0. ECE 476 POWER SYSTEM ANALYSIS
Lecture 14 Power Flow
Professor Tom Overbye
Department of Electrical and Computer Engineering

1. Announcements
Homework 7 is 6.46, 6.49, 6.52, 11.19, 11.21, 11.27; due date is October 30
Potential spring courses: ECE 431 and ECE 398RES (Renewable Electric Energy Systems)
If interested you can still sign up for a power lunch.

2. 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

3. The N-R Power Flow: 5-bus Example
Type V
per unit 
degrees PG
per
unit QG PL QL
QGmax
QGmin 1
Swing
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 3

4. 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, 1 = 0
P1, Q1 2
P2 = PG2-PL2 = -8
Q2 = QG2-QL2 = -2.8
V2, 2 3
V3 = 1.05
P3 = PG3-PL3 = 4.4
Q3, 3 4
P4 = 0, Q4 = 0
V4, 4 5
P5 = 0, Q5 = 0
V5, 5
Table 4. Input data
and unknowns 4

5. Time to Close the Hood: Let the Computer Do the Math! (Ybus Shown)5

6. Ybus Details
Elements of Ybus connected to bus 2 6

7. Here are the Initial Bus Mismatches7

8. And the Initial Power Flow Jacobian8

9. And the Hand Calculation Details!9

10. Five Bus Power System Solved10

11. 37 Bus Example Design Case
This is Design Case 2 From Chapter 6 11

12. Good Power System Operation
Good power system operation requires that there be no reliability violations 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.
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) 12

13. Looking at the Impact of Line Outages
Opening one line (Tim69-Hannah69) causes an overload. This would not be allowed 13

14. 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 14

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
In a planning context additions to the grid can be considered
In the next example we look at how to remove the existing contingency violations while serving new load. 15

16. An Unreliable Solution
Case now has nine separate contingencies with reliability violations 16

17. A Reliable Solution
Previous case was augmented with the addition of a 138 kV Transmission Line 17

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 system slack, distributed across multiple generators by participation factors or by economics 18

19. 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 19

20. Generation Change Example 2
Display repeats previous case except now the change in generation is picked up by other generators using a participation factor approach 20

21. Voltage Regulation Example: 37 Buses
Display shows voltage contour of the power system, demo will show the impact of generator voltage set point, reactive power limits, and switched capacitors 21

22. Real-sized Power Flow Cases
Real power flow studies are usually done with cases with many thousands of buses
Buses are usually group in to 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 22

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, known as sparse matrix/vector methods, can be used to store the values and solve the power flow
Without these techniques large systems would be essentially unsolvable. 23

24. Eastern Interconnect Example
Example, which models the Eastern Interconnect contains about 43,000 buses. 24

25. Solution Log for 1200 MW Gen Outage
In this example we simulated the loss of a 1200 MW generator in Northern Illinois. This caused a generation imbalance in the associated balancing authority area, which was corrected by a redispatch of local generation. 25

26. “DC” Power Flow
The “DC” power flow makes the most severe approximations:
completely ignore reactive power, assume all the voltages are always 1.0 per unit, ignore line conductance
This makes the power flow a linear set of equations, which can be solved directly

27. Power System Control
A major problem with power system operation is the limited capacity of the transmission system
lines/transformers have limits (usually thermal)
no direct way of controlling flow down a transmission line (e.g., there are no valves to close to limit flow)
open transmission system access associated with industry restructuring is stressing the system in new ways
We need to indirectly control transmission line flow by changing the generator outputs

28. DC Power Flow Example 28

29. DC Power Flow 5 Bus Example
Notice with the dc power flow all of the voltage magnitudes are 1 per unit. 29

30. Indirect Transmission Line Control
What we would like to determine is how a change in
generation at bus k affects the power flow on a line
from bus i to bus j.
The assumption is
that the change
in generation is
absorbed by the
slack bus

31. Power Flow Simulation - Before
One way to determine the impact of a generator change is to compare a before/after power flow.
For example below is a three bus case with an overload

32. Power Flow Simulation - After
Increasing the generation at bus 3 by 95 MW (and hence
decreasing it at bus 1 by a corresponding amount), results
in a 31.3 drop in the MW flow on the line from bus 1 to 2.

33. Analytic Calculation of Sensitivities
Calculating control sensitivities by repeat power flow solutions is tedious and would require many power flow solutions. An alternative approach is to analytically calculate these values

34. Analytic Sensitivities

35. Three Bus Sensitivity Example

36. Balancing Authority Areas
An balancing authority area (use to be called operating areas) has traditionally represented the portion of the interconnected electric grid operated by a single utility
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-flow

37. Area Control Error (ACE)
The area control error (ace) is the difference between the actual flow out of an area and the scheduled flow, plus a frequency component
Ideally the ACE should always be zero.
Because the load is constantly changing, each utility must constantly change its generation to “chase” the ACE.

38. Automatic Generation Control
Most utilities use automatic generation control (AGC) to automatically change their generation to keep their ACE close to zero.
Usually the utility control center calculates ACE based upon tie-line flows; then the AGC module sends control signals out to the generators every couple seconds.

39. Power Transactions
Power transactions are contracts between generators and loads to do power transactions.
Contracts can be for any amount of time at any price for any amount of power.
Scheduled power transactions are implemented by modifying the value of Psched used in the ACE calculation

40. PTDFs
Power transfer distribution factors (PTDFs) show the linear impact of a transfer of power.
PTDFs calculated using the fast decoupled power flow B matrix

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

42. Nine Bus PTDF Example, cont'd
Figure now shows percentage PTDF flows from A to I

43. Nine Bus PTDF Example, cont'd
Figure now shows percentage PTDF flows from G to F

44. WE to TVA PTDFs

45. 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
LODFs are used extensively in real-time operations
LODFs are state-independent but do dependent on the assumed network topology

46. Flowgates
The real-time loading of the power grid is accessed via “flowgates”
A flowgate “flow” is the real power flow on one or more transmission element for either base case conditions or a single contingency
contingent flows are determined using LODFs
Flowgates are used as proxies for other types of limits, such as voltage or stability limits
Flowgates are calculated using a spreadsheet

47. NERC Regional Reliability Councils
NERC is the North American Electric Reliability Council

48. NERC Reliability Coordinators