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gaurav ranjan
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YBUS FORMATION FOR LOAD-FLOW
STUDIES
Gaurav Ranjan
Narender Singh
N Moses Binny
BY:-
ABSTRACT
Load flow is an important power system analysis component to ensure the
system upgrades, future upgrades and present distribution equipment meeting
the present and future requirements.
Load flow study in power system is the steady state solution of power system
network.
The information obtained from load flow solution is used for the continuous
monitoring of current state of the system and for analyzing the effectiveness of
future system expansion to meet increased load demand.
The main objective of the load flow is to find the voltage magnitude of each bus
and its angle when the powers generated and loads are specified.
INTRODUCTIONLoad flow studies can be used to obtain the voltage magnitudes and angles
at each bus in the steady state.Once the bus voltage magnitudes and their angles are computed using the
load flow, the real and reactive power flow through each line can be computed.
This project deals with Ybus formation using different methods for load
flow analysis.
Formation of Ybus plays a vital role in solving any load flow problem.
Ybus matrix is sparse matrix that is why it is preferred over Z matrix
for load flow solutions.
Building up and modification of Y bus is easy because of different
available simple methods which are discussed here.
CLASSIFICATION OF BUSESLoad Buses: In these buses no generators are connected and hence the
generated real power PGi and reactive power QGi are taken as zero.
Voltage Controlled Buses: These are the buses where generators are connected.
slack or swing buses : Usually this bus is numbered 1 for the load flow studies. This bus sets the angular reference for all the other buses.
GAUSS SEIDEL METHOD ALGORITHM FOR LOAD-FLOW SOLUTION
With the load profile known at each bus (i.e. PDi and QDi known), allocate PGi and QGi to all generating stations
Assembly of bus admittance matrix YBUS Iterative computation of bus voltagesCurrent at the ith bus Pi-jQi/ Vi
*= Ii
= ∑nk=1 Yik Vk
= Yi1V1 + Yi2V2 + Yi3V3 +…+ YinVn
For (r+1)th iteration, the voltage becomesVi
(r+1)=Ai/Vi(r) -∑k=1 i-1 Bi
k Vk(r+1) -∑k=i+1 n Bi
k Vk(r)
Ai=Pi-jQi/Yii
Bik=Yik/Yii
Computation of slack bus power Si
*=Pi-jQi
Computation of line flows
The current fed by bus i into the line can be expressed as Iik=Iik1+Iik0=(Vi-Vk)Yik+ViYik0
The power fed into the line from bus i is,Sik=Pik+jQik=Vi I*
ik=Vi(V*i-V*
k)Y*ik+ViV*
iY*ik0
Similarly the power fed into the line from bus k isSki=Vk(V*
k-V*i)Y*
ik+VkV*kY*
ki0
NEWTON -RAPHSON (NR) METHODConsider a set of n non-linear algebraic equations
fi (x1, x2,……, xn) =0; i=1,2,3, ….., n
fi (x10 +∆x1
0, x20 +∆x2
0,… …….. xn0+∆xn
0) =0;
Taylor series expansionfi (x1
0, x20,………..xn
0)+[(∂fi/∂x1)0 ∆x10 +
(∂fi/∂x2)0 ∆x20+……….+(∂fi/∂xn)0 ∆xn
0]+ higher order terms=0
Neglecting higher order terms we can write above equation in matrix form
Or in vector matrix form f0+J0∆x0=0J0 is known as the jacobian matrix
the above Eq can be written as f0≈ [-j0] ∆x0
Update values of x are then
x1=x0+∆x0
or, in general form of x (r+1)th iterationx(r+1) =x(r) +∆x(r)
Iterations are continued till Eq is satisfied to any desired accuracy, i.e,
fi(x(r)) <ε (A specified value);
fip=Pi (specified)-Pi(calculated)=∆Pi
fiQ=Q(specified)-Qi(calculated)=∆Qi
•Where ∆P and ∆Q are the real and reactive power mismatch at each bus. j is the
jacobian matrix, j represents the sensitivity measurement of the real and reactive
power with respect to the bus voltage angle and magnitude.
Bus type Number of buses
Qualities specified
Number of available equations
Number of δi |Vi| state variables
Slacki=1
1 δi , |Vi| 0 0
Voltage controlled(i=2,3,….Ng+1)
Ng Pi, |Vi| Ng-1 Ng-1
Load (Ng+2,…..N)
N-Ng-1 Pi, Qi 2(N-Ng) 2(N-Ng)
Total N 2N 2N-Ng 2N-Ng
DECOUPLED LOAD FLOW METHOD
The decoupled power flow method is an approximate version of Newton-Raphson procedure.
The approximation of the Newton-Raphson procedure only affects the iteration approach it does not reduce the accuracy of the final solution.
The principle underlying the decoupled approach is based on two observations:
Change in the voltage angle delta at a bus primarily affects the flow of real power P in the transmission lines and leaves the flow of reactive power Q relatively unchanged.
Change in the voltage magnitude at a bus primarily affects the flow of reactive power Q in the transmission lines and leaves the flow of real power relatively unchanged.
A well designed an properly operated power transmission system:
The angular differences between typical buses of the system are usually so small that
The line susceptances are many times larger than the line conductance
so that
The reactive power Qi injected into any bus i of the system during normal operation is much less than the reactive power which would flow if all lines from that bus were short-circuited to reference.
That is
After simplifying:
THE SOLUTION STRATEGYCalculate the initial mismatch PSolve for Update the angles and use them to calculate
mismatch Solve for and update the magnitude ,and Repeat the iteration until all mismatches are
within specified tolerances.
PRIMITIVE NETWORK
• The voltage relation for fig (a) can be written as
Vrs+ers=zrsirs (or) V+E=[Z]I
• Similarly, the current relation for fig (b) can be written as
irs+jrs=yrsvrs (or) I+J=[Y]V
• Primitive network is defined as representation of network in the form of impedance or admittance.
FORMATION OF Y bus BY
DIRECT METHOD
Diagonal values is brought up by adding the branches connected to point (or) node
Others are brought by taking negative of the value between the two nodes considered.
YBus can also be obtained
by other methods
• Bus admittance and Bus impedance matrix.
• Branch admittance and Branch impedance matrix.
• Loop admittance and Loop impedance matrix.
BUS ADMITTANCE AND BUS IMPEDANCE MATRIX
i+j=[Y]VATi+ATj=AT[Y]V as,(IBus=ATj , ATi=0 )
0+IBus=AT[Y]V
(J*)TV= (IBus*)TEBus
(J*)TAEBus=(j*)TV
V=AEBus
YBus=AT[Y]AThe bus impedance matrix can be obtained as
ZBus=Y-1Bus= [AT[Y]A]-1
NOTE:-SAME IS THE CAES WITH THE ‘BRANCH ADMITTANCE AND BRANCH IMPEDANCE MATRIX’ THE ONLY DIFFERENCE IS ‘B’ MULTIPLIED WITH PRIMITIVE NETWORK PARAMETERS.
LOOP ADMITTANCE AND LOOP IMPEDANCE
MATRIX v+e= [Z]iCTv+CTe=CT[Z]I as,(CTv=0, ELoop
=CTe ) (ILoop*)TELoop= (i*)Te
(ILoop*)TCT= (i*)T
i=CILoop
ELoop=CT[Z]CILoop
ZLoop=CT[Z] CLoop admittance matrix can be obtained from
YLoop=ZLoop-1= [CT[Z]C]-1
ATTRIBUTES
GAUSS SEIDEL NEWTON RAPHSON
FAST DECOUPUED NEWTON RAPHSON
HOW THE PROGRAM IS Easy Quite complex Less complex when compared to NR
STORAGE REQUIREMENT Minimum Maximum 40% Less then Newton Raphson
PROGRAMMING Easy Tough Less tough
CONVERGENCE Linear convergence Quadratic convergence Geometric convergence
SENSITIVITY PROPERTIES Not present Present Present
SYSTEM SIZE Time Increases linearly Size hardly matters convergence is sure in 5 to 6 iterations
TYPE OF SYSTEM System may or may not converge
Sure to converge No convergence problem
COMPARISION BETWEEN THE THREE METHODS
Ybus FORMATION 14 BUS % | From | To | R | X | B/2 |
% | Bus | Bus | pu | pu | pu |
linedata = [1 2 0.01938 0.05917 0.0264
1 5 0.05403 0.22304 0.0246
2 3 0.04699 0.19797 0.0219
2 4 0.05811 0.17632 0.0170
2 5 0.05695 0.17388 0.0173
3 4 0.06701 0.17103 0.0064
4 5 0.01335 0.04211 0.0
4 7 0.0 0.20912 0.0
4 9 0.0 0.55618 0.0
5 6 0.0 0.25202 0.0
6 11 0.09498 0.19890 0.0
6 12 0.12291 0.25581 0.0
6 13 0.06615 0.13027 0.0
7 8 0.0 0.17615 0.0
7 9 0.0 0.11001 0.0
9 10 0.031810.08450 0.0
9 14 0.12711 0.27038 0.0
10 11 0.08205 0.19207 0.0
12 13 0.22092 0.19988 0.0
13 14 0.17093 0.34802 0.0 ];
fb = linedata(:,1); % From bus number...tb = linedata(:,2); % To bus number...r = linedata(:,3); % Resistance, R...x = linedata(:,4); % Reactance, X...b = linedata(:,5); % Ground Admittance, B/2...z = r + i*x; % Z matrix...y = 1./z; % To get inverse of each element...nbus = max(max(fb),max(tb)); % no. of buses...nbranch = length(fb); % no. of branches...ybus = zeros(nbus,nbus); % Initialise YBus... % Formation of Off Diagonal Elements... for k=1:nbranch ybus(fb(k),tb(k)) = -y(k); ybus(tb(k),fb(k)) = ybus(fb(k),tb(k)); end % Formation of Diagonal Elements.... for m=1:nbus for n=1:nbranch if fb(n) == m | tb(n) == m ybus(m,m) = ybus(m,m) + y(n) + b(n); end end end ybus
OUTPUTybus = Columns 1 through 6
6.0760 -19.4981i -4.9991 +15.2631i 0 0 -1.0259 + 4.2350i 0
-4.9991 +15.2631i 9.6039 -30.3547i -1.1350 + 4.7819i -1.6860 + 5.1158i -1.7011 + 5.1939i 0
0 -1.1350 + 4.7819i 3.1493 - 9.8507i -1.9860 + 5.0688i 0 0
0 -1.6860 + 5.1158i -1.9860 + 5.0688i 10.5364 -38.3431i -6.8410 +21.5786i 0
-1.0259 + 4.2350i -1.7011 + 5.1939i 0 -6.8410 +21.5786i 9.6099 -34.9754i 0 + 3.9679i
0 0 0 0 0 + 3.9679i 6.5799 -17.3407i
0 0 0 0 + 4.7819i 0 0
0 0 0 0 0 0
0 0 0 0 + 1.7980i 0 0
0 0 0 0 0 0
0 0 0 0 0 -1.9550 + 4.0941i
0 0 0 0 0 -1.5260 + 3.1760i
0 0 0 0 0 -3.0989 + 6.1028i
0 0 0 0 0 0
Columns 7 through 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 4.7819i 0 0 + 1.7980i 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.9550 + 4.0941i -1.5260 + 3.1760i 0 -19.5490i 0 + 5.6770i 0 + 9.0901i 0 0 0 0 + 5.6770i 0 - 5.6770i 0 0 0 0 0 + 9.0901i 0 5.3261 -24.2825i -3.9020 +10.3654i 0 0 0 0 -3.9020 +10.3654i 5.7829 -14.7683i -1.8809 + 4.4029i 0 0 0 0 -1.8809 + 4.4029i 3.8359 - 8.4970i 0 0 0 0 0 0 4.0150 - 5.4279i 0 0 0 0 0 -2.4890 + 2.2520i 0 0 -1.4240 + 3.0291i 0 0 0
Columns 13 through 14 0 0 0 0 0 0 0 0 0 0 -3.0989 + 6.1028i 0 0 0 0 0 0 -1.4240 + 3.0291i 0 0 0 0 -2.4890 + 2.2520i 0 6.7249 -10.6697i -1.1370 + 2.3150i -1.1370 + 2.3150i 2.5610 - 5.3440i
YBUS formation using singular transformation
ydata=[1 1 2 1/(0.05+j*0.15) 0 0
2 1 3 1/(0.1+j*0.3) 0 0
3 2 3 1/(0.15+j*0.45) 0 0
4 2 4 1/(0.10+j*0.30) 0 0
5 3 4 1/(0.05+j*0.15) 0 0];
elements=max(ydata(:,1))
yprimitive=zeros(elements,elements)
for i=1:elements,yprimitive(i,i)=ydata(i,4)
if(ydata(i,5)~=0)
j=ydata(i,5)
ymutual=ydata(i,6)
yprimitive(i,j) =ymutual
end
end
buses=max(max(ydata(2,:)),max(ydata(3,:)))
A=zeros(elements,buses);
for i=1:elements,
if(ydata(i,2)~=0)
A(i,ydata(i,2))=1
end
if ydata(i,3)~=0
A(i,ydata(i,3))=-1
end
end
YBUS=A'*yprimitive*A
OUTPUT elements = 5 yprimitive = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0yprimitive = 2.0000 - 6.0000i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
yprimitive = 2.0000 - 6.0000i 0 0 0 0 0 1.0000 - 3.0000i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 yprimitive = 2.0000 - 6.0000i 0 0 0 0 0 1.0000 - 3.0000i 0 0 0 0 0 0.6667 - 2.0000i 0 0 0 0 0 0 0 0 0 0 0 0
yprimitive = 2.0000 - 6.0000i 0 0 0 0 0 1.0000 - 3.0000i 0 0 0 0 0 0.6667 - 2.0000i 0 0 0 0 0 1.0000 - 3.0000i 0 0 0 0 0 0 yprimitive = 2.0000 - 6.0000i 0 0 0 0 0 1.0000 - 3.0000i 0 0 0 0 0 0.6667 - 2.0000i 0 0 0 0 0 1.0000 - 3.0000i 0 0 0 0 0 2.0000 - 6.0000i
buses = 1.0000 - 3.0000i A = 1 0 0 0 0 A = 1 -1 0 0 0 0 0 0 0 0
A = 1 -1 1 0 0 0 0 0 0 0 A = 1 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0
A = 1 -1 0 1 0 -1 0 1 0 0 0 0 0 0 0 A = 1 -1 0 1 0 -1 0 1 -1 0 0 0 0 0 0
A = 1 -1 0 1 0 -1 0 1 -1 0 1 0 0 0 0 A = 1 -1 0 0 1 0 -1 0 0 1 -1 0 0 1 0 -1 0 0 0 0
A = 1 -1 0 0 1 0 -1 0 0 1 -1 0 0 1 0 -1 0 0 1 0 A = 1 -1 0 0 1 0 -1 0 0 1 -1 0 0 1 0 -1 0 0 1 -1
YBUS = 3.0000 - 9.0000i -2.0000 + 6.0000i -1.0000 + 3.0000i 0 -2.0000 + 6.0000i 3.6667 -11.0000i -0.6667 + 2.0000i -1.0000 + 3.0000i -1.0000 + 3.0000i -0.6667 + 2.0000i 3.6667 -11.0000i -2.0000 + 6.0000i 0 -1.0000 + 3.0000i -2.0000 + 6.0000i 3.0000 - 9.0000i
CONCLUSIONLoad flow study comprises the magnitude and phase angle of
load bus voltages, reactive power at generator buses, the real and reactive power flow on transmission lines, and other variable being specified.
The Ybus matrix forms the network models for load flow studies.
Because of sparsity the minimal storage is required.
The alternative approach is of great theoretical and practical significance particularly in the case of mutual coupling and phase shifting transformers.
REFERENCESBOOKS: Stagg,G.W and A.H.El-Abiad, Computer Method in Power System analysis Nargrath, I.J.and D.P.Kothari, Power System Engineering Weedy,B.M. and B.J.Cory, Electrical power Systems, 4 th Ed., John Wiley, NEW
YORK,1998 Nargrath, I.J.and D.P.Kothari, Modern Power System Analysis, Third Edition Tata
McGraw-Hill, New Delhi Power system analysis by Hadi Saadat –TMH Edition MATLAB ® and its Tool Boxes user’s manual and –Mathworks, USA
PAPERS: Tinney,W.F., and C.E.Hart, ‘Power flow Solution By Newton’s Method’, IEEE Trans.,
November 1967,No.11, PAS-86:1449 Stott, B.,’ Decoupled Newton Load Flow’, IEEE Trans., 1972, PAS-91,1955 Stott, B., ‘ Review of Load Flow Calculation Method’, Proc. IEEE, July1974, PAS-93
THANKS YOU
Gaurav RanjanNarender SinghN Moses Binny
BY:-
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