Download Load Flow Study using Tellegen`s Theorem

Load Flow Study using
Tellegen’s Theorem
Load Flow
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The load-flow study is an important tool
involving numerical analysis applied to a power
system
It focuses on various forms of AC power (ie:
reactive, real, and apparent) rather than voltage
and current.
It analyses the power systems in normal steadystate operation.
Many software implementations perform other
types of analysis, such as short-circuit fault
analysis and economic analysis.
In particular, some programs use linear
programming to find the optimal power flow, the
conditions which give the lowest cost per kilowatt
generated.
Importance of load-flow studies
• Great importance of power flow or loadflow studies is in the planning the future
expansion of power systems as well as in
determining the best operation of existing
systems.
• The principal information obtained from
the power flow study is the magnitude
and phase angle of the voltage at each
bus and the real and reactive power
flowing in each line.
Classification Of Load Flows
• Classical Load Flow Method
and
• Exclusive Load Flow Method
Load Flow methods are
categorized as
Classical load Flow Methods
• Gauss – Siedal(GS) Method
• Netwon- Raphson (NR) Method
• Fast Decoupled Load Flow Method
Exclusive load Flow Methods are
categorized into
• Forward Sweeping Method
• Dist Flow Method
TYPES OF LOAD FLOWS
Newton- Raphson Method
• Newton's method can often converge remarkably
quickly, especially if the iteration begins
"sufficiently near" the desired root.
• In NR method, the changes in real power (P) are
very much influenced by the changes in load
angle only and no influence due to the voltage
magnitude changes.
• Similarly the changes in reactive power are very
much influenced by changes in voltage
magnitudes and no change takes place due to
load angle changes.
Fast Decoupled Load Flow
• It is reliable and fastest method in
obtaining convergence.
• This method with branches of high (R/X)
ratios, could not solve problems with
regard to non- convergence and long
execution time.
Gauss- Seidel load flow method
• The iteration process begins with a flat
voltage profile assumption to all the buses
expect the slack bus.
• The bus voltages are updated and the
convergence check is made on updated
voltages and the iteration process is
continued till the tolerance value is
reached.
Load Flow study methods of
Distribution networks
• Efficient load flow method is one of the most
important and highly demanded software in the
power industry
• The analysis of a distribution network has become
an important area of activity for present day
power systems engineer.
• Due to the high (R/X) ratios of the branches of the
network made the researches to develop an
exclusive Load flow methods.
• The conservation of power principle at a node
level was the main principle used in the load flow
methods.
Principle Involved In Exclusive Load
Flow Methods
The principle says that, at any
node the power fed into the
node is equal to the sum of the
power dissipated in the series
branch connected to that node
and the power fed to the load
connected to that node.
Forward Sweeping Method
• The bus and branch numbering is simple and
direct.
• The node voltages are computed iteratively.
• Initially branch power flows are assumed to be
zero.
• The branch power flows are computed using the
power flow equations and terminal node voltages
are updated and in turn branch power losses are
updated.
• A tolerance check is made to the updated
branches power losses and iteration process is
continued till the convergence is obtained.
• Since the iteration process is carried in the
forward direction of the power flow, the method
is named “Forward Sweeping Method”.
Dist Flow Method
• The distribution network is reduced to a single branch
network and by using the power flow equation, the real
and reactive powers injected into the reduced network
are determined in an iterative way.
• A convergence check is applied to these powers injected
into the reduced network and the iteration process is
continued till the tolerance is met.
• If the convergence is not met, a new equivalent network
is determined with new parameters and the process is
continued till the convergence is achieved.
• The node voltages and branch power losses are
computed.
• The main advantage of this exclusive method is the
efficiency achieved by avoiding repeated computations of
node voltage magnitudes.
Thought for the development of
Tellegen theorem
The majority of classical load flow
solution methods have problems,
like
• Non- convergence
• High memory requirement
• Large computation time
So, to overcome some of these
problems
• The exclusive load flow methods for distribution
networks are studied.
• Power conservation principle is used in all
exclusive load flow methods
• If the same principle is applied to distribution
networks a better load flow can be developed.
• Conservation principle, when it is used to find a
new load flow solution could also be real and
near to the practical solution.
• The power conservation principle for the entire
network is designated as theorem called
Tellegen theorem.
Why Tellegen’s Theorem?
• Based solely on network
topology and kirchoff’s laws
• Is a power conservation
theorem.
• States that vectors of flows
and forces are orthogonal.
• Simple to code, less storage &
execution time.
The Theorem
• Consider an arbitrary lumped network whose
graph G has b branches and nt nodes.
• Suppose that to each branch of the graph we
assign arbitrarily a branch potential
difference Wk and a branch current Fk for
k= 1 ,2 … b and suppose that they are
measured with respect to arbitrarily picked
associated reference directions.
• If the branch potential differences W1 , W2
, W3 , … , Wb satisfy all the constraints
imposed by KVL and if the branch currents
F1 , F2 , F3 … Fb satisfy all the contraints
imposed by KCL, then
The Theorem..
• The Tellegen theorem is
extremely general.
• It is valid for any lumped
network that contains any
elements, linear or nonlinear,
passive or active, time-varying
or time-invariant. The
generality follows from the
fact that Tellegen's theorem
depends only on the two
Kirchhoff laws.
Algorithm
•
Make an initial guess of all unknown voltage magnitudes and
angles. It is common to use a "flat start" in which all voltage
angles are set to zero and all voltage magnitudes are set to
1.0 p.u.
•
solve the power balance equations using the most recent
voltage angle and magnitude values.
•
linearize the system around the most recent voltage angle and
magnitude values
•
solve for the change in voltage angle and magnitude
•
update the voltage magnitude and angles
•
check the stopping conditions, if met then terminate, else go
to step 2.
Load flow results of Tellegen theorem method:
Voltage magnitude in pu , branch real reactive power loss
– 18 bus , 440 V distribution network with main feeder.
Nu
m
b
e
rNode voltage
Real Power
Loss(KW)
Reactiv Power
Loss(KVAR
)
1
1
32.7106
13.6115
2
0.981
33.0682
14.0692
3
0.9598
18.0104
7.4944
4
0.9476
10.5739
4.3655
5
0.9399
35.2935
14.6862
6
0.912
22.6388
9.4204
7
0.8927
10.0685
4.1896
8
0.8832
15.4245
6.4184
9
0.8672
16.5198
6.8742
10
0.8481
9.4573
2.6736
11
0.8361
3.5163
0.9941
12
0.8309
7.5706
2.14
13
0.8177
4.7139
0.9773
14
0.8075
2.1479
0.552
15
0.8015
1.0375
0.4400
16
0.7973
0.6101
0.115
17
0.7936
0.1019
0.0288
18
0.7924
0.0000
0.0000
Number of iterations: 4
Execution time: 2.65
Memory Requirement: 3190
Total Real power loss: 223.4637KW
Total Reactive power toss: 89.0503KVAR
Result
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INPUT CASE FILE
% this function gives data for r, x, pl and ql,nb,epslon,vr of 18
bus system
function [r,x,pl,ql,nb,epslon,vr] = case18bus
r =[1.8216 2.2270 1.3662 0.9180 3.6432 2.7324 1.4573 2.7324
3.6432 2.7520 1.3760 4.1280 4.1280 3.0272 2.7520 4.1280
2.7520 0.000]
x =[0.7580 0.9475 0.5685 0.3790 1.5160 1.1370 0.6064
1.1370 1.5160 0.7780 0.3890 1.1670 0.8558 0.7780 1.1670
0.7780 0.7780 0.0000]
pl =[0.00 140.00 80.0 80.0 100.0 80.0 90.0 90.0 80.0 90.0
80.0 80.0 90.0 70.0 70.0 70.0 60.0 60.0]
ql =[0.00 90.0 50.0 60.0 60.0 50.0 40.0 40.0 50.0 50.0 50.0
40.0 50.0 40.0 40.0 40.0 30.0 30.0]
nb=[18];
epslon=[0.0001];
vr=[440];
Output
Tellegen’s Therom results are as follows:
• The number of iterations are = 10
• The total load = 1410.00000kw
• Total reactive load = 810.00000kvar
• Total real power losses = 223.46564kw
• Total reactive power losses = 89.05121kvar
ans =
Columns 1 through 9
1.0000 0.9810 0.9598 0.9476 0.9399 0.9120 0.8927 0.8832 0.8672
Columns 10 through 18
0.8481 0.8361 0.8309 0.8177 0.8075 0.8015 0.7973 0.7936 0.7924
The above mentioned array indicate the voltage at different nodes.
Applications
• The classical application area for network
theory and Tellegen's theorem is electrical
circuit theory. It is mainly in use to design
filters in signal processing applications.
• A more recent application of Tellegen's
theorem is in the area of chemical and
biological processes.
• The assumptions for electrical circuits
(Kirchhoff laws) are generalized for
dynamic systems obeying the laws of
irreversible thermodynamics.
.
Applications..
• Topology and structure of reaction networks
(reaction mechanisms, metabolic networks) can
be analyzed using the Tellegen theorem.
• Another application of Tellegen's theorem is to
determine stability and optimality of complex
process systems such as chemical plants or oil
production systems.
• The Tellegen theorem can be formulated for
process systems using process nodes, terminals,
flow connections and allowing sinks and sources
for production or destruction of extensive
quantities