Cross-Bonded Cable

It is often observed in practice that the number of conductors changes at a boundary, as shown in Figure 1.53, where phases a and b are short-circuited at node 1. In the case of a cross-bonded cable, three-phase metallic sheaths are rotated at every cross-bonding point. In such a case, it is required to reduce the order of an impedance matrix and/or to rotate the matrix elements. 

Single-point bonding as part of a cross-bonded cable. 

Figure 3.6 illustrates a major section of a cross-bonded cable. The bold solid line and the broken line express the core and sheath, respectively. The sheaths are grounded through grounding impedance Zg at both sides of the major section. The core and sheath voltages and Vk and currents Ij and jjJ at the kth cross-bonded node are related as in the following equations ... 

The voltage difference between the terminals of the major section (AV) gives an equivalent impedance of a cross-bonded cable. It is obtained (3.13) by applying the following relations ... 

In this same manner, the equivalent admittance of a cross-bonded cable can be obtained from the current difference (AI) ... 

The core admittance submatrix [F ] is a diagonal matrix determined by the capacitances between the cores and the sheath, because a sheath encloses a core. The admittance submatrix of the cores for the cross-bonded cable is identical to the solidly bonded cable ... 

In this section, we derive theoretical formulas of the sequence currents for the majority of underground cable systems that is, a cross-bonded cable that has more than two major sections. We also derive theoretical formulas for a solidly bonded cable considering the increased use of submarine cables. 

Cross-bonded cable and its equivalent model (a) a cross-bonded cable system with m-major sections and (b) an equivalent four-conductor system. 

Setup for measuring sequence currents for a cross-bonded cable (a) zero-sequence current and (b) positive-sequence current. 

Equation 3.71 shows that the positive-sequence current can be approximated by the coaxial mode current. It also shows that, in a manner similar to that of a cross-bonded cable, the positive-sequence current remains unaffected by the substation-grounding resistance Rg. 

The calculation process in the case of a cross-bonded cable using the proposed formulas is shown as follows (the 6x6 impedance matrix Z is obtained using cable constants [1,11,12]) ... 

Table 3.1 shows that the positive-sequence impedance is smaller for a solidly bonded cable than for a cross-bonded cable, and the positive-sequence current is larger for a solidly bonded cable. Because of this size differential, the return current flows only through the... 

The impedance calculation in lEC 60909-2 assumes solid bonding. As a result, if the positive-sequence impedance of a cross-bonded cable is derived based on lEC 60909-2, it might be smaller than the actual positive-sequence impedance. 

The phase angle of the zero-sequence current mentioned in Table 3.1 demonstrates that grounding resistance at substations in both cross-bonded and solidly bonded cables significandy affects the zero-sequence current. As a result, there is litde difference in the zero-sequence impedance of the cross-bonded cable and the solidly bonded cable. The results indicate the importance of obtaining an accurate grounding resistance at the substations to derive accurate zero-sequence impedances of cable systems. 

The capacitance matrix looks similar to the impedance matrix in Table 3.3a-2. The capacitance between the core and sheath of the homogeneous model is identical to that of the solidly bonded cable. The equivalent capacitance Q4 of the cross-bonded cable in Table 3.3b-2 is given as the sum of the elements as shown in Equations 3.44 and 3.45. 

The reduced transformation matrix of a cross-bonded cable is shown in Table 3.3b-3. The composition of the top left 3x3 matrix (the first three modes) is similar to that of an overhead transmission line. The current of the third mode returns from the equivalent sheath instead of the earth. The fourth mode expresses the equivalent earth-return mode of the cross-bonded cable system. 

Modes 1-3 in the solidly bonded cable shown in Table 3.3a-4 are coaxial modes and are the same as mode 3 in the homogeneous cross-bonded cable model. Although the attenuations of the inter-core modes (modes 1 and 2) shown in Table 3.3b-4 are almost identical to that of the coaxial mode of the solidly bonded cable, the velocities are lower. The velocity of the coaxial mode is determined by the permittivity of the main insulator = 2.3 shown in Table 3.2 ... 

Figure 3.14d shows the transient response of the core voltage in a solidly bonded cable. It shows a stair-like waveform with a length of 70 ps. This length is determined by the round-trip time shown in Equation 3.89. Sheath voltages of the solidly bonded cable are much smaller than those of the cross-bonded cable. The results indicate that not all cross-bonded cables can be simplified by a solidly bonded cable from the viewpoint of not only the sheath voltages but also the core voltages. 

Cross-bonded cable, (a) Sheath cross-bonding and (b) core cross-bonding. 

Homogeneous Model of a Cross-Bonded Cable 3.2.3.1 Homogeneous Impedance and Admittance... 

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