Lecture #22 EEE 574 Dr. Dan Tylavsky Newton-Raphson Power Flow Implementation

N-R Power Flow Implementation © 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 Converge Check

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 Optimal Ordering: –Find optimal ordering based on the structure of Y bus 4 Permute Input Data: –Either permute the input data and work with permuted info. OR –When needing input data use bus ordering to retrieve appropriate information. 4 Permute Results for Printing

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 Initial Estimate of bus voltages: –1/0 0 4 Line Flow and Mismatches: –Save line flows for use in Jacobian. 4 Let’s look in detail at the following subprograms –Bus Type Switching –Constructing the Jacobian (Data Structure) –Factorizing the Jacobian –Solving for ,  V

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 Bus Type Switching: –Recall that the expression for reactive power flowing down a transmission line is: j X E/  V/0 0   As |E| increases, so does Q 12. As |E| decreases, -Q 12 increases.

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky –Assuming E is the generator side: If gen. hits max VAR limit, (because of low |V|) release control of |E|. If gen. hits min VAR limit, (because of high |V|) release control of |E|. If gen. (on max VAR limits) has Q 12 <Q 12 max, reacquire control of |E|. If gen. (on min VAR limits) has Q 12 >Q 12 min, reacquire control of |E|.

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 Bus Type Conversion for P-V/P-Q buses: Is PV Bus on VAR Limits? No Is Q G >Qmax? Y Chg to P-Q Q=Qmax Is Q G <Qmin? Y N Chg to P-Q Q=Qmin P-V Type Retained N Yes Is Q g on max limit? Y Y Is |E| >V SP ? Chg P-V |E|=V Sp N P-Q Type Retained N N Is Q G on min limit? Error Chg P-V |E|=V Sp N P-Q Type Retained Y Is |E| <V SP ?

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 The matrix that we constructed assumed all buses were P-Q type buses. 4 For a P-V bus we need the  P eqn. and V k =V k Sp. –Use V k =V k Sp in all equations –Set diagonal entry of  Q k eqn to L kk =1. –Zero  Q k =0. –Zero L ik =0(k  i). –Zero J ik =0, J ii =0.

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 To model bus 2 as a P-V bus we make the following changes: From modified  Q 2 eqn we get: May be set to zero.

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 We can retain the slack bus in the matrix or delete it. If we retain the slack bus: –Use V k =V k Sp in all equations. –Use  k =V k Sp in all equations. –Set diagonal entry of  P k,  Q k eqn to L kk =1, H kk =1. –Zero all other H, N, J, L entries for bus k.

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 To model bus 1 as a slack bus we make the following changes: From modified  P 1  Q 1 eqn we get: May be set to zero.

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 Jacobian Data Structure.

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 LU Factorization. –Symbolic Factorization is unchanged (treating matrix as a partitioned matrix of 2X2 blocks). –Numerical Factorization Replace numerical statements in the factorization routine to: – account for the data structure –AND to account for whether you: »Invert the diagonal 2X2 blocks. (Easier to program but slower to execute.) »Perform LDU factorization of the diagonal blocks. (More complex to program but faster to execute.)

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 Approach #1: Blocks LU factorization. (I.e., inverting the diagonal block. 6 Multiplications 8 Multiplications –Consider # mults. needed to factorize 1st block row. Be careful of mult order!

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky 4 Approach #2: Traditional LU factorization. –Consider # mults. needed to factorize 1st block row. 6 Multiplications

N-R Power Flow Implementation © Copyright 1999 Daniel Tylavsky –Numerical Forward/Backward Substitution Replace numerical statements in the forward backward substitution routine: – to account for the data structure –AND to account for whether you: »Invert the diagonal 2X2 blocks. »Perform LDU factorization of the diagonal blocks.

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