A newton approach for long term stability studies in power systems B.I. Lima Lopes, A.C. Zambroni de Souza * Federal University of Itajuba, Electrical Engineering, AvBPS 1303, Pinheirinho, 37500903 Itajuba, MG, Brazil article info Keywords: Transient and long term stability Quasi-dynamic model Jacobian matrix abstract This paper deals with the problem of transient and long term stability of power systems. The issue of assessing both horizons of analysis is particularly focused. This is because the long term stability may be studied by a simplified algebraic model which also captures some dynamic characteristics. Such an approach is called quasi-dynamic model. The idea of analyzing the transient period and migrating to the quasi-dynamic model is addressed in this paper. The theoretical foundation is presented and some tests are carried out in order to validate the approach. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Studying a power system response following a disturbance has been an important issue for engineers and researchers for a long time, since it regards the ability of a system to operate in a stable manner. For this sake, one is primarily concerned about the transient response to a fault. Several approaches, like SIME [1,2] and Lyapunov-based techniques [3–5] have been proposed. However, many approaches lie on integrating the differential–algebraic set of equations along time evolution. For this purpose, Ref. [6] presents several techniques, which are also discussed in [7]. In particular, the trapezoidal rule of integration with a variable step size is very effective, and commonly used. The scope of a power system stability became wider in the 1980s, when the problem of voltage stability became an issue. Such a problem, commonly associated with stressed systems, present some differences with respect to the transient stability problem, since it may be triggered as a consequence of successive load variations. In many cases, a static system model is enough to provide good results for voltage collapse assessment, like load margin calculation [8], critical buses identification [9] and control actions determination [10]. As a consequence, assessing voltage security in a power system may not be com- promised if a power flow model is employed. This is because voltage collapse, in contrast with transient stability, may be associated with a larger time frame, from seconds to minutes. This analysis provides an insightful set of information about the system security, as long as the system parameter varies from a stable operating point. Therefore, system security may be analyzed by the means of static analysis, like proposed in [11], where many important issues, like buses classification, load margin and control actions are addressed. Ref. [12] focuses on the contingency screening based on neural networks. If a dy- namic model is employed, other kinds of study may be carried out. Ref. [13] addresses the problem of dynamic risk assess- ment by the means of different scenarios, whereas Ref. [14] proposes a secondary voltage control with the help of a hierarchical coordinated voltage control. If oscillations are a point of concern, Hopf bifurcation, may also be identified and prevented, just like proposed in [15], where some preventive control actions are considered. Another kind of study may be necessary if it is assumed that a system has experimented a fault and reached a post-fault equilibrium point associated 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.10.021 * Corresponding author. E-mail address: zambroni@unifei.edu.br (A.C. Zambroni de Souza). Applied Mathematics and Computation 215 (2010) 3327–3334 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc