• Solution of a linear system by triangular factorization and subsequent forward
and back substitution is very popular because of the many advantages of the
• Ability to preserve sparsity of the matrix
11. 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
• If Rs = 2.5, then 12,500 nonzero elements need to be stored.
12. 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
• Power System Modeling, Analysis and Control By A. P. Sakis Meliopoulos
• “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