ING eléctrica

1. Lecture 10
Transformers, Generators, Load, Ybus
Professor Tom Overbye
Department of Electrical and
Computer Engineering
ECE 476
POWER SYSTEM ANALYSIS

2. 1
Announcements
 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

3. 2
Load Tap Changing Transformers
 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"

4. 3
LTCs and Circulating Vars
slack
1 1.00 pu
2 3
40.2 MW
40.0 MW
1.7 Mvar
-0.0 Mvar
1.000 tap 1.056 tap
24.1 MW
12.8 Mvar
24.0 MW
-12.0 Mvar
A
MVA
1.05 pu
0.98 pu
24 MW
12 Mvar
64 MW
14 Mvar
40 MW
0 Mvar
0.0 Mvar
80%
A
MVA

5. 4
Phase Shifting Transformers
 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.

6. 5
Phase Shifter Example 3.13
slack
Phase Shifting Transformer
345.00 kV 341.87 kV
0.0 deg
216.3 MW 216.3 MW
283.9 MW 283.9 MW
1.05000 tap
39.0 Mvar 6.2 Mvar
93.8 Mvar
125.0 Mvar
500 MW
164 Mvar 500 MW
100 Mvar

7. 6
ComED Control Center

8. 7
ComED Phase Shifter Display

9. 8
Phase Shifting Transformer Picture
230 kV 800 MVA Phase Shifting
Transformer During factory testing
Source: Tom Ernst, Minnesota Power
Costs about $7 million,
weighs about 1.2
million pounds

10. 9
Autotransformers
 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 auto-
transformers 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!

11. 10
Could it Happen Tomorrow?
 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

12. 11
Could It Happen Tomorrow?
• 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

13. 12
Load Models
 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
– 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
– constant power: Si = Pi + jQi
– constant impedance: Si = |V|2 / Zi

14. 13
Generator Models
 Engineering models depend upon application
 Generators are usually synchronous machines
 For generators we will use two different models:
– 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

15. 14
Power Flow Analysis
 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)
– 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

16. 15
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

17. 16
Linear Power System Elements
Resistors, inductors, capacitors, independent
voltage sources and current sources are linear
circuit elements
1
V = R I V = V =
Such systems may be analyzed by superposition
j L I I
j C



18. 17
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

19. 18
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 f(x) = x2 + 2
two solutions where f(x) = 0 no solution f(x) = 0

20. 19
Multiple Solution Example 3
The dc system shown below has two solutions:
where the 18 watt
load is a resistive
load
2
2
Load
Load
Load
The equation we're solving is
9 volts
I 18 watts
1 +R
One solution is R 2
Other solution is R 0.5
Load Load
R R
 
 
 

 
 
 
What is the
maximum
PLoad?