lecture 14
TRANSCRIPT
EE 369POWER SYSTEM ANALYSIS
Lecture 14Power Flow
Tom Overbye and Ross Baldick
1
AnnouncementsRead 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
400 MVA15 kV
400 MVA15/345 kV
T1
T2800 MVA345/15 kV
800 MVA15 kV
520 MW
80 MW40 Mvar
280 MVAr 800 MW
Line 3 345 kV
Line
2
Line
1345 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
Bus Type|V| per unit
θdegrees
PG
perunit
QG
perunit
PL
perunit
QL
perunit
QGmax
perunit
QGmin
perunit
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
Rper unit
Xper unit
Gper unit
Bper unit
MaximumMVA
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
Bus-to-Bus
R perunit
Xperunit
Gc
perunit
Bm
perunit
MaximumMVA
per unit
MaximumTAP
Settingper 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
Let the Computer Do the Calculations! (Ybus Shown)
6
Selected Ybus Details
02321 YY
2424 24
1 1 0.89276 9.919640.009 0.1
Y j per unitR jX j
2525 25
1 1 1.78552 19.839320.0045 0.05
Y j per unitR jX j
24 2522
24 24 25 25
1 12 2B BY j j
R jX R jX
288.0
272.1)83932.1978552.1()91964.989276.0( jjjj
unitperj 624.845847.284590.2867828.2
Entries of Ybus relating to elements connected to bus 2.Note that resistances, inductive reactances, and admittancescome from Table 2; subscripts on them refer to line from-to.Subscripts on Ybus correspond to entries of that matrix.
7
Here are the Initial Bus Mismatches
8
And the Initial Power Flow Jacobian
9
Five Bus Power System Solved
slack
One
Two
ThreeFourFiveA
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 MW 114 Mvar
520 MW 337 Mvar
800 MW 280 Mvar
80 MW 40 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 device. This is known as n-1 reliability.
11
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
slack
Metropolis Light and Power Electric Design Case 2SLACK345
SLACK138
RAY345
RAY138
RAY69
FERNA69A
MVA
DEMAR69
BLT69
BLT138
BOB138
BOB69
WOLEN69
SHI MKO69
ROGER69
UI UC69
PETE69HI SKY69
TI M69
TI M138
TI M345
PAI 69
GROSS69
HANNAH69
AMANDA69
HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138
J O138 J O345
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1.03 pu
1.02 pu
1.03 pu
1.03 pu
1.01 pu
1.00 pu1.01 pu
1.00 pu1.02 pu
1.01 pu
1.00 pu
1.01 pu1.01 pu
1.01 pu1.01 pu
1.02 pu
1.00 pu
1.00 pu
1.02 pu
0.99 pu
0.99 pu
1.00 pu
1.02 pu
1.00 pu1.01 pu
1.01 pu
1.00 pu 1.00 pu
1.01 pu
1.02 pu 1.02 pu
1.02 pu 1.03 pu
A
MVA
1.02 pu
A
MVA
A
MVA
LYNN138
A
MVA
1.02 pu
A
MVA
1.00 pu
A
MVA
System Losses: 10.70 MW 220 MW 52 Mvar
12 MW 3 Mvar
20 MW 12 Mvar
124 MW 45 Mvar
37 MW 13 Mvar
12 MW 5 Mvar
150 MW 0 Mvar
56 MW 13 Mvar
15 MW 5 Mvar
14 MW 2 Mvar
38 MW 3 Mvar
45 MW 0 Mvar
25 MW 36 Mvar
36 MW 10 Mvar
10 MW 5 Mvar
22 MW 15 Mvar
60 MW 12 Mvar
20 MW 28 Mvar
23 MW 7 Mvar
33 MW 13 Mvar 15.9 Mvar 18 MW
5 Mvar
58 MW 40 Mvar
60 MW 19 Mvar
14.2 Mvar
25 MW 10 Mvar
20 MW 3 Mvar
23 MW 6 Mvar 14 MW
3 Mvar
4.9 Mvar
7.3 Mvar
12.8 Mvar
28.9 Mvar
7.4 Mvar
0.0 Mvar
55 MW 25 Mvar
39 MW 13 Mvar
150 MW 0 Mvar
17 MW 3 Mvar
16 MW -14 Mvar
14 MW 4 Mvar
KYLE69A
MVA
13
Looking at the Impact of Line Outages
slack
Metropolis Light and Power Electric Design Case 2SLACK345
SLACK138
RAY345
RAY138
RAY69
FERNA69A
MVA
DEMAR69
BLT69
BLT138
BOB138
BOB69
WOLEN69
SHI MKO69
ROGER69
UI UC69
PETE69HI SKY69
TI M69
TI M138
TI M345
PAI 69
GROSS69
HANNAH69
AMANDA69
HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138
J O138 J O345
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1.03 pu
1.02 pu
1.03 pu
1.03 pu
1.01 pu
1.00 pu1.01 pu
1.00 pu1.02 pu
1.01 pu
1.00 pu
1.01 pu1.01 pu
1.01 pu1.01 pu
1.02 pu
1.01 pu
1.00 pu
1.02 pu
0.90 pu
0.90 pu
0.94 pu
1.01 pu
0.99 pu1.00 pu
1.00 pu
1.00 pu 1.00 pu
1.01 pu
1.01 pu 1.02 pu
1.02 pu 1.03 pu
A
MVA
1.02 pu
A
MVA
A
MVA
LYNN138
A
MVA
1.02 pu
A
MVA
1.00 pu
A
MVA
System Losses: 17.61 MW 227 MW 43 Mvar
12 MW 3 Mvar
20 MW 12 Mvar
124 MW 45 Mvar
37 MW 13 Mvar
12 MW 5 Mvar
150 MW 4 Mvar
56 MW 13 Mvar
15 MW 5 Mvar
14 MW 2 Mvar
38 MW 9 Mvar
45 MW 0 Mvar
25 MW 36 Mvar
36 MW 10 Mvar
10 MW 5 Mvar
22 MW 15 Mvar
60 MW 12 Mvar
20 MW 40 Mvar
23 MW 7 Mvar
33 MW 13 Mvar 16.0 Mvar 18 MW
5 Mvar
58 MW 40 Mvar
60 MW 19 Mvar
11.6 Mvar
25 MW 10 Mvar
20 MW 3 Mvar
23 MW 6 Mvar 14 MW
3 Mvar
4.9 Mvar
7.2 Mvar
12.8 Mvar
28.9 Mvar
7.3 Mvar
0.0 Mvar
55 MW 32 Mvar
39 MW 13 Mvar
150 MW 4 Mvar
17 MW 3 Mvar
16 MW -14 Mvar
14 MW 4 Mvar
KYLE69A
MVA
80%A
MVA
135%A
MVA
110%A
MVA
Opening one line (Tim69-Hannah69) causes overloads. This would not be acceptableunder NERCstandards.
14
Contingency Analysis
Contingencyanalysis providesan automaticway of lookingat all the contingencies ina specified “contingency set.” In this example thecontingency setis all the single line/transformeroutages
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 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 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
slack
Metropolis Light and Power Electric Design Case 2SLACK345
SLACK138
RAY345
RAY138
RAY69
FERNA69A
MVA
DEMAR69
BLT69
BLT138
BOB138
BOB69
WOLEN69
SHI MKO69
ROGER69
UI UC69
PETE69HI SKY69
TI M69
TI M138
TI M345
PAI 69
GROSS69
HANNAH69
AMANDA69HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138
J O138 J O345
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1.02 pu
1.01 pu
1.02 pu
1.03 pu
1.01 pu
1.00 pu1.01 pu
1.00 pu1.02 pu
1.01 pu
1.00 pu
1.01 pu1.01 pu
1.01 pu1.01 pu
1.02 pu
0.99 pu
1.00 pu
1.02 pu
0.97 pu
0.97 pu
0.99 pu
1.02 pu
1.00 pu1.01 pu
1.01 pu
1.00 pu 1.00 pu
1.01 pu
1.02 pu 1.02 pu
1.02 pu 1.03 pu
A
MVA
1.02 pu
A
MVA
A
MVA
LYNN138
A
MVA
1.02 pu
A
MVA
1.00 pu
A
MVA
System Losses: 14.49 MW 269 MW 67 Mvar
12 MW 3 Mvar
20 MW 12 Mvar
124 MW 45 Mvar
37 MW 13 Mvar
12 MW 5 Mvar
150 MW 1 Mvar
56 MW 13 Mvar
15 MW 5 Mvar
14 MW 2 Mvar
38 MW 4 Mvar
45 MW 0 Mvar
25 MW 36 Mvar
36 MW 10 Mvar
10 MW 5 Mvar
22 MW 15 Mvar
60 MW 12 Mvar
20 MW 40 Mvar
23 MW 7 Mvar
33 MW 13 Mvar 15.9 Mvar 18 MW
5 Mvar
58 MW 40 Mvar
60 MW 19 Mvar
13.6 Mvar
25 MW 10 Mvar
20 MW 3 Mvar
23 MW 6 Mvar 14 MW
3 Mvar
4.9 Mvar
7.3 Mvar
12.8 Mvar
28.9 Mvar
7.4 Mvar
0.0 Mvar
55 MW 28 Mvar
39 MW 13 Mvar
150 MW 1 Mvar
17 MW 3 Mvar
16 MW -14 Mvar
14 MW 4 Mvar
KYLE69A
MVA
96%A
MVA
Case now has nine separate contingencies having reliability violations(overloads in post-contingency system).
17
A Reliable Solution:no line outages result in overloads
slack
Metropolis Light and Power Electric Design Case 2SLACK345
SLACK138
RAY345
RAY138
RAY69
FERNA69A
MVA
DEMAR69
BLT69
BLT138
BOB138
BOB69
WOLEN69
SHI MKO69
ROGER69
UI UC69
PETE69HI SKY69
TI M69
TI M138
TI M345
PAI 69
GROSS69
HANNAH69
AMANDA69HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138
J O138 J O345
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1.03 pu
1.01 pu
1.02 pu
1.03 pu
1.01 pu
1.00 pu1.01 pu
1.00 pu1.02 pu
1.01 pu
1.00 pu
1.01 pu1.01 pu
1.01 pu1.01 pu
1.02 pu
1.00 pu
0.99 pu
1.02 pu
0.99 pu
0.99 pu
1.00 pu
1.02 pu
1.00 pu1.01 pu
1.01 pu
1.00 pu 1.00 pu
1.01 pu
1.02 pu 1.02 pu
1.02 pu 1.03 pu
A
MVA
1.02 pu
A
MVA
A
MVA
LYNN138
A
MVA
1.02 pu
A
MVA
A
MVA
System Losses: 11.66 MW 266 MW 59 Mvar
12 MW 3 Mvar
20 MW 12 Mvar
124 MW 45 Mvar
37 MW 13 Mvar
12 MW 5 Mvar
150 MW 1 Mvar
56 MW 13 Mvar
15 MW 5 Mvar
14 MW 2 Mvar
38 MW 4 Mvar
45 MW 0 Mvar
25 MW 36 Mvar
36 MW 10 Mvar
10 MW 5 Mvar
22 MW 15 Mvar
60 MW 12 Mvar
20 MW 38 Mvar
23 MW 7 Mvar
33 MW 13 Mvar 15.8 Mvar 18 MW
5 Mvar
58 MW 40 Mvar
60 MW 19 Mvar
14.1 Mvar
25 MW 10 Mvar
20 MW 3 Mvar
23 MW 6 Mvar 14 MW
3 Mvar
4.9 Mvar
7.3 Mvar
12.8 Mvar
28.9 Mvar
7.4 Mvar
0.0 Mvar
55 MW 29 Mvar
39 MW 13 Mvar
150 MW 1 Mvar
17 MW 3 Mvar
16 MW -14 Mvar
14 MW 4 Mvar
KYLE69A
MVA
Kyle138A
MVA
Previous case was augmented with the addition of a 138 kV Transmission Line
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
slack
SLACK345
SLACK138
RAY345
RAY138
RAY69
FERNA69
A
MVA
DEMAR69
BLT69
BLT138
BOB138
BOB69
WOLEN69
SHI MKO69
ROGER69
UI UC69
PETE69
HI SKY69
TI M69
TI M138
TI M345
PAI 69
GROSS69
HANNAH69
AMANDA69
HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138
J O138 J O345
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
0.00 pu
-0.01 pu
0.00 pu
0.00 pu
0.00 pu
-0.03 pu-0.01 pu
0.00 pu0.00 pu
0.00 pu
-0.03 pu
-0.01 pu0.00 pu
0.00 pu0.00 pu
0.00 pu
0.00 pu
0.00 pu
0.00 pu
-0.002 pu0.00 pu
0.00 pu
0.00 pu
0.00 pu0.00 pu
0.00 pu
0.00 pu 0.00 pu
0.00 pu
0.00 pu 0.00 pu
0.00 pu 0.00 pu
A
MVA
-0.01 pu
A
MVA
A
MVA
LYNN138
A
MVA
0.00 pu
A
MVA
0.00 pu
A
MVA
162 MW 35 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
-157 MW -45 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 2 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW
0 Mvar
0 MW 3 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 4 Mvar
0 MW 0 Mvar
0 MW 0 Mvar -0.1 Mvar 0 MW
0 Mvar
0 MW 0 Mvar 0 MW
0 Mvar
-0.1 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar 0 MW
0 Mvar
-0.1 Mvar
0.0 Mvar
-0.1 Mvar
-0.2 Mvar
0.0 Mvar
0.0 Mvar
0 MW 51 Mvar
0 MW 0 Mvar
0 MW 2 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
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
Generation Change Example 2
slack
SLACK345
SLACK138
RAY345
RAY138
RAY69
FERNA69
A
MVA
DEMAR69
BLT69
BLT138
BOB138
BOB69
WOLEN69
SHI MKO69
ROGER69
UI UC69
PETE69
HI SKY69
TI M69
TI M138
TI M345
PAI 69
GROSS69
HANNAH69
AMANDA69
HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138
J O138 J O345
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
0.00 pu
-0.01 pu
0.00 pu
0.00 pu
0.00 pu
-0.03 pu0.00 pu
0.00 pu0.00 pu
0.00 pu
-0.03 pu
-0.01 pu-0.01 pu
0.00 pu0.00 pu
0.00 pu
0.00 pu
0.00 pu
0.00 pu
-0.003 pu0.00 pu
0.00 pu
0.00 pu
0.00 pu0.00 pu
0.00 pu
0.00 pu 0.00 pu
0.00 pu
0.00 pu 0.00 pu
0.00 pu 0.00 pu
A
MVA
0.00 pu
A
MVA
A
MVA
LYNN138
A
MVA
0.00 pu
A
MVA
0.00 pu
A
MVA
0 MW 37 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
-157 MW -45 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW
0 Mvar
42 MW -14 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
99 MW -20 Mvar
0 MW 0 Mvar
0 MW 0 Mvar -0.1 Mvar 0 MW
0 Mvar
0 MW 0 Mvar 0 MW
0 Mvar
-0.1 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar 0 MW
0 Mvar
0.0 Mvar
0.0 Mvar
-0.1 Mvar
-0.2 Mvar
-0.1 Mvar
0.0 Mvar
19 MW 51 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
0 MW 0 Mvar
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
Voltage Regulation Example: 37 Buses
Display shows voltage contour of the power system
slack
SLACK345
SLACK138
RAY345
RAY138
RAY69
FERNA69
A
MVA
DEMAR69
BLT69
BLT138
BOB138
BOB69
WOLEN69
SHI MKO69
ROGER69
UI UC69
PETE69
HI SKY69
TI M69
TI M138
TI M345
PAI69
GROSS69
HANNAH69
AMANDA69HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138
J O138 J O345
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVAA
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
1.03 pu
1.01 pu
1.02 pu
1.03 pu
1.01 pu
1.00 pu1.00 pu
0.99 pu1.02 pu
1.01 pu
1.00 pu
1.01 pu1.01 pu
1.01 pu1.01 pu
1.02 pu
1.00 pu
1.00 pu
1.02 pu
0.997 pu0.99 pu
1.00 pu
1.02 pu
1.00 pu1.01 pu
1.00 pu
1.00 pu 1.00 pu
1.01 pu
1.02 pu 1.02 pu
1.02 pu 1.03 pu
A
MVA
1.02 pu
A
MVA
A
MVA
LYNN138
A
MVA
1.02 pu
A
MVA
1.00 pu
A
MVA
219 MW 52 Mvar
21 MW 7 Mvar
45 MW 12 Mvar
157 MW 45 Mvar
37 MW 13 Mvar
12 MW 5 Mvar
150 MW 0 Mvar
56 MW 13 Mvar
15 MW 5 Mvar
14 MW 2 Mvar
38 MW 3 Mvar
45 MW 0 Mvar
58 MW 36 Mvar
36 MW 10 Mvar
0 MW 0 Mvar
22 MW 15 Mvar
60 MW 12 Mvar
20 MW 9 Mvar
23 MW 7 Mvar
33 MW 13 Mvar 15.9 Mvar 18 MW
5 Mvar
58 MW 40 Mvar 51 MW
15 Mvar
14.3 Mvar
33 MW 10 Mvar
15 MW 3 Mvar
23 MW 6 Mvar 14 MW
3 Mvar
4.8 Mvar
7.2 Mvar
12.8 Mvar
29.0 Mvar
7.4 Mvar
20.8 Mvar
92 MW 10 Mvar
20 MW 8 Mvar
150 MW 0 Mvar
17 MW 3 Mvar
0 MW 0 Mvar
14 MW 4 Mvar
1.010 pu 0.0 Mvar
System Losses: 11.51 MW
22
Automatic voltage regulation system controls voltages.
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
Peoria
Rockford
Nor th Chi cago
Abbot t Labs Par kU. S. N Tr ai ni ng
O l d El m
Deerf i el d
Nor thbr ook
Lakehurst
Waukegan
Zi on
G urnee
Anti och
Pl easant
Round Lake
Zi on (138 kV)
Lake Zur i ch
Lesthon
Apt aki si c
Buf fal o G r oove
Wheel i ng
Pr ospect Hei ght sPal at i ne
Ar l i ngton
M ount Pr ospect
Pr ospect
G ol f M i l l
Des Pl ai nes
El mhurst
I t asca
Garfi el d
Tol l w ay
W407 ( Fer mi )
Wi l son
Bar ri ngt on
Dundee
Si l ver Lake
Cherry Val ley
Wempl eton
Nelson
H-471 (NW Steel )
Paddock
Ponti ac Mi dpoint
Br ai dw ood
State Li ne
Shefi el d
Chi ave
Munster
St. J ohn
El ect r i c Junct i on
Pl ano
La Sal le
Lombard
Li sle
Col li ns
Dresden
Lockport
East Frankfort
Goodi ngs Grove
Li ber tyvi l l e345 kV
Li ber tyvi l l e138 kV
Lake George
Dunacr
Green Acres
Schahfer
Tower Rd
Babcock
Hei ghts
Prairie
Racine
Mi chi gan City
El wood
Dequi ne
Loui sa
East Mol i ne
Sub 91
Walcott
Davenport
Sub 92
Rock Crk.
Salem
GI LMAN
WATSEKA 17GODLND
ELPASO T
MI NON K T
OGLESBY
1556A TPOTTAWA T
OGLSBY M
OGLES; T
HENNEPI N
ESK TAP
LTV TP NLTV TP E
HENNE; T
LTV STL
PRI NC TP
PRI NCTN
RI CHLAN D
KEWAN IP
S ST TAP
GALESBRG
NORMA; BNORMA; R
R FAL; R
MONMO UTH
GALESBR5
KEWAN;
HALLOCK
CAT MO SSFARGO
SPNG BAY
E PEORIA
RSW EAST
PI ONEERC
RADN OR
CAT TAP
CAT SUB1
SB 18 5
E MO LI NE
SB 43 5
SB 112 5
KPECKTP5
SO. SUB 5
SB 85 5
SB 31T 5
SB 28 5
SB 17 5
SB 49 5
SB 53 5
SB 47 5SB 48 5
SB A 5
SB 70 5
SB 79 5
SB 88 5
SB 71 5
BVR CH65 BVR CH 5 ALBAN Y 5
YO RK 5
SAVANNA5
GALEN A 5
8TH ST. 5
LO RE 5
SO .GVW.5
SALEM N 5
ALBAN Y 6
GARDE;
H71 ;BTH71 ; B
H71 ; R
R FAL; B
NELSO; R
NELSO;RT
STERL; B
D IXON;BT
MECCORD3
CO RD O;
Quad Ci ties
LEECO;BP
Byron
MARYL; B
MENDO; T
STI LL;RT
B427 ;1T
LANCA; R
PECAT; B
FREEP;
ELERO;BT ELERO;RT
LENA ; RLENA ; B
H440 ;RT
H440 ; R
STEWA; B
H445 ;3B
Roscoe
Pi erpont
S PEC; R
FO RD A; R
Harl em
Sand Park
NWT 138
BLK 138
RO R 138
JAN 138
ALB 138
NOM 138
DAR 138
HLM 138POT 138 MRE 138
COR 138 D IK 138
BCH 138
Sabrooke
Bl awkhawk
Al pine
E. Rockford
Charl es
Bel vi dere
B465
Marengo
WI B 138
WBT 138ELK 138
NLG 138
NLK GV T
SGR CK5
BRLGTN1
BRLGTN2
SGR CK4
UNI VRSTY
UN IV NEU
WHTWTR5
WHTWTR4
WHTWTR3
SUN 138
VI K 138
LBT 138
TI CHI GNPARIS WE
ALBERS-2
C434
El mw ood
Ni l es
Evanst on
Devon
Rose Hi l l
Skoki e
Nor thw est
Dr i ver
Ford City
Hayford
Sawyer
Nort hri dge
Hi ggi nsDes Pl ai nes
Fr ankl i n Par k
O ak Par k
Ri dgel and
D799
G al ew ood
Y450
Congr ess
Rockw el lCl ybour n
Quarry
Lasal l e
State
Cr osby
Ki ngsbur y
Jeff erson
Ohio
Taylor
Cl int
Dekov
Fi sk
Crawford
Uni versi ty
Ri ver
Z-494
Washi ngton ParkHarbor
Cal umet
Hegewi sch
Z-715
South Holl and
Evergreen
Damen
Wall ace
Beverl y
G3851
Z-524G3852
Wi ldwood
Harvey
Green Lake
Sand Ri dge
Chicago H ei ghts
Burnham
Lansi ng
F-575
F-503Glenw ood
Bl oomPark ForestMatteson
Country Club H i ll s
Al t G E
Natoma
Woodhil lU. Park
Moken
M cHenr y
Cr yst al Lake
Al gonqui n
Hunt l ey
P Val
Woodstock
Blue I sl and
G394
Al si p
Crestwood
K-319 #1
K-319 #2
Bradl ey
Kankakee
Davi s Creek
Wi lmington
Wi lton Center
Frankfort
N Len
Brigg
O akbr ook
Dow ners Groove
Woodri dge
W604
W603
Bol ingbrook
Sugar Grove
W. De Kal b G l i dden
N Aurora
El gi n
Hanover
Spaul di ngBar t l et t
Hof fm an Estat es
S. Schaumberg
Tonne
LandmBusse
Schaumber g
How ar d
Berkel ey
Bel l w ood
La G r ange
Chur ch
Addi son
Nor diG l endal e
G l en El l yn
But te
Yor k Cent er
D775
Bedford Park
Cl earning
Sayre
Bri dgevi ew
Ti nley Park
Roberts
Pal osRomeo
Wi l l ow
Bur r Ri dge
Jo456
J 322
South El gi n Wayne
West Chi cago
Aur or a
Warr envi l l e
W507
Montgomery
Oswego
Wolf Creek
Frontenac
W600 ( Naper vi l l e)
W602
W601J307
Sandw ich
Wat erman
J 323
Mason
J-371
J-375
J-339
Streator
Marseil l esLasal l e
N LASAL
Mendota
J370
Shore
Goose Lake
J-305
J-390J -326
Plainfi el d
J -332
Archer
Bell Road
Wi l l Co.
Hi ll crest Rockdale
Jol i et
Kendra
Crete
Upnor
LAKEVI EW
BAI N 4
Kenosha
SO MERS
ST RI TA
BI G BEN D
MUKWONGO
NED 138
NED 161
LAN 138
EEN 138
CASVI LL5
TRK RI V5
LIBERTY5
ASBURY 5
CN TRGRV5
JULIAN 5
MQO KETA5
E CALMS5
GR MND 5
DEWI TT 5
SBHYC5
SUB 77 5
SB 74 5SB 90 5
SB 78 5
DAVN PRT5
SB 76 5
SB 58 5
SB 52 5
SB 89 5
IPSCO 5
IPSCO 3
NEWPORT5
HWY61 5
WEST 5
9 SUB 5
TRI PP
Z-100Orlan
Kenda
MPWSPLI T
WYO MIN G5
MT VERN5
BERTRAM5
PCI 5
SB J I C 5
SB UI C 5
-0.40 deg
2.35 deg
-13. 3 deg -13. 4 deg
McCook
-1. 1 deg
1. 9 deg
0. 6 deg
93%B
MVA105%
B
MVA
Example, which models the Eastern Interconnectcontains about 43,000 buses. 25
Solution Log for 1200 MW OutageIn this example thelosss of a 1200 MWgenerator in NorthernIllinois was simulated. This caused a generation imbalancein the associated balancing authorityarea, which wascorrected by a redispatch of localgeneration.
26
Interconnected OperationPower 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 AreasA “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 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 f
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 ControlMost 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 TransactionsPower 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 Bus 1
Bus 3Home Area
266 MW133 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 MWAGC ONAVR ON
AGC ONAVR ON
Net tie-line flow is close to zero
Generationis automaticallychanged to matchchange in load
34
100 MW Transaction between areas in Eastern or Western
Bus 2 Bus 1
Bus 3Home Area
Scheduled Transactions
225 MW113 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 MWAGC ONAVR ON
AGC ONAVR ON
100.0 MW
Scheduled100 MWTransaction from Left to Right
Net tie-lineflow is now100 MW
35
PTDFsPower 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:
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 Pθ
P
ticipate in the power transfer. 36
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
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
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
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 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
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 lk
P kP LODF P
lk
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
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