1
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)
2
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(Zbus is bonafide record of the work doneby D.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.
3
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
4
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 matrix
presents 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.
5
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
6
CONTENTSNOMENCLATURE
LIST OF TABLES
LIST OF FIGURES
Page NoCHAPTER 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 102.3 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 192.6 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 283.5 Algorithm For Transmission Network Cost Allocation Using
avgbusZ Technique 29
7
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 615.2 avg
busZ technique results 685.3 Modified avg
busZ technique results 755.4 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
8
NOMENCLATURE
jkC -Cost of line jk ($/h)
Ii -Nodal current (A)
Ijk - Current through the line jk (A)
n -Number of buses
PGi - Active power consumed by the generator located at bus i (W)
PDi - Active power consumed by the load located at bus i (W)
Pjk - Active power flow through line jk (W)
Sjk - Complex power flow through line jk calculated at bus j (VA)
Vj - Nodal voltage at bus j (V)
yjk - Series admittance of the -equivalent circuit of line jk (S)shjky - Shunt admittance of the - equivalent circuit of line jk (S)
Zbus - Impedance matrix (ohm)
Zij - Element ij of the impedance matrix (ohm)ijka - Electrical distance between bus i and line jk (adimensional)
DiC - Total transmission cost allocated to the load located at bus i ($/h)GiC - Total transmission cost allocated to the generator located at bus i ($/h)DijkC - Transmission cost allocated to the generator located at bus i ($/h)
GijkC - Transmission cost allocated to the generator located at bus i ($/h)
ijkP - Active power flow through the line jk associated with the nodal current i(W)
rjk - Cost rate for line jk ($/W & h)GijkU - 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).
ijkU - Usage of line associated with nodal current (W).
Ujk - Usage of line jk (W).
9
LIST OF TABLESPage No
Table 2.1 Converged Voltages of Zbus technique 19
Table 2.2 Bus Currents of Zbus technique 19
Table 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 avgbusZ technique 35
Table 3.2 Bus Currents of avgbusZ technique 35
Table 3.3 Powerflow Contributions P(i,k)in Pjk>0 direction of avgbusZ technique 35
Table 3.4 Powerflow Usage Contributions U(k,i)in Pjk>0 direction of avgbusZ 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 of avgbusZ technique 37
Table3.11 Powerflow Contributions P1(k,i)in Pjk<0 direction of avgbusZ technique 38
Table3.12 Powerflow Usage Contributions U1(k,i)in Pjk<0direction of avgbusZ technique 38
Table3.13 Powerflow Usage Of Line usage1(k)in Pjk<0 direction of avgbusZ technique 38
10
Table3.14 Powerflow Contributions Ug1(i,k)in Pjk<0 direction of avgbusZ technique 38
Table3.15 Powerflow Contributions Ud1(i,k) in Pjk<0 direction of avgbusZ technique 39
Table3.16 Generator Cost Contributions cg1(k,i) in Pjk<0 direction of avgbusZ technique 39
Table3.17 Load Cost Contributions cd1(k,i) in Pjk<0 direction of avgbusZ technique 39
Table3.18 Total generation and load costs and Total cost1 for all the buses in Pjk<0
direction of avgbusZ technique 39
Table3.19 Average Generator Cost Contributions cgavg(k,i) of avgbusZ technique 40
Table3.20 Average Load Cost Contributions cdavg(k,i) of avgbusZ technique 40
Table3.21 Total average generation and load costs and Total avgcost for all the buses
of avgbusZ technique 40
Table 4.1 Converged voltages of modified avgbusZ technique 54
Table 4.2 Bus Currents of modified avgbusZ technique 54
Table 4.3 Powerflow Contributions S1(i,k) in Pjk>0 direction of modified avgbusZ
technique 54
Table 4.4 Powerflow Contributions S2(i,k) in Pjk>0 direction of modified avgbusZ
technique 54
Table 4.5 Powerflow Contributions S3(i,k) in Pjk>0 direction of modified avgbusZ
technique 55
Table 4.6 Powerflow Contributions S4(i,k) In Pjk>0 Direction of modified avgbusZ
technique 55
Table 4.7 Powerflow Contributions Ug(i,k) in Pjk>0 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 avgbusZ
11
technique 56
Table 4.11 Total generation and load costs and Total cost for all the buses of modifiedavgbusZ technique 56
Table 4.12 Powerflow Contributions S11(i,k) in Pjk<0 direction of modified avgbusZ
technique 57
Table 4.13 Powerflow Contributions S21(i,k) in Pjk<0 direction of modified avgbusZ
technique 57
Table 4.14 Powerflow Contributions S31(i,k) in Pjk<0 direction of modified avgbusZ
technique 57
Table 4.15 Powerflow Contributions S41(i,k) in Pjk<0 direction of modified avgbusZ
technique 58
Table 4.16 Powerflow Contributions Ug1(i,k)IN Pjk<0 direction of modified avgbusZ
technique 58
Table 4.17 Powerflow Contributions Ud1(i,k)IN Pjk<0 direction of modified avgbusZ
technique 58
Table 4.18 Generator Cost Contributions cg1(k,i) in Pjk<0 direction of modified avgbusZ
technique 58
Table 4.19 Load Cost Contributions cd1(k,i) in Pjk<0 direction of modified avgbusZ
technique 58
Table 4.20 Total generation and load costs and Total cost for all the buses
in Pjk <0 direction of modified avgbusZ technique 59
Table 4.21 Average Generator Cost Contributions cgavg(k,i) of modified avgbusZ
technique 59
Table 4.22 Average Load Cost Contributions cdavg(k,i) of modifiedavgbusZ technique 59
Table 4.23 Total average generation and load costs and Total avgcost for all the buses
of modified avgbusZ technique 59
Table 5.1 Generator Cost Contributions Cg(k,i) in Pjk>0 Direction of Zbus technique 61
Table 5.2 Generator Cost Contributions Cg(k,i) in Pjk>0 Direction of Zbus technique 62
12
Table 5.3 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique 63
Table 5.4 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique 64
Table 5.5 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique 65
Table 5.6 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique 66
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 5.11 Average Load Cost Contributions Cdavg(k,i) of avgbusZ technique 71
Table 5.12 Average Load Cost Contributions Cdavg(k,i) of avgbusZ technique 72
Table 5.13 Average Load Cost Contributions Cdavg(k,i) of avgbusZ technique 73
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 avgbusZ
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 modifiedavgbusZ 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
13
LIST OF FIGURES Page No
Fig 1.1 Expansion in Centralized Systems 5
Fig 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
14
CHAPTER 1INTRODUCTION
1.1 DeregulationIn Eighties, almost all electric power utilities throughout the world were operated
with an organizational model in which one controlling authority—the utility—operated
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 power
network 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.
15
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 point
of 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.
16
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 of
power 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.
17
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 are
responsible 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
18
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
19
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.
20
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 company’s 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 Kirchhoff’s 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 Kirchhoff’s 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.
21
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 treating“close 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.
22
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
23
CHAPTER 2TRANSMISSION 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 susceptanceshjky 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
jv shjky sh
jky kv
- -
Fig. 2.1 equivalent - circuit of line jk .
24
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 jkV 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 ii
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
( ) shjk j k jk j jkI V V y V y= − + (2.3)
Substituting (2.2) in (2.3) and rearranging
1( )
nsh
jk ji ki jk ji jk ii
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 andigenQ respectively) and the active and
reactive load demands at bus i ( iloadP and
iloadQ respectively ) as
*
( ) ( )i i i igen load gen load
ii
P P j Q QI
V− − −
= (2.5)
25
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
iN
I
ijkJK IaI ∑ =
=1 (2.6)
Where
( ) shjkjijkkiji
ijk yzyzza +−= (2.7)
Observe that the magnitude of parameterijka provides a measure of the electrical
distance between bus i and line jk .
Substituting (2.6) in (2.1)
( ) ∑∑ ====
n
i iijkj
n
i iijkjjk IaVIaVS
1***
1 (2.8)
Then, the active power through line jk is
{ }∑ =ℜ=
n
i iijkjjk IaVP
1** (2.9)
or, equivalently
{ }∑ =ℜ=
n
i iijkjjk 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
ijkj
ijk IaVP ℜ= (2.11)
26
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 theni
jk
N
ijk UU ∑ ==
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
isijk
Gijk UU = (2.14)
On the other hand, if bus contains only demand, the usage allocated to demand pertaining
to line jk isijk
Dijk 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
( )[ ] ijkDiGiGi
Gijk UPPPU += (2.16)
and the usage allocated to the demand at bus pertaining to line jk is
( )[ ] ijkDiGiDi
Dijk UPPPU += (2.17)
27
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 kj k
j k
Cr
U= (2.19)
Using the per unit cost rate, we can write,GijkC , the allocated cost of line jk to the
generator ‘i' located at bus ‘i' isGi Gijk jk jkC r U= (2.20)
In the same way, we can write,DijkC , the allocated cost of line jk to the demand ‘i'
located at bus ‘i' isDi Dijk jk jkC r U= (2.21)
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 Gijk
j k nlineC 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 ijk
j k nlineC C
∈
= ∑ (2.23)
28
2.4 Algorithm For Transmission Network Cost Allocation Using
Zbus Technique
Algorithm1. (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
qi YVV θδ −∑
=
Qcal(i)= )sin(1
iqiqiqq
n
qi YVV θδ −∑
=
(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) Calculate maxP∆ and maxQ∆ form [ ∆ p] and [ ∆ Q] vectors
(h) Is maxP∆ ≤∈ 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
29
(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=1020;
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∆+= δδ & Vi=Vi+{ ∆ X(i+n)}Vi
5. One iteration over
Advance iteration count k1=k1+1
If (k1< itermax) then goto step 2(b) else print problem is not converged in
“itermax” iterations, Stop.
6. Print problem is converged in ‘iter’no. 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 using 1−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 ,
GijkU Di
jkU and ijkU using the equations (2.12),(2.16),(2.17).
End of Do loop
b) Find usage allocated to the line jk
1
ni
jk jki
U U=
=∑
End of if
30
B) Do for each bus, 1 to n
a) Determine the contributions of generators and loads paying for
using the line jk , jkr , GijkC , Di
jkC using equations (2.19), (2.20) and (2.21)
b) Find the factor per unit usage cost rate rjk interchanging ‘from bus’
and ‘to bus’
jk
jkjk U
Cr =
c) Find the generation i cost contributions for using line jk
interchanging ‘from bus’ and ‘to bus’Gijkjk
Gijk UrC =
d) Find the load cost contributions for using line jk
interchanging ‘from bus’ and ‘to bus’Dijkjk
Dijk 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 Gijk
j k nline
C C∈
= ∑
10. Find the cost of contribution of load i using all the lines in the network
( , )
Di Dijk
j k nline
C C∈
= ∑
31
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, jkC is considered to be proportional
to its series reactance jkx i.e. 1000jk jkC 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
32
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
33
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 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
34
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 972 35.8984 21.3386 31.8153 7.9477 973 49.6443 13.5066 13.4857 20.3634 974 17.285 63.8273 0 15.8877 975 8.209 20.8313 33.0189 34.9408 97
35
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 and 4
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 2–4. 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.
36
CHAPTER 3TRANSMISSION NETWORK COST ALLOCATION
USING avgbusZ TECHNIQUE
3.1 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.
3.2 Background:
The equivalent circuit of a line having a line with primitive admittance jky and half line
charging susceptanceshjky connected between the buses j and k is shown in Fig.3.1
[10]. jv and kv represent the nodal voltages of buses j and k respectively.
j k
+ Sjk jky +
jkI
jv shjky sh
jky kv
- -
Fig. 3.1 equivalent -circuit of line jk .
37
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 jkV I= (3.1)
The voltage at node j in terms of the elements of bus impedance matrix Zbus and the nodal
current iI is given by
1
n
j j i ii
V Z I=
= ∑ (3.2)
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
( ) shjk j k jk j jkI V V y V y= − + (3.3)
Substituting (3.2) in (3.3) and rearranging
1( )
nsh
jk ji ki jk ji jk ii
I Z Z y Z y I=
= − + ∑ (3.4)
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 andigenQ respectively) and the active and
reactive load demands at bus i ( iloadP and
iloadQ respectively ) as
*
( ) ( )i i i igen load gen load
ii
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
iN
I
ijkjk IaI ∑ =
=1 (3.6)
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
38
contribution of each generator and load on the line flow jk.this aspect is considered in
proposing new technique.
Where
( ) shjkjijkkiji
ijk yzyzza +−= (3.7)
Observe that the magnitude of parameterijka provides a measure of the electrical
distance between bus i and line jk .
Substituting (3.6) in (3.1)
( ) ∑∑ ====
n
i iijkj
n
i iijkjjk IaVIaVS
1***
1 (3.8)
Then, the active power through line jk is
{ }∑ =ℜ=
n
i iijkjjk IaVP
1** (3.9)
or, equivalently
{ }∑ =ℜ=
n
i iijkjjk 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
ijkj
ijk IaVP ℜ= (3.11)
39
3.3 Transmission Cost AllocationFollowing (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 theni
jk
N
ijk UU ∑ ==
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
isijk
Gijk UU = (3.14)
On the other hand, if bus contains only demand, the usage allocated to demand pertaining
to line jk isijk
Dijk 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
( )[ ] ijkDiGiGi
Gijk UPPPU += (3.16)
and the usage allocated to the demand at bus pertaining to line jk is
( )[ ] ijkDiGiDi
Dijk UPPPU += (3.17)
40
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 kj k
j k
Cr
U= (3.18)
Using the per unit cost rate, we can write,GijkC , the allocated cost of line jk to the
generator ‘i' located at bus ‘i' isGi Gijk jk jkC r U= (3.19)
In the same way, we can write,DijkC , the allocated cost of line jk to the demand ‘i'
located at bus ‘i' isDi Dijk jk jkC r U= (3.20)
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 Gijk
j k nlineC C
∈
= ∑ (3.21)
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 ijk
j k nlineC C
∈
= ∑ (3.22)
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 and i
kja .
However, (3.7) shows that distance parameters are not generally symmetrical with
respect to line indexes, i.e., ikj
ijk 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.
42
3.6 Algorithm For Transmission Network Cost Allocation UsingavgbusZ Technique
Algorithm1 (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
qi YVV θδ −∑
=
Qcal(i)= )sin(1
iqiqiqq
n
qi YVV θδ −∑
=
(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) Calculate maxP∆ and maxQ∆ form [ ∆ p] and [ ∆ Q] vectors
(h) Is maxP∆ ≤∈ 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
(c) Form diagonal elements Hpp, Npp, Mpp & Lpp
(c) Form off – diagonal elements: Hpq, Npq, Mpq & Lpp
(d) Form right hand side vector(mismatch vector)
43
B[i]= ∆ P[i] , B[i+n]= ∆ Q[i] for i=1 to n
(f) Modify the elements
For p=slack bus; Hpp=1e20=1020; Lpp=1e20=1020;
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∆+= δδ & Vi=Vi+{ ∆ X(i+n)}Vi
5. One iteration over
Advance iteration count k1=k1+1
If (k1< itermax) then goto step 2(b) else print problem is not converged in
“itermax” iterations, Stop.
6. Print problem is converged in ‘iter’no. of iterations.
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 )
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 ,
GijkU Di
jkU and ijkU 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
44
b) Find usage allocated to the line jk
1
ni
jk jki
U U=
=∑
c) Find usage allocated to the line by interchanging the ‘from bus’
and ‘to bus’
11 1
ni
jk jki
U U=
=∑
Else
Assign ‘from bus’ as ‘to bus’ and ‘to bus’ as ‘from bus’ and
repeat steps 1), 2) & 3)
End of if
B) Do for each bus, 1 to n
a) Determine the contributions of generators and loads paying for
using the line jk , jkr , GijkC , Di
jkC using equations (3.19), (3.20) and (3,21)
b) Find the factor per unit usage cost rate r1jk interchanging ‘from bus’
and ‘to bus’
11
jkjk
jk
Cr
U=
c) Find the generation i cost contributions for using line jk
interchanging ‘from bus’ and ‘to bus’
1 1 1Gi Gijk jk jkC r U=
e) Find the load cost contributions for using line jk
interchanging ‘from bus’ and ‘to bus’
1 1 1Di Dijk jk jkC r U=
End of bus Do loop
End of line Do loop
45
9. Find the cost of contribution of generator i using all the lines in the network
( , )
Gi Gijk
j k nline
C C∈
= ∑
10. Find the cost of contribution of generator i using all the lines in the network
interchanging ‘from bus’ and ‘to bus’
( , )
1 1Gi Gijk
j k nline
C C∈
= ∑
11. Find the cost of contribution of load i using all the lines in the network
( , )
Di Dijk
j k nline
C C∈
= ∑
12. Find the cost of contribution of load i using all the lines in the network
interchanging ‘from bus’ and ‘to bus’
( , )
1 1Di 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
12
Gi Gijk jkGi
jk
C CCavg
+=
B) Find the average cost contribution of load i using the line jk
12
Di Dijk jkDi
jk
C CCavg
+=
End of bus loop
End of line loop
46
14. Find the average cost contribution of generator i using all the lines in the
network
∑ ==
sforalllinejkGijk
Gi CavgCavg
15. Find the average cost contribution of load i using all the lines in the
network
∑ ==
sforalllinejkDijk
Di CavgCavg
47
3.6 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.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, jkC is considered to be proportional
to its series reactance jkx i.e. 1000jk jkC 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. 3. 2 Four Bus System
48
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
Table 3.2 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 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 0.3111
5 0.1907 0.484 0.7672 -0.8119
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 5
1 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
50
Table 3.8 Generator Cost Contributions cg(k,i)in Pjk>0 direction,equ(3.19)
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 3.9 Load Cost Contributions cd(k,i) in Pjk>0 direction,equ(3.20)
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
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
51
Table3.11 Powerflow Contributions P1(k,i)in Pjk<0 direction
Line\Bus 1 2 3 4
1 0.8486 -0.7536 -0.5032 -0.1879
2 -0.3081 0 -1.25 -0.3164
3 -0.3815 0.2391 -0.2538 -0.8763
4 0.1907 -0.7231 -0.5134 -0.8119
5 0.3081 0 -1.25 0.3164
Table3.12 Powerflow Usage Contributions U1(k,i)in Pjk<0 direction
Line\Bus 1 2 3 4
1 0.8486 0.7536 0.5032 0.1879
2 0.3081 0 1.25 0.3164
3 0.3815 0.2391 0.2538 0.8763
4 0.1907 0.7231 0.5134 0.8119
5 0.3081 0 1.25 0.3164
Table3.13 Powerflow Usage Of Line usage1(k)in Pjk<0 direction
Line USAGE1
1 2.293242
2 1.874441
3 1.750674
4 2.239097
5 1.874441
Table3.14 Powerflow Contributions Ug1(i,k)in Pjk<0 direction
Bus\Line 1 2 3 4 5
1 0.849 0.308 0.381 0.191 0.308
2 0.754 0 0.239 0.723 0
3 0 0 0 0 0
4 0 0 0 0 0
52
Table3.15 Powerflow Contributions Ud1(i,k) in Pjk<0 direction
Bus\Line 1 2 3 4 5
1 0 0 0 0 0
2 0 0 0 0 0
3 0.503 1.25 0.254 0.513 1.25
4 0.188 0.316 0.876 0.812 0.316
Table3.16 Generator Cost Contributions cg1(k,i) in Pjk<0 direction
Line\Gen GEN-1 GEN-2
1 35.8924 31.8763
2 15.9432 0
3 21.1366 13.2477
4 8.263 31.3261
5 15.9432 0
Table3.17 Load Cost Contributions cd1(k,i) in Pjk<0 direction
Table3.18 Total generation and load costs and Total cost1 for all the buses
in Pjk<0 direction
Bus No CG1 CD1 COST1
1 97.17843 0 97.17843
2 76.45012 0 76.45012
3 0 186.9605 186.9605
4 0 124.411 124.411
Line\Load LOAD-3 LOAD-4
1 21.285 7.9464
2 64.686 16.3708
3 14.0631 48.5526
4 22.2404 35.1705
5 64.686 16.3708
53
Table3.19 Average Generator Cost Contributions cgavg(k,i)
Line\Gen GEN-1 GEN-2
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)
Line\Load LOAD-3 LOAD-4
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 Zbus and Zbus average
methods 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 and Zbus average allocate most of the
usage of line 5 (between buses 3 and 4) to buses 3 and 4.
54
Also note 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.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 and 4
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 2–4. 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.
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 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 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 to
demands. 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.
56
CHAPTER 4TRANSMISSION NETWORK COST ALLOCATION
USING MODIFIED avgbusZ TECHNIQUE (newly proposing technique)
4.1 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. 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:
The equivalent circuit of a line having a line with primitive admittance jky and half line
charging susceptanceshjky connected between the buses j and k is shown in Fig. 4.1
[10]. jv and kv represent the nodal voltages of buses j and k respectively.
j k
+ Sjk jky +
jkI
jv shjky sh
jky kv
- -
Fig. 4.1. equivalent -circuit of line jk .
57
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 jkV I= (4.1)
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 ii
V Z I=
= ∑ (4.2)
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
( ) shjk j k jk j jkI V V y V y= − + (4.3)
Substituting (4.2) in (4.3) and rearranging
1( )
nsh
jk ji ki jk ji jk ii
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
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.
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 andigenQ respectively) and the active and
reactive load demands at bus i ( iloadP and
iloadQ respectively ) as
*
( ) ( )i i i igen load gen load
ii
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
58
1
[( ) ( )]n
i i i i ijk jk gen load gen load
iS Factor P P j Q Q
=
= − + −∑ (4.6)
where
*( ) shj ji ki jk ji jki
jki
V Z Z y Z yFactor
V
− + = (4.7)
Thus , the active and reactive power flow jkS through any line jk is represented as a
function of the power generation and load at all buses
i.e , ,i i igen load genP P Q and
iloadQ ; i = 1,2,3…..n
Eq. (6) can be rewritten as
1( 1 2 3 4 )
ni i i i
jk jk jk jk jki
S S S S S=
= + + +∑ (4.8)
Where
1 * ; 2 *
3 * ; 4 *
i i i i i ijk jk gen jk jk load
i i i i i ijk 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 ijkF a c 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 ijk jk jk jkS 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 ijk jk jk jk jkS S S S S= + + + (4.9)
59
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' , GijkU , 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 and igenQ .
4.3 Transmission Cost Allocation:
Thus, usage of line jk by generator ‘i' can be written as
| ( 1 )| | ( 3 ) |Gi i ijk jk jkU S S= ℜ + ℜ (4.10)
Similarly, the usage of line jk by any demand ‘i’,DijkU 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 andiloadQ .
Hence, the usage of line jk by demand ‘i’ can be written as
| ( 2 )| | ( 4 ) |Di i ijk jk jkU S S= ℜ + ℜ (4.11)
The usage of line by bus ‘i' , ijkU , is then given by
i Gi Dijk jk jkU U U= + (4.12)
The total usage of line jk , jkU , by all buses is then
1
ni
jk j ki
U U=
= ∑ (4.13)
The complex power flow components through line jk due to individual power
generations 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
60
out. Let jkC in $/h, represents the total annualized line cost including operation,
maintenance and building costs [8]. cost of each line, jkC is considered to be
proportional to its series reactance jkx i.e. 1000jk jkC x= × $/h[8].
Then the per unit usage cost rate j kr can be written as
j kj k
j k
Cr
U= (4.14)
Using the per unit cost rate, we can write,GijkC , the allocated cost of line jk to the
generator ‘i' located at bus ‘i' isGi Gijk jk jkC r U= (4.15)
In the same way, we can write,DijkC , the allocated cost of line jk to the demand ‘i'
located at bus ‘i' isDi Dijk jk jkC r U= (4.16)
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 Gijk
j k nlineC C
∈
= ∑ (4.17)
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 ijk
j k nlineC C
∈
= ∑ (4.18)
It is to be noted that complex power flow equation (4.8) 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 (4.8) leads to electrical distance parameters ijkFactor and
ikjFactor . However, (4.7) shows that distance parameters are not generally symmetrical
61
with respect to line indexes, i.e., i ijk kjFactor 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 direction
of 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 avgbusZ technique (newly proposing technique)
Algorithm1 (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
qi YVV θδ −∑
=
Qcal(i)= )sin(1
iqiqiqq
n
qi YVV θδ −∑
=
(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,
62
(g) Calculate maxP∆ and maxQ∆ form [ ∆ p] and [ ∆ Q] vectors
(h) Is maxP∆ ≤∈ 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
(d) Form diagonal elements Hpp, Npp, Mpp & Lpp
(c) Form off – diagonal elements: Hpq, Npq, Mpq & Lpp
(d) Form right hand side vector(mismatch vector)
B[i]= ∆ P[i] , B[i+n]= ∆ Q[i] for i=1 to n
(g) Modify the elements
For p=slack bus; Hpp=1e20=1020; Lpp=1e20=1020;
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∆+= δδ & Vi=Vi+{ ∆ X(i+n)}Vi
5. One iteration over
Advance iteration count k1=k1+1
If (k1< itermax) then goto step 2(b) else print problem is not converged in
“itermax” iterations, Stop.
6. Print problem is converged in ‘iter’no. of iterations.
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 using 1−busY )
63
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 ijkFactor , 1 , 2 ,i i
jk jkS S 3 , 4i ijk jkS S ,
GijkU 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 , 41i i
jk jkS S
1GijkU , 1Di
jkU and 1ijkU by interchanging the ‘from bus’
and ‘to bus’ and repeating step a)
End of Do loop
b) Find usage allocated to the line jk
1
ni
jk jki
U U=
=∑
c) Find usage allocated to the line by interchanging the ‘from bus’
and ‘to bus’
11 1
ni
jk jki
U U=
=∑
Else
Assign ‘from bus’ as ‘to bus’ and ‘to bus’ as ‘from bus’ and
repeat steps 1), 2) & 3)
End of if
B) Do for each bus, 1 to n
a) Determine the contributions of generators and loads paying for
using the line jk , jkr , GijkC , Di
jkC using equations (4.14), (4.15) and (4.16)
b) Find the factor per unit usage cost rate r1jk interchanging ‘from bus’
and ‘to bus’
11
jkjk
jk
Cr
U=
64
c) Find the generation i cost contributions for using line jk
interchanging ‘from bus’ and ‘to bus’
1 1 1Gi Gijk jk jkC r U=
f) Find the load cost contributions for using line jk
interchanging ‘from bus’ and ‘to bus’
1 1 1Di Dijk jk jkC r U=
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 Gijk
j k nline
C C∈
= ∑
10. Find the cost of contribution of generator i using all the lines in the network
interchanging ‘from bus’ and ‘to bus’
( , )
1 1Gi Gijk
j k nline
C C∈
= ∑
11. Find the cost of contribution of load i using all the lines in the network
( , )
Di Dijk
j k nline
C C∈
= ∑
12. Find the cost of contribution of load i using all the lines in the network
interchanging ‘from bus’ and ‘to bus’
( , )
1 1Di Dijk
j k nline
C C∈
= ∑
65
13. Do for all lines
Do for all the buses
A) Find the average cost contribution of generator i using the line jk
12
Gi Gijk jkGi
jk
C CCavg
+=
B) Find the average cost contribution of load i using the line jk
12
Di Dijk jkDi
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
∑ ==
sforalllinejkGijk
Gi CavgCavg
15. Find the average cost contribution of load i using all the lines in the
network
∑ ==
sforalllinejkDijk
Di CavgCavg
66
4.5 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. 4.2 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, jkC is considered to be proportional
to its series reactance jkx i.e. 1000jk jkC 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. 4. 2 Four Bus System
67
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
Table 4.2 Bus Currentsibus(1)=2.48868+i0.915759ibus(2)=2.34015+i0.824762ibus(3)=-2.33872+i0.402353ibus(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
Table 4.4 Powerflow Contributions S2(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 0.752+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-0.000000E(2)=1.048494+j0.056220E(3)=1.038233+j-0.178619E(4)=1.049926+j-0.121420
68
Table 4.5 Powerflow Contributions S3(i,k) in Pjk>0 direction,equ(4.8)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 51 0.339 0.849 0.916 0.339 0.2072 1.25 0.519 0.257 1.25 0.5343 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 02 0 0 0 0 03 0 0.752 0.249 0 0.7674 0.311 0.188 0.376 0.311 0.812
69
Table 4.9 Generator Cost Contributions cg(k,i) in Pjk>0 direction,equ(4.15)
Line\Gen GEN-1 GEN-21 17.285 63.82732 35.6653 21.83313 49.4319 13.86394 17.285 63.82735 8.6695 22.3176
Table 4.10 Load Cost Contributions cd(k,i) in Pjk>0 direction,equ(4.16)
Line\Load LOAD3 LOAD41 0 15.88772 31.6061 7.89553 13.4279 20.27634 0 15.88775 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 cost1 128.3367 0 128.33672 185.6693 0 185.66933 0 77.10707 77.107074 0 93.88696 93.88696
70
Table 4.12 Powerflow Contributions S11(i,k) in Pjk<0 direction
Bus/Line
1 2 3 4 5
1 0.849+j-0.000 -0.329+j0.056 -0.398+j0.045 0.199+j-0.023 0.329+j-0.056
2 -0.765+j0.039 0.000+j-0.000 0.252+j-0.042 -0.760+j0.127 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 4.13 Powerflow Contributions S21(i,k) in Pjk<0 direction
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 -0.503+j-0.088 -1.250+j 0.000 -0.254+j-0.014 -0.513+j-0.030
-1.250+j 0.000
4 -0.188+j-0.022 -0.316+j 0.017 -0.876+j-0.000 -0.812+j 0.000 0.316+j-0.017
Table 4.14 Powerflow Contributions S31(i,k) in Pjk<0 direction
Bus/Line
1 2 3 4 5
1 -0.000+j-0.312 0.021+j 0.121 0.017+j 0.146 -0.008+j-0.073 -0.021+j-0.121
2 0.011+j 0.224 0.000+j 0.000 -0.012+j-0.074 0.037+j 0.223 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
71
Table 4.15 Powerflow Contributions S41(i,k) in Pjk<0 direction
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 4.16 Powerflow Contributions Ug1(i,k)IN Pjk<0 direction
Bus\Line 1 2 3 4 51 0.849 0.349 0.415 0.207 0.3492 0.776 0 0.264 0.798 03 0 0 0 0 04 0 0 0 0 0
Table 4.17 Powerflow Contributions Ud1(i,k)IN Pjk<0 direction
Bus\Line 1 2 3 4 51 0 0 0 0 02 0 0 0 0 03 0.503 1.25 0.254 0.513 1.254 0.188 0.316 0.876 0.812 0.316
Table 4.18 Generator Cost Contributions cg1(k,i) in Pjk<0 direction
Line\Gen GEN-1 GEN-21 35.5409 32.51642 17.6803 03 22.2418 14.15494 8.6317 33.20725 17.6803 0
Table 4.19 Load Cost Contributions cd1(k,i) in Pjk<0 direction
Line\Load LOAD-3 LOAD-41 21.0748 7.86792 63.2998 16.023 13.6111 46.99224 21.3689 33.7922
72
5 63.2998 16.02
Table 4.20 Total generation and load costs and Total cost for all the busesin Pjk <0 direction
Bus No CG1 CD1 COST11 101.7749 0 101.77492 79.87851 0 79.878513 0 182.6543 182.65434 0 120.6923 120.6923
Table 4.21 Average Generator Cost Contributions Cgavg(k,i)
Line\Gen GEN-1 GEN-21 26.413 48.17182 26.6728 10.91663 35.8368 14.00944 12.9584 48.51725 13.1749 11.1588
Table 4.22 Average Load Cost Contributions Cdavg(k,i)
Line\Load LOAD-3 LOAD-41 10.5374 11.87782 47.453 11.95773 13.5195 33.63424 10.6844 24.83995 47.6864 24.9799
Table 4.23 Total average generation and load costs and Total avgcost for all the buses
Bus No CG CD TOTALAVGCOST
1 115.0558 0 115.05582 132.7739 0 132.77393 0 129.8807 129.88074 0 107.2896 107.2896
73
4.6 Conclusions:
The “modified 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
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.
From the above cost allocation tables, it is clear that, the proposed new method
allocates a significant part of the usage of any line to the buses which are directly
connected to it. This is evident from the results of line 5 ,line 1 and line 4. the proposed
method allocate most of the usage of line 5 (between buses 3 and 4) to buses 3 and 4. In
the proposed new approach, the factors S1,S2,S3,S4 for forward flows and
S11,S21,S31,S41 for reverse direction flows have offered very useful information about
the contribution of generator buses and load buses towards the line flow of line jk.
Rigorous calculations without any approximations are possible and cost allocation results
would be more reliable and accurate compared to other methods.It is also noted that
proposed usage based method allocate the transmission cost of line 1(between buses 1 &
2) and line 4 (between buses 2 & 4) to all the buses where as Zbus method allocates zero
cost for bus 3.
Better cost allocation has been observed with modified Zbus average approach
compared to Zbus and Zbus average methods.
74
CHAPTER 5
RESULTS-IEEERTS 24BUS SYSTEM AND CONCLUSIONSIn this project work a software package has been developed for three algorithms:
1. Zbus based
2. Zbus average based
3. Modified Zbus average based transmission cost allocation.
The third method is a new approach proposed in this project which is accurate and
offered very promising results compared to other two methods. The basic calculation of
all three methods is illustrated, step-by-step, for a 4-buses test system. The software
package has been tested on IEEE RTS-24 bus system. The results are reported in this
chapter. The input data of the 24 -bus system is reported in appendix of this thesis.
5.1 Zbus technique Results:
The Contributions of Generators in Forward Power flow Direction are reported inTable 5.1
Table 5.1 Generator Cost Contributions Cg(k,i) in Pjk>0 Direction of Zbus technique
LINE\
GEN
GEN
1
GEN
2
GEN
7
GEN
13
GEN
15
GEN
16
GEN
18
GEN
21
GEN
22
GEN
23
1 0.7487 0.8363 0.1668 0.0328 0.2025 0.1133 0.1905 1.6169 0.9663 1.815
2 5.0854 5.6053 1.4917 0.4735 4.185 2.1864 3.8471 32.6249 19.2029 30.4166
3 6.78 7.3634 1.6495 0.1434 0.3169 0.0226 0.2194 1.392 0.2927 4.3349
4 3.7852 5.3342 2.685 0.4072 1.5066 0.8639 1.4996 11.9357 6.8788 18.0482
5 7.2876 9.8898 2.0525 0.4321 2.6979 1.4042 2.5466 20.8026 11.9398 22.9481
6 1.0155 0.8952 3.3997 0.1858 2.3564 1.0147 1.9529 17.5389 10.2822 2.7046
7 1.0514 1.327 0.913 0.0514 1.9537 1.1252 1.6831 16.4305 10.6746 9.6374
8 2.5175 3.5746 2.2181 0.3397 1.2862 0.7374 1.276 10.2029 5.8986 15.1844
9 5.7097 6.0786 1.8782 0.1989 0.0129 0.189 0.0921 1.1361 1.1429 7.0168
10 0.4164 0.446 1.4864 0.2361 0.9839 0.5737 0.9622 7.9148 4.6796 11.045
11 0 0 40.1096 0 0 0 0 0 0 0
12 0.4867 0.6213 20.2469 0.3032 1.5718 0.7979 1.5128 11.896 6.6119 14.5572
75
The Contributions of Generators in Forward Power flow Direction are reported inTable 5.2
Table 5.2 Generator Cost Contributions Cg(k,i) in Pjk>0 Direction of Zbus technique
LINE\
GEN
GEN
1
GEN
2
GEN
7
GEN
13
GEN1
5
GEN
16
GEN1
8
GEN
21
GEN
22
GEN
23
13 0.0262 0.0593 25.195 0.1887 1.0895 0.4821 1.0456 7.7682 4.001 8.825
14 1.3784 1.834 3.6991 0.2802 0.5364 0.4817 0.6474 5.0429 3.1767 6.6427
15 1.5383 2.0519 4.0091 0.3785 0.0445 0.0593 0.0899 0.212 0.3339 17.396
16 0.1283 0.1445 0.1174 0.3523 1.5703 1.0828 1.5323 13.7365 8.8505 16.5803
17 0.1297 0.1423 0.069 0.3988 1.4092 0.9748 1.3702 12.3649 8 21.7699
18 0.2463 0.3189 0.5066 0.953 0.1948 0.1518 0.2011 1.7827 1.171 21.2532
19 0.1749 0.2355 0.4794 0.0531 0.9034 0.6679 0.8962 8.1401 5.3566 4.1353
20 0.2505 0.3317 0.6126 0.8201 0.8242 0.6018 0.8227 7.364 4.797 10.2632
21 0.3026 0.4254 0.7815 0.0175 1.4519 1.22 1.4167 14.0643 9.8966 34.406
22 0.1939 0.2755 0.555 0.3796 1.2563 1.0494 1.2242 12.1369 8.5256 34.1715
23 0.2018 0.2916 0.5663 0.0533 1.2825 1.1262 1.2408 12.7593 9.1876 6.8785
24 0.0326 0.0352 0.0747 0.0306 0.8117 0.2554 0.2449 4.2469 2.0801 2.8473
25 0.2251 0.3086 0.2702 0.0141 0.6747 0.0553 1.6593 23.7125 14.4716 0.2534
26 0.2251 0.3086 0.2702 0.0141 0.6747 0.0553 1.6593 23.7125 14.4716 0.2534
27 0.5473 0.7078 0.3772 0.0475 1.2023 0.7856 1.0301 10.8055 7.4297 6.9974
28 0.1102 0.1564 0.2221 0.0313 0.1975 0.1721 1.0044 8.0954 7.5969 2.3767
29 0.0045 0.0013 0.0656 0.0356 0.45 0.38 0.4349 4.3855 3.1094 6.1403
30 0.0123 0.0201 0.0561 0.0119 0.2137 0.0923 0.9725 7.0825 1.3768 1.0804
31 0.2037 0.2879 0.2391 0.0117 0.5372 0.0393 0.212 15.2865 82.5741 0.3507
32 0.0678 0.0893 0.0358 0.0078 0.4805 0.1236 1.7238 14.4758 3.1589 1.1103
33 0.0678 0.0893 0.0358 0.0078 0.4805 0.1236 1.7238 14.4758 3.1589 1.1103
34 0.0841 0.1209 0.2685 0.0818 0.713 0.587 0.6951 6.8341 4.7698 11.5888
35 0.0841 0.1209 0.2685 0.0818 0.713 0.587 0.6951 6.8341 4.7698 11.5888
36 0.067 0.0945 0.1801 0.0477 0.3643 0.3053 0.3552 3.5245 2.4781 6.4798
37 0.067 0.0945 0.1801 0.0477 0.3643 0.3053 0.3552 3.5245 2.4781 6.4798
38 0.0985 0.1391 0.1156 0.0057 0.2596 0.019 0.1025 7.3881 57.1613 0.1695
76
The Contributions of loads in Forward Power flow Direction are reported inTable 5.3
Table 5.3 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique
LINE\LOAD LD-1 LD-2 LD-3 LD-4 LD-5 LD-6 LD-7 LD-8 LD-9
1 0.4701 0.4716 1.2265 0.5368 0.7893 0.2462 0.0869 0.3784 0.2296
2 3.1931 3.1611 33.0204 1.8625 4.7767 4.3675 0.7769 3.3941 8.3068
3 4.2572 4.1526 9.2438 5.861 16.135 0.9425 0.8591 3.7246 2.666
4 2.3767 3.0083 3.3178 16.969 3.453 3.664 1.3985 6.1629 12.7388
5 4.5759 5.5774 16.799 9.6788 6.3284 18.8057 1.069 4.7252 11.2395
6 0.6376 0.5048 35.3106 2.463 0.4308 0.8557 1.7707 7.7714 13.3119
7 0.6602 0.7484 11.0976 1.4091 1.0901 1.4528 0.4755 2.0278 2.8423
8 1.5808 2.0159 3.1262 15.6109 2.1483 2.0191 1.1553 5.091 9.7108
9 3.5851 3.428 7.1124 4.808 20.8505 1.5319 0.9782 4.2794 1.3335
10 0.2615 0.2515 3.2471 1.0872 1.1004 1.9558 0.7742 3.4052 3.2762
11 0 0 0 0 0 0 20.8904 0 0
12 0.3056 0.3504 7.9938 3.3321 0.4182 2.164 10.5453 46.1185 12.2192
13 0.0164 0.0334 6.4433 2.7572 1.3694 4.1143 13.1226 57.8444 11.4203
14 0.8655 1.0343 8.3953 5.129 1.2242 1.2272 1.9266 8.2638 16.9764
15 0.9659 1.1572 10.1767 5.6673 1.2722 1.1234 2.0881 8.9347 18.7899
16 0.0806 0.0815 3.6815 0.7653 0.9526 2.5037 0.0612 0.2798 3.7787
17 0.0814 0.0803 3.617 0.8426 0.9742 2.5587 0.0359 0.1754 4.1099
18 0.1546 0.1798 0.4812 0.4519 0.4544 0.8949 0.2638 1.1308 1.0699
77
The Contributions of loads in Forward Power flow Direction are reported inTable 5.4
Table 5.4 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique
LINE\LOAD LD-10 LD-13 LD-14 LD-15 LD-16 LD-18 LD-19 LD-20
1 0.5774 0.0304 0.6417 0.2986 0.0731 0.1586 0.6619 0.4338
2 1.4353 0.4398 10.6353 6.1704 1.4106 3.2027 12.1888 7.5391
3 10.9098 0.1332 1.6291 0.4673 0.0146 0.1827 0.5282 0.7781
4 0.1612 0.3782 6.5075 2.2214 0.5573 1.2485 5.7803 4.112
5 1.6897 0.4014 8.1421 3.9778 0.9059 2.1201 8.4599 5.503
6 2.7396 0.1726 0.587 3.4744 0.6546 1.6258 3.9605 1.3833
7 1.4269 0.0478 3.1761 2.8806 0.7259 1.4012 5.013 2.6775
8 0.7249 0.3155 5.47 1.8965 0.4758 1.0623 4.894 3.4673
9 10.6898 0.1848 2.584 0.019 0.122 0.0767 1.5361 1.4253
10 3.8974 0.2193 3.9601 1.4507 0.3701 0.8011 3.6526 2.5457
11 0 0 0 0 0 0 0 0
12 4.8144 0.2816 5.2117 2.3175 0.5148 1.2594 5.1193 3.4282
13 7.7313 0.1752 3.1697 1.6064 0.311 0.8705 3.2268 2.1065
14 0.1413 0.2603 8.8925 0.7909 0.3107 0.5389 2.659 1.6446
15 0.2802 0.3515 0.8081 0.0656 0.0383 0.0748 2.8091 3.2838
16 4.5466 0.3272 8.6103 2.3153 0.6986 1.2757 5.9934 3.9527
17 4.6608 0.3704 4.8536 2.0778 0.6289 1.1407 6.349 4.8146
18 1.33 0.8852 5.821 0.2872 0.0979 0.1674 3.515 4.0356
78
The Contributions of loads in Forward Power flow Direction are reported inTable 5.5
Table 5.5 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique
LINE\LOAD LD-1 LD-2 LD-3 LD-4 LD-5 LD-6 LD-7 LD-8 LD-9
19 0.1098 0.1328 0.6244 0.362 0.3984 0.855 0.2497 1.0634 0.9124
20 0.1573 0.1871 0.301 0.493 0.5238 1.0896 0.3191 1.3629 1.2127
21 0.19 0.2399 0.9969 0.5771 0.6369 1.3475 0.407 1.6891 1.4615
22 0.1218 0.1554 1.0481 0.3903 0.4436 0.9629 0.2891 1.2004 1.0076
23 0.1267 0.1644 1.0745 0.3937 0.4488 0.9689 0.295 1.2114 1.019
24 0.0205 0.0198 1.1152 0.0056 0.0329 0.1395 0.0389 0.1568 0.0438
25 0.1414 0.1741 1.4422 0.2986 0.2617 0.4244 0.1407 0.5714 0.6622
26 0.1414 0.1741 1.4422 0.2986 0.2617 0.4244 0.1407 0.5714 0.6622
27 0.3437 0.3992 6.0695 0.6626 0.4814 0.5615 0.1965 0.8259 1.3085
28 0.0692 0.0882 0.0407 0.1795 0.1863 0.3672 0.1157 0.4697 0.4386
29 0.0028 0.0007 0.5802 0.0191 0.04 0.1191 0.0342 0.1404 0.0767
30 0.0078 0.0114 0.239 0.0305 0.0403 0.0952 0.0292 0.1172 0.0869
31 0.1279 0.1623 1.1915 0.2548 0.2245 0.3631 0.1245 0.4935 0.5727
32 0.0426 0.0504 0.7867 0.0746 0.0504 0.049 0.0187 0.0779 0.1441
33 0.0426 0.0504 0.7867 0.0746 0.0504 0.049 0.0187 0.0779 0.1441
34 0.0528 0.0682 0.6655 0.1798 0.2106 0.4691 0.1398 0.5816 0.4737
35 0.0528 0.0682 0.6655 0.1798 0.2106 0.4691 0.1398 0.5816 0.4737
36 0.042 0.0533 0.2746 0.1304 0.1456 0.3113 0.0938 0.3894 0.3328
37 0.042 0.0533 0.2746 0.1304 0.1456 0.3113 0.0938 0.3894 0.3328
38 0.0618 0.0785 0.5759 0.1232 0.1085 0.1755 0.0602 0.2385 0.2768
79
The Contributions of loads in Forward Power flow Direction are reported inTable 5.6
Table 5.6 Load Cost Contributions Cd(k,i) in Pjk>0 Direction of Zbus technique
LINE\LOAD LD-10 LD-13 LD-14 LD-15 LD-16 LD-18 LD-19 LD-20
19 1.3531 0.0494 8.229 1.332 0.4309 0.7461 2.7998 1.3093
20 1.6864 0.7618 5.0714 1.2152 0.3882 0.6849 3.4695 2.3885
21 2.1629 0.0162 3.2444 2.1407 0.7871 1.1794 8.6588 7.2818
22 1.5689 0.3526 0.9345 1.8523 0.677 1.0191 8.1002 7.1081
23 1.5892 0.0495 7.9214 1.8909 0.7266 1.0329 4.386 2.113
24 0.299 0.0284 0.9669 1.1968 0.1648 0.2039 1.1885 0.7194
25 0.5656 0.0131 0.1155 0.9947 0.0357 1.3814 0.1255 0.007
26 0.5656 0.0131 0.1155 0.9947 0.0357 1.3814 0.1255 0.007
27 0.4338 0.0441 2.3143 1.7726 0.5069 0.8576 3.4041 1.8874
28 0.5744 0.0291 0.8216 0.2911 0.111 0.8362 0.8522 0.5663
29 0.2271 0.0331 0.9119 0.6635 0.2451 0.3621 2.9018 1.6352
30 0.1675 0.011 0.3678 0.3151 0.0596 0.8096 0.4262 0.2671
31 0.496 0.0109 0.1444 0.7921 0.0254 0.1765 0.0597 0.0379
32 0.0201 0.0073 0.3669 0.7084 0.0797 1.4351 0.5199 0.2947
33 0.0201 0.0073 0.3669 0.7084 0.0797 1.4351 0.5199 0.2947
34 0.7747 0.076 1.2115 1.0513 0.3787 0.5787 4.301 3.044
35 0.7747 0.076 1.2115 1.0513 0.3787 0.5787 4.301 3.044
36 0.5028 0.0443 0.567 0.5371 0.197 0.2957 2.2527 1.9336
37 0.5028 0.0443 0.567 0.5371 0.197 0.2957 2.2527 1.9336
38 0.2397 0.0053 0.0698 0.3828 0.0123 0.0853 0.0288 0.0183
80
The individual generator/load costs for using all the lines in the network and total cost inforward power flows are reported in Table 5.7
Table 5.7 Cost For Individual Generators/Loads And Total Cost in Pjk>0 Direction of Zbus
technique
Bus No CG CD TOTAL
COST
1 41.3522 25.9653 67.3176
2 50.6569 28.5682 79.2251
3 0 184.485 184.485
4 0 84.0901 84.0901
5 0 68.6698 68.6698
6 0 59.9801 59.9801
7 117.549 61.2232 178.772
8 0 175.917 175.917
9 0 145.662 145.662
10 0 72.2811 72.2811
11 0 0 0
12 0 0 0
13 7.15621 6.64702 13.8032
14 0 116.219 116.219
15 35.7738 52.7455 88.5193
16 20.812 13.4271 34.2391
17 0 0 0
18 39.1378 32.5822 71.7199
19 0 127.021 127.021
20 0 89.022 89.022
21 383.247 0 383.247
22 352.952 0 352.952
23 378.857 0 378.857
24 0 0 0
81
5.2 avgbusZ technique results :
The average Contributions of Generators are reported in Table 5.8
Table 5.8 Average Generator Cost Contributions Cgavg(k,i) of Zbus average method
LINE\
GEN
GEN
1
GEN
2
GEN
7
GEN
13
GEN
15
GEN
16
GEN
18
GEN
21
GEN
22
GEN
23
1 1.0584 1.4225 0.1836 0.0251 0.1274 0.0641 0.1139 0.9855 0.5826 1.0933
2 5.3927 5.9318 1.2554 0.4511 4.1783 2.1341 3.8358 32.2454 18.7824 29.3645
3 6.5615 7.0593 1.7605 0.1657 0.218 0.077 0.1241 0.7898 0.4634 5.2913
4 3.5983 5.0471 2.7274 0.4194 1.5511 0.8723 1.5453 12.1758 6.9383 18.363
5 7.5152 10.0346 1.8561 0.4191 2.6807 1.3122 2.535 20.0821 11.1289 21.7555
6 0.9692 0.8291 3.5701 0.2084 2.3423 0.9786 1.9346 17.2185 9.977 1.9537
7 1.0347 1.269 0.9236 0.055 2.0301 1.0698 1.7673 16.2605 10.0645 9.6072
8 2.6043 3.6646 2.2015 0.34 1.2724 0.7161 1.2655 9.9983 5.7092 14.9588
9 5.8857 6.2671 1.8333 0.1893 0.0633 0.1573 0.0598 0.6577 0.8745 6.4938
10 1.1231 1.4259 0.9379 0.1658 0.8058 0.4305 0.7784 6.2384 3.555 7.9423
11 0.0305 0.0365 38.5351 0.0095 0.0407 0.0225 0.0397 0.319 0.1835 0.4361
12 0.3948 0.5045 21.059 0.2863 1.4958 0.7281 1.442 11.1015 6.0159 13.4793
13 0.1226 0.1729 24.4106 0.2131 1.1821 0.5321 1.1377 8.4817 4.3991 9.881
14 1.3604 1.786 3.7184 0.2871 0.5403 0.4672 0.6554 4.9819 3.0636 6.5694
15 1.5065 1.977 4.0144 0.3889 0.053 0.0518 0.083 0.2656 0.3679 17.1791
16 0.1252 0.1379 0.1156 0.3688 1.6077 1.0478 1.5786 13.6456 8.5166 16.5465
17 0.126 0.1346 0.0654 0.421 1.4473 0.9377 1.4174 12.2616 7.6465 21.7209
18 0.3349 0.4274 0.6718 0.967 0.1017 0.0898 0.1126 0.9794 0.654 20.3868
The Contributions of Generators are reported in
82
The average Contributions of Generators are reported in Table 5.8
Table 5.9 Average Generator Cost Contributions Cgavg(k,i ) of Zbus average method
LINE\
GEN
GEN
1
GEN
2
GEN
7
GEN
13
GEN
15
GEN
16
GEN
18
GEN
21
GEN
22
GEN
23
19 0.1999 0.2651 0.5312 0.0608 0.8994 0.6511 0.8957 8.0083 5.2044 3.889
20 0.3436 0.4478 0.7886 0.8433 0.7643 0.555 0.7685 6.7962 4.394 9.5734
21 0.48 0.6381 1.1382 0.0378 1.3875 1.0836 1.3761 12.8656 8.6841 33.7182
22 0.3591 0.4813 0.87 0.4586 1.1775 0.9306 1.1638 10.998 7.4789 33.5097
23 0.232 0.3224 0.6425 0.0665 1.3185 1.0638 1.2922 12.4819 8.6094 6.5393
24 0.0274 0.0281 0.0828 0.0316 0.8071 0.2558 0.2414 4.184 2.0328 2.8867
25 0.2893 0.3922 0.3864 0.0286 0.7137 0.0521 1.5705 21.865 12.9979 1.1077
26 0.2893 0.3922 0.3864 0.0286 0.7137 0.0521 1.5705 21.865 12.9979 1.1077
27 0.5991 0.7646 0.4712 0.0397 1.2058 0.7411 1.036 10.4884 7.0189 6.4817
28 0.12 0.1678 0.2413 0.0346 0.1899 0.1739 1.0101 7.8887 7.2881 2.4801
29 0.006 0.008 0.0758 0.0373 0.4477 0.3716 0.4339 4.3186 3.0352 6.1873
30 0.0184 0.0284 0.0656 0.0129 0.2079 0.0945 0.9644 6.9476 1.3219 1.1352
31 0.488 0.6815 0.7662 0.0797 0.3793 0.325 0.3081 10.5486 69.9922 5.1344
32 0.0455 0.0635 0.0704 0.015 0.4401 0.1481 1.7214 13.8608 2.8669 1.5654
33 0.0455 0.0635 0.0704 0.015 0.4401 0.1481 1.7214 13.8608 2.8669 1.5654
34 0.0486 0.0734 0.2101 0.0754 0.7492 0.6138 0.7293 7.1663 4.9958 11.2918
35 0.0486 0.0734 0.2101 0.0754 0.7492 0.6138 0.7293 7.1663 4.9958 11.2918
36 0.0562 0.08 0.1628 0.046 0.3752 0.3116 0.3659 3.6129 2.5307 6.3782
37 0.0562 0.08 0.1628 0.046 0.3752 0.3116 0.3659 3.6129 2.5307 6.3782
38 0.2355 0.3295 0.3376 0.0294 0.3461 0.0956 0.1882 7.614 49.8338 1.7586
83
The average Contributions of loads are reported in Table 5.8
Table 5.10 Average Load Cost Contributions Cdavg(k,i) of Zbus average method
LINE\LOAD LD-1 LD-2 LD-3 LD-4 LD-5 LD-6 LD-7 LD-8 LD-9
1 0.6646 0.8022 1.2497 1.0773 1.0016 0.8395 0.0956 0.4185 0.5102
2 3.3861 3.3452 33.7739 2.2226 5.2429 5.072 0.6539 2.8649 7.9768
3 4.12 3.9811 8.7652 5.6637 16.0448 1.1503 0.9169 3.9943 2.2917
4 2.2594 2.8463 3.5079 16.6219 3.2589 3.3607 1.4205 6.2792 12.6535
5 4.7189 5.659 17.179 10.1288 6.5551 20.8938 0.9667 4.3113 11.2526
6 0.6086 0.4676 35.7829 2.6166 0.3345 1.079 1.8594 8.1849 13.8081
7 0.6497 0.7157 11.4744 1.4543 1.127 1.5068 0.4811 2.084 2.9057
8 1.6352 2.0667 3.0216 16.135 2.2829 2.2154 1.1466 5.0679 9.8123
9 3.6957 3.5343 7.5158 5.016 21.4331 1.4148 0.9549 4.1771 1.5829
10 0.7052 0.8041 3.696 1.793 1.3508 11.5488 0.4885 2.1599 2.9665
11 0.0191 0.0206 0.1476 0.058 0.0554 0.1044 20.0703 0.0751 0.1313
12 0.2479 0.2845 7.7597 3.2555 0.6142 2.5862 10.9682 48.2077 12.1924
13 0.077 0.0975 6.6819 2.8456 1.1436 3.6384 12.7138 56.1448 11.4339
14 0.8542 1.0072 8.438 5.1879 1.2318 1.2302 1.9367 8.3643 17.1536
15 0.9459 1.1149 10.2023 5.7194 1.2746 1.1155 2.0908 9.0225 18.9383
16 0.0786 0.0778 3.7516 0.7848 0.9693 2.5648 0.0602 0.2781 3.8571
17 0.0791 0.0759 3.6947 0.8678 0.994 2.631 0.0341 0.1688 4.2104
18 0.2103 0.241 0.8724 0.6161 0.6121 1.1931 0.3499 1.5085 1.4446
84
The average Contributions of loads are reported in Table 5.8
Table 5.11 Average Load Cost Contributions Cdavg(k,i) of Zbus average method
LINE\LOAD LD-10 LD-13 LD-14 LD-15 LD-16 LD-18 LD-19 LD-20
1 0.3662 0.0234 0.3949 0.1879 0.0413 0.0948 0.3313 0.2446
2 2.2407 0.419 10.2613 6.1605 1.3768 3.1933 11.9228 7.3157
3 11.093 0.1539 1.9723 0.3214 0.0497 0.1033 0.8648 1.0032
4 0.3245 0.3895 6.6277 2.287 0.5628 1.2864 5.8905 4.1853
5 2.6349 0.3893 7.7379 3.9524 0.8465 2.1104 8.108 5.2362
6 3.0901 0.1936 0.3146 3.4536 0.6314 1.6106 3.7605 1.2222
7 1.4585 0.051 3.1911 2.9931 0.6902 1.4713 4.9971 2.6673
8 0.5548 0.3158 5.3964 1.8761 0.462 1.0535 4.8137 3.4132
9 10.733 0.1758 2.3984 0.0934 0.1015 0.0498 1.3421 1.2996
10 2.3759 0.154 2.843 1.1881 0.2778 0.648 2.7395 1.8577
11 0.1458 0.0088 0.1566 0.0599 0.0145 0.0331 0.1457 0.1008
12 5.4568 0.266 4.8353 2.2055 0.4697 1.2005 4.7638 3.179
13 6.9915 0.198 3.5513 1.7429 0.3433 0.9472 3.5685 2.3479
14 0.1217 0.2667 8.872 0.7966 0.3014 0.5456 2.6352 1.6273
15 0.3153 0.3612 0.8353 0.0781 0.0335 0.0691 2.7589 3.2371
16 4.6341 0.3426 8.6276 2.3704 0.676 1.3142 5.9804 3.9424
17 4.7641 0.391 4.8618 2.134 0.605 1.18 6.3305 4.7994
18 1.7506 0.8982 6.2681 0.15 0.058 0.0937 3.1846 3.8234
85
The average Contributions of loads are reported in Table 5.8
Table 5.12 Average Load Cost Contributions Cdavg(k,i) of Zbus average method
LINE\LOAD LD-1 LD-2 LD-3 LD-4 LD-5 LD-6 LD-7 LD-8 LD-9
19 0.1255 0.1495 0.5384 0.4122 0.4475 0.9519 0.2766 1.1847 1.0285
20 0.2158 0.2526 0.3587 0.6635 0.6891 1.4049 0.4107 1.7624 1.6053
21 0.3014 0.3599 0.648 0.9246 0.9757 2.0093 0.5928 2.5199 2.2636
22 0.2255 0.2714 0.6046 0.6944 0.7392 1.5321 0.4531 1.9202 1.7108
23 0.1457 0.1818 1.0054 0.4682 0.523 1.1254 0.3347 1.4029 1.1935
24 0.0172 0.0158 1.0952 0.0078 0.0404 0.1533 0.0431 0.1746 0.062
25 0.1816 0.2212 1.676 0.4084 0.3684 0.6241 0.2012 0.8269 0.9159
26 0.1816 0.2212 1.676 0.4084 0.3684 0.6241 0.2012 0.8269 0.9159
27 0.3762 0.4312 6.4697 0.7675 0.5781 0.7304 0.2454 1.0406 1.534
28 0.0753 0.0946 0.082 0.1988 0.205 0.4035 0.1257 0.5152 0.4828
29 0.0038 0.0045 0.5604 0.0287 0.0494 0.1369 0.0395 0.1631 0.0993
30 0.0115 0.016 0.2154 0.0395 0.0489 0.1109 0.0342 0.1376 0.1079
31 0.3064 0.3843 1.756 0.7113 0.6824 1.2381 0.3991 1.6179 1.6611
32 0.0285 0.0358 0.6169 0.0658 0.0629 0.1135 0.0367 0.1486 0.1534
33 0.0285 0.0358 0.6169 0.0658 0.0629 0.1135 0.0367 0.1486 0.1534
34 0.0305 0.0414 0.8094 0.1238 0.1571 0.371 0.1094 0.4547 0.3444
35 0.0305 0.0414 0.8094 0.1238 0.1571 0.371 0.1094 0.4547 0.3444
36 0.0353 0.0451 0.3175 0.1139 0.1299 0.2827 0.0848 0.3522 0.2947
37 0.0353 0.0451 0.3175 0.1139 0.1299 0.2827 0.0848 0.3522 0.2947
38 0.1479 0.1858 1.0469 0.3276 0.306 0.5363 0.1758 0.7084 0.7548
86
The average Contributions of loads are reported in Table 5.8
Table 5.13 Average Load Cost Contributions Cdavg(k,i) of Zbus average method
LINE\LOAD LD-10 LD-13 LD-14 LD-15 LD-16 LD-18 LD-19 LD-20
19 1.4909 0.0565 8.2476 1.3262 0.4201 0.7457 2.7365 1.2568
20 2.1369 0.7833 4.8595 1.127 0.3581 0.6398 3.2313 2.2265
21 3.1026 0.0351 2.6067 2.0457 0.6991 1.1456 8.2822 7.0786
22 2.3826 0.4259 0.6603 1.7361 0.6004 0.9689 7.7343 6.9128
23 1.8089 0.0618 8.1441 1.944 0.6863 1.0758 4.2919 2.038
24 0.3187 0.0294 0.9818 1.19 0.165 0.201 1.199 0.7278
25 0.8559 0.0266 0.4158 1.0522 0.0336 1.3074 0.2778 0.2033
26 0.8559 0.0266 0.4158 1.0522 0.0336 1.3074 0.2778 0.2033
27 0.6585 0.0369 2.1403 1.7779 0.4781 0.8625 3.2537 1.7724
28 0.6251 0.0322 0.8607 0.28 0.1122 0.8409 0.8826 0.589
29 0.2529 0.0347 0.8868 0.66 0.2397 0.3612 2.9128 1.6447
30 0.1903 0.012 0.3875 0.3065 0.061 0.8028 0.4422 0.2792
31 1.8245 0.074 1.8072 0.5592 0.2097 0.2565 1.6484 1.1611
32 0.1667 0.0139 0.5273 0.6488 0.0956 1.433 0.6585 0.3971
33 0.1667 0.0139 0.5273 0.6488 0.0956 1.433 0.6585 0.3971
34 0.6331 0.07 1.3656 1.1046 0.396 0.6071 4.4476 2.9807
35 0.6331 0.07 1.3656 1.1046 0.396 0.6071 4.4476 2.9807
36 0.4612 0.0427 0.6116 0.5532 0.201 0.3046 2.2911 1.9591
37 0.4612 0.0427 0.6116 0.5532 0.201 0.3046 2.2911 1.9591
38 0.7734 0.0273 0.6261 0.5103 0.0617 0.1567 0.5054 0.3812
87
The individual average generator/load costs for using all the lines in the network andtotal average cost reported in Table 5.7
Table 5.14 Average Cost For Individual Generators/Loads and Total Avg Cost of Zbus average
method
BUS
NO:
CGAVG(I) CDAVG(I) TOTAL
COSTAVG
1 43.7322 27.4597 71.1919
2 53.5085 30.1763 83.6848
3 0 188.735 188.735
4 0 88.7323 88.7323
5 0 72.2478 72.2478
6 0 77.2602 77.2602
7 117.51 61.2032 178.713
8 0 180.024 180.024
9 0 151.042 151.042
10 0 78.8505 78.8505
11 0 0 0
12 0 0 0
13 7.44293 6.91334 14.3563
14 0 117.195 117.195
15 35.4253 52.2317 87.657
16 20.2819 13.0851 33.367
17 0 0 0
18 38.8784 32.3663 71.2447
19 0 126.608 126.608
20 0 88.6508 88.6508
21 364.838 0 364.838
22 320.599 0 320.599
23 377.002 0 377.002
24 0 0 0
88
5.3 Modified avgbusZ technique results :
The average Contributions of Generators are reported in Table 5.15
Table 5.15 Average Generator Cost Contributions Cgavg(k,i)of Modified Zbus average method
LINE\
GEN
GEN
1
GEN
2
GEN
7
GEN
13
GEN
15
GEN
16
GEN
18
GEN
21
GEN
22
GEN
23
1 2.092 2.2894 0.2677 0.2874 0.258 0.1564 0.4713 0.4701 0.3118 0.522
2 12.6363 11.2804 2.1561 5.7334 9.404 5.9291 17.6071 17.4092 11.4801 15.8537
3 14.5713 12.9648 2.8358 2.2263 0.2661 0.1488 0.5712 0.5449 0.2986 2.6812
4 8.5644 9.9267 4.7126 5.2465 3.3567 2.4947 6.8889 6.6101 4.3524 9.969
5 17.5622 19.7619 3.1142 4.8653 5.5603 3.6705 10.8311 10.5594 6.8069 11.4396
6 2.5106 1.7661 6.6436 2.9489 5.8467 2.9054 9.8227 10.0785 6.5408 1.1434
7 2.3777 2.316 1.6461 0.7487 5.0274 2.895 8.8188 8.9247 6.0439 5.2734
8 6.3267 7.3654 3.8785 4.3417 2.8151 2.0855 5.7634 5.5346 3.6474 8.2804
9 13.7611 12.2483 3.0636 2.5619 0.1484 0.3643 0.6828 0.5723 0.4603 3.4177
10 2.9972 3.2709 1.7308 2.2129 1.8915 1.3172 3.7526 3.6393 2.3858 4.6222
11 0.0443 0.0447 39.2035 0.0711 0.0536 0.0383 0.1074 0.1039 0.0685 0.142
12 1.0304 1.1572 38.4848 3.5782 3.3016 2.2583 6.596 6.3754 4.0721 7.7403
13 0.3507 0.4823 45.256 2.5691 2.5504 1.7167 5.1235 4.9387 3.0845 5.7527
14 3.4791 3.7008 7.1417 4.028 1.2894 1.425 3.2519 2.9464 2.0669 3.8854
15 3.98 4.2252 7.9959 5.8057 0.3911 0.241 0.3601 0.4844 0.2256 10.5224
16 0.32 0.3034 0.2113 5.2675 4.0128 2.9699 7.9636 7.7406 5.3415 9.3864
17 0.3264 0.3023 0.1237 6.1541 3.6617 2.6943 7.2466 7.0497 4.8607 12.4888
18 0.6578 0.6754 1.0042 11.7117 0.2119 0.199 0.4695 0.4422 0.321 9.3162
19 0.4758 0.5068 0.973 0.8825 2.2927 1.7719 4.5891 4.4543 3.1491 2.1635
89
The average Contributions of Generators are reported in Table 5.16
Table 5.16 Average Generator Cost Contributions Cgavg(k,i) Modified Zbus average method
LINE\
GEN
GEN
1
GEN
2
GEN
7
GEN
13
GEN
15
GEN
16
GEN
18
GEN
21
GEN
22
GEN
23
20 0.5912 0.6184 1.0441 9.1055 1.4011 1.1002 2.8346 2.7418 1.9348 3.8621
21 1.302 1.378 2.4354 0.6737 4.2628 3.3204 8.4482 8.2361 5.9456 21.6369
22 0.8434 0.8978 1.6276 7.4154 3.1951 2.4814 6.3023 6.155 4.4619 18.7514
23 0.5209 0.57 1.1794 1.0389 3.5077 2.7024 6.8452 6.7097 4.8982 3.558
24 0.0516 0.039 0.148 0.509 2.0959 0.5916 1.262 2.0793 1.0481 1.4415
25 0.6035 0.6278 0.7185 0.5354 2.0345 0.1264 8.9535 11.5243 7.0479 0.6042
26 0.6035 0.6278 0.7185 0.5354 2.0345 0.1264 8.9535 11.5243 7.0479 0.6042
27 1.2968 1.2842 0.8491 0.6278 3.2168 1.8732 5.5057 5.6297 3.9809 3.4877
28 0.2407 0.2595 0.4275 0.5913 0.5169 0.4039 5.3266 3.9535 3.7783 1.2686
29 0.014 0.0171 0.1521 0.6756 1.308 0.9961 2.5196 2.4821 1.8302 3.5717
30 0.0287 0.034 0.0924 0.1816 0.4515 0.1682 4.0666 2.7001 0.5241 0.4604
31 1.425 1.5176 2.0419 2.1589 1.8217 1.0861 2.7637 8.9663 56.4575 3.8959
32 0.0697 0.0742 0.0997 0.2158 0.9806 0.2659 7.5031 5.4568 1.1479 0.6474
33 0.0697 0.0742 0.0997 0.2158 0.9806 0.2659 7.5031 5.4568 1.1479 0.6474
34 0.1178 0.1396 0.4176 1.3372 2.1614 1.6525 4.1859 4.1155 3.0216 6.4852
35 0.1178 0.1396 0.4176 1.3372 2.1614 1.6525 4.1859 4.1155 3.0216 6.4852
36 0.134 0.149 0.3254 0.829 1.0957 0.8422 2.1242 2.0881 1.5376 3.6861
37 0.134 0.149 0.3254 0.829 1.0957 0.8422 2.1242 2.0881 1.5376 3.6861
38 0.704 0.7479 0.9399 0.8751 1.6072 0.3295 1.7812 6.3656 41.4254 1.4084
90
The average Contributions of loads are reported in Table 5.17
Table 5.17 Average Load Cost Contributions Cdavg(k,i) Modified Zbus average method
LINE\LOAD LD-1 LD-2 LD-3 LD-4 LD-5 LD-6 LD-7 LD-8 LD-9
1 1.3148 1.2914 0.5741 0.488 0.4638 0.3865 0.1396 0.1967 0.2303
2 7.9786 6.3732 18.2347 1.2953 2.8921 2.7986 1.1239 1.5834 4.3066
3 9.195 7.3197 4.4114 2.8507 8.1468 0.5796 1.4786 2.0907 1.1527
4 5.3974 5.6035 1.9045 9.0713 1.7815 1.8303 2.4607 3.4226 6.869
5 11.1308 11.1729 9.0322 5.3398 3.491 11.1489 1.6297 2.2674 5.9164
6 1.5871 0.997 20.9452 1.5317 0.1957 0.6977 3.4654 4.8455 8.0825
7 1.5392 1.3162 6.7941 0.8555 0.6673 0.9026 0.8605 1.2063 1.6534
8 3.9864 4.1575 1.6726 9.0478 1.2701 1.2262 2.0251 2.8171 5.4315
9 8.7189 6.9215 3.9551 2.6397 11.3112 0.7446 1.5957 2.2601 0.8329
10 1.9117 1.8524 2.1727 1.0628 0.7953 6.9183 0.9039 1.2655 1.7362
11 0.0281 0.0253 0.048 0.0189 0.018 0.0346 20.4439 0.0244 0.0427
12 0.6618 0.6566 4.4577 1.8703 0.3538 1.488 20.0693 28.1343 7.0057
13 0.235 0.2761 3.8918 1.6574 0.667 2.1211 23.6252 32.8418 6.6608
14 2.2023 2.092 5.0271 3.2433 0.7385 0.7276 3.7314 5.2585 10.6215
15 2.5236 2.3894 6.347 3.7159 0.7919 0.6833 4.1785 5.8868 12.1862
16 0.2044 0.1719 2.2335 0.4849 0.573 1.5758 0.1116 0.1577 2.3195
17 0.2085 0.1713 2.2292 0.5459 0.5968 1.6422 0.0665 0.097 2.5792
18 0.4195 0.3825 0.4017 0.2985 0.2969 0.5974 0.5249 0.7392 0.6891
91
The average Contributions of loads are reported in Table 5.18
Table 5.18 Average Load Cost Contributions Cdavg(k,i) Modified Zbus average method
LINE\LOAD LD-10 LD-13 LD-14 LD-15 LD-16 LD-18 LD-19 LD-20
1 0.1791 0.2647 0.1898 0.3975 0.1057 0.3715 0.1612 0.1175
2 1.2099 5.2636 5.5996 14.559 4.0336 13.8121 6.4371 3.9497
3 5.5849 2.0621 1.0181 0.4819 0.1151 0.3992 0.4823 0.5171
4 0.2372 4.8073 3.6502 5.2533 1.7156 5.3515 3.1979 2.2721
5 1.3847 4.4403 4.1392 8.7242 2.5354 8.3886 4.2633 2.7533
6 1.8885 2.7136 0.2077 9.0035 1.9631 7.7476 2.2011 0.7154
7 0.8297 0.6904 1.7808 7.6436 1.9295 7.0401 2.7429 1.4641
8 0.4061 3.9786 3.0299 4.4028 1.4333 4.4798 2.6646 1.8894
9 5.6484 2.3697 1.2663 0.2816 0.2449 0.5263 0.7168 0.684
10 1.3759 2.0269 1.6782 2.9541 0.9045 2.9199 1.5961 1.0816
11 0.0475 0.0651 0.0517 0.0835 0.0263 0.0837 0.0474 0.0328
12 3.1384 3.261 2.8308 5.2143 1.5705 5.0782 2.7357 1.8255
13 4.0753 2.3298 2.1175 4.0656 1.207 3.9095 2.0777 1.367
14 0.0719 3.7003 5.2868 2.012 0.9705 2.5415 1.5586 0.9624
15 0.3552 5.3461 0.6256 0.6317 0.1888 0.2221 1.6898 1.9827
16 2.7622 4.8638 4.9203 6.1352 1.9923 6.3223 3.3925 2.2364
17 2.885 5.687 2.8102 5.5985 1.8076 5.7528 3.6398 2.7595
18 0.85 10.8579 2.8868 0.3204 0.1321 0.3763 1.4539 1.7469
92
The average Contributions of loads are reported in Table 5.19
Table 5.19 Average Load Cost Contributions Cdavg(k,i) Modified Zbus average method
LINE\LOAD LD-1 LD-2 LD-3 LD-4 LD-5 LD-6 LD-7 LD-8 LD-9
19 0.3033 0.2871 0.3645 0.2444 0.2654 0.586 0.5094 0.7156 0.6051
20 0.3777 0.3504 0.1649 0.2849 0.296 0.6249 0.5464 0.7681 0.6813
21 0.8375 0.7822 0.5028 0.6447 0.6806 1.4564 1.277 1.7892 1.5631
22 0.5438 0.51 0.412 0.4238 0.4513 0.9736 0.8541 1.1952 1.0351
23 0.3381 0.3244 0.6703 0.2824 0.3149 0.7078 0.6203 0.8649 0.716
24 0.0359 0.0226 0.6558 0.0045 0.0232 0.0952 0.0784 0.1081 0.0406
25 0.4071 0.3605 1.0476 0.2581 0.233 0.4093 0.3794 0.5256 0.5645
26 0.4071 0.3605 1.0476 0.2581 0.233 0.4093 0.3794 0.5256 0.5645
27 0.8628 0.7348 4.0103 0.4707 0.3566 0.455 0.4455 0.6236 0.9032
28 0.1609 0.1487 0.0412 0.1185 0.1219 0.2511 0.2258 0.3126 0.2841
29 0.0098 0.0098 0.384 0.0187 0.032 0.0946 0.0803 0.1114 0.0668
30 0.0192 0.0195 0.1081 0.0188 0.0231 0.0555 0.049 0.0674 0.0515
31 0.97 0.8734 1.6086 0.6517 0.6224 1.1751 1.081 1.4913 1.4884
32 0.0474 0.0427 0.3063 0.0318 0.0304 0.0574 0.0528 0.0728 0.0727
33 0.0474 0.0427 0.3063 0.0318 0.0304 0.0574 0.0528 0.0728 0.0727
34 0.0768 0.0794 0.5571 0.0809 0.1022 0.2539 0.22 0.3059 0.2262
35 0.0768 0.0794 0.5571 0.0809 0.1022 0.2539 0.22 0.3059 0.2262
36 0.0877 0.085 0.2272 0.0754 0.0856 0.1952 0.1715 0.2384 0.1942
37 0.0877 0.085 0.2272 0.0754 0.0856 0.1952 0.1715 0.2384 0.1942
38 0.4824 0.4311 1.0015 0.3152 0.2936 0.5362 0.4981 0.6859 0.7087
93
The average Contributions of loads are reported in Table 5.20
Table 5.20 Average Load Cost Contributions Cdavg(k,i) Modified Zbus average method
LINE\LOAD LD-10 LD-13 LD-14 LD-15 LD-16 LD-18 LD-19 LD-20
19 0.8915 0.8173 4.5966 3.4733 1.1761 3.6761 1.5221 0.6991
20 0.9223 8.447 1.968 2.1267 0.7318 2.2665 1.3036 0.8982
21 2.1825 0.6179 1.7276 6.3941 2.1794 6.8333 5.3267 4.5525
22 1.4696 6.8548 0.4011 4.7817 1.6244 5.1091 4.3416 3.8785
23 1.103 0.9596 4.5137 5.2362 1.7684 5.5717 2.3387 1.1107
24 0.1912 0.4677 0.5181 3.0941 0.3825 1.0422 0.6021 0.3653
25 0.5386 0.4896 0.2363 3.0206 0.0824 7.4071 0.1505 0.1107
26 0.5386 0.4896 0.2363 3.0206 0.0824 7.4071 0.1505 0.1107
27 0.382 0.5796 1.2014 4.79 1.219 4.4863 1.7587 0.9571
28 0.3752 0.5418 0.4625 0.7673 0.2622 4.3988 0.452 0.302
29 0.1697 0.6217 0.5426 1.9385 0.6448 2.0626 1.6871 0.9521
30 0.0917 0.1659 0.1643 0.6733 0.1096 3.3797 0.1794 0.1133
31 1.6574 1.9695 1.4313 2.7443 0.7072 2.2521 1.2402 0.876
32 0.0813 0.197 0.2279 1.4651 0.1738 6.2385 0.2725 0.1642
33 0.0813 0.197 0.2279 1.4651 0.1738 6.2385 0.2725 0.1642
34 0.42 1.232 0.8308 3.2111 1.0726 3.4184 2.5568 1.7119
35 0.42 1.232 0.8308 3.2111 1.0726 3.4184 2.5568 1.7119
36 0.3084 0.7633 0.3763 1.6259 0.5459 1.7367 1.3301 1.137
37 0.3084 0.7633 0.3763 1.6259 0.5459 1.7367 1.3301 1.137
38 0.7388 0.796 0.5203 2.4115 0.2158 1.4684 0.3995 0.3023
94
The individual average generator/load costs for using all the lines in the network andtotal average cost reported in Table 5.7
Table 5.21 Average Cost For Individual Generators/Loads and total average cost Modified Zbus
average method
Bus
No:
CGAVG(I) CDAVG(I) TOTAL
COSTAVG
1 102.932 65.4225 168.355
2 103.933 58.7992 162.732
3 0 108.533 108.533
4 0 50.3883 50.3883
5 0 39.4342 39.4342
6 0 44.9461 44.9461
7 184.503 96.3466 280.849
8 0 106.113 106.113
9 0 88.5755 88.5755
10 0 45.8012 45.8012
11 0 0 0
12 0 0 0
13 100.928 92.9311 193.86
14 0.68239 65.4797 66.1621
15 88.2683 134.839 223.108
16 56.1084 37.6462 93.7545
17 0 0 0
18 194.107 155.471 349.578
19 0 70.8302 70.8302
20 0 49.6119 49.6119
21 201.267 0 201.267
22 217.313 0 217.313
23 210.789 0 210.789
24 0 0 0
95
5.4 comparison of Zbus based techniques:
The comparison of all the Zbus based techniques are reported in table 5.22
Table 5.22 comparison of all the Zbus based techniques
Zbus technique Zbus average technique Modified Zbus average techniqueBusNo
CG CD TOTALCOST
CGAVG(I) CDAVG(I) TOTALCOSTAVG
CGAVG(I) CDAVG(I) TOTALCOSTAVG
1 41.3522 25.9653 67.3176 43.7322 27.4597 71.1919 102.932 65.4225 168.3552 50.6569 28.5682 79.2251 53.5085 30.1763 83.6848 103.933 58.7992 162.7323 0 184.485 184.485 0 188.735 188.735 0 108.533 108.5334 0 84.0901 84.0901 0 88.7323 88.7323 0 50.3883 50.38835 0 68.6698 68.6698 0 72.2478 72.2478 0 39.4342 39.43426 0 59.9801 59.9801 0 77.2602 77.2602 0 44.9461 44.94617 117.549 61.2232 178.772 117.51 61.2032 178.713 184.503 96.3466 280.8498 0 175.917 175.917 0 180.024 180.024 0 106.113 106.1139 0 145.662 145.662 0 151.042 151.042 0 88.5755 88.575510 0 72.2811 72.2811 0 78.8505 78.8505 0 45.8012 45.801211 0 0 0 0 0 0 0 0 012 0 0 0 0 0 0 0 0 013 7.15621 6.64702 13.8032 7.44293 6.91334 14.3563 100.928 92.9311 193.8614 0 116.219 116.219 0 117.195 117.195 0.68239 65.4797 66.162115 35.7738 52.7455 88.5193 35.4253 52.2317 87.657 88.2683 134.839 223.10816 20.812 13.4271 34.2391 20.2819 13.0851 33.367 56.1084 37.6462 93.754517 0 0 0 0 0 0 0 0 018 39.1378 32.5822 71.7199 38.8784 32.3663 71.2447 194.107 155.471 349.57819 0 127.021 127.021 0 126.608 126.608 0 70.8302 70.830220 0 89.022 89.022 0 88.6508 88.6508 0 49.6119 49.611921 383.247 0 383.247 364.838 0 364.838 201.267 0 201.26722 352.952 0 352.952 320.599 0 320.599 217.313 0 217.31323 378.857 0 378.857 377.002 0 377.002 210.789 0 210.78924 0 0 0 0 0 0 0 0 0
The above table gives the information about the cost allocated to different
generators and loads for IEEE RTS 24 bus system for all the three Zbus based techniques.
From the above table it is concluded that the Zbus technique allocates more usage to
generators rather than demands and similarly allocates most of the cost to generators
compared to demands. The Zbus average technique avoids the allocating most of the cost
to generators than demands. Modified Zbus average technique allocates a fair allocation
96
between generators and loads when compared to both Zbus and Zbus average techniques.
other two methods not able to allocate the cost for the generators and loads which are
not directly connected to it. but this modified Zbus average method allocates the cost for
that buses which are not directly connected to it.
Better cost allocation has been observed with modified Zbus average approach
compared to Zbus and Zbus average methods.
97
5.5 Conclusions:
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.
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 Modified Zbus average method. The above methods are
compared and the new method is very effective in transmission cost allocation .
The proposed Zbusavg method method provides a methodology to apportion the
cost of the transmission network to generators and demands that use it.. This work
contributes to seek an appropriate solution to this allocation problem using an usage-
based procedure that relies on circuit theory. . In the proposed new approach, the factors
S1,S2,S3,S4 for forward flows and S11,S21,S31,S41 for reverse direction flows have
offered very useful information about the contribution of generator buses and load buses
towards the line flow of line jk. Rigorous calculations without any approximations are
possible and cost allocation results would be more reliable and accurate compared to
other methods.
For the above three techniques software is developed and implemented in
MATLAB and applied to 4-bus system and IEEE RTS-24 bus systems. 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.
98
APPENDIX
A.1 4-BUS SYSTEM:
The number of buses : 4The number of lines : 5No.of generators :2
Bus data:
Bus Type Pgen Qgen Pload Qload Vspecified
1 P-V 0 0 0 0 1.05
2 P-V 2.5 0 0 0 1.05
3 P-Q 0 0 2.5 0 1
4 P-Q 0 0 2.5 0 1
Line data :
Line Frombus
To bus R X Halflinechargingsuceptance(ycp)
Halflinechargingsuceptance(ycq)
Tap ratio
1 1 2 0.01275 0.097 0.23055 0.23055 12 1 3 0.01275 0.097 0.23055 0.23055 13 1 4 0.01275 0.097 0.23055 0.23055 14 2 4 0.01275 0.097 0.23055 0.23055 15 3 4 0.01275 0.097 0.23055 0.23055 1
Table A.1 Bus data of 4- bus system
Table A.2 Line data of 4- bus system
99
A.2 IEEE 24- BUS RELIABILITY TEST SYSTEM:
No. of buses: 24
No. of lines: 38
No. of generators: 11
Bus Data:
Bus Type Pgen Qgen Pload Qload Vspecified
1 P-V 1.72 0.282 1.08 0.22 1.035
2 P-V 1.72 0.14 0.97 0.2 1.035
3 P-Q 0 0 1.8 0.37 1
4 P-Q 0 0 0.74 0.15 1
5 P-Q 0 0 0.71 0.14 1
6 P-Q 0 0 1.36 0.28 1
7 P-V 2.4 0.516 1.25 0.25 1.025
8 P-Q 0 0 1.71 0.35 1
9 P-Q 0 0 1.75 0.36 1
10 P-Q 0 0 1.95 0.4 1
11 P-Q 0 0 0 0 1
12 P-Q 0 0 0 0 1
13 Slack 2.853 1.221 2.65 0 1.02
14 P-V 0 0.137 1.94 0.39 0.98
15 P-V 2.15 0.0005 3.17 0.64 1.014
16 P-V 1.55 0.2522 1 0.2 1.017
17 P-Q 0 0 0 0 1
18 P-V 4 1.374 3.33 0.68 1.05
19 P-Q 0 0 1.81 0.37 1
20 P-Q 0 0 1.28 0.26 1
21 P-V 4 1.082 0 0 1.05
22 P-V 3 0.297-0.2976 0 0 1.05
23 P-V 0 0 0 0 1.05
24 P-Q 0 0 0 0 1
Table A.3 Bus data of IEEE RTS 24- bus system
100
Line data:
Line Frombus
To bus R X Halflinechargingsuceptance(ycp)
Halflinechargingsuceptance(ycq)
Tapratio
1 1 2 0.003 0.014 0.2305 0.2305 12 1 3 0.055 0.211 0.0285 0.0285 13 1 5 0.022 0.085 0.0165 0.0165 14 2 4 0.033 0.127 0.017 0.017 15 2 6 0.05 0.192 0.026 0.026 16 3 9 0.031 0.119 0.016 0.016 17 3 24 0.002 0.084 0 0 1.0158 4 9 0.027 0.104 0.014 0.014 19 5 10 0.023 0.088 0.012 0.012 110 6 10 0.014 0.061 1.2495 1.2495 111 7 8 0.016 0.061 0.0085 0.0085 112 8 9 0.043 0.165 0.0225 0.0225 113 8 10 0.043 0.165 0.0225 0.0225 114 9 11 0.002 0.084 0 0 1.0315 9 12 0.002 0.084 0 0 1.0316 10 11 0.002 0.084 0 0 1.01517 10 12 0.002 0.084 0 0 1.01518 11 13 0.006 0.048 0.05 0.05 119 11 14 0.005 0.042 0.044 0.044 120 12 13 0.006 0.048 0.05 0.05 121 12 23 0.012 0.097 0.1015 0.1015 122 13 23 0.011 0.087 0.091 0.091 123 14 16 0.005 0.059 0.041 0.041 124 15 16 0.002 0.017 0.018 0.018 125 15 21 0.006 0.049 0.0515 0.0515 126 15 21 0.006 0.049 0.0515 0.0515 127 15 24 0.007 0.052 0.0545 0.0545 128 16 17 0.003 0.026 0.0275 0.0275 129 16 19 0.003 0.023 0.0245 0.0245 130 17 18 0.002 0.014 0.015 0.015 131 17 22 0.014 0.105 0.1105 0.1105 132 18 21 0.003 0.026 0.0275 0.0275 133 18 21 0.003 0.026 0.0275 0.0275 134 19 20 0.005 0.04 0.0415 0.0415 135 19 20 0.005 0.04 0.0415 0.0415 136 20 23 0.003 0.022 0.023 0.023 137 20 23 0.003 0.022 0.023 0.023 138 21 22 0.009 0.068 0.071 0.071 1
Table A.4 Line data of IEEE RTS 24- bus system
101
1
4
2 7
58
6
39 10
2411
12
131415
16 19 20
23
222117
18
Fig A.1 IEEE 24-bus Reliability Test System
102
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[8] A. J. Conejo, J.Contreras,D.A.Lima,A.P.Feltrin “Zbus Transmission network costallocation,” IEEE Transactions On Power Systems, Vol. 22, No. 1, February 2007
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