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 on MVA base Power flow equations bus admittance matrix of node-voltage equation is formulated currents can be expressed in terms of voltages resulting equation can be in terms of power in MW
Transcript
1. 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 on MVA base Power flow equations bus
admittance matrix of node-voltage equation is formulated currents
can be expressed in terms of voltages resulting equation can be in
terms of power in MW
2. 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 ij jxrZ y + == 11
3. Energy Conversion Lab BUS ADMITTANCE MATRIX Bus admittance
equations the admittance is based on bus-to- bus: see Fig.6.2 if no
connection between bus-to- bus, leave as zero node voltage equation
is in the form busbusbus VYI =
4. 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 reference node 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
5. 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 I and
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 matrix Zbus if matrix of Ybus is invertible, Ybus
should be non-singular ij 0 = = n j ijii yY ijjiij yYY == busbusbus
IYV 1 = 1 = busbus YZ
6. 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 few nearby 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 5.125.220.50.5
00.575.85.2 00.55.25.8 jj j-jjj jjj jjj Ybus
7. Energy Conversion Lab SOLUTION OF NONLINEAR ALGEBRA
EQUATIONS Techniques for iterative solution of non-linear equations
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) or f(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
8. 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
9. Energy Conversion Lab GAUSE-SEIDAL METHOD Nature of
Gause-Seidal method solution: if initial estimate x is within
convergent region, solution will converge in zigzag fashion to one
of the roots no solution: if initial estimate x is outside
convergent region, process will diverge, no solution found in some
case, an acceleration factor is added to improve the rate of
convergence: x(k+1) = x(k) +[g(x(k))-x(k)], where >1
acceleration factor should not too large to produce overshoot see
Ex.(6.3) for the acceleration factor used
10. 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 second equation 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)
is tested against x1 (k),,xn (k) for accuracy criterion nnn n n
cxxxf cxxxf cxxxf = = = ),,,( ),,,( ),,,( 21 2212 1211 ),,,( ),,,(
),,,( 21 21222 21111 nnnn n n xxxgcx xxxgcx xxxgcx += += +=
11. Energy Conversion Lab POWER FLOW SOLUTION Power Flow (Load
Flow) operating condition: balanced, single phase model quantities
used in power flow equation are: voltage magnitude |V|, phase angle
, real power P, and reactive power Q system bus classification:
slack bus (swing bus): taken as reference where |V| and V are
specified. It makes up the loss between generated power and
scheduled loads load bus (PQ bus): P and Q are specified, |V| and V
are unknown regulated bus (PV bus): P and |V| are specified, V and
Q are unknown
12. 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 equations which
must be solved by iteration methods ij 10 = == j n j ij n j ijii
VyyVI * i ii i V jQP I = ij 10 * = == j n j ij n j iji i ii VyyV V
jQP
13. 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 i y Vy V jQP V ijRe )( 10 )()(*)1( = == + k j n ij j ij n
j ij k i k i k i VyyVVP ijIm )( 10 )()(*)1( = == + k j n ij j ij n
j ij k i k i k i VyyVVQ
14. 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 or close to the magnitude of the slack bus voltage magnitude at
load buses is lower than the slack bus value voltage magnitude at
generator buses is higher than the slack bus value phase angle of
load buses are below the reference angle phase angle of generator
buses are above the reference angle
15. Energy Conversion Lab INSTRUCTIONS FOR G-S SOLUTION
Instructions for PQ bus solution real and reactive power Pi sch, Qi
sch are known starting with an initial estimate of voltage using Vi
equation Instructions for PV bus solution Pi sch, |Vi| are
specified assume Vi = |Vi|0o, solve the Qi equation as below ij )(
)(* )1( + = + ij k jijk i sch i sch i k i y Vy V jQP V ijIm )( 10
)()(*)1( = == + k j n ij j ij n j ij k i k i k i VyyVVQ
16. 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 i y Vy V jQP V { } { }(
)2)1(2)1( Re ++ = k ii k i VimagVV { } { })1()1()1( ImRe +++ += k i
k i k i VjVV { } { } { } { } ++ )()1()()1( ImIm,ReRe k i k i k i k
i VVVV
17. Energy Conversion Lab INSTRUCTIONS FOR G-S SOLUTION
Instructions for PV bus solution to accelerate the convergence,
using the following approximation after new Vi is obtained is in
the range between 1.3 to 1.7 voltage accuracy in |Vi| and is in the
range between 0.00001 to 0.00005 ( ))()()()1( k i k cali k i k i
VVVV +=+
18. 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 the specified value, typically is about
0.001pu ijRe )( 10 )()(*)1( = == + k j n ij j ij n j ij k i k i k i
VyyVVP ijIm )( 10 )()(*)1( = == + k j n ij j ij n j ij k i k i k i
VyyVVQ
19. G-S Power flow Homework For the one-line diagram shown
below, using the G-S method to determine all bus voltages
(magnitude and phase) and show the power flow solution between the
buses assume the regulated bus (#2) reactive power limits are
between 0 and 600Mvar.
20. 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 can be 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 ( ) cx dx fd x dx df
xfxxf =+ + +=+ 2)0( )0( 2 2 )0( )0( )0()0()0( !2 1 )()( cx dx df
xfxxf = +=+ )0( )0( )0()0()0( )()( )0( )0( )0()1( += dx df c
xx
21. 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 root may converge to a root
different from the expected one or diverge if the starting value is
not close enough to the root )0( )0( )0()1( += dx df c xx )()()1(
)( )( )( )()( )( kkk k k k kk xxx dx df c x xfcc += = = +
22. Energy Conversion Lab NEWTON RAPHSON METHOD FOR n VARIABLES
Newton Raphson method for solving n variables nn n nnn nn n n n n
cx x f x x f x x f xfxxf cx x f x x f x x f xfxxf cx x f x x f x x
f xfxxf = ++ + +=+ = ++ + +=+ = ++ + +=+ )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 )()( )()(
)()(
23. 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
24. 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
25. 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 involves inverse of
J(k) , a triangular factorization is used to facilitate the
computation in MATLAB, the operator (i.e., X=JC) is used to apply
the triangular factorization Newton-Raphson method converge to
solution quadratically when near a root The limitation is that it
does not generally converge to a solution from an arbitrary
starting point
26. 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 += )(
27. 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 YVVP 1 cos ( )= += n j jiijijjii YVVQ 1
sin = VJJ JJ Q P 43 21 ( ) ( ) ijsin sin += += jiijijji j i ij
jiijijji i i YVV P YVV P
28. 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 YV V P YVYV V P ( ) ( ) ijcos cos += += jiijijji j
i ij jiijijji i i YVV Q YVV Q
29. 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 YV V Q YVYV V Q )()()()( , k
i sch i k i k i sch i k i QQQPPP == )()()1()()()1( , k i k i k i k
i k i k i VVV +=+= ++
30. 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: )()()()( , k i sch i k i k i
sch i k i QQQPPP == ( )= += n j jiijijjii YVVP 1 cos ( )= += n j
jiijijjii YVVQ 1 sin ( )= += n j jiijijjii YVVP 1 cos )()( k i sch
i k i PPP =
31. 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) )()( , k i k i QP = VJJ JJ Q P 43 21 )()()1()()()1( , k i
k i k i k i k i k i VVV +=+= ++
32. 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,
the Newton-Raphson method could be further simplified Consider the
Newton-Raphson power flow equation 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 in Jacobian matrix =
VJJ JJ Q P 43 21
33. 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-i is 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
34. 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 obtain solution to
calculate PV bus, J4 can be further eliminated, only J1 is used to
obtain solution VB V Q B V P ii = = ''' , [ ] [ ] V Q BV V P B = =
11 ",'
35. Energy Conversion Lab FAST DECOUPLED POWER FLOW Comparison
between fast decouple power flow solution and Newton Raphson power
flow solution fast decoupled solution requires more iterations than
Newton Raphson solution fast decoupled solution requires less time
per iteration since decoupled solution needs less time for
iteration, the overall computation time may be less than using the
Newton Raphson method fast decoupled solution often used in fast
computation of power flow, for example, contingency analysis or on-
line control of power flow see Ex. 6.12