1. ECE 576 POWER SYSTEM DYNAMICS AND STABILITY
Lecture 35
Direct Methods for Stability Analysis
Professor M.A. Pai
Department of Electrical and Computer Engineering
© 2000 University of Illinois Board of Trustees, All Rights Reserved

2. Stability Phenomena and Tools
Large Disturbance Stability (Non-linear Model)
Small Disturbance Stability (Linear Model)
Structural Stability (Non-linear Model) Loss of stability due to parameter variations.
Tools
Simulation
Repetitive time-domain simulations are required to find critical parameter values, such as clearing time of circuit breakers.
Direct methods using Lyapunov-based theory (Also called Transient Energy Function (TEF) methods)
Sensitivity based methods.

3. TEF Techniques No repeated simulations are involved.
Limited somewhat by modeling complexity.
Energy of the system used as Lyapunov function.
Computing energy at the “controlling” unstable equilibrium point (CUEP) (critical energy).
CUEP defines the mode of instability for a particular fault.
Computing critical energy is not easy.

4. Judging Stability / Instability
Monitor Rotor Angles
(a) Stable
(b) Stable
(d) Unstable
(c) Unstable
Stability is judged by Relative Rotor Angles.

5. Mathematical Formulation
Power System undergoing disturbance (fault etc) followed by clearing of the fault has the following model
(1) PRIOR TO FAULT (Pre - fault)
(2) DURING THE FAULT (FAULT - ON OR FAULTED)
(3) AFTER THE FAULT (POST - FAULT)
state vector, is clearing time of circuit breaker. X X
Post-Fault
(line-cleared)
Faulted

6. Critical Clearing Time
In the pre fault state the system would have reached a steady state.
Hence is known. The pre fault dynamics are of no interest. .
Two systems FAULTED (1) AND POST-FAULT (2).
Initial Conditions for (2) are provided by the solution of (1) evaluated at

7. Critical Clearing Time (contd)
Assume post - fault system has a stable equilibrium point
All possible values of for differing clearing times, provide initial conditions for the post - fault system.
Will the trajectory of the post fault system starting at converge to
Largest value of (called ) for which this is true is called critical clearing time.
is different for different faults.
Let us view this pictorially:

8. Region of Attraction (RoA)
All faulted trajectories cleared before they reach boundary
of RoA will tend to
The region need not be closed. It can be open: . .

9. Methods to Compute RoA
Topic of intense research in P.S. literature since early 60’s.
stable equilibrium point (s.e.p.) of post - fault system
is generally close to pre-fault s.e.p
Surrounding this s.e.p there are a number of unstable equilibrium points (u.e.p).
Boundary of RoA is characterized via these u.e.p’s

10. Characterization of RoA
Define an energy function of the post-fault system.
Compute
over all i is one possible choice for
RoA is defined by
Extremely conservative result.
Alternative method: Depending on the fault, identify THE towards which faulted trajectory is headed. Call it (controlling u.e.p)
Then is a good estimate of RoA.

11. Lyapunov’s Method
If there exists a scalar function such that and around equilibrium point ‘0’ and then equilibrium ‘0’ is asymptotically stable. .
Thus enters directly in the computation of

12. Lyapunov’s Method in P.S.
condition can be relaxed to provided along any other solution except x=0.
This is an important aspect of the Lyapunov theory.
Early application of Lyapunov’s Method in Power Systems
Gless – Aylett
Magnusson – El - Abiad - Nagappan
Many research papers after 1964.
Transient Energy Function (TEF).
1940 – 1960

13. Multimachine internal node model

14. Constructing TEF Relative rotor angle formulation.
COI reference frame.
It is preferable since we measure angles with respect to the “mean motion” of the system.
TEF for conservative system
and the center of speed as
where
We then transform the variables to the
COI variables as
It is easy to verify

15. TEF (contd) The swing equations with become (omitting the algebra):
If one of the machines is an infinite bus, say, m whose inertia constant is very large, then and also and
The COI variables become
In the literature is simply taken as zero.
Equation is modified accordingly, and there will be only (m-1) equations after omitting the equation for machine m.

16. TEF (contd)
We consider the general case in which all are finite. We have two sets of differential equations:
Let the post fault system (2) have the stable equilibrium point at
is obtained by solving the nonlinear algebraic eqns:
Since can be expressed in terms of the other and substituted in (3), which is then equivalent to:
and

17. TEF (contd) Steps for computing the critical clearing time are:
Construct an energy or Lyapunov function for the post-fault system.
Find the critical value of for a given fault denoted by
Integrate the faulted equations, until This instant of time is called the critical clearing time .
Most of the methods differ as to how to implement steps 2 and 3.

18. TEF (contd)
Integrating the pairs of equations for each machine between
the post-fault s.e.p. to results in
This is known as the individual machine energy function.
is first integral of motion.

19. TEF (contd)
since

20. TEF (contd) contains path dependent terms. Cannot claim that is p.d.
If , then can be shown to be a Lyapunov function i.e.
Methods to compute
Potential Energy Boundary Surface (PEBS) method.
Boundary Controlling Unstable (BCU) equilibrium point method.
Other methods (Hybrid, Second-kick etc)
(a) and (b) are the important ones.

21. Equal Area Criterion and TEF
Single-machine infinite-bus system
A three-phase fault occurs at the middle of one of the lines at t=0, and is subsequently cleared at by opening the circuit breakers at both ends of the faulted line.
The pre-fault, faulted, and post-fault configurations and their reduction to a two-machine equivalent are are constructed.
The electric power during pre-fault, faulted, and post-fault states are respectively.
>
> ~
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Single-machine infinite-bus system

22. Computing system parameters
Pre-fault system and its two-machine equivalent.

23. Faulted system and its two machine equivalent.
Computing Parameters
(a)
(b)
(c)
Faulted system and its two machine equivalent.

24. Computing Computing Series of Δ-Υ transformations. General method.
The point at which the fault occurs is labeled node 4.
There are current injections at nodes 1,2 and 4 and none at node 3.
Faulted system
The nodal equation is:

25. Computing
Since the fault is at node 4 with the impedance equal to zero, Hence delete row 4 and column 4.
Node 3 is eliminated since there is no injection at 3.

26. Post-fault system and its two machine equivalent
Computing
is computed from the off-diagonal entry as:
Hence
Computing X
Post-fault system and its two machine equivalent