TABLE OF CONTENTS A continuous function can be simulated by substituting a numerical integration formula into the differential equation and rearranging the function into an appropriate form. Among the factors to be taken into account in the selection of the numerical integrator are the error due to truncated terms, its properties as a differentiator, error propagation and frequency response.
Numerical integration substitution (NIS) constitutes the basis of Dommel's EMTP [l] [3], which, as explained in the introductory chapter, is now the most generally accepted method for the solution of electromagnetic transients. The EMTP method is an integrated approach to the problems of:
forming the network differential equations
collecting the equations into a coherent system to be solved
numerical solution of the equations.
The trapezoidal integrator (described in Appendix C) is used for the numerical integrator substitution, due to its simplicity, stability and reasonable accuracy in most circumstances. However, being based on a truncated Taylor's series, the trapezoidal rule can cause numerical oscillations under certain conditions due to the neglected terms [4]. This problem will be discussed further in Chapters 5 and 9.
The other basic characteristic of Dommel's method is the discretisation of the system components, given a predetermined time step, which are then combined in a solution for the nodal voltages. Branch elements are represented by the relationship which they maintain between branch current and nodal voltage.
This chapter describes the basic formulation and solution of the numerical integrator substitution method as implemented in the electromagnetic transient programs.
