Numerical Methods Bicen Maths

[Faustina Bartsch]

Powercables play a very important role in urban development. On the other hand, the integrity and the safe operation of power cables are crucial as even a minor potential mishap can result in serious loss of life and property. Hence, early diagnosis and warning of cable faults are imperative. Addressing the current lack of practical detection technologies, this study proposes a non-destructive testing method based on electromagnetic field principles. Through scanning the power-on power cables with probes, the electric field intensity around the cables can be measured, and the weak anomaly caused by structural and material defects can be detected using Superlets transformation. Additionally, to gain a better characterization of the faults, a digital post-processing approach consisting of imaging and sonification algorithms is developed to aid in pinpointing the location of the faults. Both numerical simulation and experimental test indicate that the proposed non-destructive detection method is feasible and can achieve good accuracy in locating cable faults. With image and audio characterization, the present method has great potential applications in ensuring the safety of power cables.

The manufacturing and installation processes of power cables arranged in underground tunnels are complex, operating in demanding environments [1, 2]. Consequently, various types of faults in power cables can arise due to external forces or aging, thereby impacting the safe operation of power systems.

Underground power cable faults can broadly be categorized into two major types: circuit faults and material and structural faults. The former primarily refers to low-resistance faults, high-resistance faults, open-circuit faults, short-circuit faults, ground faults, etc. [5]. These faults mostly occur due to excessively high or low insulation resistance. The latter is prone to occur within the cable body, joints, and terminations, primarily resulting from material non-uniformity, structural defects, and damages in the cable core and protective layers [6]. Figure 1 illustrates the structure of underground power cables along with some typical defects.

The traveling wave method is a non-destructive testing technique for cables. This method involves analyzing the relationship between the propagation time of fault currents or voltages and the transmission distance to determine the fault distance [12, 13]. However, it encounters technical difficulties in identifying reflected waves, eliminating high-frequency noise, dealing with wave attenuation, and testing blind spots.

As conventional non-destructive testing techniques, X-ray methods have been used for detecting structural faults in cable lines [14,15,16]. However, because of the environmental constraints on the X-ray devices, these methods often require offline testing and involve higher costs in terms of manpower and resources. Eddy current testing is also a non-destructive method that is widely employed in flaw detection. It operates based on the electromagnetic induction principle, where an alternating current applied to the test coil generates an alternating electric field and induces eddy currents on the surface of the conductor. By detecting changes in impedance resulting from the interaction between eddy currents and the coil, it becomes possible to identify the location of conductor defects. An example of the application of this method is its use in detecting lead-sealed defects in power cables [17]. On the other hand, eddy current can only be generated on the surfaces of conductive materials, indicating that eddy current testing is ineffective in detecting nonconductive material defects.

Both the aforementioned methods for detecting non-material structural faults and material structural faults focus primarily on researching the principles of the detection methods, with limited exploration into post-detection processing methods. On the other hand, the uses of vision and hearing are the two most common ways for human beings to obtain external information. Therefore, converting detection data into image or sound can not only vividly display the characteristics of faults, providing a pleasant visual and audio experience, but also improve the accuracy of judgments. In light of this, a non-destructive online detection method for underground power cables based on electromagnetic induction principles is proposed, along with associated post-processing imaging and audio characterization techniques. Theoretical and experimental results indicate that this detection and post-processing method can accurately display the specific locations of faults.

According to the theory of electric fields, in a current-carrying straight conductor, when a defect or anomaly (hereinafter referred to as an abnormality) occurs and current flows through this anomaly, the direction of the current will change. As depicted in Fig. 2, although the current magnitude remains unchanged, there will be a significant difference in current density, causing a deviation in the axis of the current center.

Figure 3 illustrates a schematic diagram of a cable core containing an anomaly, where P represents a point moving along the cable axis. Without loss of generality, it is assumed an anomaly occurs on the conductor, and the electromagnetic field at point P is analyzed. In this situation, the cable axis can be divided into three parts longitudinally, with the second part encompassing the anomaly. As depicted, the dimensions of the anomaly are represented by \(2l\) for length and \(2\Delta\) for depth. \(r_0\) signifies the distance from point P to the cable axis, \(S\) denotes the displacement of point P, which stands for the product of velocity \(v\) and time \(t\), and \(\mu_0\) represents the dielectric constant of the material between point P and the cable. \(\theta_1\sim\theta_6\) denotes the angle as depicted in Fig. 3.

However, it is important to note that the current varies with time in Fig. 3. Therefore, according to electromagnetic field theory, an electric field will be generated around the cable. Based on Maxwell's equation given in Eq. (2), the theoretical formula for the gradient of the electric field strength in the radial direction of the cable under abnormal conditions for a current-carrying straight conductor can be derived as Eq. (3).

According to Eq. (3), the expression of the electric field strength around the defective cable can simplified into a cosine function, in which the amplitude and the phase angle contain specific information about the anomaly.

Two detection schemes were employed: Scheme one involved placing one probe on the slider while the other was grounded, with an oscilloscope measuring the relative voltage between the detection point and ground. Scheme two utilized two probes placed simultaneously on the slider, both moving at the same speed around the periphery of the cable, with an oscilloscope measuring the relative voltage between the two probes. The corresponding electric field signals are depicted in Fig. 4. The results revealed that the waveform characteristics and undulations corresponding to both detection schemes were ambiguous. In this scenario, detecting the position of the notch solely through the voltage amplitude became challenging. Hence, appropriate post-processing methods and techniques became crucial. This study employed Superlets [18] time-frequency analysis technology for post-processing imaging and audio characterization of the electric field surrounding the cable.

In this study, for higher time and frequency resolution, the electromagnetic signals generated during the probe scanning process were processed using a spectral estimator called Superlets [18], which employs wavelet sets with increasingly limited bandwidth. Similar to wavelet transforms, this method involves establishing a wavelet set with fixed central frequencies but different ranges of cycles.

Therefore, the geometric mean of the responses of each wavelet in the wavelet set to the processed signal defines the response of the SLs to that signal [18], and x is the signal to be processed.

SL is an estimator of the oscillatory component appearing in the signal at the central frequencies \(f\) of the SLs. By computing the geometric mean of the response magnitudes across individual wavelets, the intensity estimation can be obtained, and its magnitude is provided by Eq. (16). As time is discretized into a series of time intervals (e.g., \(n\)), the SL spectrum of a signal, represented as a \(1 \times n\) data series, can be obtained. By selecting the frequency range of interest, subdividing it into several frequency intervals (e.g., \(m\)), and applying the Superlets transform to the central frequency of each frequency interval as described above, can generate a \(m \times n\) matrix. The matrix reflects the intensity variation with time for each central frequency within the frequency range of interest.

The electromagnetic field data collected by the oscilloscope underwent Superlets processing, resulting in a matrix of size \(m \times n\), where \(m\) represents the number of frequency intervals contained in the matrix, and \(n\) denotes the number of time intervals. Frequency and time interval lengths were chosen based on requirements. Notably, the frequency range should be centered around the alternating current frequency and can be adjusted according to practical considerations, setting frequency upper and lower bounds accordingly.

The curve depicted in Fig. 5 illustrates the intensity variation of a specific frequency over time. This curve can be converted into an audio signal, offering a perspective different from the image to reflect the cable's defects, because the visual and audio signals can trigger different responses through different senses of human beings or sensors of specific detection systems such as an artificial intelligence model. With the two types of signals, engineers or artificial intelligence machines can choose to judge by one or both of watching and listening. For this purpose, Eq. (18) is employed in this paper to transform the image information.

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