Gramian-Based Reduction Method Applied to Large Sparse Power System Descriptor Models

This paper presents an efficient linear system reduction method that computes approximations to the controllability and observability gramians of large sparse power system descriptor models. The method exploits the fact that a Lyapunov equation solution can be decomposed into low-rank Choleski factors, which are determined by the alternating direction implicit (ADI) method. Advantages of the method are that it operates on the sparse descriptor matrices and does not require the computation of spectral projections onto deflating subspaces of finite eigenvalues, which are needed by other techniques applied to descriptor models. The gramians, which are never explicitly formed, are used to compute reduced-order state-space models for large-scale systems. Numerical results for small-signal stability power system descriptor models show that the new method is more efficient than other existing approaches.

[1]  R. Kamyar,et al.  Solving Large-Scale Robust Control Problems by Exploiting the Parallel Struc- ture of Polya’s Theorem , 1800 .

[2]  H. H. Happ,et al.  Power System Control and Stability , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[3]  Gene H. Golub,et al.  Matrix computations , 1983 .

[4]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[5]  N. Martins.  Efficient Eigenvalue and Frequency Response Methods Applied to Power System Small-Signal Stability Studies , 1986, IEEE Transactions on Power Systems.

[6]  John F. Dorsey,et al.  Reducing the order of very large power system models , 1988 .

[7]  E. Wachspress.  Iterative solution of the Lyapunov matrix equation , 1988 .

[8]  Hartmut Logemann,et al.  Multivariable feedback design : J. M. Maciejowski , 1991, Autom..

[9]  Joshua R. Smith,et al.  Transfer function identification in power system applications , 1993 .

[10]  N. Martins,et al.  New methods for fast small-signal stability assessment of large scale power systems , 1995 .

[11]  A. Semlyen,et al.  Efficient calculation of critical eigenvalue clusters in the small signal stability analysis of large power systems , 1995 .

[12]  Joe H. Chow,et al.  Power system reduction to simplify the design of damping controllers for interarea oscillations , 1996 .

[13]  Roland W. Freund,et al.  Small-Signal Circuit Analysis and Sensitivity Computations with the PVL Algorithm , 1996 .

[14]  Nelson Martins,et al.  Computing dominant poles of power system transfer functions , 1996 .

[15]  A.S. Costa,et al.  Computationally efficient optimal control methods applied to power systems , 1997, Proceedings of the 20th International Conference on Power Industry Computer Applications.

[16]  J. H. Chow,et al.  Computation of power system low-order models from time domain simulations using a Hankel matrix , 1997 .

[17]  Thilo Penzl,et al.  A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations , 1998, SIAM J. Sci. Comput..

[18]  D. Sorensen,et al.  A Survey of Model Reduction Methods for Large-Scale Systems , 2000 .

[19]  R. Freund.  Krylov-subspace methods for reduced-order modeling in circuit simulation , 2000 .

[20]  Jing-Rebecca Li.  Model reduction of large linear systems via low rank system gramians , 2000 .

[21]  Thilo Penzl.  LYAPACK A MATLAB Toolbox for Large Lyapunov and Riccati Equations , Model Reduction Problems , and Linear – Quadratic Optimal Control Problems Users , 2000 .

[22]  D. Sorensen,et al.  Approximation of large-scale dynamical systems: an overview , 2004 .

[23]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[24]  Y. Zhou,et al.  On the decay rate of Hankel singular values and related issues , 2002, Syst. Control. Lett..

[25]  T. Stykel.  Analysis and Numerical Solution of Generalized Lyapunov Equations , 2002 .

[26]  Tatjana Stykel,et al.  Gramian-Based Model Reduction for Descriptor Systems , 2004, Math. Control. Signals Syst..

[27]  Jacob K. White,et al.  Low-Rank Solution of Lyapunov Equations , 2004, SIAM Rev..

[28]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[29]  A. de Castro,et al.  Utilizing transfer function modal equivalents of low-order for the design of power oscillation damping controllers in large power systems , 2005, IEEE Power Engineering Society General Meeting, 2005.

[30]  Paul Van Dooren,et al.  Model reduction of state space systems via an implicitly restarted Lanczos method , 1996, Numerical Algorithms.

[31]  T. Stykel.  Low rank iterative methods for projected generalized Lyapunov equations , 2005 .

[32]  M.A. Pai,et al.  Model reduction in power systems using Krylov subspace methods , 2005, IEEE Transactions on Power Systems.

[33]  N. Martins,et al.  Efficient computation of transfer function dominant poles using subspace acceleration , 2006, IEEE Transactions on Power Systems.

[34]  Peter Benner,et al.  Numerical Linear Algebra for Model Reduction in Control and Simulation , 2006 .

[35]  Peter Russer,et al.  System identification and model order reduction for TLM analysis , 2007 .

[36]  Ulrike Baur,et al.  Control-Oriented Model Reduction for Parabolic Systems , 2008 .

[37]  M. T. Qureshi,et al.  Lyapunov Matrix Equation in System Stability and Control , 2008 .