A.1 Theory and analysis of distributed multi-conductor lines and cables

In order to describe accurately the performance of multi-conductor transmission lines, we used in Chapter 3 two-port nodal admittance equations that included the line’s series impedance and shunt admittance matrices. We will now briefly present the theory of distributed multi-conductor lines. The theory also applies to cables.

Consider a multi-conductor line that consists of N conductors. Each conductor, say conductor k, has a self-impedance Z_kk, with earth return, and is mutually coupled to all other conductors through mutual impedances Z_kj, with earth return. In addition, conductor k has a self-admittance Y_kk and is mutually coupled to all other conductors through mutual admittances Y_kj. Taking an element of infinitesimal length δx on the line, the series voltage developed in length δx on conductor k when currents flow in all conductors, 1, 2, …, k, …, N, is given by

     (A.1a)

The minus sign is because the change in voltage is negative with increasing x.

Similarly, the shunt displacement current due to the voltage applied to conductor k and voltages on all other conductors, 1, 2, …, N, is given by

     (A.1b)

Writing Equations (A.1) in matrix form for δx → 0, we have

     (A.2a)

     (A.2b)

where Z and Y are the series phase impedance and shunt phase admittance matrices of the line in per-unit length. Differentiating Equations (A.2a) and (A.2b) with respect to x, we obtain

     (A.3a)

     (A.3b)

where

     (A.3c)

γ is the line’s propagation coefficient matrix and γ^T is its transpose noting that Z and Y are symmetric matrices.

The general solutions to Equations (A.3a) and (A.3b) are given by

     (A.4a)

     (A.4b)

where V_i, V_r, I_i and I_r are column matrices that satisfy the boundary conditions of the line. Differentiating Equation (A.4a) with respect to x and substituting into Equation (A.2a), we obtain a different form for Equation (A.4b) as follows:

     (A.5a)

where

     (A.5b)

Y_o is the characteristic admittance matrix of the line and its inverse, the characteristic impedance matrix is given by

     (A.5c)

Also, differentiating Equation (A.5a) and substituting in Equation (A.2b), we obtain another formulation for Y_o and Z_o as follows:

     (A.5d)

Equations (A.4a) and (A.5a) are general and provide the voltage and current solutions at any point along the line. However, if we are only interested in the line’s terminal conditions, i.e. at x = 0 and at x = l, we can derive an equivalent two-port π admittance model. Substituting x = 0 and at x = l in Equations (A.4a) and (A.5a), and using V(x = 0) = V_S, V(x = l) = V_R, I(x = 0) = I_S and I(x = l) = I_R, we have

For x = 0:

     (A.6a)

     (A.6b)

For x = l:

     (A.7a)

     (A.7b)

From Equation (A.6), we obtain

     (A.8a)

     (A.8b)

Substituting Equation (A.8) into Equation (A.7), and after some matrix and hyperbolic functions manipulations, we obtain

     (A.9a)

     (A.9b)

where

     (A.10a)

     (A.10b)

Substituting Equation (A.3c), , and using Equation (A.5d), Y_o = Yγ⁻¹, in Equation (A.10), we obtain

     (A.11a)

     (A.11b)

Using Equations (A.9), the π equivalent admittance model of a multi-conductor line or cable is shown in Figure A.1.

Figure A.1 π equivalent circuit of a multi-conductor overhead line or a multi-conductor cable where A, B, Z and Y are square matrices, and V and I are column matrices

A.2 Typical data of power system equipment

A.2.2 Data

Table A.1 Typical parameters of synchronous generators

Table A.2 Typical parameters of induction generators

Table A.3 Typical parameters of induction motors

Table A.4 Typical sequence parameters of single-circuit and double-circuit overhead lines

Table A.5 Typical sequence parameters of single-circuit three-phase cables (a) Cable parameters (50 Hz)

Table A.6 Typical sequence parameters of transformers (a) Two-winding transformers

Table A.7 Typical sequence parameters of quadrature boosters and phase shifters

Table A.8 Typical sequence parameters of series reactors

Table A.9 Typical sequence parameters of shunt reactors

Table A.10 Typical sequence parameters of shunt capacitors

Table A.11 Typical positive phase sequence X/R ratios and dc time constants of power system equipment