IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 10, OCTOBER 1986 981 Static Bifurcations in Electric Power Networks: Loss of Steady-State Stability, and Voltage Collapse HARRY G. KWATNY, SENIOR MEMBER, IEEE, ARUN K. PASRIJA, MEMBER, IEEE, AND LEON Y. BAHAR Ahsrracr -This paper presents an analysis of static stability in electric power systems. The study is based on a model consisting of the classical swing equation characterization for generators and constant admittance, PV bus and/or PQ bus load representations which, in general, leads to a semi-explicit (or constrained) system of differential equations. A precise definition of static stability is given and basic concepts of static bifurcation theory are used to show that this definition does include conventional notions of steady-state stability and voltage collapse, but it provides a basis for rigorous analysis. Static bifurcations of the load flow equations are analyzed using the Liapunov-Schmidt reduction and Taylor series expan- sion of the resulting reduced bifurcation equation. These procedures have been implemented using symbolic computation (in MASYMA). It is shown that static bifurcations of the load flow equations are associated with either divergence-type instability or loss of causality. Causality issues are found to be an important factor in understanding voltage collapse and play a central role in organizing global power system dynamics when loads other than constant admittance are present. I. INTRODUCTION P ROBLEMS associated with the steady-state stability and voltage collapse of electric power systems have become increasingly important during the past decadeand have consequently received the attention of many investi- gators. To a large extent, this is due to the fact that the present-day power system operating environment substan- tially increases the difficulty of maintaining acceptable system voltage profiles. Low voltage can result in loss of stability and voltage collapse and ultimately to cascading power outages. Indeed, voltage difficulties have been asso- ciated with major incidents in Italy, France, Britain, Japan, and the U.S. (see [l]-[9] for a sample of the literature). Several factors have contributed to this situation, including the adoption of higher transmission voltages, ‘the relative decrease, in the reactive power output of large generating units, and the shift in power flow patterns associated with changing fuel costs and generation availability. The severity and extended time duration of recent major system disturbances are responsiblefor the current interest in comprehensive system restoration plans (Kafka et al. [lo], Blankenship and Trygar [ll]). Here again, steady-state Manuscript received January 29,1986. Part of this work was performed at Systems Engineering, Inc., Greenbelt, MD, with the support of the U.S. Department of Energy, Division of Electric Energy Systems. - H. G. Kwatny and L. Y. Bahar are with the Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104. A. K. Pasrija is with AT&T Engineering Research Center, Princeton, NJ. IEEE Log Number 8609855. stability and voltage collapse are vital issuesin the recon- struction of a power system in minimum time following a major system failure. Such incidents, although very rare, are of serious concern to the electric utility industry, which is actively pursuing the development of computer-aided procedures for minimizing the effect of such eventualities. In response to these concerns, there continues to be a substantial international effort to develop on-line voltage control algorithms based on some form of optimal (active and reactive) power flow formulation (Hano, et al. [12], Aldrich et al. [13], Burchet et al. [14]). This is a very difficult problem and even the structure of an appropriate performance index is not obvious (the discussions in Savulesco[15] and Capasso et al. [16] are interesting in this regard). In fact, it is likely that a multiple objective formu- lation will prove necessary. The experiencein France and Italy suggeststhat a practical control algorithm will iden- tify critical buses and maintain tight control of voltage on those buses. But how does one identify the critical buses? In our view, this and other basic questions require answers if comprehensive, on-line voltage control is to become a reality. It is common to associate steady-statestability with the ability of the transmission network to transport real power (for example, seethe discussionof transmission capacity in [32]) and to associate voltage collapse with the inability to provide reactive power at the necessarylocations within the network as describedby Laths [4], [34]. As meaningful as these interpretations may be in appropriate cir- cumstances, a clear understanding of steady-statestability and voltage collapse can only be achieved by considering them both as arising from a common, well-defined origin. The main purpose of this paper is to establish the frame- work for such a point of view. Indeed, there do not now exist precise definitions of steady-statestability or voltage collapse which are generally accepted or useful. Existing characterizations of these processesremain primarily in terms of relatively simple paradigms. One consequence of this situation is the continuing discussion on whether voltage collapse is a static or dynamic phenomenon (Tamura et al.. [26]). Perhaps the most widely held notion of steady-state stability requires the existence of an equilibrium point, i.e., a solution to the load flow equations,which is stable in the 0098-4094/86/1000-0981$01.00 01986 IEEE