# Adaptive Band-Pass Filter and VMD-Esprit Based Multi-Mode Monitoring Method for Broadband Electromagnetic Oscillation in “Double High” Power Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Adaptive Band-Pass Filter and VMD-Esprit Based Multi-Modal Monitoring Method

_{0}, amplitude A

_{0}) is calculated, and the mode with amplitude A

_{0i}(i ≤ n represents the ith mode, n represents the number of selected modes) that exceeds the set threshold A

_{set}(where A

_{set}can be set according to the actual operation of the l grid) is selected.

_{set}), the band is not filtered.

#### 2.2. Key Technical Methods

#### 2.2.1. Designations of Line Frequency Notch

_{NF}, and notch factor is ξ

_{NF}, respectively. Two factors of notch effect and dynamic are considered in the parameter design of the line frequency notch filter. The smaller the notch factor, the better the notch characteristics of the line frequency notch filter, but a smaller notch factor also leads to a decrease in frequency adaptability. According to Equation (1), when the notch factor is greater than 1.0, the characteristic equation has two unequal real poles on the negative real axis of the s-plane, which is in the overdamped state. When the notch factor is equal to 1.0, the characteristic equation has two equal real poles on the negative real axis of the s plane, which is in the critical damping condition. When Q is less than 1.0, the conjugate complex pole of the characteristic equation in the left half of the s plane is underdamped. Considering the notch characteristics and dynamic characteristics, the parameter design f

_{NF}is 50 Hz, and the notch factor ξ

_{NF}is set to 1.0.

#### 2.2.2. Adaptive Band-Pass Filter (ABPF)

_{L}and f

_{H}are the two frequency cut-off points of ABPF, and f

_{C}is the center frequency. In this paper, digital filters are used, such as infinite impulse response (IIR) filters or finite impulse response (FIR) filters. The specific design method can be referred to in the references [25]. The ABPF expression used in this paper is as follows:

_{s}is the sampling frequency, and f

_{0}is the center frequency.

_{0}is the frequency of the signal, and the parameter b is set to a smaller value (the smaller the b is, the narrower the bandwidth is, which can be set according to the actual demand) to ensure the accuracy of this frequency signal filtering. In the next VMD decomposition of the filtered signal, it directly enters the Esprit parameter identification link. Only one signal does not need VMD decomposition because it itself is the desired modal component. This step of flexible design can reduce data processing time and data processing resources.

_{L}and f

_{H}. The two parameters are determined by the two signals with the largest and smallest frequencies in the frequency band (with corresponding margins), adaptive setting of the center frequency (take the mean of the two), bandwidth, and other parameters. When there are multiple signals in the frequency band, the filter provides a more accurate signal decomposition for VMD, reducing the amount of calculation and improving the calculation speed.

#### 2.2.3. Variational Mode Decomposition (VMD)

_{k}and ω

_{k}are the kth IMF component and its center frequency, respectively.

_{t}is the gradient calculation and * is the convolution calculation symbol.

_{k}and k are as follows:

#### 2.2.4. Esprit Parameter Identification

_{0}, x

_{1}, …, x

_{N−1}, as shown in Equation (11) [44]:

_{n}are obtained by singular value decomposition of X:

_{1}, V

_{2}. There exists a unique invertible transformation matrix T such that V

_{1}= V

_{2T}. Let U be the original signal. Similarly, the original signal subspace U

_{1}, U

_{2}are obtained, which satisfies:

_{1}and U

_{2}are known quantities obtained by data matrix decomposition, so the matrix shown in Equation (14) can be constructed to estimate the signal.

_{p}, the frequency ω

_{p}and attenuation coefficient ω

_{p}of each periodic component in the signal can be estimated according to Equation (15).

## 3. Results

#### 3.1. Validation and Analysis Based on the Construct Signal

_{1}= 5 V; A

_{2}= 5 V; A

_{3}= 5 V; A

_{4}= 5 V; A

_{5}= 5 V; A

_{6}= 0.16; f

_{1}= 1.5; f

_{2}= 25; f

_{3}= 50; f

_{4}= 300; f

_{5}= 400; φ

_{1}= Π/3; φ

_{2}= Π/4; φ

_{3}= Π/2; φ

_{4}= Π/3; φ

_{5}= Π/3.

#### 3.2. Broadband Osillation in Weak Grid with Two VSCs

_{12}= 0.001 Ω, R

_{13}= 0.001 Ω, R

_{14}= 0.001 Ω, L

_{12}= 0.00029 H, L

_{12}= 0.00029 H, L

_{14}= 0.00001 H.

_{12a}is subjected to FFT analysis in the modal detection link. The spectrum is shown in Figure 12:

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Comparison of the signal after the line frequency notch and the signal without the line frequency notch.

**Figure 10.**300, 400 Hz VMD mode decomposition waveform and their original signal waveform comparison chart.

Adaptive Band-Pass Filter | Oscillation Type | Sampling Time | Sampling Frequency | Oscillation Frequency Band |
---|---|---|---|---|

ABPF-1 | Low-frequency oscillation | 10 s | 100 | 0~2.5 Hz |

ABPF-2 | subsynchronous oscillation | 2 s | 1000 | 2.5~45 Hz |

ABPF-3 | supersynchronous oscillation | 2 s | 1000 | 55~95 Hz |

ABPF-4 | wide-range-frequency oscillations | 0.1 s | 10,000 | ≥100 Hz |

Adaptive Band-Pass Filter | Parameter | Theoretical Value | Identification Value | Relative Error/% |
---|---|---|---|---|

Mode 1 | amplitude | 5 V | 4.9344 V | 1.312 |

frequency | 1.5 Hz | 1.5000 Hz | 0.000 | |

phase angle | Π/3 | 1.0524 | 0.506 | |

Mode 2 | amplitude | 5 V | 4.9006 V | 1.988 |

frequency | 25 Hz | 25.0000 Hz | 0.000 | |

phase angle | Π/4 | 0.7283 | 7.277 | |

Mode 3 | amplitude | 5 V | 4.8862 V | 2.228 |

frequency | 300 Hz | 300.0000 Hz | 0.000 | |

phase angle | Π/3 | 0.9587 | 8.442 | |

Mode 4 | amplitude | 5 V | 4.8862 V | 2.228 |

frequency | 400 Hz | 400.0000 Hz | 0.000 | |

phase angle | Π/3 | 0.9059 | 13.484 |

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**MDPI and ACS Style**

Zhong, T.; Yang, H.; Sun, C.; Liu, C.; Chen, J. Adaptive Band-Pass Filter and VMD-Esprit Based Multi-Mode Monitoring Method for Broadband Electromagnetic Oscillation in “Double High” Power Systems. *Energies* **2023**, *16*, 3110.
https://doi.org/10.3390/en16073110

**AMA Style**

Zhong T, Yang H, Sun C, Liu C, Chen J. Adaptive Band-Pass Filter and VMD-Esprit Based Multi-Mode Monitoring Method for Broadband Electromagnetic Oscillation in “Double High” Power Systems. *Energies*. 2023; 16(7):3110.
https://doi.org/10.3390/en16073110

**Chicago/Turabian Style**

Zhong, Tie, Heling Yang, Cong Sun, Chuang Liu, and Junrui Chen. 2023. "Adaptive Band-Pass Filter and VMD-Esprit Based Multi-Mode Monitoring Method for Broadband Electromagnetic Oscillation in “Double High” Power Systems" *Energies* 16, no. 7: 3110.
https://doi.org/10.3390/en16073110