LOAD FLOW ANALYSIS & LOSS ALLOCATION FOR

UNBALANCED RADIAL POWER DISTRIBUTION SYSTEMS

THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF TECHNOLOGY IN

ELECTRICAL ENGINEERING SPECIALIZATION

POWER SYSTEMS ENGINEERING

BY SIVKUMAR MISHRA

05 EE 6312

UNDER THE SUPERVISION OF

PROF. DEBAPRIYA DAS

APRIL 2007

DEPARTMENT OF ELECTRICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY

KHARAGPUR -721302 INDIA

DEPARTMENT OF ELECTRICAL ENGINEERING APRIL, 2007

CERTIFICATE The thesis entitled “Load Flow Analysis & Loss Allocation for Unbalanced Radial

Power Distribution Systems”, submitted by Sivkumar Mishra, Roll No. 05 EE 6312,

for the award of degree of Master of Technology in Electrical Engineering in Indian

Institute of Technology, Kharagpur is a record of bonafide work carried out by him under

my supervision for partial fulfillment of the requirements for the degree of Master of

Technology in Power System Engineering during the academic year 2005-2007 in the

Department of Electrical Engineering, Indian Institute of Technology, Kharagpur. Prof. Debapriya Das Date: Department Of Electrical Engineering Place: Kharagpur I.I.T,Kharagpur

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my supervisor Prof. Debapriya Das for his

constant guidance and inspiration. His invaluable guidance, critical reviews and constant

help enabled me to understand the subject and to complete the project work.

I would like to thank Prof. S.K. Das, Head of the Dept. and Prof. A. K. Sinha, our faculty

advisor for extending me all the possible facilities to carry out the project work.

I am also grateful to Prof. T. K. Basu, Prof. N. N. Kishore, Prof. A. Routray and Prof.

A.K. Pradhan from whom I have learnt so many things throughout my stay in IIT,

Kharagpur.

Finally, I would also like to thank all my family members, friends and batch mates in IIT,

Kharagpur for their support and encouragement during my project work.

SIVKUMAR MISHRA Date: Department of Electrical Engineering Kharagpur Indian Institute of Technology,Kharagpur

This Thesis is dedicated to my father

Late Pt. Nilamani Mishra

ABSTRACT

The project work consists of two parts. The first part starts with an extensive literature

survey on the distribution system load flow methods. Five well established three phase

unbalanced radial distribution system load flow methods (Implicit ZBUS Gauss method,

Topological Direct method, Ladder Network based method, Forward and Backward

Sweep method and Power Summation based FBS method) have been implemented. Two

variations of the Implicit ZBUS Gauss method have also been proposed and implemented.

A new forward backward sweep based unbalanced radial distribution system load flow

method has been proposed. A novel scheme of bus identification and multiphase data

handling is proposed in the method. Three distribution system test systems have been

used to compare the performances of these 8 methods and the proposed forward and

backward sweep method has been found to be faster compared to the other methods. In

the second part, a new loss allocation scheme has been proposed, which utilizes the load

flow results of the multi phase unbalanced radial distribution system to allocate the active

losses to the various consumers of the system connected at the different buses. For load

flows the earlier proposed forward and backward sweep method has been used in which

the system loads are modeled as composite loads. The loss allocation scheme has been

successfully implemented with the three test distribution systems.

CONTENTS PAGE NO. 1.0 Introduction 1 – 10 1.1 Distribution System Load Flow Methods- A Survey 3 1.2 Organization of the thesis 9 2.0 Modeling of distribution system components 11 - 18 2.1 Feeder Modeling 11 2.2 Load Modeling 16 2.3 Loss Formula for Unbalanced Distribution Networks 17 3.0 Implementation of some important UDSLF methods 19 - 38 3.1 ZBUS Gauss Approach 19 3.2 Topological Direct Method 26 3.3 Ladder Network based Method 31 3.4 Forward and Backward Sweep Method 35 3.5Power Summation based FBS Method 36 4.0 Proposed simple and fast UDSLF method 38 - 49 4.1 Proposed bus identification scheme 39 4.2 Multi phase data handling 42 4.3 Algorithm of the proposed method 43 4.4 Test Systems 44 5.0 Loss Allocation in Unbalanced Distribution Systems 50- 56 5.1 Methodology 50 5.2 Proposed Loss Allocation Method 53 5.3 Implementation and Flowchart of the method 55 6.0 Results and Discussion 57 - 67 6.1 Performance analysis of the three phase UDSLF methods 57 6.2 Implementation of the proposed loss allocation scheme 61 7.0 Conclusion 68 –70 7.1 Future Prospects 69 References 71 - 77 Appendix 78– 78

LIST OF FIGURES AND TABLES Figures 1 Three phase line section model 11 2 The equivalent circuit of a three phase line section 15 3 The equivalent circuit of a two phase line section 15 4 The equivalent circuit of a single phase line section 15 5 An unbalanced distribution network 20 6 A simple balanced distribution system 26 7 Currents and voltages in a simple ladder circuit 31 8 Currents and voltages in a ladder circuit with constant complex power 32 9 A radial system with multiple laterals and sub-laterals 33 10 Single phase line section with load connected at node-j between phase -a and neutral 36 11 Storing and pointer operation of the proposed vectors 40 12 Sample 8-node multi- phase unbalanced distribution system 40 13 25-node practical multi- phase unbalanced distribution system (Test System -2 ) 46 14 34-node IEEE Test System (Test System -3 47 15 Simple distribution network with 12 nodes 51 16 Storing and pointer operation of the proposed vectors 55 17 Flowchart of the proposed loss allocation method 56 18 Performance chart of all the UDSLF methods (no.of iterations) 62 19 Performance chart of all the UDSLF methods (CPU execution time) 62 20 Overhead line spacings 78

Tables 1 mf[ ], mt [ ] vectors of sample distribution network (fig.12) 40 2 adb[ ] vector of sample distribution network (fig.12) 41 3 bp[ ], bpt [ ] vectors of sample distribution network (fig.12) 42 4 Data of sample distribution network (fig.12) 43

5 Load data of sample distribution network ( Test System-1) 44 6 Feeder data of sample distribution network ( Test System-1) 45 7 Feeder data of sample distribution network ( Test System-1) 45 8 Load data of sample distribution network ( Test System-2) 46 9 Data of Test System-2 47 10 Load data of sample distribution network ( Test System-3) 48 11 Data of Test System-3 48 12 Data of Test System-2 49 13 Data of Test System-2 49 14 Data of distribution network (fig. 15) 51 15 List of all UDSLF methods 57 16 Converged voltage magnitudes (Test System-1) 57 17 Active and reactive power losses (Test System-1) 58 18 Converged voltage magnitudes (Test System-2) 58 19 Active and reactive power losses (Test System-2) 60 20 Converged voltage magnitudes (Test System-3) 60 21 Active and reactive power losses (Test System-3) 62 22 Comparison of CPU time and no. of iterations of all the 8 methods 62 23 Formation of mfs[ ], mts [], nsb[ ] and sb [ ] vectors of Test System-1 63 24 Allocated losses to the buses of Test System-1( constant P,Q case ) 64 25 Allocated losses to the buses of Test System-1( composite load case ) 64 26 Allocated losses to the buses of Test System-2( constant P,Q case ) 64 27 Allocated losses to the buses of Test System-2( composite load case ) 65 28 Allocated losses to the buses of Test System-3( constant P,Q case ) 66 29 Allocated losses to the buses of Test System-3( composite load case ) 67 30 Overhead line spacings 78

1

CHAPTER 1

INTRODUCTION

Load flow analysis is a very important and basic tool in the field of power system

engineering. Load flow studies are used to ensure that electrical power transfer from

generators to consumers through the grid system is stable, reliable and economic .This is

used in the operational as well as in the planning stages. Basically, it does the steady state

analysis of any power system. The main objective of the load flow analysis is to find out the

real and reactive powers flowing in each line along with the magnitude and phase angle of

the voltage at each bus of the system for the specific loading conditions. Certain applications,

particularly in distribution automation and optimization of a power system, require repeated

load flow solutions. In these applications it is very important to solve the load flow problem

as efficiently as possible .Since the invention and widespread use of digital computers,

beginning in the 1950’s and 1960’s, many methods for solving the load flow problem have

been developed [1]. Most of the methods have “grown up” around transmission systems and,

over the years, variations of the Newton method such as the fast decoupled method [2], have

become the most widely used. Although these classical techniques have been widely used,

there are situations when they may experience difficulties or become inefficient as in the case

of ill- conditioned or poorly initialized networks, hence, require various modifications for the

load flow analysis [3-5]. The analysis of a distribution system is an important area of activity

as distribution systems provide the final link between the bulk power system and the

consumers. A distribution circuit normally uses primary or main feeders and lateral

distributors. A main feeder originates from the substation and passes through the major load

centers. 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 and are defined as radial distribution systems. Radial systems are

2

popular because of their low cost and simple design. Distribution networks because of the

following special features fall in the category of ill conditioned power systems for generic

Newton–Raphson and fast decoupled load flow methods.

- Radial structures

- High R/X ratios of the feeders

- Multiphase, unbalanced operation

- Unbalanced distributed loads

- Large no. of nodes and untransposed feeders.

- Dispersed Generation

Radial distribution feeders are characterized by having only one path for power to flow from

the source (distribution substation) to each customer .Some distribution feeders serving

densely loaded areas operated with weakly meshed loops by closing the normally open tie

switches. Therefore, strictly speaking, distribution circuits are mainly radial in nature with

some weakly meshed loops. Conventional power flow methods show convergence problem

in solving such networks. For the converged cases also these methods are very inefficient in

respect of storage requirements and solution speed. Special power flow methods have

therefore come out over years, which exploit the special characteristics of distribution

networks, namely radiality and the presence of only one voltage controlled bus. These

alternate algorithms show better efficiency and simplicity for radially configured networks

than the traditional Gauss Siedel and Newton-Raphson methods [6-11]. All these special load

flow methods come in the category of distribution system load flow (DSLF) methods.

Moreover, due to inherent unbalance nature, the distribution systems are always analyzed on

three phase basis, thus, the distribution system load flow studies are always performed on

three phase basis with the detail modeling of the various components of the system which

includes mutual coupling between the feeders.

Electric power industries throughout the world are undergoing major restructuring process

and are adapting the deregulated market operation. The vertically integrated systems has

been restructured and unbundled to one or more generation companies, transmission

companies and a number of distribution companies. Competition has been introduced in

power systems around the world based on a premise that it will increase the efficiency of this

3

industrial sector and reduce the cost of electrical energy of all the consumers. Unlike

generation and sale of electrical energy, activities of transmission and distribution are

generally considered as a natural monopoly. The cost of transmission and distribution

activities needs to be allocated to the users of these networks. Allocation can be done through

network use tariffs, with a focus on the true impact they have on these costs. Among others,

distribution power losses are one of the costs to be allocated. Conceptually, loss allocation is

a difficult task. The main difficulty faced in allocating losses is the nonlinearity between the

losses and delivered power which complicates the impact of each user on network losses

[12]. It is impossible to calculate the exact amount of losses in advance, without running a

power flow. At the same time, even after computing the power flow solution, there is a

strong interdependence among all the users, expressed by the presence of cross terms due to

the fact that losses are a nearly quadratic function of the power flows. Hence, allocating the

losses to the market participants cannot be carried out in a straight forward way. Different

techniques have been published in the literature for allocation of losses, most of them

dedicated to transmission networks and can be classified into three broad categories – pro

rata procedures, marginal procedures and proportional sharing procedures [13-19]. Costa and

Matos [20] have addressed the allocation of losses in distribution networks with embedded

generation by considering quadratic loss allocation technique. In general, a first distinction

can be made between loss allocation methods dedicated to transmission and to distribution

systems. The difference between these two classes of methods basically lies in the role of

given to the slack node. In transmission systems, the generator located in the slack node

compensates for all the losses and is explicitly considered in the mechanism of loss

allocation. In radial distribution systems, the location of the slack node at the root node of the

distribution tree is naturally unique, and the slack node usually represents the connection to

the higher voltage network.

1.1 DISTRIBUTION SYSTEM LOAD FLOW METHODS- A SURVEY

Unlike transmission system, Distribution System Load Flow (DSLF) methods had received a

comparatively late attention. However, gradually; the tendency towards the Distribution

Automation (DA) [21-23] has led researchers to develop the so-called control functions,

which perform on line predefined tasks, either in emergency or normal conditions. These

4

application programs require robust and efficient load flow solution methods incorporating

detail modeling of special features of distribution systems. Radial distribution systems are

inherently unbalanced, owing to factors such as the occurrence of asymmetrical line spacing,

the combinations of single, double and three phase line sections and the imbalance of

customer loads; hence, solution methods based on the assumption of balanced loading are not

applicable. Yet, many DSLF methods have been proposed that solve only the line to neutral

equivalent of a balanced system. Thus, important information such as the effect of mutual

coupling on voltage drop and unbalance is lost [24]. Baran and Wu [25] have proposed a load

flow method for distribution system based on Newton Raphson formulation. Renato et al [26]

have proposed a method for obtaining load flow solutions of radial distribution networks

(RDN), which seems to be quite promising. However, it gives solution to bus magnitude

only. Chiang [27] has presented a load flow solution, modifying the method proposed by

Baran and Fu[25] for distribution system by iterative solution of three fundamental equations

representing real power, reactive power and voltage magnitude. Jasmon and Lee [28-29]

have proposed an algorithm to examine voltage stability and to detect voltage collapse, if

any, for radial distribution networks. D. Das et al.[30] have proposed a load flow solution

method by writing an algebraic equation for bus voltage magnitude. Similarly, various other

DSLF methods [31-45] are found in literature, which effectively exploit the radial nature and

overcome the ill-conditioned nature due to high R/X ratio of the distribution networks.

However, these methods [25-45] are suitable for single phase analysis of radial distribution

networks [RDN], assuming balanced operation of the RDNs. The load flow methods

proposed for distribution systems considering the unbalance operation can be grouped into

two basic categories [46-51]. The first category is Forward Backward Sweep (FBS) / Ladder

Network based methods, Loop Impedance Method and Implicit Zbus Gauss method or its

modified versions. The other class is composed of methods which require information on the

derivatives of the network equations. Newton like methods involving formation of Jacobians

and computation of power mismatches at the end of the feeder and laterals and other fast

decoupled methods, tailored specially for distribution systems, come under this second

category.

5

1.1.1 Forward Backward Sweep / Ladder Network Methods

A literature survey on the DSLF methods reveals that most of the methods for radial

distribution load flow have been predominantly based on Forward and Backward Sweeping

(FBS)/ Ladder Network based approach of the network tree representation. The obvious

reason for the above is that this method is simple to implement, fast and robust. The general

algorithm consists of two basic steps: forward sweep and backward sweep. The forward

sweep is mainly a voltage drop calculation from the sending end to the far end of a feeder or

a lateral, and the backward sweep is primarily a current, power or admittance summation

based on the voltage updates from the far end of the feeder to the sending end .An early

approach [52] to radial power flow analysis utilized linear network theory by approximating

all the constant power loads as admittances calculated corresponding to assumed bus

voltages. Network reduction techniques are used to get an equivalent circuit and new bus

voltages are calculated by ‘unfolding’ the network. The process is iterative and repeated till

convergence. The Ladder Network Approach [53-55] proposed in 1976, performs load flow

analysis on unbalanced three phase radial systems, is basically a backward sweep method,

where iterations are started with estimates of ending bus voltages and a reverse trace is then

performed for determining the various bus voltages leading to a calculated value of the

source voltage. New estimates for the ending bus voltages are determined based on the

mismatch between the calculated and specified source voltages. Iterations are continued till

the specified source voltage is obtained by calculation within a required accuracy. The

ladder approach is very simple to implement for large systems, but when there are ‘laterals’

in the network, additional sub-iterations for the lateral sections are required. Broadwater et al

[56] have presented an algorithm for solving the power flow in a multiphase unbalanced

radial distribution system. This algorithm uses a reverse trace of the system to sum up the

power followed by a forward trace solving a quadratic equation for the square of the voltage

magnitude. Thukaram et al. [57] have presented a branch current summation based forward

backward sweep load flow algorithm. Ranjan et al.[58] have proposed a power summation

based forward backward sweep load flow algorithm. Some distribution feeders serving

densely loaded areas operated with weakly meshed loops by closing the normally open tie

switches. Meshed networks are very common in distribution systems especially with the

6

implementation of distribution automation (DA). Shirmohammadi et al. [59] have proposed a

compensation method for distribution systems with weakly meshed structures. This method

starts from a network structure analysis to find the interconnection points. It then breaks

these interconnection points using the compensation method so that the meshed system

structure could be changed to simple tree type radial system. Based on the previous work

[59], a modified compensation based load flow method has been presented [60], which uses

active and reactive power as variables rather than complex currents. In [61] Cheng et al.

extended the method [59] from single phase to three phase, with the emphasis on modeling of

dispersed generation (P-V nodes), unbalanced and distributed loads and voltage regulators

and shunt capacitors with automatic local tap controls. This method is capable of addressing

these modeling challenges while still maintaining a high execution speed required for real

time applications in automated distribution systems. In [62], Haque has enhanced the

compensation method [61] for weakly meshed distribution networks considering the effects

of load and shunt admittance. Rajicic and Taleski [63] have proposed two methods based on

admittance and current summation method. A two stage load flow method for meshed

distribution networks has been proposed by Chen et al [64] to further enhance the

convergence speed of the compensation based methods. Ghosh and Das [65] also have

extended their earlier work [34] using multi port compensation technique for solving meshed

distribution networks.

1.1.2 Implicit Zbus Gauss and Loop Impedance Matrix Methods

In [66-67], a Gauss Implicit method has been proposed which uses a bi-factorized complex Y

admittance matrix based on equivalent current injection and which works well as long as the

power components can be modeled in the Y-matrix or can be converted into equivalent

current injection (ECI). The implicit Z-bus method requires the factorization of the full Y-bus

matrix, adversely affecting the performance in terms of speed. Teng [68] has proposed a

modified Gauss-Siedel method, by blending the implicit Z-bus method [66-67] and the

Gauss-Siedel method to improve the computational efficiency. This method, which factorizes

the 3 sub matrices of the Y-matrix instead of the fully Y-matrix, works efficiently with

7

radial, weakly meshed and looped networks. A topological approach [69-70] has been

proposed where two matrices are developed, viz. the bus injection to branch current (BIBC)

matrix and branch current to bus voltage (BCBV) matrix. The load flow solution is obtained

by using simple matrix multiplication of these two matrices. This method can also be

extended for weakly meshed networks. Goswami and Basu [71-72] have formulated a load

flow algorithm by writing simple loop equation and representing loads as complex

impedances. This method works for both balanced and unbalanced distribution networks and

holds good for meshed distribution networks. Wang et al. [73] have modified the method

proposed by Goswami and Basu [72] by introducing a new simple branch and node

numbering scheme. Aravindhababu et al. [74] have developed a similar method of load flow

for radial distribution systems.

1.1.3 Newton Like and Modified Fast Decoupled Methods

For last three decades, the Newton-Raphson (NR) method and its diversities are most widely

used with sparse techniques and optimal ordering for transmission networks. The number of

buses in the distribution system amounts to thousands. The NR method takes relatively much

time since it must calculate the Jacobian matrix and inverse the matrix at each iteration.

Similarly, the Fast Decoupled Newton Raphson (FDNR) method has poor convergence due

to high R/X ratio. In spite of this, due to the ease of extending Newton based methods to

applications such as distribution state estimation and the optimal power flow [75-76], various

attempts have been made to modify the Newton’s method to make it applicable to

distribution systems. In [25] Baran and Wu have proposed a methodology for solving the

radial load flow as a subroutine within the optimal capacitor sizing problem. For each branch,

three non-linear equations are written in terms of the branch power flows and bus voltages.

The number of equations is subsequently reduced by using the terminal conditions associated

with the main feeder and its laterals and the Newton –Raphson method is applied to this

reduced set. In order to improve the computational efficiency, some simplifications are made

in the Jacobian matrix. In the modified NR technique[77],the network radial structure is

explored to express the Jacobian matrix as an product of UDUT, where U is constant upper

triangular matrix depending solely on system topology and D is a diagonal matrix, the

8

elements of which are updated at every iteration. With this the conventional steps of forming

Jacobian matrix, LU factorization and forward/back substitution are replaced by

backward/forward sweeps on radial feeders with equivalent impedances. A three phase

power flow formulation is described in [78] where the Jacobian matrix is presented in

complex form, but some simplifications are introduced by neglecting the component of the

mismatch arising from voltage changes. In[79] the power flow equations are expressed as a

function of new variables that replace Vi2, Vi Vj sin θij and Vi Vj cos θij terms. The resulting

system of equations has order 3n and good convergence properties were attained when the

method was applied to balanced networks. An attempt has been made [81] to extend the

standard three phase decoupled theory [80] with certain modifications to obtain a fast

decoupled three phase load flow which is suited for DSLF. This model differs from previous

work [80] in the way the sub matrices(B’ and B”) ,used to calculate the angle and voltage

corrections, are built and evaluated. A fast decoupled power flow method has been proposed

in [82], which orders the “laterals” instead of “buses” into “layers”, thus reduces the problem

size to the no. of laterals, and then assumes initial end voltage for all laterals .The iteration

starts from the first lateral by using the method proposed in [53]. The voltage mismatch

obtained from this lateral is applied to correct not only the end voltage of this lateral but also

the end voltage of laterals of the next layer. The algorithm converges when all voltage

mismatches are within a certain tolerance. Using lateral variables instead of node variables

makes this method efficient for a given system topology , but it may add some overhead if

the system topology is changed regularly, which is common in distribution systems due to

switching operations. A Phase Decoupled load flow method [84] has been proposed which is

based on equivalent current injection (ECI) method [83], tailored for unbalanced distribution

networks with radial or weakly meshed structures. Here branch current based Newton –

Raphson method is employed. The constant Jacobian matrix of the proposed method could be

decoupled into three sub-Jacobian matrices, one for each phase and is used to execute load

flow. The method is insensitive to line parameters and significantly faster. Moon et al. [85]

have proposed a load flow algorithm based on YBUS formulation in rectangular coordinates

where only diagonal elements of the jacobian matrix are to be updated for the load flow. This

method can be applied to both radial and meshed networks. Lin et al. [86] have proposed

anew distribution system load flow method based on traditional Newton Raphson approach.

9

A constant Jacobian matrix can be obtained and decoupled both on phases and on real and

imaginary parts. In [87-88], a new sparse formulation for the solution of unbalanced three

phase power systems using the Newton –Raphson method is presented. The three phase

current injection equations are written in rectangular coordinates resulting in an order 6n

system of equations. The Jacobian matrix is composed of 6x6 block matrices and retains the

same structure as the admittance matrix. Lin and Teng [89] have proposed a current injection

based Newton Raphson distribution system load flow in rectangular coordinates. The

constant Jacobian of the proposed method can be further decoupled in to two identical sub-

jacobian matrices. Teng and Chang [90-91] have proposed a method which is based on the

NR formulation and utilizes the branch voltage as state variables. A constant Jacobian matrix

can be developed and a building algorithm for Jacobin matrix is then developed from the

observation of the constant Jacobian matrix. A solution technique is adopted, which takes the

network structure into account to avoid the time consuming LU factorization. For any power

system equipment, if its equivalent current injection or admittance matrix can be obtained, it

can be integrated into the proposed method and traditional NR method can be utilized to find

the solution. In [92], an extension of the method [77] has been made. Aravindhababu [93] has

formulated a fast decoupled Newton Raphson based distribution system load flow without

considering any of the assumptions made in [89]. Ramos et al. [94] have proposed a quasi

coupled three phase load flow. Ignoring mutual coupling in three phase load flow

computations saves a lot of computational effort but provides inaccurate results for many

applications. In this method the effect of mutual coupling is taken into account through

equivalent branch voltage sources or bus current injections rather than neglecting it. This

simple idea allows accurate enough solutions to be obtained, while much of the

computational saving associated with decoupled load flows is retained.

1.2 ORGANIZATION OF THE THESIS In the light of the above developments, two distinct but related works on the three phase

unbalanced radial networks are presented in the present thesis. The first work discusses seven

well established three phase unbalanced radial distribution load flow methods and

implements all of them. A novel three phase radial distribution load flow method is proposed

10

and the performances of all the methods are compared on three distribution test feeders. The

second work proposes a novel loss allocation scheme for unbalanced radial distribution

networks. A chapter wise organization of the thesis is given below:

• The second chapter develops models for various components of three phase

unbalanced radial distribution systems.

• In the third chapter, five well established three phase unbalanced distribution system

load flow (UDSLF) methods are implemented. Two variations of the ZBUS Gauss

Method are also proposed and implemented. The methodologies and the algorithms for

all the seven methods are presented in detail.

• In the fourth chapter, a novel algorithm to implement the forward backward sweep

based three phase unbalanced radial load flow method is proposed. A new scheme for

bus identification and multiphase data handling is presented. This chapter also includes

the description of the 3 test systems for which the load flows are carried out. The loads

are modeled as constant power type.

• In the fifth chapter, a novel loss allocation scheme is proposed for the three phase

unbalanced distribution system. This scheme utilizes the load flow results of the test

systems by the load flow method proposed in fourth chapter. The load in the three test

systems are modeled as composite loads.

• The sixth chapter presents the various results of load flows by the eight different

methods including the proposed novel algorithm for the three test distribution systems.

The results of the proposed loss allocation scheme are also presented in this chapter.

• The seventh chapter concludes the thesis with a discussion on future directions of the

current work.

11

CHAPTER 2

MODELING OF DISTRIBUTION SYSTEM COMPONENTS Distribution systems are inherently unbalanced due to the reasons stated before. To avoid

significant error arising from inherent system imbalance, rigorous distribution analysis using

detailed component models is required.

2.1 FEEDER MODELING Because distribution systems consist of single phase, two phase, and untransposed three

phase lines serving unbalanced loads, it is necessary to retain the identity of the self and

mutual impedance terms of the conductors and take into account the ground return path for

unbalanced currents. Using modified Carson’s equations [95], a 4x4 primitive impedance

matrix can be written for the three phase feeders with one neutral, shown in figure-1.

12

[ ]Primitive Impedance Matrix (2.1)

aa ab ac an

ba bb bc bnabcn

ca cb cc cn

na nb nc nn

Z Z Z ZZ Z Z Z

ZZ Z Z ZZ Z Z Z

⎡ ⎤⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

where ( 0.0953) 0.12134[ln(1/ ) 7.934] /mile (2.2)ii i iZ r j GMR= + + + Ω 0.0953 0.12134[ln(1/ ) 7.934] /mile (2.3)ik ikZ j D= + + Ω ir is the conductor resistance ( /mile)Ω

i

ik

GMR is the conductor geometric mean radius (ft.)

D is the spacing between conductor i and k (ft.)

i,k represents the three phases a,b,c and the neutral n

Thus, the receiving end voltages are related to the sending end voltages of the feeder shown

in fig. 1, as

a a' a

b b' b

c c' c

n n' n

V V IV V I

. (2.4)V V IV V I

aa ab ac an

ba bb bc bn

ca cb cc cn

na nb nc nn

Z Z Z ZZ Z Z ZZ Z Z ZZ Z Z Z

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

The a.c resistance and GMR of the conductors are taken directly from a table of conductor

data. The values for spacing between the conductors, a particular standard configuration and

phase sequence is referred to the standard configurations given in Appendix-A. Because,

distribution circuits are multi grounded (i.e the neutral is solidly grounded at every

convenient point), the voltage drop

' 0 (2.5)n nV V− =

13

The 4 x 4 primitive impedance matrix of (2.4) is reduced to a 3 x 3 matrix known as the

phase impedance matrix.. Using Kron’s reduction, each element of the phase impedance

matrix is given by

- ( . / ) (2.6)ik ik in nk nnz Z Z Z Z= Here, the grounding effect has been included by using the modified matrix element zij instead

of Zij. Thus, the phase impedance matrix is give as

[ ] (2.7)aa ab ac

abc ba bb bc

ca cb cc

z z zZ z z z

z z z

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Now, the receiving end voltages are related to the sending end voltages of the feeder shown

in fig. 1, as

a a' aa ab ac a

b b' ba bb bc b

c c' ca cb cc c

V V z z z IV V z z z . I (2.8)V V z z z I

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

The same approach is followed to model double phase and single phase line sections. In case

of a double phase line involving phases a and c, for example, Carson’s equations lead to a 3 x

3 matrix (a,c,n) which is then reduced to a 2 x 2 matrix using Kron’s reduction. The 2 x 2

matrix is then expanded to a 3 x 3 matrix by placing zeros in the row and the column of the

missing phase (b in example). The resulting matrix equation is given as:

a a' aa-n ac-n a

c c' ca-n cc-n c

V V Z 0 Z I0 0 0 0 0 . 0 (2.9)V V Z 0 Z I

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

14

Similarly, for a single phase line, Carson’s equations result in a 2 x 2 matrix. This 2 x 2 could

be reduced to a single term using Kron’s reduction. This single element is then represented in

3 x 3 matrix by placing zeros in the row and column of the missing phases.Thus for phase –a:

a a' aa-n aV V Z 0 0 I0 0 0 0 0 . 0 (2.10)0 0 0 0 0 0

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

From (2.8), the admittance matrix is solved as

a aa ab ac a a'

b ba bb bc b b'

c ca cb cc c c'

I y y y V -VI y y y . V -V (2.11)I y y y V -V

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

aa ab ac

ba bb bc

ca cb cc

y y ywhere y y y is the phase admittance matrix, which is the inverse of the phase impedance matrix.

y y y

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

From (2.11) currents are then expressed as:

' ' '

' ' '

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) (2.12-a)

a aa a a ab b b ac c c

a aa a a ab a b ab a b ac a c ac a c

I y V V y V V y V VI y V V y V V y V V y V V y V V= − + − + −

⇒ = − + − − − + − − −

' ' '

' ' '

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) (2.12-b)

b ba a a bb b b bc c c

b ba b a ba b a bb b b bc b c bc b c

I y V V y V V y V VI y V V y V V y V V y V V y V V= − + − + −

⇒ = − + − − − + − − −

' ' '

' ' '

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) (2.12-c)

c ca a a cb b b cc c c

c ca c a ca c a cb c b cb c b cc c c

I y V V y V V y V VI y V V y V V y V V y V V y V V= − + − + −

⇒ = − − − + − − − + −

The mutual coupling among phase conductors and the grounding effect of distribution line

segments are considered by (2-12-a, b and c) and the corresponding equivalent circuit is

illustrated by figure-2.By the same manner, the equivalent circuit models of two phase and

single phase line segments can be developed as shown in figure-3 and figure- 4 respectively.

15

These equivalent admittance models of the multi phase mutually coupled distribution feeders

become extremely helpful when the Y-matrix of an unbalanced three phase radial distribution

network is constructed.

Yaa

Yca Yca

Ycc

Fig. 3 The equivalent circuit of a two phase line section

-Yac -Yac

Va

Vc

Va’

Vc’

Yaa

Fig. 4 The equivalent circuit of a single phase line section Va Va’

16

2.2 LOAD MODELING Most of the electrical loads of a power system are connected to the low voltage distribution

systems. The electrical loads of a system comprise residential, commercial, industrial and

municipal loads. The loads on a distribution system are typically specified by the complex

power consumed. The specified load will be the maximum diversified demand. This demand

can be specified as kVA and power factor, kW and power factor, kW and power factor, or

kW and kVAR .Loads on a distribution feeders can be modeled as Y-connected or Δ-

connected. In case of Δ connection, it can be changed to Y- connection so that only Y-

connected case needs to be considered simplifying the formulation. Furthermore, the loads

can be three phase, two phase or single phase with any degree of unbalance. The active and

reactive load powers of a distribution system are not independent of system voltage and

frequency deviations. Also, the active and reactive power characteristics of various types of

load differ from each other. In static analysis, like load flow analysis, it is considered that the

frequency deviation is insignificant and thus only the effects of voltage deviation on the

active and reactive load powers is considered to get better and accurate results.

Thus, the loads connected in a typical unbalanced three phase distribution system can be

modeled as:

- Constant real and reactive power ( constant P-Q)

- Constant current

- Constant impedance

- Composite loads ( any combination of the above loads)

Thus, the loads can be modeled as a polynomial load:

20 0 1 2

20 0 1 2

0 1 2 0 1 2

( ) (2.13)( ) (2.14)

1 (2.15)

P P a a V a VQ Q b b V b Va a a b b b

= + +

= + ++ + = + + =

where, V is the p.u value of the node voltage magnitude ; P0, Q0 are the real and reactive power consumed at the specific node under the reference voltage.

17

a0, b0 are the parameters for constant power (constant P and Q) load component a1, b1 are the parameters for constant current (constant I) load component a2, b2 are the parameters for constant impedance (constant -Z) load component The value of a0, b0 ,a1, b1 ,a2, b2 are determined for different load types in distribution

systems. Usually experimental or experience values are used. All these loads can be

represented as an equivalent current injection into a bus, and the load at each bus is a linear

combination of the above three types.

For bus i , the corresponding load current injection ILi is computed as a function of the bus

voltage Vi .

*( ) / , i=1,2............nb (2.16)Li i i iI P Q V= − Where

Pi and Qi are constant active and reactive power loads at node i.

ILi is a complex vector.

Capacitors are often placed in distribution networks to regulate voltage levels and to reduce

real power loss. A shunt capacitor is modeled as a constant admittance matrix yck. The

corresponding current injection into bus k, Ick , is computed using the bus voltage Vk as

follows:

. (2.17)ck ck kI y V= where Ick is a complex (px1) vector, p is the no. of phases of bus ‘k’

2.3 LOSS FORMULA FOR UNBALANCED DISTRIBUTION NETWORKS As the feeders in a multiphase unbalanced radial network are mutually coupled, simple I2r

expression will not give the accurate active power loss in the corresponding feeder. Thus, a

special power loss formula is presented in this section which is used in the load flow

18

programs developed based on various methods. This formula takes into account the mutual

coupling between the feeders and can be derived as below.

Power fed into the phase-a of line (Fig. 1.) at the sending end bus is Va .( Ia)*.and at the

receiving end bus is Va’.( Ia)*. Therefore real and reactive power losses for phase-a in the

line may be written as:

* *'.( ) .( ) (2.18-a)a a a a a a aSL PL jQL V I V I= + = −

Similarly, for phase –b and phase-c

* *'

* *'

.( ) .( ) (2.18-b).( ) .( ) (2.18-c)

b b b b b a b

c c c a c a c

SL PL jQL V I V ISL PL jQL V I V I

= + = −

= + = −

The real part of the right hand side of the expression in (2.18-a, b and c) gives the real power

loss in the respective feeders of phase-a, phase –b and phase- c. Similarly, the imaginary part

of the right hand side expression presents the reactive power loss in the feeders. These loss

formulae (2.18-a,b and c) are used in the load flow programs to compute the active and

reactive power losses in the feeders of the unbalanced radial distribution networks.

19

CHAPTER 3

IMPLEMENTATION OF SOME IMPORTANT UDSLF METHODS Many unbalanced distribution system load flow (UDSLF) methods have been proposed from

time to time. These methods broadly fall into the two categories as mentioned previously. In

this chapter, five well established UDSLF methods have been chosen to be implemented in

the present work. Two variations of the first method are also proposed.

3.1 ZBUS GAUSS APPROACH [66-67] This method uses the sparse LU factored Y-bus matrix and equivalent current injections to

solve network equations. The convergence behavior of the ZBUS method is highly dependent

upon the number of voltage specified buses in the system. If the only voltage specified bus in

the system is the swing bus, the rate of convergence is very fast. The distribution system is

well suited for the Z-bus method; the only voltage specified bus in the system is the

substation bus and each co generator bus is handled as a P-Q specified bus.

3.1.1 Algorithm Development The method is based upon the principle of superposition applied to the system bus voltages:

the voltage of each bus is considered to arise from two different contributions, the specified

source voltage and equivalent current injection. The loads, co generators, capacitors and

reactors are modeled as current injection sources /sinks at their respective buses. The

superposition principle dictates that only one type of source will be considered at a time

when calculating the bus voltages. On the one hand, when swing bus voltage source is

activated, all current injections sources are disconnected from the system. On the other hand,

when all current injection sources are connected to the system, the swing bus is short

circuited to the ground. The component of each bus voltage obtained by activating only the

swing bus voltage source represents the no-load system voltage. This component can be

determined directly as equal to the swing bus voltage for every bus in the system, however,

the other component, affected by load currents, cannot be determined directly. Since load

20

currents are affected by bus voltages and vice versa, these quantities must be determined in

an iterative manner. Load flow programs based on this method are quite popular and many

utilities have been found using this approach.

3.1.2 Methodology A small unbalanced system (fig. 5) is considered to illustrate the method. The arrows at buses

represent the current injections

Considering bus-1 and bus-2, the following equation can be written

12 12 12 12 1 2

12 12 12 12 1 2

12 12 12 12 1 2

. (3.1)a aa ab ac a a

b ba bb bc b b

c ca cb cc c c

I Y Y Y V VI Y Y Y V VI Y Y Y V V

−

−

−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Similarly, for bus -2 and bus-3

23 23 23 2 3

23 23 23 2 3. (3.2)

b bb bc b b

c cb cc c c

I Y Y V VI Y Y V V

−

−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

For bus-3 and bus- 4 [ ] [ ] [ ]34 34 3 4. (3.3)c cc c cI Y V V−= For bus-2 and bus- 5 [ ] [ ] [ ]25 25 2 5. (3.4)a aa a aI Y V V= −

21

Combining 4.1, 4.2, 4.3 and 4.4

12 1 212 12 12

12 1 212 12 12

12 112 12 12

23 23 23

23 23 23

34 34

25 25

0 0 0 00 0 0 00 0 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0 00 0 0 0 0 0

a a aaa ab ac

b b bba bb bc

c cca cb cc

b bb bc

c cb cc

c cc

a aa

I V VY Y YI V VY Y YI VY Y YI Y YI Y YI YI Y

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ −⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

2

2 3

2 3

3 4

2 5

(3.5)c

b b

c c

c c

a a

VV VV VV VV V

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥

−⎢ ⎥⎢ ⎥−⎣ ⎦

The Y-matrix of (4.5) is a matrix; whose building algorithm is very simple to construct.

The loads connected at various buses can be modeled as equivalent current injections.

2 12 25

12 1 2 12 1 2 12 1 2 25 2 5

Current injected at the bus 2a ( ) ( ) ( ) ( )

CIa a a

aa a a ab b b ac c c aa a a

I I IY V V Y V V Y V V Y V V

= = −= − + − + − − −

(3.6)

2 12 23

12 1 2 12 1 2 12 1 2 23 2 3 23 2 3

Current injected at the bus 2b ( ) ( ) ( ) ( ) ( )

CIb b b

ab a a bb b b bc c c bb b b bc c c

I I IY V V Y V V Y V V Y V V Y V V

= = −= − + − + − − − − −

(3.7)

2 12 23

12 1 2 12 1 2 12 1 2 23 2 3 23 2 3

Current injected at the bus 2c ( ) ( ) ( ) ( ) ( )

CIc c c

ac a a bc b b cc c c bc b b cc c c

I I IY V V Y V V Y V V Y V V Y V V

= = −= − + − + − − − − −

(3.8)

3 23

23 2 3 23 2 3

Current injected at the bus 3b ( ) ( )

CIb b

ab a a bb b b

I IY V V Y V V

= == − + −

(3.9)

3 23 34

23 2 3 23 2 3 23 2 3 34 3 4

Current injected at the bus 3c ( ) ( ) ( ) - ( )

CIc c c

ac a a bc b b cc c c cc c c

I I IY V V Y V V Y V V Y V V

= = −= − + − + − −

(3.10)

4 34

34 3 4

Current injected at the bus 4c ( )

CIc c

cc c c

I IY V V

= == −

(3.11)

5 25

25 2 5

Current injected at the bus 5a ( )

CIa a

aa a a

I IY V V

= == −

(3.12)

22

Arranging (4.6- 4.12) in matrix form we get

2 12 12 12 25

2 12 12 12 23 23

12 12 12 23 232

23 233

23 23 343

344

255

0 0 00 00 0

0 0 0 0 00 0 0 00 0 0 0 0 00 0 0 0 0 0

CIa aa ab ac aa

CIb ab bb bc bb bc

CIac bc cc bc ccc

CIbb bcb

CI bc cc ccc

CI ccc

CI aaa

I Y Y Y YI Y Y Y Y YI Y Y Y Y YI Y Y

Y Y YIYI

YI

⎡ ⎤ −⎡⎢ ⎥⎢ ⎥ − −⎢ ⎥

− −⎢ ⎥⎢ ⎥ =⎢ ⎥

−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1 2

1 2

1 2

2 3

2 3

3 4

2 5

a a

b b

c c

b b

c c

c c

a a

V VV VV VV VV VV VV V

−⎡ ⎤⎤⎢ ⎥⎢ ⎥ −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −⎢ ⎥⎢ ⎥ −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎣ ⎦

(3.13)

Equation (4.6) can be rearranged in the following form:

12 1 2 12 1 2 12 1 2 25 1 2 25 1 52

12 25 1 2 12 1 2 12 1 2 25 1 52

( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

CIaa a a ab b b ac c c aa a a aa a aa

CIaa aa a a ab b b ac c c aa a aa

I Y V V Y V V Y V V Y V V Y V V

I Y Y V V Y V V Y V V Y V V

= − + − + − + − − −

⇒ = + − + − + − − −

(3.14) Similarly, rearranging the equations (3.7- 3.12)

12 1 2 12 23 1 2 12 23 1 22

23 1 3 23 1 3

( ) ( )( ) ( )( ) ( ) ( )

CIab a a bb bb b b bc bc c cb

bb b b bc c c

I Y V V Y Y V V Y Y V VY V V Y V V

= − + + − + + −− − − −

(3.15)

12 1 2 12 23 1 2 12 23 1 22

23 1 3 23 1 3

( ) ( )( ) ( )( ) ( ) ( )

CIac a a bc cc b b cc cc c cc

bc b b cc c c

I Y V V Y Y V V Y Y V VY V V Y V V

= − + + − + + −− − − −

(3.16)

23 1 2 23 1 2 23 1 3 23 1 33 ( ) ( ) ( ) ( )CI

bb b b bc c c bb b b bc c cbI Y V V Y V V Y V V Y V V= − − − − + − + − (3.17)

23 1 2 23 1 2 23 1 3 23 34 1 3 34 1 43 ( ) ( ) ( ) ( )( ) ( )CIbc b b bc c c bc b b cc cc c c cc c ccI Y V V Y V V Y V V Y Y V V Y V V= − − − − + − + + − − −

(3.18)

34 1 3 34 1 44 ( ) ( )CIcc c c cc c ccI Y V V Y V V= − − + − (3.19)

25 1 2 25 1 55 ( ) ( ) CIaa a a aa a aaI Y V V Y V V= − − + − (3.20)

23

Arranging (4.14- 4.20) in matrix form we get

2 12 25 12 12 25

2 12 12 23 12 23 23 23

12 12 23 12 23 23 232

23 23 23 233

3

4

5

0 0 00 00 0

0 0 00

CIa aa aa ab ac aa

CIb ab bb bb bc bc bb bc

CIac bc bc cc cc bc ccc

CIbb bc bb bcb

CIc

CIc

CIa

I Y Y Y Y YI Y Y Y Y Y Y YI Y Y Y Y Y Y YI Y Y Y Y

YI

I

I

⎡ ⎤ + −⎢ ⎥⎢ ⎥ + + − −⎢ ⎥

+ + − −⎢ ⎥⎢ ⎥ = − −⎢ ⎥

−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1 2

1 2

1 2

1 3

1 323 23 23 23 34 34

1 434 34

1 525 25

00 0 0 0 0

0 0 0 0 0

a a

b b

c c

b b

c cbc cc bc cc cc cc

c ccc cc

a aaa aa

V VV VV VV VV VY Y Y Y YV VY YV VY Y

−⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −⎢ ⎥⎢ ⎥ −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −− + −⎢ ⎥⎢ ⎥

−− ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −−⎣ ⎦ ⎣ ⎦

(3.21)

3.1.3 Algorithm-1(Zbus Gauss approach) [66-67] STEP-1 Input the data about the distribution system. STEP-2 Estimate the voltage magnitude at all nodes to be 1p.u and voltage angles to be 00,-

1200 and 1200 for phase-a , b and c respectively. Construct the Y-bus matrix of the system as per (3.21).

STEP-3 Factorize the Y-bus matrix. STEP-4 Set k=1 STEP-5 Compute bus injected current ICI

k for loads , co generators, transformers, shunt elements.

STEP-6 Compute voltage deviation due to the bus injected currents i.e ICI

k =LU[Y-bus] ΔVk STEP-7 Apply voltage superposition algorithm i.e add on no load swing bus voltage

Vk+1=VNL + ΔVk

STEP-8 Compare the present value of the voltage with that of the previous iteration, if converge, go to step-9, otherwise k=k+1 and go to step-5.

STEP-9 Calculate bus voltages, power flow, current flow and system loss.

3.1.4 Some Variations of the previous method (Proposed) The branch currents can be expressed as functions of injected bus currents e.g for the

network in figure- 5 , starting from the end buses, it can be written

24

25 5

CIa aI I= (3.22)

34 4

CIc cI I= (3.23)

23 2

CIb bI I= (3.24)

23 3 4

CI CIc c cI I I= + (3.25)

12 2 5

CI CIa a aI I I= + (3.26)

12 2 3

CI CIb b bI I I= + (3.27)

12 2 3

CI CIc c cI I I= + (3.28)

It can be observed that for end buses the current in the branch connected to the bus upstream

is equal to the current injected at that bus, whereas, for all other types of buses the branch

current is equal to the sum of all the current injections at the buses which are connected to the

bus concerned including the current injection at the bus. Now, using the matrix equation (3.5)

and (3.22-3.28), another method can be developed the algorithm of which is given in the next

section.

3.1.5 Algorithm – 2 (Proposed) STEP-1 Input the data about the distribution system. STEP-2 Estimate the voltage magnitude at all nodes to be 1 p.u and voltage angle to be 00 , -

1200 and 1200 for phase-a , b and c respectively. Construct the Y-bus matrix of the system as per matrix equation (3.5).

STEP-3 Factorize the Y-bus matrix. STEP-4 Set k=1. STEP-5 Compute bus injected current ICI

k for loads , co generators, transformers, shunt elements

STEP-6 Compute the branch currents starting from end nodes using the method explained

before.

25

STEP-7 Compute voltage drop due to the branch currents i.e I br-k = LU [Y-bus]. ∆Vk, where ∆Vk is the branch voltage drop.

STEP-8 Use the voltage dropand the previous bus voltage to find the updated

busvoltageVk+1 STEP-9 Compare the present value of the voltage with that of the previous iteration, if

converge, go to step-10, otherwise k=k+1 and go to step-5. STEP-10 Calculate bus voltages, power flow, current flow and system loss. Similarly, using the matrix equation (3.13) and (3.22-3.28), another method can be developed

the algorithm of which is given in the next section.

3.1.6 Algorithm – 3 (Proposed) STEP-1 Input the data about the distribution system. STEP-2 Estimate the voltage magnitude at all nodes to be 1 p.u and voltage angle to be 00,

-1200 and 1200 for phase-a , b and c respectively. Construct the Y-bus matrix of the system as per matrix equation (3.13).

STEP-3 Factorize the Y-bus matrix. STEP-4 Set k=1. STEP-5 Compute bus injected current ICI

k for loads , co generators, transformers, shunt elements.

STEP-6 Compute voltage drop due to the bus injected currents i.e Ik

CI =LU[ Y-bus ] ΔVk ,

ΔVk is the branch voltage drop. STEP-7 Use the voltage drop and the previous bus voltage to find the updated bus voltage STEP-8 Compare the present value of the voltage with that of the previous iteration, if

converge, go to step-10, otherwise k=k+1 and go to step-5. STEP-9 Branch currents are calculated from the injected currents at buses using the method

as explained before. STEP-10 Calculate power flow and system loss.

26

3.2 TOPOLOGICAL DIRECT APPROACH [ 69-70 ] The implicit Z-bus method or the other variations require the factorization of the full Y-bus

matrix, adversely affecting the performance in terms of speed. The arithmetic operation

number of LU factorization is approximately proportional to N3. For a large value of n, the

LU factorization will occupy a large portion of computational time. Therefore, if the LU

factorization can be avoided, the load flow method can save tremendous computational

resource. J.H.Teng proposed a method [69-70], which does not require such factorization.

The topological approach has been used in this method to tackle the problem of load flow.

Two matrices are developed, viz. the bus injection to branch current (BIBC) matrix and

branch current to bus voltage (BCBV) matrix. By using simple matrix multiplication of these

two matrices, the load flow solution is obtained.

3.2.1 Algorithm Development A simple distribution system shown in figure-6 is used as an example.I2, I3,I4,I5and I6 are the

respective current injections at the buses and B1,B2,B3,B4and B5 are the respective branch

currents.

Writing branch currents in terms of injected currents, 1 2 3 4 5 6 (3.29)B I I I I I= + + + + 2 3 4 5 6 (3.30)B I I I I= + + + 3 4 5 (3.31)B I I= +

27

4 5 (3.32)B I= Therefore, the relationship between the bus current injections and branch currents can be

expressed as:

2132

3 454

5 6

1 1 1 1 10 1 1 1 1

= . (3.33)0 0 1 1 00 0 0 1 00 0 0 0 1

IBIB

B IIBIB

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

The above matrix equation can be expressed as:

[ ] [ ] [ ]. (3.34) Where, BIBC is the bus injection to branch curent matrix The constsnt

B BIBC I=

BIBC matrix is an upper triangular matrix and contains values 0 and +1 only

The relationship between branch currents and bus voltages can be written as: 2 1 1 12 . (3.35)V V B Z= − 3 2 2 23 . (3.36)V V B Z= − 4 3 3 34 . (3.37)V V B Z= − Substituting the values of V2 & V3 in V4 4 1 1 12 2 23 3 34 . . . (3.38)V V B Z B Z B Z= − − − It can be seen that the bus voltage can be expressed as a function of branch currents, line

parameters, and the substation voltage. Similar procedures can be performed on other buses;

therefore, the relationship between branch currents and bus voltages can be expressed as:

28

1 2 12 1

1 3 12 23 2

1 4 12 23 34 3

1 5 12 23 34 45 4

1 6 12 23 36 5

0 0 0 00 0 0

. (3.39)0 00

0 0

V V Z BV V Z Z BV V Z Z Z BV V Z Z Z Z BV V Z Z Z B

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=−⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

The above matrix equation can be written in a general form as:

[ ] [ ] [ ]. (3.40) where, BCBV is the branch current to bus voltage matrix

V BCBV BΔ =

The BIBC matrix relates the branch currents to the bus current injections whereas the BCBV

matrix relates the bus voltage deviations to the branch currents of a radial distribution system

3.2.2 Algorithm Development BIBC and BCBV matrix building algorithm Observing (3.33), a building algorithm for BIBC matrix can be developed as follows: STEP 1 For a distribution system with m-branch section and n –bus, the dimension, the

dimension of the BIBC matrix is m x (n-1). STEP 2 If a line section (Bk) is located between bus i and bus j, copy the column of the ith

bus of the BIBC matrix to the column of the jth bus and fill a +1 to the position of the kth row and jth bus column.

STEP 3 Repeat procedure (2) until all line sections are included in the BIBC matrix. Similarly, observing (3.39) a building algorithm for BCBV matrix can be developed as

follows:

STEP 4 For a distribution system with m-branch section and n –bus , the dimension of the

BCBV matrix is (n-1) x m . STEP 5 If a line section (Bk) is located between bus i and bus j , copy the row of the ith bus

of the BCBV matrix to the row of the jth bus and fill the line impedance (Zij) to the position of the jth bus row and the kth column.

The algorithm can easily be expanded to a multiphase line section or bus. For example, if the

line section between bus i and bus j is a three phase line section, the corresponding branch

29

current Bi will be a 3 x 1 vector and the +1 in the BIBC matrix will be a 3 x 3 matrix.

Similarly, if the line section between bus i and bus j is a three line section, the Zij in the

BCBV matrix is a 3 x 3 impedance matrix as shown in (2.7).

3.2.3 Solution technique development The BIBC and BCBV matrices are developed based on the topological structure of

distribution systems. The BIBC matrix represents the relation ship between bus current

injections and branch currents. The corresponding variations at branch currents, generated by

the variations at bus current injections, can be calculated directly by the BIBC matrix. The

BCBV matrix represents the relation ship between branch currents and bus voltages. The

corresponding variations at bus voltages, generated by the variations at branch currents can

be calculated directly by the BCBV matrix. Combining (3.7) and (3.13) the relationship

between bus current injections and bus voltages can be expressed as

[ ] [ ] [ ] [ ]

[ ] [ ]. . (3.41)

= .

V BCBV BIBC I

DLF I

Δ =

Thus, DLF matrix relates the directly the bus voltage deviations of a radial distribution

system to the corresponding bus current injections.

3.2.3 Methodology The distribution system of fig.5 is considered. Following the building algorithm given in

section (3.2.2) the BIBC and BCBV matrices for the system are developed as:

1 0 0 0 0 0 10 1 0 1 0 0 00 0 1 0 1 1 0

(3.42)0 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

30

The branch currents and the injected node currents are related as:

12a 2a

12a 2b

12c 2c

23b 3b

23c 3c

34c 4c

25a 5a

I 1 0 0 0 0 0 1 II 0 1 0 1 0 0 0 II 0 0 1 0 1 1 0 I

= . I 0 0 0 1 0 0 0 II 0 0 0 0 1 0 0 II 0 0 0 0 0 1 0 II 0 0 0 0 0 0 1 I

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(3.43)

Similarly, the BCBV matrix is given as:

12aa 12ab 12ac

12ab 12bb 12bc

12ac 12bc 12cc

12ab 12bb 12bc 23bb 23bc

12ac 12bc 12cc 23bc 23cc

12ac 12bc 12cc 23bc 23cc 34cc

12aa 12ab 12ac 25aa

Z Z Z 0 0 0 0Z Z Z 0 0 0 0Z Z Z 0 0 0 0

BCBV= Z Z Z Z Z 0 0Z Z Z Z Z 0 0Z Z Z Z Z Z 0Z Z Z 0 0 0 Z

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3.44)

The voltage deviations are related to the branch currents as:

1 2 12aa 12ab 12ac

1 2 12ab 12bb 12bc

1 2 12ac 12bc 12cc

1 3 12ab 12bb 12bc 23bb 23bc

1 3 12ac 12bc 12cc 23bc 23cc

1 4 12ac 12bc 12cc 2

1 5

Z Z Z 0 0 0 0Z Z Z 0 0 0 0Z Z Z 0 0 0 0

= Z Z Z Z Z 0 0Z Z Z Z Z 0 0Z Z Z Z

a a

b b

c c

b b

c c

c c

a a

V VV VV VV VV VV VV V

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥

−⎢ ⎥⎢ ⎥−⎣ ⎦

12a

12a

12c

23b

23c

3bc 23cc 34cc 34c

12aa 12ab 12ac 25aa 25a

III

. (3.45)II

Z Z 0 IZ Z Z 0 0 0 Z I

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

3.2.4 Algorithm – 4 [ 69-70 ] STEP 1 Input the data about the distribution system. STEP 2 Estimate the voltage magnitude at all nodes to be 1p.u and voltage

angles to be 00 , -1200 and 1200 for phase-a , b and c respectively. STEP 3 Construct BIBC and BCBV matrices .Also the DLF matrix.

31

STEP 4 Set k=1. STEP 5 Compute bus injected current ICI

k for loads, co generators, transformers, shunt elements.

STEP 6 Compute voltage deviations due to the bus injected currents [ΔVk+1] =

[DLF] [Ik CI].

STEP 7 Use the voltage drop and the previous bus voltage to find the updated

bus voltage as [Vk+1 ] = Vk- ΔVk . STEP 8 If Voltage values converge, go to Step-9, otherwise k=k+1 and go to

step-5. STEP 9 Branch currents are calculated from the injected currents at buses using

BIBC matrix. STEP 10 Calculate power flow and system loss. 3.3 LADDER NETWORK BASED METHOD [ 53-55 ] The Ladder circuit is one of the most common configurations found in circuit analysis. A

simple ladder circuit is shown in fig. 7. Note that the series impedances are not necessarily

equal nor are the shunt admittances. When this is the case, the circuit is called the non-

recurrent ladder. In general, the ladder circuits found in radial power distribution systems are

non-recurrent. The analysis of the ladder of fig.7 proceeds simply as below, by beginning at

the receiving-end with a known (assumed) receiving-end voltage (V4). From this it is

possible to solve for all the currents and voltages in the circuit, e.g.,

I1Z1

V2

V3

Z3 V4

Y2 Y4

Fig. 7 Currents and voltages in a simple ladder circuit

I4 I2

V1

V0

32

4 4 4.I Y V= (3.46) 3 4 I I= (3.47) 3 3 3.V Z I= (3.48) 2 3 4V V V= + (3.49) 2 2. 2I Y V= (3.50) 1 2 3I I I= + (3.51) 1 1 1.V Z I= (3.52) 0 1 2V V V= + (3.53)

The equations (3.46- 3.53) above are easily seen to have good properties for automatic

computation. If the loads are modeled as constant real and reactive power, rather than

constant admittance, solution becomes little more difficult. The system of fig-8 is identical to

that of fig.-7, except that the constant admittance loads have been replaced by loads with

constant complex power.

33

Writing equations for fig. 8

( )*44

4SI V= (3.54)

3 4I I= (3.55) 3 3 3 .V Z I= (3.56) 2 3 4V V V= + (3.57)

( ) *22 2

SI V= (3.58)

1 2 3I I I= + (3.59) 1 1 1.V Z I= (3.60) 0 1 2V V V= + (3.61) Note that these equations (3.54-3.61) are identical to the preceding chain of equations (3.46-

3.53) except where the load currents are calculated. When loads are modeled as constant

complex power, as is nearly always the case, in load flow analysis, the system equations

become nonlinear. Hence, the adjustment of solutions becomes an iterative process rather

than the simple linear technique. The method explained above forms the basis of the load

flow solution technique as applied to the radial distribution networks. The real distribution

networks are much more complex in nature with multiple laterals and sub-laterals as shown

below in fig. 9.

34

The general algorithm consists of two basic steps: Forward Sweep and Backward Sweep. The

Forward Sweep is mainly a voltage drop calculation from the sending end node(s) or the

substation node(s) to the far end of a feeder or a lateral, and the Backward Sweep is primarily

a current summation based on the voltage updates from the far end of the feeder to the

sending end. This method was proposed by W.H. Kersting and D.L. Mendive [53] in 1976,

and over years has been modified by various researchers to make this approach more robust .

3.3.1 Algorithm-5 [53-55 ] STEP 1 Input the data about the distribution system. Find the end nodes. STEP 2 Choose any end node to start the backward Sweep. Estimate the voltage

magnitude of the end nodes to be 1p.u and voltage angles to be 00 , -1200 and 1200 for phase-a , b and c respectively.

STEP 3 Compute the load current at that node and currents in the various phases of the

branch upstream using KCL. Using this branch current and compute the node voltage upstream.

STEP 4 Check the type of upstream node (i.e a junction node or not). In case of a junction

node, find the nearest end node down stream and follow step-2 to compute the voltage of the junction node .This is the most recent voltage at the junction. the other case (if the upstream node is not a junction node ) use step-3 to compute the branch currents and the upstream voltages. This process is repeated till the upstream node is the substation node.

STEP 5 Compare the calculated voltage at node-1 to the specified source voltage. If the

difference between them is more than the convergence tolerance then the forward sweep is started (step-6), otherwise step-8 is followed.

STEP 6 Starting from the node-1(substation), using the branch currents computed during

the backward sweep and down stream voltages at all the nodes are updated. The forward sweep is completed when voltages at all the nodes are updated.

STEP 7 Go to step-2 and start the backward sweep with updated end node voltages. STEP 8 At this point the voltages are known at all the nodes and the currents flowing in

all branches are known. The active and reactive power losses in each of the feeders are calculated using ( 2.18-a,b and c). And thus the total active and reactive power losses in each of the phases are calculated.

35

3.4 FORWARD AND BACKWARD SWEEP METHOD [ 57 ] : Though, in principle, this method is equivalent to the Ladder network method, but there are

differences in the steps of implementation. In the Ladder network method, the bus voltages

are calculated twice in the same iteration as compared to only once in this method. During

the backward sweep, voltage values of the nodes are held constant and information about

currents are transmitted backward along the feeder along the backward sweep. Moreover, the

convergence is checked in the ladder network method by comparison between the specified

and calculated voltage values of the swing bus, whereas the difference between the values of

bus voltages at the present and previous iterations is considered for convergence in the

forward and backward sweep method.

3.4.1 Algorithm -6 [ 57 ] STEP 1 Input the data about the distribution system. STEP 2 Estimate the voltage magnitude at all nodes to be 1p.u and voltage angles

to be 00 , -1200 and 1200 for phase-a , b and c respectively. STEP 3 Set k=1. STEP 4 Compute bus injected current ICI

k for loads, co generators, transformers, shunt elements.

STEP 5 A backward sweep is started from the last node to calculate branch

currents. If it is an leaf node i.e no further branches are connected down stream, the current in the branch connected the node in the upstream is equal to the current injected at that node, otherwise the current is equal to the sum of branch currents connected to the node down stream plus the injected current of the node. All the branch currents are thus found out.

STEP 6 A forward sweep is started to calculate the voltages at each node for all

phases starting from the child node of the substation node. The current in all the branches are held constant to the value obtained in the back ward sweep.

STEP 7 A convergence of the solution is checked for the difference in voltage

magnitudes of two successive iterations. STEP 8 If Voltage values converge, go to Step-9, otherwise k=k+1 and go to step-

4. STEP 9 Calculate power flow and system loss.

36

3.5 POWER SUMMATION BASED FBS METHOD [58] : This method is an interesting variation of the previous two methods. A node in a radial

system is connected to several other nodes. However, owing to the radial nature of the

system, it is clear that a node is connected to the substation through only one line that feeds

the node. All other lines connected to the node to other neighboring nodes draw power from

the node. Figure-10 shows phase-a of a 3-phase system where lines between nodes i and j

feed the node j and all the other lines connecting node-j draw power from node –j. Thus the

total power which flows in the i-j branch of phase a is given as:

,

m,n index of all

(3.62)

a a a a a aij ij k k mn mn

where k is the index of all nodes fed through the line between nodes iand j

P jQ P jQ PL jQL+ = + + +∑ ∑

lines connected between nodes m and n fed through the

line between nodes i and j

Fig. 10 Single phase line section with load connected at node-j between phase a and neutral

37

Following equations for the branch currents can be written for all the three phases.

( )*+ (3.63 )a a

aa

ij ijij

j

P jQI aV= −

( )*+ (3.63 )b b

bb

ij ijij

j

P jQI bV= −

( )*+ (3.63 )c c

cc

ij ijij

j

P jQI cV= −

However, to obtain better starting values in the very first iteration, equations (3.63-a, b and c)

can be approximated as below:

( )*+ (3.64 )

a aa

aij ij

iji

P jQI aV= −

( )*+ (3.64 )b b

bb

ij ijij

i

P jQI bV= −

( )*+ (3.64 )c c

cc

ij ijij

i

P jQI cV= −

3.5.1 Algorithm -7 [58] STEP 1 Input the data about the distribution system. STEP 2 Make initial branch active and reactive powers to be zero. STEP 3 Set k=1 STEP 4 Starting from the end nodes, calculate the total power drawn from the ith bus

using equation (3.62) for all phases. STEP 5 Set j=2. STEP 6 A forward sweep is started. If k=1(i.e. for 1st iteration ) then the branch

currents of all phases are calculated using (3.64-a,b, c ) otherwise are calculated using equation (3.63-a,b ,c ).

38

STEP 7 The voltages at bus –j are calculated. Also compute the losses in the branch using (2.18).

STEP 8 j=j+1 STEP 9 If j is not equal to the nb (the total number of buses in the system) then

step-6 is followed, otherwise, go to step-10. STEP 10 A convergence of the solution is checked for the difference in voltage

magnitudes of two successive iterations. If converge, then go to step-12, otherwise go to step-11.

STEP 11 k=k+1. STEP 12 Calculate power flow and system loss.

39

CHAPTER 4

PROPOSED SIMPLE AND FAST UDSLF METHOD The method proposed is basically a forward and backward sweep method. However, the

novel scheme of identification of buses and the multiphase data handling makes the

algorithm simple and fast for the unbalanced radial distribution system load flow analysis.

Test results show that the proposed method is a substantial improvement over the various

versions of the forward and backward sweep approach and other popular methods for solving

unbalanced distribution system load flow.

4.1 PROPOSED BUS IDENTIFICATION SCHEME For a multiphase unbalanced radial distribution network, the tree is represented as a single

line equivalent, where a line between two buses represents only the connectivity between the

buses irrespective of the type of phase of the feeders. For such a radial distribution network:

1 , where is the no. of branches in the tree of the RDN

nb is the no. of nodes in the RDN (4.1)br brn nb n= −

A vector of dimension double the number of branches of a radial distribution network,

namely, adb [2*nbr] is introduced. This vector would store the adjacent buses of each of the

buses of the radial distribution network. Two other vectors mf [ ] and mt[ ] are introduced,

which act as pointers to the adb[ ] vector. These vectors in turn govern the reservation

allocation of memory location for each node, where mf[i] and mt[i] hold the data of starting

memory location and end memory location of bus-i in the adb[ ] vector for i = 1,2………

nb. All the buses are numbered in the increasing order down stream with the substation node

numbered as 1. A branch preceding a bus will be numbered one less than the bus number.

The storing and pointer operation of the above vectors are explained in fig. 11.

40

Fig. 11 Storing and pointer operation of the proposed vectors The above mentioned bus identification scheme is explained with reference to a sample

unbalanced distribution system of fig.12. Table-1 and Table-2 shows data stored in mf[ ],

mt[ ] and adb[ ] vectors of the sample multiphase unbalanced distribution network .

Fig. 12 Sample 8-node multi- phase unbalanced distribution system

Table-1

Bus No. mf[i] mt[i]

1 1 1 2 2 5 3 6 8 4 9 9 5 10 11 6 12 12 7 13 13 8 14 14

41

Table-2

S. No. [ s ]

adb [s] Node No.

1 2 1 2 1 3 3 4 5 5 7

2

6 2 7 4 8 8

3

9 3 4 10 2 11 6

5

12 5 6 13 2 7 14 3 8

Advantages of the Scheme

• An end bus can be easily identified. For an end bus i, mt[i] - mf[i] = 0 (4.2) Of course, the only exception is the substation bus, which is numbered as 1.

• A junction bus can be identified as:

mt[i] - mf[i] > 1 (4.3) Similarly, an intermediate bus which is not a junction Bus can be identified as : mt[i] - mf[i] = 1 (4.4) • Given a bus –i , the previous bus n , can be computed as :

( [ ]; [ ]; ) ( [ ] ) [ ]

for k mf i k mt i kif adb k i

n adb k

= <= + +<

= (4.5)

• Applying the scheme, the backward sweep to calculate the branch currents from the

ECIs becomes very fast and effective.

42

• Application of this scheme reduces a lot of memory and CPU time as it minimizes the search process in identifying the adjacent buses and branches of all the buses of a radial distribution system.

4.2 MULTIPHASE DATA HANDLING A practical unbalanced distribution system contains feeders which are multiphase in nature.

The phase impedance matrices of three phase, two phase and single phase feeders are

represented as 3 x 3, 2 x 2 and 1 x 1 matrices respectively, as given in (2.8), (2.9) and (2.10)

respectively, the null entries of these matrices have been discarded. A bus phase is assigned

to a particular bus depending upon the phase type of the feeders preceding the bus. Thus, in

fig. 12, the bus-2 is assigned a bus phase 3. Similarly, bus-3 is assigned a bus phase 2 and

rest other buses are assigned bus phase of 1. The only exception is the substation bus-1 which

is always assigned a bus phase of 3. A vector bp [i] is introduced to store the bus phases of

all the buses of the radial distribution network. All the feeders are assigned a unique

configuration code depending upon the phase sequence and configuration of line conductors

as well as depending upon the type of phase and neutral conductors etc. A vector bt [i] is

introduced to store the codes assigned to the feeders which in turn get assigned to the buses

following the feeder branches. This is a further classification of buses in addition to the bus

phase classification using bp[i] vector. The two vectors have dimensions equal to the number

of branches (nbr) of the RDN and can be formed from the usual input data of radial

distribution network.Table-3 explains the above scheme with reference to the radial

distribution network of fig.12. The various configuration codes are listed in Table-4.

Table-3

Bus i

bp [i] bt[i]

2 3 1

3 2 7

4 1 5

5 1 5

6 1 5

7 1 3

8 1 4

43

Table-4 For sample distribution system (fig. 12) and Test System -1

Conf. ID

Phasing Phase Cond.

Neutral Cond.

Spacing ID

Type

1 ABCN 336,400 336,400 500 3-Φ 4wire 2 ABCN 1/0 1/0 500 3-Φ 4wire 3 AN 1/0 1/0 510 1-Φ 2wire 4 BN 1/0 1/0 510 1-Φ 2wire 5 CN 1/0 1/0 510 1-Φ 2wire 6 ABN 1/0 1/0 505 2-Φ 3wire 7 BCN 1/0 1/0 505 2-Φ 3wire 8 ACN 1/0 1/0 505 2-Φ 3wire

4.3 ALGORTHM OF THE PROPOSED METHOD STEP - 1 : Read input data regarding the radial distribution system. STEP - 2 : Using the data bp[i],bt[i], adb[i], mf[i] and mt[i] vectors are formed. STEP - 3 : Estimate the voltage magnitude at all the buses to be 1p.u and voltage angles to be

00, -1200 and 1200 for phase –a , phase-b and phase-c respectively. STEP - 4 : Set iteration count k=1. STEP - 5 : Equivalent Current Injections (ECIs) are calculated using (2.16) for all the buses

starting from the i =2 to i = nb. STEP - 6 : A backward sweep from the bus i = nb is started to calculate the branch currents.

(i) Check for mt[i] – mf [i] • For zero value i.e for an end bus, the ECI at the bus gives the branch

current preceding the bus down stream. • Else the branch current is computed by summing up the branch

currents connected to the bus preceding it while going upstream.

(ii) While carrying out the above step, the bp[i] and bt[i] values of the bus -i are also checked to ensure computations for only the concerned phases.

STEP -7 : Set i=2.

STEP-8 : A forward sweep is started to update bus voltage magnitudes and angles. • Using (13), the bus ‘n’ which precedes the bus i is computed.

bp[i] value is checked which is followed by the checking of bt[i] value.

44

• The value of bp[i] refers to the phase of the feeders connected to the ith bus upstream and bt[i] refers to the configuration code of the feeders, enabling to decide the equation to be considered for the computation of bus voltage magnitude and angle.

STEP-9: i = i +1.

STEP-10: Check for i = nb , if yes step-11 is followed. Else step-8 is followed

STEP 11: Convergence is checked for each of the bus voltage magnitude comparing with the values of previous iteration. On convergence, step-13 is followed; else step-12 is followed.

STEP-12: k = k+1, and goto step-5 STEP-13: Power flow and System loss is calculated.

4.4 TEST SYSTEMS For all the methods explained in the previous sections, it is important to make sure the final

solution of all these methods should match with each other and with the standard result, if

available. Keeping this in view, three test systems have been chosen.

4.4.1 Test System-1 Test system-1 is an eight bus system (equivalent 13 node system), which includes the three

phase, double phase, and single phase line sections and buses as shown in fig. 12. The other

details are shown in Table-4, Table-5 , Table-6 and Table-7 . Power Factor is assumed to be

0.8. Base kV and base kVA are chosen to be: 4.16 kV and 300 kVA respectively. For details

about the spacing ID., refer to the Appendix-A. The three phase loads are all balanced.

However, the system is having unbalanced loading with respect to the substation (Table-5) .

The configuration code of the feeders is a unique number assigned to describe the spacing

model.

Table-5

Active Load Reactive Load

Phase-a 120 (kW) 90 (kVAR) Phase-a 100 (kW) 75 (kVAR) Phase-a 140 (kW) 105 (kVAR)

Total Load 360 (kW) 270 (kVAR)

45

Table-6 Sending

end Node

Sending end

node phase

Receiving end

node

Receiving end node Phase

Branch no.

Conf. ID.

Branch Length(ft.)

1 3 2 3 1 1 1000 2 3 3 2 2 7 400 3 2 4 1 3 5 500 2 3 5 1 4 5 400 5 1 6 1 5 5 400 2 3 7 1 6 3 500 3 2 8 1 7 4 500

Table-7

Conductor Type Resistance( Ω / mile )

Diameter (inch)

GMR (feet)

4/0 ACSR 0.592 0.563 0.00814 1/0 ACSR 1.12 0.398 0.00446 #2 ACSR 1.69 0.316 0.00418 #4 ACSR 2.55 0.257 0.00452

4.4.2 Test System-2 [54] The test system -2 is a 24- bus practical distribution system [54] as shown in figure-13. In the

figure, the numbers within the square brackets denote the feeder configuration ID, the details

of these configuration ID. are given in Table -2.The overhead conductors are the same as in

Test System-1 given in Table-3. The other details are shown in Table-4 . Power Factor is

assumed to be 0.8. Base kV and base kVA are chosen to be: 4.16 kV and 1000 kVA

respectively. For details about the spacing ID., refer to the Appendix A. The three phase

loads are all balanced. The load type at a bus can be identified from the configuration ID.of

the feeder connected to the bus upstream. There are no two phase loads connected to any of

the buses of the test system.

46

Table-8

Active Load Reactive Load

Phase-a 513.33 (kW) 385 (kVAR) Phase-a 473.33 (kW) 355 (kVAR) Phase-a 493.33(kW) 325 (kVAR)

Total Load 1480.00(kW) 1065 (kVAR)

Fig. 13 25-node practical multi- phase unbalanced distribution system (Test System -2 )

47

Table- 9 Sending

end node

Sending end

node phase

Receiving end

node

Receiving end

node phase

Branch no.

Conf. ID.

Branch Length(ft.)

1 3 2 3 1 1 1000 2 3 3 3 2 1 500 3 3 4 3 3 1 500 4 3 5 3 4 2 500 2 3 6 3 5 2 500 6 3 7 3 6 2 500 6 3 8 3 7 2 1000 7 3 9 3 8 2 500 9 3 10 3 9 2 500

10 3 11 2 10 6 300 11 2 12 1 11 3 200 11 2 13 1 12 4 200 7 3 14 3 13 2 500

14 3 15 3 14 2 300 14 3 16 1 15 5 300 7 3 17 3 16 2 500 3 3 18 3 17 2 500

18 3 19 3 18 2 500 19 2 20 1 19 4 400 18 3 21 1 20 5 400 21 1 22 1 21 5 400 4 3 23 3 22 2 400

23 3 24 2 23 7 400 24 2 25 1 24 4 400

4.4.3 Test System -3 [24, 96,97 ]

Fig. 14 34-node IEEE Test System (Test System -3)

48

Test system-3 is an IEEE 34-Node system ( fig.14). The base kV and the base kVA for the

system are chosen to be 24.9 kV and 300 kVA respectively. The details are as given in

Table-10, 11, 12 and 13.

Table-10

Active Load Reactive Load

Phase-a 3074.33 (kW) 2353.50 (kVAR) Phase-b 1781.00 (kW) 1371.00 (kVAR) Phase-c 1496.00(kW) 1157.25 (kVAR)

Total Load 6351.00 (kW) 4881.75 (kVAR) Table- 11

Sending End Node

Sending End

Node phase

Receiving End Node

Receiving End

Node Phase

Branch no. Branch Code

Branch Length(ft.)

1 3 2 3 1 1 2585 2 3 3 3 2 1 1730 3 3 4 3 3 1 5984 4 3 5 1 4 4 5841 4 3 6 3 5 1 1411 6 3 7 3 6 1 5256 7 3 8 3 7 1 1800 8 3 9 3 8 1 310 9 3 10 1 9 6 1710

10 1 11 1 10 6 15345 11 1 12 1 11 6 10464 9 3 13 3 12 1 3650

13 3 14 2 13 7 3031 13 3 15 3 14 1 840 15 3 16 3 15 1 6563 16 3 17 3 16 1 3802 17 3 18 1 17 4 6927 17 3 19 3 18 1 741 19 3 20 3 19 1 3781 20 3 21 1 20 5 3832 21 1 22 1 21 5 10562 20 3 23 3 22 1 1619 23 3 24 1 23 3 1620 23 3 25 3 24 1 5831

49

25 3 26 3 25 2 280 26 3 27 3 26 2 4632 27 3 28 3 27 2 3641 28 3 29 3 28 2 3812 25 3 30 3 29 1 544 30 3 31 3 30 1 3281 31 3 32 1 31 3 860 31 3 33 2 32 7 280 33 2 34 2 33 7 4860

Table-12

Conf. ID

Phasing Phase and Neutral Conductor ACSR

Spacing I Type

1 ABCN 4/0 500 3-Φ 4wire 2 ABCN 1/0 500 3-Φ 4wire 3 AN #4 510 1-Φ 2wire 4 BN #4 510 1-Φ 2wire 5 CN #2 510 1-Φ 2wire 6 AN #2 510 1-Φ 2wire 7 BCN #4 505 2-Φ 3wire

Table -13

Conductor Type Resistance ( Ω / mile )

Diameter (inch)

GMR (feet)

4/0 ACSR 0.592 0.563 0.00814

1/0 ACSR 1.12 0.398 0.00446

#2 ACSR 1.69 0.316 0.00418

#4 ACSR 2.55 0.257 0.00452

50

CHAPTER 5

LOSS ALLOCATION IN UNBALANCED DISTRIBUTION SYSTEM Different techniques have been published in the literature for allocation of losses, most of

them dedicated to transmission networks. Though some of them can be applied to the radial

distribution system, but are not fully suitable for such networks. Moreover, no method really

considers the unbalanced nature of the distribution networks. In the light of such

developments, a novel loss allocation scheme is proposed in this chapter for three phase

unbalanced radial networks. A loss formula for a balanced radial distribution system is first

developed and the loss allocation scheme is then proposed based on the formula. The scheme

is then extended to allocate loss for an unbalanced radial distribution system. Some principles

for effective loss allocation are recalled here. A loss allocation technique should be:

• Easy to understand and based on real data of the network.

• Carefully designed to avoid discrimination between users.

• Able to recover the total amount of losses.

• Consistent with the rules of competitive electricity markets.

• Economically efficient, avoiding cross-subsidization between users.

• Able to send out economic signals aimed at increasing the efficiency of loads in the network.

• Able to provide correct signals concerning the size and location of the in the network.

• Applicable to different situations, e.g., following the time evolution of the generation and load

patterns.

5.1 METHODOLOGY

For the purpose of explanation of the proposed method, consider a sample distribution

network as shown in Fig.15. In Fig.15, branch numbers are shown in (.). The branch number,

sending end and receiving end nodes are given in Table 14.

51

The load current at any node p, is given as:

*( ) , 2,3,...,−

= =p p

p

PL jQLIL p p NB

V (5.1)

Consider branch-5 of Fig. 15, number of node beyond branch-5 is one and this is node 6.

Therefore, current through branch-5 is:

(5) (6)=I IL (5.2)

Table-14

Branch No. (jj)

Sending end node IS(jj)

Receiving end node IR(jj)

Total no. of nodes beyond branch-jj N(jj)

Nodes beyond branch-jj ie(jj,i) i=1,2….N(jj)

1 1 2 11 2,3,4,5,6,7,8,9,10,11,12 2 2 3 6 3,4,5,6,10,11 3 3 4 5 4,5,6,10,11 4 4 5 2 5,6 5 5 6 1 6 6 2 7 4 7,8,9,12 7 7 8 3 8,9,12 8 8 9 1 9 9 4 10 2 10,11 10 10 11 1 11 11 8 12 1 12

Now consider branch-4. The total number of nodes beyond branch-4 is two and these nodes

are 5 and 6, respectively. Therefore, current through branch-4 is:

52

(4) (5) (6)= +I IL IL (5.3)

Similarly consider branch-3. Total number of nodes beyond branch-3 is five and these nodes

are 4, 5, 6, 10 and 11. Therefore, current through branch-3 is:

(3) (4) (5) (6) (10) (11)= + + + +I IL IL IL IL IL (5.4)

From Eqns. (5.2) – (5.4), it is clear that, if we identify the nodes beyond all the branches and

if the load currents are known, then it is extremely easy to compute the branch currents. In

the previous project work an algorithm for identifying the nodes beyond all the branches has

been proposed. A load flow algorithm based on the identification of these nodes beyond all

the branches has also been proposed. In the present work, this load flow algorithm is used to

compute the load currents and branch currents.

General expression of branch current through branch-j is given by:

( )

1( ) ( , )

N jj

kI jj IL ie jj k

=

= ∑ (5.5)

In Table-14, total number of nodes (consumers ) beyond branch-jj ( N(jj) ) and nodes

(consumers ) beyond branch jj ( ie(jj,k) ; k= 1, 2, …, N(jj)) are given for Fig. 14 for the

purpose of explanation. In (5.5), the load currents can be replaced by the following relation:

*

( ) ( )( )( )

PL i j QL iIL iV i−

= (5.6)

Thus, (5.5) modifies to

( )

*1

( , ) ( , )( ) ( , )

N jj

k

PL ie jj k jQL ie jj kI jjV ie jj k=

−= ∑ (5.7)

Real power loss of branch-jj with sending end and receiving end voltages Vi and Vj is given by: *( ) Real( ) . ( )i jPLOSS jj V V I jj= − (5.8)

53

( )*

*1

( , ) ( , )( ) Re ( ) ( ) ( , )

N jji j

k

PL ie jj k j QL ie jj kPLOSS jj al V VV ie jj k=

−⇒ = − ∑ (5.9)

( )*( )

1( ) Re ( , ) ( , )

( , )

N jji j

k

V VPLOSS jj al PL ie jj k j QL ie jj kV ie jj k=

⎧ ⎫⎛ ⎞−⎪ ⎪⇒ = −⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

∑ (5.10)

*

Let ( , ) ( , ) ( , )

i jV V A ie jj k j B ie jj kV ie jj k⎛ ⎞−

= +⎜ ⎟⎝ ⎠

(5.11)

( )

1( ) Re ( ( , ) B ( , )).( ( , ) ( , ))

N jj

kPLOSS jj al A ie jj k j ie jj k PL ie jj k jQL ie jj k

=

⎧ ⎫∴ = + −⎨ ⎬

⎩ ⎭∑ (5.12)

Hence,

( )

1( ) ( ( , ). ( , ) ( , ). ( , ))

N jj

iPLOSS jj A ie jj k PL ie jj k B ie jj k QL ie jj k

=

= +∑ (5.13)

5.2 PROPOSED LOSS ALLOCATION METHOD Using (5.13) real power loss in branch-jj can be allocated to consumers beyond branch-jj. Real power loss of branch-jj allocated to a consumer connected to node ie(jj,k) is given by:

, ( , ) ( , ). ( , ) ( , ). ( , ) 1, 2,........... 1

ploss jj ie jj k A ie jj k PL ie jj k B ie jj k QL ie jj kfor jj NB

= +

= − and k=1,2..............N(jj)

(5.14)

The global value of losses to be supported by consumer connected to node l results from the

sum of the losses allocated to it in each branch-jj of the network, which is given by

1

1

( ) ( , ) 2,3,...,NB

jj

Tploss ploss jj for NB−

=

= =∑l l l (5.15)

5.2.1 Extension to 3-Phase Unbalanced Distribution Network The loss formula in (5.13) derived for a balanced single phase radial distribution network can

be extended to write the loss formula for individual phases of a three phase unbalanced radial

distribution network. Thus, rewriting (5.13) for individual phases:

54

For phase-a, ( )

1

( ) ( ( , ). ( , ) ( , ). ( , ))a

a

N jj

a a a ai

PLOSS jj A ie jj k PL ie jj k B ie jj k QL ie jj k=

= +∑ (5.16)

For phase-b, ( )

1

( ) ( ( , ). ( , ) ( , ). ( , ))b

b

N jj

b b b bi

PLOSS jj A ie jj k PL ie jj k B ie jj k QL ie jj k=

= +∑ (5.17)

For phase-c, ( )

1

( ) ( ( , ). ( , ) ( , ). ( , ))c

c

N jj

c c c ci

PLOSS jj A ie jj k PL ie jj k B ie jj k QL ie jj k=

= +∑ (5.18)

Here, N(jj)a , N(jj)b , N(jj)c refers to the no. of subsequent nodes of the branch-jj of phase-a,

phase-b and phase-c respectively. Similarly, PL, QL, A and B terms refer to the

corresponding phases in (5.16-5.18). Based on the loss formulae (5.16-5.18) loss allocation

methods can also be extended to the three phase unbalanced distribution network:

, ( , ) ( , ). ( , ) ( , ). ( , ) 1, 2,........... 1

a a a a aploss jj ie jj k A ie jj k PL ie jj k B ie jj k QL ie jj kfor jj NB

= +

= −a and k=1,2..............N(jj)

(5.19)

, ( , ) ( , ). ( , ) ( , ). ( , )

1, 2,........... 1

b b b b bploss jj ie jj k A ie jj k PL ie jj k B ie jj k QL ie jj kfor jj NB

= +

= −b and k=1,2..............N(jj)

(5.20)

, ( , ) ( , ). ( , ) ( , ). ( , ) 1, 2,........... 1

c c c c cploss jj ie jj k A ie jj k PL ie jj k B ie jj k QL ie jj kfor jj NB

= +

= −c and k=1,2..............N(jj)

(5.21)

The total loss for each of the individual phases is:

1

1( ) ( , ) 2,3,...,

NB

a ajj

Tploss ploss jj for NB−

=

= =∑l l l (5.22)

1

1( ) ( , ) 2,3,...,

NB

b bjj

Tploss ploss jj for NB−

=

= =∑l l l (5.23)

55

1

1( ) ( , ) 2,3,...,

NB

c cjj

Tploss ploss jj for NB−

=

= =∑l l l (5.24)

5.3 IMPLEMENTATION AND FLOWCHART OF THE METHOD In order to allocate the losses to the consumers of a three phase unbalanced radial distribution

network, the load flow of the system for a particular load pattern has to be performed first.

The load flow solution gives the converged voltages at various nodes, which are used latter

on for allocating losses to the various customers. For load flow solution, the method

proposed in chapter-4 is used. The load flow method is based on forward backward sweep

approach. However, in order to implement the proposed loss allocation method some

modifications are made in the method. Four more arrays are introduced to store the data

regarding the subsequent buses to all the branches of the distribution network. The nature of

storage is similar to that of the earlier introduced vectors for load flow. In Fig.16, the storage

and pointer operation of the four arrays are clearly explained.

Fig. 16 Storing and pointer operation of the proposed vectors

56

The mfs [i] and mts [i] are the two pointer vectors which store the starting and the end

addresses of the subsequent buses to each of the buses in the vector sb [] of the radial

distribution system, whereas, sb [ ] and nsb [ ] vectors store the subsequent buses and number

of subsequent buses. These vectors, along with the other vectors introduced earlier are used

to implement the proposed loss allocation scheme in the unbalanced radial distribution

network. The complete algorithm of the proposed loss allocation scheme for three phase

unbalanced network has been presented in fig. 17.

Fig. 17 Flowchart of the proposed loss allocation method

57

CHAPTER 6

RESULT AND DISCUSSION 6.1 PERFORMANCE ANALYSIS OF THE THREE PHASE UDSLF METHODS The three phase load flow programs for all the 8 methods (Table-15) are implemented in C-

language on a Windows-XP based Pentium-4, 2.6 GHz., 256 MB RAM PC. The convergence

tolerance was 0.0001 p.u. The results for all the 3 test systems are given in the subsequent

sections. Table-15

Method Description

Method- 1 As per Algorthm-1 (Zbus Gauss approach) [65-66]

Method- 2 As per Algorthm-2 (Proposed)

Method -3 As per Algorthm-3 (Proposed) Method- 4 As per Algorithm- 4 (Topological Direct Approach) [67-68] Method- 5 As per Algorithm- 5 ( The Ladder network approach ) [53-54] Method- 6 As per Algorithm- 6 ( Forward and Backward Sweep Method ) [57]Method- 7 As per Algorithm- 7 ( Power Summation based FBS Method ) [58]Method - 8 Proposed method

6.1.1 Test System-1 Table-16 (Converged bus voltage magnitudes)

Bus No.

Method-1 |V| p.u

Method-2 |V| p.u

Method-3|V| p.u

Method-4|V| p.u

Method-5|V| p.u

Method-6|V| p.u

Method-7 |V| p.u

Method-8|V| p.u

2-a 0.9974 0.9974 0.9974 0.9974 0.9974 0.9974 0.9974 0.9974 2-b 0.9967 0.9967 0.9967 0.9967 0.9967 0.9967 0.9967 0.9967 2-c 0.9968 0.9968 0.9968 0.9968 0.9968 0.9968 0.9968 0.9968 3-b 0.9942 0.9944 0.9942 0.9942 0.9941 0.9942 0.9942 0.9942 3-c 0.9957 0.9957 0.9957 0.9957 0.9957 0.9957 0.9957 0.9957 4-c 0.9937 0.9937 0.9937 0.9937 0.9937 0.9937 0.9937 0.9937 5-c 0.9944 0.9944 0.9944 0.9944 0.9944 0.9944 0.9944 0.9944 6-c 0.9936 0.9936 0.9936 0.9936 0.9936 0.9936 0.9936 0.9936 7-a 0.9945 0.9945 0.9945 0.9945 0.9945 0.9945 0.9945 0.9945 8-b 0.9902 0.9905 0.9902 0.9902 0.9902 0.9902 0.9902 0.9902

58

Table-17

(Active and Reactive power losses)

Total Active Power Loss(kW) Total Reactive Power Loss(kVAR) Methods Phase-a Phase-b Phase-c Phase-a Phase-b Phase-c

Method-1 0.371039 0.798139 0.539481 0.398447 0.836775 0.793622 Method-2 0.371039 0.798144 0.539467 0.398447 0.836778 0.793634 Method-3 0.371039 0.787276 0.539504 0.398459 0.811493 0.793609 Method-4 0.371030 0.798120 0.539479 0.398438 0.836760 0.793590 Method-5 0.372951 0.805242 0.542278 0.400220 0.844488 0.797886 Method-6 0.371039 0.798144 0.539467 0.398447 0.836778 0.793634 Method-7 0.371470 0.800098 0.539916 0.398847 0.839489 0.794802 Method-8 0.371036 0.798094 0.539475 0.398445 0.836744 0.793590

The results in Table-16 and Table-17 basically confirm the successful implementation of the

programs developed for the 8 different methods as explained in the previous chapters. The

final converged bus voltages and the total active and reactive power losses as computed by

the 8 methods for the Test System-1 exactly match.

6.1.2 Test System-2 Similarly, in Table-18, Table-19, Table-20 and Table-21 the converged voltage magnitudes

and the total active and reactive power losses for the three phases are presented respectively

for Test System -2 and 3. In Table-22, fig. 18 and fig.19 a comparison has been made about

the no. of iterations taken and CPU execution time for all the 8 methods applied to all the 3

Test Systems.

Table-18 (Converged bus voltage magnitudes) Bus No.

Method-1 Method-2 Method-3 Method-4 Method-5 Method-6 Method-7 Method-8

2a 0.9846 0.9846 0.9835 0.9846 0.9843 0.9846 0.9844 0.9846 2b 0.9902 0.9902 0.9910 0.9902 0.9902 0.9902 0.9902 0.9902 2c 0.9869 0.9869 0.9875 0.9869 0.9867 0.9869 0.9867 0.9869 3a 0.9823 0.9823 0.9807 0.9823 0.9820 0.9823 0.9821 0.9823 3b 0.9872 0.9872 0.9885 0.9873 0.9872 0.9872 0.9872 0.9872 3c 0.9840 0.9840 0.9850 0.9840 0.9838 0.9840 0.9839 0.9840 4a 0.9806 0.9806 0.9785 0.9807 0.9803 0.9806 0.9805 0.9806 4b 0.9856 0.9856 0.9874 0.9856 0.9856 0.9856 0.9856 0.9856 4c 0.9834 0.9834 0.9847 0.9834 0.9832 0.9834 0.9833 0.9834

59

5a 0.9794 0.9794 0.9773 0.9794 0.9791 0.9794 0.9792 0.9794 5b 0.9845 0.9845 0.9863 0.9845 0.9845 0.9845 0.9845 0.9845 5c 0.9823 0.9823 0.9835 0.9823 0.9820 0.9823 0.9821 0.9823 6a 0.9738 0.9738 0.9727 0.9738 0.9732 0.9738 0.9735 0.9738 6b 0.9841 0.9841 0.9850 0.9842 0.9841 0.9842 0.9841 0.9842 6c 0.9787 0.9787 0.9794 0.9787 0.9783 0.9787 0.9785 0.9787 7a 0.9642 0.9642 0.9631 0.9642 0.9634 0.9642 0.9639 0.9642 7b 0.9792 0.9792 0.9801 0.9792 0.9792 0.9792 0.9792 0.9792 7c 0.9717 0.9717 0.9724 0.9718 0.9712 0.9717 0.9715 0.9717 8a 0.9713 0.9713 0.9703 0.9713 0.9707 0.9713 0.9710 0.9713 8b 0.9820 0.9820 0.9828 0.9820 0.9819 0.9820 0.9819 0.9820 8c 0.9764 0.9764 0.9771 0.9764 0.9760 0.9764 0.9762 0.9764 9a 0.9578 0.9578 0.9568 0.9579 0.9569 0.9578 0.9575 0.9578 9b 0.9780 0.9780 0.9788 0.9779 0.9779 0.9780 0.9779 0.9780 9c 0.9726 0.9726 0.9732 0.9726 0.9720 0.9726 0.9723 0.9726 10a 0.9538 0.9538 0.9528 0.9539 0.9528 0.9538 0.9535 0.9538 10b 0.9760 0.9760 0.9769 0.9760 0.9760 0.9760 0.9760 0.9760 10c 0.9734 0.9734 0.9740 0.9734 0.9729 0.9734 0.9731 0.9734 11a 0.9515 0.9515 0.9504 0.9515 0.9504 0.9515 0.9512 0.9515 11b 0.9756 0.9756 0.9765 0.9756 0.9756 0.9756 0.9756 0.9756 12a 0.9505 0.9505 0.9495 0.9506 0.9494 0.9505 0.9502 0.9505 13b 0.9750 0.9750 0.9759 0.9750 0.9749 0.9750 0.9750 0.9750 14a 0.9610 0.9610 0.9599 0.9610 0.9601 0.9610 0.9607 0.9610 14b 0.9756 0.9756 0.9764 0.9756 0.9755 0.9756 0.9755 0.9756 14c 0.9639 0.9639 0.9646 0.9639 0.9632 0.9639 0.9637 0.9639 15a 0.9585 0.9585 0.9574 0.9585 0.9576 0.9585 0.9582 0.9585 15b 0.9734 0.9734 0.9742 0.9734 0.9733 0.9734 0.9733 0.9734 15c 0.9616 0.9616 0.9623 0.9617 0.9609 0.9616 0.9614 0.9616 16c 0.9630 0.9630 0.9637 0.9630 0.9623 0.9630 0.9628 0.9630 17a 0.9642 0.9642 0.9631 0.9642 0.9634 0.9642 0.9639 0.9642 17b 0.9792 0.9792 0.9801 0.9792 0.9792 0.9792 0.9792 0.9792 17c 0.9717 0.9717 0.9724 0.9718 0.9712 0.9717 0.9715 0.9717 18a 0.9804 0.9804 0.9788 0.9804 0.9800 0.9804 0.9802 0.9804 18b 0.9843 0.9843 0.9856 0.9843 0.9842 0.9843 0.9842 0.9843 18c 0.9796 0.9796 0.9805 0.9796 0.9793 0.9796 0.9794 0.9796 19a 0.9788 0.9788 0.9772 0.9788 0.9784 0.9788 0.9786 0.9788 19b 0.9824 0.9824 0.9837 0.9824 0.9823 0.9824 0.9824 0.9824 19c 0.9802 0.9802 0.9811 0.9802 0.9800 0.9802 0.9800 0.9802 20b 0.9805 0.9805 0.9819 0.9805 0.9805 0.9805 0.9805 0.9805 21c 0.9765 0.9765 0.9774 0.9765 0.9761 0.9765 0.9763 0.9765 22c 0.9746 0.9746 0.9756 0.9746 0.9743 0.9746 0.9745 0.9746 23a 0.9790 0.9790 0.9765 0.9790 0.9787 0.9790 0.9789 0.9790 23b 0.9837 0.9837 0.9858 0.9837 0.9837 0.9837 0.9837 0.9837 23c 0.9828 0.9828 0.9849 0.9828 0.9826 0.9828 0.9827 0.9828 24b 0.9812 0.9812 0.9884 0.9812 0.9812 0.9812 0.9812 0.9812 24c 0.9821 0.9821 0.9852 0.9822 0.9819 0.9821 0.9820 0.9821 25b 0.9787 0.9787 0.9859 0.9787 0.9786 0.9787 0.9787 0.9787

60

Table-19 (Active and Reactive power losses)

Total Active Power Loss(kW) Total Reactive Power Loss(kVAR) Methods

Phase-a Phase-b Phase-c Phase-a Phase-b Phase-c Method-1 15.111234 8.083638 11.745946 17.148268 11.130221 15.256300 Method-2 14.949968 8.036380 11.585946 17.048268 11.060221 15.479300 Method-3 15.394802 7.694970 11.544555 17.294951 11.467065 12.882447 Method-4 15.394834 7.695039 11.544524 17.294982 11.467065 15.028660 Method-5 16.025152 7.811030 12.010990 18.083287 11.679746 15.500762 Method-6 15.394830 7.694970 11.544555 17.294979 11.467065 15.028660 Method-7 15.586686 7.724300 11.691839 17.585836 11.550025 15.202450 Method-8 15.394558 7.695130 11.544210 17.294535 11.467128 15.028670

6.1.3 Test System-3 Table-20

(Converged bus voltage magnitudes) Bus No.

Method-1 Method-2 Method-3 Method-4 Method-5 Method-6 Method-7 Method-8

2a 0.9931 0.9931 0.9928 0.9931 0.9931 0.9931 0.9929 0.9931 2b 0.9949 0.9949 0.9951 0.9949 0.9949 0.9949 0.9949 0.9949 2c 0.9938 0.9938 0.9942 0.9939 0.9938 0.9938 0.9937 0.9938 3a 0.9886 0.9886 0.9882 0.9886 0.9886 0.9886 0.9883 0.9886 3b 0.9917 0.9917 0.9920 0.9917 0.9917 0.9917 0.9915 0.9917 3c 0.9898 0.9898 0.9904 0.9898 0.9898 0.9898 0.9895 0.9898 4a 0.9733 0.9733 0.9724 0.9733 0.9734 0.9734 0.9728 0.9734 4b 0.9806 0.9806 0.9814 0.9806 0.9806 0.9806 0.9803 0.9806 4c 0.9762 0.9762 0.9776 0.9763 0.9762 0.9762 0.9755 0.9762 5b 0.9785 0.9785 0.9793 0.9785 0.9785 0.9785 0.9782 0.9785 6a 0.9699 0.9699 0.9688 0.9698 0.9699 0.9699 0.9692 0.9699 6b 0.9783 0.9783 0.9793 0.9783 0.9783 0.9783 0.9780 0.9783 6c 0.9731 0.9731 0.9746 0.9731 0.9730 0.9730 0.9723 0.9730 7a 0.9575 0.9575 0.9600 0.9575 0.9576 0.9576 0.9568 0.9576 7b 0.9703 0.9703 0.9717 0.9703 0.9703 0.9703 0.9700 0.9703 7c 0.9618 0.9618 0.9640 0.9618 0.9618 0.9618 0.9608 0.9618 8a 0.9534 0.9534 0.9517 0.9534 0.9536 0.9536 0.9526 0.9536 8b 0.9677 0.9677 0.9692 0.9677 0.9677 0.9677 0.9671 0.9677 8c 0.9581 0.9581 0.9605 0.9581 0.9580 0.9580 0.9570 0.9580 9a 0.9527 0.9527 0.9509 0.9527 0.9529 0.9529 0.9519 0.9529 9b 0.9672 0.9672 0.9687 0.9672 0.9672 0.9672 0.9667 0.9672 9c 0.9575 0.9575 0.9599 0.9575 0.9574 0.9574 0.9564 0.9574 10a 0.9498 0.9498 0.9480 0.9488 0.9505 0.9505 0.9490 0.9505 11a 0.9301 0.9301 0.9283 0.9291 0.9347 0.9347 0.9292 0.9347 12a 0.9211 0.9211 0.9194 0.9201 0.9275 0.9275 0.9203 0.9275 13a 0.9484 0.9484 0.9463 0.9484 0.9486 0.9486 0.9476 0.9486 13b 0.9612 0.9612 0.9630 0.9612 0.9612 0.9612 0.9606 0.9612 13c 0.9505 0.9505 0.9534 0.9505 0.9504 0.9504 0.9493 0.9504 14b 0.9602 0.9602 0.9668 0.9602 0.9602 0.9602 0.9596 0.9602 14c 0.9496 0.9496 0.9536 0.9497 0.9496 0.9496 0.9485 0.9496

61

15a 0.9475 0.9475 0.9453 0.9475 0.9476 0.9476 0.9466 0.9476 15b 0.9600 0.9600 0.9618 0.9600 0.9600 0.9600 0.9594 0.9600 15c 0.9490 0.9490 0.9519 0.9490 0.9489 0.9489 0.9478 0.9489 16a 0.9406 0.9406 0.9383 0.9406 0.9408 0.9408 0.9398 0.9408 16b 0.9510 0.9510 0.9530 0.9510 0.9510 0.9510 0.9503 0.9510 16c 0.9380 0.9380 0.9413 0.9380 0.9380 0.9380 0.9367 0.9380 17a 0.9370 0.9370 0.9345 0.9370 0.9372 0.9372 0.9362 0.9372 17b 0.9461 0.9461 0.9483 0.9461 0.9462 0.9462 0.9454 0.9462 17c 0.9320 0.9320 0.9355 0.9320 0.9320 0.9320 0.9307 0.9320 18b 0.9436 0.9436 0.9457 0.9436 0.9436 0.9436 0.9429 0.9436 19a 0.9364 0.9364 0.9339 0.9364 0.9366 0.9366 0.9355 0.9366 19b 0.9454 0.9454 0.9475 0.9453 0.9454 0.9454 0.9447 0.9454 19c 0.9309 0.9309 0.9344 0.9309 0.9308 0.9308 0.9295 0.9308 20a 0.9334 0.9334 0.9308 0.9334 0.9336 0.9336 0.9326 0.9336 20b 0.9416 0.9416 0.9439 0.9416 0.9416 0.9416 0.9409 0.9416 20c 0.9253 0.9253 0.9290 0.9253 0.9252 0.9252 0.9239 0.9252 21c 0.9218 0.9218 0.9255 0.9218 0.9217 0.9217 0.9204 0.9217 22c 0.9163 0.9163 0.9201 0.9164 0.9163 0.9163 0.9150 0.9163 23a 0.9321 0.9321 0.9294 0.9321 0.9323 0.9323 0.9313 0.9323 23b 0.9401 0.9401 0.9425 0.9401 0.9402 0.9402 0.9394 0.9402 23c 0.9238 0.9238 0.9276 0.9239 0.9238 0.9238 0.9225 0.9238 24a 0.9315 0.9315 0.9288 0.9315 0.9317 0.9317 0.9306 0.9317 25a 0.9284 0.9284 0.9255 0.9284 0.9285 0.9285 0.9275 0.9285 25b 0.9352 0.9352 0.9377 0.9352 0.9352 0.9352 0.9344 0.9352 25c 0.9191 0.9191 0.9232 0.9192 0.9191 0.9191 0.9178 0.9191 26a 0.9282 0.9282 0.9253 0.9282 0.9284 0.9284 0.9273 0.9284 26b 0.9349 0.9349 0.9375 0.9349 0.9350 0.9350 0.9342 0.9350 26c 0.9189 0.9189 0.9230 0.9189 0.9189 0.9189 0.9175 0.9189 27a 0.9255 0.9255 0.9225 0.9255 0.9256 0.9256 0.9246 0.9256 27b 0.9319 0.9319 0.9344 0.9319 0.9319 0.9319 0.9311 0.9319 27c 0.9161 0.9161 0.9202 0.9161 0.9160 0.9160 0.9147 0.9160 28a 0.9251 0.9251 0.9222 0.9251 0.9253 0.9253 0.9243 0.9253 28b 0.9311 0.9311 0.9336 0.9311 0.9311 0.9311 0.9304 0.9311 28c 0.9155 0.9155 0.9196 0.9155 0.9155 0.9155 0.9141 0.9155 29a 0.9251 0.9251 0.9221 0.9250 0.9252 0.9252 0.9242 0.9252 29b 0.9305 0.9305 0.9331 0.9306 0.9306 0.9306 0.9298 0.9306 29c 0.9152 0.9152 0.9193 0.9152 0.9152 0.9152 0.9138 0.9152 30a 0.9283 0.9283 0.9253 0.9283 0.9284 0.9284 0.9274 0.9284 30b 0.9350 0.9350 0.9376 0.9350 0.9350 0.9350 0.9343 0.9350 30c 0.9190 0.9190 0.9231 0.9190 0.9189 0.9189 0.9176 0.9189 31a 0.9280 0.9280 0.9249 0.9279 0.9281 0.9281 0.9271 0.9281 31b 0.9343 0.9343 0.9370 0.9343 0.9344 0.9344 0.9336 0.9344 31c 0.9183 0.9183 0.9226 0.9183 0.9182 0.9182 0.9169 0.9182 32a 0.9277 0.9277 0.9247 0.9277 0.9279 0.9279 0.9269 0.9279 33b 0.9342 0.9342 0.9432 0.9342 0.9342 0.9342 0.9335 0.9342 33c 0.9183 0.9183 0.9226 0.9183 0.9182 0.9182 0.9169 0.9182 34b 0.9331 0.9331 0.9468 0.9331 0.9332 0.9332 0.9324 0.9332 34c 0.9185 0.9185 0.9228 0.9185 0.9184 0.9184 0.9171 0.9184

62

Table-21 (Active and Reactive power losses)

Total Active Power Loss(kW) Total Reactive Power Loss(kVAR) Methods Phase-a Phase-b Phase-c Phase-a Phase-b Phase-c

Method-1 115.943846 79.444200 102.476016 142.961777 94.649264 135.228467Method-2 114.055699 84.115979 108.801359 136.638388 96.504539 151.608413Method-3 117.938465 76.444200 99.476016 141.961777 94.649264 129.228267Method-4 113.939202 84.101915 108.786088 138.245934 96.528831 151.585329Method-5 117.026123 84.195225 108.930167 126.159003 96.105784 150.089677Method-6 117.026854 84.195372 108.930114 126.189816 96.312690 151.764357Method-7 117.058110 86.390984 112.804878 140.827444 98.834521 157.082981Method-8 117.026854 84.195372 108.930114 126.189816 96.312690 151.764357

Table-22 (Comparision of the no.of iterations & CPU time for all the 8methods) Convergence tolerance .0001

IEEE -34 Bus system 25-Bus system 8-node system Methods No. of

Iterations CPU time (seconds)

No. of Iterations

CPU time (seconds)

No. of Iterations

CPU time (seconds)

Method-1 4 0.120 3 0.088 2 0.002 Method-2 4 0.094 3 0.047 2 0.002 Method-3 4 0.141 3 0.057 2 0.002 Method-4 5 0.063 4 0.016 2 0.002 Method-5 5 0.027 4 0.015 2 0.002 Method-6 3 0.016 3 0.015 2 0.002 Method-7 3 0.016 3 0.016 2 0.002 Method-8 3 0.012 2 0.011 2 0.002

Fig.18 Fig.19

63

6.2 IMPLEMENTATION OF THE PROPOSED LOSS ALLOCATION SCHEME

6.2.1 Test System-1 As already discussed, the proposed FBS method (method-8) has been used for load flow

analysis, where the composite load models have been implemented. In (2.13 – 2.15) , the

values of a0,a1,a2,b0,b1 and b2 have been considered as given below to represent the

composite nature of the load.

a0= b0= 0.4 a1= b1= 0.3 (6.1) a2= b2= 0.3 The value of P0 and Q0 are taken as the values of given constant P and Q values of the load.

If the constant P and Q models are to be considered, then a0= b0= 1, a1= b1= 0 and a2= b2= 0

are set. In Table-23, the values stored in the mfs[], mts[], nsb[] and sb[] vectors for the test

sytem-1 as computed by the program are presented.

Table-23 (Formation of mfs[ ], mts[ ],nsb [ ] & sb [] vectors)

Node No. ( i )

mfs [ i ] mts [i] nsb [ i ]

2 10 16 7 3 7 9 3 4 6 6 1 5 4 5 2 6 3 3 1 7 2 2 1 8 1 1 1

sb [ ] vector sb [1] 8 sb [5] 6 sb [9] 8 sb [13] 8 sb [2] 7 sb [6] 4 sb [10] 2 sb [14] 5 sb [3] 6 sb [7] 3 sb [11] 3 sb [15] 6 sb [4] 5 sb [8] 4 sb [12] 4 sb [16] 7 In Table-24, 25,26,27,28 and 29, the allocated active power losses to the various buses using

the loss allocation scheme proposed, are presented for the three test systems considering two

different load models. In the first case, the loads are modeled as constant power loads

64

whereas in the latter case the composite loads have been considered with load composition as

described in (6.1).

Table-24 (Constant P,Q Load)

Node No. Loss Allocated(kW) Phase-a

Loss Allocated(kW) Phase-b

Loss Allocated(kW) Phase-c

2 0.076694 0.113514 0.078759 3 0.000000 0.000000 - 4 - - 0.183477 5 - - 0.174681 6 - - 0.103481 7 0.294929 - - 8 - 0.687582 -

Total Loss 0.371623 0.801096 0.540398

Table-25 (Composite load a0= b0= 0.4 , a1= b1= 0.3, a2= b2= 0.3)

Node No. Loss Allocated(kW) Phase-a

Loss Allocated(kW) Phase-b

Loss Allocated(kW) Phase-c

2 0.076164 0.112438 0.078279 3 0.000000 0.000000 - 4 - - 0.181645 5 - - 0.173064 6 - - 0.102430 7 0.292067 - - 8 - 0.676141 -

Total Loss 0.368231 0.788579 0.535418

6.2.2 Test System-2 Table-26

(Constant P,Q Load)

Node No. Loss Allocated(kW) Phase-a

Loss Allocated(kW) Phase-b

Loss Allocated(kW) Phase-c

2 0.000000 0.000000 0.000000 3 0.000000 0.000000 0.000000 4 0.000000 0.000000 0.000000 5 0.613903 0.359618 0.421616 6 0.000000 0.000000 0.000000 7 0.000000 0.000000 0.000000 8 0.973274 0.532386 0.763904 9 2.304830 0.000000 0.000000 10 0.000000 0.696534 0.000000 11 1.823625 0.000000 -

65

12 2.790733 - - 13 - 0.720862 - 14 0.000000 0.000000 1.931400 15 4.992988 3.063675 4.617106 16 - - 1.323284 17 0.000000 0.000000 0.000000 18 0.579770 0.424480 0.535988 19 0.662366 0.000000 0.000000 20 - 0.828019 - 21 - - 0.650379 22 - - 1.079012 23 0.959916 0.000000 0.000000 24 - 0.000000 0.409703 25 - 1.154647 -

Total Loss 15.701406 7.780221 11.732390

Table-27 (Composite load a0= b0= 0.4 , a1= b1= 0.3, a2= b2= 0.3)

Node No. Loss Allocated(kW) Phase-a

Loss Allocated(kW) Phase-b

Loss Allocated(kW) Phase-c

2 0.000000 0.000000 0.000000 3 0.000000 0.000000 0.000000 4 0.000000 0.000000 0.000000 5 0.588274 0.353573 0.400213 6 0.000000 0.000000 0.000000 7 0.000000 0.000000 0.000000 8 0.919648 0.521097 0.720333 9 2.143575 0.000000 0.000000 10 0.000000 0.679870 0.000000 11 1.684640 0.000000 - 12 2.575396 - - 13 - 0.703192 - 14 0.000000 0.000000 1.798606 15 4.650280 2.958997 4.290127 16 - - 1.231436 17 0.000000 0.000000 0.000000 18 0.556187 0.416022 0.508790 19 0.634906 0.000000 0.000000 20 - 0.806745 - 21 - - 0.616897 22 - - 1.022610 23 0.000000 0.000000 24 - 0.000000 0.388895 25 - 1.122101 -

Total Loss 14.672452 7.561592 10.977906

66

6.2.3 Test System-3 Table-28

(Constant P,Q Load)

Node No. Loss Allocated(kW) Phase-a

Loss Allocated(kW) Phase-b

Loss Allocated(kW) Phase-c

2 0.275314 0.184005 0.236047 3 0.457451 0.304756 0.392168 4 1.703049 1.124449 1.459000 5 - 1.778958 - 6 1.554142 0.994392 1.350722 7 1.852089 1.180259 1.625020 8 2.034059 1.287129 1.791679 9 4.129336 2.609815 3.639610 10 6.676705 - - 11 10.016303 - - 12 23.169165 - - 13 2.275569 1.585686 2.089888 14 - 2.977639 3.848247 15 4.025753 2.847382 3.724544 16 4.433426 3.444826 4.309089 17 4.732981 3.836091 4.710027 18 - 4.920301 - 19 2.869447 2.340483 2.873195 20 4.615476 3.880234 4.761832 21 - - 9.263170 22 - - 13.054650 23 2.465390 4.149677 5.066936 24 6.240955 - - 25 2.195604 1.876686 2.239107 26 2.643415 5.091733 6.067129 27 19.628426 17.019880 20.017008 28 2.995952 2.622701 3.063573 29 1.398387 4.322931 5.026576 30 3.520374 3.008379 3.588588 31 3.111413 2.655956 3.165386 32 8.737969 - - 33 - 3.712270 4.418522 34 - 3.520074 4.144557

Total Loss 127.758139 83.276689 115.926281

67

Table-29 (Composite load a0= b0= 0.4 , a1= b1= 0.3, a2= b2= 0.3)

Node No. Loss Allocated(kW) Phase-a

Loss Allocated(kW) Phase-b

Loss Allocated(kW) Phase-c

2 0.259704 0.176893 0.218953 3 0.429568 0.292044 0.362267 4 1.575582 1.066347 1.329808 5 - 1.687752 - 6 1.440413 0.939254 1.221553 7 1.686282 1.107350 1.460637 8 1.844385 1.204373 1.604564 9 3.741694 2.440907 3.257494 10 6.029620 - - 11 8.840397 - - 12 20.246409 - - 13 2.053279 1.471988 1.857161 14 - 2.762009 3.418980 15 3.628999 2.639243 3.304742 16 3.968627 3.159088 3.780596 17 4.221095 3.498570 4.107389 18 - 4.475436 - 19 2.557445 2.132668 2.502761 20 4.100870 3.520727 4.125134 21 - - 7.999561 22 - - 11.216855 23 2.187196 3.759510 4.382842 24 5.532885 - - 25 1.939428 1.691430 1.927360 26 2.334534 4.587859 5.221209 27 17.283041 15.281864 17.172305 28 2.637240 2.352645 2.626250 29 1.230832 3.875372 4.307244 30 3.109156 2.711069 3.088537 31 2.746043 2.392081 2.722596 32 7.707969 - - 33 - 3.342904 3.800041 34 - 3.165525 3.561265

Total Loss 113.332683 75.734907 100.578099

68

CHAPTER 7

CONCLUSION In the beginning of this project work, an extensive literature survey about the distribution

system load flow methods has been made. Five well established three phase unbalanced

distribution system load flow (UDSLF) methods have been successfully implemented. These

methods are :

1 Implicit ZBUS Gauss method [66-67]

2 Topological Direct method [69-70]

3 Ladder Network based Method [ 53-55]

4 Forward Backward Sweep(FBS) based method [ 57]

5 Power Summation based FBS method [58]

Two variations of the Implicit ZBUS Gauss method [66-67] has been proposed and

implemented. A new algorithm based on FBS has been proposed, where a novel scheme of

bus identification and multiphase data handling is proposed. In Table-22, a comparison of all

the 8 methods has been presented. It can be concluded that forward backward sweep based

methods are usually very fast as these methods effectively exploits the radial nature of the

distribution systems. The new scheme of bus identification and multiphase data handling has

made the proposed method (method-8) even faster than other FBS based methods as it

minimizes the search process to find the adjacent buses of any bus of the system. Based on

this new three phase unbalanced distribution system load flow method, a loss allocation

scheme for unbalanced distribution system has been proposed. This loss allocation scheme

successfully allocates the active losses to various buses of the three test systems considered

in the project work. Composite load modeling has been implemented in the load flow

program and allocations have been made accordingly.

69

7.1 FUTURE PROSPECTS Recent technology improvements in micro turbines, fuel cells and energy storage devices

have provided the opportunity for dispersed generation at the distribution level. With the

possibility of significant penetration of distributed generation, more studies are needed on

dynamic analysis of distribution systems. For dynamic simulations considering network

effects, [98-99] load flow calculations must be performed at appropriate time steps. The

system operational conditions may vary widely during dynamic events. Together, this

requires that the load flow algorithm be more robust and faster than that required for static

studies considering multiple feeding sources.

Flexible AC Transmission Systems (FACTS) devices are playing a leading role in efficiently

controlling the line power flow and improving voltage profiles of the power system network.

The major objectives of FACTS devices installed on a distribution feeder are to improve

voltage profiles, correct power factor and reduce line losses. Modifications and extensions to

standard distribution load flow algorithms with FACTS devices [100-101] can an interesting

direction to the present work.

Distribution Management Systems (DMSs) have been subjected to significant changes in

their functions as well as in their computing architectures and characteristics. To cope with

the urging issues of power quality and distribution system reliability and to fully exploit the

introduction of automation, power electronics devices and dispersed generation, classical

DMS functions are revised and some new tasks introduced. Modern DMSs adopt open

system architectures,based on local and wide area networks and distributed computation. In

this scenario, the Object Oriented (OO) methods are particularly appealing [102-108],

because OO modeling and programming allow to fully exploiting the advantages of the new

architectures and guarantees flexibility, expansibility and easiness of maintenance of DMS

functions and software packages.

Three phase four wire distribution networks are widely adopted in modern power distribution

system. A multi grounded three phase four wire service has higher sensitivity for fault

protection than a three phase three wire service. The return current is due to both the

70

unbalanced load and nonlinear characteristics of electrical equipments through the

distribution feeder. However, the neutral wire in most of power flow software is usually

merged into phase wires using Kron’s reduction. Since the neutral wire and ground wire are

not explicitly represented, neutral wire currents and voltages remain unknown. In some

applications, like power quality and safety analysis, loss analysis, etc. knowledge of the

neutral wire and ground currents and voltages could be of special interest. Thus, exclusive

three phase four wire distribution load flow methods [109-111] can be developed.

71

REFERENCES [1] B. Stott, “Review of Load flow calculation methods,” Proc. of IEEE, vol-2, no.7 , pp

916 -929, July 1974. [2] B. Stott and O. Alsac, “ Fast Decoupled Load Flow,” IEEE-PAS-93, pp 859-869,

May/ June 1974 . [3] S. Iwmato and Y.Tamura , “ A load flow calculation method for ill conditioned

power systems,” IEEE Trans. on Power Apparatus and Systems,vol. PAS-100, no. 4, pp 1736-1743, 1981.

[4] S.C.Tripathy, G. D. Prasad, O. P. Mallick, G.S.Hope, “ Load flow solutions for ill conditioned power systems by a Newton like method,” IEEE Trans. on Power Apparatus and Systems,vol. PAS-101, no.10,pp 3648-3657, 1982.

[5] D. Rajicic and A. Bose, “A modification to the fast decoupled power flow for networks with high R/X ratios,” IEEE Trans. Power Systems, vol-3,no.2, pp 743-745, May-1988.

[6] W.H. Kersting and S.A. Seeker., “ A program to study the effects of mutual coupling and unbalanced loading on a distribution system,IEEE PES Winter meeting-1975.

[7] S.A.Rajgopalan, “ A new computational algorthim for load flow study for radial distribution sysytem,” Computer and Electrical engineering, vol. 5 , Oxford Pengamon, pp 225-231, 1978.

[8] R. A. Stevens, D. T. Rizy and S. L. Purucker, “Performance of conventional power flow routines for real time distribution automation application,” Proc 18th Southeastern Symposium on Systems Theory, IEEE Computer society, pp196-200, 1986.

[9] C.E. Lin , Y.W. Huang and C.L Huang, “Distribution system load flow calculation with microcomputer implementation,” Electrical Power System Research, vol-13,pp 139-145, 1987.

[10] A. H. Khan, R.P. Broadwater and A.Chandrasekharan, “Comparative study of three radial power flow methods,”Proceedings of the IASTED International Symposium on High Technology in the power industry, Arizona,March-1988.

[11] A.O.Ekwue,R.B.I Johnson and B.J.Cory, “Improved load flow method for distribution systems,” Proceedings of the IASTED International Symposium on High Technology in the power industry, Arizona,March-1988.

[12] A. Gómez, J. M. Riquelme, T. González, and E. Ruiz, “Fair allocation of transmission power losses,” IEEE Trans. Power System., vol. 15, pp.184–188, Feb. 2000.

[13] J. J. Gonzalez and P. Basagoiti, “Spanish power exchange market and information system. Design concepts, and operating experience,” Proc. of the 1999 IEEE Power Industry Computer Applications Conference, Santa Clara, USA, pp. 245–252, May 1999.

[14] J. W. Bialek, S. Ziemianek, and N. Abi-Samra, “Tracking-based loss allocation and economic dispatch,” in Proceedings of the 13th Power Systems Computation Conference, Trondheim, Norway, pp. 375–381, July 1999.

[15] F. D. Galiana, A. J. Conejo, and I. Kockar, “Incremental transmission loss allocation under pool dispatch,” IEEE Trans. Power Syst., vol. 17, pp. 26– 33, Feb. 2002.

72

[16] J. W. Bialek, “Identification of source-ink connections in transmission networks,” in Proc. 4th Int. IEE Conf. Power System Control and Management, London, U.K., Apr. 16–18, pp. 200–204, 1996.

[17] D. Kirschen, R. Allan, and G. Strbac, “Contributions of individual generators to loads and flows,” IEEE Trans. on Power Systems, vol. 12, no. 1, pp. 52–60, Feb. 1997.

[18] D. Kirschen and G. Strbac, “Tracing active and reactive power between generators and loads using real and imaginary currents,” IEEE Trans. Power Syst., vol. 14, pp. 1312–1319, Nov. 1999.

[19] A. J. Conejo, F. D. Galiana, and I.Ivana Kockar, “Z-bus loss allocation,” IEEE Trans. Power Syst., vol. 16, pp. 105–110, Feb. 2001.

[20] P. M. Costa and M. A. Matos, “Loss allocation in distribution networks with embedded generation”, IEEE Trans. Power Syst., vol. 19, pp. 384–389, Feb. 2004.

[21] W.M.Lin and M.S.Chen, “An overall distribution automation structure,” Electrical Power System Research, vol-10,no.1,pp 7-19, 1986.

[22] “ Distribution Automation : A practical tool for shaping a more profitable future,” Special Report, Electrical World, pp 43- 50,1986.

[23] M-S Chen and T-H Chen, “ Application of three phase load flow to power system distribution automation,”Proc. IEE Int. Conf. on Advances in Power System Control,Operation and Management,pp 472-478,Honkong, Nov-1991.

[24] C.S Chen., C.T Hsu and Y.H.Yan, “Optimal distribution feeder capacitor placement considering mutual coupling effect of conductors,” IEEE Trans. Power Delivery, vol-10, no. 2,pp 987-994, April- 1995.

[25] M.E. Baran and F.F. Wu, “Optimal sizing of capacitors placed on a radial distribution system,” IEEE Trans. Power Delivery, vol. 4, no.1, pp , Jan- 1989.

[26] C.G. Renatoe, “New method for the analysis of distribution networks” IEEE Trans. Power Delivery,vol. 5, no.1, pp391-396, Jan- 1990.

[27] H. D. Chiang., “A fast decoupled load flow method for distribution power network: Algorithms analysis and convergence study,” Electrical Power and Energy Systems, vol.13, no.3, pp 130-138, June-1991.

[28] G.B Jasmon. and L.H.L Lee “Distribution network reduction for voltage stability analysis and load flow calculations,” Electrical Power and Energy Systems, vol.13, no.1, pp 9-13, Jan-1991.

[29] G. B. Jasmon and L.H.L Lee” Stability of load flow techniques for distribution system voltage stability analysis,” IEE Proc. Genr. Trans. Distri., vol. 138, no.6,pp 479-484, 1991.

[30] D. Das, H.S Nagi and D.P. Kothari , “Novel method for solving radial distribution networks,” IEE-Part C, vol. 141,no.4, pp 291-298, 1994.

[31] A.Chandrasekharan and R.P. Broadwater, “A new formulation of load flow equations in balanced radial distribution systems”,Canadian Electrical Engineering Journal,vol. 12,no. 2, pp147-151, Oct. 1987.

[32] L. Jun and W. Song, “ A simple algorithm of load flow for distribution system,”Proc. IEE Int. Conf. on Advances in Power System Control,Operation and Management,pp 467-471,Honkong, Nov-1991.

[33] D. Das , D.P. Kothari ,A. Kalam, “Simple and efficient method for radial distribution networks,” Elect. Power and Enery Systems, vol. 17, no.5,pp 335-346, 1995.

73

[34] S. Ghosh. and D. Das, “ Method for load flow solution of radial distribution netwoks" IEE Proc. Gen. Trans. Distri.,vol.146,no.6, Nov-1999.

[35] S.Jovanovic and F. Milicevic, “Triangular distribution load flow,”IEEE Power Engg. Review,vol.20, pp60-62, May-2000.

[36] S. Mok, S. Elangovan.,C. Longjian and M. M. Salama , “A new approach for power flow analysis of balanced radial distribution systems,”Electric Machines Power Systems, vol.28,no.4, pp 325-340,2000.

[37] E.R.E Romero Ramos and A.G.A Gomez Exposito, “A general purpose load flow technique for balanced distribution networks,”IASTED Conference, Marbella, Spain,September-2000.

[38] A.Augugliaro.,L.Dusonchet., M.G Ippolito. and E. RivaSanseverino, “ An efficient iterative method for load flow solution in radial distribution system,” IEEE Porto Powertech Conference,10-13 Sept.,Portugal 2001.

[39] S.F Mekhamer, S.A Soliman, M.A. Moustafa.,M.E El-Hawary, “Load flow solution of radial distribution feeders: a new approach,” IEEE Porto Powertech Conference,10-13 Sept.,Portugal 2001.

[40] M.Jin,Xu Jianyuan,W.Shengui and Lin Xin,“Calculation and analysis for line losses in distribution networks,”Proc. IEEE ICPST POWERCON, pp 2537-2541,Oct 13-17,2002.

[41] M. Afsari. ,S.P Singh.,G.S. Raju. and G.K.Rao, “A fast power flow solution of radial distribution networks”, Electric Power Components and Systems,vol. 30,no.10,pp 1065-1074,Oct- 2002.

[42] S.F. Mekhamer., S.A. Soliman., M. A. Moustafa., M.E El-Hawary, “Load flow solution of radial distribution feeders: a new contribution,” Electrical Power & Energy Systems, vol. 24 , pp 701-707, 2002.

[43] L.Li., G. Zhizhong and W. Chengmin “A new topology structure and load flow solution for radial distribution netwoks,”Proc.ICPST POWERCON,vol-2,pp1064-1067,2002.

[44] R. Ranjan and D. Das , “Simple and efficient computer algorithm to solve radial distribution networks”,Electric Power Components and Systems,vol. 31,no.1,pp 95-107, 2003.

[45] S. Sivanagaraju , M.S.Giridhar and E. Jagadeesh Babu, “ A novel Load Flow Technique for radial distribution Systems,” Proc. National Power Systems Conference (NPSC), vol-2, pp 140-144 ,2004.

[46] H.Rudnick and M. Munoz, “ Influence of modeling in load flow analysis of three phase distribution systems,” Proc. IEEE colloquium in South America31 Aug-15 Sept., pp 173-176,1990.

[47] M.S. Srinivas and M. L. Kothari, “On some aspects of Distribution Load Flow,”IEEE TENCON, vol. II, pp 510-513,1998.

[48] M.S. Srinivas, “Distribution Load Flows: A brief review,” Proc. IEEE PES Winter Meeting,pp 942-945, Jan-1999.

[49] M.S. Srinivas, M.Sharma, S. S. Dey and L. L .Lai “New Findings on radial distribution system load flow algorithms,” Proc. IEEE PES Winter Meeting, Singapore- 2000.

74

[50] A. G. Bhutad, S. V. Kulkarni and S. A. Kharparde, “ Three phase load flow methods for radial distribution networks ,”IEEE TENCON Conference , Bangalore , pp 781-785, 15-17 October -2003.

[51] G. D. Patil and S. V. Kulkarni , “ Performance Assessment of Load Flow Techniques for Unbalanced Distribution Systems,” Proc. National Power Systems Conference (NPSC), vol-2, pp ,2004.

[52] R.Berg and E.S.Hawkins., “ Mechanized calculation of unbalanced load flow on radial distribution circuits”, IEEE Trans. PAS-86,pp 415-422, 1967.

[53] W.H. Kersting and D.L. Mendive, “An application of ladder network theory to the solution of three phase radial load flow problem”, IEEE PES Winter meeting-1976.

[54] W.H. Kersting , “A method to teach the design and operation of distribution system,”IEEE Trans. on Power Apparatus System,vol.103,no.7, pp 1945-1952,July-1984.

[55] W.H. Kersting. and W.H. Philips, “A radial three phase power flow program for the PC,”Presented at Frontiers Power Conference, Stillwater,UK,October-1987.

[56] R.P. Broadwater, A. Chandrasekharan,C.T. Huddle and A.H. Khan, “Power flow analysis of unbalanced multiphase radial distribution systems,” Electrical Power System Research, vol-14,pp 23-33, 1988.

[57] D. Thukaram, H.M.W. Banda and J.Jerome, “ A robust three phase power flow algorithm for radial distribution systems,”Electrical Power System Research, vol-50,no.3,pp 227-236, June-1999.

[58] R.Ranjan, B. Venkatesh, A. Chaturvedi and D.Das, “Power flow solution of 3-phase unbalanced radial distribution network,”Electric Power Components and Systems, vol. 32, pp 421-433,2004.

[59] D.Shirmohammadi ,H.W.Hong.,A. Semlyen., G.X. Luo., “A compensation based power flow method for weakly meshed distribution and transmission networks,” IEEE Trans. Power Systems, vol-3,no.2, pp 753-762, May-1988.

[60] Luo G.X. and Semlyen A., “Efficient load flow for large weakly meshed networks”,IEEE Trans. Power Systems, vol-5., pp 1309-1316, Nov-1990.

[61] Cheng C.S., Shirmohammadi D., “A three phase power flow method for real time distribution system analysis,” IEEE Trans. Power Systems, vol-10,no. 2, pp 671-679, May-1995.

[62] M.H. Haque, “Efficient load flow method for distribution systems with radial or mesh configuration,” IEE Proc. Gen. Trans. Distri.,vol.143,no.1, Jan-1996.

[63] D.Rajicic and R.Taleski, “Two novel methods for radial and weakly meshed networks analysis,” Electrical Power System Research, vol-49,pp 79-87, March-1998.

[64] Chen G.J.,Li K.K., Chung T.S. and Tang G.Q., “ An efficient two stage load flow method for meshed distribution networks,” Proc. APSCOM, pp 537-542,2000.

[65] S.Ghosh and D.Das, “An Approach for load flow solution of meshed distribution networks,” IE(India) Journal-EL,vol-84,pp 66-70,Sept-2003.

[66] D.I.Sun., S.Abe., R.R. Shoults., M-S Chen, “ Calculation of energy losses in a distribution system,”IEEE Trans. on Power Apparatus and Systems,vol. PAS-99, no.4,pp 1347-1356, 1980.

75

[67] T.H. Chen., M-S Chen, K.J.Hwang., P. Kotas. and E.A.Chebli., “Distribution sytem power flow analysis-A rigid approach”, IEEE Trans. Power Delivery, vol-6., pp 1146-1152, July-1991.

[68] J.H. Teng, “A modified Gauss-Siedel algorithm of three phase power flow analysis in distribution networks,” Elect. Power and Enery Systems, vol. 24, no.1,pp 97-102, 2000.

[69] J.H. Teng., “ A network-topology based three phase load flow for distribution systems,” Proc. National Science Council ,ROC(A), vol-24, no. 4, pp 259-264, 2000.

[70] J.H. Teng, “ A direct approach for distribution system load flow solutions,” IEEE Trans. Power Delivery, vol-18, no-3, 882-887, July-2003.

[71] S.K. Goswami and S.K Basu.,“A loop based method for distribution power flow solution,” Proc. XVI Annual Convention and Exhibition of IEEE in India,pp 59-61,Jan 22-25,1990.

[72] S.K. Goswami and S.K. Basu. “ Direct solution of distribution systems,”IEE Proceedings-C, vol. 138 , no.1, Jan-1991.

[73] S.Wang, Y.Liu and T.Ruan, “ Distribution system power flow based on loop impedance equations,”Proc. Intenational Conference on Energy Management and Power Delivery, 3-5 March, Singapore. 1998.

[74] P.Aravindababu, S.Ganapathy and K. R. Nayar, “A novel technique for the analysis of radial distribution systems,” Elect. Power and Enery Systems, vol. 23, ,pp 167-171,2001.

[75] C.N. Lu., J.H. Teng. and E.Liu., “Distribution System state estimation”, IEEE Trans. Power Systems, vol. 10,no.1,pp 229-240, Feb-1995.

[76] A.P.S.Meliopoulos and F.Zhang, “Multiphase power flow and state estimation for power distribution systems”, IEEE Trans. Power Systems, vol. 11, no. 2, pp 939-946, May-1996.

[77] F.Zhang.and C.S.Cheng, “A modified Newton method for radial distribution load flow analysis”, IEEE Trans. Power Systems, vol-12. no.1, 389-397, Feb-1997.

[78] H.L. Nguyen , “Newton-Raphson method in complex form”, IEEE Trans. Power Systems, vol-12. no.3, 1355-1359, Aug-1997.

[79] A.G. Exposito.,E.R. Ramos., “Reliable load flow technique for radial distribution networks”, IEEE Trans. Power Systems, vol-14, no.3,pp 1063-1069,Aug-1999.

[80] J. Arrillga and C.P. Arnold., “Computer modeling of electrical power systems”, John Wiley and sons. NewZealand, 1983.

[81] A.V. Garcia. and M.G. Zago, “Three phase fast decoupled power flow for distribution networks”, IEE Proc. Genr. Trans. Distri., vol 143, no.2, 188-192, March-1996.

[82] R. D. Zimmerman. and H.D. Chiang, “Fast Decoupled Power Flow for Unbalanced Radial Distribution Systems,” IEEE Trans. Power Systems, vol. 10, no. 4,pp 2045-2052, Nov-1995.

[83] J.H. Teng, “ Current based power flow solutions for distribution systems,” IEEE ICPST Confernce, Bejing, China,pp 414-418,1994.

[84] W. M. Lin. And J.H Teng, “Phase decouple load flow method for radial and weakly meshed distribution networks”, IEE Proc. Genr. Trans. Distri., vol. 143, no.1,Jan-1996.

76

[85] Y.H. Moon., B.K. Choi , B.H. Cho.,S.H. Kim., B.N. Ha. and J.H. Lee., “Fast and reliable distribution system load flow algorithm based on the y-bus formulation,”Proc. IEEE PES Summer Meeting, vol. 1,pp 238-242, July-1999.

[86] W. M. Lin,Y.S.Su,H.C.Chin and J.H.Teng, “Three phase unbalanced distribution power flow solutions with minimum data preparation,” IEEE Trans. Power Systems, vol-14, no.3, pp 1178-1183,Aug-1999.

[87] S. Carneiro Jr.,J.R.L. Pereira and P.A.N Garcia., “Unbalanced distribution system power flow using current injection method,” Proc. IEEE Power Engineering Society Winter Meeting, vol. 2, Singapore, 2000.

[88] P.A.N. Garcia, J.R.L.Pereira,S.Carneiro Jr., P.A.Costa and N.A.Martins “Three phase power flow calculations using current injection method,” IEEE Trans. Power Systems, vol. 15,no. 2,pp 508-514, 2000.

[89] W.M.Lin and J. H.Teng, “ Three phase distribution network fast decoupled power flow solutions”, Elect. Power and Energy Syst..,vol. 22, 375-380, 2000.

[90] J. H. Teng. and C. Y. Chang, “A branch voltage based three phase load flow method for unbalanced distribution systems”, ”, IEEE ICPST POWERCON, pp 601-607, Dec-2000.

[91] J. H. Teng. and C. Y. Chang., “A novel and fast three phase load flow for unbalance radial distribution systems”, IEEE Trans. Power Systems, vol-17, no.4, 1238-1244, Nov-2002.

[92] S. Kebaili., F.Adjeroud and K. Zehar.,“Extension of the modified Newton method for radial distribution system load flow,”Proc. Canadian Conf. on Electrical and computer engineering, pp 781-784,2001.

[93] P. Aravindhababu, “ A new fast decoupled power flow method for distribution systems,” Electric Power Components and Systems, vol. 31, no.9, pp 869-878,2003.

[94] E. R. Ramos, A. G. Exposito and G. A.Cordero, “Quasi coupled three phase radial load flow”, IEEE Trans. Power Systems, vol-19, no.2,pp 776-781,May-2004.

[95] W. H. Kersting, Distribution System Modeling and Analysis, CRC Press,2000 . [96] W. H . Kersting and L. Willis, “Radial distribution test feeders,” IEEE Trans. Power

Systems, vol-6,no. 3, pp 975-985, Aug-1991. [97] W. H . Kersting , “Radial distribution test feeders,”Proc. IEEE Power Engg. Society

Winter meeting , vol-2,, pp 908-911, 2001. [98] Y.Zhu and K.Tomsovic., “Adaptive power flow method for distribution systems with

dispersed generation,” IEEE Trans. Power Delivery, vol-17, no-3, 822-827, July-2002.

[99] P.S.N.Rao and R.S. Deekshit, “Radial load flow for systems having distributed generation and controlled Q sources ,” Electric Power Components and Systems,vol. 33,no. 6, pp 641-655, 2005.

[100] P.Yan and Arun Sekar, “Analysis of radial distribution systems with embedded series FACTS devices using a fast line flow based algorithm,” IEEE Trans. Power Delivery, vol. 20, no.4, pp 1775-1782, Nov-2005.

[101] P.Tripathy,S.N.Singh and S.C.Srivastava , “ Power flow analysis with optimally placed D-STATCOM in distribution system,” Proc. National Power Systems Conference (NPSC), vol-2, pp 128-133 ,2004.

77

[102] A. Losi and M. Russo, “An Object Oriented approach to Load Flow in distribution systems”Proc. IEEE Power Engineering Society Summer Meeting, vol. 4, pp 2332-2337, July-2000.

[103] A. Losi and M. Russo, “ Object Oriented Load Flow for radial and weakly meshed distribution systems” IEEE Trans. Power Systems, vol. 18, no.4, pp 1263-1274, Nov-2003.

[104] A. Losi and M. Russo, “Load Modeling Impact on object oriented distribution load flow,”Proc. IEEE Bologna Power Tech, Bologna, Italy,June-2003.

[105] A. Losi and M. Russo, “ Dispersed Genereration modeling for Object Oriented Distribution Load Flow ” IEEE Trans. Power Delivery, vol. 20, no.2, pp 1532-1538, April-2005.

[106] M. P. Selvan and K. S. Swarup, “Distribution System Load Flow using object oriented methodology,” IEEE ICPST POWERCON-2004, Singapore , Nov 21-24, 2004.

[107] M. P. Selvan and K. S. Swarup, “Object oriented modeling of distribution systems with dispersed generation for power flow analysis,” Proc. National Power Systems Conference (NPSC), vol-2, pp122-127 ,2004.

[108] M. P. Selvan and K. S. Swarup, “Modeling and analysis of unbalanced distribution system using object oriented methodology,”Electric Power Systems Research,vol.76 , no.11 , pp 968-979 ,2006.

[109] R. M. Ciric, L. F. Ochao and A. Padilha, “Power flow in distribution networks with earth return,” Elect. Power and Energy Systems,vol. 26,no.5, 373-380, 2004.

[110] D.P.R Penido,L.R.Araujo, J.LR Pereira,P.A.N. Garcia, and S.Carneiro Jr, “Four wire Newton Raphson power flow based on the current injection method,” Proc. IEEE PES Power System Conference and Exposition,2004.

[111] M.Monfared,A.M.Daryani and M.Abedi, “Three phase asymmetrical load flow for four wire distribution networks,” Proc. IEEE PES Power System Conference and Exposition,Oct 29-Nov1, pp 1899-1903,2006.

78

APPENDIX-A Standard overhead line spacings [96-97] with spacing ID is given in Fig.20 and Table-30. Table-30 (Overhead Line spacings)

Fig. 20 Standard overhead line spacings