Transmission Network Cost Allocation Using Bus Impedance Mat

8/8/2019 Transmission Network Cost Allocation Using Bus Impedance Mat

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A DISSERTATION

SUBMITTED TO THE FACULTY OF ENGINEERING

OF

NATIONAL INSTITUTE OF TECHNOLOGY, WARANGAL (A.P)

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE AWARD OF THE DEGREE OF

MASTER OF TECHNOLOGY

IN

POWER SYSTEMS ENGINEERING

BY

D. Veera Nageswara Rao(061725)

Under the esteemed guidance of

Prof.M.Sydulu

DEPARTMENT OF ELECTRICAL ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY

WARANGAL-506 004(A.P)

MAY-2008

TRANSMISSION NETWORK

COST ALLOCATION USING BUS IMPEDANCE

MATRIX (ZBUS)


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DEPARTMENT OF ELECTRICAL ENGINEERINGNATIONAL INSTITUTE OF TECHNOLOGY

WARANGAL-506004

CERTIFICATE

This is to certify that the dissertation work entitled

Transmission Network Cost Allocation using Bus Impedance

Matrix(Zbusis bonafide record of the work donebyD.Veera nageswara rao

(Roll No. 061725) and submitted in partial fulfillment of the

requirements for the award of degree of Master of Technology in

Electrical Engineering with specialization in Power Systems

Engineering , from National Institute of Technology, Warangal.

Dr. M. Sydulu Dr.D.M.Vinod Kumar

Professor (Thesis Advisor) Professor and Head of the Department

Head Power System Section Dept. of Electrical Engineering

Dept. of Electrical Engineering National Institute of Technology

National Institute of Technology Warangal.

Warangal.


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ACKNOWLEDGEMENT

I write this acknowledgement with great honor, pride and pleasure to pay my respects to

all who enabled me either directly or indirectly in reaching this stage.

I am indebted forever to my guide Dr. M. Sydulu, Professor, Department of Electrical

Engineering, for his suggestions, guidance and inspiration in carrying out this project

work.

I express my profound thanks to Dr.D.M.Vinod Kumar, Professor and Head of Electrical

Engineering Department, for providing me with all the facilities to carry out this project

work.

I take this opportunity to convey my sincere thanks to all my class mates who have

directly and indirectly contributed for the successful completion of this work.

D.VEERA NAGESWARA RAO


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SYNOPSIS

With the introduction of restructuring into the electric power industry, the price of

electricity has became the focus of all activities in the power market. In general, the price

of a commodity is determined by supply and demand.

In the present open access restructured power system market, it is necessary to

develop an appropriate pricing scheme that can provide the useful economic information

to market participants, such as generation, transmission companies and customers.

However, accurately estimating and allocating the transmission cost in the transmission

pricing scheme is a challenging task although many methods have been proposed.

The purpose of the methodology is to allocate the cost pertaining to the

transmission lines of the network to all the generators and demands. Once a load flow

solution is available, the proposed method determines how line flows depend on nodal

currents. This result is then used to allocate network costs to generators and demands.

This work addresses the problem of allocating the cost of the transmission

network to generators and demands. This work proposes three methods using bus

impedance matrix Zbus. The three techniques are Zbus method , Zbusavg method and a

newly proposed technique. The new method is very effective in transmission cost

allocation A physically-based network usage procedure is proposed..

The techniques presented in this work is related to the allocation of the cost of

transmission losses based on the Zbus. It should be emphasized that all transmission lines

must be modeled including actual shunt admittances. Doing so, the impedance matrixpresents an appropriate numerical behavior.A salient feature of the proposed techniques

are its embedded proximity effect, which implies that a generator/demand uses mostly the

lines electrically close to it. This is not artificially imposed but a result of relying on

circuit theory.


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The proposed method provides a methodology to apportion the cost of the

transmission network to generators and demands that use it. How to allocate the cost of

the transmission network is an open research issue as available techniques embody

important simplifying assumptions, which may render controversial results. This work

contributes to seek an appropriate solution to this allocation problem using an usage-

based procedure that relies on circuit theory.

This new procedure exhibits desirable apportioning properties and is easy to

implement and understand. Case studies on 4-bus system and IEEE 24-bus system are

used to illustrate the working of the proposed techniques. Relevant and important

conclusions are finally drawn


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CONTENTS

NOMENCLATURE

LIST OF TABLES

LIST OF FIGURES

Page No

CHAPTER 1: INTRODUCTION 1

1.1 Deregulation 11.2 Independent System Operator (ISO) 21.3 Open Access Same time Information System (OASIS) 31.4 Transmission Use of System Tariffs (TUSTs) 41.5 power wheeling costs 61.6 Literature Review 71.7 Contributions 81.8 Outlines of the Thesis 9

CHAPTER 2: TRANSMISSION NETWORK COST ALLOCATION

USING ZBUS TECHNIQUE 10

2.1 Problem Statement 102.2 Background 10

2.3 Transmission Cost Allocation 132.4 Algorithm For Transmission Network Cost Allocation Using

Zbus Technique 152.5 Case study 4 bus system 18

2.5.1 step by step results 19

2.6 Conclusions 22

CHAPTER 3: TRANSMISSION NETWORK COST ALLOCATION

USING avgbusZ TECHNIQUE 23

3.1 Problem Statement 233.2 Background 233.3 Transmission Cost Allocation 263.4 Effect of Flow Directions 28

3.5 Algorithm For Transmission Network Cost Allocation Usingavg

busZ Technique 29


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3.6 Case study 4 bus system 343.6. 5.1 step by step results 35

3.7 Conclusions 42

CHAPTER 4: A NEW APPROACH FOR TRANSMISSION

NETWORK COST ALLOCATION USING

MODIFIED avgbusZ TECHNIQUE

4.1 Problem Statement 434.2 Background 434.3 Transmission Cost Allocation 464.4 Algorithm for Transmission network cost allocation Using

modified

avg

busZ technique (newly proposing technique) 484.5 Case Study - 4 - Bus System 53

4.5.1 Step By Step Results 4 bus system 544.6 Conclusions 60

CHAPTER 5: RESULTS-IEEE RTS 24 BUS SYSTEM AND

CONCLUSIONS 61

5.1 Zbus technique Results 61

5.2avg

bus

Z technique results 68

5.3 Modified avgbus

Z technique results 75

5.4 comparison of Zbus based techniques 825.5 conclusions 84

APPENDIX 85

A.1 4-Bus System Data 85A.2 IEEE 24- Bus Reliability Test System 86

REFERENCES 89


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NOMENCLATURE

jkC -Cost of line jk ($/h)

Ii -Nodal current (A)

I jk -Current through the line jk(A)

n -Number of buses

PGi - Active power consumed by the generator located at bus i (W)

PDi - Active power consumed by the load located at bus i (W)

Pjk - Active power flow through line jk (W)

Sjk - Complex power flow through line jk calculated at bus j (VA)

Vj - Nodal voltage at bus j (V)

yjk - Series admittance of the -equivalent circuit of line jk (S)sh

jky - Shunt admittance of the - equivalent circuit of line jk (S)

Zbus - Impedance matrix (ohm)

Zij - Element ij of the impedance matrix (ohm)

i

jka - Electrical distance between bus i and line jk (adimensional)

DiC - Total transmission cost allocated to the load located at bus i ($/h)

GiC - Total transmission cost allocated to the generator located at bus i ($/h)

Di

jkC - Transmission cost allocated to the generator located at bus i ($/h)

Gi

jkC - Transmission cost allocated to the generator located at bus i ($/h)

i

jkP - Active power flow through the line jk associated with the nodal current i(W)

rjk - Cost rate for line jk ($/W & h)

Gi

jkU - Usage of line jk allocated to the generator located at bus (W).

DijkU - Usage of line jk allocated to the generator located at bus (W).

i

jkU - Usage of line associated with nodal current (W).

Ujk - Usage of line jk (W).


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LIST OF TABLES

Page No

Table 2.1 Converged Voltages of Zbus technique 19

Table 2.2 Bus Currents of Zbus technique 19Table 2.3 Powerflow Contributions P(i,k)in Pjk>0 direction of Zbus technique 19

Table 2.4 Powerflow Usage Contributions U(k,i)in Pjk>0 direction of Zbus technique 20

Table 2.5 Powerflow Usage of Line usage(k)in Pjk>0 direction of Zbus technique 20

Table 2.6 Powerflow Contributions Ug(i,k)in Pjk>0 direction of Zbus technique 20

Table 2.7 Powerflow Contributions Ud(i,k)in Pjk>0 direction of Zbus technique 20

Table 2.8 Generator Cost Contributions cg(k,i)in Pjk>0 direction of Zbus technique 21

Table 2.9 Load Cost Contributions cd(k,i) in Pjk>0 direction of Zbus technique 21

Table 2.10 total generation and load costs and Total cost for all the buses in Pjk>0

direction of Zbus technique 21

Table 3.1 Converged Voltages of avgbus

Z technique 35

Table 3.2 Bus Currents of avgbus

Z technique 35

Table 3.3 Powerflow Contributions P(i,k)in Pjk>0 direction of avgbus

Z technique 35

Table 3.4 Powerflow Usage Contributions U(k,i)in Pjk>0 direction of avgbus

Z technique 36

Table 3.5 Powerflow Usage of Line usage(k)in Pjk>0 direction of avgbusZ technique 36

Table 3.6 Powerflow Contributions Ug(i,k)in Pjk>0 direction of avgbusZ technique 36

Table 3.7 Powerflow Contributions Ud(i,k)in Pjk>0 direction of avgbusZ technique 36

Table 3.8 Generator Cost Contributions cg(k,i)in Pjk>0 direction of avgbusZ technique 37

Table 3.9 Load Cost Contributions cd(k,i) in Pjk>0 direction of avgbusZ technique 37

Table3.10 Total generation and load costs and Total cost for all the buses in Pjk>0

direction ofavg

busZ technique 37

Table3.11 Powerflow Contributions P1(k,i)in Pjk


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Table3.14 Powerflow Contributions Ug1(i,k)in Pjk0 direction of modified avgbusZ

technique 55

Table 4.8 Powerflow Contributions Ud(i,k) in Pjk>0 direction of modified avgbusZ

technique 55

Table 4.9 Generator Cost Contributions cg(k,i) in Pjk>0 direction of modified avgbusZ

technique 56

Table 4.10 Load Cost Contributions cd(k,i) in Pjk>0 direction of modified avgbus

Z


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technique 56

Table 4.11 Total generation and load costs and Total cost for all the buses of modified

avg

busZ technique 56

Table 4.12 Powerflow Contributions S11(i,k) in Pjk


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Table 5.3 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique 63




Table 5.7 Cost For Individual Generators/Loads And Total Cost in Pjk>0 Direction of

Zbus technique 67

Table 5.8 Average Generator Cost Contributions Cgavg(k,i) of avgbusZ technique 68

Table 5.9 Average Generator Cost Contributions Cgavg(k,i ) of avgbusZ technique 69

Table 5.10 Average Load Cost Contributions Cdavg(k,i) of avgbusZ technique 70




Table 5.14 Average Cost For Individual Generators/Loads and Total Avg Cost of avgbusZ

technique 74

Table 5.15 Average Generator Cost Contributions cgavg(k,i) of modified avgbusZ

technique 75

Table 5.16 Average Generator Cost Contributions cgavg(k,i) of modified

avg

busZ

technique 76

Table 5.17 Average Load Cost Contributions cdavg(k,i) of modified avgbusZ technique 77

Table 5.18 Average Load Cost Contributions cdavg(k,i) of modifiedavg

busZ

technique 78

Table 5.19 Average Load Cost Contributions Cdavg(k,i) of modified avgbusZ technique 79

Table 5.20 Average Load Cost Contributions Cdavg(k,i) of modified avgbusZ technique 80

Table 5.21 Average Cost For Individual Generators/Loads and total average cost of

modified avgbusZ technique 81


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LIST OF FIGURES

Page No

Fig 1.1 Expansion in Centralized Systems 5Fig 1.2 Expansion in Competitive Environment 5

Fig. 2.1.Equivalent circuit of line jk of Zbus technique 10

Fig. 2. 2 Four Bus System of Zbus technique 18

Fig. 3.1 Equivalent circuit of line jk of avgbusZ technique 23

Fig. 3. 2Four Bus System of avgbusZ technique 34

Fig. 4.1.Equivalent circuit of line jk of modified avgbusZ technique 43

Fig. 4. Four Bus System of modified avgbusZ technique 53

Fig A.1 IEEE 24-bus Reliability Test System 85


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CHAPTER 1

INTRODUCTION

1.1 Deregulation

In Eighties, almost all electric power utilities throughout the world were operated

with an organizational model in which one controlling authoritythe utilityoperated

the generation, transmission, and distribution systems located in a fixed geographic area

and it refers to as vertically integrated electric utilities(VIEU). Economists for some time

had questioned whether this monopoly organization was efficient. With the example of

the economic benefits to society resulting from the deregulation of other industries such

as telecommunications and airlines, electric utilities are also introducing privatization in

their sectors to improve efficiency. During the nineties many electrical utilities and powernetwork companies world wide have been forced to change their ways of doing business

from vertically integrated mechanism to open market system. This kind of process is

called as deregulation or restructuring or unbundling.

Deregulation word refers to un-bundling of electrical utility or restructuring of

electrical utility and allowing private companies to participate. The aim of deregulation is

to introduce an element of competition into electrical energy delivery and thereby allow

market forces to price energy at low rates for the customer and higher efficiency for the

suppliers and the necessity for deregulation is

(i) To provide cheaper electricity.

(ii) To offer greater choice to the customer in purchasing the economic Energy.

(iii) To give more choice of generation.

(iv) To offer better services with respect to power quality i.e. Constant voltage,

Constant frequency and uninterrupted power supply.


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The benefits that the customers and government will get with the deregulated power

systems are

(i) Cheaper Electricity

(ii) Efficient capacity expansion planning at GENCO level, Transco level

and disco level.

(iii) Pricing is cost effective rather than a set tariff.

(iv) More choice of generation.

(v) Better service is possible.

1.2 Independent System Operator(ISO)

In deregulated power systems TRANSCOs, GENCOs, DISCOs are under

different organizations. To maintain the coordination between them there will be one

system operator in all types of deregulated power system models, generally called

Independent System Operator (ISO).

In deregulated environment, all the GENCOs and DISCOs make the transactions

ahead of time, but by the time of implementations, there may be congestion in some of

the transmission lines. Hence, ISO has to relieve that congestion so that the system is

maintained in secure state.

Cost free means:

(i) Out-aging of congested lines.

(ii) Operation of transformer taps/phase shifters.

(iii) Operation of FACTS devices particularly series devices.

Non-cost-free means:

(i) Re-dispatch of generation in a manner different from the natural settling pointof the market. Some generators back down while others increase their output.

The effect of this is that generators no longer operate at equal incremental

costs.

(ii) Curtailment of loads and the exercise of (not-cost-free) load interruption

options.


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In the deregulated power system the challenge of congestion management for the

transmission system operator (ISO) is to create a set of rules that ensure sufficient control

over producers and consumers (generators and loads) to maintain and acceptable level ofpower system security and reliability in both the short term (real-time operation) and the

long term while maximizing market efficiency. The rules must be robust, because there

will be many aggressive entities seeking to exploit congestion to create market power and

increased profits for themselves at the expense of market efficiency. The rules should

also be fair in how they affect participant, and they should be transparent, that is, it

should be clear to all participants why a particular outcome has occurred.

As deregulation of the electric system becomes an important issue in many

countries, the transmission congestion management, which the ISO has to perform more

frequently, is challenging.

1.3 Open Access Same time Information System (OASIS):

Power transaction between a specific seller bus/area and a buyer bus/area can be

committed only when sufficient Available Transfer Capacity (ATC) is available for that

interface to ensure the system security. The information about the ATC is to be

continuously updated and made available to the market participants through the Internet-

based system such as Open Access Same time Information System (OASIS).

In a Deregulated Power Structure, Power producers and customers share a

common Transmission network for wheeling power from the point of generation to the

point of consumption.


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1.4 Transmission Use of System Tariffs (TUSTs):

In many countries worldwide important changes in the electric sector have

occurred, through a process whose main characteristic is the substitution of a centralized

environment, where a planning institute is responsible by the system expansion, for a

competitive environment in generation (G) and retailing. In turn, the transmission (T) and

distribution (D) sectors remain under regulation due to their characteristics as natural

monopolies. The implementation of a competitive environment in the generation area is

conceptually straightforward: agents freely decide to construct generating units and

compete for energy sales contracts with utilities and customers. The decision on plant

type and size will typically depend on investment and fuel costs, duty cycle, availability

rates etc. However, the plant sitting decision also depends on the transmission cost

associated to energy transport from generation to load centers. For obvious reasons, it is

neither feasible nor economical to build independent transmission systems for each

generation-load pair. The transmission network then becomes a service to which all

generators and customers have access and it becomes necessary to develop rules which

allow the shared use of the transmission system. This transmission service cost is

allocated among generators and consumers though transmission use of system tariffs

(TUSTs).

Therefore, TUSTs play an important role in this new environment, where they areresponsible for a fair allocation of the transmission costs among the agents as well as for

providing efficient economic signals, i.e. induce private agents to build generation

facilities at sites that will lead to the best overall use of the generation-transmission

system.

For example, Fig.1.1 depicts a centralized process of expansion, where the

planner aims at conciliating both expansion and operation planning decisions of the

system. In this figure, variable x represents the decisions on the generation projects to be

built while variable y is related to transmission investments decisions. Variables I(x) and

O(x) represent the investment and operation expenses associated with the decisions x and

y while D(x,y) represents the redispatch cost of the generating system x considering the

transmission projects y. The single node dispatch represents the optimal operation


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without considering transmission constraints, which are strongly influenced by the

reinforcements in the grid.

GenerationExpansion

Single nodeDispatch

TransmissionExpansion

Redispatch Gen.and Transm.

+

X

MIN

Y

I(X)

D(X,Y)

O(X)

I(Y)

Fig 1.1 Expansion in Centralized Systems

GenerationExpansion

Single nodeDispatch

TransmissionExpansion

Redispatch Gen.and Transm.

+

X

MIN

Y

I(X)

D(X,Y)

O(X)

I(Y)+

MIN

T(X)

Fig 1.2 Expansion in Competitive Environment


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In this process, all steps of the study are known and, through an analysis of investment

costs and their impacts in operation costs, the planner decides which is the optimal

planning in global terms, i.e., generation and transmission.

Figure 1.1 - Expansion in Centralized Systems Figure 1.2 Expansion in

Competitive Environment In processes based on competitive schemes in generation,

TUSTs play a fundamental role in the expansion of system. As it can be seen in Fig. 2,

studies of transmission system expansion could be illustrated as a black box where

investors have access only to its results through the TUSTs. In this sense, TUSTs shall

signal the impacts of transmission costs in electric sector in a fair and efficient way and

these signals are important to induce the generation investors correctly, and to allow an

optimal expansion of the electric sector.

Given the acquired importance of TUSTs, many methods to allocate transmission

costs among network users have been discussed and developed in a worldwide context. In

general, it can be said that each method has its own advantages and disadvantages and

there is no consensus related to the most appropriate method to be adopted. However, as

a general guideline, the transmission tariff structure should be efficient i.e. it should

induce generation investments that lead to the overall best use of the transmission system

and fair i.e. it should not create cross-subsidies from one market agent to the other.

1.5 Power wheeling costs:

In a Deregulated Power Structure, Power producers and customers share a common

Transmission network for wheeling power from the point of generation to the point of

consumption. They are given by

1. Rolled-In-Embedded Method or Postage Stamp Method:

The rolled-in method assumes that the entire transmission system is used in wheeling,

irrespective of the actual transmission facilities that carry the transaction. The cost of

wheeling as determined by this method is independent of the distance of the power

transfer.


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2. Contract Path Method:

The second traditional method, called the contract path method, is based upon the

assumption that the power transfer is confined to flow along a specified electrically

continuous path through the wheeling companys transmission system. Note that

changes in flows in facilities that are not within the identified path are ignored. The

embedded capital costs, correspondingly, are limited to those facilities that lie

along the assumed path.

1.6 Literature Review:

A brief description of the most significant proposals reported in the technical

literature on the allocation of the cost of the transmission network among generators and

demands follows.1.In the traditional pro rata method, both generators and loads are charged a flat

rate per megawatt-hour, disregarding their respective use of individual

transmission lines.

2.Other more elaborated methods are flow-based .

These methods estimate the usage of the lines by generators and demands and

charge them accordingly. Some flow-based methods use the proportional sharing

principle which implies that any active power flow leaving a bus is proportionally

made up of the flows entering that bus, such that Kirchhoffs current law is

satisfied.

3.Other methods that use generation shift distribution factors , are dependent on the

selection of the slack bus and lead to controversial results.

4.The usage-based method uses the so-called equivalent bilateral exchanges (EBEs).

To build the EBEs, each demand is proportionally assigned a fraction of each

generation, and conversely, each generation is proportionally assigned a fraction

of each demand, in such a way as both Kirchhoffs laws are satisfied.

The technique presented in this project is related to the allocation of the cost of

transmission losses based on Zbus matrix approach. It should be emphasized that all

transmission lines must be modeled to include actual shunt admittances and taps.


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Doing so, the impedance matrix presents an appropriate behavior of all the elements of

the transmission network.

A salient feature of the proposed technique is its embedded proximity effect,

which implies that a generator/demand uses mostly the lines electrically close to it. This

is not artificially imposed but a result of relying on circuit theory.

This proximity effect does not take place if the equivalent bilateral exchanges

(EBE) principle is used, as this principle allocates the production of any

generator/demand proportionally to all loads/generators, which implies treatingclose by

and far away lines in same manner .the proximity effect is ignored.

Other techniques require stronger assumptions, which diminish their practical

interest. Applying the proportional sharing principle implies imposing that principle, and

using the pro-rata criterion implies disregarding altogether network locations.

Particularly, it should be noted that the proposed methodology simply relies on circuit

laws in identifying the contribution factors, while the proportional sharing technique

relies on the proportional sharing principle.

1.7 Contributions:

The contributions of this project are stated below. The proposed techniques:1) uses the contributions of the nodal currents to line power flows to apportion the

use of the lines;

2) shows a desirable proximity effect; that is, the buses electrically close to a line

retain a significant share of the cost of using that line;

3) is slack independent.

4) does not require an a priori definition of the proportion in which to split

transmission costs between generators and demands.

Specifically, the main contribution of this project is a physical-based technique to

identify how much an individual power injection uses the network.


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1.8 Outlines of the Thesis:

Chapter 2 discusses the problem of transmission network cost allocation and

presents the solution methodology using Zbus. A detailed algorithm is presented and a

case study on 4 - bus system is considered and explained in detail by giving step by step

results and drawn some conclusions.

Chapter 3 covers the problem of transmission network cost allocation and

presents the solution methodology using avgbusZ technique . A detailed algorithm is

presented and a case study on 4 - bus system is considered and explained in detail by

giving step by step results and relevant conclusions are reported..

Chapter 4 presents a new technique which is based on bus impedance matrix,

discusses the problem of transmission network cost allocation and indicates the solution

methodology using modified avgbusZ technique . A detailed algorithm is presented and a

case study on 4 - bus system is considered and explained in detail by giving step by step

results. The effectiveness of the new technique is investigated and the salient features of

it are summarized.

Chapter 5 gives the results of the above three techniques performed on IEEE

RTS 24 - bus system

Finally, Appendix presents the Input data of 4- bus and IEEE RTS 24- bus systems


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CHAPTER 2

TRANSMISSION NETWORK COST ALLOCATION

USING ZBUS TECHNIQUE

2.1 Problem Statement:

The methodology starts from a converged load flow solution which gives the

entire information pertaining to the network such as bus voltages, complex line flows,

slack bus power generation etc. The purpose of the methodology presented in this work is

to allocate the cost pertaining to the transmission lines of the network to all the generators

and demands. Once a load flow solution is available, the proposed method determines

how line flows depend on nodal currents. This result is then used to allocate network

costs to generators and demands.

2.2 Background:

The equivalent circuit of a line having a line with primitive admittance jky and half line

charging susceptancesh

jky connected between the buses j and k is shown in Fig.2.1

[10]. jv and kv represent the nodal voltages of buses j and k respectively.

j k

+ Sjk jky +

jkI

jvsh

jkysh

jky kv

- -

Fig. 2.1 equivalent - circuit of line jk.


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From the load flow solution we can write expression for the complex line flow jkS in

terms of the node voltage and the line current jkI through the line jk as

*

jkS j jk V I= (2.1)

The voltage at node j in terms of the elements of bus impedance matrix Zbus and the nodal

current iI is given by ( from Vbus =Zbus Ibus )

1

n

j j i i

i

V Z I=

= (2.2)

where jiZ is the element ji of Zbus and n is the total number of buses.

Current through the line jk can be written as

( ) sh jk j k jk j jk

I V V y V y= + (2.3)

Substituting (2.2) in (2.3) and rearranging

1

( )n

sh

jk ji ki jk ji jk i

i

I Z Z y Z y I =

= + (2.4)

At this stage,we wish to make equ(2.4) as dependent on Pgen, Qgen, Pload and Qload of the

bus-i. This would help in building up the relevant mathematical support in identifying the

contribution of each generator and load on the line flow jk.this aspect is considered in

proposing new technique.

From the load flow analysis, the nodal current can be written as a function of active and

reactive power generations at bus i (i

genP and

i

genQ respectively) and the active and

reactive load demands at bus i (i

loadP and iloadQ respectively ) as

*

( ) ( )i i i igen load gen load

i

i

P P j Q QI

V

= (2.5)


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Note that the first term of the product in (2.4) is constant, as it depends only on network

parameters. Thus, (2.4) can be written as

i

N

I

i

jkJKIaI == 1 (2.6)

Where

( shjkjijkkiji

i

jk yzyzza += (2.7)

Observe that the magnitude of parameteri

jka provides a measure of the electrical

distance between bus i and line jk .

Substituting (2.6) in (2.1)

( == ==n

i i

i

jkj

n

i i

i

jkjjk IaVIaVS 1**

*

1(2.8)

Then, the active power through line jk is

=

=n

ii

i

jkjjk IaVP1

**(2.9)

or, equivalently

{ = =n

i i

i

jkjjk IaVP 1**

(2.10)

Note that the terms in the summation represent contribution due to each bus - Ii Thus, the

active power flow through any line can be identified as function of the nodal currents in

a direct way. Then, the active power flow through line jk due to the nodal current Ii is

**i

i

jkj

i

jk IaVP = (2.11)


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2.3. Transmission Cost Allocation:

Following (2.11), we define the usage of line jk due to nodal current as the absolute

value of the active power flow component ijkP , i.e.,

ijk

ijk PU = (2.12)

That is, we consider that both flows and counter-flows do use the line.

The total usage of line jk is then

i

jk

N

ijkUU == 1 (2.13)

Then, we proceed to allocate the use of transmission line jk to any generator and

demand. Without loss of generality, we consider at most a single generator and a single

demand at each node of the network.

Then, the usage of line jk apportioned to the generator or demand located at bus is stated

below.

If bus i contains only generation, the usage allocated to generation pertaining to line jk

is

i

jk

Gi

jk UU = (2.14)

On the other hand, if bus contains only demand, the usage allocated to demand pertaining

to line jk is

i

jk

Di

jk UU = (2.15)

Else, if bus i contains both generation and demand, the usage allocated to the generation

at bus pertaining to line jk is

([ ijkDiGiGiGi

jk UPPPU += (2.16)

and the usage allocated to the demand at bus pertaining to line jk is

([ ijkDiGiDiDi

jk UPPPU += (2.17)


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The complex power flow components through line jk due to individual power

generations and load demands have been found out. Having found the contributions of

individual generators and demands in each of the line flows and the usage of line by those

generations and demands, allocation of transmission cost among generators and demands

can be found out. Let jkC in $/h, represents the total annualized line cost including

operation, maintenance and building costs [8].

Then the per unit usage cost rate j kr can be written as

j k

j k

j k

Cr

U= (2.19)

Using the per unit cost rate, we can write,

Gi

jkC , the allocated cost of line jk to the

generator i' located at bus i' is

Gi Gi

jk jk jk C r U= (2.20)

In the same way, we can write,Di

jkC , the allocated cost of line jk to the demand i'

located at bus i' is

Di Di


The total transmission network cost,GiC , allocated to generator i' is the sum of the

individual cost components of each line due to that generator.

( , )

G i Gi

jk

j k n lin e

C C

= (2.22)

where nline represents the set of all transmission lines present in the system.

Similarly, the total transmission cost, DiC , allocated to the demand i' is given as

( , )

D i D i

jk

j k n li ne

C C

= (2.23)


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2.4 Algorithm For Transmission Network Cost Allocation Using

Zbus Technique

Algorithm

1. (a) Read the system line data and bus data

Line data: From bus, To bus, line resistance, line reactance, half-line charging

Susceptance and off nominal tap ratio.

Bus data: Bus no, Bus itype, Pgen, Qgen, Pload, Qload, and Shunt capacitor data.

(b) Form Ybus using sparsity technique.

2. (a) k1=1 iteration count

(b) Set maxP =0.0 , maxQ =0.0

(c) Cal Pshed(i),Qshed(i), for i=1 to n.

Where Pshed(i) = Pgen(i)- Pload(i)

Qshed(i) = Qgen(i)- Qload(i)

(d) Calculate Pcal(i)= )cos(1

iqiqiqq

n

q

iYVV

=

Qcal(i)= )sin(1

iqiqiqq

n

q

iYVV

=

(e) Calculate P(i)=Pshed(i) Pcal(i)

Q(i)=Qshed(i) - Qcal(i) for i=1 to n

Set Pslack=0.0, Qslack=0.0,

(g) CalculatemaxP and maxQ form [ p] and [ Q] vectors

(h) IsmaxP and maxQ

If yes, go to step no. 6

3. Form Jacobian elements:

(a) Initialize A[i][j]=0.0 for i=1 to 2n , j=1 to 2n

(b) Form diagonal elements Hpp, Npp, Mpp & Lpp

(c) Form off diagonal elements: Hpq, Npq, Mpq & Lpp


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(d) Form right hand side vector(mismatch vector)

B[i]= P[i] , B[i+n]= Q[i] for i=1 to n

(e) Modify the elements

For p=slack bus; Hpp=1e20=1020; Lpp=1e20=10

20;

4. Use Gauss Elimination method for following

[A] [ X] = [B]

Update the phase angle and voltage magnitudes i=1 to n

For itype=1 &2, calculate iii X+= & Vi=Vi+{ X(i+n)}Vi

5. One iteration over

Advance iteration count k1=k1+1If (k1< itermax) then goto step 2(b) else print problem is not converged in

itermax iterations, Stop.

6. Print problem is converged in iterno. of iterations.

a. Calculate line flows

b. Bus powers, Slack bus power.

c. Print the converged voltages, line flows and powers.

7. Form the bus impedance matrix Zbus. (Zbus is calculated using1

busY )

8. Do for all the lines in the system, 1 to nline

A) If the active power flow direction is from bus to to bus

a) Do for all the buses from 1 to n

i) Calculate ijka ,

Gi

jkU Di

jkU and i

jkU using the equations (2.12),(2.16),(2.17).

End of Do loop

b) Find usage allocated to the line jk

1

ni

jk jk

i

U U=

=

End of if


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B) Do for each bus, 1 to n

a) Determine the contributions of generators and loads paying for

using the line jk ,jk

r , Gijk

C , Dijk

C using equations (2.19), (2.20) and (2.21)

b) Find the factor per unit usage cost rate rjkinterchanging from bus

and to bus

jk

jk

jkU

Cr =

c) Find the generation i cost contributions for using line jk

interchanging from bus and to bus

Gi

jkjk

Gi

jk UrC =d) Find the load cost contributions for using line jk


Di

jkjk

Di

jk UrC =

End of bus Do loop

End of line Do loop

9. Find the cost of contribution of generator i using all the lines in the network

( , )

Gi Gi

jk

j k nline

C C

=

10. Find the cost of contribution of load i using all the lines in the network

( , )

Di Di

jk

j k nline

C C

=


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2.5 Case study 4 bus system:

The proposed usage based technique has been illustrated with the help of a sample

four bus, 5 line system shown in Fig. 2.2 All the lines have equal per unit resistance,reactance and half line charging susceptance of 0.01275, 0.097, 0.4611 respectively. For

the sake of simplicity either a single generator or a single load demand of 250 MW has

been taken at each bus. Finally, cost of each line,jk

C is considered to be proportional

to its series reactancejk

x i.e. 1000 jk jk

C x= $/h[8].

250.0 MW 500 MW

Line 5

3 4

63.0 MW

Line 2 Line 3 Line 4

191.7 MW 190.0 MW

129.2 MW

Line 1

60.0 MW

1 2

261.3 MW 250.0 MW

Fig. 2.2 Four Bus System


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2.5.1 Step By Step Results 4 bus system:

Detailed line data and bus data are given in appendix.

The slack bus power is 261.311351 MW+j -96.154655 MVAR

The total loss =11.311357 MW

Table 2.1 Converged Voltages

E(1)=1.050000+j-0.000000

E(2)=1.048494+j0.056220

E(3)=1.038233+j-0.178619

E(4)=1.049926+j-0.121420

Table 2.2 Bus Currents

ibus(1)=2.48868+i0.915759

ibus(2)=2.34015+i0.824762

ibus(3)=-2.33872+i0.402353

ibus(4)=-2.34969+i0.27173

Table 2.3 Powerflow Contributions P(i,k)in Pjk>0 direction,equ(2.11)

Line\Bus 1 2 3 4

1 -0.3385 1.25 0 -0.3111

2 0.8486 0.5044 0.752 -0.1879

3 0.916 0.2492 -0.2488 0.3757

4 0.3385 1.25 0 0.3111

5 0.1907 0.484 0.7672 -0.8119


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Table 2.4 Powerflow Usage Contributions U(k,i)in Pjk>0 direction,equ(2.12)

Line\Bus 1 2 3 4

1 0.3385 1.25 0 0.3111

2 0.8486 0.5044 0.752 0.1879

3 0.916 0.2492 0.2488 0.3757

4 0.3385 1.25 0 0.3111

5 0.1907 0.484 0.7672 0.8119

Table 2.5 Powerflow Usage of Line usage(k)in Pjk>0 direction,equ(2.13)

Line Usage

1 1.89966

2 2.29286

3 1.78977

4 1.89966

5 2.25381

Table 2.6 Powerflow Contributions Ug(i,k)in Pjk>0 direction,equ(2.14) to ,equ(2.17)

Bus\Line 1 2 3 4 5

1 0.339 0.849 0.916 0.339 0.191

2 1.25 0.504 0.249 1.25 0.484

3 0 0 0 0 0

4 0 0 0 0 0

Table 2.7 Powerflow Contributions Ud(i,k)in Pjk>0 direction,equ(2.14) to ,equ(2.17)

Bus\Line 1 2 3 4 5

1 0 0 0 0 02 0 0 0 0 0

3 0 0.752 0.249 0 0.767

4 0.311 0.188 0.376 0.311 0.812


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Table 2.8 Generator Cost Contributions cg(k,i)in Pjk>0 direction,equ(2.20)

Line\Gen GEN-1 GEN-2

1 17.285 63.8273

2 35.8984 21.3386

3 49.6443 13.5066

4 17.285 63.8273

5 8.209 20.8313

Table 2.9 Load Cost Contributions cd(k,i) in Pjk>0 direction,equ(2.21)

Line\Load LOAD-3 LOAD-4

1 0 15.8877

2 31.8153 7.9477

3 13.4857 20.3634

4 0 15.8877

5 33.0189 34.9408

Table 2.10 total generation and load costs and Total cost for all the buses

in Pjk>0 direction,equ(2.22) and ,equ(2.23)

Bus CG CD TOTAL COST

1 128.3219 0 128.3219

2 183.331 0 183.331

3 0 78.31983 78.31983

4 0 95.02724 95.02724

Table 2.11. relationship between the line costs and reactance of the line

Line\Bus 1 2 3 4 Cjk=1000*Xjk=971 17.285 63.8273 0 15.8877 97

2 35.8984 21.3386 31.8153 7.9477 97

3 49.6443 13.5066 13.4857 20.3634 97

4 17.285 63.8273 0 15.8877 97

5 8.209 20.8313 33.0189 34.9408 97


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From above tables it can be noted that, for all the lines, the Zbus method have the

property that they allocate a significant amount of the cost of each line to the buses

directly connected to it. For lines 1, 2, 3, and 5, the two buses with the highest line usage

are these at the ends of the corresponding line. Taking into account that the power

injected and extracted at each bus is very similar, the results reflect the location of each

bus in the network. For instance, the Zbus allocate most of the usage of line 5 (between

buses 3 and 4) to buses 3 and 4.

Note also that, for line 4 (between buses 2 and 4), the results provided by the zbus

method are somewhat different, since the allocation to bus 1, not directly connected to

line 4, is also relevant. This happens, mostly, because the power injected at bus 1 is

greater than the power extracted at bus 4: 261.3 and 250.0 MW, respectively. In addition,

the absolute values of the electrical distance terms 124a and4

24a are identical, as well as

the values of z12 and z24 , which makes buses 1 and 4 being at the same electrical distance

to line 24. Nevertheless, the cost allocated to bus 4 is significant and similar to the cost

allocated to bus 1.

2.6 Conclusions:

The busZ technique to allocate the cost of the transmission network to generators

and demands are based on circuit theory. This technique generally behave in a similar

manner as other techniques previously reported in the literature. However, they exhibit a

desirable proximity effect according to the underlying electrical laws used to derive them.

This proximity effect is more apparent on peripheral rather isolated buses. For these

buses, other techniques may fail to recognize their particular locations.

The busZ technique allocates a higher line usage to generators versus demands.

Thus, we conclude that the proposed methods are appropriate for the allocation of the

cost of the transmission network to generators and demands, complement existing

methods, and enrich the available literature.


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36

CHAPTER 3


USING avgbusZ TECHNIQUE




slack bus power generation etc. The purpose of the methodology presented in this work is

to allocate the cost pertaining to the transmission lines of the network to all the generators

and demands. Once a load flow solution is available, the proposed method determines

how line flows depend on nodal currents. This result is then used to allocate network

costs to generators and demands.

3.2 Background:



jky connected between the buses j and k is shown in Fig.3.1


j k

+ Sjk jky +

j kI

jvsh

jkysh

jky kv

- -

Fig. 3.1 equivalent -circuit of line jk.


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37



*

jkS j jk V I= (3.1)


current iI is given by

1

n

j j i i

i

V Z I=

= (3.2)




I V V y V y= + (3.3)


1

( )n

sh

jk ji ki jk ji jk i

i

I Z Z y Z y I =

= + (3.4)



genP andi



loadP and

i

loadQ respectively ) as

*


i

i

P P j Q QI

V

= (3.5)

Note that the first term of the product in (3.4) is constant, as it depends only on network

parameters. Thus, (3.4) can be written as

i

N

I

i

jkjkIaI == 1 (3.6)

At this stage, we wish to make equ(2.4) as dependent on Pgen, Qgen, Pload and Qload of the



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38

contribution of each generator and load on the line flow jk.this aspect is considered in

proposing new technique.

Where

( shjkjijkkiji

i

jk yzyzza += (3.7)

Observe that the magnitude of parameteri

jka provides a measure of the electrical

distance between bus i and line jk .

Substituting (3.6) in (3.1)

( == ==n

i i

i

jkj

n

i i

i

jkjjk IaVIaVS 1**

*

1(3.8)

Then, the active power through line jk is

==n

i i

i

jkjjk IaVP 1**

(3.9)

or, equivalently

{=

=n

i i

i

jkjjk IaVP 1**

(3.10)

Note that the terms in the summation represent contribution due to each bus - Ii .Thus, the

active power flow through any line can be identified as function of the nodal currents in a

direct way. Then, the active power flow through line jk due to the with nodal current Ii is

**

i

i

jkj

i

jk IaVP = (3.11)


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39

3.3 Transmission Cost Allocation

Following (3.11), we define the usage of line jk due to nodal current as the absolute

value of the active power flow component ijkP , i.e.,

ijk

ijk PU = (3.12)

That is, we consider that both flows and counter-flows do use the line.

The total usage of line jk is then

i

jk

N

ijkUU == 1 (3.13)

Then, we proceed to allocate the use of transmission line jk to any generator and

demand. Without loss of generality, we consider at most a single generator and a single

demand at each node of the network.

Then, the usage of line jk apportioned to the generator or demand located at bus is stated

below.

If bus-i contains only generation, the usage allocated to generation pertaining to line jk

is

i

jk

Gi

jk UU = (3.14)

On the other hand, if bus contains only demand, the usage allocated to demand pertaining

to line jk is

i

jk

Di

jk UU = (3.15)

Else, if bus i contains both generation and demand, the usage allocated to the generation

at bus pertaining to line jk is

([ ijkDiGiGiGi

jk UPPPU += (3.16)

and the usage allocated to the demand at bus pertaining to line jk is

([ ijkDiGiDiDi

jk UPPPU += (3.17)


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The complex power flow components through line jk due to individual power

generations and load demands have been found out. Having found the contributions of

individual generators and demands in each of the line flows and the usage of line by those

generations and demands, allocation of transmission cost among generators and demands

can be found out. Let jkC in $/h, represents the total annualized line cost including

operation, maintenance and building costs [8].


j k

j k

j k

Cr

U= (3.18)

Using the per unit cost rate, we can write,

Gi



Gi Gi





Di Di




( , )

G i Gi

jk

j k n lin e

C C

= (3.21)



( , )

D i D i

jk

j k n li ne

C C

= (3.22)


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41

3.4 Effect of Flow Directions:

It is to be noted that complex power flow equation (3.1) can be written either in

the direction of active power flow i.e. 0jkP or in the direction of active power counter

flows [3]. This way to write (3.1) leads to electrical distance parameters ijka andi

kja .

However, (3.7) shows that distance parameters are not generally symmetrical with

respect to line indexes, i.e., ikji

jk aa , which results in different usage allocations

depending on whether (3.1) is written in the direction of the active power flows or

counter-flows [see (3.10)( 3.11)]. The proposed usage based technique takes the average

value of allocated cost (usage) obtained

1) with (3.1) written in the direction of the active power flows and

2) with (3.1) written in the direction of the active power counter-flows.


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42

3.6 Algorithm For Transmission Network Cost Allocation Using

avg

busZ Technique

Algorithm

1 (a) Read the system line data and bus data






(b) Set maxP =0.0 , maxQ =0.0(c) Cal Pshed(i),Qshed(i), for i=1 to n.




iqiqiqq

n

q

iYVV

=

Qcal(i)= )sin(1

iqiqiqq

n

q

iYVV

=





(h) IsmaxP and maxQ




(c) Form diagonal elements Hpp, Npp, Mpp & Lpp




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43


(f) Modify the elements


20;


[A] [ X] = [B]




Advance iteration count k1=k1+1

If (k1< itermax) then goto step 2(b) else print problem is not converged initermax iterations, Stop.


d. Calculate line flows

e. Bus powers, Slack bus power.

f. Print the converged voltages, line flows and powers.

7. Form the bus impedance matrix Zbus. ( Zbus is calculated using

1

busY )




i) Calculate ijka ,

Gi

jkU Di

jkU and i

jkU using the equations given in equ(3.12) to equ(3.17)

ii) Obtain the values of ikja ,

1GijkU , 1Di

jkU and 1ijkU by interchanging the from bus

and to bus and repeating step a)

End of Do loop


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44


1

ni

jk jk

i

U U=

=

c) Find usage allocated to the line by interchanging the from bus

and to bus

1

1 1n

i

jk jk

i

U U=

=

Else

Assign from bus as to bus and to bus as from bus and

repeat steps 1), 2) & 3)

End of if



using the line jk ,jkr , Gi

jkC , Di

jkC using equations (3.19), (3.20) and (3,21)

b) Find the factor per unit usage cost rate r1jkinterchanging from bus

and to bus

11

jk

jk

jk

Cr

U=



1 1 1Gi Gi jk jk jk C r U=

e) Find the load cost contributions for using line jk


1 1 1 Di Di jk jk jk C r U=

End of bus Do loop

End of line Do loop


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45


( , )

Gi Gi

jk j k nline

C C=



( , )

1 1Gi Gijk j k nline

C C

=


( , )

Di Di

jk

j k nline

C C

=



( , )

1 1 Di Dijk j k nline

C C

=

13. Do for all lines

Do for all the buses

A) Find the average cost contribution of generator i using the line jk

1

2

Gi Gi

jk jk Gi

jk

C CCavg

+=

B) Find the average cost contribution of load i using the line jk

1

2

Di Di

jk jk Di

jk

C CCavg

+=

End of bus loop

End of line loop


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46

14. Find the average cost contribution of generator i using all the lines in the

network

== sforalllinejkGi

jk

GiCavgCavg

15. Find the average cost contribution of load i using all the lines in the

network

== sforalllinejkDi

jk

DiCavgCavg


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47

3.6 Case study 4 bus system:


four bus, 5 line system shown in Fig.3.2 All the lines have equal per unit resistance,reactance and half line charging susceptance of 0.01275, 0.097, 0.4611 respectively. For

the sake of simplicity either a single generator or a single load demand of 250 MW has

been taken at each bus. Finally, cost of each line,jk

C is considered to be proportional

to its series reactancejk

x i.e. 1000 jk jk

C x= $/h[8].

250.0 MW 500 MW

Line 5

3 4

63.0 MW


191.7 MW 190.0 MW

129.2 MW

Line 1

60.0 MW

1 2

261.3 MW 250.0 MW

Fig. 3. 2 Four Bus System


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3.6.1 Step By Step Results - 4 bus system:

Detailed line data and bus data are given in appendix.

The slack bus power is 261.311351 MW+j -96.154655 MVAR

The total loss =11.311357 MW

Table 3.1 Converged Voltages

E(1)=1.050000+j-0.000000

E(2)=1.048494+j0.056220

E(3)=1.038233+j-0.178619

E(4)=1.049926+j-0.121420


ibus(1)=2.48868+i0.915759

ibus(2)=2.34015+i0.824762

ibus(3)=-2.33872+i0.402353

ibus(4)=-2.34969+i0.27173

Table 3.3 Powerflow Contributions P(i,k)in Pjk>0 direction,equ(3.11)

Line\Bus 1 2 3 4

1 -0.3385 1.25 0 -0.3111

2 0.8486 0.5044 0.752 -0.1879

3 0.916 0.2492 -0.2488 0.3757

4 0.3385 1.25 0 0.31115 0.1907 0.484 0.7672 -0.8119


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49

Table 3.4 Powerflow Usage Contributions U(k,i)in Pjk>0 direction,equ(3.12)

Line\Bus 1 2 3 4

1 0.3385 1.25 0 0.3111

2 0.8486 0.5044 0.752 0.1879

3 0.916 0.2492 0.2488 0.3757

4 0.3385 1.25 0 0.3111

5 0.1907 0.484 0.7672 0.8119

Table 3.5 Powerflow Usage of Line usage(k)in Pjk>0 direction,equ(3.13)

Line Usage

1 1.89966

2 2.29286

3 1.78977

4 1.89966

5 2.25381

Table 3.6 Powerflow Contributions Ug(i,k)in Pjk>0 direction,equ(3.14)to ,equ(3.16)

Bus\Line 1 2 3 4 5

1 0.339 0.849 0.916 0.339 0.191

2 1.25 0.504 0.249 1.25 0.484

3 0 0 0 0 0

4 0 0 0 0 0

Table 3.7 Powerflow Contributions Ud(i,k)in Pjk>0 direction,equ(3.15)to ,equ(3.17)

Bus\Line 1 2 3 4 51 0 0 0 0 0

2 0 0 0 0 0

3 0 0.752 0.249 0 0.767

4 0.311 0.188 0.376 0.311 0.812


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Table 3.8 Generator Cost Contributions cg(k,i)in Pjk>0 direction,equ(3.19)


1 17.285 63.8273

2 35.8984 21.3386

3 49.6443 13.5066

4 17.285 63.8273

5 8.209 20.8313



1 0 15.8877

2 31.8153 7.9477

3 13.4857 20.3634

4 0 15.8877

5 33.0189 34.9408

Table3.10 Total generation and load costs and Total cost for all the buses

in Pjk>0 direction,equ(3.21) and equ(3.11)

Bus CG CD TOTAL COST

1 128.3219 0 128.3219

2 183.331 0 183.331

3 0 78.31983 78.31983

4 0 95.02724 95.02724


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51

Table3.11 Powerflow Contributions P1(k,i)in Pjk


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52

Table3.15 Powerflow Contributions Ud1(i,k) in Pjk


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53

Table3.19 Average Generator Cost Contributions cgavg(k,i)


1 26.5887 47.8518

2 25.9208 10.6693

3 35.3905 13.3772

4 12.774 47.5767

5 12.0761 10.4156

Table3.20 Average Load Cost Contributions cdavg(k,i)


1 10.6425 11.917

2 48.2506 12.1592

3 13.7744 34.458

4 11.1202 25.5291

5 48.8524 25.6558

Table3.21 Total average generation and load costs and Total avgcost1 for all the buses

Bus No CGAVG(I) CDAVG(I) TOTAL COSTAVG(i)

1 112.7502 0 112.7502

2 129.8906 0 129.8906

3 0 132.6402 132.6402

4 0 109.7191 109.7191

From above tables it can be noted that, for all the lines, the Z bus and Zbus average

methods have the property that they allocate a significant amount of the cost of each lineto the buses directly connected to it. For lines 1, 2, 3, and 5, the two buses with the

highest line usage are these at the ends of the corresponding line. Taking into account that

the power injected and extracted at each bus is very similar, the results reflect the location

of each bus in the network. For instance, the Zbus and Zbus average allocate most of the

usage of line 5 (between buses 3 and 4) to buses 3 and 4.


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54

Also note that , for line 4 (between buses 2 and 4), the results provided by the Z bus

method are somewhat different, since the allocation to bus 1.bus that is not directly

connected to line 4, is also relevant. This happens, mostly, because the power injected at

bus 1 is greater than the power extracted at bus-4, 261.3 and 250.0 MW, respectively. In

addition, the absolute values of the electrical distance terms 124a and4

24a are identical, as

well as the values of z12 and z24 , which makes buses 1 and 4 being at the same electrical

distance to line 24. Nevertheless, the cost allocated to bus 4 is significant compared to

the cost allocated to bus 1. It should also be noted that for line 4.The Zbus average

approach allocated the highest portion of line usage to buses 2 and 4, which are the

terminal buses of line 4.

It may be noted that the Zbus based approach usually allocates higher transmission cost to

generator buses compared to load buses. Comparing the methods Zbus and Zbus average

methods, it can be concluded that the Zbus average method smoothes the trend of the zbus

one (as well as of other methods)and avoids allocation of higher portion of usage to

generating buses compared to demand buses. In view of the results are significantly

different.


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55

3.7 Conclusions:

The avgbusZ technique to allocate the cost of the transmission network to generators

and demands are based on circuit theory. This technique generally behave in a similarmanner as other techniques previously reported in the literature. However, they exhibit a

desirable proximity effect according to the underlying electrical laws used to derive them.

This proximity effect is more apparent on peripheral rather isolated buses. For these

buses, other techniques may fail to recognize their particular locations.

The avgbusZ approach smoothes the trend of the method (as well as of other

techniques) and avoids to allocate a higher line usage to generators compared todemands. We have performed extensive numerical simulations on IEEE-RTS-24 bus

system and encountered neither numerical induced ill-conditioning nor unreasonable

results. Thus, we conclude that the proposed methods are appropriate for the allocation of

the cost of the transmission network to generators and demands, complement existing

methods, and enrich the available literature.


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CHAPTER 4


USING MODIFIED avgbusZ TECHNIQUE (newly proposing technique)




slack bus power generation etc. This paper presents a comprehensive methodology that

finds the coefficients of the power generations and load demands in the complex line

flow. Once the coefficients are determined, next step is to find the allocation of

transmission cost pertaining to individual generators and loads.

4.2 Background:



jky connected between the buses j and k is shown in Fig. 4.1


j k+ Sjk jky +

j kI

jv sh

jky sh

jky

kv

- -

Fig. 4.1. equivalent -circuit of line jk.


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*

jkS j jk V I= (4.1)


current iI is given by ( from Vbus=Zbus Ibus )

1

n

j j i i

i

V Z I=

= (4.2)




I V V y V y= + (4.3)


1

( )n

sh

jk ji ki jk ji jk i

i

I Z Z y Z y I =

= + (4.4)

At this stage, we wish to make equ(2.4) as dependent on Pgen, Qgen, Pload and Qload of the


contribution of each generator and load on the line flow jk.



genP and

i



loadP and

i

loadQ respectively ) as

*


i

i

P P j Q QI

V

= (4.5)

Substituting the values jkI and iI from (4.4) and (4.5) in (4.1) and rearranging


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1

[( ) ( )]n

i i i i i

jk jk gen load gen load

i

S Factor P P j Q Q=

= + (4.6)

where

*( )sh

j ji ki jk ji jk i

jk

i

V Z Z y Z yFactor

V + = (4.7)

Thus , the active and reactive power flow jkS through any line jkis represented as a

function of the power generation and load at all buses

i.e , ,i i i

gen load genP P Q andi

loadQ ; i = 1,2,3..n

Eq. (6) can be rewritten as

1

( 1 2 3 4 )n

i i i i

jk jk jk jk jk

i

S S S S S=

= + + + (4.8)

Where

1 * ; 2 *

3 * ; 4 *

i i i i i i

jk jk gen jk jk load

i i i i i i

jk jk gen jk jk load

S Factor P S Factor P

S jFactor Q S jFactor Q

= =

= =

Note that, for a converged load flow solution, the magnitude of parameter ijk

Fac tor

provides a measure of the electrical distance between bus i and line jk.

Eq. (4.6) clearly illustrates the fact that complex power flow through any line depends on

the power generations (active and reactive) and demands (active and reactive). The

components 1 , 2 , 3 & 4i i i i

jk jk jk jk S S S S represent the contribution/share of each of the

power generation and demand to the complex power flow through the line jk. Hence

the complex power flow through a line j k can be split up into individual components

associated to power generations and demands at a particular bus as shown below. Thus,

the component of complex power flow due to bus i through a line j k associated with

the bus power generation and demand at bus i can be written as

1 2 3 4i i i i i jk jk jk jk jk

S S S S S= + + + (4.9)


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This approach can be considered as new contribution in the area of transmission cost

aloocation among generators and load buses. Following the information reported in

reference [8], we consider that both flows and counter-flows do use the line. The usage of

line jk by any generator i' ,Gi

jk

U , is defined as the sum of the absolute value of the

active power flow components due to active and reactive power generation of the

generator i' , i.e.,i

genP andi

genQ .

4.3 Transmission Cost Allocation:

Thus, usage of line jkby generator i' can be written as

| ( 1 ) | | ( 3 ) |

Gi i i

jk jk jk U S S= +

(4.10)

Similarly, the usage of line jk by any demand i,Di

jkU is defined as the sum of the

absolute value of the active power flow components due to active and reactive parts of

demand i' i.e.,i

loadP andi

loadQ .

Hence, the usage of line jk by demand i can be written as

| ( 2 )| | ( 4 ) | Di i i jk jk jk U S S= + (4.11)

The usage of line by bus i' ,i

jkU , is then given by

i Gi Di

jk jk jk U U U= + (4.12)

The total usage of line jk, jkU , by all buses is then

1

ni

jk j k

i

U U=

= (4.13)

The complex power flow components through line jk due to individual powergenerations and load demands have been found out directly without much additional

complexity and computation Having found the contributions of individual generators and

demands in each of the line flows and the usage of line by those generations and

demands, allocation of transmission cost among generators and demands can be found


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out. Let jkC in $/h, represents the total annualized line cost including operation,

maintenance and building costs [8]. cost of each line,jk

C is considered to be

proportional to its series reactancejk

x i.e. 1000 jk jk C x= $/h[8].


j k

j k

j k

Cr

U= (4.14)

Using the per unit cost rate, we can write,Gi



Gi Gi





Di Di




( , )

G i Gijk

j k n lin e

C C

= (4.17)



( , )

D i D i

jk

j k n li ne

C C

= (4.18)

It is to be noted that complex power flow equation (4.8) can be written either in thedirection of active power flow i.e. 0jkP or in the direction of active power counter

flows [3]. This way to write (4.8) leads to electrical distance parameters ijk

Factor and

i

kjFactor . However, (4.7) shows that distance parameters are not generally symmetrical


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with respect to line indexes, i.e., i i jk kj

Factor Factor , which results in different usage

allocations depending on whether (4.8) is written in the direction of the active power

flows or counter-flows [see (4.10)( 4.11)]. The proposed usage based technique takes

the average value of allocated cost (usage) obtained 1) with (4.8) written in the directionof the active power flows and 2) with (4.8) written in the direction of the active power

counter-flows.

4.4 Algorithm for Transmission network cost allocation

Using modified

avg

busZ technique (newly proposing technique)Algorithm

1 (a) Read the system line data and bus data






(b) SetmaxP =0.0 , maxQ =0.0

(c) Cal Pshed(i),Qshed(i), for i=1 to n.




iqiqiqq

n

q

iYVV

=

Qcal(i)= )sin(1

iqiqiqq

n

q

iYVV

=





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(h) IsmaxP and maxQ




(d) Form diagonal elements Hpp, Npp, Mpp & Lpp




(g) Modify the elements


20;


[A] [ X] = [B]




Advance iteration count k1=k1+1

If (k1< itermax) then goto step 2(b) else print problem is not converged in

itermax iterations, Stop.


g. Calculate line flows

h. Bus powers, Slack bus power.

i. Print the converged voltages, line flows and powers.

7. Form the bus impedance matrix Zbus. ( Zbus is calculated using1

busY )


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i) Calculate ijk

Factor , 1 , 2 ,i i jk jk

S S 3 , 4i i jk jk

S S ,

Gi

jkU Di

jkU and ijkU using the equations given equ(4.7) to equ(4.12)

ii) Obtain the values of 1ijkFactor , 11 ,i

jkS 21ijkS , 31 , 41

i i

jk jk S S

1GijkU , 1Di

jkU and 1ijkU by interchanging the from bus

and to bus and repeating step a)

End of Do loop


1

ni

jk jk

i

U U=

=

c) Find usage allocated to the line by interchanging the from bus

and to bus

1

1 1n

i

jk jk

i

U U=

=

Else

Assign from bus as to bus and to bus as from bus and

repeat steps 1), 2) & 3)

End of if



using the line jk ,jkr , GijkC , DijkC using equations (4.14), (4.15) and (4.16)

b) Find the factor per unit usage cost rate r1jkinterchanging from bus

and to bus

11

jk

jk

jk

Cr

U=


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1 1 1Gi Gi jk jk jk C r U=

f) Find the load cost contributions for using line jkinterchanging from bus and to bus

1 1 1 Di Di jk jk jk C r U=

End of bus Do loop

End of line Do loop


( , )

Gi Gijk

j k nline

C C

= 10. Find the cost of contribution of generator i using all the lines in the network


( , )

1 1Gi Gijk j k nline

C C

=


( , )

Di Di

jk j k nlineC C

=



( , )

1 1 Di Dijk

j k nline

C C

=


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13. Do for all lines

Do for all the buses

A) Find the average cost contribution of generator i using the line jk

1

2

Gi Gi

jk jk Gijk

C C

Cavg

+

=

B) Find the average cost contribution of load i using the line jk

1

2

Di Di

jk jk Di

jk

C CCavg

+=

End of bus loop

End of line loop

14. Find the average cost contribution of generator i using all the lines in the

network

== sforalllinejkGi

jk

GiCavgCavg

15. Find the average cost contribution of load i using all the lines in the

network

== sforalllinejkDi

jk

Di CavgCavg


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4.5 Case Study - 4 - Bus System:


four bus, 5 line system shown in Fig. 4.2 All the lines have equal per unit resistance,

reactance and half line charging susceptance of 0.01275, 0.097, 0.4611 respectively. Forthe sake of simplicity either a single generator or a single load demand of 250 MW has

been taken at each bus. Finally, cost of each line, jkC is considered to be proportional

to its series reactance jkx i.e. 1000 jk jk C x= $/h[8].

250.0 MW 500 MW

Line 5

3 4

63.0 MW


191.7 MW 190.0 MW

129.2 MW

Line 1

60.0 MW

1 2

261.3 MW 250.0 MWFig. 4. 2 Four Bus System


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4.5.1 Step By Step Results 4 bus system:

Detailed line data and bus data are given in appendix.The slack bus power is 261.311351 MW+j -96.154655 MVARThe total loss =11.311357 MW

Table 4.1 Converged voltages


ibus(1)=2.48868+i0.915759

ibus(2)=2.34015+i0.824762ibus(3)=-2.33872+i0.402353

ibus(4)=-2.34969+i0.27173

Table 4.3 Powerflow Contributions S1(i,k) in Pjk>0 direction,equ(4.8)

Bus/Line

1 2 3 4 5

1 -0.332+j-0.019 0.849+j-0.000 0.916+j 0.000 0.332+j 0.019 0.199+j-0.023

2 1.250+j-0.000 0.512+j-0.026 0.253+j-0.013 1.250+j-0.000 0.509+j-0.085

3 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

4 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000


Bus/Line

1 2 3 4 5

1 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

2 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

3 0.000+j 0.000 0.752+j 0.132 -0.249+j-0.043

0.000+j 0.000 0.767+j 0.045

4 -0.311+j-0.054 -0.188+j-0.022 0.376+j 0.044 0.311+j 0.054 -0.812+j 0.000

E(1)=1.050000+j-0.000000

E(2)=1.048494+j0.056220E(3)=1.038233+j-0.178619

E(4)=1.049926+j-0.121420


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Bus/Line

1 2 3 4 5

1 -0.007+j 0.122 -0.000+j-0.312 0.000+j-0.337 0.007+j-0.122 -0.008+j-0.073

2 -0.000+j-0.367 -0.008+j-0.150 -0.004+j-0.074 -0.000+j-0.367 -0.025+j-0.149

3 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

4 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

Table 4.6 Powerflow Contributions S4(i,k) In Pjk>0 Direction,equ(4.8)

Bus/Line

1 2 3 4 5

1 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

2 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

3 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

4 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000 0.000+j 0.000

Table 4.7 Powerflow Contributions Ug(i,k) in Pjk>0 direction,equ(4.10)

Bus\Line 1 2 3 4 5

1 0.339 0.849 0.916 0.339 0.2072 1.25 0.519 0.257 1.25 0.534

3 0 0 0 0 04 0 0 0 0 0

Table 4.8 Powerflow Contributions Ud(i,k) in Pjk>0 direction,equ(4.11)

Bus\Line 1 2 3 4 51 0 0 0 0 0

2 0 0 0 0 0

3 0 0.752 0.249 0 0.767

4 0.311 0.188 0.376 0.311 0.812


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Table 4.9 Generator Cost Contributions cg(k,i) in Pjk>0 direction,equ(4.15)


1 17.285 63.8273

2 35.6653 21.8331

3 49.4319 13.86394 17.285 63.8273

5 8.6695 22.3176


Line\Load LOAD3 LOAD4

1 0 15.8877

2 31.6061 7.8955

3 13.4279 20.27634 0 15.8877

5 32.073 33.9399

Table 4.11 Total generation and load costs and Total cost for all thebuses,equ(4.17)andequ(4.18)

Bus no CG CD cost

1 128.3367 0 128.33672 185.6693 0 18

Transmission Network Cost Allocation Using Bus Impedance Mat

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