1. EE369 POWER SYSTEM ANALYSIS
Lecture 2
Complex Power, Reactive Compensation, Three Phase
Tom Overbye and Ross Baldick

2. Reading and Homework Read Chapters 1 and 2 of the text.
HW 1 is Problems 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 2.12, 2.14, 2.17, 2.19, 2.24, 2.25 and Case Study Questions A., B., C., D. from the text; due Thursday 9/5.

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),
where:
v(t) and i(t) are the instantaneous voltage and current as a function of time t,
w is the angular frequency (2πf, with f the frequency in Hertz),
Vmax and Imax are the magnitudes of voltage and current sinusoids,
qv and qI are angular offsets of the peaks of sinusoids from a reference waveform.
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”.)
Also note that the convention in power engineering is that the magnitude of the phasor is the RMS voltage of the waveform:
contrasts with circuit analysis.

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, cont’d

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

12. Complex Power, cont’d

13. 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 real power and conservation of reactive power follows since S = VI*.

14. Conservation of Power Example
Power flowing from source to load at bus
Earlier we found
I = 20-6.9 amps
= 1600W + j1200VAr

15. Power Consumption in Devices

16. Example
First solve
basic circuit I

17. Example, cont’d Now add additional reactive power load
and re-solve, assuming
that load voltage is
maintained at 40 kV.
Need higher source voltage to maintain load voltage magnitude when
reactive power load is added to circuit. Current is higher.

18. 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

19. 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.
Supply voltage magnitude and line current is lower with compensation.

20. 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 use of smaller wires, or alternatively, supply more load over the same wires,
Voltage drop on the line is less.
Reactive compensation is used extensively throughout transmission and distribution systems.
Capacitors can be used to “correct” a load’s power factor to an arbitrary value.

21. Power Factor Correction Example

22. Distribution System Capacitors

23. 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.
Single phase transmission used for electric trains in Europe.

24. Balanced 3 -- Zero Neutral Current

25. Advantages of 3 Power
Can transmit more power for same amount of wire (twice as much as single phase).
Total torque produced by 3 machines is constant, so less vibration.
Three phase machines use less material for same power rating.
Three phase machines start more easily than single phase machines.

26. Three Phase - Wye Connection
There are two ways to connect 3 systems:
Wye (Y), and
Delta ().

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

28. Wye Connection, cont’d
We call the voltage across each element of a wye connected device the “phase” voltage.
We call the current through each element of a wye connected device the “phase” current.
Call the voltage across lines the “line-to-line” or just the “line” voltage.
Call the current through lines the “line” current.

29. Delta Connection
Ica Ic
Iab
Ibc Ia Ib

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

31. Three Phase Example, cont’d

32. Delta-Wye Transformation

33. Delta-Wye Transformation Proof+

34. Delta-Wye Transformation, cont’d

35. Three Phase Transmission Line