1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 29
Small Signal Analysis (SSA) of Power Systems
Professor M.A. Pai
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. References Basic Text References
P.W. Sauer & M.A. Pai ‘Power system dynamics and stability’. Prentice Hall 1998.
References
P.Kundur. ‘Power system stability and control’. Mc Graw Hill 1994.
K.R. Padiyar. ‘Power system dynamics and control’ John Wiley and sons (Asia) 1996.
M. Ilic and J. Zaborszky. ‘Dynamics and control of large electric power systems’ Wiley – Interscience 2000.
Graham Rogers. ‘Power system oscillations’ Kluwer Academic Publishers 2000.
“Analysis & control of P.S. oscillations”. CIGRE Committee 38 Task Force 07 of Advisory group 01, Dec 1996.

3. Oscillations – Why SSA?
Problems of Low Frequency Oscillations Cause ? Unknown
Local Model 1-3 Hz.
Inter Area Oscillations Hz.
Torsional Mode Oscillations Associated with T-G Rotational Components.
Control Mode Oscillation Due to poorly tuned Power System Stabilizers
not to be discussed ~
long line
Area 1
Area 2
Tie line * * *

4. Techniques for SSA Both DAE and DE models are used.
In DAE models, the network equations can be either
- Power Balance Form OR
-Current Balance Form
Industry favors a numerical approach (DAE).
In DAE Models, after linearizing, the algebraic equations are eliminated to compute eigen values.
Eigen values, eigen vectors of the linear system give lots of insight.

5. Basic Linearization Technique
Power Balance form
The vector y includes both the and vectors.
(1) is of dimension 7m and (2) is of dimension 2(n+m)
Bus 1 is slack, 2…m are P-V, m+1…m are P-Q type.
This partitioning is done to separate load flow variables into vector .Vector consists of other algebraic variables.
Objective is to show presence of bifurcation phenomena in voltage stability.

6. Linearization Linearizing Eliminate to get where .
is the Load Flow Jacobian.
is the Algebraic Jacobian (Note: contains stator algebraic variables!)
Define

7. 2 axis model &
IEEE Type I
Exciter
Linearized
Model
i = 1,2 … m

8. Linearization (contd)
The m machine system
where are block diagonal matrices.
These are the D.E’s.
Now the Stator and Algebraic equations.

9. Linearization – Stator Equations
Compactly
In matrix notation

10. Linearization of Network equations
Power Balance
Equations at
all Gen. Buses
i=1,2 … m
Power Balance
Equations at
all P-Q Buses
i=m+1,… n

11. Generator Equations (Power Balance)
After
linearization
The various submatrices can be easily identified. The elements are partial derivatives evaluated at the operating point.
Matrix form:
Note:

12. P-Q Bus Power Balance Equations
Matrix Form:
Note:

13. Overall DAE Model
Systematic elimination now follows to obtain a reduced DE model only.

14. Reducing DAE Model to DE Model
Step 1
From eqn.(2)
Step 2
Substitute in (3)
involves taking a series of 2x2 inverses of
Equation (6) becomes

15. Reducing DAE Model Step 3 Eliminate from (1).
Step4 Reorder the variables in as follows:
Step 5 Equations (8) – (10) are

16. Reducing DAE Model (contd)
Voltage dependency of loads is absorbed in Diagonal elements of which is structurally similar to Load Flow Jacobian
This model is used in voltage stability studies to see effect of loads on eigen values of
(Algebraic and Load Flow Jacobian)

17. Reference angles
As in non-linear simulation, the order of D.E can be reduced to 7m-1 instead of 7m.
This mainly suppresses the zero eigen value in the original system.
To do it systematically proceed as follows.
Select one machine angle as reference angle.

18. Reference angles (contd)
Define
Angles appear as differences.
Retain the notation
Bus angles

19. Reference angles (contd)
Delete D.E corresponding to in (11).
Delete the column corresponding to in in (11).
Replace
This means “mechanically” put a-1 in the inter sections of the rows corresponding to and corresponding to

20. Numerical example
Two machine system (classical model)

21. Numerical example (contd)
Linearizing around

22. Numerical example (contd)
In state space form

23. Numerical example (contd)
Choose as reference.
Define