The application of traditional synchronous measurement methods is limited by frequent fluctuations of electrical signals and complex frequency components in distribution networks. Therefore, it is critical to find solutions to the issues of multifrequency parameter estimation and synchronous measurement estimation accuracy in the complex environment of distribution networks. By utilizing the multifrequency sensing capabilities of discrete Fourier transform signals and Taylor series for dynamic signal processing, a multifrequency signal estimation approach based on HT-IpDFT-STWLS (HIpST) for distribution networks is provided. First, by introducing the Hilbert transform (HT), the influence of noise on the estimation algorithm is reduced. Second, signal frequency components are obtained on the basis of the calculated signal envelope spectrum, and the interpolated discrete Fourier transform (IpDFT) frequency coarse estimation results are used as the initial values of symmetric Taylor weighted least squares (STWLS) to achieve high-precision parameter estimation under the dynamic changes of the signal, and the method increases the number of discrete Fourier. Third, the accuracy of this proposed method is verified by simulation analysis. Data show that this proposed method can accurately achieve the parameter estimation of multifrequency signals in distribution networks. This approach provides a solution for the application of phasor measurement units in distribution networks.

PMU estimates the amplitude and phase of voltage and current waveform signals in the power grid, providing the power grid with time-stamped phasor and frequency information [

Most existing practical and low-complexity synchronization measurement methods are based on discrete Fourier transform (DFT) and Taylor series and achieve synchronous measurement estimation via different improvements in these series. Indeed, because it is easy to understand and simple to calculate, DFT has been widely used in PMU [

To improve the accuracy of parameter estimation under dynamic signal changes, the Taylor series and its improved algorithm are further proposed in references [

For wideband phasor estimation, the two methods perform differently. Zhan et al. [

Upon analyzing the waveform signals of voltage and current in power distribution networks, it has been determined that the fundamental frequency is the dominant frequency of the voltage or current signal, whereas the frequencies of its harmonics remain undetermined. DFT has a number of restrictions, but it can still identify the frequency components that are present in the signal. STWLS offers certain benefits when the signal fluctuates dynamically, but it also has some drawbacks when the frequency component of the signal is unknown. Therefore, a high-precision estimation algorithm called HT-IpDFT-STWLS (HIpST) for multifrequency signal parameters of the distribution network is proposed. This algorithm aims to meet the demand for multifrequency signal parameter characteristics in the practical application of distribution networks, taking into account the actual scene of strong signal dynamic characteristics and rich harmonic content.

In data processing, HT is used to minimize the impact of noise on estimate performance [

HIpST extends the frequency sensing range of PMU in the distribution network to 5500 Hz on the basis of harmonic frequency, expanding the frequency sensing range of PMU compared to the literature technique.

By utilizing high-performance embedded processors, the frequency sensing range will be further extended without taking into account the PMU cost of the distribution network.

In order to accomplish a high-precision estimate of multifrequency signal parameters, this approach combines the multifrequency estimation capabilities of the IpDFT algorithm with the high-performance dynamic estimation qualities of the STWLS algorithm. The IpDFT and STWLS algorithms are first examined, and the distribution network voltage or current waveform signal is modeled. Next, the HIpST multifrequency signal estimating approach is suggested. The study examined the dynamic changes in the distribution network voltage or current waveform signal to assess the performance of the HT-IpDFT, STWLS, and HIpST in order to verify the correctness of the suggested technique. The estimated performance of HIPST was evaluated when the signal was dynamic and contained harmonics. This approach strikes a balance between the competing demands of high-precision signal parameter estimation, making it appropriate for deployment in embedded platforms.

The distribution network is directly connected to the user side, and its dynamic process is complex. In particular, the signal amplitude and phase will be modulated due to the influence of large-capacity source-load switching, line faults, protection misoperation, etc. It is necessary to carry out steady-state and dynamic modeling of distribution network signals, especially the construction of dynamic signal models, as a data source for algorithm feasibility verification.

Since the distribution network is directly connected to the user, its voltage and current fluctuate frequently. In particular, the distributed energy is connected to the converter interface, and its harmonics and interharmonics are enhanced in the signal. In the normal operation of the distribution network, the influence of interharmonics, harmonics, DC components, noise, and so on is considered. The steady-state signal model can be described as [

However, the steady-state signal cannot fully reflect the distribution network signal, the signal amplitude, and the phase caused by the switching of the impact load and the short-circuit fault in the distribution network. Therefore, the dynamic signal model of the distribution network can be described as [

The steady and dynamic models of the distribution network, as shown in

The IpDFT algorithm is a variant of the DFT algorithm, which can achieve a certain accuracy of frequency estimation. The process is as follows: sampling the continuous signal

However, due to the fluctuation of the fundamental frequency of the signal, the frequency obtained by

The signal estimation correction value can be calculated according to

The calculation process of IpDFT is composed of

STWLS can realize the parameter estimation of dynamic signals, and some operations can be completed offline with low computational complexity. According to the basic form of Taylor expansion, the single-frequency signal is modeled, as shown in

The frequency value of the signal can be calculated by using

Through the analysis of IpDFT and STWLS, it can be seen that when the signal parameters change dynamically, the

1) The frequency component of the distribution network signal is perceived, and the envelope spectrum of the signal after the Hilbert transform is calculated to determine the signal frequency spectrum line value.

2) High-precision estimation of multifrequency signal parameters: IpDFT is used to calculate the frequency value of the known frequency spectral line value, which is used as the initial frequency value of the STWLS to calculate the amplitude and phase of different frequency signals.

The distribution network signal has strong dynamic characteristics and multiple frequencies. Accurately estimating the signal frequency value plays an important supporting role in the application of distribution network state estimation and fault location. The following factors must be taken into account for the algorithm:

1) Response time: A dynamic change time scale of the distribution network signal is small, and the signal interception length must be as short as possible. However, the faster the time length of the intercepted signal is, the lower the operation accuracy will be. The trade-off between the length and accuracy of the signal interception must be considered when building the algorithm.

2) The computational complexity: The general algorithm has high computational complexity and high calculation accuracy, but the algorithm complexity is high, and its applicability on the embedded platform is low.

3) Algorithm calculation time: The algorithm

A multifrequency signal estimating approach of HT-IpDFT-STWLS is developed, taking into account the reaction time, computational complexity, and application of the algorithm as well as the respective benefits of the IpDFT and STWLS algorithms. Based on appropriately increasing the complexity, the application scenarios of the algorithm are expanded. The proposed algorithm introduces the Hilbert transform (HT) to convert the real signal into an analytical signal. The spectrum has only positive frequency components, which reduces the influence of negative spectral line interference to a certain extent. Even in the case of a relatively short window length, it can also provide accurate synchronization estimates [

Considering the complexity of the signal frequency components in the DN, the proposed algorithm estimates the initial frequency values of the signal

Therefore, the parameter values of each frequency in the signal can be calculated using

Step 1: The collected signal

Step 2: Calculate the envelope spectrum of the signal. The envelope spectrum is sensitive to the frequency in the signal, and the total number of spectral lines

Step 3: According to the determined spectral line number

Step 4: The correction value calculated in step 3 is brought into

Step 5: Bring the results of step 4 into

Complex procedures on an embedded platform take a lot of time. According to the Euler rule,

Step 6: The calculated results are brought into

Step 7: According to the calculation results of step 5, the exact value of the signal parameters is calculated by

Step 8: Determine whether the number of calculations m is greater than

The HIpST algorithm is optimized based on IpDFT and STWLS, focusing on the analysis of multifrequency signal parameter estimation under the superposition of noise and harmonic signals. The parameters of the proposed algorithm are set as follows: the sampling frequency is 12800 Hz, the window is 4 cycles, and the

The number of STWLS calculation points is odd, and the last sampling point needs to be added during the calculation. After actual testing, it does not affect the accuracy of signal estimation. To determine the effectiveness of the algorithm, three indicators are used to judge [

First, the collected signal is processed by the HT, and the real signal is converted into an analytical signal, which can expand the spectrum range and enhance the energy of the positive spectrum line under the condition of limited points. According to the basic parameter setting, the fundamental frequency range is 50 ± 0.1 Hz, and the frequency error is calculated. The results are shown in

At the set frequency offset of −0.1–0.1 Hz, the signal is processed by HT, and the frequency estimation value and frequency error of the proposed algorithm and STWLS algorithm are calculated. The frequency error is the proposed algorithm’s estimated value minus the STWLS algorithm’s estimated value. From

The

The steady-state signal parameters of the DN were set as follows: (1) the fundamental frequency

In the case of a fundamental frequency shift of −2.5–2.5 Hz, it can be observed from

Through the data analysis of

Signal parameter | Estimation result | Error | |||
---|---|---|---|---|---|

Amplitude/pu | Amplitude/pu | Amplitude/pu | |||

50.05 | 1.00 | 50.0527 | 1.0000 | 0.0027 | 0.0000 |

100.00 | 0.10 | 100.0115 | 0.0996 | 0.0115 | −0.0004 |

150.00 | 0.08 | 150.0105 | 0.0800 | 0.0105 | 0.0000 |

200.00 | 0.06 | 200.0227 | 0.0599 | 0.0227 | −0.0001 |

250.00 | 0.05 | 250.0158 | 0.0501 | 0.0158 | 0.0001 |

300.00 | 0.04 | 300.0466 | 0.0398 | 0.0466 | −0.0002 |

350.00 | 0.03 | 350.0117 | 0.0300 | 0.0117 | 0.0000 |

400.00 | 0.02 | 400.0103 | 0.0198 | 0.0103 | −0.0002 |

450.00 | 0.015 | 449.9826 | 0.0150 | −0.0174 | 0.0000 |

500.00 | 0.01 | 500.0216 | 0.0099 | 0.0216 | −0.0001 |

1500.0 | 0.01 | 1500.0735 | 0.0101 | 0.0735 | 0.0001 |

2500.0 | 0.01 | 2500.0613 | 0.0100 | 0.0613 | 0.0000 |

3500.0 | 0.01 | 3500.0107 | 0.0101 | 0.0107 | 0.0001 |

5500.0 | 0.01 | 5499.9680 | 0.0099 | 0.1680 | −0.0001 |

The phase of the signal model is set to −30°, the frequency is 100 Hz, and the phase of other frequency signals is set to 30°.

Signal frequency | Estimation result | Error | ||
---|---|---|---|---|

Phase/rad | Phase/rad | Phase/rad | Phase/° | |

50.05 | 0.5236 | 0.5240 | −0.0004 | −0.0229 |

100.00 | −0.5236 | −0.5325 | 0.0089 | 0.5098 |

150.00 | 0.5236 | 0.5269 | −0.0033 | −0.1883 |

200.00 | 0.5236 | 0.5216 | 0.0020 | 0.1124 |

250.00 | 0.5236 | 0.5202 | 0.0034 | 0.1925 |

300.00 | 0.5236 | 0.5338 | −0.0102 | −0.5844 |

350.00 | 0.5236 | 0.5295 | −0.0059 | −0.3389 |

400.00 | 0.5236 | 0.5180 | 0.0056 | 0.3226 |

450.00 | 0.5236 | 0.5279 | −0.0043 | −0.2466 |

500.00 | 0.5236 | 0.5303 | −0.0067 | −0.3866 |

1500.0 | 0.5236 | 0.5126 | −0.0090 | −0.5157 |

2500.0 | 0.5236 | 0.5273 | 0.0037 | 0.2120 |

3500.0 | 0.5236 | 0.5286 | 0.0050 | 0.2865 |

5500.0 | 0.5236 | 0.5275 | 0.0029 | 0.1662 |

The dynamic signal model of the distribution network was built concerning

As shown in

As seen from the

To verify the effect of the proposed algorithm on harmonic parameter estimation, the harmonic signal of 100–5500 Hz is superimposed in the signal dynamic model, and its parameters are listed in

Signal parameter | Estimation result | Error | |||
---|---|---|---|---|---|

Amplitude/pu | Amplitude/pu | Amplitude/pu | |||

51.50 | 1.00 | 51.5030 | 0.9997 | 0.0030 | −0.0003 |

100.00 | 0.10 | 99.8705 | 0.0996 | −0.1295 | −0.0004 |

150.00 | 0.08 | 150.0376 | 0.0797 | 0.0376 | −0.0003 |

200.00 | 0.06 | 200.0588 | 0.0601 | 0.0588 | 0.0001 |

250.00 | 0.05 | 250.0145 | 0.0497 | 0.0145 | −0.0003 |

300.00 | 0.04 | 300.1103 | 0.0398 | 0.1103 | −0.0002 |

350.00 | 0.03 | 350.0864 | 0.0299 | 0.0864 | −0.0001 |

400.00 | 0.02 | 400.1418 | 0.0199 | 0.1418 | −0.0001 |

450.00 | 0.02 | 450.0729 | 0.0148 | 0.0729 | −0.0002 |

500.00 | 0.01 | 500.1026 | 0.0099 | −0.1026 | −0.0001 |

1500.0 | 0.01 | 1499.9909 | 0.0100 | −0.0091 | 0.0000 |

2500.0 | 0.01 | 2499.9604 | 0.0099 | −0.0396 | −0.0001 |

3500.0 | 0.01 | 3500.0968 | 0.0100 | 0.0968 | 0.0000 |

5500.0 | 0.01 | 5500.0636 | 0.0099 | 0.0636 | −0.0001 |

Signal frequency | Estimation result | Error | ||
---|---|---|---|---|

Phase/rad | Phase/rad | Phase/rad | Phase/° | |

50.05 | 0.5236 | 0.5235 | −0.0001 | −0.0057 |

100.00 | −0.5236 | −0.5276 | −0.0040 | −0.2091 |

150.00 | 0.5236 | 0.5269 | 0.0033 | 0.1891 |

200.00 | 0.5236 | 0.5225 | −0.0011 | −0.0630 |

250.00 | 0.5236 | 0.5269 | 0.0033 | 0.1891 |

300.00 | 0.5236 | 0.5248 | 0.0012 | 0.0688 |

350.00 | 0.5236 | 0.5209 | −0.0027 | −0.1547 |

400.00 | 0.5236 | 0.5265 | 0.0029 | 0.1662 |

450.00 | 0.5236 | 0.5258 | 0.0022 | 0.1261 |

500.00 | 0.5236 | 0.5193 | −0.0043 | −0.2464 |

1500.0 | 0.5236 | 0.5253 | 0.0017 | 0.0974 |

2500.0 | 0.5236 | 0.5233 | −0.0003 | −0.0172 |

3500.0 | 0.5236 | 0.5256 | 0.0020 | 0.1146 |

5500.0 | 0.5236 | 0.5219 | −0.0017 | −0.0974 |

To ensure the comparability of the algorithm estimation results, the dynamic signal phase parameters are set as in

Aiming at the strong dynamic characteristics and complex frequency components of distribution network signals, a multifrequency measurement signal estimation method for distribution networks based on HIpST is proposed. Compared with the same type of algorithm, this algorithm can realize the parameter estimation of distribution network signals and meet the existing standard requirements, which is suitable for the application requirements of different scenarios of distribution networks. Through simulation experiments, the following conclusions are drawn:

1) The proposed algorithm introduces HT, which can improve the signal frequency estimation accuracy by 2–3 times under the same calculation conditions. The proposed algorithm is still effective under different noise levels.

2) When the fundamental frequency offset is small, the estimation performance of the HIpST algorithm is more than 5 times higher than that of STWLS and HT-IpDFT. The proposed algorithm fully satisfies the 2–50 harmonic parameter estimation, and theoretically, the frequency parameter estimation below 6000 Hz can be realized.

3) Based on the multifrequency characteristics of IpDFT and the dynamic estimation advantages of STWLS, the multifrequency parameter estimation is realized based on appropriately increasing the calculation process of the algorithm, and the application scenario of the STWLS algorithm is expanded. The proposed method has the possibility of application in practical scenarios. The next research goal of the project will be to explore the application of multifrequency measurement parameters in different scenarios of distribution networks, and the deployment of synchronous measurement devices in distribution networks.

We thanked Tongliao Power Supply Company of State Grid East Inner Mongolia Electric Power Company Limited for giving financial support for this study. We would like to thank three anonymous reviewers and the editor for their comments.

This project is supported by the State Grid Corporation of China Headquarters Management Science and Technology Project (No. 526620200008).

The authors confirm contribution to the paper as follows: study conception and design: Bin Liu, Shuai Liang; project administration: Renjie Ding; analysis and interpretation of results: Shuguang Li; draft manuscript preparation: Bin Liu, Shuguang Li. All authors reviewed the results and approved the final version of the manuscript.

The original contributions presented in the study are included in the article. Further inquiries can be directed to the corresponding author.

The authors declare that they have no conflicts of interest to report regarding the present study.