r~'lu TT E RWO RT H ~I~E I N E M A N N Electrical Power & Eneroy Systems, Vol. 17, No. I, pp. 61~8, 1995 Copyright ~, 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0142-0615/95 $10.00+ 0.00 Conditions for saddle-node bifurcations in AC/DC power systems C A Cafiizares University of Waterloo, Department of Electrical and Computer Engineering, Waterloo, Ontario N2L 3G1, Canada Saddle-node bifurcations are dynamic instabilities of differential equation models that have been associated with voltage collapse problems in power systems. This paper presents the conditions needed for detectin9 these types of bifurcations usin9 power flow equations for a dynamic model of AC/DC systems, represented by differential equations and algebraic constraints. Two methods typically used to detect saddle-node bifurcations, namely, direct and parameterized continuation methods, are briefly analysed ,from the point of view of numerical robustness. Keywords: saddle-node bifurcations, voltage stability, A C/DC power systems, numerical robustness I. Introduction Hopf bifurcations and saddle-node bifurcations, or turning points, have been recognized as some of the reasons, albeit not the only ones, for voltage stability problems in a variety of power system models 1-8. Local bifurcations are detected by monitoring the eigenvalues of the current operating point. As certain parameters in the system change slowly, allowing the system to recover quickly and maintain a stable operating point, the system eventually turns unstable, either due to one of the eigenvalues becoming zero (saddle-node, transcritical, pitchfork bifurcations), or due to a pair of complex eigenvalues crossing the imaginary axes of the complex plane (Hopf bifurcation). The instability of the system is reflected on the state variables, usually represented by frequency, angles and voltages, by an oscillatory behaviour or a continuous change. For a PQ load model these bifurcations can be associated to the power transfer limit of the transmission system8; in other instances the Received 25 November 1993; revised 26 April 1994; accepted 8 June 1994 bifurcations appear due to voltage control problems, such as fast acting automatic voltage regulators (AVR) in the generator 11, or voltage dependent current order limiters (VDCOL) in HVDC links 7. In all cases these bifurcations occur on very stressed systems, i.e., the region of stability for the current operating point (stable equilibrium point or sep) is small, hence, the system is not able to withstand small perturbations and becomes unstable. Although there are reports of these bifurcations occurring in unstressed systems 12, this cannot be considered typical, because power system controls are designed so that eigenvalues of several operating points are well into the left-half complex plane. Some voltage collapse problems can also be associated with voltage control devices such as under-load tap changer (ULTC) transformers or AVRs 13-16. In some of these cases, but not all, the voltage controls force the eigenvalues to jump instantaneously into the unstable region, making the system immediately unstable. This phenomena is not directly associated with a bifurcation, since the eigenvalues do not go through zero or the imaginary axis. Nevertheless, transcritical bifurcation theory can be used to explain the phenomena when AVR limits are assumed to apply gradually 14. The particular problem of voltage collapse in power systems has been generally associated with saddle-node bifurcations 1-8. These type of instabilities are usually local area voltage problems due to lack of reactive power support, that yield a system-wide stability problem characterized by sudden voltage drops. Saddle-node bifurcations are well defined instabilities in system models fully represented by differential equations 1'5"6'~7'18. However, bifurcations in systems that are also modelled with algebraic constraints have been recently addressed and new results are regularly being reported in the literature 5'6'7' 19. This paper focuses also on this subject, showing the requirements for having equivalency of typical saddle-node bifurcation conditions between differential equation models and mixed models, i.e. systems represented by both differential equations and