Lecture 14Power Flow
Professor Tom Overbye
Department of Electrical andComputer Engineering
ECE 476POWER SYSTEM ANALYSIS
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 ECE398RES (Renewable Electric Energy Systems)
If interested you can still sign up for a powerlunch.
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400 MVA
15 kV
400 MVA
15/345 kV
T1
T2
800 MVA
345/15 kV
800 MVA
15 kV
520 MVA
80 MW
40 Mvar
280 Mvar
800 MW
Line 3345 kV
Line 2
Line 1
345 kV100 mi
345 kV200 mi
50 mi
1
4
3
2
5
Single-line diagram
The N-R Power Flow: 5-bus Example
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Bus
Type
V
perunit
degrees
PG
per
unit
QG
per
unit
PL
per
unit
QL
per
unit
QGmax
per
unit
QGmin
per
unit
1
Swing
1.0
0
0
0
2
Load
0
0
8.0
2.8
3
Constantvoltage
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
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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, 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
The N-R Power Flow: 5-bus Example
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Time to Close the Hood: Let theComputer Do the Math! (Ybus Shown)
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Ybus Details
Elements of Ybus connected to bus 2
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Here are the Initial Bus Mismatches
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And the Initial Power Flow Jacobian
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And the Hand Calculation Details!
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Five Bus Power System Solved
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37 Bus Example Design Case
This is Design Case 2 From Chapter 6
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Good Power System Operation
•Good power system operation requires that there be noreliability violations for either the current condition orin the event of statistically likely contingencies
•Reliability requires as a minimum that there be notransmission line/transformer limit violations and that busvoltages be within acceptable limits (perhaps 0.95 to 1.08)
•Example contingencies are the loss of any single device. Thisis known as n-1 reliability.
•North American Electric Reliability Corporation nowhas legal authority to enforce reliability standards (andthere are now lots of them). Seehttp://www.nerc.com for details (click on Standards)
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Looking at the Impact of Line Outages
Opening one line (Tim69-Hannah69) causes an overload.This would not be allowed
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Contingency Analysis
Contingencyanalysis providesan automaticway of lookingat all thestatisticallylikelycontingencies. Inthis example thecontingency set
Is all the singleline/transformeroutages
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Power Flow And Design
•One common usage of the power flow is to determinehow the system should be modified to removecontingencies problems or serve new load
•In an operational context this requires working with theexisting electric grid
•In a planning context additions to the grid can be considered
•In the next example we look at how to remove theexisting contingency violations while serving new load.
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An Unreliable Solution
Case now has nine separate contingencies with reliabilityviolations
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A Reliable Solution
Previous case was augmented with the addition of a138 kV Transmission Line
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Generation Changes and The Slack Bus
•The power flow is a steady-state analysis tool, so theassumption is total load plus losses is always equal tototal generation
•Generation mismatch is made up at the slack bus
•When doing generation change power flow studies onealways needs to be cognizant of where the generation isbeing made up
•Common options include system slack, distributed acrossmultiple generators by participation factors or by economics
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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
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Generation Change Example 2
Display repeats previous case except now the change ingeneration is picked up by other generators using aparticipation factor approach
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Voltage Regulation Example: 37 Buses
Display shows voltage contour of the power system, demowill show the impact of generator voltage set point,reactive power limits, and switched capacitors
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Real-sized Power Flow Cases
•Real power flow studies are usually done with caseswith many thousands of buses
•Buses are usually group in to various balancing authorityareas, with each area doing its own interchange control
•Cases also model a variety of different automaticcontrol devices, such as generator reactive power limits,load tap changing transformers, phase shiftingtransformers, switched capacitors, HVDC transmissionlines, and (potentially) FACTS devices
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Sparse Matrices and Large Systems
•Since for realistic power systems the model sizes arequite large, this means the Ybus and Jacobian matricesare also large.
•However, most elements in these matrices are zero,therefore special techniques, known as sparsematrix/vector methods, can be used to store the valuesand solve the power flow
•Without these techniques large systems would be essentiallyunsolvable.
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Eastern Interconnect Example
Example, which models the Eastern Interconnectcontains about 43,000 buses.
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Solution Log for 1200 MW Gen Outage
In this example wesimulated the lossof a 1200 MWgenerator in NorthernIllinois. This causeda generation imbalancein the associatedbalancing authorityarea, which wascorrected by aredispatch of localgeneration.
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“DC” Power Flow
The “DC” power flow makes the most severeapproximations:
–completely ignore reactive power, assume all the voltagesare always 1.0 per unit, ignore line conductance
This makes the power flow a linear set of equations,which can be solved directly
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Power System Control
A major problem with power system operation isthe limited capacity of the transmission system
–lines/transformers have limits (usually thermal)
–no direct way of controlling flow down a transmissionline (e.g., there are no valves to close to limit flow)
–open transmission system access associated with industryrestructuring is stressing the system in new ways
We need to indirectly control transmission line flowby changing the generator outputs
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DC Power Flow Example
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DC Power Flow 5 Bus Example
Notice with the dc power flow all of the voltage magnitudes are1 per unit.
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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
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Power Flow Simulation - Before
One way to determine the impact of a generator changeis to compare a before/after power flow.
For example below is a three bus case with an overload
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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.
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Analytic Calculation of Sensitivities
Calculating control sensitivities by repeat powerflow solutions is tedious and would require manypower flow solutions. An alternative approach is toanalytically calculate these values
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Analytic Sensitivities
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Three Bus Sensitivity Example
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Balancing Authority Areas
An balancing authority area (use to be calledoperating areas) has traditionally representedthe portion of the interconnected electric gridoperated by a single utility
Transmission lines that join two areas areknown as tie-lines.
The net power out of an area is the sum of theflow on its tie-lines.
The flow out of an area is equal tototal gen - total load - total losses = tie-flow
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Area Control Error (ACE)
The area control error (ace) is the differencebetween the actual flow out of an area andthe scheduled flow, plus a frequencycomponent
Ideally the ACE should always be zero.
Because the load is constantly changing, eachutility must constantly change its generationto “chase” the ACE.
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Automatic Generation Control
Most utilities use automatic generationcontrol (AGC) to automatically change theirgeneration to keep their ACE close to zero.
Usually the utility control center calculatesACE based upon tie-line flows; then theAGC module sends control signals out to thegenerators every couple seconds.
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Power Transactions
Power transactions are contracts betweengenerators and loads to do powertransactions.
Contracts can be for any amount of time atany price for any amount of power.
Scheduled power transactions areimplemented by modifying the value of Pschedused in the ACE calculation
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PTDFs
Power transfer distribution factors (PTDFs) showthe linear impact of a transfer of power.
PTDFs calculated using the fast decoupled powerflow B matrix
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Nine Bus PTDF Example
Figure shows initial flows for a nine bus power system
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Nine Bus PTDF Example, cont'd
Figure now shows percentage PTDF flows from A to I
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Nine Bus PTDF Example, cont'd
Figure now shows percentage PTDF flows from G to F
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WE to TVA PTDFs
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Line Outage Distribution Factors (LODFS)
LODFs are used to approximate the change in theflow on one line caused by the outage of a secondline
–typically they are only used to determine the change inthe MW flow
–LODFs are used extensively in real-time operations
–LODFs are state-independent but do dependent on theassumed network topology
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Flowgates
The real-time loading of the power grid is accessedvia “flowgates”
A flowgate “flow” is the real power flow on one ormore transmission element for either base caseconditions or a single contingency
–contingent flows are determined using LODFs
Flowgates are used as proxies for other types oflimits, such as voltage or stability limits
Flowgates are calculated using a spreadsheet
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NERC Regional Reliability Councils
NERCis theNorthAmericanElectricReliabilityCouncil
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NERC Reliability Coordinators
