ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 19 Multi-machine Dynamic Models Professor Pete Sauer Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

Multi-machine dynamic models m – synchronous machines n – system nodes (buses) (excluding reference) b - branches

The center of inertia speed The synchronous rotating reference frame Common Reference Frame One of the machines The center of inertia speed The synchronous rotating reference frame

The synchronous rotating reference frame

To see the reason for this choice, consider:

Suppose we have VDi, VQi as time varying quantities with , then:

Define

Then:

If the steady-state values are: then,

(keep Idi, Iqi in rest of model)

Transformation between reference frames

Balanced 3-phase line

Network transients

m=1 (one machine) n=3 (three buses) b=4 (four branches)

Branch-loop incidence matrix

Systematically create independent loops: b-n independent loops Systematically create independent loops: Form a tree Add b-n links to form b-n loops This creates the “basic loop” matrix cb (b x b-n)

A tree: all nodes, but no loops

Add one tree link (branch c) and orient the loop with the direction of link.

Add second tree link (branch d) and orient the loop with the direction of the link (new loop may not go through an old link).