60 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005 Transient Stabilization of Multimachine Power Systems With Nontrivial Transfer Conductances Romeo Ortega, Martha Galaz, Alessandro Astolfi, Yuanzhang Sun, and Tielong Shen Abstract—In this paper, we provide a solution to the long-standing problem of transient stabilization of multima- chine power systems with nonnegligible transfer conductances. More specifically, we consider the full -dimensional model of the -generator system with lossy transmission lines and loads and prove the existence of a nonlinear static state feedback law for the generator excitation field that ensures asymptotic stability of the operating point with a well-defined estimate of the domain of attraction provided by a bona fide Lyapunov function. To design the control law we apply the recently introduced interconnection and damping assignment passivity-based control methodology that endows the closed-loop system with a port-controlled Hamil- tonian structure with desired total energy function. The latter consists of terms akin to kinetic and potential energies, thus has a clear physical interpretation. Our derivations underscore the deleterious effects of resistive elements which, as is well known, hamper the assignment of simple “gradient” energy functions and compel us to include nonstandard cross terms. A key step in the construction is the modification of the energy transfer between the electrical and the mechanical parts of the system which is obtained via the introduction of state-modulated interconnections that play the role of multipliers in classical passivity theory. Index Terms—Nonlinear systems, passivity-based control, power systems, stability. I. INTRODUCTION T RADITIONAL analysis and control techniques for power systems are undergoing a major reassessment in recent years; see [29] for an excellent tutorial account. This worldwide trend is driven by multiple factors including the adoption of new technologies, like flexible ac transmission systems, which offer improvements in power angle and voltage stability but give rise to many modeling and control issues that remain to be resolved. Manuscript received February 3, 2004; revised September 13, 2004. Rec- ommended by Associate Editor M. Reyhanoglu. This work was supported in part by the GEOPLEX program of the European Commission with reference code IST-2001-34166, http://www.geoplex.cc, and by the joint Sino-French Laboratory in Informatics, Automation and Applied Mathematics (LIAMA), http://liama.ia.ac.cn/. Part of this work was carried out while R. Ortega and M. Galaz were visiting Tsinghua University. The warm hospitality of this institu- tion is gratefully acknowledged. The work of Martha Galaz was supported in part by the CONACyT of Mexico. R. Ortega and M. Galaz are with the Laboratorie des Signaux et Sys- témes, Supelec, 91192 Gif-sur-Yvette, France (e-mail: ortega@lss.supelec.fr; galaz@lss.supelec.fr). A. Astolfi is with the Electrical Engineering Department, Imperial College, London SW7 2BT, UK (e-mail: a.astolfi@ic.ac.uk). Y. Sun is with the Department of Electrical Engineering, Tsinghua University, Beijing, 10084, China (e-mail: yzsun@mail.tsinghua.edu.cn). T. Shen is with the Department of Mechanical Engineering, Sophia Univer- sity, Tokyo 102-8554, Japan (e-mail: tetu-sin@sophia.ac.jp). Digital Object Identifier 10.1109/TAC.2004.840477 Also, the ever increasing utilization of power electronic con- verters is drastically modifying the energy consumption profile, as well as the underlying distributed generation. The new dereg- ulated market, on the other hand, has seen the emergence of separate entities for generation that imposes more stringent re- quirements on the dynamic behavior of voltage regulated units and the task of coordinating a large number of (small and large) active and reactive control units in the face of significant load uncertainty. It is, by now, widely recognized that the existing methods and tools to approach power systems should be re- visited to ensure reliable and secure planning—with the recent dramatic blackouts in North America and Italy providing com- pelling evidence of this fact. In this paper, we study the fundamental problem of transient stability of power systems whose reliable assessment has be- come a major operating constraint, particularly in regions that rely on long distance transfers of bulk power. Transient stability is concerned with a power system’s ability to reach an accept- able steady-state following a fault, e.g., a short circuit or a gen- erator outage, that is later cleared by the protective system op- eration; see [2], [9], [13], [25], and the tutorial paper [4] for more details. The fault modifies the circuit topology—driving the system away from the stable operating point—and the ques- tion is whether the trajectory will remain in the basin of attrac- tion of this (or other) equilibrium after the fault is cleared. The key analysis issue is then the evaluation of the domain of attrac- tion of the system’s operating equilibrium, while the control ob- jective is the enlargement of the latter. Transient stability analysis dates back to the beginning of the electric age [5] with the problem originally studied via numer- ical integration and, starting in the 1947 seminal paper [15], with Lyapunov-like methods. A major open problem in this area is the derivation of Lyapunov functions for transmission systems that are not lossless, i.e., with transfer conductances between busses. 1 While the transmission system itself can be modeled as being lossless without loss of accuracy, the clas- sical network reduction of the load busses induces transfer con- ductances between the rest of the system busses, rendering the negligible transfer conductances assumption highly unsatisfac- tory [2], [13]. Although considerable efforts have been made to find Lyapunov functions for lossy line systems, to the best of the authors’ knowledge this research has unfortunately been in vain. Actually, in [18] it is claimed that, even for the simple 1 More precisely, the conductances represent partial losses caused by the line and the loads in the nodes. For the sake of simplicity, in this paper we say the line is lossless or lossy if the conductances are neglected or not, respectively. 0018-9286/$20.00 © 2005 IEEE