1. Lecture #23 EEE 574 Dr. Dan Tylavsky Decoupled Power Flow Algorithms

2. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky 4 We know that for the full Newton power flow we interleave  P,  P and ,  Vas shown.

3. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky –An alternate ordering which will still preserve the quasi-diagonal dominance property is: –Symbolically we can write this as: H N L J VV PP QQ –The draw back to this ordering is the increased fill.

4. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky 4 Consider the following 3 bus system: –Conventional Ordering: –No Fill –New Ordering: –Much Fill

5. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky 4 Claim: –There exists only weak coupling between: –  P and  V, –  Q and  ; –(Said another way, changes in  P have little effect on  Q and vice versa.) –hence N and J can be ignore. Recall the Jacobian is a linear approximation, ignoring N and J, simply makes the approximation less accurate.

6. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky For typical power system branches: – X/R >> 1. –  ik <20 0. Let’s investigate how this allows us to approximate the real and reactive power flow equations. –Let’s show that this approximation is reasonable. –Recall the equations for power flow through a transmission line: R ik +j X ik Vi/iVi/i V/  k i k P ik +j Q ik

7. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky –Starting with the real power flow equation: For X/R >> 1: 0 0 For  ik < 20 0 : (  i -  i)

8. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky –For the reactive power flow equation: For  ik < 20 0 : 1 For X/R >> 1: 0 (Recall B ik <0)

9. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky –These equations imply:

10. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky –The decoupled equations become: –Where: –There are various ways of handling the iteration scheme. A popular way is:

11. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky Y Begin as with Newton-Raphson q=0 Is q>3? Y Perform bus type switching Did buses Switch Types? N Converged? |  P q max|, |  Q q max |<  ? Y Create Output N N Solve Update Bus Angle  q+1 =  q +   q N Is q>3? Y Perform bus type switching Did buses Switch Types? N Converged? |  P q max|, |  Q q max |<  ? Y Create Output N N Solve Update Bus Angle V q+1 = V q +  V q N q=q+1

12. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky –The decoupled algorithm looses quadratic convergence and resorts to linear (geometric convergence. Convergence Tolerance Decoupled Full Newton (-Raphson) Log(max(  P,  )) Iteration Number

13. Decoupled P.F. Algorithms © Copyright 1999 Daniel Tylavsky –The advantages of the decoupled algorithm: Less calculations: –Full Newton O(N 3 /3) –Decoupled O(2*(N/2) 3 /3)=O(N 3 /12) –Disadvantages: Convergence is unreliable. –Improved Convergence through Fast Decoupled Power Flow Algorithm.

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