EE 369
POWER SYSTEM ANALYSIS
Lecture 14
Power Flow
Tom Overbye and Ross Baldick
1
Announcements
Read Chapter 12, concentrating on sections
12.4 and 12.5.
Homework 11 is 6.24, 6.26, 6.28, 6.30 (see
figure 6.18 and table 6.9 for system), 6.38,
6.42 (note in Ybus in problem 6.34 should
have Y32 = Y23 = j5, not j2 as stated), 6.43,
6.46, 6.49, 6.50; due Tuesday 11/24. Note
that HW is due on Tuesday because
Thanksgiving is on Thursday.
2
The N-R Power Flow: 5-bus Example
1
T1
5
T2
800 MVA
4 345/15 kV
Line 3
345 kV
50 mi
345 kV
100 mi
Line 1
400 MVA
15/345 kV
Line 2
400 MVA
15 kV
345 kV
200 mi
3
520 MW
800 MVA
15 kV
40 Mvar 80 MW
2
280 MVAr
800 MW
Single-line diagram
3
The N-R Power Flow: 5-bus Example
Table 1.
Bus input
data
θ
degrees
PG
per
unit
QG
per
unit
PL
per
unit
Bus
Type
|V|
per
unit
1
Slack
1.0
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 2.
Line input data
QL
per
unit
QGmax
per
unit
QGmin
per
unit
0


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
Bus-toBus
4
The N-R Power Flow: 5-bus Example
Table 3.
Transformer
input data
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
—
Bus-toBus
Bus
Table 4. Input data
and unknowns
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
5
Let the Computer Do the Calculations!
(Ybus Shown)
6
Selected Ybus Details
Entries of Ybus relating to elements connected to bus 2.
Note that resistances, inductive reactances, and admittances
come from Table 2; subscripts on them refer to line from-to.
Subscripts on Ybus correspond to entries of that matrix.
Y21  Y23  0
Y24 
1
1

 0.89276  j 9.91964 per unit
R24  jX 24 0.009  j 0.1
Y25 
1
1

 1.78552  j19.83932 per unit
R25  jX 25 0.0045  j 0.05
Y22 
1
1
B
B

 j 24  j 25
R24  jX 24 R25  jX 25
2
2
 (0.89276  j 9.91964)  (1.78552  j19.83932)  j
1.72
0.88
j
2
2
 2.67828  j 28.4590  28.5847  84.624 per unit
7
Here are the Initial Bus Mismatches
8
And the Initial Power Flow Jacobian
9
Five Bus Power System Solved
One
395 MW
114 Mvar
A
MVA
Five
Four
A
MVA
Three
520 MW
A
MVA
337 Mvar
slack
1.000 pu
0.000 Deg
0.974 pu
-4.548 Deg
0.834 pu
-22.406 Deg
A
A
MVA
MVA
1.019 pu
-2.834 Deg
80 MW
40 Mvar
1.050 pu
-0.597 Deg
Two
800 MW
280 Mvar
10
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 such as overloads past
capacity) 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 capacity limit
violations and that bus voltages be within
acceptable limits (perhaps 0.95 to 1.08)
• Example contingencies are the loss of any single
11
device. This is known as n-1 reliability.
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)
• Consider impact of line contingency on 37 bus
design example case.
12
37 Bus Example Design Case
Metropolis Light and Power Electric Design Case 2
SLA C K3 4 5
A
MVA
A
MVA
1 .0 3 pu
sla ck
System Losses: 10.70 MW
1 .0 2 pu
T IM 3 4 5
A
A
MVA
MVA
A
SLA C K1 3 8
1 .0 2 pu
MVA
RA Y 1 3 8
A
A
1 .0 3 pu
A
MVA
MVA
T IM 1 3 8
1 .0 0 pu
3 3 MW
1 3 M var
A
A
1 .0 2 pu
1 5 .9 M var
2 3 MW
7 M var
1 .0 1 pu
M O RO 1 3 8
3 7 MW
A
FERNA 6 9
A
1 .0 0 pu
DEM A R6 9
KYLE69
A
A
2 0 MW
1 2 M var
UIUC 6 9
1 .0 0 pu
1 2 .8 M var
A
MVA
MVA
2 5 MW
3 6 M var
A M A NDA 6 9
5 6 MW
1 .0 1 pu
A
MVA
MVA
MVA
SH IM KO 6 9
7 .4 M var
5 5 MW
2 5 M var
1 5 MW
5 M var
A
P A T T EN6 9
A
MVA
1 .0 1 pu
A
A
MVA
MVA
2 3 MW
6 M var
1 0 MW
5 M var
LA UF1 3 8
BUC KY 1 3 8
RO GER6 9
2 M var
A
MVA
SA V O Y 6 9
1 .0 2 pu
A
3 8 MW
3 M var
JO 1 3 8
MVA
A
MVA
1 4 MW
1 4 MW
3 M var
1 .0 2 pu
1 .0 1 pu
MVA
MVA
4 5 MW
0 M var
WEBER6 9
2 2 MW
1 5 M var
1 .0 1 pu
A
1 .0 0 pu
LA UF6 9
1 .0 2 pu
A
MVA
1 .0 0 pu
A
MVA
MVA
A
7 .3 M var
1 .0 2 pu
MVA
A
3 6 MW
1 0 M var
MVA
0 .0 M var
1 .0 0 pu
A
BLT 6 9
MVA
2 0 MW
2 8 M var
MVA
A
MVA
6 0 MW
1 2 M var
LY NN1 3 8
1 4 MW
4 M var
A
BLT 1 3 8
1 .0 0 pu
1 .0 1 pu
H A LE6 9
MVA
A
A
1 3 M var
1 6 MW
-1 4 M var
A
MVA
A
2 0 MW
3 M var
1 .0 0 pu
BO B6 9
1 2 4 MW
4 5 M var
A
A
2 5 MW
1 0 M var
MVA
A
MVA
1 .0 2 pu
A
A
MVA
BO B1 3 8
A
MVA
MVA
MVA
MVA
MVA
H O M ER6 9
A
1 .0 1 pu
2 8 .9 M var
1 .0 0 pu
WO LEN6 9
4 .9 M var
5 8 MW
4 0 M var
MVA
1 4 .2 M var
0 .9 9 pu
1 .0 1 pu
1 2 MW
3 M var
MVA
A
1 3 M var
MVA
A
P ET E6 9
A
3 9 MW
1 3 M var
H A NNA H 6 9
6 0 MW
1 9 M var
0 .9 9 pu
GRO SS6 9
MVA
MVA
1 2 MW
5 M var
RA Y 6 9
1 7 MW
3 M var
A
H ISKY 6 9
MVA
1 .0 2 pu
MVA
MVA
A
A
1 .0 3 pu
MVA
A
MVA
1 8 MW
5 M var
P A I6 9
1 .0 1 pu
T IM 6 9
A
MVA
A
MVA
A
MVA
MVA
1 .0 0 pu
2 2 0 MW
5 2 M var
RA Y 3 4 5
1 .0 1 pu
A
MVA
SA V O Y 1 3 8
JO 3 4 5
A
1 5 0 MW
0 M var
MVA
A
MVA
1 5 0 MW
0 M var
A
MVA
1 .0 2 pu
A
1 .0 3 pu
MVA
13
Looking at the Impact of Line Outages
Metropolis Light and Power Electric Design Case 2
SLA C K3 4 5
A
MVA
A
MVA
1 .0 3 pu
1 .0 2 pu
sla ck
System Losses: 17.61 MW
T IM 3 4 5
A
A
MVA
MVA
A
SLA C K1 3 8
1 .0 2 pu
MVA
RA Y 1 3 8
A
A
1 .0 3 pu
A
MVA
MVA
T IM 1 3 8
1 .0 1 pu
3 3 MW
1 3 M var
A
A
1 .0 2 pu
1 6 .0 M var
2 3 MW
7 M var
1 .0 1 pu
MVA
1 .0 0 pu
0 .9 0 pu
GRO SS6 9
3 7 MW
A
FERNA 6 9
A
1 .0 0 pu
DEM A R6 9
KYLE69
A
A
2 0 MW
1 2 M var
UIUC 6 9
1 .0 0 pu
1 2 .8 M var
A
MVA
MVA
2 5 MW
3 6 M var
A M A NDA 6 9
110%
5 6 MW
1 .0 1 pu
MVA
MVA
5 5 MW
3 2 M var
A
3 6 MW
1 0 M var
MVA
0 .9 9 pu
1 5 MW
5 M var
1 .0 0 pu
A
MVA
2 3 MW
6 M var
A
80%
A
A
1 0 MW
5 M var
MVA
LA UF1 3 8
BUC KY 1 3 8
RO GER6 9
2 M var
A
MVA
SA V O Y 6 9
1 .0 2 pu
A
3 8 MW
9 M var
JO 1 3 8
MVA
A
MVA
1 4 MW
1 4 MW
3 M var
1 .0 1 pu
1 .0 0 pu
MVA
MVA
4 5 MW
0 M var
WEBER6 9
2 2 MW
1 5 M var
1 .0 1 pu
P A T T EN6 9
A
MVA
A
A
MVA
1 .0 0 pu
LA UF6 9
1 .0 1 pu
1 .0 2 pu
MVA
MVA
A
7 .2 M var
1 .0 0 pu
0 .0 M var
2 0 MW
4 0 M var
SH IM KO 6 9
7 .3 M var
MVA
MVA
6 0 MW
1 2 M var
MVA
A
BLT 6 9
MVA
A
A
MVA
A
1 .0 1 pu
H A LE6 9
MVA
LY NN1 3 8
1 4 MW
4 M var
A
BLT 1 3 8
1 .0 0 pu
A
135%
1 3 M var
1 6 MW
-1 4 M var
A
MVA
A
2 0 MW
3 M var
0 .9 4 pu
BO B6 9
1 2 4 MW
4 5 M var
A
A
2 5 MW
1 0 M var
MVA
A
MVA
1 .0 2 pu
A
A
MVA
BO B1 3 8
A
MVA
MVA
MVA
MVA
MVA
H O M ER6 9
A
1 .0 1 pu
2 8 .9 M var
1 .0 0 pu
WO LEN6 9
4 .9 M var
5 8 MW
4 0 M var
MVA
1 1 .6 M var
0 .9 0 pu
1 .0 1 pu
1 2 MW
3 M var
P ET E6 9
MVA
A
1 3 M var
MVA
A
MVA
A
3 9 MW
1 3 M var
H A NNA H 6 9
6 0 MW
1 9 M var
1 2 MW
5 M var
RA Y 6 9
1 7 MW
3 M var
A
H ISKY 6 9
M O RO 1 3 8
1 .0 2 pu
MVA
MVA
MVA
1 .0 3 pu
MVA
A
A
1 8 MW
5 M var
P A I6 9
1 .0 1 pu
T IM 6 9
A
MVA
A
MVA
A
MVA
MVA
Opening
one line
(Tim69Hannah69)
causes
overloads.
This would
not be
acceptable
under NERC
standards.
2 2 7 MW
4 3 M var
RA Y 3 4 5
1 .0 1 pu
A
MVA
SA V O Y 1 3 8
JO 3 4 5
A
1 5 0 MW
4 M var
MVA
A
MVA
1 5 0 MW
4 M var
A
MVA
1 .0 2 pu
A
1 .0 3 pu
MVA
14
Contingency Analysis
Contingency
analysis provides
an automatic
way of looking
at all the
contingencies in
a specified
“contingency set.”
In this example the
contingency set
is all the single
line/transformer
outages
15
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 redispatch 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 add a
new line in order to remove the existing
contingency violations while serving new load.
16
An Unreliable Solution:
some line outages result in overloads
Metropolis Light and Power Electric Design Case 2
SLA C K3 4 5
A
MVA
A
MVA
1 .0 2 pu
Case now
has nine
separate
contingencies
having
reliability
violations
(overloads in
post-contingency
system).
sla ck
System Losses: 14.49 MW
1 .0 2 pu
T IM 3 4 5
A
A
MVA
MVA
A
SLA C K1 3 8
1 .0 1 pu
MVA
RA Y 1 3 8
A
A
1 .0 3 pu
A
MVA
MVA
T IM 1 3 8
0 .9 9 pu
3 3 MW
1 3 M var
A
A
1 .0 2 pu
1 5 .9 M var
2 3 MW
7 M var
1 .0 1 pu
0 .9 7 pu
3 7 MW
A
FERNA 6 9
A
1 .0 0 pu
DEM A R6 9
KYLE69
A
2 0 MW
1 2 M var
UIUC 6 9
1 .0 0 pu
1 2 .8 M var
A
MVA
2 5 MW
1 0 M var
1 2 4 MW
4 5 M var
A
5 6 MW
A
MVA
MVA
5 5 MW
2 8 M var
A
3 6 MW
1 0 M var
MVA
1 5 MW
5 M var
MVA
1 .0 1 pu
A
A
MVA
MVA
2 3 MW
6 M var
A
P A T T EN6 9
1 0 MW
5 M var
LA UF1 3 8
BUC KY 1 3 8
RO GER6 9
2 M var
A
MVA
SA V O Y 6 9
1 .0 2 pu
A
3 8 MW
4 M var
JO 1 3 8
MVA
A
MVA
1 4 MW
1 4 MW
3 M var
1 .0 2 pu
1 .0 1 pu
MVA
MVA
4 5 MW
0 M var
WEBER6 9
2 2 MW
1 5 M var
1 .0 1 pu
A
1 .0 0 pu
LA UF6 9
A
A
MVA
A
1 .0 2 pu
1 .0 2 pu
MVA
MVA
A
7 .3 M var
1 .0 0 pu
0 .0 M var
1 .0 0 pu
SH IM KO 6 9
7 .4 M var
MVA
MVA
2 0 MW
4 0 M var
A
MVA
6 0 MW
1 2 M var
LY NN1 3 8
1 4 MW
4 M var
MVA
BLT 6 9
1 .0 1 pu
H A LE6 9
MVA
A
A
MVA
BLT 1 3 8
1 .0 0 pu
A
A
A
1 3 M var
1 6 MW
-1 4 M var
A
MVA
1 .0 1 pu
A M A NDA 6 9
BO B6 9
MVA
A
0 .9 7 pu
MVA
A
MVA
A
2 5 MW
3 6 M var
MVA
MVA
1 .0 2 pu
A
MVA
MVA
2 0 MW
3 M var
0 .9 9 pu
BO B1 3 8
A
MVA
A
MVA
A
1 .0 1 pu
2 8 .9 M var
1 .0 0 pu
WO LEN6 9
4 .9 M var
5 8 MW
4 0 M var
MVA
1 3 .6 M var
H O M ER6 9
1 .0 1 pu
1 2 MW
3 M var
MVA
A
1 3 M var
MVA
A
P ET E6 9
A
3 9 MW
1 3 M var
H A NNA H 6 9
6 0 MW
1 9 M var
1 2 MW
5 M var
GRO SS6 9
MVA
MVA
MVA
RA Y 6 9
1 7 MW
3 M var
A
H ISKY 6 9
96%
M O RO 1 3 8
1 .0 2 pu
MVA
MVA
A
A
1 .0 2 pu
MVA
A
MVA
1 8 MW
5 M var
P A I6 9
1 .0 1 pu
T IM 6 9
A
MVA
A
MVA
A
MVA
MVA
1 .0 0 pu
2 6 9 MW
6 7 M var
RA Y 3 4 5
1 .0 1 pu
A
MVA
SA V O Y 1 3 8
JO 3 4 5
A
1 5 0 MW
1 M var
MVA
A
MVA
1 5 0 MW
1 M var
A
MVA
1 .0 2 pu
A
1 .0 3 pu
MVA
17
A Reliable Solution:
no line outages result in overloads
Metropolis Light and Power Electric Design Case 2
SLA C K3 4 5
A
MVA
A
MVA
1 .0 2 pu
sla ck
System Losses: 11.66 MW
1 .0 2 pu
T IM 3 4 5
A
A
MVA
MVA
A
SLA C K1 3 8
1 .0 1 pu
MVA
RA Y 1 3 8
A
A
1 .0 3 pu
A
MVA
MVA
T IM 1 3 8
1 .0 0 pu
Previous
case was
augmented
with the
addition of a
138 kV
Transmission
Line
3 3 MW
1 3 M var
A
A
1 .0 2 pu
1 5 .8 M var
2 3 MW
7 M var
1 .0 1 pu
M O RO 1 3 8
3 7 MW
A
FERNA 6 9
A
1 .0 0 pu
Kyle138
KYLE69
A
A
2 0 MW
1 2 M var
UIUC 6 9
1 .0 0 pu
1 2 .8 M var
A
MVA
MVA
2 5 MW
3 6 M var
MVA
5 6 MW
2 5 MW
1 0 M var
MVA
MVA
5 5 MW
2 9 M var
MVA
MVA
1 5 MW
5 M var
1 .0 1 pu
A
A
MVA
MVA
2 3 MW
6 M var
A
A
1 0 MW
5 M var
LA UF1 3 8
BUC KY 1 3 8
RO GER6 9
2 M var
A
MVA
SA V O Y 6 9
1 .0 2 pu
A
3 8 MW
4 M var
JO 1 3 8
MVA
A
MVA
1 4 MW
1 4 MW
3 M var
1 .0 2 pu
1 .0 1 pu
MVA
MVA
4 5 MW
0 M var
WEBER6 9
2 2 MW
1 5 M var
1 .0 1 pu
P A T T EN6 9
A
MVA
A
A
MVA
1 .0 0 pu
LA UF6 9
1 .0 2 pu
1 .0 2 pu
MVA
MVA
A
7 .3 M var
1 .0 0 pu
0 .0 M var
1 .0 0 pu
SH IM KO 6 9
7 .4 M var
MVA
A
3 6 MW
1 0 M var
A
2 0 MW
3 8 M var
A
BLT 6 9
MVA
6 0 MW
1 2 M var
MVA
MVA
A
1 .0 1 pu
H A LE6 9
MVA
LY NN1 3 8
1 4 MW
4 M var
A
BLT 1 3 8
1 .0 0 pu
A
A
2 0 MW
3 M var
1 .0 0 pu
A
A
MVA
1 .0 1 pu
1 3 M var
1 6 MW
-1 4 M var
A
A M A NDA 6 9
BO B6 9
1 2 4 MW
4 5 M var
A
A
0 .9 9 pu
A
MVA
1 .0 2 pu
A
MVA
MVA
MVA
MVA
MVA
BO B1 3 8
A
MVA
MVA
A
1 .0 1 pu
DEM A R6 9
A
WO LEN6 9
4 .9 M var
5 8 MW
4 0 M var
2 8 .9 M var
1 4 .1 M var
H O M ER6 9
1 .0 1 pu
1 2 MW
3 M var
MVA
M VA
A
1 3 M var
MVA
A
P ET E6 9
A
3 9 MW
1 3 M var
H A NNA H 6 9
6 0 MW
1 9 M var
0 .9 9 pu
GRO SS6 9
A
MVA
1 2 MW
5 M var
RA Y 6 9
1 7 MW
3 M var
MVA
H ISKY 6 9
MVA
1 .0 2 pu
MVA
MVA
A
A
1 .0 3 pu
MVA
A
MVA
1 8 MW
5 M var
P A I6 9
1 .0 1 pu
T IM 6 9
A
MVA
A
MVA
A
MVA
MVA
0 .9 9 pu
2 6 6 MW
5 9 M var
RA Y 3 4 5
1 .0 1 pu
A
MVA
SA V O Y 1 3 8
JO 3 4 5
A
1 5 0 MW
1 M var
MVA
A
MVA
1 5 0 MW
1 M var
A
MVA
1 .0 2 pu
A
1 .0 3 pu
MVA
18
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
Generation Change Example 1
SLA C K3 4 5
A
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
MVA
A
Slack bus
MVA
0 .0 0 pu
1 6 2 MW
3 5 M var
RA Y 3 4 5
sla ck
0 .0 0 pu
T IM 3 4 5
A
A
MVA
MVA
A
SLA C K1 3 8
-0 .0 1 pu
A
MVA
RA Y 1 3 8
A
0 .0 0 pu
A
MVA
T IM 1 3 8
0 .0 0 pu
MVA
0 MW
0 M var
A
A
A
0 .0 0 pu
-0 .1 M var
0 MW
0 M var
MVA
MVA
MVA
MVA
-0 .0 1 pu
RA Y 6 9
0 .0 0 pu
T IM 6 9
P A I6 9
0 .0 0 pu
0 MW
0 MW
0 M var
A
MVA
A
0 M var
MVA
A
A
0 MW
0 M var
0 .0 0 pu
GRO SS6 9
A
MVA
FERNA 6 9
MVA
A
MVA
H ISKY 6 9
MVA
MVA
-0 .1 M var
A
MVA
0 .0 0 pu
WO LEN6 9
A
A
0 MW
0 M var
0 .0 0 pu
0 MW
0 M var
A
M O RO 1 3 8
0 MW
0 M var
-0 .0 1 pu
-0 .0 3 pu
A
P ET E6 9
DEM A R6 9
MVA
H A NNA H 6 9
0 MW
0 M var
0 MW
0 M var
-0 .2 M var
MVA
MVA
0 MW
0 M var
A
0 .0 0 pu
0 .0 0 pu
0 .0 0 pu
-0 .1 M var
0 MW
MVA
0 MW
0 M var
MVA
0 .0 0 pu
BLT 1 3 8
-0 .0 3 pu
0 MW
0 M var
MVA
A
A
A
H O M ER6 9
0 MW
0 M var
MVA
SH IM KO 6 9
0 .0 M var
MVA
MVA
0 .0 0 pu
A
A
H A LE6 9
0 .0 0 pu
A
BLT 6 9
-0 .0 1 pu
A
0 MW
0 M var
0 .0 0 pu
LY NN1 3 8
A
A
MVA
A M A NDA 6 9
0 M var
0 MW
0 M var
MVA
MVA
A
-0 .0 0 2 pu
BO B6 9
MVA
-1 5 7 M W
-4 5 M var
A
A
A
MVA
A
A
MVA
UIUC 6 9
-0 .1 M var
BO B1 3 8
A
MVA
MVA
MVA
MVA
0 MW
5 1 M var
A
MVA
0 MW
0 M var
A
MVA
0 MW
0 M var
A
A
0 MW
0 M var
MVA
MVA
MVA
A
0 .0 M var
A
A
0 .0 0 pu
0 .0 M var
0 .0 0 pu
MVA
0 .0 0 pu
P A T T EN6 9
MVA
MVA
A
MVA
0 .0 0 pu
LA UF6 9
0 .0 0 pu
0 MW
4 M var
0 .0 0 pu
A
A
MVA
MVA
0 MW
0 M var
0 MW
0 M var
0 MW
0 M var
LA UF1 3 8
0 .0 0 pu
0 MW
0 M var
WEBER6 9
0 .0 0 pu
BUC KY 1 3 8
RO GER6 9
0 M var
0 MW
0 M var
A
MVA
SA V O Y 6 9
0 .0 0 pu
0 MW
3 M var
A
A
MVA
0 MW
JO 1 3 8
MVA
0 .0 0 pu
A
MVA
SA V O Y 1 3 8
JO 3 4 5
A
0 MW
2 M var
MVA
A
MVA
0 MW
2 M var
A
MVA
0 .0 0 pu
A
0 .0 0 pu
MVA
20
Generation Change Example 2
SLA C K3 4 5
A
MVA
A
MVA
0 .0 0 pu
0 MW
3 7 M var
RA Y 3 4 5
sla ck
0 .0 0 pu
T IM 3 4 5
A
A
MVA
MVA
A
SLA C K1 3 8
-0 .0 1 pu
A
MVA
RA Y 1 3 8
A
0 .0 0 pu
A
MVA
T IM 1 3 8
0 .0 0 pu
MVA
0 MW
0 M var
A
A
A
0 .0 0 pu
-0 .1 M var
0 MW
0 M var
MVA
MVA
MVA
MVA
0 .0 0 pu
RA Y 6 9
T IM 6 9
P A I6 9
0 .0 0 pu
0 MW
0 MW
0 M var
A
0 .0 0 pu
MVA
A
0 M var
MVA
A
A
0 MW
0 M var
0 .0 0 pu
GRO SS6 9
A
MVA
FERNA 6 9
MVA
A
MVA
H ISKY 6 9
MVA
MVA
0 .0 M var
A
MVA
0 .0 0 pu
WO LEN6 9
A
A
0 MW
0 M var
0 .0 0 pu
0 MW
0 M var
A
M O RO 1 3 8
0 MW
0 M var
0 .0 0 pu
-0 .0 3 pu
A
P ET E6 9
DEM A R6 9
MVA
H A NNA H 6 9
0 MW
0 M var
0 MW
0 M var
-0 .2 M var
MVA
MVA
0 MW
0 M var
A
0 .0 0 pu
0 .0 0 pu
0 .0 0 pu
-0 .1 M var
-1 5 7 M W
-4 5 M var
A
0 MW
-0 .0 0 3 pu
0 MW
0 M var
MVA
0 .0 0 pu
BLT 1 3 8
-0 .0 3 pu
MVA
0 MW
0 M var
MVA
A
A
A
H O M ER6 9
0 MW
0 M var
MVA
SH IM KO 6 9
-0 .1 M var
MVA
MVA
-0 .0 1 pu
A
A
H A LE6 9
0 .0 0 pu
A
BLT 6 9
-0 .0 1 pu
A
0 MW
0 M var
0 .0 0 pu
LY NN1 3 8
A
A
A M A NDA 6 9
0 M var
0 MW
0 M var
MVA
MVA
A
MVA
BO B6 9
MVA
A
A
MVA
A
A
MVA
UIUC 6 9
-0 .1 M var
BO B1 3 8
A
MVA
MVA
MVA
MVA
1 9 MW
5 1 M var
A
MVA
0 MW
0 M var
A
MVA
0 MW
0 M var
A
A
0 MW
0 M var
MVA
MVA
MVA
A
0 .0 M var
A
A
0 .0 0 pu
0 .0 M var
0 .0 0 pu
MVA
0 .0 0 pu
P A T T EN6 9
MVA
MVA
A
MVA
0 .0 0 pu
LA UF6 9
0 .0 0 pu
9 9 MW
-2 0 M var
0 .0 0 pu
A
A
MVA
MVA
0 MW
0 M var
0 MW
0 M var
0 MW
0 M var
LA UF1 3 8
0 .0 0 pu
0 MW
0 M var
WEBER6 9
0 .0 0 pu
BUC KY 1 3 8
RO GER6 9
0 M var
0 MW
0 M var
A
MVA
SA V O Y 6 9
0 .0 0 pu
A
A
MVA
0 MW
4 2 MW
-1 4 M var
JO 1 3 8
MVA
0 .0 0 pu
A
MVA
SA V O Y 1 3 8
JO 3 4 5
A
0 MW
0 M var
MVA
A
MVA
0 MW
0 M var
A
MVA
0 .0 0 pu
A
0 .0 0 pu
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.
MVA
21
Voltage Regulation Example: 37 Buses
Automatic voltage regulation system controls voltages.
SLA C K3 4 5
A
MVA
A
MVA
1 .0 2 pu
System Losses: 11.51 MW
1 .0 2 pu
sla ck
T IM 3 4 5
A
MVA
MVA
A
SLA C K1 3 8
1 .0 1 pu
A
MVA
RA Y 1 3 8
A
1 .0 3 pu
A
MVA
3 3 MW
1 3 M var
A
A
T IM 6 9
1 5 .9 M var
1 8 MW
5 M var
1 .0 2 pu
RA Y 6 9
1 .0 1 pu
3 7 MW
1 7 MW
3 M var
A
MVA
A
2 3 MW
7 M var
1 .0 3 pu
MVA
P A I6 9
1 .0 1 pu
MVA
A
MVA
MVA
1 .0 2 pu
A
MVA
T IM 1 3 8
1 .0 0 pu
2 1 9 MW
5 2 M var
RA Y 3 4 5
A
GRO SS6 9
A
1 3 M var
MVA
A
MVA
FERNA 6 9
MVA
A
MVA
M O RO 1 3 8
MVA
H ISKY 6 9
4 .8 M var
MVA
2 0 MW
8 M var
1 .0 0 pu
A
P ET E6 9
DEM A R6 9
H A NNA H 6 9
5 1 MW
1 5 M var
5 8 MW
4 0 M var
2 9 .0 M var
A
MVA
1 .0 0 pu
1 2 .8 M var
0 .9 9 7 pu
MVA
5 6 MW
MVA
H O M ER6 9
5 8 MW
3 6 M var
3 3 MW
MVA
MVA
1 .0 1 pu
H A LE6 9
3 6 MW
1 0 M var
A
A
7 .2 M var
MVA
1 .0 0 pu
LA UF6 9
A
1 .0 0 pu
MVA
2 3 MW
6 M var
A
MVA
P A T T EN6 9
0 MW
0 M var
LA UF1 3 8
1 .0 1 pu
1 4 MW
1 .0 2 pu
BUC KY 1 3 8
RO GER6 9
2 M var
1 4 MW
3 M var
A
MVA
SA V O Y 6 9
1 .0 2 pu
A
3 8 MW
3 M var
JO 1 3 8
MVA
A
MVA
MVA
MVA
4 5 MW
0 M var
WEBER6 9
2 2 MW
1 5 M var
A
A
1 .0 0 pu
A
1 .0 2 pu
1 .0 1 pu
MVA
MVA
1 .0 0 pu
2 0 .8 M var
1 5 MW
5 M var
9 2 MW
1 0 M var
A
A
MVA
1.010 pu
MVA
MVA
MVA
1 .0 2 pu
BLT 6 9
1 .0 1 pu
MVA
2 0 MW
9 M var
SH IM KO 6 9
7 .4 M var
MVA
6 0 MW
1 2 M var
1 4 MW
4 M var
MVA
A
A
A
MVA
BLT 1 3 8
1 .0 0 pu
MVA
A
A
LY NN1 3 8
A
MVA
A
0.0 Mvar
1 0 M var
1 5 MW
3 M var
1 .0 0 pu
1 3 M var
0 MW
A
0 M var
A
A M A NDA 6 9
BO B6 9
1 5 7 MW
4 5 M var
MVA
MVA
0 .9 9 pu
MVA
A
MVA
1 .0 2 pu
A
A
MVA
MVA
MVA
A
A
A
BO B1 3 8
A
4 5 MW
1 2 M var
0 .9 9 pu
UIUC 6 9
1 4 .3 M var
A
A
1 .0 0 pu
MVA
1 .0 0 pu
WO LEN6 9
A
A
1 2 MW
5 M var
1 .0 1 pu
2 1 MW
7 M var
A
MVA
1 .0 1 pu
A
MVA
SA V O Y 1 3 8
JO 3 4 5
A
1 5 0 MW
0 M var
MVA
A
MVA
1 5 0 MW
0 M var
A
MVA
1 .0 2 pu
A
1 .0 3 pu
MVA
Display shows voltage contour of the power system
22
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
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
Eastern Interconnect Example
VIK 138
BIG BEN D
WH TWTR3
EEN 138
ST RITA
M UKWO N GO
WH TWTR4
SUN 138
TRIPP
WH TWTR5
UN IVRSTY
Raci ne
JAN 138
SGR CK4
UN IV N EU
LBT 138
SGR CK5
LAN 138
BRLGTN 2
SO M ERS
ALB 138
RO R 138
N LK GV T
BRLGTN 1
ALBERS-2
Paddock
PO T 138
N O M 138
M RE 138
PARIS WE
TICH IGN
H LM 138
BAIN 4
WIB 138
D AR 138
N LG 138
N ED 138
Pl easant Prai ri e
N WT 138
N ED 161
Kenosha
LIBERTY5
BCH 138
TRK RIV5
CASVILL5
BLK 138
LEN A ; B
CO R 138
WBT 138
ELK 138
LAKEVIEW
D IK 138
LEN A ; R
8TH ST. 5
LO RE
Zi on
ELERO ; RT
SO . GVW. 5
Wempl eton
PECAT; B
Zi on (138 kV)
Ant i och
Rockford
5
ELERO ; BT
ASBURY 5
M cHenr y
G ur nee
Round Lake
CN TRGRV5
Waukegan
LAN CA; R
JULIAN 5
SALEM N 5
H arl em
Sal em
FREEP;
Bel vi dere
M arengo
Woodstock
Wi l son
Roscoe
Lakehur st
P Val
GALEN A 5
Cr yst al Lake
Sand Park
Pi erpont
Li ber t yvi l l e
345 kV
Si l ver Lake
Hunt l ey
B465
FO RD A; R
Li ber t yvi l l e
138 kV
Nor t h Chi cago
Al gonqui n
S PEC; R
E. Rockf ord
U. S. N Tr ai ni ng
Al pi ne
Abbot t Labs Par k
Lest hon
Charl es
B427 ; 1T
Sabrooke
Apt aki si c
Cherry Val l ey
O l d El m
Lake Zur i ch
Buf f al o G r oove
Bar r i ngt on
Bl aw khaw k
Wheel i ng
Deer f i el d
Pal at i ne
D undee
SAVAN N A5
Pr ospect Hei ght s
Ar l i ngt on
STILL; RT
M Q O KETA5
WYO M IN G5
Pr ospect
Hof f m an Est at es
Nor t hbr ook
Hei ght s
C434
M ount Pr ospect
Tol l w ay
Schaum ber g
M T VERN 5
PCI
El m w ood
5
Byron
Hanover
S. Schaum ber g
G ol f M i l l
Busse
Landm
BERTRAM 5
Skoki e
Spaul di ng
Bar t l et t
El gi n
YO RK
Evanst on
Des Pl ai nes
Tonne
5
Ni l es
How ar d
M ARYL; B
Devon
Wayne
Sout h El gi n
Des Pl ai nes
I t asca
Hi ggi ns
Al t G E
Rose Hi l l
Nor di
G l endal e
Nor t hr i dge
M i chi gan Ci ty
West Chi cago
W407 ( Fer m i )
LEECO ; BP
W. De Kal b
H 445 ; 3B
Addi son
-0. 40 deg
Nor t hw est
Nat om a
Chur ch
G l i dden
Aur or a
2. 35 deg
El m hur st
Dr i ver
Lom bar d
Rockw el l
G al ew ood
O ak Par k
Rock Crk.
-13. 4 deg
-13. 3 deg
Fr ankl i n Par k
H 440 ; R
GR M N D 5
E CALM S5
Cl ybour n
ALBAN Y 6
BVR CH 65
D EWITT 5
BVR CH 5
ALBAN Y 5
Sugar Grove
M EN D O ; T
D IXO N ; BT
GARD E;
G l en El l yn
Ber kel ey
Congr ess
O akbr ook
N Aurora
Wat er m an
STEWA; B
Yor k Cent er
El ect r i c Junct i on
Ki ngsbur y
Cl i nt
Bel l w ood
H 440 ; RT
H 71 ; B
Cr osby
O hi o
But t e
H 71 ; BT
Y450
Jef f erson
D ekov
Tayl or
La G r ange
H 71 ; R
STERL; B
Uni versi ty
Ri dgel and
D unacr
Li sl e
H -471 (N W Steel )
M cCook
Lasal l e
Fi sk
D799
Washi ngton Park
Craw f ord
War r envi l l e
State
D775
-1. 1 deg
H arbor
Garf i el d
D ow ners Groove
Frontenac
Woodri dge
Wol f Creek
Q uad Ci ti es
M ECCO RD 3
W604
O sw ego
J307
W602
Wi l l Co.
N ELSO ; RT
W507
D avenport
Wal cott
Lockport
Kenda
H i l l crest
Rockdal e
SB 88 5
H egew i sch
Wi l dw ood
M unster
Z-524
Bl ue Isl and
Green Acres
Ti nl ey Park
South H ol l and
Jo456
Shore
3
IPSCO
Tow er Rd
Goodi ngs Grove
J322
SB 76 5
IPSCO
Lake George
Z-100
Green Lake
N O RM A; B
SB 78 5
SB JIC 5
Shef i el d
Z-715
Burnham
J-332
Bel l Road
N O RM A; R
Crestw ood
Archer
Pl ai nf i el d
SB 17 5
SB 71 5
Babcock
State Li ne
Z-494
G3851
G3852
M endota
SB 74 5
1. 9 deg
Wal l ace
Beverl y
G394
O rl an
SB 49 5
SB 90 5
SUB 77 5
D AVN PRT5
SB 89 5
D amen
Chi ave
Evergreen
Al si p
Roberts
Romeo
Pal os
SBH YC5
SB UIC 5
Sub 92
H ayf ord
Sayre
Bri dgevi ew
Bol i ngbrook
M ontgomery
R FAL; R
Cal umet
Ri ver
W603
Pl ano
N ELSO ; R
R FAL; B
0. 6 deg
Bedf ord Park
Bur r Ri dge
W601
CO RD O ;
SB 79 5
Q uarry
Wi l l ow
Sandw i ch
Sub 91
Saw yer
Ford Ci ty
Cl earni ng
W600 ( Naper vi l l e)
N el son
SB 58 5
Sand Ri dge
H arvey
J323
Lansi ng
Jol i et
J370
Gl enw ood
SB 70 5
5
Chi cago H ei ghts
F-503
Bri gg
J-371
SB A 5
M oken
J-326
J-390
SB 28 5
SB 52 5
J-375
F-575
East Frankf ort
Frankf ort
Country Cl ub H i l l s
El w ood
M atteson
N Len
SB 48 5
PRIN C TP
SB 47 5
Park Forest
Bl oom
J-339
U. Park
SB 31T 5
Woodhi l l
St. John
J-305
SB 53 5
PRIN CTN
SB 85 5
Col l i ns
LTV TP E
Wi l ton Center
LTV TP N
KEWAN ;
SB 43 5
S ST TAP
B
ESK TAP
B
Schahf er
105%
93%
H EN N E; T
SB 112 5
Crete
D resden
M ason
East M ol i ne
SB 18 5
Upnor
Goose Lake
LTV STL
KEWAN IP
E M O LIN E
H EN N EPIN
Kendra
O TTAWA T
MVA MVA
1556A TP
N LASAL
O GLES; T
O GLESBY
Lasal l e
O GLSBY M
M arsei l l es
La Sal l e
Wi l mi ngton
K-319 # 1
Loui sa
D avi s Creek
K-319 # 2
KPECKTP5
WEST
5
Bradl ey
SO . SUB 5
Streator
Br ai dw ood
H WY61 5
9 SUB 5
M IN O N K T
GALESBR5
Kankakee
GALESBRG
RICH LAN D
N EWPO RT5
M O N M O UTH
SPN G BAY
Ponti ac M i dpoi nt
D equi ne
M PWSPLIT
H ALLO CK
ELPASO T
Peoria
WATSEKA
17GO D LN D
GILM AN
FARGO
CAT M O SS
RSW EAST
RAD N O R
CAT SUB1
PIO N EERC
E PEO RIA
Example, which models the Eastern Interconnect
contains about 43,000 buses.
CAT TAP
25
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
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
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 flow
28
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  Pactual tie-line flow  Psched  10f
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
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 outside of ERCOT.
30
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
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-line flow – Psched + 10β Δf
…and then controlling the generation to bring
ACE towards zero.
32
“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
Three Bus Case on AGC:
no interchange.
Bus 2
-40 MW
8 MVR
40 MW
-8 MVR
Bus 1
1.00 PU
266 MW
133 MVR
1.00 PU
101 MW
5 MVR
150 MW AGC ON
166 MVR AVR ON
-39 MW
-77 MW
25 MVR
12 MVR
78 MW
-21 MVR
Home Area
Generation
is automatically
changed to match
change in load
100 MW
39 MW
-11 MVR
Bus 3
1.00 PU
133 MW
67 MVR
250 MW AGC ON
34 MVR AVR ON
Net tie-line flow is
close to zero
34
100 MW Transaction between
areas in Eastern or Western
Bus 2
8 MW
-2 MVR
-8 MW
2 MVR
Bus 1
1.00 PU
225 MW
113 MVR
1.00 PU
0 MW
32 MVR
150 MW AGC ON
138 MVR AVR ON
-92 MW
-84 MW
27 MVR
30 MVR
85 MW
-23 MVR
Home Area
93 MW
-25 MVR
Bus 3
Scheduled Transactions
100.0 MW
Scheduled
100 MW
Transaction from Left to Right
100 MW
1.00 PU
113 MW
56 MVR
291 MW AGC ON
8 MVR AVR ON
Net tie-line
flow is now
100 MW
35
PTDFs
Power transfer distribution factors (PTDFs)
show the linearized impact of a transfer of
power.
PTDFs can be estimated using the fast
decoupled power flow B matrix:
θ  B 1P
Once we know θ we can derive the change in
the transmission line flows to evaluate PTDFs.
Note that we can modify several elements in P,
in proportion to how the specified generators would
participate in the power transfer.
36
Nine Bus PTDF Example
Figure shows initial flows for a nine bus power system
300.0 MW
400.0 MW
A
300.0 MW
250.0 MW
B
D
71%
10%
C
71.1 MW
57%
60%
92%
0.00 deg
55%
11%
F
G
74% 250.0 MW
64%
E
150.0 MW
250.0 MW
44%
32%
24%
H
200.0 MW
I
150.0 MW
37
Nine Bus PTDF Example, cont'd
Figure now shows percentage PTDF flows for a change in transaction from A to I
300.0 MW
400.0 MW
A
300.0 MW
250.0 MW
B
D
30%
43%
C
71.1 MW
10%
57%
13%
0.00 deg
35%
2%
F
G
34% 250.0 MW
20%
E
150.0 MW
250.0 MW
34%
32%
34%
H
200.0 MW
I
150.0 MW
38
Nine Bus PTDF Example, cont'd
Figure now shows percentage PTDF flows for a change in transaction from G to F
300.0 MW
400.0 MW
A
300.0 MW
250.0 MW
B
D
18%
6%
C
71.1 MW
6%
6%
12%
0.00 deg
61%
19%
F
G
21% 250.0 MW
12%
150.0 MW
250.0 MW
E
20%
21%
H
200.0 MW
I
150.0 MW
39
WE to TVA PTDFs
40
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 precontingency 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
Line Outage Distribution Factors
(LODFs)
Pl  change in flow on line l ,
due to outage of line k .
Pk  pre-contingency flow on line k
Pl  LODFl ,k Pk ,
Estimates change in flow on line l
if outage on line k were to occur.
42
Line Outage Distribution Factors
(LODFs)
If line k initially had Pk  100 MW of flow on it,
and line l initially had Pl  50 MW flow on it,
and then there was an outage of line k ,
if LODFl ,k =0.1 then the increase in flow
on line l after a contingency of line k would be:
Pl  LODFl ,k Pk  0.1  100  10 MW
from 50 MW to 60 MW.
43
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
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