis of Closed-Loop System

#### 3.2. Analysis of the Maximum Rate of Change of Frequency of System Dynamic Frequency

_{max}of the system occurs at the initial stage of the fault. At this time, the system frequency deviation Δw is slight and approaches 0.

_{max}of the dynamic frequency response of the system can be obtained by the Laplace initial value theorem transformation calculation of formula (19):

_{max}is, the stronger the system resistance to disturbance is. With the increasing permeability of the wind power, RoCoF

_{max}will depend on the size of the virtual inertia control parameter A

_{2}.

_{1}of system transient time-domain frequency change rate, the value of virtual inertia control parameter A

_{2}can be defined as follows:

#### 3.3. Analysis of Lowest Point of System Dynamic Frequency

_{n}is related to A

_{1}and A

_{2}. The relationship between A

_{1}and A

_{2}can be determined by determining the maximum frequency deviation threshold η

_{2}in the transient time domain of the system.

#### 3.4. Analysis of Steady-State Frequency Deviation of the System

_{1}and A

_{2}parameter settings on the system’s frequency response in case of a power disturbance.

_{3}and the wind power permeability K of the system are determined, the size of the virtual droop control parameter A

_{1}can be determined.

## 4. Case Study

#### 4.1. Simulation Model Construction

#### 4.1.1. Small-Scale System

_{1}according to different simulation requirements.

#### 4.1.2. 39-Bus System

#### 4.2. Case 1 Simulation Analysis

#### 4.2.1. Scenario 1

_{1}= A

_{2}= 0, and sets the system load L

_{1}to generate 6% power step disturbance at 10 s.

#### 4.2.2. Scenario 2

_{1}to generate 10% power step disturbance at 10 s.

#### 4.3. Case 2 Simulation Analysis

_{1}to generate 10% power step disturbance at 10 s.

_{1}= ±0.5 Hz/s. With reference to the low-frequency load shedding action conditions, this section sets the maximum acceptable dynamic frequency deviation of the system as η

_{2}= 0.5 Hz. The maximum permissible frequency deviation of the power system is η

_{3}= ±0.2 Hz.

_{1}> 0. In all control mode combinations, the system’s steady-state frequency deviation requirements are met, which verifies the previous theoretical calculation.

_{2}> 2.13, the system meets the maximum frequency change rate requirements. However, as shown in Figure 12, when A

_{2}> 2.5, the maximum frequency change rate of the system meets the requirements; when A

_{2}< 1.5, the maximum frequency change rate of the system is greater than 0.5 Hz/s. It shows that the calculation formula proposed in Section 3.2 has errors within the allowable range and verifies the correctness of the theoretical calculation.

#### 4.4. Case 3 Simulation Analysis

## 5. Conclusions and Research Prospect

#### 5.1. Conclusions

#### 5.2. Research Prospects

- Establishing DFIG frequency response models under different working states (the focus of the next work).
- Simplifying the calculation of the system frequency stable indexes when the SFR model considers the complex frequency response control strategies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Classification | Parameters | Values |
---|---|---|

Synchronous machine (G1, G2, G3) | Nominal power | 100 MW |

Line-to-line voltage | 13.8 kV | |

frequency | 50 Hz | |

Reactance X_{d} | 1.3125 | |

Reactance X_{d}′ | 0.1813 pu | |

Reactance X_{d}″ | 0.107 pu | |

Reactance X_{q} | 1.2578 pu | |

Reactance X_{q}′ | 0.4 pu | |

Reactance X_{q}″ | 0.107 pu | |

Reactance Xl | 0.0742 pu | |

Inertial time constant H | 6.4 s | |

Transformer (T1, T2, T3) | Nominal power | 100 MW |

Frequency | 50 Hz | |

Winding 1 V1 (Ph-Ph) | 13.8 kV | |

Winding 1 R1 | 0.002 pu | |

Winding 1 L1 | 0 | |

Winding 2 V2 (Ph-Ph) | 230 kV | |

Winding 2 R2 | 0.002 pu | |

Winding 2 L2 | 0.0586 | |

Magnetization resistance | 500 pu | |

Magnetization resistance | 500 pu | |

Transformer (T4) | Nominal power | 100 MW |

Frequency | 50 Hz | |

Winding 1 V1 (Ph-Ph) | 575 V | |

Winding 1 R1 | 0.002 pu | |

Winding 1 L1 | 0 | |

Winding 2 V2 (Ph-Ph) | 230 kV | |

Winding 2 R2 | 0.002 pu | |

Winding 2 L2 | 0.0586 | |

Magnetization resistance | 500 pu | |

Magnetization resistance | 500 pu | |

DFIG | Nominal power | 1.5/0.9 MW |

Line-to-line voltage | 575 V | |

frequency | 50 Hz | |

Stator R_{s} | 0.00706 pu | |

Stator L1s | 0.171 pu | |

Rotor R_{r}′ | 0.005 pu | |

Rotor L1_{r}′ | 0.156 pu | |

Magnetizing inductance L_{m} | 2.9 pu | |

Inertia constant H(s) | 5.04 s | |

Power at point C | 0.73 pu | |

Wind speed at point C | 12 m/s | |

Power regulatorgains[k_{p} k_{i}] | [1 100] | |

DC bus voltage regulator gains[k_{p} k_{i}] | [0.002 0.05] | |

Grid-side converter current regulator gains[k _{p} k_{i}] | [1 100] | |

Rotor-side converter current regulator gains[k _{p} k_{i}] | [0.3 8] |

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Parameter Name | Parameter Value |
---|---|

$D$ | 2 |

$T$ | 12 s |

$a$ | 0.0812 |

$M$ | 12 s |

$\mathrm{R}$ | 0.0124 |

Parameter Name | Parameter Value |
---|---|

$D$ | 0.3 |

$T$ | 8 s |

$a$ | 0.3 |

$M$ | 4 s |

$\mathrm{R}$ | 0.05 |

Frequency Lowest Point Error | Steady-State Frequency Error | ||
---|---|---|---|

Scenario 1 | Model 1 | 0.03 Hz | 0.02 Hz |

Model 2 | 0.01 Hz | 0 | |

Scenario 2 | Model 1 | 0.05 Hz | 0.07 Hz |

Model 2 | 0.01 Hz | 0 |

Frequency Lowest Point Error | Steady-State Frequency Error | ||
---|---|---|---|

Case 3 | Model 1 | 0.04 Hz | 0.05 Hz |

Model 2 | 0.01 Hz | 0 |

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**MDPI and ACS Style**

Qi, J.; Tang, F.; Xie, J.; Li, X.; Wei, X.; Liu, Z.
Research on Frequency Response Modeling and Frequency Modulation Parameters of the Power System Highly Penetrated by Wind Power. *Sustainability* **2022**, *14*, 7798.
https://doi.org/10.3390/su14137798

**AMA Style**

Qi J, Tang F, Xie J, Li X, Wei X, Liu Z.
Research on Frequency Response Modeling and Frequency Modulation Parameters of the Power System Highly Penetrated by Wind Power. *Sustainability*. 2022; 14(13):7798.
https://doi.org/10.3390/su14137798

**Chicago/Turabian Style**

Qi, Junfeng, Fei Tang, Jiarui Xie, Xinang Li, Xiaoqing Wei, and Zhuo Liu.
2022. "Research on Frequency Response Modeling and Frequency Modulation Parameters of the Power System Highly Penetrated by Wind Power" *Sustainability* 14, no. 13: 7798.
https://doi.org/10.3390/su14137798