Equivalency of Continuation and Optimization Methods to Determine Saddle-Node and Limit-Induced Bifurcations in Power Systems

This paper presents a comprehensive and detailed study of an optimization-based approach to identify and analyze saddle-node bifurcations (SNBs) and limit-induced bifurcations (LIBs) of a power system model, which are known to be directly associated with voltage stability problems in these systems. Theoretical studies are presented, formally demonstrating that solution points obtained from an optimization model, which is based on complementarity constraints used to properly represent generators' voltage controls, correspond to either SNB or LIB points of this model. These studies are accomplished by proving that optimality conditions of these solution points yield the transversality conditions of the corresponding bifurcation points. A simple but realistic test system is used to numerically illustrate the theoretical discussions.

[1]  Claudio A. Canizares,et al.  Multiobjective optimal power flows to evaluate voltage security costs in power networks , 2003 .

[2]  Hande Y. Benson,et al.  INTERIOR-POINT METHODS FOR NONCONVEX NONLINEAR PROGRAMMING: COMPLEMENTARITY CONSTRAINTS , 2002 .

[3]  T. Cutsem.  A method to compute reactive power margins with respect to v , 1991, IEEE Power Engineering Review.

[4]  R. Seydel.  Practical bifurcation and stability analysis : from equilibrium to chaos , 1994 .

[5]  P. Kundur,et al.  Power system stability and control , 1994 .

[6]  N. Mithulananthan,et al.  Linear performance indices to predict oscillatory stability problems in power systems , 2004, IEEE Transactions on Power Systems.

[7]  Fuzhen Zhang.  The Schur complement and its applications , 2005 .

[8]  Venkataramana Ajjarapu,et al.  The continuation power flow: a tool for steady state voltage stability analysis , 1991 .

[9]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[10]  I. Dobson.  Observations on the geometry of saddle node bifurcation and voltage collapse in electrical power systems , 1992 .

[11]  Costas D. Vournas,et al.  Relationships between static bifurcations and constrained optima , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).

[12]  A. Conejo,et al.  Perturbation Approach to Sensitivity Analysis in Mathematical Programming , 2006 .

[13]  P. Kundur,et al.  Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions , 2004, IEEE Transactions on Power Systems.

[14]  G. Irisarri,et al.  Maximum loadability of power systems using interior point nonlinear optimization method , 1997 .

[15]  Carson W. Taylor,et al.  Definition and Classification of Power System Stability , 2004 .

[16]  W. Marszalek,et al.  Singularity-induced bifurcations in electrical power systems , 2005, IEEE Transactions on Power Systems.

[17]  Ricardo Riaza,et al.  On singular equilibria of index-1 DAEs , 2000 .

[18]  H. Schättler,et al.  Dynamics of large constrained nonlinear systems-a taxonomy theory [power system stability] , 1995, Proc. IEEE.

[19]  Costas D. Vournas,et al.  Maximizing Power-System Loadability in the Presence of Multiple Binding Complementarity Constraints , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[20]  Claudio A. Canizares,et al.  Point of collapse and continuation methods for large AC/DC systems , 1993 .

[21]  William Rosehart,et al.  Optimal Power Flow with Complementarity Constraints , 2006 .

[22]  D. Hill,et al.  Stability theory for differential/algebraic systems with application to power systems , 1990 .

[23]  Federico Milano,et al.  Multiobjective optimization for pricing system security in electricity markets , 2003 .

[24]  E. Vaahedi,et al.  Dynamic security constrained optimal power flow/VAr planning , 2001 .

[25]  C. Cañizares.  Conditions for saddle-node bifurcations in AC/DC power systems , 1995 .

[26]  C. Vournas,et al.  Voltage security considerations in an open power market , 2000, DRPT2000. International Conference on Electric Utility Deregulation and Restructuring and Power Technologies. Proceedings (Cat. No.00EX382).

[27]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[28]  W. Rosehart.  Optimal power flows incorporating network stability , 2002, 2002 IEEE Power Engineering Society Winter Meeting. Conference Proceedings (Cat. No.02CH37309).

[29]  N. Maratos,et al.  Bifurcation points and loadability limits as solutions of constrained optimization problems , 2000, 2000 Power Engineering Society Summer Meeting (Cat. No.00CH37134).

[30]  Ian Dobson,et al.  Towards a theory of voltage collapse in electric power systems , 1989 .

[31]  G. L. Torres,et al.  An interior-point method for nonlinear optimal power flow using voltage rectangular coordinates , 1998 .

[32]  C. Cañizares.  Calculating optimal system parameters to maximize the distance to saddle-node bifurcations , 1998 .

[33]  Michael C. Ferris,et al.  Expressing Complementarity Problems in an Algebraic Modeling Language and Communicating Them to Solvers , 1999, SIAM J. Optim..

[34]  R. Seydel.  From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis , 1988 .

[35]  F. Alvarado,et al.  Point of collapse methods applied to AC/DC power systems , 1992 .

[36]  Ian Dobson,et al.  Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encountered , 1992 .

[37]  William D. Rosehart.  Optimization of power systems with voltage security constraints , 2001 .