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

2. 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:

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

4. 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:

5. 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.

6. 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:

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

8. 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.

9. 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. 1 2 3 0.02+j0.06  P.U. All line charging susceptances are given by j0.02

10. 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

11. 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

12. The End