Electric Power Systems Research 79 (2009) 1209–1215 Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr Extending the perturbation technique to the modal representation of nonlinear systems S. Soltani a,∗ , N. Pariz b , R. Ghazi b a Department of Electrical Engineering, Science & Research Branch, Islamic Azad University (IAU), 1477893855-14515775, Tehran, Iran b Department of Electrical Engineering, ferdowsi University, 9177948944-1111, Mashhad, Iran article info Article history: Received 23 May 2008 Received in revised form 13 January 2009 Accepted 28 February 2009 Available online 3 April 2009 Keywords: Perturbation technique Modal Series Normal Form method Unified power flow controller (UPFC) abstract After a brief review of perturbation technique, using this method an approach is developed to represent and study the behavior of nonlinear dynamic power systems. For the first time in this field, perturbation technique is applied to obtain an approximate closed form expression for the zero input response of stressed power systems. In order to show the superiority of the proposed method, it has been applied to a typical nonlinear system which is a single machine infinite bus (SMIB) power system with unified power flow controller (UPFC). The accuracy and competency of this method in comparison with Modal Series method will also be validated. © 2009 Elsevier B.V. All rights reserved. 1. Introduction It is a well-known fact that nonlinear factors will increase the nonlinear modal interaction when a power system is stressed and this in turn increases the complexity of the system behavior. It is therefore necessary to study the influence of nonlinear factors on stressed power systems, a task much emphasized in the related literature [1]. To study the dynamic behavior of power systems, two approaches are commonly used. The first approach is the non- linear simulation which clearly exhibits the effects of nonlinear factors but it requires a great deal of effort and time. The other one which is restricted to a small neighborhood of the operating point is referred to as linear modal analysis. When the system under study is subjected to large disturbances, the similarity between real time response and linear modal simulation is lost. An alternative approach recently proposed is the Normal Form of vector field tech- nique (NF) [2–5]. To achieve an approximate solution, this method uses higher order terms in Taylor series expansion. Although in studying the behavior of complex nonlinear systems, NF is regarded as a useful and effective tool, but its application involves nonlin- ear transformations as well as nonlinear algebraic equations. In some conditions, such as resonance and quasi-resonance, equa- tions may result in limited valid regions and in some cases they even lead to the lack of solutions. A different recently proposed ∗ Corresponding author. Tel.: +98 9151711829. E-mail addresses: sep soltani@iaus.ac.ir (S. Soltani), n-pariz@ferdowsi.um.ac.ir (N. Pariz), rghazi@ferdowsi.um.ac.ir (R. Ghazi). approach which retains the advantages of NF while overcoming many of its drawbacks is the Modal Series (MS) method [6,7]. In this paper, in order to obtain the appropriate solutions for the zero input response of large scale nonlinear systems, a new approach based-on perturbation technique (PT), is developed to derive an approximate closed form expression. Similar to MS method, the proposed method does not have the shortcomings of previous methods and under any operating condition the solution is pos- sible. The paper is organized as follows: in Section 2, a very brief review of perturbation technique is presented. Section 3, for the first time deals with obtaining an approximate closed form expres- sion for the zero input response of nonlinear systems. Section 4 outlines the Normal Form Method. For the purpose of comparison between different solutions, the proximity measure is introduced in Section 5. The solution algorithm and the successive steps are stated in Section 6. Section 7 is devoted to a case study in which at first, according to the developed method, the mathematical formu- lation of a SMIB power system with UPFC is introduced and with regard to the controller the studied cases defined. Then by intro- duction of a second order resonance index, the valid regions and proximity measures for various operating conditions are examined. Conclusions are given in Section 8. 2. Perturbation technique One of the most common and successful tools in the analysis of nonlinear differential equations and studying the effects of parameter variations on the behavior of the system is the pertur- 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.02.011