More Related Content


Triangular factorization

  1. TRIANGULAR FACTORIZATION By Syed Zulqadar Hassan CIIT Abbottabad Campus
  2. TRIANGULAR FACTORIZATION • It involves three steps: • Step 1 Triangular Factorization • Step 2 Forward Substitution • Step 3 Back Substitution
  3. TRIANGULAR FACTORIZATION Count… • So we can write • We Know that
  4. • Finally we get TRIANGULAR FACTORIZATION Count…
  5. ADVANTAGES • Solution of a linear system by triangular factorization and subsequent forward and back substitution is very popular because of the many advantages of the method: • Efficiency • Ability to preserve sparsity of the matrix
  6. Sparsity • The fraction of zero elements (non-zero elements) in a matrix is called the sparsity (density).
  7. Sparsity in Power System • Let us analyze the requirements for a 1000 node/2000 branch circuit. • For this network, the admittance matrix Y will have approximately 5000 nonzero elements. The table of factors for this matrix will have 5000Rs nonzero elements. • If Rs = 2.5, then 12,500 nonzero elements need to be stored.
  8. Sparsity in Power System Count… • The sparsity preservation index also impacts the efficiency of the method. • This becomes obvious by considering the fact that the forward and back substitutions require as many multiply-adds as the number of non-zeros entries in the table of factors. • If Rs = 2.5, then only 12,500 multiply-adds are required in the forward and back substitution, a small number compared with the required multiply- adds for the operation inverse of Y. • The inverse of Y have 10,000,000 Multiply-adds while Factorization have 900,000 Multiply-adds.
  9. References • Power System Modeling, Analysis and Control By A. P. Sakis Meliopoulos (Page 18) • • “Triangular Factorization Method for Power Flow Analysis” by Y.Okamoto Published in journal “Electrical Engineering in Japan” Vol 96, No 1, January 1976, pp 31-35