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.


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


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


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


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.


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].


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


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


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.


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.


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.


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).


(a) (b) (c)

(d) (e) (f)

Figure 4.6: LCA for an Unbalanced WCN (∆Q = 1)


Figure 4.7: WCN Queue Charge Algorithm(∆Q = 1.)

(a) (b) (c)

(d) (e) (f) (g)


Figure 4.8: Complete Binary Tree (∆Q = 1.)

(a) (b)

(c) (d)

(e) (f)

(g) (h)


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


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.


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.


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.


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.


(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),


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


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


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,


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.


(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


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

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WIRELESSLY CHARGED NETWORKS A Thesis Project Submitted …