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

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

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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

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3 Phasor Representation

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

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

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6 RL Circuit Example

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7 Complex Power

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8 Complex Power, cont’d

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

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10 Complex Power, cont’d

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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*

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

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13 Power Consumption in Devices

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14 Example First solve basic circuit

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

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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

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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

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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.

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19 Power Factor Correction Example

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20 Distribution System Capacitors

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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

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22 Balanced 3 -- No Neutral Current

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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

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

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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)

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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

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

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

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29 Three Phase Example, cont’d

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30 Delta-Wye Transformation

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31 Delta-Wye Transformation Proof

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32 Delta-Wye Transformation, cont’d

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33 Three Phase Transmission Line

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