ECE 476 POWER SYSTEM ANALYSIS

1. ECE 476POWER SYSTEM ANALYSIS Lecture 12Power Flow Professor Tom Overbye Department of Electrical andComputer Engineering

2. Announcements • Homework 5 is due on Oct 4 • Homework 6 is due on Oct 18 • First exam is 10/9 in class; closed book, closed notes, one note sheet and calculators allowed • Power plant, substation field trip, 10/11 starting at 1pm. We’ll meet at corner of Gregory and Oak streets. • Be reading Chapter 6, but Chapter 6 will not be on the example

3. In the News • Ground was broken Monday for the new 1600 MW Prairie State Energy Campus coal-fired power plant • Project is located near Lively Grove, IL, and will be connected to grid via 3 345 kV lines (38 new miles) • Prairie State, which will cost $ 2.9 billion to build (about $1.82 per watt), should be finished by 2011 or 2012. • Output of plant will be bought primarily by municipal and coop customers throughout region (Missouri to Ohio)

4. Prairie State New 345 kV Lines

5. Gauss Iteration

6. Gauss Iteration Example

7. Stopping Criteria

8. Gauss Power Flow

9. 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 V2? SLoad = 1.0 + j0.5 p.u.

10. Gauss Two Bus Example, cont’d

11. Gauss Two Bus Example, cont’d

12. Gauss Two Bus Example, cont’d

13. Slack Bus • In previous example we specified S2 and V1 and then solved for S1 and V2. • 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.

14. Gauss with Many Bus Systems

15. Gauss-Seidel Iteration

16. 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

17. Inclusion of PV Buses in G-S

18. Inclusion of PV Buses, cont'd

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

20. Two Bus PV Example, cont'd

21. 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.

22. 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

23. Accelerated G-S Convergence

24. Accelerated Convergence, cont’d

25. 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

26. 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

27. 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

28. Newton-Raphson Method (scalar)

29. Newton-Raphson Method, cont’d

30. Newton-Raphson Example

31. Newton-Raphson Example, cont’d

32. Sequential Linear Approximations At each iteration the N-R method uses a linear approximation to determine the next value for x Function is f(x) = x2 - 2 = 0. Solutions are points where f(x) intersects f(x) = 0 axis

33. 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

34. Multi-Variable Newton-Raphson

35. Multi-Variable Case, cont’d

36. Multi-Variable Case, cont’d

37. Jacobian Matrix

38. Multi-Variable N-R Procedure

39. Multi-Variable Example

40. Multi-variable Example, cont’d

41. Multi-variable Example, cont’d

42. NR Application to Power Flow