EE 369

POWER SYSTEM ANALYSISLecture 14 Power FlowTom Overbye and Ross Baldick*

AnnouncementsRead 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.

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Single-line diagramThe N-R Power Flow: 5-bus Example*

Table 1. Bus input dataTable 2. Line input dataThe N-R Power Flow: 5-bus Example*

BusType|V| per unitdegreesPGperunitQGperunitPLperunitQLperunitQGmaxperunitQGminperunit1Slack 1.00002Load008.02.83Constant voltage1.055.20.80.44.0-2.84Load00005Load0000

Bus-to-BusRper unitXper unitGper unitBper unitMaximumMVAper unit2-40.00900.10001.7212.02-50.00450.05000.8812.04-50.002250.02500.4412.0

Table 3. Transformer input dataTable 4. Input data and unknownsThe N-R Power Flow: 5-bus Example*

Bus-to-BusR perunitXperunitGcperunitBmperunitMaximumMVAper unitMaximumTAPSettingper unit1-50.001500.02006.03-40.000750.010010.0

BusInput DataUnknowns1|V1 |= 1.0, 1 = 0P1, Q12P2 = PG2-PL2 = -8Q2 = QG2-QL2 = -2.8|V2|, 2

3|V3 |= 1.05P3 = PG3-PL3 = 4.4Q3, 34P4 = 0, Q4 = 0|V4|, 45P5 = 0, Q5 = 0|V5|, 5

Let the Computer Do the Calculations! (Ybus Shown)*

Ybus Details*

Here are the Initial Bus Mismatches*

And the Initial Power Flow Jacobian*

Five Bus Power System Solved*

37 Bus Example Design Case*

Good Power System OperationGood 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.*

Good Power System OperationNorth 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)*

Looking at the Impact of Line OutagesOpening one line (Tim69-Hannah69) causes overloads. This would not be Allowed.*

Contingency AnalysisContingency analysis provides an automatic way of looking at all the statistically likely contingencies. In this example the contingency setis all the single line/transformer outages*

Power Flow And DesignOne common usage of the power flow is to determine how the system should be modified to remove contingencies problems or serve new loadIn 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.*

An Unreliable Solution:some line outages result in overloadsCase now has nine separate contingencies having reliability violations(overloads in post-contingency system).*

A Reliable Solution:no line outages result in overloadsPrevious case was augmented with the addition of a 138 kV Transmission Line*

Generation Changes and The Slack BusThe power flow is a steady-state analysis tool, so the assumption is total load plus losses is always equal to total generationGeneration mismatch is made up at the slack busWhen doing generation change power flow studies one always needs to be cognizant of where the generation is being made upCommon options include distributed slack, where the mismatch is distributed across multiple generators by participation factors or by economics. *

Generation Change Example 1Display 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 slackSlack bus*

Generation Change Example 2Display 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.*

Voltage Regulation Example: 37 BusesDisplay shows voltage contour of the power system*Automatic voltage regulation system controls voltages.

Real-sized Power Flow CasesReal power flow studies are usually done with cases with many thousands of busesOutside 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.*

Sparse Matrices and Large SystemsSince 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. *

Eastern Interconnect ExampleExample, which models the Eastern Interconnect contains about 43,000 buses. *

Solution Log for 1200 MW OutageIn 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. *

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.*

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*

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. *

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.

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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.*

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 + 10 fand then controlling the generation to bring ACE towards zero.*

Physical power TransactionsFor 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.*

Three Bus Case on AGC:no interchange.Net tie-line flow is close to zeroGenerationis automaticallychanged to matchchange in load*

100 MW Transaction between areas in Eastern or WesternScheduled100 MWTransaction from Left to RightNet tie-lineflow is now100 MW*

PTDFsPower transfer distribution factors (PTDFs) show the linearized impact of a transfer of power.PTDFs calculated using the fast decoupled power flow B matrix:

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Nine Bus PTDF ExampleFigure shows initial flows for a nine bus power system*

Nine Bus PTDF Example, cont'dFigure now shows percentage PTDF flows for a change in transaction from A to I*

Nine Bus PTDF Example, cont'dFigure now shows percentage PTDF flows for a change in transaction from G to F*

WE to TVA PTDFs*

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 linetypically 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.

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Line Outage Distribution Factors (LODFs)*

Line Outage Distribution Factors (LODFs)*

FlowgatesThe 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 contingencyFlows 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.*

FlowgatesLimits 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.*

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