Two-Sheath Loop Short Circuit Defects Detection in High-Voltage Cable Systems Using Sheath Current Phasors

Abstract

The joint is the weak point of HV (high voltage) cable insulation systems; creep discharge between insulation layers of the cable joint, due to moisture intrusion, is one of the main defects leading to single-phase grounding. Carbonization on the insulation interface after creep discharge would lead to a short-circuit defect in the sheath loops and result in abnormal sheath current. In this study, a novel diagnostic criterion using the phasor difference of sheath currents at both ends of the same circuit is proposed. The coupling effect between the sheath and the conductor under defect conditions is considered, and the original lumped parameter model of the cable circuit is optimized. The cable parameters are further corrected using a genetic algorithm. The diagnostic criterion comprehensively accounts for the adverse effects of unequal cable segment lengths, load current fluctuations, grounding impedance, and phase voltage variations. When the phase angle fluctuation of the phasor difference is within 10° and the defect impedance is below 100 Ω, the defective joint can be accurately diagnosed by this method. The conclusion has been validated through PSCAD simulations, with a diagnostic accuracy above 97%. Even under 20 dB noise interference, the error increase remains within 2%.

1. Introduction

With the increasing integration of renewable energy into the grid and the growing deployment of electric vehicle charging infrastructure, urban underground power cable networks are expanding rapidly [1,2,3]. Early installed cable lines often operate in harsh environments, while the rapid growth of newly commissioned cables further complicates their operation and maintenance [4]. A statistical study of 231 defect cases conducted by the Southern Power Grid identified cable joints as the weak point of cable systems [5]. In prefabricated cable joints, defects mainly stem from the degradation and failure of insulation performance [6]. As surface discharges intensify, short-circuit defects may occur in the internal metal sheath. This results in abnormal sheath currents, which are a typical external manifestation of cable joint defects [7]. Current research has not thoroughly addressed this issue, and further analysis may provide new technical approaches for monitoring such defects.

Operational experience shows that most defects in combined prefabricated cable joints are caused by moisture ingress [8]. Moisture significantly reduces the surface resistance inside the joint, leading to surface discharges that eventually damage the insulation layer. Currently, partial discharge detection is a widely used method for early fault diagnosis in electrical equipment. However, partial discharge signals are often weak and unstable, and detection technologies face challenges such as noise interference and signal attenuation [9,10,11]. Power-off testing methods, such as the external insulation resistance method and the high-voltage bridge method, are limited by wiring configurations and testing conditions. These methods can only locate faults after they occur and are unsuitable for routine inspections [12,13,14]. Online testing approaches, such as the small signal discharge method, are also less effective when sufficient induced voltage cannot be generated on the cable sheath [13]. When insulation in intermediate joints degrades, a low-resistance path may form between the two metallic sheaths inside the joint. This prevents the cancellation of the three-phase sheath-induced voltages [15], producing abnormal sheath grounding currents. In contrast, the power-frequency current signal in the cable’s metallic sheath remains stable, and its phase information is highly sensitive. This provides a reliable long-term indicator [16] and serves as an effective supplementary approach for diagnosing insulation defects in cable joints.

Several studies have attempted to improve defect diagnosis methods using sheath current as a state variable. The State Grid Corporation of China and China Southern Power Grid have published the Q/GWD456 and Q/CSG1206997 standards [17,18], recommending the use of sheath current magnitude, current ratios, and relative values of three loop currents as fault diagnosis criteria. Reference [19] proposed a criterion based on the sheath current at the direct grounding point. However, it is only effective when the sheath-to-ground fault impedance is below 40 Ω and the load current is relatively large. Reference [20] proposed an improved criterion for the Q/GDW 456 standard based on operational data. However, it still focuses on analyzing effective current values and ratios. Subsequently, an online monitoring method for the relative value of cable dielectric loss was developed based on distributed sheath current measurements [21]. Later studies investigated breakdown faults in intermediate joints by using the ratio of defective sheath current to normal sheath current as a diagnostic indicator [22]. However, this method still showed limitations, particularly in misjudging defect locations. Reference [23] approached the problem from a circuit topology perspective, employing features such as the amplitude ratio and phase angle difference of sheath currents. By introducing a logistic regression algorithm, the study successfully identified defects such as new branches between sheaths, although the approach lacked validation. Although these studies provide valuable insights for monitoring cable sheath grounding currents and diagnosing intermediate joint insulation defects, they still often misjudge faults and cannot effectively differentiate fault types and locations.

To detect insulation defects in intermediate joints of HV cables, this paper proposes a recognition method based on the phase angle difference of sheath grounding current. The structure of this paper is as follows: The second section introduces a computational model considering the coupling relationship between the HV cable core voltage, core current, and sheath current, and presents a parameter correction method and a dual iterative model for the lumped parameters of the cable line. The third section presents and explains a short-circuit defect diagnosis method for two sheath loops, and analyzes the impact of short-circuit impedance, load current, phase voltage, and cable segment length on the criteria by using 110 kV cable line parameters. The proposed method is then validated through PSCAD transient simulations, and the influence of noise on the results is evaluated. Section 4 concludes the paper.

2. Defective-State HV Cable Sheath Current Calculation Model

2.1. HV Cable Sheath Current Calculation Method

Currently, the cross-bonding metal sheath connection is applied in sectioned single-core power cables both domestically and internationally. A typical cross-bonding major cable section is illustrated in Figure 1, where each cable has three segments. At both ends of the main section, terminals or joints are used to connect other cables, overhead lines, or substation equipment. The metal sheath of the main section is grounded directly and connected to grounding boxes G1 and G2. Cable joints JA1, JB1, JC1, JA2, JB2, and JC2 are used to connect different coaxial segments. The metal sheath of adjacent coaxial segments is connected to the junction boxes, and the cross-bonding connection is completed within the junction boxes J1 and J2.

The metal sheaths of different segments are interconnected and switched within the cross-bonding junction box via coaxial cables. As shown in Figure 2, after cross-bonding, the three metal sheath circuits are as follows: A1-B2-C3, B1-C2-A3, and C1-A2-B3. The current flowing through the metal sheath circuits is referred to as sheath current. I_1a, I_1b, and I_1c represent the left-side sheath ground currents, while I_2a, I_2b, and I_2c represent the right-side sheath ground currents.

In contrast, the characteristics of the power-frequency current signal from the metal sheath of the cable are stable, and the phase information is highly sensitive, providing a long-term stable indication capability [24]. This offers an effective complementary solution for diagnosing insulation defects in cable joints. In reference [22], the sheath current was simplified by decomposing it into induced and leakage components when constructing the lumped parameter model. However, the effect of the leakage component on the induced component was not considered, leading to errors in the calculation of the sheath current phase angle, which impacted diagnostic accuracy. To address this issue, this paper will analyze the effect of the leakage component on the induced component to improve diagnostic accuracy.

This study fully considers the operating behavior of HV cables in environments with intense electromagnetic fields and high voltage. The influence of the conductor is equivalently applied to the sheath. With this as a foundation, the variation pattern of the sheath current under defect conditions is further analyzed. The equivalent circuit of the cross-bonding grounding system for the HV cable sheath is shown in Figure 3. Here, I_Ln is the leakage current through the main insulation (A). Using the equivalent path of leakage current as the boundary, the sheath impedance is divided into the left sheath impedance Z_SLn and right sheath impedance Z_SRn; Z_Ln is the main insulation impedance; R_e and R_g are the grounding resistances (Ω); E_n is the phase voltage of the conductor; and E_Sn is the induced voltage of the sheath (kV). The subscript n indicates the cable number: A1, B2, C3, B1, C2, A3, C1, A2, and B3.

Based on the theory of conductive coupling, the induced voltage equations for the small cross-sectional metal sheath are derived, as shown in Equations (1)–(9).

(1)$E_{SA1}=-j\omega (I_A·L_S+(I_B+I_{SB1R}+I_{SB1L})·M_{AB}·+(I_C+I_{SC1R}+I_{SC1L})·M_{AC})·l_1$

(2)$E_{SB1}=-j\omega (I_B·L_S+(I_A+I_{SA1R}+I_{SA1L})·M_{AB}+(I_C+I_{SC1R}+I_{SC1L})·M_{BC})·l_1$

(3)$E_{SC1}=-j\omega (I_C·L_S+(I_A+I_{SA1R}+I_{SA1L})·M_{AC}+(I_B+I_{SB1R}+I_{SB1L})·M_{BC})·l_1$

(4)$E_{SA2}=-j\omega (I_A·L_S+(I_B+I_{SB2R}+I_{SB2L})·M_{AB}+(I_C+I_{SC2R}+I_{SC2L})·M_{AC})·l_2$

(5)$E_{SB2}=-j\omega (I_B·L_S+(I_A+I_{SA2R}+I_{SA2L})·M_{AB}+(I_C+I_{SC2R}+I_{SC2L})·M_{BC})·l_2$

(6)$E_{SC2}=-j\omega (I_C·L_S+(I_A+I_{SA2R}+I_{SA2L})·M_{AC}+(I_B+I_{SB2R}+I_{SB2L})·M_{BC})·l_2$

(7)$E_{SA3}=-j\omega (I_A·L_S+(I_B+I_{SB3R}+I_{SB3L})·M_{AB}+(I_C+I_{SC3R}+I_{SC3L})·M_{AC})·l_3$

(8)$E_{SB3}=-j\omega (I_B·L_S+(I_A+I_{SA3R}+I_{SA3L})·M_{AB}+(I_C+I_{SC3R}+I_{SC3L})·M_{BC})·l_3$

(9)$E_{SC3}=-j\omega (I_C·L_S+(I_A+I_{SA3R}+I_{SA3L})·M_{AC}+(I_B+I_{SB3R}+I_{SB3L})·M_{BC})·l_3$

where l₁, l₂, and l₃ are the lengths of the three cable sections. I_A, I_B, and I_C represent the load current phasors of the cable system under purely inductive load conditions. I_SnL is the current in the left-side sheath of the cable, and Z_SnR is the impedance of the right-side sheath. L_S is the inductance coefficient between the core conductor and its metal sheath. M_AB, M_AC, and M_BC are the mutual inductance coefficients between phases A and B, A and C, and B and C, respectively. For this cross-bonding sheath system, the vector form of the equations can be derived from Kirchhoff’s Current Law and Kirchhoff’s Voltage Law.

(10)$A_{\mathrm{m}}·I_S=0$

(11)$B_{\mathrm{fm}}·U_S=0$

In the equations, A_m is the circuit incidence matrix and B_fm is the fundamental loop matrix. I_S is the branch current vector and U_S is the branch voltage vector. The branch current matrix equation is:

(12)$U_{\mathrm{S}}=Z_{\mathrm{bm}}·I_S+Z_{\mathrm{bm}}·I_{\mathrm{sm}}-U_{\mathrm{sm}}$

In the equations, Z_bm is the impedance matrix of passive components, Ism is the current source phasor matrix, and the voltage source phasor matrix U_sm contains the induced voltage E_Sn and the phase voltage E_n. Since this network does not include independent current sources, we have:

(13)$I_{\mathrm{sm}}=0$

By solving the system of Equations (10)–(13), the sheath current vector is derived as:

(14)$I_S={\left[\begin{array}{c}{B}_{\mathrm{fm}}·Z_{\mathrm{bm}}\\ A_{\mathrm{m}}\end{array}\right]}^{-1}\left[\begin{array}{c}{B}_{\mathrm{fm}}\\ 0\end{array}\right]U_{\mathrm{sm}}$

2.2. Parameter Correction Based on Genetic Algorithm

The measurement data of conductor current, conductor voltage, and sheath current under normal conditions are collected, along with the factory parameters of the cable. The cable parameters are corrected by comparing the calculated sheath current with the measured values. However, the correction process can be affected by data acquisition errors and calculation errors. As a result, the parameters obtained at a single time point cannot represent the final correction result. To address this, a genetic algorithm is used to optimize the process and ensure the accuracy and reliability of the final corrected parameters. Genetic algorithms are well-suited for handling multi-parameter optimization problems and offer relatively fast convergence.

When optimizing using a genetic algorithm, the parameters to be adjusted include the impedance per unit length z, self-inductance per unit length l, mutual inductance per unit length m, capacitance per unit length c, and the three segment lengths of the cable d_n, where n represents the cross-bonding segment numbers 1, 2, and 3. The variation range for these parameters is ±10%. In the genetic algorithm, appropriate values for the maximum number of generations q, population size p_s, crossover probability p_c, and mutation probability p_m must be selected. The fitness function is based on the difference between the calculated sheath current and the actual measured values.

(15)$\tau =\min {\displaystyle \sum _{i=1}^N\left[{\left(I_{1a}^c-I_{1a}^m\right)}^2+{\left(I_{1b}^c-{{\rm I}}_{1b}^m\right)}^2\right.}+{\left(I_{1c}^c-{{\rm I}}_{1c}^m\right)}^2\left.+{\left(I_{2a}^c-{{\rm I}}_{2a}^m\right)}^2+{\left(I_{2b}^c-{{\rm I}}_{2b}^m\right)}^2+{\left(I_{2c}^c-{{\rm I}}_{2c}^m\right)}^2\right]$

In the formula, τ is the fitness function; N represents the number of samples; ${\mathit{I}}_{\mathit{x}}^{\mathit{c}}$ is the calculated value of the sheath grounding current; ${\mathit{I}}_{\mathit{x}}^{\mathit{m}}$ is the measured value of the sheath grounding current; and x denotes 1a, 1b, 1c, 2a, 2b, and 2c. The convergence condition for the line parameter correction calculation based on the genetic algorithm is the maximum evolution generation and the minimum value of the fitness function τ. Considering the error range, τ ≤ 1% is set as the convergence condition. Based on the above analysis, the process of parameter correction using the genetic algorithm is shown in Figure 4.

2.3. Calculation Model of Sheath Current Under Short-Circuit Defect in Two Sheath Loops

After the HV cable joint is invaded by moisture, the resistance of the inner surface of the joint significantly decreases, leading to discharge on the surface of the silicone rubber. As the discharge phenomenon intensifies, the epoxy preform is the first to be punctured, forming a carbonized channel, as shown in Figure 5. The electrical insulation failure between the sheath circuits causes a short-circuit defect between the two sheath circuits.

This paper builds on the sheath current derivation in Section 2.1 and the parameter correction in Section 2.2. A dual-iteration method is introduced to improve the lumped parameter iterative model proposed in [22]. The improved model solves the sheath current under joint defect conditions more effectively. The specific procedure is shown in Figure 6. First, the model parameters are corrected using cable data under normal operating conditions. Then, the first iteration is performed to obtain the induced sheath voltage E_Sn in the normal state, which is used as the input for the second iteration. In the second iteration, the relevant parameters in Equation (14) are replaced with those under the defect condition, including the impedance matrix ${\mathit{Z}}_{\mathrm{b}\mathrm{m}}^{\prime }$ after the sheath short-circuit, as well as the circuit incidence matrix ${\mathrm{A}}_{\mathrm{m}}^{\prime }$ and the basic loop matrix ${\mathit{B}}_{\mathrm{f}\mathrm{m}}^{\prime }$. The final output sheath current ${\mathit{I}}_{\mathrm{m}}^{″}$ is the accurate solution.

The initial value for the second iteration is the sheath-induced voltage under normal conditions, which ensures the convergence, speed, and stability of the iteration process, resulting in a more accurate sheath current value under defect conditions. To verify whether the double iteration model achieves the expected results, the results from the previous literature were replicated and compared. In the normal case, the phase differences of the grounded current phasors for the three sheath circuits were consistent between both models. In the case of a short-circuit between two sheath circuits, the short-circuit impedance is set to 1 Ω, as shown in Table 1. JA1 (loop1) represents the phase difference of the grounded current for sheath circuit 1 when the sheath circuit in joint JA1 is short-circuited, and so on. As can be seen, the deviation in the magnitude of the sheath current between the two iteration models is within 0.2 A, but the phase deviation can reach up to 2.4°.

3. Two-Loop Short-Circuit Discrimination Method Based on Sheath Current Phasor

The difficulty in diagnosing HV cable sheath defects lies in their physical invisibility. This paper addresses the issue of two sheath loop short circuits caused by the deterioration and breakdown of epoxy prefabricated components. First, based on the characteristic changes in sheath grounding current phasors under the two-loop short-circuit defect, it is selectively identified from among various sheath defects. Other influencing factors on the criteria are then considered. The distribution of sheath grounding current and leakage current in the cross-bonding interconnected sheath system is shown in Figure 7.

If the sheath system is considered as a whole, the sum of the currents flowing into the sheath system is zero. It can be derived that the phasor difference of the grounding current at both ends of the sheath loop is the sum of the leakage currents from the three small segments of the cable.

(16)$I_{2\mathrm{c}}-I_{1\mathrm{a}}=I_{\mathrm{LA}1}+I_{\mathrm{LB}2}+I_{\mathrm{LC}3}=I_{\mathrm{L}1}$

(17)$I_{2\mathrm{a}}-I_{1\mathrm{b}}=I_{\mathrm{LB}1}+I_{\mathrm{LC}2}+I_{\mathrm{LA}3}=I_{\mathrm{L}2}$

(18)${{\rm I}}_{2\mathrm{b}}-I_{1\mathrm{c}}=I_{\mathrm{LC}1}+I_{\mathrm{LA}2}+I_{\mathrm{LB}3}=I_{\mathrm{L}3}$

(19)${\displaystyle \sum I_{\mathrm{L}}}=I_{\mathrm{L}1}+I_{\mathrm{L}2}+I_{\mathrm{L}3}$

where, I_L1, I_L2, and I_L3 represent the phasor differences of the grounding current at both ends of sheath loops 1, 2, and 3, respectively, and ∑I_L is the total grounding current entering the sheath system. According to Equation (19), the total grounding current at both ends of the sheath system is equal to the combined leakage current from the three-phase main insulation of the HV cable. The impedance of each phase in the main insulation is nearly identical. The applied three-phase voltage is also balanced. Under these conditions, the total grounding current ∑I_L is approximately zero during normal operation. This feature enables the detection of grounding defects in the sheath system. If the grounding currents in two sheath loops are abnormal while ∑I_L remains close to zero, it can be initially inferred that the fault is within the sheath system. Possible causes include a short circuit between the two sheath loops or a misconnection in the cross-bonding configuration.

As an example, consider a short-circuit fault occurring at the left-side cross-bonding point between sheath loop 1 and loop 2, as shown in Figure 8. Define I_R as the current flowing through this defect point, with the direction from the sheath loop 1 to loop 2.

At this point, the phasor difference of the grounding current at both sides of loop 1 and loop 2 is

(20)$I_{2\mathrm{c}}-I_{1\mathrm{a}}=I_{\mathrm{L}1}+I_{\mathrm{R}}$

(21)$I_{2\mathrm{a}}-I_{1\mathrm{b}}=I_{\mathrm{L}2}-I_{\mathrm{R}}$

In the ideal case, the metal sheath of an HV cable is divided into equal-length segments and interconnected. This configuration eliminates the induced voltage on the sheath. However, in practical applications, factors such as environmental constraints, engineering conditions, and human errors make perfect segmentation difficult. As a result, the length differences between segments in operating cross-bonding HV cable systems are typically controlled within 50 m. Under normal conditions, the corresponding sheath grounding currents—I_L1, I_L2, and I_L3—are generally at the 0.1 A level. This defines the expected current range during healthy operation. When a defect current is large, the phasor differences of the grounding currents at both ends of loops 1 and 2 will appear approximately equal in magnitude but opposite in phase. A special case may also arise. Even in the absence of sheath grounding defects or short circuits, abnormal grounding currents can be observed. These are typically caused by construction errors leading to incorrect cross-bonding connections, as shown in Figure 9.

Such cross-bonding connection errors occur between the two loops. Taking the misconnection between sheath loop 1 and loop 2 as an example, the previously defined loops are altered, with I_1a and I_2a in the same loop, and I_1b and I_2c in the same loop. In other words, under the previous definition, by crossing and subtracting the grounding currents at both ends of the two loops, we obtain:

(22)$I_{2\mathrm{a}}-I_{1\mathrm{a}}=I_{\mathrm{LA}1}+I_{\mathrm{LC}2}+I_{\mathrm{LA}3}$

(23)$I_{2\mathrm{c}}-I_{1\mathrm{b}}=I_{\mathrm{LB}1}+I_{\mathrm{LB}2}+I_{\mathrm{LC}3}$

Since it is the current subtraction at both ends of the same branch, the induced currents cancel each other out. The result is the phasor sum of the leakage currents from the three small segments of the cable, as shown in Figure 10. In the phasor diagram, the phase of U_A is 0°, and the other quantities are referenced accordingly. The magnitude of the leakage current in each small segment is positively correlated with the length of the small cable segment. It can be observed that the phase angle difference of the new grounding current phasor is approximately 180°.

In summary, the analysis of the grounding current in HV cable sheaths enables effective identification and exclusion of grounding defects and sheath misconnection issues. As shown in the Figure 11, when circulating currents exceed the threshold—i.e., when the ratio of sheath grounding current to load current (K) is greater than 10%—and the total grounding current entering the sheath system (∑I_L) is approximately zero, it can be concluded that no grounding defect exists in the sheath system. Under these conditions, the difference between the grounded current at the left end of loop Y and the right end of loop X is computed as I_XY, and similarly, I_YX is obtained by reversing the direction of subtraction. These current differences are used for further diagnosis of potential misconnection or short-circuit defects.

If the magnitudes of the sheath grounding currents in the two loops are similar and their phase angles are nearly 180° apart, it indicates a misconnection between the two sheath loops. Otherwise, the phasor difference of the sheath grounding currents at both ends of the two loops, denoted as I_X and I_Y, is computed. If the resulting phasor difference also exhibits a small magnitude and an approximate 180° phase shift, it indicates a short-circuit defect between the two sheath loops with low impedance. In such cases, the grounding current phasors at both ends of the loops will have similar magnitudes and nearly opposite phase angles due to the strong coupling effect of the fault.

4. Example Analysis

The YJLW03-64/110-1×1200 type cable is selected in this study, and the relevant parameters are shown in Table 2. The sheath capacitance and metallic sheath impedance vary approximately linearly with temperature, with the operating temperature set at 70 °C. External strong electromagnetic interference is neglected, and the system is assumed to operate mainly under power-frequency steady-state and short-term transient conditions. The phase voltages in the cable system are U_A = 63.5∠0° kV, U_B = 63.5∠−120° kV, U_C = 63.5∠120° kV. The load currents in the conductors are I_A = 400∠−90° A, I_B = 400∠150° A, I_C = 400∠30° A. The ground impedance R_e and R_g are 0.2 Ω. The line adopts a three-phase in-line configuration with phases A, B, and C, where phase B is the middle phase and phases A and C are the outer phases. When correcting the cable parameters, the evolutionary generation number is set to q = 50, the population size to p_s = 1200, the crossover probability to p_c = 0.5, and the mutation probability to p_m = 0.2. Through parameter correction, the parameter error can be reduced to 0.6%.

The short-circuit fault locations of the two sheath defects may occur at the joints JB1, JC2, JA1, JB2, JC1, and JA2. The short-circuit impedance was set to 0.1 Ω. As shown in Figure 12a–g, under normal operating conditions, the sheath current phasor differences at both ends of loops 1, 2, and 3 are 0.43 A ∠−122.2°, 0.43 A ∠177.8°, and 0.43 A ∠−2.2°, respectively. Under normal conditions, the sheath current phase differences are consistent with the results in the literature [21].

When sheath circuits 1 and 2 are shorted, two defect scenarios are considered. If the fault occurs at joint JB1, the sheath current phasor difference is 111 A ∠106.7° for loop 1 and 110.9 A ∠−73.5° for loop 2, with a magnitude difference of 0.1 A and a phase angle difference of 180.2°. Similarly, if the defect is located at joint JC2, the phasor difference is 154.7 A ∠63.5° for loop 1 and 154.9 A ∠−116.7° for loop 2. Again, the amplitude difference is 0.1 A, and the phase angle difference is 180.2°.

When sheath circuits 1 and 3 are shorted, two fault locations are analyzed. If the defect occurs at joint JB1, the sheath current phasor difference is 146 A ∠61.2° for loop 1 and 146.2 A ∠−118.7° for loop 3, resulting in a magnitude difference of 0.2 A and a phase angle difference of 179.9°. If the defect occurs at joint JB2, the phasor difference is 110.7 A ∠12.2° for loop 1 and 110.6 A ∠−167.6° for loop 3, with a magnitude difference of 0.1 A and a phase angle difference of 179.8°.

When sheath circuits 2 and 3 are shorted, two fault scenarios are considered. If the defect occurs at joint JC1, the sheath current phasor difference is 115.8 A ∠101.7° for loop 2 and 115.5 A ∠−78.1° for loop 3, resulting in a magnitude difference of 0.3 A and a phase angle difference of 179.8°. If the defect occurs at joint JA2, the phasor difference for loop 1 is 102.9 A ∠−174.5°, while for loop 3 it is 103.2 A ∠5.7°, yielding a magnitude difference of 0.3 A and a phase angle difference of 180.2°. These results demonstrate that, even in the event of a short-circuit between any two sheath circuits, the fault location can still be accurately identified using the phase angle of the sheath current phasor difference.

In practical scenarios, the breakdown of the epoxy prefabricated part typically results in the formation of a carbonized channel. However, the corresponding short-circuit impedance R remains uncertain. To investigate the relationship between the sheath current phasor phase difference and the short-circuit impedance R, its value is varied from 1 to 200 Ω. The simulation results, with one representative circuit selected for analysis, are presented in Figure 13.

Figure 13a illustrates the short-circuit condition between sheath circuits 1 and 2. When the short-circuit impedance is below 115.4 Ω, the fault location can be effectively identified using the phase angle difference of the sheath current phasors. Specifically, if an insulation defect occurs at joint JB1, the phase angle difference in circuit 1 varies within two intervals: 106.7° to 180° and −180° to −169.3°. In contrast, if the insulation defect occurs at joint JC2, the phase angle variation range is 63.5° to 100.2°.

Figure 13b illustrates the short-circuit condition between sheath circuits 1 and 3. When the short-circuit impedance is below 200 Ω, the fault location can be accurately identified using the phase angle difference of the sheath current phasors. If an insulation defect occurs at joint JA1, the phase angle difference in circuit 1 falls within two distinct intervals: 61.2° to 180° and −180° to −123.7°. In comparison, when the insulation defect is located at joint JB2, the phase angle variation range is −98.7° to 12.2°.

Figure 13c illustrates the short-circuit condition between sheath circuits 2 and 3. When the short-circuit impedance is below 200 Ω, the fault location can be accurately identified using the phase angle difference of the sheath current phasors. Specifically, if an insulation defect occurs at joint JB1, the phase angle difference in circuit 2 falls within two distinct ranges: −174.5° to −180° and 136.1° to 180°. If the defect is located at joint JA2, the variation range is 63.5° to 100.2°. As the short-circuit impedance increases, the difference in sheath current phasors gradually converges toward the normal value. Compared to traditional amplitude-based methods, the phase angle-based approach offers superior accuracy and remains effective even when the short-circuit impedance exceeds 100 Ω.

5. Discussion

In engineering practice, maintaining uniform segment lengths during cable installation is often challenging. Additionally, fluctuations in load current, phase voltage, and ground impedance can occur. The following analysis uses a controlled variable approach to assess the influence of these factors.

To study the influence of the three segment lengths of the cross-bonding HV cables on the phase angle difference of the sheath current phasor, 200 random numbers between 400 and 500 m were generated, representing the lengths of three cable segments, l₁, l₂, and l₃. According to actual installation requirements, the difference in length between the two small segments does not exceed 50 m, simulating the case where the small cable segments are of unequal lengths. In the simulation, the load current RMS value is 400 A, the ground impedance at both ends of the line is 0.2 Ω, and the sheath short-circuit impedance is 0.1 Ω. It should be noted that Table 2 is based on an actual operating cable circuit. Due to engineering and construction constraints, the circuit exhibits large length differences. For this reason, it has become a key monitoring object in related studies.

Taking the short circuit between sheath circuits 1 and 2 as an example, the simulation results are shown in Figure 14. The variation in the phase angle difference of the sheath current phasor due to unequal small segment lengths is limited to between 0° and 5.3°. When the short-circuit fault occurs at JB1, the phase angle difference ranges from 105.5° to 110.8°, while for a fault at JC2, it ranges from 60.7° to 62.5°. These results indicate that the influence of unequal segment lengths is negligible, and the phase angle difference remains a reliable indicator for fault location identification within the two sheath circuits.

To investigate the influence of load current on the phase angle difference of the sheath current phasor, the phase voltage and ground impedance were kept constant. The short-circuit impedance was set to three levels (0.1 Ω, 20 Ω, and 80 Ω), and the load current varied between 300 A and 500 A. As shown in Figure 15, the short-circuit between sheath circuits 1 and 2 is analyzed as a representative example. When the short-circuit impedance is 0.1 Ω or 20 Ω, the phase angle difference of the sheath current phasors increases only slightly with rising load current. In contrast, at a short-circuit impedance of 80 Ω, the phase angle difference increases more significantly as the load current rises.

This behavior can be explained by the influence of short-circuit impedance on the dominant current components. When the short-circuit impedance is high, the phase angle difference of the sheath current phasor is primarily influenced by the leakage current component and becomes almost independent of the load current. In contrast, at lower short-circuit impedance, the phase angle difference is mainly determined by the short-circuit current component, which exhibits a positive correlation with the load current. As a result, the phase angle difference becomes more sensitive to variations in load current.

Although the phase angle difference of the sheath current phasor is significantly affected by load current at high short-circuit impedance, the load current has little impact on fault location identification when the impedance is below 115.4 Ω.

To investigate the effect of ground impedance on the phase angle difference of the sheath current phasor, the load current and phase voltage are kept constant. The ground impedance is varied from 0.1 Ω to 5.0 Ω, with the short-circuit impedance fixed at 0.1 Ω. Using the short-circuit condition between sheath circuits 1 and 2 as an example, Figure 16 illustrates the relationship between ground impedance and the sheath current phasor difference. As the ground impedance increases, the phase angle difference remains within 7°, indicating minimal impact on fault location identification. This is due to the resistive component of the sheath circuit impedance being much larger than the inductive reactance, resulting in limited phase angle variation

To examine the effect of phase voltage on the phase angle difference of the sheath current phasor, the load current and ground impedance were fixed at 400 A and 0.2 Ω, while the short-circuit impedance was set at 0.1 Ω. According to GB/T 12325, the operating voltage may fluctuate within ±10% [25]. Therefore, the line voltage was varied between 99 kV and 121 kV. As shown in Figure 17, simulation results indicate that such voltage fluctuations cause less than a 0.1° change in the sheath current phase angle difference. Overall, variations in phase voltage, ground impedance, and load current contribute no more than 7° of deviation. Moreover, when the short-circuit impedance is below 115.4 Ω, the fault location can still be accurately identified using phase angle information. In engineering applications, possible measurement errors should also be considered. To enhance robustness and reliability, it is recommended to adjust the phase angle margin appropriately in practical use.

The criterion proposed in this paper accurately identifies the location of the open joint when phase angle fluctuations are within 10° and the defect impedance is below 100 Ω. The corresponding diagnostic criteria are presented in Table 3. Initially, the short-circuit defect in the two sheath circuits is diagnosed in Figure 11. The defect location is then determined using the proposed diagnostic criterion. To validate the accuracy of this method, a three-section HV cable interconnection model with shielding was developed in PSCAD.

The simulation parameters were consistent with those listed in Table 2. Defect scenarios were modeled by varying the defect resistance from 0 to 100 Ω. The initial fault phase angle was set to 0°, the 110 kV voltage amplitude fluctuated within ±10%, the grounding impedance ranged from 0.1 to 5.0 Ω, and the load current varied between 300 and 500 A. For each type of defect, 1500 test samples were generated, and the corresponding phase information was analyzed to determine whether it satisfied the proposed diagnostic criteria. The results are shown in Table 4. The diagnostic criteria proposed in this paper achieve an overall accuracy of over 97%. Nevertheless, as the phase range specified in the criteria becomes broader, the probability of misclassification increases accordingly.

The robustness of the proposed method against measurement errors was evaluated by introducing noise into simulated current and voltage data. Results indicate that the method consistently locates fault positions with high accuracy, confirming the necessity of allowing margins in diagnostic criteria. As shown in Figure 18, at SNR levels of 20 dB and 30 dB, the diagnostic accuracy decreased by less than 2% compared to ideal conditions.

In engineering applications, 1000 datasets can be collected in intervals through an online monitoring system. A diagnosis is considered reliable when the confidence level for defect identification reaches 95%. If the probability of a certain fault type exceeds 5%, it should be continuously monitored during operation. Parameter calibration and iterative computation converge within 70–120 iterations, and the complete diagnostic process is completed in less than one minute. The defect localization step, including data processing and criterion comparison, requires approximately 30 s. The results demonstrate that the method combines robustness to noise with efficiency, making it suitable for real-time online monitoring.

6. Conclusions

This paper presents a novel diagnostic criterion for accurately identifying short-circuit faults in the cable sheath. The criterion is based on the phasor difference of sheath currents at both ends of the same circuit. The simulation and analysis results show that:

(1). Optimizing cable circuit parameters using a lumped-parameter dual-iteration model and a genetic algorithm enhances the effectiveness of the diagnostic criterion. This approach enables the precise localization of defective joints by detecting anomalies in sheath current.

(2). The criterion remains robust under various external disturbances, such as unequal cable segment lengths, load current fluctuations, ground impedance, and phase voltage variations.

(3). The method accurately diagnoses faulty joints when the phase angle fluctuation of the phasor difference is within 10° and the defect impedance is below 100 Ω. The conclusion has been validated through PSCAD simulations, with a diagnostic accuracy above 97%. Even under 20 dB noise interference, the error increase remains within 2%.

Author Contributions

Conceptualization, W.Y. and J.T.; methodology, R.S., X.G. and B.Y.; data curation, J.H. and M.J.; validation, Y.A. and Z.X.; writing—original draft preparation, H.W. and B.L.; writing—review and editing: W.Y. and J.T.; supervision: B.Y.; project administration: J.T. and Y.A.; funding acquisition: J.T. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data available on request.

Conflicts of Interest

Author Weihua Yuan, Zhanran Xia, Ruoxin Song, Xinyun Gao and Bin Yang were employed by the company Hubei Jiji Electric Power Group Company Ltd. Author Jing Tu, Yongheng Ai and Jianfeng He were employed by the company State Grid Wuhan Electric Power Supply Company. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Schematic diagram of grounding for cross-bonding interconnection of high voltage cable sheath.

Figure 2 Schematic diagram of sheath grounding current detection.

Figure 3 Equivalent circuit diagram of the sheath cross-bonding grounding system.

Figure 4 Flowchart of cable parameter correction.

Figure 5 Short-circuit defect in the epoxy resin joint preform.

Figure 6 Flowchart for sheath current calculation.

Figure 7 Schematic diagram of sheath ground current and leakage current distribution.

Figure 8 Schematic diagram of two sheath circuits short-circuiting.

Figure 9 Incorrect situation of the cross-bonding connection of two sheath circuits.

Figure 10 Phase of incorrect connection in two sheath loops.

Figure 11 Method for distinguishing short-circuit defects in two sheath loops.

Figure 12 The phase difference of the sheath current under a short-circuit fault of two sheaths at different positions.

Figure 13 (a) The variation of ∠I_2c−I_1a under the short-circuit condition of loop1 and loop2; (b) The variation of ∠I_2c−I_1a under the short-circuit condition of loop1 and loop3; (c) The variation of ∠I_2a−I_1b under the short-circuit condition of loop2 and loop3.

Figure 14 (a) The variation of ∠I_2c−I_1a under the short-circuit condition of JB1; (b) The variation of ∠I_2c−I_1a under the short-circuit condition of JC2.

Figure 15 (a) The variation of ∠I_2a−I_1b under the short-circuit condition of JB1; (b) The variation of ∠I_2a−I_1b under the short-circuit condition of JC2.

Figure 16 (a) The variation of ∠I_2a−I_1b under the short-circuit condition of JB1; (b) The variation of ∠I_2a−I_1b under the short-circuit condition of JC2.

Figure 17 (a) The variation of ∠I_2a−I_1b under the short-circuit condition of JB1; (b) The variation of ∠I_2a−I_1b under the short-circuit condition of JC2.

Figure 18 Localization error with different signal-to-noise ratios (SNRs).

Table 1

Comparison of calculation results between the two iterative models.

Defect Location	Single Iteration Model	Dual Iteration Model	Phase Angle Deviation
JA1 (loop1)	32.3∠103.7° A	32.4∠104.8° A	1.1°
JB2 (loop1)	45.4∠58.5° A	45.6∠60.1° A	1.6°
JC1 (loop1)	39.8∠59.5° A	40∠58.9° A	0.6°
JA2 (loop1)	29.5∠10.1° A	29.7∠9.5° A	0.6°
JB1 (loop2)	27.7∠−170.9° A	27.9∠−173.3° A	2.4°
JC2 (loop2)	31.7∠103.1° A	31.5∠103.6° A	0.5°

Table 2

Parameters of the 110 kV cable in simulation.

Parameter/Unit	Value
Conductor outer diameter (m)	38.9 × 10−2
Main insulation thickness (m)	16 × 10−2
Inner diameter of metallic sheath (m)	97.3 × 10−2
Outer diameter of metallic sheath (m)	99.6 × 10−2
Cable outer diameter (m)	110 × 10−2
Phase AB spacing (m)	0.27
Phase BC spacing (m)	0.27
Phase AC spacing (m)	0.54
Section 1 length/m	425
Segment 2 length/m	477
Segment 3 length/m	536
Phase and sheath mutual inductance/H·m−1	2.4 × 10−4∠90°
Mutual inductance between AB phase/H·m−1	8.2 × 10−5∠90°
Mutual inductance between AC phases/H·m−1	3.9 × 10−5∠90°
Sheath impedance/Ω·m−1	2.1 × 10−4∠78°
Grounding impedance/Ω	0.1

Table 3

Diagnosis criteria for two shielding short-circuit faults.

Defect Location	Diagnostic Criteria
JA1	The K value of Loop 1 and 2 ≥ 10%;The angle of IL1 falls within (61.2° ± 10°, 68.2° ± 10°).
JB2	The K value of Loop 1 and 2 ≥ 10%;The angle of IL1 falls within (12.2° ± 10°, 65.9° ± 10°).
JC1	The K value of Loop 1 and 3 ≥ 10%;The angle of IL1 falls within (149.1° ± 10°, 185.5° ± 10°).
JA2	The K value of Loop 1 and 3 ≥ 10%;The angle of IL1 falls within (101.7° ± 10°, 112.3° ± 10°).
JB1	The K value of Loops 2 and 3 ≥ 10%;The angle of IL2 falls within (106.7° ± 10°, 180.5° ± 10°).
JC2	The K value of Loops 2 and 3 ≥ 10%;The angle of IL2 falls within (63.5° ± 10°, 65.4° ± 10°).

Table 4

Results of simulation verification.

Status Number	Test Sample Count	Criterion Accuracy (%)
JA1	1500	99.00
JB2	1500	98.13
JC1	1500	97.73
JA2	1500	98.70
JB1	1500	97.07
JC2	1500	99.87

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