Mini Project Report submitted to the Department of Electrical and Electronics
Engineering in accordance with the academic requirement for the award of the degree.
AD SRIKANTH 05841A0232
LVS KARTHIK 05D91A0210
-1-

This is to certify that the Mini Project work titled
“Load Flow Study using Tellegen’s
Theorem” submitted by AD Srikanth and LVS Karthik of EEE, Aurora’s Technological &
Research Institute, Parvathapur, Uppal, Hyderabad in accordance with the academic requirement for the award of degree Bachelor of Technology in Electrical and
Electronics Engineering, is a bonafied work carried out by them from May 2008 to
June 2008 under our guidance and supervision. The project report has not been submitted to any other University or other Institute.
Project Coordinator Project Expert Head of the Department
2

It gives us great pleasure in acknowledging the help received from various individuals in completing the project and presenting results. We are gratefully beholden beyond words to express deep sense of reverence and gratitude to our guide Mrs. Madhuri and her meticulous guidance, valuable suggestions, constructive criticism and wholehearted co-operation throughout the mini-project..
We are thankful to Mr./ Mrs. (Name & Designation of the Head of the organization where the candidate had done the project) for allowing me to do the
Project work in (organization Name and address) .
We take immense pleasure of dedicating our thanks to our principal Dr. Alka
Mahajan for her moral help and encouragement during our project.
Project Associates
A.D.Srikanth
L.V.S. Karthik
3

1. Abstract
2. Chapter 1 Introduction
1.1 Classical Load Flow Techniques
1.1.1 Limitations of Classical Load Flow methods
1.2 Load Flow Study of Distribution Networks
1.3
Objective of the Thesis
Generalized form of Tellegen Theorem
3. Chapter 2 Load Flow Solution formulation using exclusive
methods and Tellegen Theorem method
2.1
Introduction
2.2 Description of the radial Distribution network
2.3 Load Flow problem formulation using
tellegen theorem method
2.4 Distribution Network with main feeder
4.Chapter 3 Case study and analysis of load flow solutions methods
Case study: Load flow analysis of distribution network
with main feeder using Tellegen Theorem Method
3.2 Algorithm for the Distribution networks with
main feeder using Tellegen theorem method
3.2.1 Flow chart
3.3 Software Used (
3.3.1
Introduction to MATLAB
3.3.2 Starting MATLAB
3.3.3 Entering Source code in MATLAB
5. Chapter 4 Conclusions
4.1 Introduction
4.2 Conclusions on exclusive load flow methods
and Tellegen theorem method
Appendix:
X Source Code
X.1 Result
Bibliography
4

ABSTRACT
The present day distribution networks are enormously growing to meet the rapidly increasing demand of electrical power, but with a less attention towards its optimum growth. Due to this erratic and unplanned expansion, the real power losses have become appreciable, constituting the major portion of the overall system losses and with effect that, the operating voltage profile has become very poor. Such unplanned distribution network performance can be analyzed using an efficient load flow method and a solution to optimize the growth of the network can be obtained. Hence the distribution load flow study has become a vital tool for the analysis of Automated
Distribution Systems .
The radial structure and the high resistance/reactance (R/X) ratios of the branches of the distribution networks, made the conventional load flow methods, like Fast
Decoupled Load Flow ( FDLF ) and Newton-Raphson ( NR) unsuitable to provide the reasonable solution. Due to the topological specialty of the distribution networks and the non-applicability of the conventional load flow methods, the researchers preferred to develop exclusive load flow methods, are either loop-based or branch based methods, depending on the complexity of the distribution network.
The main principle involved in the exclusive load flow methods, is the “ principle of power conservation“ at a node of the network. The principle is that , the amount of power injected in to the node is equal to the sum of the power dissipated in the series branch and the power fed to the load. The principle of power conservation at the entire network level is known through an elegant theorem called “ Tellgen Theorem”.
The new load flow solution called “Tellegen theorm method” is developed on a distribution network with main feeder having 12 and 18 buses and on an another distribution network with main feeder and laterals having 28 and 33 buses. The voltage profile had chosen for the above networks are 440V and 110V.
5

General Introduction
The electricity utility system is usually divided into three subsystems, which are generation, transmission and distribution. Further the distribution system is commonly broken down into three compenents: distribution subsations, distribution primary and secondary. At the subsation level, the voltage is reduced and the power is distributed in smaller amounts to the customers. Consequently, one substation will supply many customers with power. Thus, the number of transmission lines in the distribution systems is many times that of the transmission systems. Furthermore, most customers are connected to only one of the three phases in the distribution system. Therefore, the power flow on each of the lines is different and the system is typically ‘ unbalanced’.
This characteristic needs to be counted for in load flow studies related to distribution networks.
A distribution circuit uses primary or main feeders and lateral distributiors. The main feeder originates from the substation and passes through the major load centers. The lateral distributors connect the individual load points to the main feeder with distribution transformers at their ends. Many distribution systems used in practice have a single circuit main feeder with wide range of resistance and reactance values.
Thus a radial configured distribution network has a main feeder and a number of laterals emanate from the nodes of the main feeder.
The present day automated distribution systems are under continuous reformation.
They are performing a number of jobs, like configuration management, continuous voltage control and routing of power through feeders by effective utilization of tieswitches and sectionalizing switches.
Since the distribution networks are under continuous expansion program to meet the present demand of power, but unfortunately a less attention is paid towards to the optimum growth. As a result, the real power losses have become appreciable and voltage profile has become very poor. The load flow analysis is the solution, through which any network can be analyzed to improve its performance.
6

The bus voltage angles are omitted in the problem formulation, because the radial loads use the voltage magnitudes as the most interested variable, and voltage phase angle is usually, less important compared to the magnitude of the voltage. Besides the difference among voltage phase angles in a feeder do not exceed a few degrees.
The Tellgen Theorem principle is applied to the radial distribution network, developing an objective equation, representing the total real and reactive powers injected in to the network. The equation is observed to be containing only the node voltage magnitudes, and the series-branch parameters, resistance and reactance. A proper bus and branch indexing scheme is implemented to read and retrieve the bus and load data.
The objective equation developed using the Tellgen theorem does not have either any higher order terms or any trigonometric terms. With effect that, it is observed that, the computation time and memory requirement is very much reduced. Further, a flat voltage profile assumption, which is used to initate the iteration process, has solved many other initial parameter assumption problems, that arose in the other solution techniques. It is proved that the memory requirement is reduced by 33.78% and the execution tims is saved about 35.44 %.
7

The classical load flow study methods used are guass – seidal load flow method (GS)
Netwon- Raphson method(NR) and Fast Decoupled load flow method (FDLF).
(a) Gauss-seidal load (GS) method :
This is most primitive method of load flow study. A recursive expression for bus voltage Vp is developed from the load current flowing Ip through the bus ‘p’ and the power Sp injected to the bus.
The expression for the bus voltage magnitude Vp can be written as
Vp^(it+1) = ((Sp* / Ypp) / (1 / V*(it)p)) - Ypq / Ypp * Vq(it) - Ypq /
Ypp * Vq^(it+1)
The iteration process begins with aflat voltage profile assumption to all the buses except the slack bus . the bus injected powers and the series branch admittance parameters are known and hence the bus voltages are updated using the eqn. A convergence check is made on these updated voltages and the iteration process is continued till the tolerance value is reached.
(b) Netwon Raphson (NR) method :
This method load flow study is a land mark in the load flow solution methods . the recursive power flow eqn’s for real and reactive powers (Pp , Qp) are used as core eqn’s.
The bus voltage vector is considered in two forms mainly rectangular form
(V = e + jf)
And polar form (V=|V|angle).
If the rectangular form of voltage vector is used , the load flow method is called rectangular coordinate method and if the polar form of voltage vector is used , the load flow method is called rectangular polar coordinate method.
( c ) Fast Decoupled load flow method (FDLF) :
It is observed that , from the Netwon Raphson method , the changes , the changes in real power are very much influenced by the changes in load angle only and no influence due to the very magnitude changes . similarly the changes in reactive power
(Q) are very much influenced by the changes in voltage magnitudes and no changes
8

takes place due to the load angle changes . This formed the basis for the Fast decoupled load flow method and this method is called Approximate Netwon method.
The iteration procedure is same as the NR method and the memory and storage requirements are reduced considerably , the solution is a converged with the decoupling condition that the series branch conduction that the series branch conductance (Gpq) should be smaller series branch succeptance (Bpq).
The main limitation of the NR method is the large storage & large solution time.
It is due to the repeated Formation and triangularization of jacobian matrix. Then by making certain approximations in the jacobian element values , an approximate network method , also called Fast Decoupled Load Flow (FDLF) method came into existence. It has been observed that the computational efficiency and reliability of
FDLF method is higher than NR method and has been used as a main mathematical tool to compute load flow of transmission networks of power industry . The negative aspect of the FDLF method has been observed that the method failed to give converged solution for a network having high (R/X) ratios.
9

It is well known that the efficient load flow method is one of the most important and highly demanded software in the power industry, through which any network can be analyzed . The analysis of a distribution network has become an important area of activity for present day power system engineer.
The conservation of power principle at a node level was the main principle used in the 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 .The distribution network is considered in two configurations namely distribution network with main feeder only and the other is distribution network with main feeder and laterals.
In the case of distribution network with main feeder only, the bus and branch numbering is simple and direct. In this method , the node voltages are computed iteratively. Initially the branch power flows are assumed to be zero. The branch power flows are computed using the power flow equations and the terminal node voltages are updated and in turn the branch power losses are updated . A tolerance check is made to the updated branch power losses and the iteration power losses is continued till the convergence is obtained.
In the case of distribution network main feeder and laterals a proper bus and branch numbering scheme is used to read and retrieve the branch parameters and load values.
In this configuration each lateral is treated as a main feeder of distribution network and the iteration process is continued for each lateral. Since the iteration process is carried in the forward direction of the power flow, the method is named “ Forward
Sweeping Method”.
In another method “Dist Flow method”, the distribution network is reduced to a single branch network and by using the power flow Eqn’s , the real and reactive powers injected into the reduced network are determined in an iterative way. If convergence is not met , a new equivalent network is determined with the new parameters and the process is continued till the convergence is achieved. Then the node voltages and branch power losses are computed. The main advantage of this method is the efficiency is achieved by avoiding repeated computations of node voltage magnitudes.
10

.
:
The thought for the development of new load flow method for the distribution network is initiated, when an attempt is made to find the load flow solution for a radial distribution network using the classical load flow methods. The majority of the classical load flow methods have problems, like non-convergence, high memory requirement, and large computation time. In that crisis, to overcome some of the problems, some of the exclusive load flow methods for distribution networks are studied. It is observed that the power conservation principle is used in all the exclusive load flow methods. If the same principle is applied to distribution networks, a better load flow is developed. Also it can be understood that, the conservation principle is a known reality and when it is used to find a new load flow solution of a network, that new load solution could also be a real and near to the practical solution.
The power conservation principle for the entire network is designated as a theorem called Tellegen theorem.
Hence, a part of the objective of this thesis work is to develop and formulate an efficient solution algorithm for the distribution networks using Tellegen theorem.
The electric network properties are analyzed from the impedance functions without a detailed knowledge of the circuit diagram. But certain properties like energy consumption and reciprocity of network could not be analyzed with the basic impedance functions. Usually the inner structure of the network is always taken into consideration for the energy distribution in the network . Hence a general theorem called Tellegen theorem has come into existence .
Among the theorems of circuit theory , Tellegen theorem is typical and solely depends upon the kirchoff’s laws and the topology of the network . The theorem applies to all electrical networks that obey kirchoff’s laws whether they are linear or non-linear.
11

Time invariant or time variant , reciprocal or non – reciprocal. The excitation is arbitrary and initial conditions are also immaterial. When specific assumptions are made concerning the network elements , the excitation and the initial conditions , tellegen theorem reduces to many useful network theorems .
The general Tellegen theorem is stated as followed
In any network configuration , Imagine branch currents "I " such that for every
Node I=0, Assume branch voltages V such that for every mesh V =0 and for every branch let the Positive Direction Of 'The Current be from the "+" to the "-
" Denoting the Positive Polarity Of the Voltage Fig. 1.1, Then VI = 0 Where the
Summation Is Over All The Branches.
The theorem is represented mathematically in a basic form v b Tib = 0
Where b varies from 1 to Nb where Nb is Total Number Of branches
For any mesh , the Kirchoff s Voltage law says that the sum Of the instantaneous Voltages are
Zero i.e.. v = 0.
Let the Potentials of the arbitrary nodes 'M' and ‘N' are V m
and V n
respectively and voltage across the branch Connecting 'M' and *N' nodes Is V mn
= V m
- V n
Let the Current flowing through the branch Is I mn
Then the Product Of Voltage across the branch and Current flowing through the branch is called instantaneous power,
Vm Imn = (Vm – Vn ) Imn = Vm Imn – Vn Im
The Eqn [1.48] is applied to all the branches of the network , whose one of the node voltage is Vm and collect all the terms containing Vm. vm Imn = Vm Imn
The Kirchoff’s Current Law (KCL) Is applied to the Eqn.[1.49]
Imn = 0
The eqn.[ 1.49] Can Be Written As
Vm Imn = 0
Hence the Tellegen theorem gives the Idea that the Inner Product of the branch voltage current vectors is Zero . Specifically the sum of the products of Voltage & Current In across each branch of a circuit is always equal to Zero .
12

The network is viewed as pairs of terminal nodes and an element exists between. the terminal pairs and elements are called as branches of the network .
If positive senses of current and voltage of terminal pair is considered, as shown
I n the fig. 1.2, the Tellegen theorem can be written as
V1 i1 + V
2
i2 = V m1
i m1
+ V m2
im2. + Vm3 im3 where the term v1 I1 + v 2 i2 is the power fed in to the network through the sources and the term v m
1 i m
1 + v m
2 i m
2 + Vm3 I m
3 is the power dissipated by the series branches or loads in the network.
In general if the network has 't' terminal pairs and 'b' internal branches, the can used as the power absorbed by the internal branches is equal to the power delivered to the network through the terminal pairs.
Vi ii = Vj I j
Where "I” ranges from 1 to t and j ranges from 1 to b.
This is also called as the "Energy theorem" of the networks.
The Tellegen theorem is basically a statement of the conservation of power. If I and V are the complex. quantities representing sinusoidal branch currents and voltages of the same frequency, then it can be written as
Vi Ii* = Vj Ij*
Where the asterisk “ * “ denotes the complex conjugate of the vector quantity.
Where “ I “ ranges from 1 to t and j ranges from 1 to b.
13

The topological properties of the distribution networks are exploited to device dedicated exclusive power flow methods and at each stage the power conservation principle at each node is extensively used. The chapter presents the description of the radial distribution network and the importance of the bus and branch indexing in the load flow solution process is explained. Later the load flow problem formulation of a radial distribution network using the typical exclusive load flow methods "Forward sweeping method" and "DistFlow method" and a new load flow method “Tellegen theorem method" with necessary detailed derivations is presented.
In general, the structure of the distribution network has to be properly explained, so the load flow problem formulation becomes easier. The notation of the node and branches used and the various electrical variables used for the power flow equations.
The distribution network is considered in two configurations. They are the
Distribution network with main feeder and other distribution network with main feeder laterals.
The network has Nb nodes and Nb-j branches. The node and branch of the main feeder or laterals are indexed such that data storing and retrieval are easy. The node is identified with the number in the square brackets "[ ]" and the branch is identified with the number in normal brackets "( ) ". It can be observed that the sending end node and the corresponding branch connected to that node are identified with the same number. The branch "j" is represented with resistance and reactance Rj , X r
.The load at node "i" is represented with PLi , QLi
The power flow equations are represented as a set of equations and unknowns, by which a solution methodology can be developed. These equations and unknowns are organized through a particular node ordering. The number of equations and number of unknowns at each node are reduced with a proper bus and branch-indexing scheme. As
14

the distribution network is radial nature, the indexing of bus and branch should be proper , such that reading and retrieval of the branch and node data is easy. Hence a, proper indexing scheme is demanded by the load flow solution method.
The main feeder nodes and branches are indexed first and then lateral nodes and branches are indexed. The main feeder is treated as a "lateral 1 " (L- l) and the actual laterals are numbered from 2 to NLat (L=2,3.... NLat) .The first node of the main feeder or my lateral is defined as a "source node SN (L)” . The second node is called as the
Nodc just after the source node. LB (L)" and the last node of the main feeder or lateral js called “End node . EB (L)”.
A new load flow solution technique for distribution network using the Tellegen theorem is presented . The distribution network considered has two configurations , one is the main feeder only and the other is the main feeder laterals . The basic
Tellegen theorem principle is effectively utilized to develop the core power flow equation for the load flow method . The power summation principle at a node is also used to determine the power flowing through the terminal node . Since the node is voltage angles are omitted , a simple linear equation for the node voltage magnitudes is derived .
2.4 Distribution Network with main feeder :
The power flow equations are developed using the basic Tellegen theorem and the distribution network is treated like a ladder network .
The distribution network with main feeder is as shown It is converted in to a ladder network for obtaining the power flow equations . The distribution network with main feeder is in the ladder circuit form is as shown.
The ladder network elements are like sources, and series branches with (r, x) and with (P
L
,Q
L
).
The ladder network, as shown in the fig has single port through which the power is fed to the network and the series branch elements and load elements dissipate this power.
The expression for the complex power in terms of the node vollage magnitudes and the node currents is written using the Tellegen theorem and the eqn. [l .52]
15

The left hand side term Vj I j * is the power injected to the distribution network through the substation node. The complex power injected in the network is resolved in to two components: real power and reactive power.
V1I1* = Pp + JQ
Where
P p
= Real Power injected into the network
QP = Reactive power injected into network.
The term (V1-V2) I1 is the power consumed in the power consumed in the series branch element between the nodes 1 and 2. This is called as the series branch power loss
(V1 – V2 ) I1 * + (V2 – V3 ) I2 * + ….. + ( Vnb-1 - Vnb ) Inb* Is the summation of all the branch power loss.
( V1 – V2 ) I1 * +( V2 – V3 ) I2 *+… + ( Vnb-1 – Vnb ) Inb * = (Vi – Vi-j ) Ij *
Where the term (V1-Vi+1) I j is the general series branch "j” between the nodes the nodes “i” and"i+l"
The power loss of the branch
‘j' between the nodes the nodes "i" and "i+1” is given as
LPj = Re ( Vi – Vi+1 ) Ij *
LQj = Img ( Vi -- Vi+1 ) Ij *
The impedance of the branch "j" between the nodes "i" and "i+1" is Zj = rj + j
Xj . The current flowing through the branch "j" having potential drop of (Vj – Vi+1) is
Ij * = (Vi-Vi+1) /Zj
The complex conjugate of the current vector I j is
Ij * = (Vi-Vi+1) /Zj*
On simplification
Ij * = ( V i
– V i+1)(r j+jX j) / ( r j
2 +
X j
2
)
The eqn. [3.56] is substituted in the eqns.[3.52] and [3.53]
LPJ = Real, part of {((Vi-V i+I
)(Vi-V i+1
)) (rj +j Xj ) / ( r i
2 + xj 2 )}
LQ J = Imaginary part of {((Vj-V w
) (Vi-V i+1
)) (rj + j
Xj
)../ (rj
2
+ Xj
2
)
16

The term V2 lL2*.is the power consumed by the b load connected at the node 2. The term
V2IL2* +V3IL3* +…is the sum of the powers consumed by the loads-connected at all the nodes. Hence
V2 IL2* + V3 IL3* +...V
nb
I
Lnh
* = ViILi*
The load power at a node ‘I' is resolved in to two components real power
PIJ and reactive power Q
Li .
PLi + j QLi = Vi ILi *
The phase angles are omitted in the problem formulation. The reason for this omission is that node voltage phase angles do not vary much for the distribution network and the power flowing through the series branch are effectively dependent on the voltage magnitudes only .
Pp = ( ( Vi – Vi+1)^2 (rj /((Ri)^2 + (Xi)^2 ) ) ) + PLi
2 2 2
Qp = ( ( Vi – Vi+1) (rj /(ri + Xi ) ) ) + QLi
The eqns.[3.60] and [3.61] are the linear algebraic equations for the real and reactive powers injected in to the network ,in terms of line parameters and voltage magnitudes. It is the objective function for the load flow solution using the Tellegen theorem method .
The equations .[3.60] and [3.61] are the power flow equations used to determine the total Power injected in to the network.
The power balance equation at the node ' i+1" is written as
Pj+1 = Pi – LPj - PLi
Qj+1 = Qi – LQj – QLi
Where
2 2 2
LPj = ( ( Vi – Vi+1) (rj /(ri + Xi ) ) )
2 2 2
LQj = ( ( Vi – Vi+1) (Xj / (ri + Xi ) ) )
The Eqns.[3.62],[3.63], [3.57] and [3.58] are the power flow equations used to determine the exact power feeding through the terminal node and the branch power losses. The current flowing in the branch " j " between the nodes ‘' i " and " i+1"
17

2 2 2 2
Ij = ( ( Vi – Vi+1) / (ri + Xi ) )
The current flowing in the line ‘j' in terms of node injected powers and voltage magnitude .
Ij ^2 = (Pi^2 + Qi^2) / Vi ^2 |3.65| from the equations [3.64] and [3.65 ]
Vi+1 = ( Vi ^2 - K3 ) / Vi
Where
K3 = sqrt (( Pi ^2 + Qi ^2 ) ( ri ^2 + Xi ^2 ))
The eqns.[3.66] and [3.67] are the power flow equations used to determine the node Voltage magnitude of the terminal node.
18

In the 'Tellegen Theorem Method", the theorem principle is applied to the distribution network with main feeder, a linear expression is developed and is treated as an objective function. The function represents the real and reactive powers injected to the network.
The iteration process begins with initial assumption of flat voltage profile to a i l - the nodes of the distribution network.
The new powers fed in to the nodes are determined using the eqns. [3.62] and [3.63].
The terminal node voltage magnitude is computed using the eqns. [3.66] and 13.67]. .
The branch power losses are updated using the eqns. [3.57] and [3.58].
The updated voltages and the branch power losses are used eqn.[3.60| and [3.61 j and then powers injected in to the network are updated.
A tolerance check of 1 kW and 1 KVAR is applied on to the new powers injected in to the network. If convergence is not met, the above process is repealed till the convergence is obtained. Once the convergence is obtained the updated voltage magnitudes and the branch power losses are printed out.
19

15
16
17
18
9
10
11
12
13
14
6
7
8
3
4
5
Number Node voltage
1
2
1
0.981
0.9598
0.9476
0.9399
0.912
0.8927
0.8832
0.8672
0.8481
0.8361
0.8309
0.8177
0.8075
0.8015
0.7973
0.7936
0.7924
Number of iterations: 4
Real Power Loss(KW)
32.7106
33.0682
18.0104
10.5739
35.2935
22.6388
10.0685
15.4245
16.5198
9.4573
3.5163
7.5706
4.7139
2.1479
1.0375
0.6101
0.1019
0.0000
ReactivPower Loss(KVAR)
13.6115
14.0692
7.4944
4.3655
14.6862
9.4204
4.1896
6.4184
6.8742
2.6736
0.9941
2.14
0.9773
0.552
0.4400
0.115
0.0288
0.0000
Execution time:
Memory Requirement:
Total Real power loss:
2.65
3190
223.4637KW
Total Reactive power toss: 89.0503KVAR
20

S.No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Node Voltage
1.0000
1.0000
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9998
0.9998
0.9998
0.9998
0.9998
0.9997
0.9997
0.9997
0.9997
0.9997
Number of iterations : 4
Execution time: 2.66 minutes
Total Real power loss: 0.2521388KW
Total reactive power loss: 0.1009080KVAR
Loss(KW)
0.0399
0.039
0.021
0.0122
0.0397
0.025
0.0109
0.0167
0.0176
0.0098
0.0036
0.0078
0.0049
0.0022
0.0011
0.0006
0.0001
0
Loss(KVAR)
0.0166
0.0166
0.0088
0.005
0.0165
0.0104
0.0045
0.007
0.0073
0.0028
0.001
0.0022
0.001
0.0006
0.0004
0.0001
0
0
21

Step 1: Read the substation voltage magnitude V(1), Line data (R,X) and the load data (Pl, Ql)
Step2: Assume a flat voltage profile.
Step3: Set OldPP, OldQQ to previous iteration real and reactive powers injected in to the network at zero
Step4: Compute new powers injected NewPP, NewQQ in to the distribution network using eqns 3.60 & 3.61
Step5: Compute the differences between (OldPP and NewPP) , and ( OldQQ and
NewQQ)
Step6: Check for DelPP & DelQQ less than the tolerance
Step7: If the tolerance is not met, increment the iteration count.
a ) Set the OldPP = NewPP and OldQQ = NewQQ
b) The real and reactive powers injected and end node voltage magnitudes are calculkated by using the eqns 3.62, 3.63, 3.66 and 3.67 then go to step 5.
Step8: If tolerance is met, print the results.
22

A
Bus x = flat voltage profile
Old Pp , old
Qq R=X=0
NewPp , New
Qq are injected
OldPp+New Pp –
OldQq +New Qq
Del Pp ,Del Qq
< tolerance
Tolerance
<0
N=n+1
Print the result
Old Pp=New
Pp
OldQq=New
Qq
A
23

3.3 Software Used :-
3.3.1 Introduction to MATLAB:-
MATLAB® is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include Math and computation Algorithm development Data acquisition Modeling, simulation, and prototyping Data analysis, exploration, and visualization Scientific and engineering graphics Application development, including graphical user interface building
MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. This allows you to solve many technical computing problems, especially those with matrix and vector formulations, in a fraction of the time it would take to write a program in a scalar noninteractive language such as C or Fortran.
The name MATLAB stands for matrix laboratory. MATLAB was originally written to provide easy access to matrix software developed by the LINPACK and
EISPACK projects. Today, MATLAB engines incorporate the LAPACK and BLAS libraries, embedding the state of the art in software for matrix computation.
MATLAB has evolved over a period of years with input from many users. In university environments, it is the standard instructional tool for introductory and advanced courses in mathematics, engineering, and science. In industry, MATLAB is the tool of choice for high-productivity research, development, and analysis.
MATLAB features a family of add-on application-specific solutions called toolboxes. Very important to most users of MATLAB, toolboxes allow you to learn and apply specialized technology. Toolboxes are comprehensive collections of
MATLAB functions (M-files) that extend the MATLAB environment to solve particular classes of problems. Areas in which toolboxes are available include signal
24

processing, control systems, neural networks, fuzzy logic, wavelets, simulation, and many others.
The MATLAB System:-
The MATLAB system consists of five main parts:
Desktop Tools and Development Environment . This is the set of tools and facilities that help you use MATLAB functions and files. Many of these tools are graphical user interfaces. It includes the MATLAB desktop and Command Window, a command history, an editor and debugger, a code analyzer and other reports, and browsers for viewing help, the workspace, files, and the search path.
The MATLAB Mathematical Function Library.
This is a vast collection of computational algorithms ranging from elementary functions, like sum, sine, cosine, and complex arithmetic, to more sophisticated functions like matrix inverse, matrix eigenvalues, Bessel functions, and fast Fourier transforms.
The MATLAB Language . This is a high-level matrix/array language with control flow statements, functions, data structures, input/output, and object-oriented programming features. It allows both "programming in the small" to rapidly create quick and dirty throw-away programs, and "programming in the large" to create large and complex application programs.
Graphics.
MATLAB has extensive facilities for displaying vectors and matrices as graphs, as well as annotating and printing these graphs. It includes high-level functions for two-dimensional and three-dimensional data visualization, image processing, animation, and presentation graphics. It also includes low-level functions that allow you to fully customize the appearance of graphics as well as to build complete graphical user interfaces on your MATLAB applications.
The MATLAB External Interfaces/API.
This is a library that allows you to write
C and Fortran programs that interact with MATLAB. It includes facilities for calling routines from MATLAB (dynamic linking), calling MATLAB as a computational engine, and for reading and writing MAT-files.
25

3.3.2 Starting MATLAB :-
On Windows platforms, start MATLAB by double-clicking the MATLAB shortcut icon on your Windows desktop.
On UNIX platforms, start MATLAB by typing matlab at the operating system prompt.
You can customize MATLAB startup. For example, you can change the directory in which MATLAB starts or automatically execute MATLAB statements in a script file named startup.m.
3.3.3 Entering Source code in MATLAB:-
When you start MATLAB, the MATLAB desktop appears, containing tools
(graphical user interfaces) for managing files, variables, and applications associated with MATLAB. The following illustration shows the default desktop. You can customize the arrangement of tools and documents to suit your needs.
After entering the source code we can see a window Figure opening which shows the resultant Plot after executing the program
26

4.1 Introduction:
In this chapter, the first part presents the conclusions and remarks made on the exclusive load flow methods “Forward Sweeping Method”, “Dist Flow method” and the new load flow method “Tellegen Theorem Method”, are presented in the form tables. The conclusions are with regard to the performance criteria execution time , memory requirement and number of iterations.
4.2
Conclusions on exclusive load flow methods and “Tellegen theorem method”:
The classical load flow methods namely Netwon-Raphson (NR), Fast Decoupled
Load Flow (FDLF) have projected problems like non- convergence , high memory requirement , and large computation time. When a load flow solution is attempted for a distribution network. It is found that , the (R/X) ratio and the radial configuration of the network are found to be reasons for the non- convergence and the large computational time. The circumvent same of the aforesaid problems a modified FDLF method was proposed. In that method, the network parameters are modified by adding certain compensating devices. Even then, the load flow solution was not a fully converged one and hence the exclusive load flow methods are developed for distribution networks.
The power summation principle at a node is used as a general principle in power flow expressions formulation. The power flow equations are the mathematical expressions for the real and reactive powers flowing through the branch and the voltage magnitude of the terminal node.
In “Forward Sweeping Method”, the iteration process is initiated with the initial branch power losses assumed to be zero. The substation node voltage is known at the beginning of the iteration process. The power flow equations are used to compute the branch powers and the voltages of all the nodes of the main feeder as well as laterals.
The branch power losses are updated using the updated voltages. A tolerance check of known value is applied on the branch power losses and the iteration process is continued based on the convergence achievement.
27

The main problems faced with the method are the order of the expression for the terminal node voltage magnitude and computation of the exact power feeding through the terminal node. The iteration time has become large and as the size of the network increases, the number of iterations also increased. The exact power feeding through the terminal node computation is very tedious process and the complexity increased as the size of the network increased. In the case of network with main feeder and laterals, the solution process demanded large memory requirement and resulted in to a large computational time.
In “dist Flow method” the same power balance equations are utilized, the distribution network is reduced to a single branch network with an equivalent resistance and reactance. A convergence check is made on the real and reactive powers injected into the equivalent network and the iteration process is continued on the convergence achievement.
The main advantage of this method is that, the repeated voltage computations are eliminated unlike in other load flow methods. But the major disadvantages are that the power flow equations used are of higher order and of reduction of network into single line equivalent.
In the “ Tellegen Theorem Method” of load flow study, the basic principle is applied to the distribution network and a simple linear objective expression representing real and reactive powers injected into the network are obtained. Since the node voltage angles are omitted, a simple linear expression for the voltage magnitude is resulted and a flat voltage profile assumption has relieved from no furthur assumptions to make to obtain the solution. The main performance criteria considered for the comparison purpose are iteration count, execution time, and memory requirement.
The main problems faced in the “forward sweeping method” and “dist flow method”, namely the high order nature of the expressions and the large number of iterations, are partially reduced and the memory requirement is drastically reduced.
Since the power conservation principle is a true and real phenomenon and exists irrespective of the circuit nature, the load flow technique developed is also near to reality.
28

function tellemfeeder(casefilename)
% tellemfeeder performs load flow study of radial distribution network using tellegen’s theorem. This program works only with a main feeder.
% r and x are resistance and reactance values of xx-bus system pl and ql are active power load and reactive power.
% nb represents numer of buses
% vr represents substation voltage
% the output arguments represent
% v -- node voltages
% tploss -- total real poweer losses
% tqloss -- totl reactive power losses
[r,x,pl,ql,nb,epslon,vr] = feval(casefilename)
% the above function gives data for R , X , pl and ql, nb, epslon,vr of 18 bus system vpu(1)=1,v(1)=vr,tpload=0,tqload=0,k=1,oldpp=0.00,oldqq=0.00,counter=0; for j=1:nb tpload=tpload+pl(j); tqload=tqload+ql(j); end for j=2:nb v(j)=v(1); % assuming flat voltage profile end while(counter==0) tploss=0; tqloss=0; for i=1:nb-1 term(i)=r(i)^2+x(i)^2;
29

k1(i)=r(i)/term(i); k2(i)=x(i)/term(i); term1(i)=(v(i)-v(i+1))^2; ploss(i)=k1(i)*term1(i); qloss(i)=k2(i)*term1(i); tploss=tploss+ploss(i); tqloss=tqloss+qloss(i); end ploss(nb)=0; qloss(nb)=0; p(1)=tploss+tpload; q(1)=tqloss+tqload; newpp=p(1); newqq=q(1); delp=abs(newpp-oldpp); delq=abs(newqq-oldqq); if(delp < epslon && delq < epslon) counter=1; else k=k+1; oldpp=newpp; oldqq=newqq; for i=1:nb-1 j=i+1; p(j)=p(i)-k1(i)*term1(i)-pl(j); q(j)=q(i)-k2(i)*term1(i)-ql(j); term4=(p(i)^2+q(i)^2); k3=sqrt(term4*term(i)); v(j)=(v(i)^2-k3)/v(i); vpu(j)=v(j)/v(1); end end end clc fprintf('Tellegen’s Therom results are as followes\n the number of iterations are =
%d\n the total load = %9.5f kw\n total reactive load = %9.5fkvar \n total real power losses=%9.5fkw\n total reactive power losses =
%9.5fkvar',k,tpload,tqload,tploss,tqloss); v/440 return;
30

RESULTS
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 on the bus
/ Update and verify/
31

1.
B.D.H Tellegen “A General Network Theorem Applications” Phillips
Research Report
2.
Tripathy sc, durga Prasad. G.Mallic.OP , and Hope. G.S “Load Flow solution for ill-condioned power system by a network like method” IEEE Trans, PAS
101 October 1972.
3.
LAFM Ferreira “Tellegen Theorem and Power Systems –New Load Flow
Equations New Solution Methods” Trans. Circuits and systems vol 37 No.4
April 1990.
4.
GS Jasmom LHCC Lee “Distribution network reduction for voltage stability analysis and load flow calculations” Electric Power and Energy Systems vol.13 no -1Feb 1991.
5.
D.Das , Nagi and Kothari “New load flow technique for solving radial distribution feeder- Main feeder case “ IEE proceedings -73 October 1992.
6.
D.Das, Nagi and Kothari, A Kalam “Simple and efficient method for load flow solution of radial distribution networks” Electrical power and Energy systems Vol 17, No 5 PP 355, 346 ,1995.
7.
GS Jasmom LHCC Lee “Stability of Load flow techniques for distribution system voltage stability analysis” IEE proceedings- C vol 138 no.6
November – 1991.
8.
SF Mekhamer, SA Soliman, MA moustafa, ME EI-Hawary “Load flow solution of radial distribution feeders: a new contribution “Electrical power and Energy systems”.
32

33

34

35
Toleranc e <0