ybus formation for load flow studies

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:-