Application of the normal form of vector fields to predict interarea separation in power systems

Jyotika Thapar, Vijay Vittal, Wolfgang Kliemann, A. A. Fouad

Research output: Contribution to journalArticlepeer-review


Heavily loaded and stressed power transmission networks exhibit complex dynamic behavior when subjected to disturbances. Following large disturbances stressed systems may experience a system separation involving generators far away from the location of the disturbance. In other instances, the disturbance may only result in a few generators close to the disturbance being affected and is usually referred to as a plant mode separation case. What differentiates the above two situations? What structural characteristics of the system control the mechanism of a disturbance resulting in an interarea or a plant mode separation type behavior? This paper attempts to answer the above questions and provides an explanation for the underlying mechanism of system separation. The approach used in this paper is the normal form of vector fields. Using the nonlinear transformation obtained from the normal form method, a closed form solution of the time evolution of the system variables is obtained up to any desired order in the absence of resonance conditions. In this paper, this solution up to order two is used. In addition, the solution accounts for the modal structure, the effect of the disturbance, and the nonlinearities. This unique feature provides important qualitative information regarding system dynamic behavior. It characterizes time evolution behavior in terms of the fundamental modes of oscillation, and also indicates nonlinear interaction of modes via the higher order terms. This aspect of the approach is used to develop an analytically based index to identify the dominant modal interaction and predict the onset of complex dynamic behavior. Nonlinear interaction coefficients are then used to identify the interacting modes and explain the mechanism of separation as being caused by the nonlinear interaction of dominant modes. The proposed technique is then applied to the 50-generator WEE test system. Tests are conducted under different loading conditions, and using two different values of damping in the system equations. The system behavior is predicted using the interaction index. This prediction is then verified using time simulation results obtained from EPRI's ETMSP program. The following are among the findings reported upon in this paper: • The normal form analysis up to second order accurately predicts the structural characteristics of the system in terms of nonlinear modal interaction, and the associated machines via participation factors. • The index which is used to predict the nonlinear structural characteristics, combines both the linear and the nonlinear portion of the closed form solution of the time evolution of the system variables up to second order, and compares their relative size. • This index is developed analytically based on the closed form solution. • The analysis accurately predicts whether the system response is dominated by a plant or local behavior, or by an interarea separation type of behavior. This prediction has been verified by time simulation of the nonlinear system. • The analysis also indicates that certain frequencies observed in time simulation results could appear not only due to the fundamental mode at this frequency but also due to nonlinearly interacting modes close to resonance. This aspect of the system behavior is captured by the interaction coefficient.

Original languageEnglish (US)
Number of pages1
JournalIEEE Power Engineering Review
Issue number5
StatePublished - Dec 1 1997
Externally publishedYes

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


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