Step-by-Step Eigenvalue Analysis With EMTP discrete-time Solutions

Abstract

The present work introduces a methodology to obtain a discrete time state space representation of an electrical network using the nodal [G] matrix of the Electromagnetic Transients Program (EMTP) solution. This is the first time the connection between the EMTP nodal analysis solution and a corresponding state-space formulation is presented. Compared to conventional state space solutions, the nodal EMTP solution is computationally much more efficient. Compared to the phasor solutions used in transient stability analysis, the proposed approach captures a much wider range of eigenvalues and system operating states. A fundamental advantage of extracting the system eigenvalues directly from the EMTP solution is the ability of the EMTP to follow the characteristics of nonlinearities. The system's trajectory can be accurately traced and the calculated eigenvalues and eigenvectors correctly represent the system's instantaneous dynamics. In addition, the algorithm can be used as a tool to identify network partitioning subsystems suitable for real-time hybrid power system simulator environments, including the implementation of multi-time scale solutions. The proposed technique can be implemented as an extension to any EMTPbased simulator. Within our UBC research group, it is aimed at extending the capabilities of our real-time PC-cluster Object Virtual Network Integrator (OVNI) simulator.

A discussion about optimum time step size and maximum simulation time in EMTP-based programs

“…From (15), it is possible to calculate the minimum and maximum time values related to the system natural oscillations modes, as presented in (20) and (21),…”

“…Step-by-step eigenvalue analysis with EMTP discrete-time solutions is presented in [15,16]. Essentially, a discrete-time state-space matrix is first calculated using (31), from which discrete-time eigenvalues ðz i Þ and their respective eigenvectors ðv i Þ are calculated.…”

“…Unless the circuit equations are solved analytically (or the eigenvalues are somehow calculated for linear systems or for linearized parts of nonlinear systems [15,16]), the time simulation parameters are not known by inspection. It is possible, however, to establish ''ranges'' in which the time step size and the maximum simulation time must be contained, as presented in [17,18].…”

“…The evaluation of accuracy and stability of different integration rules (integration solvers) as functions of the time step size (or of the sampling frequency, or of the per unit of the Nyquist frequency) is clearly discussed in [13]. For an overall accurate and stable simulation, CDA takes advantage of the accuracy of the trapezoidal integration rule and the stability of the Backward Euler integration rule, for a given and fixed time step size.Unless the circuit equations are solved analytically (or the eigenvalues are somehow calculated for linear systems or for linearized parts of nonlinear systems [15,16]), the time simulation parameters are not known by inspection. It is possible, however, to establish ''ranges'' in which the time step size and the maximum simulation time must be contained, as presented in [17,18].…”

PC Implementation of a Real-Time Simulator Using ATP Foreign Models and a Sound Card

“…ATP enables the simulation of electromagnetic transient phenomena on the basis of sophisticated models of transmission lines, transformers, electric machinery, etc., as well as the components of circuits and elements that control the switches and triggers of thyristors [10,11]. Studies conducted in the 1990s that continued until 2010 [12][13][14] reported the development of real-time simulators based on EMTP [15,16].…”