1. FUNDAMENTAL OF ELECTRICAL POWER SYSTEMS (EE 270)
Chapter 3
Basic Principles

2. Objectives
Review basic concepts and establish terminology and notation encountered in electric circuit theory.
Review phasors, instantaneous power, complex power, network equations and elementary aspects of balance three-phase circuits.

3. Power System Network

4. Three-Phase Power Transformer

5. Sub-station

6. Distribution Transformer

7. Sinusoidal voltage or current at constant frequency characterized by:
Phasors
Sinusoidal voltage or current at constant frequency characterized by:
Phase angle
Maximum value

8. The root-mean-square  effective value Let the rms value of
Phasors
The root-mean-square  effective value
Let the rms value of
Voltage: Current:
And let

9. Power in Single Phase AC Circuit
Assume a single phase sinusoidal source supplying a load.
v(t) = instantaneous voltage
i(t) = instantaneous current
Find the instantaneous power p(t)

10. Power in Single Phase AC Circuit
Use trigonometric identity

11. Power in Single Phase AC Circuit

12. The power absorbed by the resistive component of the load.
Real Power
The average power, P is also referred as the active power or real power.
The power absorbed by the resistive component of the load.
Standard unit: Watt

13. The power absorbed by the reactive component of the load.
Reactive Power
The power absorbed by the reactive component of the load.
Standard unit: var (volt-ampere reactive)

14. Standard unit: VA (volt-ampere)
Complex Power
The complex power, S is the product of voltage and the conjugate of the current.
Standard unit: VA (volt-ampere)

15. Complex Power

16. Purely Capacitive Load
Phasor Diagram
Purely Resistive Load
Q=0; S=P
Purely Inductive Load
P=0
Purely Capacitive Load

17. Phasor Diagram

18. Purely Capacitive Load
Power Triangle
Purely Inductive Load
P=0, Q=+ve
Purely Capacitive Load
P=0, Q=-ve

19. Power Triangle

20. Impedance
Impedance of complex power is given by:

21. The Complex Power Balance
The sum of real and reactive power supplied by the source is equal to the sum of real and reactive powers transferred to the load.
Law of energy conservation

22. The Complex Power Balance

23. The Complex Power Balance
Example:

24. Solution
Current at each load:

25. Solution
Complex power absorbed at each load:
Total load complex power:

26. Two loads connected in parallel are
Exercise
Two loads connected in parallel are
supplied from a single-phase 240Vrms
source. The two loads draw a total real
power of 400kW at a power factor of 0.8
lagging. One of the loads draws 120kW at a
power factor of 0.96 leading. Find the
complex power of the other load.

27. Two impedances, Z1=0.8+j5.6Ω and
Exercise
Two impedances, Z1=0.8+j5.6Ω and
Z2=8-j16Ω, and a single phase motor are
connected in parallel across a 200Vrms,
60Hz supply. The motor draws 5kVA at 0.8
pf lagging. Find S1, S2 and S3 for the motor.

28. Power Factor Correction
PF = 1 } unity power factor
If PF<1,
apparent power |S|>real power P
Current increase, cost of utility increase.
Major loads of the system should be near to unity power factor.

29. Power Factor Correction
Inductive load : lagging pf
Capacitive load : leading pf
How to fix PF? Capacitor is added to the system (inductive load).
PF is mostly considered in industrial consumers (using inductive load) and not in residential and small commercial since the power factor is near unity.

30. Power Factor Correction

31. Power Factor Correction

32. Exercise
Two loads Z1=100+j0Ω and Z2=10+j20Ω are connected across a 200Vrms, 50Hz source.
Find the total real and reactive power, the power factor at the load, and the total current without C.
Find the capacitance of the capacitor connected across the loads to improve the overall power factor to 0.8 lagging.

33. Solution (a)
Current at each load:
Power at each load:

34. Solution (a) Total real and reactive power: Total current without C:
Power factor at the load:

35. Solution (b)
Power at the capacitor:

36. Solution (b)
Capacitance of the capacitor:

37. Exercise Three loads are connected in parallel across a
1400Vrms, 50Hz single-phase supply.
Load 1: Inductive load, 125kVA at 0.28 power factor
Load 2: Capacitive load, 10kW and 40kvar
Load 3: Resistive load of 15kW
Find the total kW, kvar, kVA and the supply power factor.
A capacitor of negligible resistance is connected in parallel with the above loads to improve the power factor to 0.8 lagging. Determine the kvar rating of this capacitor and the capacitance in µF.

38. Exercise Two loads are connected in parallel across a 200Vrms,
50Hz single-phase supply.
Load 1: j5.6Ω
Load 2: 8 - j16Ω
Find the total kW, kvar, kVA and the supply power factor.
A capacitor is connected in parallel with the loads. Find the kvar and the capacitance in µF to improve the overall power factor to unity.
What is the new line current?

39. Complex Power Flow
Need to consider two way current (i.e. From V1 to V2 and from V2 to V1) and two way S (i.e. From V1 to V2 and from V2 to V1).
If P is negative than the P in which the source is associated to receives/absorbs the P.
If P is positive than the P in which the source is associated to generates/delivers the P.
If Q is negative than the Q in which the source is associated to receives/absorbs the Q.
If Q is positive than the Q in which the source is associated to generates/delivers the Q.

40. Generator & Load Convention
P/Q
Characteristic
Generator +
Delivered/
Generated –
Absorbed/
Received
Load

41. Exercise
Consider two voltage sources V1=120∟–5°V and V2=100∟0°V are connected by a short line of impedance Z=1+j7Ω. Determined the real and reactive power supplied or received by each source and the power loss in the line.

42. Solution Find current which flows from V1 to V2 i.e. I12
Find S from V1 to V2 i.e. S12
Find S from V2 to V1 i.e. S21

43. Solution Evaluate the source from the previous calculated S.
Based on S1:
Source 1 receive 97.5W and delivers 363.3var.
Based on S2
Source 2 generates 107.3W and receives 294.5var.

44. Solution Power loss in the line Check!
Thus, the real power loss in the line is 9.8W and
the reactive power loss in the line is 68.6var.

45. Exercise Two single-phase ideal voltage sources are
connected by a line of impedance of 0.7+j2.4Ω.
V1=500∟16.26°V and V2=585∟0°V. Find the
complex power for each source and determine
whether they are delivering or receiving real and
reactive power. Also, find the real and the reactive
power loss in the line.

46. Power in single-phase AC
Review..
Power in single-phase AC
S, P, Q, p(t)
Phasor analysis/diagram
Power triangle
Complex power balance
Power factor correction
Complex power flow
Generator / Load Convention