Zero-Sequence Current Limitation of Parallel-Laid HV Cable Sheathing Based on Phase Sequence Optimization

Abstract

Parallel laying of high-voltage cables will generate a zero-sequence current, due to spatial electromagnetic induction, which reduces the cable’s current-carrying capacity, causing heating and corrosion of the grounding points and deteriorating grounding performance. Currently, there is a lack of effective control measures. This article establishes a calculation model for the cable sheath current under the condition of double circuit cable cross interconnection grounding, analyzes the causes of a zero-sequence grounding current in a double circuit cable sheath, and proposes an optimal phase sequence selection method, considering load changes with the goal of maximizing the probability of the cable sheath current, not exceeding the standard. The results show that when the double circuit cable is evenly distributed in the cross interconnection section, the zero-sequence grounding current will be generated on the metal sheath of the cable, causing an excessive total grounding current. By applying the proposed probability-based phase-sequence optimization, the likelihood that both circuits simultaneously satisfy the sheath-current criterion can be significantly improved; for example, under representative layouts and load distributions, the “both-within-limit” probability can reach 53.3% (horizontal layout), 76.2% (horizontal equilateral triangle layout), 90.5% (vertical layout), and 81.6% (vertical equilateral triangle layout). For different working conditions, selecting the optimal load phase sequence combination by maximizing the probability of the sheath current and not exceeding the standard within the current carrying area can help to reduce the cable sheath current.

1. Introduction

Power cables offer advantages such as high supply reliability, convenient operation and maintenance, immunity to climatic conditions, and superior safety, making them widely adopted in urban power transmission and distribution networks [1]. Concurrently, as cities continue to expand and land resources become increasingly scarce, underground power cables—which eliminate the need for overhead corridors, conserve land, and enhance urban esthetics—have seen their proportion in urban power transmission lines steadily rise [2,3]. By 2019, the total length of high-voltage cables in Beijing reached 1800 km. Newly constructed lines above 110 kV in other major cities’ power grids also predominantly utilize power cables, with Shenzhen achieving a 100% rate of using cables for its high-voltage transmission lines [4]. Compared to single-circuit configurations, parallel cable installation within the same corridor conserves land and enhances transmission capacity [5]. However, complex electromagnetic interactions between double-circuit cables affect the magnitude of grounding currents in cable sheaths [6]. Significant grounding currents increase operational losses, cause cable heating, reduce current-carrying capacity, and, in severe cases, may burn grounding wires and grounding boxes, compromising the safe and stable operation of power systems [7,8].

There have been more studies on the analysis of single-circuit cable sheath grounding currents, and fewer studies on the grounding currents of double-circuit cable sheaths laid in tunnels. Studies [9,10,11,12,13] analyzed the effect of the laying method, cross-interconnection segmentation non-uniformity, metal sheath parameters, grounding resistance and earth resistivity, and other factors on the single-circuit cable sheath grounding current. Study [14], with the condition of a cable load of 292 A, calculated the loop current in the cable sheath under the double-circuit laying straight line arrangement. Study [15] used the cable trench-laying multi-circuit cable line sheath ring current calculation software to study the cable sheath’s current influencing factors; cable laying to ensure that the metal sheath cross-connections of the three sections of equal length can effectively reduce the sheath current. Studies [16,17] analyzed the effect of uneven cross-interconnection segmentation on the sheath current of single-circuit and double-circuit cables, using a mixed arrangement, and found that uneven segmentation causes a significantly larger sheath current. Study [18] analyzed the influence of parallel and triangular laying methods on the sheath current of multi-return cables under the load current of 412 A, and proposed the optimal phase sequence combination under this condition.

The current research has largely focused on the issue of an increased sheath current resulting from asymmetric cable installation methods, and due to the mutual influence of the two circuits, even if each cable cross-interconnection segmentation is equal, the sheath-induced voltage is not balanced, and at this point, a zero-sequence sheath current is generated in each circuit of the cable sheath. Furthermore, the current research has only analyzed double-circuit cables under a fixed load current condition, without considering the impact of load current variations on the sheath’s grounding current. Due to the interaction between the two cables, the optimal phase sequence obtained under different loading conditions of the double-circuit cable may be different.

This paper establishes a calculation model for cable sheath currents in double-circuit cables, employing a cross-interconnected grounding method, and proposes an optimal phase sequence selection method that accounts for variations in load currents within the double-circuit cables.

2. Optimal Phase Sequence Selection Method When Considering Changes in Load Currents of Double Circuit Cables

2.1. Calculation Model of Zero-Sequence Sheath Current for Double-Circuit Cables

High-voltage cables rated at 110 kV and above typically employ a single-core structure [19]. The load current flowing through the cable core generates an alternating magnetic field around it, inducing an electromotive force in the metal sheath under the influence of this alternating magnetic field. To mitigate safety risks from excessive induced voltages, long cable lines require segmented, cross-interconnected grounding of the metal sheath [20]. This configuration forms a loop between the metal sheath and the earth, generating the grounding current. Additionally, the capacitive current flows when the metal sheath is grounded, though its magnitude is negligible and can be disregarded in calculations [16,17,18,19,20,21,22]. This paper focuses on the induced grounding current in cable metal sheaths. Schematic diagrams of the cross-interconnected grounding method and equivalent circuit for the cable metal sheath are shown in Figure 1 and Figure 2.

In Figure 2, R_d1 and R_d2 represent the grounding resistances at both ends of the cross-interconnection, R_e denotes the earth resistance, Z_i is the self-impedance of each phase’s metallic sheath, I_si is the induced circulating current in the metallic sheath, ${\dot{E}}_{\mathrm{s}i}^{\prime }$ represents the induced voltage generated on each cable’s metallic sheath by the sheath circulation current, E_si is the induced voltage generated by the core conductor current on each cable’s metallic sheath, and the subscript i indicates the identification number of the six cables in the double-circuit system. Let the segment lengths of the cable cross-interconnection be L₁, L₂, and L₃. Based on the equivalent circuit, the following equations can be derived:

$$\left\{\begin{array}{l}{Z}_1{\dot{I}}_{\mathrm{s}1}+(R_{\mathrm{d}1}+R_{\mathrm{d}2}+R_{\mathrm{e}}){\dot{I}}_{\mathrm{se}}+{\dot{E}}_{\mathrm{s}1}^{\prime }={\dot{E}}_{\mathrm{s}1}\hfill \\ Z_2{\dot{I}}_{\mathrm{s}2}+(R_{\mathrm{d}1}+R_{\mathrm{d}2}+R_{\mathrm{e}}){\dot{I}}_{\mathrm{se}}+{\dot{E}}_{\mathrm{s}2}^{\prime }={\dot{E}}_{\mathrm{s}2}\hfill \\ Z_3{\dot{I}}_{\mathrm{s}3}+(R_{\mathrm{d}1}+R_{\mathrm{d}2}+R_{\mathrm{e}}){\dot{I}}_{\mathrm{se}}+{\dot{E}}_{\mathrm{s}3}^{\prime }={\dot{E}}_{\mathrm{s}3}\hfill \\ Z_4{\dot{I}}_{\mathrm{s}4}+(R_{\mathrm{d}1}+R_{\mathrm{d}2}+R_{\mathrm{e}}){\dot{I}}_{\mathrm{se}}+{\dot{E}}_{\mathrm{s}4}^{\prime }={\dot{E}}_{\mathrm{s}4}\hfill \\ Z_5{\dot{I}}_{\mathrm{s}5}+(R_{\mathrm{d}1}+R_{\mathrm{d}2}+R_{\mathrm{e}}){\dot{I}}_{\mathrm{se}}+{\dot{E}}_{\mathrm{s}5}^{\prime }={\dot{E}}_{\mathrm{s}5}\hfill \\ Z_6{\dot{I}}_{\mathrm{s}6}+(R_{\mathrm{d}1}+R_{\mathrm{d}2}+R_{\mathrm{e}}){\dot{I}}_{\mathrm{se}}+{\dot{E}}_{\mathrm{s}6}^{\prime }={\dot{E}}_{\mathrm{s}6}\hfill \end{array}\right.$$

(1)

In Equation (1), the remaining terms, ${\dot{E}}_{\mathrm{s}}$, ${\dot{E}}_{\mathrm{T}}$, ${\dot{E}}_{\mathrm{T}}^{\prime }$, and Z, can be similarly derived:

$${\dot{E}}_{\mathrm{s}1}={\dot{E}}_{\mathrm{T}1}{L}_1+{\dot{E}}_{\mathrm{T}2}{L}_2+{\dot{E}}_{\mathrm{T}3}{L}_3$$

(2)

$$\begin{array}{c}{\dot{E}}_{\mathrm{T}1}=j(X_{11}{\dot{I}}_1+X_{12}{\dot{I}}_2+X_{13}{\dot{I}}_3+\\ X_{14}{\dot{I}}_4+X_{15}{\dot{I}}_5+X_{16}{\dot{I}}_6)\end{array}$$

(3)

$${\dot{E}}_{\mathrm{s}1}^{\prime }={\dot{E}}_{\mathrm{T}1,1}^{\prime }{L}_1+{\dot{E}}_{\mathrm{T}1,2}^{\prime }{L}_2+{\dot{E}}_{\mathrm{T}1,3}^{\prime }{L}_3$$

(4)

$$\begin{array}{c}{\dot{E}}_{\mathrm{T}1,1}^{\prime }=\mathrm{j}(X_{12}{\dot{I}}_{\mathrm{s}2}+X_{13}{\dot{I}}_{\mathrm{s}3}+X_{14}{\dot{I}}_{\mathrm{s}4}+\\ X_{15}{\dot{I}}_{\mathrm{s}5}+X_{16}{\dot{I}}_{\mathrm{s}6})\hfill \end{array}$$

(5)

$${\dot{I}}_{\mathrm{se}}={\dot{I}}_{\mathrm{s}1}+{\dot{I}}_{\mathrm{s}2}+{\dot{I}}_{\mathrm{s}3}+{\dot{I}}_{\mathrm{s}4}+{\dot{I}}_{\mathrm{s}5}+{\dot{I}}_{\mathrm{s}6}$$

(6)

$$Z_1=(R_1+\mathrm{j}{X}_{11})(L_1+L_2+L_3)$$

(7)

$$R_{\mathrm{e}}={\pi }^{2}f(L_1+L_2+L_3)\times {10}^{-7}$$

(8)

$$X_{ii}=4\pi f\times {10}^{-7}\ln \left(D_{\mathrm{e}}/r_{\mathrm{GMR}}\right)$$

(9)

$$X_{ij}=4\pi f\times {10}^{-7}\ln \left(D_{\mathrm{e}}/d_{ij}\right)$$

(10)

Equations (2)–(10): ${\dot{E}}_{\mathrm{Ti}}$ is the induced voltage generated per unit length on the i-th cable metal sheath by the current flowing through the conductor; ${\dot{E}}_{\mathrm{T}\mathrm{i},\mathrm{j}}^{\prime }$ is the induced voltage on the metal sheath of the i-th cable per unit length in the j-th cross-interconnection section, due to the sheath current; ${\dot{I}}_{\mathrm{i}}$ is the load current of the i-th cable; ${\dot{I}}_{\mathrm{s}\mathrm{e}}$ is the earth loop current; X_ii is the self-inductance per unit length of the metal sheath; X_ij is the mutual inductance per unit length between two cables; f is the frequency; D_e is the equivalent earth loop depth; ${\mathrm{r}}_{\mathrm{GMR}}$ is the geometric mean radius of the cable sheath; and d_ij is the center-to-center distance between the two cables [23,24,25]. For coaxial cables, the mutual inductance between the conductor and sheath equals the sheath’s self-inductance.

Equation (1) can be transformed into the relationship between the two-circuit cable sheath currents and the core currents, as follows:

$$\mathit{A}\left[\begin{array}{l}{\dot{I}}_{\mathrm{s}1}\hfill \\ {\dot{I}}_{\mathrm{s}2}\hfill \\ {\dot{I}}_{\mathrm{s}3}\hfill \\ {\dot{I}}_{\mathrm{s}4}\hfill \\ {\dot{I}}_{\mathrm{s}5}\hfill \\ {\dot{I}}_{\mathrm{s}6}\hfill \end{array}\right]=\mathit{B}\left[\begin{array}{l}{\dot{I}}_1\hfill \\ {\dot{I}}_2\hfill \\ {\dot{I}}_3\hfill \\ {\dot{I}}_4\hfill \\ {\dot{I}}_5\hfill \\ {\dot{I}}_6\hfill \end{array}\right]=\left[\begin{array}{l}{\dot{E}}_{\mathrm{s}1}\hfill \\ {\dot{E}}_{\mathrm{s}2}\hfill \\ {\dot{E}}_{\mathrm{s}3}\hfill \\ {\dot{E}}_{\mathrm{s}4}\hfill \\ {\dot{E}}_{\mathrm{s}5}\hfill \\ {\dot{E}}_{\mathrm{s}6}\hfill \end{array}\right]$$

(11)

In Equation (11), the expression for matrix A is given in Appendix Equations (A1)–(A5), and the expression for matrix B is given in Appendix Equations (A6)–(A10).

When cables are laid within a tunnel, the installation location of the cross-interconnection box is less constrained by spatial positioning. This allows for equal lengths across all three sections of the cross-interconnection, with each section designated as length L. At this point, the induced voltage generated by the load current on each cable sheath can be calculated as follows:

$$\begin{array}{c}{\dot{E}}_{\mathrm{s}1}={\dot{E}}_{\mathrm{s}2}={\dot{E}}_{\mathrm{s}3}=jL(X_{11}^{\prime }{\dot{I}}_1+X_{12}^{\prime }{\dot{I}}_2+\\ X_{13}^{\prime }{\dot{I}}_3+X_{14}^{\prime }{\dot{I}}_4+X_{15}^{\prime }{\dot{I}}_5+X_{16}^{\prime }{\dot{I}}_6)\end{array}$$

(12)

$$\begin{array}{c}{\dot{E}}_{\mathrm{s}4}={\dot{E}}_{\mathrm{s}5}={\dot{E}}_{\mathrm{s}6}=jL(X_{21}^{\prime }{\dot{I}}_1+X_{22}^{\prime }{\dot{I}}_2+\\ X_{23}^{\prime }{\dot{I}}_3+X_{24}^{\prime }{\dot{I}}_4+X_{25}^{\prime }{\dot{I}}_5+X_{26}^{\prime }{\dot{I}}_6)\end{array}$$

(13)

In Equations (12) and (13):

$$\left\{\begin{array}{cc}\begin{array}{c}{X}_{1m}^{\prime }=X_{1m}+X_{2m}+X_{3m}\\ X_{2m}^{\prime }=X_{4m}+X_{5m}+X_{6m}\end{array}& m=1,2,\dots ,6\end{array}\right.$$

(14)

From Equations (12) and (13), it can be seen that when the magnitude of the core current load or the phase sequence changes, the induced voltage generated by the core current on the metal sheath of the double-circuit cables will vary. However, the induced voltage on the sheaths of each circuit’s three phases remains equal. At this point, the sheath current expression for each cable can be transformed into the following equation:

$$\left[\begin{array}{l}{\dot{I}}_{\mathrm{s}1}\hfill \\ {\dot{I}}_{\mathrm{s}2}\hfill \\ {\dot{I}}_{\mathrm{s}3}\hfill \\ {\dot{I}}_{\mathrm{s}4}\hfill \\ {\dot{I}}_{\mathrm{s}5}\hfill \\ {\dot{I}}_{\mathrm{s}6}\hfill \end{array}\right]={\mathit{A}}^{-1}\mathit{B}\left[\begin{array}{l}{\dot{I}}_1\hfill \\ {\dot{I}}_2\hfill \\ {\dot{I}}_3\hfill \\ {\dot{I}}_4\hfill \\ {\dot{I}}_5\hfill \\ {\dot{I}}_6\hfill \end{array}\right]={\mathit{A}}^{-1}\left[\begin{array}{l}{\dot{E}}_{\mathrm{s}1}\hfill \\ {\dot{E}}_{\mathrm{s}1}\hfill \\ {\dot{E}}_{\mathrm{s}1}\hfill \\ {\dot{E}}_{\mathrm{s}4}\hfill \\ {\dot{E}}_{\mathrm{s}4}\hfill \\ {\dot{E}}_{\mathrm{s}4}\hfill \end{array}\right]$$

(15)

When the cable cross-interconnection segments are of equal length, matrix A can be transformed into the following form:

$$\mathit{A}=\left[\begin{array}{cccccc}{Z}_1^{\prime }& Z_2^{\prime }& Z_2^{\prime }& Z_3^{\prime }& Z_5^{\prime }& Z_4^{\prime }\\ Z_2^{\prime }& Z_1^{\prime }& Z_2^{\prime }& Z_4^{\prime }& Z_3^{\prime }& Z_5^{\prime }\\ Z_2^{\prime }& Z_2^{\prime }& Z_1^{\prime }& Z_5^{\prime }& Z_4^{\prime }& Z_3^{\prime }\\ Z_3^{\prime }& Z_4^{\prime }& Z_5^{\prime }& Z_7^{\prime }& Z_6^{\prime }& Z_6^{\prime }\\ Z_5^{\prime }& Z_3^{\prime }& Z_4^{\prime }& Z_6^{\prime }& Z_7^{\prime }& Z_6^{\prime }\\ Z_4^{\prime }& Z_5^{\prime }& Z_3^{\prime }& Z_6^{\prime }& Z_6^{\prime }& Z_7^{\prime }\end{array}\right]$$

(16)

The expressions for the individual elements in Equation (16) are given in Appendix Equations (A11)–(A17).

Analyzing the characteristics of matrix A in Equation (16), it can be found that the result of summing the elements of each row of rows 1~3 and rows 4~6 of matrix A are equal, and the sum of elements in each of the columns 1~3 and the columns 4~6 is equal, and since matrix A is a symmetric array, the result of summing the elements of each row of the rows 1~3 and rows 4~6 of matrix A⁻¹ is still equal, respectively. Therefore, the result of Equation (15) has the same elements in the first three rows and the same elements in the last three rows, which are as follows:

$$\left\{\begin{array}{c}{\dot{I}}_{\mathrm{s}1}={\dot{I}}_{\mathrm{s}2}={\dot{I}}_{\mathrm{s}3}\\ {\dot{I}}_{\mathrm{s}4}={\dot{I}}_{\mathrm{s}5}={\dot{I}}_{\mathrm{s}6}\end{array}\right.$$

(17)

From Equation (17), it can be seen that the three-phase sheath currents of each circuit of cables are, respectively, the same, characterized by a zero-sequence current, and the total current of the grounding line of each cable is the sum of the three-phase sheath currents.

Equation (17) indicates that, under the equal-length condition of the three cross-interconnection sections (L₁ = L₂ = L₃), the magnitudes of the three-phase sheath currents in each circuit tend to be identical, which implies that the zero-sequence component is dominant. Therefore, even if the magnitude of the single-phase sheath current is not large, the total grounding conductor current may still increase significantly, due to the summation effect.

The formation of the zero-sequence sheath current is closely related to the coupling parameters of the multi-conductor system. The key contributing factors can be summarized as follows: (1) the layout and cable spacings determine the mutual impedance and mutual inductance terms, thereby changing the superposition of induced sheath voltages among phases and affecting the zero-sequence component; (2) double-circuit coupling causes the induced sheath voltage to include contributions from both the local circuit and the adjacent circuit, so a pronounced zero-sequence component may still occur even when a single circuit is geometrically symmetric; (3) the cross-interconnection segment lengths (L₁, L₂, L₃) govern the sectional accumulation of line parameters and the integral effect of induced sheath voltages, where the equal-length condition (L₁ = L₂ = L₃) strengthens the matrix symmetry and drives the three-phase sheath currents in each circuit to become nearly identical, i.e., zero-sequence dominance; and (4) the grounding parameters affect the damping of the zero-sequence return path and thus the magnitudes of I₀ and the total grounding conductor current, I_g. Together, these factors determine the variation in the zero-sequence sheath current under different layouts, load conditions, and phase sequences, and provide the physical basis for the subsequent probability assessment and phase-sequence optimization.

After determining the parameters of each quantity in Equation (1), the calculation program can be written to calculate the change in the cable metal sheath current and grounding current under different working conditions.

2.2. Analysis of Zero-Sequence Currents in the Sheath of Parallel-Laid High-Voltage Cables

For a given circuit, let the sheath currents of phases A, B, and C be I_sA, I_sB, and I_sC, respectively. The zero-sequence sheath current can then be expressed as

$$I_0={\displaystyle \frac{I_{sA}+I_{sB}+I_{sC}}{3}}$$

(18)

Accordingly, the total grounding conductor current (corresponding to the Phase-T current measured by the monitoring platform in this paper) is the sum of the three-phase sheath currents,

$$I_g=I_{sA}+I_{sB}+I_{sC}=3I_0$$

(19)

In particular, the operational criterion commonly adopted in practice, |I_g| < 100 A, can be equivalently converted to |I₀| < 33.3 A. This criterion is used in the subsequent probability assessment and phase-sequence optimization to judge whether the sheath current exceeds the limit.

The two cables of 220 kV, Line A and Line B, are laid in parallel in the tunnel, and their arrangement is shown in Figure 3. According to the grounding current monitoring platform data, the three-phase sheath current values for each circuit of the Line A and Line B are essentially consistent; the total cable sheath grounding current is about equal to the sum of the three-phase sheath current; and the maximum grounding current reaches 208 A, exceeding the relevant electric power regulations [20,21] on the cable grounding current, which cannot be more than 100 A and belongs to the abnormal state. A field inspection of the cable section revealed that the cross-interconnection box wiring was correct, all cable segments were of equal length, and the outer sheath showed no visible damage. Power was shut off for maintenance on Line A. As shown in

Table 1. Current measurements at the joints of Line A when Line A is de-energized.

Joint Number	Phase A Current/A	Phase B Current/A	Phase C Current/A	Phase T Current/A
9	10.58	11.86	4.74	14
12	48.39	54.69	48.78	156.4
15	55.33	57.07	56.16	169.56

, on-site testing revealed that the total ground current on Line A remained as high as 170 A during the power-off state. By applying the aforementioned model to analyze the actual engineering project in this area, the validity of the model is verified.

In this study, for all investigated layouts, the cable geometric parameters, the cross-interconnection segment lengths L₁–L₃, and the grounding conditions are kept consistent with the engineering case. To ensure reproducibility, the sheath currents and the associated evaluation indices are obtained from Equations (1)–(17), following a unified procedure: first, the layout geometry, cross-interconnection segment lengths, and grounding-related parameters are specified; second, the self- and mutual impedances of the metallic sheaths (including the distance-dependent mutual-inductance terms) are calculated, based on the cable spacings, and used to assemble the impedance matrices; third, Equation (1) is solved to obtain the sheath currents of all cables; fourth, the zero-sequence sheath current, I₀, and the total grounding conductor current, I_g, are computed from the three-phase sheath currents; finally, the compliance criterion |I₀| < 33.3 A, which is equivalent to |I_g| < 100 A is applied to statistically evaluate the results under different operating conditions and phase sequences, which then serves as the basis for the subsequent probability assessment and phase-sequence optimization.

Figure 3b shows the numbering of the six cables in two circuits, labeled 1–6. Line A is arranged in a triangular pattern at the top, with 200 mm phase spacing. Line B is arranged in an isosceles triangle pattern at the bottom, with a center-to-center distance of 350 mm between cables 4 and 6. The four cable supports are evenly spaced at 500 mm intervals. The phase sequence for cables 1–6 is ABC/ABC. The cross-interconnected segment length of the cable line is 520 m, with a grounding resistance of 0.2 Ω and a soil resistivity of 300 Ω·m. Cable model for Line A: ZR-YJLW02-127/220 kV−1 × 1600 mm². Cable model for Line B: ZR-YJLW02-127/220 kV−1 × 2500 mm². Parameters for both cable circuits are shown in

Table 2. Cable parameters.

Parameter	Line A	Line B
Conductor cross-sectional area/mm2	1600	2500
Conductor outer diameter/mm	48.5	61.2
Internal shielding thickness/mm	2.0	2.0
Insulation thickness/mm	24.0	24.0
Outer shield thickness/mm	1.0	1.0
Metal sheath thickness/mm	2.6	2.8
Outer sheath thickness/mm	5.0	5.0
Cable outer diameter/mm	135.2	149.3
AC resistance/Ω/km	0.0254	0.0187
Maximum load-carrying capacity/A	1800	2400

.

Based on the above parameters, the cable sheath currents for Line A and Line B were calculated, assuming balanced three-phase currents. The load for Line A was set as I₁ and for Line B as I₂. The calculation results are shown in

Table 3. Calculation results when I₁ = 439 A and I₂ = 475 A.

Location		Current Value/A
		Measured Value	Calculated Value	Error/%
Line A	Phase A	51.7	48.8 ∠ 57.1°	5.6
	Phase B	45.4	48.8 ∠ 57.1°	7.5
	Phase C	46.1	48.8 ∠ 57.1°	5.9
	Phase T	143.8	146.3 ∠ 57.1°	1.7
Line B	Phase A	44.1	47.7 ∠ -122.2°	7.5
	Phase B	49.1	47.7 ∠ -122.2°	2.9
	Phase C	48.0	47.7 ∠ -122.2°	0.6
	Phase T	139.7	143.0 ∠ -122.2°	2.4

and

Table 4. Calculation results when I₁ = 0 A and I₂ = 688 A.

Location		Current Value/A
		Measured Value	Calculated Value	Error/%
Line A	Phase A	57.2	58.5 ∠ 56.6°	2.2
	Phase B	58.4	58.5 ∠ 56.6°	0.2
	Phase C	57.6	58.5 ∠ 56.6°	1.6
	Phase T	170.1	175.6 ∠ 56.6°	3.2
Line B	Phase A	50.1	54.7 ∠ -121.3°	9.2
	Phase B	54.3	54.7 ∠ -121.3°	0.7
	Phase C	52.1	54.7 ∠ -121.3°	5.0
	Phase T	156.2	164.1 ∠ -121.3°	5.1

.

As shown in Table 2 and Table 3, the theoretical calculated values are consistent with the measured values, verifying the accuracy of the theoretical calculation model. Calculations indicate that under the conditions of uniform cross-interconnection segment length, zero-sequence currents are generated in the metallic sheath, resulting in a relatively large total three-phase grounding current. Even when one circuit is powered off, a significant zero-sequence sheath current is still induced in the cable line of that circuit.

2.3. Optimal Phase Sequence Selection Method

The “Power Cable Line Testing Regulations” [21] stipulate that “the absolute value of grounding current in single-core cable lines shall be less than 100 A.” Since the grounding conductor current is the sum of the three-phase sheath currents, the total three-phase grounding conductor current can only be guaranteed to be less than 100 A when the zero-sequence sheath current is below 33.3 A. Excessive cable sheath currents can cause cable heating, and, in severe cases, may burn the grounding wire or even cause fires. When optimizing the phase sequence, the primary consideration should be ensuring cable sheath currents remain within limits.

Figure 4 is generated from model-based batch computations, rather than directly from field-monitoring data. Specifically, the analytical sheath-current model for double-circuit cables with cross-interconnection and grounding conditions (Equations (1)–(17)) is implemented in MATLAB R2025a (Windows version). The geometric parameters, cross-interconnection segment lengths L1–L3, and grounding conditions are kept identical to the engineering case described in Section 2.2. A grid sweep is then performed over the load –current pair (I₁, I₂) within the allowable ampacity ranges of the two circuits. For each operating point, the equation set is solved to obtain the sheath-current magnitudes of both circuits. The computed results are mapped onto the I₁–I₂ plane to form the heatmap shown in Figure 4. The boundary indicated by the white dashed line corresponds to the sheath-current compliance criterion, i.e., |I₀| < 33.3 A, which is equivalent to |I_g| < 100 A; this line separates the “within-limit” region from the “exceeding-limit” region.

Figure 4 illustrates how sheath currents vary under different load combinations for two horizontally laid cable circuits. The white dashed line represents the load current combinations corresponding to a sheath current of 33.3 A. The area bounded by the white line and the coordinate axes containing the origin corresponds to the condition where the sheath current is below 33.3 A.

We calculate the probability that the sheath current is less than 33.3 A at this time, and select the phase sequence with the highest probability that the sheath current of both lines will not exceed the standard as the optimal phase sequence. When constructing new cable lines, there is no actual load curve; the load current of two cables at any given time is uncertain. Assuming that the load currents of the two circuits are uniformly distributed within the maximum current-carrying capacity range, the probability of load currents occurring at any point within the current-carrying capacity range is equal. At this point, the area formed by the load current when the sheath current is less than 33.3 A is divided by the total area within the load-carrying capacity range; you can find out the probability that the sheath current does not exceed the standard under this phase’s sequence conditions.

For the actual cable line, we already have a prediction of the load on the line. At this time, the load current combination can be taken every hour in a year: a total of 8760 groups of data. For double-circuit cable lines with different arrangements, we calculate whether the sheath current exceeds the standard for 8760 sets of data in different phase sequences. Afterwards, we divide the number of data with an excessive sheath current by 8760, and find out the phase sequence that maximizes the probability that the sheath current will not exceed the standard when considering the distribution of the load current according to the actual load curve.

Under certain conditions, there may be a number of phase sequences that can make sure that the sheath current does not exceed the standard, and the purpose of making sure that the grounding current does not exceed the standard is to reduce the heating of the cable sheath and improve the line current-carrying capacity. Therefore, the smaller the cable sheath current value the better; at this point, the heating power expectations of the cable sheath under each phase sequence can be further calculated.

To further distinguish candidate phase sequences when the within-limit probability is identical, the Joule loss associated with sheath currents is introduced as a secondary evaluation index. This index is used for relative comparison among phase-sequence options, rather than for predicting the cable temperature rise. The calculation equation for the expected value of heating power is as follows:

$$\overline{P}={\displaystyle \underset{D}{\iint }3\times I_{\mathrm{s},mn}^2\times R\times P_{mn}}$$

(20)

In Equation (20), the integration region D is defined by the maximum current-carrying capacity of the two-circuit cable; I_s,mn is the size of the sheath current when the load current I₁ = m, I₂ = n; R is the AC resistance of the sheath of the cable in each phase; and P_mn is the probability of the occurrence of the load current, I₁ = m, I₂ = n. When the load current is distributed according to a uniform distribution, each combination of load currents has the same probability of occurrence. When the load current is distributed according to the actual load curve, it can be assumed that the hourly load current obtained according to the load curve is unchanged in one hour, and the total number of hours in which each load combination appears is divided by 8760, whereby the probability of each load combination appearing in one year is calculated, and the probability of the load combination not appearing is zero.

In summary, the optimal phase sequence selection method that accounts for load current variations can be derived, as shown in Figure 5.

3. Optimal Phase Sequence Selection Method Considering Load Variations

3.1. The Load Current Is Not Uniformly Distributed Within the Maximum Current-Carrying Capacity Range

For double-circuit cable lines, there are 36 possible phase sequence combinations. Assuming balanced three-phase load currents in each circuit, the relationship between the sheath current and load current in Equation (15) indicates that the sheath current results obtained with phase sequences BCA/BCA and CAB/CAB differ from those with ABC/ABC only in phase shift (lagging or leading by 120°), while the magnitude remains unchanged. Furthermore, since the spatial arrangement of Line A and Line B is axially symmetric, the spatial distances from Cable 1 and Cable 3 to each of the other five cables are equal. This implies identical mutual inductance between Cable 1/3 and each of the other five cables. Therefore, as shown by Equations (12)–(14), the induced voltage generated on each cable remains unchanged after swapping the load current phase sequence of cables 1 and 3. Therefore, the magnitude of the sheath current remains unaffected. Similarly, swapping the load current phase sequence of Cables 4 and 6 also does not affect the sheath magnitude. Consequently, the 36 possible phase sequence combinations can be reduced to 3 distinct combinations, as shown in

Table 5. Phase sequence combination.

Combination Number	Serial Number	Phase Sequence
		Benchmark	120° Lag	120° Ahead
1	1	ABC/ABC	BCA/BCA	CAB/CAB
	2	ABC/CBA	BCA/ACB	CAB/BAC
	3	CBA/CBA	ACB/ACB	BAC/BAC
	4	CBA/ABC	ACB/BCA	BAC/ACB
2	5	ABC/ACB	BCA/BAC	CAB/CBA
	6	ABC/BCA	BCA/CAB	CAB/ABC
	7	CBA/ACB	ACB/BAC	BAC/CBA
	8	CBA/BCA	ACB/CAB	BAC/ABC
3	9	ABC/BAC	BCA/CBA	CAB/ACB
	10	ABC/CAB	BCA/ABC	CAB/BCA
	11	CBA/BAC	ACB/CBA	BAC/ACB
	12	CBA/CAB	ACB/ABC	BAC/CBA

.

In practical operation, the load currents of a transmission line are not uniformly distributed within the maximum ampacity range. In this study, a 220 kV double-circuit cable line in a certain region is taken as an example, and its load current curves in 2021 are shown in Figure 6. Accordingly, one pair of load current values for the two circuits is sampled for each hour over the year, yielding a total of 8760 load current combinations.

In practical cable transmission lines, for construction convenience, the commonly used arrangements for double-circuit cables include four types: horizontal, equilateral triangle horizontal, vertical, and equilateral triangle vertical configurations, as shown in Figure 7. In the figure, s₁ denotes the phase-to-phase spacing of the cables, and s₂ denotes the inter-circuit spacing. The three-phase cables of Circuit 1 are labeled as No. 1, No. 2, and No. 3, and those of Circuit 2 are labeled as No. 4, No. 5, and No. 6.

Under all investigated layout configurations, the geometric parameters of the cables, the cross-interconnection segment lengths, and the grounding conditions are kept the same as those in the engineering case presented in Section 2.2. Based on the measured load current curves in 2021, one pair of load current values for the double-circuit cables is sampled hourly over the year (8760 samples in total). For each load condition, the cable sheath currents under different phase-sequence arrangements are calculated. A given operating point is judged as the “sheath current within limit” when the sheath currents of both circuits are lower than 33.3 A (corresponding to a three-phase total grounding current lower than 100 A). On this basis, the ratio of samples satisfying this criterion among the 8760 operating points is counted to obtain, for each phase sequence, the “probability that the sheath currents of both circuits remain within the limit”.

For the double-circuit cable lines in the horizontal layout and the horizontal equilateral triangle layout, the calculated probabilities for the sheath current within limit under different phase sequences are shown in Figure 8.

As shown in Figure 8a, for the horizontal layout, when Phase Sequence 6 or 8 is adopted, the sheath currents of both circuits are lower than 33.3 A for all the 8760 load samples, and the corresponding “within-limit probability” is 100%. This indicates that, under this layout, both phase sequences can effectively suppress the zero-sequence sheath current over the annual load variation range. However, since the optimal phase sequence is not unique, an additional criterion is required to compare their overall effectiveness in reducing sheath heating. Therefore, the expected value of the sheath-current heating power is introduced as a secondary evaluation index to further select the optimal option between Phase Sequences 6 and 8.

As shown in

Table 6. Expected sheath current heating power efficiency values under different phase sequence arrangements for horizontal layout.

Serial Number	Phase Sequence	Line A/W/km	Line B/W/km	Both Circuits Expected/W/km
6	ABC/BCA	5.5	5.3	10.8
8	CBA/BCA	5.1	4.8	9.9

, for the horizontal layout of the double-circuit cable lines, when Phase Sequence 8 is used, the combined expected heating power of the sheath currents of both circuits is the smallest. Therefore, it is recommended to adopt Phase Sequence 8 for the horizontal layout of double-circuit cable lines under the load distribution conditions discussed in this paper.

As shown in Figure 8b, for the horizontal equilateral triangle layout, when Phase Sequences 3, 5, 6, 7, or 8 are adopted, the probability that the sheath currents of both circuits remain within the limit also reaches 100%. Compared with the horizontal layout, the horizontal equilateral triangle layout has higher geometric symmetry, which leads to more balanced electromagnetic coupling among phases and thus effectively reduces the magnitude of the zero-sequence sheath current. Consequently, under this layout, the number of phase sequences satisfying the within-limit criterion increases significantly. Since the optimal solution is still not unique, it is likewise necessary to compare the expected sheath-current heating power under these candidate phase sequences to determine the more preferable phase-sequence scheme for engineering application.

As shown in

Table 7. Expected sheath current heating power efficiency values under different phase sequence arrangements for equilateral triangle horizontal layout.

Serial Number	Phase Sequence	Line A/W/km	Line B/W/km	Both Circuits Expected/W/km
3	CBA/CBA	17.9	14.5	32.4
5	ABC/ACB	7.9	9.8	17.7
6	ABC/BCA	2.0	1.5	3.5
7	CBA/ACB	2.0	1.5	3.5
8	CBA/BCA	10.3	7.5	17.8

, for the double-circuit cable lines arranged in the equilateral triangle horizontal layout, when Phase Sequences 6 or 7 are used, the combined expected heating power of the sheath currents of both circuits is the smallest. Since the equilateral triangle horizontal layout is axially symmetric, Phase Sequences 6 and 7 are also axially symmetric, resulting in identical outcomes for both sequences. Therefore, under the load distribution conditions discussed in this paper, it is recommended to adopt either Phase Sequence 6 or 7 for the equilateral triangle horizontal layout of the double-circuit cable lines.

When the double-circuit cables are arranged in the vertical layout or the vertical equilateral triangle layout, the spatial configuration of the two circuits is symmetrical, with respect to the vertical axis. Based on the previously derived phase-sequence equivalence, the original 12 phase-sequence arrangements can be further grouped into three phase-sequence combinations. Using the same hourly statistical method described above, the within-limit probabilities of the sheath current under different phase-sequence combinations are calculated, and the results are summarized in

Table 8. Probability of cable sheath current not exceeding the standard for vertical and equilateral triangle vertical arrangement under different phase sequences.

Combination Number	Vertical Arrangement			Equilateral Triangle Vertical Arrangement
	Line A/%	Line B/%	Both Within/%	Line A/%	Line B/%	Both Within/%
1	100.0	100.0	100.0	6.7	6.4	6.0
2	59.8	72.6	59.4	73.9	91.5	73.9
3	74.6	58.9	58.9	83.4	83.2	81.6

.

As indicated in Table 8, due to the differences in cable layout, the electromagnetic coupling between the two circuits differs significantly, and therefore, the phase-sequence combination that maximizes the within-limit probability is also layout-dependent. For the vertical layout, when the phase sequences included in Combination 1 are adopted, the within-limit probabilities for both circuits are 100%, indicating that this combination can effectively suppress zero-sequence sheath currents under the full range of annual load variations and can be prioritized in engineering practice.

For the vertical equilateral triangle layout, when the phase sequences included in Combination 3 are adopted, the probability that the sheath currents of both circuits are simultaneously within the limit reaches 81.6%, which is significantly higher than those of the other combinations. Therefore, for the vertical equilateral triangle configuration in practical projects, Combination 3 is recommended to reduce sheath currents and improve the grounding operating condition.

3.2. The Load Current Is Evenly Distributed Within the Maximum Current-Carrying Capacity Range

The calculation parameters remain consistent with Section 3.1. For double-circuit cable lines with horizontal and equilateral triangular horizontal layouts, the probability that sheath currents remain within limits under different phase sequences is shown in Figure 9.

As shown in Figure 9a, when two circuits are arranged horizontally, using Phase Sequence 6 or 7 maximizes the probability that both cable sheath currents remain within limits, reaching 53.3%. As shown in Figure 9b, when two cable circuits are arranged in a horizontal equilateral triangle configuration, using Phase Sequence 6 or 7 yields the highest probability of 76.2% that sheath currents of two circuits remain within limits. This occurs because both horizontal and equilateral triangle layouts exhibit axial symmetry between the two circuits, and Phase Sequences 6 and 7 also follow an axially symmetric arrangement. Therefore, under uniform load current conditions, the calculation results for both phase sequences are identical. Therefore, it is recommended that double-circuit cable lines with horizontal or equilateral triangle horizontal layouts adopt Phase Sequence 6 or 7 to reduce sheath currents.

Calculation parameters remain consistent with Section 3.1. Assuming uniform load distribution across two circuits, the probability of the sheath currents not exceeding the standard was calculated for vertical and equilateral triangle vertical layouts under different phase sequence combinations, as shown in

Table 9. Probability of cable sheath current not exceeding the standard when the current is evenly distributed.

Combination Number	Vertical Arrangement			Equilateral Triangle Vertical Arrangement
	Line A/%	Line B/%	Both Within/%	Line A/%	Line B/%	Both Within/%
1	94.5	94.5	90.5	15.0	15.0	14.1
2	36.3	39.2	33.7	33.8	39.2	32.0
3	39.2	36.3	33.7	39.2	33.8	32.0

.

As shown in Table 9, when two cable circuits are vertically arranged, the probability that two circuits remain within the standard is highest when using the phase sequence from Combination 1, at 90.5%. Therefore, it is recommended to adopt the phase sequence from Combination 1 for vertically arranged double-circuit cable lines to reduce the sheath current. For two cables arranged vertically in an equilateral triangle configuration, the probability that both cables remain within limits is identical and maximum when using either Phase Sequence Combination 2 or 3. This occurs because the equilateral triangle arrangement is inherently perfectly symmetrical, and the axes of symmetry for the two triangles coincide. Consequently, the results under Phase Sequence Combination 2 and Combination 3 are identical. Therefore, it is recommended that newly constructed two-circuit cable lines arranged vertically in an equilateral triangle configuration may select either Phase Sequence Combination 2 or 3.

4. Conclusions

This paper establishes a computational model for the sheath current and ground current in double-circuit cables, analyzes the influence of the load current on the zero-sequence ground current in the metallic sheath of double-circuit cables, and proposes an optimal phase sequence selection method that accounts for load variations. The following conclusions are drawn:

(1) When the lengths of the cross-interconnection segments for the two-circuit cables are equal, a zero-sequence ground current will occur on the metallic sheath, leading to an excessive total ground current.

For the investigated 220 kV double-circuit cable tunnel, on-line monitoring showed that the three-phase sheath currents in each circuit were nearly identical and the total grounding conductor current was approximately the sum of the three phases, with a maximum grounding current of 208 A, exceeding the commonly used criterion that the grounding current should not exceed 100 A. Moreover, when Line A was powered off, the measured total grounding current of Line A remained as high as 170 A, indicating that a significant zero-sequence sheath current can still be induced, even under power-off conditions, due to electromagnetic coupling between parallel circuits.

(2) The proposed model is validated by field measurements and can accurately reproduce the zero-sequence grounding-current phenomenon.

Under the measured operating cases, the calculated sheath currents and Phase-T grounding currents agree well with the on-site data. For example:

When (I₁ = 439) A and (I₂ = 475) A, Phase-T of Line A is 143.8 A (measured) vs. 146.3 A (calculated) (error 1.7%), and Phase-T of Line B is 139.7 A (measured) vs. 143.0 A (calculated) (error 2.4%).

When (I₁ = 0) A and (I₂ = 688) A, Phase-T of Line A is 170.1 A (measured) vs. 175.6 A (calculated) (error 3.2%), and Phase-T of Line B is 156.2 A (measured) vs. 164.1 A (calculated) (error 5.1%).

Across the listed phases, the maximum reported deviation is within 9.2%, supporting the model’s suitability for engineering assessment and phase-sequence optimization.

(3) The proposed probability-based phase-sequence optimization framework provides a quantitative and layout-dependent solution for suppressing zero-sequence sheath currents under load uncertainty. Based on the 2021 measured load distribution (8760 hourly samples), the horizontal layout achieves a 100% within-limit probability when Phase Sequence 6 or 8 is adopted. Since the optimum is not unique, the expected sheath-current heating power is further introduced as a secondary index; as indicated in Table 6, Phase Sequence 8 yields the minimum combined expected heating power of both circuits (9.9 W/km, compared with 10.8 W/km for Phase Sequence 6), and is therefore recommended for the horizontal layout under the investigated load distribution. For the equilateral triangle horizontal layout, the minimum combined expected heating power is obtained with Phase Sequence 6 or 7 (3.5 W/km, Table 7); due to axial symmetry, these two sequences lead to identical results, and either of them can be recommended. Moreover, exploiting the axial symmetry and phase-shift equivalence reduces the original phase-sequence permutations to a limited number of representative combinations, substantially lowering the engineering screening complexity and enabling layout-specific recommendations. For example, Combination 3 reaches the highest both-circuits-within-limit probability of 81.6% for the vertical equilateral triangle layout, while Combination 1 is recommended for the vertical layout with a maximum both-within-limit probability of 90.5% under the planning-stage uniform-load assumption.