WIRELESSLY CHARGED NETWORKS A Thesis Project Submitted …
Transcript of WIRELESSLY CHARGED NETWORKS A Thesis Project Submitted …
WIRELESSLY CHARGED NETWORKS
A Thesis Project
Submitted to the Department of
Systems and Computing Engineering
at Los Andes University
in Partial Fulfillment of the Requirements
for the Degree of
Systems and Computing Engineer
Adviser: Yezid Enrique Donoso Meisel, PhD
Andres Gomez
July 2010
<acknowledgements>
<h e i l i g e n g e i s t >
Fur o r c h e s t r i e r e n der Veranstaltungen , d i e a l l d i e s
moglich gemacht haben . Vie l en Dank .
</ h e i l i g e n g e i s t >
<fami ly>
I am e t e r n a l l y g r a t e f u l f o r the support , encouragement ,
and the words o f wisdom , even i f I sometimes f a i l e d
to l i s t e n . Thank You .
</family>
<ydonoso>
You gave me the freedom to sugges t my own work , and
gave me the time and space I needed to complete i t .
I am we l l aware not many p r o f e s s o r s do that . Thank
You .
</ydonoso>
<MC3>
Tu es , sans doute , l e p lus beau dans ma v i e .
</MC3>
<pala>
Ce t r a v a i l n ’ a u r a i t pas l a q u a l i t e qu ’ i l a sans vos
commentaires , c o n s e i l s , n i l ’ a ide en gene r a l . Merci .
Beaucoup !
</pala>
<los demas>
So l o por que no os he mencionado por nombre , no qu i e r e
d e c i r que no se a i s importante para mı . Lo s o i s . Mi
vida en Los Andes no hubiera s ido tan agradable n i
memorable s i no hubiera s ido por vo so t ro s . Grac ias !
</los demas>
</acknowledgements>
iv
Preface
In the fall semester 2009, I was a teacher’s assistant in Computer Science 101. My
“complementary class” (as it’s called in Los Andes) had about 10 students, most of
them eager to learn about their mayor. Me being in my senior year, I was able to
give them a sneak-peak into what their academic lives would be like for the following
4 years.
One of the main lessons I tried to impart on them was to research, to go beyond
what they were taught in class and try to find how that knowledge is being applied in
the world. I asked my students to search for an article that was related to computer
science in any way and make a small presentation about it, the condition being that it
came from a serious academic journal. To my surprise some of them went beyond what
I asked, and ventured into other fields, ranging from physics to electrical engineering.
One of the presentations was about a little group at MIT that was working on
wireless energy transfer, which they called WiTricity. I was absolutely intrigued
by the idea of “wireless electricity”. After reading every publication mentioning
“wireless electricity” I could get my hands on, the idea of wirelessly charging cell
phones (WiTricity’s genesis) expanded in my head. Why not any electronic device?
Could one wireless link be turned into a network of links, enabling all connected
devices to share their stored energy? This document is how I see those networks
developing in the future. Hopefully, we’ll get there someday soon.
v
Contents
Preface v
0 Abstract 1
1 Introduction 2
1.1 A Brief History of Electricity . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 How this Document is Organized . . . . . . . . . . . . . . . . . . . . 7
2 General Description 8
2.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Background Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Problem Identification and its Importance . . . . . . . . . . . . . . . 11
3 Design and Specification 12
3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Design Development 15
4.1 Information Gathering . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Design Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2.1 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2.2 Proposed Algorithms . . . . . . . . . . . . . . . . . . . . . . . 20
vi
5 Implementation 26
5.1 Implementation Description . . . . . . . . . . . . . . . . . . . . . . . 26
5.1.1 Implemented Algorithms . . . . . . . . . . . . . . . . . . . . . 26
6 Validation 31
6.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2 Result Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2.1 Single Hop WCNs . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2.2 Multi Hop WCNs . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7 Conclusions 41
7.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Bibliography 43
A List of Abbreviations 45
vii
List of Tables
4.1 Summary of Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Restriction of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.1 Simulation Results for Single-Hop WCNs . . . . . . . . . . . . . . . . 33
6.2 Simulation Results for a WCN Queue . . . . . . . . . . . . . . . . . . 34
6.3 Simulation Results for a WCN Binary Tree . . . . . . . . . . . . . . . 36
6.4 Simulation Results for a WCN Grid Tree . . . . . . . . . . . . . . . . 38
6.5 Algorithm Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 40
viii
List of Figures
1.1 Tesla’s schematic representation of his patent No. U.S 0,645,576 . . . 6
2.1 Schematic representation of the resonant coupling between two coils. [9] 10
4.1 (a) Star Topology. (b) Class diagram for a WCN node. . . . . . . . . 16
4.2 A multi-hop WCN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 A WCN Queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4 (a) Balanced Network (b) Unbalanced Network. . . . . . . . . . . . . 19
4.5 Block Diagram for Simulation of Proposed Algorithms. . . . . . . . . 20
4.6 LCA for an Unbalanced WCN (∆Q = 1) . . . . . . . . . . . . . . . . 22
4.7 WCN Queue Charge Algorithm(∆Q = 1.) . . . . . . . . . . . . . . . 23
4.8 Complete Binary Tree (∆Q = 1.) . . . . . . . . . . . . . . . . . . . . 24
4.9 3× 2 Grid Tree RR Algorithm (∆Q = 1.) . . . . . . . . . . . . . . . 25
6.1 Simulation Results for a single-hop WCN . . . . . . . . . . . . . . . . 32
6.2 Complexity of a single-hop WCN . . . . . . . . . . . . . . . . . . . . 32
6.3 Simulation Results for a WCN Queue. . . . . . . . . . . . . . . . . . 34
6.4 WCN Queue Complexity . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.5 ITC Simulation Results for Binary Trees . . . . . . . . . . . . . . . . 35
6.6 MITC Simulation Results for Binary Trees . . . . . . . . . . . . . . . 35
6.7 Binary Tree Complexity . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.8 ITC Simulation Results for Grid Trees . . . . . . . . . . . . . . . . . 37
6.9 MITC Simulation Results for Grid Trees . . . . . . . . . . . . . . . . 37
6.10 Grid Tree Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 38
ix
6.11 Simulation Results for Unbalanced Structures . . . . . . . . . . . . . 39
6.12 Unbalanced Grid Tree Complexity . . . . . . . . . . . . . . . . . . . . 39
x
Chapter 0
Abstract
This document is the result of a full semester of research, laying the foundations for
the concept of Wirelessly Charged Networks (WCNs). A WCN can be briefly defined
as a network of electronic devices that can store energy and wirelessly exchange
energy between one another. After covering the original paradigms of electrical energy
generation, transmission and use, WCN emerges as a plausible solution to small scale
wireless energy transmission networks. To this end, several transmission algorithms
are presented in different topologies. An abstract model was created in order to
represent these networks, without limiting the technology needed to implement it.
The main focus of this work is to find an algorithm that would minimize the time to
charge all the nodes belonging to a network, depending on the topology. Additionally,
an optimization scheme is presented for different algorithms based on two criteria:
(1) the time to charge for the entire network and (2) the mean time to charge of
individual nodes.
1
Chapter 1
Introduction
1.1 A Brief History of Electricity
Over the past few centuries, few discoveries have had as great an impact as the
discovery of electricity. None of our modern marvels would have been possible without
the use of electrical energy.
Origins
Electricity and Magnetism are natural forces that first intrigued humans thousands
of years ago. The ancient Chinese first learned how to construct magnetic compasses
and induce magnetism in iron. Around 600 B.C. a Greek philosopher named Thales
discovered that rubbing amber with a piece of cloth gave it the power to attract small
bits of wood, feathers, leaves, and other light objects.
Scientific Revolution
It was not until the 1600’s, however, that electricity and magnetism became the
object of a more scientific study. William Gilbert was one of the first physicians to
study electricity and magnetism with his De Magnete, Magneticisque Corporibus, et
de Magno Magnete Tellure (On the Magnet and Magnetic Bodies, and on the Great
Magnet the Earth). In the 1700’s, Benjamin Franklin famously proved the electrical
2
CHAPTER 1. INTRODUCTION 3
nature of lightning by attaching a metal key to a kite during a thunderstorm.
Late in the 18th century, the Italian Luigi Galvani published his theory of animal
electricity, De viribus electricitatis in motu mosculari commentarius (Commentary
on the Effect of Electricity on Muscular Motion) after he noticed that a frog’s legs
moved when its nerves touched an electrically charged scalpel. Galvani’s findings
attracted the attention of another famous Italian physicist: Alessandro Volta. After
experimenting further with frog legs, he would eventually discover a relationship be-
tween different chemical reactions and a generated electromotive force. His discoveries
eventually led to the first battery.
Volta’s battery proved to be more reliable than previous sources of electrical en-
ergy, allowing further research into electric phenomena. Great scientists such as Carl
Friedrich Gauss, Andre-Marie Ampere and Hans Christian Ørsted, among many oth-
ers, experimented extensively with electricity in the early 19th century.
Early Uses
“Why sir, there is a possibility that you will soon be able to tax it!” 1
Michael Faraday
Though some of the fundamental laws of electricity, it could not be yet be put
into practical use. By the 1800’s, it was well documented that electricity, somehow,
produced magnetism. In 1831, Michael Faraday discovered that magnetism could
also produce electricity. His law of induction formed the basis for most of the elec-
tric generating equipment used to this day. By the mid 1850’s, electricity was just
beginning to be used in numerous applications. Samuel Morse developed the electro-
magnetic telegraph, spawning a new field in long distance communication. Joseph
Henry constructed an electro-magnetic motor, leading to applications of electricity
for transportation purposes.
1Reply to then Britain’s Minister of Finance William Gladstone’s questioning about the usefulnessof electricity.
CHAPTER 1. INTRODUCTION 4
Electric Lighting Revolution
“I shall make electricity so cheap that only the rich can afford to burn candles”
Thomas Alba Edison
In 1808, British chemist Sir Humphrey Davy demonstrated early forms of both arc
and incandescent lighting. Incandescent lighting works by passing electrical current
through a piece of metal wire, heating it to a point that causes a white hot glow.
Though Davy’s inventions soon proved to be impractical on a large scale, they pointed
the way to the coming revolution in electrical use.
Thomas Edison invented the incandescent electric light bulb in 1879, after years
of failed prototypes. Edison realized that a practical electric lighting system would be
a significant technological advance for the world, and he had the vision to propose a
complete electrical system from generation of electrical energy to its distribution. His
proposal, based on Direct Current (DC) technology was only suitable for small scale
electrical networks. When extended beyond a mile, the system became inefficient. For
this reason, Edison’s technologies could only work with distributed energy generation,
which proved too costly at the time to be practical.
Electrical Generation
In the late 1880’s, another proposal for efficient transmission of electricity emerged.
Nikola Tesla developed the theoretical background to build a complete polyphase gen-
eration and transmission system based on Alternating Current (AC). The competition
between the two technologies became famously known as the “war of the currents”
and had such infamous moments as Edison’s electrocution of an elephant using AC
to try to scare people into thinking that it was dangerous. Unfortunately for Edison,
Tesla’s AC system was superior in both generation and transmission efficiency.
Having the technology to generate large amounts of energy, and to safely and
efficiently transmit it to urban centers, allowed the centralization of power plants.
This way, it could benefit from the economies of scale by powering many big and
small town with just one power plant. Centralized generation became the norm for
the electric power business for decades. In recent years, however, the consciousness
CHAPTER 1. INTRODUCTION 5
of finite and limited sources of energy on earth and internal disputes over the envi-
ronment, global safety has shifted back the discussion for distributed generation [7].
There is now a trend of generating power locally at distribution voltage level by using
non-conventional /renewable energy source like wind power, solar photovoltaic cells,
fuel cells, among others [2].
Wireless Freedom
“Electrical energy in great amounts can be efficiently and safely transmitted without
the use of wires to any point of the globe, however distant.”
Nikola Tesla (1899)
Around a century ago, before the electrical grid was installed. Nikola Tesla was
approved three patents related to the transmission of electrical power through what he
called the “Natural Medium”. The first patent was entitled: “System of Transmission
of Electrical Energy”. A diagram is presented in figure 1.1. In this figure, the points
D and D′ are put at a very high potential difference and at a considerable height.
The idea was to ionize the air between those two points in order to make it behave
as a conductor. Quoting Tesla himself:
The method hereinbefore described of transmitting electrical energy
through the natural media, which consist in producing between the earth
and a generator terminal elevated above the same, at a generating-station,
a sufficiently high electromotive force to render the elevated air strata
conducting, causing thereby a propagation or flow of electrical energy, by
conduction through the air strata. . . [10]
Tesla’s idea would prove to be both inefficient and unsafe − two major issues
would persist in pretty much all forthcoming attempts to solve the same problem.
The efficiency problem came from the use of high frequency currents that invariably
radiated potent electric and magnetic fields radially outward from the station, wasting
great amounts of energy. The safety concern was due not only to the bold plan of
ionizing the air above everybody’s head, but because the same electric field that were
CHAPTER 1. INTRODUCTION 6
Figure 1.1: Tesla’s schematic representation of his patent No. U.S 0,645,576
the cause of the first concern, would surely interact with any electric sensitive object,
including people and thus, would constitute a hazard for health.
The second patent approved to Tesla was called “Apparatus for Transmission of
Electrical Energy” and followed the same lines of the one just described. The third
patent was different. It was called “Art of Transmitting Electrical Energy through
the Natural Medium”. According to this work, the earth responds to electrical dis-
turbances, in the same manner as a conductor of limited size would; and so, it is
possible to establish stationary electrical waves through it. Tesla based his affir-
mation on his observations on the effects of lightning discharges on the electrical
condition of the earth. He noted that, at certain places, sensitive electrical instru-
ments failed to respond to the electrical disturbances produced by the lightning,
even when programmed to do so. He attributed the irregularity to the formation
of nodal, electrical waves on the earth. If the instrument is located in the node,
Tesla reasoned, it won’t feel the disturbance. So, he proposed to populate the globe
with wisely located transmitters, that would generate standing waves throughout the
CHAPTER 1. INTRODUCTION 7
planet, from which either power or information would be retrieved with accordingly
designed receptors[11]. Once again, Tesla’s efforts to establish a world-wide wireless
electrical grid were hampered by efficiency and safety.
In more recent years, there have been several attempts to get around this prob-
lem. Most of them focused in the close-range 2of distances. Most of them also rely
on non-radiating “evanescent” fields, thus improving the efficiency but sacrificing the
mobility. Other approaches involve either focused lasers or highly directional anten-
nas that provide reasonable efficiency but require an uninterrupted line of transmis-
sion, which, is usually unrealistic for long distance, urban areas. Even so, the rapid
massification of a myriad of portable devices (laptops, digital cameras, cell phones,
etc) makes the subject worth studying. Additionally, the nature of the problem has
changed since Tesla’s times. Today, we already have the electrical grid that efficiently
transmits electricity virtually everywhere. Low to mid-range “apparatuses” are being
developed to complement this existing grid and bring it to the wireless era.
1.2 How this Document is Organized
This document is divided into seven chapters. Chapter One briefly recounts a history
of electrical energy. Chapter Two describes the objectives for this document, as well
as some background work on the subject. Witricity is presented as a novel means for
wireless energy transmission, and the concept expanded to a network. Chapter Three
defines the problem that will be the focus of this document, along with its context
and restrictions. Chapter Four describes the model presented for WCNs, along with
several topologies and the respective algorithms. Chapter Five explains in detail the
algorithms implemented, along with their expected results. Chapter Six analyzes the
obtained results. Chapter Seven discusses the conclusions, and describes future work.
2RFID is one such example of close range devices, limited to a range of about 10 cm
Chapter 2
General Description
2.1 Objectives
For more than a hundred years, a great deal of research has gone into finding the best
way to transmit large amounts of electrical energy as far and as efficiently as possible.
Many advances have been made since the early days of Edison vs. Tesla. In recent
years, novel ways of wirelessly transmitting electrical energy have emerged. When this
technology becomes as efficient as our current power networks, it will open the door to
new and exciting applications. One of them will be networks of electronic devices that
will be able to freely and wirelessly obtain energy. This document attempts to lay
the theoretical foundations for those Wirelessly Charged Networks (WCN). Making
an analogy to current packet-based data networks, several energy routing algorithms
are proposed. The efficiency of these algorithms depends on several factors. An
optimization of these parameters is studied, and its results are presented.
2.2 Background Work
Wireless Electricity
Wireless Electricity (WE) is the process by which electrical energy is transmitted
from one device to another without the use of cables between the devices.
8
CHAPTER 2. GENERAL DESCRIPTION 9
As in any scientific endeavor, the development of WE has been a flow of con-
tinuous little steps filling the gaps between major breakthroughs. Usually, those
breakthroughs are the result of challenging long time held assumptions, followed by
an awakening to so far unknown possibilities.
In the case of WE, there have been two major breakthroughs, separated by a
one-century gap. The first being the suggestion of the idea itself, made by Nikola
Tesla in the early days of electricity; the second one being the work done by a group
of scientists at MIT using resonant “evanescent” fields to solve the problem.
The second breakthrough came when instead of the previously mentioned ap-
proaches by Tesla, an MIT team introduced the well known principle of resonant
coupling, which is the fact that two same-frequency resonant objects tend to couple
while interacting weakly with other off-resonant objects. In general terms the new
method consist of the following theoretical components [4]:
• Coupling of the objects through non-radiating evanescent fields. This solves
the problem that Tesla had of energy flowing outward from the source in all
directions. It also creates a “resonant tunnel” which targets the receiver inde-
pendently of the geometry of the surroundings.
• Coupling through the magnetic fields. This is ultra-important for safety reasons
because most materials present in typical urban environments are much more
sensitive to electric than to magnetic fields. Those material include organic
tissues such as those in the human body.
• Tuning of the parameters in order to make the coupling “strong”. This is
thought as a measure to maximize efficiency.
Under the above conditions, for mid-range situations the source behavior is similar
to such of an omnidirectional source that targets the receiver. Also these principles
can be applied in any physical system presenting resonance, either electromagnetic,
acoustic, nuclear or any other. Figure 2.1 shows a schematic representation of an
already implemented model involving resonant coils [6]. The model was implemented
as a master thesis project by Andre Kurs, a student of the same MIT group that
established the theoretical framework above expounded.
CHAPTER 2. GENERAL DESCRIPTION 10
Figure 2.1: Schematic representation of the resonant coupling between two coils. [9]
After the publication of Kurs’ paper, the field of wireless electricity has blossomed.
Many applications are appearing by the day. The most notable are those in which
energy has to be fed to robots of any size that have to go into difficult access ar-
eas. A typical example includes the nano and micro-robots, developed nowadays for
biomedical applications[13].
Wireless Data Networks
Wireless data networks can be classified according to two criteria: (i) whether a
packet in the wireless network crosses exactly one wireless hop or multiple wireless
hops, and (ii) whether there is infrastructure such as a base station in the network [5].
Wireless Sensor Networks (WSN) fall into the category of multi-hop, infrastructure-
based networks.
A WSN consists of sensor nodes deployed over a geographical area for monitoring
physical phenomena like temperature, humidity, seismic events, etc [1]. Due to their
usually remote locations, access to an electrical grid is not available. For this reason,
WSN nodes usually use batteries as their main power source, limiting their lifetime
to the battery’s energy capacity. Much research has gone into extending the lifetime
of WSN nodes. One way of achieving this is through software. This technique is
called energy conservation. The three main types of energy conservation are: (1)
Duty Cycling, (2) Data-driven, and (3) Mobility Based, explained with detail in [1].
CHAPTER 2. GENERAL DESCRIPTION 11
The other method of extending a WSN lifetime is though hardware. For starters,
a battery with a high energy density will automatically extend a WSN node’s lifetime,
as shown in [3]. Another, more sophisticated approach is called energy scavenging
(or harvesting), where a network node can recharge its battery from the ambient
by using vibrational energy, thermal energy, light, or electro-magnetic waves [12].
By constantly recharging its battery, a WSN node can extend its lifetime almost
indefinitely, without considering technical problems.
Wirelessly charging networks is now proposed as another hardware-based solution
to extending the lifetime of network nodes. Using wireless electricity as the equivalent
of a “physical-layer” for energy transmission, enables us to attack the problem anal-
ogous to the well-tested Open System Interconnection (OSI) network model. This
document can be used as a basis for developing Internetwork protocols for energy
transmission.
2.3 Problem Identification and its Importance
The first problem to solve for a WCN is how to efficiently “route” energy to charge
all network nodes. After having defined a set of nodes for a WCN, the best topology
and algorithm must be employed to minimize the time to charge for the WCN. This
time to charge can actually be two different criteria: 1) The time to charge every
node belonging to the WCN, and 2) The mean time to charge of each node belonging
to the network. This problem is fundamental for a WCN. The most probable case
for an initial setup of a WCN is having all nodes initially discharged. The times
to charge (using either of the criteria previously defined) must then be minimized.
This is especially important if WCNs are to be implemented in the renewable energy
sources (i.e. photovoltaic cells) scenario where the power source can be limited to
several hours per day.
Chapter 3
Design and Specification
3.1 Problem Definition
There are two types of nodes in a WCN: 1) Nodes that only give power, and 2)
Nodes that only consume power. The former are called P nodes, and the latter U
(user) nodes. Initially, U nodes are completely discharged and they only store power,
since it is the best-case scenario to minimize the time to charge. In order to avoid
the electro-chemical complexities of present-day storage elements (i.e. batteries), the
only information stored in each node is it’s state of charge (SOC). Additionally, the
amount of energy transfered between nodes per unit time is represented in terms of
the recipient’s SOC. This parameter will be called ∆Q from now on. Borrowing from
wireless data network terminology, WCN can be classified into the following:
1. Single-hop, infrastructure-based
2. Multi-hop, infrastructure-based
In wireless data networks, the term “infrastructure” means having base stations
with fast and reliable links to eternal networks. Since we are making an analogy
between data and energy, this “base station” with readily available energy is the P
node in the WCN. In a single-hop network, one P node can directly charge all U
nodes within the network. Alternately, in a multi-hop network, directly connected
12
CHAPTER 3. DESIGN AND SPECIFICATION 13
U nodes serve as intermediaries to other U nodes who are not. These intermediary
U nodes will only start sharing their energy after their SOC has reached a threshold
value. This parameter will be called ϕ from now on.
The objective is to minimize the charge time for both single-hop and multi-hop
WCNs. As previously mentioned, two different charge times can be used as criterion
for comparison. Say we stored all the times it took to charge each individual node
belonging to a WCN. The first time criterion is then defined as the mean of these
times. In plain terms, this is the average time it took for a U node to charge. The
second criterion is defined as the maximum of the times. This is the longest time it
took an individual U node to charge, which can also be interpreted as the time it
took to charge the entire network.
3.2 Specifications
The charging problem for WCNs can be modeled using several different data-structures.
Choosing the appropriate data-structure is crucial to represent the optimum topology
of a WCN. All of the proposed algorithms must then be implemented in the chosen
technology in order to simulate its efficiency. In order to compare the efficiencies of
different algorithms more easily, it was decided to use the number of iterations since
it is equally comparable to its runtime.
To show this, let’s say we have a simple network consisting of one P node and one
U node. If ∆Q = 1, it would mean that in just one iteration, the U node would be
completely charged. Since ∆Q is the percentage of SOC received per unit time, this
would mean that the network was charged in one unit time. On the other hand, if
∆Q = 0.1, it would take 10 iterations to fully charge the U node. As ∆Q is per unit
time, it would take 10 units of time to charge the U node.
We now redefine our efficiency criteria as the following: (1) Mean Iterations To
Charge (MITC) and, (2) Iterations To Charge (ITC). The MITC is the average num-
ber of iterations it took to charge all of the nodes belonging to a network. The ITC
is the number of iterations it took to charge the entire network.
Different algorithms will be ranked by their efficiency using the same number of U
CHAPTER 3. DESIGN AND SPECIFICATION 14
nodes. The two parameters (∆Q and ϕ) will be varied in order to find their optimum
operating point. In all cases, there will only be one central P node to charge the
entire network.
3.3 Restrictions
In order to simplify the simulation model, several simplifications were made. For
starters, using ∆Q as the transfer unit per unit time, it inherently implies that all
U nodes have the same total capacity, such that the percentage of SOC transfered
is equal in all cases. Additionally, the transmission efficiency(
PowerReceivedPowerTransmitted
)is
unitary for every case. The number itself is not important, what is important is that
it is constant for all nodes. Though this efficiency is in reality related to the distance
between nodes, this initial model for WCN assumes all nodes as equidistant.
A very important restriction imposed on every WCN node is that they can only
transmit to one other node at the same time. Additionally, it cannot both transmit
and receive energy at the same time. Though this would eventually depend on the
technology used to implement WCN, this is a fairly natural restriction present in
wireless data networks. This restriction is quite clear when using directional antennas,
but it is also present in omni-directional ones.
A unitary ∆Q would means that a node would be able to fully charge another in
one unit of time. If the unit of time is sufficiently small, it would imply having the
capacity to both transmit and receive large amounts of energy. Though this might
not be physically realistic, it is assumed possible.
Chapter 4
Design Development
4.1 Information Gathering
In order to lay the foundations of the WCN concept, large amounts of time were dedi-
cated to researching the origins of electrical energy generation and distribution. Since
the concept of WCN emerged for relatively small-scale networks, it was analogous to
present-day microgrids. The nodes of a WCN are specifically tied to anything, except
that they were battery based electronic devices. There are currently many differ-
ent charging technologies and battery management systems that had to be studied
so they could be included within WCN. In order to create an accurate simulation
model, several different data-structures were used and implemented in the technology
of choice. Due to its computational prowess, Matlab c© was chosen to simulate all of
the proposed algorithms.
4.2 Design Alternatives
4.2.1 Topologies
So far, the concept of WCN has not been tied to any topology. Before proposing
any algorithms, the different topologies must first be defined. As with wireless data
networks, WCNs can be classified into two main group: (1) Single-Hop and, (2)
15
CHAPTER 4. DESIGN DEVELOPMENT 16
Multi-Hop. They will now be along with their corresponding topologies.
Single-Hop WCNs
In a single-hop network, every U node (green) is directly connected to a P node
(blue). Since we work with just one P node, single-hop WCNs are actually centralized
networks with a star topology, as shown in figure 4.1(a). Since the flow of energy will
always be from the inside out, the most natural data-structure to use for this type of
networks is a one-level N-ary tree. Figure 4.1(b) shows the class diagram for a WCN
node. The power node will always have id = 1. The variable sons is an array of the
sons’ id’s. The variable lock will be used later on for multi-hop WCNs. When lock is
true, it means that the node’s sons are completely charged trees, meaning it can only
store, not transmit, the energy it receives.
(a) (b)
Figure 4.1: (a) Star Topology. (b) Class diagram for a WCN node.
Multi-Hop WCNs
A multi-hop network can be thought of as a hybrid network consisting of a centralized
power node and decentralized user nodes, as shown in figure 4.2. Besides those nodes
directly linked to the P nodes, U nodes must share energy between themselves. It
is important to note that energy transfer will only occur in one direction: from the
CHAPTER 4. DESIGN DEVELOPMENT 17
father to the son. For this reason, it is once again convenient to use a N-ary Tree as
our data-structure.
Figure 4.2: A multi-hop WCN.
Queues
The simplest multi-hop WCN can be seen as a simple array with the power node at
its head and more than one user node following, as shown in figure 4.3. Due to the
transmission restriction imposed on all WCN nodes, aligning them in a queue is not
a very good idea. Whenever the first user node starts transmitting to the second one,
the power node is left in an idle state. This idle state is “wasting” precious energy
that could be used to charge other nodes. For this reason, it is necessary to have at
least two user nodes directly connected to the power node, such that when the first
user node is transmitting, the second one get charged.
Figure 4.3: A WCN Queue
Charging WCN Queues is a fairly simple task. At most, any single node will only
be able to transmit to one other node. With our previously defined criteria (ITC and
MITC), there will be no difference between proposed algorithms, since transmitter
nodes cannot choose different receiver nodes. The ITC and MITC will only depend on
CHAPTER 4. DESIGN DEVELOPMENT 18
the ∆Q, ϕ variables and, naturally, the number of nodes. Figure 4.7 shows a graphical
representation of the Charge Algorithm for WCN Queues with ∆Q = ϕ = 1.
Balanced Trees
The weight of a tree is defined to be the number of user nodes it has as descendants.
A tree is said to be balanced when all of it sons have the same weight and are balanced
themselves. A balanced ternary tree is shown in figure 4.4(a). Balanced multi-hop
WCNs can be modeled by various data-structures. For the purposes of this document,
only two will be used: (1) Complete Binary Trees, and (2) Grid Tree.
Complete Binary Trees
A complete binary tree is a tree where non-leafs have two sons, and leafs are only
found in the deepest level. This data-structure was chosen because it minimized the
number of nodes per level. Though this might sound attractive, it was actually chosen
for comparison because it is the worst-case scenario for multi-hop WCNs. Intuitively,
the best topology is the one that places the most number of nodes as close as possible
to the root (P node). In a complete N-ary tree, the level farthest from the P nodes
has the most number of U nodes, binary trees give us the least number of nodes
possible per level. Figure 4.8 shows a sample charge algorithm for binary trees with
∆Q = ϕ = 1.
Grid Tree
A grid tree is now defined as a P node with N branches of M U nodes in a queue
structure. Figure 4.4(a) can then be identified to be a grid tree with N = 3 and
M = 2. To minimize the ITCs for networks with 2K nodes, we simply set N = K
and M = 2. This will be the best-case topology for a network meeting the definition
of balanced multi-hop WCNs. Figure 4.9 shows a sample algorithm for grid trees with
∆Q = ϕ = 1.
CHAPTER 4. DESIGN DEVELOPMENT 19
Unbalanced Trees
A tree is said to be unbalanced when its sons have different weights, or are not
balanced trees. An unbalanced tree can be seen in figure 4.4(b). When dealing with
this topology, a problem arises for the proposed Least Charged Algorithm (LCA).
Depending on the ∆Q, it is not insured that the P will always be able to transmit
on every iteration. Since LCA tries to charge by level, there will come a point where
only one branch is left (the deepest one) and the topology then becomes a simple
WCN Queue.
The other proposed algorithm, Most Charged Algorithm (MCA), also has this
problem, but the P node’s “idle time” can be minimized by first ordering all the
branches by weight. If it begins by charging the deepest branch first, it will minimize
the length of the resulting queue, thus minimizing the “idle time”.
The best solution to this problem comes when the P node spends time on each
branch proportional to its weight. If, for example, one branch has twice as many nodes
as the other, the P node needs to spend 23
of the time on the bigger branch, and 13
on the other. This algorithm’s use of the P node still depends on the topology. Since
unbalanced structures have a very broad definition, and can include many different
specific topologies, only trees with two branches were simulated.
(a) (b)
Figure 4.4: (a) Balanced Network (b) Unbalanced Network.
CHAPTER 4. DESIGN DEVELOPMENT 20
Table 4.1: Summary of Topologies
Single HopMulti-Hop
Balanced Unbalanced
Topology StarQueue
Grid TreeBinary TreeGrid Tree
Summary
Table 4.1 shows a summary of the proposed topologies for WCNs.
4.2.2 Proposed Algorithms
Having redefined our criteria for algorithm efficiency to be ITC and MITC, we must
propose iterative algorithms so the number of iterations can be effectively compared.
ITC will be used as the first criterion, using MITC only when ITC cannot distinguish
between two. In general terms, for each iteration a list of capable nodes is generated.
We now define a capable node as an unlocked node whose SOC is greater than or
equal to the threshold value ϕ. What varies from one algorithm to another is the
receiver each capable node chooses to transmit to. The validity of the algorithms
depend on the topologies of the WCN. To this end, three different algorithms will be
presented.
Figure 4.5: Block Diagram for Simulation of Proposed Algorithms.
CHAPTER 4. DESIGN DEVELOPMENT 21
Variable Restriction∆Q (0,1]ϕ [0,1]
Table 4.2: Restriction of Variables
Least Charged Algorithm (LCA)
For both single-hop and balanced multi-hop WCN, the first algorithm that comes to
mind a is simple round robin between all sons. For each iteration, all capable nodes
choose the son that is least charged. When all sons have the same SOC, they are
chosen by order.
LCA attempts to distribute as uniformly as possible between all sons. When
used in a tree structure, it charges the tree by levels and behaves as a Round Robin
algorithm. For unbalanced multi-hop WCNs, though, a problem arises when using
this algorithm. LCA is inefficient because the topology does not insure that the P
node will be used continuously. To illustrate this, see a sample run of LCA in figure
4.6. Since LCA tries to charge by uneven levels, there will be an iteration where the
P node will not be used, as shown in 4.6(d).
Most Charged Algorithm (MCA)
The Most Charged Algorithm takes the opposite approach as LCA. It tries to stay
with one node (or branch) until it is completely charged and only then will it move
to the next one. When used in a N-ary tree structure, the tree is charged by depth.
Intuitively, it is expected that MCA will generally lower the MITC for all networks
since, for low ∆Q’s, MCA tries to concentrate the charge on one node, while LCA
tries to spread it out evenly.
Examples
The next four pages show a graphical representation for LSA algorithms in three
different topologies (Queue, Complete Binary Tree, Grid Tree).
CHAPTER 4. DESIGN DEVELOPMENT 22
(a) (b) (c)
(d) (e) (f)
Figure 4.6: LCA for an Unbalanced WCN (∆Q = 1)
CHAPTER 4. DESIGN DEVELOPMENT 23
Figure 4.7: WCN Queue Charge Algorithm(∆Q = 1.)
(a) (b) (c)
(d) (e) (f) (g)
CHAPTER 4. DESIGN DEVELOPMENT 24
Figure 4.8: Complete Binary Tree (∆Q = 1.)
(a) (b)
(c) (d)
(e) (f)
(g) (h)
CHAPTER 4. DESIGN DEVELOPMENT 25
Figure 4.9: 3× 2 Grid Tree RR Algorithm (∆Q = 1.)
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Chapter 5
Implementation
5.1 Implementation Description
As previously mentioned, Matlab c© was chosen as the technology to implement the
proposed WCN model and algorithms. For each topology, the applicable algorithms
were implemented in a functional manner. Each function was then mapped by a
simulation algorithm, shown in 5.1.1. For ease of presentation, most parameters were
omitted from function calls with the proposed algorithms. Additionally, the code
presented is a reduced version of the original, which included more code for debugging
purposes. With little effort, the presented code can be completed and executed to
verify results1.
5.1.1 Implemented Algorithms
Generic Algorithms
Since all proposed algorithms charge various WCN topologies, all implementations
had shared components. The only difference from one algorithm to another is what
receiver node each capable node chooses to transmit to. For this reason, a generic
scheme for all proposed algorithms is now presented.
1Since Matlab passes parameters by value, every function that modifies a parameter variable mustreturn the variable and the calling function must then reassign the variable’s value. Most of theserequired reassignments were also omitted for ease of presentation.
26
CHAPTER 5. IMPLEMENTATION 27
Generic Charging Algorithm
1 function [ ITC MITC] = Algorithm (N, deltaQ , phi )
2 n o d e l i s t = Generate Topology (N) ;
3 i t e r a t i o n s = ones (N, 1 ) ;
4 while a l l c h a r g e d ( ) ˜= 1
5 c nodes = capable nodes ( ) ;
6 a l l n o d e s = charge ( c nodes ) ;
7 end
8 ITC = max( i t e r a t i o n s ) ;
9 MITC = mean( i t e r a t i o n s ) ;
10 end
This is the main algorithm that returns the results for a specific algorithm/topol-
ogy pair. The Generate Topology function in line 2 loads the specific topology that
will be simulated. The algorithm then iterates until all the nodes are charged. The
function that tests this condition (all charged) automatically increments the iterations
vector for each node that is not yet charged. This vector is then used to calculate the
simulation results: ITC and MITC.
Generic capable nodes Function
1 function [ l i s t ] = capab le nodes ( nodes )
2 l i s t (1 ) = 1 ;
3 c = 2 ;
4 for i = 2 : length ( nodes )
5 i f nodes ( i ) . sons (1 ) = = 0
6 cont inue ;
7 end
8 i f nodes ( i ) .SOC >= phi && locked son s ( nodes , i ) = = 0
9 l i s t ( c ) = i ;
10 c = c + 1 ;
11 end
12 end
13 end
CHAPTER 5. IMPLEMENTATION 28
The capable nodes function generates a list of indexes of the nodes that can trans-
mit energy. These nodes are ones whose SOC ≥ ϕ, have sons, and whose sons are not
fully charged (this last condition is verified by the locked sons function). Additionally,
the P node is always marked as capable, in line 2.
Generic Charge Function
1 function a l l n o d e s = charge ( a l l n o d e s , capable nodes , deltaQ )
2 for i = 1 : length ( capab le nodes )
3 i f charged sons ( a l l n o d e s , capab le nodes ( i ) ) = = 1
4 a l l n o d e s ( capab le nodes ( i ) ) . lockNode ( ) ;
5 cont inue ;
6 end
7 a l l n o d e s ( capab le nodes ( i ) ) . l o ck =0;
8 des t = s p e c i f i c a l g o r i t h m ( ) ;
9 t r a n s f e r ( a l l n o d e s , capab le nodes ( i ) , dest , deltaQ ) ;
10 end
11 end
The charge function iterates through all capable nodes and for each one, it de-
termines its receiver node and transfers one ∆Q from the transmitter’s SOC to the
receiver’s. It is in this function where capable nodes whose sons are all charged get
locked. More importantly, this function is where each of the proposed algorithms is
called, in line 8.
Specific Algorithms
The first proposed algorithm is the LCA. It receives the list of nodes, and iterates
through the list of sons to find the one with the lowest SOC. It is important to
note that this algorithm becomes a Round Robin (R.R.) when used in a binary tree
structure, with other structures is is not insured. To implement a simple R.R. a
special case for the P node is added, and after taking the iteration count’s modulus,
the index for the next son is found. For the topologies that required it, the R.R.
modification was made to obtain the simulation results.
CHAPTER 5. IMPLEMENTATION 29
Least Charged Algorithm
1 function k = l e a s t c h a r g e d ( a l l n o d e s , i )
2 sons = a l l n o d e s ( i ) . sons ;
3 mini = 1 . 0 ;
4 k = 0 ;
5 for j = 1 : length ( sons )
6 i f a l l n o d e s ( sons ( j ) ) .SOC < mini
7 min = a l l n o d e s ( sons ( j ) ) .SOC;
8 k = sons ( j ) ;
9 end
10 end
11 end
The second proposed algorithm, MCA, takes the opposite approach. For each
capable node, it selects the son that has the highest SOC. It was important not to
choose nodes that were already charged, otherwise it would try to continue charging
charged nodes.
Most Charged Algorithm
1 function k = most charged ( a l l n o d e s , i )
2 sons = a l l n o d e s ( i ) . sons ;
3 maxi = 0 ;
4 k = 0 ;
5 for j = 1 : length ( sons )
6 i f a l l n o d e s ( sons ( j ) ) .SOC >= maxi && a l l n o d e s ( sons ( j ) ) .SOC < 1 .0 2
7 maxi = a l l n o d e s ( sons ( j ) ) .SOC;
8 k = sons ( j ) ;
9 end
10 end
11 end
2The second conditional turned out to be trickier than it looks due to some strange Matlabapproximations. At one point, the SOC could be within 10−13 from 1.0 but the condition wouldstill be false. For this reason, an “epsilon” function was defined to make sure that the SOC waswithin 10−10 from 1.0.
CHAPTER 5. IMPLEMENTATION 30
Simulation Algorithm
The simulation algorithms starts by creating arrays for the two variables (∆Q and ϕ).
The function lspace was defined to create a vector starting from init and increments
by step until it reaches max. The simulation then iterates through both vectors and
inputs them into the each Algorithm for the selected topologies. In line 17, the
resulting ITC is stored in a result matrix. When the MITC was need, the MITC
variable was then stored in the result matrix. Using built-in Matlab functions surfc
and contour, the resulting simulations were graphed.
Simulation Algorithm
1 function s imulate ( )
2 deltaQ = l s p a c e ( q i n i t , q step , q max ) ;
3 phi = l s p a c e ( p h i i n i t , ph i s t ep , phi max ) ;
4 r e s u l t s = zeros ( length ( deltaQ ) , length ( phi ) ) ;
5 k1 = 1 ;
6 k2 = 1 ;
7 for i = 1 : length ( deltaQ )
8 k2=0;
9 for j = 1 : length ( phi )
10 [ ITC MITC] = Algorithm (N, i , j ) ;
11 r e s u l t s ( k1 , k2 ) = ITC ;
12 k2 = k2 + 1 ;
13 end
14 k1 = k1 + 1 ;
15 end
16 surfc ( phi , deltaQ , r e s u l t s ) ;
17 contour ( phi , deltaQ , r e s u l t s ) ;
18 end
Chapter 6
Validation
6.1 Methods
For each of the proposed topologies for WCNs, the applicable charging algorithms
were simulated using Matlab. The simulation algorithm enabled graphing the charge
function for the ∆Q and ϕ variables. Another additional algorithm was implemented
to measure the complexity of the algorithms in each topology. The results are now
presented.
6.2 Result Validation
6.2.1 Single Hop WCNs
As discussed earlier, a single-hop WCN is a centralized network. These networks have
a star topology, making its ITC vulnerable only to changes in ∆Q. Additionally, the
ITC is actually the same for all algorithms when fixing the number of nodes. This is
fairy natural since it is actually just a one-one relation between the P node and all
the rest. This can be expressed for N nodes by the following formula: ITC = N∆Q
1.
For this reason, simulations for the proposed algorithms are compared based only on
their MITC, which does change by algorithm, after setting the number of nodes to
1This formula shows that single-hop networks WCNs have a linear complexity
31
CHAPTER 6. VALIDATION 32
N=5. The simulation for ∆Q = ϕ = 1 is presented in figure 6.1. The results show
that ITC ∝ 1∆Q
. Additionally, it can be seen that MCA is more efficient than LCA
in terms of MITC.
(i) Simulation Results for LCA (j) Simulation Results for MCA
Figure 6.1: Simulation Results for a single-hop WCN
The complixity of both LCA and MCA were simulated for single-hop WCNs. The
results are shown in figure 6.2. It can be seen that MCA is a more efficient algorithm
in terms of MITC.
Figure 6.2: Complexity of a single-hop WCN
Summary
Table 6.1 shows numerical results for both algorithms in a single hop WCN, for both
of the proposed algorithms and different parameter values.
CHAPTER 6. VALIDATION 33
Algorithm Nodes ∆Q ITC MITCLCA 5 0.5 9 7.5MCA 5 0.5 9 6LCA 10 1.0 10 6MCA 10 1.0 10 6
Table 6.1: Simulation Results for Single-Hop WCNs
6.2.2 Multi Hop WCNs
Queues
As previously mentioned, in WCN Queues both the ITC and the MITC depend only
on two parameters: ∆Q and ϕ. Mathematically, the function can be expressed in
the following terms (using Matlab syntax): [ITC MITC] = Queue(N, ∆Q, ϕ). The
simulation results for a 5 node WCN Queue is presented in figure 6.3.
Figure 6.3(a) shows the 3d plot of the Queue function. As expected, the relation-
ship between ITC and ∆q is similar to the single-hop case. The difference is now that
the ϕ variable has relatively linear, and weaker, relationship with ITC. The optimal
case is clearly seen to be ∆Q = 1. Since in reality the value for ∆Q might actually
be low, a close-up of the function for low values was analyzed.
Figure 6.3(b) shows a 2D contour plot of the Queue function for the following
ranges: ∆Q ∈ [0.01, 0.15] and ϕ ∈ [0.01, 1]. Each color shows a region of equal ITCs,
the redder the color, the higher the ITC. It is interesting to note that the normal
tendency of increasing ITC for a fixed ∆Q value disrupts when reaching 1.
The runtime for the charge algorithm in WCN Queues was calculated by varying
the number of nodes, and graphing the ITC. Figure 6.4 shows the result with ∆Q =
ϕ = 1. It can be seen that there is a linear relationship between N and ITC.
For comparison purposes, table 6.2 shows numeric results for the WCN Queue
charge algorithm with several variable values.
CHAPTER 6. VALIDATION 34
(a) Graph of Charge Algortithm (b) Contour Plot for ∆Q < 0.15
Figure 6.3: Simulation Results for a WCN Queue.
Figure 6.4: WCN Queue Complexity
Nodes ∆Q ϕ ITC MITC5 0.5 0.5 15 11.55 1.0 1.0 8 510 0.5 0.5 35 23.8810 1.0 1.0 18 10
Table 6.2: Simulation Results for a WCN Queue
Binary Trees
The first criteria for analysis was previously chosen to be ITC. The simulation results
for both of the proposed algorithms are shown in figure 6.5. As predicted, LCA
does not change its ITC for variations in ϕ, shown in figure 6.5(a). MCA, on the
other hand, does increase its ITC for values of ϕ greater than 0.65 (for a fixed ∆Q),
CHAPTER 6. VALIDATION 35
shown in figure 6.5(b). When comparing worst-case scenarios for low ∆Q’s, it is clear
that LCA has fewer iterations than MCA (for equal number of nodes). It is then
concluded that LCA is a more efficient charging algorithm for binary trees using the
ITC criteria.
(a) ITC Result for LCA (b) ITC Result for MCA
Figure 6.5: ITC Simulation Results for Binary Trees
The second criteria for analysis was MITC. Both algorithms were tested on a 6
node tree. The result of LCA is shown in figure 6.5(a), and MCA’s in figure 6.6(b).
The simulation results for both algorithms in a 6 node tree are shown in figure 6.6.
One notable result for both LCA is that it is sensible to changes in ϕ when ∆Q < 0.5.
This is of significant value since practical implementations of WNC will have relatively
low ∆Q’s due to lower transmission and conversion efficiencies.
(a) MITC Result for LCA (b) MITC Result for MCA
Figure 6.6: MITC Simulation Results for Binary Trees
CHAPTER 6. VALIDATION 36
In order to get the runtime for the both algorithms, they were tested for MITC
by fixing the values ϕ = ∆Q = 0.5 and varying the number of nodes belonging to
the network. The results of this simulation are shown in figure 6.7. Even though the
number of nodes belonging to a tree grows exponentially with respect to its levels,
the relationship between MITC and the number of nodes is linear in nature.
Figure 6.7: Binary Tree Complexity
For comparison purposes, table 6.3 shows numerical results for both proposed
algorithms with different parameter values.
Algorithm Nodes ∆Q ϕ ITC MITCRR 8 0.5 0.5 28 19.35
MCA 8 0.5 0.5 28 17.42RR 16 1.0 1.0 30 16.5
MCA 16 1.0 1.0 30 16.5
Table 6.3: Simulation Results for a WCN Binary Tree
Grid Trees
As with binary trees, the first criteria for analysis is ITC. The simulation results for
both of the proposed algorithms are shown in figure 6.8. As was expected with all
CHAPTER 6. VALIDATION 37
Round Robin algorithms, the ITC is sensible only to changes in ∆Q, not ϕ. When
using MCA, increasing ϕ will generally lower the ITC when ϕ > 0.65.
(a) ITC Result for RR (b) ITC Result for MCA
Figure 6.8: ITC Simulation Results for Grid Trees
The two proposed algorithms had very different simulation results for MITC.
When using the RR algorithm (LCA), MITC showed a linear relationship with ϕ and
∆Q when both variables are greater than 0.5. MCA, on the other hand, it showed
little dependence on ϕ. Only for small ∆Q values did ϕ influence the MITC.
(a) MITC Result for RR (b) MITC Result for MCA
Figure 6.9: MITC Simulation Results for Grid Trees
The Grid Tree structure was also tested for running time. For Grid Trees, however,
there are now two parameters: N and M (ϕ = ∆Q = 0.5). For this reason, a 3D
graph with contour lines was made, shown in figure 6.10. For an easy analysis, fix
one variable, and see how the MITC changes with respect to the other. For example,
CHAPTER 6. VALIDATION 38
when fixing the number of columns at two, the MITC grows linearly with respect to
the number of rows. When fixing the number of rows at two, the MITC decreases
linearly with respect to the number of column, though with a small slope.
(a) RR Complexity (b) MCA Complexity
Figure 6.10: Grid Tree Complexity
For comparison purposes, table 6.4 show numerical results for both proposed al-
gorithms with different parameter values.
Algorithm Nodes ∆Q ϕ ITC MITCRR 2x3 0.5 0.5 12 8.5
MCA 2x3 0.5 0.5 12 9RR 3x2 0.5 0.5 12 8.16
MCA 3x2 0.5 0.5 12 8.5
Table 6.4: Simulation Results for a WCN Grid Tree
Unbalanced Structures
As previously mentioned, unbalanced structures have a very broad definition. For this
reason, unbalanced structures were limited to just two branches in this document.
Figure 6.11 shows the simulation results for 3 nodes in the first branch, 2 in the
second. As expected, the ITC is not sensible to changes in ϕ. The MITC, on the
other hand, is generally lower for low values of ϕ for a fiex ∆Q.
CHAPTER 6. VALIDATION 39
(a) MITC Result for RR (b) MITC Result for MCA
Figure 6.11: Simulation Results for Unbalanced Structures
The complexity of the charge algorithm for an unbalanced WCN Tree was tested
in a similar manner as balanced trees. Only a two branch tree was tested, though,
varying the number of nodes in the first branch and the second branch. The results
are shown in figure 6.12. It can be seen that while the MITC is linear with respect
to the number of nodes in the second branch, it is not so with the first branch. As
expected, the MITC is lowest when there are more nodes in the first branch, than in
the second branch.
Figure 6.12: Unbalanced Grid Tree Complexity
CHAPTER 6. VALIDATION 40
6.2.3 Summary
Table 6.5 summarizes the optimal algorithm for all topologies and both criteria, when
applicable. The runtimes shown are for fixed values of ∆Q and ϕ.
Table 6.5: Algorithm Comparison
Topology Criteria Optimal Algorithm Runtime
StarITC equal −
MITC MCA O(n)Queue − − O(n)
Binary TreeITC LCA O(n)
MITC MCA O(n)
GridITC RR O(n ·m)
MITC MCA O(n ·m)Unbalanced − − O(n)
Chapter 7
Conclusions
7.1 Discussion
For multi-hop WCNs, all algorithms are equal when ∆Q = 1 in terms of ITC. This
is fairly intuitive as having the unitary charge transfers prevent any node from re-
ceiving charge in two consecutive iterations. Therefore, there is not any difference for
balanced WCNs.
In order to minimize the MITC of balanced multi-hop networks, one should min-
imize the number of levels the N-ary Tree has. The best-case scenario for 2N user
nodes is having N branches of 2 user nodes each. With unbalanced networks, the
topology cannot insure that the P node will be transmitting in every iteration. To
minimize this “idle-time,” it is important to start by charging the heaviest branches
first.
The values for ∆Q must be chosen carefully. Since the main objective of WCN is
to efficiently use wirelessly transmitted energy, a problem arises when ∆Q is chosen
to be a non-multiple of 1. If, for example, ∆Q is chosen to be 0.99, after one iteration
the condition for being charged (SOC ≥ 1) is not met. Therefore, it must wait for
another iteration to be considered “charged,” when the SOC reaches 1.98. Since the
proposed model only deals with a primary storage component, this means that 0.98
SOC will be “wasted”. Even worse, when using certain batteries, overcharging them
could damage them permanently. This extra energy, weather used or not, does not
41
CHAPTER 7. CONCLUSIONS 42
affect the analysis presented in this document. Since we are minimizing the number
of iterations, there will simply by “plateaus” where a small range of ∆Q will have the
same number of iterations until the next multiple of 1 is reached and the ITC lowers.
7.2 Future Work
This document has presented different models in order to optimize different charging
times for WCN. The next step to design a more robust WCN would need to take into
account heterogeneous efficiencies throughout the network. Additionally, to have a
more exact approximation of charging times, the internal battery efficiency must be
taken into account, as shown in [8]. When more robust algorithms take into account
these considerations, it would then be possible to allow WCN nodes to go mobile
and still be optimally charged. Another, perhaps simpler problem would be to design
algorithms that maximize the lifetime of a WCN when all P nodes are shutdown.
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Appendix A
List of Abbreviations
ITC - Iterations To Charge. This is the number of iterations to have all nodes
belonging to a network charged
LCA - Least Charged Algorithm
MCA - Most Charged Algorithm
MITC - Mean Iterations To Charge. This value indicates the mean number of
iterations it took to charge every node in the network.
RR - Round Robin
SOC - State of Charge. This percentage represents how full the storage component
is, a unitary SOC indicated it is completely charged.
WCN - Wirelessly Charged Networks
WSN - Wireless Sensor Networks
WE - Wireless Energy
45