Digital computer calculation of power system short circuit and load flow utilising diakoptics and sparsity techniques

In part A a non-singular connection matrix is used to combine the self and mutual impedance matrix of a group of mutually coupled elements with a network bus impedance matrix; the resulting impedance matrix is then reduced by eliminating rows and columns if necessary, to give the bus impedance matrix of the interconnected network. The self impedances of the mutually coupled group of elements are added to the network bus impedance matrix in the same way as uncoupled elements, then the mutual impedances Are added followed by matrix reduction. By considering examples of the connection matrix applied to adding a single element, then to adding groups of uncoupled and coupled elements to a network, rules are devised for combining the celf impedances of branch and loop elements and group mutual imped-ances with the network bus impedance matrix. From the bus impedance matrix of power system sequence networks fault parameters are derived by simple arithmetic operations. It is shown that rules for adding a group of mutually coupled loop elements can be applied to modify a bus impedance matrix when element self and mutual impedances are changed. The derivation of an equivalent network from the bus impedance matrix is noted; the addition of two network bus impedance matrixes is considered and shown to be a special case of the more general problem of adding a self and mutual impedance matrix to a bus impedance matrix. A numerical example involving the calculation and modification of the bus impedance matrix, deriving an equivalent circuit and adding bus impedance matrixes is included. An outline of a digital computer power system short circuit programme which calculates fault parameters from the bus impedance matrix derived from randomly ordered lists of network element self and mutual impedances is given. The inverse of the connection matrix discussed in part A is used in part B to combine a network bus admittance matrix with the self and mutual admittance matrix of a group of mutually coupled elements. From this the Well known method of forming the bus admit-tance matrix from uncoupled element self admittances follows and is extended to cover self and mutual admittances of coupled elements. For a group of mutually coupled elements, the diagonal terms of the group admittance matrix are added to the bus admittance matrix in the same way as self admittances of uncoupled elements while the off-diagonal terms are added in a matrix operation either before or after the diagonal terms. A relationship is indicated between the admit-tance connection matrix and the group element bus incidence matrix. Although the presence of mutual coupling results in some loss of spal-sity, it is shown that for power systems the bus admittance matrix still has a large proportion of zero terms. By eliminating terms below the main diagonal in an optimal order, a \factored inverse\" of the admittance matrix is derived which has considerably fewer non-zero terms than the corresponding bus impedance matrix. Terms of the Impedance matrix can be obtained from the inverse as required. The numerical calculation of the bus admittance matrix of a power system zero sequence network is set out and derivation of fault impedance and current distribution factors included. A digital computer programme using the bus admittance matrix and factored inverse method for power system short circuit studies is described and a tabulation indicates the affect on computer storage requirements of the optimal factoring procedure. In part C Newton's method of power system load flow calculation using Gaussian elimination to solve the voltage correction equations is discussed. The network and problem parameters are specified in rectangular cartesian co-ordinates. As the voltage correction equation matrix has the same form as the bus admittance matrix a preferred order for the Gaussian elimination which preserves sparsity is devised by analogy with network reduction. A digital computer load flow programme is outlined and a tab-ulation included which shows that for typical power system networks the preferred elimination order retains sparsity in the matrix. Algol listings of the digital computer short circuit and load flow programmes are included in the supplement with data and corresponding calculated results for power system studies."