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.

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)

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

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:

© Copyright 1999 Daniel Tylavsky

© 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.

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

© Copyright 1999 Daniel Tylavsky

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

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

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:

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:

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

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

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:

The End