ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 2 System Structures and Electromagnetic Transients Professor Pete Sauer Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

System dynamic structure

Dynamic phenomena time scales

Two-time-scale system Slow time scale t (sec) Fast time scale

 = 0 in t scale  = 0 in  scale

System organizational structure

Voltage-levels, control, stability Frequency-control, stability Phenomena of interest Voltage-levels, control, stability Frequency-control, stability Current-thermal limits Relay action Overall S.S. stability Overall transient stability

Electromagnetic transients Transmission line Long, medium, short Partial D.E. Ordinary D.E. Phasors

Electromagnetic fields

Transmission line segment

Special case #1 (neglect shunts)

Special case #2 The lossless line wave equation

General solution

Terminal conditions Receiving end m is x = 0 Use this to eliminate f1

Look at sending end at negative time Use this to eliminate f2

Eliminating f1 and f2 gives:

Repeat this for the other end to get:

Sinusoidal steady state equivalent circuit

Numerical integration EMTP solutions Numerical integration Find: y(t)

Euler’s forward (Explicit method) Trapezoidal rule (Implicit method)

Example

Trapezoidal rule solution

For R = 200, L=0.3H, and the following voltage applied at time zero, with the exact time-domain solution for the current is:

Trapezoidal rule solution Exact solution:

t = .0002 i(.0002) = 117.3A Exact solution i(.0002) = 117.3A Solving for i(.0002) i(.0002) = 117.3A Exact solution i(.0002) = 117.3A