General

A line is a static power plant that has electrical parameters distributed along its length. The basic parameters of the line are conductor series impedance and shunt admittance. Each conductor has a self-impedance and there is a mutual impedance between any two conductors. The impedance generally consists of a resistance and a reactance. The shunt admittance consists of the conductor’s conductance to ground and the susceptance between conductors and between each conductor and earth. The conductance of the air path to earth represents the leakage current along the line insulators due to corona. This is negligibly small and is normally ignored in short circuit, power flow and transient stability analysis.

Practical calculations of multi-conductor line parameters with series impedance expressed in pu length (e.g. ω/km) and shunt susceptance in μS/km are carried out using digital computer programs. These parameters are then used to form the line series impedance and shunt admittance matrices in the phase frame of reference as will be described later. The line capacitances or susceptances are calculated from the line potential coefficients which are essentially dependent on the line and tower physical dimensions and geometry. The calculations use the method of image conductors, assumes that the earth is a plane at a uniform zero potential and that the conductor radii are much smaller than the spacings among the conductors. The self and mutual impedances depend on the conductor material, construction, tower or line physical dimensions or geometry, and on the earth’s resistivity. Figure 3.2 is a general illustration of overhead line tower physical dimensions and spacings of conductors above the earth’s surface as well as conductor images below earth used for the calculation of line electrical parameters.

Figure 3.2 A general illustration of an overhead line physical dimensions and conductor spacings relative to tower centre and earth

The fundamental theories used in the calculations of resistance, inductance and capacitance parameters of overhead lines are extensively covered in most basic power system textbooks and will not be repeated here. Practical digital computer based calculations used in industry consider the effect of earth as a lossy conducting medium. The equations used in the calculation of line parameters are presented below.

Potential coefficients, shunt capacitances and susceptances

Using Figure 3.2, given a set of N conductors, the potential V of conductor i due to conductor’s own charge and charges on all other conductors is given by

     (3.1)

where P_ij is the Maxwell’s potential coefficient expressed in km/F and Q_j is the charge in C/km. The equations assume an infinitely long perfectly horizontal conductors above earth whose effect is included using the method of electrostatic images. This method is generally valid up to a frequency of about 1 MHz. The potential of a conductor i above earth due to its own charge and an equal but negative charge on its own image enables us to calculate the self-potential coefficient of conductor i as follows:

     (3.2a)

where y_i is the height of conductor i above earth in m and r_i is the radius of conductor i in m. Clearly y_i is much greater than r_i. The potential of conductor i due to a charge on conductor j and an equal but negative charge on the image of conductor j enables us to calculate the mutual potential coefficient between conductor i and conductor j as follows:

     (3.2b)

where D_ij is the distance between conductor i and the image beneath the earth’s surface of conductor j in m, and d_ij is the distance between conductor i and conductor j in m. Equation (3.1) can be rewritten in matrix form as

     (3.3a)

where P is a potential coefficient matrix of N × N dimension. Multiplying both sides of Equation (3.3a) by P⁻¹, we obtain

     (3.3b)

where

     (3.3c)

C is line’s shunt capacitance matrix and is equal to the inverse of the potential coefficient matrix P. Under steady state conditions, the current and voltage vectors are phasors and are related by

     (3.4a)

since the line’s conductance is negligible at power frequency f. The line’s shunt susceptance matrix is given by

     (3.4b)

Series self and mutual impedances

Self impedance

The equations assume infinitely long and perfectly horizontal conductors above a homogeneous conducting earth having a uniform resistivity ρ_c(ωm) and a unit relative permeability. Proximity effect between conductors is neglected. Using Figure 3.2, the series voltage drop in V of each conductor due to current flowing in the conductor itself and currents flowing in all other conductors in the same direction is given by

     (3.5a)

where Z is the impedance expressed in Ω/km and I is the current in amps. The self-impedance of conductor i is given by

     (3.5b)

where subscript c represents the contribution of conductor i resistance and internal reactance, g represents a reactance contribution to conductor i due to its geometry, i.e. an external reactance contribution and e represents correction terms to conductor i resistance and reactance due to the contribution of the earth return path. If skin effect is neglected, i.e. assuming a direct current (dc) condition or zero frequency, the internal dc impedance of a solid magnetic round conductor, illustrated in Figure 3.3(a), is given by

Figure 3.3 Illustration of some conductor types: cross-section of a (a) solid round conductor, (b) tubular round conductor and (c) 30/7 stranded conductor (30 Aluminium, 7 inner steel stands)

     (3.6)

where μ_r and ρ_c are the relative permeability and resistivity of the conductor, respectively, and r_c is the conductor’s radius.

Where skin effect is to be taken into account, the following exact equation for the internal impedance of a solid round conductor can be used

     (3.7a)

where

is defined as the complex propagation constant and δ is skin depth or depth of penetration into the conductor and is given by

     (3.7b)

and I_i are modified Bessel functions of the first kind of order i. For calculation of line parameters close to power frequency, the following alternative equation, suitable for hand calculations using electronic calculators, is found accurate up to about 200 Hz:

     (3.8)

Equation (3.8) shows that skin effect causes an increase in the conductor’s effective ac resistance and a decrease in its effective ac internal reactance. Also, at f = 0, Equation (3.8) reduces to Equation (3.6).

In the case of a tubular or hollow conductor, illustrated in Figure 3.3(b), the dc internal impedance is given by

     (3.9)

Mathematically, the case of a solid round conductor is a special case of the hollow conductor since by setting r_i = 0, Equation (3.9) reduces to Equation (3.6). Aluminium conductor steel reinforced (ACSR) or modern gapped-type conductors can be represented as hollow conductors if the effect of steel saturation can be ignored. Saturation may be caused by the flow of current through the helix formed by each Aluminium strand that produces a magnetic field within the steel.

Where skin effect of a hollow conductor is to be taken into account, the following exact equation for the internal impedance of such a conductor can be used

     (3.10a)

     (3.10b)

where I_i and K_i are modified Bessel functions of the first and second kind of order i, respectively. Mathematical solutions suitable for digital computations that provide sufficient accuracy are available in standard handbooks of mathematical functions and also in modern libraries of digital computer programs. Equation (3.7a) that represents the case of a solid conductor can be obtained from Equation (3.10a) by substituting r_i = 0.

The external reactance of conductor i due to its geometry is given by

     (3.11)

In practice, the internal reactance of a conductor is much smaller than its external reactance except in the case of very large conductors at high frequencies. Generally, precision improvements in the former would have very small effect on the total reactance.

The contribution of the earth’s correction terms to the self-impedance of conductor i is presented after the next section.

Stranded conductors

Overhead lines with stranded conductors are extensively used and their effect is important in calculating power frequency line parameters. Figure 3.3(c) illustrates a conductor that consists of N strands. Even a tightly packed bundle of a large number of strands will still leave small unfilled gaps between the strands. The total cross-sectional area of the strands is smaller than the equivalent cross-sectional area of a solid conductor. This reduction factor is the area of the conductors divided by the area of the equivalent solid conductor and can be used to increase the equivalent conductor effective resistivity whilst keeping its overall radius the same. However, for line parameter calculations, the equivalent geometric mean radius of the conductor is calculated and for a general conductor that consists of N strands each having a radius r, this is given by

     (3.21)

The effect of stranding causes a reduction in the outer radius of the equivalent conductor of the bundle of strands. A reduction factor, termed the stranding factor, is defined as the ratio of the geometric mean radius, GMR_c, of the conductor to the conductor’s outer radius and this can be calculated for any known stranding arrangement. For example, for a homogeneous conductor that consists of seven strands arranged on a hexagonal array, it can be shown that the area reduction factor is equal to 0.77778 and the stranding factor is equal to 0.72557.

Bundled phase conductors

Figure 3.4 illustrates a general case of a phase bundle consisting of N conductors per phase with asymmetrical spacings between the conductors. Practical examples of such asymmetrical arrangement are rectangular and non-equilateral triangular bundles. Asymmetrical conductor bundles of two or more conductors can be represented by an equivalent single conductor. The radius of the equivalent conductor can be calculated by applying the GMR technique to an arbitrary set of axes and origin as shown in Figure 3.4. The GMR of the equivalent conductor of the entire bundle is given by

Figure 3.4 General asymmetrical bundle of phase conductors

     (3.22)

where N is ≥2 and is the number of conductors in the bundle. When impedance calculations are carried out, GMR_c is the geometric mean radius of one conductor in the bundle. When potential coefficient calculations are carried out, GMR_c is equal to the conductor’s outer radius.

Special cases of symmetrical conductor bundles are shown in Figure 3.5. This shows bundles of two, three, four and six conductors, with circles drawn through the centres of the individual conductors. It can be shown that the general Equation (3.22) of the equivalent GMR_Eq reduces to

Figure 3.5 Typical symmetrical bundles of phase conductors

     (3.23a)

where A is the radius of a circle through the centres of the bundled conductors and is given by

     (3.23b)

and d is the spacing between any two adjacent conductors in the symmetrical bundle.

It should be noted that for N subconductors in a bundle, the internal resistance and inductance included in Equation (3.19a) must be divided by N.

An alternative method of bundling phase conductors is to form the full phase impedance and susceptance matrices from the parameters of all conductors including bundled subconductors and earth wires. For example, for a single-circuit line with one earth wire and four conductors per phase, the dimension of the resultant phase impedance matrix is 13 × 13. The bundled conductors are effectively short-circuited by zero impedance so the voltages are equal for all the subconductors in the bundle and the sum of currents in all subconductors is equal to the equivalent phase current. For example, for phase R, the subconductor voltages are V₁ = V₂ = V₃ = V₄ = V_R whereas for the subconductor currents, I₁ + I₂ + I₃ + I₄ = I_R. Using these voltage and current constraints, the bundle conductors can then be combined using standard matrix reduction techniques to produce a 4 × 4 matrix that represents three equivalent phase conductors and one earth wire.

Average height of conductor above earth

The calculation of line parameters is based on the assumption of perfectly horizontal conductors above the earth’s plane. The average sag of phase conductors and earth wires between towers together with the height of the conductor at the tower can be used to calculate an average conductor height for use in the calculation of line’s electrical parameters. Using Figure 3.6, it can be shown that the average height of the earth wire conductor above ground is given by

Figure 3.6 Average conductor height above earth between two towers

     (3.24)

where H_t is the conductor height at the tower in m and S_av is the average conductor sag in m measured at mid-span between two suspension towers and usually assumed to apply to the entire line length, i.e. ignoring the effect of angle or tension towers. For span lengths of up to 400 to 500 m, a simple formula can be derived by assuming that the variation of conductor height with distance between the two towers is a parabola, that is

     (3.25)

where x_ms is half the span length. The average height above ground between the two towers is calculated by integrating Equation (3.25) over the span length, i.e.

Thus,

     (3.26)

The calculation of average conductor height should take into account the different sags of phase and earth wire conductors.

Phase and sequence series impedance matrices

Figure 3.8(a) shows the coupled series inductive circuit of Figure 3.7 where each phase and earth wire conductors, and earth are represented as equivalent self and mutually coupled impedances.

Figure 3.8 Series impedance and shunt susceptance circuits of Figure 3.7: (a) three-phase coupled series impedance circuit; (b) three-phase coupled shunt capacitance circuit; (c) reduced three-phase shunt capacitance circuit with earth wires eliminated and (d) three-phase shunt capacitance circuit in sequence terms

To derive a general formulation of the series phase impedance matrix for such a line, the phase R series impedance voltage drop equation can be written as

     (3.27a)

     (3.27b)

With the phases at the receiving end all earthed, I_R + I_Y + I_B = I_E + I_S1 + I_S2 and using Equation (3.27b) in Equation (3.27a), we have

or

     (3.28a)

where

     (3.28b)

The self and mutual phase impedances defined in Equation (3.28b) as well as those of the earth wires include the effect of earth impedance Z_EE. We can write similar equations for ΔV_Y, ΔV_B, ΔV_S1, ΔV_S2 and combine them all to obtain the following series voltage drop and partitioned series phase impedance matrix:

     (3.29)

In large-scale power system short-circuit analysis, we are interested in the calculation of short-circuit currents on the faulted phases R, Y or B or a combination of these but generally not in the currents flowing in the earth wires. One exception is the calculation of earth fault return currents covered in Chapter 10. For the elimination of the earth wires, the partitioned Equation (3.29) is rewritten as follows:

     (3.30a)

where

     (3.30b)

Z_AA consists of the self and mutual impedances of phase conductors R, Y and B with earth return. Z_AS consists of the mutual impedances between the phase conductors R, Y and B, and earth wires S₁ and S₂, with earth return. noting that the individual impedances are symmetric, i.e. Z_RS1-E = Z_S1R-E. Z_SS consists of the self and mutual impedances of the earth wires S₁ and S₂ with earth return. The earth return effect is shown in Equation (3.28b). Expanding Equation (3.30a), we obtain

     (3.31a)

     (3.31b)

In the majority of line installations, the earth wires are bonded to the tower tops and are only partially earthed by the footing resistance of each tower to which they are connected but solidly earthed at substations. However, it is usual to assume zero tower footing resistances and that the earth wires are at zero voltage at all points. Thus, using ΔV_S1S2 = 0 in Equation (3.31b) and substituting the result in Equation (3.31a), we obtain

     (3.32a)

where

     (3.32b)

and

     (3.32c)

Equation (3.32b) shows that the self and mutual phase impedances of the phase conductors of matrix Z_AA are reduced by the presence of the earth wires. Equation (3.32c) indicates the elements of the phase impedance matrix of the single-circuit three-phase line with both earth wires eliminated and including the effect of the earth impedance. It is noted that the mathematical elimination of the earth wires only eliminates their presence from the full matrix but not their effects which are included in the modified elements of the reduced matrix, i.e. Z_RR(S-E) –; Z_RR–E.

Although far less common, segmented earth wires may be used to prevent circulating currents in earth wires and associated I²R losses. This is a ‘T’ arrangement where the earth wires are bonded to the top of the middle tower but insulated at the adjacent towers on either side. This is equivalent to I_S1 = I_S2 = 0 in Equation (3.29) and hence the reduced phase impedance matrix Z_RYB(S–E) is directly obtained from Equation (3.29) by deleting the last two rows and columns that correspond to the earth wires. This results in Z_RYB(S–E) = Z_AA. The impedance matrix of Equation (3.32c) is symmetric about the diagonal and in the case of asymmetrical spacings between the conductors, the self or diagonal terms are generally not equal to each other, and neither are the mutual or off-diagonal terms. Currents flowing in any one conductor will induce voltage drops in the other two conductors and these may be unequal even if the currents are balanced. This is because the mutual impedances, which are dependent on the physical spacings of the conductors, are unequal. Rewriting the voltage drop equation using Equation (3.32c) and dropping the S and E notation for convenience, we can write

     (3.33)

Assuming balanced three-phase currents, i.e. I_Y = h²I_R and I_B = hI_R where h = e^j2π/3, we obtain from Equation (3.33)

     (3.34a)

     (3.34b)

     (3.34c)

Equation (3.34) describes the per-phase or single-phase representation of the three-phase system when balanced currents flow. However, the three per-phase equivalent impedances are clearly unequal and a single per-phase representation cannot be used.

The sequence impedance matrix of the phase impedance matrix given in Equation (3.33) can be calculated, assuming a phase rotation of RYB, using V_RYB = HV^PNZ and I_RYB = HI^PNZ where H is the sequence to phase transformation matrix given in Chapter 2. Therefore,

(3.35)

where the nine sequence impedance elements of this matrix are as given by Equation (2.26a) in Chapter 2.

The conversion to the sequence reference frame still produces a full and even asymmetric sequence impedance matrix that includes intersequence mutual coupling. Where this intersequence coupling is to be eliminated, the circuit has to be perfectly transposed. Transposition is dealt with in Section 3.2.4.

Phase and sequence shunt susceptance matrices

Figure 3.8(b) shows the shunt capacitance circuit of Figure 3.7 involving phase conductors and earth. To derive the shunt phase susceptance matrix, we use the potential coefficients calculated from the line’s dimensions. Thus, the voltage on each conductor to ground as a function of the electric charges on all the conductors is given by

(3.36a)

which can be written as

     (3.36b)

Again, with the earth wires at zero voltage, they are eliminated from Equation (3.36b) as follows

     (3.37a)

where

     (3.37b)

Again, P_RYB(S) is the reduced potential coefficient matrix that includes the effects of the eliminated earth wires. Equation (3.37b) shows that the self and mutual potential coefficients of the phase conductors of the matrix P_AA are reduced by the presence of the earth wires. To derive the shunt phase capacitance matrix of the line, multiplying Equation (3.37a) by , we obtain

     (3.38a)

where

     (3.38b)

and C_RYB(S) is the shunt phase capacitance matrix.

Expanding Equation (3.38b), and noting that the capacitance matrix elements include the effect of the eliminated earth wires, we have

     (3.38c)

In Equation (3.38c), the elements of this capacitance matrix are increased by the presence of the earth wires which reduce the potential coefficients. Equation (3.38c) is illustrated in the reduced shunt capacitance equivalent shown in Figure 3.8(c) after the elimination of the earth wires but not their effects. Using Equations (3.4b) and (3.38c) and dropping the S notation for convenience, the nodal admittance matrix of Figure 3.8(c) is given by

     (3.39a)

or

     (3.39b)

The negative signs for the off-diagonal capacitance or susceptance terms are due to the matrices being in nodal form. For example, from Figure 3.8(c), and using susceptances instead of capacitances, the injected current into node R is given by

     (3.40a)

where

     (3.40b)

and similarly for I_Y and I_B. The off-diagonal terms represent shunt susceptances between two-phase conductors, e.g. R and Y, etc. The diagonal terms, e.g. that for conductor R, represent the sum of the shunt capacitances between conductor R and all other conductors including earth as shown in Figure 3.8(c).

The shunt susceptance matrix of Equation (3.39a) is symmetric about the diagonal but in the case of asymmetrical spacings between the conductors, the self that is diagonal terms are generally not equal to each other, and neither are the mutual or off-diagonal terms. Therefore, as for the series phase impedance matrix, the sequence shunt susceptance matrix is given by

(3.41)

where the nine sequence susceptance elements of this matrix are as given by Equation (2.26b) in Chapter 2 but with Z replaced by B.

As for the series sequence impedance matrix, the intersequence mutual coupling present can be eliminated by assuming the line to be perfectly transposed. This is dealt with in the Section 3.2.4.

An alternative method to obtain the reduced 3 × 3 capacitance matrix of Equation (3.38c) is to calculate the inverse of the 5 × 5 potential coefficient matrix of Equation (3.36a) which gives a 5 × 5 shunt capacitance matrix. The 3 × 3 shunt capacitance matrix can then be directly obtained by simply deleting the last two rows and columns that correspond to the earth wires. The reader is encouraged to prove this statement.

We have presented in this section the general case of an untransposed single-circuit three-phase line with two earth wires. The cases where the line has only one earth wire or no earth wires become special cases from a mathematical viewpoint. The 5 × 5 matrices of Equations (3.29) and (3.36a) become 4 × 4 matrices in the case of one earth wire and 3 × 3 matrices in the case of no earth wires and the rest of the analysis is similar. If the earth wires are identical and are symmetrical with respect to the three-phase circuit, they can be initially analytically replaced by an equivalent single earth wire whose equivalent impedance is half the sum of the self-impedance of one earth wire and the mutual impedance between the earth wires. However, analytical calculations are not necessary because of the extensive use in industry of digital computer calculations of line parameters or constants.

Phase and sequence series impedance matrices

Figure 3.10 illustrates a typical double-circuit line with two earth wires. For circuit A, the phase conductors are numbered 1, 2 and 3 and occupy positions t₁, m₁ and b₁. For circuit B, the phase conductors are numbered 4, 5 and 6 and occupy positions t₂, m₂, b₂. The earth wires are numbered 7 and 8.

Figure 3.10 Typical double-circuit three-phase overhead line with two earth wires

The general formulation of the series phase impedance matrix for this line can be derived from the voltage drops across each conductor in a manner similar to that presented in Section 3.2.2. Let Z_ii be the self-impedance of conductor i with earth return and Z_ij be the mutual impedance between conductors i and j with earth return. The series voltage drops across all phase and earth wire conductors, denoted C1 to C8, are given by

(3.57a)

or

     (3.57b)

where

     (3.57c)

Z_AA consists of the self and mutual impedances of circuit A phase conductors 1, 2 and 3. Z_BB consists of the self and mutual impedances of circuit B phase conductors 4, 5 and 6. consists of the mutual impedances between the phase conductors of circuit A and the phase conductors of circuit B. consists of the mutual impedances between circuit A phase conductors 1, 2 and 3 and the earth wires 7 and 8. consists of the mutual impedances between circuit B phase conductors 4, 5 and 6 and the earth wires 7 and 8. Z_SS consists of the self and mutual impedances of conductors 7 and 8 that represent the earth wires.

To eliminate the two earth wires, we set ΔV_S = 0 in Equation (3.57b), and after a little matrix algebra, we obtain

     (3.58a)

where

and

(3.58b)

Equation (3.58b) is the series phase impedance matrix of the double-circuit line containing the impedance elements of the two circuits A and B with earth return, and with both earth wires eliminated. In the general case of asymmetrical spacings between the conductors within each circuit and between the two circuits, the self-impedance matrices of each circuit, Z′_AA of circuit A and Z′_BB of circuit B, and the mutual impedance matrices between the two circuits, Z′_AB and Z′_BA, are not balanced within themselves. Currents flowing in any one conductor will induce voltage drops in the five other conductors and these may be unequal even if the currents are balanced. The sequence impedance matrix of the 6 × 6 phase impedance matrix of Equation (3.58b) can be calculated by applying the phase-to-sequence transformation matrix to each voltage and current matrix vector in Equation (3.58a). Assuming an electrical phase sequence of R, Y, B for conductors 1, 2, 3 of circuit A and similarly for conductors 4, 5, 6 of circuit B, we can apply V_RYB = HV^PNZ and I_RYB = HI^PNZ to circuit A and circuit B voltage and current vectors. Therefore, the sequence voltage drop equation is given by

     (3.59a)

where

     (3.59b)

Equation (3.59b) shows that both the self-impedance matrices of each circuit and the mutual impedance matrices between the two circuits must be pre-and-post multiplied by the appropriate transformation matrix viz Z^PNZ = H⁻¹Z′_PhaseH. In each case of Equation (3.59b), this conversion will produce, in the general case of asymmetrical spacings, a full asymmetric sequence matrix. Even if the two circuits are identical, the inter-circuit sequence matrices will not be equal (the reader is encouraged to prove this statement). Consequently, in order to derive appropriate sequence impedances for a double-circuit line, including equal inter-circuit parameters, for use in large-scale power frequency steady state analysis, certain transposition assumptions need to be made. This is dealt with in Section 3.2.6.

Phase and sequence shunt susceptance matrices

We now derive the shunt phase susceptance matrix of our double-circuit line using the potential coefficients calculated from the line’s physical dimensions or geometry. The voltage on each conductor to ground is a function of the electric charges on all conductors, thus

(3.60a)

or

     (3.60b)

where

     (3.60c)

To eliminate the earth wires, we set V_S = 0 in Equation (3.60b), and after a little matrix algebra, we obtain

     (3.61a)

where

and

(3.61b)

The Maxwell’s or capacitance coefficient matrix of the double-circuit line is given by . Dropping the S notation for convenience and remembering that this has the form of a nodal admittance matrix, we have

(3.62a)

or

     (3.62b)

and using B_Phase = ωC_Phase, the nodal shunt phase susceptance matrix is given by

     (3.62c)

Again, in the general case of asymmetrical spacings between the conductors within each circuit and between the two circuits, the self-susceptance matrices of each circuit, B_AA of circuit A and B_BB of circuit B, and the mutual susceptance matrices between the two circuits, B_AB and B_BA, are not balanced within themselves. The sequence susceptance matrix of Equation (3.62c) is calculated by applying the phase-to-sequence transformation matrix to each voltage and current matrix vector. Thus, using V_RYB = HV^PNZ and I_RYB = HI^PNZ, we obtain

     (3.63a)

where

     (3.63b)

This conversion will produce, in each case, a full asymmetric sequence matrix and even if the two circuits are identical, the inter-circuit sequence matrices will not be equal. Consequently, in order to derive appropriate sequence susceptances for a double-circuit line, including equal inter-circuit parameters, certain transposition assumptions need to be made. This is dealt with in Section 3.2.6.

An alternative method to obtain the 6 × 6 capacitance or susceptance matrix is to calculate the inverse of the 8 × 8 potential coefficient matrix of Equation (3.60a) giving a 8 × 8 shunt capacitance matrix including, explicitly, the earth wires. The required 6 × 6 shunt phase capacitance matrix can be directly obtained by simply deleting the last two rows and columns that correspond to the earth wires. The reader is encouraged to prove this statement.

Lines with three coupled circuits

Consider a line with three mutually coupled circuits 1, 2 and 3 erected on the same tower as illustrated in Figure 3.15(a). The transposed and balanced 9 × 9 series phase impedance matrix that retains PPS, NPS and ZPS inter-circuit mutual coupling can be derived using the technique presented for double-circuit lines. Each circuit is perfectly transposed within itself and the inter-circuit mutual impedance matrices between any two circuits result inequal self and equal mutual impedances. However, the inter-circuit mutual coupling impedances between circuits 1 and 2, circuits 1 and 3, and circuits 2 and 3 are assumed unequal to maintain generality. It can be shown that the resultant balanced phase impedance matrix is given by

(3.90)

The elements of this matrix can be calculated as for a double or two mutually coupled circuits given in Section 3.2.6 by averaging the appropriate self and mutual phase impedance terms of the original untransposed matrix.

For the ideal transposition the inter-circuit impedance matrices have all nine elements equal thereby retaining only ZPS inter-circuit mutual coupling. The ideal transposition assumption results in Z_S12 = Z_M12 = A₁₂ for circuits 1 and 2, Z_S13 = Z_M13 = A₁₃ for circuits 1 and 3, and Z_S23 = Z_M23 = A₂₃ for circuits 2 and 3. The sequence impedance matrix can be calculated based on knowledge of the electrical phasings of the three circuits. The above series impedance matrix assumes that all circuits are phased RYB and the corresponding sequence impedance matrix can be calculated using

     (3.91)

giving

(3.92a)

where

     (3.92b)

In the ideal transposition case where only ZPS inter-circuit mutual coupling is retained, we have .

The 9 × 9 transposed and balanced shunt phase susceptance matrix that retains PPS, NPS and ZPS inter-circuit mutual coupling, or ZPS inter-circuit mutual coupling only, is similar in form to the series phase impedance matrix. Similarly, the sequence susceptance matrix is similar in form to the sequence impedance matrix.

Lines with four coupled circuits

We now briefly outline for the most interested of readers the balanced phase impedance matrix for an unusual case of identical four circuits or two double-circuit lines running in close proximity to each other. In deriving such a matrix, it is assumed that each circuit is perfectly transposed and hence represented by a self and a mutual impedance. The inter-circuit mutual impedance matrices between any two circuits are also assumed balanced reflecting a transposition assumption similar to that for double-circuit lines. Further, the inter-circuit mutual coupling matrices between any two pair of circuits are assumed unequal to maintain generality. It can be shown that the balanced series phase impedance matrix of such a complex multi-conductor system with transpositions that retain PPS/NPS and ZPS inter-circuit mutual coupling is given by

(3.93)

Alternatively, an ideal transposition assumption results in C = D, E = F, G = H, J = K, L = M and U = W. The sequence impedance matrix that corresponds to Equation (3.93) can be calculated with all circuits phased RYB using

     (3.94)

giving

(3.95a)

where

     (3.95b)

In the ideal transposition case where only ZPS inter-circuit mutual coupling is retained, we have .

Solidly bonded cables

Solid bonding of cable sheaths is where both ends of the cable are bonded and connected to earth as shown in Figure 3.19.

Figure 3.19 Solidly bonded three-phase cable

Under PPS voltage conditions, voltages are induced in the sheaths and because these are solidly bonded and earthed at both ends, circulating currents flow such that the sheaths are at zero voltage along their entire length. However, these currents can cause significant sheath losses and heating which can adversely affect the thermal rating of the cable’s core conductor. This arrangement is most suitable for three-core cables and is not usually used at voltages above 66 kV. Under ZPS conditions, the sheaths provide an excellent path for the earth fault return current.

Single-point bonded cables

Bonding the cable sheaths to earth at one end and using sheath voltage limiters at the other end is a good arrangement for short cable lengths. This limits the sheath voltages and prevents circulation of sheath currents during normal operation because of the absence of a closed sheath circuit. A similar arrangement that provides similar benefits and allows doubling the cable length is bonding and earthing the cable sheath at the cable mid-point and using sheath voltage limiters at the ends. Because no sheath current can flow, no sheath losses occur. However, in practical installations, a transposed earth continuity conductor earthed at both ends is also used in order to provide a return path for the current under ZPS earth fault conditions. Under PPS conditions, the transposed earth continuity conductor carries no current. Figure 3.20 illustrates these three earthing arrangements.

Figure 3.20 Single-point bonded three-phase cable: (a) end-point bonded cable; (b) mid-point bonded cable and (c) end-point bonded cable with a transposed earth continuity conductor

Cross-bonded cables

For long cable circuits, cross-bonding is commonly used. This method aims at minimising the total induced voltages in the sheath in order to minimise circulating currents and losses. The cable circuit is divided into major and minor sections so that each major section consists of three minor sections of equal length. Figure 3.21 illustrates a major section of a cross-bonded cable.

Figure 3.21 (a) Cross-bonded three-phase cable; (b) 275 kV oil-filled paper insulated cable and (c) 400 kV cross-linked polyethylene (XLPE) cable

The best arrangement is where the cores of the three minor sections within each major section are perfectly transposed but the sheaths are not. The sheaths at both ends of the major section are solidly bonded and earthed but at the other two positions within the major section, they are bonded to sheath voltage limiters. The bonding and earthing at the ends of a major section eliminate the need for the earth continuity conductor required in single-point bonded and earthed arrangements.

Under PPS conditions, voltages are induced in the sheaths, but because the cores are perfectly transposed, the resulting voltages in each minor section are separated by 120° and thus sum to zero. Therefore, no sheath currents flow. Under ZPS conditions, e.g. earthed faults, the cross-bonding does not affect the excellent path which the sheaths provide for the return current. The earlier and rather less satisfactory cross-bonding method of transposing the sheaths but not the cores cannot achieve a good balance of induced sheath voltages unless the cables are laid in a trefoil formation. Therefore, the cores of cross-bonded cables laid in flat formations are generally transposed. Cross-bonding with core transposition is a general practice in the UK at 275 and 400 kV.

General

Single-core and three-core cables are characterised by close coupling between the core conductor and the conducting sheath. Also, for buried self-contained single-core cables, there is a conducting earth path between the three adjacent conductors. In pipe-type cables, the steel pipe is also a conducting medium. Therefore, a single circuit self-contained three-phase cable is a multi-conductor system that consists of three cores and three sheaths (six conductors) together with the earth path. If the three cables are in a steel pipe, induced voltages in the pipe will cause currents to circulate. For single-core submarine cables where each has a conducting armour, a three further conductors are present. Therefore, for N three-phase cable circuits, the number of coupled conductors is 6N for core–sheath single–core or three–core cables, and 9N for core–sheath–armour single-core cables. For core–sheath–armour single-circuit three-core cables, the total number of coupled conductors is seven.

Similar to an overhead line, the basic electrical parameters of cables are the self and mutual impedances between conductors, and conductor shunt admittances. For power frequency steady state analysis, the small conductance of the insulation is usually neglected. Practical calculations of multi-conductor cable parameters with series impedance expressed in ω per-unit length and shunt susceptance in μ S unit length, are carried out using digital computer programs. These parameters are then used to form the cable series impedance and shunt admittance matrices in the phase frame of reference as will be described later. The cable capacitances or susceptances are calculated from the cable physical dimensions and geometry. The self and mutual impedances are calculated from the conductor material, construction, physical dimensions and earth resistivity. The significant variety of available cable designs and installations cannot be covered here. The equations of the parameters for some widely used cable designs are presented below.

Shunt capacitances and susceptances

For single-core screened cables where the phases are laid parallel to each other and to the earth’s surface, the earth the screens are connected to acts as an electrostatic shield so that there is no electrostatic coupling between the phases. This also applies to three-phase three-core screened cables where each core is individually screened. Therefore, there is no mutual capacitance among the three phases and the individual cable capacitance is independent of the spacings between the phases. Figure 3.22 is a cross-section of a general cable showing a hollow core conductor, core insulation, sheath/screen conductor, sheath insulation, armour conductor and a further insulation layer such as a plastic sheath.

Figure 3.22 Cross-section of one phase of a self-contained armoured cable

Using Figure 3.22, the capacitance between the core and sheath of a screened single core or a screened three-core cable is given by

     (3.96a)

and the corresponding susceptance is given by

     (3.96b)

The capacitance between the sheath and armour, where present, is given by

     (3.97a)

and the corresponding susceptance is given by

     (3.97b)

Similarly, the capacitance between the armour and earth is given by

     (3.98a)

and the corresponding susceptance is given by

     (3.98b)

where

r_ic = inner radius of core tubular conductor

r_oc = outer radius of core conductor = inner radius of core conductor insulation assuming no core semiconductor tape

r_is = outer radius of core conductor insulation = inner radius of metallic sheath assuming no tape over insulation

r_os = outer radius of metallic sheath = inner radius of sheath insulation

r_ia = inner radius of metallic armour = outer radius of sheath insulation

r_oa = outer radius of metallic armour = inner radius of metallic armour’s plastic sheath

t_ps = thickness of metallic armour’s plastic sheath

ε_cs, ε_sa and ε_ae are the relative permittivities of core-to-sheath, sheath-to-armour and armour-to-earth insulation.

The capacitance between the armour and earth would be redundant if there were no armour insulation or where the armour is directly earthed. Similarly, this capacitance would not exist for cables that have no armour and in this case, the cable would generally be characterised by two capacitances only. The relative permittivity of insulation is in general a complex number with frequency dependent real and imaginary parts, i.e. ε_r = ε′(f) –jε″(f). However, for power frequency steady state analysis, only the real part of ε_r is of significance and this is equivalent to assuming zero shunt conductance. Typical values for the real part of ε_r are: 4 for solid-impregnated paper, 3.5 for oil-filled-impregnated paper, 3.7 for oil-pressure pipe-type, 3.5 for gas-pressure, 4 for butyl rubber (BR), 3 for EPR, 2.5 for XLPE, 2.3 for polyethylene (PE) and 8 for polyvinyl chloride (PVC).

For three-phase pipe-type cables, the cables inside the pipe may be laid in a touching triangular or a cradle formation as illustrated in Figure 3.23. The core-to-sheath capacitance of each cable phase can be calculated using Equation (3.96a).

Figure 3.23 Pipe-type three-phase cables

Mutual capacitances between the sheaths of the three phases and between the sheath of each phase and the pipe can be calculated using

     (3.99a)

and

     (3.99b)

where q is the inner radius of the steel pipe, ε_p is the permittivity of insulation inside the pipe, d_i is the distance between cable i core centre and pipe centre, r_i is the outer radius of cable i core conductor and θ_ik is the angle subtended between the pipe’s centre and the centres of cables i and k. The infinite series term in Equation (3.99a) is usually very small and for most d_i and q distances encountered in practical installations, it may be neglected with insignificant loss of accuracy. It should be noted that the mutual sheath capacitance would be short-circuited where the sheaths are in contact with each other, and the pipe’s inside wall. This applies for both cradle and triangular formations.

For three-core unscreened or belted cables, there is no simple reasonably accurate formula that can be used because of the effect of two insulation layers that have to be considered namely the core insulation and the belt insulation. For these cables, it is recommended that resort is made either to empirical design data from the manufacturers or better still to field test measurements. This is dealt with in Section 3.6.3.

Series self and mutual impedances

For a three-phase screened cable buried underground and having armour protection, the core, sheath and armour will each have a self-impedance with earth return. Electromagnetic coupling, and hence mutual impedances, exist between the core, sheath and armour within each phase. Even though the three phases are usually laid parallel to each other and the earth’s surface, the earth does not act as an electromagnetic shield between them. Therefore, mutual impedances exist between all the conductors of all the phases. Figures 3.22 and 3.24 are used in the calculation of self and mutual impedances of underground cables for use in steady state power frequency analysis.

Figure 3.24 Illustration of spatial dimensions of two underground cable phases

The self-impedance of a core conductor with earth return is given by

     (3.100a)

where

     (3.100b)

and D_erc is the depth of equivalent earth return conductor given in Equation (3.15) and μ_c is the relative permeability of the core conductor.

The self-impedance of a sheath with earth return is given by

     (3.101a)

where

     (3.101b)

and μ_s is the relative permeability of the sheath conductor.

The self-impedance of an armour with earth return is

     (3.102a)

where

     (3.102b)

and μ_a is the relative permeability of the armour conductor.

The mutual impedance between core or sheath or armour i, and core or sheath or armour j, with earth return, is given by

     (3.103)

where S_ij is the distance between the centres of cables i and j if the conductors belong to different cables. If the conductors belong to the same cable, S_ij is the geometric mean distance between the two conductors, e.g. the GMD between the core and the sheath of cable j is given by S_jj = (r_os + r_is)/2 which is sufficiently accurate for practical cable dimensions.

General

Having calculated the basic phase parameters in Section 3.3.3, we will now derive the cable’s phase and sequence impedance matrices. We use the suffixes C, S and A to denote core, sheath and armour conductors respectively for cable phases 1, 2 and 3. Figure 3.25 shows six practical cable layouts for three-core and single-core cables. Pipe-type cables were shown in Figure 3.23.

Figure 3.25 Typical practical single- and three-core cable layouts

Cables with no armour

A three-phase single-core or shielded three-core cable has a total of six conductors; three cores and three sheaths as shown in Figure 3.26a.

Figure 3.26 Multi-conductor cables with sheaths and armours: (a) cable conductors, no armour and (b) cable conductors, with armour

The series voltage drop per-unit length across the cores and sheaths is calculated from the cable’s full impedance matrix, core and sheath conductor currents and is given by

(3.113)

where Z_CC and Z_SS are the core and sheath self-impedances with earth return, respectively, and Z_CS is the mutual impedance between core and sheath with earth

return. Equation (3.113) can be simplified depending on the cable layout and spacings between the phases as shown in Figure 3.25.

Three-core cables, three single-core cables in touching trefoil layout and three-single-core cables in equilateral layout

These three cable layouts, shown in Figure 3.25 are fully symmetrical. From the configuration of these cables, a number of practical equalities can be deduced as follows:

     (3.114)

Therefore, the cable full impedance matrix of Equation (3.113) can be written as

(3.115a)

or

     (3.115b)

where

     (3.115c)

It is interesting to observe that the matrices Z_CC, Z_CS and Z_SS are all balanced matrices since all diagonal terms are equal and all off-diagonal terms are equal. This is due to the symmetrical cable layouts. Where the cable sheaths are solidly bonded and earthed at both ends, the sheaths can be eliminated. Therefore, the phase impedance matrix involving the cores only can be calculated by setting V_S = 0 in Equation (3.115b). The resultant core or phase impedance matrix is given by

     (3.116)

Because the individual impedance matrices Z_CC, Z_CS and Z_SS are all balanced, the resultant phase impedance matrix Z_Phase, that includes the effect of the eliminated sheaths, will also be balanced. The reader is encouraged to prove this statement. Therefore, the sequence impedance matrix is given by Z^PNZ = H⁻¹Z_PhaseH or

     (3.117a)

where

     (3.117b)

Single-core cables in a trefoil layout

Using Figure 3.25 and following similar steps, it can be shown that the cable full impedance matrix of Equation (3.113) can be written as

(3.118a)

where

     (3.118b)

The calculation of the phase impedance matrix involving the cores only and the corresponding sequence impedance matrix follows the same steps as above. However, in this case, the phase impedance matrix will not be balanced and as a result, the sequence impedance matrix will contain off-diagonal intersequence mutual terms.

Single-core cables in a flat layout

The general asymmetrical flat layout of single-core three-phase cables is shown in Figure 3.25. From the configuration of the cable, we can write the following equalities

     (3.119)

Therefore, the cable full impedance matrix of Equation (3.113) can be written as

(3.120a)

For cross-bonded cables shown in Figure 3.21, the cores of the minor sections are transposed so that each occupies the three positions, but the sheaths are not. This is illustrated in Figure 3.27.

Figure 3.27 Core transposition of cross-bonded cables

Equation (3.120a) is the full impedance matrix of the first minor section. The full impedance matrices of the second and third minor sections are given by

(3.120b)

Using Equation (3.120) and all impedances are in per unit length i.e. ω/km, the full impedance matrix of a cross-bonded cable with transposed cores in each major section is given by

or

(3.121a)

or including core and sheath voltage vectors

     (3.121b)

As expected, the core matrix Z_CC is balanced, due to the perfect core transpositions assumed but the sheath matrix Z_SS is not because the sheaths are untransposed. The mutual impedance matrix between the cores and the sheaths Z_CS is also unbalanced.

In both cases of a solidly bonded cable where the sheaths are bonded and earthed at the cable ends, and the case of a cross-bonded cable where the sheaths are bonded and earthed at the ends of major sections, the phase impedance matrix involving the cores only can be calculated by setting V_S = 0 in Equation (3.121b). Therefore, the core or phase impedance matrix takes the following general form

     (3.122)

The elements of the matrix Z_Phase are the self-impedances of each phase (core) and the mutual impedances between phases (cores) with the sheaths and earth return included. Generally, for solidly bonded cables, the self or diagonal terms of Z_Phase are not equal among each other nor are the off-diagonal mutual terms. In other words, the matrix is generally not balanced. This might also be expected to be the case for cross-bonded cables but in practical cable spacings and layouts, the core transposition eliminates the unbalance. The general sequence impedance matrix equation that corresponds to Equation (3.122), calculated using Z^PNZ = H⁻¹Z_PhaseH, is given by

     (3.123)

where Z^PP, Z^NN and Z^ZZ are the PPS, NPS and ZPS impedances of the cable and Z^PP = Z^NN. The off-diagonal terms are intersequence mutual terms which are normally small in comparison with the diagonal terms for solidly bonded cables and practically negligible for cross-bonded cables.

For a cable that is solidly bonded and earthed at its centre and a transposed earth continuity conductor is used, two full impedance matrices would need to be built for each half of the cable including the earth continuity conductor then combined into a single matrix. The calculation of the phase impedance matrix of such a cable makes use of the fact that no current can flow in the sheath that is I_S = 0. The derivation of the phase impedance matrix for such a cable is left for the reader.

Cables with armour

Some land based cables and almost all submarine cables have metallic armour. The armour acts as a third conductor in addition to the core and the sheath.

Single-core armoured cables

Using Figure 3.26(b)(i) for single-core armoured cables, the series voltage drop per-unit length is given by

(3.124)

In order to control the voltages between the sheath and the armour, the sheath is usually bonded to the armour at a number of points along the route. The armour covering is not normally an electric insulation so that the armour, and the sheath bonded to it, are effectively earthed. Therefore, the calculation of the cable’s phase (core) impedance matrix requires the elimination of both sheath and armour. Writing Equation (3.124) in partitioned matrix form, we have

(3.125)

The sheaths and armours are eliminated by setting V_S = V_A = 0 in Equation (3.125) and combining their matrices as follows:

     (3.126a)

Therefore, the phase (core) impedance matrix of the cable with sheaths, armours and earth, or sea water in the case of a submarine cable, constituting the current return path, is obtained as follows:

     (3.126b)

For single-core submarine cables where the individual phases are laid physically far apart from each other, the mutual electromagnetic coupling between the phases may be so weak that it can be neglected. Therefore, from the configuration of such three-phase submarine cables, a number of practical equalities can be deduced as follows:

     (3.127)

and all mutual terms between the three phases of the cables are zero. Therefore, Equation (3.124) simplifies to

(3.128)

Using Equation (3.126b), the phase (cores) impedance matrix is given by

     (3.129a)

where Z_α = d – (a²f + b²e − 2abc) or using Equation (3.127),

     (3.129b)

Z_α is the impedance of each core or phase including the effect of the combined sheath, armour and sea water return. The corresponding PPS/NPS and ZPS sequence impedances for the cable are all equal to Z_α which is a direct result of the absence of mutual coupling between the three phases.

In the case where the distances between the phases are not so large that the mutual coupling between them cannot be ignored, three further impedance parameters need to be defined to account for the interphase coupling. Let the mutual impedance parameter between phases 1 and 2 be g such that

The mutual impedances between phases 1 and 3, and phase 2 and 3 are similarly defined. The analysis for calculating the cable phase impedance matrix proceeds as above.

Three-core armoured cables

Using Figure 3.26(ii) for three-core armoured cables, there are three cores and three sheaths but only one armour. Thus, the series voltage drop per-unit length is given by

(3.130)

Using the equalities of Equation (3.114) for the case with no armour, and the new equalities

we can write

(3.131)

As for the single-core cable, where the sheaths and armours are bonded and earthed, we can now eliminate the sheaths and armour by combining the matrices that correspond to them by setting V_S = V_A = 0. Writing Equation (3.131) in partitioned form, we have

     (3.132a)

The phase (core) impedance matrix of the cable with sheath, armour and earth, or sea water in the case of a submarine cable, constituting the current return path, is given by

     (3.132b)

The phase (core) impedance matrix will virtually be balanced. The corresponding sequence impedance matrix calculated using Z^PNZ = H⁻¹Z_PhaseH will be a diagonal matrix, i.e. having virtually no intersequence terms. The diagonal terms are the required PPS/NPS and ZPS impedances.

Screened cables with no armour

For screened three-core and screened single-core cables, we have already established that there is no capacitance between one cable phase and another. From Figure 3.22, we note that there are two capacitances, or susceptances, corresponding to the core insulation and the sheath insulation layers and these are illustrated in Figure 3.28(a).

Figure 3.28 Equivalent capacitances of screened cables: (a) no armour and (b) with armour

For each phase, the core and sheath current/voltage equations can be written as I_C = jB_CS(V_C – V_S) and I_C + I_S = jB_SEV_S where B_CS and B_SE are the core-to-sheath and sheath-to-earth susceptances, respectively. Therefore, writing similar equations for the three cores and three sheaths, the full shunt susceptance matrix is given by

(3.133)

For solidly bonded and cross-bonded cables, the sheaths can be eliminated using V_S1 = V_S2 = V_S3 = 0. The resultant shunt phase (cores) susceptance matrix is given by

     (3.134)

that is the PPS, NPS and ZPS susceptances are all equal, and equal to B_CS.

For cables that are bonded and earthed at one end only or at the cable centre, there will also be a capacitance or susceptance between the earth continuity conductor and earth with I_ECC = jB_ECC-EV_ECC. This increases the dimension of the full admittance matrix given in Equation (3.133) by one row and one column, i.e. from 6 × 6 to 7 × 7. The calculation of the 3 × 3 phase susceptance matrix for such cables, the constraint that no sheath current can flow, i.e. I_S1 = I_S2 = I_S3 = 0 is applied. The resultant phase (cores) susceptance matrix is given by

     (3.135a)

This result shows that the cable-to-sheath capacitance and the sheath-to-earth capacitance of each core are effectively in series as can be seen from Figure 3.28(a).

The corresponding shunt sequence susceptance matrix of Equations (3.134) and (3.135a) is given by

     (3.135b)

that is the PPS, NPS and ZPS susceptances are all equal to the relevant phase susceptances.

Screened cables with armour

From Figure 3.22, we note that there are three capacitances, or susceptances, corresponding to the core insulation layer, the sheath insulation layer and the armour coating where this is made of insulation material. These are illustrated in Figure 3.28(b). For each phase, the core, sheath and armour current/voltage equations can be written as

and

Therefore, writing similar equations for the three cores, three sheaths and three armours, the full shunt susceptance matrix B is given by

(3.136)

If the sheath and armour are bonded together and earthed at both ends, then circulating currents can flow through the sheath and armour in parallel with earth return, or sea water return in the case of a submarine cable. The phase (core) susceptance matrix is easily obtained by deleting the last six rows and columns that correspond to the sheath and armours. The earthing of the sheath and armour, in effect, short circuits the sheath-to-armour and armour-to-earth capacitances as can be seen in Figure 3.28(b). The PPS, NPS and ZPS susceptances in equal to the phase susceptance B_CS. If the sheath and armour are unearthed or earthed at one point only, then no sheath or armour current would flow and the equivalent core or phase capacitance is equal to the three capacitances of Figure 3.28(b) in series that is

The PPS, NPS and ZPS susceptances are equal to B_Phase.

Unscreened or belted cables

Belted cables are three-core cables where each core has an insulation layer but no screen or metallic sheath. There is also a belt insulation layer around all three cores and a single metallic sheath cover as illustrated in Figure 3.29(a).

Figure 3.29 Three-core belted cable and capacitance equivalent circuits: (a) cross-section of a three-phase belted cable; (b) capacitance equivalent circuit and (c) capacitance equivalent circuit with earthed sheath

There are capacitances among the three cores, between each core and sheath and between the sheath and earth. Figure 3.29(b) shows the equivalent capacitance circuit of the cable where C_CC is the core-to-core capacitance, C_CS is the core-to-sheath capacitance and C_SE is the sheath-to-earth capacitance. Where the sheath is solidly earthed, as is usually the case, the equivalent capacitance circuit reduces to that shown in Figure 3.29(c). The injected currents into the three cores can be expressed as follows:

or

     (3.137)

and similarly for the two remaining cores. The nodal phase susceptance matrix is

     (3.138a)

and the corresponding sequence susceptance matrix is given by

     (3.138b)

where

     (3.138c)

It is interesting to note that the ZPS susceptance of a belted cable is smaller than its PPS/NPS susceptance. This contrasts with the equal PPS/NPS and ZPS susceptances of screened three-core and single-core cables. It is useful to calculate the core-to-core cable susceptance in terms of the sequence susceptances giving B_CC = (B^P – B^Z)/3.

Overhead lines

The PPS, NPS and ZPS impedances and susceptances of perfectly transposed single-circuit overhead lines were derived in Section 3.2.4. Recalling the sequence impedance and susceptance matrices of Equations (3.50a) and (3.54a), denoting 1 and 2 as the sending and receiving ends of the line or cable circuit and connecting 1/2 the susceptance at each end, as shown in Figure 3.33, we can write

Figure 3.33 (a) PPS/NPS and (b) ZPS π models for single-circuit overhead line or cable

and

     (3.143a)

For the PPS, NPS and ZPS circuits, the total injected current at ends 1 and 2 are given by

     (3.143b)

Therefore, using Equation (3.143), we obtain three separate PPS, NPS and ZPS sequence Π models for the line or cable as follows:

     (3.144)

As the PPS and NPS impedances and susceptances are equal, only two sequence Π models are needed as shown in Figure 3.32. For convenience, we will only refer to the PPS and ZPS sequence parameters and models in subsequent sections.

Cables

The sequence impedance matrices of single-circuit cables of various sheath earthing arrangements were derived in Section 3.3.4. The sequence impedance matrix of Equation (3.123), combined with the circuit sequence currents and voltages, gives

     (3.145)

The mutual intersequence terms are normally much smaller than the self-terms for solidly bonded cables and practically zero for cross-bonded cables. For the former, it is usual practice to ignore the mutual terms and use the self-terms only for the PPS, NPS and ZPS impedances of the cable. In the case of submarine cables where the three phases are laid a significant distance apart, the sequence impedance matrix is diagonal with no mutual intersequence terms. This is also the case for three-core cables, and single-core cables in touching trefoil or equilateral arrangements. The sequence susceptance matrix of all shielded and belted cables is diagonal and hence the PPS, NPS and ZPS susceptances are equal to the phase terms unaffected by any intersequence terms. Therefore, as in the case of single-circuit overhead lines, PPS and ZPS sequence Π models can be derived as shown in Figure 3.33.

General

Measurements of overhead lines power frequency electrical parameters or constants can be made at the time of commissioning when the line construction has been completed so that the tests replicate the line in normal service operation. The parameters that may be measured are phase conductor self and mutual impedances with earth return, and self and mutual susceptances. The series phase impedance and shunt phase susceptance matrices are then constructed and the sequence impedances and susceptances calculated. There are numerous designs of overhead lines used in practical installations but we will illustrate the techniques that can be used for single-circuit line with one earth wire and double-circuit line with one earth wire. The technique can be extended for application to any other line construction.

Single-circuit overhead lines with one earth wire

Figures 3.39(a) illustrates a technique for phase impedance measurements where the three-phase conductors are earthed at the far end of the line but kept free from earth at the near measurement end.

Figure 3.39 Measurement of the electrical parameters of a single-circuit overhead line: measurement of self and mutual (a) series phase impedances with earth return and (b) shunt phase susceptances

A test voltage is applied to one-phase conductor and the current flowing through it is measured. The induced voltages to earth on the remaining conductors are also measured. In addition to the calculation of the impedances from the measured voltages and currents, a wattmeter or phase angle meter can be used to find the phase angle of these voltages with respect to the injected current. This allows the calculation of the resistance and reactance components of the impedance using R = P/I² (P is measured phase power in Watts) and X = in the case of a wattmeter, but R = Z × cos φ and X = Z × sin φ in the case of a phase angle meter measurement. The process is repeated for all other conductors. Where earth wire(s) are present, the measured impedances should represent values with the earth wire in parallel with earth return path to represent normal service operation.

Using the single-circuit phase impedance matrix including the effect of earth wire that is with the earth wire mathematically eliminated, we have

     (3.157a)

     (3.157b)

     (3.157c)

With the voltage source applied to conductor 1, V₁ is known and the measured values are I₁, V₂ and V₃ with I₂ = I₃ = 0. Using the measured values in Equations (3.157), we obtain

     (3.158a)

Applying the voltage source to conductor 2 then conductor 3, we obtain

     (3.158b)

and

     (3.158c)

For sequence impedance calculations, and with the line assumed to be perfectly transposed, the self and mutual impedances per phase are calculated as the average of the measured self and mutual impedances as follows:

     (3.159a)

and

     (3.159b)

Therefore, the PPS and ZPS impedances are given by

     (3.160a)

     (3.160b)

Figure 3.39(b) illustrates a technique for phase susceptance measurements where the three-phase conductors are earthed through milliammeters at the near measurement end of the line but kept free from earth, i.e. open circuit at the far end. A test voltage is applied in series with one-phase conductor and the currents flowing through all conductors are measured. The process is repeated for all other conductors. As in the case of impedance measurements the measured susceptances should include the effect of the earth wire. Using the single-circuit phase susceptance matrix including the effect of earth wire that is with the earth wire mathematically eliminated, we have

     (3.161a)

     (3.161b)

     (3.161c)

With the test voltage source applied to conductor 1, V₁ is known and the measured values are I₁, I₂ and I₃ with V₂ = V₃ = 0. Using the measured values in Equations (3.161), we obtain

     (3.162a)

Applying the test voltage source to conductor 2 then conductor 3, we obtain

     (3.162b)

and

     (3.162c)

For sequence susceptance calculations and with the line assumed to be perfectly transposed, the self and mutual susceptances perphase are calculated as the average of the measured self and mutual susceptances as follows:

     (3.163a)

and

     (3.163b)

Therefore, the PPS and ZPS impedances are given by

     (3.164a)

     (3.164b)

The measured voltages and currents inherently take into account the distributed nature of the line’s impedances and susceptances. However, we have used lumped linear impedance and susceptance matrix models for the line which will introduce small errors in the calculated impedances and susceptances depending on line length. The calculated impedances and susceptances can be corrected for the line length effect if the equivalent Π circuit of a distributed multi-conductor line derived in Appendix 1 is used. The impedance and susceptance constants now become impedance and susceptance matrices and the general nodal admittance equations for a multi-conductor line are given by

     (3.165a)

     (3.165b)

where

     (3.165c)

and I_S and I_R are the line’s sending and receiving end current vectors, respectively; and V_S and V_R are the line’s sending and receiving end voltage vectors, respectively.

In the impedance measurement technique, the voltage and current vectors V_S and I_S are known from the measurements, V_R = 0 but I_R is unknown. Therefore, from Equation (3.165), we can write Expanding the hyperbolic tangent into an infinite series, we obtain

     (3.166)

In the susceptance measurement technique, the voltage and current vectors V_S and I_s are known from the measurements, I_R = 0 but V_R is unknown. Therefore, from Equation (3.165), we can write Using the values of A and B from Equation (3.165) and expanding the hyperbolic tangent into an infinite series, we obtain

     (3.167)

We can use the voltages and currents obtained from the test measurements to calculate to a first approximation of the elements of the impedance and susceptance matrices using the approximate lumped parameter equations as follows:

     (3.168)

Z₁ and Y₁ can now be equated to the series impedance and susceptance values of Equations (3.166) and (3.167) and used to obtain a second approximation as follows:

     (3.169a)

     (3.169b)

Similarly, the new values of the impedance and susceptance matrices can again be used to obtain an improved approximation and the process can be repeated until the desired level of accuracy is obtained. In general, we can write

     (3.170a)

     (3.170b)

Double-circuit overhead lines

The technique presented above for a single-circuit overhead line with one earth wire can be extended to double-circuit overhead lines with any number of earth wires. Figure 3.40 shows the impedance and susceptance test arrangements.

Figure 3.40 Measurement of electrical parameters of a double-circuit overhead line: measurement of self and mutual (a) series phase impedances with earth return and (b) shunt phase susceptances

The measured phase impedance and susceptance matrices are given as

(3.171a)

and

     (3.171b)

In Figure 3.40(a), we apply a voltage source V₁ to conductor 1 and measure I₁, V₂, V₃, V₄, V₅ and V₆. In Figure 3.40(b), we apply a voltage source V₁ to conductor 1 and measure I₁, I₂, I₃, I₄, I₅ and I₆. From these tests, and using Equation (3.171), the self and mutual phase impedances and susceptances are calculated as follows:

     (3.172a)

     (3.172b)

In both impedance and susceptance tests, the voltage source is applied to each phase conductor in turn in order to measure all the self and mutual impedances and susceptances of Equation (3.171). The balanced phase and resultant sequence impedances can be calculated assuming an appropriate number of phase transpositions such as three, six or ideal nine transpositions. The reader is encouraged to derive the balanced phase and sequence parameters. The sequence susceptance parameters can be similarly calculated.

General

Measurements of power frequency electrical parameters or constants of cables can be made at the time of commissioning when the cable installation has been completed so that the tests replicate the cable in normal service operation. The relevant parameters that are usually measured are the conductor dc resistances at prevailing ambient temperature, the PPS/NPS and ZPS impedances and sometimes the PPS/NPS/ZPS susceptances. As there are numerous designs and layouts of cables in use in practical installations, we will illustrate the techniques that can be used for a typical single-circuit single-core three-phase coaxial cable with core and sheath conductors. An illustration of the PPS/NPS susceptance and ZPS susceptance measurement of a three-core belted cable is also presented.

Measurement of dc resistances

The dc resistances of the cores and sheaths (screens) are measured using a dc voltage source as illustrated in Figure 41.3(a) and (b). In the first case, the dc voltage source V is applied between cores 1 and 2 with the third core floating and remote ends of the three cores short-circuited. The dc source V is next applied between cores 1 and 3, then 2 and 3. The measured core loop resistance in each case is calculated from R_Cc(loop) = V/I and the three measured loop resistances are given by

     (3.173a)

Therefore, the individual core resistances are given by

and

     (3.173b)

The test procedure for the sheath or screen dc resistances is similar to that for the core dc resistances but the dc voltage source is now applied between two sheaths at a time as shown in Figure 3.41(b).

Figure 3.41 Measurement of dc resistances of (a) core and (b) sheath/screen of a three-phase cable

Measurement of PPS/NPS impedance

The measurement of the cable’s PPS/NPS impedance per phase requires a three-phase power frequency source. On each of the single cores, measurements of voltage, current and phase power or phase angle are made. The sheath’s bonding and earthing must correspond to that in actual circuit operation. Figure 3.42(a) illustrates the PPS/NPS impedance measurement for a cable with a sheath that is solidly bonded at both ends. From the measurements, the following calculations for each phase are made.

Figure 3.42 Measurement of (a) PPS/NPS and (b) ZPS impedances of a three-phase cable

If a wattmeter is used to measure input power, we have

     (3.174)

If a phase angle meter is used to measure the angle between the input voltage and current

     (3.175)

where P^P is the measured phase power in Watts. The three values obtained for each phase are then averaged and this is particularly important in the case of untransposed cable systems. The final PPS/NPS cable parameters are given as

     (3.176)

Measurement of ZPS impedance

The measurement of the cable’s ZPS impedance per phase requires a single-phase power frequency source. The three cores at the test end are bonded together and at the remote end are bonded together and to the sheaths. Single-phase currents are injected into the cable cores to return via the sheath and earth. Figure 3.42(b) illustrates the test circuit. The sheaths are solidly bonded at both ends as in actual cable operation. The calculations of the ZPS impedance, resistance and reactance use similar equations to those used for the PPS/NPS impedance except that the current should now be divided by 3, i.e. the impedance equation becomes

     (3.177)

Measurement of earth loop impedances

As in the case of an overhead line, and if required, the earth loop impedances for each core and sheath may also be measured using a single-phase power frequency source. The technique is similar to that presented in Section 3.7.2.

Measurement of PPS/NPS/ZPS susceptance

The measurement of a screened cable core-to-sheath, and if required, sheath-to-earth susceptances uses a single-phase power frequency source. The three cores at the remote end are left floating, i.e. open circuit. The applied voltage V, charging current I and phase angle φ between applied voltage and current, the latter measured using a low power factor meter, are measured. This is usually called the power factor and capacitance test of the dielectric. The calculations made are

     (3.178)

B is the PPS/NPS/ZPS susceptance per phase. G is very small and is usually neglected in power frequency steady state analysis.

For three-core belted cables, the core-to-core and core-to-sheath capacitances can be calculated from two measurements. The core to sheath, i.e. ZPS susceptance can be measured by short-circuiting the three cores and applying a voltage source V between them and the earthed sheath, see Figure 3.29(c). From the measured injected current I, we can write

     (3.179)

The core-to-core capacitance can be obtained from a measurement where a voltage source V is applied between two cores with the third core short-circuited to the sheath. Let the measured susceptance be B_Meas. Therefore, using Figure 3.29(c), we can write

Therefore, the core-to-core susceptance is calculated as

     (3.180)

Using Equation (3.138c), the PPS susceptance of the cable is given by

     (3.181)