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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
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Example continued Exact x(.1) Forward Euler
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Heun’s
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Fourth-order Runge-Kutta
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Roundoff and truncation errors
Consider the truncation error of Forward Euler: Local Truncation Error (LTE)
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Approximate LTE:
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Try on our example (Forward Euler 2nd step)
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The exact solution at .2 The exact change in the solution from .1 .2
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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
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(at t = .2 sec)
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Error in LTE estimate: (over estimate) (under estimate)
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Choosing a step size Approximate 2nd derivatives:
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Choose a maximum acceptable LTE of .001
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Tradeoff between round-off and truncation errors:
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Implicit methods Backward Euler (1st order Adams-Moulton) Trapezoidal (2nd order Adams-Moulton) Gear’s algorithms
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Look at polynomial solutions up to order n:
Using multi-steps up to order k:
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Suppose the exact solution is (n=2):
Find ’s and ’s to give exact answer.
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Try using no previous times (k = 0)
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(trapezoidal rule)
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