1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 18
Numerical Integration
Professor Pete Sauer
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. Order k (uses k “previous values”):
Multi-step methods
Order k (uses k “previous values”):
When coefficients are chosen properly, this gives an exact answer for any problem whose solution is an nth degree polynomial in t.

3. nth order Adams-Bashforth family (k = n-1):
Explicit methods have
nth order Adams-Bashforth family (k = n-1):
1 = 2 = … = k = 0 (for k  0)
Solve for 0 and the  coefficients (n+1 unknowns) so that the answer is exact for problems with time polynomial solutions of degree n.
n=1 gives 0 = 1 and 0 = 1 (forward Euler)

4. nth order Adams-Moulton family (k=n-2):
Backward Euler
Implicit methods have
nth order Adams-Moulton family (k=n-2):
1 = 2 = … = k = 0 (for k0)
Solve for 0 and the  coefficients (n+1 unknowns) so that the answer is exact for problems with time polynomial solutions of degree n.
n=1 gives 0 = 1 and -1 =1 (backward Euler)

5. Our earlier example (linear)

6. Backward Euler solution with time step of 0.1

7. Exact:

8. Trapezoidal rule
n=2 gives 0 = 1, -1 = ½ and 0 = ½
(Trapezoidal rule)

9. Trapezoidal rule with time step of 0.1

10. Exact:

11. (and better than B. Euler)
Example single time step summary
F. Euler Heun RK B. Euler Trapezoid Exact
Best implicit
Best explicit
(and better than B. Euler)

12. nth order Gear family (k = n-1):
Gear’s algorithms
nth order Gear family (k = n-1):
Solve for the  coefficients and -1 (n+1 unknowns) so that the answer is exact for problems with time polynomial solutions of degree n.
n=1 gives 0 = 1 and -1 =1 (backward Euler)

13. n=2 gives 0 = 4/3, 1 = -1/3, and -1 =2/3

14. Variable time step
Monitor LTE and increase time step when LTE is small
Waveform relaxation: Use different step sizes for different states (i.e. keep some states constant while integrating other states)

15. Differential Algebraic Equations
Where g cannot be explicitly solved for z in terms of x.

16. Partitioned Explicit (PE) solution
Solve
for
Integrate
for one t
Update z by solving
Integrate
for one t
constant

17. Simultaneous Implicit (SI) solution
Backward Euler
Solve for x(ti+t) and z(ti+t) using Newton’s method