1. ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu 9/2/2015 1 Lecture 4: Newton-Raphson Power Flow

2. Announcements No class next Monday (labor day) Homework 1 due next Wednesday, in class A reference book on linear algebra: The Matrix Cookbook http://www.math.uwaterloo.ca/~hwolkowi/matrixcookb ook.pdf http://www.math.uwaterloo.ca/~hwolkowi/matrixcookb ook.pdf 2

3. Newton-Raphson Method 3 To solve a set of nonlinear equations denoted by N-R proceeds by iterative linearization

4. Power Flow Variables 4

5. N-R Power Flow Solution 5

6. Power Flow Formulation 6

7. Jacobian Matrix 7

8. Power Flow Jacobian 8

9. Power Flow Jacobian, cont’d 9

10. Two Bus Newton-Raphson Example 10

11. Two Bus Example, cont’d 11

12. Two Bus Example, cont’d 12 2 2

13. Two Bus Example, First Iteration 13

14. Two Bus Example, Convergence 14

15. 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 PowerWorld Case Name: Bus2_Intro 15

16. Two Bus Case, Low Voltage Solution 16

17. Low Voltage Solution, cont'd Low voltage solution 17

18. Practical Power Flow Software Most commercial software packages have built in defaults to prevent convergence to low voltage solutions. – One approach is to automatically change the load model from constant power to constant current or constant impedance when the load bus voltage gets too low – In PowerWorld these defaults can be modified on the Tools, Simulator Options, Advanced Options page; note you also need to disable the “Initialize from Flat Start Values” option – The PowerWorld case Bus2_Intro_Low is set solved to the low voltage solution – Initial bus voltages can be set using the Bus Information Dialog 18

19. NR Initialization A textbook starting solution for the NR is to use “flat start” values in which all the angles are set to the slack bus angle, all the PQ bus voltages are set to 1.0, and all the PV bus voltages are set to their PV setpoint values This approach usually works for small systems It seldom works for large systems. The usual approach for solving large systems is to start with an existing solution, modify it, then hope it converges – If not make the modification smaller – More robust methods are possible, but convergence is certainly not guaranteed!! 19

20. PV Buses 20

21. Three Bus PV Case Example 21

22. Modeling Voltage Dependent Load 22

23. Voltage Dependent Load Example 23

24. Voltage Dependent Load, cont'd 24

25. Voltage Dependent Load, cont'd With constant impedance load the MW/Mvar load at bus 2 varies with the square of the bus 2 voltage magnitude. This if the voltage level is less than 1.0, the load is lower than 200/100 MW/Mvar PowerWorld Case Name: Bus2_Intro_Z 25

26. Generator Reactive Power Limits 26

27. Generator Reactive Limits, cont'd 27

28. Switching Bus Status 28 How do we want to code up the N-R algorithm?

29. 400 MVA 15 kV 400 MVA 15/345 kV T1 T2 800 MVA 345/15 kV 800 MVA 15 kV 520 MVA 80 MW40 Mvar 280 Mvar800 MW Line 3 345 kV Line 2Line 1 345 kV 100 mi 345 kV 200 mi 50 mi 143 2 5 Single-line diagram The N-R Power Flow: 5-bus Example This five bus example is taken from Chapter 6 of Power System Analysis and Design by Glover, Sarma, and Overbye, 5 th Edition, 2011 29

30. BusType V per unit  degrees P G per unit Q G per unit P L per unit Q L per unit Q Gmax per unit Q Gmin per unit 1Swing1.00  00  2Load  008.02.8  3Constant voltage 1.05  5.2  0.80.44.0-2.8 4Load  0000  5  0000  Table 1. Bus input data Bus-to- Bus R’ per unit X’ per unit G’ per unit B’ per unit Maximum MVA per unit 2-40.00900.10001.7212.0 2-50.00450.05000.8812.0 4-50.002250.02500.4412.0 Table 2. Line input data The N-R Power Flow: 5-bus Example 30

31. Bus-to- Bus R per unit X per unit G c per unit B m per unit Maximum MVA per unit Maximum TAP Setting per unit 1-50.001500.02006.0— 3-40.000750.010010.0— Table 3. Transformer input data BusInput DataUnknowns 1 V 1 = 1.0,  1 = 0 P 1, Q 1 2P 2 = P G2 -P L2 = -8 Q 2 = Q G2 -Q L2 = -2.8 V 2,  2 3V 3 = 1.05 P 3 = P G3 -P L3 = 4.4 Q 3,  3 4P 4 = 0, Q 4 = 0 V 4,  4 5P 5 = 0, Q 5 = 0 V 5,  5 Table 4. Input data and unknowns The N-R Power Flow: 5-bus Example 31

32. Five Bus Case Ybus PowerWorld Case Name: Bus5_GSO 32

33. Ybus Calculation Details Elements of Y bus connected to bus 2 33

34. Here are the Initial Bus Mismatches 34

35. And the Initial Power Flow Jacobian 35

36. And the Hand Calculation Details 36

37. Five Bus Power System Solved 37

38. Quasi-Newton Power Flow Methods 38

39. Dishonest Newton Method 39

40. Dishonest N-R Example, cont’d We pay a price in increased iterations, but with decreased computation per iteration 40

41. Numerical differentiation 41

42. Secant Method 42 Fig. 3.8 of Crow’s book, 3 rd edition