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A Hybrid Hilbert-Huang Method for Monitoring Distorted Time-Varying Waveforms^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Hilbert—Huang Method. A Timeline

## 3. Hybrid Hilbert–Huang

#### 3.1. Versatile DFT and Masking Signals to Improve EMD

- (h1)
- Perform DFT over the $i\left[p\right]$ (corresponding to $i\left(t\right)$ during ${T}_{w}$) to estimate the frequencies of its components ${f}_{1},{f}_{2},{f}_{3},\dots \dots ,{f}_{n}$ and corresponding amplitudes ${A}_{1},{A}_{2},{A}_{3},\dots \dots ,{A}_{n}$, where ${f}_{1}$ is the fundamental and ${f}_{1}<{f}_{2}{f}_{3}\dots \dots {f}_{n}$. The resulting spectrum is corresponding to the image of a model signal representing the stationary equivalent of the original signal over the ${T}_{w}$ window.
- (h2)
- Amplitudes threshold selection ${A}_{imp}$ for retaining the meaningful frequency components. At this stage and using this selection tools, only the components possessing more than ${\delta}_{e}$ percentage from the total signal energy are kept and processed. The contribution here stands for this flexible new tool capable of selecting only the meaningful components out of the DFT spectrum associated with the stationary modal equivalent of the original signal. The value of ${\delta}_{e}$ was set to 1% for the components’ energy to be a good compromise for the associated uncertainty of the overall chain of measurements in the distribution grids (instrument transformers, smart meters, Intelligent Electronic Devices, etc.).
- (h3)
- Based on the component analysis and the retained frequencies ${f}_{k}$ and their associated amplitudes, corresponding masking signals are created using the following formula:$$mas{k}_{k}\left[p\right]={M}_{k}\xb7\mathrm{sin}(2\pi \left({f}_{k}+{f}_{k-1}\right)\xb7p$$

#### 3.2. Enhanced EMD with Masking Signals

- (a1)
- Identification of local extrema (minima and maxima) for the digitized distorted signal $x\left[p\right]$.
- (a2)
- Perform cubic spline interpolation for the maxima and the minima to obtain the envelopes ${e}_{max}\left(p\right)$ and ${e}_{min}\left(p\right)$.
- (a3)
- With the new envelopes, the mean of the two is computed as:$$m\left[p\right]=\frac{{e}_{max}\left[p\right]+{e}_{min}\left[p\right]}{2}$$
- (a4)
- The previous computed mean is subtracted from the original signal as$${s}_{1}\left[p\right]=x\left[p\right]-m\left[p\right]$$
- (a5)
- ${s}_{1}\left[p\right]$ is considered an IMF if its number of local extrema is equal of differs at most with one from the number of zero crossings, and the average of ${s}_{1}\left[p\right]$ is zero. If those conditions are not met, then ${s}_{1}\left[p\right]$ is not an IMF, and the steps from b1 to b4 will be subsequently repeated using ${s}_{1}\left[p\right]$ instead of $x\left[p\right]$, until the new ${s}_{k}\left[p\right]$ satisfies the conditions of an IMF.
- (a6)
- A residue is defined as$${r}_{1}\left[p\right]=x\left[p\right]-{s}_{1}\left[p\right]$$
- (a7)
- If this residue does not satisfy the condition of being below a threshold tolerance of error, then the method will repeat the steps from b1 to b6 for ${r}_{1}\left[p\right]$ to compute the next IMF and a new residue.

- (h4)
- The next step in the overall framework of the hybrid Hilbert–Huang is performing EMD over two new signals based on the original digitized signal and the masking signals from step “(h3)”, resulting in their IMFs, based on the algorithm described from (a1) to (a7).$$i{M}_{1}\left[p\right]=i\left[p\right]+mas{k}_{n}\left[p\right]$$$$i{M}_{2}\left[p\right]=i\left[p\right]-mas{k}_{n}\left[p\right]$$The computed IMFs are stored as $IM{F}_{a}\left[p\right]$ and $IM{F}_{b}\left[p\right]$. It can be then constructed:$${s}_{1}\left[p\right]=\frac{IM{F}_{a}\left[p\right]+IM{F}_{b}\left[p\right]}{2}$$
- (h5)
- ${s}_{1}\left[p\right]$ is named as the first IMF subtracted from the original digitized signal $i\left[p\right]$. For the calculated residue as for the formula in (9), the first sift stage will be applied and the first component of the signal will be identified.$${r}_{1}\left[p\right]=i\left[p\right]-{s}_{1}\left[p\right]$$
- (h6)
- The sequence (h4)–(h5) will be iteratively followed using the residues replacing the original $i\left[p\right]$ signal, until (N-1) IMFs associated with the mono-component described by the ${f}_{n},{f}_{n-1},\dots \dots ,{f}_{3},{f}_{2}$ will be computed.

#### 3.3. Post—Processing Method

_{b}= 20 ms. This is related to achieving an “acceptable” resolution in time and frequency.

## 4. The Ability of the Method to Separate Components

## 5. Demonstration

_{s}= 50 kHz. The comparison with 10 kHz sampling frequency was made because of the computational burden associated with the EMD, and Hilbert Transform applied over non-stationary waveforms linearly decreases with a factor k, where f

_{s}* = f

_{s}/k. As acknowledged in [19], the computational resources associated with the method are significant, thus there was the proposition of using the method over smaller data windows, in the 1s range, and usual PQ sampling frequencies (10 kHz).

_{s}and T

_{b}. The final identification is not based on the Hilbert-Huang Transform but on the DFT applied over the identified steady-state time intervals. The performances of the HHT method were previously extensively studied [10,11,12,13,14,15,16,17,18,19,20,21,23,24,25,26,27,28,29,30,31] and compared with other time-frequency analysis tools such as wavelet transform, S-transform, short-time Fourier Transform. This paper uses HHT as starting point in the analysis of non-stationary, non-linear power signals for in light of the intended application: Identification of quasi-steady-state intervals using frequency domain information After the QSSI was performed over the results of Hilbert Spectrum over the entire $\left({T}_{w}\right)$ under study, DFT will be computed over the Hanning enhanced time windows, such as in Figure 8. The results in Figure 7a show good similarity with those provided by the PQ analyzer (f

_{s}= 50 kHz) in Figure 10 above.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Framework for decomposing distorted signal into high-frequency components using Hilbert–Huang method (adapted from [21]).

**Figure 3.**(

**a**) Original digitized signal described in Table 1, (

**b**) Discrete Fourier Transform (DFT) spectrum of the original signal-range 100 Hz to 550 Hz.

**Figure 4.**(

**a**) First quasi-steady-state interval enhanced with a Hanning window, (

**b**) second interval enhanced with a Hanning window, (

**c**) last interval enhanced with a Hanning window.

**Figure 5.**(

**a**) Instantaneous frequencies and (

**b**) amplitudes, of the mono-components extracted using the hybrid Hilbert-Huang method, of the synthetic signal described in Table 1.

**Figure 6.**(

**a**) Original digitized current signal of a microwave oven, (

**b**) DFT spectrum of the original signal-range 100 Hz to 500 Hz.

**Figure 7.**(

**a**) Hilbert Spectrum results for the signal digitized at 50 kHz, (

**b**) Hilbert Spectrum results for the signal digitized at 10 kHz.

**Figure 8.**Quasi-steady-state intervals enhanced with Hanning windows: (

**a**) First interval, (

**b**) second interval, (

**c**) third interval subjected to EMD for a second time, (

**d**) fourth interval.

**Figure 9.**(

**a**) Instantaneous frequencies and (

**b**) amplitudes, of the mono-components extracted using the hybrid Hilbert-Huang method, of the microwave oven current described in Figure 6a.

**Figure 10.**Time-varying frequency content of mono-component functions of the original signal, as reported by ELSPEC [27], fs = 50 kHz.

Time [s] | Component Amplitude (% of RMS of Signal) | ||||
---|---|---|---|---|---|

150 Hz | 250 Hz | 350 Hz | 430 Hz | 450 Hz | |

0–0.253 | 23% | 9% | 19% | 0% | 0% |

0.253–0.450 | 14% | 0% | 5% | 0% | 5% |

0.450–1 | 0% | 15% | 0% | 15% | 0% |

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**MDPI and ACS Style**

Plamanescu, R.; Dumitrescu, A.-M.; Albu, M.; Suryanarayanan, S.
A Hybrid Hilbert-Huang Method for Monitoring Distorted Time-Varying Waveforms. *Energies* **2021**, *14*, 1864.
https://doi.org/10.3390/en14071864

**AMA Style**

Plamanescu R, Dumitrescu A-M, Albu M, Suryanarayanan S.
A Hybrid Hilbert-Huang Method for Monitoring Distorted Time-Varying Waveforms. *Energies*. 2021; 14(7):1864.
https://doi.org/10.3390/en14071864

**Chicago/Turabian Style**

Plamanescu, Radu, Ana-Maria Dumitrescu, Mihaela Albu, and Siddharth Suryanarayanan.
2021. "A Hybrid Hilbert-Huang Method for Monitoring Distorted Time-Varying Waveforms" *Energies* 14, no. 7: 1864.
https://doi.org/10.3390/en14071864