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Aiming at the operation and maintenance requirements of the fault location of high-temperature superconducting cables, a fault location method of high-temperature superconducting cables based on the improved time-frequency domain reflection method and EEMD noise reduction is proposed. Considering the cross-term interference problem in the traditional time-frequency domain reflection method, this paper introduces the affine transformation to project the time-frequency distribution of the self-term and the cross term and further highlights the characteristic differences between the two through coordinate transformation, and the particle swarm algorithm is employed to solve the optimal stagger angle of the affine transformation. The unscented particle filter is adopted to separate the cross term, and EEMD noise reduction is introduced to solve the signal noise problem. Finally, two software programs, PSCAD and MATLAB, are employed for joint simulation to build a model of high-temperature superconducting cable. The simulation example shows that the proposed method in this paper can eliminate the cross-term interference of the traditional time-frequency domain reflection method, effectively locate the fault of the high-temperature superconducting cable, and improve the positioning accuracy.

In recent years, with the continuous development of social economy, conventional urban cable transmission capacity is small, covers a large area, and has high line loss, so the shortcomings such as the difficulty of grid expansion are increasingly prominent, which restricts the development of smart grid [

At present, there have been many studies on cable fault location, mainly focusing on the low-voltage pulse reflection method based on traveling wave theory [

Signals are often noisy during transmission, and the accuracy of fault location can be improved by using suitable noise reduction methods. In literature [

This paper proposes a fault location method for high-temperature superconducting cables based on the improved time-frequency domain reflection method and EEMD noise reduction. Firstly, the affine transformation is introduced to solve the cross-term interference problem of time-frequency domain reflection method, and the particle swarm algorithm is applied to solve the problem of optimal stagger angle accuracy, and the cross-term separation is achieved by the unscented particle filter algorithm. Secondly, the signal is denoised by EEMD to make the positioning results more accurate. Finally, the simulation environment is built to verify the validity and accuracy of the model in this paper.

As a cable fault non-destructive detection method, TFDR sends an incident signal to the cable and uses a digital oscilloscope to pick up the folded reflection signal generated by the incident signal at the point of impedance discontinuity, so as to locate the high-temperature superconducting cable fault location, and the schematic diagram is described in Figure

TFDR schematic.

The incident signal is usually a linearly modulated signal with Gaussian envelope that provides localized information in the time and frequency domains, and its mathematical expression is_{0} is the time center of the signal, _{0} is the center frequency of the signal. The signal pulse width becomes narrower when

The incident signal produces folded reflection at the point of impedance discontinuity, and the reflected signal is correlated with the incident signal in the time-frequency domain. TFDR uses the Wigner–Ville distribution (WVD) to analyze the signal in the time-frequency domain, that is,

TFDR calculates the time-frequency correlation function of the incident and reflected signals._{sr} (_{r} and _{s} match, and the peak position can be used to detect the fault location. Suppose that the time difference between the incident signal reflected at the head point of the HTS cable and the local peak reflected at the fault point is Δ

Due to the attenuation of the incident signal as it propagates in the high-temperature superconducting cable, both the time center and frequency center of the signal are shifted, requiring a time compensation for the time difference Δ_{ω} is the frequency offset and

Conventional TFDR faces the cross-term interference problem, thus causing misclassification. References [

The affine transformation is a two-dimensional planar transformation method widely used in the field of image processing with homogeneity and union, and the features of the transformed figure still retain their original characteristics [

The key of affine transformation is to obtain the affine transformation matrix, and the result matrix after affine transformation can be obtained by the product of this matrix and the time-frequency distribution matrix. The solution process is usually based on the Fourier transform to find the matching expression of the spectral integral curve in logarithmic polar coordinates and then to find the extreme value point of the curve or to directly fit the curve after the transformation non-linearly. In the literature [

The optimal stagger angle

Particle swarm algorithm is a common artificial intelligence optimization algorithm, which has the advantages of simplicity, easy implementation, and fast convergence compared to the genetic algorithms and simulated annealing methods [^{m} (^{m} (_{th} generation, respectively, _{1} and _{2} are the local and global learning factors, respectively, _{1} and _{2} are random numbers between 0 and 1, ^{m} denotes the historical optimal position of particle ^{m} represents the global optimal position of the whole particle swarm at each iteration.

In this paper, the particle swarm algorithm is used to solve the optimal stagger angle, and the affine transformation matrix can be obtained. In order to maintain the diversity of the population, this paper draws on the idea of genetic algorithm and introduces the variation operator into the particle swarm algorithm, where the particle positions may be mutated at each iteration, and its expression is as follows:^{m} is the mutation operator after the _{th} iteration, _{1} and _{2} are both random numbers in the range of [0, 1], and px^{m+1} (_{th} iteration after mutation. At the beginning of the iteration, ^{m} takes a larger value to strengthen the global search ability of the particle, while at the end of the iteration, ^{m} takes a smaller value to strengthen the local search ability of the particle.

After affine transformation, the self-term is distributed in the low-frequency part and the cross term is distributed in the high-frequency part, so the cross term can be eliminated by low-pass filtering. This method does not affect the self-term resolution, which is theoretically more advantageous than the window addition method.

The unscented particle filter method is based on Bayesian estimation theory and is very suitable for non-Gaussian non-linear systems with high estimation accuracy and fast computational efficiency and is widely used in modern signal processing [

Initialization is performed using the following expression:

where _{0}^{i} denotes the initial state variable at the number of iterations _{0}^{i} is the covariance matrix, and

Calculate the sigma point set, conduct one-step prediction, and calculate the covariance matrix:

Calculate the observed predictions:

Calculate the system covariance matrix according to the weighted values:

Calculate the Kalman gain and update the system state and covariance matrices:

Determine whether resampling is required, continuously update the particle state until the iteration termination condition is satisfied, calculate the normalized weights for the particles, and update the particles:

where

Incident signals may be noisy when transmitted in high-temperature superconducting cables, and the accuracy of fault location can be improved by using suitable noise reduction methods. Empirical mode decomposition is a widely used adaptive time-frequency signal processing method, which can decompose the trends in the original signal at different scales one by one to obtain a finite number of intrinsic mode functions (IMFs), which contain local features at different time scales. The decomposition process is as follows:

Use the method of cubic spline interpolation to find out all the maximal and minimal values of the signal sequence _{max} (_{min} (

Determine whether the deviation de (_{1}(_{1} (_{1} (

_{1} (

The EMD has the shortage of mode mixing, which limits the noise reduction effect. To overcome this problem, EEMD introduces Gaussian white noise and achieves a better improvement effect. The decomposition steps are as follows:

Add a zero-mean Gaussian white noise sequence to the original signal

Perform EMD on the new sequence

Repeat steps (1) and (2) several times, with different white noise added each time.

Derive the mean value of the IMF obtained several times as the IMF of the EEMD.

In this paper, we propose a fault location method for high-temperature superconducting cables. Firstly, incident signals are sent to the cables, and the folded reflection signals generated by the incident signals at the impedance discontinuities can be obtained by the digital oscilloscope. Secondly, the time-frequency domain reflection method is applied to locate cable faults. The affine transformation is introduced to solve the cross-term interference problem of the time-frequency domain reflection method. The particle swarm algorithm is used to improve the accuracy of the stagger angle. Additionally, the unscented particle filter algorithm is employed to achieve cross-term separation and EEMD is introduced to denoise the signal to make the localization results more accurate. The steps of the method in this paper are as follows, and the flowchart is shown in Figure

Set up the simulation environment and model of high-temperature superconducting cable and set the appropriate parameters.

Apply the TFDR method to send incident signal to the cable and use the digital oscilloscope to pick up the fold reflection signal generated by the incident signal at the impedance discontinuity point.

The affine transformation is conducted to solve the cross-term interference problem.

Particle swarm algorithm is used to improve the accuracy of the stagger angle.

The unscented particle filter algorithm is employed to filter out the high-frequency cross term.

The filtered transformed signal is inverse affine transformed back to the original coordinate system.

Denoise the signal by EEMD.

Locate the fault point based on the denoised folded reflection signal.

Algorithm flowchart.

In this paper, two software programs, PSCAD and MATLAB, are used for joint simulation. Since PSCAD does not have a model of high-temperature superconducting cable at the moment, it needs to be built manually. This study borrows the simulation construction method from the literature [

Calculation diagram of high-temperature superconducting cable model.

As can be seen from Figure

During actual operation, insulation faults and short-circuit faults may occur in high-temperature superconducting cables. In this paper, the HTS model is established by PSCAD and MATLAB to simulate a 50 m long high-temperature superconducting cable with a three-phase short-circuit fault at 20 m from the head end. Since the high-temperature superconducting cable normally operates in a liquid nitrogen cryogenic environment, this paper assumes that the cable temperature during steady-state operation of the high-temperature superconducting cable is 80 K.

In order to verify the accuracy and validity of the method in this paper, the following six models are selected for comparison with the proposed method in this paper.

Case 1: the PWVD-TFDR model of Zheng [

Case 2: the GRNN-TFDR model of Rafinia and Moshtagh [

Case 3: the affine transformation-TFDR model of Liu et al. [

Case 4: the method proposed in this paper and that without EEMD noise reduction.

Case 5: the affine transformation-TFDR-particle swarm-Fourier filtering-EEMD noise reduction model is considered in this case.

Case 6: the affine transformation-TFDR-unscented particle filter-EEMD noise reduction is considered.

The localization polar spectrum of the above six control group models is shown in Figures

The PWVD-TFDR cannot completely eliminate the cross-term interference, and there are more interference peaks.

GRNN-TFDR has slightly less interference peaks than pseudo-Wigner distribution-TFDR, but the localization accuracy is not as good as the latter.

The affine transformation-TFDR eliminates the interference peaks to a great extent, and there is only one small interference peak in the figure, and the localization accuracy is better than that of the pseudo-Wigner distribution-TFDR and GRNN-TFDR, which verifies the advantage of the affine transformation idea in removing the cross-term interference.

Compared with affine transformation-TFDR, affine transformation-TFDR-particle swarm-unscented particle filter further reduces the interference peaks and improves the localization accuracy, which verifies that particle swarm and unscented particle filter are beneficial to the accuracy improvement of the algorithm.

Compared with the affine transformation-TFDR, the affine transformation-TFDR-particle swarm-Fourier filter-EEMD noise reduction further reduces the interference peaks and improves the localization accuracy, which verifies that the processing of particle swarm search for the stagger angle and EEMD noise reduction can improve the algorithm accuracy.

Compared with the affine transformation-TFDR, the affine transformation-TFDR-unscented particle filter-EEMD noise reduction further reduces the interference peaks and improves the localization accuracy, and it is verified that the unscented particle filter and EEMD noise reduction are beneficial to the algorithm accuracy improvement.

Positioning result of the PWVD-TFDR model.

Positioning result of the GRNN-TFDR model.

Positioning result of the affine transformation-TFDR model.

Positioning result of the affine transformation-TFDR-particle swarm-unscented particle filter model.

Positioning result of the affine transform-TFDR-particle swarm-Fourier filter-EEMD noise reduction model.

Positioning result of the affine transformation-TFDR-unscented particle filter-EEMD noise reduction model.

Positioning result of the proposed method.

The proposed method of this paper basically eliminates all the interference peaks and performs better compared with the 4th, 5^{th}, and 6th control groups, which verifies that the three measures introduced in this paper, namely, particle swarm search for stagger angle, unscented particle filter for high-frequency cross terms, and EEMD noise reduction, are effective in improving the fault location accuracy.

The comparison of the positioning accuracy of each algorithm is shown in Table

Comparison of positioning accuracy of various algorithms.

Cases | Fault point (m) | Absolute error (m) | Relative error (%) |
---|---|---|---|

Case 1 | 19.37 | 0.63 | 3.15 |

Case 2 | 19.18 | 0.82 | 4.10 |

Case 3 | 19.65 | 0.35 | 1.75 |

Case 4 | 19.87 | 0.13 | 0.65 |

Case 5 | 19.81 | 0.19 | 0.95 |

Case 6 | 19.83 | 0.17 | 0.85 |

The proposed method in this paper | 20.01 | 0.01 | 0.05 |

In this paper, a fault location method for high-temperature superconducting cables is proposed, which can improve the fault location accuracy by effectively solving the problems of self-term resolution reduction and incomplete cross-term removal that may be caused when removing cross-term interference.

This paper combines PSCAD and MATLAB software programs to build a high-temperature superconducting cable model, adopts the time-frequency domain reflection method to locate cable faults, introduces the affine transformation, applies particle swarm algorithm to improve the accuracy of the affine transformation stagger angle, realizes the cross-term separation according to the unscented particle filter algorithm, and finally performs EEMD denoising on the signal. The simulation results show that this method has good fault location accuracy for high-temperature superconducting cables, which has certain reference significance for engineering practice.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This study was supported by the Tongli High-Temperature Superconducting DC Cable Demonstration Project of Jiangsu Suzhou.