Complex Power, Reactive Compensation, Three Phase

1
ECE 476POWER SYSTEM ANALYSIS

• Lecture 2
• Complex Power, Reactive Compensation, Three Phase
  
• Professor Tom Overbye
• Department of Electrical andComputer Engineering

2
Reading and Homework

• For lectures 2 through 3 please be reading
  Chapters 1 and 2
• HW 1 is 2.7, 12, 21, 26 due Thursday 9/4

3
Vertical Monopolies

• Within a particular geographic market, the
  electric utility had an exclusive franchise

In return for this exclusive franchise, the
utility had the obligation to serve all existing
and future customers at rates determined
jointly by utility and regulators It was a cost
plus business
4
Vertical Monopolies

• Within its service territory each utility was the
  only game in town
• Neighboring utilities functioned more as
  colleagues than competitors
• Utilities gradually interconnected their systems
  so by 1970 transmission lines crisscrossed North
  America, with voltages up to 765 kV
• Economies of scale keep resulted in decreasing
  rates, so most every one was happy

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Current Midwest Electric Grid
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History, contd -- 1970s

• 1970s brought inflation, increased fossil-fuel
  prices, calls for conservation and growing
  environmental concerns
• Increasing rates replaced decreasing ones
• As a result, U.S. Congress passed Public
  Utilities Regulator Policies Act (PURPA) in 1978,
  which mandated utilities must purchase power from
  independent generators located in their service
  territory (modified 2005)
• PURPA introduced some competition

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History, contd 1990s 2000s

• Major opening of industry to competition occurred
  as a result of National Energy Policy Act of 1992
• This act mandated that utilities provide
  nondiscriminatory access to the high voltage
  transmission
• Goal was to set up true competition in generation
  
• Result over the last few years has been a
  dramatic restructuring of electric utility
  industry (for better or worse!)
• Energy Bill 2005 repealed PUHCA modified PURPA

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State Variation in Electric Rates
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The Goal Customer Choice
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The Result for California in 2000/1
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The California-Enron Effect
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August 14th, 2003 Blackout
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2007 Illinois Electricity Crisis

• Two main electric utilities in Illinois are ComEd
  and Ameren
• Restructuring law had frozen electricity prices
  for ten years, with rate decreases for many.
• Prices rose on January 1, 2007 as price freeze
  ended price increases were especially high for
  electric heating customers who had previously
  enjoyed rates as low as 2.5 cents/kWh
• Current average residential rate (in cents/kWh)
  is 10.4 in IL, 8.74 IN, 11.1 WI, 7.94 MO, 9.96
  IA, 19.56 CT, 6.09 ID, 14.03 in CA, 10.76 US
  average

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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)

Root Mean Square (RMS) voltage of sinusoid
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Phasor Representation
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Phasor Representation, contd
(Note Some texts use boldface type for
complex numbers, or bars on the top)
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Advantages of Phasor Analysis
(Note Z is a complex number but not a
phasor)
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RL Circuit Example
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Complex Power
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Complex Power, contd
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Complex Power
(Note S is a complex number but not a
phasor)
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Complex Power, contd
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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 power follows since S VI

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Conversation of Power Example
Earlier we found I 20?-6.9? amps
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Power Consumption in Devices
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Example
First solve basic circuit
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Example, contd
Now add additional reactive power load and resolve
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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
Transmission lines are shown as a single line
Generators are shown as circles
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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
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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 I2 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 loads
  power factor to an arbitrary value.

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Power Factor Correction Example
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Distribution System Capacitors
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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

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Balanced 3? -- No Neutral Current
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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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Three Phase - Wye Connection

• There are two ways to connect 3? systems
• Wye (Y)
• Delta (?)

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Wye Connection Line Voltages
-Vbn
(a 0 in this case)
Line to line voltages are also balanced
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Wye Connection, contd

• 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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Delta Connection
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Three Phase Example

• Assume a ?-connected load is supplied from a 3?
  13.8 kV (L-L) source with Z 100?20?W

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Three Phase Example, contd
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Delta-Wye Transformation
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Delta-Wye Transformation Proof
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Delta-Wye Transformation, contd
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Three Phase Transmission Line