1. 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 ECE 576 POWER SYSTEM DYNAMICS AND STABILITY

2. Lecture 41 – Page 1 Introduction 1. Nonlinear model of multimachine system. 2. Linear model of dynamic devices, network and their interconnection. 3. Derivation of 4. Computation of rotor motors (eigenvalues) one at a time (AESOPS). 5. Generalization of AESOPS to compute any desired eigenvalue(s). 6. Voltage stability. *AESOPS= Analysis of Essentially Spontaneous Oscillations in Power Systems *

3. Lecture 41 – Page 2 Nonlinear Model 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. SYSTEM I = YV ~~ ~ D L L L D D

4. Lecture 41 – Page 3 Differential-Algebraic Model Differential Equations

5. Lecture 41 – Page 4 Differential-Algebraic Model (contd) Differential Equations continued

6. Lecture 41 – Page 5 Linearization Linearize around operating point ‘0’ and assuming T mi and V ref,i =constant where

7. Lecture 41 – Page 6 Linearization (contd) Finally From (13) and (14) we can express From (9) size of matrices

8. Lecture 41 – Page 7 Stator Algebraic Equations (-) may be V or I Substitute (12) in (11) for voltage variables V di and V qi. Then linearize to express Stator Algebraic Equations and the network to machine transformation is

9. Lecture 41 – Page 8 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. Lecture 41 – Page 9 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. Lecture 41 – Page 10 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. Lecture 41 – Page 11 Network Equations (contd) Eliminating load buses and separating into real and imaginary parts yields

13. Lecture 41 – Page 12 Overall model where

14. Lecture 41 – Page 13 Compact form More compactly (22), (23), (24) are These equations form the basic math model from which computation of A system 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. Lecture 41 – Page 14 Obtaining A system To obtain A system Equate (26) and (27) to express Substitute (28) in (25) The eigenvalues of A system 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. Lecture 41 – Page 15 Structure of SSSP of EPRI SSSP MASS MAM AESOPS PEALS (Multi Area Small Signal Program) (Program for Eigenvalue Analysis of Large Systems) (Analysis of Essentially Spontaneous Oscillations of Power Systems) (Computes only rotor angle mode) (Modified Arnoldi Method) (Computes up to 5 Eigenvalues at time)

17. Lecture 41 – Page 16 Single Machine Infinite Bus Case ~

18. Lecture 41 – Page 17 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 K A 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. Lecture 41 – Page 18 AESOPS Algorithm In multimachine case we get Change of notation ω s =ω o, K D (s), K s (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 ω ο. Solve ΔT m (s)=0 iteratively. The zeros of ΔT m (s) are obtained by Newton’s method. Assume initial guess as s o. Then Heuristic argument

20. Lecture 41 – Page 19 AESOPS Algorithm (contd) In AESOPS algorithm is very complicated and is approximately 4H e where Suppose eigenvalue of machine #2 is sought. Then Δω ο =ω ο. Next approximation is Generally

21. Lecture 41 – Page 20 How to obtain ΔT m (s) from SSSP model

22. Lecture 41 – Page 21 How to obtain ΔT m (s) from SSSP model

23. Lecture 41 – Page 22 Computing Rotor Angle Mode Suppose we want to compute rotor angle mode of machine 2 (say). Then set Δω 2 =ω ο + and. Apply a torque ΔT m2 only at machine 2. Assume (initial guess), Δω 1 (s) and Δω 3 (s) will be computed. Then equations for machine 2 are

24. Lecture 41 – Page 23 Laplace Domain Since Δω 2 =ω o, Δδ 2 =, then from the third d.e in (43), in Laplace domain Substitute (45) in (44) are known functions of s. From (43)

25. Lecture 41 – Page 24 Laplace Domain (contd) For machines (1) and (2) Substituting in Laplace domain of (50)

26. Lecture 41 – Page 25 Network Equations The network equations are Equating right hand sides of (47) and (53) with those of (54), we get

27. Lecture 41 – Page 26 Algorithm Set n = 0 1. Assume an initial value of complex frequency s n for machine 2. 2. Compute 3. Solve (55) for 4. From (52) compute (by assumption). 5. Compute 6. From (48) compute 7. From (37) and using (38) compute 8. Iterate till