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  • Topic 1: Basics of Power Systems

    A.H.MohsenianRad(UofT) 1NetworkingandDistributedSystems

    ECE5332:CommunicationsandControlforSmart

    Spring2012

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 2

    TheFourMainElementsinPowerSystems:

    PowerProduction/Generation

    PowerTransmission

    PowerDistribution

    PowerConsumption/Load

    Ofcourse,wealsoneedmonitoringandcontrolsystems.

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 3

    PowerProduction:

    DifferentTypes:

    Traditional

    Renewable

    Capacity,Cost,CarbonEmission

    StepupTransformers

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 4

    PowerTransmission:

    HighVoltage(HV)TransmissionLines

    SeveralHundredMiles

    SwitchingStations

    Transformers

    CircuitBreakers

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 5

    ThePowerTransmissionGridintheUnitedStates:

    www.geni.org

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 6

    MajorInterconnectionsintheUnitedStates:

    www.geni.org

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 7

    PowerDistribution:

    MediumVoltage(MV)TransmissionLines(

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 8

    PowerConsumption:

    Industrial

    Commercial

    Residential

    DemandResponse

    ControllableLoad

    NonControllable

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 9

    Generation Transmission Distribution Load

  • Power Systems

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 10

    PowerSystemControl:

    DataCollection:Sensors,PMUs,etc.

    DecisionMaking:Controllers

    Actuators:CircuitBreakers,etc.

  • Power Grid Graph Representation

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 11

    Nodes:Buses

    Links:TransmissionLines

    Generator

    Load

  • Power Grid Graph Representation

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 12

    Nodes:Buses

    Links:TransmissionLines

    Generator

    Load

    Buses (Voltage)

  • Power Grid Graph Representation

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 13

    Nodes:Buses

    Links:TransmissionLines

    Generator

    Load

    TransmissionLines(PowerFlow,Loss)

  • Power Grid Graph Representation

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 14

    Nodes:Buses

    Links:TransmissionLines

    Generator

    Load

    Consum

    ers

  • Power Grid Graph Representation

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 15

    Nodes:Buses

    Links:TransmissionLines

    Generator

    Load

    10MW 3MW

    7MW

  • Transmission Line Admittance

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 16

    Admittanceyisdefinedastheinverseofimpedancez:

    z=r+jx (r:Resistance,x:Reactance)

    y=g+jb (g:Conductance,b:Susceptance)

    y=1/z

    Parametergisusuallypositive

    Parameterb:

    Positive:Capacitor

    Negative:Inductor

  • Transmission Line Admittance

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 17

    Forthetransmissionlineconnectingbusi tobusk:

    Addmitance:yik

    Example:

    yik =1 j4(perunit)

    Notethat,yii isdenotedbyyi andindicates:

    Susceptance foranyshuntelement(capacitor)togroundatbusi.

  • Y-Bus Matrix

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 18

    Wedefine:

    Ybus =[Yij ]where

    DiagonalElements:

    OffdiagonalElements:

    NotethatYbas matrixdependsonthepowergridtopologyandtheadmittanceofalltransmissionlines.

    Nisthenumberofbussesinthegrid.

    N

    ikkikiii yyY

    ,1

    ijij yY

  • Y-Bus Matrix

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 19

    Example:Foragridwith4buses,wehave:

    Afterseparatingtherealandimaginaryparts:

    4342414434241

    3434323133231

    2423242321221

    1413121413121

    yyyyyyyyyyyyyyyyyyyyyyyyyyyy

    Ybus

    BjGYbus

  • Bus Voltage

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 20

    LetVi denotethevoltageatbusi:

    Notethat,Vi isaphasor,withmagnitude andangle.

    Inmostoperatingscenarioswehave:

    iii VV

    jiji VV

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 21

    LetSi denotethepowerinjection atbusi:

    Si =Pi +jQi

    GenerationBus: Pi >0

    LoadBus: Pi

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 22

    UsingKirchhofflaws,ACPowerFlowEquations become:

    Doweknowallnotationshere?

    Ifweknowenoughvariables,wecanobtaintherestofvariablesbysolvingasystemofnonlinearequations.

    N

    jjkkjjkkjjkk

    N

    jjkkjjkkjjkk

    BGVVQ

    BGVVP

    1

    1

    )cos()sin(

    )sin()cos(

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 23

    TheACPowerFlowEquationsarecomplicatedtosolve.

    Next,wetrytosimplifytheequationsinthreesteps.

    Step1:Formostnetworks,G

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 24

    Step2:Formostneighboringbuses:.

    Asaresult,wehave:

    15 to10 ji

    1)()(

    jk

    jkjk

    CosSin

    N

    jkjjkk

    N

    jjkkjjkk

    BVVQ

    BVVP

    1

    1)(

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 25

    Step3:Inperunit,|Vi|isverycloseto1.0(0.95to1.05).

    Asaresult,wehave:.

    Pk hasalinearmodelandQk isalmostfixed.

    1ji VV

    kN

    kjj

    kjkk

    N

    jkjk

    N

    jjkkjk

    bBBBQ

    BP

    ,11

    1)(

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 26

    Step3:Inperunit,|Vi|isverycloseto1.0(0.95to1.05).

    Asaresult,wehave:.

    Pk hasalinearmodelandQk isalmostfixed.

    1ji VV

    kN

    kjj

    kjkk

    N

    jkjk

    N

    jjkkjk

    bBBBQ

    BP

    ,11

    1)(

    DCPowerFlowEquations

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 27

    Giventhepowerinjectionvalues atallbuses,wecanuse

    toobtainthevoltageangles atallbuses.

    LetPij denotethepowerflow frombusi tobusj,wehave:

    N

    jjkkjk BP

    1)(

    )( jiijij BP

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 28

    Example:Obtainpowerflowvaluesinthefollowinggrid:

    puPg 21 puPg 22

    puPg 14 puPl 43

    puP l 12 1014 jy 1013 jy

    1034 jy

    1023 jy

    1012 jy

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 29

    First,weobtaintheYbusmatrix:

    44434241

    34333231

    24232221

    14131211

    4342414434241

    3434323133231

    2423242321221

    1413121413121

    BBBBBBBBBBBBBBBB

    jBjj

    bbbbbbbbbbbbbbbbbbbbbbbbbbbb

    jYbus

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 30

    Next,wewritethe(active)powerflowequations:

    Thiscanbewrittenas:

    44342413432421414

    43433432312321313

    42432322423211212

    41431321211413121

    BBBBBBPBBBBBBPBBBBBBP

    BBBBBBP

    4

    3

    2

    1

    434241434241

    343432313231

    242324232121

    141312141312

    4

    3

    2

    1

    BBBBBBBBBBBBBBBBBBBBBBBB

    PPPP

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 31

    Fromthelasttwoslides,wefinallyobtain:

    Therefore,thevoltageanglesareobtainedas:

    4

    3

    2

    1

    1

    4

    3

    2

    1

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 32

    However,thelastmatrixinthepreviousslideissingular!

    Therefore,wecannottaketheinverse.

    Thesystemofequationswouldhaveinfinitesolutions.

    Theproblemisthatthefouranglesarenotindependent.

    Whatmattersistheangular/phasedifference.

    Wechooseonebus(e.g.,bus1)asreferencebus:.01

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 33

    Weshouldalsoremovethecorrespondingrows/columns:

    Theangulardifferences(withrespectto):

    4

    3

    2

    1

    201001010301010010201010101030

    14

    12

    4

    3

    2

    2010010301001020

    14

    1

    1

    025.015.0

    025.0

    14

    1

    2010010301001020 1

    4

    3

    2

    025.015.0025.0

    14

    13

    12

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 34

    Finally,thepowerflowvaluesarecalculatedas:

    puPg 21 puPg 22

    puPg 14 puPl 43

    puP l 12

    25.1)025.015.0(10)(25.1)15.0025.0(10)(

    25.0)025.00(10)(5.1)15.00(10)(

    25.0)025.00(10)(

    433434

    322323

    411414

    311313

    211212

    BPBPBPBPBP

    0.25

    1.25

    1.250.25 1.5

  • Power Flow Equations

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 35

    Whatifthegeneratorconnectedtobus1isrenewable?

    Whatifthecapacity oftransmissionlink(1,3)is1pu?

    Whatifwecanapplydemandresponse toloadbus3?

    Whatifoneofthetransmissionlinesfails?

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 36

    Intheexamplewediscussedearlier,wehad:

    Inparticular,wehad:

    However,generationlevelsandassumedgiven.

    Q:Whatifthegeneratorshavedifferentgenerationcosts?

    PowerSupply=PowerLoad

    llggg PPPPP 32421

    ggg PPP 421 ,,

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 37

    Forthermalpowerplants,generationcostisquadratic:

    Example:agridwiththreepowerplants:

    Eachpowerplanthassomeminandmaxgenerationlevels.

    GenerationCost=C(P)=a1 +a2 xP+a3 xP2

    C1(P1)=561+7.92xP1 +0.001562x(P1)2 150MWP1 600MW

    C2(P2)=310+7.85xP2 +0.001940x(P2)2 100MWP2 400MW

    C3(P3)=78+7.97xP3 +0.004820x(P3)2 50MWP3 200MW

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 38

    Forthermalpowerplants,generationcostisquadratic:

    Example:agridwiththreepowerplants:

    Eachpowerplanthassomeminandmaxgenerationlevels.

    GenerationCost=C(P)=a1 +a2 xP+a3 xP2

    C1(P1)=561+7.92xP1 +0.001562x(P1)2 150MWP1 600MW

    C2(P2)=310+7.85xP2 +0.001940x(P2)2 100MWP2 400MW

    C3(P3)=78+7.97xP3 +0.004820x(P3)2 50MWP3 200MW

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 39

    Forthermalpowerplants,generationcostisquadratic:

    Example:agridwiththreepowerplants:

    Eachpowerplanthassomeminandmaxgenerationlevels.

    GenerationCost=C(P)=a1 +a2 xP+a3 xP2

    C1(P1)=561+7.92xP1 +0.001562x(P1)2 150MWP1 600MW

    C2(P2)=310+7.85xP2 +0.001940x(P2)2 100MWP2 400MW

    C3(P3)=78+7.97xP3 +0.004820x(P3)2 50MWP3 200MW

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 40

    Forthermalpowerplants,generationcostisquadratic:

    Example:agridwiththreepowerplants:

    Eachpowerplanthassomeminandmaxgenerationlevels.

    GenerationCost=C(P)=a1 +a2 xP+a3 xP2

    C1(P1)=561+7.92xP1 +0.001562x(P1)2 150MWP1 600MW

    C2(P2)=310+7.85xP2 +0.001940x(P2)2 100MWP2 400MW

    C3(P3)=78+7.97xP3 +0.004820x(P3)2 50MWP3 200MW

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 41

    Forthermalpowerplants,generationcostisquadratic:

    Example:agridwiththreepowerplants:

    Eachpowerplanthassomeminandmaxgenerationlevels.

    GenerationCost=C(P)=a1 +a2 xP+a3 xP2

    C1(P1)=561+7.92xP1 +0.001562x(P1)2 150MWP1 600MW

    C2(P2)=310+7.85xP2 +0.001940x(P2)2 100MWP2 400MW

    C3(P3)=78+7.97xP3 +0.004820x(P3)2 50MWP3 200MW

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 42

    WeshouldselectP1,P2,andP3 to:

    Meettotalload Pload =850MW

    Minimizethetotalcost ofgeneration

    EconomicDispatchProblem:

    85020050400100600150 subject to

    CCC minimize

    321

    3

    2

    1

    321 , , 321

    PPPPPP

    PPPPPP

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 43

    Istheformulatedproblemaconvexprogram?Why?

    Convexprogramscanbesolvedefficiently.

    AnusefulsoftwareisCVXforMatlab (http://cvxr.com/cvx).

    Theoptimal economicdispatchsolution:

    P1 =393.2MW

    P2 =334.6MW

    P3 =122.2MW

    Q:Dotheysatisfyallconstraints?

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 44

    Istheformulatedproblemaconvexprogram?Why?

    Convexprogramscanbesolvedefficiently.

    AnusefulsoftwareisCVXforMatlab (http://cvxr.com/cvx).

    Theoptimal economicdispatchsolution:

    P1 =393.2MW

    P2 =334.6MW

    P3 =122.2MW

    MinimumCost =3916.6+3153.8+1123.9=8194.3

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 45

    Whatifwehavetosatisfytopologyconstraints?

    1P

    3P MW 400

    2P

    MW 45020013 P

    3

    21

    4

    3

    2

    400450

    2010010301001020

    P

    P

    202010)( 33311313 BP

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 46

    Thesame optimalsolutionsarestillvalid:

    MW 393.21 P

    MW 122.2 3 P MW 400

    MW 334.6 2 P

    MW 450

    805.3830.19685.15

    4

    3

    2

    203 20013 P

    156.8

    160.3

    41.438.1 198.3

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 47

    Thesame optimalsolutionsarestillvalid:

    MW 393.21 P

    MW 122.2 3 P MW 400

    MW 334.6 2 P

    MW 450

    805.3830.19685.15

    4

    3

    2

    203 20013 P

    156.8

    160.3

    41.438.1 198.3

    Whatif 17013 P

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 48

    Thentheeconomicdispatchproblembecomes:

    17

    400450

    2010010301001020

    850 20050 400100 600150 subject to

    CCC minimize

    3

    3

    21

    4

    3

    2

    321

    3

    2

    1

    321 , ,, , , 321321

    P

    P

    PPPPPP

    PPPPPP

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 49

    Thentheeconomicdispatchproblembecomes:

    17

    400450

    2010010301001020

    850 20050 400100 600150 subject to

    CCC minimize

    3

    3

    21

    4

    3

    2

    321

    3

    2

    1

    321 , ,, , , 321321

    P

    P

    PPPPPP

    PPPPPP StillaConvex

    Program?

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 50

    Thenewoptimalsolutionsareobtainedas:

    Thetotalgenerationcostbecomes:$8,233.66>$8,194.3

    Here,wehadtosacrificecostforimplementation.

    MW 2801 P

    MW 170 3 P MW 400

    MW 400 2 P

    MW 450

    110

    170

    600 170

  • Economic Dispatch Problem

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 51

    Thenewoptimalsolutionsareobtainedas:

    Thetotalgenerationcostbecomes:$8,233.66>$8,194.3

    Here,wehadtosacrificecostforimplementation.

    MW 2801 P

    MW 170 3 P MW 400

    MW 400 2 P

    MW 450

    110

    170

    600 170

    Whatif 15013 P

  • Unit Commitment

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 52

    EconomicDispatchissolvedafewhoursahead ofoperation.

    Ontheotherhand,weneedtodecideaboutthechoiceofpowerplantsthatwewanttoturnon forthenextday.

    ThisisdonebysolvingtheUnitCommitment problem.

    Weparticularlydecideonwhichslowstarting powerplantsweshouldturnonduringthenextdaygivenvariousconstraints.

    ThemathematicalconceptsaresimilartotheEDproblem.

  • References

    Dr. Hamed Mohsenian-Rad Texas Tech UniversityCommunications and Control in Smart Grid 53

    W. J. Wood and B. F. Wollenberg, Power Generation,Operation, and Control, John Wiley & Sons, 2nd Ed., 1996.

    J. McCalley and L. Tesfatsion, "Power Flow Equations", LectureNotes, EE 458, Department of Electrical and ComputerEngineering, Iowa State University, Spring 2010.

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