Method for Detecting High-Voltage (HV) Cable Faults

Introduction

With the fast development of the economy and the quick extension of the power grid's size, the number of power cables has expanded dramatically in the last 20 years due to their small footprint and excellent dependability. At the same time, the cables were laid in different ways, and the working circumstances and surroundings were fairly diverse, resulting in varying aging rates of cables in different laying portions, which leads to significant changes in the electrical properties of the cables. Furthermore, when replacing a cable in the case of a failure, the method commonly utilized is to remove the problem section and use a cable junction at either end to attach the new cable to the original.

Currently, HV cable fault location is mostly based on two ideas. The first is the distance protection concept, which is based on impedance measurement , and the second is the traveling wave ranging principle, which is based on traveling wave measurement. The primary premise of distance protection is to determine the line's impedance parameters, since when the line fails, the impedance parameters change. However, because there is not always a linear connection between the measured impedance and the distance to the problem spot, it is difficult to precisely detect the fault using the impedance-distance relationship. The transient traveling wave arrival time of the fault voltage or current is analyzed to achieve traveling wave ranging.

The major challenge of traveling wave ranging is the removal of noise and the extraction of the wave head.

Continuous research has been conducted in recent years to increase fault location accuracy and the application breadth of the impedance-based approach. The impedance-based approach was enhanced by taking into account the mutual inductance impact of double-circuit lines, making it appropriate for fault finding in medium-voltage distribution networks with double-circuits. Under fault situations, the electrical properties of nodes in distributed networks were studied, and fault localization criteria for distinct fault sections of distributed networks were provided. A fault line selection approach for dispersed networks based on the directional vector properties of fault currents during fault situations was developed.

Theoretical Foundations of UQ Methods

In contrast to traditional model computations, UQ's input parameters are its conceivable probability distributions. A particular distribution UQ technique that eventually achieves the output is essentially separated into two categories through the propagation of model uncertainty: a probabilistic framework and a non-probabilistic framework. Under the probabilistic framework, UQ models the computational model's input parameters as random variables and assigns variable types to them; statistical methods fit the probability distribution types of model parameters; and statistical analysis determines the distribution probability of the model output in a given interval. The MCS approach , the PCE method, the surrogate model method, and the dimensionality reduction method are the most often used probabilistic UQ methods.

MCS Method

The MCS technique is the most basic sample computation method. Multiple samples are necessary to repeat the calculation for handling unknown issues, and sample dependency is substantial [20]. In theory, the MCS uncertainty analysis approach is straightforward and focuses on sample generation.

First, generate N samples xi = xi1, xi2,..., xid(i = 1,..., N) based on the distribution type of the random input variable X = [X1, X2,..., Xd]; then substitute xi into Y = g(X) in turn to solve yi(i = 1,..., N); finally, calculate its related statistical information for N outputs, such as mean value, standard deviation, probability function distribution, etc.

The mean convergence rate of MCS is 1/N, which means that correct results can be produced only when there are a sufficient number of samples. As a result, the MCS approach is often employed only as a benchmark in uncertainty quantification. Its progress is hampered by the vast number of calculations and the lengthy running time.

The PCE Method

In recent years, the PCE approach has become a prominent tool for quantifying uncertainty. It conducts finite-order truncation expansion on stochastic issues, represents the original uncertainty problem with a sequence of deterministic coefficient polynomial equations, and provides the precise solution of the stochastic problem by solving the coefficients of the polynomial equations. The chaotic polynomial model's output Y = g(X) is given in (1).

𝑌=𝑐0𝐻0+∑𝑞1=1𝑝𝑐𝑞1𝐻1(𝜉𝑞1)+∑𝑞1=1𝑝∑𝑞2=1𝑞1𝑐𝑞1𝑞2𝐻2(𝜉𝑞1,𝜉𝑞2)+…+∑𝑞1=1𝑝∑𝑞2=1𝑞1… ∑𝑞𝑑=1𝑞𝑑−1𝑐𝑞1𝑞2…𝑞𝑑𝐻𝑝(𝜉𝑞1,𝜉𝑞2,…,,𝜉𝑞𝑑)

In (1), p is the polynomial model's order; c0 is a constant coefficient; cq1q2...qd =cq is the q-th order polynomial coefficient to be computed; x = (xq1, xq2,..., xqd) dimension standard random variable; Hp(x) is a p-th order Hermite orthogonal polynomial. The PCE technique offers high precision and quick convergence speed, however the solution procedure of cq becomes tedious for large model calculation problems.

The UDRM Method

UDRM decomposes the original function, using the mean point as the single variable's reference point [22]. UDRM is primarily implemented through the three processes listed below.

First, locate a set of reference points and approximate deconstruct the original function g(X) at the mean point mi of each single variable into the form of summation of multiple single variable functions, as shown in (2).

𝑔(𝑋)≈𝑔̂ (𝑋)=𝑔̂ (𝑋1,…,𝑋𝑑)=∑𝑖=1𝑑𝑔(𝜇1,…,𝜇𝑖−1,𝑋𝑖,𝜇𝑖+1,…,𝜇𝑑)−(𝑑−1)𝑔(𝜇1,…,𝜇𝑑)

In (2), d is the variable dimension; mi is the mean value corresponding to the variable of the i-th dimension; Xi is the sole variable;g(μ1, …, μi−1, Xi, μi+1, …, μd) is the function value of the variable Xi; g(1,..., d) is the function value of g(X) at mX. (2) shows that the d summation terms on the right-hand side of the equal sign minus the d1 constant terms are equal to the equal sign's left-hand end term.

The second phase of UDRM is to solve the r-th order statistical moment of ĝ(X), that is, to directly integrate (2) to yield (3) after decomposing the single variable of g(X).

𝑚𝑟≈𝐸[𝑔̂ 𝑟(𝐗)].

in (3), E(⋅) is the mathematical expectation operator .

The r-order statistical moment of g(X) is produced using the binomial theorem expansion, as shown in (4) and (5), where i = 1,..., r.

𝑚𝑟≈∑𝑖=0𝑟(𝑟𝑖)𝑆𝑖𝑑[−(𝑑−1)𝑔(𝜇1,…,𝜇𝑑)]𝑟−𝑖

𝑆𝑖𝑑=∑𝑘=0𝑖(𝑖𝑘)𝑆𝑘𝑑−1𝐸[𝑔𝑖−𝑘(𝜇1,…,𝜇𝑑−1,𝑋𝑑)]

The third phase in UDRM is to solve the d one-dimensional integrals explicitly. As stated in (6), use a Gaussian interpolation product formula to calculate higher-order integrals.

𝐸[𝑔ℎ(𝜇1,…,𝜇𝑗−1,𝑋𝑗,𝜇𝑗+1,…,𝜇𝑑)]=

∫𝑔ℎ(𝜇1,…,𝜇𝑗−1,𝑋𝑗,𝜇𝑗+1,…,𝜇𝑑)𝑓𝑋𝑗(𝑥𝑗)𝑑𝑥𝑗≈∑𝑖=1𝑚𝜔𝑗𝑖[𝑔(𝜇1,…,𝜇𝑗−1,𝑋𝑗,𝜇𝑗+1,…,𝜇𝑑)]ℎ

In (6), h = ik; fXj(xj) is the marginal probability density function of the j-th dimension variable Xj, which can be computed or given depending on the specified random variable type; g(m1,..., mj1, lji, mj+1,..., md) is the j-th dimension variable's single variable function value.

Model for HV Cable Fault Location

A fault finding criterion for high-voltage cross-connected cables based on the determination of the phase angle of the sheath circulation. Figure 1 depicts the model and cable construction parameters utilized in this work.

Figure 1 shows a schematic representation of the sheath current and structure of an HV cable. In the equivalent circuit, Zc stands for the equivalent impedance of the core conductor, Ri for the equivalent resistance of the main insulation, Ci for the equivalent capacitor of the main insulation, ZmL and ZmR for the equivalent impedances of the left and right halves of the metal sheath, respectively, ZmLk for the equivalent ground impedance of the metal sheath on the left side, and ZmRk for the equivalent capacitor of the main insulation.

As seen in Figure 1, a typical HV cable has at least 5 layers. The metal sheath is grounded, and the core conductor is used to carry the load current. Cross-bonded cables are typically used to ground cables longer than 1.2 km, as seen in Figure 2. The fault examined in this work is mostly a short-circuit defect brought on by the failure of the primary insulation, and its location is primarily determined by examining the grounding current of the sheath since it is comparatively safer and simpler to monitor the sheath current.

Figure 2: A cross-bonded HV cable schematic diagram.

Figure 3 depicts the equivalent circuit model of the sheath current of the A1-B2-C3 section in a main section under typical operating circumstances. The sheath current calculation expression is shown in (7), where Im1 denotes the sheath current of the A1-B2-C3 section, Ua1 denotes the induced electromotive force of the sheath section A1, Ub2 denotes the induced electromotive force of the sheath section B2, Uc3 denotes the induced electromotive force of the sheath section C3, and Zma1 denotes the equivalent impedance of the The comparable induced electromotive force of neighboring lines is included in Ua1, Ub2, and Uc3.

Figure 3: Under normal conditions, the equivalent circuit of sheath current in the A1-B2-C3 portion.

Because half of the leakage current flows in the same direction as the sheath circulation current and the other half flows in the opposite direction, it may be ignored when calculating the sheath grounding current in normal operation. When a short-circuit defect develops, the equivalent circuit of the sheath current changes, as does the technique of calculating the components in the related transmission line equation. The fault current passes from the core conductor into the metal sheath and subsequently to ground through the metal sheath during a short-circuit fault.

Between the core conductor and the metal sheath, a new current loop is produced, which has its own loop self-inductance as well as mutual inductance with other current loops.

Figure 4 depicts the equivalent circuit for a short-circuit fault in section A1, where Ua stands for the equivalent voltage source of the A-phase line, UA1S for the induced electromotive force of the metal sheath section from G1 to the fault location, UA1R for the induced electromotive force of the metal sheath section from the fault location to J1, Rf for the fault resistance, and If for the fault current of t.

Figure 4: Sheath current equivalent circuit when a short circuit fault develops in section A1.

IS and IR are expressed in (7) and (8), respectively, where ZA1S represents the equivalent impedance of the metal sheath section from G1 to the fault position and ZA1R represents the equivalent impedance of the metal sheath section from the fault position to J1. The phase difference P(section) between IS and IR may be used to compute the position of the fault site for the specific solution technique of each parameter. P(A1), for example, reflects the phase difference between IS and IR in the A1 region, while sheath currents are measured at G1 and J1.

𝐼𝑆=𝑈𝑎+𝑈𝐴1𝑆𝑍𝐴1𝑆+𝑅𝑓+𝑅𝑔1

𝐼𝑅=𝑈𝑎+𝑈𝐴1𝑅+𝑈𝐵2+𝑈𝐶3𝑍𝐴1𝑅+𝑍𝐵2+𝑍𝐶3+𝑅𝑓+𝑅𝑔2

Case Study of HV Cable Fault Location

The high-voltage cables were laid in three-phase tunnels in a zigzag pattern with a phase spacing of 0.3 m and a height of 0.5 m above the ground in a simple power system consisting of a voltage source, a transformer, a complete cross-connected section of high-voltage cables, and three-phase balanced loads. The entire length of the cross-connection section was 1500 m, each tiny portion was 500 m long, and the grounding resistance of the metal sheath's direct grounding point was 0.1. The resistivity of the soil was 100 m. The core conductor's cross-section was 800 mm2, and the rated current carrying capability was 976 A. Table 1 displays the structural parameters.

Table 1. Parameters of cross-sectional structure of the cable 1.

The three-phase voltage source was adjusted at Ua = 63.510° kV, Ub = 63.51120° kV, and Uc = 63.51120° kV, the three-phase balanced load was run at maximum load, and a short-circuit fault was set in the A1 section. The conductivity of the penetrating major breakdown channel was around 103 cm, according to studies on the breakdown properties of cross-linked polyethylene materials. The resistance of the breakdown channel of the primary insulation of the cable, according to this estimate, was just a few milliohms, depending on the equivalent cross-sectional dimension of the breakdown channel.

Furthermore, given the possibility of a minor contact resistance at the cross-bonded site, the transition resistance Rf was set to 0.01 and the fault point distance Lx was specified as an independent variable in this situation.

Because the cable section A1 was 500 m long, the variation range of Lx was (0, 500), allowing the fault point to cross any position in the section. Figure 5 depicts the computed phase difference P(A1) of the sheath current at both ends of the section A1.

Figure 5 shows the phase difference between the fault current IS and the fault current IR when the fault occurs in cable segment A1.

Analysis of Fault Location Using the UQ Method

Simulation Using the Three UQ Methods

At the moment, the uncertainty factors of the real fault site restrict the accuracy of fault location. This study focuses on three typical uncertainty factors: sheath resistivity per unit length, unequal capacitance distribution, and line section length.

To quantify the uncertainty in HV cable fault location, three UQ approaches were used: the MCS method, the PCE method, and the UDRM method. The change rate of the examined uncertainty factors was 50% random and uniform, and the sheath current and fault localization criterion output changes were computed and evaluated.

(1) The MCS approach. The MCS approach is commonly used as a standard for uncertainty quantification methodologies. The MCS approach for calculating the uncertainty of the resistivity per unit length of the HV cable fault finding issue consists mostly of the following steps: (a) N samples are randomly generated based on the given distribution type; (b) the N samples are substituted into the HV cable fault location calculation model in turn, and the calculation is repeated to obtain the corresponding N sample output values; and (c) the relevant statistical characteristics of the sheath current, such as mean, standard deviation, probability density distribution, and so on, are calculated.

First, the connection between the MCS mean convergence rate and N was computed under typical operating conditions, as illustrated in Figure 6a. When the sample count hits 300, the outcome tends to converge. The mean convergence rate of the fault localization criterion was obtained under short-circuit fault conditions when the short-circuit point was 50-500 m, as shown in Figure 6b, and when the number of samples approaches 300, the result starts to converge. As a result, the sample size for the follow-up research was set at 500.

Figure 6 shows the MCS mean convergence rate. (a) Regular operation. (b) A short-circuit fault has occurred.

(2) The PCE approach. The PCE technique focuses on the building of a polynomial model, which is subsequently quantitatively analyzed. The high-voltage cable fault localization model is stated as a polynomial model in (1), and N sample sites are chosen in the standard random space and translated into the original random space. After that, choose a suitable technique for calculating the PCE coefficient and the probability and statistical features of the sheath current value. Because the truncation order of the PCE technique might affect the computation results, the second-order expansion (p = 2), third-order expansion (p = 3), and fourth-order expansion (p = 4) PCE methods were employed for simulation.

(3) UDRM. The parameters in the calculation example in Section 3.2 are utilized as reference values when utilizing the UDRM approach, and the fault localization model of the HV cable is represented as indicated in (2). The number of integration nodes m corresponding to each dimension's random input variables is then calculated. Substitute the specified uncertainty factor range into the single-variable element computation of (6) and calculate the relevant 1-dimensional Gaussian nodes and weights based on the random input variable type. As a result, the statistical moment of the model's output voltage may be computed.

Results Quantitative Analysis

Parameters such as mean and standard deviation are commonly utilized to represent the output uncertainty information in order to quantitatively assess the uncertainty quantification outcomes. When there are a large number of examples, as illustrated in (9), m is the arithmetic mean of the samples.

𝜇=∫Ω𝑋𝑓𝑋(𝑋)𝑑𝑋

The standard deviation s denotes the degree to which the random variable distribution X deviates from its mean. The lower the standard deviation, the more concentrated the sample point distribution.

10.

As illustrated in (11), covariance (Cov) is used to characterize the connection between variables. X and Y are the mean values of variables X and Y, respectively.

11.

The coefficient of variation (Cvar) indicates the degree of dispersion of the model's predicted values, also known as the "dispersion coefficient," and shows the model's stability. (12) depicts the expression of Cvar. Table 2 shows the model stability quantification table, with different Cvar values representing different stability circumstances.

12.

𝐶𝑣𝑎𝑟=|𝜎𝜇|

Table 2 shows the quantification of model stability.

The mean absolute error (Mape) is used to differentiate the model's prediction accuracy; its equation is presented in (13), where N is the number of samples, u is the actual value, and û is the model projected value.

13.

𝑀𝑎𝑝𝑒=1𝑁∑𝑖=1𝑁|𝑢−𝑢̂ |𝑢

Under a 50% amplitude variation range of three uncertainty factors, the resistivity of the sheath per unit length, the equivalent grounding resistance on both sides, and the length of the cable sections, the probability distribution of the sheath current Im1 (under normal operation) and the fault location criterion P(A1) at a distance of 50 m from the head end of the short-circuit fault point were calculated. The statistical parameter findings are provided in Table 3, Table 4, Table 5, Table 6, Table 7, and Table 8, and the probability density distribution plot of the results is displayed in Figures 7, 8, 9, 10, 11, and 12.

Figure 7 depicts the probability density of the derived values of Im1 using the five different approaches under 50% uncertainty of sheath resistivity during normal operation.

Figure 8 shows the probability distribution of the derived data of Im1 using the five different approaches with a 50% uncertainty of grounding resistance in normal operation.

Figure 9 shows the probability distribution of the estimated data of Im1 using the five different approaches with a 50% uncertainty of the cable length in normal operation.

Figure 10 depicts the probability distribution of the computed data of P (A1) using the five different approaches with a 50% uncertainty of sheath resistivity in a short-circuit fault state.

Figure 11 shows the probability distribution of the computed data of P (A1) using the five different approaches with a 50% uncertainty in grounding resistance under a short-circuit fault state.

Figure 12 shows the probability distribution of the computed data of P (A1) using the five different approaches with a 50% uncertainty in the cable length under a short-circuit fault situation.

Table 3 shows the evaluation of several UQ approaches under 50% uncertainty of sheath resistivity in normal operation.

Table 4 shows the evaluation of several UQ approaches under 50% uncertainty of grounding resistance in typical operation.

Table 5: Evaluation of various UQ algorithms under 50% uncertainty in cable length in typical operation.

Table 6 shows the evaluation of several UQ approaches with 50% uncertainty of sheath resistivity under a short-circuit fault.

Table 7 shows the evaluation of several UQ approaches with 50% uncertainty of sheath resistivity under a short-circuit fault.

Table 8: Evaluation of several UQ techniques with 50% uncertainty in cable length under short-circuit fault.

All of the computations reported in the research were done in MATLAB, and the simulations were run on an AMD Ryzen Threadripper 3970X 32-Core, 64 GB RAM machine. When the sample size is sufficient, the five UQ approaches have good impacts on the quantification of the unknown components, as shown in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, and Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, and Figure 12. The mean values of the five approaches are quite similar, but the dispersion are significantly different.

In general, the distribution of UDRM findings is very concentrated, whereas the distribution of MCS results is somewhat dispersed. Changes in sheath resistivity make the waveform flatter under 50% amplitude fluctuation range of three uncertainty factors, whereas changes in cable length make the waveform more spread out by different techniques, whether in normal operation or short-circuit fault circumstances. All of the distributions of findings from these five approaches are significantly concentrated in the 50% amplitude variation region of grounding resistance. The probability density of the estimated data of P(A1) utilizing the five different approaches is the most dispersed under 50% uncertainty of the cable length under short-circuit fault situation.

The waveforms in normal operation are flatter than the waveforms in short-circuit fault situation.

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Method for Detecting High-Voltage (HV) Cable Faults