IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006 1267 Local Voltage-Stability Index Using Tellegen’s Theorem Ivan ˇ Smon, Student Member, IEEE, Gregor Verbiˇ c, Member, IEEE, and Ferdinand Gubina, Senior Member, IEEE Abstract—In the paper, the Tellegen’s theorem and adjoint net- works are used to derive a new, local voltage-stability index. The new approach makes it possible to determine the Thevenin’s pa- rameters in a different way than adaptive curve-fitting techniques, from two consecutive phasor measurements. The new index was rigorously tested on different test systems. The results were obtained on a static two-bus test system and on the dynamic Belgian–French 32-bus test system that includes full dynamic models of all power-system components crucial to the voltage instability analysis. The results show advantages of the proposed index: it is simple, computationally very fast, and easy to implement in the wide-area monitoring and control center or locally in a numerical relay. Index Terms—Adjoint networks, local phasors, Tellegen’s the- orem, Thevenin’s equivalent, voltage-stability index. I. INTRODUCTION P OWER systems are forced to operate ever closer to their load limits because of the demands of deregulated elec- tricity markets. As a result, several blackouts have occurred due to voltage instability. This means that voltage stability has be- come a matter of serious concern for system operators and is a subject of a lot of investigation due to its importance in terms of the system security and power quality. Significant efforts are still being directed toward definitions, classifications, new con- cepts, practices, and tools for solving the voltage-stability and security-analysis problems [1], [2]. The voltage-instability problem is characterized by voltage uncontrollability at certain locations in a power network after a disturbance. The problem occurs more frequently in stressed networks with reduced stability margins and/or reduced re- active-power reserves. The voltage instability is basically a dynamic phenomenon with rather slow dynamics and a time domain ranging from a few seconds to some minutes or more. Although the voltage instability is a complex problem, it is very important that system operators use quick, simple, and correct methods to calculate the distance to the worst case: the voltage collapse. Many system-oriented approaches and long-term voltage-sta- bility methods are based on static models because of the high dimensionality and complexity of stability problems. For Manuscript received April 27, 2005; revised January 17, 2006. This work was supported by the Ministry of Higher Education, Science, and Technology of the Republic of Slovenia. Paper no. TPWRS-00254-2005. The authors are with the Laboratory of Power Systems and High Voltage, Faculty of Electrical Engineering, University of Ljubljana, SI-1000 Ljubljana, Slovenia (e-mail: ivo.smon@fe.uni-lj.si). Digital Object Identifier 10.1109/TPWRS.2006.876702 this reason, control-actions, time-range, quasi-steady-state time-domain simulations are used [3] to simplify matters. The results of such studies can also be used for screening purposes to identify critical cases that require a more detailed or dynamic analysis [4]. Most methods are based on executing a large number of power flows using conventional models and the P-V or Q-V curves at selected buses [5], [6]. Sensitivity methods based on linearization around an operating point (the solved power-flow case) are usually repeated at several points along the P-V or Q-V curves. Together with modal analysis [7], all these methods use the system’s Jacobian matrix. To avoid uncertainty of the load-flow convergence, the continuation power flow [8] allows tracing of the solution path passing through the loading margin; this is the most widely accepted index for determining the distance of the power system state to the voltage collapse. Methods to assess the voltage stability of a power system, such as bifurcations, direct methods, energy functions, etc., can be found in the literature [2]. The main idea behind local methods is that the local phasors contain enough information to directly detect the voltage- stability margin using their measurements. This concept is attractive, since real-time measurements of the voltage and current phasors at the system buses are already available from the phasor measurement units (PMUs) installed in many power systems [9]. In addition to the benefits of small computational effort and simplicity, local methods also give a very good insight into the voltage-collapse process and can easily be used for online system monitoring. The local method proposed by Strmˇ cnik and Gubina [10] is based on the power-system decomposition on reactive-power radial transmission paths modeled as two-bus equivalents. In addition to proximity to the voltage collapse, the method also considers exhaustion of the generator’s reactive-power reserves. However, two indicators mean that the determination of the problem is not straightforward. Another weakness of this method is a complex identification of critical buses in the system and the fact that the electric distance of the reac- tive-power source from the affected load bus is not considered when calculating the generator’s reactive-power reserve. Verbiˇ c and Gubina [11] proposed a local method based on the fact that, in vicinity of the voltage collapse, the entire increase in the apparent power loading at the sending end of a line is due to the supply of transmission losses. The main shortfall of this method lies in additional checking if the line is loaded below its natural loading. Vu et al. [12] proposed a local method based on the power- transfer impedance-matching principle. The measured data are 0885-8950/$20.00 © 2006 IEEE Authorized licensed use limited to: UNIVERSITY OF SYDNEY. Downloaded on August 09,2010 at 01:09:49 UTC from IEEE Xplore. Restrictions apply.