Lecture 2 Complex Power, Reactive Compensation, Three Phase Dr. Youssef A. Mobarak Department of Electrical Engineering EE 351 POWER SYSTEM ANALYSIS

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

2 Review of Phasors Goal of phasor analysis is to simplify the analysis of constant frequency ac systems v(t) = V max cos(  t +  v ) i(t) = I max cos(  t +  I ) Root Mean Square (RMS) voltage of sinusoid

3 Phasor Representation

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

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

6 RL Circuit Example

7 Complex Power

8 Complex Power, cont’d

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

10 Complex Power, cont’d

11 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*

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

13 Power Consumption in Devices

14 Example First solve basic circuit

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

16 Power System Notation Power system components are usually shown as “one-line diagrams.” Previous circuit redrawn C:\Program Files (x86)\PowerWorld\SimulatorGSO17\5th Ed. Book Cases\Chapter2\Problem2_32.pwb Arrows are used to show loads Generators are shown as circles Transmission lines are shown as a single line

17 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

18 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 I 2 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.

19 Power Factor Correction Example

20 Distribution System Capacitors

21 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

22 Balanced 3  -- No Neutral Current

23 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

24 Three Phase - Wye Connection There are two ways to connect 3  systems – Wye (Y) – Delta (  )

25 Wye Connection Line Voltages V an V cn V bn V ab V ca V bc -V bn Line to line voltages are also balanced (α = 0 in this case)

26 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

27 Delta Connection I ca IcIc I ab I bc IaIa IbIb

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

29 Three Phase Example, cont’d

30 Delta-Wye Transformation

31 Delta-Wye Transformation Proof

32 Delta-Wye Transformation, cont’d

33 Three Phase Transmission Line