APPLICATION OF COMPUTER BASED SIMULATION IN GAS
NETWORK SYSTEM USING GRAPH THEORY ALGORITHMS AND
NUMERICAL METHODS
POH HONG HWEE
UNIVERSITI TEKNOLOGI MALAYSIA
To my beloved father, mother, brothers and sister.
ACKNOWLEDGEMENTS
I would to take this opportunity to express my appreciation to my supervisors,
Prof. Dr Zulkelfi Yaacob and Assoc. Prof. Dr. Rahmat Mohsin, who spent their
precious time to guide, encourages, inspires and even helps me to solve my problems
patiently during my research.
I would like to thank my family for their unselfish support, patience and
encouragement during the whole period of the research. I thank them all and
dedicate this thesis to them with all my love.
Last but not least, I would like to express my thanks to all other friends that
support and help me in this research. Thank you.
ABSTRACT
The existing gas network software is highly depending on the Breadth First
Search (BFS) and the Depth First Search (DFS) in performing gas network
computation of any topology. This is causing the other graph theory algorithms
remain least known to users. Apart from that, there are many numerical methods that
can be used in performing the gas pipeline network analysis. Thus, the hypothesis of
the method states that the Newton Gauss Elimination method is faster and more
accurate than the Newton Gauss Seidel method. The objectives of this research are
to determine the fastest graph theory algorithm , and to determine the fastest and the
most accurate numerical method, in the development of gas network computer
simulator. The developed computer simulator applies the Newton Nodal Method in
performing the network analysis. Object oriented approach was used in designing
the structure of the computer simulator. Two case studies and one extended case
study were performed. Case study 1 and 2 involve 5 nodes low pressure gas pipeline
network and 6 nodes medium pressure gas pipeline network respectively. The
extended case study 1 involves 7 nodes high pressure gas pipeline network. The
flow results from the computer simulator in the case study 1 and 2 were compared
with GaPIS version 2.3. While in the extended case study, two flow equations for
high pressure gas network system were compared for the accuracy. In conclusion,
the BFS is the fastest graph theory with 5.37 milliseconds for case study 1, 5.48
milliseconds for case study 2 and 5. 62 milliseconds for extended case study 1. The
Newton Gauss Elimination is the fastest numerical method with 34.86 milliseconds
for case study 1, 46.35 milliseconds for case study 2 and 50.26 milliseconds for
extended case study 1. The combination of BFS and Newton Gauss Elimination
gives the fastest speed, with 40.23 milliseconds in case study 1, 51.83 milliseconds in
case study 2 and 55.87 milliseconds in extended case study 1. The average
accurac ies of both numerical methods for case study 1, case study 2 and extended
case study 1 are 99.22%, 95.64% and 91.71% respectively.
ABSTRAK
Perisian komputer gas sekarang amat bergantung kepada Breadth First Search
(BFS) dan Depth First Search (DFS) untuk mewakili sistem rangkaian gas dalam
komputer. Ini menyebabkan teori graf yang lain kurang diketahui oleh pengguna. Di
samping itu, terdapat pelbagai kaedah berangka boleh digunakan dalam
menyelesaikan analisis rangkaian talian gas. Oleh itu, hipotesis tentang kaedah ini
menyatakan bahawa Newton Gauss Elimination adalah lebih cepat dan lebih jitu
daripada kaedah Newton Gauss Seidel. Objektif kajian ini adalah untuk menentukan
teori graf yang paling pantas, dan menentukan kaedah berangka yang paling pantas
dan paling jitu dalam membangunkan suatu perisian simulasi rangkaian gas. Perisian
simulasi ini mengaplikasikan kaedah Newton Nodal dalam menyelesaikan analisis
rangkaian. Pendekatan berorientasi objek telah digunakan dalam mereka struktur
utama perisian simulasi. Dua kajian kes dan satu kajian kes tambahan telah
dijalankan. Kajian kes 1 dan 2 masing-masing melibatkan 5 nod talian rangkaian gas
bertekanan rendah dan 6 nod talian rangkaian gas bertekanan sederhana. Kajian kes
tambahan 1 melibatkan 7 nod talian rangkaian gas bertekanan tinggi. Nilai aliran
daripada perisian simulasi bagi kajian kes 1 dan 2 dibandingkan dengan keputusan
daripada GaPIS versi 2.3. Manakala, untuk kajian kes tambahan, dua persamaan
aliran bagi rangkaian gas bertekanan tinggi telah dibandingkan untuk menilai
kejituannya. Pada kesimpulannya, BFS ialah teori graf yang terpantas, dengan 5.37
milisaat untuk kajian kes 1, 5.48 milisaat untuk kajian kes 2 dan 5.62 milisaat untuk
kajian kes tambahan 1. Newton Gauss Elimination ialah kaedah berangka yang
terpantas dengan 34.86 milisaat untuk kajian kes 1, 46.35 milisaat untuk kajian kes 2
and 50.26 milisaat untuk kajian kes tambahan 1. Penggabungan BFS dengan Netwon
Gauss Elimination memberikan kelajuan terpantas, dengan 40.23 milisaat bagi kajian
kes 1, 51.83 milisaat bagi kajian kes 2 and 55.87 milisaat bagi kajian kes tambahan 1.
Kejituan purata bagi kedua-dua kaedah berangka untuk kajian kes 1, kajian kes 2 dan
kajian kes tambahan 1 ialah 99.22%, 95.64 dan 91.71%.
TABLE OF CONTENTS
CHAPTER
TITLE
TITLE PAGE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
TABLE OF CONTENTS
1
2
PAGE
vii
x
LIST OF TABLES
xiv
LIST OF FIGURES
xvi
LIST OF ABBREVIATION
xix
INTRODUCTION
1.1 Introduction
1
1.2 Problem Statement
2
1.3 Objectives
3
1.4 Scope of Study
3
LITERATURE REVIEW
2.1 Introduction
5
2.2 Gas Network
5
2.3 Object Oriented
6
2.3.1 Object
8
2.3.2 Message
9
2.3.3 Class
9
2.3.4 Domain
9
2.3.5 Object Oriented Decomposition
9
2.3.6 Notation
10
2.3.7 Object Oriented Development
11
2.3.7.1 Encapsulation
11
2.3.7.2 Inheritance and Reuse
11
2.3.7.3 Polymorphism
11
2.4 Graph Theory
2.4.1
Depth First Search
11
12
2.4.2 Breadth First Search
13
2.4.3
13
Spanning Tree Algorithm
2.4.4 Dijkstra’s Minimum Path Algorithms
15
2.4.5
Strongly Connected Components
16
2.4.6
Application of Graph Theory Algorithms In
16
Gas Network System
2.4.7 Basic Analysis of Graph Theory Algorithms
19
2.5 Network Analysis
20
2.6 Common Flow Equations
21
2.6.1
Selection of Flow Equations
22
2.6.2
Kirchhoff’s First Law
25
2.6.3
Kirchhoff’s Second Law
27
2.6.4 Newton Nodal Method
28
2.6.5 Newton Loop Method
32
2.6.6 Newton Nodal Method Versus Newton Loop
36
Method
2.7 Numerical Solution Of Linear Algebraic Equation
2.7.1
Gauss Elimination Method
37
2.7.2
Gauss Seidel Method
39
2.8 Commercial Software
3
36
40
METHODOLOGY
3.1 Introduction
42
3.2 Structural Design Method
43
3.5 Object Oriented Approach for Computer simulator
44
Design
3.4 Procedure
45
3.5 Computer Program Selection
46
3.6 Formulae and Methods Selection
46
3.7 Computer Simulator Development
47
3.7.1 Graphical User Interface(GUI)
47
3.8 Results Testing
48
3.9 Modification
49
3.10 Results
49
3.11 Process Flow Diagram for Computer Simulator
49
3.11.1 Graph/Drawing Type Selection
51
3.11.2 Draw the Graph/Drawing
51
3.11.3 Algorithms Selection
51
3.11.4 Graph Results
51
3.11.5 Data Input For Network System and Network
52
System Calculation
3.11.6 Calculation Results
4
5
53
COMPUTER SIMULATOR FEATURES
4.1 Introduction
56
4.2 Properties Information
56
4.3 User Interface Features
58
4.4 Computer Simulator Tools
59
4.5 Dialog Boxes
62
4.6 Main Menu
65
4.7 Simple User Guide
67
4.8 Hardware and Software Requirements
70
RESULTS AND DISCUSSION
5.1 Introduction
71
5.2 Analysis on Case Studies
71
5.3 Speed Analysis For Gas Network System
78
5.3.1 Speed test without N etwork Analysis
79
5.3.2 Speed test With Network Analysis
84
5.3.3 Speed Test Analysis for Case Study 1
86
5.3.4 Speed Test Analysis for Case Study 2
92
5.3.5 Speed Test Analysis for Extended Case Study
97
1
5.3.6 Discussion on Speed test with Network
102
Analysis
5.4 Accuracy Analysis For Gas Network System
103
5.4.1
Accuracy Analysis For Case Study 1
103
5.4.2
Accuracy Analysis For Case Study 2
106
5.4.3
Accuracy Analysis For Extended Case Study
109
1
5.4.4
6
REFERENCE
Discussion on the Accuracy Analysis
112
CONCLUSION AND RECOMMENDATION
6.1 Conclusion
113
6.2 Recommendation
115
116
LIST OF TABLES
TABLES
2.1
TITLE
Main kinds of programming styles and kinds of abstractions
PAGE
6
used
2.2
Minimum Spanning Tree algorithm
18
2.3
Basic Analysis on Graph Theory Algorithms
20
2.4
Guidelines to Selection of a flow equation for Distribution
23
System Calculation
2.5
Flow Equations
24
2.6
Limitation and Assumption Made for the Six Flow Equations
25
2.7
The Summary of Commercial Software Studied
41
4.1
Hardware and Software Requirements for User and Developer
70
5.1
Case studies pr operties
72
5.2
Input Data for Study Case 1
73
5.3
Input Data for Study Case 2
75
5.4
Input Data for Study Case 3
77
5.5
Speed Analysis Results without Network Analysis in Three
80
Case Studies
5.6
One Iteration Time for the Case Study 1 with Gauss
84
Elimination and Gauss Seidel Method
5.6.1
Total Iteration Time for the Case Study 1 with Gauss
84
Elimination and Gauss Seidel Method
5.7
One Iteration Time for the Case Study 2 with Gauss
85
Elimination and Gauss Seidel Method
5.7.1
Total Iteration Time for the Case Study 2 with Gauss
85
Elimination and Gauss Seidel Method
5.8
One Iteration Time for the Case Study 3 with Gauss
Elimination and Gauss Seidel Method
85
5.8.1
Total Iteration Time for the Case Study 3 with Gauss
86
Elimination and Gauss Seide l Method
5.9
Accuracy Analysis in Case Study 1
104
5.10
Accuracy Analysis in Case Study 2
107
5.11
Accuracy Analysis in Case Study 3
110
LIST OF FIGURES
FIGURES
TITLE
PAGE
2.1
Example of a Graph
17
2.2
Example of a Weighted Graph
18
2.3
Graph of gas network system for Newton Nodal Method
30
2.4
Graph of gas network system for Newton Loop Method
34
3.1
Algorithms Decomposition For Computer simulator Design
43
3.2
Overview of the basic structure of the computer simulator
44
based on the object-oriented concept
3.3
Procedure of computer simulator development
45
3.4
Process Flow Diagram for Computer Simulator
50
3.5
Flow chart of Process Occurs inside Algorithms Selection
54
3.6
Flow chart of Processes Occur in Network System
55
Calculation
4.1
Main Screen of Computer Simulator with a Sample Graph
58
4.2
General Toolbars
59
4.3
Drawing and Algorithms Toolbars
60
4.4
Algorithms Buttons
61
4.5
Results Display Column
61
4.6
Information Dialog box
62
4.7
Input Data Dialog Box for Node
63
4.8
Input Data Dialog Box for Pipe
64
4.9
Results Dialog Box
65
4.10
Converter
67
5.1
Schematic diagram for case study 1
74
5.2
Schematic diagram for case study 2
76
5.3
Schematic diagram for Extended Case Study 1
78
5.4
Computation Time for Case Study 1 versus Graph Theory
81
Algorithms
5.5
Computation Time for Case Study 2 versus Graph Theory
82
Algorithms
5.6
Computation Time for Extended Case Study 1 versus Graph
83
Theory Algorithms
5.7
One Iteration Time for case study 1 versus the Newton Gauss
89
Elimination and Gauss Seidel Method
5.8
Total Iteration Time for case study 1 versus the Newton
90
Gauss Elimination and Gauss Seidel Method
5.9
Computation Time for case study 1 versus the Graph Theory
91
Algorithms and Numerical Methods
5.10
One Iteration Time for case study 2 versus the Newton Gauss
94
Elimination and Gauss Seidel Method
5.11
Total Iteration Time for case study 2 versus the Newton
95
Gauss Elimination and Gauss Seidel Method
5.12
Computation Time for case study 2 versus the Graph Theory
96
Algorithms and Numerical Methods
5.13
One Iteration Time for Extended Case Study 1 versus the
99
Newton Gauss Elimination and Gauss Seidel Method
5.14
Total Iteration Time for Extended Case Study 1 versus the
100
Newton Gauss Elimination and Gauss Seidel Method
5.15
Computation Time for Extended Case Study 1 versus the
101
Graph Theory Algorithms and Numerical Methods
5.16
Newton Nodal Method (flow comparison case study 1–
105
Newton Gauss Elimination)
5.17
Newton Nodal Method (flow comparison case study 1–
105
Newton Gauss Seidel Method)
5.18
Newton Nodal Method (flow comparison case study 2–
108
Newton Gauss Elimination)
5.19
Newton Nodal Method (flow comparison case study 2–
108
Newton Gauss Seidel Method)
5.20
Newton Nodal Method (flow comparison extended case study
1– Newton Gauss Elimination)
111
5.21
Newton Nodal Method (flow comparison extended case study
1– Newton Gauss Seidel Method)
111
LIST OF ABBREVIATION
BFS
-
Breadth-First Search
DFS
-
Depth-First Search
MST
-
Minimum Spanning Tree
path
-
Edge, branch, chord
SCC
-
Strongly Connected Component
vertex
-
Node, point
E
-
Efficiency factor
D
-
Internal pipe Diameter
i
-
Initial vertex
K
-
Constant flow equation
L
-
Length
m
Q
-
Flow rate
m3/hr
S
-
Specific gravity of gas
T
-
Temperature
K
p1
-
Pressure at sending node
bar/kPa
p2
-
Pressure at receiving node
bar/kPa
Pn
-
Pressure at standard Temperature kPa
Tn
-
Standard Temperature
ε a,i
-
Percentage approximation
mm
K
CHAPTER 1
INTRODUCTION
1.1
Introduction
Natural gas is one of the most important energy sources used in the world,
besides oil and coal. Normally natural gas is supplied to users through natural gas
pipeline system. Many pipeline systems are built throughout the world, from the gas
reservoir to the end users, in order to support the highly increasing demand for
natural gas. In America alone, from 1996 to 1998, at least 78 pipeline construction
projects were completed adding approximately 11.7 billion cubic feet per day of
capacity [1]. Meanwhile in Malaysia, as in January 2004, network of gas pipeline
covering a total of 1,193.9 kilometres (831.7 kilometres completed) is constantly
expanding to reach a larger population [2 ].
Natural gas pipeline systems become very complex as years pass by. The
complexity of the gas pipeline systems makes people think ways to solve it. Program
development is one of the solutions. With the help of computer program, people can
gain information regarding the construction site and the pipeline chosen; design the
gas pipeline systems and performing the network analysis before the construction of
the actual pipeline systems.
This research is to develop a program that can help the users to design the
pipeline system and solve the network analysis. Besides that, users can also know
the pressure of certain point along the pipeline systems and load or flow rate of the
pipe leg.
1.2
Problem Statement
Graph theory (graph theory algorithms) is a branch of mathematics concerned
about how networks can be encoded and their properties measured [3]. This means
in gas engineering term, graph theory is used to represent the gas network
computation of any topology before the network analysis can be performed.
Generally, graph theory algorithms are used in mathematic field.
The existing software (user friendly software) nowadays are highly
depending on one or two graph theory algorithms, especially the Breadth First Search
(BFS) and Depth First Search (DFS) in order to represent gas network computation
of any topology, before working on a network calculation. So, lacks of studies on
other graph theory algorithms, led to these graph theory algorithms remain unknown
to users. These studies like the way of graph theory algorithms affect on network
configuration, way of graph theory algorithms work on gas network system and so
on. T his research helps users to understand vividly about the graph theory
algorithms, the way of graph theory algorithms work in the gas network system and
compare all the graph theory algorithms in order to determine the fastest graph
theory algorithms.
There are many numerical methods that can be used in solving the network
analysis. The commonly used numerical methods are Newton Gauss Elimination
method and Newton Gauss Seidel method. Thus, hypothesis of Newton Gauss
Elimination method is faster than and more accurate than Newton Gauss Seidel
method exists. This research helps users to understand both numerical methods and
to determine the fastest and more accurate numerical method on steady state gas
network system.
1.3
Objectives
The main objective of the research is to determine the fastest graph theory
algorithms selected based on the computation time obtained in the development of
gas network computer simulator.
In addition, the other objective of this research is to determine the fastest and
the most accurate numerical methods in the development of gas network computer
simulator. The numerical methods will be tested for their speed of execution
(computation time) and checked their accuracy based on the flow results obtained
from the computer simulator.
1.4
Scope of Study
In this research, a computer simulator for gas pipeline network system will be
developed. This computer simulator can be used to simulate natural gas distribution
network systems.
In order to ensure the success of the research, a right path is needed to be
carried out step by step. The most important step is to choose the right computer
program, which will be used to develop the computer simulator , and the
programming language . MS Visual C++ is chosen due to its speed of execution,
simple to use for programmers and developers and it is suitable for the creation of
scientific, mathematical and statistical applications. C++ programming language is
chosen because it is simple to understand and to code, and it is the programming
language for MS Visual C++.
After choosing the right computer program and the programming language,
studies of the existing software in the market will be done before the design of the
structure of the gas network program. Object oriented approach will be used in the
designing of the structure of the program.
Next, studies on different graph theory algorithms and numerical methods
will be done due to different graph theory algorithms and numerical methods will be
implemented into the computer simulator. There are five graph theory algorithms
that will be studied, which are Breadth First Search (BFS), Depth First Search (DFS),
Minimum Spanning Tree (MST), Strongly Connected Components (SCC) and
Dijkstra’s Minimum Path. Meanwhile, two numerical methods are studied, which
are the Newton Gauss Elimination method and the Newton Gauss Seidel method.
The final step is to study the way of performing the network analysis. In
performing the network analysis, Newton Nodal method in steady state condition and
several of flow equations will be used. A network is in a steady state when the
values of the quantities characterizing the flow of gas in the system are independent
of time and the system is described by a set of nonlinear algebraic equations [4]. In
steady state analysis, the pressure of the nodes and the flow rate in the pipes must
satisfy the flow equations and the value of load node and source node must fulfil the
two Kirchhoff’s laws, which are the Kirchhoff’s first law and Kirchhoff’s second law
(see Chapter 2).
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
Literature revie w is a very important part in this research. It works as the
foundation of this research, which is to determine the most suitable way to develop
the computer simulator. In order to develop the simulator, studies on the concepts of
gas network, graph theory algorithms, object oriented and network analysis are
needed.
2.2
Gas Network
In general, gas network system is built with the purpose of transferring
natural gas in high capacity, which normal storage tanks cannot supply this capacity
of gas. So, a gas network system is able to supply the high capacity and high
pressure of gas directly to the users of the gas.
Gas distribution network can be classified in two different types namely tree
or loop [5, 6]. The concept of the tree and loop as mathematical entities was firstly
proposed by Kirchhoff in the connection with the definition of fundamental circuits
used in the analysis of electrical circuit [7]. The tree type pipeline system happens
where the pipeline system is not in the loop condition. While for the loop type
system, the pipeline system contains loops.
It is much easier to solve the calculations for the tree type system compare to
the loop type pipeline system. This is due to the looped nature, gas flow and
direction of each pipe cannot easily be calculated. By calculating the pressure drop
of the pipeline based on appropriate formulations can easily solve the calculations of
the tree type pipeline system. While for the loop type pipeline system, a trial and
error method is needed to solve the network calculation, with the help of specified
formulations and numerical methods.
2.3
Object Oriented
There are about five main kinds of programming styles. Table 2.1 shows the
kinds of programming styles, as well as kinds of abstractions used [8].
Table 2.1: Main kinds of programming styles and kinds of abstractions used
Programming Styles
Abstractions
Procedure oriented
Algorithms
Object oriented
Classes and Objects
Logic oriented
Goals, often expressed in a predicate calculus
Rule oriented
If-then rules
Constraint oriented
Invariant relationships
Procedure oriented is the oldest programming style while object oriented is
the latest programming style, often used by programmers nowadays.
Object-oriented programming is intended to promote greater flexibility and
maintainability in programming, and is widely popular in large -scale software
engineering [9]. In this research, the computer simulator developed can be
considered as a large -scale engineering project because it contains many equations,
graph theory algorithms, numerical methods and formulations that needed to be code.
Besides that, the computer simulator developed must be able to provide platform for
future enhancement so that the computer simulator can become commercial software.
The computer simulator developed must be user-friendly and contains Graphical
User Interface (GUI) too. Object oriented is suitable in developing the Graphical
User Interface (GUI).
Due to these reasons, object oriented approach is chosen for design the basic
structure of the simulator . B y using this approach, an architectural framework of the
simulator can be formed. In order to use this approach, firstly, one has to understand
the meaning of object oriented programming, object or iented analysis and object
oriented design.
The meanings of these three important phases of object oriented are as shown
as below [8, 10, and 11].
“Object oriented programming is a method of implementation in which
programs are organized as cooperative collections of objects, each of which
represents an instance of some classes, and whose classes are all members of a
hierarchy of classes united via inheritance relationships.”
“Object oriented analysis is a method of analysis that examines requirements
from the perspective of the classes and object found in the vocabulary of the problem
domain”
“Object oriented design is a method of design encompassing the process of
object oriented decomposition and a notation for depicting both logical and physical
as well as static and dynamic models of the system under design”
Secondly, one has to understand the meanings of object, message, class,
domain, object oriented decomposition and notation. The object-oriented approach
for the simulator design is show n in Chapter 3.
2.3.1
Object
An object has state, behaviour, and identity; the structure and behaviour of
similar object are defined in their common class; the terms instance and object are
interchangeable. The state of an object encompasses all of the properties of the
object and the current values of each of the properties. The behaviour is how an
object acts and reacts, in term of its state changes and message passing. The identity
is that property of an object which distinguishes it from all objects [8]. Most objects
can be categorized into five categories [12] :
i.
Tangible objects. Abstractions of the actual existence of some thing in the
physical world. For example, pipe, pump, valve and tank in a juice bottling
plant.
ii.
Roles. Abstractions of the purpose or assignment of a person, piece of
equipment, or organization. For example, piston, inlet valve and outlet valve
in a car engine.
iii.
Incidents. Abstractions of some happening or occurrence. For example,
accident in plant.
iv.
Interactions. Objects that result from associations between other objects. For
example, connections of two pipes.
v.
Specification objects. Objects that used to represent rules, standards, or
quality criteria (as opposed to the tangible object or role that meets these
standards). For example, a compound represents the composition of a
chemical, but not a particular sample of that compound.
An object can only act as an actor, server or agent. An actor is object that can
operate upon other objects and never operated upon by other objects. A server is
object that can only operated upon by other objects. An agent is object that can both
operate upon other objects and operated upon by other objects [8].
2.3.2
Message
Message passing only happens when an object relates with other objects [13,
14]. There are two types of messages, which are the unidirectional message and
bidirectional message. Unidirectional message is only a one -way message while the
bidirectional message is a two-way message. Examples of unidirectional message
and bidirectional message are relationships between a general with a soldier and
relationships between husband and wife respectively.
2.3.3
Class
A class is a set of objects that share a common structure and common
behaviour [8]. For example, a metal chair and a wooden chair can be grouped into
the same class, called Chair.
2.3.4
Domain
A domain is a separate real, hypothetical, or abstract world inhabited by a
distinct set of objects that behave according to rules and policies characteristic of the
domain [12]. For example, a gas distribution system domain consists of depressurize
station, pipeline system, various type of gas users as objects.
2.3.5
Object Oriented Decomposition
When designing a complex simulator system, it is essentia l to decompose it
into smaller parts. There are two ways of decompositions, that are the algorithms
decomposition and the object oriented decomposition [11] . The algorithms
decomposition is a method of building top-down structured design, wherein each
module in the system denotes a major step in some overall process. This method is
based upon algorithms view that highlights the ordering of event.
On the other hand, object oriented decomposition is a method which
decomposes the system according to the key abstraction in the problem domain. In
other word, object oriented decomposition is based upon objects. Both ways of
decompositions are useful. However, only one of the ways can be used when
designing a complex simulator system. In this research, s ince object oriented
approach is used, so, object oriented decomposition is chosen in designing the
structure of the gas network system. The reasons of choosing this way of
decomposition is it can reduce the risk of building complex simulator system and it is
able to evolve over time, because of its stable intermediate form.
2.3.6
Notation
Notation is a very important method in the process of simulator development.
The notation for object oriented development includes four basic diagrams and two
supplementary diagrams [8]. The four basic diagrams are class diagrams, object
diagrams and process diagrams while the two supplementary diagrams are state
transition diagrams and interaction diagrams. A class diagram is used to show the
existence of classes and their relationships in the logical design of a system. An
object diagram is used to show the existence of objects and their relationships in the
logical design of a system. A module diagram is used to show the allocation of
classes and objects to module s in the physical design of a system. A process diagram
is used to show the allocation of processes to processors in the physical design of a
system. A state transition diagram is used to show the state space of an instance of
give class, the events that cause a transition from one state to another, ant the actions
that result from a state change. An interaction diagram is used to trace the execution
of a scenario in the same context as an object diagram.
2.3.7
Object Oriented Development
Object orie nted development includes encapsulation, inheritance and
polymorphism.
2.3.7.1 Encapsulation
The property of being a self -contained unit is called encapsulation [15]. With
encapsulation, data hiding can be accomplished. Data hiding is the highly valued
characteristic that an object can be used without the user knowing or caring how it
works internally [15].
2.3.7.2 Inheritance and Reuse
Inheritance, also called specialization or derivation, which is supported by
C++ [16, 17] , is a way to declare a new type which is an extension of an existing
type [15]. The new subclass is said to derive from the existing type and is sometimes
called a derived type. It inherits all qualities from the existing types but adds to them
or modifies them as needed [15].
2.3.7.3 Polymorphism
Polymorphism is a function and a class which enables different objects to
respond differently to the same function call [18].
2.4
Graph Theory
A graph G = (X, A) is a collection of points or vertices (node) x1 , x2 … xn
(denoted by the set X) and a collection of lines (edge/branch) a 1 , a2 … an (denoted by
the set A) joining all or some of these points [19, 20]. Graph can be divided into
directed graph and undirected graph. If the lines in A have direction – which is
usually shown by an arrow – they are called arcs and the resulting graph is called a
directed graph [19]. Dendrite is a connected network graph Gn without loops [5].
Tree branch is the branches belong to a dendrite and those do not belong to a
dendrite are called chords.
2.4.1
Depth First Search (DFS)
The procedure of Depth First Search for both directed graph and undirected
graph is shown below [21].
1) Start the search at a vertex v and initialize the search path to v.
2) If there is an unexplored edge (branch) at the vertex at the head of the search
path leading to an unexplored vertex w, extend the search path to w, which
becomes its new head and repeat step (2).
3) If there is an unexplored edge at the head of the search path leading to an
explored vertex w, mark that edge as explored and repeat (2) for the next edge
incident at the head of the search path.
4) If there are no further unexplored edges at the head of the search path, retract
the search path to the previous vertex by removing the current head of the
path and repeat step (2).
5) Stop once the search path is empty.
The Depth First Search method visits all the vertices. It always advances
from its last explored vertex, making the path drive deeper into the graph before it
explores closer region of the graph.
2.4.2
Breadth First Search (BFS)
The Breadth First Search method is like the Depth First Search Method.
Breadth First Search is a traversal through a graph that touches all the vertices
reachable from a particular vertex. However, unlike DFS method, BFS explores all
the neighbours of the particular vertex before proceeds to the neighbours of its
neighbours. This method is liked a wave of emanating from the stone drop into the
pool of water [22].
The algorithm for BFS is given as:
At first no vertices of the graph are labelled.
1. Label vertex s with 0.
2. i ? 0
3. Find all unlabeled vertices adjacent to at least one vertex labelled i. If none
are found, stop.
4. Label all the vertices found in (3) with i+1.
5. If vertex t is labelled, stop.
Depth First Search is useful in identifying structural properties of graphs. On
the other hand, for problems, such as maximum flow, where the structural
characteristics of depth first search are not needed, breadth first search may lead to
better performance [21].
2.4.3
Spanning Tree Algorithms
For a connected, undirected graph, a spanning tree of that graph is a sub-
graph which is a tree and connects all the vertices together. A minimum spanning
tree is a spanning tree with weight less than or equal to the weight of every other
spanning tree.
Initially all edges are uncoloured and all buckets are empty [23].
Step 1: Select any edge that is not a loop. Colour this edge blue and place both its
endpoints into an empty bucket.
Step 2: Select any uncoloured edge that is not a loop. (If no such edge exists, stop
the algorithm; no spanning tree exits.) One of four different situations must occur:
i.
ii.
Both endpoints of this edge are in the same bucket.
One endpoint of this edge is in a bucket, the other endpoint is not in any
bucket.
iii.
Neither endpoint is in any bucket.
iv.
Each endpoint is in a different bucket.
If item (i) occurs, colour the edge orange (not in the tree) and return to Step 2. If (ii)
occurs, colour the edge blue (in the tree) and assign the un-bucketed endpoint to the
same bucket as the other endpoint. If (iii) occurs, colour the edge blue and assign
both endpoints to an empty bucket. If (iv) occurs, colour the edge blue and combine
the contents of both buckets into one bucket, leaving the other bucket empty. Go to
Step 3.
Step 3: If all the vertices of the graph are in one bucket, stop the algorithm since the
blue edges form a spanning tree. Otherwise, return to Step 2.
(Note that each time a step of the algorithm is performed an edge is indelibly
coloured. If there are only a finite number of edges in the graph, then the algorithm
must stop after a finite number of steps)
If the algorithm does not terminate with a spanning tree, then no spanning tree exists
for the graph for the following reason: The algorithms will terminate with two sets
(buckets) of vertices that have no edge joining a member of one set to a member of
the other set. Otherwise, such an edge would have been coloured blue by the
algorithm and the tow buckets would have been merged together. Hence the
algorithm does what it is supposed to do, construct a spanning tree.
The minimum spanning tree algorithm examines the edges in order of
ascending weight (smallest first, largest last). If two or more edges have the same
weight, order them arbitrarily. This algorithm is used in to determine the total
weight, which represents cost, length and others of the links used is a minimum; and
all the points are connected together.
2.4.4
Dijkstra’s Minimum Path Algorithms
Dijkstra’s algorithm solves the shortest path problem for a directed graph
with nonnegative edge weights. For example, if the vertices of the graph represent
gas wells and edge weights represent distances between pairs of wells connected by a
pipeline, this algorithm can be used to find the shortest route between two gas wells.
Step 1: Initially all arcs and vertices are uncoloured [23]. Assign a number d(x) to
each vertex x to denote the length of the shortest path from s to x that uses only
coloured vertices as intermediate vertices. Initially, set d(s) = 0 and d(x) = 8 for all x
? s. Let y denote the last vertex to be coloured.
Colour vertex s and let y = s.
Step 2: For each uncoloured vertex x, redefine d(x) as follow:
d ( x ) = min{ d ( x ), d ( y ) + a ( y , x )}
where a ( y, x) = length of the arc or edge from y to any neighbour x of w.
If d(x) = 8 for all uncoloured x, then stop because no path exists from s to any
uncoloured vertex. Otherwise, colour the uncoloured vertex x with the smallest
value of d(x). Also colour the arc directed into vertex x from a colo ured vertex that
determined the value of d(x) in the above minimization. Let y = x.
Step 3: If vertex t has been coloured stop because a shortest path from s to t has been
discovered. This path consists of the unique path of coloured arcs form s to t. If
vertex t has not been colo ured yet, repeat Step 2.
2.4.5
Strongly Connected Components
A graph is said to be strongly connected if for any two vertices xi and xj, there
is at least one path going to vertices xi and xj [21]. For a strongly connected graph,
vertex xj is reachable from vertex xi and vice versa for any xi and xj , the strong
component containing a given vertex xi is unique and xi will appear in the set of
vertices of one and only one strong component.
For example: Considering a gas piping network. The network is strongly
connected if the gas flows in one direction between every two positions. The network
is weakly connected if the gas flows in both directions between every two positions.
The network is disconnected if there is no possible way for the gas to flow from one
point to another.
Strongly connected components algorithms use DFS or BFS method in order
to check whether the graph tested is strongly connected.
2.4.6
Application of Graph Theory Algorithms In Gas Network System
As stated, graph theory is very useful in order to represent the gas network
computation of any topology. Thus, graph theory can check the connectivity of the
gas network system.
6
3
5
2
1Figure 2.1: Example of a Graph.
4
5
Based on Figure 2.1, using Breadth First Search method, to reach node 5
1
from node 1, firstly all the neighbouring nodes of node 1, which4are node 2 and node
3
3 will be checked. After that node 2 will go to node 4 while node 3 will go to node 4
2
and node 5. Lastly the method stops due to the desired node is reached. This means
this graph is a connected graph.
For the same figure, now using the Depth First Search method, to reach node
5 from node 1, firstly node 1 will go to node 2 through edge 1. This is followed
respectively by the nodes, which are from node 2 to node 4 through edge 3, from
node 4 to node 3 through edge 4 and from node 3 to node 5 through edge 6. After all
the nodes are touched, the method stops. The graph now is also a connected graph.
Now, saying the graph shown in Figure 2.1 is changed to a weighted graph
like Figure 2.2 as shown in the following page.
3
7
5
3
9
1
4
2
6
Note:
Weight
2
Figure 2.2: Example of a Weighted Graph.
The summary of using the Minimum Spanning Tree algorithm is indicated in
Table 2.2 as below.
Table 2.2: Minimum Spanning Tree Algorithm.
Edges
Weight
Bucket 1
Bucket 2
Colour?
(1, 2)
2
1, 2
Empty
v
(1, 3)
3
1, 2, 3
Empty
v
(2, 4)
6
1, 2, 3, 4
Empty
v
(3, 5)
7
1, 2, 3, 4, 5
Empty
v
(4, 5)
9
1, 2, 3, 4, 5
Empty
×
Since all the vertices are in the bucket, the algorithm stops at edge (3, 5).
Edge (4, 5) is uncoloured. Four edges (1, 2), (1, 3), (2, 4) and (3, 5) form a minimum
spanning tree of the graph.
Using strongly connected components algorithm, if DFS method is chosen,
the graph is strongly connected. The strongly connected components number is 1.
Using Dijkstra’s algorithm, the steps used in carrying out the analysis are
given as (from vertex 1 to vertex 5, so s=1; t=5):
i.
Initially only vertex 1 is coloured, d(1) = 0, and d(x) = 8 for all x ? 1
ii. Set y = 1
d(2) = min {d(2),d(1) + a(1,2)} = min {8,0+2} = 2
d(3) = min {d(3),d(1) + a(1,3)} = min {8,0+3} = 3
d(4) = min {d(4),d(1) + a(1,4)} = min {8,0+8} = 8
d(5) = min {d(5),d(1) + a(1,5)} = min {8,0+8} = 8
As a result, d(2) = 2
iii. Colour vertex 2 and edge (1,2)
iv. Proceed with y = 2, y = 3, y = 4, y = 5 until vertex 5 is coloured.
v. Final shortest path arborescence consists of edge (1,2), (1,3), (3,5) and (2,4).
vi. Shortest path consists of (1,3) and (3,5) with length 3+7 = 10
Understanding the effects of different graph theory algorithms on the gas network
system is very important. By using graph theory algorithms, any graph in the gas
network system drawn can be proved whether the graph drawn is connected or not.
If the graph drawn is not connected, the algorithms will tell the defects to users.
Different graph theory algorithms have different computation time. If the
graph drawn is very large, it means the computation time become longer. The longer
the computation time means users have to wait longer time. Besides that, the
shortest path for the starting vertex to the ending vertex can be determined. This
helps users to find out the total minimum length and cost.
2.4.7
Basic Analysis on Graph Theory Algorithms
After studying all the five graph theory algorithms, a basic analysis on all the
graph theory algorithms has been done. This basic analysis is done with the purpose
of helping the user to have a clear picture on all the five graph theory algorithms that
will be used in the development of gas network computer simulator. Table 2.3 is the
summary of analysis on graph theory algorithms based on the difference in the
methods used and their applications.
Table 2.3 : Basic Analysis of Graph Theory Algorithms.
Algorithms
Analysis
Method
BFS
Uses a queue
Application
Used to find the shortest path in unweighted graph
DFS
Uses a stack
Used to find cycles, to produce
topological ordering or to determine
which nodes are reachable from a
given node
MST
Dijkstra’s
Examining the edge in the
Used to obtain optimal solution and
order of ascending weight
a minimum total cost in a graph
Used shortest path algorithm Used to find the shortest path from
Shortest
the given node to the destination
Path
node in weighted graph
SCC
Uses DFS method
Used to find the strong connected
components in directed graph
2.5
Network Analysis
Natural gas network analysis is used to determine the amount of gas, flow
rate of the gas and the pressure used in a pipeline system. Besides that, it also can
used to determine the usage of gas by consumers especially during the peak times.
By making the network analysis, time of maintenance and failure of pipeline can be
measured. The effect of new load on the pipeline also can be predicted. Generally,
network analysis can be divided into three type, that are Network tracing, Network
routing and Network Allocation [24]. Network tracing is done in order to trace the
location of the particular pipeline in the system. Network routing is done in order to
find out the optimum path or the shortest path, which costs less for the system.
Network allocation is done in order to relocate all the pipelines into the main support.
2.6
Common Flow Equations
Common flow equation can be expressed in a general form [4]. For any pipe
k, the pipe flow equation from node i to node j can be expressed as
φ[(Q n ) k ] = K k (Q nm1 ) k
where φ[(Qn ) k ] k = the flow function for pipe k
Kk
= the pipe constant for pipe k
(Qn ) k = the flow in pipe k
m1
= the flow exponent = 2 for low pressure networks
= 1.848 for medium pressure networks
= 1.854 for high pressure networks.
For the low pressure version of the flow equation,
φ[(Qn ) k ] = K k (Qn2 ) k = pi − p j = ∆pk
where ∆ pk = the pressure drop for pipe k,
pi = the absolute pressure at node i, (i = the sending node of pipe k)
p j = the absolute pressure at node j. (j = the receiving node of pipe k)
For the medium and high pressure version of the flow equation
φ[(Qn ) k ] = K k (Qnm1 ) k = Pi − P j = ∆Pk
where Pi = pi2 , Pj = p 2j
The equations for low pressure and for medium and high pressures can be
rearranged:
1
φ[ ∆pk ] = (Qn ) k = ( ∆pk / K k ) 2
(2. 1)
1
φ[ ∆Pk ] = (Qn ) k = (∆Pk / K k ) m 1
(2.2)
Equations (2.1) and (2.2) can be rearranged to the form as below after taking
account of the fact that a change of the flow direction of the gas stream may take
place in the network.
(Qn ) k = S ij (
(Qn ) k = S ij (
S ij ( p i − p j )
Kk
S ij ( p i − p j )
Kk
1
)2
(2.3)
1
) m1
(2.4)
where S ij = 1 if Pi > Pj ( p i > p j ) , S ij = −1 if Pi < Pj ( p i < p j )
2.6.1
Selection of Flow Equations
Flow equations are required to calculate the pressure drop in the gas network
system. There are many flow equations that can be used in gas industry. Majority of
them are developed based on the results of gas flow experiments. Thus, they are
only capable to a limited range of flow and pipe surface conditions Table 2.4 shows
the guidelines to the selection of a flow equation for the distribution system
calculation [25, 26].
.
Six flow equations are mainly used in this research and applied into the
computer simulator. They are Lacey’s equation, Pole’s equation, Cox’s equation,
Polyflo equation, Panhandle ‘A’ equation and Weymouth equation [27, 28, 29, 30
and 31]. These equations are shown in Table 2.5. Lacey’s and Pole’s equation are
flow equations for low pressure network. Cox’s and Polyflo equation are flow
equations for medium pressure network. Panhandle ‘A’ and Weymouth equation are
flow equations for high pressure network. For Weymouth, length L and D is in
meter (m) and pressure P is in Pascal (Pa).
Table 2.4: Guidelines to Selection of a flow equation for Distribution System
Calculation
Type of
Predominant Equation
Piping
Type
Used
High pressure
Partially
Panhandle A
utility supply
turbulent
Range of Capacity
Relatively good, slightly
optimistic appr oximation for
mains
Smooth pipe Flow Law at
Reynolds numbers >30000
High pressure
Fully
utility supply
turbulent
Weymouth
Good approximation to Fully
Turbulent Flow Law for clears
mains
rough commercial pipe of 10
to 30 inch diameter
Medium and
Partially
High pressure
turbulent
Panhandle A
Relatively good, slightly
optimistic approximation for
distribution
Smooth pipe Flow Law at
Reynolds numbers >30000
Medium and
Partially
High pressure
turbulent
Weymouth
Very conservative for pipe of
less than 20 inch diameter
distribution
Medium and
Partially
High pressure
turbulent
Cox’s
Pressure range >5 psi, Velocity
<20m/s in all pipes
distribution
Low pressure
Partially
distribution
turbulent
Pole’s
Good approximation to
Smooth pipe Flow Law for
pipe of 4 inch diameter or
smaller
Table 2.5: Flow Equations
Flow
Equation
K value Calculation
Equations
Lacey’ s/Pole’s
Cox’s
Polyflo
Panhandle ‘A’
Weymouth
Q = 7.1 × 10 −3
K = 11 .7 × 10 3
( p1 − p 2 ) D 5
SL
K = 206 .22527 × 10 3
2
Q = 1 .69 × 10 −3
Q = 7.57 × 10
−4
Q = 7.57 × 10 −4
L
D5
( p1 − p 2 ) D 5
SL
2
Tn
Pn
( p1 − p 2 ) D 5
fSLT
K = 27.24
Tn
Pn
( p1 − p 2 ) D 5
fSLTZ
K = 18.43
2
2
2
2
T
( p1 − p 2 )
Q = 11854124 .6 n D 8 / 3
Pn
SLT
2
2
L
D5
L
E D 4.848
2
L
2
E D 4 .854
K = 2590000
L
D 16 / 3
Assumptions have been made for the six flow equations above. These
assumptions have been made in order to avoid confus ion to the users of this
computer simulator and to help users to understand the reasons of picking the six
flow equations above. Table 2.6 shows the limitation and assumptions made for the
six flow equations.
The fraction factor, f can be calculated using the equations given in the Table
2.6. There is no fraction factor for Cox’s and Weymouth, because the value is
already inserted into the flow equations. In this research, the specific of gas is
assumed to be 0.589 and the compressibility factor, Z is assumed 0.95. These two
values are subjected to change, depend on the type of natural gas (natural gas
properties) and the type of pipe used. The temperature and pressure is assumed as
standard temperature and standard pressure, normally used by international gas users
in solving network analysis. The efficiency factor normally varies between 0.8 and 1
for most gas pipes [32]. The actual flow in a pipe will be 80% of the flow predicted
[4]. So, the efficiency factor, E in this research is assumed to be 0.8.
Table 2.6: Limitation and Assumption Made for the Six Flow Equations
Flow
Pressure
Equations
Range
(bar
gauge)
Assumption Made
Fraction factor, f
Specific gravity of gas, S
Temperature, T and Normal Temperature, T n
Normal Pressure, P n
Compressibility factor, Z
Efficiency factor, E
Lacey’ s/Pole’s
0–
f = 0.0044 (1 +
0.075
12
0 .276 D
S = 0.589
Cox’s
0.75 -7
Polyflo
0.75 - 7
S = 0.589
1
 SQ 
= 11.98 × E

f
 D
0 .076
S = 0.589
T = 288K
Panhandle ‘A’
>7
1
 SQ 
= 14.94 × E 

f
 D 
0 .073
S = 0.589
Tn = T = 288K, Pn = 1.01325 bar
Z = 0.95, E = 0.8
Weymouth
>7
S = 0.589
Tn = T = 288K, Pn = 101.325 kPa
2.6.2
Kirchhoff’s First Law
Kirchhoff’s first law states that the algebraic sum of the flows at any node is
zero [4]. This means that any load at any node is equal to the sum of the branch flows
into and out of the node.
Q1 + Q2 + Q3 ... + Qn = Load
Load − Q = 0
Load = Q
(2.5)
where Q1 + Q2 + Q3 ... + Qn = Q
Q
= total flow in the branches, depend on the flow direction
Load = demand in the load nodes,
The nodal equation, equation (2.5) can be expressed in matrix form:
L = A1 Q
where: L
(2.6)
= vector of loads at the nodes,
Q
= vector of flows in the branches ,
A1
= reduced branch-nodal incidence matrix.
The pressure drop in the branches can be related to the nodal pressures.
The nodal pressure drop equation can be expressed in matrix form:
∆P = - AT P
where: ? P
P
T
A
(2.7)
= vector of pressure drops in the branch,
= vector of nodal pressures,
= transpose of branch-nodal incidence matrix.
From equation (2.2),
Q = φ ' ( ∆P)
where φ ’ (?P) = vector of pressure drop functions .
Substituting for ? P from equation (2.7), equation (2.8) becomes
(2.8)
Q = φ ' (- AT P)
(2.9)
Substituting for Q from equation (2.9), equation (2.6) becomes
L = A1 [φ ' ( − A T P )]
2.6.3
(2.10)
Kirchhoff’s Second Law
Kirchhoff’s second law states that the pressure drop around any closed loop is
zero [4]. This means that there is no pressure drop around the loop since the closed
loop starts and finishes at the same node.
∆P1 + ∆P2 + ∆P3 ... + ∆Pn = 0
∆PT = 0
(2.11)
where ∆P1 + ∆P2 + ∆P3 ... + ∆Pn = ∆PT
∆PT
= pressure drop in the branches, depend on the loop direction
The loop equation, equation (2.11) can be expressed in matrix form:
B∆P = 0
where: ∆P
B
(2.12)
= vector of pressure drops in the branches,
= branch loop incidence matrix
Rearrange equation (2.8) , give
∆P = φ (Q) .
where φ ’ (Q) = vector of flow functions.
(2.13)
Substituting for ? P from equation (2.13) , equation (2.12) becomes
B[φ (Q)] = 0
2.6.4
(2.14)
Newton Nodal Method
The set of nodal equations that describes a gas network is shown as below [4,
33] :
L = A1 [φ ' ( − A T P )]
(2. 10)
where: L = vector of loads at the load nodes, of dimension n1,
A1 = reduced branch-nodal incidence matrix,
AT = transpose of branch nodal incidence matrix,
P = vector of nodal pressure, of dimension n,
φ ' ( ∆P) = vector of pressure drop functions, of dimension m =
= φ ' ( − AT P)
This equation can be rearranged as below :
A1 [F ’ (-A TP)] – L = 0
(2. 15)
Equation (2. 15) is a mathematical representation of Kirchhoff’s first Law. In
the nodal method, an initial approximation is made to the nodal pressure. This
approximation is then successively corrected until the final solution reached. At the
iteration the left hand side of equation (2.15) is not equal to zero. The pressures are
only approximations of their true values and the flows calculated from these
pressures are not balanced at each node. The imbalance at a node is the nodal error,
which is a function of all the nodal pressures and is denoted as f (P 1) where P1 is the
vector of squared pressures for high and medium pressure ga s network at the load
nodes of dimension n1.
There is a nodal error for each node and this set of errors is represented by:
 f1 ( P1 , P2 ,...Pn 1 ) 
 f ( P , P ,...P ) 
n1 
F (P1 ) =  2 1 2
........................ 


 f n 1 ( P1 , P2 ,...Pn1 )
where F denotes a vector of functions.
The nodal errors are incorporated into equation (2.6):
F (P1 ) = A1 [φ ' (− AT P )] − L
(2. 16)
The errors F(P1 ) will tend to zero as the pressure approach their true values.
The Newton nodal method solves the set of equations (2.16) iteratively until the
nodal errors are less than a specified tolerance. The iterative scheme for correcting
the approximations to the nodal pressures is (see next page)
P1k +1 = P1k + (∂P1 ) k
(2. 17)
where k = number of iterations.
Term ∂P1 is computed from the following equation:
J k (∂P1 ) k = −[ F (P1 )]k
(2. 18)
The matrix J is the nodal Jacobi matrix and is given by
∂ f1 
 ∂f 1 ∂f 1
 ∂P ∂P L ∂P 
n1
 1 2

 ∂f 2 ∂f 2
∂f 2 
L
J =  ∂P1 ∂P2 ∂Pn1 
M



 ∂f n1 ∂f n1 L ∂f n1 
 ∂P1 ∂P2 ∂Pn1 
where f i stands for fi(P1).
(2. 19)
L2
L3
4
L4
5
3
2
4
2
1
3
1
Figure 2.3: Graph of gas network system for Newton Nodal Method.
Assuming the flow equation is as shown as equation (2.4) and based on the
chart (2.1) (see above). Hence the nodal equations are:
f 2 = S 12 [ S12
f 3 = S 13[ S 13
( P1 − P2 ) 1 / m1
(P − P2 ) 1 / m1
]
+ S 32 [S 32 3
]
− L2
K1
K4
( P1 − P3 ) 1 / m1
( P − P2 ) 1 / m1
( P − P4 ) 1 / m1
]
− S 32 [ S 32 3
]
− S 34 [S 34 3
]
− L3
K2
K4
K5
f 4 = S 14 [ S14
( P1 − P4 ) 1 / m1
(P − P4 ) 1 / m1
]
+ S 34 [S 34 3
]
− L4
K3
K5
(2. 20)
The nodal Jacobi matrix is
 ∂f 2
 ∂P
 2
 ∂f
J= 3
 ∂P2
 ∂f 4

 ∂P2
∂f 2 ∂f 2 
∂ P3 ∂P4 
∂f 3 ∂ f 3 

∂ P3 ∂P4 
∂f 4 ∂f 4 

∂ P3 ∂P4 
After differentiating equation (2. 20) with respect to P 2 and considering the
answer gained with equation (2.4), one new equation which is equation (2.21) is
formed as shown as the following.
 Q1
Q
Q

+ 4 ) − 4
0
(

∆P4
 ∆P1 ∆P4


1  Q4
Q2
Q4
Q5
Q5
J=−
(
+
+
) −
−

m1  ∆P4
∆P2 ∆P4 ∆P5
∆ P5


Q3
Q5 
Q4
−
(
+
)
0
∆P4
∆ P3 ∆P5 

(2. 21)
The corrections to the nodal pressures are computed from equation (2. 18).
Since the Jacobi matrix J is negative due to the (-1/m1) term in equation (2. 21), the
correction to the nodal pressure will be positive if the error at the node is positive,
and will be negative if the nodal error is negative.
2.6.5
Newton Loop Method
The set of loop equations that describes a gas network is shown as below [4].
B[φ (Q )] = 0
(2.14)
Where: B = branch loop incidence matrix,
0 = zero vector, of dimension k,
∆P = vector of pressure drop in the branches, of dimension m,
φ ' (Q) = vector of flow functions, of dimension m.
= ∆P
Equation (2.14) is a mathematical representation of Kirchhoff’s second Law
which states that the sum of the pressure drops around any loop is zero. The loop
method requires that a set of loops in the network be defined. An initial
approximation is made to the branch flows ensuring that a flow balance exists at each
node. Since the branch flows are approximations to their true values, a loop flow is
introduced. This loop flow is the flow correction to be added to the branch flow
approximations to yield the true values. In general, the branch flows are a function
of the initial approximations and of all the loops flows, given like equation below.
Q = Q 0 + BT q
(2. 22)
where q = vector of loop flows of dimension k, (k is the number of loops)
Q = the branch flows,
Q0 = initial branch flow approximations.
In the loop method, the iteration the left hand side of equation (2.14) will not
be zero. The branch flows are only approximations of their true values and the
pressure drops calculated from these flows will not summate to zero around each
loop. This introduces a loop error into each loop which is a function of all loop
flows and is denoted as f (q). There is a loop error for each loop and this set of errors
is represented by:
 f 1 ( q1 , q 2 ,...q k ) 


f 2 ( q1 , q 2 ,...q k ) 

F (q ) =
........................


 f k ( q1 , q 2 ,...q k ) 
where F denotes a vector of functions.
The loop errors are incorporated into equation (2.13):
F ( q ) = B[φ (Q)]
(2. 23)
After substituting from equation (2. 22) into equation (2. 23), equation (2. 24)
is formed.
F (q) = B[φ (Q 0 + B T q)]
(2. 24)
The errors F(q) will tend to zero as the loop flow s approach their true values.
The Newton Loop method solves the set of equations (2.16) iteratively until the loop
errors are less than a specified tolerance.
The iterative scheme for correcting the approximations to the loop flow is
q k +1 = q k + (∂q) k
(2. 25)
where k = number of iterations.
The correction to the loop flows are computed from the following equation:
J k (∂q) k = −[ F ( q)]k
The matrix J is the loop Jacobi matrix and is given by
(2. 26)
 ∂f 1
 ∂q
 1
 ∂f 2

J =  ∂q1
M

 ∂f k
 ∂q1
∂ f1
∂f 
L 1
∂q 2 ∂q k

∂f 2
∂f 2 
L

∂q 2 ∂q k 


∂f k
∂f k 
L
∂q 2 ∂q k 
(2. 27)
where f i stands for fi(q).
Q4
3
Q5
2
4
qA
qB
Q2
Q1
Q3
1
Figure 2.4: Graph of gas network system for Newton Loop Method.
Based on equation ∆Pk = K k (Q nm1 ) k and chart (2.2) (see above), the loop
equations (2.28) are:
f A = − S1 K1 [ S1 (Q10 − q A )] m1 + S 2 K 2 [S 2 (Q20 + q A + qB )] m1 + S 4 K 4 [S 4 (Q40 − q A )] m1
(2. 28)
f B = S 2 K 2 [S 2 (Q 20 + q A + q B )] m1 − S 3 K 3 [S 3 (Q30 − q B )]m1 + S 5 K 5 [S 5 (Q 50 + q B )] m1
(2.29)
where Sk = +1, if Qk is positive,
Sk = -1, if Qk is negative,
The Jacobi matrix is
 ∂f A
 ∂q
A
J =
 ∂f B
 ∂q
 A
∂f A 
∂q B 

∂f B 
∂q B 
The differential equation for equation (2.28) with respect to qA is gained and
as shown as the following.
δf A
= m1 K1 ( S1 Q1 ) m1 −1 + m1 K 2 (S 2 Q2 ) m1 −1 + m1 K 4 (S 4 Q4 ) m1 −1
δq A
The differential equation for equation (2.29) with respect to qB is gained and
as shown as the following.
δf B
= m1 K 2 (S 2 Q2 ) m1 −1 + m1 K 3 ( S 3 Q3 ) m1 −1 + m1 K 5 ( S 5 Q5 ) m1 −1
δq B
The differential equation for equation (2.28) with respect to q B and the
differential equation for equation (2.29) with respect to q B are gained and as shown in
the following page.
δf A δf B
=
= m1 K 2 (S 2 Q2 ) m1 −1
δq A δ q B
Since S k and Qk have the same sign for any pipe k, the term m1 K k ( S k Qk ) m1 −1
is always positive and can be written as m1 K k Qk
m1 −1
. Hence
{K 1 Q1 m1 −1 + K 2 Q2
J = m1 
 K 2 Q2 m1 −1

m1 −1
+ K 4 Q4
m1 −1
} K 2 Q2
{K 2 Q2
m1 −1
m1 −1
+ K 3 Q3
m1 −1


m −1
+ K 5 Q5 1 }
(2. 30)
The corrections to the loop pressures are computed from equation (2.26).
Since the Jacobi matrix J is positive, the correction to a loop flow will be negative if
the loop error is positive and will be positive if the loop error is negative.
2.6.6
Newton Nodal Method Versus Newton Loop Method
Newton Nodal method and Newton Loop method have advantage and
disadvantage. The advantage of the nodal method is this method is suitable in
solving set of equations. Meanwhile the advantage of the loop method over the
nodal method is its good convergence characteristic. The disadvantage of the nodal
method is the poor convergence characteristic while the disadvantage of loop method
is the difficulty to define the loop in gas network.
Newtown Nodal Method will be implemented into the gas network simulator
as the basic method to solve the network analysis.
2.7 Numerical Solution Of Linear Algebraic Equation
Numerical methods for solving systems of linear equations fall into two
general classes; they are the direct methods and the iterative methods. Direct
methods lead to an exact solution in a finite number of steps if a round off error is
not involved. Iterative method leads to an approximation that is acceptably close to
the exact solution by performing an infinite number of arithmetic operations.
2.7.1
Gauss Elimination Method
Most of the direct methods developed are based on Gaussian elimination,
which yields a system of equations with a triangular coefficient matrix allowing their
easy solution.
Gauss elimination method is method that used to solve a pair of simultaneous
equations, which consists of two steps. Assume a linear system of 3x3 equations as
shown as the matrix form below.
a11 a12
a
a
 21 22
a31 a32
a13   x1  b1 
a23  ×  x2  = b2 
    
a33   x3  b3 
(2. 31)
The first step is to eliminate one of the unknowns from the equations. This
step is called Forward elimination. The result of this elimination step is the forming
of one equation with one unknown. While the second step is solving the equation
directly and the result back substituted into one of the original equations to solve the
remaining unknown. The advantage of this method is it having a higher precision
and disadvantage is possible division by zero [34].
Assuming equations (2. 31) has initial values as below.
 3 − 0 .1 − 0 .2  x1 
0.1
7
− 0.3 ×  x2  =

0 .3 − 0.2 10   x3 
 7 .85 
− 19 .3


 71 .4 
Separate the above equations into three equations.
3x1 − 0.1x2 − 0.2x3 = 7.85
(2. 32)
0.1x1 + 7x2 − 0.3x3 = −19.3
(2.33)
0 .3 x1 − 0 .2 x2 + 10 x3 = 71 .4
(2. 34)
Use the Gauss elimination method to solve the equations (2.32) to (2.34) .
1)
Forward elimination
The procedure is multiply equation (2. 32) by 0.1/3 and subtracts the result
from equation (2. 33) to get
7 .00333 x 2 − 0 .293333 x3 = − 19.5617
Then multiply equation (2.32) by 0.3/3 and subtracts the result form equation
(2. 34) to eliminate x1. After these operations, the set of equations is
3 x1 − 0.1x 2 − 0.2 x 3 = 7.85
(2. 35)
7 .00333 x 2 − 0 .293333 x3 = − 19.5617
(2. 36)
− 0 .190000 x2 + 10 .0200 x 3 = 70 .6150
(2.37)
Remove x 2 from equation (2.37). To accomplish this, multiply equation
(2. 36) by -0.190000/7.0033 and subtract the result from equation (2. 37). An upper
triangular is formed after x2 from the equation (2.37).
3x1 − 0.1x2 − 0.2x3 = 7.85
2)
(2.38)
7.00333 x2 − 0.293333 x3 = − 19.5617
(2.39)
10 .0200 x3 = 70 .0843
(2. 40)
Back substitution
Equation (2. 40) can be solved by solving the x3.
x3 =
70 .0843
= 7 .00003
10 .0200
(2. 41)
Use equation (2.41) to solve equation (2. 39) to get x2.
x2 =
− 19.5617 + 0.293333 (7.00003 )
= −2.50000
7.00333
(2. 42)
Finally, solve x1 by subtract equation (2. 41) and equation (2. 42) into
equation (2.38).
x1 =
2.7.2
7.85 + 0.1(−2.50000 ) + 0.2(7.00003 )
= 3.00000
3
Gauss Seidel Method
This method is the most generally used iterative methods. Assume the
equation (2.31) is given. If the diagonal elements are all nonzero, the first equation
can be solved for x1 , the second for x2 and the third for x3 to yield
x1 =
x2 =
x3 =
b1 − a12 x2 − a13 x3
a11
b2 − a 21x1 − a 23x3
a 22
b3 − a31x1 − a32 x2
a33
(2. 43)
(2. 44)
(2. 45)
Now assume that x2 and x3 are zero. Subtract these zeros into equation (2.35)
to calculate new value of x1=b 1/a 11. Then subtract this new value of x1 along with the
previous guess of zero for x3 into equation (2. 44) to calculate new value of x2. This
process is repeated for equation (2. 45) to calculate new estimate for x3. After these
operations, return to equation (2. 43) and repeat the entire procedure until the solution
converges closely enough to the true values. Convergence can be checked using the
criterion ε a, i below [6, 35],
ε a ,i =
ε a, i =
Current approximation − previous approximation
× 100 %
Current approximation
xi j − xij −1
xi j
× 100 %
for all i, where j and j-1 are the present and previous iterations.
2.8
Commercial Software Studied
There is a variety of commercial gas software in the market. Some of them
can be used to perform the network analysis. While some of them can be used for
pipe sizing. In this research, commercial software has been studied in the purpose to
find the problems facing in the software. The problems are mentioned in Chapter 1.
The summary of the commercial software studied is shown in table in next page [5,
36, 37, and 38].
Commercial software, GaPIS is selected as the comparing tool for the
accuracy in this research. The latest version of the software is v2.3. The reason of
choosing this software is it suits the requirement of the computer simulator
developed. This software contains flow equations for low and medium pressure
distribution network system. It is widely used by local researchers and users, in
solving the network analysis.
Table 2.7: The Summary of Commercial Software Studied.
Commercial
Type
Usage
Software
Contains
Graph Theory
Numerical
Algorithms
Methods
GaPIS
User
Gas network
Friendly
Yes
BFS, DFS
Yes
GASNETT
User
Gas network
Friendly
Yes
BFS, DFS
Yes
GASWorkS
User
Gas network
Friendly
Yes
Yes
PIPEFLO
User
Fluid and Gas
Friendly network
Yes
Yes
STANET
User
Fluid and Gas
Friendly network
Yes
BFS, DFS
Yes
CHAPTER 3
METHODOLOGY
3.1
Introduction
To set up the basic structure of the computer simulator, one approach called
object oriented approach is used. After setting up the structure, the concept of graph
theory is used to check the connectivity of the pipeline system drawn in the computer
simulator. Network analysis is done after the pipeline system is drawn. Only one
method is studied in order to solve the gas network problems - the Newton Nodal
Method. This method for gas pipeline system is in steady state.
As stated, there are two types of decomposition or methods used in the
computer simulator design. The first one is called algorithms decomposition or
structural design, and another one is called object oriented design. In this research,
object oriented approach or decompos ition is chosen. The reasons are stated in
Chapter 2.
3.2
Structural Design Method
If structural design method or algorithm decomposition is chosen, the
computer simulator design of this research will be shown as Figure 3.1.
Gas Network Software
Draw Graph
User Input
…
Gauss Elimination
BFS
DFS
SCC
MST
Dijkstra’s
Check Input
Matrix factorization
Output
Inverse Matrix
.
.
.
Jacobi Matrix
Newton Nodal
Figure 3.1: Algorithms Decomposition for Computer simulator Design.
Based on Figure 3.1, the design for the computer simulator developed is too
complicated and looks confusing. The structural design for this research is very
large. So, the disadvanta ges of using the algorithms decomposition method for this
research are the computer simulator design is too large and much unorganized. More
time and efforts needed in order to develop the computer simulator, comparing if
using the object oriented approac h.
3.3
Object Oriented Approach for Computer simulator Design
Gas Network Simulator
Drawing
Graph Theory
Algorithms
Network Analysis
Help
Calculation
(Formulation)
User Manual
Itera tion (Matrix)
About
Simulation
Figure 3.2: Overview of the basic structure of the computer simulator based on the
object-oriented concept.
Unlike the algorithms decomposition method, using object oriented approach
or decomposition, the computer simulator is divided into two main parts (object),
which are the Drawing, and Network Analysis, and one additional part (object),
which is Help. The Graph Theory Algorithms (Attribute) is inside the Drawing.
While the Calculation, Iteration and Simulation (Attributes) are inside the Network
Analysis. (In this research, Newton Nodal Method is used as the basic method of
calculation, Jacobi Matrix is used as the basic matrix and Gauss Elimination Method
is used to solve the simulation.) The User Manual and About (Attributes) are inside
the Help. Inside the Graph Theory Algorithms, all the algorithms needed in the
computer simulator, like the BFS and DFS methods are called Functions or Events.
There exist c onnections between the objects.
3.4
Procedure
COMPUTER
PROGRAM
SELECTION
FORMULAE
AND METHOD
SELECTION
COMPUTER SIMULATOR
DEVELOPMENT USING
OBJECT ORIENTED
APPROACH
(Graph Theory Algorithms,
Network Analysis)
RESULTS
TESTING
SUCCESS
NO
MODIFICATION
RESULTS
YES
Figure 3.3: Procedure of computer simulator development.
3.5
Computer Program Selection
There are many computer compilers in the market, such as Borland C++
Builder, Delphi and so on, which can be used in writing the C++ programming
language. In this research, the computer compiler used in developing the computer
simulator is Microsoft Visual C++ 6.0.
3.6
Formulae and Methods Selection
In order to make the computer simulator become user-friendly to the end
users, literature review is the most important procedure in this research. This
procedure can be divided into three segments. The first segment is the study of
object oriented concepts and graph theory algorithms. By doing this study, the
structure of the computer simulator can be designed using object oriented approach,
which has been discussed in the early of this chapter. The concepts of graph theory
algorithms are studied and applied on the computer simulator, in the purpose to
check the connectivity of the graph or pipelines drawn by the users.
The second segment is the selection of the formulation for the gas network
analysis. So, in this research, only one formulation is selected to do the network
analysis - Newton Nodal Method. In future, the Newton Loop Method will be
applied into the computer simulator. Thus, users can choose to use, either Newton
Nodal Method or Newton Loop Method, for the network analysis calculation.
The last segment is the selection of the numerical methods like Newton
Gauss Elimination and Newton Gauss Seidel methods in solving the network
analysis problems, especially the matrix equations. The root of equation will be
determined by convergence numerical methods.
3.7
Computer Simulator Development
After the formulas and methods selection, the next procedure is to implement
the studies, which have been done by the developer, into the computer simulator.
This procedure is called computer simulator development. Generally this computer
simulator development can be divided into two parts; so called the graphic part and
the logic part. Both parts have their own functions and they can interact with each
others. The graphic part is normally called graphical user interface. Its main
purpose is to reduce users’ complication in using the computer simulator. While the
logic part is part which dealing with the engine of the computer simulator.
3.7.1
Graphical User Interface (GUI)
It is built in order to simplify all the works needed in this simulation. In the
computer simulator developed, there are eight things should be observed.
1.
Dialog Boxes
More than two dialog boxes are created for data inserting.
2.
Drawing Sheet
The graph or pipeline system is drawn in this sheet. Graph theory algorithms can be
directly tested on the drawing sheet and coloured node or edge will be shown.
3.
Menu
Particular task can be executed by clicking specified menu.
4.
Push Buttons
There are about five push buttons for doing different task. When a button is clicked,
it will carry out the logic coded by the developer or programmer.
5.
Radio Buttons
Like the push button, there are about ten radio buttons for doing different task. Most
of these buttons are involved in the graph classification, construction and graph
algorithms. When a button is clicked, it will carry out the logic coded by the
developer or programmer. The difference between the push button and the radio
button is normally there is one of the radio button is pre-selected.
6.
Tool Boxes
The tool boxes work as a tool in helping the user besides choosing the radio buttons.
7.
Result Boxes
The result boxes show the results of the graph theory algorithms and time of
execution.
8.
List Boxes
Like the result boxes, the list boxes are used to display the results of network
analysis calculation in list. They are put inside a specific result box.
3.8
Results Testing
The purpose of doing this testing is to check for any error that may occur in
the computer simulator. A test plan with a variety of test cases will be done and the
test plan will be shown in the coming chapter. If there is no error in the computer
simulator, the results obtained from the simulation of the computer simulator may be
compared with the data s imulated from any commercial computer simulator
available in the market. Thus, the computer simulator can be published in the
market.
3.9
Modification
Modification or trouble shooting is undergoing if there are mistakes in the
computer simulator. The mistakes can be syntax errors, internal errors and others.
All the mistakes in the computer simulator will be identify and corrected.
3.10
Results
This is the final result for the computer simulator. When arriving in this
stage, the computer simulator can be considered become perfect.
3.11
Process Flow Diagram for Computer Simulator
The process flow diagram for the computer simulator is shown in Figure 3.4.
This diagram is different than the diagram of developing the computer simulator.
The diagram shows and explains all the processes involved in the computer
simulator. Generally, the diagram can be separated into two segments. The first
segment is the drawing of graph or network system without network analysis or
simulation. The second segment is the drawing of graph with simulation.
The first segment involves the procedure to run the tests on graph theory
algorithms. It consists of drawing the graph or drawing, selecting of graph theory
algorithms, input the data that are the start point (node) or end point only for BFS
and Dijkstra’s Shortest Path algorithms, run the test on algorithms and obtain results.
The first segment is only suitable for undirected graph and directed graph without
performing the network analysis.
The second segment mainly involves the procedure to run the network
analysis on drawing or network system. It consists of drawing the graph or drawing,
selecting of graph theory algorithms, input the data that are the start point (node) or
end point only for BFS and Dijkstra’s Shortest Path algorithms, input data for the
network system (for network system calculation), run the test on algorithms, obtain
graph results, network system calculation and obtain the calculation results. The
second segment is only suitable for directed graph which needs to perform the
network analysis.
START
Undirected/
Directed
GRAPH/DRAWING
TYPE
SELECTION
Directed &
Simulation
DRAW THE
GRAPH/DRAWING
DRAW THE
GRAPH/DRAWING
ALGORITHMS
SELECTION
ALGORITHMS
SELECTION
RUN ALGORITHMS
DATA INPUT FOR
NETWORK SYSTEM
& RUN ALGORITHMS
GRAPH
RESULTS
END
GRAPH
RESULTS
NETWORK SYSTEM
CALCULATION
CALCULATION
RESULTS
END
Figure 3.4: Process Flow Diagram for Computer Simulator.
3.11.1 Graph/Drawing Type Selection
User of this computer simulator needs to select the type of graph or drawing
(network system) prefer. The type of graph can be whether directed, undirected or
directed with simulation. If user choose graph without simulation, the computer
simulator will only run the graph theory algorithms.
3.11.2 Draw the Graph/Drawing
After choosing the type of graph, user can draw the graph on the drawing area
in the computer simulator.
3.11.3 Algorithms Selection
After the graph is done, the next stage is to select the graph theory
algorithms. There are five graph theory algorithms to choose. Details about the
graph theory algorithms are shown in Chapter 2. Figure 3.5 shows the process
occurs inside the Algorithms Selection. User can only select one of the graph theory
algorithms each time to obtain the graph results. If user selects BFS and Dijkstra’s
Minimum path (Yes path in Figure 3.5), user needs to select the start and end point
(start node and end node) at the graph drawn, before the results of the algorithm can
display. An error message will display if user doesn’t do so. If user chooses
algorithms rather than the both algorithms stated above, whether is DFS, MST and
SCC, the results of the graph theory algorithms will directly display on the graph
results in the computer simulator
3.11.4 Graph Results
The graph results consist of all the results of the desired graph theory
algorithms and the computation time taken in calculating the algorithms. For graph
without simulation (uncheck or unclick the “Simulation” button, see Chapter 4,
section 4.4), the computer simulator will stop after all the calculation and all the
results have been displayed on the computer simulator. While for graph with
simulation, it will display all the results on the computer simulator and later proceeds
to the next stage, that is the network system calculation, or in other word, network
analysis.
3.11.5 Data Input For Network System and Network System Calculation
This stage is to perform the network analysis on the graph drawn. The results
will display after the calculation. However before that, all the data needed for the
calculation must be entered into the computer simulator.
User needs to enter the total node, the total branch, the total chord, tolerance,
properties, and methods and flow equation into the computer simulator. Otherwise,
errors will occur in the computer simulator and user needs to restart the computer
simulator.
If user fulfils all the required above, the process will go into the calculation of
K value process and the Initial Chord Flow Calculation process. To calc ulate the K
value, the formulas for the low, medium and high pressure gas network are as follow
(see Chapter 2 for details),
Pole formula (low pressure), K =
11 .7 × 10 3 L
D5
Cox’s formula (medium pressure), K =
206 .2253 × 10 3 L
D5
Weymouth formula (high pressure), K =
25900 × L
D16 / 3
To calculate the chord flow rate, equations (2.2) and (2.4) are needed.
After that, the data calculated will go to Formation of Jacobi Matrix process.
The Jacobi matrix formed depends on the size of the nodes. The matrix can be
formed using equation (2.12). The next process is Numerical Methods Calculation
process. In this process, there are two numerical methods in the computer simulator
and only one of them can be selected. Details about the two numerical methods are
as shown in Chapter 2.
Next, the calculation will proceed to New Branch Flow Calculation process
and New Chord Flow Calculation process. New flow rate for the branch and chord
will be calculated using equations (2.2) and (2.4). They are to calculate the new
branch flow rate and chord flow rate. The calculation will stop if the specified
tolerance (user input) is achieved. Otherwise, it will go back to the K value
Calculation process to recalculate the new K value using the formulas above and
continue all the processes above until the specified tolerance is achieved. The final
results of network system calculation will display after the specified tolerance is
achieved. Figure 3.6 show all the processes mentioned here.
3.11.6 Calculation Results
The results of network system calculation are display here. They include
time of one iteration and total iteration, number of iteration, branch flow rate, and
chord flow rate and so on.
Choose
Yes
Start/End
Point
Selection
BFS
No
No
Yes
Yes
DFS
No
Yes
MST
No
Yes
SCC
No
Start/End
Point
Selection
Dijkstra
Yes
No
No
Calculation
Graph Results
Figure 3.5: Flow chart of Process Occurs inside Algorithms Selection.
Start
Data Input & Check
Yes
Error
No
K value Calculation
Initial Chord flow
Calculation
Formation of Jacobi
Matrix
Numerical Methods
Calculation
New Branch flow
Calculation
New Chord flow
Calculation
No
Achieve
Tolerance?
Yes
Calculation Results
Figure 3.6: Flow chart of Processes Occur in Network System Calculation.
CHAPTER 4
COMPUTER SIMULATOR FEATURES
4.1
Introduction
The computer simulator developed can be used for network simulation for
steady state gas flow. The computer simulator uses the Newton Nodal Method in
order to do the network analysis. The computer simulator can be used to solve
piping problem and to check the connectivity of a graph drawn. To solve the piping
problem, firstly user is required to draw the piping system on the drawing sheet,
subsequently check the “Simulate” button and click at the system to enter the
necessary data. User can view the results obtained and the data entered in a special
results box. To check the connectivity of the piping system or the graph drawn, user
is required to choose the necessary information given on the main frame and click on
the “Run” button to simulate the graph theory algorithms. The results will be shown
on the same main frame with the time execution.
4.2
Properties Information
In order for the system to work properly, the following are the essential
information required:
(i)
General Information
a) Fluid Type (Gas)
b) Friction Factor Equation
(ii)
Node Information
a) Demands
b) Elevations
c) Pressures
(iii)
Gas Properties
a) Compressible factor
b) Specific gravity
c) Temperature
(iv)
Pipe Information
a) Diameter
b) Length
(v)
Flow Equations
a) Lacey
b) Pole
c) Polyflo
d) Cox’s
e) Pandhandle
f) Weymouth
4.3
User Interface Features
The developed application is a user friendly application. It consists of a
Drawing Area at the center, one menu and three toolbars on the top, one toolbar on
the left, four buttons and a results display column on the right. The main screen of
computer simulator with a sample graph is shown in Figure 4.1.
Figure 4.1: Main Screen of Computer Simulator with a Sample Graph.
4.4
Computer Simulator Tools
These tools are very useful to the user because it can help user easily to draw
the pipeline system.
(i)
Graph Toolbar
The Graph toolbar consists of tool buttons “Node” (Vertices) and “Edge”
(Branch).
(a)
Node (Vertices)
By clicking this tool button, user can draw the node of the pipeline system.
The node drawn in the drawing sheet is a small black dot.
(b)
Edge (Branch)
This tool button will not work if there is no node on the drawing sheet. At
least a node is needed (self-looping). User can draw edges which can connect many
nodes.
Figure 4.2: General Toolbars.
(ii)
Drawing and Algorithms Toolbars
These toolbars consists of nine radio buttons and one check button. The radio
buttons are “Directed”, “Undirected”, “Vertices”, “Edges”, “Breadth-First Shortest
Path Search”, “Depth-First Search”, “Strongly Connected Components”, “Minimum
Spanning Tree” and “Dijkstra’s Minimum Path” buttons (see Chapter 2 for details).
Meanwhile, the check button is “Simulation” button. Each of them has their own
functions. Figure 4.3 shows the display of the Drawing and Algorithms Toolbars.
(iii)
Algorithms Buttons
Only four buttons needed to do the graph theory algorithms. They are the
“Clear Drawing” button, the “Clear All Edges” button, the “Run Algorithms” button
and the “Reset All” button. The “Clear Drawing” button is used to clear all the
drawing drawn. The “Clear All Edges” button is used to clear all the edges drawn,
including the data input of the edge. By clicking the “Run Algorithms” button, the
computer simulator will run on the selected algorithm and the results will be shown
in the Results Display Column. The “Reset All” button is to reset all the results
calculated and displayed.
Figure 4.3: Drawing and Algorithms Toolbars.
Figure 4.4: Algorithms Buttons.
(iv)
Results Display Column
The Results Display Column is used to display the algorithms summary and
the computation time.
Figure 4.5: Results Display Column
4.5
Dialog Boxes
Dialog boxes are very useful in this software. There are three types of dialog
boxes involved in this software. One is the Information dialog box, second is the
Input dialog box and the third is the Results dialog box. Information dialog box
gives the required information to user. Input dialog box allow user to input the
necessary data into the simulation of the software. There are two types of input
dialog boxes. They are Input Data for Node and Input Data for Pipe. Results dialog
box shows the results of the calculation of the network problem.
(i)
Information Dia log Boxes
After checking the Dijkstra’s Shortest Path algorithm or Breadth First Search
algorithms, if user forgets to specify the start and end (finish) vertices of the graph
drawn, this type of dialog boxes will display. The purpose of showing the dia log box
is to remind user to specify the start and end vertices. One of these dialog boxes is
shown in Figure 4.6. The sentences on the dialog box in Figure 4.6 say “Dijkstra’s
Shortest Path Algorithm requires a Start Vertex and a Finish Vertex”.
Figure 4.6: Information Dialog box.
(ii)
Input Data for Node
In this dialog box, user has to insert the data of the node number, pressure and
the flow rate of the specified node. Besides that, user has to specify whether the
node is a source node or a load node by clicking the radio button on the left hand side
of the dialog box. Figure 4.7 below shows the Input data for node dialog box.
Figure 4.7: Input Data Dialog Box for Node .
(iii)
Input Data for Pipe
For this dialog box, it contains data like the branch number, length,
equivalent length and diameter of the pipe drawn by user. User need to enter all the
data of the pipe. The graphical dialog box for data input on pipe for this software is
illustration in Figure 4.8 below.
Figure 4.8: Input Data Dialog Box for Pipe .
(iv)
Results Dialog Box
The Results Dialog box is used to show the calculation of the network
problem. Before viewing the results, user has to enter all the input data for the nodes
and branches involved. This also means user has to draw the network system on the
drawing sheet. By clicking the push button “Simulation” and selecting the radio
button “Directed”, user can enter to the Results Dialog box. Before running the
simulation, user need to enter some information required, such as the total of nodes,
branch, type of method, tolerance and so on. To run the simulation, user just need to
click the “Calculate” button. All the results will be displayed with the time of
execution. The Results Dialog box is shown in Figure 4.9.
Figure 4.9: Results Dialog Box.
4.6
Main Menu
(i)
File
(a)
New
Through this button, user can open a new file to draw a network
system and do the calculation of the network system.
(b)
Open
Through this button, any saved file in Data File type can be opened.
(c)
Save/Save As
It can help user to save the modified file in Graph File type.
(c)
Print
It can help user to print the file.
(d)
Exit
Through this button, user can exit the file.
(ii)
View
(a)
Converter
Through this button, user can convert the temperature, mass, pressure
and length. The Converter is shown in the following page.
Figure 4.10: Converter.
(b)
Show Results
Through this button, user can display the Results Dialog box.
However, to make the button works, user need to click the Push
button “Simulation” and select the Radio button “Directed”. Details
about the Results Dialog box are displayed in the above section.
4.7
Simple User Guide
User needs to know the simplest way of using the computer simulator. In
order to reduce the time consuming in using this computer simulator and avoid any
unwanted error occurs, user have to follow these steps in next page:
(i)
For Network Simulation
(a)
Choose the classification of the piping system; select the
“Directed” and “Simulation” button for solving the network
simulation.
(b)
Draw the network piping system using the tools provided in
the computer simulator. While drawing, user can enter the data
of the nodes and the pipelines just by double -click at the node
when user is drawing the node and the edge (pipeline) when
user is drawing the edge. If any error occurred in the drawing,
user can simply delete the drawing by clicking the “Clear All”
button or the “Clear All Edges” button. Besides that, user can
delete the edge by right-click the edge and choose “Delete
Edges”.
(c)
User can even move the node or edge drawn by just drag the
node or edge.
(d)
Select the graph theory algorithms needed for the network
system. User needs to set the start and end point (choose
“Vertices” radio button, then right-click at the specified node)
only for Breadth-First Shortest Path Search and Dijkstra’s
Minimum Path algorithm.
(e)
Click the “Run Algorithms” button to see the algorithm results
and the computation time.
(f)
Click the “Show Results” button to display the data of nodes
and the pipelines.
(g)
Enter all the initial condition needed, the properties and the
methods chosen in the Results Dialog Box. To display the
network simulation results, just click the “Calculate” Button.
(ii)
For Non-Network Simulation
(a)
Choose the classification of the graph (drawing), select the
“Directed” or “Undirected” button. Leave the “Simulation”
button unchecked or blank.
(b)
Draw the graph using the tools provided in the computer
simulator. User can delete the node or edge by just double click at the node or edge. “Clear Drawing” and “Clear All
Edges” buttons also can be used.
(c)
User can even move the node or edge drawn by just drag the
node or edge.
(d)
Select the graph theory algorithms needed. User needs to set
the start and end point (choose “Vertices” radio button, then
right -click at the specified node) only for Breadth-First
Shortest Path Search and Dijkstra’s Minimum Path algorithm.
(e)
User can enter the weight of the graph by right-click the edge
after choosing the “Edges” radio button.
(f)
Click the “Run Algorithms” button to see the algorithm results
and the computation time.
4.8
Hardware & Software Requirements
In order for the computer simulator to operate smoothly and efficiently, it
needs the following hardware and software requirements.
Table 4.1: Hardware and Software Requirements for User and Developer.
Requirements
User
Developer
Operating System
Windows 98/2000/XP
Windows 98/2000/XP
Software
None
Microsoft Visual C++ 6.0,
MS Word
CPU
Pentium 3 (650 MHz) or
Pentium 3 (650 MHz)
higher
RAM
196 MB RAM
196 MB RAM
(16 bit recommended)
(16 bit recommended)
Hard Disc
30 MB Hard Disc
40 GB Hard Disc
Screen Display
256-colour SVGA
256-colour SVGA
(16 bit recommended)
(16 bit High Colour)
CD-ROM
40x or higher CD-ROM
40x or higher CD-ROM
Video
GeForce2 MX/MX400 display
NVIDIA GeForce2
card
MX/MX400 display card
Direct-8x or higher
Direct-9x
Mouse, keyboard
Mouse, keyboard
Other
CHAPTER 5
RESULTS AND DISCUSSIONS
5.1
Introduction
The developed gas network computer simulator is able to run five graph
theory algorithms and examine the effects of graph theory algorithms on gas network
system. Besides that, this computer simulator is able to run the Newton Nodal
Method for the network analysis. In this chapter, comparison on its accuracy and
speed can be done based on various different network configurations. The accuracy
test is compared with the simulated results obtained by using the commercially
software, GAPIS. The speed analysis is done on the various graph theory
algorithms, number of iteration and various numerical methods such as the Newton
Gauss Elimination and Newton Gauss Seidel method.
5.2
Analysis on Case Studies
Two case studies and one extended case study have been performed in this
research. For each case study, different conditions have been used. The set
conditions are different pressure, length, diameter, flow rate and the arrangement of
the network system. Different flow equations are applied into these case studies. In
case study 1, the network is in low pressure, Pole’s (Lacey’s) equation is used. In
case study 2, Cox’s equation is used since the network is in medium pressure. In
extended case study 1, the network is in high pressure and the Weymouth and
Panhandle ‘A’ equation is used. The reason of using different conditions is to check
the capacity of the computer simulator developed. The computer simulator
developed must be able to perform network analysis in low, medium and high
pressure. Table 5.1 below shows these case studies properties.
Table 5.1: Case studies properties
Case
Study
Network Condition
Initial
Number Number
Pressure
of Node
(bar
Number
Pressure
Equation
of Loop
Range
Used
3
Pressure
Pole
of
Branch
gauge )
1
0.03
5
7
(low)
0-75 mbar
gauge
2
1
6
9
4
(medium)
Pressure
Cox
29.4 kPa
and above
Extended
8
case
(high)
study 1
7
11
5
Pressure
Weymouth
more than
and
7 bars
Panhandle
‘A’
i)
Case Study 1
The network is a simple low pressure network. The initial pressure of this
network is 30 mbar and the specific gravity of gas is 0.589. The input data and the
network schematic diagram are shown in Table 5.2 and Figure 5.1 respectively.
Table 5.2: Input Data for Study Case 1
Node
Status
Pressure
Load (m3h -1 )
(mbar gauge)
1
Source
30
-
2
Load
-
250
3
Load
-
100
4
Load
-
180
5
Load
-
120
Diameter (mm)
Length (mm)
Branch
Sending
Receiving
Node
Node
1
1
2
150
680
2
1
3
100
500
3
1
4
150
420
4
1
5
100
600
5
3
2
100
600
6
3
4
100
340
7
5
4
100
420
Figure 5.1: Schematic diagram for case study 1.
ii)
Case Study 2
The network is a medium pressure network with the initial pressure of 1 bar
and the specific gravity of gas is 0.589. Cox’s equation is used. The efficiency
factor , E is set to be 0.8. The input data and the network schematic diagram are
shown in Table 5.3 and Figure 5.2 respectively.
Table 5.3: Input Data for Study Case 2
Node
Status
Pressure
Load (m3h-1 )
(bar gauge)
1
Source
1
-
2
Load
-
250
3
Load
-
100
4
Load
-
180
5
Load
-
120
6
Load
-
150
Diameter (mm)
Length (mm)
Branch
Sending
Receiving
Node
Node
1
1
2
150
680
2
1
3
100
500
3
1
4
150
420
4
1
5
100
600
5
1
6
150
600
6
3
2
100
600
7
3
4
100
340
8
5
4
100
420
9
6
5
100
600
Figure 5.2: Schematic diagram for case study 2 .
iii)
Extended Case Study 1
This is the extended study of high pressure. The network is a high pressure
network. The initial pressure of 8 bars and the specific gravity of ga s are 0.589. The
efficiency factor is 0.8. The input data and the network schematic diagram are
shown in Table 5.4 and Figure 5.3 respectively. The flow results obtained from
Weymouth equation in the computer simulator are compared with the flow results of
Panhandle ‘A’ equation in the computer simulator, during the accuracy test.
Table 5.4 : Input Data for Extended Case Study 1
Node
Status
Pressure
Load (m3h-1 )
(bar gauge)
1
Source
8
-
2
Load
-
250
3
Load
-
100
4
Load
-
180
5
Load
-
120
6
Load
-
150
7
Load
-
200
Diameter (mm)
Length (mm)
Branch
Sending
Receiving
Node
Node
1
1
2
150
420
2
1
3
100
500
3
1
4
200
680
4
1
5
200
700
5
1
6
150
450
6
1
7
100
500
7
3
2
150
500
8
3
4
100
420
9
5
4
100
340
10
6
5
150
650
11
7
6
100
250
Figure 5.3: Schematic diagram for extended case study 1.
5.3
Speed Analysis for Gas Network System
The speed analysis is done on the all the case studies in order to determine the
fastest graph theory algorithms for gas network system and the fastest numerical
method for gas network system using the Newton Nodal Method. This analysis is
carried out using the five graph theory algorithms, on three different case studies.
There are two type of speed analysis carried out in this research. The first one is
speed analysis without the network analysis and the second one is speed analysis
with the network analysis.
5.3.1
Speed Test without Network Analysis
The speed analysis is done for gas networks system without performing the
network analysis. The purpose of undergoing this analysis is to determine the fastest
graph theory algorithm, before performing the network analysis and determine the
effect of graph theory algorithms on different gas network configuration. All the
results are recorded and tabled as shown in Table 5.5. For BFS and Dijkstra’s
Shortest Path Algorithms, the start and end nodes (points) must be defined before
running the algorithms. For case study 1, the start node is 1 and end node is 5. For
case study 2, the start node is 1 and the end node is 6 while for extended study case
study 1, the start node is 1 and the end node is 7.
Three bar graphs are plotted as shown in Figure 5.4 to Figure 5.6. From these
graphs, the fastest graph theory algorithm is Breadth First Search. The slowest graph
theory algorithm is Strongly Connected Component. BFS is the fastest because of
the method used. As stated in Chapter 2, BFS explores all the neighbouring nodes
before proceeds to their respective neighbouring nodes. For these three case studies,
using BFS, starting from node 1, it will go to node 2, node 3, node 4, node 5(case
study 1), node 6(case study 2) and node 7(extended study case study 1). Since all the
nodes are “visited” by node 1, it stops. The time taken is becoming shorter as the
ending node is directly connected to the starting node, node 1.
Strongly Connected Component uses DFS method to find the strongly
connect components. For case study 1, starting from node 1, it goes to nodes 2 using
branch 1:2 and 3:2, node 3 using branch 1:3, node 4 using branch 1:4, 5:4 and branch
3:4, and node 5 using branch 1:5. Similar case also happens in case study 2 and
extended study case study 1. Since there are re-“visited” incident happens and
calculation needed to be done to obtain the strongly connected component, the time
taken is longer, even longer than DFS.
After comparing with three case studies, the effect of graph theory algorithms
on different network configuration is the computation time. The larger the network,
the time taken is longer. The reason is the number of nodes and branches increase.
Table 5.5: Speed Analysis Results without Network Analysis in Three Case
Studies.
Algorithms
Case
Time (msec)
Study
1
2
Average
3
4
5
(msec)
Breadth
1
5.37
5.47
5.35
5.34
5.33
5.37
First
2
5.5
5.46
5.46
5.47
5.51
5.48
3
5.59
5.62
5.59
5.63
5.65
5.61
Depth First
1
6.74
6.78
6.74
6.76
6.72
6.75
Search
2
7.5
7.56
7.52
7.48
7.48
7.51
(DFS)
3
7.63
7.56
7.55
7.57
7.53
7.57
Strongly
1
10.49
10.39
10.53
10.37
10.40
10.44
Connected
2
12.05
11.82
11.89
11.88
11.90
11.91
3
12.76
12.76
12.84
12.77
12.86
12.80
Minimum
1
5.46
5.44
5.43
5.43
5.43
5.44
Spanning
2
5.7
6.97
5.60
5.58
5.67
5.90
3
5.77
5.79
5.78
5.78
5.76
5.78
Dijkstra’s
1
5.36
5.37
5.45
5.49
5.36
5.41
Shortest
2
5.56
5.59
5.51
5.51
5.49
5.53
Path
3
5.72
5.64
5.62
5.68
5.68
5.67
Search
(BFS)
Component
(SCC)
Tree
(MST)
12
The number of:
Node
5
Branch
7
5.44
5.41
MST
Dijiktra's
10.44
10
Computation Time(ms)
8
6.75
6
5.37
4
2
0
BFS
DFS
SCC
Graph Theory Algorithms
Without Network Analysis
Figure 5.4: Computation Time for Case Study 1 versus Graph Theory
Algorithms.
14
The number of:
11.91
12
Node
6
Branch
9
Computation Time(ms)
10
8
6
7.51
5.90
5.53
5.48
4
2
0
BFS
DFS
SCC
MST
Dijiktra's
Graph Theory Algorithms
Without Network Analysis
Figure 5.5: Computation Time for Case Study 2 versus Graph Theory
Algorithms.
14
The number of:
12.80
12
Node
7
Branch
11
5.78
5.67
MST
Dijiktra's
Computation Time(ms)
10
8
6
7.57
5.62
4
2
0
BFS
DFS
SCC
Graph Theory Algorithms
Without Network Analysis
Figure 5.6: Computation Time for Extended case study 1 versus Graph Theory
Algorithms.
5.3.2 Speed Analysis with Network Analysis
The speed analysis is also done for gas network system with network
analysis. The network analysis is using the Newton Nodal Method. Newton Gauss
Elimination Method and Newton Gauss Seidel Method are using numerical solutions
to solve Linear Algebraic Equation in the network analysis [5]. The purpose of
undergoing this speed analysis is to determine the fastest numerical method and the
fastest method in performing the network analysis. This speed analysis can be
divided into two sections. The first section is analysis on One Iteration T ime for the
three case studies with different numerical methods that are the Newton Gauss
Elimination and Gauss Seidel Method. The second analysis is analysis on Total
Iteration T ime for the three case studies with the Newton Gauss Elimination and
Gauss Seidel Method. All the results of the speed analysis for gas network system
with network analysis is tabled and shown in tables below.
Table 5.6: One Iteration Time for the Case Study 1 with Gauss Elimination and
Gauss Seidel Method
Numerical Method
Gauss Elimination
Gauss Seidel
Tolerance
0.1
0.01
0.001
0.1
0.01
0.001
1
4.90
4.66
4.89
6.57
7.14
8.59
Time (msec)
2
4.64
4.88
4.72
6.64
7.10
8.22
3
4.64
4.89
4.88
6.42
7.04
8.09
Average
Time (msec)
4.73
4.81
4.83
6.54
7.09
8.30
Table 5.6.1: Total Iteration Time for the Case Study 1 with Gauss Elimination
and Gauss Seidel Method
Numerical Method
Tolerance
Gauss Elimination
0.1
0.01
0.001
0.1
0.01
0.001
Gauss Seidel
1
29.16
32.48
35.21
40.37
51.09
57.34
Time (msec)
2
31.84
32.48
34.42
39.90
57.74
57.92
3
27.76
34.93
34.95
39.89
55.26
55.17
Average
Time (msec)
29.59
33.30
34.86
40.05
54.69
56.81
Table 5.7: One Iteration Time for the Case Study 2 with Gauss Elimination and
Gauss Seidel Method
Numerical Method
Tolerance
Gauss Elimination
0.1
0.01
0.001
0.1
0.01
0.001
Gauss Seidel
1
6.62
6.46
6.51
8.12
8.39
9.45
Time (msec)
2
6.08
6.42
6.50
7.94
8.73
9.40
3
6.26
6.34
6.58
8.78
8.77
9.09
Average
Time (msec)
6.32
6.41
6.53
8.28
8.63
9.31
Table 5.7.1: Total Iteration Time for the Case Study 2 with Gauss Elimination
and Gauss Seidel Method
Numerical Method
Tolerance
Gauss Elimination
0.1
0.01
0.001
0.1
0.01
0.001
Gauss Seidel
1
26.73
40.32
45.21
30.74
48.99
66.88
Time (msec)
2
25.85
38.31
49.61
39.74
49.47
64.11
3
25.65
39.92
44.23
35.13
50.25
65.55
Average
Time (msec)
26.08
39.52
46.35
35.20
49.57
65.52
Table 5.8: One Iteration Time for the extended study case study 1 with Gauss
Elimination and Gauss Seidel Method
Numerical Method
Tolerance
Gauss Elimination
0.1
0.01
0.001
0.1
0.01
0.001
Gauss Seidel
1
7.77
8.01
7.76
9.04
11.25
10.99
Time (msec)
2
7.57
7.74
7.80
8.81
11.01
11.21
3
7.56
7.79
7.74
9.02
10.88
10.69
Average
Time (msec)
7.63
7.85
7.77
8.96
11.05
10.96
Table 5.8.1: Total Iteration Time for the extended study case study 1 with
Gauss Elimination and Gauss Seidel Method
Numerical Method
Tolerance
Gauss Elimination
0.1
0.01
0.001
0.1
0.01
0.001
Gauss Seidel
5.3.3
1
23.60
46.04
52.31
34.37
51.85
73.90
Time (msec)
2
29.20
42.30
48.52
35.27
50.23
71.34
3
23.55
44.34
49.94
34.82
50.01
71.83
Average
Time (msec)
25.45
44.23
50.26
34.82
50.70
72.35
Speed Test Analysis for Case Study 1
Based on the results obtained, three bar graphs have been plotted in order to
provide a clear picture of the analysis. Figure 5.7 shows the one iteration
computation time for case study 1 versus the Newton Gauss Elimination and Gauss
Seidel Method. Figure 5.8 shows t he total iteration computation time for case study
1 versus the Newton Gauss Elimination and Gauss Seidel Method. Figure 5.9 shows
the computation time for case study 1 versus the five graph theory algorithms and
two numerical methods.
Based on Figure 5.7 and Figure 5.8, Newton Gauss Elimination can provide
faster results than Newton Gauss Seidel Method in case study 1. For tolerance 0.1,
Newton Gauss Elimination is 1.81 milliseconds faster than Newton Gauss Seidel
Method in one iteration and 10.47 milliseconds in total iteration. For tolerance 0.01,
Newton Gauss Elimination is 2.28 milliseconds faster than Newton Gauss Seidel
Method in one iteration and 21.40 milliseconds in total iteration. While for tolerance
0.001, Newton Gauss Elimination is 3.45 milliseconds faster than Newton Gauss
Seidel Method in one iteration and 21.95 milliseconds in total iteration.
Newton Gauss Elimination is a direct method while Newton Gauss Seidel
Method is an iterative method. This means Newton Gauss Elimination pr ovides a
straightforward solution while Newton Gauss Seidel Method generates a sequence of
successive approximations to the exact solution.
The main reason of Newton Gauss Elimination is faster than Newton Gauss
Seidel for both graphs is its short sequence of calculation. Newton Gauss
Elimination consists of two phases, which are the Forward Elimination and the Back
substitution. The Forward Elimination is to eliminate one of the unknowns from the
equations in the Jacobi Matrix and the Back substitution is to solve the equations
directly. T hen the results obtained, after the Forward Elimination are back
substituted into the original equations in the Jacobi Matrix, to solve the remaining
unknown. The calculation stops until the final results are closely to the true values.
Meanwhile, Newton Gauss Seidel Method has a similarity with the technique
of simple fixed point iteration. Newton Gauss Seidel Method solves the unknowns
from the equations in the Jacobi Matrix, using initial guesses that assume one or
more the unknowns (n-1) are zero. For this case study, the number of unknowns, n is
4 because of 4 load nodes. So, 3 unknowns are initially assumed zero. The result of
the last unknown obtained is substituted into the equations in Jacobi Matrix, in order
to solving other unknowns. These calculation steps are repeated until the final
results are closely to the true values (depend on the tolerance inserted by user of the
computer simulator) and may sometimes rises up to 100. Due to the “trial and error”
method and these steps, the time taken to obtain a good result becomes longer. The
reuses of Jacobi matrix for Newton Gauss Seidel Method increases when all the old
equations in the Jacobi Matrix are frequently replaced by the new equations each
time the calculation steps are repeated, more than the reuses of Jacobi matrix for
Newton Gauss Elimination method. The reason is the total number of iteration for
Newton Gauss Elimination method is 7 times, comparing to the total number of
iteration for Newton Gauss Seidel method is 414 (see Figure 5.9). So, Newton Gauss
Seidel Method is slower than Newton Gauss Elimination at one iteration and total
iteration.
In order to obtain more accurate results, the tolerance must be smaller. Thus,
the time taken to obtain the result is longer. Because of this reason, the total iteration
time for tolerance 0.001 has been chosen to plot the graph in Figure 5.9.
Figure 5.9 is plotted in order to determine the fastest method in performing
the analysis on graph theory algorithms and numerical methods for case study 1. It
can be considered as the combination of Figure 5.4 and Figure 5.8 because the same
values for graph theory algorithms and numerical methods for tolerance 0.001 are
obtained, after many tests on the computation time have been carried out.
As stated, BFS is the fastest graph theory algorithms while Newton Gauss
Elimination is the fastest numerical method from previous results. For Figure 5.9,
still BFS is the fastest graph theory algorithms while Newton Gauss Elimination is
the fastest numerical method. The fastest method in performing the analysis on
graph theory algorithms and numerical methods for case study 1 is the combination
of BFS and Newton Gauss Elimination with the total time 40.2328 milliseconds.
However, the combination of MST and Newton Gauss Elimination, and the
combination of Dijkstra’s Shortest Path Method and Newton Gauss Elimination also
are two good ways to obtain fast results for case study 1, with the time difference of
only 0.07 milliseconds and 0.03 milliseconds respectively, comparing with the time
of the combination of BFS and Newton Gauss Elimination. Both the combination of
MST with Newton Gauss Elimination and the combination of Dijkstra’s Shortest
Path with Newton Gauss Elimination can provide fast results because the case study
1 is an un-weighted graph/drawing. These two graph theory algorithms are suitable
for weighted graph.
One Iteration, 5 nodes
9
8.30
8
7.09
7
6.54
Computation Times (ms)
6
Tolerance
5
4.73
4.81
4.83
0.1
0.01
0.001
4
3
2
1
0
Gauss Elimination
Gauss Seidel
Numerical Methods
Figure 5.7: One Iteration Time for case study 1 v ersus the Newton Gauss
Elimination and Gauss Seidel Method.
Total Iteration, 5 nodes
60
56.81
54.69
50
40.05
Computation Times (ms)
40
Tolerance
34.86
33.30
30
0.1
0.01
29.59
0.001
20
10
0
Gauss Elimination
Gauss Seidel
Numerical Methods
Figure 5.8: Total Iteration Time for case study 1 versus the Newton Gauss
Elimination and Gauss Seidel Method.
80
Computation Time for Case Study 1 versus Graph Theory
Algorithms and Numerical Methods
Lege nd
Netwon Gauss
Seidel (0.001)
70
Newton Gauss
Elimination
(0.001)
Dijkstra's
Shortest Path
60
MST
Computation Time (msec)
50
SCC
40
56.81
DFS
56.81
56.81
56.81
56.81
30
BFS
34.86
34.86
34.86
34.86
34.86
Total Iteration:
20
Newton Gauss
Elimination
Method (0.001)
7
10
10.44 10.44
5.37 5.37
6.75 6.75
5.44
5.44
5.41
5.41
Newton Gauss
Seidel Method
(0.001)
414
0
Graph Theory Algorithms and Numerical Methods
Figure 5.9: Computation Time for case study 1 versus the Graph Theory
Algorithms and Numerical Methods.
5.3.4
Speed Test Analysis for Case Study 2
Based on the results obtained, like in case study 1, three bar graphs have been
plotted in order to provide a clear picture of t he analysis. Figure 5.10 shows the one
iteration computation time for case study 2 versus the Newton Gauss Elimination and
Gauss Seidel Method. Figure 5.11 shows the total iteration computation time for
case study 2 versus the Newton Gauss Elimination and Gauss Seidel Method. Figure
5.12 shows the computation time for case study 2 versus the five graph theory
algorithms and two numerical methods.
Based on the both graphs, still Newton Gauss Elimination can provide faster
results than Newton Gauss Seide l Method in case study 2, although different flow
equation is used. For tolerance 0.1, Newton Gauss Elimination is 1.96 milliseconds
faster than Newton Gauss Seidel Method in one iteration and 9.13 milliseconds for
total iteration. For tolerance 0.01, Newton Gauss Elimination is 2.22 milliseconds
faster than Newton Gauss Seidel Method in one iteration and 10.05 milliseconds in
total iteration. While for tolerance 0.001, Newton Gauss Elimination is 2.78
milliseconds faster than Newton Gauss Seidel Method in one iteration and 19.17
milliseconds in total iteration.
Like in case study 1, the main reason of Newton Gauss Elimination is faster
than Newton Gauss Seidel for both graphs is its short sequence of calculation.
Newton Gauss Elimination eliminates one of the unknowns from the equations in the
Jacobi Matrix and solves the equation directly. The result obtained is substituted into
the original equations in the Jacobi Matrix to solve the remaining unknown. The
calculation stops until the final results are closely to the true values.
Meanwhile, Newton Gauss Seidel Method solves the unknowns from the
equations in the Jacobi Matrix, using initial guesses that assume all the unknowns are
zero. For this case study, the number of unknowns, n is 5 because there are 5 load
nodes. So, the number of unknowns is 4. They are initially assumed zero. The last
unknown obtained is substituted into the equations in the Jacobi Matrix, in order to
solve other unknowns. These “trial and error” calculation steps a re repeated until the
final results are closely to the true values. The reuses of Jacobi matrix for Newton
Gauss Seidel Method increases when all the old equations in the Jacobi Matrix are
frequently replaced by the new equations each time the calculation steps are
repeated, more than the reuses of Jacobi matrix for Newton Gauss Elimination
method. The reason is the total number of iteration for Newton Gauss Elimination
method is 7 times, comparing to the total number of iteration for Newton Gauss
Seidel method is 377 (see Figure 5.9). So, the time taken to obtain a result becomes
longer.
From the graphs, the smaller the tolerance, the longer the time taken. To
obtain a more accurate result, the tolerance must be smaller. Consequently, the time
taken to obtain the result is longer.
Like case study 1, Figure 5.12 is the combination of Figure 5.5 and Figure
5.11 because the computation time results obtained for case study 2 are still the same
although many tests on the computation time have been carried out. The fastest
method in performing the analysis on graph theory algorithms and numerical
methods for case study 2 is the combination of BFS and Newton Gauss Elimination
with the total time 51.83 milliseconds, slightly slower than in case study 1. The
reason is the graph becomes larger.
The combination of MST with Newton Gauss Elimination and the
combination of Dijkstra’s Shortest Path with Newton Gauss Elimination also can
provide fast results because the case study 2 is also an un-weighted graph like case
study 1. Different computation time will be shown if the graph is a weighted graph.
One Iteration, 6 nodes
10
9.31
9
8.63
8.28
8
Tolerance
7
Computation Times (ms)
6.32
6.41
6.53
6
0.1
0.01
5
0.001
4
3
2
1
0
Gauss Elimination
Gauss Seidel
Numerical Methods
Figure 5.10: One Iteration Time for case study 2 versus the Newton Gauss
Elimination and Gauss Seidel Method.
Total Iteration, 6 nodes
70
65.52
60
49.57
50
46.35
Computation Times (ms)
Tolerance
39.52
40
0.1
35.20
0.01
0.001
30
26.08
20
10
0
Gauss Elimination
Gauss Seidel
Numerical Methods
Figure 5.11: Total Iteration Time for case study 2 versus the Newton Gauss
Elimination and Gauss Seidel Method
Computation Time for Case Study 2 versus Graph Theory Algorithms and
Numerical Methods
90
Legend
80
Newton Gauss
Seidel Method
(0.001)
Newton Gauss
Elimination
(0.001)
70
Computation Time (msec)
Dijkstra's
Shortest Path
60
MST
50
SCC
65.52
DFS
65.52
40
65.52
65.52
65.52
46.35
30
BFS
46.35
46.35
46.35
46.35
Total Iteration:
20
Newton Gauss
Elimination
Method (0.001)
7
10
11.91 11.91
5.48 5.48
7.51 7.51
5.90 5.90 5.53 5.53
0
Newton Gauss
Seidel Method
(0.001)
377
Graph Theory Algorithms and Numerical Methods
Figure 5.12: Computation Time for case study 2 versus the Graph Theory
Algorithms and Numerical Methods.
5.3.5
Speed Test Analysis for Extended Case Study 1
Based on the results obtained, like in case study 1 and 2, three bar graphs
have been plotted in order to provide a clear picture of the analysis. Figure 5.13
shows the one iteration computation time for extended case study 1 versus the
Newton Gauss Elimination and Gauss Seidel Method. Figure 5.14 shows the total
iteration computation time for extended case study 1 versus the Newton Gauss
Elimination and Gauss Seidel Method. Figure 5.15 shows the computation time for
extended case study 1 versus the five graph theory algorithms and two numerical
methods.
Based on the both graphs, still Newton Gauss Elimination can provide faster
results than Newton Gauss Seidel Method in extended case study 1. For tolerance
0.1, Newton Gauss Elimination is 1.32 milliseconds faster than Newton Gauss Seidel
Method in one iteration and 9.37 milliseconds for total iteration. However, for
tolerance 0.01, Newton Gauss Elimination is 3.20 milliseconds faster than Newton
Gauss Seidel Method in one iteration and 6.4711 milliseconds in total iteration.
While for tolerance 0.001, Newton Gauss Elimination is 3.19 milliseconds faster
than Newton Gauss Seidel Method in one iteration and 22.09 milliseconds in total
iteration.
Like case study 1 and 2, the reason Newton Gauss Elimination is faster than
Newton Gauss Seidel Method is short sequence of calculation for Newton Gauss
Elimination.
For this case, there is a bit different of time between the tolerance 0.01 and
the tolerance 0.001 comparing with the other two case studies. For tolerance 0.001,
the time taken for both numerical methods in one iteration is faster than time taken
for tolerance 0.01 in one iteration. The reason for this incident happens is the
internal diameter and the length of network system inserted during the data input is
too small, causing the pressure drop becomes small. Thus, the results and time are
easily obtained since the tolerance is small.
Like case study 1 and 2, Fig ure 5.15 is the combination of Figure 5.6 and
Figure 5.14 because the computation time results obtained for extended case study 1
are still the same although many tests on the computation time have been carried out.
The fastest method in performing the analysis on graph theory algorithms and
numerical methods for extended case study 1 is the combination of BFS and Newton
Gauss Elimination with the total time 55.87 milliseconds, 4.04 milliseconds slower
than case study 2 and 15.64 milliseconds slower than case study 1. The reason is the
graph becomes larger than the graphs in case study 1 and case study 2.
The graph of extended case study 1 is an un-weighted graph. Thus, the
computation times obtained from the combination of MST and Newton Gauss
Elimination and the combination of Dijkstra’s Shortest Path and Newton Gauss
Elimination is almost as fast as the computation time obtained from the combination
of BFS and Newton Gauss Elimination.
One Iteration, 7 nodes
12
11.05
10.96
10
8.96
Computation Times (ms)
8
7.63
7.85
7.77
Tolerance
0.1
0.01
6
0.001
4
2
0
Gauss Elimination
Gauss Seidel
Numerical Methods
Figure 5.13: One Iteration Time for extended study case study 1 versus the
Newton Gauss Elimination and Gauss Seidel Method
Total Iteration,7 nodes
80
72.35
70
60
50.70
50.26
Tolerance
Computation Times (ms)
50
44.23
0.1
0.01
40
0.001
34.82
30
25.45
20
10
0
Gauss Elimination
Gauss Seidel
Numerical Methods
Figure 5.14: Total Iteration Time for extended study case study 1 versus the
Newton Gauss Elimination and Gauss Seidel Method.
Computation Time for Case Study 2 versus Graph Theory Algorithms and
Numerical Methods
90
Legend
80
Newton Gauss
Seidel Method
(0.001)
Newton Gauss
Elimination
(0.001)
70
Dijkstra's
Shortest Path
Computation Time (msec)
60
MST
SCC
50
72.35
72.35
72.35
72.35
72.35
DFS
40
50.26
BFS
50.26
30
50.26
50.26
50.26
20
10
12.80 12.80
5.62 5.62
7.57 7.57
5.78 5.78 5.67 5.67
0
Graph Theory Algorithms and Numerical Methods
Figure 5.15: Computation Time for Extended Case Study 1 versus the Graph
Theory Algorithms and Numerical Methods.
5.3.6
Discussion on Speed Test with Network Analysis
Generally, there are many factors that can influence in the calculation of
computation time for the speed test analysis with network analysis. These factors are
the flow equations used, the network arrangement (how the network is connected,
like one node to many nodes and so on) and network configuration (number of
nodes, branch and loop), graph theory algorithms used, numerical methods used, the
size of Jacobi Matrix and so on. However, for these three case studies, the factors
that have been focused and analysed in this research are the network configuration,
the size of Jacobi Matrix, graph theory algorithms used and the numerical methods
used. The computation time for the flow equations used in the computer simulator
and the network arrangement are not studied in this research, due to the objective of
this research.
For all the case studies, to obtain the fastest computation time in the computer
simulator, smaller network, smaller size of Jacobi Matrix, Breadth First Search and
Newton Gauss Elimination must be used. Smaller network means the number of
nodes, branch and the number of loop involved are small, like in case study 1. When
smaller network is used, the size of Jacobi Matrix formed is smaller because the size
of the matrix only depends on the number of node, especially the load nodes (if
Newton Nodal Method is used). Thus, the total computation times for the graph
theory algorithms and the numerical methods are faster. This is the situation happens
in case study 1, Figure 5.9, where the computation for case study 1 is fas ter than case
study 2, Figure 5.12.and extended case study 1, Figure 5.15.
Breadth First Search and Newton Gauss Elimination method are the fastest
graph theory algorithms and numerical methods in this research. The reason of
Breadth First Search is the fastest because of the start node (source node) are directly
connected to the end node (last load node). Different graph theory algorithms have
different computation time on different network configuration. This means that
Breadth First Search is fastest in case study 1 (network with 5 nodes), faster in case
study 2 (network with 6 nodes) and fast in extended case study 1(network with 7
nodes). Newton Gauss Elimination method is faster than Newton Gauss Seidel
method because of its short sequence of calculation
5.4
Accuracy Analysis for Gas Network System
The accuracy analysis is done on the two case studies in order to determine
how accuracy the flow rate results is, obtained from the computer simulator
developed, comparing with other commercial software, GaPIS in the market and to
determine the most accurate numerical methods. For extended case study 1, the flow
results obtained from the Weymouth equation is compared with the flow results
obtained from the Panhandle ‘A’ equation, in the computer simulator.
5.4.1
Accuracy Analysis for Case Study 1
In case study 1, using the tolerance of 0.001 for both Newton Gauss
Elimination and Newton Gauss Seidel Method, accuracy analysis on the flow is
done. From the results obtained in Table 5.9, the average accuracy results on the
flow rate for both numerical methods are 99.22%. Thus, the flow results obtained
from the computer simulator developed is very close to the flow results from the
commercial software, GaPIS. The reason is the same flow equation, w hich is the
Pole’s equation and the same numerical methods are used.
Graph of flow comparison for Newton Gauss Elimination is plotted based on
the accuracy analysis results obtained. The graph is shown in Figure 5.16. From the
graph, line s are plotted for the flow results obtained from GaPIS and the computer
simulator. The flow results obtained from computer simulator are in the same shape
like the flow results obtained from GAPIS. T his means the flow results obtained
from the computer simulator is as accurate as the flow results obtained from GaPIS .
However, all the flow results are not perfectly accurate, comparing with GaPIS. This
is due to different specific gravity of gas is used. Same things happen for Newton
Gauss Seidel Method. The flow results obtained from the computer simulator is as
accurate as the flow results obtained from GaPIS. The graph of flow comparison for
Newton Gauss Seidel Method is shown in Figure 5.17. Based on the tables and
graphs, both numerical methods provide flow results with same accuracy.
For this case study, the computer simulator developed can provide accurate
flow results. So, the Pole’s equation and the numerical methods in this computer
simulator can be used to perform the network analysis. The computer simulator
developed is suitable to be used in small gas network system with low pressure.
Table 5.9 : Accuracy Analysis in Case Study 1
Newton Gauss Elimination
Tolerance=0.001
GASNETWORK
Result
SYTEMV1
No
GaPIS
Percentage
Flowrate
Branch
Send
Receive
Flowrate (m3/h)
(m3/h)
100%
1
1
2
223.218
223.771
99.75
2
1
3
89.7936
89.125
99.26
3
1
4
253.9282
254.525
99.77
4
1
5
83.06012
82.5798
99.42
5
3
2
26.7819
26.2295
97.94
6
3
4
36.9884
37.1045
99.69
7
5
4
36.9399
37.4202
98.72
Average:
99.22
Newton Gauss Seidel Method
Tolerance=0.001
GASNETWORK
Result
SYTEMV1
No
GaPIS
Percentage
Flowrate
Branch
Send
Receive
Flowrate(m3/h)
(m3/h)
100%
1
1
2
223.218
223.771
99.75
2
1
3
89.7936
89.125
99.26
3
1
4
253.9282
254.525
99.77
4
1
5
83.06012
82.5798
99.42
5
3
2
26.7819
26.2295
97.94
6
3
4
36.9883
37.1045
99.69
7
5
4
36.9398
37.4202
98.72
Average:
99.22
Gas Network System Versus GaPIS
Flowrate(m3/hr)
300
250
200
150
100
50
0
1
2
3
4
5
6
7
Branch
GASNETWORKSYTEMV1
GaPIS
Figure 5.16: Newton Nodal Method (flow comparison case study 1– Newton
Gauss Elimination).
Gas Network System Versus GaPIS
Flowrate(m3/hr)
300
250
200
150
100
50
0
1
2
3
4
5
6
7
Branch
GASNETWORKSYTEMV1
GaPIS
Figure 5.17: Newton Nodal Method (flow comparison case study 1– Newton
Gauss Seidel Method).
5.4.2
Accuracy Analysis for Case Study 2
In case study 2, the average accuracy results on the flow rate for both
numerical methods are 95.64%. Thus, the flow results obtained from the computer
simulator developed is close to the flow results from the commercial software ,
GaPIS. The reason is the same flow equation, which is the Cox’s equation, and the
same numerical methods are used. Table 5.10 shows the accuracy test for case study
2.
Figure 5.18 shows the graph of flow comparison for Newton Gauss
Elimination. From the graph, like in case study 1, lines are plotted for the flow
results obtained from GaPIS and the computer simulator. The flow results obtained
from computer simulator are stil l in the same shape like the flow results from
GAPIS, but compare with case study 1; the shape is not that alike starting from
branch 6. This means the flow results obtained from the computer simulator is just
as good as the flow results obtained from GaP IS, but not more accurate than in case
study 1. This is due to different specific gravity of gas is used and different Cox’s
coefficient is used in the computer simulator, comparing with GaPIS. The flow
result of branch 6 has significant different from the other flow results and the
accuracy of flow results from branch 6 to 9 (chords) are between 84 to 97%. The
cause of these phenomena is the flow results starting from branch 6, are affected by
the flow results starting from branch 1 to 5 (branches), when the chord flow
calculation process takes place (see Chapter 3, section 3.11.5). Slightly changes in
the flow results (comparing with GaPIS) starting from branch 1 to 5, will cause
different in the pressure drop (pressure drop become bigger). Thus, it will cause
major changes in the flow results from branch 6 to 9. Same things happen for
Newton Gauss Seidel Method. The flow results obtained from the computer
simulator is as good as the flow results obtained from GaPIS . The graph of flow
comparison for Newton Gauss Seidel Method is shown in Figure 5.19. Based on the
tables and graphs, both numerical methods provide flow results with same accuracy.
For this case study, the Cox’s equation used in the computer simulator can
provide accurate flow results and suitable to be used in gas network system with
medium pressure.
Table 5.10: Accuracy Analysis in Case Study 2
Newton Gauss Elimination
Tolerance=0.001
GASNETWORK
Result
SYTEMV1
No
GaPIS
Percentage
Flowrate
Branch
Send
Receive
Flowrate(m3/h)
(m3/h)
100%
1
1
2
219.7473
214.086
97.42
2
1
3
86.8908
88.4701
98.21
3
1
4
238.1071
241.271
98.69
4
1
5
73.3305
73.3538
99.97
5
1
6
181.9232
182.891
99.47
6
3
2
30.2526
35.914
84.24
7
3
4
43.3617
47.449
91.39
8
5
4
14.6949
13.8272
94.10
9
6
5
31.9223
32.819
97.27
Average:
95.64
Newton Gauss Seidel Method
Tolerance=0.001
GASNETWORK
Result
SYTEMV1
No
GaPIS
Percentage
Flowrate
Branch
Send
Receive
Flowrate(m3/h)
(m3/h)
100%
1
1
2
219.7473
214.086
97.42
2
1
3
86.8908
88.4701
98.21
3
1
4
238.1071
241.271
98.69
4
1
5
73.3305
73.3538
99.97
5
1
6
181.9232
182.891
99.47
6
3
2
30.2526
35.914
84.24
7
3
4
43.3617
47.449
91.39
8
5
4
14.6949
13.8272
94.10
9
6
5
31.9223
32.819
97.27
Average:
95.64
Gas Network System Versus GaPIS
Flowrate(m3/hr)
300
250
200
150
100
50
0
1
2
3
4
5
6
7
8
9
Branch
GASNETWORKSYTEMV1
GaPIS
Figure 5.18: Newton No dal Method (flow comparison case study 2– Newton
Gauss Elimination).
Gas Network System Versus GaPIS
Flowrate(m3/hr)
300
250
200
150
100
50
0
1
2
3
4
5
6
7
8
9
Branch
GASNETWORKSYTEMV1
GaPIS
Figure 5.19: Newton Nodal Method (flow comparison case study 2– Newton
Gauss Seidel Method).
5.4.3
Accuracy Analysis for Extended Case Study 1
In extended study case study 1, the flow results obtained from the Weymouth
equation in the computer simulator are compared with the flow results obtained from
the Panhandle ‘A’ equation in the computer simulator. The test is performed under
the tolerance of 0.001 for numerical methods, Newton Gauss Elimination and
Newton Gauss Seidel method. T he average accuracy results on the flow rate for both
numerical methods are 91.71%. Table 5.11 shows the accuracy test for extended
case study 1.
Figure 5.20 shows the graph of flow comparison for Newton Gauss
Elimination. From the graph, like in case study 1 and 2, two lines are plotted for the
flow results obtained from both flow equations in the computation simulator. If the
Panhandle ‘A’ works as the control system, the Weymouth equation is in same shape
with Panhandle ‘A’ equation. However, the flow results obtained from Weymouth
equation become far different, starting from branch 3. This difference in the result is
caused by the length input and the diameter input for that particular pipe is too small
for Weymouth equation to calculate. Same things also happen for Newton Gauss
Seidel Method. The graph of flow comparison for Newton Gauss Seidel Method is
shown in Figure 5. 21. Based on the tables and graphs, both numerical methods
provide flow results with same accuracy.
For this case study, the Weymouth equation can provide the flow results
almost as same as the flow results obtained from the Panhandle equation. Thus, the
computer simulator can provide good results for gas network system w ith high
pressure.
Table 5.11 : Accuracy Analysis in Extended Case Study 1
Result (Tolerance=0.001)
Weymouth
Panhandle ‘A’
Newton Gauss Elimination
No
Percentage
Flowrate
Flowrate
Branch
Send
Receive
(m3/h)
(m3/h)
100%
1
1
2
218.5398
218. 7038
99.93
2
1
3
69.8479
70.8589
98.57
3
1
4
229.5347
224.4258
97.77
4
1
5
221.2319
219.8126
99.36
5
1
6
167.4098
171.2337
97.77
6
1
7
92.9194
94.2278
98.61
7
3
2
47.9851
41.4026
86.28
8
3
4
60.7721
59.1699
97.36
9
5
4
10.6862
4.3045
40.28
10
6
5
89.6693
84.3084
94.02
11
7
6
107.0805
105.767
98.77
Average:
91.70
Newton Gauss Seidel Method
No
Flowrate
Flowrate
Branch
Send
Receive
(m3/h)
(m3/h)
100%
1
1
2
218.5398
218.7038
99.93
2
1
3
69.8478
70.8589
98.57
3
1
4
229.5346
224.4267
97.77
4
1
5
221.2318
219.811
99.36
5
1
6
167.4098
171.2333
97.77
6
1
7
92.9194
94.2277
98.61
7
3
2
47.9844
41.4029
86.28
8
2
4
60.772
59.1698
97.36
9
3
4
10.6862
4.3095
40.33
10
4
6
89.6693
84.3088
94.02
11
4
5
107.0805
105.767
98.77
Average:
91.71
Flowrate(m3/hr) GASNETWORKSYSTEM
V1
Weymouth equation versus Panhandle 'A' equation
250
200
150
100
50
0
1
2
3
4
5
6
7
8
9
10
11
Branch
Weymouth
Panhandle 'A'
Figure 5.20: Newton Nodal Method (flow comparison extended case study 1–
Newton Gauss Elimination)
Flowrate(m3/hr) GASNETWORKSYSTEM
V1
Weymouth equation versus Panhandle 'A' equation
250
200
150
100
50
0
1
2
3
4
5
6
7
8
9
10
11
Branch
Weymouth
Panhandle 'A'
Figure 5.21: Newton Nodal Method (flow comparison extended case study 1–
Newton Gauss Seidel Method).
5.4.4
Discussion on the Accuracy Analysis
For the three case studies, based on the flow results, both numerical methods
provide flow results with same accuracy. This means Newton Gauss Elimination
method is as accurate as Newton Gauss Seidel method. Both numerical methods
have the same accuracy because of the same flow equation used and the most
importantly, the same mathematical concept used that is to find the solutions in these
case studies based on the same tolerance used, although the mathematical steps to
find the solutions are different. However, Newton Gauss Elimination method is
faster than Newton Gauss Seidel method in performing network analysis (already
explained in the previous sections). That is why users of this method prefer using it
than other numerical methods due to its speed and accuracy. Newton Elimination
method is one of the earliest methods for solving simultaneous equations; it remains
among the most important algorithms in use today and is the basic for linear equation
solving a many popular software packages [5].
In this research, the bigger initial pressure and the larger the network, the
accuracy of the flow results decreases. The reasons are different flow equations, and
different fraction factor used in the computer simulator. As the initial pressure used
become bigger, different flow equations are applied like Pole is only suitable for low
pressure distribution. The flow equations have different fraction factors. It is hard to
determine the fraction factors especially for high pressure network with high flow
rate. Thus the accuracy of the flow results decreases.
CHAPTER 6
CONCLUSION AND RECOMMENDATION
6.1
Conclusion
A computer simulator has been successfully developed. This computer
simulator is called GASNETWORKSYSTEMV1. The objectives of this research
have been achieved.
Object oriented approach has been used in the development of the structure of
this computer simulator. Object oriented is suitable to develop Graphical User
Interface (GUI). The user interface is needed to develop a user friendly program.
This computer simulator has GUI, which will easily help users to navigate in this
computer simulator.
The computer simulator uses Newton Nodal Method in order to perform the
network analysis. The reason of choosing this method is it is a simple method that
suitable in solving set of equations, when comparing with Newton Loop method.
The method is used as the basic method to solve the network analysis. In future,
other methods such as the Newton Loop Method, Newton Loop-Nodal Method and
so on can be implemented into this computer simulator.
The flow equations used in this computer simulator are Pole’s (Lacey),
Polyflo, Cox’s, Panhandle ‘A’ and Weymouth. These flow equations selected have
their limitation like the pressure range. Assumptions have been made on these flow
equations so that they are suitable to be used in this computer simulator.
Graph theory algorithms are used to represent the gas network computation of
any topology, before performing the network analysis. The graph theory algorithms
have effect on gas network system, especially the network configuration. The effect
is the computation time. The larger the gas network system, this means more nodes
and branches involve, and the computation time taken is longer. From the results
obtained (speed test analysis), BFS is the fastest graph theory algorithms that are
5.37ms for case 1, 5.48 ms for case 2 and 5.62 ms for extended case study 1.
In this research, Newton Gauss Elimination is the fastest numerical method
giving us 34.86 ms for case study 1 (tolerance 0.001), 46.35 ms for case study 2
(tolerance 0.001) and 50.26 ms for extended case study 1 (tolerance 0.001), that can
be used to solve the network analysis with good speed. The combination of BFS and
Newton Gauss Elimination is the fastest way to obtain the network analysis results,
with the total time of 40.23 milliseconds in case study 1, 51.83 milliseconds in case
study 2 and 55.87 milliseconds in extended case study 1. Both numerical methods
used in computer simulator developed have the same accuracy, when the flow results
of low pressure network and medium pressure network are compared with the
commercial software, GaPIS and when Panhandle ‘A’ and Weymouth equation are
compared with each other for high pressure network. For low pressure network,
accuracy of 99.22% is obtained when Pole’s equation used. For medium pressure
network, accuracy of 95.64% is obtained when Cox’s equation used. For high
pressure network, accuracy of 91.71% is obtained. However, the accuracy is
decreasing as the network system become larger and the initial pressure become
bigger. This is due to different flow equations are used in the network calculation.
Lastly, this computer simulator is able to perform the network analysis for
low, medium and high pressure distribution network system. This computer
simulator can be used as a teaching aid in the gas reticulation course. Future
enhancement is needed in order to commercialize this computer simulator.
6.2
Reco mmendations
After referring to the result and conclusion obtained from the three case
studies, the following recommendations are given for future work.
1. To further develop the computer simulator into a real gas engineering
application, which able user to perform calculation involving pump, elbow
and others.
2. To incorporate Graphical Information System (GIS) in the computer
simulator. It enables the computer simulator to consider the landscape factor
for specific area.
3. To further develop the computer simulator into a computer application that
can link with other useful program such as AUTOCAD, Matlab and so on.
4. To further develop the computer simulator with addition of some new flow
equations like Institute of Gas Technology (IGT) equation, Spitzglass
equation, and other methods such as the Newton Loop Method, Newton
Loop-Nodal Method and so on.
REFERENCE
1. Energy Information Administration (EIA) Natural Gas (1998). 109-127.
2. Gas Malaysia’s Business (2004). (www.gasmalaysia.com)
3. Rodrigue, J. P. (2005). Graph Theory: Definition and Properties.
(http://people.hofstra.edu/geotrans/eng/ch2en/meth2en/ch2m1e n.html)
4. Osiadacz, A.J. Simulation and Analysis of Gas Network . First Edition.
London: D and F.N. Spon Ltd. 1 – 176; 1987.
5. Yang, K.W. Convergence Characteristics of Steady State Gas Network
System Using Numerical Loop -Node Method. Master Thesis. Universiti
Teknologi Malaysia; 2001
6. Steven, C. C. and Raymond, P. C. Numerical Methods for Engineers. Third
Edition. Singapore: McGraw-Hill. 118-328; 1998
7. Kirchhoff, G. Annalen der Physik and Chemie. 72: 497. 1847.
8. Booch, G. Object Oriented Analysis and Design – With Applications. Second
Edition, California: The Benjamin/Cummings Publishing Company, Inc.
1994
9. Object Oriented Programming (2006).
(www.www.en.wikipedia.org/wiki/Object-oriented_programming)
10. Kim, W., Lochovsky, Frederick, H. Object Oriented Concepts, Databases
and Applications. USA: Addison Wesley Publishing, Company, Inc. 1989
11. Whitmire, S.A. Object Oriented Design Measurement. USA: John Wiley &
Son Inc . 1997
12. Shlaer, S. and Mellor, S.J. Object Lifecycles Modeling the World in States.
New Jersey: Prentice Hall, Inc. 1992
13. Taylor, D.A. Object Oriented Technology: A Manager Guide. USA:
Addison Wesley Publishing, Company, Inc. 1990
14. Lewandowski, A. Object Oriented Modelling of The Natural Gas Pipeline
Network. Journal of Pipeline Simulation Interest Group, 1994. 94(2): 1-30.
15. Liberty, J. SAMS Teach Yourself C++ in 21 Days. Fourth Edition. United
States: Sams Publishing. 2001
16. Svenk, G. Object Oriented Programming – Using C++ Engineering and
Technology. New York: Delmar Learning. 2003
17. Muller, P. Introduction to Object-Oriented Programming Using C++.
Berlin, Germany: Globewide Network Academy (GNA). 1996
18. Hubbard, J.R. Programming with C++. Second Edition. Singapore:
McGraw -Hill Book Co. 2000
19. Christofides, N. Graph Theory. New York: Academic Press. 1975
20. Wilf, H.S. Algorithms and Complexity. Internet Edition. University of
Pennsylvania, Philadelphia. 1994
21. McHugh, J.A. Algorithmic Graph Theory. New Jersey, USA: Prentice Hall.
1990
22. Siek, J.G., Lee, L.Q., Lumsdaine, A. The Boost Graph Library: User Guide
and Reference Manual. USA: Addison Wesley Publishing, Company, Inc.
2001
23. Minieka, E. Optimization Algorithms for Networks and Gra ph. New York,
USA: Marcel Dekker. 1978
24. Puah, H. Implementasi Sistem Maklumat Geografi Untuk Pemetaan Sistem
Utiliti Bawah Tanah Dengan Mengaplikasikan Sistem Maklumat (GIS).
Undergraduate Thesis. Universiti Teknologi Malaysia; 2002
25. Wilson, G.G. Gas Distribution . First Edition. Chicago: IGT. 59-84. 1982.
26. Ohashi, T. Natural Gas Utilization for Industrial Users. Tokyo: Tokyo Gas
Co. Ltd. 1-390. 1990.
27. Abdel-Alim Hashem. Part Two: Steady State Flow of Gas through Pipes.
Oil and Gas Pipeline Design, Maintenance and Repair, 2000. PE 607: 1-71.
28. Dahl, H.J., Romo, F., Tomasgard, A. Optimisation Model for Rationing
Efficient Allocation of Capacity in a Natural Gas Transportation Network.
SINTEF Report, 2003. IAEE: 4-5.
29. Schroeder, D.W., Jr. A Tutorial on Pipe Flow Equations. Journal of Pipeline
Simulation Interest Group, 2001. 01(1): 1-18. .
30. Piggott, J., Kurschat, T., Revell, N. Taking the Rough with the Smooth – a
new look at transmission factor formulae. Journal of Pipeline Simulation
Interest Group, 2002. 02(2): 2-21.
31. Menon, E. S., P.E. Gas Pipeline Hydraulic. Online Course. 2006
32. Aylmer, S. Interactive Gas Flow Analysis. PhD. Thesis. University of
Manchester: Institute of Science and Technology. 1980.
33. Hoeven, T.V.D. Some Mathematical Aspects of Gas Network Simulation.
Journal of Pipeline Simulation Interest Group, 1992. 92(6): 1-15.
34. Ferziger, J.H. Numerical Methods for Engineering Application . New York:
Wiley. 1981
35. Hartman-Baker, R.J. Introduction to Iterative Methods. 2002
36. Fisher-Uhrig Engineering. STANET: Network Analysis of Gas and Water
Networks with Net Database. Berlin: User Guide – Version 7.3. 2003
37. Bean, B.B. GASWorkS 9.0 Demonstration Guide. Colorado: User Guide –
Version 9.0. 1997.
38. Hardee, R. Fluid Flow Software vs. Spreadsheets for Piping System Design
& Analysis. Engineered Software, Inc . PIPEFLO: White Paper. 2000.
39. The American Society of Mechanical Engineers. Gas Transmission and
Distribution Piping Syste m. U.S.A.: American National Standard ASME
B31.8. 1990
40. Greg, V. Object Oriented Programming. Berkeley, California: McGraw Hill.
1991
41. Andrus, S. J. Steady State Simulation of Pipeline Network Using a
Spreadsheet Model. Journal of Pipeline Simulation Interest Group, 1994.
94(1): 1-13.
42. Vetterling, W.T., Teukolsky, S.A., Press , W.H., Flannery, B.P. Numerical
Recipes Example Book (C++). Second Edition. USA: Cambridge
University Press. 2002
43. Perry, G. and Spencer, I. Visual C++ in 12 Easy Lessons. Indianapolis:
Sams Publishing. 1995
44. Deitel, H. M. and Deitel, P. J. C++ How to Program. New Jersey, USA:
Prentice Hall. 1997
45. Gottfried, B.S. Schaum’s Outline of Theory and Problems of Programming
with C. Second Edition. Singapore: McGraw -Hill. 1996
46. Jones, R. M. Introduction to MFC Programming With Visual C++. New
Jersey, USA: Prentice Hall. 2000
47. Zulkefli Yaacob. Modelling and Simulation of Transient Gas Flow. PhD.
Thesis. University of Salford; 1996