An Understanding of Zero Sequence Impedance Characteristics of Transmission Circuits with Special Consideration to Cables (Part 2)

Introduction

Part 1 of the paper introduced the reader to a concept whereby zero-sequence fault currents involving ground return current can be analysed in a simple manner by the application of Carson's equations.

In this second part, the techniques for the calculation of the geometric factors mentioned in Part 1 are explained. Such geometric reduction techniques allow complex cable arrangements to be reduced to a simple form thus allowing positive, negative and zero sequence impedances to be evaluated.

Positive & Negative Sequence Reactance

To determine the positive and negative sequence inductive reactance of three-phase cables and transmission lines it is first necessary to develop a few concepts that greatly simplify the problem. First, the total inductive reactance of a conductor carrying current is considered as the sum of two components:

(1) The inductive reactance due to the flux within a radius of one foot from the conductor centre, including the flux inside the conductor (self inductance).

(2) The inductive reactance due to the flux external to a radius of one foot and out to some finite distance (mutual inductance).

This concept was first given in Wagner and Evans book on Symmetrical Components (1933). It can be shown most easily by considering a two-conductor ('a' & 'b') single-phase circuit with the current flowing out in one conductor and returning in the other. Each conductor being of radius 'r' and axially separated by a distance 'd'.

The self inductance (H/m) of one wire is:

The mutual inductance (H/m) of one wire is:

The total external flux linkages of one wire is the sum of linkages with conductor 'b' due to the current in conductor 'a,' and the linkages with conductor 'a' due to the current in conductor 'b'. Adding the two together gives the classic inductance formula for a single round straight wire in the two-conductor single-phase circuit:

Giving the inductive reactance (/km): where f = frequency (Hz)

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Substituting GMD for 'd' and GMR for 're-¼ ' the equation for reactance in a single-phase circuit becomes:

In a three-phase circuit, assuming the conductors are symmetrically spaced, voltages are balanced and currents are balanced (no earth return current) then the positive or negative sequence reactance per phase is three times the value of that for one conductor of a single-phase circuit as derived above. Care needs to be taken in the derivation of the GMD values because the axial spacings between conductors in a multicore cable and single-core cable arrangements obviously differ. Values given for single-core cables by manufacturers assume a trefoil formation.

For transmission lines, as the inductive reactance resulting from the equivalent return conductor is dependent on line design, manufacturer's data sheets only quote the inductive reactance from magnetic flux to a 0.3048 metre (1 foot) radius. This value of self-inductance is determined from the formula:

where GMRc, in mm, is the geometric mean radius of the conductor

Geometry of Cables

The space relationship among sheaths and conductors in a cable circuit is a major factor in determining reactance. Geometric factors have come into universal use for defining the cross-section geometry of a cable circuit, and these are covered in following paragraphs.

Calculations involving zero-sequence reactance can be simplified if the conductors comprising a three-phase circuit are considered as a group and converted to a single equivalent conductor.

Geometric Mean Radius (GMR) of Conductors

The GMR is a term applied to a conductor, or group of conductors, that is used for the expression of self-inductance.

In the following, the lower case 'r' is used to designate the radius a single strand of an homogenous conductor and the upper case 'R' is used to designate the radius an entire group of strands. From the foregoing, it will be seen that the GMR of a circular solid conductor of radius 'r' is:

This is the radius of a theoretical tubular conductor having an infinitely thin wall that is inductively equivalent to a circular solid conductor. The GMR varies with stranding and also shape or make-up of the conductor.

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Now consider a 7-stranded circular conductor formed by 6 strands equi-spaced around a central strand. The axial spacing between adjacent strands is '2r'. Assuming that all strands are of the same metal and assuming uniform distribution of current over the cross section, there are 7 equal currents in the 7 strands, and the GMR of the entire current is the 49th root of the product of the 49 individual GMR among the 7 strands.

The first term in the large radical is the GMR among the 6 outer strands. Mutual distances (there & back) between pairs of these strands are equal to corresponding mutual distances between their centres. For 6 strands there are 6 x 5 distances, and the product of them all is equal to the 30 th

power of their GMR. The second term is the mutual distances (there & back) from each of the 6 outer strands to the inner strand. The last part of the equation is the expression for the GMR of each of the 7 individual strands. Notice that the sum of the powers within the radical equates to the overall root value (30+12+7). The above equation simplifies to:

Similarly, the GMR for a 19-stranded conductor is derived from the following formula:

[here R = 5r]

A 19-stranded conductor consists of a 7-stranded conductor with a further outer layer of 12 strands. There are 19 equal currents in the 19 strands, and the GMR of the entire current is the 361st root of the product of the 361 individual GMR among the 19 strands. The first two terms contain the GMR among strands of the outer and middle layers respectively. The middle term is the mutual distances from the outer layer to all other strands. The fourth term is the mutual distances between the 6 strands of the middle layer to the centre strand, and the final term under the radical represents the GMR of the 19 individual strands.

Similarly, for a 37 stranded conductor: [here R = 7r]

Similarly, for a 61 stranded conductor: [here R = 9r]

Similarly, for a 91 stranded conductor: [here R = 11r]

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For a 3-stranded conductor where there is no central strand:

The above GMR values are for single circular conductors. For a three-core cable the GMR of the three circular conductors as a group is the cube route of the product of the GMR for a single conductor and the square of the GMD between conductor centres. Segmental, hollow-core conductors etc., have different GMR values to those of circular conductors.

Geometric Mean Distance (GMD)

The GMD is a distance representing the equivalent spacing among conductors, and is used for the expression of mutual inductance. The total flux-linkages surrounding a conductor can be divided into two components, one extending inward from a cylinder of 12" radius, and the other extending outward from this cylinder to the current return path beyond which there are no net flux-linkages.

The flux-linkages per unit conductor current between the 12" cylinder and the return path are a function of the separation between the conductor and its return. The return path can in many cases be a parallel group of conductors, so that a geometric mean of all the separations between the conductor and each of its returns must be used in calculations. GMD, therefore, is a term that can be used in the expression for external flux-linkages, not only in the simple case of two adjacent conductors (may be same or different sizes) where it is equal to the distance between conductor centres, but also in the more complex case where two circuits each composed of several conductors are separated by an equivalent GMD.

The positive or negative sequence reactance of a three-phase circuit depends on separation among phase conductors. If the conductors are equilaterally spaced the GMD among conductors for that circuit is the distance from one conductor centre to another. Otherwise the GMD between three asymmetrically spaced circular conductors is the 6th root of the product of the square of their axial distances, simplified to the cube root of the product of their axial spacings.

The zero-sequence reactance of a three-phase circuit may depend on spacing among conductors and sheath as well as among conductors. A distance that represents the equivalent spacing between a conductor or a group of conductors and the enclosing sheath can be expressed as a GMD. Also, the equivalent separation between cable conductors and the sheath of a nearby cable, or the equivalent separation between two nearby sheaths, can be expressed as a GMD. Because these and other versions of GMD may be used successively in a single problem, care must be taken to identify and distinguish among them during calculations.

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The GMD of the sheath (armour or braid) of a multicore cable is the same as the GMR of the sheath and is the mean radius of the sheath. This also applies to lead sheaths and conductor screens. For three single-core cables the GMD (& GMR) of the three sheaths as a group is the cube route of the product of the GMR of a single sheath and the square of the GMD between cable centres. Where a cable has double armour, the overall GMD (& GMR) of the armour is the square root of the product of the GMD of each armour layer.

Worked Example for Multicore Cable

Data for 4c 185mm2 0.6/1kV PVC/PVC/GSWA/PVC cable:

Conductor stranding 37

Conductor self-radius 0.7678

Conductor radius (mm) rc = 8.465

Insulation diameter (mm) di = 21.28

Armour outer diameter (mm) da = 52.2

Armour wire diameter (mm) dw = 2.5

Conductor resistance at 20oC (/km) Rc = 0.0991

Armour resistance at 20oC (/km) Ra = 0.531

Frequency (Hz) f = 60

Geometric Factors for Phase Conductors:

GMR of one conductor (mm): GMRc = 6.5

Axial spacing of conductors (mm): Dab = Dac = di

GMD of phase conductors (mm): GMD3c = 23.89

GMR of phase conductors (mm): GMR3c = 15.48

Geometric Factors for Cable Armour:

GMR of armour (mm): GMRa= 24.85

GMD of armour (mm): GMDa = GMRa

Zero Sequence Impedance:

Resistivity of earth (-m): e = 10 (wet conditions)

Equivalent depth of earth (m): De = 268.8

The equivalent circuit for earth fault current is (refer to Part 1 of paper):

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Note that the armour branch is purely resistive since GMRa = GMDa. This is always the case for earth fault currents that have no neutral path.

Zero-sequence impedance of conductor branch Zco = 0.099 + i0.107 /km

Zero-sequence impedance of armour branch Zao = 1.593 /km

Zero-sequence impedance of earth branch Zm = 0.178 + i2.101 /km

For earth return current in both armour and earth paths, zero-sequence impedance is:

Zo1 = 1.097 + i0.813 /km

For earth return current in armour only, zero-sequence impedance is:Zo2 = 1.692 + i0.107 /km

Return path impedance, Zrp = Zm(Zao – Zm)/Zao = 0.998 + i0.706 /km

The current through the earth and armour will split according to the impedances of the branches.

Percentage of fault current through the ground = |Zrp/Zm| = 58%

Percentage of fault current through the armour = |Zm/Zao| = 76.7%

Increasing the ground resistivity (dry conditions) decreases the percentage fault current through the ground. Changing the ground resistivity to 0.9 x 10-6 -m (offshore steel) greatly increases the percentage fault current through the ground path, and that percentage would be even greater if the cable had braided armour instead of wire armour.

Positive and Negative Sequence Impedances:

Positive and negative sequence impedances are derived from:

Z1 = Z2 = 0.99 + i0.098

The leakage reactance of 0.098 /km would be the value quoted by manufacturers. For a three-core cable the reactance would be 0.089 /km, and sometimes manufacturers quote this value for both three and four-core cables – clearly inaccurate.

Worked Example for Single-Core Cables in Umbilical

Data for 3 x 1c 50mm2 12/20kV XLPE/SCR/PVC/SSWA/PE:

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Conductor stranding 19

Conductor self-radius 0.7576

Conductor radius (mm) rc = 4.25

Insulation diameter (including semi-conducting screen) (mm) di = 23

Metallic screen outside diameter (mm): ds = 23.2

Armour outer diameter (mm) da = 31

Armour wire diameter (mm) dw = 1.6

Conductor resistance at 20oC (/km) Rc = 0.387

Resistance of screen at 20oC (/km): Rs = 1.54

Armour resistance at 20oC (/km) Ra = 7.4

Frequency (Hz) f = 50

Geometric Factors for Phase Conductors:

GMR of one conductor (mm): GMRc = 3.22

Axial spacing of conductors (mm): Dab = 75 Dbc = 100 Dac = 75

GMD of phase conductors (mm): GMD3c = 82.55

GMR of phase conductors (mm): GMR3c = 28

Geometric Factors for Cable Armour:

GMR of armour (mm): GMRa= 14.7

GMD of armour (mm): GMDa = GMRa

GMR of 3 armours (mm): GMR3a= 46.44

GMD of 3 armours (mm): GMD3a = GMR3a

Zero Sequence Impedance:

Resistivity of earth (-m): e = 10 (wet conditions)

Equivalent depth of earth (m): De = 294.4

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The equivalent circuit for earth fault current is (refer to Part 1 of paper):

Zero-sequence impedance of conductor branch Zco = 0.387 + i0.095 /km

Zero-sequence impedance of armour branch Zao = 1.277 /km

Zero-sequence impedance of earth branch Zm = 0.148 + i1.65 /km

For earth return current in both armour/screen and earth paths, zero-sequence impedance is:

Zo1 = 0.863 + i1.609 /km

For earth return current in armour/screen only, zero-sequence impedance is:

Zo2 = 7.786 + i0.095 /km

Return path impedance, Zrp = Zm(Zao – Zm)/Zao = 0.476 + i1.514 /km

The current through the earth and armour/screen will split according to the impedances of the branches.

Percentage of fault current through the ground = |Zrp/Zm| = 95.8%

Percentage of fault current through the armour/screen = |Zm/Zao| = 21.4%

Positive and Negative Sequence Impedances:

Positive and negative sequence impedances are derived from:

Z1 = Z2 = 0.387 + i0.204

The reactance value for a trefoil formation (based on 36.4 mm o/d) would be 0.152 /km.

The value of the positive and negative sequence impedance calculated for single core cables, and for screened conductors in multicore cables, does not include the effect of sheath/screen currents that would flow when the sheath/screens are bonded at both ends. For the purists, the incremental change is small and can be ignored. Sheath/screen currents result in losses and flow in a direction opposite to that in the conductor, thus tending to limit the flux to the region between the conductor and the sheath/screen. The resistance value increases and the leakage reactance

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decreases. If the sheath/screen is isolated from earth at one end, then no sheath/screen currents flow and there would be no changes to the impedance value.

Part 3 of the paper will cover the geometric reduction techniques allowing complex cable arrangements to be reduced to a simple form.

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