# Wavelet Packet Decomposition for IEC Compliant Assessment of Harmonics under Stationary and Fluctuating Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed WPD Method

#### 2.1. Wavelet Transforms in Power Systems

#### 2.2. Characteristics of the Proposed Method

## 3. Compliance with IEC Accuracy Requirements

## 4. Validation of the Method

#### 4.1. Test Waveforms

- Constant modulation: instantaneous reduction of the amplitude from 100% to 20%, 85 $\mathrm{m}$$\mathrm{s}$ after the beginning of the signal. This is the modulation proposed in IEC 61000-4-7.
- Linear modulation: amplitude linearly reduced from 100% to 75% in the last 115 $\mathrm{m}$$\mathrm{s}$ of the signal. This modulation represents a motor start.
- Rectangular modulation: flicker-type modulation of 5% amplitude and $8.8$ $\mathrm{Hz}$ modulation frequency.

#### 4.2. Validation for Stationary Conditions

#### 4.3. Validation for Non-Stationary Conditions

#### 4.4. Computational Effort

## 5. Analysis of Real Signals

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Stationary | Constant | Linear | Rectangular | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

ID | ${\mathit{\epsilon}}_{\mathbf{max}}$ (%) | Order | ${\mathit{\epsilon}}_{\mathbf{max}}$ (%) | Order | ${\mathit{\epsilon}}_{\mathbf{max}}$ (%) | Order | ${\mathit{\epsilon}}_{\mathbf{max}}$ (%) | Order | |||

1 | 0.048 | 2 | 0.079 | 1 | 0.041 | 2 | 0.048 | 2 | |||

2 | 0.416 | 5 | 0.504 | 5 | 0.281 | 3 | 0.486 | 5 | |||

3 | 0.416 | 5 | 0.514 | 5 | 0.271 | 5 | 0.479 | 5 | |||

4 | 0.374 | 5 | 0.458 | 5 | 0.232 | 5 | 0.428 | 5 | |||

5 | 0.322 | 4 | 0.372 | 5 | 0.285 | 4 | 0.327 | 4 | |||

6 | 0.319 | 5 | 0.390 | 5 | 0.184 | 4 | 0.341 | 5 | |||

7 | 0.348 | 5 | 0.428 | 5 | 0.213 | 4 | 0.394 | 5 | |||

8 | 0.589 | 7 | 0.625 | 7 | 0.446 | 7 | 0.678 * | 5 | |||

9 | 0.363 | 5 | 0.443 | 5 | 0.221 | 5 | 0.406 | 5 | |||

10 | 0.366 | 5 | 0.449 | 5 | 0.225 | 5 | 0.418 | 5 | |||

11 | 0.566 | 7 | 0.667 | 5 | 0.437 | 2 | 0.671 | 5 | |||

12 | 0.401 | 5 | 0.488 | 5 | 0.258 | 5 | 0.462 | 5 | |||

13 | 0.397 | 5 | 0.484 | 5 | 0.254 | 5 | 0.476 | 5 |

## References

- Kalair, A.; Abas, N.; Kalair, A.R.; Saleem, Z.; Khan, N. Review of harmonic analysis, modeling and mitigation techniques. Renew. Sustain. Energy Rev.
**2017**, 78, 1152–1187. [Google Scholar] [CrossRef] - Sharma, H.; Rylander, M.; Dorr, D. Grid impacts due to increased penetration of newer harmonic sources. IEEE Trans. Ind. Appl.
**2016**, 52, 99–104. [Google Scholar] [CrossRef] - Liang, X. Emerging Power Quality Challenges Due to Integration of Renewable Energy Sources. IEEE Trans. Ind. Appl.
**2017**, 53, 855–866. [Google Scholar] [CrossRef] - Moeed Amjad, A.; Salam, Z. A review of soft computing methods for harmonics elimination PWM for inverters in renewable energy conversion systems. Renew. Sustain. Energy Rev.
**2014**, 33, 141–153. [Google Scholar] [CrossRef] - Electromagnetic Compatibility (EMC)—Part 4-7: Testing and Measurement Techniques—General Guide on Harmonics and Interharmonics Measurements and Instrumentation, for Power Supply Systems and Equipment Connected Thereto. IEC 61000-4-7:2002+AMD1:2008.
- IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems. In IEEE Std. 519-2014; IEEE, 2014.
- Clarkson, P.; Wright, P.S. A wavelet based method of measuring fluctuating harmonics for determining the filter time constant of IEC standard harmonic analyzers. IEEE Trans. Instrum. Meas.
**2005**, 54, 488–491. [Google Scholar] [CrossRef] - Jain, S.K.; Singh, S.N. Harmonics estimation in emerging power system: Key issues and challenges. Electr. Power Syst. Res.
**2011**, 81, 1754–1766. [Google Scholar] [CrossRef] - Mostarac, P.; Malarić, R.; Mostarac, K.; Jurčević, M. Noise reduction of power quality measurements with time-frequency depth analysis. Energies
**2019**, 12, 1052. [Google Scholar] [CrossRef] - Wang, C.; Yang, R.; Yu, Q. Wavelet transform based energy management strategies for plug-in hybrid electric vehicles considering temperature uncertainty. Appl. Energy
**2019**, 256, 113928. [Google Scholar] [CrossRef] - Telesca, L.; Guignard, F.; Helbig, N.; Kanevski, M. Wavelet Scale Variance Analysis of Wind Extremes in Mountainous Terrains. Energies
**2019**, 12, 3048. [Google Scholar] [CrossRef] - Zhu, H.; Li, X.; Sun, Q.; Nie, L.; Yao, J.; Zhao, G. A power prediction method for photovoltaic power plant based on wavelet decomposition and artificial neural networks. Energies
**2016**, 9, 11. [Google Scholar] [CrossRef] - Avdakovic, S.; Lukac, A.; Nuhanovic, A.; Music, M. Wind speed data analysis using wavelet transform. World Acad. Sci. Eng. Technol.
**2011**, 51, 829–833. [Google Scholar] - Karmacharya, I.M.; Gokaraju, R. Fault Location in Ungrounded Photovoltaic System Using Wavelets and ANN. IEEE Trans. Power Deliv.
**2018**, 33, 549–559. [Google Scholar] [CrossRef] - Yoon, W.K.; Devaney, M.J. Reactive power measurement using the wavelet transform. IEEE Trans. Instrum. Meas.
**2000**, 49, 246–252. [Google Scholar] [CrossRef] - Morsi, W.G.; El-Hawary, M.E. On the application of wavelet transform for symmetrical components computations in the presence of stationary and non-stationary power quality disturbances. Electr. Power Syst. Res.
**2011**, 81, 1373–1380. [Google Scholar] [CrossRef] - Deokar, S.A.; Waghmare, L.M. Integrated DWT-FFT approach for detection and classification of power quality disturbances. Int. J. Electr. Power Energy Syst.
**2014**, 61, 594–605. [Google Scholar] [CrossRef] - De Apráiz, M.; Barros, J.; Diego, R.I. A real-time method for time-frequency detection of transient disturbances in voltage supply systems. Electr. Power Syst. Res.
**2014**, 108, 103–112. [Google Scholar] [CrossRef] - Latran, M.B.; Teke, A. A novel wavelet transform based voltage sag/swell detection algorithm. Int. J. Electr. Power Energy Syst.
**2015**, 71, 131–139. [Google Scholar] [CrossRef] - Alves, D.K.; Costa, F.B.; de Araujo Ribeiro, R.L.; de Sousa Neto, C.M.; de Oliveira, T. Power Measurement Using the Maximal Overlap Discrete Wavelet Transform. IEEE Trans. Ind. Electron.
**2017**, 64, 3177–3187. [Google Scholar] [CrossRef] - Hamid, E.Y.; Yokoyama, N.; Kawasaki, Z.I. Rms and Power Measurements: A Wavelet Packet Transform Approach. IEEJ Trans. Power Energy
**2002**, 122, 599–606. [Google Scholar] [CrossRef] - Eren, L.; Unal, M.; Devaney, M.J. Harmonic Analysis Via Wavelet Packet Decomposition Using Special Elliptic Half-Band Filters. IEEE Trans. Instrum. Meas.
**2007**, 56, 2289–2293. [Google Scholar] [CrossRef] - Tiwari, V.K.; Umarikar, A.C.; Jain, T. Fast Amplitude Estimation of Harmonics Using Undecimated Wavelet Packet Transform and Its Hardware Implementation. IEEE Trans. Instrum. Meas.
**2017**, 1–13. [Google Scholar] [CrossRef] - Diego, R.I.; Barros, J. Global Method for Time-Frequency Analysis of Harmonic Distortion in Power Systems Using the Wavelet Packet Transform. Electr. Power Syst. Res.
**2009**, 79, 1226–1239. [Google Scholar] [CrossRef] - Barros, J.; Diego, R.I. Analysis of Harmonics in Power Systems Using the Wavelet Packet Transform. IEEE Trans. Instrum. Meas.
**2008**, 57, 63–69. [Google Scholar] [CrossRef] - Bruna, J.; Melero, J.J. Selection of the Most Suitable Decomposition Filter for the Measurement of Fluctuating Harmonics. IEEE Trans. Instrum. Meas.
**2016**, 65, 2587–2594. [Google Scholar] [CrossRef] - Mallat, S.G. A Wavelet Tour of Signal Processing, 2nd ed.; Academic Press: Cambridge, MA, USA, 1999. [Google Scholar]
- NPL. Power Quality Waveform Library. Available online: http://resource.npl.co.uk/waveform/ (accessed on 30 October 2019).
- Brekke, K.; Seljeseth, H.; Mogstad, O. Rapid Voltage Changes—Definition and Minimum Requirements. In Proceedings of the 20th International Conference on Electricity Distribution (CIRED), Prague, Czech Republic, 8–11 June 2009. [Google Scholar] [CrossRef]
- Tiwari, V.K.; Jain, S.K. Hardware Implementation of Polyphase-Decomposition-Based Wavelet Filters for Power System Harmonics Estimation. IEEE Trans. Instrum. Meas.
**2016**, 65, 1585–1595. [Google Scholar] [CrossRef]

**Figure 2.**

**Top**: errors of the proposed method with steady-state single frequency signals.

**Bottom**: comparison of the error produced by the conventional FFT, the grouped IEC, and the WPD methods in case of single frequency signals, under fluctuating conditions (as explained in the text). The grey lines show the limits provided by IEC 61000-4-7 for stationary conditions.

**Figure 3.**Analysis of the IEC 61000-3-2 limits waveform under stationary conditions. The comparison is made between the IEC, the grouped IEC, and the proposed WPD methods. The upper graph shows the spectrum of the RMS content of each IEC frequency band for the three methods, compared to the real content. The middle graph shows the errors for each frequency band. The difference between the errors is shown in the bottom graph.

**Figure 5.**Schematic illustration of the employed modulation patterns. From left to right: constant modulation, linear modulation, and flicker-type modulation.

**Figure 6.**Analysis of the IEC 61000-3-2 limits waveform under non-stationary conditions (constant modulation). The upper graph shows the spectrum of the RMS content of each IEC frequency band for the three methods, compared to the real content. The middle graph shows the errors for each frequency band. The difference between the errors is shown in the bottom graph.

**Figure 7.**RMSE for each assessed waveform under fluctuating conditions, for grouped IEC and WPD, in the case of constant modulation.

**Figure 8.**RMSE for each assessed waveform under fluctuating conditions, for grouped IEC and WPD, in the case of linear modulation (motor-start type).

**Figure 9.**RMSE for each assessed waveform under fluctuating conditions, for grouped IEC and WPD, in the case of flicker-type rectangular modulation.

**Figure 10.**Analysis of real waveforms with a stationary (

**a**) and non-stationary (

**b**) character. The top graphs show the waveforms in the time domain, the middle graphs the harmonic content, and the bottom graphs the absolute difference between the proposed method and the IEC strategy.

Level Number | Number of Nodes | Samples/Node | Δt (ms) | Bw/Node (Hz) |
---|---|---|---|---|

1 | 2 | 640 | 0.312 | 1600 |

2 | 4 | 320 | 0.625 | 800 |

3 | 8 | 160 | 1.250 | 400 |

4 | 16 | 80 | 2.500 | 200 |

5 | 32 | 40 | 5.000 | 100 |

6 | 64 | 20 | 10.000 | 50 |

7 | 128 | 10 | 20.000 | 25 |

7* | 128/2 = 64 | 10 + 10 = 20 | N/A | 25 + 25 = 50 |

**Table 2.**Validation test waveforms, taken from [28], showing the total number of harmonic components included n, the most intense harmonic order H, and the calculated THD.

ID | Description | n | H | THD (%) |
---|---|---|---|---|

1 | IEC 61000-3-2 voltage limits for the power amplifier | 38 | 3 | 1.26 |

2 | Voltage waveform for an accounting operation building | 4 | 5 | 2.80 |

3 | Voltage waveform for an apartment building | 6 | 5 | 1.95 |

4 | Voltage waveform for a commercial and residential load | 3 | 5 | 2.44 |

5 | Voltage waveform of fluorescent lights with electronic ballast | 5 | 5 | 3.74 |

6 | Voltage waveform of fluorescent lights with magnetic ballast | 3 | 5 | 2.69 |

7 | Voltage waveform for an industrial and residential load | 3 | 5 | 3.00 |

8 | Possible limits of voltage harmonics of a single phase feeder | 49 | 3 | 6.72 |

9 | Voltage waveform for a machining plant | 4 | 5 | 2.00 |

10 | Voltage waveform for a residential load | 3 | 5 | 2.55 |

11 | Square wave | 24 | 3 | 47.30 |

12 | Voltage waveform for a supermarket | 4 | 5 | 2.42 |

13 | Voltage waveform for a welded pipes plant | 4 | 5 | 2.40 |

Computational Time (FFT) (s) | Computational Time (WPD) (s) | Ratio WPD/FFT |
---|---|---|

3.6 × 10^{−4} | 5.2 × 10^{−2} | 148 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lodetti, S.; Bruna, J.; Melero, J.J.; Sanz, J.F.
Wavelet Packet Decomposition for IEC Compliant Assessment of Harmonics under Stationary and Fluctuating Conditions. *Energies* **2019**, *12*, 4389.
https://doi.org/10.3390/en12224389

**AMA Style**

Lodetti S, Bruna J, Melero JJ, Sanz JF.
Wavelet Packet Decomposition for IEC Compliant Assessment of Harmonics under Stationary and Fluctuating Conditions. *Energies*. 2019; 12(22):4389.
https://doi.org/10.3390/en12224389

**Chicago/Turabian Style**

Lodetti, Stefano, Jorge Bruna, Julio J. Melero, and José F. Sanz.
2019. "Wavelet Packet Decomposition for IEC Compliant Assessment of Harmonics under Stationary and Fluctuating Conditions" *Energies* 12, no. 22: 4389.
https://doi.org/10.3390/en12224389