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ECE 476
POWER SYSTEM ANALYSIS
Lecture 10
Transformers, Generators, Load, Ybus
Professor Tom Overbye
Department of Electrical and
Computer Engineering
Announcements
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Be reading Chapter 6.
HW 3 is due now.
HW 4 is 3.4, 3.10, 3.14, 3.19, 3.23, 3.60; due September 29
in class.
First exam is October 11 during class. Closed book, closed
notes, one note sheet and calculators allowed
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Load Tap Changing Transformers
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LTC transformers have tap ratios that can be varied
to regulate bus voltages
The typical range of variation is 10% from the
nominal values, usually in 33 discrete steps
(0.0625% per step).
Because tap changing is a mechanical process, LTC
transformers usually have a 30 second deadband to
avoid repeated changes.
Unbalanced tap positions can cause "circulating
vars"
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LTCs and Circulating Vars
slack
64 MW
14 Mvar
1.00 pu
1
24.1 MW
12.8 Mvar
40.2 MW
1.7 Mvar
1.000 tap
A
A
80%
1.056 tap
MVA
MVA
40.0 MW
-0.0 Mvar
24.0 MW
-12.0 Mvar
0.98 pu
2
3
1.05 pu
0.0 Mvar
24 MW
12 Mvar
40 MW
0 Mvar
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Phase Shifting Transformers
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Phase shifting transformers are used to control the
phase angle across the transformer
Since power flow through the transformer depends
upon phase angle, this allows the transformer to
regulate the power flow through the transformer
Phase shifters can be used to prevent inadvertent
"loop flow" and to prevent line overloads.
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Phase Shifter Example 3.13
345.00 kV
500 MW
341.87 kV
283.9 MW
39.0 Mvar
283.9 MW
6.2 Mvar
slack
164 Mvar
Phase Shifting Transformer
216.3 MW
125.0 Mvar
500 MW
100 Mvar
216.3 MW
0.0 deg
93.8 Mvar
1.05000 tap
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ComED Control Center
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ComED Phase Shifter Display
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Phase Shifting Transformer Picture
Costs about $7 million,
weighs about 1.2
million pounds
230 kV 800 MVA Phase Shifting
Transformer During factory testing
Source: Tom Ernst, Minnesota Power
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Autotransformers
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Autotransformers are transformers in which the
primary and secondary windings are coupled
magnetically and electrically.
This results in lower cost, and smaller size and
weight.
The key disadvantage is loss of electrical isolation
between the voltage levels. Hence autotransformers are not used when a is large. For
example in stepping down 7160/240 V we do not
ever want 7160 on the low side!
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Could it Happen Tomorrow?
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Geomagnetic disturbances (GMDs) impact the
power grid by causing geomagenetic induced dc
currents (GICs) that can push the transformers into
saturation.
Saturated
transformers
have high
harmonics which
leads to high
reactive losses and
heating
Image from Ed Schweitzer June 2011 JASON Presentation
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Could It Happen Tomorrow?
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A 1989 storm caused a major blackout in Quebec.
Much larger storms have occurred in the past, such
as in 1859, which knocked out much of the
telegraph system in the Eastern US
A 2010 Metatech Report
indicated an 1859 type
event could destroy
hundreds of EHV
transformers, crippling
our grid for months!
Metatech R-319, Figure 4.11
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Load Models
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Ultimate goal is to supply loads with electricity at
constant frequency and voltage
Electrical characteristics of individual loads matter,
but usually they can only be estimated
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actual loads are constantly changing, consisting of a large
number of individual devices
only limited network observability of load characteristics
Aggregate models are typically used for analysis
Two common models
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constant power: Si = Pi + jQi
constant impedance: Si = |V|2 / Zi
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Generator Models
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Engineering models depend upon application
Generators are usually synchronous machines
For generators we will use two different models:
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a steady-state model, treating the generator as a constant
power source operating at a fixed voltage; this model
will be used for power flow and economic analysis
a short term model treating the generator as a constant
voltage source behind a possibly time-varying reactance
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Power Flow Analysis
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We now have the necessary models to start to
develop the power system analysis tools
The most common power system analysis tool is the
power flow (also known sometimes as the load flow)
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power flow determines how the power flows in a network
also used to determine all bus voltages and all currents
because of constant power models, power flow is a
nonlinear analysis technique
power flow is a steady-state analysis tool
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Linear versus Nonlinear Systems
A function H is linear if
H(a1m1 + a2m2) = a1H(m1) + a2H(m2)
That is
1) the output is proportional to the input
2) the principle of superposition holds
Linear Example: y = H(x) = c x
y = c(x1+x2) = cx1 + c x2
Nonlinear Example: y = H(x) = c x2
y = c(x1+x2)2 ≠ (cx1)2 + (c x2)2
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Linear Power System Elements
Resistors, inductors, capacitors, independent
voltage sources and current sources are linear
circuit elements
1
V = R I V = j L I V =
I
j C
Such systems may be analyzed by superposition
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Nonlinear Power System Elements
Constant power loads and generator injections are
nonlinear and hence systems with these elements can
not be analyzed by superposition
Nonlinear problems can be very difficult to solve,
and usually require an iterative approach
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Nonlinear Systems May Have
Multiple Solutions or No Solution
Example 1: x2 - 2 = 0 has solutions x = 1.414…
Example 2: x2 + 2 = 0 has no real solution
f(x) = x2 - 2
two solutions where f(x) = 0
f(x) = x2 + 2
no solution f(x) = 0
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Multiple Solution Example 3
The dc system shown below has two solutions:
where the 18 watt
load is a resistive
load
The equation we're solving is
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 9 volts 
I RLoad  
RLoad  18 watts

 1 +R Load 
What is the
One solution is R Load  2
maximum
Other solution is R Load  0.5
PLoad?
2
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