1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 17
Numerical Integration
Professor Pete Sauer
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. Example continued
Exact x(.1)
Forward Euler

3. Heun’s

5. Fourth-order Runge-Kutta

12. Roundoff and truncation errors
Consider the truncation error of Forward Euler:
Local Truncation Error (LTE)

13. Approximate LTE:

15. Try on our example (Forward Euler 2nd step)

16. The exact solution at .2
The exact change in the solution from .1  .2

17. The approximate change
The “actual” local truncation error
(-1%)
Note: This is not the “true” LTE because x at the previous time step was not exact

18. (at t = .2 sec)

19. Error in LTE estimate:
(over estimate)
(under estimate)

20. Choosing a step size
Approximate 2nd derivatives:

21. Choose a maximum acceptable LTE of .001

22. Tradeoff between round-off and truncation errors:

23. Implicit methods
Backward Euler (1st order Adams-Moulton)
Trapezoidal (2nd order Adams-Moulton)
Gear’s algorithms

24. Look at polynomial solutions up to order n:
Using multi-steps up to order k:

25. Suppose the exact solution is (n=2):
Find ’s and ’s to give exact answer.

26. Try using no previous times (k = 0)

27. (trapezoidal rule)