load flow analysis - i: solution of load flow and related problems using gauss-seidel method 5
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EXPERIMENT 5 LOAD FLOW ANALYSIS - I: SOLUTION OF LOAD FLOW AND RELATED PROBLEMS
USING GAUSS-SEIDEL METHOD
AIM: (i) To understand, the basic aspects of steady state analysis of power systems that are
required for effective planning and operation of power systems.
(ii) To understand, in particular, the mathematical formulation of load flow model in complex form and a simple method of solving load flow problems of small sized system using Gauss-Seidel iterative algorithm
SOFTWARE REQUIRED The software required is Matlab 7.5 THEORETICAL BACKGROUND
Need For Load Flow Analysis
Load Flow analysis, is the most frequently performed system study by electric utilities. This analysis is performed on a symmetrical steady-state operating condition of a power system under “normal” mode of operation and aims at obtaining bus voltages and line / transformer flows for a given load condition. This information is essential both for long term planning and next day operational planning. In long term planning, load flow analysis, helps in investigating the effectiveness of alternative plans and choosing the “best” plan for system expansion to meet the projected operating state. In operational planning, it helps in choosing the “best” unit commitment plan and generation schedules to run the system efficiently for the next day’s load condition without violating the bus voltage and line flow operating limits.
Load flow analysis is the most frequently performed system study by electric utilities. This analysis is performed on a symmetrical steady-state operating condition of a power system under ‘normal’ mode of operation and aims at obtaining bus voltages and line/transformer flows for a given load condition. This information is essential both for long term planning and next day operational planning. In long term planning, load flow analysis helps in investigating the effectiveness of alternative plans and choosing the ‘best’ plan for system expansion to meet the projected operating state. In operational planning, it helps in choosing the ‘best’ unit commitment plan and generation schedules to run the system efficiently for them next day’s load condition without violating the bus voltage and line flow operating limits.
The Gauss seidal method is an iterative algorithm for solving a set of non- linear algebraic
equations. The relationship between network bus voltages and currents may be represented by either loop equations or node equations. Node equations are normally preferred because the number of independent node equation is smaller than the number of independent loop equations.
The network equations in terms of the bus admittance matrix can be written as, busbusbus VYI (1) For a n bus system, the above performance equation can be expanded as,
n
p
nnnpnn
pnpppp
np
np
n
p
V
V
VV
YYYY
YYYY
YYYYYYYY
I
I
II
2
1
21
21
222212
111211
2
1
(2)
where n is the total number of nodes. Vp is the phasor voltage to ground at node p. Ip is the phasor current flowing into the network at node p. At the pth bus, current injection:
n
pqq
qpqpppn
qqpq
npnpppppp
VYVYVY
VYVYVYVYI
11
2211 .........................
(3)
npVYIY
Vn
pqq
qpqppp
p ,....2;1
1
(4)
At bus p , we can write Pp – jQp = pp IV
Hence, the current at any node p is related to P, Q and V as follows:
p
ppp V
jQPI
)( ( for any bus p except slack bus s) (5)
Substituting for Ip in Equation (4),
npVYV
jQPY
Vn
pqq
qpqp
pp
ppp .....,2;1
1*
(6)
Ip has been substituted by the real and reactive powers because normally in a power system these quantities are specified. In case of PV bus,
1)(1 .|| ki
specifiedi
ki VV
)sin.(cos|| 11)(1 k
iki
specifiedi
ki VV
Acceleration factor: ( To speed up the convergence, the node voltage (Vi(k+1)) of the succeeding iteration can be modified(accelerated) by multiplication factor called as Acceleration factor.
ki
kii VVV 1
i
ki
ki VVV .1
Algorithm: Step 1: Read the input data. Step 2: Find out the admittance matrix. Step 3: Choose the flat voltage profile 1+j0 to all buses except slack bus. Step 4: Set the iteration count p = 0 and bus count i = 1. Step 5: Check the slack bus, if it is the generator bus then go to the next step otherwise go to next step 7. Step 6: Before the check for the slack bus if it is slack bus then go to step 11 otherwise go to next step. Step 7: Check the reactive power of the generator bus within the given limit. Step 8: If the reactive power violates a limit then treat the bus as load bus. Step 9: Calculate the phase of the bus voltage on load bus Step 10: Calculate the change in bus voltage of the repeat step mentioned above until all the bus voltages are calculated. Step 11: Stop the program and print the results
Flowchart:
Yes
Read the input data values
Start
Form Y Bus matrix
Set flat voltage profile 1+j0 except slack bus
Set iteration count, p=0
Set the bus count, i = 1
Check for slack bus
Check for Gen bus
It is a load bus calculate
n
jkik
j
kkik
i
ii
ii
pical VYVY
VjQP
YV
1
1
1*
1 1
Calculate
pk
n
ikik
pk
i
kik
pi VYVYipQ V 1
1
1
*1 Im
Check
min1 QQ p
i
Set Qi=Qi min
Check
max1 QQ p
i
Set Qi=Qi max
A
Y
No
Yes
No
No
No
Yes
Yes
D
E
C
B
RESULT: The given set of load flow equations for a given power system were solved using Gauss-Seidal method and verified using Matlab 7.5.
Treat this as gen bus & calculate Vpi
n
ik
pkik
i
k
pkik
i
i
ii
pi VYVY
VjQP
YV
1
1
1
1*
1 1
Calculate the change in voltage 1 piV
Increment the bus count
Check ni
Check 1p
iV
Print the result
Stop
Increment iteration count P = P+1
Yes
Yes
No
No
B
E
D
C
A
1 2
3
-j3
-j4 -j5
Problems:1 A three bus power system is shown. (Refer book Power System Analysis by K.B. Hemalatha, S.T. Jayachrista – Page no: 4.11) The relevant p.u line admittances are indicated on the diagram and bus data are given in table.
Determine the voltages at buses 2 and 3 after1st iteration using Gauss – Seidal method. Take the acceleration factor α = 1.6. clc clear all busdata=[1 1 1.02 0 0 0 0 0 0 0 2 3 1 0 0.25 0.15 0.5 0.25 0 0 3 3 1 0 0 0 0.6 0.3 0 0]; linedata=[ 1 2 0 -3i 1 3 0 -4i 2 3 0 -5i]; fb=linedata(:,1); tb=linedata(:,2); r=linedata(:,3); x=linedata(:,4); y=r+x; nbus=max(max(fb),max(tb)); nbranch=length(fb); ybus=zeros(nbus,nbus); for i=1:nbranch m=fb(i); n=tb(i); ybus(m,m)=ybus(m,m)+y(i); ybus(n,n)=ybus(n,n)+y(i); ybus(m,n)=-y(i); ybus(n,m)=ybus(m,n); end ybus bus=busdata(:,1); type=busdata(:,2); v=busdata(:,3); th=busdata(:,4);
Bus no Type Generation Load Bus voltage PG QG PL QL V
1 Slack - - - - 1.02 0o 2 PQ 0.25 0.15 0.5 0.25 - - 3 PQ 0 0 0.6 0.3 - -
genmw=busdata(:,5); genmvar=busdata(:,6); loadmw=busdata(:,7); loadmvar=busdata(:,8); qmin=busdata(:,9); qmax=busdata(:,10); p=genmw-loadmw; q=genmvar-loadmvar; vprev=v; toler=1; dv=0; iteration=1; while(toler>0.1) for i=2:nbus sumyv=0; for k=1:nbus if i~=k sumyv=sumyv+(ybus(i,k)*v(k)); end end if(type(i)==2) q(i)=-imag(conj(v(i)))*(sumyv+bus(i,i)*v(i)); if(q(i)>qmax(i) ||q(i)<qmin(i)) if q(i)<qmin(i) q(i)=qmin(i); else q(i)=qmax(i); end type(i)=3; end end v(i)=(1/ybus(i,i))*((p(i)-j*q(i))/conj(v(i))-sumyv); dv(i)=v(i)-vprev(i); v(i)=vprev(i)+(1.6*dv(i)); if type(i)==2 v(i)=pol2cart(abs(vprev(i)),angle(v(i))); end end iteration=iteration+1; toler=max(abs(abs(v)-abs(vprev))); vprev=v; end iteration; v OUTPUT: ybus = 0 - 7.0000i 0 + 3.0000i 0 + 4.0000i 0 + 3.0000i 0 - 8.0000i 0 + 5.0000i 0 + 4.0000i 0 + 5.0000i 0 - 9.0000i v = 1.0200 0.9920 - 0.0500i 0.9538 - 0.1511i
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