Simulation of sheath voltage, losses and loss factor of high voltage underground cable using MATLAB/Simulink

Received Jul 2, 2021 Revised Jan 15, 2022 Accepted Jan 24, 2022 In this paper, 22 equations for high voltage cable sheaths are simulated in one model. The model outputs are represented by cable sheath voltages, circulating currents, losses and factors, eddy currents, losses, and factors in both tides laying states (trefoil and flat) when grounding the sheaths from a single point, two points, or cross-link. These values depend on the cable manufacturing's specific factors. The other factors affecting these values are specific to the laying and operation: the load current, the length of the cable to be laid out, the spacing between the cables, and the power frequency. This research aims to reduce or eliminate the losses of the cable sheath. These two types of currents cause losses that may sometimes equal the losses of the conductor of the cable carrying the load current. Which reduces the capacity of the cable and reduces the heat dissipation of the cable into the soil and damages it. Electricians are at risk of electrocution due to the high voltages of the sheaths when there is no current in the sheaths. Therefore, these currents and voltages must be eliminated by making a new model that studies the effect of all these factors on them.


INTRODUCTION
The danger of electric shock and fault currents is reduced with electric shock and fault currents reduced with electric shock, and fault currents reduced with insulating material for high voltage underground cables compared to the overhead transmission lines [1], [2]. Single-core cables have a high current capacity, so they are more likely to be used for high power transmission than three core cables [3], [4]. The cable insulation material is covered with metal tape or wires (sheath) to return fault and capacitive current, reduce the mechanical effects on the main conductor and prevent the electric field effect outside this sheath [5]. An induced voltage is generated at the metal sheath due to the passage of load current that magnetic product flux, which penetrates this sheath as explained in (one end bonding). A current flow between the ground and the sheath when it is grounded at (both ends); its value depends mainly on the sheath voltage and its impedance, causing so-called circulating losses [6]. Overheating the soil surrounding the cable due to conductor and circulating sheath currents may cause the soil surrounding the cable due to conductor and circulating sheath currents may cause the insulation's thermal breakdown [7]. The factors affecting circulating currents are discussed, and several methods of connecting cable sheaths are presented to reduce the risk of the sheath (voltages and losses) [8]− [13]. Therefore, the sheath current value must be reduced on the metallic sheath to prevent its negative effects, and the metallic sheath must be grounded to reduce the sheath current effect. The equal spacing between the phases of power cables causes similar sheath induced voltages. Where M is the mutual inductance and depends on the spacing between axes of any adjacent cables (S) and the mean radius of the circle formed by the sheath conductor as, 1,2 = 2,3 = 1,3 = = 2 * 10 −7 ln ( ℎ ) H −1 (2) either the spacing in the flat formation arrangement is different because of the outer cables as,  (3), the middle cable has equal spacing from the outer two cables, and its sheath voltage is similar to those of trefoil formation arrangement.

SHEATH LOSSES OF SINGLE-CORE UNDERGROUND CABLES
Sheath losses are current-dependent and can be divided into two categories according to the type of bonding [4], [24], [25]. Sheath circulating losses appear due to the sheath bonding from both ends as a result of passing a current between the sheath and the ground. Eddy losses occur regardless of bonding type (one point, two points, or cross bonding).

Sheath circulating losses
Two points bonding causes the flowing of sheath circulating currents, leading to circulating losses, as explained in Figure 2. The voltages above and the impedance represented by the sheath resistance and sheath reactance define the circulating current in trefoil and flat arrangement. The sheath impedance of threephase cables in the trefoil arrangement is the same because of the equal spacing. Based on this, circulating sheath current per phase of the three cables can be written as (4). From (4), the sheath circulating loss and factor are explained: Where the factor in (6) is the ratio of sheath circulating loss to the conductor load loss in Wm −1 . However, this is not true in the flat arrangement at two-point sheath bonding where the sheath circulating equations are more complicated than trefoil one because the different spacing between the cables results in unequal sheath voltages, impedances, and currents. These currents of this arrangement can be calculated from (7)-(9).
The sheath losses per meter for each phase and the loss factor are written in (10) Three-phase arrangement with sheaths cross-bonded for long-run length and spacing of cables can be allowed by eliminating exaggerated sheath voltage and circulating current using cross-bonding methods. Current-carrying capacity is improved after increasing cables spacing with this method because of each cable's thermal independence. The induced voltage is canceled by dividing the cable run into three parts (sections) and cross-connecting the sheaths, as explained in the Figure 3.
In the first part, the sheath end of the first cable is cut and connected to the sheath beginning of the second cable in the middle part. It was connecting the second cable's sheath end-point with the third cable's beginning sheath point in the last section to make the directional summation of induced voltages in all parts equal to zero without flowing any circulating current except the eddy currents. This method is most likely to be employed in very high voltage cables because its expansiveness [4], [23]− [25].

Sheath eddy losses
Whether single or three cores, different bonding methods and cable types will not prevent eddy current losses from occurring. In solid bonding (two ends) for sheaths of single-core cable, the circulating current losses are enormous compared to these losses, and therefore, they can be neglected if the cable conductor layers have small segments. The flowing eddy current is due to the different voltages on the sides of the sheath. The density of the non-uniform current in the cable conductors makes (ep) on the outer surface of the sheath lower than on the inner one because of their convergence from each other. This indicates that the divergent cables eliminate this type of currents [4], [22], [23]. The sheath eddy current and loss factor will be calculated by (13) and (14).
Where: δ SE : Eddy loss factor of the sheath ISE: Sheath eddy-current in A Nevertheless, in the flat arrangement, their values calculated by (15) to (16).
Where: δ SE1 , δ SE3 , Eddy loss factor of the sheath in two outer cables δ SE2 , Eddy loss factor of the sheath in middle cable ISE1, ISE3: Sheath eddy-current in two outer cables in A ISE2: Sheath eddy-current in middle cable in A In the sheath cross-banded case, the eddy loss factor and currents calculated by (19) and (20).
: The electrical resistivity of sheath material at operating temperature (Ω.m) D.S.: The external diameter of the cable sheath (mm) : The thickness of sheath (mm) Δ1 and Δ2 are factors whose values depend on the types of cable layouts formation. and β1 are factors whose values depend on the cable parameters. For (m ≤ 0.1, Δ1 and Δ2 can be neglected) Where: (m) is a factor that depends on power frequency and metallic sheath resistance. These values can be calculated from [4], [23], [24].

SIMULATION AND RESULTS
The (1) to (23) have been simulated by the method of mathematics simulation. All equations include dependent and non-dependent variables. The independent variables were utilized as variable inputs to show the effect on all equation's outputs. Figure 4 illustrates all of these inputs with their sub-inputs values. The  Table 1 are taken from [26]. When the value of the load current changes within the range (200-500), the cable sheathes' voltages will change at the trefoil and flat arrangements as in Figure 20 in the case of the sheaths being grounded from one end only. The sheathes of the three-phase for the trefoil, and the mid-phase one in the flat arrangement will increase within the range (108.56-304) V, while the limits for those of the outer phases of the flat arrangement are (169.8-475.4) V.
The circulating currents results when the sheaths are grounded at both ends are shown in Figure 21. The results show the difference in the rate of increase of sheath currents at low and high load currents concerning cable sheaths for flat and trefoil arrangement. This is because the difference in distances between the power cables in a flat one causes a difference (increase) in the reactance value on the outer cable sheaths, thus an apparent increase in the sheath voltage circulating currents.  The increase in the sheath circulating currents increases the sheath circulating losses and factors, as shown in Figure 22 and within the range in Table 2. Figures 23 to 26 in addition to the Table 3  for currents and their losses indicate that eddy currents decrease as the resultant magnetic flux which induces the sheath voltage is more uniform on that sheath. This talk crystallizes its result in the external cables of flat layout formation compared to those of the middle cable and the three cables of trefoil arrangement where they carry the lowest sheath eddy currents and then cause the smallest losses.  The

CONCLUSION
In (1) to (22) were simulated using the MATLAB program in this paper. The elements in the Math Operation List were used to solve these equations. These equations' outputs are mainly related to the cables' trefoil and flat arrangements when their sheaths are connected to the ground from one point, two points, or cross-connected. Keep attention to load current, cable length, spacing, and power frequency values before proceeding with any laying method. When the cable's sheath is connected from two points, the load on the cable should be reduced to less than the permissible limit when connected from one point because its capacity will be minor. Otherwise, the cable will be broken down. The eddy current losses can be ignored because it is minimal compared to the circulating current losses. It is not recommended to set the phase far apart from each other because this leads to an increase in the sheath voltage and the circulating currents exponentially. The sheath voltage can go up to 900 volts if one end is connected and the currents can arrive to 160 amps if two ends are connected. This is very risky on the cable condition and the safety of the electricians at maintenance. It is better to work at a frequency of 50 Hz. Finally, the best way to reduce the sheath voltages without any circulating current will be determined, in addition, a second path to return the fault current without weakening the cable connection points caused by all of the above methods. This process can be done by cutting the sheaths of the long cables at every joint. First, all the points at the beginning of every section are connected to the external insulated conductor. Then the cable is grounded from the two ends to guarantee passing the fault current through the external cable without causing damage to the joints. The total voltage of the sheath divided by the number of sections equals the sheath voltage of each section. Thus, if any sheath fault happens, the circulating current will flow in the defective section.