Load Flow Study using Tellegen’s Theorem

Mini Project Report submitted to the Department of Electrical and Electronics

Engineering in accordance with the academic requirement for the award of the degree.

SUBMITTED BY

AD SRIKANTH 05841A0232

LVS KARTHIK 05D91A0210

Department of Electrical and Electronics Engineering

Aurora’s Technological and Research Institute

(Affiliated to JNTU)

Parvathapur, Uppal, HYDERABAD – 500 039

-1-

CERTIFICATE

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

ACKNOWLEDGEMENTS

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

INDEX

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

1.4 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

3.1 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 (MATLAB) 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 methodAppendix: 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

Chapter 1

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 tie-

switches 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

6

voltage profile has become very poor. The load flow analysis is the solution, through

which any network can be analyzed to improve its performance.

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

1.1 Classical Load Flow Techniques:

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

8

(Q) are very much influenced by the changes in voltage magnitudes and no changes

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).

1.1.1 Limitations of the classical load Flow methods :

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

1.2 Load flow study of Distribution networks : 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

10

method is the efficiency is achieved by avoiding repeated computations of node

voltage magnitudes.

.

1.3 Objective of the Thesis :

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.

1.4 A Generalized form of 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 Vm and Vn respectively and voltage

across the branch Connecting 'M' and *N' nodes Is Vmn = Vm - Vn

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

12

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 .

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 = Vm1 i m1 + Vm2 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 vm1 i m1 + vm2 i m2 + Vm3 Im3 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 powerdelivered 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

Chapter 2

Load Flow Solution formulation using exclusive methods and Tellegen Theorem method

2.1 Introduction: 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.

2.2 Description of the radial Distribution network: 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 , Xr .The load at node "i"

is represented with PLi , QLi

14

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

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)”.

2.3 Load Flow problem formulation using Tellegen theorem method : 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 (PL,QL).

15

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]

V1 I1* = ( V1 – V2 ) I 1 * + V2 IL2* + (V3 IL3* +….. + (Vnb-1 – Vnb ) IL nb * + Vnb ILnb

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 Pp = 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

16

Ij * = (V i – V i+ 1) ( r j+ jX j ) / ( r j2 + 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 i2 + xj2)}

LQ J = Imaginary part of {((Vj-Vw) (Vi-Vi+1)) (rj + j Xj )../ (rj2 + Xj2 )

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* +...VnbILnh* = ViILi*

The load power at a node ‘I' is resolved in to two components real power PIJ and reactive power QLi .

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

17

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" 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

Chapter 3

3.1 Case study: Load flow analysis of distribution network with main feeder using “Tellegen Theorem Method” :

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

Table 3.12 – 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.

Number Node voltage Real Power Loss(KW) ReactivPower Loss(KVAR)1 1 32.7106 13.61152 0.981 33.0682 14.06923 0.9598 18.0104 7.49444 0.9476 10.5739 4.36555 0.9399 35.2935 14.68626 0.912 22.6388 9.42047 0.8927 10.0685 4.18968 0.8832 15.4245 6.41849 0.8672 16.5198 6.874210 0.8481 9.4573 2.673611 0.8361 3.5163 0.994112 0.8309 7.5706 2.1413 0.8177 4.7139 0.977314 0.8075 2.1479 0.55215 0.8015 1.0375 0.440016 0.7973 0.6101 0.11517 0.7936 0.1019 0.028818 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

20

Table – Load flow results of Tellegen theorem method: Voltage magnitude in pu, branch Real and Reactive Power loss -18bus , 11Kv – distribution network and main feeder

S.No Node Voltage Real power Loss(KW)

Reactive power Loss(KVAR)

1 1.0000 0.0399 0.0166 2 1.0000 0.039 0.0166 3 0.9999 0.021 0.0088 4 0.9999 0.0122 0.005 5 0.9999 0.0397 0.0165 6 0.9999 0.025 0.0104 7 0.9999 0.0109 0.0045 8 0.9999 0.0167 0.007 9 0.9998 0.0176 0.0073 10 0.9998 0.0098 0.0028 11 0.9998 0.0036 0.001 12 0.9998 0.0078 0.0022 13 0.9998 0.0049 0.001 14 0.9997 0.0022 0.0006 15 0.9997 0.0011 0.0004 16 0.9997 0.0006 0.0001 17 0.9997 0.0001 0 18 0.9997 0 0

Number of iterations : 4

Execution time: 2.66 minutes

Total Real power loss: 0.2521388KW

Total reactive power loss: 0.1009080KVAR

21

3.2 Algorithm for the Distribution networks with main feeder using Tellegen theorem method.

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

3.3 Flowchart

23

Start

Bus x = flat voltage profile

Old Pp , old Qq R=X=0

NewPp , New Qq are injected

OldPp+New Pp – OldQq +New QqA

Del Pp ,Del Qq < tolerance

Tolerance <0

N=n+1

Old Pp=New Pp OldQq=NewQq

Print the result

3.3 Software Used :- MATLAB

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.

24

A

stop

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

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.

25

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.

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

26

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

Chapter 4

Conclusions 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

27

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.

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

28

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.

SOURCE CODE

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

29

% 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;

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;

30

elsek=k+1;oldpp=newpp;oldqq=newqq;for i=1:nb-1j=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);endendendclc

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/440return;

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

31

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/

Bibliography

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.

32

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”.

33

34

35