Calculation Method for Sheath Induced Voltage and Circulating Current of a Multi-Circuit Cable System

Abstract

The present invention discloses a calculation method for sheath induced voltage and circulating current of a multi-circuit cable system. First, this method calculates the calculation models of the series impedance matrix and the shunt admittance matrix per unit length of the cable line; subsequently, according to the telegraph equation in the frequency domain, series approximation is performed on the multi-circuit micro-segment cable lines under different grounding methods to construct the node admittance matrix of the micro-segment cable lines; then, a variety of cascade formulas are derived to obtain the complete node admittance matrix of the multi-circuit full-line cable system; according to the boundary conditions and the complete node admittance matrix of the multi-circuit cable system, the cable line micro-segments are subdivided, and the calculation models of the induced voltage and circulating current of the metal sheath of the multi-circuit cable under different grounding methods are constructed segment by segment. The present invention can accurately describe the distribution of sheath induced voltage and circulating current in the multi-circuit cable system, laying a solid foundation for the subsequent research on cable loss reduction measures.

Description

Technical Field

[0001] The present invention belongs to the field of power transmission, and particularly relates to a method for calculating the sheath induced voltage and circulating current of a multi-circuit cable system. Background Art

[0002] With the development of China's power industry, in accordance with the requirements of urban power planning specifications, the number of urban power transmission lines and high-voltage power cables is increasing nowadays. With the rapid increase of the population, the urban and industrial land is expanding day by day. With the steady progress of China's urbanization process, the radiation area of urban power transmission lines in various regions is becoming wider and wider, and the urban power supply system is bound to face a new revolution.

[0003] There are two main forms of power transmission in the urban power grid, namely overhead transmission lines and underground multi-circuit power cable systems. Overhead transmission lines are generally supported by poles and towers to lay multiple conductor lines side by side in the air, which is easy to repair and construct. However, with the increasing shortage of urban land area and corridor resources, this disadvantage has doomed the overhead lines to gradually withdraw from the stage of the urban power grid. However, most of the underground multi-circuit power cables are laid underground, which has little impact on the urban occupied area and land resources. Therefore, the urban power grid is gradually transitioning from traditional overhead lines to power cables, and in the near future, power cables will become an indispensable part of the urban power grid. In addition, due to the extremely unbalanced resources and economy in China, the land resource problem in the East China region needs to be solved urgently. The land resource problem will inevitably bring about the development transformation of the trend from transmission lines to underground power cables

[0004] In summary, since buried power cables are widely used in cities, it is necessary to study the sheath induced voltage and circulating current distribution of the cable system. In addition, due to the complex structure of multiple parallel cables in the same phase, the electromagnetic coupling between different cables is enhanced, bringing many new problems to the safe operation and relay protection of the cables, which are mainly manifested in the following two aspects: (1) The uneven load distribution of parallel cables in the same phase and the increase of sheath circulating current affect the safe operation of the cables; (2) The increase of induced voltage endangers the safety of equipment and personnel. Therefore, a set of algorithms that can accurately reflect the sheath induced voltage and circulating current of the multi-circuit cable system is needed, which also provides a solid foundation for subsequent research on cable system losses. Summary of the Invention

[0005] The purpose of the present invention is to solve the problem of calculating the steady-state characteristics of the multi-circuit cable grounding system, and provide a method for calculating the sheath induced voltage and circulating current of the multi-circuit cable system, which also provides a solid foundation for subsequent research on cable system losses.

[0006] In order to achieve the above purpose, the present invention adopts the following technical solutions:

[0007] The present invention provides a method for calculating the sheath induced voltage and circulating current of a multi-circuit cable system, which includes the following steps:

[0008] Step 1: Based on the geometric structure of the multi-circuit cable and the material parameters of each dielectric layer, considering the electromagnetic coupling characteristics between the multi-circuit cables, the influence of the multi-layer conductor model and the laying environment, calculate the calculation models of the series impedance matrix and the parallel admittance matrix per unit length of the cable line;

[0009] Step 2: Combine the series impedance matrix and the parallel admittance matrix models per unit cable length and perform series approximation on the multi-circuit micro-element cable lines under different grounding methods according to the telegraph equation in the frequency domain to construct the nodal admittance matrix of the micro-element cable lines;

[0010] Step 3: Derive the cascade formulas for symmetric, asymmetric, and those with grounding nodes in the middle, and combine with the nodal admittance matrix of the cable line micro-elements to obtain the complete nodal admittance matrix of the multi-circuit cable system;

[0011] Step 4: According to the boundary conditions and the complete nodal admittance matrix of the multi-circuit full-line cable system, subdivide the cable line micro-elements, and gradually construct the calculation formulas for the induced voltage and circulating current of the metal sheath of the multi-circuit cable lines under different grounding methods, so as to realize the calculation of the sheath induced voltage and circulating current of the multi-circuit cable system.

[0012] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0013] (1) The characteristics of multi-circuit cables are very different from those of traditional cables. Considering the electromagnetic coupling characteristics between multi-circuit cables, the influence of the multi-layer conductor model and the laying environment, it is more capable of accurately evaluating the steady-state characteristics of the electrical grounding system of multi-circuit cables.

[0014] (2) Existing electromagnetic simulation software can only solve the current and voltage values at specific position points, and cannot simulate the induced voltage along the entire sheath of the cable and the circulating current values. However, the present invention can solve the distribution of the induced voltage and circulating current of the metal sheath of the entire cable system based on the multi-conductor algorithm, providing a solid foundation for the subsequent research on the cable system loss.

[0015] (3) Based on the cascade algorithm, the present invention derives the cascade formulas for symmetric, asymmetric, and those with grounding nodes in the middle, avoiding the increase in the order of the nodal admittance matrix after cascading, greatly reducing the calculation time required, and improving the working efficiency of solving the nodal admittance matrix. Description of the Drawings

[0016] Figure 1 is a schematic flow chart of the calculation method of the present invention.

[0017] Figure 2 is an equivalent circuit model diagram of a single multi-circuit cable system.

[0018] Figure 3 It is a schematic diagram of the structure of a three-phase multi-circuit cable system.

[0019] Figure 4 It is a schematic diagram of the equivalent capacitance circuit model of a single multi-circuit cable system.

[0020] Figure 5 It is a schematic diagram of the classical cable system model under different grounding methods.

[0021] Figure 6(a) is a schematic diagram of the induced voltage distribution on the sheath of a multi-circuit cable.

[0022] Figure 6(b) is a schematic diagram of the sheath circulating current distribution of a multi-circuit cable system. Specific implementation manners

[0023] To describe the present invention more specifically, the technical solution of the present invention will be described in detail below in conjunction with the drawings and specific implementation manners.

[0024] As Figure 1 shown, a method for calculating the induced voltage and circulating current of the sheath of a multi-circuit cable system includes the following steps:

[0025] Step 1: Based on the geometric structure of the multi-circuit cable and the material parameters of each dielectric layer, and considering the electromagnetic coupling characteristics between multi-circuit cables, the influence of the multi-layer conductor model and the laying environment, calculate the calculation models of the series impedance matrix and parallel admittance matrix per unit length of the cable line:

[0026] The equivalent circuit model of a single multi-circuit cable is as Figure 2 shown. First, considering the influence of the internal electromagnetic coupling characteristics of the cable, solve the loop impedance matrix of the single multi-circuit cable, and its specific expression is as follows:

[0027]

[0028] where u1 is the voltage between the cable core and the sheath, u2 is the voltage between the sheath and the ground, i1 is the current of the cable core - inner insulation layer - sheath layer loop, i2 is the current of the sheath layer - outer insulation layer - ground loop, Z 11 is the self-impedance per unit length of loop 1, Z 12 and Z 21 are the mutual impedances per unit length between loop 1 and loop 2, and Z 22 is the self-impedance per unit length of loop 2.

[0029] Then, determine the self-impedance and mutual impedance of the series impedance matrix per unit length of the cable line, and use the following expression:

[0030] where Z 11 The specific expression is as follows:

[0031] z 11 = z c1 + z i1 + z s1 (2)

[0032]

[0033]

[0034]

[0035] In equations (2), (3), (4), and (5), Z c1 is the internal impedance of the cable core per unit length; Z i1 is the series impedance per unit length of the cable core of the first loop caused by the magnetic field; Z s1 is the impedance of the inner surface of the sheath per unit length of the first loop; ρ c , ρ s are the resistivities of the cable core and the sheath respectively, μ c , μ s are the permeabilities of the cable core and the sheath respectively; I0 and I1 are the Bessel functions of the first kind of order 0 and order 1 respectively; ω is the angular frequency, R1 is the conductor radius, R2 is the inner insulation radius, and R3 is the sheath radius.

[0036] The mutual impedance per unit length Z 12 and Z 21 between loop 1 and loop 2 are specifically expressed as follows:

[0037]

[0038] Similarly, the expression of the self-impedance Z22 per unit length of loop 2 is as follows

[0039] z 22 = z s2 + z i2 + z g (7)

[0040] where, Z s2 is the impedance of the outer surface of the sheath per unit length of the second loop, Z i2 is the series impedance per unit length of the cable core of the second loop caused by the magnetic field, Z g is the sum of the external reactance and the lossy ground impedance. The specific expressions of Zs2 and Zi2 are as follows:

[0041]

[0042]

[0043] Where R4 is the outer insulation radius.

[0044] The specific expression of Zg is:

[0045]

[0046] Where h is the burial depth and σ is the soil conductivity.

[0047] Then, the series impedance matrix per unit length of a single cable line is determined, and its specific expression is as follows:

[0048]

[0049] where u s is the voltage between the sheath and the ground, u c is the voltage between the cable core and the ground, i c is the current flowing through the cable core, i s is the current flowing through the metal sheath, and the series impedance matrix coefficient Z per unit length of a single cable line cc , Z cs , Z ss From (2) and (3), we can get:

[0050]

[0051] Finally, the impedance matrix of a single power cable is extended to a three-phase cable system. The schematic diagram of the three-phase cable structure is as follows: Figure 3 As shown, the calculation formula of the series impedance matrix is obtained:

[0052]

[0053] where u s1 、u s2 、u s3 are the sheath-to-ground voltages of phases A, B, and C respectively; u c1 、u c2 、u c3 are the voltages of the three-phase cable cores A, B, and C to the ground; i c1 、i c2 、i c3 are the currents flowing through the three-phase cable cores A, B, and C respectively; i s1 、i s2 、i s3 are the currents flowing through the metal sheaths of the three phases A, B, and C respectively; cc1 、z cc2 、z cc3 、z ss1 、z ss2 、z ss3 are the series self-impedance per unit length of the three-phase cable lines A, B, and C; z cci j, zcsi j, z ssi j is the series mutual impedance per unit length of the three-phase cable line, where the subscripts i, j = 1, 2, 3.

[0054] Among them, z cc1 , z cc2 , z cc3 , z ss1 , z ss2 , z ss3 It is calculated according to the impedance matrix formula of a single overhead cable. At the same time, since the distance between the cores of phase A and phase B is almost equal to the distance between the core of phase A and the sheath of phase B, there is:

[0055] z ccij ≈z csij ≈z ssij (14)

[0056] The specific expression of Zccij is:

[0057]

[0058] Among them, h2 is the buried depth, d1 is the distance between cable conductors, μ0 is the magnetic permeability, ω is the angular frequency, and σ is the soil conductivity.

[0059] Then establish the shunt admittance matrix per unit length of the cable line. The equivalent capacitance circuit model of a single power cable is as Figure 4 shown.

[0060] First, solve the shunt admittance matrix of a single multi-circuit cable. Its specific expression is as follows:

[0061]

[0062] Among them, u c and u s are the voltages of the core and the sheath to the ground respectively, i c is the incident current of the core, is is the incident current of the sheath, Y cs is the admittance between the core and the sheath; Y sg is the admittance between the sheath and the ground, which consists of two parts, namely the admittance of the outer insulation layer Y sg1 and the admittance of air Y sg2 . The specific expressions are as follows:

[0063]

[0064] Among them:

[0065]

[0066] Among them, ε0 is the vacuum permittivity, εin is the relative permittivity of the inner insulation layer, ε ex is the relative permittivity of the outer insulation layer, R1 is the conductor radius, R2 is the inner insulation radius, R3 is the sheath radius, R4 is the outer insulation radius, and h is the burial depth.

[0067] Finally, the admittance matrix of a single power cable is extended to a three-phase power cable to obtain the calculation formula for the parallel admittance matrix, and the specific formula is as follows:

[0068]

[0069] where i c1 、i c2 、i c3 are the currents flowing through the cores of phases A, B, and C respectively, i s1 、i s2 、i s3 are the currents flowing through the metal sheaths of phases A, B, and C respectively, u s1 、u s2 、u s3 are the voltages of the sheaths of phases A, B, and C to the ground; Y ss1 、Y ss2 、Y ss3 are the parallel self-admittances per unit length of the cable lines of phases A, B, and C respectively, u c1 、u c2 、u c3 are the voltages of the cores of phases A, B, and C to the ground respectively; Y cs1 、Y cs2 、Y cs3 are the parallel mutual-admittances per unit length of the cable lines of phases A, B, and C respectively.

[0070] where Y cs1 、Y cs2 、Y cs3 are calculated according to the formula for a single overhead cable. And Y ss1 、Y ss2 、Y ss3 are calculated according to the following formula.

[0071]

[0072] where ε0 is the vacuum permittivity, ε ex is the relative permittivity of the outer insulation layer, R3 is the sheath radius, and R4 is the outer insulation radius.

[0073] Step 2: Perform series approximation on the multi-circuit micro-segment cable lines under different grounding methods, and the process of constructing the node admittance matrix of the micro-segment cable lines is as follows:

[0074] First, according to the telegraph equation in the frequency domain, the frequency-domain equation of the transmission line is obtained:

[0075]

[0076] Where \(U_x\) and \(I_x\) are the voltage vector and current vector of the end node of the \(\Delta x\)-long line respectively, \(Z\) is the series impedance matrix per unit length of the cable system, and \(Y\) is the loop impedance matrix of the multi-circuit cable.

[0077] Then, solve the differential equation of formula (19) and combine the boundary conditions at the beginning and end:

[0078] At the beginning, that is, when \(x = 0\), \(U_x=U\) S , \(I_x = I_s\);

[0079] At the end, that is, when \(x = l\), \(U_x=U\) R , \(I_x = I\) R ;

[0080] Substituting into the differential equation, we can obtain:

[0081]

[0082] Where \(u\) R and \(i\) R are the voltage vector and current vector of the end node of the line respectively, and the expression of \(\Omega\) is:

[0083]

[0084] Finally, by introducing hyperbolic functions through matrix transformation, the nodal admittance matrix of the infinitesimal cable line is obtained:

[0085]

[0086] Perform Laurent series expansion on the hyperbolic functions \(\coth(\Omega l)\) and \(\csch(\Omega l)\):

[0087]

[0088] Where:

[0089]

[0090]

[0091] Where:

[0092]

[0093] Where \(B_n\) is the Bernoulli number.

[0094] Step 3: Derive the cascade formulas for symmetric, asymmetric, and those with a grounding node in the middle, and combine with the node admittance matrix of the cable line micro-element segment to obtain the complete node admittance matrix of the multi-loop full-line cable system;

[0095] The cable systems studied are divided into three major parts: single-ended grounding, double-ended grounding, and cross-bonding grounding. Their structures are as Figure 5 shown, where (a) is single-ended grounding, (b) is double-ended grounding, and (c) is cross-bonding grounding. This invention will be elaborated with the symmetric cascade algorithm, asymmetric cascade algorithm, and cascade algorithm including grounding nodes used in the cascades of single-ended grounding, double-ended grounding, and cross-bonding grounding:

[0096] First, for the cascade of the symmetric node admittance matrices Ya and Yb, the specific method is as follows:

[0097]

[0098]

[0099] Where: Ya and Yb are symmetric node admittance matrices, Is and I M " are the sending-end currents of the corresponding cable segments of Ya and Yb respectively, and I M ' and I R are the receiving-end currents of the corresponding cable segments of Ya and Yb respectively, Us and U M are the voltages to ground at the first ends of the corresponding cable segments of Ya and Yb, and U M ' and U R are the voltages to ground at the tail ends of the corresponding cable segments of Ya and Yb, and YS and Ym are the quarter sub-matrices of Ya and Yb respectively.

[0100] Since the currents flowing into the middle node of the two cascaded cable segments are equal in magnitude and opposite in direction, the specific relationship is as follows:

[0101] I' M = -I” M (31)

[0102] The cascade formula for the symmetric node admittance matrix is obtained:

[0103]

[0104] Subsequently, for the cascade of the asymmetric node admittance matrices Y1 and Y2, the specific method is as follows:

[0105]

[0106]

[0107] Where: Y1 and Y2 are asymmetric nodal admittance matrices, E, F, G, and H are quarter sub-matrices in the Y1 nodal admittance matrix, and P, Q, R, and S are quarter sub-matrices in the Y2 nodal admittance matrix.

[0108] Since the magnitudes of the currents flowing into the intermediate node between two cascaded cable segments are equal and the directions are opposite, the cascading formula for the symmetric nodal admittance matrix is obtained:

[0109]

[0110] Then, for the cascading that includes a grounded node in the middle, the specific method is as follows:

[0111]

[0112]

[0113] Where:

[0114]

[0115] Where: Y A , Y B are the nodal admittance matrices that include a grounded node in the middle, A, B, C, and D are quarter sub-matrices in the Y1 nodal admittance matrix, and T, W, L, and O are quarter sub-matrices in the Y2 nodal admittance matrix; A pq , B pq , C pq , D pq , T pq , W pq , L pq , O pq are the intermediate matrices in the nodal admittance matrices of nodes A, B, C, D, T, W, L, and O respectively, where the subscripts p, q = 1, 2; A 11 , B 11 , C 11 , D 11 , T 11 , W 11 , L 11 , O 11 are (n - 1)×(n - 1) order matrices, A 12 , B 12 C 12 , D 12 , T 12 , W 12 , L 12 , O 12 are (n - 1)×1 order matrices, A 21 , B 21 , C 21 , D 21 , T 21 , W21 , L 21 , O 21 is a 1×(n - 1) order matrix, A 22 , B 22 , C 22 , D 22 , T 22 , W 22 , L 22 , O 22 is a 1×1 order matrix, and n is the number of multi - conductor cables;

[0116] For Y A 、Y B , by solving the simultaneous equations of the admittance matrix of the intermediate ground node, Y A 、Y B The admittance matrix of the nodes after cascading the admittance matrix of the nodes:

[0117] <**********]]

[0118] Where: <**********]]

[0119] K1 = (D 11 + T 11 ) -1 C 11 (40)

[0120] K2 = (D 11 + T 11 ) -1 W 11 (41)

[0121] Finally, the cascading algorithms for symmetric, asymmetric, and intermediate ground - node - included are obtained by using equations (32), (35), and (39).

[0122] Step 4: According to the boundary conditions and the complete node admittance matrix of the multi - loop full - line cable system, subdivide the micro - element segments of the cable line, and construct the calculation formulas for the induced voltage and circulating current of the metal sheath of the multi - loop cable line under different grounding methods section by section, so as to realize the calculation of the induced voltage and circulating current of the multi - loop cable system sheath:

[0123] First, the node admittance matrix of the j - th micro - element in the i - th section of the cable is obtained as:

[0124]

[0125] Where: U s,i,j 、U R,i,j 、I S,i,j 、I R,i,j respectively represent the voltage and current values at the sending - end and receiving - end of the j - th micro - element in the i - th section of the cable. Y s,i,j 、Y m,i,jFor a quarter sub-matrix in a symmetric matrix.

[0126] By transforming its matrix, we can obtain:

[0127]

[0128] Finally, the induced voltage and circulating current distribution of the multi-loop cable in the differential element section are solved using the above formula. Combining with the cascade algorithm, the calculation formulas for the induced voltage and circulating current of the multi-loop cable system of the whole-line cable system are derived, realizing the calculation of the sheath induced voltage and circulating current of the multi-loop cable system.

[0129] Figures 6(a) and 6(b) are schematic diagrams of the induced voltage and sheath circulating current distribution of single - end grounded, double - end grounded, and cross - connected grounded sheaths. The distribution curves are obtained by the present invention. First, based on the multi - circuit cable geometric structure and the material parameters of each dielectric layer, considering the electromagnetic coupling characteristics between multi - circuit cables, the influence of the multi - layer conductor model and the laying environment, the calculation models of the series impedance matrix and the parallel admittance matrix per unit length of the cable line are calculated; then, combining the series impedance matrix and the parallel admittance matrix models per unit cable length and according to the telegraph equation in the frequency domain, a series approximation is made for the multi - circuit micro - element cable lines under different grounding methods to construct the nodal admittance matrix of the micro - element cable lines; subsequently, the cascade formulas for symmetric, asymmetric, and those containing grounding nodes in the middle are deduced, and combined with the nodal admittance matrix of the cable line micro - element to obtain the complete nodal admittance matrix of the multi - circuit full - line cable system; finally, according to the boundary conditions and the complete nodal admittance matrix of the multi - circuit full - line cable system, the cable line micro - element is subdivided, and the calculation models of the induced voltage and circulating current of the metal sheath of the multi - circuit cable line under different grounding methods are constructed section by section. It can be seen from Figure 6(a) that for the single - end grounded cable, the induced voltage of the sheath shows a linear increasing trend with the distance starting from the grounding point, and the maximum value of the induced voltage of the sheath appears at the end of the cable system, reaching 332.32V. For the cross - connected grounding system of the cable, the change curve of the sheath voltage shows an "M" - shaped distribution law, and the maximum value of the induced voltage of the sheath always appears at the cable cross - connection joint, with a value of 114.55V. For the double - end grounded cable, the induced voltage also shows a gradually increasing trend with the distance starting from the two grounding points, and the maximum value of the induced voltage of the sheath appears at the mid - point of the cable system, reaching 0.56V. Compared with the single - end grounded and cross - connected grounded cable systems, if the double - end grounding is adopted, the induced voltage will be significantly reduced. It can be seen from Figure 6(b) that for the single - end grounded cable, the sheath circulating current shows a linear decreasing trend with the distance starting from the grounding point, and the maximum value of the induced voltage of the sheath appears at the head of the cable system, reaching 15.98A. For the cross - connected grounding system of the cable, the two cross - connection joints divide the sheath circulating current curve into three parts, and the change curve of the sheath circulating current shows a trend of first decreasing and then increasing, and the maximum value of the sheath circulating current appears at the end of the cable system, which is 7.5A. For the double - end grounded cable, it shows a linear increasing trend with the distance starting from the grounding point, and the maximum value of the induced voltage of the sheath appears at the end of the cable system, reaching 1246.63A. Compared with the single - end grounded and cross - connected grounded cable systems, if the double - end grounding is adopted, the induced voltage will be significantly increased. And compared with the simulation results of the PSCAD electromagnetic software, the induced voltage and circulating current of the sheath of the multi - circuit cable system simulated by the calculation method of the present invention are within the error allowable range. In summary, the calculation method of the present invention can calculate the induced voltage and circulating current of the sheath of the multi - circuit cable system.

Claims

1. A method for calculating sheath induced voltage and circulating current of a multi-circuit cable system, characterized in that: The following steps are involved: Step 1: Based on the geometric structure of the multi-circuit cable and the material parameters of each dielectric layer, and considering the electromagnetic coupling characteristics between the multi-circuit cables, the influence of the multi-layer conductor model and the laying environment, the series impedance matrix and parallel admittance matrix calculation model per unit length of the multi-circuit cable line are calculated; Step 2: Combining the series impedance matrix and parallel admittance matrix models under unit cable length and using the frequency-domain telegraph equation, a series approximation is performed on the multi-circuit infinitesimal segment cable lines under different grounding methods to construct the node admittance matrix of the multi-circuit infinitesimal segment cable lines. Step 3: Derive the cascade formulas for symmetric, asymmetric, and intermediate grounded nodes, and combine them with the node admittance matrix of the infinitesimal segments of the multi-circuit cable line to obtain the complete node admittance matrix of the multi-circuit cable system; Step 4: Based on the boundary conditions and the complete node admittance matrix of the multi-circuit cable system, the multi-circuit cable line is subdivided into micro-element segments. The calculation formulas for the induced voltage and circulating current of the metal sheath of the multi-circuit cable line under different grounding methods are constructed segment by segment to realize the calculation of the induced voltage and circulating current of the sheath of the multi-circuit cable system; The derivation process of the symmetric, asymmetric and intermediate grounded cascade formulas in step 3 is as follows: First, for the symmetric node admittance matrix Y a and Y b The specific method of cascading is as follows: Where: I s , I M "Y a 、Y b Corresponding to the current at the sending end of the cable segment, I M '、I R Y a 、Y b The corresponding cable segment receiving end current, U s 、U M Y a 、Y b The voltage between the first end of the corresponding cable segment and the ground, U M '、U R Y a 、Y b The voltage at the end of the corresponding cable segment to ground, Y S 、Y m Y a 、Y b One-quarter submatrix; Since the currents flowing into the intermediate nodes of the two cascaded cable segments are equal in magnitude and opposite in direction, the specific relationship is as follows: I' M =-I" M The cascade formula of the symmetric node admittance matrix is obtained: Then, for the cascade of asymmetric node admittance matrices Y1 and Y2, the specific method is as follows: Where: Y1 and Y2 are asymmetric node admittance matrices, E, F, G, H are one-quarter sub-matrices in the Y1 node admittance matrix, and P, Q, R, S are one-quarter sub-matrices in the Y2 node admittance matrix; Since the currents flowing into the intermediate nodes of the two cascaded cable segments are equal in magnitude and opposite in direction, the cascade formula of the symmetric node admittance matrix is obtained: Then, for the cascade with a ground node in the middle, the specific method is as follows: in: Where: Y A 、Y B is the admittance matrix containing the ground node in the middle, A, B, C, D are one quarter of the sub-matrices in the admittance matrix of the Y1 node, T, W, L, O are one quarter of the sub-matrices in the admittance matrix of the Y2 node; A pq 、B pq 、C pq 、D pq 、T pq 、W pq , L pq , O pq are the intermediate matrices in the admittance matrices of nodes A, B, C, D, T, W, L, and O, respectively, where the subscripts p and q are 1 and 2 respectively; A 11 ,B 11 ,C 11 ,D 11 ,T 11 ,W 11 ,L 11 ,O 11 A is a (n-1)×(n-1) matrix. 12 ,B 12 C 12 ,D 12 ,T 12 ,W 12 ,L 12 ,O 12 A is a (n-1)×1 matrix. 21 ,B 21 ,C 21 ,D 21 ,T 21 ,W 21 ,L 21 ,O 21 is a 1×(n-1) matrix, A 22 ,B 22 ,C 22 ,D 22 ,T 22 ,W 22 ,L 22 ,O 22 is a 1×1 matrix, and n is the number of multi-circuit cables; For Y A 、Y B Solve the admittance matrix for the middle ground node and get Y A 、Y B The node admittance matrix after cascading the node admittance matrix is: Where K1 and K2 are Y A，B The intermediate matrix of is expressed as follows: K1=(D 11 +T 11 ) -1 C 11 K2=(D 11 +T 11 ) -1 W 11 So far, the cascade formulas of symmetric, asymmetric and intermediate ground nodes have been constructed through the cascade algorithm; The process of constructing the calculation formulas for the induced voltage and circulating current of the metal sheath of the multi-circuit cable line under different grounding methods section by section in step 4 is as follows: First, the admittance matrix of the jth infinitesimal node in the i-th cable segment is obtained as: Among them: U s,i,j 、U R,i,j , I S,i,j , I R,i,j They represent the voltage and current values Y at the sending and receiving ends of the jth infinitesimal element in the i-th cable segment. s,i,j 、Y m,i,j For one quarter submatrix in a symmetric matrix; By transforming the matrix, we can get: Finally, the above formula is used to solve the induced voltage and circulating current distribution of the multi-circuit cable in the infinitesimal section. Combined with the cascade algorithm, the calculation formula of the induced voltage and circulating current of the multi-circuit cable system of the entire cable system is derived, and the calculation of the induced voltage and circulating current of the sheath of the multi-circuit cable system is realized.

2. The method for calculating sheath induced voltage and circulating current of a multi-circuit cable system according to claim 1, characterized in that: The process of establishing the series impedance matrix per unit length of the cable line in step 1 is as follows: First, considering the influence of the electromagnetic coupling characteristics inside the cable, Get the loop impedance matrix of a single multi-loop cable: Where u1 is the voltage between the cable core and the sheath, u2 is the voltage between the sheath and the earth, i1 is the current in the cable core-inner insulation layer-sheath layer loop, i2 is the current in the sheath layer-outer insulation layer-earth loop, Z 11 is the self-impedance per unit length of loop 1, Z 12 and Z 21 is the mutual impedance per unit length between loop 1 and loop 2, Z 22 is the self-impedance per unit length of loop 2; Then, determine the self-impedance of the series impedance matrix per unit length of the cable line using the following expression: where Z c1 is the internal impedance of the cable core per unit length; Z i1 Z is the series impedance of the cable core per unit length of the first loop caused by the magnetic field; i2 is the series impedance of the cable core per unit length of the second loop caused by the magnetic field; Z s1 is the inner surface impedance of the sheath per unit length of the first loop; Z s2 is the inner surface impedance of the sheath per unit length of the second loop; Z g It is the sum of external impedance and lossy earth impedance; Then, the series impedance matrix per unit length of a single cable line is determined, and its specific expression is as follows: where u s is the voltage between the sheath and the ground, u c is the voltage between the cable core and the ground, i c is the incident current on the cable core, i s is the incident current on the sheath layer, and the series impedance matrix coefficient Z per unit length of a single cable line cc , Z cs , Z ss It can be expressed as follows: Finally, the impedance matrix of a single power cable is extended to the three-phase cable system, and the calculation formula of the series impedance matrix is obtained: where u s1 、u s2 、u s3 are the sheath-to-ground voltages of phases A, B, and C respectively; u c1 、u c2 、u c3 are the voltages of the three-phase cable cores A, B, and C to the ground; i c1 、i c2 、i c3 are the currents flowing through the three-phase cable cores A, B, and C respectively; i s1 、i s2 、i s3 are the currents flowing through the metal sheaths of the three phases A, B, and C respectively; cc1 、z cc2 、z cc3 、z ss1 、z ss2 、z ss3 are the series self-impedance per unit length of the three-phase cable lines A, B, and C; z ccij 、z csij 、z ssij is the series mutual impedance per unit length of the three-phase cable line, where subscripts i, j = 1, 2, 3.

3. The method for calculating sheath induced voltage and circulating current of a multi-circuit cable system according to claim 1, characterized in that: The process of establishing the parallel admittance matrix per unit length of the cable line in step 1 is as follows: First, solve the parallel admittance matrix of a single multi-loop cable. Its specific expression is as follows: Among them, u c and u s are the voltages of the cable core and sheath to ground, i c is the incident current on the cable core, i s is the incident current on the sheath layer, Y cs Y is the admittance between the cable core and the sheath layer; sg It is the admittance between the sheath layer and the earth, which consists of two parts: the outer insulation layer admittance Y sg1 and air admittance Y sg2 , the specific expression is as follows: Finally, the admittance matrix of a single power cable is extended to a three-phase power cable, and the calculation formula of the parallel admittance matrix is obtained. The specific formula is as follows: where i c1 、i c2 、i c3 are the currents flowing through the three-phase cable cores A, B, and C respectively; i s1 、i s2 、i s3 are the currents flowing through the metal sheaths of the three phases A, B, and C, u s1 、u s2 、u s3 are the sheath-to-ground voltages of phases A, B, and C respectively; Y ss1 、Y ss2 、Y ss3 are the parallel self-admittance per unit length of the three-phase cable lines A, B, and C respectively; u c1 、u c2 、u c3 are the voltages of the three-phase cable core to ground, A, B, and C respectively; Y cs1 、Y cs2 、Y cs3 They are the parallel mutual admittance per unit length of three-phase cable lines A, B and C respectively.

4. The method for calculating sheath induced voltage and circulating current of a multi-circuit cable system according to claim 1, characterized in that: The specific construction process of the node admittance matrix of the microelement segment cable line in step 2 is as follows: First, according to the telegraph equation in the frequency domain, the transmission line frequency domain equation is obtained: Where Ux and Ix are the voltage vector and current vector of the end node of the Δx-long line, Z is the series impedance matrix per unit length of the cable system, and Y is the loop impedance matrix of the multi-loop cable; Then, according to the boundary conditions at the beginning and end, we can get: At the beginning, that is, when x=0, Ux=U S , Ix=Is; At the end, that is, when x=l, Ux=U R , Ix=I R ; Substituting into the differential equation we obtain: where u R and i R are the voltage vector and current vector of the line end node respectively, where Ω is expressed as: Finally, the hyperbolic function is introduced through matrix transformation to obtain the node admittance matrix of the infinitesimal segment cable line:

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