Nonlinear Koopman Modes and Coherency Identification of Coupled Swing Dynamics

We perform modal analysis of short-term swing dynamics in multi-machine power systems. The analysis is based on the so-called Koopman operator, a linear, infinite-dimensional operator that is defined for any nonlinear dynamical system and captures full information of the system. Modes derived through spectral analysis of the Koopman operator, called Koopman modes, provide a nonlinear extension of linear oscillatory modes. Computation of the Koopman modes extracts single-frequency, spatial modes embedded in non-stationary data of short-term, nonlinear swing dynamics, and it provides a novel technique for identification of coherent swings and machines.

[1]  B. O. Koopman,et al.  Hamiltonian Systems and Transformation in Hilbert Space. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[2]  K. S. Rao,et al.  Coherency Based System Decomposition into Study and External Areas Using Weak Coupling , 1985, IEEE Transactions on Power Apparatus and Systems.

[3]  C. Caramanis.  What is ergodic theory , 1963 .

[4]  Aurel Wintner,et al.  Harmonic Analysis and Ergodic Theory , 1941 .

[5]  Karl Petersen.  Ergodic Theory , 1983 .

[6]  K. R. Padiyar,et al.  ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILITY , 1990 .

[7]  A.R. Messina,et al.  Extraction of Dynamic Patterns From Wide-Area Measurements Using Empirical Orthogonal Functions , 2007, IEEE Transactions on Power Systems.

[8]  Michael C. Mackey,et al.  Chaos, Fractals, and Noise , 1994 .

[9]  J. H. Chow,et al.  Large-scale system testing of a power system dynamic equivalencing program , 1998 .

[10]  Joe H. Chow,et al.  New Algorithms for Slow Coherency Aggregation of Large Power Systems , 1995 .

[11]  A.H.M.A. Rahim,et al.  Identification of coherent generators using energy function , 1990 .

[12]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[13]  Robin Podmore,et al.  Identification of Coherent Generators for Dynamic Equivalents , 1978, IEEE Transactions on Power Apparatus and Systems.

[14]  Yoshihiko Susuki,et al.  Nonlinear Koopman modes of coupled swing dynamics and coherency identification , 2010, IEEE PES General Meeting.

[15]  A. R. Messina,et al.  Nonlinear, non-stationary analysis of interarea oscillations via Hilbert spectral analysis , 2006, IEEE Transactions on Power Systems.

[16]  I. Mezić.  Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , 2005 .

[17]  Felix F. Wu,et al.  Chaos in a simple power system , 1993 .

[18]  I. Mezić,et al.  Spectral analysis of nonlinear flows , 2009, Journal of Fluid Mechanics.

[19]  M. Pai.  Energy function analysis for power system stability , 1989 .

[20]  I. Mezic,et al.  Nonlinear Koopman Modes and a Precursor to Power System Swing Instabilities , 2012, IEEE Transactions on Power Systems.

[21]  Joe Chow,et al.  A Sparsity-Based Technique for Identifying Slow-Coherent Areas in Large Power Systems , 1984, IEEE Transactions on Power Apparatus and Systems.

[22]  M. Hayashi,et al.  Construction of power system transient stability equivalents using the Lyapunov function , 1981 .

[23]  A.G. Phadke,et al.  Synchronized phasor measurements in power systems , 1993, IEEE Computer Applications in Power.

[24]  B. Feeny,et al.  On the physical interpretation of proper orthogonal modes in vibrations , 1998 .

[25]  Joe H. Chow,et al.  Area decomposition for electromechanical models of power systems , 1980, Autom..

[26]  S. Shankar Sastry,et al.  Coherency for interconnected power systems , 1981 .

[27]  P. Varaiya,et al.  Nonlinear oscillations in power systems , 1984 .

[28]  Dan S. Henningson,et al.  Reduced-order models for flow control: balanced models and Koopman modes , 2010 .

[29]  B. Chaudhuri,et al.  Coherency identification in power systems through principal component analysis , 2005, IEEE Transactions on Power Systems.

[30]  Mohammad Shahidehpour,et al.  The IEEE Reliability Test System-1996. A report prepared by the Reliability Test System Task Force of the Application of Probability Methods Subcommittee , 1999 .

[31]  Andrzej Banaszuk,et al.  Comparison of systems with complex behavior , 2004 .

[32]  Pablo A. Parrilo,et al.  Model reduction for analysis of cascading failures in power systems , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[33]  Takashi Hikihara,et al.  Coherent Swing Instability of Power Grids , 2011, J. Nonlinear Sci..

[34]  M. Mackey,et al.  Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics , 1998 .

[35]  G. J. Rogers,et al.  Slow coherency based network partitioning including load buses , 1993 .

[36]  Joe H. Chow,et al.  Time-Scale Modeling of Dynamic Networks with Applications to Power Systems , 1983 .

[37]  Takashi Hikihara,et al.  Global swing instability of multimachine power systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[38]  George Troullinos,et al.  Coherency and Model Reduction: A State Space Point of View , 1989, IEEE Power Engineering Review.

[39]  B. Bowler,et al.  An Analysis of Interarea Dynamics of Multi-Machine Systems , 1981, IEEE Transactions on Power Apparatus and Systems.