Multiple Parallel Circuits

When cable circuits are installed in parallel, one circuit will impact the other circuit. This impact will be directly, where the heat generated by the cable may heat the other cable circuit. These effects are considered in the installation specific sections of this document. Additionally, the magnetic field generated by one circuit will impact on the other circuit, altering the losses generated by the conductors and the sheaths.

7.1.3.1 Conductor ac Resistance

When calculating the current rating of a cable circuit, the currents in parallel circuits will generate magnetic fields in the conductors of the cables in question that will cause some redistribution of the current within the conductor and an increase in its apparent AC resistance. This is a similar effect to the other phase cables within the same cable circuit and this is calculated as the proximity effect.

However, given that generally the spacing between cable circuits is significantly greater than those between cables within the same circuit and that the proximity effect is generally small, the effect of parallel circuits on the AC resistance of the conductor of the cable can generally be neglected.

If there is a concern that the current rating accuracy may be affected, the generalised computation method (filament heat source simulation) described in Section 7.1.3.2 can be used.

7.1.3.2 Impact on Sheath Loss for Specially Bonded Cable System

When there are several circuits in a proximity to each other, the induced voltages and the corresponding eddy currents in sheaths caused by the parallel circuits should be considered. IEC 60287-1-2 gives methods to calculate sheath eddy current losses for two identical single core parallel cable circuits in flat spaced formation.

However, the methods to calculate the sheath losses in IEC 60287 apply to limited circuit configurations and, in addition to being limited to two cable circuits, the methods to calculate eddy currents for single core cables are limited to three phase systems with flat or trefoil formations.

Analytical methods for alternative arrangements can be derived, for example Jackson (1975), but can become very complex. An alternative generalised method to calculate losses due to induced voltages can be applied and is briefly described below. It should be noted that this method will not only calculate the losses generated by the eddy currents, but all the losses generated by induced currents such as circulating losses in the sheath and proximity effects in conductors.

Such systems can conveniently be analyzed with the application of the filament heat source simulation (f. h. s. s.) method published by Anders (1997). The term conductor is used to denote any metallic component of the cable. Applying the method of filament heat source simulation, the conductors are replaced by a large number of smaller cylindrical sub-conductors or filaments. The number of filaments should be large enough so that the current density can be assumed uniform throughout each filament cross section. The size of the filaments is calculated such that the sum total cross-sectional area of the filaments equals the total conductor cross-sectional area. Helically wound wires are replaced by tubes with equivalent resistances.

The expressions describing the electrical connections of the filaments are as follows:

I

  B I

E B E  

B = connection matrix, B



E = column vector of filament longitudinal voltage drop, ^{E E E}





t

I^c = column vector of total conductor currents,





c

c = superscript referring to conductor quantities n = number of filaments

The value of NC will depend on the number of cables per phase and on whether or not the cables have metallic armour or sheath. The presence of neutral cables and earth conductors will further increase NC. In above method it is assumed that there is no cross-filament current: the contact resistance is ignored.

We can see that the connection matrix B is such that the sum of the filament currents in each conductor equals the total conductor current, and the longitudinal voltage drops in each filament of a conductor are equal to the conductor longitudinal voltage drop.

Our aim is to express filament currents as a function of phase conductor currents and the geometry of the system. The longitudinal voltage drop in a filament is given by:

ER  G I

Where:

s_ii = geometric mean radius of filament [m]

sij = geometric mean distance between filaments i and j [m]

μ0 = permeability of air, H/m

μr = relative permeability of the material, (μr=1 for nonmagnetic materials)

Since both E and I are complex quantities, their separate components must be determined. In effect, therefore, there are 2n equations and 4n unknown quantities to be found. After some manipulations, we obtain:

For systems where all the conductor currents are known, evaluation of the above equations represents the required solution. For systems in which total conductor currents are not known, calculations must be performed to determine the unknown values of the currents.

These equations can now be used to determine the sheath and armour loss factors by suitably specifying the matrix boundary conditions. If the sheaths are solidly bonded, the sheath and armour filaments are solidly bonded. Equation 22 yields both the circulating and approximate eddy currents after observing the boundary conditions that the voltage drops in all sheath filaments are equal and the sum of all sheath filament currents is zero. If the sheaths are bonded at one end only, the filaments representing the sheath of each cable are bonded together, but not those belonging to different cables. The boundary conditions now require the sum of sheath filament currents in each cable be equal to zero. Thus, the eddy currents and standing voltages can be computed from the above equations.

For solidly bonded systems, for example, we proceed as follows. Let us suppose that the first i-1 entries in the vector I^crepresent known cable conductor currents. From Kirchhoff’s first and second laws, we have:

Where:

E0 = the sheath longitudinal voltage drop [V]

Defining:

This constitutes a set of NC+1 equations in NC+1 unknowns. Some reduction in computational effort can be obtained by noting that the longitudinal voltage drops of the central conductors are not of interest.

Similar equations can be set up for single-point bonded systems. The sheath voltages in this case are different and are to be computed.

The loss factor for a particular conductor (a sheath, armour, or pipe) composed of filaments k to m is equal to:

Where j is the index of central conductor filaments belonging to the same cable as the sheath or armour. The current represents the rms values. Please note that the f.h.s.s. method assumes that all the conductors are straight (and parallel). As it is, it is not directly applicable to the calculation of losses in the screen of single-core cables composed of a bundle of wires or in the metal screen of 3-core cable (because of the twisting of the conductors).

7.1.3.3 Impact on Sheath Loss for Solidly Bonded and Earthed Cable System

When there are several circuits in a proximity to each other, the induced voltages and the circulating currents in sheaths caused by the parallel circuits should be considered. IEC 60287-1-3 gives methods to calculate sheath circulating current losses for any given formation.

7.1.3.4 Impact on Armour Losses

The magnetic field from one circuit will modify the losses generated by the armour on another circuit.

For multicore cables that are armoured, the spacing between the cores will be much smaller than the spacing between circuits, so the effect is likely to be negligible. For single core cables, in most cases the cables will be installed much closer together than the circuits so it is anticipated that the effect will be small and the effect can be neglected.

7.1.3.5 Thermal Impact of Multiple Circuits

The thermal impact of circuits on one another will depend on how they are installed and hence are covered in the sections on each specific type of installation.