1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 41
Selective Eigenvalue Analysis in Power Systems
Professor M.A. Pai
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. Introduction Nonlinear model of multimachine system.
Linear model of dynamic devices, network and their interconnection.
Derivation of
Computation of rotor motors (eigenvalues) one at a time (AESOPS).
Generalization of AESOPS to compute any desired eigenvalue(s).
Voltage stability.
*AESOPS= Analysis of Essentially Spontaneous Oscillations in Power Systems *

3. Nonlinear Model L D D ~ ~ SYSTEM I = YV L L ~ D
D Dynamic Devices (synchronous machine, induction machines, SVC)
L Constant power, const. imp. or nonlinear voltage dependent loads.
Consider a 3-machine 9-bus system
Syn. M/C: Consider two axis model, i.e., one damper winding in Q axis. Also IEEE Type 1 Exciter. No turbine Governor dynamics. Loads are constant impedance.

4. Differential-Algebraic Model
Differential Equations

5. Differential-Algebraic Model (contd)
Differential Equations continued

6. Linearization
Linearize around operating point ‘0’ and assuming Tmi and Vref,i=constant
where

7. Linearization (contd)
From (9)
From (13) and (14) we can express
size of matrices
Finally

8. Stator Algebraic Equations
and the network to machine transformation is
(-) may be V or I
Substitute (12) in (11) for voltage variables Vdi and Vqi. Then linearize to express

9. Stator Algebraic Equations (contd)
Substituting (13), (15) and (16) in (18) gives the overall structure for linearized syn. model as
For each syn. machine it is

10. Stator Algebraic Equations (contd)
The stator algebraic equations are expressed in network reference frames as follows. Substitute (12) in (11) both for voltage and current variables and then linearize to obtain
(18) and (19) therefore constitute the device equation for each machine.

11. Network Equations The network equations are I = YV
After linearization, static non-linear loads will modify the diagonal elements of Y matrix. We then get

12. Network Equations (contd)
Eliminating load buses and separating into real and imaginary parts yields

13. Overall model
where

14. Compact form More compactly (22), (23), (24) are *
These equations form the basic math model from which computation of Asystem or eigenvalues proceed.
Bench Mark paper “A comprehensive computer program package for S.S. stability analysis of power systems” Kundur et al, IEEE PWRS November 1990. * *

15. Obtaining Asystem To obtain Asystem Equate (26) and (27) to express
Substitute (28) in (25)
The eigenvalues of Asystem can be computed using any Q-R algorithm or any eigenvalue routine. This can be done for small medium sized system on a P.C.

16. Structure of SSSP of EPRI
PEALS
MASS
(Multi Area Small
Signal Program)
(Program for Eigenvalue
Analysis of Large Systems)
AESOPS
MAM
(Analysis of Essentially
Spontaneous Oscillations
of Power Systems)
(Modified Arnoldi Method)
(Computes up to 5
Eigenvalues at time)
(Computes only
rotor angle mode)

17. Single Machine Infinite Bus Case~

18. Single Machine Infinite Bus Case (contd)
Suppose Then with H(s)=0 obtain char. roots of
with the values of s=jω compute H(jω) to compute Re(H(jω)) and Im(H(jω)) to obtain synchronizing and damping torque contribution due to exciter and field winding. In heavily loaded, high KA situation we may have negative damping which leads to oscillation. This can be damped by power system stabilization. This is local mode of oscillation.

19. AESOPS Algorithm In multimachine case we get
Change of notation ωs=ωo , KD(s), Ks(s) ratio of polynomials.
Basis of AESOPS Algorithm
If a particular eigenvalue is to be found, then for that machine Δω 0 but frequency is assumed to be ωο. Solve ΔTm(s)=0 iteratively. The zeros of ΔTm(s) are obtained by Newton’s method. Assume initial guess as so. Then
Heuristic argument

20. AESOPS Algorithm (contd)
Next approximation is
Generally
In AESOPS algorithm is very complicated and is approximately 4He where
Suppose eigenvalue of machine #2 is sought. Then Δωο=ωο .

21. How to obtain ΔTm(s) from SSSP model

22. How to obtain ΔTm(s) from SSSP model

23. Computing Rotor Angle Mode
Suppose we want to compute rotor angle mode of machine 2 (say). Then set Δω2=ωο+ and Apply a torque ΔTm2 only at machine 2. Assume (initial guess), Δω1(s) and Δω3(s) will be computed. Then equations for machine 2 are

24. Laplace Domain
Since Δω2=ωo, Δδ2= , then from the third d.e in (43), in Laplace domain
Substitute (45) in (44)
are known functions of s. From (43)

25. Laplace Domain (contd)
For machines (1) and (2)
Substituting in Laplace domain of (50)

26. Network Equations The network equations are
Equating right hand sides of (47) and (53) with
those of (54), we get

27. Algorithm
Set n = 0
Assume an initial value of complex frequency sn for machine 2.
Compute
Solve (55) for
From (52) compute (by assumption).
From (48) compute
From (37) and using (38) compute
Iterate till