# Improved Parameter Identification Method for Envelope Current Signals Based on Windowed Interpolation FFT and DE Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principles of WIFFT and DE Algorithms

#### 2.1. Estimation of Harmonic Parameters by WIFFT

_{m}, A

_{m}and θ

_{m}are the frequency, amplitude and phase parameter of m-th harmonic, respectively. k

_{m}refers to the number of local maximum spectral line. $\left|X({k}_{\mathrm{m}})\right|$ and $\mathrm{angle}[X({k}_{\mathrm{m}})]$ are the amplitude and phase of spectral line k

_{m}in the weighted signal, and β

_{m}is the spectral peak ratio of two adjacent spectral lines.

#### 2.2. Principle of DE Algorithm

**X**) be the target function, the crossover rate is C, the scaling factor is F and the evolution generation is t. The steps of DE are as follows [27].

**K**

_{i}(t)) ≤ ε or t = t

_{max}, then output

**K**

_{i}(t) as the optimal solution. Otherwise,

**X**

_{i}(t + 1) =

**K**

_{i}(t) and return to step (2).

## 3. An improved Method Based on WIFFT and DE

_{m}, A

_{m}and θ

_{m}represent the frequency, amplitude and phase of the harmonic signal, respectively. a is the slope of the oblique envelope, and b is the intercept.

_{0}also corresponds to the DC parameter of FFT and the envelope parameter is given by linear fitting. In order to improve the estimation precision and the iterative efficiency, the pre-estimate value (a

^{*},b

^{*}) obtained by the pretreatment of the envelope points can be used as the initial iterative value. The calculation process of the pre-estimate value is as follows.

_{1}t + p

_{2}.

_{1}and p

_{2}. It is effective to eliminate the noise interference in the envelope due to the mean square error minimum criterion is employed. According to Formula (6), p

_{1}and p

_{2}take this form.

^{*}and b

^{*}are:

^{*},b

^{*}) and the harmonic parameters (A

_{m},θ

_{m}) can be used to form the initial population of DE iteration. The initial population is

**X**=

**X**

_{min}+ rand()·(

**X**

_{max}−

**X**

_{min}), where

**X**

_{min}= 0.5·[a

^{*},b

^{*},A

_{m},θ

_{m}] and

**X**

_{max}= 1.5·[a

^{*},b

^{*},A

_{m},θ

_{m}].

**X**) of the DE algorithm, and it is defined as

_{c}(n) is the reconstructed sequence by the identified parameters of the envelope signal, and N is the number of sequence. The smaller RMSE value indicates that the deviation of the actual waveform and the reconstructed signal waveform which is obtained by the combination algorithm is smaller, and the precision of parameter identification is also higher when using this combination method. Finally, the flow chart of the proposed method is shown in Figure 2.

## 4. Simulation of Parameter Identification for Envelope Current Signal

#### 4.1. Simulation of Oblique Envelope Signal with 3 Times Harmonic

_{0}= 49.8 Hz, the phase θ are 60, 45 and 30, respectively, and the amplitude A

_{m}are 100A, 5A and 10A, respectively. The envelope parameters are a = 0.5, b = 1 and the DC component is B

_{0}= 0.2. The sampling frequency is f

_{s}= 5000 Hz, the sampling interval time T

_{s}is 0.2 ms, and the total number of data points is N = 2000. The analysis is as follows.

#### 4.1.1. The Analysis of Hanning WIFFT

#### 4.1.2. The Analysis of WIFFT and Envelope Parameters Estimation

_{1}and p

_{2}are given by linear fitting of the envelope, and the envelope parameters a

^{*}and b

^{*}are given by Formula (8). For the simulation model, the harmonic amplitude parameter in the Formula (8) can be a set value which is able to investigate the performance of the fitting algorithm. Meanwhile, in the analysis of the actual measured signal, the amplitude parameter should be the estimated values computed by WIFFT algorithm. In addition, the following formula

#### 4.1.3. The Analysis of the Improved Method Based on WIFFT and DE Algorithms

_{m}and phase parameters θ

_{m}. Set the aforementioned estimated value as the initial value of iteration, and DE algorithm parameters are as follows. The upper and lower bounds of the independent variables are 1.5 and 0.5 times of the initial value, respectively, the population size is 60, crossover factor is 0.4, crossover rate is 0.9 and the number of evolution generations is 100. Because the amplitude and envelope parameters are involved in the iteration, and they meet Formula (11) at the same time, it will result in different descriptions (a, b, A

_{m}) of the same signal. Therefore, the following Formula (13) is proposed to calculate the envelope parameter error when analyzing the parameter identification.

_{_cal}is the optimal value given by the differential evolution algorithm, a

_{_exact}is the set value, norm(A

_{m_cal}) is the amplitude norm of the optimal estimation which obtained by the DE algorithm, and norm(A

_{m_exact}) is the norm of the set harmonic amplitude value. The dynamic parameter identification in this example is shown in Table 3.

#### 4.2. Analysis of Envelope Current Signal of an Electric Locomotive

_{s}= 5000 Hz, and the total number of data points is N = 5000. The parameters of the DE algorithm are selected as for the previous example.

## 5. Results and Discussion

^{*}decreases from 30.2041% to 0.0176% and the relative error of b

^{*}decreases from 16.9166% to 7.1951 × 10

^{−4}%. Besides, the simulation results indicate that the improved method has good adaptability to noise.

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Cataliotti, A.; Cosentino, V.; Nuccio, S. The Measurement of Reactive Energy in Polluted Distribution PowerSystems: An Analysis of the Performance of Commercial Static Meters. IEEE Trans. Power Deliv.
**2008**, 23, 1296–1301. [Google Scholar] [CrossRef] - Brito, V.H.F.; Kume, G.Y.; Quinalia, M.S.; Sachetti, M.A.; Silva, R.P.B.; Souza, W.A.; Silva, L.C.P. Analysis of the influence of non-linear loads on the measurement and billing of electrical energy compared with the CPT. In Proceedings of the 2016 IEEE 17th International Conference on Harmonics and Quality of Power, Belo Horizonte, Brazil, 16–19 October 2016; pp. 617–622. [Google Scholar] [CrossRef]
- De Vasconcellos, A.B.; Carvalho, B.C.; Martins, W.C.; Anabuki, E.T.; Marques, L.T. The influence of the non-linearity of electric loads on capacitive compensation. In Proceedings of the 2012 IEEE 15th International Conference on Harmonics and Quality of Power, Hong Kong, China, 17–20 June 2012; pp. 880–886. [Google Scholar] [CrossRef]
- Gallardo-Lozano, J.; Milanés-Montero, M.I.; Guerrero-Martínez, M.A.; Romero-Cadaval, E. Electric vehicle battery charger for smart grids. Electr. Power Syst. Res.
**2012**, 90, 18–29. [Google Scholar] [CrossRef] - Gallo, D.; Landi, C.; Langella, R.; Testa, A. On the Accuracy of Electric Energy Revenue Meter Chain Under Non-Sinusoidal Conditions: A Modeling Based Approach. In Proceedings of the 2007 IEEE Instrumentation & Measurement Technology Conference, Warsaw, Poland, 1–3 May 2007; pp. 1–6. [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] - Dai, Y.; Xue, Y.; Zhang, J. A continuous wavelet transform approach for harmonic parameters estimation in the presence of impulsive noise. J. Sound Vib.
**2016**, 360, 300–314. [Google Scholar] [CrossRef] - Garanayak, P.; Panda, G. Fast and accurate measurement of harmonic parameters employing hybrid adaptive linear neural network and filtered-x least mean square algorithm. IET Gener. Transm. Distrib.
**2016**, 10, 421–436. [Google Scholar] [CrossRef] - Guellal, A.; Larbes, C.; Bendib, D.; Hassaine, L.; Malek, A. FPGA based on-line Artificial Neural Network Selective Harmonic Elimination PWM technique. Int. J. Electr. Power Energy Syst.
**2015**, 68, 33–43. [Google Scholar] [CrossRef] - Sun, X.; Sun, L.; Zhao, S. Harmonic Estimation Algorithm based on ESPRIT and Linear Neural Network in Power System. Telkomnika
**2016**, 14, 47–55. [Google Scholar] [CrossRef] - Nikolić, M.; Jovanović, D.P.; Lim, Y.L.; Bertling, K.; Taimre, T.; Rakic, A.D. Approach to frequency estimation in self-mixing interferometry: Multiple signal classification. Appl. Opt.
**2013**, 52, 3345–3350. [Google Scholar] [CrossRef] [PubMed] - Su, T.; Yang, M.; Jin, T.; Flesch, R.C.C. Power harmonic and interharmonic detection method in renewable power based on Nuttall double-window all-phase FFT algorithm. IET Renew. Power Gener.
**2018**, 12, 953–961. [Google Scholar] [CrossRef] - Jin, T.; Chen, Y.; Flesch, R.C.C. A novel power harmonic analysis method based on Nuttall-Kaiser combination window double spectrum interpolated FFT algorithm. J. Electr. Eng.
**2018**, 68, 435–443. [Google Scholar] [CrossRef] - Weishi, M.; Jianhua, W.; Qing, K. Harmonic and inter-harmonic detection based on synchrosqueezed wavelet transform. In Proceedings of the 2016 IEEE Information Technology, Networking, Electronic and Automation Control Conference, Chongqing, China, 20–22 May 2016; pp. 428–432. [Google Scholar] [CrossRef]
- Liu, Z.; Geng, X.; Xie, Z.; Lu, X. The multi-core parallel algorithms of wavelet/wavelet packet transforms and their applications in power system harmonic analysis and data compression. Int. Trans. Electr. Energy Syst.
**2016**, 25, 2800–2818. [Google Scholar] [CrossRef] - Murugan, A.S.S; Kumar, V.S. Determining true harmonic contributions of sources using neural network. Neurocomputing
**2016**, 173, 72–80. [Google Scholar] [CrossRef] - Nascimento, C.F.; Jr, A.A.O.; Goedtel, A.; Dietrich, A.B. Harmonic distortion monitoring for nonlinear loads using neural-network-method. Appl. Soft Comput. J.
**2013**, 13, 475–482. [Google Scholar] [CrossRef] - Wen, H.; Zhang, J.; Meng, Z.; Guo, S.; Li, F.; Yang, Y. Harmonic Estimation Using Symmetrical Interpolation FFT Based on Triangular Self-Convolution Window. IEEE Trans. Ind. Inform.
**2015**, 11. [Google Scholar] [CrossRef] - Testa, A.; Gallo, D.; Langella, R. On the Processing of Harmonics and Interharmonics: Using Hanning Window in Standard Framework. IEEE Trans. Power Deliv.
**2004**, 19, 28–34. [Google Scholar] [CrossRef] - Barros, J.; Diego, R.I. On the Use of the Hanning Window for Harmonic Analysis in the Standard Framework. IEEE Trans. Power Deliv.
**2006**, 21, 538–539. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Differential Evolution—A Simple and Efficient Heuristic for global Optimization over Continuous Spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Rahnamayan, S.; Tizhoosh, H.R.; Salama, M.M.A. Opposition-Based Differential Evolution. IEEE Trans. Evolut. Comput.
**2008**, 12, 64–79. [Google Scholar] [CrossRef][Green Version] - Wen, H.; Dai, H.; Teng, Z.; Yang, Y.; Li, F. Performance Comparison of Windowed Interpolation FFT and Quasisynchronous Sampling Algorithm for Frequency Estimation. Math. Probl. Eng.
**2014**, 1–7. [Google Scholar] [CrossRef] - Chen, K.F.; Mei, S.L. Composite Interpolated Fast Fourier Transform with the Hanning Window. IEEE Trans. Instrum. Meas.
**2010**, 59, 1571–1579. [Google Scholar] [CrossRef] - Gosh, A.; Das, S.; Mallipeddi, R.; Das, A.K.; Dash, S.S. A Modified Differential Evolution with Distance-based Selection for Continuous Optimization in Presence of Noise. IEEE Access
**2017**, 5, 26944–26964. [Google Scholar] [CrossRef] - Cai, Y.; Zhao, M.; Liao, J.; Wang, T.; Tian, H.; Chen, Y. Neighborhood guided differential evolution. Soft Comput.
**2017**, 21, 4769–4812. [Google Scholar] [CrossRef] - Deb, A.; Roy, J.S.; Gupta, B. A Differential Evolution Performance Comparison: Comparing How Various Differential Evolution Algorithms Perform in Designing Microstrip Antennas and Arrays. IEEE Anten. Propag. Mag.
**2018**, 60, 51–61. [Google Scholar] [CrossRef]

**Figure 3.**Simulation results of WIFFT: (

**a**) Comparison of original signal and reconstructed signal; (

**b**) Absolute error of original signal and reconstructed signal.

**Figure 4.**Simulation results of WIFFT and envelope parameters estimation method: (

**a**) Comparison of original signal and reconstructed signal; (

**b**) Absolute error of original signal and reconstructed signal.

**Figure 5.**Simulation results of the improved method based on WIFFT and DE algorithm: (

**a**) Comparison of the original signal and reconstructed signal; (

**b**) Absolute error of original signal and reconstructed signal.

**Figure 6.**Analysis results of envelope current signal of an electric locomotive when employing the WIFFT and envelope parameter estimation method: (

**a**) The comparison of original signal and the reconstructed signal; (

**b**) Absolute error of original signal and reconstructed signal.

**Figure 7.**Analysis results of envelope current signal of an electric locomotive when employing the improved method based on WIFFT and DE: (

**a**) Comparison of the original signal and reconstructed signal; (

**b**) Absolute error of original signal and reconstructed signal.

Parameter | f_{0} | A_{1} | A_{2} | A_{3} | θ_{1} | θ_{2} | θ_{3} |
---|---|---|---|---|---|---|---|

Calculated value | 49.7987 | 109.9965 | 5.4999 | 11.0005 | 59.9290 | 44.7660 | 29.5623 |

Relative error (%) | 0.0026 | 9.9965 | 9.9974 | 10.0050 | 0.1184 | 0.5200 | 1.4589 |

Parameter | Valuation (A_{m} Is the Set Value) | Relative Error (%) | Valuation (A_{m} Is the Estimated Value) | Relative Error (%) |
---|---|---|---|---|

a^{*} | 0.7161 | 43.22 | 0.6510 | 30.2041 |

b^{*} | 0.9139 | 8.6111 | 0.8308 | 16.9166 |

Parameter | a^{*} | b^{*} | A_{1} | A_{2} | A_{3} | θ_{1} | θ_{2} | θ_{3} |
---|---|---|---|---|---|---|---|---|

Valuation | 0.3829 | 0.7656 | 130.6214 | 6.5348 | 13.0577 | 1.0490 | 0.7874 | 0.5246 |

Error (%) | 0.0176 | 7.1951 × 10^{−4} | / | / | / | 0.1723 | 0.2580 | 0.1865 |

Parameter | a^{*} (%) | b^{*} (%) | RMSE | max[x(n) − x_{c}(n)] |
---|---|---|---|---|

40dB | 0.6669 | 0.0576 | 0.7762 | 2.5009 |

50dB | 0.1529 | 0.0284 | 0.2557 | 1.0617 |

60dB | 0.0568 | 0.0117 | 0.1082 | 0.3385 |

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**MDPI and ACS Style**

Su, X.; Zhang, H.; Chen, L.; Qin, L.; Yu, L. Improved Parameter Identification Method for Envelope Current Signals Based on Windowed Interpolation FFT and DE Algorithm. *Algorithms* **2018**, *11*, 113.
https://doi.org/10.3390/a11080113

**AMA Style**

Su X, Zhang H, Chen L, Qin L, Yu L. Improved Parameter Identification Method for Envelope Current Signals Based on Windowed Interpolation FFT and DE Algorithm. *Algorithms*. 2018; 11(8):113.
https://doi.org/10.3390/a11080113

**Chicago/Turabian Style**

Su, Xiangfeng, Huaiqing Zhang, Lin Chen, Ling Qin, and Lili Yu. 2018. "Improved Parameter Identification Method for Envelope Current Signals Based on Windowed Interpolation FFT and DE Algorithm" *Algorithms* 11, no. 8: 113.
https://doi.org/10.3390/a11080113