ECE 476 POWER SYSTEM ANALYSIS

1. ECE 476POWER SYSTEM ANALYSIS Lecture 13 Power Flow Professor Tom Overbye Department of Electrical andComputer Engineering

2. Announcements • Be reading Chapter 6, also Chapter 2.4 (Network Equations). • HW 5 is 2.38, 6.9, 6.18, 6.30, 6.34, 6.38; do by October 6 but does not need to be turned in. • First exam is October 11 during class. Closed book, closed notes, one note sheet and calculators allowed. Exam covers thru the end of lecture 13 (today) • An example previous exam (2008) is posted. Note this is exam was given earlier in the semester in 2008 so it did not include power flow, but the 2011 exam will (at least partially)

3. Multi-Variable Example

4. Multi-variable Example, cont’d

5. Multi-variable Example, cont’d

6. Possible EHV Overlays for Wind AEP 2007 Proposed Overlay

7. NR Application to Power Flow

8. Real Power Balance Equations

9. Newton-Raphson Power Flow

10. Power Flow Variables

11. N-R Power Flow Solution

12. Power Flow Jacobian Matrix

13. Power Flow Jacobian Matrix, cont’d

14. 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 SBase = 100 MVA.

15. Two Bus Example, cont’d

16. Two Bus Example, cont’d

17. Two Bus Example, First Iteration

18. Two Bus Example, Next Iterations

19. Two Bus Solved Values Once the voltage angle and magnitude at bus 2 are known we can calculate all the other system values, such as the line flows and the generator reactive power output

20. Two Bus Case Low Voltage Solution

21. Low Voltage Solution, cont'd Low voltage solution

22. Two Bus Region of Convergence Slide shows the region of convergence for different initial guesses of bus 2 angle (x-axis) and magnitude (y-axis) Red region converges to the high voltage solution, while the yellow region converges to the low voltage solution

23. PV Buses • Since the voltage magnitude at PV buses is fixed there is no need to explicitly include these voltages in x or write the reactive power balance equations • the reactive power output of the generator varies to maintain the fixed terminal voltage (within limits) • optionally these variations/equations can be included by just writing the explicit voltage constraint for the generator bus |Vi| – Vi setpoint = 0

24. Three Bus PV Case Example

25. The N-R Power Flow: 5-bus Example T2 800 MVA 345/15 kV T1 1 5 4 3 520 MVA Line 3 345 kV 50 mi 400 MVA 15 kV 800 MVA 15 kV 400 MVA 15/345 kV 345 kV 100 mi 40 Mvar 80 MW 345 kV 200 mi Line 2 Line 1 2 280 Mvar 800 MW Single-line diagram

26. The N-R Power Flow: 5-bus Example Table 1. Bus input data Table 2. Line input data

27. The N-R Power Flow: 5-bus Example Table 3. Transformer input data Table 4. Input data and unknowns

28. Time to Close the Hood: Let the Computer Do the Math! (Ybus Shown)

29. Ybus Details Elements of Ybus connected to bus 2

30. Here are the Initial Bus Mismatches

31. And the Initial Power Flow Jacobian

32. And the Hand Calculation Details!

33. Five Bus Power System Solved

34. 37 Bus Example Design Case

35. Good Power System Operation Good power system operation requires that there be no reliability violations for either the current condition or in the event of statistically likely contingencies Reliability requires as a minimum that there be no transmission line/transformer limit violations and that bus voltages be within acceptable limits (perhaps 0.95 to 1.08) Example contingencies are the loss of any single device. This is known as n-1 reliability. North American Electric Reliability Corporation now has legal authority to enforce reliability standards (and there are now lots of them). See http://www.nerc.com for details (click on Standards)

36. Looking at the Impact of Line Outages Opening one line (Tim69-Hannah69) causes an overload. This would not be allowed

37. Contingency Analysis Contingencyanalysis providesan automaticway of lookingat all the statisticallylikely contingencies. Inthis example thecontingency set Is all the single line/transformeroutages

38. Power Flow And Design One common usage of the power flow is to determine how the system should be modified to remove contingencies problems or serve new load In an operational context this requires working with the existing electric grid In a planning context additions to the grid can be considered In the next example we look at how to remove the existing contingency violations while serving new load.

39. An Unreliable Solution Case now has nine separate contingencies with reliability violations

40. A Reliable Solution Previous case was augmented with the addition of a 138 kV Transmission Line

41. Generation Changes and The Slack Bus The power flow is a steady-state analysis tool, so the assumption is total load plus losses is always equal to total generation Generation mismatch is made up at the slack bus When doing generation change power flow studies one always needs to be cognizant of where the generation is being made up Common options include system slack, distributed across multiple generators by participation factors or by economics

42. Generation Change Example 1 Display shows “Difference Flows” between original 37 bus case, and case with a BLT138 generation outage; note all the power change is picked up at the slack

43. Generation Change Example 2 Display repeats previous case except now the change in generation is picked up by other generators using a participation factor approach

44. Voltage Regulation Example: 37 Buses Display shows voltage contour of the power system, demo will show the impact of generator voltage set point, reactive power limits, and switched capacitors

45. Solving Large Power Systems • The most difficult computational task is inverting the Jacobian matrix • inverting a full matrix is an order n3 operation, meaning the amount of computation increases with the cube of the size size • this amount of computation can be decreased substantially by recognizing that since the Ybus is a sparse matrix, the Jacobian is also a sparse matrix • using sparse matrix methods results in a computational order of about n1.5. • this is a substantial savings when solving systems with tens of thousands of buses

46. Newton-Raphson Power Flow • Advantages • fast convergence as long as initial guess is close to solution • large region of convergence • Disadvantages • each iteration takes much longer than a Gauss-Seidel iteration • more complicated to code, particularly when implementing sparse matrix algorithms • Newton-Raphson algorithm is very common in power flow analysis