Design of a Portable Power Generation System usingCompressed Air Energy Storage Concept

Francisco Bandeira Brás Monteiro

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisors: Prof. Edgar Caetano FernandesProf. Luís Rego da Cunha de Eça

Examination Committee

Chairperson: Prof. Carlos Frederico Neves Bettencourt da SilvaSupervisor: Prof. Edgar Caetano Fernandes

Member of the Committee: Prof. Mário Manuel Gonçalves da Costa

December 2018

ii

Acknowledgments

I would like to start by thanking both my supervisors, Professor Edgar Fernandes and Professor Luıs

Eca, for all their support and advice during the last year, whose guidance was essential to the conclusion of

this work.

I would also like to thank Professors Aires Santos and Jose Aguilar Madeira, for all the assistance

provided in several parts of this work.

To all my colleagues who entered in IST with me and made this whole path alongside me, I would like to

address them my gratitude. I would have never reached this point without all their support, help, and above

all, their friendship.

To the IN+ staff and to my laboratory colleagues, where I have spent the last two years, I would like to

thank you for all you have taught me. To my friends Goncalo Secca Serra and Tiago Oliveira, a special word

to remind all the help they gave me during the course of this work. To my friend Francisco Vieira, thank you

for reading my thesis and for all your recommendations.

To all my friends from Missao Paıs ESEL, my deepest gratitude for the amazing week we had and for all

the bonds of friendship that I gained there. Without any doubt, all of you helped me to face this semester

and the challenges that this work gave me.

At last, I would like to thank my family, especially to my parents, brother and sisters, for all the support I

was given during these last 5 years in IST. A special word to my grandfather Jose, for everything he taught

me during his life, probably one of the main reasons that brought me to this course, and to my grandfather

Antonio for the example that he is to me, everyday in my life.

Since this work marks the end of an important part of my life, a special thank to everyone in IST. I think I

can say without any doubt that I leave this University much more prepared to deal with adversity than I was

when I arrived.

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Resumo

A recente tendencia de desenvolvimento de um sistema de producao de energia global mais sus-

tentavel levou ao aparecimento de novas configuracoes para producao de energia, as quais seguem as

tendencias actuais de reducao de emissoes e de escala, relativamente as formas convencionais. Entre

estas, a utilizacao de ar comprimido como forma de armazenamento de energia tem merecido o interesse

de muitos investigadores nos ultimos anos. O presente trabalho pretende explicar o desenvolvimento de

um sistema portatil de producao de energia, que utilizando os excessos de producao de energia nas horas

de baixo consumo, bem como fontes de energia renovaveis, comprime ar atmosferico para reservatorios

de alta pressao. Este ar comprimido sera depois utilizado para a producao de energia, por expansao numa

turbina radial de pequenas dimensoes. Por forma a desenhar o sistema, sera obtido um modelo matematico

da sua operacao em primeiro lugar. Com este modelo, procurar-se-a optimizar o desempenho do sistema.

Nunca esquecendo o objectivo de portabilidade, e fulcral que o sistema possa produzir tanta energia quanto

possıvel, ao mesmo tempo que deve apresentar as menores dimensoes possıveis.

Foram aqui obtidas varias configuracoes optimas para o sistema, com rendimentos da ordem dos 30%,

e densidades energeticas proximas de 90 kJ/kg, valor ainda abaixo dos apresentados pelas baterias con-

vencionais, mas com a vantagem de ter um impacto ambiental reduzido. Os custos de investimento sao

tambem reduzidos, da ordem dos 10,000 e, e podem ser recuperados em poucos anos de operacao, caso

o sistema seja operado de forma diaria.

Palavras-chave: Producao de Energia Descentralizada, Energias Renovaveis, Armazenamento de

Energia atraves de Ar Comprimido, Optimizacao Multi-Objectivo

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Abstract

The recent tendency to develop a more sustainable power system has led to the development of new

power generation systems, that are intended to be greener and smaller in size than the conventional ones.

Among these, the use of compressed air as a form of energy storage has gained the interest of many re-

searchers in recent years. This work intends to explain the development of a portable power generation

system, that uses energy production excesses from off-peak consumption hours, as well as RES, to com-

press the air and store it in high-pressure tanks. The stored compressed air will later be used to expand

through a small turbine, so that the desired electricity can be produced. To design this system, a simple

mathematical model will be initially obtained. This model intends to provide a tool to optimize the perfor-

mance of the system. Keeping in mind the objective of portability, it is important to have a system that can

produce the maximum amount of energy possible, while its size is kept the smallest possible.

Several optimal configurations were here obtained, with efficiencies around 30%, and a stored energetic

density close to 90 kJ/kg, this one still lower than the typical value of a battery, but having the advantage of

not being hazardous to the environment. The investment cost of this system is also low, roughly 10,000 e,

which can be recovered within few years of operation, if the system is continuously used.

Keywords: Distributed Power Generation, Renewable Energy Sources, Compressed Air Energy Stor-

age, Multi-Objective Optimization

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Distributed Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Energy Storage Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Compressed Air Energy Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Expansion Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 System Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 State of Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6.1 Theoretical Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6.2 Experimental Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Objective of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Mathematical Model 15

2.1 Preliminaries of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Numerical Methods to solve ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Thermodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Tank Charging Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Tank Discharging Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 Energy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Optimization of System Operation 34

3.1 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Direct Multi-Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Objective Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

ix

3.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Results and Discussion 41

4.1 Pareto Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Optimal Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Large System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Small System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Cost Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Conclusions and Future Work 51

5.1 Guidelines for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Bibliography 55

A MATLAB Routine - Simulator 61

B Pareto Front Points 65

C Cost Model 68

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List of Tables

1.1 Summary of DPG Technologies [12, 13, 15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Summary of Energy Storage Technologies [1, 15, 18]. . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Summary of main Expansion Devices used in the Literature [26, 32, 33]. . . . . . . . . . . . . 10

2.1 Values of Parameters used to test the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Independent Variables for MOO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 Comparison of Large and Small Optimal Systems . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Large System Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Small System Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4 Cost of Large System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5 Cost of Small System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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List of Figures

1.1 Forecast of Changes in Primary Energy Demand from 2016 to 2040 [3]. . . . . . . . . . . . . 1

1.2 Portuguese Load Diagram of 26-06-2018 [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Energy Density of Several Technologies and Fuels [7]. . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Power Outputs vs Energy Stored of different Energy Storage Technologies [8]. . . . . . . . . 3

1.5 Energy Storage Technology Cost per Unit Power vs Cost per Unit Energy [11]. . . . . . . . . 4

1.6 Principle of CAES system: Schematic layout of Huntorf plant in Germany [20]. . . . . . . . . 7

1.7 Principle of CAES system: Schematic layout of McIntosh plant in Alabama, USA [19]. . . . . 7

1.8 CAES plant with isolated tank [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.9 Schematics of the prototype of Isothermal CAES plant [25]. . . . . . . . . . . . . . . . . . . . 8

1.10 Schematics of Distributed CAES System [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.11 System Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.12 Schematic of System Charging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Configuration that will be used in the Mathematical Model. . . . . . . . . . . . . . . . . . . . . 15

2.2 Thermodynamic System for Heat Transfer study. . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Compression energy spent per unit mass of air stored for adiabatic and isothermal compression. 21

2.4 Stored Mass of Air function of final tank pressure for adiabatic and isothermal compression. . 21

2.5 Thermodynamic System for Tank Charging Process. . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Total Compression Work as a function of final tank pressure and volume. . . . . . . . . . . . 24

2.7 Thermodynamic System for Tank Discharging Process . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Pressure Evolution during discharge of tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.9 Turbine Power Production during Tank Discharge . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10 Total Energy Produced function of tank pressure and volume. . . . . . . . . . . . . . . . . . . 28

2.11 Total Energy Produced function of tank discharge pressure, for initial pressure p0 = 100 bar

and volume V = 1000L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.12 Total Operating time function of Discharge Pressure for initial pressure p0 = 100 bar and

volume V = 1000L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.13 Maximum Turbine Power, function of Discharge Pressure for initial pressure p0 = 100 bar and

volume V = 1000L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.14 Total Compression Work and Total Expansion Work as a function of p0. . . . . . . . . . . . . 31

2.15 Stored Energy Density as a function of p0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

xiv

3.1 Pareto front for a two-objective problem [62]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Direct-Search Algorithm [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Implementation of the MOO Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Pareto front obtained without restrictions to the objective functions. . . . . . . . . . . . . . . . 41

4.2 Quasi-linear shape of the final part of a typical Pareto front [62]. . . . . . . . . . . . . . . . . . 42

4.3 Pareto front obtained with restrictions to the objective functions. . . . . . . . . . . . . . . . . . 42

4.4 Pareto front obtained with restrictions to the objective functions split in different zones. . . . . 44

4.5 Power developed by the Large System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6 Power developed by the Small System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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Nomenclature

Roman symbols

m Mass Flow Rate.

Wc Compressor Power Input.

Wt Turbine Power Output.

Ae Exit Duct Cross Section.

cp Constant Pressure Specific Heat.

cv Constant Volume Specific Heat.

de Exit Duct Diameter.

K Valve Head Loss Coefficient.

L Exit Duct Length.

m Mass of air inside the tank.

p Pressure.

p0 Tank Initial Pressure.

pe Discharge Pressure.

patm Atmospheric Pressure.

R Air Constant.

T Temperature.

Tatm Atmospheric Temperature.

U Flow Velocity.

Uc Overall Heat Transfer Coefficient.

ux, uy, uz Velocity vector Cartesian Components.

V Tank Volume.

xvii

v Specific volume of air inside the tank.

Wc Total Compression Work.

Wt Total Energy Produced.

Greek symbols

∆t Total Operating Time.

η System Overall Efficiency.

ηc Compressor Efficiency.

ηt Turbine Efficiency.

κ Thermal conductivity coefficient.

µ Molecular viscosity coefficient.

∇ Laplacian Operator.

ν Kinematic viscosity.

ρ Density.

ρen Stored Energy Density.

σcv Entropy Generation in a Control Volume.

Acronyms

1D One-Dimensional.

A-CAES Adiabatic Compressed Air Energy Storage.

AA-CAES Advanced Adiabatic Compressed Air Energy Storage.

BDF Backward Differentiation Formula.

CAES Compressed Air Energy Storage.

CFD Computational Fluid Dynamics.

CHP Combined Heat and Power.

CPG Centralized Power Generation.

D-CAES Distributed Compressed Air Energy Storage.

DPG Distributed Power Generation.

ERSE Entidade Reguladora do Sector Electrico (Portuguese Electricity Sector Regulator).

ESA Energy Storage Association.

xviii

IEA International Energy Agency.

MT Micro-Turbine.

NS Navier Stokes.

ODE Ordinary Differential Equation.

RES Renewable Energy Source.

UMGT Ultra-Micro Gas Turbine.

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Chapter 1

Introduction

1.1 Background and Motivation

The increase in electricity consumption, mainly due to the increase in total world population, has led to

an excessive consumption of conventional fossil fuels and CO2 emissions. The International Energy Agency

(IEA) estimates that by 2040, the global energy demand will have grown by 40%. However, despite the

increase in consumption, the primary energy mix will be somehow different from what it is nowadays. The

changing scenario is mainly driven by four major forces: the rapid development and falling costs of clean

energy, the growing electrification of energy (that is, electricity as end-use energy, which is estimated to

have increased from 12% in 2008 to 34% in 2025 [1]), the shift into more service oriented economy and

the cleaner energy mix in China, which accounts for almost 23% of world energy consumption [2]. In figure

1.1, the IEA forecast of changes in world primary energy demand from 2016 to 2040 are shown. It can be

seen that the total energy demand will increase in the next years, being India the largest contributor for this

growth in demand, whose share of global energy will rise up to 11% in 2040.

Figure 1.1: Forecast of Changes in Primary Energy Demand from 2016 to 2040 [3].

1

Considering also the expected growing relevance of Renewable Energy Sources (RES), the objective

of IEA is to create a sustainable and more environmentally friendly power generation system, based on

Distributed Power Generation System (DPG), instead of traditional Centralized Power Generation System

(CPG), as the latter must account for more than 20% of losses in transmission and distribution systems [4].

The integration of DPG with RES, particularly solar photovoltaic (PV) or wind energy, is gaining the interest

of several power generation projects, as it will allow for more financial solutions, with energy savings and

lower emissions, besides the growing interest in the concept of portability, since that there have been several

developments of compact power systems, to feed portable electronic devices, many of them used in military

applications [5].

However, DPG by itself cannot meet the energy demand, as due to significant changes in weather

conditions, that is, the weather unpredictability, the power production system would not be neither continuous

nor reliable. These fluctuating sources increase the difficulty in stabilizing the power network, providing an

additional problem of supply-demand imbalance, as the demand itself is also highly variable throughout

the day. The Portuguese Load Diagram, from 26th June 2018, in figure 1.2 shows how the consumption

of electricity (black line) evolves during a typical summer day. The mismatch between peak consumption

hours, that extend until late, and the peak of solar PV production (about midday time) increase the difficulty

of integrating DPG in the Power Generation System.

Figure 1.2: Portuguese Load Diagram of 26-06-2018 [6].

To overcome this difficulty in using DPG technologies, there is a great necessity to implement energy

storage technologies, so that the energy supply can be kept equal to the demand levels, providing control-

lable operation. Electricity storage can be achieved effectively, transforming first the electricity into another

form of storable energy, and later transform it back to electrical energy, whenever required.

The use of batteries has been the traditional solution. However, batteries still face nowadays some

problems that bring the necessity of developing alternative systems. Batteries have a low energy density

(approximately 0.5 MJ/kg, when compared to the typical value of 40 MJ/kg for most hydrocarbons, as easily

observed in figure 1.3), they are often too heavy for some of the desired portable applications, the recharging

time must also be taken into account, since it is much higher than the one of a hydrocarbon-based power

2

system, as well as the fact that most of these batteries chemistry relies on materials that are difficult to

recycle and can be environmentally hazardous [5].

Figure 1.3: Energy Density of Several Technologies and Fuels [7].

One possible alternative to reduce the negative impact of batteries, is the storage of energy through

the form of compressed air systems, also known as Compressed Air Energy Storage (CAES), which can

be an adequate solution for long term and high capacity energy storage. The peaks in energy production,

that exceed instantaneous demand, can be used to power compressors, and store atmospheric air in high-

pressure vessels.

The CAES technology has been studied in recent years and it is considered to be suitable for small power

generation systems [1], being able to store amounts of energy of the order of 10 kWh, as well as producing

an output power of more than 10 kW (figure 1.4), which makes CAES suitable for a portable system.

Figure 1.4: Power Outputs vs Energy Stored of different Energy Storage Technologies [8].

It is also important to consider the cost factor, as like any other investment, a storage system will only be

interesting if the total gains or savings exceed, or at least match, the total expenses. It is usual to consider

that the total costs are obtained by summing the investment costs and the operational costs, being the

3

latest spread over the system life. According to [1, 9], both theses costs are proportional to the total energy

produced, Wut, and to the installed capacity, Pd, considering also the operational costs as proportional to

the investment costs, reaching up to 40% of its value [10]. The total cost can then be given by the following

equation:

Ct = (ac1 + c2)Pd (1.1)

where a = Wut/Pd is defined as the system autonomy (ratio between system energy capacity and discharge

power), c1 and c2 being the coefficients that relate the total costs to the energy produced, Wut, and installed

capacity, Pd, respectively.

Considering then the cost factor, it is easily observed in figure 1.5 that CAES systems, among other

energy storage technologies, are those that present simultaneously one of the lowest cost per unit power,

$/kW, as well as one of the lowest cost pert unit energy $/kWh.

Figure 1.5: Energy Storage Technology Cost per Unit Power vs Cost per Unit Energy [11].

1.2 Distributed Power Generation

According to [12], a DPG system is defined as an electric power source, that is connected either directly

to the distribution network, or on the consumer side of the meter. The main advantages of on-site electricity

productions, according to [13], include the reduction of power losses in transmission and distribution systems

(in the transmission system between 4% and 5% of electrical energy is wasted, and at the end of the

distribution network about 10% - 15% is wasted [14]), the reduction of power system oscillations, as the

DPG can easily meet demand peaks, enhancing the power system stability, reliability and security, as well

as the power transfer capacity, increasing at the same time, the band of operation. Some of DPG most

important applications, according to [12], include emergency backup, as many services in which power

outage cannot be tolerated, are willing to pay premium to maintain the power supply without any failure.

Besides this, for peak-load shaving, it is economically more interesting to install a DPG system for peak

periods in demand, instead of starting the production in a traditional thermal plant. DPG systems can also

4

be presented as a good solution when the concept of portability is involved: DPG systems can be used to

generate power in remote places where no other sources, like the grid, are available.

The main drivers that are responsible for pushing then development of DPG nowadays can be grouped

in three main categories, according to [15]

i. Environmental Drivers: almost 25% of gaseous emissions are originated in the electricity generation

sector [16], therefore, DPG units based on RES can be a promising solution to reduce the environ-

mental impact of power generation, while offering a cost-effective solution.

ii. Market Drivers: the market players are looking for DPG systems with lowest financial risk, thus, DPG

units with lower capital installation costs, as well as lower maintenance costs, are to be favoured by

the electricity market.

iii. National Regulation Drivers: the focus of nowadays energy policy, as previously mentioned, is to keep

electricity generation secure and sustainable, lowering gaseous emissions. The DPG systems with

lower starting time can therefore be an effective solution in emergency situations and help to overcome

power capacity shortfalls at lower prices.

Typically, the DPG systems are grouped in two main categories: non-renewable and renewable tech-

nologies. The first one includes systems that are based on traditional power generation concepts, but with

smaller sizes, using technologies such as internal combustion energies, Stirling engines, and steam or gas

turbines. The second category, and gaining more and more importance nowadays, includes all the systems

powered by a RES, including here fuel cells, small-scale hydro plants, solar PV or wind energy. Since these

sources cannot be controlled by the user, to make DPG systems operations more continuous and reliable,

it is imperative to give special attention to energy storage systems. Most of the technologies mentioned, as

well as some of their main characteristics are summarized in table 1.1.

Table 1.1: Summary of DPG Technologies [12, 13, 15].

Technology Power Range Efficiency (%) Capital Cost ($/kW) ApplicationICE 5 kW - 10 MW up to 43 300 - 1400 EmergencyMicroturbines 20 - 500 kW 20 - 80 600 - 2600 Emergency, DPG and CHPStirling Engine 2 - 10 kW up to 30 - DPGFuel Cells up to 1 MW 35 - 80 - DPGSolar PV 20 W - 100 kW 10 - 20 2500 - 7000 DPG and CHPSmall Scale Hydro 5 - 10 MW 60 - 80 3400 - 10,000 DPGWind Turbine 1 kW - 3 MW 20 - 50 800 - 3500 DPG

1.3 Energy Storage Technologies

To enable power generation in small scale, using RES, there is a great necessity of developing reliable

and efficient energy storage technologies, so that power fluctuations can be managed, providing continuous

supply of energy. The energy storage technologies are usually grouped in three main categories: mechani-

cal, thermal and chemical. Based on their application they can also be classified into small power generation

5

systems (usually isolated locations for emergencies) and medium DPG systems for independent electricity

generation [1].

For small DPG systems, fuell cells, flywheels, batteries and compressed air storage are some of the

most used technologies. Particularly, for long storage duration with small discharge time, the use of CAES

and fuel cells seems to be more adequate, when compared with batteries. According to [14, 17], for small

power generations systems, super capacitors, batteries and small CAES systems are presented as the

best choices available. In table 1.2, the most used storage technologies are presented, for both small and

medium-large power applications.

Table 1.2: Summary of Energy Storage Technologies [1, 15, 18].

Technology Power(MW)

Energy Density(kWh/m3)

Efficiency(%)

DischargeTime

StorageDuration

Capital Cost($/kW)

PHS 100 - 5000 0.5 - 1.5 70 - 80 1 - 24 hours Hours - mon 600 - 2000Large CAES 5 - 300 3 - 6 40 - 55 1 - 24 hours Hours - mon 400 - 800Small CAES 0.003 - 3 0.5 - 2.5 up to 40 1 hour Hours - mon 200 - 250Lead AcidBattery 0 - 20 15 - 70 75 - 85 Sec - hours Min - days 300 - 800

NickelCadmiumBattery

0 - 40 20 - 80 60 - 85 Sec - hours Min - days 500 - 1500

LithiumBattery 0 - 0.1 200 - 500 85 - 95 Min - hours Min - days 1500 - 4000

Capacitor 0 - 0.05 10 - 30 60 - 75 Sec - 1 hour Sec - hours 200 - 750Flywheel 0 - 0.250 20 - 80 90 - 95 Sec - min Sec - min 250 - 750HT TES 0 - 60 120 - 300 30 - 60 1 - 24 hours Min - days 200 - 300Fuel Cells 0 - 50 500 - 3000 20 - 50 Sec - 24 hours Hours - months + 10,000

1.3.1 Compressed Air Energy Storage

The Compressed Air Storage is an old concept, that started being explored during the second half of

the twentieth century, even though not being widely implemented worldwide. The CAES systems can be

grouped into several categories, according to the way the compressed air is used to produce energy. From

all the existing categories, presented in [1, 15, 19], four main categories can be highlighted here:

i. Conventional CAES: consists of a typical Brayton cycle, that uses nearly two thirds of available power

in the air compression process. It separates compression and expansion processes in time, to use

electrical power excess during off-peak consumption hours, to compress air into underground large salt

or rock caverns. The system then relies on the implementation of CAES with a conventional Brayton

cycle, where turbine and compressor are decoupled into separate operation phases. This separation

of processes allows the CAES plant to generate three times more energy with the same amount of

fuel, when compared to a typical Brayton cycle. The burning of fuel with compressed air increases fuel

consumption and CO2 emissions, as well as the significant heat losses during compression phase,

that lead to a decrease in overall efficiency. However, it is important to remember that the input energy

for compression was free [19].

There are already several conventional CAES plants spread worldwide, the first one being imple-

6

mented in 1978, in Huntorf, Germany, that by using a large underground cavern was able to generate

290 MW for 4 hours (figure 1.6). Later, a similar plant was installed in McIntosh, Alabama - USA, which

was able to produce continuously an output power of 110 MW, for 26 hours (figure 1.7).

Figure 1.6: Principle of CAES system: Schematic layout of Huntorf plant in Germany [20].

Figure 1.7: Principle of CAES system: Schematic layout of McIntosh plant in Alabama, USA [19].

ii. Adiabatic CAES (A-CAES): this configuration of CAES system intends to decrease fuel consumption

from conventional CAES, keeping the air at high temperatures before the expansion. This is mostly

achieved through the implementation of a thermal energy storage (TES) technology, whose main func-

tion is to recover lost heat during compression, and later use it to preheat the air, before entering the

turbine. The thermodynamic analysis of an A-CAES system proved that it could improve the overall

performance of the conventional system, and a result of 30% increase in overall efficiency can be

obtained [21, 22]. It was also investigated the use of an insulated vessel, shown in figure 1.8, and

the results showed that it could only be implemented for short periods of time [15], which could be

adequate for a small system, where there is only need of storage for short periods of time.

iii. Isothermal CAES (I-CAES): I-CAES try to prevent temperature increase in the compressors during

charging and temperature drop during discharging in the expansion devices [23]. It is usually obtained

7

Figure 1.8: CAES plant with isolated tank [22].

when both compression and expansion processes are slow, and allow to have significant heat transfer

between the air and the surroundings, keeping it an almost constant temperature. This system, de-

picted in figure 1.9, relies on the injection of water or oil during the compression process, and these

hot liquids are then re-injected during expansion phase to reheat the air entering the turbine [24]. The

I-CAES system is still under development, and further improvements on the heat transfer are required,

to obtain higher levels of performance.

Figure 1.9: Schematics of the prototype of Isothermal CAES plant [25].

iv. Distributed CAES (D-CAES): the small scale CAES system, presented in figure 1.10, is regarded as

an adequate solution for DPG based on CAES, using high-pressure air vessels. The air can be com-

pressed using peaks on energy supply, or even RES (particularly solar PV or wind), being stored in

pressurized cylinders at up to 300 bar, allowing to achieve efficiencies up to 50% [1]. This solution also

presents some advantages, when compared to other DPG systems: i) safe and environmental friendly

technology, ii) can be generated with RES for cost effective electricity supply, iii) less installation restric-

tions and iv) it is of simpler manufacturing. On the other hand, there are also some disadvantages:

since it is not widely used, the efficiencies of both compression and expansion processes are low,

leading to relatively low overall efficiency.

8

Figure 1.10: Schematics of Distributed CAES System [15].

1.4 Expansion Devices

The expansion device, or simply expander, is used for power recovery, by extracting the energy of a

working fluid and converting it into mechanical work. The small expander is a vital part in a D-CAES system,

and it will affect heavily its performance, in terms of efficiency and also cost of power generation [26, 27].

The choice of an expander is then of extreme importance when designing these systems. To choose an

adequate expander, several parameters have to be considered: the cycle operating conditions, the system

size and the working fluid properties [28]. The expanders for small applications are usually grouped in two

main categories: velocity or dynamic type expanders and volume or displacement expansion devices.

The first ones, also known as turbines or turbochargers intend to convert kinetic energy of the passing

fluid into a form of mechanical work, by means of a rotating shaft with a set of blades. The turbines used

in power generation systems are divided into axial and radial turbines, and despite the small scales here

considered, they operate in a similar way to the conventional turbines. The axial turbines are more common

in high flow rate applications, with low pressure ratio, and due to the axial flow through the blades, the axial

turbines have lower aerodynamic losses and higher efficiency [29]. The radial turbines are the best choice

when the application requires low mass flow rate and high pressure ratio [30]. For small output powers, the

radial turbine presents less mechanical stresses, which enables its design to be simpler, more robust and

more efficient (in a similar power range), when compared to the axial turbines.

Regarding the volume type expanders, they are more common in micro-combined heat and power (CHP)

systems. They are used when the application requires low mass flow rate, low rotational speed and high

inlet pressure [31]. There are four main volume expanders used:

i. Scroll Expander: common to all positive displacement devices, the scroll expander has a fixed volu-

metric ratio [26]. Compared to turbines, it has lower performance levels due to friction and pressure

losses [31].

ii. Screw expander: consists of a pair of male-female screw rotors in one case, and the energy is trans-

ferred as result of volume change between both rotors. Screw expanders with small powers are still

under development and difficult to be found in the market [28].

iii. Reciprocating Piston Expander: piston expander is more complex compared to other expanders be-

9

cause of the complicated valve system for both intake and exhaust processes. Also, the overall cycle

performance is affected by losses generated due to friction between piston and cylinder [26].

iv. Rotary Vane Expander: also known as air motor expander, it uses vanes driven by compressed air.

According to [28], it is easy to manufacture, with a lower production cost, when compared to other

expansion devices.

An overview of these expanders can be found in table 1.3, where details of typical values of several

parameters, such as power, can be found.

Table 1.3: Summary of main Expansion Devices used in the Literature [26, 32, 33].

Technology Power(kW)

Efficiency(%)

Rotation Speed(rpm) Pressure Ratio

Radial Turbine 0.15 - 30 30 - 85 up to 100,000 1.5 - 5Scroll Expander 0.5 - 3.5 10 - 80 500 - 5000 1.5 - 10.5Screw Expander 0.5 - 15.5 25 - 70 400 - 3500 2 - 10Rotary VaneExpander 0.8 - 2.2 17.5 - 55 300 - 4500 2 - 24

ReciprocatingPiston Expander 0.2 - 20 10.5 - 65 100 - 200 2.1 - 2.4

1.5 System Configuration

With all the previous considerations, the scope of this work will consist of designing a DPG system, using

CAES, as depicted in figure 1.11. Considering the overlap of these two concepts, these systems can also

be referred as D-CAES, as mentioned earlier. A small compressed air tank, which will be filled using RES

or excesses of energy production, will be connected to a small turbine, so that small amounts of power can

be produced, for a certain period of time.

As mentioned earlier, the charging of the tank will be made using excess of energy from off-peak con-

sumption hours, as well as RES (solar PV or wind) - figure 1.12, which allows the system to be charged

without any extra cost.

Among all the expansion devices previously explored, the choice of the small turbine, also known as

Micro Turbine (MT), was due to its higher power-to-weight ratio and efficiency. The turbines present also

more compact sizes, they are more reliable, with lower operation and maintenance costs, lower noise levels

and more flexible in terms of the fuel to be used [14].

Considering again the work of [14], a D-CAES system can be applied in several ways. The first one, peak

shaving, intends to use a D-CAES to reduce costs of peak production and improve the overall efficiency, as

well as reduce investments in bulk transmission. The system can also be used for backup generation,

providing an uninterruptible power supply (UPS). It can also be used as a standby generator, that is, a

backup electrical system that operates whenever required. The last and most important of the applications

hereby mentioned is related to the concept of portability. As mentioned in [5], there has been an increase

in the demand for portable electrical devices, especially in military applications. Therefore, the D-CAES

10

system can be used for remote power, that is, in locations where the construction of networks is not possible

or not economically viable, that is the cost of grid construction is higher than the one of a DPG system. The

fuel flexibility provided by the utilization of a MT is also a determining factor for use in remote applications,

which causes significant reduction of the investment costs.

Figure 1.11: System Configuration

Figure 1.12: Schematic of System Charging

1.6 State of Art

Besides some works already mentioned, the present section intends to focus on small scale power gen-

eration systems that use CAES concept. By making some research in the literature, it is easy to understand

that there has been some research lately on D-CAES systems, however, there is still lack of available in-

formation. Since there is a clear split in the literature between theoretical and experimental research, this

section will be divided in the same way.

1.6.1 Theoretical Research

The vast majority of theoretical research on D-CAES systems intended to find ways of improving overall

efficiency, integrating them with RES [34, 35]. It was concluded by [Paloheimo2009] that the low efficiency

11

of these systems is mainly due to the low efficiency levels of the turbines, requiring further research on the

development of small scale turbines with higher efficiency [36, 37, 38, 39].

A detailed energy and exergy analysis of small CAES system was performed by [40], and by testing

different compression and expansion processes for a constant pressure vessel, it was shown the potential

of a high system efficiency, up to 60%, for quasi-isothermal processes in both compression and expansion

devices. A variable configuration system was studied by [41], with an input power of 500 kW, based on RES,

being able to produce 14, 400 MJ of energy, by storing compressed air at 200 bar, for charging duration of 8

hours. The system overall efficiency could reach up to 72%, even though this value was overestimated due

to simplifications assumed in the model.

In order to reduce the exergy losses caused by the throttling valve and improve system performance,

[42] proposed a new EA-CAES system, where the E stands for an additional ejector. By modifying the

throttling strategy, the thermodynamic analysis shown that the efficiency can improve almost by 2%, and the

profit by more than 21%, compared to conventional A-CAES systems. A study from [43] also shown that

there is an optimum value for the expander inlet pressure to obtain maximum values of both performance

and stored energy density. In this study it was also shown that a greater heat transfer coefficient, between

tank and environment, which leads to quasi-constant air temperature in the tank during discharge, that is,

quasi-isothermal process, is advantageous for the cycle, and can improve system efficiency.

1.6.2 Experimental Research

It is easy to find some hybrid small CAES systems in the literature, with power ranges from 2 kW to

almost 1 MW [34]. All the small D-CAES systems are based on scroll expanders [34], which have low

efficiencies compared to turbines [44].

A detailed analysis of a mini CAES cycle was carried out by [45], for a system with a tank of 270 L,

storing air at 11 bar, and using a small turbine rotating at 800 RPM, which allowed to obtain a voltage of 8

V. The experiments shown once again, the great necessity for developing more efficient expansion devices.

The coupling of a wind turbine with small CAES was studied by [46], whose main focus was to improve

the conversion efficiency of the wind energy into compressed air by developing a controller that adapts the

gearbox to the wind speed. The results obtained showed that the amount of stored energy is dependent

on the maximum pressure in the tank, which in turn, is controlled by the operating wind speed. In War-

wick, England, a hybrid CAES system, also integrating a wind turbine, was designed by [34, 44], being the

experimental rig still under construction, and intended to optimize the production of electricity for low wind

speeds.

A pilot Advanced Adiabatic CAES (AA-CAES) plant was recently built by [47, 48], with the objectives of

evaluating the potential of rock caverns to store compressed air, withstanding stresses, and also to inves-

tigate the performance of a thermal energy storage TES, using a packed bed of rocks placed inside the

caverns. The estimated efficiency was between 63% and 74%.

Regarding small scale expanders, a small turbine, with 10 mm diameter, was produced by [38] in Bel-

gium, and by using compressed it was able to produce a shaft power of almost 30 W, with an efficiency of

18.5%, being consistent with [34] necessity of improving the efficiency of the expanders. A later experiment

12

from [15], in 2017, was able to produce a larger turbine for compressed air, but still of small dimensions,

whose inlet pressures varied between 1.2 and 1.6 bar, and it was able to produce a maximum output power

of 700 W, with almost 60% efficiency.

1.7 Objective of the Thesis

As earlier mentioned, the objective of this work is to develop a portable power generation system, with

the configuration shown in section 1.5. This portable system is intended to use compressed air stored in a

tank, to produce small amounts of energy by expanding it through a turbine.

Even though the energy used to power the system is be obtained without any cost, that is, it is intended

to use either wastes of energy production or RES to be stored, it is important to consider the overall per-

formance of the system, allowing it to produce the maximum amount of energy possible, with the lowest

possible cost.

The first step taken to design the system consists of defining a model that allows to obtain the values of

the performance variables, such as total energy produced or stored energy density, which will be functions of

the values of both system dimensions and operating parameters. At this point, it is important to see what are

the variables that have the strongest influence in the performance of the system. The goal here is to obtain

a system with the best performance possible, which considering the interest in portability, makes necessary

to obtain a system capable of producing the maximum amount of energy, at the same time that its size is

kept as small as possible.

1.8 Thesis Outline

To help the reader to follow the course of this work, the thesis is organized in five chapters. Chapter 1

provides an overview on the concept of D-CAES and on the main goals of this work, exploring some existent

solutions and possible applications for the system. Chapter 2 will present the mathematical model of the

system and how it was obtained. Besides the formulation, also the numerical methods used to obtain the

solutions will be presented, as well as some results from the model that show the influence of some variables

in the performance of the system.

After obtaining the system mathematical model, an optimization algorithm will be used to obtain the

best performance of the system, being the procedure described in chapter 3. The next chapter, chapter 4,

presents the analysis of the results obtained, as well as the discussion of the several system configurations

and its operating parameters, that allow to obtain the best performance levels. Besides this, a simple

economic analysis to the system will be here made, as it will allow to compare the total investment costs to

the savings from producing electricity with the system, instead of using electricity directly from the grid. At

last, in chapter 5, a summary of the main conclusions of this work is presented, as well as some guidelines

for future work, based on the results obtained in this thesis.

13

14

Chapter 2

Mathematical Model

In this chapter, the system model is presented, based on the configuration previously described in chap-

ter 1. The main goal of this model is to obtain a simple way of evaluating the performance of the system,

especially in terms of efficiency, total energy produced and total operating time, which will be done consid-

ering the system depicted in figure 2.1.

Figure 2.1: Configuration that will be used in the Mathematical Model.

The model will be based on simple one-dimensional Thermodynamics and Fluid Mechanics equations,

using the Ideal Gas Model, and it is intended to obtain ordinary differential equations (ODEs) to study both

compression and expansion processes evolution in time. These ODEs will be solved numerically using a

MATLAB routine, producing as an output, the time evolution of the tank pressure, which will then be used to

obtain the other variables that characterize the system operation.

This chapter starts by providing a quick overview on some Thermodynamics concepts, as well as some

basics of Numerical Methods, to understand how the model and the results are obtained. After obtaining

the solution of the ODEs, some results obtained with the model will be presented, in order to discuss the

relevance of some design parameters, as well as their influence in the system overall performance.

15

2.1 Preliminaries of Thermodynamics

Since most of the equations obtained in this chapter are based on simple Thermodynamics and Fluid

Mechanics, it is important to provide a quick overview on basic conservation principles, presenting the basic

equations for mass, momentum, energy and entropy conservation, obtained in basic course books [49, 50].

The differential form of a mass balance is simply given by the following equation, where the subscripts i

and e stand for inlet and exit of the control volume, respectively.

dm

dt=∑i

mi(t)−∑e

me(t) (2.1)

Regarding momentum conservation, also known as Newton 2nd Law, the differential form is given by

the Navier-Stokes (NS) equations. Since this model will be one-dimensional (1D), the NS equation along x

direction, with zero velocity in other directions (uy = uz = 0) and neglecting the effect of gravity, is simply

given by the following equation, where U represents the flow velocity along x, and ν the kinematic viscosity

of the fluid.∂U

∂t+ U

∂U

∂x= −1

ρ

∂p

∂x+ ν∇2U (2.2)

The 1st Law of Thermodynamics can be expressed as an energy balance, whose differential form is

given by the following equation.

dE

dt= Q− W +

∑i

mi

(h+

U2

2+ gz

)i

−∑e

me

(h+

U2

2+ gz

)e

(2.3)

where Q and W represent, respectively, the heat and work exchanged with the system.

Finally, the equation that states the 2nd Law of Thermodynamics, the entropy conservation principle, can

be described using the following equation.

dS

dt=

∫A

(q

T

)b

dA+∑i

misi −∑e

mese + σcv (2.4)

where σcv designates the time rate of entropy production due to irreversibilities present in the control volume.

The first term on the right-hand side of the equation,∫A

(qT

)bdA represents the time rate of entropy transfer

that accompanies the heat transfer.

Besides these equations, the fluid used in this model will be air, which due to the low values of pressure

and temperature (below the critical values [49]) will be modelled as a perfect gas with constant properties,

being the equation of state given by the following equation.

pV = mRT (2.5)

where R = R0

Mair= 287 J/kg.K is the air constant.

16

2.2 Numerical Methods to solve ODEs

ODEs appear frequently in mathematical models of science, engineering, or even economy. There are

only few situations when the analytical solution can be found, which requires the use of numerical methods

to seek approximate solutions. A typical first-order ODE problem intends to determine a function, u, that

satisfies:du

dt= f(t, u) with t ∈ Ω ⊂ R (2.6)

To determine a unique solution, an additional condition is required, that usually specifies the value of u

in a single point of the domain, u(t0) = u0. From this point, the solution u is determined for the remaining

points of the domain. Since the variable t usually represents time, these problems are called initial value

problems.

The most common way to solve these equations is to split the continuous time domain in several discrete

points, separated by a time step, ∆ti = ti+1−ti. Assuming now a constant time step, ∆t = h, and integrating

both members of equation 2.6, between t and t+ h, the following is obtained:

u(t+ h) = u(t) +

∫ t+h

t

f(ξ, u(ξ))dξ (2.7)

It is easy to notice the presence of the function u in the integral, which makes impossible to calculate the

integral explicitly, since u is the function we are trying to determine. Denoting by Fh(t, u) an approximate

value for the integral, all the values of u can be approximated as follows:

ui+1 = ui + hFh(ti, ui) (2.8)

To solve the integral numerically, and therefore obtain the values of the function u in the other points of

the domain, the simplest way is to use the Euler methods. In these methods, the domain is divided in N

points, equally spaced by h. The integral is calculated using the rectangle rule of integration, being here two

options available to do that:

i. Explicit Euler method: the integral is approximated using the previously known point (n)

∫ tn+1

tn

f(ξ, u(ξ))dξ ≈ hf(tn, u(tn)) (2.9)

ii. Implicit Euler method (Backward Differentiation Formulas - BDF): the integrated is approximated using

the value of the function in the next instant of time (n+ 1)

∫ tn+1

tn

f(ξ, u(ξ))dξ ≈ hf(tn+1, u(tn+1)) (2.10)

The second method stability does not depend on the time step used [51]. However, it involves the solution

of an equation in each iteration, which if non-linear can increase the difficulty of the problem [52].

The Euler methods are however only first-order accurate [51]. In case of a non-stiff ODE, the Runge-

Kutta methods can be applied. These methods are usually considered a simple step method, but involve

17

several intermediate calculations in each iteration. These methods sacrifice the efficiency of the Euler

methods in order to increase precision of the calculated solution [52]. One of the most popular methods, the

4th order Runge-Kutta method, requires four calculations of f in each time step. The fifth order method, for

instance, is no longer effective, since it requires six calculations of the function f per time step [51].

Sometimes, it is necessary to solve an ODE of order higher than one. The most usual way to solve these

problems is to transform a higher-order ODE in a system of first-order ODEs. Consider now the problem:

d(n)u

dt(n)= f

(t, u(t),

du

dt, ...,

d(n−1)u

dt(n−1)

)(2.11)

with initial conditions:

u(t0) = u0

u′(t0) = u′0

...

u(n−1)(t0) = u(n−1)0

(2.12)

Considering u1 ≡ u, the transformation to apply is the following:

u′1(t) = u2(t)

u′2(t) = u3(t)

...

u′n−1(t) = un(t)

u′n(t) = f(t, u1(t), u2(t), ..., un−1(t))

(2.13)

In systems of ODEs, it is usual to have components of the solution with total distinct behaviours: rapid

transients overlapping with slower evolutions. This phenomena results in stiffness of the system, where

sometimes it is the less relevant component of the solution is the one that determines the time step. Accord-

ing to [53], a problem is considered to be stiff, if the solution being sought varies slowly, but there are nearby

solutions that vary rapidly, so that the numerical method must take small steps to obtain satisfactory results.

Stiffness is then mainly a matter of efficiency: a non-stiff solver could solve a stiff problem, however it would

take a long time to do it!

The most popular methods for the solution of stiff problems are the BDF formulas [54]. These have a

very simple form when the step size is kept constant and backwards differences are used.

Other alternative for stiff systems is the use of Rosenbrock formulas. These are classified as linearly

implicit because the computation of un+1 requires the solution of a system of linear equations. These

formulas require, however, the evaluation of the derivative of f at the beginning of the step. Due to the

necessity of knowing the value of the derivative (Jacobian), some alternative formulas were developed: the

modified Rosenbrock formulas. These can be studied in detail in [54, 55, 56]. The main point in these

modified formulas is that they do not depend on having an accurate Jacobian [54].

The MATLAB solver to use for the ODE presented in section 2.5, which required the ability to deal with a

stiff ODE, will be based in this modified Rosenbrock formulas. It is the ode23s solver, and it was chosen for

18

being more efficient and stable to solve stiff problems than BDF formulas. This solver is particularly effective

when Jacobians have eigenvalues close to the imaginary axis [54], and among all the solvers tested, it was

the one that proved to be more efficient.

2.3 Thermodynamic Model

When modelling processes of charging or discharging a tank with compressed air, there are two limit

situations for the processes: adiabatic or isothermal. The first one is usually related to fast processes,

where heat transfer does not have enough time to take place. The second, isothermal, is usually applied

when the process is long in time, and therefore heat transfer can appear, leading to a quasi-equilibrium

between the temperature of the stored air and the ambient temperature.

To choose which is the most adequate solution for this problem in particular, a heat transfer study was

performed to the tank. The objective here is to determine a characteristic time for heat transfer to take place,

and then, compare it to intended operating time of the system. Should the heat transfer time be higher than

the operating time, and the process can be considered fast, leading to the use of the adiabatic assumption.

On the other hand, if the characteristic time of heat transfer is lower than the operating time of the system,

the assumption used will be the isothermal.

Consider now the thermodynamic system of figure 2.2, where a non-adiabatic tank is filled with com-

pressed air. The tank initial temperature is given by T0, and the objective is to determine the law of temper-

ature evolution in the tank, until it reaches the ambient temperature, Tamb, of the surroundings. Therefore, a

heat transfer from the tank to the surroundings must take place, Q < 0.

Figure 2.2: Thermodynamic System for Heat Transfer study.

Performing a simple energy balance, and neglecting effects of kinetic and potential energy, the following

result is obtained from equation 2.3:

Q =d

dt(um) = mcv

dT

dt(2.14)

If the heat transfer rate is related to an overall heat transfer coefficient, Uc, it would then be given by:

Q = −AUc (T − Tatm) (2.15)

19

where the term A is the surface area of the tank. The global heat transfer coefficient, Uc, which includes

both internal and external convection, as well as conduction on the walls of the tank [57], can be given by:

1

Uc=

1

hout+L

k+

1

hin(2.16)

where the differences between inner and outer surface areas have been neglected.

A differential equation for the tank temperature can therefore be obtained. Rearranging the terms it is

given by:

− AUcmcv

dt =dT

T − Tamb(2.17)

Integrating both sides of the equation, an exponential law for the temperature evolution can be found,

given by the following:

T (t) = Tamb + (T0 − Tamb) e(−AUcmcv

t) (2.18)

It can be noticed the presence of the term AUc/(mcv) in the exponent, which can be related as the

inverse of a characteristic decay time, tdec, that is, the time it would took for the temperature to lower to

approximately 36.8% (1/e) of its initial value. The choice of the model to use will then be related to this

decay time: if it assumes low values, order of seconds, the model used will be the isothermal, since the

objective is to have high operation time of the system. Otherwise, if this decay time assumes high values,

the assumption used will be adiabatic.

To estimate a decay time, consider now a cubic tank of 1m3 volume and surface area 6m2, which will be

assumed equal for both inner and outer surfaces, since the thickness of the tank should be relatively small

(order of 1 cm). The values for convective heat transfer coefficients will be assumed 25 W/m2K, typical

for natural convection of air [57], and the thermal conductivity will be kept equal to 20 W/mK, typical of

stainless-steel. With these values, the value of the decay time can be estimated:

tdec =mcvAUc

=ρV cvAUc

(2.19)

and it results in a characteristic time of few seconds (order of tens of seconds).

With these results, and with the objective of having operating time of the system above half an hour, 1800

s, the model chosen was the isothermal. The long operating time will then assure that heat transfer will take

place and keep the air temperature almost constant. This was verified experimentally in [58], where it was

seen that the long operating times for both charge and discharge processes approximate the processes of

isothermal, instead of adiabatic.

Besides the time factor, it must also be taken into account the difficulty in having a perfectly adiabatic tank.

Even if the processes could be modelled as adiabatic, if we consider that the air can sometimes be stored

for a long time (weeks or even months), the heat transfer would eventually occur and lead to a reduction

of both temperature and pressure of air. It was also mentioned in chapter 1, that the best performance for

CAES systems was obtained with high heat transfer coefficients, which would approximate the process to

an isothermal evolution [43].

To achieve a quasi-isothermal process, if it does not happen naturally, it would be necessary to cool the

20

air, by using a heat exchanger. The advantage of storing air at ambient temperature, that is, the quasi-

isothermal process, is to avoid the heat losses during storage, as well as it allows to spend less energy

per unit mass to compress the air (figure 2.3), because for the same final pressure, it is possible to store

more air in the tank, if the temperature is kept lower (figure 2.4). The disadvantage of isothermal storage is

related to the lower temperature of the air when it expands through the turbine, which results in lower values

of power produced, and therefore may lead to a lower energetic density of the system.

Figure 2.3: Compression energy spent per unit mass of air stored for adiabatic and isothermal compression.

Figure 2.4: Stored Mass of Air function of final tank pressure for adiabatic and isothermal compression.

21

2.4 Tank Charging Process

The process of compressing the air into the pressure tank will have an energetic cost which must be

quantified. The first step in this section will be to obtain an equation that allows to quantify the total work

spent in the compression process. Considering the system of figure 2.5, with the tank initial conditions being

the same as the atmospheric, Tatm and patm, the desired equation is easily obtained by performing transient

mass and energy balances to the system, and integrating them between the initial and the final conditions,

when the tank is filled up to the desired pressure.

Figure 2.5: Thermodynamic System for Tank Charging Process.

Considering now the mentioned system, the mass balance from equation 2.1 simplifies as follows:

dm

dt= mi (2.20)

The energy balance, equation 2.3, neglecting the effects of kinetic and potential energy changes is simply

given by:d(um)

dt= Q− W + mihi (2.21)

where the term hi refers to the atmospheric air enthalpy, since the air used to fill the tank is extracted from

the atmosphere.

Rearranging the latest equation, and applying the ideal gas model, that is, du = cvdT and dh = cpdT ,

the following equation is easily obtained:

d

dt(mcvT ) = Q− W +

dm

dtcpTatm (2.22)

Recalling now that the temperature of the gas in the tank, T , will remain constant, and considering

constant specific heats, as earlier mentioned, the previous equation can be further simplified:

Q− W =dm

dt(cvT − cpTatm) (2.23)

Assuming now that the temperature in the tank will remain constant and approximately equal to the

22

ambient temperature, T ≈ Tatm, and applying the ideal gas law (equation 2.5), the following can be written:

Q− W =V

RT

dp

dtT (cv − cp)⇒ −W = −V dp

dt− Q (2.24)

where the relation cp − cv = R was here used.

Integrating in time the previous equation, the total compression work can be obtained:

−Wc = −V (p2 − p1)−Q (2.25)

where Wc =∫ 2

1W is the total energy spent in the process. Since the convention here used states that

the work done in the system is negative, −Wc = |Wc|, and so the total compression work will be simply

expressed by Wc, representing its absolute value.

The remaining unknown is the heat transferred from the system. This can be obtained using an entropy

balance (equation 2.4), assuming a reversible process, which for the present system simplifies as follows:

d

dt(sm) =

Q

T+ misi (2.26)

Integrating the equation in time, and rearranging the terms (remember that s1 = si since the initial

condition in the tank is equal to the atmospheric), one obtains:

Q

T= m2(s2 − si) =

p2V

RT

[cp ln

(T2

Tatm

)−R ln

(p2

patm

)]⇒ Q = −p2V ln

(p2

p1

)(2.27)

since the temperature in the tank is assumed to be kept constant and equal to the atmospheric.

Inserting this result in equation 2.25, the final expression for the total compression work is obtained:

Wc = V

[(p2 − p1) + p2 ln

(p2

p1

)](2.28)

It is important to discuss here the assumption of reversible process, since it may not be evident at first

sight. For a heat transfer process to be considered reversible, it should occur with infinitesimal temperature

and pressure differences. Since the temperature in the tank must remain constant and equal to the am-

bient temperature, there is almost no spontaneous entropy generation due to temperature gradients [49].

However, there is a pressure increase in the tank during the process, which could lead to the generation of

entropy. Even considering this, we can still consider the process to be reversible, if the process is slow, that

is, if the pressure evolution in the tank is slow enough to avoid high pressure variations in time, which would

lead to entropy generation. Therefore, considering a quasi-equilibrium between the tank and the ambient,

and small pressure increases in each instant, the approximation of reversible process can be used. This will

not represent exactly the reality, but still would provide a reference to be used, when calculating the value of

total compression work used to fill the tank.

Using now the result from equation 2.28, and varying the tank final pressure from 10 to 100 bar, and

the tank volume from 200 to 1000 L, the evolution of total compression work with these two variables can

be observed in figure 2.6. It is easily concluded that if we use small power compressors, with powers of

23

the order of 10 kW, the total time to fill the tank would reach the order of hours (thousands of seconds) in

some cases (the total compression energy is of the order of MJ), which is much higher than the decay time

obtained in section 2.3, validating then the assumption of isothermal evolution, in case of using small power

compressors, which is the objective if we consider once again the idea of portability.

Figure 2.6: Total Compression Work as a function of final tank pressure and volume.

2.5 Tank Discharging Process

After having the compressed air stored in the tank, it will be discharged at a constant pressure, to expand

through a turbine and produce the desired shaft power. In this section, the model of the discharging process

is explained in a detailed way. Considering the system of figure 2.7, the model will have several important

assumptions:

i. The air in the tank will be at constant temperature and in equilibrium with the environment: T = 290K.

ii. The velocity in the tank is negligible, that is, we have stagnation conditions in the tank.

iii. The flow will be assumed to be 1D, that is, any changes along y and z will be neglected.

Starting with the NS equation along x direction, equation 2.2, and noticing that U ∂U∂x = 1

2∂U2

∂x , integrating

the NS equation along x direction, from 1 to 2 (figure 2.7), allows to write:

∫∂U

∂tdx+

U22 − U2

1

2+

∫dp

ρ=

∫ν∇2Udx (2.29)

which is equivalent to the non-steady Bernoulli equation, when the right-hand side term is zero, that is,

the situation of inviscid flow. This term has units of (m/s)2 and can be expressed in terms of a head loss

24

Figure 2.7: Thermodynamic System for Tank Discharging Process

coefficient in the pressure regulator [50]:

∫ν∇2Udx = −Kv

U2

2(2.30)

In fact, this head loss in the valve is a function of the pressure difference upstream and downstream of the

valve, however, to simplify the model and the solution of the ODE, its value will be kept constant during the

operation of the system.

Since the velocity in the tank is negligible, and considering a constant cross section on the exit duct, the

velocity should remain constant along the pipe, as well as its time variation, leading to:

∫ 2

1

∂U

∂tdx ≈ LdU

dt(2.31)

where L refers to the length of the duct.

With the perfect gas law (equation 2.5), and considering the constant temperature of air in the tank, it is

possible to rewrite: ∫ pe

p

dp

ρ= RT

∫ ρe

ρ

dρ

ρ= RT ln

(ρeρ

)= RT ln

(pep

)(2.32)

where the index e refers to the exit condition, that is, downstream of the pressure regulator.

Substituting these results in equation 2.29, the following equation is obtained:

LdU

dt+RT ln

(pep

)+

(1 +Kv

2

)U2 = 0 (2.33)

It is important now to relate both pressure and flow velocity, which can be easily done by applying a mass

balance to the system, as follows:

dm

dt= −m⇒ V

dρ

dt+ ρeAeU = 0 (2.34)

From the previous equation, with dρdt = V

RTdpdt , it is easy to obtain both velocity and its derivative:

U = − V

peAe

dp

dt⇒ dU

dt= − V

peAe

d2p

dt2(2.35)

where V is the volume of the tank and Ae is the cross section of the exit duct.

25

Substituting these results in equation 2.33, the final ODE is obtained.

d2p

dt2− RTpeAe

LVln

(pep

)+

(1 +Kv)V

2LpeAe

(dp

dt

)2

= 0 (2.36)

Since it is a second-order differential equation, two initial conditions are required. The first is the initial

pressure of the tank, p0, being the second condition applied to the first derivative of the pressure at t = 0,

where it is stated that dpdt

∣∣∣t=0

= 0, since the value of the velocity U in t = 0 must be zero.

Observing the equation, it is noticed that there is a logarithmic term that complicates its solution, since

it makes the equation non-linear. Therefore, as already mentioned in section 2.2, and following the recom-

mendations from [odesuite], the solver ode23s was chosen. To test this model, another MATLAB routine

was defined, and some test values for the variables were assumed (table 2.1).

Table 2.1: Values of Parameters used to test the Model

Parameter Units ValueTank Pressure bar 100Tank Volume L 1000Discharge Pressure bar 2.5Exit Duct Length m 0.5Exit Duct Diameter cm 1Valve Head Loss Coefficient - 20

Running the MATLAB routine, the discharge of the tank, until the pressure in the tank matched the

discharge pressure, pe, shown the evolution depicted in figure 2.8.

Figure 2.8: Pressure Evolution during discharge of tank

It is easily seen that the the pressure starts to decay almost linearly, but as it approximates the value of

the discharge pressure, imposed by the pressure regulator, its derivative tends to zero. It is also observed

that with these parameters, the total discharge time reaches almost one hour (3600 s), which is also several

26

orders of magnitude over the decay time for heat transfer, validating once again the hypothesis of isothermal

process.

2.6 Energy Production

After obtaining the pressure evolution in the tank, and considering the constant value of the temperature,

it is easy to obtain the time evolution of the total mass of air inside the tank, just by using the Ideal Gas Law.

Knowing the value of the mass of air in the tank in each instant, the mass flow rate leaving the tank can

be calculated, as follows:

m =mt+1 −mt

∆t(2.37)

where the superscripts t + 1 and t are two different time instants, separated by an interval of time ∆t, that

matches the time step used by the numerical solver of the ODE.

The turbine output power in each instant of time can be calculated by:

Wt(t) = m(t)cp (T1 − T2) = m(t)cpT1

(1− T2

T1

)(2.38)

where 1 and 2 refer to the inlet and outlet of the turbine, respectively.

Using an isentropic efficiency, ηt, the real power produced by the turbine can be related to the ideal value

of power, that would be produced if the expansion process in the turbine was isentropic. This consideration

allows to write the power produced by the turbine as a function of its expansion ratio.

Wt(t) = ηtWt,s(t) = ηtm(t)cp

[1−

(patmpe

) γ−1γ

](2.39)

With the results from section 2.5, the turbine power can be obtained from t = 0 until the end of the

discharge process, leading to the results depicted in picture 2.9, which were obtained for the same conditions

of figure 2.8 (table 2.1). It is easily observed that the turbine power slowly decreases at the beginning, but

then falls quickly until zero, when the operation stops, that is, when the pressure time derivative starts to

tend to zero.

Having now the turbine power, the total energy produced, Wt, can be obtained by integrating the turbine

power, that is:

Wt =

∫Wtdt (2.40)

Since the solver used, ode23s, uses a 2nd order precision method, the integration rule chosen should be

also 2nd order, and therefore, the trapezoidal rule of integration will be hereby used, leading to the following

result:

Wt =

∫Wtdt ≈

N−1∑i=1

W i+1t + W i

t

2∆ti (2.41)

where N refers to the total number of points where the pressure was calculated.

27

Figure 2.9: Turbine Power Production during Tank Discharge

2.7 Model Results

Having now a mathematical formulation for the operation of the system, it is interesting to explore the

influence of some parameters in the system performance. It was seen that some of the variables have more

influence than others, being presented in this section some of the most important results collected. All the

results presented will use the values of the variables presented in table 2.1, changing only the value of the

variable whose effect is intended to be studied.

For example, it is easy to notice the influence of both tank pressure and volume in the total amount of

energy produced, depicted in figure 2.10. As expected, the increase in both tank pressure and volume leads

to the production of larger amounts of energy.

Figure 2.10: Total Energy Produced function of tank pressure and volume.

28

This leads to a problem in designing the system: the objective is to have a system as small as possible,

producing large amounts of energy. A compromise must be found between the size of the system and the

amount of energy that we intend to produce!

It can also be observed that the increase in the value of the discharge pressure, similarly to the tank

pressure and volume, also leads to an increase in the total amount of energy produced (figure 2.11).

Figure 2.11: Total Energy Produced function of tank discharge pressure, for initial pressure p0 = 100 barand volume V = 1000L.

Another important variable to consider is the total operating time: it must be as high as possible, to allow

the production of energy for the maximum time possible. However, it decreases with the increase of the

discharge pressure (Figure 2.12), also decreasing the total amount of energy produced, as previously seen.

Figure 2.12: Total Operating time function of Discharge Pressure for initial pressure p0 = 100 bar and volumeV = 1000L.

29

Besides the total energy produced, it is also important to consider the power provided by the system,

since it would not be interesting to have a system operating for a large time, but providing low values of power

that would not be enough to feed the electrical devices. It was here observed that the parameter with most

influence on the maximum power obtained in the turbine was the discharge pressure: the higher its value,

the higher the maximum mass flow rate, as well as the expansion ratio in the turbine (ratio between inlet

and exit pressures). It was then observed, a quasi-linear evolution for the maximum power, by increasing

the value of the discharge pressure (figure 2.13).

Figure 2.13: Maximum Turbine Power, function of Discharge Pressure for initial pressure p0 = 100 bar andvolume V = 1000L.

Considering now the fact that we are designing a power production system, using a form of energy

storage, there are two important concepts that must be considered: the total efficiency of the system, η, and

the energetic density of the stored energy, ρenergetic. Both these variables are respectively defined as:

i. System Efficiency:

η =Wt

Wc(2.42)

where Wt represents the total work produced, and Wc the total work consumed to fill the tank, as

described previously in this chapter.

ii. Stored Energy Density:

ρenergetic =Wt

mstored= Wt

RT

p0V(2.43)

where p0 and V represent the stored air pressure and the tank volume, respectively.

As it can be observed in figure 2.14, an increase in the pressure of the tank leads to a larger increase in

the required compression work, than in the total energy produced, which can be translated as a decrease

in the system efficiency. Therefore, to have a more efficient system, this should be designed reducing the

tank pressure, which can be equivalent to say that less mass of air must be stored in the tank. Regarding

30

the system characteristic dimensions, this could be translated in designing a smaller system, if we consider

the stored mass of air as a measure of the system size.

Figure 2.14: Total Compression Work and Total Expansion Work as a function of p0.

However, when dealing with the concept of portability, sometimes it is more important to consider the

capacity of the system to produce as much energy as possible, while keeping the smallest size possible,

even if it has a low efficiency. Therefore, it is very important to consider the amount of stored energy per unit

mass, ρenergetic, which contrary to the system efficiency, increases with the increase in the value of the tank

pressure p0, as depicted in figure 2.15. It is then concluded that, in order to increase the energetic density

of stored energy, the system must be designed with larger dimensions, that is, in other words, increasing the

tank pressure.

Figure 2.15: Stored Energy Density as a function of p0.

31

It can also be observed in the previous figure that for lower values of pressure, this increase is more

accentuated, as with the increase of the tank pressure, the value of ρenergetic tends to stabilize. It is also

easy to conclude that these values are lower than the values presented by other energy storage technologies

(section 1.3), however, it must be remembered that this system intends to use wastes of produced energy,

and unlike some of the other options mentioned, it has low environmental impact. Comparing the values

of energy produced per volume with the values presented in table 1.2, it can be seen that the values here

obtained are within the range of the systems presented in the literature. For a 1m3 tank, with a pressure of

100 bar, the energy produced per unit volume was approximately 1.7 kWh/m3.

It is interesting to notice, that by increasing the size of the system, the system efficiency tends to de-

crease, while the energetic density of the air tends to increase. It is therefore necessary to find a compromise

between these two variables: we cannot maximize both at the same time as we would like, and therefore, a

compromise between these two variables must be found, which will be the objective of the following chap-

ters.

32

33

Chapter 3

Optimization of System Operation

In the previous chapter, the mathematical model of the system was defined. By observing the results

shown in chapter 2, it could be seen that some parameters have more influence than others in the system

performance, as well as that some of the most important performance variables have distinct behaviours

when we vary the system operating conditions. It would be now important to find out what should be the

values of those parameters that assure the best operating conditions of the system.

In this chapter, it is presented the Optimization of the system parameters that allowed to obtain the

best possible conditions. It begins with a short presentation of Multi-Objective Optimization (MOO) and

a description of the algorithm used: the Direct Multi-Search (DMS) method. After that, the choice of the

independent variables is explained, as well as the objective functions and the restrictions attached to them.

The chapter then ends with a brief explanation of how the DMS algorithm was here implemented, to obtain

the desired results.

3.1 Multi-Objective Optimization

Many engineering problems involve the optimization of more than one function, MOO, and so they have

more than one optimal solution, known as Pareto optimal solutions. In a MOO problem, two or more objective

functions are supposed to be simultaneously optimized. According to [59], a MOO problem is defined as:

min f(x) = (f1(x), f2(x), ..., fm(x))

s.t. x ∈ Ω(3.1)

where m (≥ 2) is the number of total objective functions. The feasible region, Ω, corresponds to the set

of points that satisfy the problem constraints, that is, the solution domain. Note that in case of trying to

maximize an objective function, fk, what is performed instead is the minimization of the symmetric of the

function, −fk.

For many engineering problems, however, both objectives and constraints do not have associated a

mathematical function. They are then treated as black-box functions [60], which are often evaluated by

software tools, which with given inputs return the values of the objective functions.

34

A point x∗ in the feasible design space Ω, is called a Pareto optimal solution, if there is no other point x

in the set Ω that reduces at least one objective function, without increasing another one [61], that is, if there

is no point x ∈ Ω such that f(x) ≤ f(x∗) with at least one f i(x) ≤ f i(x∗). The set of all Pareto optimal

points is then called the Pareto optimal set.

Another important concept in MOO problems is related to dominance. According to [61], a vector of

objective functions, f∗ = f(x∗) in the feasible space Ω is nondominated, if and only if, there is no other

vector f in the set Ω such that f ≤ f∗, with at least one fi ≤ f∗i . Otherwise, f∗ is dominated.

Graphically, the concept of Pareto dominance is represented in figure 3.1, which shows a Pareto front

for a two-objective optimization problem. It is easily seen that the points over the line designated as Pareto

front, are better solutions than the points in the sub-optimal plan (blue squares). Taking as example point a,

it can be said that it dominates point b, since it verifies simultaneously two conditions: a is no worse than b

in all objectives and a is better than b in at least one objective, resulting in the situation that b is dominated

by a [62].

Figure 3.1: Pareto front for a two-objective problem [62].

It is easily noticed that the definitions of nondominanted solution and Pareto optimal point are quite

similar. The difference, however, is reflected in the fact that the concept of Pareto optimal point is related

to the space of independent variables, while the concept of nondominance is related to the points in the

criterion space, that is, the space of the objective functions. It is usual, however, to use the concept of

Pareto optimality for both spaces [63].

3.2 Direct Multi-Search Algorithm

There are several algorithms in the literature that can be applied to MOO problems (see [64]). The so

called derivative-free methods, that is, methods that do not require the explicit knowledge of the derivatives of

the objective functions, are particularly appropriated for situations when the computation of these derivatives

is expensive, unreliable or even impossible [65]. Therefore, these methods seem to be the most adequate

for black-box problems.

35

The Direct Multi-Search Algorithm (DMS), from Custodio et al. [65], is an extension from single to multi-

objective optimization, from a class of derivative-free methods, called direct search. Given a current iterate,

a poll center, the poll step evaluates the values of objective functions at some neighbour points, defined

by a step size parameter. The acceptance criteria for the new points is related to the concept of Pareto

dominance, which then requires updating a list of feasible and nondominated points. This list is updated

in each iteration, saving only the nondominated points, reason why it represents an approximation of the

Pareto front, and from which the poll centers for the next iterations will be chosen. An iteration is considered

to be successful if only the list changes, that is, if a new feasible nondominated point was found and added

to the list. In case of an unsuccessful iteration, the corresponding step size parameter is decreased and the

search is performed again in a region closer to the poll center.

As previously mentioned, this method is an extension of direct-search algorithms, to MOO problems.

Graphically, a direct-search algorithm proceeds as depicted in the sequence of figures 3.2. The difference

to DMS is related to the existence of more than one objective function. The DMS, similarly to the direct-

search algorithms, evaluates the function in several points of the domain, at a certain distance from the

initial point (figure a). In case of finding a new nondominated point, this will be the center for the next poll

(figure b), and so on (figure c). When a poll cannot find a new minimum, the step size is reduced (figure d),

and then the algorithm goes on (figures e, f and g), until the step size cannot be further reduced and so, the

minimum is found (figure h).

Figure 3.2: Direct-Search Algorithm [66].

The way the DMS algorithm deals with constraints is similar to other derivative-free methods. It uses

the concept of extreme barrier function, that is, a function that assumes an infinite value in case of finding a

point that does not belong to the feasible region Ω, as follows:

fΩ(x) =

f(x) if x ∈ Ω

(+∞, ..., +∞) otherwise(3.2)

36

The algorithm also presents several options for initialization of the variables: line sampling, random

sampling, Latin hypercube sampling or with a user defined list. It was hereby verified, after several tests

performed with the algorithm, that all of these options provided the same results, but the most efficient

sampling method was line sampling, and so it was chosen.

Regarding the stopping criteria, it had two options:

i. Maximum number of iterations: 20, 000;

ii. Minimum step size: 10−1.

As soon as one of these two conditions was verified, the algorithm would stop immediately. A more

detailed description about the DMS algorithm, particularly about its mathematical formulation can be found

in Custodio et al. [65].

3.3 Independent Variables

By looking at the model presented in the chapter 2, it is easy to understand what are the independent

variables that control the operation of the system. Therefore, seven main variables were chosen as the

independent variables that will control the operation: the tank pressure p0 and its volume V , the exit duct

diameter de and length L, the head loss in the pressure regulator Kv, the discharge pressure pe and the

isentropic efficiency of the turbine ηt. All of these variables will assume discrete values, and the choice of

the values that they will assume was based on typical values found in the literature for portable systems, as

well as values based on analysis made to the model.

The list of variables, as well as their range of variation is presented in table 3.1. These will be the values

that the optimization algorithm will use to search for the optimal solutions.

Table 3.1: Independent Variables for MOO

Variable Units Range of Variation IncrementsTank Pressure bar 20 - 100 5Tank Volume L 200 - 2000 200Discharge Pressure bar 1.5 - 5 0.5Exit Duct Length m 0.5 - 1.5 0.25Exit Duct Diameter cm 0.5 - 2.5 0.5Valve Head Loss Coefficient - 0.5 - 24.5 2Turbine Efficiency (%) - 60 - 90 5

These variables will then be used by the MOO algorithm, as it will use values from this list to calculate

new values of the objective functions in each iteration, and then evaluate whether the solution found is

optimal or not, proceeding to the following iterations.

3.4 Objective Functions

After defining which variables will influence the performance of the system, it is important to decide what

should be the objective functions to evaluate. Considering the idea of portability, it is important to keep

37

the size of the system the minimum possible. However, the system must be able to produce the maximum

amount of energy possible. As previously seen in section 2.7, these are conflicting objectives, that is, to

maximize the amount of produced energy, it is also necessary to maximize the size of the system. Therefore,

the two objective functions here considered were the total energy produced, Wt, which is intended to be

the maximum possible, while the second function was the mass of stored air in the tank, mstored, which

is intended to be kept as low as possible. The choice of this objective function considered the fact that

minimizing the mass of air would allow to minimize also the tank volume, as well as the storage pressure,

which makes easier to compress the air and store it.

At this point, one may think why not maximize instead the stored energy density. The reason for not

doing so, is related to the fact that it would only give a single point, with maximum energetic density in the

system. The idea of performing a MOO analysis is to obtain several points, and several configurations, from

which a system can be designed, depending on the intended application.

Other variables that could have been here considered as objective functions would be either the system

efficiency or the investment costs. However, considering that this system must be portable, it is more

important to increase its energetic density, rather than its efficiency, and so the efficiency was discarded as

an objective function. Regarding the cost of the system, it was seen that it increased with the size of the

system, and therefore, minimizing the total investment cost was almost equivalent to minimize the size of

the system, as it will be seen later in chapter 4.

In addition to the objective functions previously mentioned, some restrictions to the system had to be

imposed. The first restriction imposed was related to the total operating time, that is, the tank discharge

time, since it would not be viable to have a system that could operate only for a few seconds. Therefore,

a minimum of half an hour (1800 s) was required, discarding all the solutions that did not comply with this

constraint. The second and last constraint imposed was related to the turbine power. Despite being a

portable system, it should be able to provide enough power to feed several electronic devices, for example,

in a military field, as mentioned in chapter 1. Therefore, if the maximum power produced by the turbine was

below 1 kW, the solution would be considered unacceptable, and therefore discarded.

The optimization problem is then formulated as follows:

min f(x) = (−Wt(x),mstored(x))

s.t. ∆t ≥ 1800s, Wt ≥ 1kW, x ∈ Ω(3.3)

where x denotes the independent variables and x ∈ Ω intends to define that those variables must assume

the values presented in table 3.1, that is, to the feasible region.

3.5 Implementation

After defining the variables that control the system performance, as well as the objective functions, the

next step is the implementation of the algorithm, which was performed using a MATLAB routine. The first

part consisted of developing a function, designated as Simulator (see Appendix A), which by providing the

values of the independent variables, was able to calculate the values of the objective functions, as well as of

38

the two restrictions.

With this routine, the program with the DMS algorithm, provided by the authors of Custodio et al. [65],

chose the point to make the initialization, and then proceeded with the algorithm described in section 3.2. If

an iteration was successful, the program saved that point in a list, and moved forward until one of the stop-

ping criteria was verified. Basically, the DMS algorithm was used to update the values of the independent

variables from one iteration to another, and decide if the values of the objective functions found were optimal

or not. The process flow chart is represented in figure 3.3, which shows the path to follow if we intend to

perform a MOO analysis.

Figure 3.3: Implementation of the MOO Procedure

It can be seen in figure 3.3, that all the optimal solutions found were saved, and the non-optimal dis-

carded. The final output of the program was a list with the optimal Pareto points, as well as the values

assumed by the independent variables in those points. The results obtained with the DMS algorithm, as well

as their analysis and discussion, will be presented in the following chapter.

39

40

Chapter 4

Results and Discussion

After defining the optimization analysis to perform, described in chapter 3, it is now important to look at

the results obtained with the algorithm and discuss their meaning. This chapter starts with the presentation

of the results provided by the DMS algorithm, and then, the discussion of what they represent in terms of the

system configuration. A comparison between the several types of systems obtained will also be made. To

conclude the chapter, a simple economic analysis will be performed, that intends to make a rough estimate

of what would be the required investment to obtain a system like the ones presented here.

4.1 Pareto Fronts

The first test performed consisted of applying the MOO algorithm to the objective functions defined in

chapter 3, without imposing any of the constraints mentioned, that is, neither applying restrictions to the

operating time, nor to the output power. By running the algorithm several times without the constraints

applied to the objective functions, a Pareto front with several points was obtained, as depicted in figure 4.1.

Figure 4.1: Pareto front obtained without restrictions to the objective functions.

41

The first interesting aspect to notice is the unusual linear shape of the Pareto front, instead of the typical

hyperbolic shape. This can be easily explained if we think that due to the aim of designing a portable

system, we are only considering small dimension systems (section 3.3), and therefore, the Pareto front

obtained shows only the final part of the typical curve, which is usually quasi-linear, as shown in figure 4.2).

Figure 4.2: Quasi-linear shape of the final part of a typical Pareto front [62].

It is also interesting to notice that there is a great concentration of optimal points in the lower right corner

of figure 4.1. Since these are points that represent small systems, it is likely that they will not obey to the

restrictions defined.

Therefore, as expected, after imposing the restrictions in both operating time and power, only a few

optimal points remained (figure 4.3). It is concluded that several of the points obtained without the restrictions

were able to obtain an optimal compromise between the total size of the system and its ability to produce

energy, but they would do it only either for short periods of time or with low values of power.

The values of all the variables corresponding to the optimal points can be found in Appendix B.

Figure 4.3: Pareto front obtained with restrictions to the objective functions.

42

Comparing now the two Pareto fronts, from figures 4.1 and 4.3, the first thing to notice is that most of the

points that disappeared, when the restrictions were introduced, were located in the lower right corner, that

is, the region of lower mass of stored air, which means small systems. As previously mentioned, this was

already expected, since the smaller systems are likely to present more difficulty in operating for long time,

at the same time that produce a high output power.

Analysing the results obtained, it is seen that, as expected, the turbine efficiency is always kept as high

as possible, that is, at 90 %. Besides this, it was interesting to notice that the exit duct diameter was always

equal to 1 cm, and the duct length varied only between two values: 1 and 1.5 m. It was also interesting to

notice that the tank pressure was always kept above 50 bar, and the discharge pressure assumed values

laying between 3.5 and 5 bar, being higher for the larger systems.

Regarding the overall efficiency of the system, its value was almost always close to 30 %. It is im-

portant to mention that this value is above the real one: the compression process was modelled as being

reversible, which means that in the real situation more energy will be required to compress the air than the

one calculated here.

It is also important to refer here the importance of having performed a MOO analysis, since at this point

one may think why not to perform a sensitivity analysis instead. It would be simpler than the application of

a complex MOO algorithm, and perhaps the same results could have been obtained. However, if we take

a look at the results obtained with the algorithm, it is observed that in order to obtain the Pareto front from

figure 4.3, hundreds of different configurations were analysed (in the last simulation performed, 264 different

combinations were analysed). If we decided instead to vary one variable each time, and try to look for the

best configurations by trial and error, the probability of finding all the optimal points from figure 4.3 would be

very low.

4.2 Optimal Configurations

As earlier mentioned, since we are here designing a portable system, all the obtained configurations

have relatively small dimensions. However, by looking at figure 4.3 and to the results from the algorithm, it

is easily seen that the optimal points can be split in several regions, and then, a comparison between what

would be the typical configuration of the system, in each of these regions, can be made. Therefore, the

Pareto front obtained will be split in three main regions (figure 4.4), from which it is interesting to study in

detail the upper one and the lower one. The criteria used here to decide how to split the regions established

that the lower region contained the points with stored energy density below 85 kJ/kg and storing masses of

air below 130 kg, as well as a discharge pressure below 5 bar, while the upper region would contain points

with both stored energy density above 89.5 kJ/kg and stored mass of air above 190 kg. In fact, these are

all small systems: the aim of this work is to develop a portable system. Nevertheless, it is interesting to

distinguish these two regions and observe some distinct behaviours of the systems from each region.

The region from the left (region 1) depicts systems that present a larger capacity of producing energy, but

that as a consequence, present larger dimensions. Therefore, these will be designated as large systems.

On the other hand, the region from the right (region 2), represents systems with smaller dimensions, that do

43

not require as large amounts of air as the previous ones, but that for this reason are not able to produce as

large amounts of energy as the large systems. The systems from region 2 will be then called small systems,

from now on. Note again that the designation of large and small systems is just a way of distinguishing both

regions: in fact all the systems present in the Pareto front are somehow small, since the idea of this work is

to develop a portable system!

Figure 4.4: Pareto front obtained with restrictions to the objective functions split in different zones.

Analysing now the large systems in detail, it is easily seen that they require higher values in three of the

operating variables: tank pressure, tank volume and discharge pressure. As seen in section 2.7, the increase

of these three variables led to an increase in the amount of energy produced by the system, therefore it was

already expected that the values assumed by these variables would be higher in this situation.

It is also interesting to notice that the efficiency of these systems present values between 30 and 31 %,

while the smaller systems can present either lower or higher values: their efficiencies present values ranging

from 27 to 32 %.

The large systems from this Pareto front are able to operate for more than 2100 s, that is, 35 minutes,

while the small systems here obtained can go up to 32.5 minutes. The power produced by large systems

was almost double of the small systems: the first could present as an output a power of 13.6 kW, while the

latest could no go beyond 7.8 kW. It is interesting to notice that the operating time of both large and small

systems are relatively close, which could not be expected initially, due to the difference in the size of the

systems. This has an obvious consequence: in order to operate for the same time as the large systems,

the output power of the small systems must be smaller, which is reflected in lower values of the discharge

pressure.

One of the most important variables to study, as previously mentioned in this work, was the energetic

density of the stored air. It was also verified that the stored energetic density of the large systems was higher

than the one of the small systems, reaching values up to 90 kJ/kg, that is, approximately 3 kWh/m3, within

the range of the systems presented in the literature (chapter 1). On the other hand, the energetic density of

44

the small systems did not go beyond 84.6 kJ/kg (roughly 2.1 kWh/m3).

A summary of the comparison between large and small systems, including the range of variation of their

operating variables, as well as of some of the system performance indicators can be found in table 4.1.

Table 4.1: Comparison of Large and Small Optimal Systems

Variable Large Systems (region 1) Small Systems (region 2)Tank Pressure [bar]: 85 - 100 50 - 75Tank Volume [L]: 1600 - 2000 1000 - 1600Discharge Pressure [bar]: 5 3.5 - 4.5

Systems PerformanceEnergy Produced [MJ] 17.4 - 21.7 6.9 - 10.7Operating Time [min]: 31.1 - 35.6 30.3 - 32.5Output Power [kW]: 11.5 - 13.6 5.0 - 7.8Efficiency [%]: 30.1 - 31.2 27.1 - 31.7Stored Energy Density [kJ/kg]: 89.5 - 90.3 71.9 - 84.6

As mentioned, the energetic density is probably the most important parameter in these systems: for

reasons related to the portability, more than an efficient system, it is important to have a system that can

produce as much energy as possible, with the smallest dimensions possible. Following this idea, two sys-

tems were chosen to be studied in more detail: the systems from region 1 and 2 that presented the best

energetic density of each region. Taking a look at figure 4.4, it is easy to notice two filled red circles, one

in region 1 and one in region 2, that represent respectively the large system and the small system to be

studied in the next two subsections.

4.2.1 Large System

Starting with the large system, it is observed that both the value of the tank pressure and discharge

pressure assume the highest values possible, respectively 100 bar and 5 bar. The remaining of the operating

parameters of this system are shown in table 4.2.

Table 4.2: Large System Configuration

Variable Units ValueTank Pressure: bar 100Tank Volume: L 1800Discharge Pressure: bar 5Exit Duct Length: m 1Exit Duct Diameter: cm 1Valve Head Loss Coefficient: - 4.5Turbine Efficiency (%): - 90

This system stores a mass of air of 216.3 kg in the tank, being able to produce a total amount of energy

of 19.54 MJ, while running for a total time of 1904 s (31.7 minutes). The maximum output power of this

system is of 13.6 kW, decreasing in time, as seen in figure 4.5, since the mass flow rate leaving the tank is

decreasing in time, as seen in chapter 2.

45

The efficiency presented by this system has a value of 30.1 %, while the energetic density is of 90.33

kJ/kg, or in an equivalent way, 3.01 kWh/m3.

Analysing both values of these two last parameters, efficiency and stored energy density, a comparison

between them and the results from literature presented in chapter 1 can be established. It is easy to verify,

that the stored energy density here obtained was slightly above the literature values, which is of course an

improvement. Regarding the overall efficiency of the system, its value of approximately 30 % is within the

range of values found in literature (table 1.2 - Small CAES).

Figure 4.5: Power developed by the Large System.

4.2.2 Small System

Performing now exactly the same analysis for the small system, it is seen that both the values of the

operating variables are not as high as in the case of the large system, as expected. All the operating

parameters of this system are then shown in table 4.3.

Table 4.3: Small System Configuration

Variable Units ValueTank Pressure: bar 75Tank Volume: L 1400Discharge Pressure: bar 4.5Exit Duct Length: m 1.5Exit Duct Diameter: cm 1Valve Head Loss Coefficient: - 10.5Turbine Efficiency (%): - 90

In order to operate this system, a mass of air of 126.2 kg is stored in the tank, allowing the system

to produce an amount of energy of 10.68 MJ, and running for a total time of 1855 s (30.9 minutes). The

maximum power developed by the system is of 7.8 kW, also decreasing in time, as seen in figure 4.5.

46

This system has an overall efficiency of 30.7 %, while the energetic density presents a value of 84.65

kJ/kg, or in an equivalent way, 2.12 kWh/m3. Comparing also the values of these variables with the results

from literature (chapter 1 - table 1.2), it is easy to see, that both stored energy density and the overall

efficiency of the system are within the expected ranges.

Figure 4.6: Power developed by the Small System.

4.3 Cost Analysis

When considering the severe market competition for energy production solutions, it becomes evident the

need of evaluating the investment costs for the proposed system. In order to do that, a simple model is to

be developed in this section. The model will divide the total investment cost in three components, each one

associated to the devices that compose the system: tank, turbine and pressure regulator. The total cost will

be obtained by summing the costs of the individual parts.

To obtain the costs of the individual parts, a simple market research was performed. By defining the

parameters that most influenced the price of each device, some functions were adjusted to the typical

market prices, so that the total cost of the system, depending on its configuration, could be estimated. The

market data analysed, as well as the fitting curves obtained are presented in Appendix C.

Regarding the tank, and considering typical values of tanks that could be easily transported, it was seen

that the volume had the highest influence in the price. Therefore, a simple quadratic function was adjusted to

the market values, and it allowed to estimate the tank cost, f1(V ), which is given by the following equation.

f1(V ) = −0.0016 V 2 + 5.1434 V + 633.49 (4.1)

where the tank volume, V , is measured in liters.

In a similar way, for the turbine a function was defined, using the power as the variable that controlled

the cost, f2(Wt). It is important to notice that there are only few values available, and most of them for

47

higher-power turbines (1 MW), which can make these values not precise. Due to that, instead of adjusting a

curve to the market values, an average value of the cost per kilowatt was found, allowing to obtain the cost

of the turbine as follows.

f2(W ) = 460 Wt (4.2)

where the turbine power, Wt, is obviously measured in kilowatts.

Finally, the cost of the pressure regulator was obtained by performing a market research on gas valves.

It was seen the their costs were mostly dependent on their size, that is, on their diameter, which allowed to

obtain a function f3(de), given by the following equation.

f3(de) = 12.749 d2e + 57.573 de + 515.32 (4.3)

where the valve diameter, de, is measured in centimetres.

The total cost of the system could then be easily obtained as follows.

C = f1(V ) + f2(Wt) + f3(de) (4.4)

As previously mentioned, the energy used to compress the air is to be supplied without any cost, that

is, it is intended to use wastes of energy production from off-peak consumption hours. Since the operation

and maintenance costs are usually a function of the amount of fuel used to produce energy, this parcel of

the cost can be here neglected. Therefore, the total cost of the system will be equal to the investment cost,

given by equation 4.4.

Considering now the two systems mentioned in the previous section, that is the large system and the

small system, an estimate of their costs will be performed here. In order to evaluate what would be the total

time to recover the investment cost, an utilization of the system 8 times a day will be hereby considered.

It means that eight tanks of air will be emptied to produce energy everyday, which can seem to be rather

exaggerated, but will provide the reference for an average value: there will be times when these systems

will be operated almost continuously, and there will be others when they will be used only few times a day.

To perform the estimate of what are the savings in electricity, an average cost of electricity of 0.1633e/kWh

will be here considered, being this the typical value of the liberalized market according to ERSE [67].

Starting with the large system, the total investment cost obtained was of approximately 11,550e. The

cost of each component of the system is shown in detail in table 4.4.

Table 4.4: Cost of Large System

Device Variable Units Value Cost (e)Tank Volume L 1800 4,707.61Turbine Power kW 13.6 6,256.00Valve Diameter cm 1 585.64

Total 11,549.25

Considering the utilization of the system described above, and the price of the kilowatt of electricity, in

order to recover the investment cost of this system, this would have to produce an amount of energy of

48

254,606 MJ, which is equivalent to perform 13,057 cycles of operation. This may seem a large number, but

if we consider now the daily rate of utilization of the system, it would allow the system to operate 2190 times

per year, and so the investment cost could be recovered in approximately four and a half years, which can

be considered a small time compared to the typical lifetimes of power production systems.

Performing the same analysis for the small system, the total investment cost obtained was of approxi-

mately 8,870e. The cost of each component of the system is shown in detail in table 4.5.

Table 4.5: Cost of Small System

Device Variable Units Value Cost (e)Tank Volume L 1400 4,698.25Turbine Power kW 7.8 3,588.00Valve Diameter cm 1 585.64

Total 8,871.89

Thinking now the same way as before, to recover the investment cost of the small system, this would

have to produce a total of 195,567 MJ, that is, complete a total of 18,278 cycles of operation. Considering

the daily rate of utilization, this means that the investment cost could be recovered in six years and three

months, which can also be considered a short period, compared to the lifetime of the system (on the order

of decades).

49

50

Chapter 5

Conclusions and Future Work

As explained in detail in chapter 1, there is nowadays a great tendency to develop a more sustainable

power generation system, leading to the appearance of new technologies that are smaller in size than

the conventional power plants. The growing interest in Distributed Power Generation has also brought the

necessity of developing adequate forms of energy storage. Among all of them, the use of compressed air

has been one of the solutions that has gained the interest of many researchers in recent years, in particular,

the integration of this technology with Renewable Energy Sources, which would provide a simple way of

storing energy without emission of pollutants.

The aim of this work was then to design a small system to produce electrical energy on-site, using

compressed air to expand through a small turbine. This system could have several applications, but due to

its small size, that is, the aim of portability, its main one wpuld be the production of energy in remote places,

where network construction is not possible or economically viable.

To design this system, the first step taken consisted of obtaining a model that allowed to estimate the

values of the several variables that measure the system performance. Since it was seen that these variables

were dependent on several other parameters, a Multi-Objective Analysis was also performed, in order to

obtain the best operating conditions for the system.

With respect to all the results obtained in this work, it can be concluded that there is a clear potential

for the development of portable power generation systems using Compressed Air Energy Storage. These

systems can use wastes of energy production from off-peak consumption hours to compress atmospheric

air and store it in high-pressure tanks, to be later used to expand through a turbine and produce electricity.

Therefore, the system presented in this work can show two major advantages, all of them in accordance

with the trends from the International Energy Agency for the future of the power generation system: the

first one is related to its portability, that is, the ease of transporting the system from one place to another,

whenever required; the second related to the fact that it is environmentally friendly, since it is intended to use

excesses of energy production to be stored in the form of compressed air, and later produce energy without

combustion, which means, without the emission of pollutants - clean energy.

Surprisingly, this work also showed that the concept of portability can be somehow difficult to implement.

The first results obtained showed that the smaller the system, the worse would be its performance in terms

of energy produced per unit mass of air stored. It is then easy to understand that there was a clear conflict

51

here between the two main objectives of this system: on the one hand, it was intended to design a system

that capable of producing as much energy as possible, while on the other hand the system should have the

smallest size possible.

However, after performing the referred Multi-Objective Optimization analysis, some interesting configu-

rations could be obtained: all of them showing a compromise between the system capacity of producing

energy and its total size. All these systems presented overall efficiencies of approximately 30% and were

capable of producing energy continuously for slightly more than 30 minutes. Despite the relatively low value

of the system overall efficiency, it is important to mention that the necessary energy to supply the system,

that is, to compress the air, is to be obtained without any cost: it is supposed to use wastes of energy

production to supply the compressors and store the air in high-pressure tanks.

An energetic density slightly above 90 kJ/kg could be obtained in some of those optimal systems. Despite

being a lower value than the one presented by the conventional batteries and other energy storage tech-

nologies, the use of compressed air has the advantage of not being harmful to the environment, following

the trend of reducing the emission of pollutants.

After a general overview on all the optimal solutions obtained, two of those were chosen to be studied in

detail. One system from what was defined as the region of large systems, and one from the region of small

systems. It was interesting to notice that both systems had similar efficiencies, but the large system had a

better performance in terms of energy produced, per unit mass of stored air, reaching a value slightly above

90 kJ/kg. Both the systems were able to operate continuously for more than 30 minutes, being the large

system capable of producing an output power of more than 13.6 kW, while the small one could produce only

7.8 kW.

These systems also presented low investment costs, of the order of 10,000e, being their operation and

maintenance costs here neglected. With the current market price of the electricity, it is easy to estimate that

this initial investment can be recovered within few years of operation, should the system operate only few

times a day.

To sum up, the proposed system can be seen as a good alternative to other existing technologies, without

emission of pollutants, and a relatively low cost, with the advantage of being portable. They are also easy

to operate, and the small size here obtained make them suitable for remote power production, as they can

be easily transported from one place to another.

5.1 Guidelines for Future Work

The work here developed led to several interesting results, as mentioned above. However, it would be

interesting to perform some further analysis, as well as testing other configurations, before moving finally to

the construction of the system. Among all these studies to be performed, some will have more impact on

the final product to obtain, being here listed:

i. It was here decided in chapter 2, that the it was better to assume the tank temperature to remain con-

stant during the system operation. However, a more detailed heat transfer study must be performed,

in order to understand how this can be implemented in reality, possibly using the ideas of Thermal

52

Energy Storage in Adiabatic-Compressed Air Energy Storage Systems (see chapter 1).

ii. The second idea for future work would be the construction of an experimental rig, consisting of a high-

pressure tank and a pressure regulator in the exit, to study experimentally the processes of charge

and discharge of the tank, and then validate if the results obtained with the model were close to the

real ones.

iii. In the end, it will be necessary to design a small radial turbine with a high efficiency. Despite the high

values of isentropic efficiencies of commercial large turbines, the efficiencies of small-scale turbines

still need to be improved. Therefore, as a last suggestion for future work, it would be interesting to

design a small radial turbine for compressed air, with the optimal operating conditions of the system,

and optimize its design to obtain an efficiency close to the value of 90%, considered in this work.

53

54

Bibliography

[1] H. Ibrahim, A. Ilinca, and J. Perron. Energy storage systems—characteristics and comparisons. Re-

newable and sustainable energy reviews, 12(5):1221–1250, 2008.

[2] BP Review of World Energy. https://www.bp.com/en/global/corporate/energy-economics/

statistical-review-of-world-energy/country-and-regional-insights/china.html, .

Accessed: 26-03-2018.

[3] World Energy Outlook 2017. https://www.iea.org/weo2017/, . Accessed: 20-03-2018.

[4] The World Bank. Electric power transmission and distribution losses: https://www.iea.org/

weo2017/, . Accessed: 03-05-2018.

[5] Y. Ju and K. Maruta. Microscale combustion: technology development and fundamental research.

Progress in Energy and Combustion Science, 37(6):669–715, 2011.

[6] REN - Portuguese Load Diagram from 26-06-2018. http://www.centrodeinformacao.ren.pt/PT/

InformacaoExploracao/Paginas/EstatisticaDiariaDiagrama.aspx, . Accessed: 28-06-2018.

[7] A. C. Fernandez-Pello. Micropower generation using combustion: issues and approaches. Proceedings

of the combustion institute, 29(1):883–899, 2002.

[8] A. C. Barbier. Emerging energy storage technologies in europe, 2013.

[9] B. Multon, J. Aubry, P. Haessig, and H. B. Ahmed. Systemes de stockage d’energie electrique, 2013.

[10] J. Anzano, P. Jaud, and D. Madet. Stockage de l’electricite dans le systeme de production electrique.

Techniques de l’ingenieur, traite de Genie Electrique D, 4030:09, 1989.

[11] Energy Storage Association. http://energystorage.org/, . Accessed: 24-03-2018.

[12] A. Thornton and C. R. Monroy. Distributed power generation in the united states. Renewable and

Sustainable Energy Reviews, 15(9):4809–4817, 2011.

[13] B. Singh and J. Sharma. A review on distributed generation planning. Renewable and Sustainable

Energy Reviews, 76:529–544, 2017.

[14] G. B. Gharehpetian and S. M. M. Agah. Distributed Generation Systems: Design, Operation and Grid

Integration. Butterworth-Heinemann, 2017.

55

[15] A. Bahr Ennil. Optimization of small-scale axial turbine for distributed compressed air energy storage

system. PhD thesis, University of Birmingham, 2017.

[16] United States Environmental Protection Agency. Global Greenhouse Emissions Data: https://www.

epa.gov/ghgemissions/global-greenhouse-gas-emissions-data, . Accessed: 28-06-2018.

[17] S. M. Schoenung. Characteristics and technologies for long-vs. short-term energy storage. United

States Department of Energy, 2001.

[18] H. Chen, T. N. Cong, W. Yang, C. Tan, Y. Li, and Y. Ding. Progress in electrical energy storage system:

A critical review. Progress in natural science, 19(3):291–312, 2009.

[19] H. Ibrahim, K. Belmokhtar, and M. Ghandour. Investigation of usage of compressed air energy stor-

age for power generation system improving-application in a microgrid integrating wind energy. Energy

Procedia, 73:305–316, 2015.

[20] B. Castellani, A. Presciutti, M. Filipponi, A. Nicolini, and F. Rossi. Experimental investigation on the

effect of phase change materials on compressed air expansion in caes plants. Sustainability, 7(8):

9773–9786, 2015.

[21] G. Grazzini and A. Milazzo. A thermodynamic analysis of multistage adiabatic caes. Proceedings of

the IEEE, 100(2):461–472, 2012.

[22] S. Lemofouet and A. Rufer. Hybrid energy storage system based on compressed air and super-

capacitors with maximum efficiency point tracking (mept). IEEJ Transactions on Industry Applications,

126(7):911–920, 2006.

[23] M. Budt, D. Wolf, R. Span, and J. Yan. A review on compressed air energy storage: Basic principles,

past milestones and recent developments. Applied Energy, 170:250–268, 2016.

[24] K. E. Stahlkopf, S. E. Crane, E. P. Berlin Jr, A. P. ABKENAR, et al. Compressed air energy storage

system utilizing two-phase flow to facilitate heat exchange, May 7 2013. US Patent 8,436,489.

[25] A. Rogers, A. Henderson, X. Wang, and M. Negnevitsky. Compressed air energy storage: Thermody-

namic and economic review. In PES General Meeting— Conference & Exposition, 2014 IEEE, pages

1–5. IEEE, 2014.

[26] J. Bao and L. Zhao. A review of working fluid and expander selections for organic rankine cycle.

Renewable and Sustainable Energy Reviews, 24:325–342, 2013.

[27] B. Saadatfar, R. Fakhrai, and T. Fransson. Waste heat recovery organic rankine cycles in sustainable

energy conversion: A state-of-the-art review. The Journal of MacroTrends in Energy and Sustainability,

1(1):161–88, 2013.

[28] G. Qiu, H. Liu, and S. Riffat. Expanders for micro-chp systems with organic rankine cycle. Applied

Thermal Engineering, 31(16):3301–3307, 2011.

[29] S. Yahya. Turbines compressors and fans. Tata McGraw-Hill Education, 1987.

56

[30] C. Hall and S. L. Dixon. Fluid mechanics and thermodynamics of turbomachinery. Butterworth-

Heinemann, 2013.

[31] V. Lemort, S. Quoilin, C. Cuevas, and J. Lebrun. Testing and modeling a scroll expander integrated into

an organic rankine cycle. Applied Thermal Engineering, 29(14-15):3094–3102, 2009.

[32] M. Imran, M. Usman, B.-S. Park, and D.-H. Lee. Volumetric expanders for low grade heat and waste

heat recovery applications. Renewable and Sustainable Energy Reviews, 57:1090–1109, 2016.

[33] W. He and J. Wang. Optimal selection of air expansion machine in compressed air energy storage: A

review. Renewable and Sustainable Energy Reviews, 87:77–95, 2018.

[34] X. Luo and J. Wang. Overview of current development on compressed air energy storage. School of

Engineering, University of Warwick,, Coventry, UK, 2013.

[35] G. Venkataramani, P. Parankusam, V. Ramalingam, and J. Wang. A review on compressed air energy

storage–a pathway for smart grid and polygeneration. Renewable and sustainable energy reviews, 62:

895–907, 2016.

[36] A. Epstein, S. Jacobson, J. Protz, C. Livermore, J. Lang, and M. Schmidt. Shirtbutton-sized, microma-

chined, gas turbine generators. In 39th Power sources Conference, Cherry Hill, NJ, 2000.

[37] S. Han, J. Seo, J.-Y. Park, B.-S. Choi, and K. H. Do. Design and simulation of 500w ultra-micro gas

turbine generator. Proc. PowerMEMS, Leuven, pages 247–250, 2010.

[38] J. Peirs, D. Reynaerts, and F. Verplaetsen. A microturbine for electric power generation. Sensors and

Actuators A: Physical, 113(1):86–93, 2004.

[39] C. Soares. Microturbines: applications for distributed energy systems. Elsevier, 2011.

[40] Y. Kim and D. Favrat. Energy and exergy analysis of a micro-compressed air energy storage and air

cycle heating and cooling system. Energy, 35(1):213–220, 2010.

[41] G. Grazzini and A. Milazzo. Thermodynamic analysis of caes/tes systems for renewable energy plants.

Renewable energy, 33(9):1998–2006, 2008.

[42] L. X. Chen, P. Hu, P. P. Zhao, M. N. Xie, D. X. Wang, and F. X. Wang. A novel throttling strategy

for adiabatic compressed air energy storage system based on an ejector. Energy conversion and

management, 158:50–59, 2018.

[43] C. Guo, Y. Xu, X. Zhang, H. Guo, X. Zhou, C. Liu, W. Qin, W. Li, B. Dou, and H. Chen. Performance

analysis of compressed air energy storage systems considering dynamic characteristics of compressed

air storage. Energy, 135:876–888, 2017.

[44] H. Sun, X. Luo, and J. Wang. Management and control strategy study for a new hybrid wind turbine

system. In Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE

Conference on, pages 3671–3676. IEEE, 2011.

57

[45] A. Khamis, Z. M. Badarudin, A. Ahmad, A. Ab Rahman, and N. A. Bakar. Development of mini scale

compressed air energy storage system. In Clean Energy and Technology (CET), 2011 IEEE First

Conference on, pages 151–156. IEEE, 2011.

[46] D. Shaw, J.-Y. Cai, and C.-T. Liu. Efficiency analysis and controller design of a continuous variable

planetary transmission for a caes wind energy system. Applied energy, 100:118–126, 2012.

[47] L. Geissbuhler, V. Becattini, G. Zanganeh, S. Zavattoni, M. Barbato, A. Haselbacher, and A. Stein-

feld. Pilot-scale demonstration of advanced adiabatic compressed air energy storage, part 1: Plant

description and tests with sensible thermal-energy storage. Journal of Energy Storage, 17:129–139,

2018.

[48] V. Becattini, L. Geissbuhler, G. Zanganeh, A. Haselbacher, and A. Steinfeld. Pilot-scale demonstration

of advanced adiabatic compressed air energy storage, part 2: Tests with combined sensible/latent

thermal-energy storage. Journal of Energy Storage, 17:140–152, 2018.

[49] M. J. Moran, H. N. Shapiro, D. D. Boettner, and M. B. Bailey. Fundamentals of engineering thermody-

namics. John Wiley & Sons, 2010.

[50] F. M. White. Fluid mechanics. 5th. Boston: McGraw-Hill Book Company, 2003.

[51] H. Pina. Metodos numericos. Escolar Editora, Rua do Vale Formoso, 37, 1995.

[52] E. Suli. Numerical solution of ordinary differential equations. 2014.

[53] C. B. Moler. Numerical Computing with MATLAB: Revised Reprint, volume 87. Siam, 2008.

[54] L. F. Shampine and M. W. Reichelt. The matlab ode suite. SIAM journal on scientific computing, 18(1):

1–22, 1997.

[55] L. F. Shampine. Implementation of rosenbrock methods. ACM Transactions on Mathematical Software

(TOMS), 8(2):93–113, 1982.

[56] R. Scraton. Some l-stable methods for stiff differential equations. International Journal of Computer

Mathematics, 9(1):81–87, 1981.

[57] T. L. Bergman, F. P. Incropera, D. P. DeWitt, and A. S. Lavine. Fundamentals of heat and mass transfer.

John Wiley & Sons, 2011.

[58] J. C. Dutton and R. E. Coverdill. Experiments to study the gaseous discharge and filling of vessels.

International Journal of Engineering Education, 13(2):123–134, 1997.

[59] K. Miettinen. Nonlinear multiobjective optimization, volume 12. Springer Science & Business Media,

2012.

[60] G. Chen, X. Han, G. Liu, C. Jiang, and Z. Zhao. An efficient multi-objective optimization method for

black-box functions using sequential approximate technique. Applied Soft Computing, 12(1):14–27,

2012.

58

[61] J. Arora. Introduction to optimum design. Elsevier, 2004.

[62] A. Alarcon-Rodriguez, G. Ault, and S. Galloway. Multi-objective planning of distributed energy re-

sources: A review of the state-of-the-art. Renewable and Sustainable Energy Reviews, 14(5):1353–

1366, 2010.

[63] Y. Cui, Z. Geng, Q. Zhu, and Y. Han. Multi-objective optimization methods and application in energy

saving. Energy, 125:681–704, 2017.

[64] G. Chiandussi, M. Codegone, S. Ferrero, and F. E. Varesio. Comparison of multi-objective optimization

methodologies for engineering applications. Computers & Mathematics with Applications, 63(5):912–

942, 2012.

[65] A. L. Custodio, J. A. Madeira, A. I. F. Vaz, and L. N. Vicente. Direct multisearch for multiobjective

optimization. SIAM Journal on Optimization, 21(3):1109–1140, 2011.

[66] Direct Search Method. https://en.wikipedia.org/wiki/Pattern_search_(optimization)#/

media/File:Direct_search_BROYDEN.gif. Accessed: 24-08-2018.

[67] ERSE Liberalized Market Energy Prices. http://www.erse.pt/pt/Simuladores/Documents/Pre%C3%

A7osRef_BTN.pdf, . Accessed: 10-09-2018.

59

60

Appendix A

MATLAB Routine - Simulator

61

function [time, pressure, W_c, W_t, max_power_t, eta, rho, C] =

Simulator(p_0, V, p_e, d_e, L, K, eta_t) % Main Function of System Model % Calculate objective functions to apply then optimization algorithm % Francisco Bandeira Brás Monteiro, IST 78137 % Master Thesis in Mechanical Engineering % Supervisors: Professor Edgar Fernandes, Professor Luís Eça % Delivery: 15 October 2018

%% Pre-defined variables (atmospheric conditions, constants) p_atm = 1.01325*10^5; % Pa T_atm = 290; % K R = 287; % J/kg.K gamma = 1.4; c_p = 1005; % J/kg.K

%% Charging Process - Total Compression Work and Pressure Evolution % Function to obtain value of total compression work required to fill the % tank (in kJ)

W_c = ((p_0*V)*(log(p_0/p_atm) - 1) + p_atm*V)/1000; m_stored = (p_0*V)/(R*T_atm);

%% Discharge of the tank % Function to obtain discharge of the tank (p vs time) [t_disch, p_disch] = solver_discharge(V, p_0, p_e, d_e, K, L);

time = t_disch; pressure = p_disch;

% Calculate mass flow rate in time mass = (pressure*10^5).*(V/(R*T_atm));

for i = 1:length(mass)-1 m_dot(i) = abs(mass(i+1)-mass(i))/(time(i+1)-time(i)); end

%% Turbine Power Production % Turbine inlet pressure is constant from t = 0 to t_disch and then drops % from p_e until p_atm (from t_disch to the end) pres_turbine = zeros(length(pressure),1); pres_turbine(1:length(p_disch)) = p_e; % Pa

Power_t = zeros(length(pres_turbine)-1,1);

% TurbinePower as function of time for i = 1:length(pres_turbine)-1 A = (p_atm/pres_turbine(i+1))^((gamma-1)/gamma); Power_t(i) = (eta_t/100)*m_dot(i)*c_p*T_atm*(1-A)/10^3; % kW end

62

% Function to obtain total energy produced (integral of power over time) W_t = trapz(time(2:end),Power_t); % in kJ (trapezoidal rule)

%% System Final Variables % Total efficiency of the system eta = W_t/W_c * 100;

% Energetic density of stored energy rho = W_t/m_stored; % kJ/kg

% Cost function f_vol = -0.016*(V*1000)^2 + 5.1434*(V*1000) + 633.49; % euros f_power = 460*max(Power_t); % euros/kW; f_valve = 12.749*d_e^2 + 57.573*d_e + 515.32; % euros (diameter) C = f_vol + f_power + f_valve;

end

63

64

Appendix B

Pareto Front Points

65

Tank Pressure (bar)

Tank Volume (𝒎𝟑)

Exit Diameter (cm)

Exit Length (m) K valveDischarge Pressure

(bar)Turbine

EfficiencyStored Mass of Air

(kg)Energy Produced

(kJ)Compression

Work (kJ)Stored Energy Density (kJ/kg)

System Efficiency

Operating Time (s)

Maximum Power (kW)

50 1.6 1 1 10.5 3.5 90 96.1191878 6920.44122 23353.0002 71.9985403 29.6340563 1877.95532 5.05968227

70 1.2 1 1.5 10.5 3.5 90 100.925147 7424.38842 27298.3809 73.5633152 27.1971749 1835.73327 5.37139038

55 1.6 1 1 10.5 4 90 105.731107 8258.52716 26510.8177 78.1087745 31.1515368 1822.58217 6.24625393

65 1.4 1 1.5 10.5 4 90 109.335576 8644.18537 28908.996 79.0610493 29.9013683 1817.89095 6.44297575

60 1.6 1 1 10.5 4 90 115.343025 9068.58005 29741.4631 78.6226997 30.4913716 1951.18227 6.34947945

70 1.4 1 1.5 10.5 4 90 117.746005 9352.97912 31848.1111 79.4335156 29.3674532 1927.16191 6.52834772

65 1.6 1 1 10.5 4.5 90 124.954944 10471.9792 33038.8525 83.8060416 31.6959532 1894.28612 7.58161502

75 1.4 1 1.5 10.5 4.5 90 126.156434 10678.6797 34837.2688 84.646334 30.6530334 1854.47538 7.78285754

70 1.6 1 1.5 10.5 5 90 134.566863 11878.9219 36397.8413 88.2752387 32.6363365 1848.31555 8.84199193

85 1.4 1 1.5 10.5 5 90 142.977292 12794.3642 40952.7654 89.4852884 31.2417588 1883.62542 9.1626735

100 1.2 1 1.5 10.5 5 90 144.178782 13024.1667 43225.6764 90.3334498 30.1306256 1835.73327 9.42271334

90 1.4 1 1.5 10.5 5 90 151.387721 13594.2831 44073.6032 89.7977921 30.8445012 1970.48083 9.25496361

80 1.6 1 1.5 8.5 5 90 153.7907 13707.749 43283.5747 89.1324961 31.6696324 1865.39832 9.97230355

95 1.4 1 1.5 8.5 5 90 159.79815 14394.6459 47233.3499 90.0801787 30.4755981 1869.06691 10.2777992

85 1.6 1 1 8.5 5 90 163.402619 14621.9437 46803.1605 89.4841449 31.2413596 1956.72139 10.0811343

100 1.4 1 1.5 8.5 5 90 168.208579 15194.5634 50429.9558 90.3316793 30.1300351 1946.55228 10.3672385

90 1.6 1 1 6.5 5 90 173.014538 15536.7451 50369.8322 89.8002285 30.8453381 1818.67601 11.4602219

75 2 1 1 6.5 5 90 180.223477 15991.3494 49767.5269 88.7306676 32.1320958 1969.57627 11.0915835

95 1.6 1 1 6.5 5 90 182.626457 16450.9388 53980.9714 90.0797129 30.4754405 1898.06832 11.5672832

100 1.6 1 1 6.5 5 90 192.238376 17365.1294 57634.2352 90.3312326 30.1298861 1976.75125 11.6679436

90 1.8 1 1 6.5 5 90 194.641355 17478.1538 56666.0613 89.7967124 30.8441304 2046.05651 11.4602225

85 2 1 1 4.5 5 90 204.253274 18277.7393 58503.9506 89.4856613 31.241889 1860.93483 13.2492042

95 1.8 1 1 4.5 5 90 205.454764 18507.5415 60728.5928 90.0808585 30.4758281 1828.51626 13.5076754

100 1.8 1 1 4.5 5 90 216.268173 19536.0082 64838.5146 90.3323312 30.1302525 1904.31879 13.6252215

95 2 1 1 4.5 5 90 228.283071 20563.2194 67476.2142 90.0777237 30.4747675 2031.7279 13.5076761

100 2 1 1 4.5 5 90 240.297969 21705.9533 72042.794 90.3293247 30.1292497 2115.95295 13.6252222

67

Appendix C

Cost Model

68

Tank Volume (L) Cost (€)

1 227.1247068 1749.02678

2 302.8329424 1904.88065

3 454.2494136 2664.23535

4 757.082356 3812.35886

5 908.4988272 4053.93237

6 1514.164712 4555.26232

7 1892.70589 4875.62861

y = -0.0016x2 + 5.1434x + 633.49R² = 0.9848

0

1000

2000

3000

4000

5000

6000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Tan

k C

ost

(€

)

Tank Volume (L)

Tank Cost as function of Volume

Source: https://hansontank.com/airreceivers-660-15000gallonv.html

Turbine Model Power (kW) Power (MW) Cost (M€) Cost per kW (€/kW)

SOLAR SATURN 1080 1.08 0.692683873 641.3739563

TURBOMECA M 1086 1.086 0.779269357 717.5592605

RUSTON HURRICANE 1575 1.575 0.952440325 604.7240159

ALLISON 501KB5 3725 3.725 1.558538714 418.3996547

ALLISON 501KH 3740 3.74 1.818295166 486.1751781

RUSTON TB5000 3830 3.83 1.47195323 384.3219921

GE LM500 3880 3.88 1.645124198 424.0010819

SOLAR CENTAUR 3880 3.88 1.47195323 379.3693891

RUSTON TYPHOON 3945 3.945 1.731709682 438.963164

DRESSER DC990 4200 4.2 1.731709682 412.311829

SOLAR TAURUS 4370 4.37 1.645124198 376.4586265

RUSTON TYPHOON 4550 4.55 1.818295166 399.6253112

ALLISON 570KA 4610 4.61 2.251222587 488.3346175

ALLISON 571KA 5590 5.59 2.424393555 433.7018882

RUSTON TORNADO 6215 6.215 2.510979039 404.0191535

SOLAR MARS 8840 8.84 3.723175816 421.1737349

NUOVO PIGNONE PGT10 9980 9.98 4.502445173 451.1468109

SOLAR MARS 10000 10 3.982932269 398.2932269

Average 459.9973828

Source: http://nyethermodynamics.com/trader/kwprice.htm

Size (in) Size (cm) Cost (€)

0.1875 0.47625 607.830098

0.375 0.9525 607.830098

0.625 1.5875 607.830098

0.75 1.905 607.830098

1 2.54 658.915534

1.25 3.175 781.001067

1.5 3.81 1042.48923

2 5.08 1153.31865

2.5 6.35 1439.05075

3 7.62 1647.72176

y = 12.749x2 + 57.573x + 515.32R² = 0.9716

200

400

600

800

1000

1200

1400

1600

1800

0 1 2 3 4 5 6 7 8

Val

ve C

ost

(€

)

Diameter (cm)

Valve Cost as function of Duct Diameter

Source: Ohio Valve Company (OVC) - Carbon, Forged Steel, Stainless Steel, Forged Stainless Steel Gate, Globe, & Check ValvePrice List

72