1. ECE 476 POWER SYSTEM ANALYSIS
Lecture 23
Transient Stability
Professor Tom Overbye
Department of Electrical and Computer Engineering

2. Announcements Be reading Chapter 11 and Chapter 12 thru 12.3
HW 10 is 11.4, 11.7, 11.10, 11.19, 11.20; due Dec 1 in class.
Project is due Thursday Dec 1 in class.

3. Power System Time Scales
Lightning Propagation
Switching Surges
Stator Transients and
Subsynchronous Resonance
Transient Stability
Governor and Load Frequency Control
Boiler/Long-Term Dynamics
10-7
10-5
10-3
0.1 10
103
105
Time (Seconds)
Voltage Stability
Power Flow
Image source: P.W. Sauer, M.A. Pai, Power System Dynamics and Stability, 1997, Fig 1.2, modified

4. Power Grid Disturbance Example
Figures show the frequency change as a result of the sudden loss of a large amount of generation in the Southern WECC
Green is bus quite close to location of generator trip while blue and red are Alberta buses. Black is BPA.
Time in Seconds
Frequency Contour

5. Frequency Response for Gen. Loss
In response to rapid loss of generation, in the initial seconds the system frequency will decrease as energy stored in the rotating masses is transformed into electric energy
Solar PV has no inertia, and for most new wind turbines the inertia is not seen by the system
Within seconds governors respond, increasing power output of controllable generation
Solar PV and wind are usually operated at maximum power so they have no reserves to contribute

6. Generator Electrical Model
The simplest generator model, known as the classical model, treats the generator as a voltage source behind the direct-axis transient reactance; the voltage magnitude is fixed, but its angle changes according to the mechanical dynamics

7. Generator Mechanical Model
Generator Mechanical Block Diagram

8. Generator Mechanical Model, cont’d

9. Generator Mechanical Model, cont’d

10. Generator Mechanical Model, cont’d

11. Generator Swing Equation

12. Single Machine Infinite Bus (SMIB)
To understand the transient stability problem we’ll first consider the case of a single machine (generator) connected to a power system bus with a fixed voltage magnitude and angle (known as an infinite bus) through a transmission line with impedance jXL

13. SMIB, cont’d

14. SMIB Equilibrium Points

15. Transient Stability Analysis
For transient stability analysis we need to consider three systems
Prefault - before the fault occurs the system is assumed to be at an equilibrium point
Faulted - the fault changes the system equations, moving the system away from its equilibrium point
Postfault - after fault is cleared the system hopefully returns to a new operating point

16. Transient Stability Solution Methods
There are two methods for solving the transient stability problem
Numerical integration
this is by far the most common technique, particularly for large systems; during the fault and after the fault the power system differential equations are solved using numerical methods
Direct or energy methods; for a two bus system this method is known as the equal area criteria
mostly used to provide an intuitive insight into the transient stability problem

17. SMIB Example
Assume a generator is supplying power to an infinite bus through two parallel transmission lines. Then a balanced three phase fault occurs at the terminal of one of the lines. The fault is cleared by the opening of this line’s circuit breakers.

18. SMIB Example, cont’d
Simplified prefault system

19. SMIB Example, Faulted System
During the fault the system changes
The equivalent system during the fault is then
During this fault no
power can be transferred
from the generator to
the system

20. SMIB Example, Post Fault System
After the fault the system again changes
The equivalent system after the fault is then

21. SMIB Example, Dynamics

22. Transient Stability Solution Methods
There are two methods for solving the transient stability problem
Numerical integration
this is by far the most common technique, particularly for large systems; during the fault and after the fault the power system differential equations are solved using numerical methods
Direct or energy methods; for a two bus system this method is known as the equal area criteria
mostly used to provide an intuitive insight into the transient stability problem

23. Transient Stability Analysis
For transient stability analysis we need to consider three systems
Prefault - before the fault occurs the system is assumed to be at an equilibrium point
Faulted - the fault changes the system equations, moving the system away from its equilibrium point
Postfault - after fault is cleared the system hopefully returns to a new operating point

24. Transient Stability Solution Methods
There are two methods for solving the transient stability problem
Numerical integration
this is by far the most common technique, particularly for large systems; during the fault and after the fault the power system differential equations are solved using numerical methods
Direct or energy methods; for a two bus system this method is known as the equal area criteria
mostly used to provide an intuitive insight into the transient stability problem

25. Numerical Integration of DEs

26. Examples

27. Euler’s Method

28. Euler’s Method Algorithm

29. Euler’s Method Example 1

30. Euler’s Method Example 1, cont’d
xactual(t)
x(t) Dt=0.1
x(t) Dt=0.05 10
0.1
9.048 9
9.02
0.2
8.187
8.10
8.15
0.3
7.408
7.29
7.35 …
1.0
3.678
3.49
3.58
2.0
1.353
1.22
1.29

31. Euler’s Method Example 2

32. Euler's Method Example 2, cont'd

33. Euler's Method Example 2, cont'd
x1actual(t)
x1(t) Dt=0.25 1
0.25
0.9689
0.50
0.8776
0.9375
0.75
0.7317
0.8125
1.00
0.5403
0.6289 …
10.0
-3.129
100.0
0.8623
-151,983

34. Euler's Method Example 2, cont'd
Below is a comparison of the solution values for x1(t)
at time t = 10 seconds Dt
x1(10)
actual
0.25
-3.129
0.10
0.01
0.001