Electric Power Systems Research 103 (2013) 92–104 Contents lists available at SciVerse ScienceDirect Electric Power Systems Research jou rn al hom epage: www.elsevier.com/locate/epsr On the periodic steady-state analysis of induction machines interfaced through VSCs using the Poincaré map method and a Voltage-Behind-Reactance model Norberto Garcia a,∗ , Enrique Acha b a Facultad de Ingeniería Eléctrica, Universidad Michoacana de San Nicolás de Hidalgo, Ciudad Universitaria, 58030 Morelia, Mexico b Department of Electrical Energy Engineering, Tampere University of Technology, Tekniikankatu 8 B, FI-33720 Tampere, Finland a r t i c l e i n f o Article history: Received 31 October 2012 Received in revised form 21 February 2013 Accepted 10 May 2013 Available online 14 June 2013 Keywords: Induction machine VSC Periodic steady-state Newton methods Poincaré map Harmonics a b s t r a c t This paper presents the most efficient method to date to determine the periodic steady-state solution of three-phase induction machines based on the amalgamation of the Poincaré map method and the so-called Voltage-Behind-Reactance (VBR) representation. An acceleration procedure based on the dis- cretization of the dynamic equations with the Poincaré map and the application of Newton’s method allows locating periodic solutions. A per-unit VBR formulation, suitable for the Newton-based accelera- tion procedure, is used in this paper to ensure highly efficient solutions. To test further the robustness and versatility of the per-unit VBR model, it is interfaced to a voltage source converter (VSC) and the results show that the high efficiency of the new model remains unabated – this applies to both small and large induction machines. The method is particularly useful to carry out harmonic-oriented analyses where the computational effort reduces dramatically compared to cases when more traditional induction machine models and solution approaches are employed. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Research work on induction machine modelling, analysis and design continues to be areas of great interest to electrical engi- neers owing to the proliferation of new applications involving this kind of rotating machinery. The use of induction machines to gen- erate electricity using the wind as the primary energy resource has increased very considerably over the past two decades, and continues to receive a great deal of attention by the global power engineering community, particularly, the newer breed of highly controllable induction machines based on the so-called Doubly-Fed Induction Generator (DFIG) technology. More recently, an emerg- ing power transmission technology based on a variable frequency transformer applies a wound rotor induction machine to inter- connect power systems in an asynchronous manner. The use of power electronics plays a key role in the integration of such tech- nology applications to ensure high performances at their intended operational regime. However, the voltage and current harmonics produced by the associated power electronic converters have the capacity to impair the performance and efficiency of the rotating machinery, if left unchecked. Hence, it is an essential engineering requirement to assess the harmonic impact in both the induction ∗ Corresponding author. Tel.: +52 443 3279728; fax: +52 443 3279728. E-mail addresses: gbarriga@umich.mx (N. Garcia), enrique.acha@tut.fi (E. Acha). machine and the power grid with a view to apply corrective actions if and when the need arises. The study of the dynamic behavior of an induction machine may be carried out with a state-of-the-art model based on a Voltage- Behind-Reactance representation of the induction machine which has shown to exhibit a superior performance than the classical induction machine model expressed in full phase coordinates, as reported in [1]. While the mechanical equations involved in the VBR model of the induction machine are described by the tradi- tional single rigid body system, the electric subsystem makes use of a state variable approach for an efficient, direct interface with the external circuits [2]. This novel formulation has been used for electromagnetic transient studies [3], extended to include the sat- uration characteristics of this machine [4]. No related information seems to exist concerning the use of this efficient VBR induction machine model to the solution of harmonic-oriented problems. In principle, any Electromagnetic Transients Program (EMTP) may be used to derive the periodic steady-state waveforms, which involves integrating the associated system of equations until a specified tolerance is reached. Despite the simplicity of the approach, conver- gence to the steady-state may be quite slow when solving lightly damping components and the determination of the steady-state becomes difficult to establish [5]. To circumvent such a problem, accelerated time domain solutions for the derivation of periodic steady-state current and voltage waveforms have been put forward [6], with a great deal of research progress having been achieved 0378-7796/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2013.05.008