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
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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
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
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
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.
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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
[ ] [ ] [ ]. (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:
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
• 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
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
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:
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:
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)
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
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APPENDIX-A Standard overhead line spacings [96-97] with spacing ID is given in Fig.20 and Table-30. Table-30 (Overhead Line spacings)