1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 40
Structure Preserving Energy Function
Angle and Voltage Stability
Professor M.A. Pai
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. Angle Stability Model With classical model for machines we have~ ~
Trans.
Network
m Machines
n Physical buses ~
Model
With classical model for machines we have
and 2n network equations
swing equations (m)

3. Trans Network 1 2 m m+1 m+(m+1) - + m+(m+2) m+2 - + Trans. Network m+m
(m+n)
- +

4. Trans Network (contd)
Let rotor angles, velocities and bus angles with respect to syn. ref. frame be
Define
Internal voltages
Network voltages

5. Trans Network (contd) Augmented includes transient reactances.
Swing eqns.
Actually the R.H. side is i
i+m

6. Network Equations Let load be represented as i Network equations
Swing equations plus (1) and (2) constitute a differential-algebraic model (DAE)
Assume for all i so that there is no frequency dependent load. i

7. Objective To construct the first integral of motion for the system ie
is the first integral or energy function.
Introduction of C.O.I. facilitates constructing V. Define

8. Swing Equations Swing equations become Where
Steps to get the first integral
1. Multiply (1) by and sum over m equations

9. Swing Equations (contd)
2. Multiply real power flow equations (slide 4) by and sum over all n physical buses.
3. Divide reactive power equation (slide 4) by
Multiply (2) by and sum over n physical buses

10. Swing Equations (contd)
(1), (2) and (3) of previous slide and (3) of slide # 7 give
Problem term!
Consider simpler case constant. Then,

11. Swing Equations (contd)
The first integral is V=V+K where K= constant.
K is chosen so that V=0 at s.e.p. Thus,
Thus

12. Swing Equations (contd)
since
It can be shown that
Define a vector
int. nodes
physical buses
= injected power at nodes

13. Swing Equations (contd)
Then
This is a compact expression relates to reactive power in network.

14. Voltage Stability – Use of Energy Functions
As the system gets loaded Q reserves in the system are not sufficient locally or globally.
Hence there is voltage depression and if not controlled may lead to voltage instability or collapse.
Two possibilities
System gets progressively loaded till the point of voltage collapse.
System presently is stable but close to the point of voltage collapse. A small disturbance will then lead to voltage collapse.

15. What is the Role of System and Load Dynamics? View point 1
Buses
View point 1
Since voltage stability is a slow phenomenon both S and L are considered static. Hence voltage stability is a static problem. Techniques
1. Det. J. (J=load flow Jacobian)
2. Min. sing. value of J
3. Condition number of J
4. Continuation method to find saddle node bifurcation
5. Use of sensitivities from OPF

16. View Point 2
Since voltage profile is controlled by excitation dynamics, modeling of system dynamics is important.
Finding both saddle node and Hopf bifurcation by eigenvalue monitoring.
Sensitivity of voltage collapse due to injected Q at GEN, SVC, TAP-changer dynamics, etc.

17. View Point 4
All of the above! There is consensus that some sort of dynamics does play a part.
Hence, an energy function approach is desirable (Demarco, Overbye)
Single machine case ~
SEP – high voltage solution
UEP – low voltage solution

18. View Point 3
Load dynamics are crucial in determining voltage collapse.
Load modeling in a dynamic sense (D.E.’s for motors, etc)
Fast changing or cyclic loads.

19. Voltage Energy Margin For a slowly varying load Hence
(energy at UEP)-(energy at SEP) voltage energy margin (VEM)
For a slowly varying load
Hence
Hence derived from SPM (transient stability).
and assume UEPV values. Hence,

20. Voltage Energy Margin (contd)
As in angle stability we need efficient algorithms to find UEP solutions.
VEM a
Voltage collapse point
If VEM>0, system
is voltage stable a
Voltage collapse point

21. Conclusion
A unified picture is now possible for both angle and voltage stability using same energy function. Effects of dynamic modeling on system and load size are needed to be studied for both angle and voltage stability.
For angle stability with s.p. models, controlling u.e.p. is not defined easily. However PEBS method works.
Sensitivity analysis of voltage energy margin may provide clues to preventing voltage collapse.

22. Coherency, synchrony, slow coherency
Podmore, R. ‘Identification of coherent generators for dynamic equivalents’ IEEE Trans. Power Appar. Syst. Vol PAS-97 (July/August 1978) pp
Pai, M A and Adgaonkar, R P. ‘Identification of coherent generators using weighted eigenvectors’ Proc. IEEE PES Winter Meeting New York, USA A (February 1979).
Slow coherency
Chow, JH (Ed.). ‘Time scale modeling of dynamic networks with applications to power systems’ Lecture notes in control and information sciences Vol 46, SpringerVerlag,USA(1982).
Synchrony
4. G.Ramaswamy, G.C. Verghese, Luis Rouco, C. Vialas, C.L. Demarco. ‘Synchrony, aggression and multi-area eigenvalues analysis’ IEEE Trans. Power Systems 1995.

23. Linearizing Swing Equations
Assume faulted period of the order of few cycles.
Assume as step input=accelerating power at
Put (1) in state space form

24. Linearizing Swing Equations (contd)
Solution of post-fault system
Mi are right eigenvectors of A
ki determined from values of X(t) at t = tcl and left eigenvectors of A.
There are (n-1) complex pairs of eigenvalues and one negative real eigenvalue pil and qil represent amplitudes of complex modes.

25. Coherency and Synchrony
Two generators i and j are classified as coherent if where is a small number.
dij is the norm of the row vectors corresponding to δi and δj in the right eigenvector matrix M.
Synchrony
Uses the angle between the two row vectors as the criterion.