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

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.

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

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

5. The N-R Power Flow: 5-bus Example
Bus-to-Bus R
per
unit X Gc Bm
Maximum
MVA
per unit
TAP
Setting
1-5
0.02
6.0 —
3-4
0.01
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 5

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

7. Ybus Details
Elements of Ybus connected to bus 2 7

8. Here are the Initial Bus Mismatches8

9. And the Initial Power Flow Jacobian9

10. Five Bus Power System Solved10

11. 37 Bus Example Design Case11

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 for details (click on Standards) 13

14. Looking at the Impact of Line Outages
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
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
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
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
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
Automatic voltage regulation system controls voltages.
Display shows voltage contour of the power system 22

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
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.

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 flow

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.

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.

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.

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-line flow – Psched + 10β Δf
…and then controlling the generation to bring ACE towards zero.

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.

34. Three Bus Case on AGC: no interchange.
Generation
is automatically
changed to match
change in load
Net tie-line flow is
close to zero

35. 100 MW Transaction between areas in Eastern or Western
Scheduled
100 MW
Transaction from Left to Right
Net tie-line
flow is now
100 MW

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:

37. Nine Bus PTDF Example
Figure shows initial flows for a nine bus power system

38. Nine Bus PTDF Example, cont'd
Figure now shows percentage PTDF flows for a change in transaction from A to I

39. Nine Bus PTDF Example, cont'd
Figure now shows percentage PTDF flows for a change in transaction from G to F

40. WE to TVA PTDFs

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.

42. Line Outage Distribution Factors (LODFs)

43. Line Outage Distribution Factors (LODFs)

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.

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.