Lecture #21 EEE 574 Dr. Dan Tylavsky Newton-Raphson Power Flow Algorithm II

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky 4 During the last lecture, you (we) derived the functional form of the Jacobian entries:

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky 4 We laid out the mismatch equation ordering  P then  Q and ordered the variable as  then  V to use LU factorization w/o pivoting:

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky 4 Recall that to avoid pivoting we needed positive definiteness –numerically intensive to prove 4 Or diagonal dominance –easy to prove, but an overly restrictive requirement. 4 Let’s investigate the approximate magnitude of each Jacobian entry.

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky 4 From our experience with power system data we know that for most branches: 4 Also at the start of the iteration process:

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky 4 Entering these approximations into our Jacobian we get: Jacobian is close diagonally dominant and it is not surprising that no pivoting is required. Changing the order of the eqn’s or variables will destroy the near diagonal dominance property.

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky –Think-Pair-Square: Construct the Jacobian matrix for the following power system assuming bus 1 is the slack bus, all bus voltages are approximately 1/0 0 and the generator bus shown is not on VAR limits j0.06  P.U. All line charging susceptances are given by j0.02

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky Determine optimal ordering for minimal fill. Permute input data. Assume Flat Bus Voltage Profile E=1/0 0 Assume Gen’s not on VAR limits. Iteration Index=q=0 Calculate Line Flows, Gen. Power and Mismatches Is q>3? Y Did buses Switch Types? N Perform bus type switching Converged? |  P q max|, |  Q q max |<  ? N Calculate Jacobian Entries and Solve Mismatch Eqn. Read Input, Construct Y bus Update Bus Voltage V q+1 =V q +  V q,  q+1 =  q +   q q=q+1 Y Create Output N Y

N-R Power Flow Algorithm II © Copyright 1999 Daniel Tylavsky Main Steering Routine Read Input SolveWrite Output Optimal Order Permute Input Data Initial Estimate of Bus Voltages Line Flows & Mismatches Bus Type Switching Construct Jacobian Factorize Jacobian Solve for  q,  V q Update  q+1, V q+1 Permute Output

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