Main

DISTRIBUTION SYSTEM VOLTAGE REGULATION FOR DIFFERENT STATIC LOADS

DISTRIBUTION SYSTEM VOLTAGE REGULATION
FOR DIFFERENT STATIC LOAD MODELS

CONTENTS :
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ABSTRACT …………………………………………………………………………..7
1. INTRODUCTION

1.1. Introduction to electrical power system……………………………….8

(a) Generation system……………………………………………………….9

(b) Transmission system……………………………………………………10

(c) Distribution system……………………………………………………..10

1.2. Brief overview of distribution system………………………………….11

1.3. Distribution system configuration……………………………………..12

1.4. Primary distribution system…………………………………………….13

1.5. Secondary distribution system………………………………………...15

1.6. Literature survey…………………………………………………………..16

2. LOAD FLOW ANALYSIS

2.1. Proposed Method…………………………………………………………..21

2.2 .Solution methodology……………………………………………………..22

2.3. Explanation of the proposed algorithm……………………………….25

2.4. Static load models………………………………………………………….27

2.5 .Algorithm for Load flow computation………………………………....30

3. EXAMPLES………………………………………………………….37

4.SUMMARY AND FUTURE SCOPE

(i) Conclusion……………………………………………………………….....40

(ii) Future scope………………………………………………………………..41

5. References………………………………………………………………………..42

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6. Appendix A……………………………………………………………………….43

7. Appendix B………………………………………………………………………45

LIST OF FIGURES:

Fig 1.1 A single line diagram of a distribution substation………..12

Fig 1.2 Primary distribution feeder……………………………………..14

Fig 1.3 Service drops in distribution systems………………………..15

Fig 2.1 Radial main feeder…………………………………………………22

Fig 2.2 Electrical Equivalent of figure…………………………………..22

Fig 2.3 Flow Chart for the Algorithm of radial

distribution network having laterals…………………………33

Fig 3.1 34 node radial distribution system……………………………37

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LIST OF TABLES:

Table1 Details of the numbering scheme of figure 3.1……………..24

Table 2 Non-zero integer values of F(i)………………………………….25

Table 3 Voltage magnitude (p.u.) of each node for 34………………38
node radial distribution network for CP,CI,CZ load models

Table 4 Power losses for CP,CI,CZ load models……………………..39

Table 5 Voltage regulation for CP,CI,CZ load models………………39

Table 6 Line Data of 34 Node Radial Distribution Network……….43

Table 7 Load Data of 34 Node Radial Distribution Network………44

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Nomenclature:
NB = total number of nodes.

(j) = branch number, j=1, 2,………, NB-1

PL(i)= real power load of ith node

QL(i)= reactive power load of ith node

|V(i)|=voltage magnitude of ith node

R(j)= resistance of jth branch

X(j)=reactance of jth branch

I(j)=current flowing through branch j

P(i+1)=total real power load fed through node i+1

Q(i+1)= total reactive power load fed through node i+1

δ(i+1)=voltage angle of node i+1

LP(j)=real power loss of branch j

LQ(j)= reactive power loss of branch j

NL=total number of laterals

[L]=lateral number, L=1, 2,.…., NL

SN(L)=source node of lateral L

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EB(L)= end node of lateral L

LB(L)=node, just ahead of source node of lateral L

F(i)= integer variable

TP(L)= total real power load fed through the node LB(L) of lateral L

TQ(L)= total reactive power load fed through the node LB(L) of lateral L

SPL(L)= sum of real power loads of all the nodes of lateral L which have

just been left plus the sum of real power losses of all the

branches of lateral L which have just been left except the real

power loss in branch {LB(L)-1} of lateral L

SQL(L)= sum of reactive power loads of all the nodes of lateral L which

have just been left plus the sum of reactive power losses of all

the branches of lateral L which have just been left except the

reactive power loss in branch {LB(L)-1} of lateral L

PS(L)= sum of the real power loads of all the nodes(except source nodes)

of all the laterals which have just been left plus the sum of real

power losses of all the branches of all the laterals which have

just been left.

QS(L)= sum of the reactive power loads of all the nodes(except source

nodes) of all the laterals which have just been left plus the sum of

reactive power losses of all the branches of all the laterals which

have just been left.

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ABSTRACT:

Voltage regulation computations for distribution systems are strongly
dependent on power flow solutions. The classical constant power load model
is typically used in power flow studies of transmission or distribution
Systems; however, the actual load of a distribution system cannot just be
modeled using constant power models, requiring the use of constant
current, constant impedance, exponential or a mixture of all these load
models to accurately represent the load. This paper presents a study of
voltage regulation of a distribution system using different Static load
models.

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CHAPTE-1

INTRODUCTION

Voltage regulation is an important subject in electrical distribution
engineering. It is the utilities responsibility to keep the customer voltage
within specified tolerances. The performance of a distribution system and
quality of the service provided are not only measured in terms of
frequency of interruption but in the maintenance of satisfactory voltage
levels at the customers’ premises. A high steady-state voltage can reduce
light bulb life and reduce the life of electronic devices. On the other hand,
a low steady-state voltage leads to low illumination levels, shirking of
television pictures, slow heating of heating devices, motor starting
problems, and overheating in motors. However, most equipment and
appliances operate satisfactorily over some reasonable range of voltages,
hence; certain tolerances are allowable at the customer’s end. Thus, it is
common practice among utilities to stay within preferred voltage levels
and ranges. The steady-state voltage regulations should be within +6%
to−13% for satisfactory operation of various electrical devices. Voltage
regulation calculations depend on the power flow solutions of a System.
Most of the electrical loads of a power system are connected to low
voltage or Medium-voltage distribution systems rather than to a high-
voltage transmission system. The loads connected to the distribution
system are certainly voltage dependent; thus, these types of load
characteristics should be considered in load flow studies to get accurate
results and to avoid costly errors in the analysis of the system. For
example, in voltage regulation improvement studies, possible under- or
over-compensation can be avoided if more accurate results of load flow
solutions are available. However, most conventional load flows use a
constant power load model, which assumes that active and reactive
powers are independent of voltage changes. In reality, constant power
load models are highly questionable in distribution systems, as most

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nodes are not voltage controlled; therefore, it is very important to
consider better load models in these types of load flow problems. In this
paper, distribution system voltage regulation and the effect of shunt
capacitor compensation on this regulation for different static load models
are studied.
1.1 Introduction to electrical power system:
The electric power system is a network of interconnected
components which generate electricity by converting different forms of
energy, (potential energy, kinetic energy, or chemical energy are the most
common forms of energy converted) to electrical energy.
The electric power system consists of three main subsystems:
1. Generation system,
2. Transmission system, and
3. Distribution system.
Electricity is generated at the generating station by converting a primary
source of energy to electrical energy. The voltage output of the generators
is then stepped up to appropriate transmission levels using a step-up
transformer. The transmission subsystem then transmits the power close
to the load centers. The voltage is then stepped down to appropriate
levels. The distribution subsystem then transmits the power close to the
customer where the voltage is stepped-down to appropriate levels or use
by a residential, industrial, or commercial customer.
1.1 (a) Generation system:
Generation plants consist of one or more generating units that convert
mechanical energy into electricity by turning a prime mover coupled to
an electric generator. Generators produce line-to-line voltages between 11
kv and 30 kv. The ability of generation plants to supply all of the power
demanded by a customers is referred to as system adequacy. Three
conditions must be met to ensure system adequacy.
1. Available generation capacity must be greater than demanded load
plus system losses.
2. The system must be able to transport demanded power to customers
without overloading equipment.
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3. Customers must be served within an acceptable voltage range.

1.1 (b) Transmission system:
Electric power transmission is the bulk transfer of electrical power, a
process in the delivery of electricity to consumers. Transmission systems
transport electricity over long distances from generation substations to
transmission or distribution substations. Typical voltage levels include
69 kv, 115 kv, 138 kv, 161 kv, 230 kv, 345 kv, 500 kv, 765 kv, and 1100
Kv. Transmission substations are transmission switching stations with
transformers that step down voltage to sub transmission levels. Sub
transmission systems transport electricity from transmission substations
to distribution substations. Typical voltage levels include 34.5kv, 46 kv,
69 kv, 115 kv, 138 kv, 161 kv, and 230 kv.

1.1 (c) Distribution systems:
Distribution substations are nodes for terminating and reconfiguring sub
transmission Lines plus transformers that step down voltage to primary
distribution levels.
Primary distribution systems: deliver electricity from distribution
substations to distribution transformers. Voltages range from 4.16 kv to
34.5 kv with the most common being 15-kv class (e.g., 12.47 kv, 13.8
kv).
Distribution transformers: Convert primary distribution voltages to
utilization voltages. Typical sizes range from 5 kva to 2500 kva.
Secondary distribution systems: deliver electricity from distribution
transformers to customer service entrances. Voltages are typically
120/240v single phase, 120/208v three phase, or 277/480v three phase.

1.2 Brief overview of distribution system:
Distribution systems deliver power from bulk power systems to
retail customers. To do this, distribution substations receive power from
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sub transmission lines and step down voltages with power transformers.
These transformers supply primary distribution systems made up of
many distribution feeders. Feeders consist of a main 3φ trunk, 2 φ and 1
φ laterals, feeder interconnections, and distribution transformers.
Distribution transformers step down voltages to utilization levels and
supply secondary mains or service drops. Distribution planning
departments at electric utilities have historically concentrated on
capacity issues, focusing on designs that supply all customers at peak
demand within acceptable voltage tolerances without violating equipment
ratings. Capacity planning is almost always performed with rigorous
analytical tools such as power flow models. Reliability, although
considered important, has been a secondary concern usually addressed
by adding extra capacity and feeder ties so that certain loads can be
restored after a fault occurs. Distribution systems begin at distribution
substations. An elevation and corresponding one-line diagram of a simple
distribution substation is shown in figure.

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Figure 1.1: A single-line diagram of a distribution substation
The substation’s source of power is a single overhead sub transmission
line that enters from the left and terminates on a take-off (dead-end)
structure. The line is connected to a disconnect switch, mounted on this
same structure, capable of visibly isolating the substation from the sub
transmission line. Electricity is routed from the switch across a voltage
transformer through a current transformer to a circuit breaker. This
breaker protects a power transformer that steps voltage down to
distribution levels. High voltage components are said to be located on the
“high side” or “primary side” of the substation.

1.3 Distribution system configuration:
The design of the distribution system mainly depends on the
chosen classification of single or three phase, radial or loop network,
overhead line or underground cables. The essential factors to be kept in
mind while planning a distribution system are:
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1) Safety: the safety factor requires the distributors to be laid following:
(i) Proper clearances.
(ii) Voltage safe enough to be used for consumer’s gadgets.
2) Smooth and even flow of power: a steady, uniform, non-fluctuating
flow of power is necessary to feed loads of all categories of consumers.
3) Economy: the third factor is economy. This usually calls for use of
higher
Voltage to ensure minimum losses while distribution power.

1.4 Primary distribution system:
Primary distribution systems consist of feeders that deliver power from
distribution substations to distribution transformers. A feeder begins
with a feeder breaker at the distribution substation. Many will exit the
substation in a concrete duct bank (feeder get-away) and be routed to a
nearby pole. At this point, underground cable transitions to an overhead
three-phase main trunk. The main trunk is routed around the feeder
service territory and may be connected to other feeders through
normally-open tie points. Underground main trunks are possible, even
common in urban areas, but cost much more than overhead
construction. Lateral taps off of the main trunk are used to cover most of
a feeder’s service territory. These taps are typically 1φ, but may also be 2
φ or 3 φ. Laterals can be directly connected to main trunks, but are more
commonly protected by fuses, recloses, or automatic sectionalizes.
Overhead laterals use pole-mounted distribution transformers to serve
customers and underground laterals use pad mount transformers. An
illustrative feeder showing different types of laterals and devices is shown
in figure.

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Figure 1.2 primary distribution feeder
There are two type of distribution line exists in primary distribution
systems overhead lines and underground lines. In overhead lines, wires
carry load current in an overhead system. Major classifications are by
insulation, size, stranding, material, impedance, and capacity. Lines
without an insulated cover are called bare conductors and all other lines
are referred to as insulated conductors. Insulated conductors are further
classified into covered conductor, tree wire, spacer cable, and aerial
cable. Covered conductor and tree wire have a thin covering of insulation
that cannot withstand phase to ground voltages, but reduce the
probability of a fault if vegetation bridges two conductors. Spacer cable
has increased insulation that allows conductors to be arranged in a small
triangular configuration. Aerial cable has fully rated insulation capable of
withstanding phase to ground voltages.

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1.5 SECONDARY DISTRIBUTION SYSTEMS:
Secondary systems connect distribution transformers to customer
service entrances. They can be extremely simple, like overhead service
drop, and extremely complex, like a secondary network. Customers are
connected to distribution systems via service drops. In general service is
typically 1Φ 3-wire 120/240V, 3 Φ 4-wire 120/208V, or 3 Φ 4-wire
277/480V. Customers close to a distribution transformer are able to
have service drops directly connected to transformer secondary
connections. Other customers are reached by routing a secondary main
for service drop connections. These two types of service connections are
shown in Figure.3 systems utilizing secondary mains are characterized
by a small number of large distribution transformers rather than a large
number of small distribution transformers. This can be cost effective for
areas with low load density and/or large lot size, but increases ohmic
losses and results in higher voltage drops. Increased line exposure tends
to reduce reliability while fewer transformers tend to increase reliability.

Figure 1.3 : Service Drops in Distribution System

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Distribution Feeders:
There are three basic types of distribution system designs: Radial,
Loop, or Network. As one might expect, one can use combinations of
these three systems, and this is frequently done. The Radial distribution
system is the cheapest to build, and is widely used in sparsely populated
areas. A radial system has only one power source for a group of
customers. A power failure, short-circuit, or a downed power line would
interrupt power in the entire line, which must be fixed before power can
be restored. A loop system, as the name implies, loops through the
service area and returns to the original point. The loop is usually tied
into an alternate power source. By placing switches in strategic locations,
the utility can supply power to the customer from either direction. If one
source of power fails, switches are thrown (automatically or manually),
and power can be fed to customers from the other source. The loop
system provides better continuity of service than the radial system, with
only short interruptions for switching. In the event of power failures due
to faults on the line, the utility has only to find the fault and switch
around it to restore service. The fault itself can then be repaired with a
minimum of customer interruptions. The loop system is more expensive
than the radial because more switches and conductors are required, but
the resultant improved system reliability is often worth the price.
Network systems are the most complicated and are interlocking loop
systems. A given customer can be supplied from two, three, four, or more
different power supplies. Obviously, the big advantage of such a system
is added reliability. However, it is also the most expensive. For this
reason it is usually used only in congested, high load density municipal
or downtown areas.

1.6 Literature Survey :
In the literature, there are a number of efficient and reliable load
flow solution techniques, such as; Gauss-Seidel, Newton-Raphson and
Fast Decoupled Load Flow. Hitherto they are successfully and widely
used for power system operation, control and planning. However, it has
repeatedly been shown that these methods may become inefficient in the
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analysis of distribution systems with high R/X ratios or special network
structures. Accordingly, a number of methods proposed in the literature
[12-28] specially designed for the solution of power flow problem in radial
distribution networks. The methods developed for the solution of ill-
conditioned radial distribution systems may be divided into two
categories.
The first type of methods is utilized by proper modification of existing
methods such as, Newton-Raphson. On the other hand, the second group
of methods is based on forward-backward sweep processes using
Kirchhoff’s Laws or making use of the well-known bi-quadratic equation
which, for every branch, relates the voltage magnitude at the receiving
end to the voltage at the sending end and the branch power flow for
solution of ladder networks. Shirmohammadi et al. [12] had presented a
compensation based
power flow method for radial distribution networks and extended it for
weakly meshed structure using a multi-port compensation technique and
basic formulations of Kirchhoff’s Laws. The radial part is solved by a
straightforward two step procedure in which the branch currents are first
computed (backward sweep) and then the bus voltages are updated
(forward sweep). In the improved version [13], branch power flow was
used instead of branch complex currents for weakly meshed
transmission and distribution
systems by Luo. Baran and Wu [14], proposed a methodology for solving
the radial load flow for analyzing the optimal capacitor sizing problem. In
this method, for each branch of the network three non-linear equations
are written in terms of the branch power flows and bus voltages. The
number of equations was subsequently reduced by using terminal
conditions associated with the main feeder and its laterals, and the
Newton-Raphson method is applied to this reduced set. The
computational efficiency is improved by making some simplifications in
the jacobian. Consequently, numerical properties and convergence rate of
this algorithm have been studied using the iterative solution of three
fundamental equations representing real power, reactive power and
voltage magnitude by Chiang [15]. G. Renato [16] made use of well-
known bi-quadratic equation which, for every branch, relates the voltage
magnitude at the receiving  end to the voltage at the sending end and
branch power flow. Only voltage magnitudes are computed, bus phase
angles do not appear in the formulation which was also used by Das et
al. in [17]. Jasmon [18] proposed a load flow technique which, for every
branch, leads to a pair of quadratic equations relating power flows at
both ends with the voltage magnitude at the sending end for the voltage
stability analysis of radial networks. Haque [19] had formulated the load
flow problem of the distribution system in terms of three sets of recursive
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equations and analyzed load flow results for various voltage dependent
load models. The effects of various load models on the convergence
pattern of the method are also studied. The effect of voltage-dependency
of load on the results and convergence characteristics of power flow
solution were also analysed [20], where the proposed method was also
based on Kirchhoff’s Laws. Liu et al.[21] had proposed Ratio-Flow method
which is based on forward-backward ladder equation for complex
distribution system by using voltage ratio for convergence control. This
method were applied with standard Newton-Raphson method for complex
distribution systems, which have multiple sources or relatively strong
connected loops with extended long radial feeders including laterals, to
solve the load flow problem. 11 R. Ranjan et al. [22] had proposed a new
method to solve radial distribution networks. They had used simple
algebraic recursive expression of voltage magnitude and the proposed
algorithm used the basic principle of circuit theory. D. Zimmerman and
H. D. Chiang [23] formulated load flow problem as a function of the bus
voltages and equations are solved by Newton’s method. The method has
been compared with classical Newton-Raphson and Forward-Backward
sweep methods by using a number of test cases. Although required
iteration number considerable favoured from classical methods for small
tolerances, no results has been provided on the accuracy of the solution
in terms of bus voltage magnitudes or angles. The results provided in [23]
suggest that undertaken comparisons only cover network structures
which are inherently convergent i.e. Solutions can also be obtained using
classical Newton Raphson method. J.Jerome et al.[25],had proposed
forward-backward substitution method which is based on the Kirchhoff’s
Laws. In backward substitution, each branch current is calculated by
Kirchhoff’s current law
(KCL). Using these currents, the node voltages are calculated by
Kirchhoff’s Voltage Law in forward substitution at each iteration. The
voltage magnitudes at each bus in an iteration are compared with their
values in the previous iteration. If the error is within the tolerance limits,
the procedure is stopped. Ladder network theory shown in ref. [26] is
similar to the Forward-Backward Substitution method. In Ladder
network theory, the currents in each branch are computed by KCL. In
addition to the branch currents, the node voltages are also computed by
KVL in each iteration. Thus magnitude of the swing bus voltage is also
determined. The calculated value of swing bus is compared with its
specified value. If the error is within the limit, the procedure is stopped.
Otherwise, the forward and backward calculations are repeated as in
forward-backward substitution method. The aim of this paper is to
compare the convergence ability of distribution system load flow methods
which are widely used for distribution systems analysis. The method,
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analysed in this section, are classical Newton-Raphson method [2], Ratio-
Flow [21], Forward Backward Substitution method [25] and Ladder
Network Theory [26], The convergence ability of methods were also
evaluated for different tolerance values, different voltage levels, different
loading conditions and different R/X ratios, under the wide range
exponents of loads. Algorithms had been implemented with Matlab codes.
12 A few researchers [29–32] had tried to incorporate composite load
model in their algorithms. The most recent of these is the work of Mok et
al. [33], which included composite loads and solves the networks by
ladder network theory. However, their convergence was not efficient and
takes a high number of iterations. Chiang [34] had also proposed three
different algorithms for solving radial distribution networks based on the
method proposed by Baran and Wu .He had proposed decoupled, fast
decoupled & very fast decoupled distribution load-flow algorithms. In fact
decoupled and fast decoupled distribution load-flow algorithms proposed
by Chiang [34] were similar to that of Baran and Wu [l4]. However, the
very fast decoupled distribution load flow proposed by Chiang [ 16] was
very attractive because it did not require any Jacobian matrix
construction and factorisation. Renato [12] had proposed one method for
obtaining a load-flow solution of radial distribution networks. He has
calculated the electrical equivalent for each node summing all the loads
of the network fed through the node including losses and then, starting
from the source node, the receiving-end voltages of all the nodes are
calculated. Goswami and Basu [35] had presented a direct method for
solving radial and meshed distribution networks. However, the main
limitation of their method is that no node in the network is the junction
of more than three branches, i.e. one incoming and two outgoing
branches. Jasmon and Lee [18] had proposed a new load-flow method for
obtaining the solution of radial distribution networks. They have used the
three fundamental equations representing real power, reactive power and
voltage magnitude derived in [35]. They have solved the radial
distribution network using these three equations by reducing the whole
network into a single he equivalent. Das et al. [36] had proposed a load-
flow technique for solving radial distribution networks by calculating the
total real and reactive power fed through any node. They have proposed a
unique node, branch and lateral numbering scheme which helps to
evaluate exact real and reactive power loads fed through any node.
Accordingly, there are a number of reported studies in the literature [17–
28] specially designed for solution of power flow problem in radial
distribution systems (RDS). Methods developed for the solution of ill-
conditioned radial distribution systems may be divided into two
categories. The first group of methods is based on the forward-backward
sweep process 13 for solution of ladder networks. On the other hand, the
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second group of methods is utilized by proper modification of existing
methods such as Newton-Raphson.

CHAPTER-2

LOAD FLOW ANALYSIS
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2.1 Proposed method:

The load flow of distribution system is different from that of
transmission system because it is radial in nature and has high R/X
ratio. Convergence of load flow is utmost important. Literature survey
shows that the following works had been carried out on load flow studies
of electric power distribution systems. The literature survey of radial
distribution networks has already been presented in Chapter 1 .
In this method of load flow analysis the main aim is to reduce the
data preparation and to assure computation for any type of numbering
scheme for node and branch. If the nodes and branch numbers are
sequential, the proposed method needs only the starting node of feeder,
lateral(s) and sub lateral(s) only. The proposed method needs only the set
of nodes and branch numbers of each feeder, lateral(s) and sub-lateral(s)
only when node and branch numbers are not sequential. The proposed
method computes branch power flow most efficiently and does not need
to store nodes beyond each branch. The voltage of each node is
calculated by using a simple algebraic equation. Although the present
method is based on forward sweep ,it computes load flow of any
complicated radial distribution networks very efficiently even when
branch and node numbering scheme are not sequential.
A 34-node radial distribution networks with constant
power(CP),constant current (CI) and constant impedance (CZ) load
modelling are considered.

2.2 Solution methodology:

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1 I(1) 2 I(2) 3 I(3) 4 I(nb) nb
R(1)+j*Q(1) R(2)+jQ(2) R(3)+jQ(3) R(nb-1)+jQ(nb-1)

P(2)+j*Q(2) P(3)+j*Q(3) P(4)+j*Q(4) P(nb)+j*Q(nb)

Fig 2.1. Radial main feeder

|V(1)| ∟δ (1) I(1) |V(2)|∟δ(2)

1 R(1)+j*X(1)

P(2)+j*Q(2)

Fig. 2.2 Electrical equivalent of fig 1

Consider a distribution system consisting of a radial main feeder
only. The one line diagram of such a feeder comprising n nodes and n-1
branches is shown in Fig. 2.1. Fig. 2.2 shows the electrical equivalent of
Fig. 2.1. From Fig. 2.2, the following equations can be written

_________________________(1)

P(2)-j*Q(2)=V*(2)I(1) ____________________________________________(2)

From eqns. 1 and 2 we have

|V(2)|=[{P(2)R(1)+Q(2)X(1)-0.5|V(1)|2)2-- (R2(1)+X2(1))
(P2(2)+Q2(2))}1/2

-(P(2)R(1)+Q(2)X(1)-0.5|V(1)|2)]1/2 ________________ (3)

Eqn. 3 can be written in generalized form

|V(i+1)|=[{P(i+1)R(i)+Q(i+1)X(i)-0.5|V(i)|2)2 - (R2(i)+X2(i))
(P2(i+1)+Q2(i+1))}1/2

-(P(i+1)R(i)+Q(i+1)X(i)-0.5|V(i)|2)]1/2 _____________________ (4)

Eqn. 4 is a recursive relation of voltage magnitude. Since the substation
voltage magnitude |V(1)| is known, it is possible to find out voltage

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magnitude of all other nodes. From Fig. 2.2 the total real and reactive
power load fed through node 2 are given by

P(2)= + __________________________________ (5)

Q(2)= +

It is clear that total load fed through node 2 itself plus the load of all
other nodes plus the losses of all branches except branch 1.

LP(1)=(R(1)*[P2(2)+Q2(2)])/(|V(2)|2) _____________________________ (6)

LQ(1)=(X(1)*[P2(2)+Q2(2)])/(|V(2)|2)

Eqn. 5 can be written in generalized form

P(i+1)= + for i=1, 2,……, NB-1 _______________(7)

Q(i+1)= + for i=1, 2,……, NB-1

Eqn. 6 can also be written in generalized form

LP(i)=(R(i)*[P2(i+1)+Q2(i+1)])/(|V(i+1)|2) _____________________________(8)

LQ(1)=(X(i)*[P2(i+1)+Q2(i+1)])/(|V(i+1)|2)

Initially, if LP(i+1) and LQ(i+1) are set to zero for all I, then the initial
estimates of P(i+1) and Q(i+1) will be

P(i+1)= for i=1, 2,……, NB-1 ______________________________(9)

Q(i+1)= for i=1, 2,……, NB-1

Eqn. 9 is a very good initial estimate for obtaining the load flow solution
of the proposed method.

The convergence criteria of this method is that if the difference of real
and reactive power losses in successive iterations in each branch is less
than 1 watt and 1 var, respectively, the solution has converged.

Technique of lateral, node and branch numbering:

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Fig.3.1 shows single line diagram of a radial
distribution feeder with laterals. First, we will number the main feeder as
lateral 1 (L=1) and number the nodes and branches of lateral 1 (main
feeder). For lateral 1, source node SN(1)=1, node just ahead of source
node LB(1)=2 and end node EB(1)=12. For lateral 1 there are 12 nodes
and 11 branches. Next we will examine node 2 it does not have any
lateral.

Next, we will examine node 3 of lateral 1. It also has one lateral. The
lateral number is 2. For lateral 2, it is seen that source node SN(2)=3,
node just ahead of source node LB(2)=13 and end node EB(1)=16. For
lateral 2 there are 5 nodes including source node (node 3). The remaining
nodes are numbered as 13, 14, 15 and 16. The branch numbers of
lateral 2 is shown inside brackets(.). Next, we will examine node 4, 5. It
does not have laterals. Next, we will examine node 6 of lateral 1. The
lateral numbered as 3. For lateral 3, source node SN(3)=6, node just
ahead of source node LB(3)=17 and end node EB(3)=27. For lateral 3
there are 11 nodes including source node (node 6). The remaining nodes
are numbered as 17, 18, 19,……….., 27. The branch numbers of lateral 2
is shown inside brackets (.). Similarly we have to examine each node of
lateral 1 and lateral, source node, node just ahead of source node, end
node and branch numbering have to be completed by using above
mentioned technique. Details are given in table. 2.

Table1 : Details of the numbering scheme of figure 3.1

Laterals Source node Node just End node
number SN(L) ahead of EB(L)
source node
LB(L)
Lateral 1 1 2 12
Lateral 2 3 13 16
Lateral 3 6 17 27
Lateral 4 9 28 30
Lateral 5 10 31 34

Any numbering each lateral and nodes we follow the steps described
below. Generalized expressions for TP(L) and TQ(L) are given below:

TP(L)= for L=1,2,….NL _________________ (10)

24


TQ(L)=,j=LB(L)-NN(L)-QL,j.+. for L=1,2,….,NL

Where

NN(1)=EB(1)

NN(2)=EB(2)

…. ….

NN(L)=EB(L)

Now we will define one integer variable F(i),i=1,2,…,NB-1,the meaning of
which is as follows:

From Fig. ,it can be seen that four laterals are connected with different
nodes of lateral 1(main feeder). Laterals are connected with node i.e. two
laterals are connected with node therefore only one lateral is connected
with node i.e. similarly other values of F(i) can easily be obtained. From
Table

Table2 : Non Zero integer values of F(i)

Source node F(i)
SN(L)
3 F(3)=1
6 F(6)=1
9 F(9)=1
10 F(10)=1

It is clear that F(i) is positive only at the source nodes {i=SN(L),L>1}.other
values of F(i) are zeros.

2.3 Explanation of the proposed algorithm:
From Fig. it is seen that for L = 1, total real and reactive power
loads fed through node 2 are TP(1) and TQ(1) (eqn. 10). At any iteration
voltage magnitude of node 2 can easily be obtained by using eqn. 4 {P(2)
= TP(1) and Q(2) = T Q ( 1 ) } . After solving the voltage magnitude of node
2 one has to obtain the voltage magnitude of node 3 and so on. Before
25


proceeding to node 3, we will define here four more variables which are
extremely important for obtaining exact load feeding through nodes 3, 4,
..., EB(1) of lateral 1 or in general obtaining exact load feeding through
LB(L) + 1, LB(L) + 2, ... ., EB(L) of lateral L. It is seen from the flow chart
(Fig. 6) that
SPL(1) = 0 + PL(2) + LP(2) = PL(2) + LP(2)

SQL(1) = 0 + QL(2) + LQ(2) = QL(2) + LQ(2)

where
SPL(1) = real power load of node 2 which has just been left plus real
power loss of branch 2 which has just been left.
SQL(1) = reactive power load of node 2 which has just been left plus
reactive power loss of branch 2 which has just been left.
Next, we have to obtain the value of K (Fig. 6). In this case K = 0 + F(2)
= 0. K =0 indicates that we have no laterals . After that we have to check
whether F(2) is positive or not? But in this case F(2) < 0. Therefore it will
compute PS(1) and QS(1) (Fig.6)
PS(1)=0.0
QS(1)=0.0

PS(1)=0+ =TP(2)

QS(1)=0+ =TQ(2)

TP(2), TP(NL) and TQ(2), TQ(NL) can easily be computed from eqn. 10 and
P1 = P1 + F(2) = 1 + 0 = 1. Therefore, real and reactive power loads fed
through the node 3 are given as:
P(3) = TP(1) - PS(1) - SPL(1)
= TP(1) - PL(2) - LP(2)
Q(3) = TQ(1) - QS(1) - SQL(1)
= TQ(1) - QL(2) - LQ(2)

26


After computing P(3) and Q(3), eqn. 4 has to be solved to obtain the
voltage magnitude at node 3. Before obtaining the voltage magnitude of
node 4, computer logic will perform the following computations:
SPL(1)=PL(2)+LP(2)+PL(3)+LP(3)
SQL(1)=QL(2)+LQ(2)+QL(3)+LQ(3)
and k=0+F(3)=0+1=1.
Next it will check whether F(3) is positive or not? But Total real and
reactive loads fed through the node 4 are: F(3) = 1, therefore
P(4) = TP(1) - PS(1) - SPL(1)

PS(1)=0+ = TP(2)

QS(1)=0+ = TQ(2)

P(4)=TP(1)-PS(1)-SPL(1)
=TP(1)-TP(2)-PL(2)-LP(2)-PL(3)-LP(3)
Q(4)=TQ(1)-PQ(1)-SQL(1)
=TQ(1)-TQ(2)-PQ(2)-LQ(2)-QL(3)-LQ(3)
and solve eqn. 4 for obtaining the voltage magnitude of node 4. For
lateral 1 (L = 1, main feeder) similar computations have to be repeated for
all the nodes. At any iteration, after solving the voltage magnitudes of all
the nodes of lateral 1 one has to obtain the voltage magnitudes of all the
nodes of laterals 2, and so on. Before solving voltage magnitudes of all
the nodes of lateral 2 the voltage magnitude of all the nodes of lateral 1 is
stored in the name of another variable, say V1, i.e. I Vl(J) I = 1 V(J) I for J
= P2 to EB(1) (Fig. 6). For lateral 1 (main feeder) P2 = 1 and EB(1) = 12.
For lateral 2, P2=EB(L)+1=12+1=13. L=L+1=1+1=2, K2 = SN(L) =
SN(2) = 3, |V(EB(1))|=|V(K2)| or |V(12)| = |V(3)| and solve the voltage
magnitudes of all the nodes of lateral 2 using eqn. 4. The proposed
computer logic will follow the same procedure for all the laterals. This will
complete one iteration. After that it will compute total real and reactive
power losses and update the loads. This iterative process continues until
the solution converges.

27


2.4 STATIC LOAD MODELS:
In power flow studies, the common practice is to represent the
composite load characteristic as seen from power delivery points. In
transmission system load flows, loads can be represented by using
constant power load models, as voltages are typically regulated by
various control devices at the delivery points. in distribution systems,
voltages vary widely along system feeders as there are fewer voltage
control devices; therefore, the v-i characteristics of load are more
important in distribution system load flow studies. Load models are
traditionally classified into two broad categories: static models and
dynamic models. Dynamic load models are not important in load flow
studies. Static load models, on the other hand, are relevant to load flow
studies as these express active and reactive steady state powers as
functions of the bus voltages (at a given fixed frequency). These are
typically categorized as follows:
Constant impedance load model (constant z): A static load model
where the power varies with the square of the voltage magnitude. It is
also referred to as constant admittance load model.
Constant current load model (constant I): A static load model where
the power varies directly with voltage magnitude.
Constant power load model (constant p): A static load model where the
power does not vary with changes in voltage magnitude. It is also known
as constant MVA load model.
Exponential load model: A static load model that represents the power
Relationship to voltage as an exponential equation in the following way:
P=Po (V/Vo)a
Q=Qo (V/Vo)b
Where Po and Qo stand for the real and reactive powers consumed
at a reference Voltage Vo. The exponents a and b depend on the type of
load that is being Represented, e.g., for constant power load models
a=b=0, for constant current Load models a=b=1 and for constant
impedance load models a=b=2. It is interesting to note that none of these
loads has a zero exponent, polynomial load model. A static load model
that represents the power-voltage relationship as a polynomial equation
of voltage magnitude. It is usually referred to as the ZIP model, as it is
made up of three different load models: constant impedance (Z), constant
current (I ) and constant power (P). The real and reactive power
characteristics of the ZIP load model are given by
28


P=Po [ ap(V/V0)2+bp(V/Vo)+cp ]
Q=Qo [ aq(V/V0)2+bq(V/Vo)+cq ]

Where ap+bp+cp=aq+bq+cq=1, and Po and Qo are the real and reactive
Power consumed at a reference voltage Vo. In this paper, three types of
static Load models, i.e., constant power, constant current and constant
impedance, Are considered to demonstrate their effect on voltage
regulation calculations in Distribution systems. The studies presented in
this paper can be readily extended to other load models as well.

29


2.5 ALGORITHM FOR LOAD FLOW COMPUTATION:

The complete algorithm for load flow calculation of radial distribution
network is shown in below.

Step1 : Read the system voltage magnitude |v(i)|, line parameters and

load data.

Step2 : Read base KV and base MVA.

Step3 : Read total number of nodes nb,

Step4 : compute per unit values of load powers at each node i.e. pl(i)

And ql(i) for i=1, 2, 3,…nb, as well as resistance and reactance

of each branch i.e. r(j) and x(j) for j=1, 2, 3,……..nb-1.

Step5 : By examine the radial feeder network note down the lateral

number l, source node sn(l), node just ahead of source node

lb()l, end node eb(l).

Step6 : Read the nonzero integer value f(i), i.e. whether node consists of

lateral or not. If yes f(i)=1, otherwise f(i)=0, for i=1, 2, 3,…nb

Step7 : Initialize the branch losses lp(i)=0.0, lq(i)=0.0 for i=1, 2, 3,.nb-1

Step8 : set iteration count IT=1, ε(0.0001).

Step9 : compute TP(l) and TQ(l) by using eqn. 10

Step10 : compute TP(1)=sum(TP), TQ(1)=sum(TQ).

Step11 : set the losses ploss(i)=lp(i), qloss(i)=lq(i) for i=1, 2, 3,…..nb-1

Step12 : l=1, p2=1

Step13 : for i=1

Step14 : set k=0, p1=1

Step15 : initialize spl(l)=0.0, sql(l)=0.0, ps(l)=0.0, qs(l)=0.0
30


Step16 : k=k+f(i)

Step17 : If f(i) is greater than zero go to next step otherwise go to step20

Step18 : compute ps(l) and qs(l) by using the formulae are

ps(l)=ps(l)+TP(l+i3), qs(l)=qs(l)+TQ(l+i3).

Step19 : p1=p1+f(i)

Step20 : compute node real power and reactive powers by using eqn. 7

Step21 : solve the eqn. 4 for |v(i+1)|

Step22 : i is incremented by i+1

Step23 : If i is not equal to eb(l) go to next step otherwise go to step26

Step24 : compute spl(l), sql(l) by using eqns.

SPL(l)=SPL(l)+PL(i)+LP(i)

SQL(l)=SQL(l)+QL(i)+LQ(i)

Step25 : Then go to step 16

Step26 : |v1(j)|=|v(j)| for j=p2 to eb(l).

Step27 : If i is not equal to nb then go to next step otherwise go to

step32

Step28 : set k1=eb(l), p2=eb(l+1)

Step29 : l is incremented by l+1.

Step30 : set k2=sn(l)

Step31 : set |v(k1)|=|v(k2)| then go to step step5.

Step32 : compute lp(i), lq(i) by using eqn.8 for i=1, 2, 3,…nb-1

Step33 : compute dp(i) and dq(i) by using eqns

dp(i)=lp(i)-ploss(i)

dq(i)=lq(i)-qloss(i) for i=1, 2, 3,…nb-1

Step34 : If (max |(dp(i))| & max|(dq(i))|) is less than not equal ε go to

next step otherwise go to step36

31


Step35 : IT is incremented by IT+1, then go to step8

Step36 : write voltage magnitudes and feeder losses.

Step37 : stop

32


FLOW CHART:
START

Read S/S voltage
magnitude |v(i)|, line
parameters and load
data.

Initialize LP(i)=0

LQ(i)=0 for i=1,2…
NB-1

IT=1

Compute TP(L) and
TQ(L) by using eqn.

From (A)

TP(1)=sum(TP)

TQ(1)=sum(TQ

Set PLOSS(i)=LP(i)

QLOSS(i)=LQ(i)

For 1=1,2,…NB-1

Set
L=1,i=1,P2=1

From(B)
33


K=0,P1=1

Initialize

SPL(L)=0.0,SQL(L)=0.0

PS(L)=0.0,QS(L)=0.0

K=K+F(i)

From (C)

Is
F(i)>0
?

PS(L)=PS(L)+
no
QS(L)=QS(L)+

P1=P1+F(i)

yes

P(i+1)=TP(L)-PS(L)-SPL(L)
34
Q(i+1)=TQ(L)-QS(L)-SQL(L)


Solve eqn.
4 for |
V(i+1)|

i=i+1

SPL(L)=SPL(L)+PL(i)+LP(i)
Is
i==EB(L) SQL(L)=SQL(L)+QL(i)+LQ(i)
yes yes

no

|V1(J)|=|V(J)|
for J=p2 to To (C)
EB(L)

35


Is no
i==NB

K1=EB(L)
yes
Compute LP(i) and LQ(i)
for i=1,2,…NB-1 by
using eqn. 8
p2=EB(L)+1

Compute L=L+1

DP(i)=LP(i)-PLOSS(i)

DQ(i)=LQ(i)-QLOSS(i)

K2=SN(L)

|V(k1)|=|
V(K2)|

IT=IT+1 is max(|DP(i)|
&max|
DQ(i)|)<ε

no To (B)

no
Write voltage
magnitudes and feeder
losses
To (A)
36

stop


yes

Fig. 2.3 Flow chart for radial distribution network having laterals.

CHAPTER-3

EXAMPLES

One example has been considered to demonstrate the effectiveness of
the proposed method. The first example is 34 node radial distribution
network (nodes have been renumbered with Substation as node 1) shown
in Figure 3.1. Data for this system are available in [9] shown in Appendix
A. Real and reactive power losses of this system for CP, CI, CZ load
modelling is shown in Table 2.1. The minimum voltage occurs at node
number 27 in all cases. Base values for this system are 11 kV and 1
MVA respectively.
● 34
37


Fig. 3.1 : 34 node radial distribution network

● 30 ● 33

● 29 ● 32

● 28 ● 31

1 2 3 4 5 6 7 8 9 10 11 12
S/S ● ● ● ● ● ● ● ● ● ● ●

● 13 ● 17

14 ● ● 18

15 ● ● 19

16 ● 20 ● ● ● ● ● ● ● ●

21 22 23 24 25 26 27

Table 3: Voltages for different static load model.

Node Voltages of Voltages of Voltages of
number constant power constant current constant
load model load model impedance load
model
1 1.0000 1.0000 1.0000
2 0.9940 0.9942 0.9945
3 0.9888 0.9893 0.9897
4 0.9817 0.9825 0.9833
5 0.9756 0.9767 0.9777
6 0.9699 0.9712 0.9725
7 0.9658 0.9673 0.9688
8 0.9636 0.9652 0.9667
9 0.9611 0.9628 0.9644
38


10 0.9599 0.9617 0.9633
11 0.9595 0.9612 0.9629
12 0.9593 0.9611 0.9628
13 0.9885 0.9889 0.9894
14 0.9882 0.9886 0.9891
15 0.9881 0.9886 0.9890
16 0.9881 0.9885 0.9890
17 0.9654 0.9670 0.9685
18 0.9617 0.9635 0.9652
19 0.9576 0.9596 0.9615
20 0.9543 0.9565 0.9586
21 0.9515 0.9538 0.9561
22 0.9482 0.9507 0.9532
23 0.9455 0.9482 0.9508
24 0.9430 0.9459 0.9486
25 0.9418 0.9447 0.9475
26 0.9413 0.9443 0.9471
27 0.9412 0.9442 0.9470
28 0.9655 0.9670 0.9685
29 0.9653 0.9668 0.9683
30 0.9652 0.9667 0.9682
31 0.9596 0.9613 0.9630
32 0.9594 0.9612 0.9629
33 0.9592 0.9610 0.9627
34 0.9591 0.9609 0.9626

Base voltage=11kv Base MVA=1MVA

Table 4 : Power losses of different static load models.

Type of load model Real power Reactive power
losses (per losses (per unit)
unit)
Constant power load 0.2276 0.0668

Constant current load 0.2066 0.0607

Constant impedance 0.1877 0.0553
load

39


Table 5: voltage regulation for different static load models

Type of load Voltage regulation(in %)
model
Constant power load 6.2497
Constant current load 5.9143
Constant impedance load 5.5996

CHAPTER-4

SUMMARY AND FUTURE SCOPE

40


Conclusion:
A novel load flow technique, named “FORWARD SWEEPING
METHOD”, has been proposed for solving radial distribution networks. It
completely exploits the radial feature of the distribution network. A
unique lateral, node and branch numbering scheme has been suggested
which helps to obtain the load flow solution of the radial distribution
network. The forward sweeping method always guarantees convergence of
any type of practical radial distribution network with a realistic R/X
ratio.

In this thesis work a method of load flow analysis has been proposed
for radial distribution networks based on the forward sweeping method to
identify the set of branches for every feeder, lateral and sub-lateral
without any repetitive search computation of each branch current.
Effectiveness of the proposed method has been tested by an example 34-
node radial distribution network with constant power load, constant
current load, constant impedance load for each of this example. The
power convergence has assured the satisfactory convergence in all these
cases. The proposed method consumes less amount of memory compared
to the other due to reduction of data preparation. Several Indian rural
distribution networks have been successfully solved using the proposed
forward sweeping method.

This paper demonstrates how voltage regulation calculations in
distribution system vary with different static loads models. Systems with
constant power load models presenting high voltage along a feeder, and
thus high voltages regulation, followed by systems with constants
impedance load models. Hence it is important to choose the load models
more suitable for a given system in order to obtain accurate results.

Future Scope of Work:

The following are the scopes of future work

(a) Fuzzy load-flow analysis.

(b) Load-flow analysis using Genetic Algorithms

41


BIBLIOGRAPHY:
REFERENCES :

42


1. T. Gonen, Electric Power Distribution System Engineering (McGraw
Hill, New York, 1986).
2. G. T. Heydt, Electric Power Quality, 2nd edn (Stars in a Circle
Publications, West LaFayette, IN, 1991).
3. M. E. El-Hawary and L. G. Dias, ‘Incorporation of load models in load-
flow studies: form of models effects’, IEE Proc. C, 134(1) (1987), 27–
30.
4. P. S. R. Murty, ‘Load modelling for power flow solution’, J. Inst. Eng.
(India), Part EL , 58(3) (1977) 162–165.
5. M. H. Haque, ‘Load flow solution of distribution systems with voltage
dependent load models’, Int. J. Electric Power System Res., 36 (1996),
151–156.
6. T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power
Systems, Power Electronics and Power System Series, Kluwer, 1998.
7. J. D. Glover and M. Sarma, Power System Analysis and Design, 2nd
edn (PWS Publishing Company, Boston, 1993).
8. C. G. Renato, ‘New method for the analysis of distribution networks’,
IEEE T rans. Power Delivery, 5(1) (1990), 391–396.
9. D. Das, H. S. Nagi, and D. P. Kothari, ‘Novel methods for solving radial
distribution networks’, IEE Proc. Generation T ransmission and
Distribution, 141(4) (1994).
10. M. M. A. Salama and A. Y. Chikhani, ‘A simplified network approach
to the var control problem for radial distribution systems’, IEEE T
rans. Power Delivery, 8(3) (1993), 1529–1535.

APPENDIX A :
43


Table: line data for 34 node radial distribution network.

Bran
ch Sendi Receivi R(oh X(oh CSA/length(mm2
No. ng ng end ms) ms) /km)
end
1 1 2 0.195 0.080 185/0.6
2 2 3 0.195 0.080 185/0.55
3 3 4 0.299 0.083 120/0.55
4 4 5 0.299 0.083 120/0.55
5 5 6 0.299 0.083 120/0.55
6 6 7 0.524 0.090 70/0.6
7 7 8 0.524 0.090 70/0.4
8 8 9 0.524 0.090 70/0.6
9 9 10 0.524 0.090 70/0.4
10 10 11 0.524 0.090 70/0.25
11 11 12 0.524 0.090 70/0.2
12 3 13 0.524 0.090 70/0.3
13 13 14 0.524 0.090 70/0.4
14 14 15 0.524 0.090 70/0.2
15 15 16 0.299 0.083 70/0.1
16 6 17 0.299 0.083 120/0.6
17 17 18 0.378 0.086 120/0.55
18 18 19 0.378 0.086 95/0.55
19 19 20 0.378 0.086 95/0.5
20 20 21 0.524 0.090 95/0.5
21 21 22 0.524 0.090 70/0.5
22 22 23 0.524 0.090 70/0.5
23 23 24 0.524 0.090 70/0.6
24 24 25 0.524 0.090 70/0.4
25 25 26 0.524 0.090 70/0.25
26 26 27 0.524 0.090 70/0.2
27 9 28 0.524 0.090 70/0.3
28 28 29 0.524 0.090 70/0.3
29 29 30 0.524 0.090 70/0.3
30 10 31 0.524 0.090 70/0.3
31 31 32 0.524 0.090 70/0.4
32 32 33 0.524 0.090 70/0.3
33 33 34 0.524 0.090 70/0.2

44


Table: Load data for 34 node radial distribution system.

Node PL(KW) QL(kVA
number R)
1 0.0 0.0
2 230 142.5
3 0 0
4 230 142.5
5 230 142.5
6 0 0
7 0 0
8 230 142.5
9 230 142.5
10 0 0
11 230 142.5
12 137 84
13 72 45
14 72 45
15 72 45
16 13.5 7.5
17 230 142.5
18 230 142.5
19 230 142.5
20 230 142.5
21 230 142.5
22 230 142.5
23 230 142.5
24 230 142.5
25 230 142.5
26 230 142.5
27 137 85
28 75 48
29 75 48
30 75 48
31 57 34.5
32 57 34.5
33 57 34.5
34 57 34.5

45


APPENDIX B:

clc;
clear;
%format long;
basekv=11;
basemva=1;
m=[1 1 2 0.000967 0.000396
2 2 3 0.000886 0.000363
3 3 4 0.001359 0.000373
4 4 5 0.001235 0.000343
46


5 5 6 0.001235 0.000343
6 6 7 0.00259 0.000446
7 7 8 0.00173 0.0002975
8 8 9 0.00259 0.000446
9 9 10 0.00173 0.0002975
10 10 11 0.00108 0.0001859
11 11 12 0.000866 0.0001487
12 3 13 0.00129 0.0002231
13 13 14 0.00173 0.0002975
14 14 15 0.000866 0.0001487
15 15 16 0.000433 0.00007438
16 6 17 0.001482 0.0004115
17 17 18 0.001359 0.000377
18 18 19 0.001718 0.000391
19 19 20 0.001562 0.000355
20 20 21 0.001562 0.000355
21 21 22 0.002165 0.000372
22 22 23 0.002165 0.000372
23 23 24 0.00259 0.000416
24 24 25 0.00173 0.0002975
25 25 26 0.00108 0.0001859
26 26 27 0.000866 0.0001487
27 7 28 0.00129 0.0002321
28 28 29 0.00129 0.0002321
29 29 30 0.00129 0.0002321
30 10 31 0.00129 0.0002321
31 31 32 0.000866 0.0001487
32 32 33 0.00129 0.0002321
33 33 34 0.00173 0.000295];
br=max(m(:,1));
r=m(:,4);
x=m(:,5);
nb=max(m(:,3));
v(1)=1;
v1=ones(1,nb);
a=[1 0 0
2 0.32 0.1425
300
4 0.23 0.1425
5 0.23 0.1425
600
700
8 0.32 0.1425
47


9 0.23 0.1425
10 0 0
11 0.23 0.1425
12 0.137 0.084
13 0.072 0.045
14 0.072 0.045
15 0.072 0.045
16 0.0135 0.0075
17 0.23 0.1425
18 0.23 0.1425
19 0.23 0.1425
20 0.23 0.1425
21 0.23 0.1425
22 0.23 0.1425
23 0.23 0.1425
24 0.23 0.1425
25 0.23 0.1425
26 0.23 0.1425
27 0.137 0.085
28 0.075 0.048
29 0.075 0.048
30 0.075 0.048
31 0.057 0.0345
32 0.057 0.0345
33 0.057 0.0345
34 0.057 0.0345];
pa=a(:,2);
qa=a(:,3);
b=[1 1 2 12;2 3 13 16;3 6 17 27;4 7 28 30;5 10 31 34];
l=b(:,1);
u=max(b(:,1));
sn(l)=b(:,2);
lb(l)=b(:,3);
eb(l)=b(:,4);
c=[3 1;6 1;7 1;10 1;34 0];
i=c(:,1);
f(i)=c(:,2);
v2=ones(1,nb);
for g=1:3
for i=1:nb
pl(i)=pa(i)*v2(i)^(g-1);
ql(i)=qa(i)*v2(i)^(g-1);
end
48


for i=1:nb-1
lp(i)=0.0;
lq(i)=0.0;
end
t=1;diff=1;diff1=1;
while (diff>0.000001 && diff1>0.000001)
for l=1:u
p1=0;p2=0;q1=0;q2=0;
for j=lb(l):eb(l)
p1=p1+pl(j);
q1=q1+ql(j);
end
for j=lb(l):eb(l)-1
p2=p2+lp(j);
q2=q2+lq(j);
end
tp(l)=p1+p2;
tq(l)=q1+q2;
end
tp(1)=sum(tp);
tq(1)=sum(tq);
for i=1:nb-1
ploss(i)=lp(i);
qloss(i)=lq(i);
end
p2=1;
for l=1:u
k=0;p1=1;
spl(l)=0;sql(l)=0;ps(l)=0;qs(l)=0;
for i=lb(l)-1:eb(l)
if (i<=eb(l)-1)
k=k+f(i);
if (f(i)>0)
for i3=p1:k
ps(l)=ps(l)+tp(l+i3);
qs(l)=qs(l)+tq(l+i3);
end
p1=p1+f(i);
end
p(i+1)=tp(l)-ps(l)-spl(l);
q(i+1)=tq(l)-qs(l)-sql(l);
v(i+1)=sqrt(sqrt((p(i+1)*r(i)+q(i+1)*x(i)-0.5*v(i)^2)^2-
(r(i)^2+x(i)^2)*(p(i+1)^2+q(i+1)^2))-(p(i+1)*r(i)+q(i+1)*x(i)-0.5*v(i)^2));
49


if ((i+1)~=eb(l))
spl(l)=spl(l)+pl(i+1)+lp(i+1);
sql(l)=sql(l)+ql(i+1)+lq(i+1);
end
end
end
for j=p2:eb(l)
v1(j)=v(j);
end
if (i~=nb)
k1=eb(l);
p2=eb(l)+1;
k2=sn(l+1);
v(k1)=v(k2);
end
end
for i=1:nb
h(i,g)=v1(i);
end
for i=1:nb-1
lp(i)=r(i)*((p(i+1)^2+q(i+1)^2))/(v(i+1)^2);
lq(i)=x(i)*((p(i+1)^2+q(i+1)^2))/(v(i+1)^2);
di(i)=lp(i)-ploss(i);
di1(i)=lq(i)-qloss(i);
end
for i=1:nb-1
lp1(i,g)=lp(i);
lq1(i,g)=lq(i);
end
diff=max(di(1,:));
diff1=max(di1(1,:));
t=t+1;
end
if(g==1)
for i=1:nb
v2(i)=v1(i);
end
end
end
mincp=min(h(:,1));
minci=min(h(:,2));
mincz=min(h(:,3));
cpreg=((v(1)-mincp)/(mincp))*100;
50


cireg=((v(1)-minci)/(minci))*100;
czreg=((v(1)-mincz)/(mincz))*100;

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