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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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