Evaluation of Factorization Methods for Thévenin Equivalent Computations in Real-Time Stability Assessment

Abstract

Thévenin equivalents are used by a range of power system stability indicators, such as the L-index for voltage stability and the aperiodic small signal rotor angle stability indicator. This paper investigates the effect of using different factorization methods for computing coefficients for wide-area Thévenin equivalents. Direct and incomplete factorization methods are compared with respect to runtime, accuracy and amount of fill-in. The paper introduces a proof that the block triangular form of bus admittance matrices will have no non-zero entries in the off-diagonal. KLU factorization is found to perform almost twice as fast as the standard LU factorization with no cost of accuracy. It is, however, shown that the largest computational workload is associated with dense matrix multiplications. An incomplete method reduces the fill-in of coefficient matrices at the cost of accuracy in Thévenin voltages. It is shown, that inaccuracies are amplified as the L-index approaches the stability limit.

A Memory-Efficient Parallelizable Method for Computation of Thévenin Equivalents Used in Real-Time Stability Assessment

“…P, Q are permutations stored as vectors, R is a scaling matrix optimally stored as a vector, L, U are complex sparse matrices and F = 0 in this context [14]. The memory requirements for storing the the coefficient matrix (either in full or sparse format) compared to storing the sparse factorization can be seen in Table VI.…”

“…In [8], [14] a method for computing Thévenin equivalents is described, which uses coefficients for super-position. Thévenin equivalents consist of a Thévenin impedance Z th and Thévenin voltage V th .…”

“…The Schur complement is generally considered to be dense [10], [13] and therefore it is computationally inefficient to determine. In reality all the coefficients for super-position and not just the Schur complement can be dense [14], which means that computing the coefficients is computational expensive.…”

“…In [14] different factorization methods are analysed to optimize the method for finding Thévenin equivalents. It is identified, that the Clark Kent LU factorization (KLU) factorization is the most efficient factorization method, however as in [8] it is determined that the greatest share of the execution time is spend on matrix multiplications due to the density of the matrices involved.…”

“…This paper introduces a new method for computing Thévenin equivalents, opposed to the reference method [8], [14]. The method builds on the ideas given in [10] of developing a factor-solve method, where the method here will compute the Thévenin equivalents for the entire system instead of only the Thévenin impedances for generators.…”