# A Time-Frequency Analysis Method for Low Frequency Oscillation Signals Using Resonance-Based Sparse Signal Decomposition and a Frequency Slice Wavelet Transform

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## Abstract

**:**

## 1. Introduction

## 2. Resonance-Based Sparse Signal Decomposition

#### 2.1. Q factor, Damping and LFO’s Resonance

#### 2.1.1. Signal Resonance and Q-Factor

_{c}is the resonant frequency, B

_{w}is the half-power bandwidth. Figure 1 illustrates the concept of signal resonance. Pulses 1 and 3, essentially a single cycle in duration, are low-resonance pulses because they do not exhibit sustained oscillatory behavior. Pulses 2 and 4, whose oscillations are more sustained, are high-resonance pulses. A low Q-factor wavelet transform is suitable for the efficient representation of pulses 1 and 3. The efficient representation of Pulses 2 and 4 calls for a wavelet transform with higher Q-factor.

#### 2.1.2. LFO’s Resonance

#### 2.2. Tunable Q-Factor Wavelet Transform

_{0}(n) has a sampling rate of αf

_{s}where f

_{s}is the sampling rate of the input signal x(n). Likewise, the sub-band signal v

_{1}(n) has a sampling rate of βf

_{s}. LPS and HPS represent low-pass scaling and high-pass scaling, respectively.

_{0}(ω) and H

_{1}(ω) are given by [12]:

#### 2.3. Morphological Component Analysis

_{1}+ x

_{2}+ n, with x, x

_{1}, x

_{2}∈ R

^{N}

_{1}and x

_{2}individually. Assuming that x

_{1}and x

_{2}can could be sparsely represented in bases (or frames) Φ

_{1}and Φ

_{2}respectively, they can be estimated by minimizing the objective function:

_{1}and w

_{2}, subject to the constraint:

## 3. Frequency Slice Wavelet Transform

#### 3.1. The Definition of Frequency Slice Wavelet Transform

_{s}is frequency resolution ratio. Δω

_{p}is the width of frequency window of FSF. As stated above in Equation (11), FSWT has another important property: FSWT can be controlled by the frequency resolution ratio η

_{s}of the measured signal.

#### 3.2. Frequency Slice Wavelet Inverse Transform

_{1}, t

_{2}, ω

_{1}, ω

_{2}) is as below[14]:

## 4. Procedure of the LFO Time-Frequency Analysis Method Using RSSD and FSWT

- (1)
- Decompose the LFO signal with noise by RSSD and get a high-resonance component, a low-resonance component and a residual. The high-resonance component is de-noised LFO signal.
- (2)
- Transform the high-resonance component by FSWT, and compute the FSWT coefficients using Equation (9), and obtain the full band of its time-frequency distribution. The 3D map expression and dominant mode of the LFO signal can be obtained. After that, search for the regions of interest, called modal domains, in which the main energy is concentrated. Chose a frequency slice [ω
_{i}, ω_{i}+1] to get an accurate analysis of the time-frequency feature. Separate and extract the LFO mode components through reconstructing signals by inverse FSWT. - (3)

## 5. Simulation

#### 5.1. Denoised RSSD

#### 5.1.1. Example 1

_{i}is the amplitude, f

_{i}is the oscillation frequency, σ

_{i}is damping factor, Φi is the phase shift, λ(t) is white noise and σ its standard deviation. We set a f

_{s}= 1000 Hz, T

_{s}= 10 s and σ = 0.1. A signal simulated with Equation (15) is described in Table 1. There are two oscillation modes of the simulated signal. One is an inter-area oscillation mode in low frequency and the other is an intra-area oscillation mode in high frequency.

_{1}=1, Q

_{2}= 4 and γ

_{1}= γ

_{2}= 3.

#### 5.1.2. Example 2

#### 5.2. FSWT Analysis

#### 5.2.1. Full Band Time-Frequency Distribution Analysis by FSWT

#### 5.2.2. Fine Analysis of Frequency Slices Based on FSWT

#### 5.3. Identification of the Parameters of LFO Mode Components by HT

#### 5.4. Impact of Noise on the Accuracy of FSWT-HT

#### 5.5. Comparison with Low Pass Filtering

#### 5.6. Comparison with Other Parameter Identification Methods

## 6. Engineering Application

_{1}=1, Q

_{2}= 4, ${\mathsf{\gamma}}_{1}={\mathsf{\gamma}}_{2}=3$.

## 7. Concluding Remarks

- (1)
- RSSD can remove most noise of LFO signals which improved accuracy of LFO modal parameter identification.
- (2)
- FSWT enables the shape of characteristics of LFO modal signal in time-frequency domain to be clearly visible.
- (3)
- RSSD-FSWT can analyze the LFO signal from multiple aspects. Due to full band time-frequency distribution analysis by FSWT, the dominant mode of LFO can be determined, and a 3D map expression of the LFO signal can be obtained. According to fine analysis of frequency slices based on FSWT, we can separate and extract LFO’s modal components. Combining with the Hilbert transform, the parameters of the LFO mode components could be identified accurately.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**The resonance property of the LFO signal. (

**a**) Power angle oscillation waveform of G4 relative to G1; (

**b**) FFT frequency spectrum.

**Figure 7.**Full band time-frequency distribution analysis by FSWT. (

**a**) Fourier spectrum; (

**b**) The 2D maps of FSWT coefficients; (

**c**) The 3D maps of FSWT coefficients.

**Figure 8.**Fine analysis of frequency slice [0, 0.8] Hz based on FSWT. (

**a**) FSWT 2D map; (

**b**) FSWT 3D map; (

**c**) Comparison of the reconstructed signal with the ideal signal.

**Figure 9.**Fine analysis of frequency slice [0.8, 2] based on FSWT. (

**a**) FSWT 2D map; (

**b**) FSWT 3D map; (

**c**) Comparison of the reconstructed signal with the ideal signal.

**Figure 10.**Full band time-frequency distribution and Fine analysis of frequency slice based on FSWT. (

**a**) FSWT 2D map (σ = 0.5); (

**b**) Reconstructed signal of [0, 0.8] Hz; (

**c**) FSWT 2D map (σ = 1); (

**d**) Reconstructed signal of [0, 0.8] Hz; (

**e**) FSWT 2D map (σ = 1.5); (

**f**) Reconstructed signal of [0, 0.8] Hz; (

**g**) FSWT 2D map (σ = 2); (

**h**) Reconstructed signal of [0, 0.8] Hz.

**Figure 13.**Fine analysis of the simulated signal with colored noise based on FSWT. (

**a**) Frequency slice [0, 0.8] Hz; (

**b**) Frequency slice [0, 0.8] Hz.

**Figure 17.**Full band time-frequency distribution analysis by FSWT. (

**a**) FFT spectrum; (

**b**) FSWT 3D map.

**Figure 18.**Fine analysis of frequency slices based on FSWT. (

**a**) Reconstructed signal of [0, 0.2] Hz; (

**b**) Reconstructed signal of [0.2, 0.5] Hz; (

**c**) Reconstructed signal of [0.8, 2] Hz; (

**d**) Reconstructed signal = (

**a**) + (

**b**) + (

**c**).

LFO Mode | Amplitude (A_{i}) | Oscillation Frequency (f_{i}) | Damping Ratio (ξ_{i}) | Damping Factor (σ_{i}) | Phase Shift (Φ_{i}) |
---|---|---|---|---|---|

1 | 1 | 0.5 | 0.02 | 0.0628 | 60° |

2 | 0.5 | 1.1 | 0.07 | 0.4837 | 45° |

Mode | True Value | FSWT-HT | ||
---|---|---|---|---|

Identification Result | Error (%) | |||

1 | Oscillation frequency (Hz) | 0.5000 | 0.4980 | 0.40 |

Damping ratio | 0.0200 | 0.0196 | 2.00 | |

2 | Oscillation frequency (Hz) | 1.1000 | 1.1101 | 0.92 |

Damping ratio | 0.0700 | 0.0666 | 4.86 |

Mode | Standard Deviation (σ) | Oscillation Frequency (Hz) | Error (%) | Damping Ratio | Error (%) |
---|---|---|---|---|---|

1 | 0.1 | 0.4980 | 0.400 | 0.0188 | 6.000 |

2 | 1.1101 | 0.92 | 0.0657 | 6.143 | |

1 | 0.5 | 0.4978 | 0.440 | 0.0190 | 5.000 |

2 | 1.1031 | 0.282 | 0.0637 | 9.000 | |

1 | 1 | 0.4968 | 0.640 | 0.0181 | 9.500 |

2 | 1.1131 | 1.20 | 0.0613 | 12.429 | |

1 | 1.5 | 0.4975 | 0.500 | 0.0171 | 14.500 |

2 | 1.1156 | 1.418 | 0.0571 | 18.428 | |

1 | 2 | 0.4977 | 0.460 | 0.0170 | 15.000 |

2 | 1.1221 | 2.009 | 0.0470 | 32.857 |

Method | Mode | Oscillation Frequency (Hz) | Error (%) | Damping Ratio | Error (%) |
---|---|---|---|---|---|

Prony | 1 | 0.4502 | 9.960 | 0.0240 | 20.00 |

2 | 1.1616 | 5.609 | 0.0580 | 17.14 | |

SSI | 1 | 0.4766 | 4.680 | 0.0222 | 11.00 |

2 | 1.1573 | 5.209 | 0.0753 | 7.571 | |

EMD-SSI | 1 | 0.4860 | 2.800 | 0.0214 | 7.000 |

2 | 1.1502 | 4.727 | 0.0743 | 6.142 | |

RSSD-SSI | 1 | 0.5060 | 1.200 | 0.0206 | 3.000 |

2 | 1.1302 | 2.745 | 0.0730 | 2.857 | |

RSSD-FSWT | 1 | 0.4978 | 0.440 | 0.0192 | 4.000 |

2 | 1.1030 | 0.272 | 0.0667 | 4.714 |

Mode | Oscillation Frequency (Hz) | Damping Ratio | |
---|---|---|---|

1 | RSSD-SSI | 0.5356 | 0.0556 |

RSSD-FSWT | 0.5244 | 0.0542 | |

EMD-SSI | 05354 | 0.0571 | |

2 | RSSD-SSI | 0.2510 | 0.0541 |

RSSD-FSWT | 0.2508 | 0.0538 | |

EMD-SSI | 0.2658 | 0.0562 | |

3 | RSSD-SSI | 0.1220 | 0.0149 |

RSSD-FSWT | 0.1215 | 0.0145 | |

EMD-SSI | 0.1199 | 0.0123 | |

4 | RSSD-SSI | - | - |

RSSD-FSWT | - | - | |

EMD-SSI | 0.0411 | 0.0993 |

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**MDPI and ACS Style**

Zhao, Y.; Li, Z.; Nie, Y.
A Time-Frequency Analysis Method for Low Frequency Oscillation Signals Using Resonance-Based Sparse Signal Decomposition and a Frequency Slice Wavelet Transform. *Energies* **2016**, *9*, 151.
https://doi.org/10.3390/en9030151

**AMA Style**

Zhao Y, Li Z, Nie Y.
A Time-Frequency Analysis Method for Low Frequency Oscillation Signals Using Resonance-Based Sparse Signal Decomposition and a Frequency Slice Wavelet Transform. *Energies*. 2016; 9(3):151.
https://doi.org/10.3390/en9030151

**Chicago/Turabian Style**

Zhao, Yan, Zhimin Li, and Yonghui Nie.
2016. "A Time-Frequency Analysis Method for Low Frequency Oscillation Signals Using Resonance-Based Sparse Signal Decomposition and a Frequency Slice Wavelet Transform" *Energies* 9, no. 3: 151.
https://doi.org/10.3390/en9030151