The feasible region plays a fundamental role in solving optimal power flow (OPF) problems. OPF solutions do not exist if the corresponding feasible region is an empty set. Any solution algorithm proposed for solving OPF problems will fail if the feasible region does not exist. In addition, the geometry and characteristics of feasible region indeed affect the performance of OPF solution methods. In this talk, a complete characterization of the feasible region of OPF problems will be presented. This mathematical characterization can lead to the development of numerical methods for computing feasible solutions. It will be shown that the feasible region of a general OPF problem equals the union of regular stable equilibrium manifolds of a non-hyperbolic dynamical system that is constructed from the set of equality and inequality constraints of the OPF problem. It will be further shown that the non-hyperbolic dynamical system is completely stable. This complete stability property allows one to locate multiple points lying on stable equilibrium manifolds via trajectories of the non-hyperbolic dynamical system, which can be useful for practical applications. Bifurcation analysis of feasibility region due to varying loading conditions will be presented to explain several challenges encountered in practical applications. The notion of a local feasible region will be introduced for large-scale OPF problems for visualization of feasible components and local optimal solutions. A numerical method for identifying the existence/non-existence of feasible solution for an OPF problem will be presented and illustrated on large-scale OPF problems.
Hsiao-Dong Chiang (M’87–SM’91–F’97) received the Ph.D. degree in electrical engineering and computer sciences from the University of California at Berkeley, Berkeley, CA, USA. Since 1998, he has been a Professor in the School of Electrical and Computer Engineering at Cornell University, Ithaca, NY, USA. His current research interests include nonlinear systems, power networks, nonlinear computation, nonlinear optimization, and their practical applications. He and his team members have published more than 400 referred papers with the H factor of 51. He holds 18 U.S. and overseas patents and several consultant positions. He is the author of two books: Direct Methods for Stability Analysis of Electric Power Systems: Theoretical Foundations, BCU Methodologies, and Applications (Hoboken, NJ, USA: Wiley, 2011) and (with Luis F. Alberto) Stability Regions of Nonlinear Dynamical Systems: Theory, Estimation, and Applications (Cambridge, UK: Cambridge Univ. Press, 2015). He has served as an associate editor for three different IEEE transactions and journals, and served as a board member for IEEJ Japan, and is the founder and President of Bigwood Systems, Inc., Ithaca, NY, USA.