1. Lecture 24 Transient Stability Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS

2. 1 Announcements Be reading Chapter 13. HW 11 is not turned in but should be done before final. HW 11 is 13.1, 13.7, 13.8, 13.18, and the special problem (see website for complete assignment) Final is Tuesday Dec 16 from 7 to 10pm in EL 165 (note this is NOT what the web says). Final is comprehensive. One new note sheet, and your two old note sheets are allowed

3. 2 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

4. 3 Generator Mechanical Model Generator Mechanical Block Diagram

5. 4 Generator Mechanical Model, cont’d

6. 5

7. 6

8. 7 Generator Swing Equation

9. 8 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 jX L

10. 9 SMIB, cont’d

11. 10 SMIB Equilibrium Points

12. 11 Transient Stability Analysis For transient stability analysis we need to consider three systems 1. Prefault - before the fault occurs the system is assumed to be at an equilibrium point 2. Faulted - the fault changes the system equations, moving the system away from its equilibrium point 3. Postfault - after fault is cleared the system hopefully returns to a new operating point

13. 12 Transient Stability Solution Methods There are two methods for solving the transient stability problem 1. 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 2. 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

14. 13 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.

15. 14 SMIB Example, cont’d Simplified prefault system

16. 15 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

17. 16 SMIB Example, Post Fault System After the fault the system again changes The equivalent system after the fault is then

18. 17 SMIB Example, Dynamics

19. 18 Transient Stability Solution Methods There are two methods for solving the transient stability problem 1. 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 2. 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

20. 19 Transient Stability Analysis  For transient stability analysis we need to consider three systems 1. Prefault - before the fault occurs the system is assumed to be at an equilibrium point 2. Faulted - the fault changes the system equations, moving the system away from its equilibrium point 3. Postfault - after fault is cleared the system hopefully returns to a new operating point

21. 20 Transient Stability Solution Methods  There are two methods for solving the transient stability problem 1. 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 2. 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

22. 21 Numerical Integration of DEs

23. 22 Examples

24. 23 Euler’s Method

25. 24 Euler’s Method Algorithm

26. 25 Euler’s Method Example 1

27. 26 Euler’s Method Example 1, cont’d tx actual (t) x(t)  t=0.1x(t)  t=0.05 010 0.19.04899.02 0.28.1878.108.15 0.37.4087.297.35 ………… 1.03.6783.493.58 ………… 2.01.3531.221.29

28. 27 Euler’s Method Example 2

29. 28 Euler's Method Example 2, cont'd

30. 29 Euler's Method Example 2, cont'd tx 1 actual (t) x 1 (t)  t=0.25 011 0.250.96891 0.500.87760.9375 0.750.73170.8125 1.000.54030.6289 ……… 10.0-0.8391-3.129 100.00.8623-151,983

31. 30 Euler's Method Example 2, cont'd tt x 1 (10) actual-0.8391 0.25-3.129 0.10-1.4088 0.01-0.8823 0.001-0.8423 Below is a comparison of the solution values for x 1 (t) at time t = 10 seconds

32. 31 Transient Stability Example  A 60 Hz generator is supplying 550 MW to an infinite bus (with 1.0 per unit voltage) through two parallel transmission lines. Determine initial angle change for a fault midway down one of the lines. H = 20 seconds, D = 0.1. Use  t=0.01 second. EaEa

33. 32 Transient Stability Example, cont'd

34. 33 Transient Stability Example, cont'd

35. 34 Transient Stability Example, cont'd

36. 35 Transient Stability Example, cont'd

37. 36 Equal Area Criteria  The goal of the equal area criteria is to try to determine whether a system is stable or not without having to completely integrate the system response. System will be stable after the fault if the Decel Area is greater than the Accel. Area