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

Example continued Exact x(.1) Forward Euler

Heun’s

Fourth-order Runge-Kutta

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

Approximate LTE:

Try on our example (Forward Euler 2nd step)

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

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

(at t = .2 sec)

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

Choosing a step size Approximate 2nd derivatives:

Choose a maximum acceptable LTE of .001

Tradeoff between round-off and truncation errors:

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

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

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

Try using no previous times (k = 0)

(trapezoidal rule)