1. Newton-Raphson Power Flow Algorithm
Lecture #20
EEE 574
Dr. Dan Tylavsky
Instructional Objectives
At the end of this lecture:
a) you will be able to describe in lay terms the information contained in an electrical schematic diagram,
b) you will be able to describe the difference between a discrete and a continuous system,
c) you will be able to describe the difference between a digital or binary discrete system and an arbitrary discrete system.

2. Formulate the Newton-Raphson Power-Flog Algorithm
© Copyright 1999 Daniel Tylavsky
Formulate the Newton-Raphson Power-Flog Algorithm
Treat all buses as P-Q type buses.
Handle bus-type switching (i.e., P-Q to P-V and vice versa) by modifying the Jacobian.
Define the PG-PL=P (injected into the bus)

3. The real and reactive power balance equations are:
© Copyright 1999 Daniel Tylavsky
The real and reactive power balance equations are:

4. Working with the real power balance eqn. Taylor’s expansion gives:
© Copyright 1999 Daniel Tylavsky
Working with the real power balance eqn.
Taylor’s expansion gives:

5. © Copyright 1999 Daniel Tylavsky

6. © Copyright 1999 Daniel Tylavsky
Writing the equation for all buses while interleaving the  & V variables gives:
The order of derivative is chosen to be  then V because the  derivative of the P function is not near zero under normal conditions.

7. Apply Taylor’s theorem.
© Copyright 1999 Daniel Tylavsky
We can perform a similar derivation for the reactive power balance equation.
Apply Taylor’s theorem.

8. © Copyright 1999 Daniel Tylavsky

9. © Copyright 1999 Daniel Tylavsky
Writing the  Q equation for all buses while interleaving the  & V variables gives:

10. Finally interleaving the P & Q equations gives:
© Copyright 1999 Daniel Tylavsky
Finally interleaving the P & Q equations gives:

11. Let’s find analytical expressions for each of the Jacobian entries:
© Copyright 1999 Daniel Tylavsky
Let’s find analytical expressions for each of the Jacobian entries:

12. Let’s find analytical expressions for each of the Jacobian entries:
© Copyright 1999 Daniel Tylavsky
Let’s find analytical expressions for each of the Jacobian entries:

13. Recall the definition of QCalc at bus i:
© Copyright 1999 Daniel Tylavsky
Recall the definition of QCalc at bus i:

14. TEAMS: find the remaining derivatives:
© Copyright 1999 Daniel Tylavsky
TEAMS: find the remaining derivatives:

15. Using the H, N, J, L notation we have for the mismatch equation:
© Copyright 1999 Daniel Tylavsky
Using the H, N, J, L notation we have for the mismatch equation:

16. The End