1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 30
Modal Analysis
Professor M.A. Pai
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. Modal Analysis - Comments
Modal analysis or analysis of small signal stability through eigenvalue analysis is at the core of all current software.
In Modal Analysis one looks at:
Eigenvalues
Eigenvectors (left or right)
Participation factors
Mode shape
Power System Stabilizer (PSS) design in a multi-machine context is done using modal analysis method.

3. Eigenvalues, Right Eigenvectors.
x = Ax
Eigenvalues of A are the roots of the characteristic equation:
Assume as distinct (no repeated eigenvalues).
For each eigenvalue there exists an eigenvector such that:
is called a right eigenvector.

4. Left Eigenvectors
For each eigenvalue there exists a left eigen vector such that:
Equivalently, the left eigenvector is the right eigen vector of i.e.

5. Left Eigenvectors (contd)
Right and left eigenvectors are orthogonal i.e.
We can normalize the eigenvectors so that:
In the future we will assume are normalized.

6. Example
Right Eigenvectors

7. Example (contd) Left eigenvectors We would like to make
This can be done in many ways.

8. Example (contd) It can be verified that .
Left and right eigenvectors are used in computing participation factor matrix.

9. Modal Matrices V is known as the Modal Matrix.
This is called a similarity transformation.
V is known as the
Modal Matrix.

10. Modal Matrices (contd)
Transformation
represents magnitude of excitation of mode resulting from initial conditions.

11. Numerical example

12. Numerical example (contd)

13. Numerical example (contd)

14. Mode Shape, Sensitivity and Participation Factors
are original state variables, are transformed variables so that each variable is associated with only one mode.
From (1) Right Eigenvector gives the “mode shape” i.e. relative activity of state variables when a particular mode is excited.
For example the degree of activity of state variable in mode is given by the element of the the Right Eigenvector

15. Mode Shape, Sensitivity and Participation Factors (contd)
The magnitude of elements of give the extent of activities of n state variables in mode and angles of elements (if complex) give phase displacements of the state variables with regard to the mode.
From (2) the Left Eigenvector identifies which combination of original state variables display only the mode.
To summarize:
element of measures activity of in mode.
element weights the contribution of the activity in the mode.

16. Eigenvalue Sensitivity

17. Eigenvalue Sensitivity (contd)

18. Eigenvalue Sensitivity (contd)
Sensitivity of Eigenvalue to the element is equal to product of element of left eigenvector and element of right eigenvector.
If j=k then we get sensitivity with respect to diagonal element