The document outlines a power system analysis lecture focusing on power flow analysis and homework assignments. It includes technical details such as bus configurations, line and transformer data, and emphasizes the importance of reliability standards, particularly NERC's enforcement of compliance. Additionally, the analysis utilizes a five-bus example to illustrate computational methods and system operations.

1. EE 369
POWER SYSTEM ANALYSIS
Lecture 14
Power Flow
Tom Overbye and Ross Baldick
1

2. Announcements
Read Chapter 12, concentrating on sections
12.4 and 12.5.
Homework 12 is 6.43, 6.48, 6.59, 6.61,
12.19, 12.22, 12.20, 12.24, 12.26, 12.28,
12.29; due Tuesday Nov. 25.
2

3. 400 MVA
15 kV
400 MVA
15/345 kV
T1
T2
800 MVA
345/15 kV
800 MVA
15 kV
520 MVA
80 MW40 Mvar
280 MVAr 800 MW
Line 3
345 kV
Line2
Line1
345 kV
100 mi
345 kV
200 mi
50 mi
1 4 3
2
5
Single-line diagram
The N-R Power Flow: 5-bus Example
3

4. Bus Type
|V|
per
unit
θ
degrees
PG
per
unit
QG
per
unit
PL
per
unit
QL
per
unit
QGmax
per
unit
QGmin
per
unit
1 Slack 1.0 0   0 0  
2 Load   0 0 8.0 2.8  
3 Constant
voltage
1.05  5.2  0.8 0.4 4.0 -2.8
4 Load   0 0 0 0  
5 Load   0 0 0 0  
Table 1.
Bus input
data
Bus-to-
Bus
R
per unit
X
per unit
G
per unit
B
per unit
Maximum
MVA
per unit
2-4 0.0090 0.100 0 1.72 12.0
2-5 0.0045 0.050 0 0.88 12.0
4-5 0.00225 0.025 0 0.44 12.0
Table 2.
Line input data
The N-R Power Flow: 5-bus Example
4

5. Bus-to-
Bus
R
per
unit
X
per
unit
Gc
per
unit
Bm
per
unit
Maximum
MVA
per unit
Maximum
TAP
Setting
per unit
1-5 0.00150 0.02 0 0 6.0 —
3-4 0.00075 0.01 0 0 10.0 —
Table 3.
Transformer
input data
Bus Input Data Unknowns
1 |V1 |= 1.0, θ1 = 0 P1, Q1
2 P2 = PG2-PL2 = -8
Q2 = QG2-QL2 = -2.8
|V2|, θ2
3 |V3 |= 1.05
P3 = PG3-PL3 = 4.4
Q3, θ3
4 P4 = 0, Q4 = 0 |V4|, θ4
5 P5 = 0, Q5 = 0 |V5|, θ5
Table 4. Input data
and unknowns
The N-R Power Flow: 5-bus Example
5

6. Let the Computer Do the Calculations!
(Ybus Shown)
6

7. Ybus Details
02321 == YY
24
24 24
1 1
0.89276 9.91964
0.009 0.1
Y j per unit
R jX j
− −
= = = − +
+ +
25
25 25
1 1
1.78552 19.83932
0.0045 0.05
Y j per unit
R jX j
− −
= = = − +
+ +
24 25
22
24 24 25 25
1 1
2 2
B B
Y j j
R jX R jX
= + + +
+ +
2
88.0
2
72.1
)83932.1978552.1()91964.989276.0( jjjj ++−+−=
unitperj °−∠=−= 624.845847.284590.2867828.2
Elements of Ybus connected to bus 2
7

8. Here are the Initial Bus Mismatches
8

9. And the Initial Power Flow Jacobian
9

10. Five Bus Power System Solved
slack
One
Tw o
ThreeFourFive
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1.000 pu 0.974 pu
0.834 pu
1.019 pu
1.050 pu
0.000 Deg - 4.548 Deg
-22.406 Deg
-2.834 Deg
- 0.597 Deg
395 M W
114 M var
520 M W
337 M var
800 M W
280 M var
80 M W
40 M var
10

11. 37 Bus Example Design Case
slack
Metropolis Light and Pow er Electric Design Case 2
SL A C K 3 4 5
SL A C K 1 3 8
R A Y 3 4 5
R A Y 1 3 8
R A Y 6 9
FER N A 6 9
A
MVA
D EM A R 6 9
B L T 6 9
B L T 1 3 8
B O B 1 3 8
B O B 6 9
W O L EN 6 9
SH I M K O 6 9
R O GER 6 9
U I U C 6 9
P ET E6 9
H I SK Y 6 9
T I M 6 9
T I M 1 3 8
T I M 3 4 5
P A I 6 9
GR O SS6 9
H A N N A H 6 9
A M A N D A 6 9
H O M ER 6 9
L A U F6 9
M O R O 1 3 8
L A U F1 3 8
H A L E6 9
P A T T EN 6 9
W EB ER 6 9
B U C K Y 1 3 8
SA V O Y 6 9
SA V O Y 1 3 8
JO 1 3 8 JO 3 4 5
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1 .0 3 p u
1 .0 2 p u
1 .0 3 p u
1 .0 3 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 0 p u
1 .0 2 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 2 p u
1 .0 0 p u
1 .0 0 p u
1 .0 2 p u
0 .9 9 p u
0 .9 9 p u
1 .0 0 p u
1 .0 2 p u
1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 0 p u 1 .0 0 p u
1 .0 1 p u
1 .0 2 p u
1 .0 2 p u
1 .0 2 p u
1 .0 3 p u
A
MVA
1 .0 2 p u
A
MVA
A
MVA
L Y N N 1 3 8
A
MVA
1 .0 2 p u
A
MVA
1 .0 0 p u
A
MVA
Syst em Losses: 10.70 MW
2 2 0 M W
5 2 M v a r
1 2 M W
3 M v a r
2 0 M W
1 2 M v a r
1 2 4 M W
4 5 M v a r
3 7 M W
1 3 M v a r
1 2 M W
5 M v a r
1 5 0 M W
0 M v a r
5 6 M W
1 3 M v a r
1 5 M W
5 M v a r
1 4 M W
2 M v a r
3 8 M W
3 M v a r
4 5 M W
0 M v a r
2 5 M W
3 6 M v a r
3 6 M W
1 0 M v a r
1 0 M W
5 M v a r
2 2 M W
1 5 M v a r
6 0 M W
1 2 M v a r
2 0 M W
2 8 M v a r
2 3 M W
7 M v a r
3 3 M W
1 3 M v a r
1 5 .9 M v a r 1 8 M W
5 M v a r
5 8 M W
4 0 M v a r
6 0 M W
1 9 M v a r
1 4 .2 M v a r
2 5 M W
1 0 M v a r
2 0 M W
3 M v a r
2 3 M W
6 M v a r 1 4 M W
3 M v a r
4 .9 M v a r
7 .3 M v a r
1 2 .8 M v a r
2 8 .9 M v a r
7 .4 M v a r
0 .0 M v a r
5 5 M W
2 5 M v a r
3 9 M W
1 3 M v a r
1 5 0 M W
0 M v a r
1 7 M W
3 M v a r
1 6 M W
- 1 4 M v a r
1 4 M W
4 M v a r
KYLE6 9
A
MVA
11

12. Good Power System Operation
• Good power system operation requires that
there be no “reliability” violations (needing to
shed load, have cascading outages, or other
unacceptable conditions) for either the current
condition or in the event of statistically likely
contingencies:
• Reliability requires as a minimum that there be no
transmission line/transformer limit violations and
that bus voltages be within acceptable limits
(perhaps 0.95 to 1.08)
• Example contingencies are the loss of any single
device. This is known as n-1 reliability. 12

13. Good Power System Operation
• North American Electric Reliability Corporation
now has legal authority to enforce reliability
standards (and there are now lots of them).
• See http://www.nerc.com for details (click on
Standards)
13

14. Looking at the Impact of Line Outages
slack
Metropolis Light and Pow er Electric Design Case 2
SL A C K 3 4 5
SL A C K 1 3 8
R A Y 3 4 5
R A Y 1 3 8
R A Y 6 9
FER N A 6 9
A
MVA
D EM A R 6 9
B L T 6 9
B L T 1 3 8
B O B 1 3 8
B O B 6 9
W O L EN 6 9
SH I M K O 6 9
RO GER6 9
U I U C 6 9
P ET E6 9
H I SK Y 6 9
T I M 6 9
T I M 1 3 8
T I M 3 4 5
P A I 6 9
GR O SS6 9
H A N N A H 6 9
A M A N DA 6 9
H O M ER 6 9
L A U F6 9
M O R O 1 3 8
L A U F1 3 8
H A L E6 9
P A T T EN 6 9
W EB ER 6 9
B U C K Y 1 3 8
SA V O Y 6 9
SA V O Y 1 3 8
JO 1 3 8 JO 3 4 5
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1 .0 3 p u
1 .0 2 p u
1 .0 3 p u
1 .0 3 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 0 p u
1 .0 2 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 2 p u
1 .0 1 p u
1 .0 0 p u
1 .0 2 p u
0 .9 0 p u
0 .9 0 p u
0 .9 4 p u
1 .0 1 p u
0 .9 9 p u
1 .0 0 p u
1 .0 0 p u
1 .0 0 p u 1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 2 p u
1 .0 2 p u
1 .0 3 p u
A
MVA
1 .0 2 p u
A
MVA
A
MVA
L Y N N 1 3 8
A
MVA
1 .0 2 p u
A
MVA
1 .0 0 p u
A
MVA
Syst em Losses: 17.61 MW
2 2 7 M W
4 3 M v a r
1 2 M W
3 M v a r
2 0 M W
1 2 M v a r
1 2 4 M W
4 5 M v a r
3 7 M W
1 3 M v a r
1 2 M W
5 M v a r
1 5 0 M W
4 M v a r
5 6 M W
1 3 M v a r
1 5 M W
5 M v a r
1 4 M W
2 M v a r
3 8 M W
9 M v a r
4 5 M W
0 M v a r
2 5 M W
3 6 M v a r
3 6 M W
1 0 M v a r
1 0 M W
5 M v a r
2 2 M W
1 5 M v a r
6 0 M W
1 2 M v a r
2 0 M W
4 0 M v a r
2 3 M W
7 M v a r
3 3 M W
1 3 M v a r
1 6 .0 M v a r 1 8 M W
5 M v a r
5 8 M W
4 0 M v a r
6 0 M W
1 9 M v a r
1 1 .6 M v a r
2 5 M W
1 0 M v a r
2 0 M W
3 M v a r
2 3 M W
6 M v a r 1 4 M W
3 M v a r
4 .9 M v a r
7 .2 M v a r
1 2 .8 M v a r
2 8 .9 M v a r
7 .3 M v a r
0 .0 M v a r
5 5 M W
3 2 M v a r
3 9 M W
1 3 M v a r
1 5 0 M W
4 M v a r
1 7 M W
3 M v a r
1 6 M W
- 1 4 M v a r
1 4 M W
4 M v a r
KYLE6 9
A
MVA
8 0 %
A
MVA
1 3 5 %
A
M VA
1 1 0 %
A
M VA
Opening
one line
(Tim69-
Hannah69)
causes
overloads.
This would
not be
Allowed.
14

15. Contingency Analysis
Contingency
analysis provides
an automatic
way of looking
at all the
statistically
likely
contingencies. In
this example the
contingency set
is all the single
line/transformer
outages
15

16. Power Flow And Design
• One common usage of the power flow is to
determine how the system should be modified
to remove contingencies problems or serve new
load
• In an operational context this requires working with
the existing electric grid, typically involving re-
dispatch of generation.
• In a planning context additions to the grid can be
considered as well as re-dispatch.
• In the next example we look at how to remove
the existing contingency violations while serving
new load. 16

17. An Unreliable Solution:
some line outages result in overloads
slack
Metropolis Light and Pow er Electric Design Case 2
SL A C K 3 4 5
SL A C K 1 3 8
R A Y 3 4 5
R A Y 1 3 8
R A Y 6 9
FER N A 6 9
A
MVA
D EM A R 6 9
B L T 6 9
B L T 1 3 8
B O B 1 3 8
B O B 6 9
W O L EN6 9
SH I M K O 6 9
RO GER 6 9
U I U C 6 9
P ET E6 9
H I SK Y 6 9
T I M 6 9
T I M 1 3 8
T I M 3 4 5
P A I 6 9
GR O SS6 9
H A N NA H 6 9
A M A N DA 6 9
H O M ER 6 9
L A U F6 9
M O R O 1 3 8
L A U F1 3 8
H A L E6 9
P A T T EN 6 9
W EB ER6 9
B U CK Y 1 3 8
SA V O Y 6 9
SA V O Y 1 3 8
JO 1 3 8 JO 3 4 5
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1 .0 2 p u
1 .0 1 p u
1 .0 2 p u
1 .0 3 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 0 p u
1 .0 2 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 2 p u
0 .9 9 p u
1 .0 0 p u
1 .0 2 p u
0 .9 7 p u
0 .9 7 p u
0 .9 9 p u
1 .0 2 p u
1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 0 p u 1 .0 0 p u
1 .0 1 p u
1 .0 2 p u
1 .0 2 p u
1 .0 2 p u
1 .0 3 p u
A
MVA
1 .0 2 p u
A
MVA
A
MVA
L Y N N1 3 8
A
MVA
1 .0 2 p u
A
MVA
1 .0 0 p u
A
MVA
Syst em Losses: 14.49 MW
2 6 9 M W
6 7 M v a r
1 2 M W
3 M v a r
2 0 M W
1 2 M v a r
1 2 4 M W
4 5 M v a r
3 7 M W
1 3 M v a r
1 2 M W
5 M v a r
1 5 0 M W
1 M v a r
5 6 M W
1 3 M v a r
1 5 M W
5 M v a r
1 4 M W
2 M v a r
3 8 M W
4 M v a r
4 5 M W
0 M v a r
2 5 M W
3 6 M v a r
3 6 M W
1 0 M v a r
1 0 M W
5 M v a r
2 2 M W
1 5 M v a r
6 0 M W
1 2 M v a r
2 0 M W
4 0 M v a r
2 3 M W
7 M v a r
3 3 M W
1 3 M v a r
1 5 .9 M v a r 1 8 M W
5 M v a r
5 8 M W
4 0 M v a r
6 0 M W
1 9 M v a r
1 3 .6 M v a r
2 5 M W
1 0 M v a r
2 0 M W
3 M v a r
2 3 M W
6 M v a r 1 4 M W
3 M v a r
4 .9 M v a r
7 .3 M v a r
1 2 .8 M v a r
2 8 .9 M v a r
7 .4 M v a r
0 .0 M v a r
5 5 M W
2 8 M v a r
3 9 M W
1 3 M v a r
1 5 0 M W
1 M v a r
1 7 M W
3 M v a r
1 6 M W
- 1 4 M v a r
1 4 M W
4 M v a r
K YLE6 9
A
MVA
9 6 %
A
MVA
Case now
has nine
separate
contingencies
having
reliability
violations
(overloads in
post-contingency
system).
17

18. A Reliable Solution:
no line outages result in overloads
slack
Metropolis Light and Pow er Electric Design Case 2
SL A C K 3 4 5
SL A C K 1 3 8
R A Y 3 4 5
R A Y 1 3 8
R A Y 6 9
FER N A 6 9
A
MVA
D EM A R 6 9
B L T 6 9
B L T 1 3 8
B O B 1 3 8
B O B 6 9
W O L EN 6 9
SH I M K O 6 9
RO GER6 9
U I U C 6 9
P ET E6 9
H I SK Y 6 9
T I M 6 9
T I M 1 3 8
T I M 3 4 5
P A I 6 9
GR O SS6 9
H A N N A H 6 9
A M A N D A 6 9
H O M ER 6 9
L A U F6 9
M O R O 1 3 8
L A U F1 3 8
H A L E6 9
P A T T EN 6 9
W EB ER 6 9
B U C K Y 1 3 8
SA V O Y 6 9
SA V O Y 1 3 8
JO 1 3 8 JO 3 4 5
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1 .0 3 p u
1 .0 1 p u
1 .0 2 p u
1 .0 3 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 0 p u
1 .0 2 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 2 p u
1 .0 0 p u
0 .9 9 p u
1 .0 2 p u
0 .9 9 p u
0 .9 9 p u
1 .0 0 p u
1 .0 2 p u
1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 0 p u 1 .0 0 p u
1 .0 1 p u
1 .0 2 p u
1 .0 2 p u
1 .0 2 p u
1 .0 3 p u
A
MVA
1 .0 2 p u
A
MVA
A
MVA
L Y N N 1 3 8
A
MVA
1 .0 2 p u
A
MVA
A
MVA
Syst em Losses: 11.66 MW
2 6 6 M W
5 9 M v a r
1 2 M W
3 M v a r
2 0 M W
1 2 M v a r
1 2 4 M W
4 5 M v a r
3 7 M W
1 3 M v a r
1 2 M W
5 M v a r
1 5 0 M W
1 M v a r
5 6 M W
1 3 M v a r
1 5 M W
5 M v a r
1 4 M W
2 M v a r
3 8 M W
4 M v a r
4 5 M W
0 M v a r
2 5 M W
3 6 M v a r
3 6 M W
1 0 M v a r
1 0 M W
5 M v a r
2 2 M W
1 5 M v a r
6 0 M W
1 2 M v a r
2 0 M W
3 8 M v a r
2 3 M W
7 M v a r
3 3 M W
1 3 M v a r
1 5 .8 M v a r 1 8 M W
5 M v a r
5 8 M W
4 0 M v a r
6 0 M W
1 9 M v a r
1 4 .1 M v a r
2 5 M W
1 0 M v a r
2 0 M W
3 M v a r
2 3 M W
6 M v a r 1 4 M W
3 M v a r
4 .9 M v a r
7 .3 M v a r
1 2 .8 M v a r
2 8 .9 M v a r
7 .4 M v a r
0 .0 M v a r
5 5 M W
2 9 M v a r
3 9 M W
1 3 M v a r
1 5 0 M W
1 M v a r
1 7 M W
3 M v a r
1 6 M W
- 1 4 M v a r
1 4 M W
4 M v a r
KYLE6 9
A
MVA
Ky le1 3 8
A
M V A
Previous
case was
augmented
with the
addition of a
138 kV
Transmission
Line
18

19. Generation Changes and The Slack
Bus
• The power flow is a steady-state analysis tool,
so the assumption is total load plus losses is
always equal to total generation
• Generation mismatch is made up at the slack bus
• When doing generation change power flow
studies one always needs to be cognizant of
where the generation is being made up
• Common options include “distributed slack,” where
the mismatch is distributed across multiple
generators by participation factors or by economics.19

20. Generation Change Example 1
slack
SL A C K 3 4 5
SL A C K 1 3 8
R A Y 3 4 5
R A Y 1 3 8
R A Y 6 9
FER N A 6 9
A
MVA
D EM A R 6 9
B L T 6 9
B L T 1 3 8
B O B 1 3 8
B O B 6 9
W O L EN 6 9
SH I M K O 6 9
R O GER6 9
U I U C 6 9
P ET E6 9
H I SK Y 6 9
T I M 6 9
T I M 1 3 8
T I M 3 4 5
P A I 6 9
GR O SS6 9
H A N N A H 6 9
A M A N D A 6 9
H O M ER 6 9
L A U F 6 9
M O R O 1 3 8
L A U F 1 3 8
H A L E6 9
P A T T EN 6 9
W EB ER 6 9
B U C K Y 1 3 8
SA V O Y 6 9
SA V O Y 1 3 8
JO 1 3 8 JO 3 4 5
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
0 .0 0 p u
- 0 .0 1 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
- 0 .0 3 p u
- 0 .0 1 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
- 0 .0 3 p u
- 0 .0 1 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
-0 .0 0 2 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u 0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
A
MVA
- 0 .0 1 p u
A
MVA
A
MVA
L Y N N 1 3 8
A
MVA
0 .0 0 p u
A
MVA
0 .0 0 p u
A
MVA
1 6 2 M W
3 5 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
- 1 5 7 M W
- 4 5 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
2 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
3 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
4 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
- 0 .1 M v a r 0 M W
0 M v a r
0 M W
0 M v a r0 M W
0 M v a r
- 0 .1 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r 0 M W
0 M v a r
- 0 .1 M v a r
0 .0 M v a r
- 0 .1 M v a r
- 0 .2 M v a r
0 .0 M v a r
0 .0 M v a r
0 M W
5 1 M v a r
0 M W
0 M v a r
0 M W
2 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
Display shows
“Difference
Flows”
between
original
37 bus case,
and case with
a BLT138
generation
outage;
note all the
power change
is picked
up at the slack
Slack bus
20

21. Generation Change Example 2
slack
SL A C K 3 4 5
SL A CK 1 3 8
RA Y 3 4 5
R A Y 1 3 8
RA Y 6 9
FER NA 6 9
A
MVA
D EM A R6 9
B L T 6 9
B L T 1 3 8
B O B 1 3 8
B O B 6 9
W O L EN6 9
SH I M K O 6 9
RO GER 6 9
UI U C6 9
P ET E6 9
H I SK Y 6 9
T I M 6 9
T I M 1 3 8
T I M 3 4 5
P A I 6 9
GR O SS6 9
H A N NA H 6 9
A M A N DA 6 9
H O M ER 6 9
L A UF6 9
M O RO 1 3 8
L A UF1 3 8
H A L E6 9
P A T T EN6 9
W EB ER 6 9
B U CK Y 1 3 8
SA V O Y 6 9
SA V O Y 1 3 8
JO 1 3 8 JO 3 4 5
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
0 .0 0 p u
-0 .0 1 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
- 0 .0 3 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
- 0 .0 3 p u
- 0 .0 1 p u
- 0 .0 1 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
-0 .0 0 3 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u 0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
0 .0 0 p u
A
MVA
0 .0 0 p u
A
MVA
A
MVA
L Y N N 1 3 8
A
MVA
0 .0 0 p u
A
MVA
0 .0 0 p u
A
MVA
0 M W
3 7 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
- 1 5 7 M W
- 4 5 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
4 2 M W
- 1 4 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
9 9 M W
- 2 0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
- 0 .1 M v a r 0 M W
0 M v a r
0 M W
0 M v a r0 M W
0 M v a r
- 0 .1 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r 0 M W
0 M v a r
0 .0 M v a r
0 .0 M v a r
- 0 .1 M v a r
- 0 .2 M v a r
- 0 .1 M v a r
0 .0 M v a r
1 9 M W
5 1 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
0 M W
0 M v a r
Display repeats previous case except now the change in
generation is picked up by other generators using a
“participation factor” (change is shared amongst generators) approach.
21

22. Voltage Regulation Example: 37 Buses
Display shows voltage contour of the power system
slack
SL A C K 3 4 5
SL A C K 1 3 8
RA Y 3 4 5
R A Y 1 3 8
RA Y 6 9
FERN A 6 9
A
MVA
D EM A R6 9
B L T 6 9
B L T 1 3 8
B O B 1 3 8
B O B 6 9
W O L EN6 9
SH I M K O 6 9
R O GER 6 9
UI UC 6 9
P ET E6 9
H I SK Y 6 9
T I M 6 9
T I M 1 3 8
T I M 3 4 5
P A I 6 9
GR O SS6 9
H A NN A H 6 9
A M A N D A 6 9
H O M ER 6 9
L A U F6 9
M O RO 1 3 8
L A U F1 3 8
H A L E6 9
P A T T EN 6 9
W EB ER 6 9
B U C K Y 1 3 8
SA V O Y 6 9
SA V O Y 1 3 8
JO 1 3 8 JO 3 4 5
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1 .0 3 p u
1 .0 1 p u
1 .0 2 p u
1 .0 3 p u
1 .0 1 p u
1 .0 0 p u
1 .0 0 p u
0 .9 9 p u
1 .0 2 p u
1 .0 1 p u
1 .0 0 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 1 p u
1 .0 2 p u
1 .0 0 p u
1 .0 0 p u
1 .0 2 p u
0 .9 9 7 p u
0 .9 9 p u
1 .0 0 p u
1 .0 2 p u
1 .0 0 p u
1 .0 1 p u
1 .0 0 p u
1 .0 0 p u 1 .0 0 p u
1 .0 1 p u
1 .0 2 p u
1 .0 2 p u
1 .0 2 p u
1 .0 3 p u
A
MVA
1 .0 2 p u
A
MVA
A
MVA
L Y N N1 3 8
A
MVA
1 .0 2 p u
A
MVA
1 .0 0 p u
A
MVA
2 1 9 M W
5 2 M v a r
2 1 M W
7 M v a r
4 5 M W
1 2 M v a r
1 5 7 M W
4 5 M v a r
3 7 M W
1 3 M v a r
1 2 M W
5 M v a r
1 5 0 M W
0 M v a r
5 6 M W
1 3 M v a r
1 5 M W
5 M v a r
1 4 M W
2 M v a r
3 8 M W
3 M v a r
4 5 M W
0 M v a r
5 8 M W
3 6 M v a r
3 6 M W
1 0 M v a r
0 M W
0 M v a r
2 2 M W
1 5 M v a r
6 0 M W
1 2 M v a r
2 0 M W
9 M v a r
2 3 M W
7 M v a r
3 3 M W
1 3 M v a r
1 5 .9 M v a r 1 8 M W
5 M v a r
5 8 M W
4 0 M v a r5 1 M W
1 5 M v a r
1 4 .3 M v a r
3 3 M W
1 0 M v a r
1 5 M W
3 M v a r
2 3 M W
6 M v a r 1 4 M W
3 M v a r
4 .8 M v a r
7 .2 M v a r
1 2 .8 M v a r
2 9 .0 M v a r
7 .4 M v a r
2 0 .8 M v a r
9 2 M W
1 0 M v a r
2 0 M W
8 M v a r
1 5 0 M W
0 M v a r
1 7 M W
3 M v a r
0 M W
0 M v a r
1 4 M W
4 M v a r
1 .0 1 0 p u
0.0 M va r
Syst em Losses: 11.51 MW
22
Automatic voltage regulation system controls voltages.

23. Real-sized Power Flow Cases
• Real power flow studies are usually done with
cases with many thousands of buses
• Outside of ERCOT, buses are usually grouped into
various balancing authority areas, with each area
doing its own interchange control.
• Cases also model a variety of different
automatic control devices, such as generator
reactive power limits, load tap changing
transformers, phase shifting transformers,
switched capacitors, HVDC transmission lines,
and (potentially) FACTS devices. 23

24. Sparse Matrices and Large Systems
• Since for realistic power systems the model
sizes are quite large, this means the Ybus and
Jacobian matrices are also large.
• However, most elements in these matrices are
zero, therefore special techniques, sparse
matrix/vector methods, are used to store the
values and solve the power flow:
• Without these techniques large systems would be
essentially unsolvable.
24

25. Eastern Interconnect Example
Pe o r ia
R o ck f o r d
Nor th Chi cago
Abbott Labs Park
U. S. N Tr ai ni ng
O l d El m
Deerfi el d
Nor thbr ook
Lakehur st
W a u k e g a n
Zi on
G ur nee
Anti och
P l e a s a n t
Round Lake
Z i o n ( 1 3 8 k V )
Lake Zur i ch
Lesthon
Aptaki si c
Buf fal o G roove
Wheel i ng
Pr ospect Hei ghts
Pal ati ne
Arl i ngt on
M ount Prospect
Pr ospect
G ol f Mi l l
Des Pl ai nes
El mhur st
I tasca
G a r f i e l d
Tol l w ay
W407 (Fermi )
Wi l son
Bar ri ngt on
D u n d e e
S i l v e r L a k e
C h e r r y V a l l e y
W e m p l e t o n
N e l s o n
H - 4 7 1 ( N W S t e e l )
Pa d d o c k
P o n t i a c M i d p o i n t
Br ai dw ood
S t a t e L i n e
S h e f i e l d
C h i a v e
M u n s t e r
S t . J o h n
El ectr i c Juncti on
Pl a n o
L a S a l l e
Lombar d
L i s l e
Co l l i n s
D r e s d e n
L o c k p o r t
Ea s t F r a n k f o r t
Go o d i n g s G r o v e
Li bert yvi l l e
345 kV
Li bertyvi l l e
138 kV
L a k e G e o r g e
D u n a c r
G r e e n Ac r e s
S c h a h f e r
T o w e r R d
B a b c o c k
Hei ght s
P r a i r i e
R a c i n e
M i c h i g a n C i t y
E l w o o d
D e q u i n e
L o u i s a
E a s t M o l i n e
S u b 9 1
W a l c o t t
D a v e n p o r t
Su b 9 2
R o c k C r k .
S a l e m
G I L M A N
W A T SE K A 1 7 G O D L N D
E L PA S O T
M I N O N K T
O G L ES BY
1 5 5 6 A T P
O T T A W A T
O G L SB Y M
O G L E S; T
H E N N E P I N
E S K T A P
L T V T P N
L T V T P E
H E N N E ; T
L T V S T L
P RI N C T P
P R I N C T N
RI C H L A N D
KE W A N I P
S S T T AP
G AL E S BR G
N O RM A ; BN O R M A ; R
R F A L ; R
M O N M O U T H
GA L ES BR 5
KE W A N ;
H A L L O C K
C A T M O S S
F AR GO
S P N G B AY
E P EO RI A
R SW E AS T
P I O N E ER C
RA D N O R
C AT T A P
C A T S U B 1
SB 1 8 5
E M O L I N E
S B 4 3 5
S B 1 1 2 5
K PE C KT P5
S O . SU B 5
S B 8 5 5
S B 3 1 T 5
S B 2 8 5
S B 1 7 5
SB 4 9 5
SB 5 3 5
S B 4 7 5
SB 4 8 5
SB A 5
S B 7 0 5
S B 7 9 5
S B 8 8 5
SB 7 1 5
B VR C H 6 5 B V R C H 5
A L B A N Y 5
YO R K 5
S AV A N N A 5
G A L EN A 5
8 T H ST . 5
L O RE 5
S O . GV W . 5
SA L EM N 5
A L B A N Y 6
G A RD E ;
H 7 1 ; B T
H 7 1 ; B
H 7 1 ; R
R F A L ; B
N E L S O ; R
N EL SO ; R T
S T E RL ; B
D I X O N ; B T
M E CC O R D 3
C O R D O ;
Q u a d Ci t i e s
L E E C O ; B P
B y r o n
M AR YL ; B
M E N D O ; T
S T I L L ; R T
B4 2 7 ; 1 T
L A N C A; R
PE C A T ; B
F RE EP ;
E L E R O ; BT EL E R O ; R T
L E N A ; R
L EN A ; B
H 4 4 0 ; R T
H 4 4 0 ; R
S T E W A ; B
H 4 4 5 ; 3 B
R o s c o e
P i e r p o n t
S PE C; R
F O RD A ; R
H a r l e m
S a n d Pa r k
N W T 1 3 8
B L K 1 3 8
R O R 1 3 8
J A N 1 3 8
A L B 1 3 8
N O M 1 3 8
D A R 1 3 8
H L M 1 3 8
P O T 1 3 8
M RE 1 3 8
CO R 1 3 8 D I K 1 3 8
B C H 1 3 8
Sa b r o o k e
B l a w k h a w k
A l p i n e
E . R o c k f o r d
C h a r l e s
B e l v i d e r e
B4 6 5
M a r e n g o
W I B 1 3 8
W B T 1 3 8
EL K 1 3 8
N L G 1 3 8
N L K G V T
S G R C K5
B RL GT N 1
B RL GT N 2
S G R CK 4
U N I V R S T Y
U N I V N E U
W H T W T R 5
W H T W T R 4
W H T W T R 3
S U N 1 3 8
V I K 1 3 8
L B T 1 3 8
T I C H I GN
PA R I S W E
A L B E R S - 2
C434
El m wood
Ni l es
Evanston
Devon
Rose Hi l l
Skoki e
Nort hw est
Dr i ver
F o r d C i t y
H a y f o r d
S a w y e r
Nor thri dge
Hi ggi nsDes Pl ai nes
Fr ankl i n Park
Oak Par k
Ri dgel and
D799
G al ew ood
Y450
Congr ess
Rockw el l
Cl ybour n
Q u a r r y
L a s a l l e
St a t e
Crosby
Ki ngsbury
J e f f e r s o n
O h i o
T a y l o r
C l i n t
D e k o v
F i s k
Cr a w f o r d
U n i v e r s i t y
R i v e r
Z - 4 9 4
W a s h i n g t o n P a r k
H a r b o r
Ca l u m e t
H e g e w i s c h
Z - 7 1 5
S o u t h H o l l a n d
E v e r g r e e n
D a m e n
W a l l a c e
B e v e r l y
G 3 8 5 1
Z - 5 2 4
G3 8 5 2
W i l d w o o d
H a r v e y
Gr e e n L a k e
S a n d Ri d g e
C h i c a g o H e i g h t s
B u r n h a m
L a n s i n g
F - 5 7 5
F - 5 0 3
G l e n w o o d
B l o o m
P a r k F o r e s tM a t t e s o n
C o u n t r y Cl u b H i l l s
Al t G E
Natoma
W o o d h i l l
U . P a r k
M o k e n
M cHenry
Cr ystal Lake
Al gonqui n
Huntl ey
P V a l
W o o d s t o c k
Bl u e I s l a n d
G 3 9 4
A l s i p
C r e s t w o o d
K- 3 1 9 # 1
K - 3 1 9 # 2
B r a d l e y
K a n k a k e e
D a v i s Cr e e k
W i l m i n g t o n
W i l t o n Ce n t e r
F r a n k f o r t
N L e n
B r i g g
O akbrook
D o w n e r s G r o o v e
W o o d r i d g e
W 6 0 4
W 6 0 3
Bo l i n g b r o o k
S u g a r Gr o v e
W. De Kal b G l i dden
N Au r o r a
El gi n
Hanover
Spaul di ng
Bart l et t
Hof fman Est at es
S. Schaumber g
Tonne
Landm
Busse
Schaum berg
How ar d
Ber kel ey
Bel l w ood
La G range
Chur ch
Addi son
Nor di
G l endal e
Gl en El l yn
But te
York Cent er
D775
Be d f o r d P a r k
C l e a r n i n g
Sa y r e
B r i d g e v i e w
T i n l e y Pa r k
Ro b e r t s
P a l o s
R o m e o
W i l l ow
Bur r Ri dge
J o 4 5 6
J 3 2 2
South El gi n Wayne
West Chicago
Auror a
Warr envi l l e
W 5 0 7
M o n t g o m e r y
O s w e g o
W o l f C r e e k
F r o n t e n a c
W 600 ( Napervi l l e)
W 6 0 2
W 6 0 1
J 3 0 7
S a n d w i c h
Water man
J 3 2 3
M a s o n
J - 3 7 1
J - 3 7 5
J - 3 3 9
St r e a t o r
M a r s e i l l e s
L a s a l l e
N L A SA L
M e n d o t a
J 3 7 0
S h o r e
G o o s e L a k e
J - 3 0 5
J - 3 9 0
J - 3 2 6
P l a i n f i e l d
J - 3 3 2
A r c h e r
B e l l R o a d
Wi l l Co.
H i l l c r e s t R o c k d a l e
Jol i et
K e n d r a
C r e t e
U p n o r
L A K EV I E W
B A I N 4
Kenosha
S O M E R S
S T R I T A
BI G B EN D
M U K W O N G O
N ED 1 3 8
N E D 1 6 1
L A N 1 3 8
E E N 1 3 8
CA S VI L L 5
T RK R I V5
L I B E R T Y 5
A S B U R Y 5
C N T RG RV 5
J U L I A N 5
M Q O KE T A5
E C A L M S 5
G R M N D 5
D E W I T T 5
SB H Y C5
S U B 7 7 5
S B 7 4 5
S B 9 0 5
S B 7 8 5
D A V N P R T 5
S B 7 6 5
SB 5 8 5
S B 5 2 5
S B 8 9 5
I PS C O 5
I PS CO 3
N EW P O R T 5
H W Y 6 1 5
W E S T 5
9 S U B 5
T R I P P
Z - 1 0 0
O r l a n
Ke n d a
M P W S P L I T
W YO M I N G5
M T V ER N 5
B E R T R A M 5
P CI 5
S B J I C 5
S B UI C 5
- 0 . 4 0 d e g
2 . 3 5 d e g
- 1 3 . 3 d e g
- 1 3 . 4 d e g
M c C o o k
- 1 . 1 d e g
1 . 9 d e g
0 . 6 d e g
9 3 %
B
MVA
1 0 5 %
B
MVA
Example, which models the Eastern Interconnect
contains about 43,000 buses. 25

26. Solution Log for 1200 MW Outage
In this example the
losss of a 1200 MW
generator in Northern
Illinois was simulated.
This caused
a generation imbalance
in the associated
balancing authority
area, which was
corrected by a
redispatch of local
generation.
26

27. Interconnected Operation
Power systems are interconnected across
large distances.
For example most of North America east of
the Rockies is one system, most of North
America west of the Rockies is another.
Most of Texas and Quebec are each
interconnected systems.
27

28. Balancing Authority Areas
A “balancing authority area” (previously called a
“control area”) has traditionally represented the
portion of the interconnected electric grid
operated by a single utility or transmission
entity.
Transmission lines that join two areas are
known as tie-lines.
The net power out of an area is the sum of the
flow on its tie-lines.
The flow out of an area is equal to
total gen - total load - total losses = tie-line flow28

29. Area Control Error (ACE)
The area control error is a combination of:
the deviation of frequency from nominal, and
the difference between the actual flow out of an
area and the scheduled (agreed) flow.
That is, the area control error (ACE) is the
difference between the actual flow out of an
area minus the scheduled flow, plus a
frequency deviation component:
ACE provides a measure of whether an area
is producing more or less than it should to
satisfy schedules and to contribute to
controlling frequency. 29
actual tie-line flow schedACE 10P P fβ= − + ∆

30. Area Control Error (ACE)
The ideal is for ACE to be zero.
Because the load is constantly changing,
each area must constantly change its
generation to drive the ACE towards zero.
For ERCOT, the historical ten control areas
were amalgamated into one in 2001, so the
actual and scheduled interchange are
essentially the same (both small compared
to total demand in ERCOT).
In ERCOT, ACE is predominantly due to
frequency deviations from nominal since
there is very little scheduled flow to or from
other areas. 30

31. Automatic Generation Control
Most systems use automatic generation
control (AGC) to automatically change
generation to keep their ACE close to zero.
Usually the control center (either ISO or
utility) calculates ACE based upon tie-line
flows and frequency; then the AGC module
sends control signals out to the generators
every four seconds or so.
31

32. Power Transactions
Power transactions are contracts between
generators and (representatives of) loads.
Contracts can be for any amount of time at
any price for any amount of power.
Scheduled power transactions between
balancing areas are called “interchange” and
implemented by setting the value of Psched
used in the ACE calculation:
ACE = Pactual tie-lineflow – Psched + 10β Δf
…and then controlling the generation to bring
ACE towards zero.
32

33. “Physical” power Transactions
• For ERCOT, interchange is only relevant over
asynchronous connections between ERCOT
and Eastern Interconnection or Mexico.
• In Eastern and Western Interconnection,
interchange occurs between areas connected
by AC lines.
33

34. Three Bus Case on AGC:
no interchange.
Bus 2 Bus 1
Bus 3Home Area
266 MW
133 MVR
150 MW
250 MW
34 MVR
166 MVR
133 MW
67 MVR
1.00 PU
-40 MW
8 MVR
40 MW
-8 MVR
-77 MW
25 MVR
78 MW
-21 MVR
39 MW
-11 MVR
-39 MW
12 MVR
1.00 PU
1.00 PU
101 MW
5 MVR
100 MW
AGC ON
AVR ON
AGC ON
AVR ON
Net tie-line flow is
close to zero
Generation
is automatically
changed to match
change in load
34

35. 100 MW Transaction between
areas in Eastern or Western
Bus 2 Bus 1
Bus 3Home Area
Scheduled Transactions
225 MW
113 MVR
150 MW
291 MW
8 MVR
138 MVR
113 MW
56 MVR
1.00 PU
8 MW
-2 MVR
-8 MW
2 MVR
-84 MW
27 MVR
85 MW
-23 MVR
93 MW
-25 MVR
-92 MW
30 MVR
1.00 PU
1.00 PU
0 MW
32 MVR
100 MW
AGC ON
AVR ON
AGC ON
AVR ON
100.0 MW
Scheduled
100 MW
Transaction from Left to Right
Net tie-line
flow is now
100 MW
35

36. PTDFs
Power transfer distribution factors (PTDFs)
show the linearized impact of a transfer of
power.
PTDFs calculated using the fast decoupled
power flow B matrix:
1
Once we know we can derive the change in
the transmission line flows to evaluate PTDFs.
Note that we can modify several elements in ,
in proportion to how the specified generators would
par
−
∆ = ∆
∆
∆
θ B P
θ
P
ticipate in the power transfer. 36

37. Nine Bus PTDF Example
10%
60%
55%
64%
57%
11%
74%
24%
32%
A
G
B
C
D
E
I
F
H
300.0 MW
400.0 MW 300.0 MW
250.0 MW
250.0 MW
200.0 MW
250.0 MW
150.0 MW
150.0 MW
44%
71%
0.00 deg
71.1 MW
92%
Figure shows initial flows for a nine bus power system
37

38. Nine Bus PTDF Example, cont'd
43%
57%
13%
35%
20%
10%
2%
34%
34%
32%
A
G
B
C
D
E
I
F
H
300.0 MW
400.0 MW 300.0 MW
250.0 MW
250.0 MW
200.0 MW
250.0 MW
150.0 MW
150.0 MW
34%
30%
0.00 deg
71.1 MW
Figure now shows percentage PTDF flows for a change in transaction from A to I
38

39. Nine Bus PTDF Example, cont'd
6%
6%
12%
61%
12%
6%
19%
21%
21%
A
G
B
C
D
E
I
F
H
300.0 MW
400.0 MW 300.0 MW
250.0 MW
250.0 MW
200.0 MW
250.0 MW
150.0 MW
150.0 MW
20%
18%
0.00 deg
71.1 MW
Figure now shows percentage PTDF flows for a change in transaction from G to F
39

40. WE to TVA PTDFs
40

41. Line Outage Distribution Factors
(LODFs)
• LODFs are used to approximate the change in
the flow on one line caused by the outage of a
second line
– typically they are only used to determine the
change in the MW flow compared to the pre-
contingency flow if a contingency were to occur,
– LODFs are used extensively in real-time
operations,
– LODFs are approximately independent of flows but
do depend on the assumed network topology.
41

42. Line Outage Distribution Factors
(LODFs)
42
,
change in flow on line ,
due to outage of line .
pre-contingency flow on line
,
Estimates change in flow on line
if outage on line were to occur.
l
k
l l k k
P l
k
P k
P LODF P
l
k
∆ =
=
∆ ≈

43. Line Outage Distribution Factors
(LODFs)
43
,
If line initially had 100 MW of flow on it,
and line initially had 50 MW flow on it,
and then there was an outage of line ,
if =0.1 then the increase in flow
on line after a continge
k
l
l k
k P
l P
k
LODF
l
=
=
,
ncy of line would be:
0.1 100 10 MW
from 50 MW to 60 MW.
l l k k
k
P LODF P∆ ≈ = × =

44. Flowgates
• The real-time loading of the power grid can be
assessed via “flowgates.”
• A flowgate “flow” is the real power flow on
one or more transmission elements for either
base case conditions or a single contingency
– Flows in the event of a contingency are
approximated in terms of pre-contingency flows
using LODFs.
• Elements are chosen so that total flow has a
relation to an underlying physical limit. 44

45. Flowgates
• Limits due to voltage or stability limits are
often represented by effective flowgate limits,
which are acting as “proxies” for these other
types of limits.
• Flowgate limits are also often used to
represent thermal constraints on corridors of
multiple lines between zones or areas.
• The inter-zonal constraints that were used in
ERCOT until December 2010 are flowgates
that represent inter-zonal corridors of lines. 45