1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 14
Damping torques and steady-state initial conditions
Professor Pete Sauer
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. Damping torques
Friction and windage
Stator currents (load)
Field current (excitation)
Damper windings

3. Look for an integral manifold:

7. Substitute this into the torque equation
Substitute this into the torque equation. If the time constant is zero (zero-order integral manifold), you get the torque equation of the one-axis model. Since we are keeping the next term (first-order integral manifold), there is an additional torque term:

8. This can be approximated by a “constant D” damping torque term:
That is: using a zero-order integral manifold does not capture any of the effects of the damper winding. However, a first-order integral manifold does capture some of the damper winding effects.

9. Steady state Assume rated speed:
Assume open circuit with no saturation.
Assume the terminal voltage is 1.0 per unit.
Using the full model:
The + and – come from the terminal voltage square root.
Choose the positive solution

11. Apply resistive load Change:
Id and Iq cannot change instantaneously but, Vd and Vq can. They will change from 0 and 1to 0 and 0 when the resistive load is added.
Could find the new steady state condition by integrating until things come to rest. What about an analytical steady-state solution?

12. New steady state (assuming it is stable).
Using the full model, solve:

13. (With Vref given from the open-circuit solution)

14. (With Pc given from the open-circuit solution)
(Assume TFW = 0)

15. 20 equations in 20 unknowns – does the solution require iteration?
After the solution is found, the angle delta is a time varying state:
The steady-state value of delta cannot be found without integrating the full set of equations.

16. Recall the transformation back to abc and the definition of delta:
The machine will have a new steady-state frequency.

17. General steady state when =s

18. Steady-state phasor diagram

21. Example

22. Find: all synchronous machine dynamic states

27. Check:

28. (close to .615)