1. Lecture 11 Power Flow Professor Tom Overbye Special Guest Appearance by Professor Sauer! Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS

2. 1 Announcements Homework #5 is 3.12, 3.14, 3.19, 3.60 due Oct 2nd (Thursday) First exam is 10/9 in class; closed book, closed notes, one note sheet and calculators allowed Start reading Chapter 6 for lectures 11 and 12

3. 2 Power Flow Analysis When analyzing power systems we know neither the complex bus voltages nor the complex current injections Rather, we know the complex power being consumed by the load, and the power being injected by the generators plus their voltage magnitudes Therefore we can not directly use the Y bus equations, but rather must use the power balance equations

4. 3 Power Balance Equations

5. 4 Power Balance Equations, cont’d

6. 5 Real Power Balance Equations

7. 6 Power Flow Requires Iterative Solution

8. 7 Gauss Iteration

9. 8 Gauss Iteration Example

10. 9 Stopping Criteria

11. 10 Gauss Power Flow

12. 11 Gauss Two Bus Power Flow Example A 100 MW, 50 Mvar load is connected to a generator through a line with z = 0.02 + j0.06 p.u. and line charging of 5 Mvar on each end (100 MVA base). Also, there is a 25 Mvar capacitor at bus 2. If the generator voltage is 1.0 p.u., what is V 2 ? S Load = 1.0 + j0.5 p.u.

13. 12 Gauss Two Bus Example, cont’d

14. 13 Gauss Two Bus Example, cont’d

15. 14 Gauss Two Bus Example, cont’d

16. 15 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.

17. 16 Gauss with Many Bus Systems

18. 17 Gauss-Seidel Iteration

19. 18 Three Types of Power Flow Buses There are three main types of power flow buses – Load (PQ) at which P/Q are fixed; iteration solves for voltage magnitude and angle. – Slack at which the voltage magnitude and angle are fixed; iteration solves for P/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

20. 19 Inclusion of PV Buses in G-S

21. 20 Inclusion of PV Buses, cont'd

22. 21 Two Bus PV Example Consider the same two bus system from the previous example, except the load is replaced by a generator

23. 22 Two Bus PV Example, cont'd

24. 23 Generator Reactive Power Limits The reactive power output of generators varies to maintain the terminal voltage; on a real generator this is done by the exciter To maintain higher voltages requires more reactive power Generators have reactive power limits, which are dependent upon the generator's MW output These limits must be considered during the power flow solution.

25. 24 Generator Reactive Limits, cont'd During power flow once a solution is obtained check to make generator reactive power output is within its limits If the reactive power is outside of the limits, fix Q at the max or min value, and resolve treating the generator as a PQ bus – this is know as "type-switching" – also need to check if a PQ generator can again regulate Rule of thumb: to raise system voltage we need to supply more vars

26. 25 Accelerated G-S Convergence

27. 26 Accelerated Convergence, cont’d

28. 27 Gauss-Seidel Advantages Each iteration is relatively fast (computational order is proportional to number of branches + number of buses in the system Relatively easy to program

29. 28 Gauss-Seidel 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

30. 29 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

31. 30 Newton-Raphson Method (scalar)

32. 31 Newton-Raphson Method, cont’d

33. 32 Newton-Raphson Example

34. 33 Newton-Raphson Example, cont’d

35. 34 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

36. 35 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

37. 36 Multi-Variable Newton-Raphson

38. 37 Multi-Variable Case, cont’d

39. 38 Multi-Variable Case, cont’d

40. 39 Jacobian Matrix

41. 40 Multi-variable Example, cont’d

42. 41 Multi-variable Example, cont’d