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Aps_chap9 (Power Flow)

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    Energy Conversion Lab

    POWER FLOW ANALYSIS

    Power flow analysis assumption steady-state

    balanced single-phase network

    network may contain hundreds of nodes and

    branches with impedance X specified in per unit onMVA base

    Power flow equations

    bus admittance matrix of node-voltage equation isformulated

    currents can be expressed in terms of voltages

    resulting equation can be in terms of power in MW

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    Energy Conversion Lab

    BUS ADMITTANCE MATRIX

    Nodal solution nodal solution is based on

    the Kirchhoffs current law

    impedance is converted to

    admittance

    Bus admittance equations

    the impedance diagram:see Fig.6.1

    ijijij

    ijjxrZ

    y+

    ==11

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    Energy Conversion Lab

    BUS ADMITTANCE MATRIX

    Bus admittanceequations

    the admittance isbased on bus-to-

    bus: see Fig.6.2 if no connection

    between bus-to-bus, leave as zero

    node voltageequation is in theform

    busbusbus VYI =

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    Energy Conversion Lab

    BUS ADMITTANCE MATRIX

    Node-voltage matrix Ibus=YbusVbus

    Ibus is the vector of injected currents

    Vbus is the vector of the bus voltage from referencenode

    Ybus is the bus admittance matrix

    =

    n

    i

    n

    i

    V

    V

    V

    V

    I

    I

    I

    I

    2

    1

    2

    1

    nnnin2n1

    iniii2i1

    2n2i2221

    1n1i1211

    YYYY

    YYYY

    YYYY

    YYYY

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    Energy Conversion Lab

    BUS ADMITTANCE MATRIX

    Node-voltage matrix diagonal element Yii: sum of admittance connected to bus i

    off-diagonal matrix Yij: negative of admittance between nodes Iand j

    when the bus currents are known, bus voltages are unknown,bus voltage can be solved as

    inverse of bus admittance matrix is known as impedance matrixZbus

    if matrix of Ybus is invertible, Ybus should be non-singular

    ij0

    ==

    n

    j

    ijii yY

    ijjiij yYY ==

    busbusbus IYV1

    =

    1= busbus YZ

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    Energy Conversion Lab

    BUS ADMITTANCE MATRIX

    Node-voltage matrix admittance matrix is symmetric along the leading diagonal,

    which result in an upper diagonal nodal admittance matrix

    a typical power system network, each bus is connected by a fewnearby bus, which cause many off-diagonal elements are zero

    many zero off-diagonal matrix is called sparse matrix the bus admittance matrix in Fig.(6.2) by inspection is

    =

    5.125.1200

    5.125.220.50.500.575.85.2

    00.55.25.8

    jj

    j-jjjjjj

    jjj

    Ybus

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    Energy Conversion Lab

    SOLUTION OF NONLINEAR ALGEBRA EQUATIONS

    Techniques for iterative solution of non-linearequations

    Gauss-Seidal

    Newton-Raphson

    Quasi-Newton

    Gause-Seidal method consider a nonlinear equation f(x)=0

    rearrange f(x) so that x=g(x), f(x)=x-g(x) orf(x)=g(x)-x

    guess an initial estimate of x = x(k)

    use iteration, obtain next x value as x(k+1) = g(x(k))

    criteria for stop iteration: |x(k+1)-x(k)|

    is the desired accuracy

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    Energy Conversion Lab

    GAUSE-SEIDAL METHOD

    Nature of Gause-Seidal method see Ex.(6.2) and Fig.(6.3) Gause-Seidal method needs many iterations to

    achieve desired accuracy

    no guarantee for the convergence, depend on the

    location of initial x estimate

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    Energy Conversion Lab

    GAUSE-SEIDAL METHOD

    Nature of Gause-Seidal method

    solution: if initial estimate x is within convergentregion, solution will converge in zigzag fashion to oneof the roots

    no solution: if initial estimate x is outside convergentregion, process will diverge, no solution found

    in some case, an acceleration factor is added toimprove the rate of convergence:

    x(k+1) = x(k) +[g(x(k))-x(k)], where >1

    acceleration factor should not too large to produceovershoot

    see Ex.(6.3) for the acceleration factor used

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    Energy Conversion Lab

    GAUSE-SEIDAL METHOD

    Extend one variable to n variable equations using Gause-

    Seidal method consider the system of n equations in n variables and solving for

    one variable from each equation in one time of iteration

    the updated variable x1(k+1) calculated in first equation in

    Eq.(6.12) is used in the calculation of x2(k+1) in the secondequation

    Ex: in the 2nd iteration x2(k+1) = c2+g2(x1

    (k+1)+x2(k)+x3

    (k)++xn(k))

    at n iteration to complete n variables, the x1(k+1),,xn

    (k+1) istested against x

    1

    (k),,xn

    (k) for accuracy criterion

    nnn

    n

    n

    cxxxf

    cxxxf

    cxxxf

    =

    =

    =

    ),,,(

    ),,,(

    ),,,(

    21

    2212

    1211

    ),,,(

    ),,,(

    ),,,(

    21

    21222

    21111

    nnnn

    n

    n

    xxxgcx

    xxxgcx

    xxxgcx

    +=

    +=

    +=

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    Energy Conversion Lab

    POWER FLOW SOLUTION

    Power Flow (Load Flow)

    operating condition: balanced, single phase model

    quantities used in power flow equation are: voltagemagnitude |V|, phase angle , real power P, andreactive power Q

    system bus classification:

    slack bus (swing bus): taken as reference where |V| andV are specified. It makes up the loss between generatedpower and scheduled loads

    load bus (PQ bus): P and Q are specified, |V| and V areunknown

    regulated bus (PV bus): P and |V| are specified, V and Qare unknown

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    Energy Conversion Lab

    POWER FLOW EQUATION

    Power flow formulation consider bus case in Fig.(6.7)

    current flow into bus i:

    express Ii in terms of P,Q:

    the power flow equation becomes

    the power flow problem results in algebraic nonlinear equationswhich must be solved by iteration methods

    ij10

    = ==

    j

    n

    j

    ij

    n

    j

    ijii VyyVI

    *

    i

    iii

    V

    jQPI =

    ij10

    * =

    ==j

    n

    j

    ij

    n

    j

    iji

    i

    ii VyyVVjQP

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    Energy Conversion Lab

    GAUSS-SEIDEL POWER FLOW EQUATION

    Gauss-Seidel power flow solution solving Vi: for PQ bus, assume P,Q are known

    solving Pi: for slack bus, assume V is known

    solving Qi: for PV bus, assume |V| is known

    ij

    )(

    )(*)1(

    +

    =

    +

    ij

    k

    kijk

    i

    sch

    i

    sch

    i

    k

    iy

    VyV

    jQP

    V

    ijRe)(

    10

    )()(*)1(

    = ==

    + kj

    n

    ijj

    ij

    n

    j

    ij

    k

    i

    k

    i

    k

    i VyyVVP

    ijIm )(

    10

    )()(*)1(

    =

    ==

    + kj

    n

    ijj

    ij

    n

    j

    ij

    k

    i

    k

    i

    k

    i VyyVVQ

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    Energy Conversion Lab

    GAUSS-SEIDEL POWER FLOW EQUATION

    Instructions for Gauss-Seidel solution there are 2(n-1) equations to be solved for n bus

    voltage magnitude of the buses are close to 1pu orclose to the magnitude of the slack bus

    voltage magnitude at load buses is lowerthan the slackbus value

    voltage magnitude at generator buses is higherthanthe slack bus value

    phase angle of load buses are below the referenceangle

    phase angle of generator buses are above thereference angle

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    Energy Conversion Lab

    INSTRUCTIONS FOR G-S SOLUTION

    Instructions for PQ bus solution

    real and reactive power Pisch, Qi

    sch are known

    starting with an initial estimate of voltage using Viequation

    Instructions for PV bus solution

    Pisch, |Vi| are specified

    assume Vi = |Vi|0o, solve the Qi equation as below

    ij

    )(

    )(*

    )1(

    +

    =

    +

    ij

    k

    jijk

    i

    sch

    i

    sch

    i

    ki

    y

    Vy

    V

    jQP

    V

    ijIm )(

    10

    )()(*)1(

    =

    ==

    + kj

    n

    ijj

    ij

    n

    j

    ij

    k

    i

    k

    i

    k

    i VyyVVQ

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    Energy Conversion Lab

    INSTRUCTIONS FOR G-S SOLUTION

    Instructions for PV bus solution

    when Qi(k+1) is available, solve Vi using equation below

    since |Vi| is specified, keep imaginary part of Vi,calculate real part of Vi

    solve Vi

    stopping criteria

    ij

    )(

    )(*

    )(

    )1( +

    =

    +

    ij

    k

    kijk

    i

    k

    i

    sch

    i

    k

    iy

    VyV

    jQP

    V

    { } { }( )2)1(2)1(Re ++ = kiiki VimagVV

    )1()1()1( ImRe +++ += kik

    i

    k

    i VjVV

    { } { } { } { } ++ )()1()()1( ImIm,ReRe kikikiki VVVV

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    Energy Conversion Lab

    INSTRUCTIONS FOR G-S SOLUTION

    Instructions for PV bus solution

    to accelerate the convergence, using the followingapproximation after new Vi is obtained

    is in the range between 1.3 to 1.7

    voltage accuracy in |Vi| and is in the range between0.00001 to 0.00005

    ( ))()()()1( kikcalikiki VVVV +=+

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    Energy Conversion Lab

    INSTRUCTIONS FOR G-S SOLUTION

    Instructions for V,

    slack bus solution solve Pi

    solve Qi

    accuracy: the largest PQ is less than thespecified value, typically is about 0.001pu

    ijRe )(

    10

    )()(*)1(

    =

    ==

    + kj

    n

    ij

    j

    ij

    n

    j

    ij

    k

    i

    k

    i

    k

    i VyyVVP

    ijIm )(

    10

    )()(*)1(

    =

    ==

    + kj

    n

    ij

    j

    ij

    n

    j

    ij

    k

    i

    k

    i

    k

    i VyyVVQ

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    G-S Power flow Homework

    For the one-line diagram shown below, using the G-S methodto determine all bus voltages (magnitude and phase) andshow the power flow solution between the buses assume theregulated bus (#2) reactive power limits are between 0 and

    600Mvar.

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    Energy Conversion Lab

    NEWTON RAPHSON METHOD

    Newton Raphson method for solving one variable

    consider the solution of one-dimensional equation f(x)=c

    assume x = x(0)+x(0)

    f(x)=f(x(0)+x(0))=c

    use Taylors series expansion

    assume x(0) is very small, higher order terms of expansion canbe neglected, Taylor series becomes

    assume f(x(0))=c-c(0), the equation becomes c(0)(df/dx)(0)x(0)

    the new approximation of x

    ( ) cxdxfdx

    dxdfxfxxf =+

    +

    +=+

    2)0(

    )0(

    2

    2)0(

    )0(

    )0()0()0(

    !21)()(

    cxdx

    df

    xfxxf =

    +=+

    )0(

    )0(

    )0()0()0(

    )()(

    )0(

    )0()0()1(

    +=

    dx

    df

    cxx

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    Energy Conversion Lab

    NEWTON RAPHSON METHOD

    Newton Raphson method for solving one variable

    the new approximation of x

    Newton Raphson algorithm

    for more information, see Ex.(6.4)

    Newtons method converges faster than Gauss-Seidal, the rootmay converge to a root different from the expected one ordiverge if the starting value is not close enough to the root

    )0(

    )0()0()1(

    +=

    dx

    df

    cxx

    )()()1(

    )(

    )()(

    )()( )(

    kkk

    k

    kk

    kk

    xxx

    dx

    df

    cx

    xfcc

    +=

    =

    =

    +

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    Energy Conversion Lab

    NEWTON RAPHSON METHOD FOR n VARIABLES

    Newton Raphson method for solving n variables

    nn

    n

    nnnnn

    n

    n

    n

    n

    cx

    x

    fx

    x

    fx

    x

    fxfxxf

    cxxfx

    xfx

    xfxfxxf

    cxx

    fx

    x

    fx

    x

    fxfxxf

    =

    ++

    +

    +=+

    =

    ++

    +

    +=+

    =

    ++

    +

    +=+

    )0(

    )0(

    )0(

    2

    )0(

    2

    )0(

    1

    )0(

    1

    )0()0()0(

    2)0(

    )0(

    2)0(2

    )0(

    2

    2)0(1

    )0(

    1

    2)0(2

    )0()0(2

    1

    )0(

    )0(

    1)0(

    2

    )0(

    2

    1)0(

    1

    )0(

    1

    1)0(

    1

    )0()0(

    1

    )()(

    )()(

    )()(

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    Energy Conversion Lab

    NEWTON RAPHSON METHOD FOR n VARIABLES

    Rearrange in matrix form

    The matrix can be written as

    C(k) = J(k) X(k)

    =

    )0(

    )0(

    2

    )0(

    1

    )0()0(

    2

    )0(

    1

    )0(

    2

    )0(

    2

    2

    )0(

    1

    2

    )0(

    1

    )0(

    2

    1

    )0(

    1

    1

    )0(

    )0(

    22

    )0(

    11

    n

    n

    nnn

    n

    n

    nn x

    x

    x

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    fc

    fc

    fc

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    Energy Conversion Lab

    NEWTON RAPHSON METHOD FOR n VARIABLES

    The Newton-Raphson algorithm for n-dimensional case is

    X(k+1) = X(k) +X(k) = X(k) + [J(k)]-1C(k)

    where

    =

    )(

    )(

    22

    )(11

    )(

    k

    nn

    k

    k

    k

    fc

    fc

    fc

    C

    =

    )()(

    2

    )(

    1

    )(

    2

    )(

    2

    2

    )(

    1

    2

    )(

    1

    )(

    2

    1

    )(

    1

    1

    )(

    k

    n

    n

    k

    n

    k

    n

    k

    n

    kk

    k

    n

    kk

    k

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    x

    f

    J

    =

    )(

    )(

    2

    )(

    1

    )(

    k

    n

    k

    k

    k

    x

    x

    x

    X

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    Energy Conversion Lab

    NEWTON RAPHSON METHOD FOR n VARIABLES

    The Newton-Raphson algorithm

    J(k) is called the Jacobian matrix

    solution to X(k+1) is inefficient because it involvesinverse of J(k) , a triangular factorization is used tofacilitate the computation

    in MATLAB, the operator \ (i.e., X=J\C) is used toapply the triangular factorization

    Newton-Raphson method converge to solutionquadratically when near a root

    The limitation is that it does not generally converge toa solution from an arbitrary starting point

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    Energy Conversion Lab

    LINE FLOWS AND LOSSES

    Complex power flow between bus i,j

    for line model, see Fig. 6.8

    current flow from bus i to bus j

    current flow from bus j to bus i

    complex power Sij from bus i to j and Sji from j to i

    power loss in the line i-j

    for more Gauss-Seidel method examples, see Ex. (6.7)and Ex. (6.8)

    iijiijilij VyVVyIII 00 )( +=+=

    jjijijjlji VyVVyIII 00 )( +=+=

    ** jijjiijiij IVSIVS ==

    jiijjiL SSS += )(

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    Energy Conversion Lab

    NEWTON-RAPHSON POWER FLOW

    Real power flow in terms of Vi , , and Yij

    Reactive power flow

    Newton-Raphson matrix form: C(k) = J(k) X(k)

    diagonal and off-diagonal elements of J1

    ( )=

    +=n

    j

    jiijijjii YVVP1

    cos

    ( )=

    +=n

    j

    jiijijjii YVVQ1

    sin

    =

    VJJ

    JJ

    Q

    P

    43

    21

    ( )

    ( ) ijsin

    sin

    +=

    +=

    jiijijji

    j

    i

    ij

    jiijijji

    i

    i

    YVVP

    YVVP

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    Energy Conversion Lab

    NEWTON-RAPHSON POWER FLOW

    Newton-Raphson matrix form: C(k) = J(k) X(k)

    diagonal and off-diagonal elements of J2

    diagonal and off-diagonal elements of J3

    =

    VJJ

    JJ

    Q

    P

    43

    21

    ( )( ) ijcos

    coscos2

    +=

    ++=

    jiijiji

    j

    i

    ij

    jiijijjiiiii

    i

    i

    YVV

    P

    YVYVV

    P

    ( )

    ( ) ijcos

    cos

    +=

    +=

    jiijijji

    j

    i

    ij

    jiijijji

    i

    i

    YVVQ

    YVVQ

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    Energy Conversion Lab

    NEWTON-RAPHSON POWER FLOW

    Newton-Raphson matrix form: C(k) = J(k) X(k)

    diagonal and off-diagonal elements of J4

    power residuals

    Pi(k)

    Qi(k)

    new estimates for bus voltages

    =

    VJJ

    JJ

    Q

    P

    43

    21

    ( )( ) ijsin

    sinsin2

    +=

    +=

    jiijiji

    j

    i

    ij

    jiijijjiiiii

    i

    i

    YVV

    Q

    YVYVV

    Q

    )()()()( , kisch

    i

    k

    i

    k

    i

    sch

    i

    k

    i QQQPPP ==

    )()()1()()()1( , kik

    i

    k

    i

    k

    i

    k

    i

    k

    i VVV +=+= ++

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    Energy Conversion Lab

    NEWTON-RAPHSON POWER FLOW

    Procedure for Newton-Raphson method:

    PQ bus: set |Vi(0)|=1.0, i

    (0)=0.0

    PV bus: set i(0)=0.0

    set PQ bus equation for J matrix elements:

    set PV bus equation for J matrix elements:

    )()()()( , kisch

    i

    k

    i

    k

    i

    sch

    i

    k

    i QQQPPP ==

    ( )=+=

    n

    jjiijijjii YVVP 1 cos

    ( )=

    +=n

    j

    jiijijjii YVVQ1

    sin

    ( )=

    +=n

    j

    jiijijjii YVVP1

    cos

    )()( k

    i

    sch

    i

    k

    i PPP =

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    Energy Conversion Lab

    NEWTON-RAPHSON POWER FLOW

    Procedure for Newton-Raphson method:

    use above equation to calculate Jacobian matrix (J1, J2,J3, J4)

    solve |V| and using Newton-Raphson matrix

    update |V| and by

    repeat the calculation until

    for example: see Ex.(6.10)

    )()( , kik

    i QP

    =

    VJJ

    JJ

    Q

    P

    43

    21

    )()()1()()()1( , kik

    i

    k

    i

    k

    i

    k

    i

    k

    i VVV +=+= ++

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    Energy Conversion Lab

    FAST DECOUPLED POWER FLOW

    Fast decoupled power flow solution:

    the algorithm is based on Newton-Raphson method

    when transmission lines has a high X/R ratio, theNewton-Raphson method could be further simplified

    Consider the Newton-Raphson power flowequation

    P are less sensitive to |V| and most sensitive to

    Q is less sensitive to and most sensitive to |V|

    we can reasonably eliminate J2 and J3 elements inJacobian matrix

    =

    VJJ

    JJ

    Q

    P

    43

    21

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    Energy Conversion Lab

    FAST DECOUPLED POWER FLOW Consider the Newton-Raphson power flow equation

    the power flow equation reduces to

    P = J1 = [P/], Q = J4|V| = [Q/|V|]|V|

    Pi/i = -Qi - |Vi|2Bii, Bii = |Yii|sinii is the imaginary part of the

    diagonal elements

    since Bii >> Qi, Pi/i (diagonal elements of J1) can be further

    reduced to

    Pi/

    i = - |Vi|Bii (|Vi|

    2

    |Vi| )

    off diagonal element of J1: Pi/i = - |Vi||Vj|Yijsin(ij-i+j), since j-iis quite small, ij-i+j = ij, J1 = Pi/j = - |Vi||Vj|Bij

    since |Vj|

    1, off diagonal elements of J1 =

    Pi/

    j = - |Vi|Bij

    =

    VJ

    J

    Q

    P

    4

    1

    0

    0

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    Energy Conversion Lab

    FAST DECOUPLED POWER FLOW

    Consider the Newton-Raphson power flow

    equation similarly, diagonal elements of J4: Qi/|Vi| = - |Vi|Bii off diagonal elements of J4: Qi/|Vj| = - |Vi|Bij therefore, P and Q has the following forms

    B and B are the imaginary part of Ybus the updated and |V| can be obtained from

    to calculate PQ bus, use simplified J1 and J4 to obtainsolution

    to calculate PV bus, J4 can be further eliminated, onlyJ

    1is used to obtain solution

    VBV

    QB

    V

    P

    ii=

    =

    ''' ,

    [ ] [ ] V

    QBV

    V

    PB

    =

    = 11 ",'

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