Prediction of dynamic voltage stability status based on Hopf and limit induced bifurcations using extreme learning machine Mohammad Hossein Velayati, Nima Amjady ⇑ , Issa Khajevandi Department of Electrical Engineering, Semnan University, Semnan, Iran article info Article history: Received 12 November 2013 Received in revised form 1 January 2015 Accepted 9 January 2015 Keywords: Hopf Bifurcation (HB) Limit Induced Bifurcation (LIB) Forecast process Extreme learning machine abstract Evaluation of voltage stability status considering its dynamic boundaries is a key issue for saving global stability of power systems. However, this evaluation is a computationally demanding task and its imple- mentation is very hard (if not impossible) for on-line environments such as dispatching centers of power systems. In this paper, a new viewpoint for the problem based on modeling it as a forecast process is pro- posed, which can be implemented with a low computation burden for practical power systems. For this purpose, a voltage stability classification model considering Hopf and limit induced bifurcations is pro- posed and a new forecast strategy to predict voltage stability class label based on the proposed classifi- cation is suggested. The suggested forecast strategy is composed of an information theoretic feature selection technique, extreme learning machine (ELM) as the forecast engine and a line search procedure to fine-tune the settings. The effectiveness of the proposed classification model and forecast strategy is extensively illustrated on the New England 39-bus and IEEE 145-bus test systems. Ó 2015 Elsevier Ltd. All rights reserved. Introduction Voltage stability is an important subset of power system stabil- ity and refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a distur- bance from a given initial operating condition [1]. Various factors such as load models, parameters of the control devices (e.g., exciter of generators), loading conditions of the system, reactive support of the system and contingencies are effective on the voltage stabil- ity boundaries. Static analysis tools, such as power flow based methods, are widely used to evaluate the static voltage stability margin [2]. However, static voltage stability analysis neglects the transient and dynamic behaviors of power system. On the other hand, voltage instability is essentially a dynamic phenomenon under large or small disturbances. Thus, some important bound- aries of voltage stability feasible region, such as HB, saddle node bifurcation (SNB) and LIB, may not be correctly analyzed by the static analysis methods [3]. HB is the onset of oscillatory behavior in a nonlinear system where, at the point of oscillation, the pair of critical eigenvalues crosses the imaginary axis of the complex plane from the left to the right [3]. SNB indicates singularity of dynamic algebraic Jacobian of power system and is associated with voltage collapse problems [3,4]. Reactive power capacity of power system generators is an important feature that has great impacts on the introduced voltage stability boundaries [4–7]. When a tran- sition occurs due to the limitation of reactive power sources, espe- cially generators, complex nonlinear behaviors occur and may even lead to an immediate instability [4–6]. This type of bifurcation is called LIB. In spite of HB and SNB that is mainly triggered from con- sumers’ area in power system, LIB is induced by lack of reactive power generation and transfer limitation of the system [5]. In [5], impacts of generator reactive reserve on structure- induced bifurcation are studied and it is shown that generation reactive power margins have significant impacts on the bifurcation points and especially LIB. In [7], using modal analysis of load flow Jacobian and ‘‘ranking coefficients’’, the generators are divided into ‘‘important’’ and ‘‘less-important’’ ones and then voltage stability margin is improved by increasing and decreasing reactive power generation at the important and less-important generators, respec- tively. Sensitivity analysis based on linear approximation is used in [8] to increase voltage stability margin considering HB boundary. In this method, sensitivity of the voltage stability margin with respect to different parameters of power system, such as active power gen- eration and voltage set-point of generators and voltage set-point in pilot buses in secondary voltage control scheme, is determined. In [9], a unified expression function method for describing piecewise, continuous and non-differential functions is proposed to investigate the impact of the exciter voltage limits on power system small sig- nal stability region. In [10], steady state analysis of voltage stability using a new methodology based on the predictor–corrector method http://dx.doi.org/10.1016/j.ijepes.2015.01.005 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +98 021 84781957; fax: +98 021 88672145. E-mail address: amjady@semnan.ac.ir (N. Amjady). Electrical Power and Energy Systems 69 (2015) 150–159 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes