1. ECE 476 Power System Analysis Lecture 12: Power Flow Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign overbye@illinois.edu

2. Announcements Read Chapter 6 H6 is 6.19, 6.30, 6.31, 6.34, 6.38, 6.45. It does not need to be turned in, but will be covered by an in-class quiz on Oct 15. 1

3. Transmission Line Corridors from the Air Image Source: Jamie Padilla

4. Slack Bus In previous example we specified S 2 and V 1 and then solved for S 1 and V 2. We can not arbitrarily specify S at all buses because total generation must equal total load + total losses We also need an angle reference bus. To solve these problems we define one bus as the "slack" bus. This bus has a fixed voltage magnitude and angle, and a varying real/reactive power injection. 3

5. Stated Another Way From exam problem 4.c we had This Y bus is actually singular! So we cannot solve This means (as you might expect), we cannot independently specify all the current injections I 4

6. Gauss with Many Bus Systems 5

7. Gauss-Seidel Iteration 6

8. Three Types of Power Flow Buses There are three main types of power flow buses – Load (PQ) at which P and Q are fixed; iteration solves for voltage magnitude and angle. – Slack at which the voltage magnitude and angle are fixed; iteration solves for P and Q injections – Generator (PV) at which P and |V| are fixed; iteration solves for voltage angle and Q injection special coding is needed to include PV buses in the Gauss-Seidel iteration (covered in book, but not in slides since Gauss-Seidel is no longer commonly used) 7

9. Accelerated G-S Convergence 8

10. Accelerated Convergence, cont’d 9

11. Gauss-Seidel Advantages/Disadvantages Advantages – Each iteration is relatively fast (computational order is proportional to number of branches + number of buses in the system – Relatively easy to program Disadvantages – Tends to converge relatively slowly, although this can be improved with acceleration – Has tendency to miss solutions, particularly on large systems – Tends to diverge on cases with negative branch reactances (common with compensated lines) – Need to program using complex numbers 10

12. Newton-Raphson Algorithm The second major power flow solution method is the Newton-Raphson algorithm Key idea behind Newton-Raphson is to use sequential linearization 11

13. Newton-Raphson Method (scalar) 12

14. Newton-Raphson Method, cont’d 13

15. Newton-Raphson Example 14

16. Newton-Raphson Example, cont’d 15

17. Sequential Linear Approximations Function is f(x) = x 2 - 2 = 0. Solutions are points where f(x) intersects f(x) = 0 axis At each iteration the N-R method uses a linear approximation to determine the next value for x 16

18. Newton-Raphson Comments When close to the solution the error decreases quite quickly -- method has quadratic convergence f(x (v) ) is known as the mismatch, which we would like to drive to zero Stopping criteria is when  f(x (v) )  <  Results are dependent upon the initial guess. What if we had guessed x (0) = 0, or x (0) = -1? A solution’s region of attraction (ROA) is the set of initial guesses that converge to the particular solution. The ROA is often hard to determine 17

19. Multi-Variable Newton-Raphson 18

20. Multi-Variable Case, cont’d 19

21. Multi-Variable Case, cont’d 20

22. Jacobian Matrix 21

23. Multi-Variable Example 22

24. Multi-variable Example, cont’d 23

25. Multi-variable Example, cont’d 24

26. NR Application to Power Flow 25

27. Real Power Balance Equations 26

28. Newton-Raphson Power Flow 27

29. Power Flow Variables 28

30. N-R Power Flow Solution 29

31. Power Flow Jacobian Matrix 30

32. Power Flow Jacobian Matrix, cont’d 31

33. Two Bus Newton-Raphson Example For the two bus power system shown below, use the Newton-Raphson power flow to determine the voltage magnitude and angle at bus two. Assume that bus one is the slack and S Base = 100 MVA. 32

34. Two Bus Example, cont’d 33

35. Two Bus Example, cont’d 34

36. Two Bus Example, First Iteration 35