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

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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. 2

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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 ), where: v(t) and i(t) are the instantaneous voltage and current as a function of time t, is the angular frequency (2πf, with f the frequency in Hertz), V max and I max are the magnitudes of voltage and current sinusoids, v and I are angular offsets of the peaks of sinusoids from a reference waveform. Root Mean Square (RMS) voltage of sinusoid: 3

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Phasor Representation 4

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Phasor Representation, contd (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. 5

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

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RL Circuit Example 7

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Complex Power 8

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Complex Power, contd 9

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

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Complex Power, contd 12

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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 Kirchhoffs 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*. 13

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Conservation of Power Example Earlier we found I = amps = 1600W + j1200VAr Power flowing from source to load at bus 14

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Power Consumption in Devices 15

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Example First solve basic circuit I 16

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

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Power System Notation Power system components are usually shown as one-line diagrams. Previous circuit redrawn. Arrows are used to show loads Generators are shown as circles Transmission lines are shown as a single line 18

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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. 19

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Reactive Compensation, contd 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 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 loads power factor to an arbitrary value. 20

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Power Factor Correction Example 21

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Distribution System Capacitors 22

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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. 23

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Balanced 3 -- Zero Neutral Current 24

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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. 25

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

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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) 27

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Wye Connection, contd 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. 28

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Delta Connection I ca IcIc I ab I bc IaIa IbIb 29

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Three Phase Example Assume a -connected load, with each leg Z = is supplied from a kV (L-L) source 30

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Three Phase Example, contd 31

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Delta-Wye Transformation 32

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Delta-Wye Transformation Proof

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Delta-Wye Transformation, contd 34

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Three Phase Transmission Line 35

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