# Hybrid Approach for Detecting and Classifying Power Quality Disturbances Based on the Variational Mode Decomposition and Deep Stochastic Configuration Network

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## Abstract

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## 1. Introduction

## 2. Variation Mode Decomposition

#### 2.1. The Structure of VMD

- (i)
- Obtain the unilateral frequency spectrum of each mode by computing the associated analytic signal by means of Hilbert transform (in which ${j}^{2}=-1$):$$\text{}\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\times {u}_{k}\left(\mathrm{t}\right)\text{}$$
- (ii)
- Shift the frequency spectrum of each mode to baseband by multiplying an exponential tuned with estimated center frequency:$$\text{}[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\times {u}_{k}\left(\mathrm{t}\right)]{e}^{-j{\omega}_{k}t}\text{}$$
- (iii)
- Calculate the bandwidth of each mode by means of the squared ${L}^{2}$-norm of the gradient. The constrained variational problem is as follows:$$\text{}mi{n}_{\{{u}_{k}\},\left\{{\omega}_{k}\right\}}\{[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\times {u}_{k}\left(\mathrm{t}\right)]{e}^{-j{\omega}_{k}t}\}\text{}$$$$\mathrm{s}.\mathrm{t}.\text{}\sum {u}_{k}=f\text{}$$$$\mathrm{L}\left(\left\{{\mathrm{u}}_{\mathrm{k}}\right\},\text{}\left\{{\mathsf{\omega}}_{\mathrm{k}}\right\},\text{}\mathsf{\lambda}\right):=\alpha {{\displaystyle \sum}}_{k=1}^{K}\Vert {\partial}_{t}[(\delta \left(t\right)+\frac{j}{\pi t})\times {u}_{k}\left(t\right)]{e}^{-j{\omega}_{k}t}{\Vert}_{2}^{2}+\Vert f\left(t\right)-{{\displaystyle \sum}}_{k=1}^{K}{u}_{k}{\left(t\right)\Vert}_{2}^{2}+\langle \mathsf{\lambda}(\mathrm{t}),\text{}f\left(t\right)-{{\displaystyle \sum}}_{k=1}^{K}{u}_{k}\left(t\right)\rangle \text{}$$

#### 2.2. The Computation of VMD

- (i)
- Initialize the $\{{\widehat{u}}_{k}^{1}\}$, $\{{\omega}_{k}^{1}\}$, $\{{\widehat{\lambda}}^{1}\}$, and $n=0$;
- (ii)
- Update the ${u}_{k}$ and ${\omega}_{k}$ repeatedly according to (6), (7);
- (iii)
- Update dual ascent $\lambda $ according to$${\widehat{\lambda}}^{n+1}={\widehat{\lambda}}^{n}+\tau (\widehat{f}\left(\omega \right)-{{\displaystyle \sum}}_{k}{\widehat{u}}_{k}^{n+1}\left(\omega \right))\text{}$$
- (iv)
- Repeat step (2), (3), until convergence: ${{\displaystyle \sum}}_{k}\Vert {\widehat{u}}_{k}^{n+1}-{\widehat{u}}_{k}^{n}{\Vert}_{2}^{2}/\Vert {\widehat{u}}_{k}^{n}{\Vert}_{2}^{2}<\epsilon $.

#### 2.3. Determination of VMD Parameters

#### 2.4. PQ Disturbances Analysis

#### 2.5. Flicker Separation

## 3. Power Disturbance Detection and Classification based on VMD and DSCN

#### 3.1. Deep Stochastic Configuration Networks Algorithm

#### 3.2. Disturbance Detection and Classification

_{ini_1}= 50 Hz) were chosen to obtain accurate modes of flicker. In addition, instantaneous frequency (IF) of modes was used to assess if flicker is existed as per IEEE Std. 1159–2009 [8]. Then, for the other power disturbances, another group of parameters of VMD was applied to decompose stationary and non-stationary types of disturbances. The fundamental component that contains stationary disturbance like sag, swell, and interruption is separated using K = 2 and ω

_{ini_1}= 50 Hz, while the non-stationary disturbances, e.g., harmonic, transient, spike, and notch, were all included in the remaining component. A small value of regularization factor α (α = 200) is chosen to retain multi-frequency modes of non-stationary disturbance. Afterwards, several types of statistical information (mean, variance, and kurtosis) are extracted from instantaneous amplitude (IA) of decomposed modes as disturbance features. Finally, the DSCN method is applied to classify stationary and non-stationary disturbances by means of mean, variance, and kurtosis.

## 4. Results and Discussion

#### 4.1. Synthetic Signal

_{ini_1}= 50 Hz. The results are illustrated in Figure 8.

#### 4.2. Real World Signal

_{ini_1}= 50 Hz. The results are shown in Figure 10.

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The magnitude and frequency spectrum of the grid voltage, (

**a**) magnitude, and (

**b**) frequency spectrum.

**Figure 4.**The magnitude and frequency spectrum of the grid voltage, (

**a**) magnitude, and (

**b**) frequency spectrum.

**Figure 7.**The flowchart of the proposed power quality disturbance (PQD) detection and classification method.

**Figure 8.**The decomposed voltage results using VMD: (

**a**) interruption with oscillation, (

**b**) sag with harmonics, (

**c**) swell and notch, and (

**d**) swell with spike.

**Figure 9.**The results of classification: (

**a**) training confusion matrix, (

**b**) test confusion matrix, and (

**c**) accuracy and root mean square error (RMSE) curve of training.

**Figure 10.**The decomposed results of four real-time signals by using VMD, (

**a**) phase A, (

**b**) phase B, and (

**c**) phase C.

**Table 1.**Results of variational mode decomposition (VMD) for harmonics and interharmonics extraction.

Synthetic Signal | Results of VMD | ||
---|---|---|---|

Freq. (Hz) | Amp. (pu) | Freq. (Hz) | Amp. (pu) |

50 | 1 | 50.03 | 1.01 |

150 | 0.45 | 150.07 | 0.452 |

250 | 0.3 | 250 | 0.31 |

350 | 0.06 | 350.16 | 0.056 |

367 | 0.12 | 367.19 | 0.124 |

450 | 0.06 | 450.16 | 0.063 |

Parameter | Synthetic Signal | Results of VMD |
---|---|---|

$\text{}{A}_{0}\text{}\left(pu\right)$ | 1 | 0.999 |

${f}_{0}$(Hz) | 50 | 50.03 |

$\text{}{A}_{n}\text{}\left(pu\right)$ | 0.4 | 0.049 + 0.05 = 0.099 |

$\text{}{f}_{0}-{f}_{n}\text{}\left(\mathrm{Hz}\right)$ | 41 | 41.02 |

$\text{}{f}_{0}+{f}_{n}\text{}\left(\mathrm{Hz}\right)$ | 59 | 59.03 |

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## Share and Cite

**MDPI and ACS Style**

Cai, K.; Alalibo, B.P.; Cao, W.; Liu, Z.; Wang, Z.; Li, G.
Hybrid Approach for Detecting and Classifying Power Quality Disturbances Based on the Variational Mode Decomposition and Deep Stochastic Configuration Network. *Energies* **2018**, *11*, 3040.
https://doi.org/10.3390/en11113040

**AMA Style**

Cai K, Alalibo BP, Cao W, Liu Z, Wang Z, Li G.
Hybrid Approach for Detecting and Classifying Power Quality Disturbances Based on the Variational Mode Decomposition and Deep Stochastic Configuration Network. *Energies*. 2018; 11(11):3040.
https://doi.org/10.3390/en11113040

**Chicago/Turabian Style**

Cai, Kewei, Belema Prince Alalibo, Wenping Cao, Zheng Liu, Zhiqiang Wang, and Guofeng Li.
2018. "Hybrid Approach for Detecting and Classifying Power Quality Disturbances Based on the Variational Mode Decomposition and Deep Stochastic Configuration Network" *Energies* 11, no. 11: 3040.
https://doi.org/10.3390/en11113040