Multifrontal Solver for Online Power System Time-Domain Simulation

This paper proposes the application of unsymmetric multifrontal method to solve the differential algebraic equations (DAE) encountered in the power system dynamic simulation. The proposed method achieves great computational efficiency as compared to the conventional Gaussian elimination methods and other linear sparse solvers due to the inherent parallel hierarchy present in the multifrontal methods. Multifrontal methods transform or reorganize the task of factorizing a large sparse matrix into a sequence of partial factorization of smaller dense frontal matrices which utilize the efficient Basic linear algebra subprograms 3 (BLAS 3) for dense matrix kernels. The proposed method is compared with the full Gaussian elimination methods and other direct sparse solvers on test systems and the results are reported.

[1]  X. Li Direct Solvers for Sparse Matrices , 2006 .

[2]  I. Duff,et al.  Multifrontal QR Factorization in a Multiprocessor Environment , 1996 .

[3]  Michael T. Heath,et al.  A Cartesian Parallel Nested Dissection Algorithm , 1992, SIAM J. Matrix Anal. Appl..

[4]  J.J. Sanchez-Gasca,et al.  Variable time step, implicit integration for extended-term power system dynamic simulation , 1995, Proceedings of Power Industry Computer Applications Conference.

[5]  A. Bihain,et al.  The mixed Adams-BDF variable step size algorithm to simulate transient and long term phenomena in power systems , 1994 .

[6]  I. Duff,et al.  The Use of multiple fronts in Gaussian elimination , 1994 .

[7]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[8]  Jeremy Johnson,et al.  PERFORMANCE ANALYSIS OF LOAD FLOW COMPUTATION USING FPGA 1 , 2005 .

[9]  Fernando L. Alvarado,et al.  Testing of Trapezoidal Integration With Damping for the Solution of Power Transient Problems , 1983, IEEE Power Engineering Review.

[10]  R. Bacher,et al.  Using automatic code differentiation in power flow algorithms , 1999 .

[11]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[12]  Timothy A. Davis,et al.  Algorithm 837: AMD, an approximate minimum degree ordering algorithm , 2004, TOMS.

[13]  Probability Subcommittee,et al.  IEEE Reliability Test System , 1979, IEEE Transactions on Power Apparatus and Systems.

[14]  Timothy A. Davis,et al.  Algorithm 836: COLAMD, a column approximate minimum degree ordering algorithm , 2004, TOMS.

[15]  M. SIAMJ.,et al.  IMPROVED SYMBOLIC AND NUMERICAL FACTORIZATION ALGORITHMS FOR UNSYMMETRIC SPARSE MATRICES , 2002 .

[16]  Timothy A. Davis,et al.  A column approximate minimum degree ordering algorithm , 2000, TOMS.

[17]  James S. T'ien,et al.  Development of Direct Multifrontal Solvers for Combustion Problems , 2008 .

[18]  Vipin Kumar,et al.  PSPASES: An Efficient and Scalable Parallel Sparse Direct Solver , 1999, PPSC.

[19]  Jack J. Dongarra,et al.  A set of level 3 basic linear algebra subprograms , 1990, TOMS.

[20]  P. Hood,et al.  Frontal solution program for unsymmetric matrices , 1976 .

[21]  Iain S. Duff A review of frontal methods for solving linear systems , 1996 .

[22]  Iain S. Duff,et al.  The Multifrontal Solution of Unsymmetric Sets of Linear Equations , 1984 .

[23]  Timothy A. Davis,et al.  A combined unifrontal/multifrontal method for unsymmetric sparse matrices , 1999, TOMS.

[24]  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 .

[25]  Eliza Varney,et al.  Institute Of Electrical And Electronic Engineers, Inc , 2010 .

[26]  Iain S. Duff,et al.  A Note on the Work Involved in No-fill Sparse Matrix Factorization , 1983 .

[27]  John R. Gilbert,et al.  Sparse Matrices in MATLAB: Design and Implementation , 1992, SIAM J. Matrix Anal. Appl..

[28]  Pontus Matstoms,et al.  Sparse QR factorization in MATLAB , 1994, TOMS.

[29]  Bruce M. Irons,et al.  A frontal solution program for finite element analysis , 1970 .

[30]  M. Berzins,et al.  An adaptive theta method for the solution of stiff and nonstiff differential equations , 1992 .

[31]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[32]  Timothy A. Davis,et al.  A column pre-ordering strategy for the unsymmetric-pattern multifrontal method , 2004, TOMS.

[33]  Timothy A. Davis,et al.  An Unsymmetric-pattern Multifrontal Method for Sparse Lu Factorization , 1993 .

[34]  Jennifer A. Scott,et al.  The design of a new frontal code for solving sparse, unsymmetric systems , 1996, TOMS.

[35]  Steven M. Hadfield On The LU Factorization Of Sequences Of Identically Structured Sparse Matrices Within A Distributed , 1994 .

[36]  Anshul Gupta,et al.  Recent advances in direct methods for solving unsymmetric sparse systems of linear equations , 2002, TOMS.

[37]  Iain S. Duff,et al.  MA57---a code for the solution of sparse symmetric definite and indefinite systems , 2004, TOMS.

[38]  John K. Reid,et al.  The design of MA48: a code for the direct solution of sparse unsymmetric linear systems of equations , 1996, TOMS.

[39]  Jennifer A. Scott,et al.  MA42 - A new frontal code for solving sparse unsymmetric systems , 1993 .

[40]  Steven M. Hadfield,et al.  THE USE OF GRAPH THEORY IN A PARALLEL MULTIFRONTAL METHOD FOR SEQUENCES OF UNSYMMETRIC PATTERN SPARSE MATRICES , 1995 .

[41]  Joseph W. H. Liu,et al.  The Multifrontal Method for Sparse Matrix Solution: Theory and Practice , 1992, SIAM Rev..

[42]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[43]  Qiming Chen,et al.  The probability, identification, and prevention of rare events in power systems , 2004 .

[44]  Patrick Amestoy,et al.  Vectorization of a Multiprocessor Multifrontal Code , 1989, Int. J. High Perform. Comput. Appl..

[45]  Patrick R. Amestoy,et al.  An unsymmetrized multifrontal LU factorization , 2000 .

[46]  Roger Grimes,et al.  The influence of relaxed supernode partitions on the multifrontal method , 1989, TOMS.

[47]  Iain S. Duff,et al.  MA27 -- A set of Fortran subroutines for solving sparse symmetric sets of linear equations , 1982 .

[48]  Jennifer A. Scott,et al.  A Comparison of frontal software with other sparse direct solvers , 1997 .