ECE 476 POWER SYSTEM ANALYSIS

1. ECE 476POWER SYSTEM ANALYSIS Lecture 2 Complex Power, Reactive Compensation, Three Phase Professor Tom Overbye Department of Electrical andComputer Engineering

2. Announcements • For lectures 2 through 3 please be reading Chapters 1 and 2

3. Review of Phasors Goal of phasor analysis is to simplify the analysis of constant frequency ac systems v(t) = Vmax cos(wt + qv) i(t) = Imax cos(wt + qI) Root Mean Square (RMS) voltage of sinusoid

4. Phasor Representation

5. Phasor Representation, cont’d (Note: Some texts use “boldface” type for complex numbers, or “bars on the top”)

6. Advantages of Phasor Analysis (Note: Z is a complex number but not a phasor)

7. RL Circuit Example

8. Complex Power

9. Complex Power, cont’d

10. Complex Power (Note: S is a complex number but not a phasor)

11. Complex Power, cont’d

12. Conservation of Power • At every node (bus) in the system • Sum of real power into node must equal zero • Sum of reactive power into node must equal zero • This is a direct consequence of Kirchhoff’s current law, which states that the total current into each node must equal zero. • Conservation of power follows since S = VI*

13. Conversation of Power Example Earlier we found I = 20-6.9 amps

14. Power Consumption in Devices

15. Example First solve basic circuit

16. Example, cont’d Now add additional reactive power load and resolve

17. Power System Notation Power system components are usually shown as “one-line diagrams.” Previous circuit redrawn Arrows are used to show loads Transmission lines are shown as a single line Generators are shown as circles

18. Reactive Compensation Key idea of reactive compensation is to supply reactive power locally. In the previous example this can be done by adding a 16 Mvar capacitor at the load Compensated circuit is identical to first example with just real power load

19. Reactive Compensation, cont’d • Reactive compensation decreased the line flow from 564 Amps to 400 Amps. This has advantages • Lines losses, which are equal to I2 R decrease • Lower current allows utility to use small wires, or alternatively, supply more load over the same wires • Voltage drop on the line is less • Reactive compensation is used extensively by utilities • Capacitors can be used to “correct” a load’s power factor to an arbitrary value.

20. Power Factor Correction Example

21. Distribution System Capacitors

22. Balanced 3 Phase () Systems • A balanced 3 phase () system has • three voltage sources with equal magnitude, but with an angle shift of 120 • equal loads on each phase • equal impedance on the lines connecting the generators to the loads • Bulk power systems are almost exclusively 3 • Single phase is used primarily only in low voltage, low power settings, such as residential and some commercial

23. Balanced 3 -- No Neutral Current

24. Advantages of 3 Power • Can transmit more power for same amount of wire (twice as much as single phase) • Torque produced by 3 machines is constrant • Three phase machines use less material for same power rating • Three phase machines start more easily than single phase machines

25. Three Phase - Wye Connection • There are two ways to connect 3 systems • Wye (Y) • Delta ()

26. Vcn Vab Vca Van Vbn Vbc Wye Connection Line Voltages -Vbn (α = 0 in this case) Line to line voltages are also balanced

27. Wye Connection, cont’d • Define voltage/current across/through device to be phase voltage/current • Define voltage/current across/through lines to be line voltage/current

28. Ic Ica Ib Iab Ibc Ia Delta Connection

29. Three Phase Example Assume a -connected load is supplied from a 3 13.8 kV (L-L) source with Z = 10020W

30. Three Phase Example, cont’d

31. Delta-Wye Transformation

32. Delta-Wye Transformation Proof

33. Delta-Wye Transformation, cont’d

34. Three Phase Transmission Line