Underground cable modeling and sheath grounding fault simulation

. The modeling of single core cable has an important influence on the calculation of cable sheath cross-connected grounding current and voltage. A suitable equivalent circuit can greatly reduce the complexity of the model. In this paper, the theoretical calculation of π type equivalent circuit of cable is analyzed, and the equivalent circuit of cable sheath grounding system is analyzed. Then the calculation equation of sheath circulation and induced voltage is obtained. Finally, by simulating two kinds of sheath grounding faults, two kinds of sheath grounding fault location are summarized.


Introduction
In recent years, underground cable has been widely used in urban high voltage transmission and distribution system because of its advantages of not occupying the ground and not affecting urban traffic.However, which has created new problems, such as cable shaft water caused by short circuit sheath, road construction cable and other faults.Therefore, it is necessary to analyze the circuit model of underground cable to lay a foundation for further analysis of cable sheath grounding system.
The cable is modelled as a four-phase distributed parameter line, and the self-impedance and mutual impedance of the three cable sheaths and the ground wire are taken into account.An analysis of a simple distribution system shows that the voltage of a grounded system can be easily calculated using π equivalent circuits [1].The reference [2] describes the single-core cable as an extensible numerical model to calculate real-time cable ratings for underground medium voltage cables.The reference [3] analyzes the circuit model of in-phase multiple parallel lines, proposes the calculation method of line sequence impedance, and verifies the current distribution coefficient of the cable with the actual line.
High voltage single core XLPE (cross-linked polyethylene) cable is often used in urban power transmission and distribution system to meet the demand of long-distance and large capacity electric energy.The cable structure is shown in Figure 1.When the core transmits electrical energy, the metal sheath of the cable generates an induced voltage.When the metal sheath is grounded, an induced current is generated on the sheath.As a result, the induced voltage generated by the cable metal sheath will increase with the increase of cable length, and too high induced voltage will breakdown PE outer sheath, serious will destroy XLPE main insulation.When the sheath is grounded, and too high induced current will produce a lot of heat loss.Which not only affect the cable life, but also limit the cable current-carrying capacity.For this reason, some scholars proposed to use the sheath cross-connection grounding to solve the above problems.

XLPE insulation
Outer semi-conductive shielding layer The statistics show that the main faults in the high voltage cable system are connector, cable well connection box and terminal cable accessories.The reference [4] summarizes the existing special connection technologies for medium and high voltage cables, and compares their advantages and disadvantages under single-phase grounding faults.The ATP-EMTP simulation is performed to calculate lightning overvoltage of metal sheath when lightning current intrudes into cable conductor along overhead line, and the influence of various factors on cable metal sheath overvoltage is analyzed and summarized [5].Based on the double π model, the current value along the line is obtained by using the electrical volume of the lead core and the front end of the sheath in [6].According to the characteristics that the front and back current fault components at the corresponding fault point are equal in non-fault phase, a single-end fault location method including the cable metal sheath is proposed.

Metal-sheath PE Sheath
The length of a small transmission line is negligible compared with the transmission wavelength and the total length of the line.The earth is taken as the voltage reference value [7], and the equivalent model is shown in the figure below.In the dx length, there are two loops, namely cable core -sheath (loop 1) and sheath -earth (loop 2).The two loop currents are I1 and I2 respectively.At length x, the loop voltage is U1 and U2, and at x + dx, the voltage increments are dU1 and dU2.

Series impedance
In the frequency domain, where Z ii is the unit length self-impedance of loop i; Z ij is the mutual impedance between loops i and j.
where Z core-out is the outer impedance of the core; Z core/sheath-insulation is the impedance in the insulation between the on-line core and the sheath of the changing magnetic field; Z sheath-in is the inner impedance of the sheath.
where Z sheath-out is the outer impedance of the sheath; Z sheath/earth-insulation is the impedance of the changing magnetic field in the insulation between the sheath and the earth; Z earth is the earth circuit impedance.Mutual impedance Z 12 , Z 21 reflect the coupling between loop 1 and loop 2 due to the common sheath, and The voltage in (1) has no common reference point, and the current in the sheath is divided into two components, which is inconvenient to use.Using actual current in conductors is more consistent with engineering measurements.The voltages of the core and sheath to the ground are set as Uc and Us, and the currents as Ic and Is as follows: Substitute ( 5) into (1) to obtain ( 6) -( 9): ss 22

Parallel admittance
As shown in figure 3, under the rated frequency, the conductance G of the main insulation is much greater than the susceptance, and ignoring the conductance has little influence on the calculation results.
where C is the unit length insulation shunt capacitance, where D i represents the outer diameter of insulation, D C represents the diameter of wire core, and the value of ε is 2.3, which represents the relative dielectric constant of XLPE, ε 0 =8.886×10-12F/m.

Equivalent grounding model of 110kV cable sheath for cross-connection
The cross-connection installation of cable sheath is shown in figure 2. The cable divided 3X (X is an integer) segments, and each three sections is a complete main section.The two sides of the sheath of the main section are grounded through the direct grounding box, and the middle section is connected to the cross-interconnection box by coaxial cable for grounding.There are three cross-connection loops: A1-B2-C3, B1-C2-A3, and C1-A2-B3 respectively.When the cable is in normal operation, its sheath is cross-connected and the induced voltage and current are very small.Therefore, the voltage and current at the beginning and end of the sheath can be detected to judge whether the cable running state and grounding state are good or not.The figure 4 shows the equivalent circuit models of cross-connected grounding of cable sheath, which include equivalent circuit of sheath A1-B2-C3, B1-C2-A3, C1-A2-B3 respectively.I l is the leakage current.It is the current from wire core through the main insulation to the sheath.

I m3
Fig. 4. Equivalent circuit model of cross-connected grounding of cable sheath.
Where, Z i is the equivalent impedance of the main insulation, U is the voltage of the cable system, and the above parameters are all vectors.
The influence of the sheath on the performance of the main insulation in a short time is ignored and the main insulation is assumed to be constant in a short time.The leakage current is mainly determined by the system voltage and the main insulation impedance.Based on the above, the leakage current can be considered unchanged and regarded as an ideal current source, as shown in the figure 4.
In the figure 4, USAi, USBi and USCi (i = 1,2,3) are the induced voltage of the metal sheath of each cable segment; Imi (i = 1,2,3) are the loop circulation of the sheath; ZmAi, ZmBi and ZmCi (i = 1,2,3) are the equivalent impedance of the metal sheath of each cable segment; IlAi, IlBi and IlCi (i = 1,2,3) are the leakage current of each cable segment.ILAi, IRAi, ILBi, IRBi, ILCi, IRCi (i = 1,2,3) are the leakage current components respectively.R e (0.5Ω) and R g (0.5Ω) are the ground resistance at both ends of the sheath.The leakage current component is matched according to the sheath loop impedance.
Due to the cable arrangement and laying mode, the induced voltage vector sum on the cable sheath is not vector zero after the cable sheath is completely cross-connected and grounded.Therefore, the sheath circulation is formed in the sheath loop as follows: Sheath circulation is mainly determined by sheath induced voltage and sheath loop impedance, while sheath induced voltage is mainly determined by cable length, laying mode, load current and sheath circulation.So sheath induced voltage and sheath circulation are coupled to each other.
The coefficient L s of mutual inductance from wire core to sheath is: The coefficient M of mutual inductance between phases is: where D c is the cable core diameter, S is the interphase distance.Sheath induced voltage is: where M AB is the mutual inductance coefficient between the two phases AB; M AC is the mutual inductance coefficient between AC and AC; M BC is the mutual inductance coefficient between BC two phases; l 1 , l 2 and l 3 are the lengths of three cables.

Sheath short circuit fault simulation
In this paper, the electromagnetic transient software ATP-EMTP is used to establish a simulation circuit to simulate the grounding fault of the sheath.The simulated step length ΔT=5×10 -7 s, the total T=0.05s, and the resistivity is 100Ωm.The three-phase single-core cable is laid horizontally, under 1m to the ground, and the cable spacing is 0.3m.The total length of the cable is 1500m, and each section is 500m.The system voltage is 110kV, and the cable end load is replaced by 457A constant current source.The cable structure parameters in figure 1 are as follows:

Sheath single-phase grounding fault
The fault point is set in the cable phase a, and the distance between the fault point and the starting point is taken as the variable.The ground current of the cable sheath head, end and fault place are measured respectively.The simulation results are as Table 2. Considering the convenience of actual field measurement, the ratio of the larger value of the ground current at the head or end of the sheath to the core is taken as the circulation ratio I%.As can be seen from Table 2, when the fault distance is 550m, the amplitude of grounding current at the head end of phase A is 29.38A, while in normal operation, the amplitude of grounding current at the head end of phase A is only 2.875A, with an increase of nearly 10 times.This greatly increases the heat generated on the sheath and seriously affects the cable life.The figure 5 shows the fault circulation distribution diagram.The maintainer can find the approximate location of the possible point of short-circuit fault according to the distribution diagram, which can effectively reduce the time of checking and improve the efficiency of fault maintenance.

Sheath three-phase grounding short circuit
Based on the above characteristics, the failure criteria of the joint sheath are shown in Table 3.
The cable sheath cross-interconnection is achieved through the cross-interconnection box, which is set on the road not far from the cable and is often invaded by water for a long time in the rainy season.Thus, the metal part of the connecting box may be soaked and short circuit between the different phase sheaths.Figure 6 shows the simulation results of J 1 , J 2 , J 1 and J 2 connection boxes when they are immersed in water.Take the short circuit at J 1 as an example, the ABC three-phase current at the beginning is greater than 20A, and I A > I B > I C shows a decreasing trend, and the terminal ABC three-phase current is greater than 30A, I B > I A > I C is inverted "V" shape.In addition, the terminal B contact ground current up to 36A.

The fault location
Point of failure criteria J1 At the begin, ABC is a decreasing trend, and the A-phase current is the largest.At the end, ABC is V-shaped, and the B phase current is the largest J2 At the begin, ABC is V-shaped, and the A-phase current is the largest.At the end, ABC is V-shaped, and the C-phase current is the largest J1, J2 At the begin, ABC is V-shaped, and the A-phase current is the largest.At the end, ABC is V-shaped , and the A-phase current is the largest In this paper, the cross-connection grounding model of cable and its sheath is established, and the complete section of 110kV cross-connection cable is simulated.The main conclusions are as follows: 1) Based on cross-interconnection model, two kinds of common short-circuit faults of cable sheath are studied, including single-phase short-circuit of sheath and three-phase shortcircuit of cross-interconnection.
2) The ground current of the sheath at both ends of the complete cable section under the condition of single-phase short circuit is obtained through simulation, and the circulation ratio is given, which can quickly determine the approximate position of the possible point of short circuit fault and narrow the fault screening range.
3) The grounding current of the sheath at both ends of the complete cable section under the condition of three-phase short circuit is obtained through simulation.Fault location can be realized by comparing the current values of the first and the end and changing trend.

Fig. 6 .
Fig. 6.Measurement of earthing current of head and end sheath.

Table 1
Cable parameters.

Table 2 .
Output current value of the sheath a-phase short-circuit measurement point.