Applied Mathematical Modeling and Analysis in Renewable Energy [1 ed.] 0367746980, 9780367746988

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Acknowledgments
About the Editors
Contributors
PART I: Mathematical Modeling and Simulation Results
Chapter 1: Mathematical Modeling of Transitional Fluid Phase Flows in Coal Bed Methane Reservoir
Chapter 2: AI-Based Detection and Localization of Gastrointestinal Polyps by Using Deep Learning, Transfer Learning and the Fusion of These Techniques
Chapter 3: Mathematical Modeling of an EOQ for a Multi-item Inventory System with Selling Price and Price Break Sensitive Demand
Chapter 4: Fair Allocation of Items: A Comprehensive Study
Chapter 5: Hierarchical Demand Response Controller
PART II: Generalized Mathematical Ideas and Their Applications
Chapter 6: A General Class of Polynomials Inspired by a General Lagrange Inversion Pair Due to Gessel and Stanton
Chapter 7: Squeezing Graphs
Chapter 8: Finding the Surface Area and Volume of the Hyperspheres Using Simple Calculus
PART III: Mathematical Modeling in Renewable Energy
Chapter 9: Analysis of RSM Method for Optimization of Ultrasound-Assisted KOH Catalyzed Biodiesel Production from Waste Cotton-Seed Cooking Oil
Chapter 10: Energy Data Analysis of an Educational Institution in India
Chapter 11: Factors to Consider: A Review of Smart Grid Implementation in India
Chapter 12: Integration and Modeling of Small-Scale Pumped Storage
Index

Citation preview

Applied Mathematical Modeling and Analysis in Renewable Energy

Mathematical Engineering, Manufacturing, and Management Sciences Series Editor: Mangey Ram Professor, Assistant Dean (International Affairs), Department of Mathematics, Graphic Era University, Dehradun, India

The aim of this new book series is to publish the research studies and articles that bring up the latest development and research applied to mathematics and its applications in the manufacturing and management sciences areas. Mathematical tool and techniques are the strength of engineering sciences. They form the common foundation of all novel disciplines as engineering evolves and develops. The series will include a comprehensive range of applied mathematics and its application in engineering areas such as optimization techniques, mathematical modeling and simulation, stochastic processes and systems engineering, safety-critical system performance, system safety, system security, high assurance software architecture and design, mathematical modeling in environmental safety sciences, finite element methods, differential equations, reliability engineering, etc. Non-Linear Programming A Basic Introduction Nita H. Shah and Poonam Prakash Mishra Applied Soft Computing and Embedded System Applications in Solar Energy Rupendra Kumar Pachauri, J. K. Pandey, Abhishek Sharma, Om Prakash Nautiyal, Mange Ram Differential Equations in Engineering Research and Applications Edited by Nupur Goyal, Piotr Kulczycki, and Mangey Ram Sustainability in Industry 4.0 Challenges and Remedies Edited by Shwetank Avikal, Amit Raj Singh, Mangey Ram Applied Mathematical Modeling and Analysis in Renewable Energy Edited by Manoj Sahni and Ritu Sahni For more information about this series, please visit: https://www.routledge. com/Mathematical-Engineering-Manufacturing-and-Management-Sciences/ book-series/CRCMEMMS

Applied Mathematical Modeling and Analysis in Renewable Energy

Edited by

Manoj Sahni and Ritu Sahni

First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 selection and editorial matter, Manoj Sahni and Ritu Sahni; individual chapters, the contributors CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf. co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 9780367746988 (hbk) ISBN: 9780367747008 (pbk) ISBN: 9781003159124 (ebk) DOI: 10.1201/9781003159124 Typeset in Times by KnowledgeWorks Global Ltd.

Contents Preface......................................................................................................................vii Acknowledgments......................................................................................................ix About the Editors.......................................................................................................xi Contributors............................................................................................................ xiii

PART I  Mathematical Modeling and Simulation Results Chapter 1 Mathematical Modeling of Transitional Fluid Phase Flows in Coal Bed Methane Reservoir................................................................3 Subhashini Nainar and Suresh Kumar Govindarajan Chapter 2 AI-Based Detection and Localization of Gastrointestinal Polyps by Using Deep Learning, Transfer Learning and the Fusion of These Techniques................................................... 21 Yogesh Chaudhari, Umme Salma Pirzada, and Darshee Baxi Chapter 3 Mathematical Modeling of an EOQ for a Multi-item Inventory System with Selling Price and Price Break Sensitive Demand.......... 35 Abhijit Barman and Pijus Kanti De Chapter 4 Fair Allocation of Items: A Comprehensive Study............................. 47 Manisha Bansal and Purnima Bindal Chapter 5 Hierarchical Demand Response Controller........................................ 57 Dima Kayyali, Hussam Nosair, Amit V. Sant, and Hannah Michalska

PART II  G  eneralized Mathematical Ideas and Their Applications Chapter 6 A General Class of Polynomials Inspired by a General Lagrange Inversion Pair Due to Gessel and Stanton........................... 79 Manisha Dalbhide-Ubale v

vi

Contents

Chapter 7 Squeezing Graphs.............................................................................. 101 Ved Suthar Chapter 8 Finding the Surface Area and Volume of the Hyperspheres Using Simple Calculus...................................................................... 125 Shantanu, Ritu Sahni, and Manoj Sahni

PART III  Mathematical Modeling in Renewable Energy Chapter 9 Analysis of RSM Method for Optimization of UltrasoundAssisted KOH Catalyzed Biodiesel Production from Waste Cotton-Seed Cooking Oil.................................................................. 133 Suvik Oza, Pravin Kodgire, and Surendra Singh Kachhwaha Chapter 10 Energy Data Analysis of an Educational Institution in India.............................................................................................. 149 Chandana Sasidharan and Aman Aggarwal Chapter 11 Factors to Consider: A Review of Smart Grid Implementation in India.................................................................... 163 Atmiya Patel, Vipul N. Rajput, Kartik S. Pandya, and Dipayan Guha Chapter 12 Integration and Modeling of Small-Scale Pumped Storage.............. 181 Jyoti Gupta and Arun Kumar Index....................................................................................................................... 193

Preface Mathematics is the backbone for the development of scientific and technical fields. It makes our life orderly and prevents chaos. This is a subject which provides power of reasoning, creativity, spatial thinking, and critical thinking; basically, it provides the way to solve worldly problems related to day-to-day marketing, maintaining credit card bills, electricity bills, banking, medical insurance, business, computer simulation, education system, statistical data in various cases, engineering problems, biological systems, and more. The mathematical techniques become the strength of everyone who works in the field of engineering, sciences, business, medical profession, politics, etc. to deal with various problems and perform different tasks. This book highlights the latest mathematical techniques, novel research in the area of mathematical modeling, and also their applications. The purpose of this book is to provide both the latest techniques discovered and the advance study of various physical phenomena. Each of its chapters contain some advanced technique for solving practical applications. It mainly focuses on the mathematical modeling of renewable energy, its benefits, and its impact on environment and society. Due to increases in population, we have a shortage of the energy generated by fossil fuel; society is facing challenges in electricity generation. So, there is a need to generate power from other sources so that we can have a better quality of life. Many technologies are developed for the generation of energy, including solar, wind, hydro power energy, biomass, biofuel energy, etc., which are longlasting natural resources. This book contains mathematical modeling of various renewable energy systems, and provides methods for creating integrated environments through effective energy management and control. Basically, it provides a brief overview of new emerging technologies for the generation and management of energy systems. It is a collection of novel advancements, both with conceptual and mathematical works containing various physical problems, a methodology for generations and smart storage of renewable energy. This book is written considering readers with mathematical backgrounds, and with a basic knowledge of physics, chemistry, engineering, statistics, fluid mechanics, thermodynamics, heat transfer, and renewable energy. It is important for scientists, researchers, students, teachers, and more who have the scope to modify these results, and can use it for making their projects by introducing new results in this area. Teachers can use these results for explaining applications of various mathematical tools. As many chapters are based on modeling of energy storage, this book also benefits both manufacturing and energy industries. In conclusion, the text explores various challenging areas dealing with various physical problems, their modeling, and novel, advanced mathematical techniques used in day-to-day life and in the energy sector. It paves the way to dealing with other problems using advanced techniques mentioned in different chapters, and so, it is very useful in present and future scenarios. We hope that all readers benefit from this book and succeed with the great effort made in this book. vii

Acknowledgments The material in this book reflects the research of many authors. We are immensely grateful to all the authors and express our sincere appreciation for their contributions. We also acknowledge our colleagues at Pandit Deendayal Energy University (formerly known as Pandit Deendayal Petroleum University) for their encouragement and support. We also thank the editorial team at Taylor & Francis who accepted the manuscript and then guided us through many stages that are necessary for shaping this book. We are indebted to many people for their invaluable assistance in writing this book. We also express our gratitude to the reviewer of the chapters. We are also thankful to our family members for their support, patience, encouragement, and all possible help provided by them while we were engaged in preparing this manuscript. We would like to especially acknowledge our son, Mohit Sahni; without him and his support, we would not be able to complete this work. This book would not have been completed if God had not provided us with enough patience during this period. We shall be highly grateful to the Almighty. We have tried our best to make this book an error-free text. Nevertheless, any suggestions, comments, and feedback for further improvement of the book will be gratefully accepted. Dr. Manoj Sahni Dr. Ritu Sahni

ix

About the Editors Manoj Sahni has been working as an associate professor and head, Department of Mathematics, Pandit Deendayal Energy University, Gandhinagar for the last seven years. He has more than 17 years’ experience in academic and research fields. During the last 17 years, he has worked with Jaypee Institute of Information Technology, Noida, Navrachna University, Vadodara, and PDEU, Gandhinagar. During this period, he has published more than 60 research papers in various international journals and conference proceedings. He is a Life Member of various societies such as Indian Science Congress, Indian Mathematical Society, Allahabad Mathematical Society, and more. He has successfully coordinated various events such as international conferences, expert lectures, seminars, workshops, and more. He is the reviewer of various reputed international journals. His research areas include continuum mechanics, functionally graded materials, fuzzy sets, and numerical methods. Ritu Sahni has been an assistant professor, Department of Physical Sciences, Institute of Advanced Research, Gandhinagar for the last six years. Previously, she worked at Jaypee Institute of Information Technology, Noida and Navrachna University, Vadodara. Her total teaching and research experience is approximately 12 years. She has published more than 35 research papers in various international journals and conference proceedings. She is a Life Member of various prestigious societies such as Indian Science Congress, Indian Mathematical Society, and more. Currently, she is working in the area of fixed-point theory and fuzzy modeling.

xi

Contributors Aman Aggarwal Teri School of Advanced Studies New Delhi, India Manisha Bansal Indraprastha College for Women University of Delhi Delhi, India Abhijit Barman Department of Mathematics National Institute of Technology Silchar, India Darshee Baxi School of Science Navrachana University Vadodara, India Purnima Bindal P.G.D.A.V. College University of Delhi Delhi, India Yogesh Chaudhari Computer Science and Engineering Navrachana University Vadodara, India Manisha Dalbhide-Ubale Indus University Ahmedabad, India Pijus Kanti De Department of Mathematics National Institute of Technology Silchar, India Suresh Kumar Govindarajan Indian Institute of Technology (IIT)-Madras Chennai, India

Dipayan Guha Motilal Nehru National Institute of Technology Prayagraj, India Jyoti Gupta Department of Mechanical Engineering Indian Institute of Technology Kanpur, India Surendra Singh Kachhwaha Mechanical Engineering Department Center for Biofuel and Bioenergy Studies Pandit Deendayal Petroleum University Gandhinagar, India Dima Kayyali McGill University Montreal, Canada Pravin Kodgire Chemical Engineering Department Center for Biofuel and Bioenergy Studies Pandit Deendayal Petroleum University Gandhinagar, India Arun Kumar Department of Hydro and Renewable Energy Indian Institute of Technology Roorkee, India Hannah Michalska McGill University Montreal, Canada xiii

xiv

Subhashini Nainar Indian Institute of Technology (IIT)-Madras Chennai, India Hussam Nosair New York Independent System Operator New York, USA Suvik Oza Chemical Engineering Department Center for Biofuel and Bioenergy Studies Pandit Deendayal Petroleum University Gandhinagar, India Kartik S. Pandya Charotar University of Science and Technology Anand, India Atmiya Patel Dr. Jivraj Mehta Institute of Technology Anand, India Umme Salma Pirzada School of Engineering and Technology Navrachana University Vadodara, India

Contributors

Vipul N. Rajput Dr. Jivraj Mehta Institute of Technology Anand, India Manoj Sahni Pandit Deendayal Energy University (PDEU) Gandhinagar, India Ritu Sahni Institute of Advanced Research (IAR) Gandhinagar, India Amit V. Sant Pandit Deendayal Petroleum University Gandhinagar, India Chandana Sasidharan Teri School of Advanced Studies New Delhi, India Shantanu Indian Institute of Science Education and Research (IISER) Pune, India Ved Suthar Indus University Ahmedabad, India

Part I Mathematical Modeling and Simulation Results

1

Mathematical Modeling of Transitional Fluid Phase Flows in Coal Bed Methane Reservoir Subhashini Nainar and Suresh Kumar Govindarajan

CONTENTS 1.1 Introduction....................................................................................................... 3 1.1.1 Conceptual Model..................................................................................4 1.2 Permeability Model...........................................................................................4 1.2.1 Auxiliary Equations...............................................................................6 1.2.2 Variable Coefficients.............................................................................6 1.3 Single-Phase Water Flow...................................................................................7 1.3.1 Mathematical Model.............................................................................. 7 1.3.2 Initial and Boundary Conditions...........................................................7 1.3.3 Numerical Model................................................................................... 8 1.4 Multiphase Gas and Water Flow........................................................................8 1.4.1 Mathematical Model.............................................................................. 8 1.4.2 Initial and Boundary Conditions......................................................... 10 1.4.3 Methodology to Solve for Multiphase Flow........................................ 10 1.4.4 Numerical Model................................................................................. 13 1.4.5 Water Saturation to be Solved Explicitly Using Equation 1.17........... 13 1.5 Verification and Validation.............................................................................. 15 1.5.1 Verification and Validation Results..................................................... 15 1.6 Single-Phase Gas Flow.................................................................................... 17 1.6.1 Numerical Model................................................................................. 19 1.7 Conclusion....................................................................................................... 19 Nomenclature............................................................................................................ 19 References.................................................................................................................20

1.1 INTRODUCTION Coal formation under average reservoir pressure and temperature contains a specified volume of gas adsorbed per unit ton of coal formation. Extracting methane from an unconventional coal formation involves fluid flow stages such as dewatering, multiphase flow, and gas flow. Every fluid flow stage includes either one or many fluid phases. DOI: 10.1201/9781003159124-1

3

4

Applied Mathematical Modeling and Analysis in Renewable Energy

Very few earlier works have provided for equations covering all fluid flow stages. The study mainly considers stress dependency of cleat permeability and sorption strains in the coal when modeling. At each period, stress dependency is either a function of cleat compressibility or sorption strain or both. Work done by Jishan Liu et al. (2011) focused on the multiphase fluid flow stage alone, while Nie et al. (2011) gave equations only for gas flow but included stress dependency effects. Lili (2012) started analysis after the dewatering stage, and therefore, omitted the single-phase water flow. Clarkson and Qanbari (2016) semi-analytically demonstrated equations for multiphase and single-phase gas. The literature is a reference for verification and validation. It has assumed cleat permeability to be pressure dependent. The objective of the present work is to understand how an equation is written for different flow stages when fluid is assumed to flow through a cleat in the onedimensional unconventional coal gas reservoir. Having modeled the fluid flow stages mathematically, they need to be numerically solved to obtain values of dependent variables. Here, an effort has been made to explain the numerical methodology adopted in solving the transition effects between the various fluid flow stages.

1.1.1  Conceptual Model A three-dimensional coal formation is shown. It comprises cleat network interwoven between porous matrices. The gas is adsorbed on the coal grains contrary to being present in pores as observed in the conventional oil/gas reservoirs. Face cleats are also shown in the figure. The three-dimensional coal formation (Fig. 1.1a) is reduced to a two-dimensional model (Fig. 1.1b). The radius is varied dynamically with production time. The study length is the limiting length used for the analysis. Payzone width is used in the computation of drained volume of gas. The two-dimensional model is further reduced to one-dimensional single cleat flow analysis (Fig. 1.1c). It is divided into nodes along the study length. The number of nodes increases with time. The varying wellbore pressure is indicated at the initial node (Fig. 1.1d). The time versus drained distance is plotted for the cleat. With an increase in production time, the drained distance increases away from the wellbore center, and so does the number of nodes. The coal formation is heterogeneous and assumed to be isotropic. Onedimensional model is studied. The flow behavior of a single cleat and the associated coal matrix around is studied. The equations are framed, considering all the phases starting from single-phase water flow, then multiphase gas and water flow, followed by single-phase gas flow. The verified and validated code for the dewatered coal is further extended to include the single-phase gas flow equations.

1.2  PERMEABILITY MODEL k n = k i × e −(3× cfn ×(σ −σ i )n ) (1.1) (Li et al., 2017)

Mathematical Modeling of Transitional Fluid Phase Flows

5

FIGURE 1.1  (a) Three-dimensional coal formation comprising cleat and matrix. (b) Twodimensional coal formation with study length, pay zone thickness, and radius varies dynamically. (c) One-dimensional model and the varying drained distance for the changing wellbore pressure. (d) One-dimensional model divided as nodes along study length with the drained distances increasing for each time increment.

6

Applied Mathematical Modeling and Analysis in Renewable Energy



  −ν   Pn  E × εL Pi × ( Pn − Pi ) + × −   3 × (1 − ν )  ( Pn + PL ) ( Pi + PL )    (1 − ν ) 

(σ − σ i )n =  0.36  ×   

  − (1 + ν )  2 × E × ε L  Pn Pi   P P + 0.64  ×   × − + × − ( )    n i  9 × (1 − ν )  ( Pn + PL ) ( Pi + PL )     3 × (1 − ν )   (1.2) (Li et al., 2017) k  φn = φi × √3  n  (1.3)  ki 





(porosity-permeability cubic relation)

1.2.1 Auxiliary Equations

Sw + Sg = 1 (1.4)



k rg + k rw = 1 (1.5)

1.2.2  Variable Coefficients  φi × Swi   φiv ( P ) × Swiv   wpiv = ( 4 × L W × X iv × h ) ×     (1.6)  −   ( 5.615 × Bwi )   5.615 × Bwiv ( P )   (Clarkson and Qanbari, 2016)

(



gpiv =

wpiv (1.7) t + dt

qw iv =



)

( 4 × L W × X iv × h ) ×   φi × Sgi  −  φiv ( P ) × Sg (Sw )  + VL × P − VL × Piv  (1.8) 1000

  

(B ) gi

 

(B



iv

giv

( P ))

qgiv =





P + PL

Piv + PL 

(Clarkson and Qanbari, 2016)

gpiv (1.9) t + dt

Bwn = e(.0000026  ×(14.7− Pn )) (1.10) (Clarkson and Qanbari, 2016)

cfn =

(

cfi × 1 − e −α (σ −σ i )n

α (σ − σ i )n



) (1.11) (Robertson, 2006)

Mathematical Modeling of Transitional Fluid Phase Flows

cl n =



7

Sgn (1.12) Pn

ct n = cl n + cfn (1.13)



Equations 1.1–1.3 represent the empirical relations for stress-dependent cleat permeability and porosity. The sum of fluid saturations add up to unity, while the fluid relative permeability values also sum up to one as shown in equations 1.4 and 1.5. The fluid flow rate is shown in equations 1.7 and 1.9, while the material balance is shown in equations 1.6 and 1.8 for water and gas, respectively. Equations 1.10–1.13 are the empirical fluid properties.

1.3  SINGLE-PHASE WATER FLOW The water in the matrix is ignored since dewatered cleat water is sufficient to lower down the reservoir pressure to desorption pressure. Overburden stress is constant, and only the fluid pressure varies with time. Overburden stress is vertical, and the sides of reservoir length are assumed to be constrained horizontally. Any deformation in the vertical direction exerts stress horizontally. The horizontal stress includes only shrinkage stress, when there is no injection of supercritical carbon dioxide (sCO2), and includes both swelling and shrinkage stresses, when sCO2 is injected. Dependent variablesa: Pnt + dt Variable coefficients: k nt+ dt , cfnt + dt, ϕ nt+ dt, Bwt+ndt , wpivt+ dt , qw

t + dt iv

Independent variable: Xinv t + dt , Ltw+ dt , t Constant coefficients: Sw, h, dt, dx, T

Note: a Unknown.

1.3.1  Mathematical Model Water flow through cleats:

φiv ( P ) × cfiv ( P ) × Swiv ( P ) ∂P k iv ( P ) ∂2 P = × 2 (1.14) × Bwiv ( P ) ∂ t µwiv ( P ) × Bwiv ( P ) ∂x

(Derived from mass continuity equation) Equation 1.14 is a non-linear partial differential equation applicable in case of water flow in the cleats of coal formation. The equation is diffusive and, thus, parabolic. Since the equation is parabolic dominant, the solution converges after some time. The equation is non-linear due to the presence of six variable coefficients.

1.3.2 Initial and Boundary Conditions 1. Boundary condition to the right (Dirichlet):  P( N ) t = Pi 2. Initial condition (Dirichlet): Pn 0 =  Pi 3. Boundary condition at left on the first node (Dirichlet): P0 t =  Pwf

8

Applied Mathematical Modeling and Analysis in Renewable Energy

1.3.3 Numerical Model The coefficients in the equation are discretized using backward time central space (BTCS) method to obtain a tri-diagonal matrix of form AX=B. Tri-diagonal matrix algorithm (TDMA) approach is used to solve the pressure at each node at the next time level. A flowchart (Fig. 1.2) explains the complete procedure followed from defining the dependent variable to calculating them for a single-phase water flow in cleat. The variable is average reservoir pressure. The iterative loop solves for the non-linearity in the flow equation. Both reservoir and fluid properties mentioned in the list vary with time.

1.4  MULTIPHASE GAS AND WATER FLOW Dependent variablesa: Pnt + dt , Swn t + dt

Independent variable: Xinv t + dt , Ltw+ dt , t Constant coefficients: h, dt, dx, T

Variable coefficients: k nt+ dt , cf , ϕ , B , ct nt+ dt , cl nt+ dt , wpivt+ dt , gpivt+ dt , k rg t + dt, k rw t + dt, Sgn t + dt t + dt n

t + dt n

t + dt wn

Note: a Unknown.

1.4.1  Mathematical Model Water flow through cleats:

µwiv ( P ) × φiv ( P )  ∂P ∂Sw  +  ct iv ( Sw , P ) × Swiv ×  k iv ( P ) × k rw ( Sw )  ∂t ∂t 

=

∂2 P dP  + cl iv ( Sw , P ) ×   2  ∂x dx 

2

(1.15)

2



dP  cl iv ( Sw , P ) ×   ≈ 0 ( Since compressibility of  water is negligible ) (1.16)  dx 

It can be written as:

µwiv ( P ) × φiv ( P )  ∂P ∂Sw  ∂2 P + (1.17)  ct iv ( P ) × Swiv × = k iv ( P ) × k rw ( Sw ) ∂t ∂ t  ∂x 2



Gas flow through cleats: Mass continuity equation:



(

∂ ρg (VPL+ ×PLP) +

ρg ×φiv ( P )× Sg ( Sw ) Bgiv ( P )

∂t

)= k

iv ( P ) × k rg ( Sw ) × ∇ ρg∇P (1.18) µg iv ( P ) × Bgiv ( P )

(

)

Mathematical Modeling of Transitional Fluid Phase Flows

9

FIGURE 1.2  The coding procedure is followed with the use of assumptions and calculation of dependent variables for single-phase water flow.

10

Applied Mathematical Modeling and Analysis in Renewable Energy

µ g × Bgiv



iv

k iv × k rg

(

∂ ρg (VPL+ ×PLP) + × ∂t

ρg ×φiv × Sg Bgiv

∂2 P dP  × ρg + ρg × cl iv ×    dx  ∂x 2 =



µg iv × Bgiv k iv × k rg

(

∂  ×  

ρg ×φiv × Sg Bgiv

∂t

) = ∇ ( ρ ∇P ) (1.19) g

2

) + ∂( ρ

VL × P g ( P + PL )

∂t

)  (1.20)  

µgiv ( p ) × Bgiv ( P ) k iv ( p ) × k rg ( sw )  ct iv ( sw , P ) × φiv ( P ) × sg ( sw ) ∂p φiv ( P ) ∂sg VL × PL ∂p  (1.21) iv × + × +   2 × Bgiv ( P ) ∂ t Bgiv ( P ) ∂ t ( P + PL ) ∂t   =



∂2 p dp  + cl iv ( Sw , P ) ×   2  ∂x dx 

2

∂P  V × PL  φiv ( P ) ∂Sg ×  ct iv ( Sw , P ) × φiv ( P ) × Sg iv ( Sw ) + L × + ∂t  ( P + PL )2  Bgiv (, P ) ∂t 2 k iv ( P ) × k rg ( sw )  ∂2 P  dP   cl S , P = + × ( )   iv w  dx   µg iv ( P ) × Bgiv ( P )  ∂x 2

(1.22)

Equation 1.17 is the water flow equation deduced from the original equation 1.15 after neglecting the compressibility term (equation 1.16). Equation 1.22 is derived from initial equation 1.18 for gas flow through stages of differentiation (equations 1.19–1.21), considering gas compressibility. It is the non-linear partial differential equation applicable for combined gas and water flow in the cleats of coal formation. The equation is both diffusive and advective. It is highly non-linear due to the second degree advective part, and thus, is hyperbolic dominant. Implicit pressure and explicit saturation (IMPES) are used to solve the pressure-saturation coupling. The non-linear term is linearized and solved using the TDMA approach.

1.4.2 Initial and Boundary Conditions Initial condition: Pressure at initial time (t = Nt; time period of single-phase water flow) is equal to previous time nodal pressures. Boundary condition: Same as for single-phase water flow stage.

1.4.3  Methodology to Solve for Multiphase Flow IMPES To solve for pressure at node ‘n’ at time ‘t+dt’ implicitly and to proceed to obtain for the value of water saturation explicitly at node ‘n’ for time ‘t+dt.’

11

Mathematical Modeling of Transitional Fluid Phase Flows

Steps: 1. Step 1: Multiply equation 1.6 by

µwiv × k rg , µgiv × k rw

we get:

µw iv × Bgiv  ct iv × φiv × Sg iv ∂P φiv ∂Sg VL × PL ∂P  × + × + ×   k iv × k rw  Bgiv ∂ t Bgiv ∂ t ( P + PL )2 ∂ t 



=

µw iv × k rg

µg iv × k rw

2 ∂ P µw × k rg dP  × cl iv ×   × 2 + iv  dx  ∂x µgiv × k rw

(1.23)

2

2. Step 2: Add equations 1.17 and 1.23.

µwiv × φiv  ∂P ∂Sw  +  ct iv × Swiv ×  k iv × k rw  ∂t ∂t 

+

=



µwiv × Bgiv  ct iv × φiv × Sg iv ∂P φiv ∂Sg VL × PL ∂P  × + × + ×   ∂ t Bgiv ∂ t ( P + PL )2 ∂ t  (1.24) k iv × k rw  Bg iv

2 ∂2 P µw iv × k rg ∂2 P µwiv × k rg Sg iv  dP  × × + × +   ∂x 2 µg iv × k rw ∂x 2 µg iv × k rw Piv  dx 

µwiv × φiv µwiv ∂P × ct iv × Swiv × + × ct iv × φiv × Sgiv ∂ t k iv × k rw k iv × k rw µwiv ∂Sg ∂P µwiv × φiv ∂Sw × + × φiv × × + ∂t ∂t ∂ t k iv × k rw k iv × k rw (1.25) µw × Bgiv VL × PL ∂P ∂2 P µwiv × k rg × = = + + iv k iv × k rw (P + PL )2 ∂ t ∂ x 2 µgiv × k rw ×





2 ∂2 P µwiv × k rg Sgiv  dP  × × +   ∂ x 2 µgiv × k rw Piv  dx 

dSwiv dSgiv + = 0 (1.26) dt dt

µwiv  VL × PL  ∂P  φiv × ct iv + Bgiv × × k iv × k rw  ( P + PL )2  ∂t  µw × k rg  ∂2 P µwiv × k rg Sgiv  dP  2 =  1 + iv × + × ×  µgiv × k rw  ∂x 2 µgiv × k rw Piv  dx  

(1.27)

12



Applied Mathematical Modeling and Analysis in Renewable Energy

µwiv  µwiv × k rg VL × PL  = a (1.28)  φiv × ct iv + Bgiv × 2  = b;  µg iv × k rw k iv × k rw  ( P + PL )  2

b×



Sg ∂P ∂2 P dP  = (1 + a ) × 2 + a × iv ×   (1.29) Piv  dx  ∂t ∂x

Equation 1.26 is the water-gas saturation relation. Equation 1.27 is splitting equation 1.25 into time and distance terms. Equation 1.29 is the decoupled form of non-linear partial differential equation (PDE) in terms of p and respective coefficients. 3. Step 3: Substitute ‘P’ and linearize equation 1.29. P = →+ v (1.30)





P

b×

∂(→+ v) P

∂t

= (1 + a ) ×

∂2 (→+ v) P

∂x 2

2

 d (→+ v)  Sg + a × iv ×  P  (1.31)  Piv  dx 2

) ∂2 (→+ v)  ∂(→) ∂(v )  Sgiv  d (→ d (v )  P P P a + + × × b× +  = (1 + a ) ×   (1.32) Piv  dx dx  ∂ t  ∂x 2  ∂ t



 ∂(→)   ∂2 (→)  ∂(v ) ∂2 (v) p p  = (1 + a ) ×   + + b× ∂t  ∂x 2   ∂ t  ∂x 2  

(1.33)

 d (→)  2 d (→) 2 sgiv   d (v )  d (v )   p p  +2× +a × ×  × +  dx   piv   dx  dx dx   

d (→) Sg iv  d (v )  (1.34)  ∂(v )   ∂2 (v )  P 1 a a 2 b× = + × + × × × × ( )      ∂ t  Piv  dx dx  ∂x 2 

(



)

 (v nt+ dt − v nt )   (v nt ++dt1 − 2 × v nt+ dt + v nt +−dt1 )  b×  = (1 + a ) ×   dt dX 2    d (→)  (v nt ++dt1 − v nt +−dt1 )  p + a × cl iv ×  2 × × dx 2 × dX  

(1.35)

13

Mathematical Modeling of Transitional Fluid Phase Flows









(v nt+ dt − v nt ) = G  × (v nt ++dt1 − 2 × v nt+ dt + v nt +−dt1 ) + H × cl iv × ( P [ N ] − P [ 0 ]) × (v nt ++dt1 − v nt +−dt1 )

(1.36)

µwiv  µwiv × k rg VL × PL  = a (1.37)  φiv × ct iv + Bg iv × 2  = b;  µg iv × k rw k iv × k rw  ( P + PL )  H=

a × dt

( N ) × b × dx 2

; G =

((1 + a ) × dt )

(

( b × dX 2 ) (1.38)

(

v nt+ dt × (1 + 2 × G ) − v nt ++dt1 G + H × cl iv × ( P [ N ] − P [ 0 ])

(

)

− v nt +−dt1 G − H × cl iv × ( P [ N ] − P [ 0 ]) = v nt

))

(1.39)

Pressure at next time level is defined by the pressure at present time and error (equation 1.30). Equation 1.30 is substituted into equation 1.29 and the steps from Equations 1.31–1.38 are followed. Equation 1.39 is the final form of PDE to be solved using the TDMA approach to solve ∈ v nt+ dt , and equate the value with the average reservoir pressure at node n and next time level.

1.4.4 Numerical Model The non-linear PDE is linearized. The coefficients in the linearized equation discretized using BTCS. The dependent variable at the next time step is solved using the TDMA approach.

1.4.5 Water Saturation to be Solved Explicitly Using Equation 1.17 Equation 1.42 is the water saturation equation, and equation 1.45 is the gas saturation equation.



µwiv ( P ) × φiv ( P )   P t + dt − Pnt   Swt+ dt − Swt   cfiv ( Sw , P ) × Swiv ×  n  +       k iv ( P ) × k rw ( Sw )  dt dt  P t + dt − 2 × Pnt + dt + Pnt −+1dt  =  n +1   dX 2

(1.40)

(Swt+ dt ) × (1 + cfiv × ( Pnt + dt − Pnt ))

 k × k rw  Pnt ++1dt − 2 × Pnt + dt + Pnt −+1dt   =  Swt + dt × iv ×   (1.41) µwiv × φiv  dX 2  

14



Applied Mathematical Modeling and Analysis in Renewable Energy

(Swt+ dt

(S + dt × )= t w

k iv × k rw µw ×φiv iv

×

(

Pnt ++1dt − 2 × Pnt + dt + Pnt +−1dt dX 2

(1 + cf × ( P iv

t + dt n

− Pnt )

)

) × 0.0158) (1.42)

Sgt+ dt = (1 − Swt+ dt ) (1.43)

A flowchart (Fig. 1.3) explains the complete procedure followed from defining the dependent variables to calculating them for a multiphase water and gas flow in cleat.

FIGURE 1.3  The coding procedure is followed with the use of assumptions and calculation of dependent variables for multiphase gas and water flow.

Mathematical Modeling of Transitional Fluid Phase Flows

15

The variables are average reservoir pressure and water saturation. Two iterative loops are present. The flow equation consists of the coupled pressure-saturation term. The first iterative loop solves the functions of pressure, while the relative permeabilities are solved in the second loop.

1.5  VERIFICATION AND VALIDATION Verification: The permeability and porosity relation, as mentioned in the reference journal, is pressure-dependent. Therefore, to verify the code with semi-analytic (semi-numerical) results, the following empirical relationships are used:

φn = φi × e( 0.0003×( Pn − Pi )) (1.44)



φ  k n = k i ×  n  (1.45)  φi 

3

Equations 1.44 and 1.45 represent the pressure-dependent cleat porosity term, and the permeability derived from it using the cubic relation, respectively.

1.5.1  Verification and Validation Results In the plots of cumulative production and the rate of gas and water against the number of days, data (semi-numerical model) could not be possible as the initial input values of some variables differ. The missing initial input data for the calculation obtained from different works done by the authors such as Clarkson and Qanbari et al. (2016), Zhang et al. (2008), and Robertson (2006) are used. These works are referred to for similar coal formation (Manville). From the above verification and validation plots, cumulative water produced shows a trend similar to both field values and semi-analytic numerical values from the reference paper (Fig. 1.4(a)). Water is produced both during the single-phase and multiphase flow stages. Initially, the increase in production is approximately linear since water saturation is assumed constant throughout the single-phase flow. With the onset of multiphase flow, the plot becomes non-linear since water produced decreases with a decrease in water saturation in cleats. Cumulative gas (Fig. 1.4(b)) increases continuously with time due to an increase in gas saturation in cleats. The increase is non-linear because there is no proportional relation of production with gas saturation, and is also dependent on other variables. The water rate shows a linearly increasing trend during the initial stages of production, followed by a reducing trend when gas flow starts (Fig. 1.4(c)). Due to the decreasing water rate in multiphase flows, the cumulative water produced, during this period, depicts a non-linear increase. The gas rate increases with time following the dewatering process (Fig. 1.4(d)). The code for the single-phase water followed by multiphase water and gas flow is verified and validated and used for further calculations.

16

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 1.4  (a) Validating numerically obtained cumulative water results with field values and verifying the same with semi-numerical results values of reference Clarkson et al. (2016). (b) Validating numerically obtained cumulative gas results with field values. (c) Validating numerically obtained water rate results with field values and verifying the same with seminumerical results values of reference Clarkson et al. (2016). (d) Validating numerically obtained gas rate results with field values and verifying the same with semi-numerical results values of reference Clarkson et al. (2016).

Mathematical Modeling of Transitional Fluid Phase Flows

17

FIGURE 1.4  (Continued)

1.6  SINGLE-PHASE GAS FLOW



∂P  V × PL  φiv ( P ) ∂Sg ×  ct iv ( Sw , P ) × φiv ( P ) × Sgiv ( Sw ) + L × + ∂t  ( P + PL )2  Bgiv ( P ) ∂t =

k iv ( P ) × k rg ( sw )  ∂ P dP   + cl iv ( Sw , P ) ×     dx   µgiv ( P ) × Bgiv ( P )  ∂x 2



2

(1.46)

2

Sgiv = 1; k rg = 1; k rw = 0 (1.47)

2  ∂2 P V × PL  k iv ( P ) ∂P   dP   (1.48) cl P ×  ct iv ( P ) × φiv ( P ) + L = + × ( )    iv  dx   ∂t  ( P + PL )2  µgiv ( P ) × Bgiv ( P )  ∂x 2

Substituting cl iv ,  ∂2 P 1  dP  2  ∂P  V × PL  k iv ( P ) ×  ct iv ( P ) × φiv ( P ) + L + ×   (1.49) 2 = ∂t  ( P + PL )  µgiv ( P ) × Bgiv ( P )  ∂x 2 Piv  dx  

 VL × PL  k iv ( P ) = a (1.50)  φiv × ct iv + 2  = b;  µ P ( ) × Bgiv ( P ) P + P ( L)  giv 



 ∂2 P 1  dP  2  ∂P ×b= a× 2 + ×   (1.51) ∂t Piv  dx    ∂x



∂P ∂2 P a  dP  ×b=a× 2 + ×   (1.52) ∂t ∂x Piv  dx 

2

18

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 1.5  The coding procedure is followed with the use of assumptions and calculation of dependent variables for single-phase gas flow.

Mathematical Modeling of Transitional Fluid Phase Flows

19

Equation 1.46 is the gas flow equation, the same as equation 1.22. Equation 1.47 shows the conditions of single-phase gas flow that modifies equations 1.46–1.48. Equation 1.49 is the final equation for gas flow, while equation 1.50 is the splitting of equation 1.49 into a and b terms. It is the non-linear partial differential equation. It is highly non-linear due to the second degree advective part, and thus is hyperbolic dominant. IMPES is used to solve the pressure-saturation coupling. The non-linear term is linearized and solved using the TDMA approach. Equations 1.51 and 1.52 represent the PDE and the coefficients.

1.6.1 Numerical Model The methodology of linearization and solving linearized equations by BTCS is the same as that used for the multiphase flow. A flowchart (Fig. 1.5) explains the complete procedure followed from defining the dependent variable to calculating them for a single-phase gas flow in the cleat. The variable is average reservoir pressure. The iterative loop solves for the non-linearity in the flow equation. Both reservoir and fluid properties mentioned in the list vary with time.

1.7 CONCLUSION Every fluid flow PDE consists of the mass and momentum conservation equation. It is non-linear or linear equation in nature. If non-linear, then the cause is understood and solved numerically. A fully implicit numerical method, finite difference, is used to convert the partial differential equation to an algebraic form. Iterative methods, in case of single-phase water flow, are adopted to solve for the non-linearity. IMPES is used to decouple pressure-saturation, followed by the methodology to linearize the pressure terms in case of multiphase flow problems. Pressure term linearization is used in PDE for a single-phase gas flow equation.

NOMENCLATURE i ( suffix ): Initial value of  variable. iv ( suffix ) : Averaged value of  variable at time t in the area of  Xinv. n ( suffix ) :  n th  nodal values of  the parameter of  points N. t ( suffix ):  t th  values of  the parameter of  points Nt. w( suffix ): Water phase. g( suffix ): Gas phase. pwf : Flowing well bore pressure  ( psi ) . P: Pressure at previous time level  ( psia ) . k: Permeability ( mD ) . φ : Fracture porosity. PL : Langmuir pressure  ( psi ) . VL : Langmuir Volume ( Mscf ) . b: A constant value related to desorption pressure. E: Young’s modulus of  coal  ( psi ) .

20

Applied Mathematical Modeling and Analysis in Renewable Energy

cf : Cleat compressibility  ( psi −1 ) . ct: Total compressibility  ( psi −1 ) . cl: Fluid compressibility  ( psi −1 ) . ε L : Langmuir volumetric strain. σ : Effective horizontal stress  ( psi ) . ν : Poisson’s ratio of  coal. rb Bw : Fomation volume factor of  water ( stb ). µw : Water viscosity (cp). Sw : Water saturation. N : Number of  points along X direction. t: Time one dt before present investigating time ( t + dt ) ( days ) . Xinv: Drained distance at time ( t + dt )(ft). wp: Cumulative water produced ( stb ) . ct: Total compressibility  ( psi −1 ) . h: Thickness of  formation ( ft ) . Nt: Total transient phase time  ( days ) . dt: Time increment within time Nt  ( days ) . dx: Space interval in distance of  xinv ( ft ) . qw: Water flow rate ( stb / d ) . gp: Cumulative gas produced ( Mscf ). qg: Gas flow rate ( Mscf / d ). rb Bg: Fomation volume factor of  gas ( stb ). µg: Gas viscosity (cp). Sg : Gas saturation.

REFERENCES Li, C., Wang, Z., Shi, L., Feng, R., (2017). ‘Analysis of Analytical Models Developed under the Uniaxial Strain Condition for Predicting Coal Permeability during Primary Depletion.’ Energies, Vol. 10, No. 11, pp. 1849. Clarkson, C.R., Qanbari, F. (2016). ‘A Semi-analytical Method for Forecasting Wells Completed in Low Permeability, Undersaturated CBM Reservoirs.’ Journal of Natural Gas Science and Engineering, Vol. 30, pp. 19–27, doi.org/10.1016/j.jngse.2016.01.040. Lili, X. (2012). ‘Numerical well testing of coal bed methane reservoir.’ PhD thesis, HeriotWatt University. Jishan, L., Chen, Z., Elsworth, D., Qu, H., Chen, D. (2011). ‘Interactions of Multiple Processes during CBM Extraction: A Critical Review.’ International Journal of Coal Geology, Vol. 87, No. 3, pp. 175–189, doi.org/10.1016/j.coal.2011.06.004. Nie, R., Ying, F., Meng, J., Chun, G., Yong, L. (2011). ‘Modeling Transient Flow Behavior of a Horizontal Well in a Coal Seam.’ International Journal of Coal Geology, Vol. 92, pp. 54–68, doi.org/10.1016/j.coal.2011.12.005. Zhang, H., Liu, J., Elsworth, D. (2008). ‘How Sorption Induced Matrix Deformation Affects Gas Flow in Coal Seams: A New FE Model.’ International Journal of Rock Mechanics and Mining Sciences, Vol. 45, No. 8, pp. 1226–1236, doi.org/10.1016/ j.ijrmms.2007.11.007. Robertson, P.E. (2006).‘Measurement and modeling of sorption-induced strain and permeability changes in coal,’ PhD thesis, Colorado School of Mines.

2

AI-Based Detection and Localization of Gastrointestinal Polyps by Using Deep Learning, Transfer Learning and the Fusion of These Techniques Yogesh Chaudhari, Umme Salma Pirzada, and Darshee Baxi

CONTENTS 2.1 2.2 2.3 2.4 2.5

Introduction: Background and Driving Forces................................................ 21 Deep Neural Network (DNN)......................................................................... 23 Transfer Learning............................................................................................24 Polyps and Computer-Aided Diagnosis...........................................................25 Dataset and Experimental Setup.....................................................................25 2.5.1 Dataset.................................................................................................25 2.5.2 Data Preparation..................................................................................26 2.5.3 CNN Architecture............................................................................... 27 2.5.4 VGG16 and ND VGG19......................................................................28 2.6 Results and Discussion.................................................................................... 30 2.7 Conclusion and Future Work........................................................................... 32 References................................................................................................................. 32

2.1  INTRODUCTION: BACKGROUND AND DRIVING FORCES As per the World Health Organization (WHO) statistics, all around 1,800,000 individuals are diagnosed and around 862,000 individuals lost their life in the year 2018 globally due to colorectal cancer (CRC). Among the various diagnosis methods, colonoscopy is the most reliable norm for screening of CRC. The petite formation of cells and enlargements of lesions in the inward covering of colon or rectum DOI: 10.1201/9781003159124-2

21

22

Applied Mathematical Modeling and Analysis in Renewable Energy

progresses to malignant tumor that gradually leads to CRC. These protuberances are termed as polyp, which when missed or ignored, can prompt CRC. Almost 90% of CRC cases result from the unnoticed development of adenomatous polyps. For effective diagnosis and treatment, early identification of colonic polyps utilizing colonoscopy is necessary. The ADR is the effective assessment parameter to measure the performance of endoscopist [1]. The colored portion in the Fig. 2.1 shows the portion of intestine examined in the colonoscopy. The 1993 milestone National Polyp Study [2] showed that the frequency of colorectal malignant growth could be decreased by colonoscopy of adenomatous polyps, which has been affirmed in many follow-ups and check-ups. Adenoma detection rate (ADR) – the expertise level of screening colonoscopies – has become a key quality measure. Higher ADRs are related with lower post-colonoscopy colorectal tumors and lower colorectal malignant growth mortality. In spite of the fact that ADR ought to be perfect, studies show that ADR shifts generally among colonoscopists

FIGURE 2.1  Anatomy of the gastrointestinal system. (©2017 American Cancer Society, Inc., Surveillance Research.)

AI-Based Detection and Localization of Gastrointestinal Polyps

23

(7–53%), and 20–30% of adenomas could be missed during screening colonoscopies. The ADR performance targets are 25% in general, ≥30% for men and ≥20% for women [3]. Automated examination for detection and localization of polyps may significantly increase ADR and control the high missed diagnosis rate. A few extra procedures and apparatuses are under analysis for improving an endoscopist’s capability to recognize adenomas with the objective to expand productivity and lessening costs. The convolutional neural network (CNN)-based diagnosis systems adapt exceptionally fast for the computer-aided diagnosis (CAD) for detection, localization and grouping of polyps. The CAD helps image investigation that junctions both expanded polyp recognizable proof and histopathologic separation without adjustments to the imaging instruments or the standard method. The point when CNN is implemented in fusion with feature extractor and transfer learning techniques, the general execution of the CAD system is improved in terms of learning time and accuracy [4]. We have made an attempt to implement the transfer learning using Visual Geometry Group (VGG)16 and VGG19 techniques to further improve the performance of deep learning architecture implemented using CNN architecture and to study its effect on performance with respect to the number of epochs.

2.2  DEEP NEURAL NETWORK (DNN) The DNN is designed by cascading multiple layers of artificial neural network (ANN). The application of deep learning techniques to recognize objects and their use in the field of computer vision have been profoundly studied since 2000, and the outcomes are well accepted [5]. Deep learning algorithms are designed to learn visual feature hierarchies that can be implemented using either unsupervised learning algorithms (if labeled data samples are relatively low) or supervised learning algorithms (if labeled data samples are relatively sufficient). The successes of supervised learning CNN-based architectures demonstrated in 2012 in ImageNet competition garnered the growing attention of computer vision community that outperformed other approaches in terms of recognition accuracy. The work published in ref. [6] validates the use of deep learning for auto-detection of polyps during colonoscopy, and the results are promising though the authors have cautioned related to the risk of major complications remain similar to conventional colonoscopy. A CNN architecture is perfectly suitable for visual applications as its design and development is inspired by the natural mechanism of ‘visual perception and attention’ [7]. The discovery by Hubel & Wiesel in 1959 [8] further motivated Kunihiko Fukushima who designed the neocognitron – a neural network-based self-organizing model to mimic visual pattern recognition mechanism in 1980 [9]. The work LeNet-5 published by Yann Le Cun in refs. [10] and [11] was capable of recognizing handwritten character and classifying when two-dimensional images are given as an input to the model. Zhang X. et al. [12] have discussed the use of CNN architecture for polyp detection and reported the finding that the implantation can help in reducing the polyp miss rate.

24

Applied Mathematical Modeling and Analysis in Renewable Energy

2.3  TRANSFER LEARNING The professor and data scientist Andrew Ng stated in NIPS 2016 that after supervised learning, transfer learning will be the next driver of commercial success of machine learning (ML). Transfer learning relaxes the hypothesis that the training data must be independent and identically distributed with the test data, which motivates us to use transfer learning to solve the problem of insufficient training data. Transfer learning is normally an approach wherein a model trained on one problem is tuned in a way to work on other similar problem. The transfer of learning in CNNs showed great potential for celiac disease classification based on endoscopic images [13]. The authors used four CNNs that were pre-trained on the ImageNet database. Three different transfer learning strategies were explored to classify the endoscopic images of the celiac disease. Full fine-tuning of the CNNs achieved the highest classification accuracies, although the small amount of training data available led to overfitting [14]. The Fig. 2.2 shows the difference in difference in knowledge transfer in (a) conventional machine learning and in (b) transfer learning.

FIGURE 2.2  Conventional machine learning (a) vs transfer learning and (b) in progress.

AI-Based Detection and Localization of Gastrointestinal Polyps

25

2.4  POLYPS AND COMPUTER-AIDED DIAGNOSIS In ref. [15], the authors studied the feasibility of polyp recognition and classification of two types of polyps – hyperplasias and adenomas. The study uses image segmentation and is based on the use of visual feature of vascularization for computer-based classification. The approaches designed with feature extraction and that are based on more polyp features like shape, size and color of polyps will lead to improved diagnosis. Figure 2.3 shows the growth of polyps from stage 1 to stage four of malignancy. The larger polyp is in metastasis stage as it spreads to nearby tissues of its origin.

2.5  DATASET AND EXPERIMENTAL SETUP For the experimental setup, the public dataset CVC-Clinic DB has been explored [16]. The polyp positive and negative images are prepared using annotation. The cropped images are grouped for the purpose of training, testing and validation. In each of these categories, the images are further sub-grouped as positive and negative.

2.5.1 Dataset The dataset utilized in the present work is CVC-Clinic DB that consists of frames extracted from videos captured through colonoscopy and is published for Endoscopic

FIGURE 2.3  Polyp (colorectal cancer) growth. (©2005, Terese Winslow, U.S. Government has certain rights.)

26

Applied Mathematical Modeling and Analysis in Renewable Energy

TABLE 2.1 Description of Dataset Sr. No. 1

Particular Original images

2

Polyp mask

3 4 5 6 7

Sequences Dimensions Size of RGB image Size of polyp mask Size of database

Details 612 frames extracted from colonoscopy videos 612 images corresponding each to original images Monochrome image representing polyp with ‘white’ and ‘black’ otherwise 29 384 by 288 324 kB 108 kB 258 MB

Vision Challenge. The CVC-Clinic DB database consists of two sets of images: (1) colored Red Green Blue (RGB) images and (2) corresponding monochrome polyp masks. The description of dataset is provided in Table 2.1. Figure 2.4 shows 3 random images taken from the dataset along with the corresponding monochrome mask for each of these images.

2.5.2 Data Preparation The images prepared using annotation are further processed with image augmentation to create transformed images to increase the data samples. These images are

FIGURE 2.4  Top row – RGB polyp frames extracted from sequence and bottom row – corresponding monochrome polyp mask. (CVC-Clinic DB database.)

AI-Based Detection and Localization of Gastrointestinal Polyps

27

FIGURE 2.5  Data hierarchy – (a) superset consisting of three subsets and (b) each subset with two more subsets.

grouped into three sets, i.e., training, validation and testing, each with two subsets: positive and negative. The details of number of samples according to data hierarchy given in Fig. 2.5 are shown in Table 2.2.

2.5.3  CNN Architecture The CNN architecture is designed (as shown in Fig. 2.6) for the training and testing of the samples from dataset. The CNN model has been implemented with Keras. A CNN architecture has three main parts: • A convolutional layer to extracts features • A pooling layer to reduce the dimensionality with no loss of features or patterns • A fully connected (dense) layer to generate a prediction Three systems are designed to detect and locate the polyp as below: • Using CNN architecture with layers as discussed in Fig. 2.7 • Using VGG16 pre-trained model • Using VGG16 pre-trained model with fusion of fully connected (FC) layer TABLE 2.2 Number of Data Samples in Each Set Sr. No. 1 2 3 4 5 6

Set Train Validation Testing

Subset Positive Negative Positive Negative Positive Negative

Samples 1820 1820 600 600 100 100

Total 3640 1200 200

Percentage 36.11 72.22 36.11 11.90 23.81 11.90 1.98 3.97 1.98

28

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 2.6  Major blocks of proposed system for polyp detection.

The polyp positive and polyp negative images are cropped from original images using semi-automated approach using the script for annotations, and hence the polyp masks are not used in the processing. The cropped images are given as an input to image augmentation to generate three more files with rotation of parent file in angles 90°, 180° and 270°. In the architecture, 64 filters are used with each convolution kernel of size 3 × 3, and to reduce spatial dimensionality, the pooling kernels used are of size 2 × 2. The stride is used to enhance the performance of the architecture. Out of available activation functions such as step, sigmoid, tanh, ReLu, ELU and Leaky ReLu, the current implementation makes use of ReLu. To prevent an overfitting in the model, dropout technique is incorporated.

2.5.4  VGG16 and ND VGG19 Various pre-trained CNN models are available that are trained on ImageNet dataset inside the Keras library. These pre-trained models can be utilized for prediction, feature extraction and fine-tuning. Few pre-trained models are listed below: • VGG16 • VGG19

FIGURE 2.7  Sequential representation of CNN architecture.

AI-Based Detection and Localization of Gastrointestinal Polyps

• • • •

29

ResNet50 Inception V3 Xception GoogLeNet

In comparison to VGG16, VGG19 performs better with more memory requirement. VGG16 and VGG19 models are designed with convolution layers, max pooling layers and fully connected layers, with total 16 and 19 layers, respectively. With the reference to the work published in ref. [17], VGG16 and VGG19 are explored and also modified [18] to test the performance for the diagnosis of gastrointestinal polyps. The diagrammatic representation of (a) VGG16 and (b) VGG19 is shown in Fig. 2.8.

FIGURE 2.8  Layered architecture of (a) VGG16 and (b) VGG19.

30

Applied Mathematical Modeling and Analysis in Renewable Energy

The layers are termed as ‘3 × 3 conv, X’, where ‘conv’ are convolutional layers with 3 × 3 filters and X represents number of filters. MaxPool layer reduces dimensions by the factor of 2. At the end, FC layer is with 4096 and 1000 units. Softmax outputs one of a 1000 classes from labeled data provided by ImageNet.

2.6  RESULTS AND DISCUSSION In the processing of input images, the precaution is taken due to varying dimensionality of samples in training and validation set. We have experimented with the following ten different implantations as detailed in Table 2.3. After execution of the above model, we obtained the results as shown in Fig. 2.9. The models are arranged in the increasing order of accuracy. It is observed during experimentation that as the number of layers in a model increases, the accuracy also increases, with the exception at model number 5. The structure of model numbers 8, 9 and 10 is similar, and improvement is achieved by reducing the drop rate. For the VGG implantation, we have used the batch size of 64 and trained the network for same number of epochs, i.e., 100. The training and validation loss (VL) results obtained for VGG16 implementation are shown in Fig. 2.5. For VGG16 implementation, the overfitting is observed as VL is far greater than training loss (TL) as shown in Fig. 2.10(a). The drop rate is set to 0.92 to reduce overfitting and the obtained result is shown in Fig. 2.10(b). From Table 2.4, we can observe that the difference between VL and TL is significantly reduced by increasing the drop rate. The third implementation of VGG16 with training of last convolutional block with fully connected layer is underfitting for more number (150) of epochs. With fine-tuning and working with hyperparameters like ‘learning rate’ and ‘momentum’, better results are obtained with model accuracy of 93.35.

TABLE 2.3 Details of Different Combinations of Architectures Implemented Sr. No. 1 2 3 4 5 6 7 8 9 10

Convolutional Layers 1 1 1 2 1 2 3 3 3 3

Number of Filters Used 32 0 0 32 0 0 64 0 0 64 128 0 128 0 0 32 64 0 32 32 64 64 64 128 64 64 128 64 64 128

Kernel Size 5×5 3×3 3×3 3×3 3×3 3×3 3×3 3×3 3×3 3×3

Number of Epochs 100 100 100 100 100 100 100 100 100 100

AI-Based Detection and Localization of Gastrointestinal Polyps

FIGURE 2.9  The accuracy plot for different CNN architectures (as per Table 2.3).

FIGURE 2.10  (a) Loss plot for drop rate = 0.6 and (b) loss plot for drop rate = 0.92.

TABLE 2.4 Effect of Drop Rates on Losses and Overfitting Drop Rate Sr. No. 1 2 3 4 5

Loss/Accuracy TL Training accuracy VL Validation accuracy VL – TL

0.60 0.003 1.000 1.129 0.867 1.127

0.92 0.075 0.973 0.702 0.844 0.627

31

32

Applied Mathematical Modeling and Analysis in Renewable Energy

2.7  CONCLUSION AND FUTURE WORK As the detection of polyps depends on shape, size and color of the polyp, it is necessary to remove training samples from dataset with blur shapes and colors. It is important to remove these inefficient samples before the data augmentation phase. The kernel of size 5 × 5 results into more information loss, and hence the kernel of 3 × 3 is ideal for implantation. The direct relationship is observed between more layers and more accuracy, but this results into increased computational complexity (execution time) of the models. The small size dataset results into overfitting, and it is recommended to use dataset larger than that used in this work. Better results are obtained with fusion of VGG16 with training of last convolutional block and fully connected layer. In the future work, we plan to study the effect of epochs, drop rates and feedback of error on models, and test the performance with more evaluation metrics listed and described in refs. [19] and [20].

REFERENCES









1. Rex DK, Schoenfeld PS, Cohen J, et al. Quality indicators for colonoscopy, Gastrointest Endosc 2015;81:31–53. 10.1016/j.gie.2014.07.058. 2. Winawer SJ, Zauber AG, Ho MN, O’Brien MJ, Gottlieb LS, Sternberg SS, Waye JD, Schapiro M, Bond JH, Panish JF, et al., Prevention of colorectal cancer by colonoscopic polypectomy. The National Polyp Study Workgroup. N Engl J Med 1993 December 30;329(27):1977–1981. 10.1056/NEJM199312303292701. 3. Chao WL, Manickavasagan H, Krishna SG. Application of artificial intelligence in the detection and differentiation of colon polyps: A technical review for physicians, Diagnostics 2019 August 20;9(3):99. 10.3390/diagnostics9030099. PMID: 31434208; PMCID: PMC6787748. 4. Tan C, Sun F, Kong T, Zhang W, Yang C, Liu C. (2018) A Survey on Deep Transfer Learning. In: Kůrková V., Manolopoulos Y., Hammer B., Iliadis L., Maglogiannis I. (eds.) Artificial Neural Networks and Machine Learning – ICANN 2018. ICANN 2018. Lecture Notes in Computer Science, vol. 11141. Springer, Cham. https://doi. org/10.1007/978-3-030-01424-7_27. 5. Deng Li, Yu Dong, Deep learning: Methods and applications, Found Trends Signal Process 2014;7(3–4): 197–387. http://dx.doi.org/10.1561/2000000039. 6. Wang P, Xiao X, Liu J, Li, L, Tu, M, He, J, Hu, X, Xiong, F, Xin, Y, Liu, X, A prospective validation of deep learning for polyp auto-detection during colonoscopy: 2017 international award, Am J Gastroenterol 2017 October;112:S106–S110. 10.1038/ ajg.2014.297. 7. Jiuxiang G, Zhenhua W, Jason K, Lianyang M, Amir S, Bing S, Ting L, Xingxing W, Li W, Gang W, Jianfei C, Tsuhan C, et al. Recent advances in convolutional neural networks. 2015 arXiv:1512.07108. 8. Hubel DH, Wiesel TN, Receptive fields and functional architecture of monkey striate cortex, J Physiol 1968;195(1):215–243. doi: 10.1113/jphysiol.1968.sp008455. PMID: 4966457; PMCID: PMC1557912. 9. Fukushima K, Miyake S. (1982) Neocognitron: A Self-organizing Neural Network Model for a Mechanism of Visual Pattern Recognition. In: Amari S., Arbib M.A. (eds) Competition and Cooperation in Neural Nets. Lecture Notes in Biomathematics, vol. 45, pp. 267–285. Springer, Berlin, Heidelberg. https://doi. org/10.1007/978-3-642-46466-9_18

AI-Based Detection and Localization of Gastrointestinal Polyps

33

10. LeCun Y, Boser B, Denker J, Henderson D, Howard R, Hubbard W, Jackel L, et al. Handwritten digit recognition with a back-propagation network, in: Proceedings of the Advances in Neural Information Processing Systems (NIPS), 1989, pp. 396–404. 11. Lecun Y, Bottou L, Bengio Y, Haffner P, et al., Gradient-based learning applied to document recognition, in: Proceedings of IEEE 1998;86 (11):2278–2324. 12. Zhang X, Chen F, Yu T, An J, Huang Z, et al. Real-time gastric polyp detection using convolutional neural networks, PLoS One 2019;14(3):e0214133. https://doi.org/10.1371/ journal.pone.0214133. 13. Wimmer G, Vécsei A, Uhl A, "CNN transfer learning for the automated diagnosis of celiac disease," 2016 Sixth International Conference on Image Processing Theory, Tools and Applications (IPTA), Oulu, 2016, pp. 1–6, doi: 10.1109/ IPTA.2016.7821020. 14. Choi J, Shin K, Jung J, Bae H-J, Kim DH, Byeon J-S, Kim N, et al., Convolutional neural network technology in endoscopic imaging: Artificial intelligence for endoscopy, Clin Endosc 2020;53(2):117–126. Publication Date (Web): 2020 March 30 (Focused Review Series: Application of Artificial Intelligence in GI Endoscopy). https://doi. org/10.5946/ce.2020.054. 15. Stehle T, Auer R, Gross S, Behrens A, Wulff J, Aach T, Winograd R, Trautwein C, Tischendorf J, Classification of colon polyps in NBI endoscopy using vascularization features, in: Proceedings SPIE 7260, Medical Imaging 2009: Computer-Aided Diagnosis, 72602S, 27 February 2009, https://doi.org/10.1117/12.808103. 16. Bernal J, Sánchez FJ, Fernández-Esparrach G, Gil D, Rodríguez C, Vilariño F, WM-DOVA maps for accurate polyp highlighting in colonoscopy: Validation vs. saliency maps from physicians, Comput. Med. Imaging Graph 2015;43:99–111. 17. Hasan MM, Islam N, Rahman MM, et al., Gastrointestinal polyp detection through a fusion of contourlet transform and Neural features, J King Saud Univ Comp & Info Sci 2020, https://doi.org/10.1016/j.jksuci.2019.12.013. 18. Wang W, Tian J, Zhang C, Luo Y, Wang X, Li J, et al., An improved deep learning approach and its applications on colonic polyp images detection. BMC Med Imaging 2020;20:83. https://doi.org/10.1186/s12880-020-00482-3. 19. Chaudhari Y, Senthil Kumar R, Baxi D, et al., AI based automated diagnosis of polyps in colonoscopy: An introduction for researchers and physicians in healthcare, Int J Eng Res Appl ISSN: 2248–9622 2019 December;9(12) (Series-II):06–12. 20. Kothoke P, Deshpande A, Chaudhari Y, et al., Analysis and determination of partial discharge type using statistical techniques and back propagation method of artificial neural network for phase-resolved data, Int. J. Eng. Res Technol ISSN: 2278-0181 2019 August;8(8):430–438.

3

Mathematical Modeling of an EOQ for a Multi-item Inventory System with Selling Price and Price Break Sensitive Demand Abhijit Barman and Pijus Kanti De

CONTENTS 3.1 Introduction..................................................................................................... 35 3.2 Literature Review............................................................................................ 36 3.3 Fundamental Assumptions and Notations....................................................... 37 3.4 Model Formulation.......................................................................................... 37 3.5 Solution Methodology..................................................................................... 38 3.6 Numerical Analysis.........................................................................................40 3.7 Conclusion....................................................................................................... 42 References.................................................................................................................44

3.1 INTRODUCTION Inventory management improves business tasks with the successful progression of merchandise and enterprises. Now a days, common people have very little time to buy their essential products from different retail shops. For this reason, the importance of shopping malls, supermarkets, or any multi-item inventory increases dayby-day. Retailers deal with several items in a multi-item inventory, and use the various concepts of multi-item inventory strategies to maximize their profit such as quantity discount, discount in selling price, lucky draw, etc. This chapter discusses a multi-item profit maximization inventory model for a retailer under sales price and price break-dependent demand. The idea of mathematical modeling on multi-product storing policy with backorders subject to budget constraints was first introduced by Miller [1]. A broad review of such results can be found in the works of Hartley and Thomas [2], Rosenblatt and Rothblum [3], etc. Bhattacharya [4] introduced a new method of multi-item inventory DOI: 10.1201/9781003159124-3

35

36

Applied Mathematical Modeling and Analysis in Renewable Energy

model in case of deteriorating items. A stochastic multi-item lot-sizing model has been formulated in ref. [5] based on the expected demand values. Hill and Pakkala [6] established a base stock inventory policy for multi-item environments. In this chapter, we, the authors, have reflected a retailer or vendor inventory system considering the multi-product storing capacity.

3.2  LITERATURE REVIEW With the application of a genetic algorithm, Guchhait et al. [7] developed an economic order quantity (EOQ) model for breakable items under time-varying breakability rate. This model is demonstrated in both crisp and fuzzy environments. Pal et al. [8] established a multi-item inventory model to find the optimal sales price, order quantity when the demand fluctuates due to sales price or price break strategy [8]. Supply chain (SC) coordination and promotional effect are incorporated in a multi-item EOQ model by Cárdenas-Barrón and Sana [9]. Ghosh et al. [10] described a limited storage space as constraints in their stockbased multi-item inventory model. An application of price-sensitive demand with the Stackelberg-Nash equilibrium game approach has been introduced in a cost-minimization SC model by Taleizadehand Noori-Daryan [11]. Alfares and Ghaithan [12] explored an inventory model under quantity discounts and timevarying holding costs. Jana and Das [13] proposed a non-linear mixed inventory problem with an instant price discounts policy using a hybrid optimization algorithm. Mixed-integer programming is used to present a multi-item, multi-period EOQ model with a quantity discount, and is partially backlogged in Alfares and Turnadi [14]. Liao and Deng [15] have studied some EOQ models with deterministic market demand. Barman et al. [16] have established a multi-item EOQ model with product deterioration, where demand fluctuates due to price, time, and stock of the products. Two-layered SC coordination underprice elasticity in demand policy, centralized market decision, and green manufacturing decision, has been developed by Barman et al. [17–19]. Wangsa and Wee [20] minimized the total cost in a vendor-buyer two-echelon SC incorporating a quantity discount policy. Even though sufficient literature is available in multi-item inventory models, significantly less literature is available on selling price-dependent multi-item EOQ models under price break. This chapter advocates an inventory mechanism for different items under the restriction of price break policy. The retailer, vendor, or seller offers a value markdown on offering price to the client when the seller’s total absolute revenue reaches his/her target sell value, known as price break level. In the assumption, the product’s demand rate increases quadratically with a decrease in the value of sales price that depends algebraically on the level of the price break. Here, we have incorporated some demand elasticity parameters and drawn a variation with geometrical interpretation to understand the changes in overall profit with respect to the changes of different parameters. In practical, retailers face different kinds of market situations directly or indirectly. By implementing our model, a vendor or seller can positively overcome the possible losses (if any) during low market demand by controlling the sales price of the item and changing the price break level. In the end, a profit

Mathematical Modeling of an EOQ for a Multi-item Inventory System

37

maximization inventory model has been analyzed by optimizing sales price, ordering lot size, and price break level using the Lagrangian multiplier. The rest of the chapter is organized as follows: In section 3.3, we have stated the basic assumptions and notations for two models under study. Section 3.4 gives the mathematical formulation of both the models with and without discount. In section 3.5, we have given solution methodology with the sufficient condition. Section 3.6 is composed of numerical discussions. Finally, conclusion has been given in section 3.7.

3.3  FUNDAMENTAL ASSUMPTIONS AND NOTATIONS Assumptions: The following assumptions are employed to develop the following inventory models: 1. The model is considered for n number of different type of items. 2. The economic variable involved with the model is ordering lot size, selling price, and price break level. 3. The demand rate varies inversely with the selling price and price break parameter. 4. When the vendor’s total revenue is greater than the price break level, then the vendor gives some price discount on selling price to consumers. 5. Shortages are not allowed. Notations: Table 3.1 shows the notations used to develop the model.

3.4  MODEL FORMULATION Now a days, people give more importance to supermarkets or shopping malls to save their time in daily life. Understanding this customer behavior, most of the retail sector retailers or vendors use different types of strategies to attract customers’ attention. This model explores a profit maximization price back policy in business strategies in a multi-item inventory setting. The practical phenomenon of price break policy states that after selling, some fixed target retailers or vendors give a discount TABLE 3.1 Notations Mi pi B Ri and Ri′ Ci Oi Qi ai , bi, ci , α , β , and a I

Production cost per item for i th product Selling price per item for i th product Price break level Demand rate before and after price break Carrying cost per item per unit time for i th product Ordering cost per order Ordering quantity in each lot Demand elasticity parameters Total profit/ Sales revenue

38

Applied Mathematical Modeling and Analysis in Renewable Energy

on the selling price of the products, which is called price break level. In this chapter, we have established two-stage inventory models before and after the level of the price break. We have considered that the demand rate at time t depends inversely on the non-linear pattern of selling price, and depends algebraically on the level of price break. Now, two cases will arise: (i) Total expenses of the retailers at time t are less than the price break level, and (ii) total expenses of the retailers at time t are greater than the price break level. Here, we have described the two cases by Model 1 and Model 2, respectively. Model 1: Suppose at time t, the total sales value does not exceed the price break level. In this instance, the discount will not be allowed. For a > 1 and ai >> bi >> ci, the demand rate is assumed as Ri = ai − bi pi − ci pi 2 + α a − β B, i = 1, 2, ..., n. Then, the total unit time profit ( I ) of the retailer or vendor (= Sales revenue – Inventory carrying cost – Ordering cost – Total production cost) per unit time, i.e., n

I=







∑  p R − 12 C Q − QR O − M R  i

i

i

i

i

i

i

i

i

1

subject to constraint is n

∑ p R < B (3.1)



i

i

1

Model 2: Similarly, suppose the total sales value at time t exceeds the price break level. In this case, the discount will not be allowed. For a > 1 and ai >> bi >> ci, the demand rate is assumed as Ri ' = ai − bi (1 − µi ) pi − ci (1 − µi )2 pi 2 + α a − β B , i = 1, 2,...n . Then, the total unit time profit ( I ) of the retailer or vendor per unit time is calculated by n

I=







∑ (1 − µ ) p R′ − 12 C Q − QR′ O − M R′ i

i

i

i

i

i

i

i

i

i

1

subject to constraint n

∑ (1 − µ ) p R′ > B (3.2)



i

i

i

1

3.5  SOLUTION METHODOLOGY Model 1: For i = 1, 2,...n , the Lagrangian function (L), formulated from equation (3.1), is described as n



L=

∑ 1

   1 Ri  pi Ri − 2 CiQi − Q Oi − M i Ri  + λ  B − i   



n

∑p R  (3.3) i

1

i

Mathematical Modeling of an EOQ for a Multi-item Inventory System

39

where λ indicates the Lagrangian multiplier, and the number of unknowns in the Lagrangian function L are 2n + 2, i.e., Q1, Q2, …,Qn, p1, p2 , …, pn, B, λ. Now, differentiating partially with respect to Qi , pi , B, λ . ∂L 1 R O = − Ci + i2 Oi = 0 ⇒ Qi 2 = 2 Ri i (3.4) ∂Qi 2 Qi Ci





O  ∂L = (1 − λ )  Ri − pi ( bi + 2ci pi )  +  i + M i  ( bi + 2ci pi ) = 0 ∂ pi  Qi 



⇒ (1 − λ ) ( ai − bi pi − ci pi 2 + α a − β B ) − pi ( bi + 2ci pi )  +



  CiOi + M  ( bi + 2ci pi ) = 0 (3.5)  i 2 −β B  2 ( ai − bi pi − ci pi + α a ) 





∂ L  = ∂B  

n

∑ 1

   Oi −β B − p − − M 1 λ ( ) i i    ( −αβ a ln(a) ) + λ = 0 (3.6) Qi   

∂L = B− ∂λ

n

∑p ( a − b p − c p i

i

i

i

i

i

2

+ α a − β B ) ≤ 0 (3.7)

1

Now, for a particular value of λ, equations (3.4), (3.5), (3.6), and (3.7) give a nonlinear system of equations with 2n+2 number of unknowns, including λ. Using software Mathematica, we can easily solve these non-linear equations. Model 2: After discount policy, using the demand function Ri′, the generalized form of Lagrangian function (L′) is calculated from equation (3.2) as follows: n

L′ =



i

i

i

i

i

i

 +λ 

i

i

i

i

1





∑ (1 − µ ) p R′ − 12 C Q − QR′ O − M R′ n

∑ 1

 (1 − µi ) pi Ri′  − B  

(3.8)

Equation (3.8) contains 2n+2 numbers of unknown with Lagrangian multiplier λ. Differentiating partially with respect to Qi , pi , B, λ , we get ∂L′ O = 0 ⇒ Qi2 = 2 Ri′ i (3.9) ∂Qi Ci





∂L′ 2 = (1 + λ ) (1 + µi )  ai − bi (1 − µi ) pi − ci (1 − µi ) pi 2 + α a − β B  ∂ pi 2 − pi (1 − µi ) bi + (1 − µi ) ci 2 pi  + 

(

)

40

Applied Mathematical Modeling and Analysis in Renewable Energy

  CiOi   M + i  2 a − b (1 − µ ) p − c (1 − µ )2 p 2 + α a − β B  i i i i i i i  

(







)

(3.10)

(1 − µi ) bi + (1 − µi )2 ci 2 pi = 0    ∂L′ = ∂B

n





∑ (1 + λ ) (1 + µ ) p − OQ − M  (αβ a i

i

i

−β B

i

1

∂L = B− ∂λ

i

n

∑ p ( a − b (1 − µ ) p − c (1 − µ ) i

i

i

i

i

i

2

i

1

ln(a) ) + λ = 0 (3.11)

)

pi 2 + α a − β B > 0 (3.12)

Optimality criteria for Model 1 and Model 2: To justify the optimality at

{Qi* , pi* , B* , λ * }, we have to check the corresponding Hessian matrix of the objec-

tive function combining with 2n+2 number of unknowns p1, p2, …, pn, Q1, Q2, …, Qn, B, and λ . If the principle minors of the Hessian matrix alternate their sign starting with negative sign, i.e., (−1) j ∆ j ≥ 0, where ∆ j is the j th order cofactor of the Hessian matrix, then the objective function is strictly concave, and {Qi* , pi* , B* , λ * } for i = 1, 2, …, n are optimal solutions.

3.6  NUMERICAL ANALYSIS In this section, an illustration of theoretical results has been discussed with two numerical examples for two different scenarios, i.e., Model 1, and Model 2. Then, we have optimized the optimal selling price, lot size, and price break label. Numerical 1: Consider a supermarket problem with three different items and the corresponding parametric values given in Table 3.2. Solution: Using software Mathematica, we solve the non-linear system of equations 3.4–3.7 and we get the value of selling price, ordering lot size, and price break level. We check the optimality condition from the Hessian matrix that the principle TABLE 3.2 Numerical values of example 1 Parameters ai , bi

Item 1

Item 2

Item 3

a2 = 146 , b2 = 1.1 c2 = 0.006 α = 5000

a3 = 129, b3 = 0.9 c3 = 0.004 α = 5000

β

a1 = 170 ,b1 = 1.0 c1 = 0.005 α = 5000 β = 0.0005

β = 0.0005

β = 0.0005

Ci Mi Oi

$0.5 $9 $150

$0.6 $7 $200

$0.4 $8 $140

ci α

41

Mathematical Modeling of an EOQ for a Multi-item Inventory System

TABLE 3.3 Sensitivity Analysis with Respect to Parameter a (before Price Break) a 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6

p1 103.92 88.36 80.55 75.61 72.18 69.71 67.90 66.57 65.60 64.90 64.40 64.03 63.77

p2 88.56 74.16 66.97 62.44 59.33 57.09 55.46 54.26 53.39 52.76 52.32 51.99 51.76

p3 102.9 84.92 76.01 70.45 66.63 63.91 61.93 60.49 59.44 58.69 58.15 57.77 57.49

Q1 294.30 259.72 245.50 238.16 234.03 231.62 230.22 229.40 228.94 228.69 228.56 228.50 228.48

Q2 298.64 259.82 243.41 234.72 229.69 226.68 224.85 223.74 223.08 222.68 222.46 222.33 222.26

Q3 280.34 241.26 224.39 215.27 209.88 206.57 204.50 203.22 202.42 201.93 201.63 201.45 201.34

B 39845.8 25387.8 20194.2 17555.0 16001.6 15016.2 14365.9 13928.1 13630.8 13428.2 13289.8 13195.0 13129.7

I 36147.3 22524.5 17650.9 15173.1 13710.4 12778.5 12160.3 11741.7 11455.8 11259.9 11125.3 11032.7 10968.6

minors alternate their sign starting with negative values. For a = 2.4, if the optimal selling price p1* = 67.90 , p2* = 55.46, and p3* = 61.93, the order quantity Q1* = 230.22, Q2* = 224.85, and Q3* = 204.50, then the price break level B = 14365.9 and the optimal profit I = $12160.3. For understanding the behavior of demand elasticity parameter a, the following sensitivity table (Table 3.3) is drawn. With the increase in a values, selling price, order quantity, price break level, and the overall profit decrease. Numerical 2: Consider another supermarket problem with three different items and the corresponding parametric values are given in Table 3.4. Solution: For same value a = 2.4, the optimal solutions from equations (3.9), (3.10), (3.11), and (3.12) are p1* = 70.42 , p2* = 66.35, p3* = 61.85, Q1* = 314.39, Q2* = 317.02, Q1* = 315.92,B = 15857.1, and overall profit I = $11805.2. The changes in optimal solutions with respect to parameter a are listed in Table 3.5.

TABLE 3.4 Numerical values of example 2 Parameters

Item 1 a1 = 161, b1 = 1.0

Item 2 a2 = 166 , b2 = 1.1

Item 3 a3 = 135, b3 = 0.9

ci α

c1 = 0.005 α = 5000

c2 = 0.006 α = 5000

c3 = 0.004 α = 5000

β Ci Mi Oi Price discount (µi %)

β = 0.0005 $0.5 $9 $300 10%

β = 0.0005 $0.6 $7 $350 12%

β = 0.0005 $0.45 $8 $320 9%

ai ,bi

42

Applied Mathematical Modeling and Analysis in Renewable Energy

TABLE 3.5 Sensitivity Analysis with Respect to Parameter a (after Price Break) a 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6

p1 108.90 91.63 83.07 77.81 74.33 71.98 70.42 69.40 68.74 68.32 68.05 67.88 67.77

p2 101.07 85.38 77.62 72.90 69.81 67.73 66.35 65.45 64.88 64.51 64.28 64.13 64.04

p3 98.95 82.29 74.01 68.93 65.58 63.34 61.85 60.87 60.25 59.85 59.60 59.44 59.34

Q1 400.94 352.42 333.00 323.44 318.42 315.77 314.39 313.71 313.38 313.23 313.16 313.14 313.14

Q2 403.69 355.83 336.39 326.64 321.41 318.55 317.02 316.21 315.79 315.59 315.49 315.44 315.41

Q3 420.46 363.56 340.01 328.03 321.49 317.89 315.92 314.86 314.31 314.02 313.87 313.80 313.76

B 41006.0 26392.4 21221.9 18666.2 17227.8 16373.5 15857.1 15544.9 15356.7 15243.3 15174.6 15132.4 15106.2

I 32936.3 20656.4 16327.0 14183.0 12970.7 12246.1 11805.2 11536.7 11373.8 11275.0 11214.7 11177.6 11154.3

3.7 CONCLUSION This chapter deals with multi-item inventory models considering the demand function totally depends on selling price and price break level. To attract more number of customers to supermarkets / malls, selling prices are reduced to some extent by offering some price discount. In this chapter, attention has been focused on the order quantity and selling price under the level of price break. We have developed the constraint optimization problem by using Lagrangian method. To maintain the profit in a positive way, the vendor can compare his selling price and price break level with any other existing model, and can change them according to his suitability. From the graphical sensitivity analysis, our study reveals that: 1. As the initial value of the parameter ‘a’ increase, the value of the price break level decreases accordingly. 2. As the price break level B increases, the demand of inventory decreases accordingly for all the items. 3. Total profit of the retailer changes inversely with the changes in parameter a, as shown in Figs. 3.1 and 3.2, respectively. 4. The model shows maximum profit when the parameter ‘a’ is minimum. From the economic viewpoint, it is clear that the profit of the vendor increases with the increase in demand of the item. The model can be extended in various environments such as floor space constraints, budget constraints under SC strategies, etc. Further, the same problem can be worked using fuzzy logic by introducing uncertainties in economic parameters such as demand, lead time, etc.

Mathematical Modeling of an EOQ for a Multi-item Inventory System

FIGURE 3.1  Impact of total profit on the parametric value a (before price break).

FIGURE 3.2  Impact of total profit on the parametric value a (after price break).

43

44

Applied Mathematical Modeling and Analysis in Renewable Energy

REFERENCES



1. Miller, B.L.: A multi-item inventory model with joint backorder criterion. Operations Research 19 (6), 1467–1476 (1971). 2. Hartley, R., &Thomas, L.C.: The deterministic, two-product, inventory system with capacity constraint. The Journal of the Operational Research Society 33 (11), 1013–1020 (1982). 3. Rosenblatt, M.J., & Rothblum, U.G.: On the single resource capacity problem for multiitem inventory systems. Operations Research 38 (4), 686–693 (1990). 4. Bhattacharya, D.K.: On multi-item inventory. European Journal of Operational Research 162, 786–791 (2005). 5. Brandimarte, P.: Multi-item capacitated lot-sizing with demand uncertainty. International Journal of Production Research 44 (15), 2997–3022 (2006). 6. Hill, R.M., & Pakkala, T.P.M.: Base stock inventory policies for a multi-item demand process. International Journal of Production Economics 109, 137–148 (2007). 7. Guchhait, P., Maiti, M.K., & Maiti, M.: Multi-item inventory model of breakable items with stock-dependent demand under stock and time-dependent breakability rate. Computers and Industrial Engineering 59, 911–920 (2010). 8. Pal, B., Sana, S.S., & Chaudhuri, K.: Multi-item EOQ model while demand is sales price and price break sensitive. Economic Modelling 29(6), 2283–2288 (2012). 9. Cárdenas-Barrón, L.E., & Sana, S.S.: Multi-item EOQ inventory model in a two-layer supply chain while demand varies with promotional effort. Applied Mathematical Modelling 39(21), 6725–6737 (2015). 10. Ghosh, S.K., Sarkar, T., & Chaudhuri, K.: A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand. American Journal of Mathematical and Management Sciences 34(2), 147–161 (2015). 11. Taleizadeh, A.A., & Noori-Daryan, M.: Pricing, manufacturing and inventory policies for raw material in a three-level supply chain.  International Journal of Systems Science 47(4), 919–931 (2016). 12. Alfares, H.K., & Ghaithan, A.M.: Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts. Computers & Industrial Engineering 94, 170–177 (2016). 13. Jana, D.K., & Das, B.: A two-storage multi-item inventory model with hybrid number and nested price discount via hybrid heuristic algorithm.  Annals of Operations Research 248(1–2), 281–304. (2017). 14. Alfares, H.K., & Turnadi, R.: Lot sizing and supplier selection with multiple items, multiple periods, quantity discounts, and backordering.  Computers & Industrial Engineering 116, 59–71 (2018). 15. Liao, H., & Deng, Q.: EES-EOQ model with uncertain acquisition quantity and market demand in dedicated or combined remanufacturing systems.  Applied Mathematical Modelling 64, 135–167 (2018). 16. Barman, A., Das, R., & De, P.K. (2020, December). Pricing and Inventory Decisions of Multi-item Deteriorating Inventory System under Stock, Time and Price Sensitive Demand Policy. In  2020 9th International Conference System Modeling and Advancement in Research Trends (SMART) (pp. 453–458). IEEE. 17. Barman, A., Das, R., & De, P.K. (2020, June). Pricing and inventory policy for deteriorating item in a two-echelon supply chain: a stackelberg duopoly game approach. In  2020 8th International Conference on Reliability, Infocom Technologies and Optimization (Trends and Future Directions)(ICRITO)  (pp. 796–800). IEEE.

Mathematical Modeling of an EOQ for a Multi-item Inventory System

45

18. Barman, A., Das, R., & De, P.K. (2020). An analysis of retailer’s inventory in a twoechelon centralized supply chain co-ordination under price-sensitive demand.  SN Applied Sciences 2(12), 1–15. 19. Barman, A., Das, R., & De, P.K. (2021). Optimal pricing and greening decision in a manufacturer retailer dual-channel supply chain. Materials Today: Proceedings. 20. Wangsa, I.D., & Wee, H.M.: Integrated inventory system with freight costs and two types of quantity discounts.  International Journal of Logistics Systems and Management 35(1), 119–147 (2020).

4

Fair Allocation of Items A Comprehensive Study Manisha Bansal and Purnima Bindal

CONTENTS 4.1 Introduction..................................................................................................... 47 4.2 Fair Allocation of Divisible Goods.................................................................. 48 4.2.1 Criterion of Fairness for Divisible Goods........................................... 49 4.2.2 Current State of the Art....................................................................... 50 4.2.2.1 Cut-and-Choose Protocol for m = 2 Agents......................... 50 4.2.2.2 Allocation is Envy-Free........................................................ 51 4.2.2.3 Allocation is Proportional..................................................... 51 4.2.2.4 For m = 3 Agents.................................................................. 51 4.2.2.5 α-Approximately Envy-Free................................................. 52 4.3 Fair Division of Indivisible Goods.................................................................. 53 4.3.1 Criterion of Fairness for Indivisible Goods......................................... 53 4.3.2 Types of Valuations............................................................................. 53 4.3.3 Current State of the Art....................................................................... 54 4.3.3.1 Algorithms Based on Ordinal Ranking................................ 54 4.3.3.2 Algorithms Based on Valuations/Cardinality....................... 54 4.4 Concluding Remarks and Directions for Future Work.................................... 55 References................................................................................................................. 55

4.1 INTRODUCTION The idea of fair division came into existence due to lack of equality in distribution of goods amongst the participating agents. This problem has persisted in the society for a long time. It also finds a reference in the bible, the book of genesis, where Abraham and Lot were assigned a task of allocating a piece of land amongst themselves. They followed the following procedure to allocate the land to themselves. One of them, say Abraham, divides the land into two parts, which are of equal value to him. Lot then chooses a piece of land, which he thinks is of more value. The allocation of land done in this manner is both proportional and envy-free. It is proportional because each one of them get at least half of the value of the total land; envy-free because the value of the land piece received by each one is not less than the piece of the land the other person has received. This is a well-known fair allocation protocol, known as divide-and-choose protocol. There are many different domains where the fair allocation finds applications. The problem has been widely studied in economics and has been helpful in making DOI: 10.1201/9781003159124-4

47

48

Applied Mathematical Modeling and Analysis in Renewable Energy

decisions regarding allocation of funds by the government or division of property amongst heirs or the allocation of bandwidth in telecommunication system. In recent years, with the advent of cloud computing, many more avenues of resource sharing have arisen. In cloud computing jobs, different types of resources are required such as CPU, memory, disk, network bandwidth etc. There may be many such jobs competing for resources. The system may want to use the resources in an optimum manner. But it is also important that the resources are allocated to jobs in a fair manner so that every job is satisfied. Optimal usage of resources and fairness of allocation are two conflicting objectives that a system of resources and agents may wish to achieve. In some situations, it is the fairness that is more desirable like division of property among heirs. Optimal utilization of resources may be more desirable for some applications like utilization of CPU in a computing environment, where an optimum scheduling is the aim. In cloud computing settings, optimal utilization of resources and customer satisfaction both are important to be achieved. In such scenarios, a trade-off between fairness and optimal utilization is required. Fair-allocation criteria studied in literature include envy-freeness (EF), proportionality and equitability. Social welfare criteria which have found mention in the current research works related to this field are utilitarian social welfare, egalitarian social welfare, Nash social welfare and Pareto optimality. All of the above allocation criteria are defined a little differently for divisible and indivisible version of the problem. In case of indivisible items, envy-free allocations may not even exist. Therefore, other notions of fairness have been introduced for such settings: envy-free up to one good (EF1), envy-free up to least valued good (EFX). Rest of the chapter is organized as follows: in section 4.2, we discuss the fair allocation of divisible goods. First, some preliminaries related to the problem are discussed. Next, we present some algorithms that find allocations which fulfill the criteria of proportionality and/or envy-freeness. Then, some results related to approximation algorithms for the problem are presented. In section 4.3, the discussion is related to the fair allocation of indivisible goods. The last section 4.4 gives the conclusion of the chapter.

4.2  FAIR ALLOCATION OF DIVISIBLE GOODS The problem of dividing a resource amongst m agents is popularly known as cake-cutting problem. The resource (cake) can be modeled as a line segment [0, 1], as done by Procaccia in [1]. Let N = {1, 2, 3, …, n} represents the set of agents. The agents have their preferences for different pieces of the cake which are defined by a valuation function, vi (.). Given a subinterval I ⊆ [0, 1] and agent i ∈ N, the function maps the subinterval to the value assigned by agent i to I. Agents have distinct valuations for a subinterval but equal entitlements. Agents can get contiguous or non-contiguous pieces of the cake depending on the allocation protocol. Preferences of agents for the pieces they want are expressed as cardinal valuations. The valuations specify how much a particular subinterval is valued by an agent. The utility/valuations of each agent for the whole cake is 1, i.e., the valuations are normalized.

49

Fair Allocation of Items

FIGURE 4.1  Interval [x, y] ⊆ [0, 1].

The valuation function vi (.) is considered to satisfy the following properties: 1. Non-negativity: vi(I) ≥ 0 ∀ i ∈ N & I ⊆ [0, 1]. 2. Divisibility: ∀ I = [x, y] ⊆ [0, 1] and 0 ≤ α ≤ 1, ∃ z ∈ [x, y] such that as shown in Fig. 4.1.

(

)

v i ([ x, z ])  =  α  v i [ x, y ]   ∀  i  ∈ N 3. Additivity: vi (I U I’) = vi(I) + vi(I’) ∀ i ∈ N and ∀ I, I’⊆ [0, 1], where I and I’ are disjoint intervals. 4. Normalized: vi ([0, 1]) = 1 and vi (nothing) = 0 ∀ i ∈ N.

An allocation of the cake is defined by a division of interval [0, 1] into a set of disjoint sub-intervals, which may be contiguous or non-contiguous, {I1, I 2, I3, …, I n}, where I i is allocated to agent i. The aim is to find an allocation which satisfies certain criteria. We may be interested in finding an allocation which satisfies an individual agent’s preferences or we may wish to have an allocation which aims for social welfare.

4.2.1 Criterion of Fairness for Divisible Goods 1. Envy-freeness (EF): In an envy-free division of a divisible item, every agent feels that his share is at least as good as the share of the other agent, i.e., no agent has a preference for a piece of cake that is received by another agent, i.e.,

( )

v i ( I i )  ≥  v i I j   ∀  i,  j  ∈ N 2. Proportionality: Every agent’s share is at least equal to 1/n portion of the whole cake according to his valuation, i.e.,



1 v i ( I i )  ≥     ∀  i  ∈ N n 3. Equitability: Every two agents give exactly the same value to their own piece of cake or subinterval, i.e.,



( )

v i ( I i )  =  v j I j   ∀  i,  j  ∈ N

50

Applied Mathematical Modeling and Analysis in Renewable Energy

Social welfare objectives are as follows: 1. Utilitarian social welfare: The objective of this allocation criterion is to maximize the sum of the utilities of all agents. In such allocations, there may be some agents who receive a piece of cake which is of very less value as compared to the value of the piece obtained by some other agents. 2. Egalitarian social welfare: The objective of this allocation is to maximize the utility of worst-off player. 3. Nash social welfare (NSW): The aim of this social welfare criterion is to maximize the geometric mean of utilities of all agents. Since 1940s, various cake cutting protocols have been written, satisfying different criteria of fairness or social welfare.

4.2.2 Current State of the Art We will discuss some cake-cutting protocols and the fairness properties satisfied by these protocols. For two agents, we discuss cut-and-choose protocol. For three agents, Selfridge-Conway Protocol [2] is discussed. When the number of agents is four, a protocol given by Haris Aziz and Simon Macanzie [3] provides an envy-free allocation. The same authors gave a protocol [4] in 2016 that finds an envy-free allocation. 4.2.2.1  Cut-and-Choose Protocol for m = 2 Agents When there are two players or agents, cut-and-choose protocol gives an envy-free and proportional solution. It is a two-step protocol: Step 1: The cake is cut into two pieces by player 1 such that both the pieces are equally valued by him, see Fig. 4.2.

v1 ([ 0, x ]) = v1 ([ x, 1]) Step 2: Player 2 chooses one of the intervals (pieces) that are of more value for him, and the remaining piece will be allocated to player 1. Let v2 ([0, x]) ≥ v2 ([x, 1]). So, player 2 selects the left interval [0, x] and the right interval [x, 1] is assigned to player 1.

FIGURE 4.2  Interval [0, 1] divided into two parts.

51

Fair Allocation of Items

4.2.2.2  Allocation is Envy-Free Player 1 is not envious of player 2 because both intervals (pieces) are of equal value to him, whereas player 2 is also getting a piece which is more valuable to him. So, both the players are satisfied with the pieces of their choice, i.e.,

v1  ([ 0, x ])  =  v1  ([ x, 1])



and v 2  ([ 0, x ]) ≥  v 2  ([ x, 1])

4.2.2.3  Allocation is Proportional Player 1 will get a piece of value exactly 1/2 as he is the cutter and both pieces are of equal value for him, and player 2 will get at least 1/2 of the total value as he is selecting the piece first.

1 v1 ([ x,1])  =    v1 ([ 0,1]) 2



1 v 2 ([ 0, x ])  ≥    v 2 ([ 0,1]) 2

In fact, when the valuations are additive, for m = 2, envy-freeness ↔ proportionality. When the number of agents, m > 2, envy-freeness ⇒ proportionality but not vice versa. 4.2.2.4  For m = 3 Agents Steinhaus gave a protocol for three agents, which satisfies the proportionality but not envy-free [2]. Banach-Knaster proposed a proportional protocol for arbitrary n [5]. Selfridge-Conway gave envy-free protocol for three players. This protocol is divided into two stages: Stage 1: Division of the whole cake Step 1: Player 1 divides the interval into three equal parts according to his own valuations. Step 2: Player 2 trims his most valuable part equal to his second most valuable part. Trimmed part of the trimmed piece is kept aside. Step 3: Player 3 chooses his piece first. So, player 3 is not envious to player 1 and player 2 as he is the first to choose a preferred piece. Step 4: Player 2 chooses a piece. He will choose the trimmed piece, if it is left. Otherwise, he will choose the second most valuable piece. He is also not envious to player 1 and player 3 because player 1 has not selected any piece till now, and player 3 selects a piece which is either player 2’s second or third choice. Step 5: Player 1 gets the remaining piece. He will not be envious to other players because all pieces are of equal value to him.

52

Applied Mathematical Modeling and Analysis in Renewable Energy

Stage 2: Division of trimmed piece Step 1: From player 2 and player 3, one who did not choose the trimmed piece will divide the trimmed piece into three equal parts. Other player who gets the trimmed part in stage1 chooses one part of the trimmings. Step 2: Player 1 gets the last trimmed part. This allocation gives non-contiguous pieces to the players. Haris Aziz and Simon Macanzie gave discrete and bounded envy-free protocol for four players in 2016 [3], and for any number of players in 2017 [4]. These protocols are envy-free as well as proportional. These protocols are also based on similar idea that we have explained for three agents. In many applications, it is also important to find an envy-free allocation which is contiguous. The existence of such allocations is known, but finding one efficiently has been a challenge. No algorithms are known which can find contiguous envyfree allocations. The approximation algorithms for finding envy-free allocation are known which are efficient. There are two types of approximation algorithms which give guarantee on the quality of result produced by the approximation algorithms. One is multiplicative or alpha-approximation factor or approximation ratio, i.e., the solution lies within the multiplicative bound of the solution, and the other is additive in which solution is a bounded amount less or more than the optimum. They are also known as relative and absolute performance guarantee, respectively. Various approximation algorithms have been defined which give approximate results and either multiplicative or additive performance guarantee. The objective of the approximation algorithms is not just to achieve single notion of fairness but to maximize the social welfare. Social welfare measures the goodness of division of the cake for the entire society. 4.2.2.5  α-Approximately Envy-Free An α-approximate envy-free allocation is one for which vi (Ii) ≥ α1 vi (Ij) for each pair of agents i, j and α ≥ 1. In 2012, Aumann et al. [6] developed an algorithm to maximize the social welfare. The algorithm gives (8 + O (1))-approximation performance guarantee for the utilitarian social welfare, and for egalitarian welfare, they have shown that it is hard to approximate the optimum to any factor less than 2. Arunachaleswaran et al. [7] gave an approximation algorithm which divides the cake into contiguous pieces such that each agent gets a piece without envying each other within a factor of (3 + O (1)). This algorithm guarantees (3 + O (1))-approximation factor with multiplicatively bounded envy, and (3 + O (1))-approximation ratio that maximizes NSW as much as possible. They also proved that maximizing NSW is APX-hard in the context of cake division. Goldberg et al. (2019) [8] developed two algorithms which divide the cake into n contiguous pieces with low envy amongst the agents in polynomial time. The first algorithm divides the cake in such a manner that no agent has envy more than 1/3 toward any other agent. In this algorithm, it might be possible that few agents will be empty-handed, i.e., they don’t get anything. The second algorithm gives an allocation with uniform valuation over single sub-interval and a lower envy of

Fair Allocation of Items

53

at most 1/4. These algorithms have unbounded multiplicative envy, i.e., some agents may be empty-handed.

4.3  FAIR DIVISION OF INDIVISIBLE GOODS Fair allocation of indivisible items allocates m indivisible or non-shareable items amongst n agents in the form of bundles. Each agent has different preferences for different bundles of goods. Preferences can be ordinal, i.e., goods are ranked 1, 2, 3, …, etc., by each agent according to their preferences, or cardinal, i.e., non-negative valuations of goods are provided by each agent for every item. There algorithms are based on different types of valuations and ordinal/cardinal preferences.

4.3.1  Criterion of Fairness for Indivisible Goods 1. Envy-free: Each agent is satisfied with the bundle of items he receives over a bundle allocated to any other agent. 2. Proportionality: Each agent’s bundle value is at least 1/n of the total value of items. 3. Pareto optimal: Pareto optimality or Pareto efficiency is a state where an allocation of goods cannot be changed without reducing the utility of some agent. 4. Maximin share (MMS): According to MMS, an agent divides n items into m bundles such that his minimum valued share is guaranteed. 5. Envy-free up to one good (EF1): Given an allocation A, we say that it is EF1 provided that for every agent a, b ∈ [m], ∃ a good g ∈ Ab s.t., va (Aa) ≥ va (Ab\{g}). Aa, Ab are the bundles allocated to the agents a and b, respectively. 6. Envy-free up to least valued good (EFX): An allocation I is said to be EFX if and only if for each agent a, b ∈ [m], and for each good g ∈ Ib, va (Ia) ≥ va (Ib\{g}). 7. Borda maximin (BMX): It involves maximizing the minimum borda score of goods received by each agent, where borda score is a sum of points given to the goods in the bundle where 0 is assigned to the agents’ lowest ranked item, 1 to next lowest, and so on. 8. Maximin aware (MMA): According to MMA, an agent is aware that he is not getting the worst valued bundle (set of goods), but he is not aware about the bundles allocated to other agents.

4.3.2 Types of Valuations Given an agent a ∈ N and a set of goods S ⊆ [n], valuation for the set S is denoted by va [S], and we assume that va (∅) = 0. 1. Additive: For a subset of items S, additive valuation for an agent a is va[S] = ∑ gε S va[g] ∀ goods g ∈ [n]. 2. Binary additive: Given an additive valuation v, it is said to be binary additive if v(g) ∈ {0, 1} for each item g ∈ [n].

54

Applied Mathematical Modeling and Analysis in Renewable Energy

3. Sub-additive (SA): A valuation is sub-additive if v (S U T) ≤ v (S) + v(T) for any S, T ⊆ [n]. 4. Submodular (SM): A valuation is submodular if for any S, T ⊆ [n] and g ∈ [n], v (S U {g}) – v(S) ≥ v (T U{g}) – v (T).

4.3.3  Current State of the Art 4.3.3.1  Algorithms Based on Ordinal Ranking Steven J. Brams and Daniel L. King [9] have defined two efficient algorithms based on ordinal preferences given at least one maximin and BMX allocations which are disjoint and envy-ensuring. They have also shown that there may exist inefficient maximin and BMX allocations at the same time. Next, they have shown that if each agent receives two items, then there always exists one efficient envy-ensuring allocation of goods. Agents may get unequal number of goods in above allocations because equal number of goods may not maximize the borda score of all agents. Steven J. Brams et al. [10] gave algorithms for two agents who have defined ranking for even number of indivisible items. They give the algorithm to find an allocation which is maximin, Pareto optimal and envy-free, but not BMX. Each agent gets an equal number of items. In 2014, they gave the algorithms which find the envy-free and Pareto optimal but not maximin allocation. Nicolas Dupuis-Roy and Frédéric Gosselin [11] did an experiment and divided ten indivisible goods amongst two pairs of agents using genetic algorithm. Its results are more satisfactory as compared to the results of fair allocation algorithms. 4.3.3.2  Algorithms Based on Valuations/Cardinality Procaccia and Wang [12] in 2014 developed a polynomial time algorithm for constant number of players when the valuations are additive. It gives a solution that is 2/3-approximate maximin fair allocation. Amanatidis et al. [13] gave the same result given by Procaccia and Wang, but for any number of agents instead of constant number of agents. Siddharth Barman and Sanath Kumar Krishna Murthy [14] in 2017 developed a simple and efficient algorithm with similar approximation guarantees. They have also shown that if valuations are monotone, non-negative and submodular, then there exists a 1/10 approximation algorithm for the problem, which uses roundrobin algorithm. Hau Chan et al. [15] in 2019 gave a polynomial time algorithm, which gives a 1/2-approximate MMA1 or exactly MMX allocation. When all agents have sub-additive valuations, allocation is 1/2-approximate EFX. Caragiannis et al. [16] in 2016 established the result that under additive valuations, there exists an EF1 and Pareto-efficient allocation. They also showed that an allocation which maximizes NSW is EF1 and Pareto-efficient. In 2018, Siddharth Barman et al. [17] developed the pseudo polynomial algorithm to find EF1 and Pareto-efficient allocation. The algorithm takes polynomial time if the valuations are bounded. They also developed the 1.45-approximation algorithm to attain the NSW objective. In 2019, Siddharth Barman et al. [18] developed a polynomial time, 1/2-approximate procedure, which is EF1 for the problem of maximizing social welfare. They

Fair Allocation of Items

55

also gave a bi-criteria approximation algorithm for maximizing the social welfare under MMS constraint, where each agent gets (1/2 – ε) times his maximin share for a fixed constant ε > 0.

4.4 CONCLUDING REMARKS AND DIRECTIONS FOR FUTURE WORK The fair-allocation problem has a number of applications in the field of economics, political science and computer science. Being inter-disciplinary in nature, the ideas can be gathered from the literature of other fields apart from the field of computer science. Both indivisible and divisible allocation problems are currently being studied and better algorithms satisfying the envy-freeness criteria and having practical applications are being developed. The results presented in the chapter do not provide a complete list of algorithms for the problem due to paucity of the space. For each criterion of fairness, a number of algorithms have been designed. This work, however, provides a sufficient starting point for understanding the problem and finding the pointers to the current research in the area.

REFERENCES









1. Ariel D. Procaccia. Cake cutting algorithms. 7 2004. doi: 10.1184/R1/6604007.v1. URL https://kilthub.cmu.edu/articles/CakeCuttingAlgorithms/6604007. 2. Steven J. Brams and Alan D. Taylor. An envy-free cake division protocol. The American Mathematical Monthly, 102(1):9–18, 1995. 3. Haris Aziz and Simon Mackenzie. A discrete and bounded envy-free cake cutting protocol for four agents. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC, 2016, Cambridge, MA, June 18–21, 2016, pp. 454–464. doi: 10.1145/2897518.2897522. URL https://doi. org/10.1145/2897518.2897522. 4. Haris Aziz and Simon Mackenzie. A discrete and bounded envy-free cake cutting protocol for any number of agents. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS, 2016, October 9–11 2016, Hyatt Regency, New Brunswick, NJ, pp. 416–427. doi:10.1109/FOCS.2016.52. URL https://doi.org/10.1109/ FOCS.2016.52. 5. Bronislaw Knaster. Sur le probleme du partage pragmatique de h. steinhaus. In Annales de la Societ_e Polonaise de Mathematique, 19:228–230, 1946. 6. Yonatan Aumann, Yair Dombb, and Avinatan Hassidim. Computing socially efficient cake divisions. In International conference on Autonomous Agents and Multi-Agent Systems, AAMAS ‘13, Saint Paul, MN, May 6–10, 2013, pp. 343–350. URL http:// dl.acm.org/citation.cfm?id=2484976. 7. Eshwar Ram Arunachaleswaran, Siddharth Barman, Rachitesh Kumar, and Nidhi Rathi. Fair and efficient cake division with connected pieces. In Web and Internet Economics-15th International Conference, WINE 2019, New York, NY, December 10–12, 2019, pp. 57–70. doi: 10.1007/978-3-030-35389-6\_5. URL https://doi. org/10.1007/978-3-030-35389-6\_5. 8. Paul W. Goldberg, Alexandros Hollender, and Warut Suksompong. Contiguous cake cutting: Hardness results and approximation algorithms. CoRR, abs/1911.05416, 2019. URL http://arxiv.org/abs/1911.05416.

56













Applied Mathematical Modeling and Analysis in Renewable Energy

9. Steven Brams and Daniel L. King. Efficient fair division: Help the worst off or avoid envy? Rationality and Society, 17(4):387–421, 11 2005. ISSN 1043–4631. doi: 10.1177/1043463105058317. 10. Steven J. Brams, D. Marc Kilgour, and Christian Klamler. An algorithm for the proportional division of indivisible items. MPRA Paper 56587, University Library of Munich, Germany, May 2014. URL https://ideas.repec.org/p/pra/mprapa/56587.html. 11. Nicolas Dupuis-Roy and Frédéric Gosselin. An empirical evaluation of fair division algorithms. In Proceedings of the Annual Meeting of the Cognitive Science Society, volume 31, 2009. 12. Ariel D. Procaccia and Junxing Wang. Fair enough: Guaranteeing approximate maximin shares. In ACM Conference on Economics and Computation, EC ‘14, Stanford, CA, June 8–12, 2014, pp. 675–692. doi: 10.1145/2600057.2602835. URL https://doi. org/10.1145/2600057.2602835. 13. Georgios Amanatidis, Evangelos Markakis, Afshin Nikzad, and Amin Saberi. Approximation algorithms for computing maximin share allocations. ACM Transactions on Algorithms, 13(4):52:1–52:28, 2017. doi: 10.1145/3147173. URL https:// doi.org/10.1145/3147173. 14. Siddharth Barman and Sanath Kumar Krishna Murthy. Approximation algorithms for maximin fair division. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC ‘17, Cambridge, MA, June 26–30, 2017, pp. 647–664. doi: 10.1145/3033274.3085136. URL https://doi.org/10.1145/3033274.3085136. 15. Hau Chan, Jing Chen, Bo Li, and Xiaowei Wu. Maximin-aware allocations of indivisible goods. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS ‘19, Montreal, QC, Canada, May 13–17, 2019, pp. 1871–1873. URL http://dl.acm.org/citation.cfm?id=3331947. 16. Ioannis Caragiannis, David Kurokawa, Herve Moulin, Ariel D. Procaccia, Nisarg Shah, and Junxing Wang. The unreasonable fairness of maximum Nash welfare. In Proceedings of the 2016 ACM Conference on Economics and Computation, EC ‘16, Maastricht, The Netherlands, July 24–28, 2016, pp. 305–322. doi: 10.1145/2940716.2940726. URL https://doi.org/10.1145/2940716.2940726. 17. Siddharth Barman, Sanath Kumar Krishnamurthy, and Rohit Vaish. Finding fair and efficient allocations. In Proceedings of the 2018 ACM Conference on Economics and Computation, Ithaca, NY, June 18–22, 2018, pp. 557–574. doi: 10.1145/3219166.3219176. URL https://doi.org/10.1145/3219166.3219176. 18. Siddharth Barman, Ganesh Ghalme, Shweta Jain, Pooja Kulkarni, and Shivika Narang. Fair division of indivisible goods among strategic agents. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS ‘19, Montreal, QC, Canada, May 13–17, 2019, pp. 1811–1813. URL http://dl.acm.org/ citation.cfm?id=3331927.

5

Hierarchical Demand Response Controller Dima Kayyali, Hussam Nosair, Amit V. Sant, and Hannah Michalska

CONTENTS 5.1 Introduction..................................................................................................... 57 5.2 Modeling TCLS............................................................................................... 59 5.3 Dr Objectives With Respect to Timescale....................................................... 59 5.3.1 Load-Following DR Controller...........................................................60 5.3.2 Voltage Feedback DR Controller......................................................... 61 5.4 Methodology.................................................................................................... 61 5.4.1 High-Level Optimizer (HLO).............................................................. 62 5.4.1.1 Mathematical Program of HLO............................................ 62 5.4.1.2 Net Load Predictor................................................................64 5.4.2 Low-Level Controller (LLC)...............................................................64 5.4.2.1 Mathematical Program of LLC............................................64 5.5 Results and Discussion.................................................................................... 65 5.5.1 Validation Example............................................................................. 65 5.5.2 Simulation Results of Case Studies.....................................................66 5.5.3 HLO Case Study.................................................................................. 67 5.5.4 LLC Case Study................................................................................... 67 5.6 Conclusion....................................................................................................... 72 Nomenclature............................................................................................................ 72 References................................................................................................................. 73

5.1 INTRODUCTION The continuous increase in the integration of renewable energy sources into the power system is challenging the reliability and security of power grid. The challenge is beyond a simple supply-demand balance; even if increasing the generation capacity is affordable, the capacity of power grid itself is questionable [1]. Conventionally in power systems, the practice aims to adjust energy supply in order to meet the demand, which is generally achieved via increasing installed capacity or purchasing reserves [2]. While such an approach maintains the reliability of system, given the availability of sufficient energy supply, the practice suffers environmental and economic drawbacks in addition to the slow response of generators. The development of variable renewable energy (VRE) and energy storage systems has been of particular interest in sustainable energy infrastructure [1]. The integration of intermittent DOI: 10.1201/9781003159124-5

57

58

Applied Mathematical Modeling and Analysis in Renewable Energy

VRE sources has introduced the issues of variability and uncertainty in the power system. The severity of these issues is correlated to the penetration level of VRE sources in the system. Variability occurs due to continuous change in supply from these sources. The imperfect knowledge about this variability limits the capability of operational strategies in terms of propagated uncertainty [3]. As a result, improving the power system flexibility is crucial for systems of high penetration of VRE such as microgrids. Flexibility of a power system is an indication of its capability to sustain the system balance throughout the variable operation conditions [4]. Microgrids operate in two modes: grid-connected mode and islanded mode. In grid-connected mode, the microgrid is dependent on the main grid to maintain the power balance of system. However, in islanded mode, the microgrid operates autonomously and should have the resources to sustain the system balance. Microgrids rely highly on VRE sources. This requires additional flexibility to maintain the system reliability. Commonly, research studies target the generation side in order to mitigate the variability and uncertainty. While investing in utilizing the supply-side resources has had major developments in elevating microgrids reliability, more recent studies investigate enhancing the power system flexibility from the demand-side perspective. Instead of altering the power supply to meet the system requirements, these studies consider controlling the power consumption, commonly referred to as demand response (DR). Being able to utilize demand resources more precisely, responsive loads will serve in enhancing the system’s flexibility and counteracting the presence of variability from the VRE sources. DR-based flexibility is manifested from utilizing loads as resources to balance the system (providing additional capacity) and having a relatively fast response capability (high ramping rate). DR practice deals with modifying the demand-side consumption patterns in response to a received information about power system conditions (such as the available generation) as well as different incentives/disincentives offered in order to maintain the system balance and security [2, 5]. Engaging consumers’ demand as responsive loads in managing the system results in benefiting both consumers and service providers. In short-term, customers respond to price signals, which leads to economic paybacks for consumers [6]. Moreover, the fast response rate of demandside resources serve well in providing ancillary services, offering a cheap and efficient reserve capacity to the system [2]. In long-term, the interaction of responsive loads facilitate shaping a smoother demand curve, hence reducing the peak demand and extending the lifetime of the system [6]. Loads are generally classified as industrial, commercial, and residential. Under different frameworks, different approaches to build load models are used. Earlier models apply statistical tools on historical measurements. Later studies model loads based on the consumers’ behavior, while others employ physical-based load models [7]. Residential loads in the framework of DR can be classified into critical and controllable loads according to the level of effect on consumers’ comfort when altering the loads operation [8]. Controllable loads can be further sub-classified into deferrable and interruptible loads, according to the suitable control method [7]. The control applied on deferrable loads can shift/schedule the operation to another time period, while the operation of interruptible loads can be controlled by adjusting their ON/ OFF switch periods or their settings [9, 10]. Examples of deferrable loads are electric

Hierarchical Demand Response Controller

59

vehicles, washers, and dryers, and examples of interruptible loads are thermostatically controlled loads (TCLs) such as heating units, which are the loads considered in this work [7, 8]. This work aims to present a more comprehensive DR controller, which is expected to reliably utilize DR resources in an islanded microgrid to accommodate variability in generation and dissipate the effect of uncertainty from the VRE sources. The work proposes a hierarchical DR controller, which addresses the aforementioned challenges.

5.2  MODELING TCLS Traditionally, DR studies aim at reducing the loads’ power consumption as a response to high electricity prices, anticipated spike, or loss of generation [10]. While this remains the fundamental approach of DR programs to maintain the system reliability, emerging DR frameworks incur that load states be considered in making the control decisions to facilitate a less disruptive function of DR program [11]. Many studies consider DR programs for large blocks of load, as in commercial loads and industrial facilities, utilizing the energy management and control system technologies [12]. Considering such large-scale loads in a DR program shows effective results in terms of power reduction in addition to the fact that such environments facilitate the automation of DR programs. More work, however, targets residential models to prompt the participation of residential loads in DR programs. In residential demand setting, TCLs represent a major portion of the total consumption [13]. In addition, TCLs can act as a storage with fast response, where such loads can be turned ON/ OFF when required without compromising the end-user comfort [14]. To this extent, TCLs become a major DR source. This is the main motivation to consider TCLs in the suggested research work, as mentioned earlier. Consideration of individual TCLs in a DR framework is of negligible effect toward the power reduction. Therefore, an aggregated TCL paradigm is essential to carry out effective DR process. TCL aggregation is, in fact, a major procedure given in many of the residential-based DR studies in literature. In the aggregation framework of TCLs, one of the key processes in DR program is to model and track the state evolution of TCLs themselves in order to carry out DR task while ensuring the end-user comfort. Different techniques are used in literature to achieve this key aspect, and they include state queuing [15–17], Fokker-Planck equations [18], and Kalman Filter [12, 19, 20]. Recent research work on TCL-based DR modeling and control has provided reasoning to consider the temperature of building mass along with the TCLs describing the appliances in those households or buildings [21–23].

5.3  DR OBJECTIVES WITH RESPECT TO TIMESCALE This section presents the relevant literature with focus on the main driver activating DR controller. The state-of-the-art literature does not contain ample research work on islanded microgrids that mainly address DR; hence, the research work on this topic is of high potential. Two distinct categories can be found in the literature, namely, long-term DR controllers and short-term DR controllers. These controllers have different mechanisms to detect changes in the net load mismatch based on the

60

Applied Mathematical Modeling and Analysis in Renewable Energy

system conditions. The long-term DR controllers are referred to as load-following DR controllers, while the short-term DR controllers are called voltage feedback DR controllers.

5.3.1 Load-Following DR Controller This type of controllers depends on the prediction of power net load, up to a predefined time horizon. The prediction of net load occurs based on the historical observation of load and generation. The DR controller then optimizes the deployment of controllable loads to maintain the systems balance, constrained by the requirements of such controllable loads. The time horizon is needed to consider sufficient information about the net load to optimally dispatch controllable loads throughout the whole horizon. Further, DR controller directly taps into the system flexibility (maintaining system balance via controllable loads) and ensures customer comfort (addressing constraints on the controllable loads). In the load-following framework, the implementation of this type of controllers depends on the prediction of power net load curve and coordination between the different responsive loads. The methodology of applying such controllers in microgrids is similar to that in larger grids. The challenges in microgrid, however, are manifested in the prediction of net load curve and in limited number of responsive loads. The prediction of net load depends on the prediction of both generation and demand. Unlike the relatively good prediction of net load in large grids, the high presence of VRE in microgrids makes the prediction of generation more challenging due to their stochastic nature, as discussed previously. Moreover, in microgrids, the demand prediction adds to the challenge of predicting the net load, since small networks consist of relatively small population; the demand curve is more sensitive to the changes in customer’s behavior and, consequently, more heteroscedastic [24]. Therefore, proper implementation of load-following DR controllers has to address the accuracy of net load prediction. In other words, the controllers in large grids can use the net load prediction on hourly time scale, while in microgrids, the prediction of net load and the scheduling of resources have to be on an intra-hourly basis. To this extent, the literature reports few load-following studies in microgrids, and they often do not consider the intermittent nature of renewable resources; scheduling of DR resources is hourly-based such as in refs. [25, 26]. Also, where smaller time intervals are considered for dispatching the DR resources, several major drawbacks such as offline prediction are used [16, 27]. The study in ref. [28] presents a formulation of a load-following DR controller for a single household (small microgrid with variable generation). The DR controller optimizes the TCLs of the house every ten minutes in an hourly time-horizon fashion. The modeled microgrid, however, is not fully islanded, meaning that the DR controller reported satisfactory results only when assistance from the grid is provided. A load-following DR controller with five-minute resolution has been suggested in ref. [27]. The proposed controller performs to only determine and enforce switching state. While it demonstrates the applicability of the proposed DR controller, the results are based on next-step load predictions which are assumed to be perfectly known. In ref. [16], a load-following DR controller is designed to control small aggregated TCLs, representing water heaters. Based on the net load prediction,

Hierarchical Demand Response Controller

61

the controller changes the states of the TCLs to provide the required flexibility. The DR controller operates on a one-minute resolution to provide minute-by-minute DR for 24 hours’ time horizon. The work, however, utilizes offline net load predictions that are generated a day-before without updates. The proposed two-level controller is capable of continuously updating the net load prediction of the system to update the deployment schedule of loads to balance power in the system.

5.3.2  Voltage Feedback DR Controller This type of controllers is triggered by the deviation of voltage. The directly measured quantity (voltage) is compared to the nominal value and activates the DR controller when the deviation exceeds a certain threshold. According to a predefined relation between the voltage deviation and the power mismatch, the DR controller adjusts the loads to ultimately eliminate this deviation. The voltage deviation-based control signal is mainly utilized in inverter-based microgrids in order to indicate power mismatch [29]. However, voltage deviation signals have been conventionally used to decide on load shedding actions in the grid. In that case, an under-voltage event triggers the controller to disconnect some of the network loads in order to maintain idle operations. Load shedding strategies, which are dependent on voltage feedback, are previously used as an extreme measure in emergency situations when the available generation is not sufficient to balance the demand. Load shedding is usually used on large industrial loads to make a fast, adequate change in the consumption when required [30, 31]. The study in ref. [32] presents a comparison of different voltage feedback DR controllers. The DR controllers are implemented at the level of a household heating system for voltage stability. The study is limited to microgrids in connected mode, and the findings indicate that the DR controllers fail in satisfying the voltage constraints. In ref. [33], a voltage feedback DR controller is proposed for incentive-based DR applications. Customer-rewards scheme is designed in order to secure the feeder voltage profile in addition to achieving peak-load reduction. A more comprehensive DR strategy is proposed in ref. [34], where voltage feedback and customer comfort information are utilized in a DR controller that aims at enhancing the voltage stability. The study demonstrates how customer comfort may be considered while voltage stability is achieved. The DR controller, however, is designed to replace ancillary services for the microgrid. Consequentially, recommendations on adopting similar DR strategies for continuous DR schemes are suggested as future work.

5.4 METHODOLOGY The proposed DR control framework optimizes the deployment of DR to provide regulation and generation following flexibility. As depicted in Fig. 5.1, the proposed framework consists of an optimizer and a controller having a hierarchical relationship. The high-level optimizer (HLO) operates on a timescale of minutes (e.g., five minutes) to optimize the DR resources set points, which are sent to the low-level controller (LLC) for tracking. The LLC operates on a timescale of seconds (e.g., 30 seconds) to track these set points, while providing load regulation by turning DR resources ON or OFF, to track and balance any power mismatch.

62

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 5.1  A hierarchical DR control framework consisting of high-level optimizer (HLO) and low-level controller (LLC).

5.4.1 High-Level Optimizer (HLO) The HLO relies on a net load predictor which predicts the expected net load trajectory over a prediction horizon of one hour to several hours. The HLO will optimize the dynamics of DR resources such that the aggregate flexibility from all DR resources tracks the net load prediction. This allows the HLO to preposition the temperature set points of TCLs over the prediction horizon, such that TCLs can turn ON and OFF to provide downward or upward flexibility, respectively, without violating the temperature dead-band and the temperature dynamics of TCLs. The optimal set points of the first period in the prediction horizon are sent to the LLC, while the rest are discarded. This process is repeated at every decision-making time step (e.g., five minutes) in a receding horizon fashion. 5.4.1.1  Mathematical Program of HLO The described mathematical program yields optimal temperature set points along a predefined planning horizon to preposition TCLs in the best way to provide flexibility, without violating users comfort (i.e., temper temperature dead-band). The following mixed-integer linear program represents optimization at a single decision time step:

min

S + ,  S − ,s + ,s −

∑c b ( k ) + c ( S i i

S

+

( k ) + S − ( k ))

i ∈

k + H1



+

∑∑ c b (t ) + c ( s i i

t = k +1 i ∈

S

+

( t ) + s − ( t )) (5.1)

Hierarchical Demand Response Controller

63

This is subject to:

∑ b (k ) P + S



i

i

+

( k ) − S − ( k ) = λ L ( k ) (5.2)

i ∈



∑ b (t ) P + s i

i

+

( t ) − s − ( t ) = λ L ( t ) ,  t = k + 1,…, k + H1 (5.3)

i ∈



Troom,  i ( t ) = To,i ( t ) + bi ( t − 1) Qi Ri



− ( To,i ( t ) + bi ( t − 1) Qi Ri − Troom,i ( t − 1)) exp− ∆t1 /R i Ci ,



t = k + 1,…, k + H1 (5.4)



Ti − ≤ Troom,i ( t ) ≤ Ti + ,  t = k + 1,…, k + H1 (5.5)



S + , S − , s + , s − ≥ 0 (5.6)

The objective in equation (5.1) is to minimize the cost of dispatching TCLs to provide flexibility to the microgrid, in addition to minimizing the cost of load shedding (S+) and VRE curtailment (S–). These costs are calculated at current time and over the prediction horizon. The integer variables bi are binary decision variables corresponding to switching ON and OFF the TCLs. Equation (5.2) ensures power balance at current time step k to meet the net load L(k) multiplied by a constant factor 0 < λ < 1. The constant λ represents the HLO’s share of meeting the overall net load L(k), in a distributed DR program. This constant is determined by the overall microgrid optimization, handled by a central microgrid controller. Equation (5.3) ensures power balance over the prediction horizon t = k + 1, …, k + H1, to meet the net load forecast L(t) multiplied by λ. The lack variables s+ and s– represent potential need for load shedding and VRE curtailment over the prediction horizon. Equation. (5.4) describes the evolution of space’s (room’s) temperature profile, where TCLs are located, over the prediction horizon, given the trajectory of ON/OFF decisions bi(t). Equation (5.5) ensures that a TCL remains constrained within its operational limits (i.e., temperature dead-band), to maintain end-user comfort. In such a case, any additional flexibility is supplied by the slack variables so as to avoid violating the end-user comfort. The optimal solution of the described mixed-integer linear program is a series of ON/OFF switching decisions over the prediction horizon. The first decision is implemented, while discarding the rest, to compute the optimal temperature set points to be sent to the LLC: sp Troom, i = To ,i ( k + 1) + bi ( k ) Qi Ri −



(T

o ,i

( k + 1) + bi ( k ) Qi Ri − Troom,i ( k )) exp− ∆t1 / RiCi (5.7)

64

Applied Mathematical Modeling and Analysis in Renewable Energy

5.4.1.2  Net Load Predictor The net load predictor uses historical net load measurements to calculate the net load forecast over the prediction horizon. The forecasted net load trajectory reflects the load-following flexibility requirement arising from variability in microgrid. A weighted moving average (WMA) predictor can be used, whereby the WMA predictor reads the past M measurements of the net load and computes the next N net load forecasts over the prediction horizon H1. The equation describing the WMA is as follows:

L ( t ) = α 1 L ( t − 1) + α 2 L ( t − 2 ) +…+ α M   L ( t − M ) (5.8)

where k is the present time step and t = k + 1, …, k + H1 is the forecast time steps. The parameters αi of the WMA predictor are learned online continuously, to adapt to variable operations of the microgrid under renewable energy integration.

5.4.2 Low-Level Controller (LLC) The LLC regulates DR resources’ power consumption on a second-to-minutes timescale to provide regulation to the microgrid by reducing the power mismatch. The DR resources are considered to be distributed across the microgrid. The amount of power for which each DR bus can compensate, whether by increasing or decreasing the consumption, is determined by the changes in voltage magnitudes between two consecutive iterations, in addition to the Jacobian matrix from the previous iteration. In this case, the compensation share is distributed among the DR buses considering their locations. Accordingly, the LLC regulates DR resource’s power consumption to nullify power mismatches over the prediction horizon, while strictly following the temperature set points determined by the HLO. This process is repeated at every control time step (e.g., 30 seconds) in a receding horizon fashion. 5.4.2.1  Mathematical Program of LLC The described mathematical program optimizes two opposing objectives: the program tries to track the temperature set points from the HLO, while eliminating the power mismatches over the prediction horizon. The following mixed-integer linear program represents the LLC at a single control time step:

min

T∆ , S + , S −

∑c

ud i

( k ) T∆ ,i ( k ) + cS ( S + ( k ) + S − ( k )) (5.9)

i ∈

This is subject to:  V2  V Pi = Pnom  Z 2 + I + P  (5.10) Vnom  Vnom 





∑ b (k ) P + S i

i ∈

i

+

( k ) − S − ( k ) = L ( k ) + ∆P ( k ) (5.11)

Hierarchical Demand Response Controller

65

Troom,i ( t ) = To,i ( t ) + bi ( t − 1) Qi Ri − To,i ( t ) + bi ( t − 1) Qi Ri − Troom,i ( t − 1) exp− ∆t2 / RiCi ,

t = k + 1,…, k + H 2 (5.12)



Ti − ≤ Troom,i ( t ) ≤ Ti + , t = k + 1,…, k + H 2 (5.13)



sp sp Troom, i ( t ) − T∆ ,i ≤ Troom,i ( t ) ≤ Troom,i ( t ) + T∆ ,i , t = k + 1,… , k + H 2 (5.14)



∆P ( k ) = α J PV (V ( k ) − V ( k − 1)) (5.15)



S + , S − , T∆ ,i ≥ 0 (5.16)

The objective in Equation (5.9) is to minimize the cost of two opposing variables, the cost of user discomfort ciud in the first part, and the cost of load shedding and generation curtailment cS in the second part. Equation (5.10) represents the ZIP model of the load, to model the effect on power consumption of the loads Pi as a function of voltage. P nom is the nominal power of the load and the parameters Z, I, and P are the active power impedance, current, and power, respectively, where (Z + I + P = 1). The TCLs are primarily resistive; hence, Z = 1, I = 0, and P = 0 with P nom = 1.5 kW [35, 36]. In equation (5.11), the integer variables bi are binary decision variables corresponding to switching ON/OFF the TCL devices. Equation (5.11) ensures power balance at current time step t to meet the net load L(k) + ΔP(k). Equation (5.12) dictates the evolution of a room’s temperature over the prediction horizon, given the trajectory of ON/OFF decisions bi(k). Equation (5.13) ensures that a TCL remains constrained within its operational limits (i.e., temperature dead-band). Equation (5.14) ensures tracking of the temperature set point over the prediction horizon, within the allowed user discomfort TΔ, which constitutes a tolerable deviation from the temperature set point. Finally, equation (5.15) predicts the power mismatch ΔP(k) and distributes it on the DR buses to compensate for it. The power mismatch ΔP(k) is determined using voltages deviation V(k) − V(k −1) and the Jacobian sub-matrix JPV that relates the change in voltage magnitudes to the change in active power. This relation is further used in distributing the shares among the DR buses to compensate for the power mismatch.

5.5  RESULTS AND DISCUSSION 5.5.1  Validation Example In order to more appropriately explain the advantage of the proposed DR framework with respect to the literature of interest, a validation example can be based on temporal resolution of available information about VRE sources. Figure 5.2 presents a 90-minute-long observations of PV array output from a site located in the province of Quebec, Canada [37]. The figure shows one-minute resolution and five-minute resolution-based PV output estimates in a relatively cloudy day (mid-day segment on 17-07-2014).

66

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 5.2  90-minute-long observations of PV array output.

Considering the literature of interest, five-minute resolution is taken for a loadfollowing DR controller, which will drive the evolution of the reserve capacity, in terms of TCLs, to meet that projection [27]. However, the estimates of one-minute resolution clearly show the significant variation in the power output estimates. Also, when the uncertainty in predictions is factored in this example, the problem becomes more challenging [24]; hence, such DR controllers will not be adequate for microgrids operating in islanded mode. To overcome this challenge, the proposed DR controller utilizes an informed LLC strategy, which responds to the variation occurring intra-time scale. The HLC in the proposed methodology generates optimum set points, which allow the LLC to not only respond to the instantaneous power imbalance in the system but also drive the TCL states to change around that appropriate set point.

5.5.2 Simulation Results of Case Studies The following results illustrate the potential effectiveness of the proposed DR control framework. The HLO and the LLC are simulated separately under various scenarios, while future work will integrate both in a full microgrid simulation testbed. Table 5.1 lists the temperature limits and the mean values of the residential space characteristics used to simulate the HLO and the LLC in this study. The characteristics are assumed to follow normal distributions, each with a coefficient of variance equal to 10%. Each residential space (room) is assumed to be equipped with a TCL (space heater).

67

Hierarchical Demand Response Controller

TABLE 5.1 Mean Values of the Physical Characteristics [35, 38] Parameter T+(°C) T-(°C) C (J/ °C) R (°/kW) Q (kW)

Value 23 19 3599.3 0.1208 400

5.5.3 HLO Case Study Figure 5.3(a) shows a ramp-up event in the net load L(t) over a one-hour prediction horizon. DR is deployed via the HLO to balance this ramp event. Figure 5.3(b) shows sample temperature profiles (dashed) and the average temperature profile of all TCLs over the prediction horizon (solid) as a result of running the HLO. It is expected to observe that there is an increase in the average temperature over the prediction horizon, knowing that the TCLs represent residential space heaters. Since the ramp-up event represents increase in the net load, the HLO turns ON more space heaters to absorb this increase in the net load, thus leading to an increase in the average temperature. Similar study is done for the ramping-down event where the TCLs were turned OFF to reduce the power consumption.

5.5.4 LLC Case Study The proposed LLC is illustrated on a modified IEEE 33-bus radial system [39]. Figure 5.4 depicts the test system schematic. The system includes a main generator at bus 1, representing the slack bus, and VRE sources at buses 9, 20, and 28. Distributed DR devices, comprising TCLs, are located at buses 5, 13, 17, 22, 29, and 33.

FIGURE 5.3  (a) A ramp-up event in the net load and (b) sample temperature profiles (dashed) and average temperature (solid) of 1000 TCLs.

68

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 5.4  Modified IEEE 33-bus radial system.

Other than the controllable TCLs, the demand, at all the buses in the system, is kept unchanged. Several TCL cluster sizes are considered in order to investigate the corresponding performance of the DR controller. The TCLs in the system are utilized to respond to the unplanned changes in power generation, caused by the VRE sources. By switching the TCLs ON/OFF, the overall consumption is changed to balance any power mismatch. This incurs less dependence on the slack variables (load shedding and generation curtailment). The slack variables here are modeled as the change of generation at the slack bus. As aforementioned, the TCLs used in this study are heating units (refer to Table 5.1). The initial states of the TCLs are randomized, turning half of the TCLs ON. The initial rooms’ temperature and their temperature set points (which are supposed to be fed from the HLO) are also randomized. Four cases are considered in the test study. In case 1, each DR bus comprises 100 TCLs (total of 600 TCLs). The user comfort level is defined such that the temperature deviation from the temperature set point is limited to 1 ºC. Random VRE generation is provided by the corresponding VRE buses throughout the time horizon. The degree of random changes in the VRE generation, the number of TCLs, and the effect of user comfort level in this case are used as a reference. Table 5.2 summarizes the cases considered to drive the results with respect to the reference case (case 1). Figure 5.5 considers case 1 to illustrate the flexibility of DR devices in acting as fast reserve resources in the system to balance the ramp events in generation. In Fig. 5.5(a), DR devices and slack variables are used in the system to change their consumption according to the available generation. While in Fig. 5.5(b), the generation is tracked using slack variables only, which represent load shedding and generation curtailment. In this case, the change in the slack variable (black line) is an x-axis mirror to the change in generation (blue line). When the generation decreases, the

Hierarchical Demand Response Controller

69

TABLE 5.2 Changes to the System Properties Case Number Case 2 Case 3 Case 4

Considered Changes from the Reference Case Number of TCLs per DR bus is reduced to 20 Tolerable deviation is reduced to 0.1 °C from set point Higher VRE generation is considered

slack variable increases to balance the mismatch and vice versa. To better show the effectiveness of the proposed DR controller deployment to reduce the dependence on slack variables, Fig. 5.6 shows the change in the amount of active power provided by the slack variables in the two cases of Fig. 5.5. Furthermore, Fig. 5.6(a) results from utilizing DR resources, and Fig. 5.6(b) corresponds to the use of slack variables only. It is clear that the utilization of DR resources increases the flexibility of system, reducing the required reserves and their cost. Figure 5.7 shows the change in total load of the system due to the LLC actions, which are triggered by the voltage deviations at load buses. In all the considered cases, the LLC successfully achieves fast-tracking behavior to the variable generation. In case 1, the DR resources are capable of balancing the change in VRE throughout the horizon and the slack variable is shown to be nullified, as in Fig. 5.7(a). Due to limited number of TCLs, the DR has limited flexibility to track the generation, as in Fig. 5.7(b). When the variable generation is increased for relatively a long period of time (from 1.5 minutes to 2.5 minutes), the flexibility of a few TCLs is quickly utilized, and the TCLs are forced to switch OFF, as the ambient temperature reaches its maximum limit. The slack variables instead compensate for the remaining power mismatch. In Fig. 5.7(c), the tolerable deviation from the temperature set point is reduced, which limits the flexibility of TCLs. The slack variable in this case is compensating for the increase in generation and decrease in the TCLs demand. In Fig. 5.7(d), the DR resources have the capacity to balance the higher change in the

FIGURE 5.5  Variable generation tracking (a) with DR and (b) without DR.

70

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 5.6  Changes in the slack variable (a) with DR and (b) without DR.

VRE for most but not all of the horizon, and therefore, the slack variable contributes to balance the mismatch in the system. The proposed controller drives the states of the TCLs with respect to voltage deviations in order to nullify the power mismatch and, consequently, bind the voltages of the buses within the operating limits (0.95–1.05 pu). Figure 5.8 illustrates the voltage stability in all four cases. Figure 5.9 compares the evolution

FIGURE 5.7  Variable generation tracking (a) reference case, (b) reduced number of TCLs, (c) limit the user comfort, and (d) increase the VRE level.

Hierarchical Demand Response Controller

71

FIGURE 5.8  Voltage magnitude at all buses (a) reference case, (b) reduced number of TCLs, (c) limit the user comfort, and (d) increase the VRE level.

FIGURE 5.9  Space temperature evolution over the horizon (a) reference case, (b) reduced number of TCLs, (c) limit the user comfort, and (d) increase the VRE level.

72

Applied Mathematical Modeling and Analysis in Renewable Energy

of the space temperature throughout the horizon for all the cases. Comparing case 2 with the reference case, it can be seen that the frequency of altering the states of the TCLs is increased since less devices are available in the system to accommodate variation ramp events. Designing a proper DR controller with sufficient number of controllable loads and adequate margin of tolerance from the end-user’s side can provide an effective source of flexibility in the system to support the high penetrations of VRE sources.

5.6 CONCLUSION The state-of-the-art DR research presents two distinct applications for utilizing controllable loads. One is meant for optimizing the DR resources, while the other is for creating fast-response solutions to changes in the network. In islanded microgrids, DR schemes should account for both aspects in order to maintain the system reliability. This chapter attempts to bridge gap between the load-following and the power mismatch DR controllers, in order to yield a more reliable DR framework. The results showcased the potential of the proposed DR framework.

NOMENCLATURE Indices i: TCL unit k: Decision time step t: Prediction horizon time step set I: Set of TCLs Parameters ci: Cost of dispatching TCL i ($/MW) cS: Cost of dispatching load shedding and VRE curtailment ($/MW) ciud: Cost of user discomfort (incentives) of TCL unit i ($/°C) Pi: Power consumption of TCL unit i (MW) Pnom: Power consumption of TCL unit at nominal voltage (MW) Qi: Equivalent heat rate of TCL unit i (W) Ri: Equivalent thermal resistance of TCL unit i (°C/W) Ci: Equivalent heat capacity of TCL unit i (J/°C) Δt: Decision time interval (minutes) T-: Dead-band minimum temperature of TCL unit i (°C) T+: Dead-band maximum temperature of TCL unit i (°C) L(k): Net load forecast at time step k (MW) L(t): Net load forecast over the prediction horizon (MW) H1: Number of time steps (length) of prediction horizon of HLO H2: Number of time steps (length) of prediction horizon of LLC Δt1: Decision time interval of HLO Δt2: Decision time interval of LLC To,i: Ambient temperature of TCL unit i (°C) ΔP: Predicted Net load power mismatch (MW) V: Actual bus voltage magnitudes (pu)

Hierarchical Demand Response Controller

73

Vnom: Nominal voltage (pu) To,i: Tolerable deviation from the temperature set point (°C) JPV: Jacobian sub-matrix relating voltage magnitudes to the active power Z: Active power constant impedance parameter (%) I: Active power constant current parameter (%) P: Active power constant power parameter (%) variables bi: ON/OFF state of TCL unit i S+; S-: Load shedding and VRE curtailment at time step k (MW) s+; s-: Load shedding and VRE curtailment over prediction horizon (MW) Troom,i: Room temperature of TCL i (°C) Tsproom,i: Room temperature set point of TCL i (°C)

REFERENCES





1. S. Behboodi, D. P. Chassin, C. Crawford, and N. Djilali, “Renewable resources portfolio optimization in the presence of demand response,” Applied Energy, vol. 162, pp. 139–148, 2016. 2. B. Li, J. Shen, X. Wang, and C. Jiang, “From controllable loads to generalized demandside resources: A review on developments of demand-side resources,” Renewable and Sustainable Energy Reviews, vol. 53, pp. 936–944, 2016. 3. E. Ela and M. O’Malley, “Studying the variability and uncertainty impacts of variable generation at multiple timescales,” IEEE Transactions on Power Systems, vol. 27, no. 3, pp. 1324–1333, Aug. 2012. 4. International Energy Agency, “Harnessing variable renewables: A guide to the balancing challenge,” OECD/IEA, Paris, France, Technical Report, 2011. 5. R. Lasseter, A. Akhil, C. Marnay, J. Stephens, J. Dagle, R. Guttromson, A. Meliopoulous, R. Yinger, and J. Eto, “The certs microgrid concept,” White paper for Transmission Reliability Program, Office of Power Technologies, US Department of Energy, vol. 2, no. 3, p. 30, 2002. 6. L. Zhongcheng and Y. Tong, “Renewable energy basing on smart grid,” in 2010 6th International Conference on Wireless Communications Networking and Mobile Computing (WiCOM), 2010. 7. P. Siano, “Demand response and smart gridsa survey,” Renewable and Sustainable Energy Reviews, vol. 30, pp. 461–478, 2014. 8. S. Shao, M. Pipattanasomporn, and S. Rahman, “Development of physical-based demand response-enabled residential load models,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 607–614, 2013. 9. Z. Wang, R. Paranjape, A. Sadanand, and Z. Chen, “Residential demand response: An overview of recent simulation and modeling applications,” in Electrical and Computer Engineering (CCECE), 2013 26th Annual IEEE Canadian Conference. IEEE, 2013, pp. 1–6. 10. N. Li, L. Chen, and S. H. Low, “Optimal demand response based on utility maximization in power networks,” in 2011 IEEE power and energy society general meeting. IEEE, 2011, pp. 1–8. 11. J. A. Short, D. G. Infield, and L. L. Freris, “Stabilization of grid frequency through dynamic demand control,” IEEE Transactions on Power Systems, vol. 22, no. 3, pp. 1284–1293, 2007. 12. B. Ramanathan and V. Vittal, “A framework for evaluation of advanced direct load control with minimum disruption,” IEEE Transactions on Power Systems, vol. 23, no. 4, pp. 1681–1688, 2008.

74

Applied Mathematical Modeling and Analysis in Renewable Energy

13. J. L. Mathieu and D. S. Callaway, “State estimation and control of heterogeneous thermostatically controlled loads for load following,” in System Science (HICSS), 2012 45th Hawaii International Conference on. IEEE, 2012, pp. 2002–2011. 14. D. S. Callaway and I. A. Hiskens, “Achieving controllability of electric loads,” Proceedings of the IEEE, vol. 99, no. 1, pp. 184–199, 2011. 15. A. Abiri-Jahromi and F. Bouffard, “Contingency-type reserve leveraged through aggregated thermostatically-controlled loadspart i: Characterization and control,” IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 1972–1980, 2016. 16. N. Lu and D. P. Chassin, “A state-queueing model of thermostatically controlled appliances,” IEEE Transactions on Power Systems, vol. 19, no. 3, pp. 1666–1673, 2004. 17. J. Kondoh, N. Lu, and D. J. Hammerstrom, “An evaluation of the water heater load potential for providing regulation service,” in 2011 IEEE Power and Energy Society General Meeting. IEEE, 2011, pp. 1–8. 18. N. Lu, “An evaluation of the HVAC load potential for providing load balancing service,” IEEE Transactions on Smart Grid, vol. 3, no. 3, pp. 1263–1270, 2012. 19. R. Malhame and C.-Y. Chong, “Electric load model synthesis by diffusion approximation of a high-order hybrid-state stochastic system,” IEEE Transactions on Automatic Control, vol. 30, no. 9, pp. 854–860, 1985. 20. J. L. Mathieu, S. Koch, and D. S. Callaway, “State estimation and control of electric loads to manage real-time energy imbalance,” IEEE Transactions on Power Systems, vol. 28, no. 1, pp. 430–440, 2013. 21. E. Vrettos, J. L. Mathieu, and G. Andersson, “Demand response with moving horizon estimation of individual thermostatic load states from aggregate power measurements,” in 2014 American Control Conference. IEEE, 2014, pp. 4846–4853. 22. W. Zhang, K. Kalsi, J. Fuller, M. Elizondo, and D. Chassin, “Aggregate model for heterogeneous thermostatically controlled loads with demand response,” in 2012 IEEE Power and Energy Society General Meeting. IEEE, 2012, pp. 1–8. 23. D. Guo, W. Zhang, G. Yan, Z. Lin, and M. Fu, “Decentralized control of aggregated loads for demand response,” in 2013 American Control Conference. IEEE, 2013, pp. 6601–6606. 24. Y. Zhang and N. Lu, “Parameter selection for a centralized thermostatically controlled appliances load controller used for intra-hour load balancing,” IEEE Transactions on Smart Grid, vol. 4, no. 4, pp. 2100–2108, 2013. 25. M. Chaouch, “Clustering-based improvement of nonparametric functional time series forecasting: Application to intra-day household-level load curves,” IEEE Transactions on Smart Grid, vol. 5, no. 1, pp. 411–419, 2014. 26. D. K. Critz, S. Busche, and S. Connors, “Power systems balancing with high penetration renewables: The potential of demand response in hawaii,” Energy Conversion and Management, vol. 76, pp. 609–619, 2013. 27. M. Mazidi, A. Zakariazadeh, S. Jadid, and P. Siano, “Integrated scheduling of renewable generation and demand response programs in a microgrid,” Energy Conversion and Management, vol. 86, pp. 1118–1127, 2014. 28. T. M. Keep, F. E. Sifuentes, D. M. Auslander, and D. S. Callaway, “Using load switches to control aggregated electricity demand for load following and regulation,” in 2011 IEEE Power and Energy Society General Meeting. IEEE, 2011, pp. 1–7. 29. D. I. H. Rodr´ıguez, J. Hinker, and J. M. Myrzik, “On the problem formulation of model predictive control for demand response of a power-to-heat home microgrid,” in 2016 Power Systems Computation Conference (PSCC). IEEE, 2016, pp. 1–8. 30. T. L. Vandoorn, B. Renders, L. Degroote, B. Meersman, and L. Vandevelde, “Active load control in islanded microgrids based on the grid voltage,” IEEE Transactions on Smart Grid, vol. 2, no. 1, pp. 139–151, 2011.

Hierarchical Demand Response Controller

75

31. J. P. Lopes, C. Moreira, and A. Madureira, “Defining control strategies for microgrids islanded operation,” IEEE Transactions on Power Systems, vol. 21, no. 2, pp. 916–924, 2006. 32. I. J. Balaguer, Q. Lei, S. Yang, U. Supatti, and F. Z. Peng, “Control for grid-connected and intentional islanding operations of distributed power generation,” IEEE Transactions on Industrial Electronics, vol. 58, no. 1, pp. 147–157, 2011. 33. C. Koch-Ciobotaru, R. Boraci, O. Pros,tean, and I. Szeidert, “Voltage control in a microgrid using demand response,” in Applied Computational Intelligence and Informatics (SACI), 2013 IEEE 8th International Symposium. IEEE, 2013, pp. 233–237. 34. C. Vivekananthan, Y. Mishra, G. Ledwich, and F. Li, “Demand response for residential appliances via customer reward scheme,” IEEE Transactions on Smart Grid, vol. 5, no. 2, pp. 809–820, 2014. 35. D. Wang, S. Parkinson, W. Miao, H. Jia, C. Crawford, and N. Djilali, “Online voltage security assessment considering comfort-constrained demand response control of distributed heat pump systems,” Applied Energy, vol. 96, pp. 104–114, 2012. 36. F. Lamberti, C. Dong, V. Calderaro, and L. F. Ochoa, “Estimating the load response to voltage changes at uk primary substations,” in IEEE PES ISGT Europe 2013. IEEE, 2013, pp. 1–5. 37. A. Bokhari, A. Alkan, R. Dogan, M. Diaz-Aguil´o, F. De Leon, D. Czarkowski, Z. Zabar, L. Birenbaum, A. Noel, and R. E. Uosef, “Experimental determination of the ZIP coefficients for modern residential, commercial, and industrial loads,” IEEE Transactions on Power Delivery, vol. 29, no. 3, pp. 1372–1381, 2014. 38. “High-resolution solar radiation datasets,” Apr 2016. Online. Available: http://www. nrcan.gc.ca/energy/renewable-electricity/solarphotovoltaic/ 18409. 39. N. Lu and Y. Zhang, “Design considerations of a centralized load controller using thermostatically controlled appliances for continuous regulation reserves,” IEEE Transactions on Smart Grid, vol. 4, no. 2, pp. 914–921, 2013.

Part II Generalized Mathematical Ideas and Their Applications

6

A General Class of Polynomials Inspired by a General Lagrange Inversion Pair Due to Gessel and Stanton Manisha Dalbhide-Ubale

CONTENTS 6.1 Introduction..................................................................................................... 79 6.2 Inverse Series Relations................................................................................... 81 6.3 Integral Representations.................................................................................. 86 6.4 Differential Equation θ -Form and Companion Matrix.............................................................................................................. 88 6.5 Particular Cases...............................................................................................90 Acknowledgments.....................................................................................................99 References.................................................................................................................99

6.1 INTRODUCTION The Lagrange inversion formula has been studied by many researchers for giving its multi-variable forms [1], q-analogs [2, 3, 14], non-commutative forms [4–7], etc. The q-Lagrange inversion pair given by I. Gessel and D. Stanton [3] unifies several other inverse series relations, for the values –1/2, – 1/3, 0, 1, 1/2 and 2 of β . This inversion pair was further generalized by the author (see [8]), and thereby extension as well as unification of certain known polynomials occurring as special cases was proved. Here, in this chapter, an attempt has been made to give an extension to the ordinary form of one of the equations of the pair in ref. [3] in the form:

F (n) =

[n / s ]

∑ k =0

DOI: 10.1201/9781003159124-6

Γ ( A + nβ + sk ) G ( k ) . (6.1) ( n − sk )!

79

80

Applied Mathematical Modeling and Analysis in Renewable Energy

The extended form (shown in equation (6.1)) inspired the construction of a general system/class of polynomials (GCP), which is denoted here by M r ( s, A, β ; x ) , r = 0,1, 2,… and is defined as:

{

}

M r ( s, A, β ; x ) =



r  s

∑ k =0

(−1)sk Γ ( A + rβ + sk ) ψ k x k . (6.2) ( r − sk )!

The above GCP serves as a generalization and unification of certain well-known polynomials, namely, Racah polynomial [9], Wilson [9], Hahn [10], Jacobi, Legendre, Bessel and Laguerre polynomials [11], when the parameters are specialized suitably. The explicit series representations of these polynomials are as follows: Racah Polynomial [9] Rn ( x ( x + c + d + 1); a, b, c, d ) =

n

∑ (−n) (1(1+ +a +a)b (1+ n+)b(+xd+)c(1+ +d c+)1) (− x) (6.3) k



k

k

k

k =0

k

k

k

Wilson Polynomial [9] Pn ( x 2 ) = (a + b)n (a + c)n

n



(a + d ) n

∑ (−n) (a +(ba++cb+) d (+a +n c−)1) (a(a++dix) ) (a − ix) (6.4) k

k

k

k =0

k

k

k

k

Hahn Polynomial [10]

Qn ( x; a, b, N ) =

n

∑ (−n)(1(1++aa) +(−bN+)n)  k(!− x) k

k

k

k =0

k

, n = 0,1, 2,…, N . (6.5)

k

Jacobi Polynomial [11]

Pn(

α ,β )

n

(x) =

∑ k =0

k

(− n) k (1 + α + β + n) k (1 + α )n  1 − x   (6.6)  (1 + α ) k   n !  k ! 2 

Legendre Polynomial [11] n



Pn ( x ) =

∑ k =0

k

(− n) k (n + 1) k  1 − x   (6.7)  k !  k ! 2 

81

A General Class of Polynomials

Bessel Polynomial [11] n



yn ( x ) =

∑ k =0

k

(− n) k (n + 1) k  − x  (6.8)  2  k!

Laguerre Polynomial [11] n

L(nα ) ( x ) =



∑ (1(−+n)α )(1 +nα!  )k ! x (6.9) k

n

k

k

k =0

In the subsequent sections, the GCP given above in equation (6.2) is taken up for investigating various properties. Inverse series relations are proved in section 6.2. Section 6.3 contains integral forms of it. The θ -form differential equation is derived in section 6.4. The companion matrix of the GCP is also given in this section. Moreover, section 6.5 gives a brief account of the particular cases of the results obtained in sections 6.2, 6.3 and 6.4. Here are some useful results and notations that are used in the sections to follow. Appell’s symbol or Pochhamer symbol [11] is defined by (α )n =



Γ (α + n ) . Γ (α ) ( λ +1)

( λ + m −1)

The symbol ∆ ( m; λ ) denotes the sequence of m parameters mλ , m ,…, m . The generalized hypergeometric function is denoted by p Fq [11] and is defined as



 a1 , a2 , …, a p ; x   = F p q  b1 , b2 , …, bq ;   

∞

∑ n= 0

(a1 )n …(a p )n x n . (b1 )n …(bq )n   n !

The following two double series relations [11] are useful in proving the Theorem in section 6.2. n



k

∑∑

A (k, j) =

k =0 j=0

n

n− j

∑∑ A ( k + j, j ) j=0 k =0

and sn



[k / s]

∑∑

A (k, j) =

k =0 j=0

n sn − sj

∑∑ A ( k + sj, j ) j=0 k =0

6.2  INVERSE SERIES RELATIONS The inverse series relation of the GCP, M r ( s, A, β ; x ) given in equation (6.2) will be obtained in this section by means of a general pair of inverse series relation, which is proved here in the form of following Theorem:

82

Applied Mathematical Modeling and Analysis in Renewable Energy

Theorem: For s = 1, 2, 3,… if F (r ) =



[r / s ]

∑α (r , k , s ) G ( k ) (6.10) k =0

and sr

G (r ) =



∑ β (r , k , s ) F ( k ) (6.11) k =0

then

α (r , k , s ) =



(−1)sk Γ ( A + sk + rβ ) (6.12) ( r − sk )!

It implies (−1) k ( A + k + kβ ) (6.13) Γ ( A + sr + kβ + 1) ( sr − k )!

β (r , k , s ) =

and

r

∑ β (r , k ,1)  F ( k ) = 0 (6.14)



k =0

if r ≠ sj, j ∈ N . The proof of the theorem uses an auxiliary result which is given here in the following form: Lemma: If ( Mβ − jβ − 1) < ( Mj − 1) , where M = sn − sr ,   s = 0, 1, 2,…, n;  r = 0, 1, 2,…, n;  j = 0, 1, 2,…, M , then

∑( −1)  Mk  Γ( ( A + M + sr + srβ + k)β + 1) (6.15) M



T (M ) =

k

 A + sr + srβ + kβ + k U ( k )



k =0

⇔

M



U (M ) =





∑( −1)  Mk  Γ ( A + srβ + Mβ + sr + k ) T ( k ). (6.16) k

k =0

Proof of the lemma is simple, and hence omitted here for brevity. Proof of the Theorem: It will be first proved that equation (6.12) ⇒ equation (6.13). In equation (6.11), denoting the RHS by tr , then in view of equation (6.13)

( −1)k ( A + k + kβ ) F ( k ) . ( sr − k )!  Γ ( A + sr + kβ + 1) k =0 sr



tr =

∑

83

A General Class of Polynomials

Now, using equation (6.10), tr takes the form [k / s]

∑∑ ( sr − k )! (( k − sj )! Γ )( A +( sr + kβ + 1)) G ( j ) . sr

tr =



(−1) k + sj   A + k + kβ   Γ A + sj + kβ

k =0 j=0

Since, sn

[k / s]

∑∑



A( k, r ) =

k =0 r =0

r

tr =



sr − sj

∑∑ j=0 k =0

n sn − sr

∑∑ A( k + sr , r ) , r =0 k =0

(−1) k ( A + k + sj + kβ + sjβ )  Γ ( A + sj + kβ + sjβ ) G ( j). k !  ( sr − sj − k )!  Γ ( A + sr + kβ + sjβ + 1)

Thus,

tr = G ( r ) +

r −1 sr − sj

∑∑ j=0 k =0

(−1) k ( A + k + sj + kβ + sjβ )  Γ ( A + sj + kβ + sjβ ) G ( j). k !  ( sr − sj − k )!  Γ ( A + sr + kβ + sjβ + 1)

Denoting r-j by N, one obtains,

 sN  ( A + k + kβ + sj + sjβ )  Γ ( A + sj + kβ + sjβ ) G ( j) (−1) k  tr = G ( r ) + Γ ( A + sr + kβ + sjβ + 1) ( sN )! k = 0  k  j=0 (6.17) r −1

∑

sN

∑

On comparing equation (6.17) with lemma, it is clear that the inner series in k above is nothing but equation (6.15) with the choice, U ( k ) = Γ ( A + sj + sjβ + kβ ) .

Taking,

 0   1, if   k = 0 T (k ) =   =   k   0, if   k ≠ 0

in equation (6.16), one gets M



U (M ) =









∑(−1)  Mk  Γ ( A + sjβ + Mβ + sj + k )  0k  k

k =0



= Γ ( A + sj + sjβ + Mβ ) .

 0 Hence, with T ( k ) =   , the choice U ( k ) = Γ ( A + sj + sjβ + kβ ) is restored. k

84

Applied Mathematical Modeling and Analysis in Renewable Energy

Therefore, from equation (6.17), tr = G ( r ) +

r −1

∑ j=0

G ( j) ( sN )!

∑( −1)  sNk  ( sN

 A + sj + sjβ + k + kβ )  Γ ( A + sj + kβ + sjβ ) Γ ( A + sr + kβ + sjβ + 1)



k

k =0

= G (r ) +



∑ ( sN( ))! T ( sN ) r −1

G j

j=0

= G (r ) +



∑ ( sN( ))!  sN0  . r −1

G j 



j=0

Thus, tr = G ( r ). This proves that equation (6.12) ⇒ equation (6.13). Now, to show that equation (6.12) implies equation (6.14), the left-hand member of equation (6.14) is denoted here by ψ r , then in the light of equation (6.10), [k / s]

r

∑∑

ψr =



k =0 j=0

( −1)k + sj ( A + k + kβ )  Γ ( A + sj + kβ ) G ( j). ( r − k )! ( k − sj )!  Γ ( A + r + kβ + 1)

Now, using the double series relation [k / s]

[r / s ] r − sj

k =0 j=0

j=0 k =0

r

∑∑ A ( k , j ) = ∑∑ A ( k + sj, j ) ,



and denoting r − sj by N , ψ r takes the form [r / s ]

ψr =



∑∑ j=0

[r / s ]



=

∑ j=0

( −1)k ( A + sj + sjβ + k + kβ ) Γ ( A + sj + sjβ + kβ ) G ( j ) k ! ( N − k )!  Γ ( A + N + sjβ + sj + kβ + 1) k =0 N

G ( j) N!

∑( −1)  Nk  ( N

k

 A + sj + sjβ + k + kβ )  Γ ( A + sjβ + sj + kβ ) . (6.18) Γ ( A + sjβ + sj + N + kβ + 1)



k =0

Now, in lemma, if M is replaced by N , then it takes the form

∑( −1)  Nk  Γ ((A + sjβ + sj + N + kβ )+ 1) U ( k ) (6.19) N



T (N ) =

k





A + sj + sjβ + kβ + k

k =0

if and only if N



U (N ) =





∑( −1)  Nk  Γ ( A + sj + sjβ + Nβ + k ) T ( k ). (6.20) k

k =0

85

A General Class of Polynomials

Now, in equation (6.18), if one replaces Γ ( A + sjβ + sj + kβ ) by U ( k ), then the inner series in k coincides with equation (6.19), whose inverse series is equation (6.20) by lemma. Here choosing,  0 T (k ) =    k

in equation (6.20), it reduces to N



U (N ) =









∑( −1)  Nk  Γ ( A + sj + sjβ + Nβ + k )  0k  k

k =0

= Γ ( A + sj + sjβ + Nβ ) .



 0 Thus, the choice for U(k) is restored with T ( k ) =   . Hence, from equation (6.18),  k one now arrives at [r / s ]

ψr =



∑

G ( j)  T ( N ) N!

[r / s ]

G ( j)  0  N !  N 

j=0

=



∑ j=0

= 0, if   N ≠ 0.



This proves that equation (6.12) implies equation (6.14), completing the proof of the first part. For proving the converse part, it will be shown that equations (6.13) and (6.14) together imply equation (6.12). Suppose that equations (6.13) and (6.14) hold true. Consider,

( −1)k ( A + k + kβ )  F ( k ) , (6.21) − Γ + + + r k A r k β !  1 ( ) ( ) k =0 r



Gr =

∑

then in view of equation (6.14),

Gr = 0 if   r ≠ sj,   j ∈ N  and (6.22)



Gsr = G ( r ) . (6.23)

86

Applied Mathematical Modeling and Analysis in Renewable Energy

Further, equation (6.21) implies that

( −1)k   Γ ( A + k + rβ ) Gk , ( r − k )! k =0 r

F (r ) =



∑

but Gr = 0 as shown in equation (6.22), hence in view of equation (6.23), one gets

( −1)k  ( A + k + kβ )  F ( k ) ( sr − k )!  Γ ( A + sr + kβ + 1) k =0 sr

G (r ) =



∑

implies that

( −1)sk   Γ ( A + sk + rβ ) Gsk − r sk ! ( ) sk = 0 r



F (r ) =



=

∑

( −1)sk   Γ ( A + sk + nβ ) Gk . r − sk ! ( ) sk = 0 r

∑

This shows that equation (6.13) ⇒  equation (6.12) subject to equation (6.14), and with that the proof completes here.

6.3  INTEGRAL REPRESENTATIONS Two well-known integrals [11], namely, (i) the beta-integral and (ii) the integral of generalized hypergeometric function p Fq ( z ), are used to derive two integral forms of the GCP M r ( s, A, β ; x ). These integrals are 1



Γ ( m )  Γ ( n ) = z m −1  (1 − z )n −1 dz ,  Re ( m ) > 0,  Re ( n ) > 0, (6.24) Γ (m + n)

∫ 0

and p

 a2 , × t a1 −1 (1 − t )b1 − a1 −1 (× p−1 Fq −1   b2 , 0  1



 a1 , a2 , …, a p ; z  Γ ( b1 ) = Fq  , , … , ; b b b a Γ ( )  1  q 2 1 Γ ( b1 − a1 )  

∫

…, a p ; zt     dt. (6.25) …, bq ;  

87

A General Class of Polynomials

For deriving the first integral form of polynomial M r ( s, A, β ; x ), replace M r by M r ( s, A, β ; x ) Γ ( 2 A + rβ ), then M r ( s, A, β ; x ) =



r  s

∑ (r − sk )! (Γ ( 2 A + rβ )) ψ x (−1)sk   Γ A + rβ + sk

k

k

k =0

[r / s ]



=

∑ (r −(sk−1))! Γψ( Ax− sk ) sk

k

k

k =0

Γ ( A + rβ + sk )  Γ ( A − sk ) Γ ( 2 A + rβ )

[r / s ]



=

1

( −1)sk ψ k x k t A+ rβ + sk −1  (1 − t ) A− sk −1 dt , r − sk !  Γ A − sk ( ) ( ) k =0 0

∑

∫

where Re ( a − sk ) > 0, Re ( A + rβ + sk ) > 0. Finally, one gets, 1



∫

M r ( s, A, β ; x ) = ξr ( s, A; x )  t A+ rβ −1  (1 − t ) A−1 dt , (6.26) 0

in which

ξr ( s, A; x ) =

[r / s ]

( −1)sk ψ k x k  t  sk  .  ( r − sk )!  Γ ( A − sk )  1 − t  k =0

∑

The next integral derivation of M r ( s, A, β ; x ) requires to put the polynomial into hypergeometric function form. For that ψ k is to be defined as ψ k = k !(1a )k , where a is a complex parameter. With this choice of ψ k , the polynomial is denoted by Fr ( x ) then

M r ( s, A, β ; x )

Γ ( A + rβ ) = r!

[r / s ]

( −1)sk   Γ ( A + sk + rβ ) k x − r sk !  k ! ( a ) ( ) k k =0

∑

Therefore,

M r ( s, A, β ; x ) =



=

[r / s ]

( −1)sk r !  Γ ( A + sk + rβ ) k x ( r − sk )!  k ! (a) k   Γ ( A + rβ ) k =0

∑ [r / s ]

∑

( −r )sk  ( A + rβ )sk x k

k =0



= Fr ( x ) .

k ! (a) k

88

Applied Mathematical Modeling and Analysis in Renewable Energy

So,

Fr ( x ) =

[r / s ]

∑ k =0



 =

(−r )sk  ( A + rβ )sk x k (a) k k !

 ∆ ( s; −r ) , ∆ ( s; A + rβ ) ; cx   , (6.27) F a;  

2s 1

where c = s 2 s is a constant. In the light of equations (6.25) and (6.27), one then obtains,



Fr ( x ) =

1

Γ (a)

Γ

( ) Γ (a − A + rβ s

A + rβ s

)∫

t

A + rβ −1 s

(1 − t )a−

A+ rβ −1 s

0

  1 + A + rβ s + A + rnβ − 1 , …, ; ctx   ∆ ( s; −r ) , × 2 s −1F0  s s  dt , (6.28)   −;

A + rβ   A + rβ   > 0,  Re  a − where s = 2, 3,…,  Re   > 0.  s   s 

6.4 DIFFERENTIAL EQUATION θ -FORM AND COMPANION MATRIX To derive the differential equation (θ -form) satisfied by the GCP M r ( s, A, β ; x ), first it is expressed in the generalized hypergeometric function form. In the defining relaΓ ( A + rβ ) tion (6.2), replace M r ( s, A, β ; x ) by r ! M r s, A, β ; x ,  then it takes the form:

(



M r ( s, A, β ; x ) =

)

r  s

∑ (− r )

sk

 ( A + rβ )sk ψ k x k . (6.29)

k =0

Now choosing ψ k = k1! in equation (6.29), and then denoting this special case by Gr ,s ( x ), the hypergeometric function form of Gr ,s ( x ) is obtained as:

 ∆ ( s; −r ) , ∆ ( s; A + rβ ) ; cx   , (6.30) Gr ,s ( x ) = 2 s F0  −;  

where c = s 2 s is a constant, and ∆ ( s; −r ) is an array of s parameters:

−r −r + 1 −r + s − 1 . ,   ,… s s s

89

A General Class of Polynomials

The differential equation in θ -form satisfied by y = p Fq  a1 ,…, a p ; b1 ,…, bq ; z 

is:  θ 



q



p

∏ (θ + b − 1) − z∏ (θ + a ) y = 0, (6.31) j

i

j =1

i =1



where θ = z dzd . Therefore, the differential equation satisfied by Gr ,s ( x ) is:  θ − cx 

in which ai =

− r + i −1 s



2s

∏ (θ + a ) G i

r ,   s

( x ) = 0, (6.32)

i =1

, for i = 1,…, s   and ai =

A + rβ + i − s −1 s

, for

i = s + 1,  s + 2,…, 2s.



If a polynomial g ( x ) ∈ [ X ], where

g ( x ) = a0 + a1 x + a2 x 2 +  + ar −1 x r −1 + x r ,

then the r × r matrix, called companion matrix [12] of g ( x ) is denoted and defined as:



0 0 C ( g ( x )) =    −a  0

1 0 0 1   − a1 − a2

 0  0   − ar −1

   . (6.33)   

Putting [r / s ] = N in GCP (equation (6.2)) and changing it to monic form M r ( s, A, β ; x ) , we find that M r ( s, A, β ; x ) =



N

∑λ x , k

k

k =0

where,

λk =

( −1)s( k − N )   Γ ( A + rβ + sk ) ( r − sN )! ψ k . (6.34) Γ ( A + rβ + sN ) ( r − sk )! ψ N

With this λ k , the companion matrix of M r ( s, A, β ; x ) takes the shape of equation (6.33). The eigenvalues of this matrix will be the roots of M r ( s, A, β ; x ). (see [12], p.39).

90

Applied Mathematical Modeling and Analysis in Renewable Energy

6.5  PARTICULAR CASES The special cases of the GCP M r ( s, A, β ; x ) (given in equation (6.2)), and of the properties discussed in sections 6.2, 6.3 and 6.4 will be presented here in this section. Further in this section, the properties of inverse series relations, integral representations and θ −form of differential equation, discussed in sections 6.2 to 6.4, are particularized. (A) Special cases of inverse series relation The special cases of general inversion pair proved in the theorem can be classified as follows: (A-1) Extended forms of inversion pairs of Gessel and Stanton. (A-2) Pairs of inverse series relations of extended versions of polynomials. (A-3) Combinatorial identities. (A-1) Extended forms of inversion pairs of Gessel and Stanton An interesting generalization and unification is seen in the theorem proved in section 6.2, as the ordinary forms of the various q-inversion pairs taken up by Gessel and Stanton [3] admit extensions through this theorem. The pairs that are obtained from the theorem by taking β = 1,  12 ,  −21 ,  −31 and 2 are stated as follows: (1) β = 1 [r / s ]

Ar =



∑ Γ ((Ar +− sksk )+! r ) B

k

k =0



⇔

sr

∑( −1)

sr − k

Br =



k =0

A + 2k

( sr − k )!  Γ ( A + sr + k + 1)

Ak

and r

∑( −1)

r−k



k =0

A + 2k

( r − k )!  Γ ( A + r + k + 1)

Ak = 0

if r ≠ sj,   j = 0, 1, 2,…. (2) β =

1 2 [r / s ]

Ar =



∑ k =0

sr



Br =

∑ k =0

( −1)sr − k

Γ ( A + sk + 2r ) Bk ( r − sk )! ⇔

A + 32k A, ( sr − k )!  Γ ( A + sr + k2 + 1) k

91

A General Class of Polynomials

and r

∑( −1)

r−k



k =0

A + 32k A = 0, ( r − k )! Γ ( A + r + k2 + 1) k

if r ≠ sj,   j = 0, 1, 2,…. (3) β =

−1 2 [r / s ]

Ar =



∑ k =0

sr

Br =



∑

( −1)sr − k

k =0

Γ ( A + sk − 2r ) Bk ( r − sk )! ⇔

A + k2 A ( sr − k )!  Γ ( A + sr − k2 + 1) k

and r

∑(−1)



r−k

k =0

A + k2 A = 0, ( r − k )! Γ ( A + r − k2 + 1) k

if r ≠ sj,   j = 0, 1, 2,…. (4) β =

−1 3 [r / s ]

Ar =



∑ k =0

sr

Br =



∑ k =0

Γ ( A + sk − 3r ) Bk ( r − sk )! ⇔

( −1)sr − k A + 23k Ak Γ ( A + sr − k3 + 1) ( sr − k )!

and r

∑( −1)

r−k



k =0

A + 23k Ak = 0, Γ ( A + r − k3 + 1) ( r − k )!

if r ≠ sj,   j = 0, 1, 2,…. (5) β = 2 [r / s ]



Ar =

∑ Γ ( A(r+−sksk−)!2r ) B

k

k =0

92

Applied Mathematical Modeling and Analysis in Renewable Energy

⇔

sr



Br =

∑( −1)

sr − k

k =0

A + 3k

( sr − k )!  Γ ( A + sr + 2k + 1)

Ak

and r



∑( −1)

r−k

k =0

A + 3k A = 0, ( r − k )!  Γ ( A + r + 2k + 1) k

if r ≠ sj,   j = 0, 1, 2,…. (A-2) Pairs of inverse series relations of extended form of polynomials Here, the extended versions of the various known polynomials occurring as the special cases of the GCP M r ( s, A, β ; x ) along with their inverse series relations will be illustrated. The polynomial M r ( s, A, β ; x ) given in equation (6.2) is reconsidered in an elegant form, already used in section 6.4 and given in equation (6.29). Making corresponding replacements in equation (6.11), it yields the inverse series of M r ( s, A, β ; x ) after some simplifications in the form:

( −1)k ( A + k + kβ ) M k ( s, A, β ; x ) (6.35) sr − k k A + k !  ! ( β ) ( ) sr + 1 k =0 sr



ψ r xr =

∑

Considering particular values of the parameters in the GCP given in equation (6.29) and its inverse given in equation (6.35), one immediately gets the inversion pairs of the extended polynomials. These pairs of inverse series relations are listed below: (1) Inversion pair of extended Racah polynomial First, the pair of Racah polynomial may be obtained in an extended form by setting β = 1, and then choosing A = 1 + a + b , x = 1 and

ψk =



Rns ( x ( x + c + d + 1); a, b, c, d ) =

[n / s ]



(− x ) k ( x + c + d + 1) k . k !(1 + a) k (1 + b + d ) k (1 + c) k

∑ (− n) k =0

sk

 (1 + a + b + n)sk  (− x ) k  ( x + c + d + 1) k (1 + a) k  (1 + c) k  (b + d + 1) k ⇔



(− x )n ( x + c + d + 1)n = n !(1 + a)n (1 + c)n (b + d + 1)n

sn

∑ ( sn −(−k1))! k(1!(1+ +a a+ +b b+ +2kk)) k

k =0

 .  Rks ( x ( x + c + d + 1); a, b, c, d )

 . sn +1

93

A General Class of Polynomials

(2) Inversion pair of extended Wilson polynomial Next, if β = 1, x = 1 and A = a + b + c + d − 1, then by selecting

ψk =



(a + ix ) k (a − ix ) k k !(a + b) k (a + c) k (a + d ) k

one gets an extended form of the Wilson’s polynomial and its inverse: Pns ( x 2 ) ( a + b ) n ( a + c) n ( a + d ) n

[n / s]

(− n)sk (a + b + c + d + n − 1)sk (a + ix ) k (a − ix ) k ( a + b ) k ( a + c) k ( a + d ) k k ! k =0 ⇔ (a + ix )n (a − ix )n ( a + b ) n ( a + c) n ( a + d ) n n ! =





∑

(−1) k  ( a + b + c + d + 2 k − 1)  Pks ( x 2 ) sn − k  (a + b) k (a + c) k (a + d ) k

sn

=

∑ k ! ( sn − k )! (a + b + c + d + k − 1) k =0

(3) Inversion pair of extended Hahn polynomial An extended form of Hahn polynomial Qn ,s ( x; a, b, N ) can now be obtained by setting β = 1,   A = 1 + a + b and x = 1 and then taking (− x ) k , where  n   =  0, 1, 2, …,  N . k !(1 + a) k (− N ) k



ψk =



Qn ,s ( x; a, b, N ) =

[n / s ]

∑ (−n) k (1! (1++a a+)b (+−nN) )  (− x) sk

k =0

sk

k

k

k



⇔



(− x ) n (−1) k  (1 + a + b + 2 k ) =  Qk ,s ( x; a, b, N ) (1 + a)n  (− N )n n ! k = 0 k ! ( sn − k )! (1 + a + b + k )sn +1

sn

∑

(4) Inversion pair of extended Jacobi polynomial An extension of the well-known Jacobi polynomial is obtained straight away by taking β = 1, A = 1 + α + β ,ψ k = k !(1+1α )k and replacing x by 1−2x . (α ,β )



Pn ,s

( x ) = (1 + α )n

[n / s ]

∑ k =0

n



 1 − x  = n !(1 + α )   n 2 

(− n)sk (1 + α + β + n)sk  1 − x    k !(1 + α ) k 2  ⇔

∑ k ! ( sn − k )! (1 (+ α + β + k ) sn

k =0

k

(−1) k   1 + α + β + 2 k ) α ,β Pk(,s ) ( x ) . +  (1 ) α sn +1 k

94

Applied Mathematical Modeling and Analysis in Renewable Energy

(5) Inversion pair of extended Legendre polynomial The inversion pair of Legendre polynomial, in an extended form, can be obtained by choosing A = β = 1,ψ k = k ! 1k ! and replacing x by ( 1−2x ) .

[n / s ]

∑

Pns ( x ) =

k =0

(− n)sk (n + 1)sk  1 − x    k !  k ! 2  ⇔

n



 1 − x  = n ! n !    2 

k

sn

∑ k =0

(−1) k ( 2 k + 1) Ps ( x ). ( sn − k )!  k !(k + 1)sn+1 k

(6) Inversion pair of extended Bessel polynomial An extension of the Bessel polynomial yn ( x ) ([11], p.293), denoted by yns ( x ) is obtained, when A = β = 1,ψ k = k1! and when x is replaced by ( −2x ).

yns ( x ) =

[n / s ]

∑ k =0

(− n)sk (n + 1)sk  − x   2  k! ⇔

n



 x  = n!  2

k

sn

∑ ( sn −(−k1))!  k( 2!(kk++11)) k

k =0

yks ( x ) .

sn +1

(7) Inversion pair of extended Laguerre polynomial Finally, with β = 0 and ψ k = k !(1+α1)k ( A)sk , one gets an extended version of the Laguerre polynomial and its inverse given as follows:

L(nα,s) ( x ) = (1 + α )n



⇔



xn = (1 + α )n

[n / s ]

− n) x ∑ k(!(1 +α) sk

k =0

k k

sn

∑ (1(−+snα))

k

k =0

L(kα,s) ( x ) .

k

(A-3) Combinatorial identities Besides yielding extended forms of polynomials and their inverse series relations, and the extensions to the ordinary versions of the inversion pairs due to Gessel and Stanton, the theorem also possesses the potential to give rise to some seemingly new combinatorial identities. Some known combinatorial identities are also contained in the theorem and some are inverted through it. First of all, setting s = 1 in the theorem, one gets r



T (r ) =

∑ k =0



Γ ( A + k + rβ ) R ( k ) (6.36) ( r − k )! ⇔

95

A General Class of Polynomials r



R (r ) =

∑ k =0

(−1)r − k ( A + k + kβ ) T (k ) Γ ( A + r + kβ + 1) ( r − k )!

b (x) Putting A = 1,   β = 0, R ( k ) = (r +2kk )!  k ! , one obtains from the first series above x 2 r . Thus, the combinatorial identity ([13], p.57), T ( r ) = ( 2r )! r

∑ (r( 2+rk)!)!b(r −( xk))!

x 2r =



2k

k =0

is obtained. The inverse of this identity follows from the second series in the pair of equation (6.36) with the same choices of the parameters, which is: b2r ( x ) = ( 2r )!



r

∑ k =0

(−1)r − k ( k + 1) x 2 k ( r + k )!( 2k )!( r + 1)

Consider now the pair of inverse series relations from the simpler Legendre class ([13], p.68), namely, n



bn =

∑(−1)

n+ k

k =0



  2n + p   2n + p      ak  −  n − k   n − k − 1   ⇔  n+ p+ k n−k k =0 n

an =



∑ 

  bk . 

This pair can be obtained from equation (6.36) when β = 1,  A = p + 1, R ( k ) = ( p+b2k k )! and T ( k ) = ak . The second pair in table 2.5 (simpler Legendre class) of ref. ([13], p.68) also occurs as a special case. For obtaining this special case, put β = 1, A = p and then ak k T ( k ) = ( −p1)+ 2 kbk . Then replacing R ( n ) by (−1)n R ( n ) , one gets R ( k ) = ( 2 k + p)! ,

an = ( 2n + p ) !

n

∑ ( p + n + kb)!( n − k )! . k

k =0



⇔

( −1)n+ k  ( p + 2n ) ( n + p + k − 1)! ak . ( p + 2k )! ( n − k )! k =0 n



bn =

∑

The inversion pair in J. Riordan ([13], p.79)

( −1)n =

n

k =0









∑  n2+kk   2kk  (−1) ⇔

k

96



Applied Mathematical Modeling and Analysis in Renewable Energy

 2n   n  =

n





 

∑  n2−nk  −  n −2kn− 1   k =0

is easily obtained from equation (6.35) by setting A = β = 1 and choosing R ( k ) = (k−! 1)k ! . Then, one gets from the first series, T ( n ) = (−1)n . On setting β = 1,   A = 2 and replacing n by n − 1 in equation (6.36), the specializam tions R* ( m ) = m( −! 1)m ! and T * ( m ) = (−1)m +1 m 2 are suggested; where R* ( m ) and T * ( m ) are obtained from R ( m ) and T ( m ) when n is replaced by n − 1. One then arrives at yet another combinatorial identity in J. Riordan ([13], p.38), whose inverse is obtained from equation (6.36) and the pair reads: k



n −1

∑ ( n(−−1)k −(1n)! +kk! )!k !

( −1)n n 2 =

k =0



1 = n !  n !

k

⇔

n −1

∑ ( n + k2+(1k)+!(1n)−k k − 1)! . 2

k =0

Another identity ([13], p.83) is also contained in the pair of equation (6.35) for β = 1 k and A = 2. With these particular values of A and β , if one sets R* ( k ) = ( −k1)! ( k + 1)!, then T * ( k ) = (−1) k +1; (where n is replaced by n–1) and the inversion pair is obtained in the form:

( −1)n+1 =

n −1

k =0

( −1)n = n !( n + 1)!

k

⇔



  nk 



∑  n k− 1   k + 1  (−1) , n −1

∑ ( n +(k−+1)1)2!((nk −+k1)− 1)! . n

k =0

Another series identity ([13], p.84) whose inverse is also constructed through the theorem in an equivalent form is as follows: n −1



fn =

∑ ( n + k + 1)!((n2−n )k! − 1)!( k + 1) k =0

⇔



( −1) 2 = 2 ( n + 1) n +1



n −1

( −1)k ( n + k )! f . ( n − k − 1)!( 2k )! k k =0

∑

For yet another series identity, inverse can be obtained from the theorem by setting s = 2, β = 0 and A = 1. Choosing F ( k ) = ( k +1)( k (+21k)! )!k !  k !2k , one gets from the theorem G ( k ) = 22 k  1k !  k ! .

97

A General Class of Polynomials

1 = 22 n   n !  n !



2n

∑ (2n − k()−!(1)k +(12)k! )k! !  k !2 k

k =0

,

⇔



( 2n )! = ( n + 1)( n + 1)!  n !  n !2n



k

[n /2]

∑ ( n − 2k()2! kk)! ! k !2

2k

,

k =0

with n

( −1)n− k ( 2k )! = 0, ( n + 1)!( n − k )!( k + 1)!  k !  k !2k k =0

∑



if n ≠ 2m,   m = 0, 1, 2,…. (B) Special cases of integrals The integral forms in view of equation (6.26) of some of the polynomial special cases mentioned above, are listed below. 1

∫

R ( x ( x + c + d + 1); a, b, c, d ) = ξ n ( s, a, b; x ))  t a + b + n



s n

0

(1 − t )a + b   dt , (6.37)

where, [n / s ]

∑ n!  k ! (1 + a()− (n)b + (d−+x)1) ( (1x ++cc+) d Γ+(1)1 + a + b − sk ) ,

ξ n ( s, a, b; x ) =



sk

k

k =0

k

k

k

k

where Re (1 + a + b + n + sk ) > 0, Re (1 + a + b − sk ) > 0. 1

P (x



s n

2

) = ∫λn ( s, a, b, c, d; x ) t a+ b+c+ d + n+ sk − 2 0

(1 − t )a + b + c + d − sk − 2 dt , (6.38)

where,

λ n ( s , a , b , c, d ; x ) =  

[n / s ]

∑ k =0



(− n)sk (a + ix ) k (a − ix ) k ( 1−t t ) , n !  k ! (a + b) k  (a + c) k  (a + d ) k   Γ ( a + b + c + d − sk − 1) sk

Re ( a + b + c + d + n + sk − 1) > 0, Re ( a + b + c + d − sk − 1) > 0.

98

Applied Mathematical Modeling and Analysis in Renewable Energy 1

∫

Qn ,s ( x; a, b, N ) = ν n ( s, a, b; x )  t a + b + n  (1 − t )a + b dt , (6.39)



0

where,

ν n ( s, a, b; x ) =



[n / s ]

∑ n!  k ! (1 + a) (− (n−)N ) (− Γx()1 + a + b − sk )  1 −t t  sk

k

k =0

k

sk

,

k

Re (1 + a + b + n + sk ) > 0, Re (1 + a + b − sk ) > 0.



1

∫

α ,β Pn(,s ) ( x ) = θ n ( s, α , β ; x )  t α + β + n  (1 − t )α + β dt , (6.40)



0

where,

θ n ( s, α , β ; x ) =

[n / s ]

∑ k =0



(− n)sk ( 1−2x ) ( 1−t t ) , kn ! (1 + α ) k   Γ (1 + α + β − sk ) k

sk

Re (1 + α + β + n + sk ) > 0, Re (1 + α + β − sk ) > 0.

Similarly, the integrals of Legendre, Bessel and Laguerre polynomials can be obtained (details of which are omitted here). In the second integral of equation (6.28), the parameter s is a natural number greater than 1. Because of the peculiar form of the extended Racah, Wilson and Hahn polynomial, we first take β = 1,  x = 1 in equation (6.28) and then choose A = 1 + a + b , and

ψk =



(− x ) k  ( x + c + d + 1) k , k ! (1 + a) k  (1 + c) k  (1 + b + d ) k

we get the hypergeometric form of extended Racah polynomial. Now, using equation (6.28), one finally obtains the integral. As an example, here the integral of the Jacobi polynomial is shown below. First considering β = 1, A = 1 + α + β , a = 1 + α and replacing x by 1−2x in the p Fq form of M r ( s, A, β ; x ) and then applying equation (6.28), one arrives at,



α ,β Pn(,s ) ( x ) =

Γ

(

1

Γ (1 + α )

1+α + β + n s

) Γ (1 + α −

 α +β +n+2 ,  ∆ ( s; − n ) , × 2 s −1F0  s    −; 

1+α + β + n s

,

)∫

t

1+α + β + n −1 s

α−

(1 − t )

1+α + β + n s

0

α +β +n+s 1− x  ; tc   2  s

   dt ,  

99

A General Class of Polynomials

(

)

(

)

where s = 2,3,…, Re(1 + α ) > 0, Re 1 + α − 1+α +s β + n > 0, Re 1+α +s β + n > 0. Similarly, the integral form of the remaining polynomials special cases can be obtained. (C) Special cases of θ -form difference equation In this section, the θ -form differential equations for extended forms of some of the polynomial special cases are listed below. All these differential equations occur as particular cases of equation (6.32) when parameters are specialized suitably. Racah polynomial 2s+2   θ θ + a θ + c θ + b + d − k )( )( )  ( (θ + ai ) Rns = 0,   i =1

∏



where θ = z dzd ,   k = s 2 s and ai = − n +s i −1 , for i = 1, 2,…, s; and ai = a + b +sn + i − s , for i = s + 1, s + 2,…, 2s; a2 s +1 = − x , and a2 s + 2 = x + c + d + 1. Wilson polynomial

2s+2   θ (θ + a + b − 1)(θ + a + c − 1)(θ + a + d − 1) − k (θ + ai ) Pns ( x 2 ) = 0,  i =1 

∏

where θ = z dzd ,  k = s 2 s ,   ai =

ai =

− n + i −1 s

, for i = 1, 2,…, s; and

a+ b+c+ d + n+i − s− 2 s

, for i = s + 1,  s + 2,…, 2s

and a2 s +1 = a + ix , a2 s + 2 = a − ix . The difference equations for Hahn, Jacobi, Legendre, Bessel and Laguerre polynomials are obtained in a similar manner.

ACKNOWLEDGMENTS The author is thankful to Dr. B. I. Dave, for his useful suggestions and guidance during the preparation of this work. The author wishes to acknowledge the useful suggestions given by the anonymous referee/s.

REFERENCES 1. Gessel, I. A combinatorial proof of the multi variable Lagrange inversion formula, J. Combin. Theory Ser. A., 45, no. 2, pp. 178–195 (1987). 2. Gasper, G., Rahman, M. Basic hypergeometric series, Cambridge University Press, Cambridge (1990). 3. Gessel, I., Stanton, D. Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc., 277, no.1, pp. 173–201 (1983). 4. Brouder, C., Frabetti, A., Krattenthaler, C. Non-commutative Hopf algebra of formal diffeomorphisms, Adv. Math., 200, no. 2, pp. 479–524 (2006). 5. Bultel, J. Combinatorial properties of the non-commutative Faa’ di Bruno algebra, J. Algebraic Combin., 38, no. 2, pp. 243–273 (2013).

100

Applied Mathematical Modeling and Analysis in Renewable Energy

6. Gessel, I. A non-commutative generalization and q-analog of the Lagrange inversion formula, Trans. Amer. Math. Soc., 257, no. 2, pp. 455–482 (1980). 7. Novelli, J., Thibon, J. Non commutative symmetric functions and Lagrange inversion, Adv. in Appl. Math., 40, no. 1, pp. 8–35 (2008). 8. Dave, B. I., Dalbhide, M. Gessel-Stanton’s inverse series and a system of q-polynomials, Bull. Sci. Math., 138, pp. 323–334 (2014). 9. Askey, R., Wilson, J. A. Some basic hypergeometric polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc., 54, pp. 1–55 (1985). 10. Gasper, G. Projection formulas for orthogonal polynomials of a discrete variable, J. Math. Anal. Appl., 45, pp. 176–198 (1974). 11. Rainville, E. D. Special functions, Macmillan, New York (1960). 12. Mignotte, M., Stefanescu, D. Polynomials: An algorithmic approach, Springer-Verlag, Berlin (1999). 13. Riordan, J. An introduction to combinatorial identities, Wiley, New York, London, Sydney (1968). 14. Garsia, A. A q-analogue of the Lagrange inversion formula, Houston J. Math., 7, no. 2, pp. 205–237 (1981).

7

Squeezing Graphs Ved Suthar

CONTENTS 7.1 Introduction................................................................................................... 101 7.2 How to Squeeze............................................................................................. 101 7.3 Squeezing the Cartesian Coordinates into Limited Space............................ 103 7.4 Squeezing Few Functions With Independent Squeezing............................... 106 7.5 Radially Squeezing........................................................................................ 106 7.6 Squeezing Few Functions With Radial Squeezing........................................ 114 7.7 Application.................................................................................................... 117 7.8 Resources....................................................................................................... 122 References............................................................................................................... 123

7.1 INTRODUCTION Normally, we use a graphing calculator/application to plot a graph (or use a traditional method of using graph paper to plot it). In all these cases, we limit our curve to specific limits of x and y (which is usually where we need to focus). By doing this, we miss out what’s happening outside the region we are seeing (mainly near infinity). In this chapter, we are going to learn to squeeze the different coordinate systems, and discuss more on other different systems which can plot understandable versions of graphs. The drawback in normal coordinate systems is that we can only explore/see a specific spot at once; we can zoom out from the graph, but it reduces the resolution (as every system of plotting the graph has a resolution limit) of the curve which we are seeing. The main goal of this chapter is to watch the characteristic of any curve from negative infinity to the infinity in a single glance without losing most of the necessary information. This can be done by squeezing the graph in a proper manner. The process is shown with the help of Desmos, which is a great graphing tool, and can help in understanding the different principles [1].

7.2  HOW TO SQUEEZE We can squeeze in two modes: 1. Linear way: By dividing the number by a specific quantity which is independent from the number (zooming out), e.g., by dividing scale by 10, we get a new scale squeezed by a factor of 10, i.e., new 10 is 1 and new 20 is 2 [2]. This can be seen in Figs. 7.1 and 7.2 where we can observe the exponential curve being linearly zoomed out. DOI: 10.1201/9781003159124-7

101

102

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 7.1  Exponential curve.

Drawbacks: a. We cannot see the characteristic of the whole graph (we will always put a limit to our field on all the axes). b. We can just zoom out by dividing the scale, and hence, we will lose information with zooming out.

FIGURE 7.2  Linearly squeezed exponential curve.

103

Squeezing Graphs

c. Most of the curves have interesting characteristics near the origin, but by scaling out linearly, we will lose most of the interesting properties of the curves. 2. Non-linear way: By dividing different numbers depending on the other variables such as x or y or both combined in a specific manner. We are mainly concerned about this type as we can visualize the whole curve without losing necessary information. There are infinitely many ways to squeeze in non-linear fashion, but we are going to talk about some specific ways that are special in visualization. Now, non-linear squeezing is further divided into two parts: a. Squeezing in infinite space: Just like normal Cartesian coordinates that extend up to infinity, this type of squeezing gives us the type of system where infinity lies on infinity. b. Squeezing in limited/definite space: If the system squeezes the whole plane (up to infinity) into a space with a definite area, we call it squeezing in limited space.

7.3 SQUEEZING THE CARTESIAN COORDINATES INTO LIMITED SPACE There are many ways to squeeze into limited space by using convergent functions. They are basically divided into two categories: 1. Independent of axes: X-axis and Y-axis are squeezed independently. 2. Dependent of axes: X-axis and Y-axis are squeezed dependent on each other. One of the independent squeezes is plotted on the plane.

FIGURE 7.3  Squeeze function.

x 1+ x

. Figure 7.3 shows the squeeze function

104

Applied Mathematical Modeling and Analysis in Renewable Energy

In this function, 0 is 0 1 is 12 2 is 23 3 is 43 and so, n is

n n +1

Another interesting thing is this function tends to 1 at infinity. Hence, the inverse of this function would give any real number in the range of – 1 to 1. Finding inverse function: Let us call the new limited value as xnew and normal value as x. By this method,  x new = 1+x x . To find inverse, let’s split the function for +ve x and –ve x. For +ve values of x, x = x

∴ x new =

x 1+ x



∴ x new (1 + x ) = x



∴ x new + x new ⋅ x = x



∴ x (1 − x new ) = x new ∴x =



x new 1 − x new

Similarly, for –ve values of x, x = − x ∴ x new =



x 1− x



∴ x new (1 − x ) = x



∴ x new − x new . x = x



∴ x + x. x new = x new



∴ x (1 + x new ) = x new



∴x =

x new 1 + x new

From this result, we can say that

x=

x new 1 − x new

105

Squeezing Graphs

FIGURE 7.4  xnew relation function. x

y

new So, by replacing x by 1− new xnew and y by 1− ynew , we can squeeze the whole 2D x-y plane. y x ∴ The new x and y are 1− x and 1− y The above Fig. 7.4 shows the plot of xnew. If we squeeze the plane, the grid will look like as shown in Fig. 7.5. Here, we can see that as the plane approaches near infinity, the scale diminishes. Hence, we can check the nature of any curve near infinity, which is quite fascinating considering how different things can be near infinity [3]. As we know how to squeeze, let’s watch a few important functions.

FIGURE 7.5  Squeezed grid.

106

Applied Mathematical Modeling and Analysis in Renewable Energy

7.4 SQUEEZING FEW FUNCTIONS WITH INDEPENDENT SQUEEZING From Figs. 7.6 to 7.16, different functions which are used frequently are shown.

7.5  RADIALLY SQUEEZING We have seen independent squeezing. Now, let’s check dependent squeezing. The squeezing factor should be dependent on both x and y. One of the ways to squeeze in this way is by squeezing the plane radially, i.e., from the distance of the point from origin. It can be obtained by applying Pythagoras theorem as shown in Fig. 7.17. Now, we derive the squeezing equation for a point on the x-y plane. Our aim here is to squeeze radially; so, polar coordinates can help us a lot. In polar coordinates,

FIGURE 7.6  y = x.

x = r cos (θ ) and  y = r sin (θ )

Squeezing Graphs

FIGURE 7.7  y = 2x.

FIGURE 7.8  y = x + 1.

107

108

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 7.9  y = x2.

FIGURE 7.10  y = x3.

Squeezing Graphs

FIGURE 7.11  y = ex.

FIGURE 7.12  y = ln (x).

109

110

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 7.13  y = sin (x).

FIGURE 7.14  y = 1x zoomed .

Squeezing Graphs

FIGURE 7.15  y = 1x .

FIGURE 7.16  y2 + x2 = 1.

111

112

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 7.17  Right angle triangle.

where, r is the distance from origin, i.e., = ( x1 , x 2 ,…, x n ), r = x 2 + y 2 ,



x x 2 + y2

cos (θ ) =



∴ cos (θ ) =



y x + y2

sin (θ ) =



2

∴ sin (θ ) =



x , r

y r

So, by squeezing, our new r (rnew) would be, rnew =



r 1+ r

Using trigonometry to get xnew and ynew, x new =



∴ x new =

r x × 1+ r r

∴ x new =



∴ x new =

Similarly, ynew =

r × cos (θ ) 1+ r

y 1+ x 2 + y 2

x 1+ r

x 1 + x 2 + y2

113

Squeezing Graphs

FIGURE 7.18  Radially squeezed grid.

By doing so to every point, our grid looks as shown in Fig. 7.18. We could also observe concentric circle plotted on the plane which is shown in Fig. 7.19. We get another fascinating view by making concentric circles with exponential difference in radii. Now, we should find an equation of xnew and x from which we can replace x in the normal equation to plot squeezed graphs. From, rnew = 1+r r , we could say that r = 1−rnew (inverse of the previous function). rnew By substituting this result, we get rnew × cos (θ ) 1 − rnew



x=



∴x =

rnew x × new 1 − rnew rnew



∴x =

x new 2 2 1 − x new + ynew

Similarly, y =

ynew 2 + y2 1− xnew new

114

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 7.19  Concentric circles.

Hence, by replacing x by radially squeezed.

x 1− x 2 + y 2

and y by

y 1− x 2 + y 2

, the graph we get will be

7.6  SQUEEZING FEW FUNCTIONS WITH RADIAL SQUEEZING The frequently used function are plotted on the squeezed plane which are shown from Figs. 7.20 to 7.27. 1. y = x

FIGURE 7.20  y = x.

Squeezing Graphs

2. y = 2x

FIGURE 7.21  y = 2x.

3. y = x + 1

FIGURE 7.22  y = x + 1.

4. y = x2

FIGURE 7.23  y = x2.

115

116

Applied Mathematical Modeling and Analysis in Renewable Energy

5. y = ex

FIGURE 7.24  y = ex.

6. y = ln (x)

FIGURE 7.25  y = ln (x).

7. y = sin (x)

FIGURE 7.26  y = sin (x).

Squeezing Graphs

8. y =

117

1 x

1 FIGURE 7.27  y = x .

7.7 APPLICATION As we have squeezed the plane into limited space, we can plot it on 3D surfaces like toroids, spheres, etc. [4]. We can connect +∞ and –∞ of Y-axis by bending the plane in third dimension and making a cylinder. Now, we can bend the whole cylinder connecting the two circles and making a toroid which contains every point of that plane. In this plane (0, 0) would be the nearest point and (0, ∞) would be the farthest point. Now, if we plot different functions, it would look as shown from Figs. 7.28 to 7.33. The surfaces are generated using different equations used for general purpose.

FIGURE 7.28  3D representation of y = x on toroidal surface (orthogonal view).

118

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 7.29  3D representation of y = x on toroidal surface (front view).

FIGURE 7.30  3D representation of y = x2 on toroidal surface (orthogonal view).

FIGURE 7.31  3D representation of y = x2 on toroidal surface (front view).

Squeezing Graphs

119

FIGURE 7.32  3D representation of y = x3 on toroidal surface (orthogonal view).

We can also try it on a sphere which is another solid [5]. Thus, the function y = x can be seen on spherical surface on Figs. 7.34 and 7.35. This method can also be used in iteration to find specific points of a function. Suppose we need to find global minima of the following:

f ( x ) = ( x − 3) ∗ ( x ) ∗ ( x + 2 ) ∗ ( x − 2 ) ∗ ( x + 3) ∗ ( x + 4 )

It is shown in Fig. 7.36. There are three minima, but we only need the global minima. As we can see, it is around x = 2.61, we can run a simple iterative program to find minima, but it would take many iterations to find. So, the key point is choosing a point that is most favorable to have a minimum. Further, after finding the general area of minima, we need to accurately pinpoint the location.

FIGURE 7.33  3D representation of y = x3 on toroidal surface (front view).

120

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 7.34  3D representation of y = x on spherical surface (front view).

FIGURE 7.35  3D representation of y = x on spherical surface (top view).

Squeezing Graphs

121

FIGURE 7.36  f ( x ) = ( x − 3) * ( x ) * ( x + 2 ) ∗ ( x − 2 ) ∗ ( x + 3) ∗ ( x + 4 ).

This is where we can use the property of the squeezed graph. The function would look as shown in Fig. 7.37. Now, we can take 40 points distributed evenly from – 1 to 1, and check for the value of the function to find the area with the global minima. By doing this, we get minima at x = 2.33, where f (x) = – 75.89. But the answer gets us to the general area where the minima is. We can increase the accuracy by taking more points like 1000 points, but we can use one of the properties of a squeezed graph to get a more accurate answer using less iteration by shifting the function on X-axis, which can be seen in Fig. 7.38 [6]. By doing this, we can have more precision over that specific area where the minima is obtained. After this, we can reiterate the program with this new shifted function to get more precise value of the required point. As it is a general iterative method, so this method can be used to find any point on the function such as maxima, minima, roots, etc., but using a squeezed graph. By applying this method, the program was able to find the global minima correct up to three decimal places in 200 iterations. Comparing this to the traditional method, we would have to iterate over small steps. In this case, minima were at 2.33, so we would need 2.33 × 1000 = 2330 iterations to find up to the same precision.

122

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 7.37  Squeezing f ( x ) = ( x − 3) ∗ ( x ) ∗ ( x + 2 ) ∗ ( x − 2 ) ∗ ( x + 3) ∗ ( x + 4 ).

This method can be used with other pre-existing fast methods to further improve its efficiency such as for solving non-linear equations [7]. We can also explore higher order equations using different approaches mentioned in ref. [8], but by adding squeezed planes, the number of iterations can be minimized too.

7.8 RESOURCES All the graphs shown in the figures of this chapter are created using Desmos (www. desmos.com). To see all the graphs shown and interact with it, follow the following link: https://www.desmos.com/calculator/h4o3cjzc8c. The program used to find the extremum was made in python (https://github.com/ GoVed/squeezingGraphs).

Squeezing Graphs

123

FIGURE 7.38  Moving the function on x-axis.

REFERENCES

1. Razieh Shahriari, The Effect of Using Technology on Students Understanding in Calculus and College Algebra, 2019. 2. Gilbert Strang and Edwin “Jed” Herman, Polar Coordinates, Calculus Volume 3, OpenStax, 2016 [https://openstax.org/details/books/calculus-volume-3]. 3. Brian Clegg, A Brief History of Infinity: The Quest to Think the Unthinkable, 2003. 4. Maks Ovsjanikov, Wilmot Li, Leonidas Guibas, and Niloy J. Mitra. 2011. Exploration of continuous variability in collections of 3D shapes. ACM Trans. Graph. 30, 4, Article 33 (July 2011), 10 pages. 5. Philip Leong and Simon Carlileb, Methods for Spherical Data Analysis and Visualization, 1998. 6. University of Washington CSE, Translation, Rotation and Scaling, pages 4–8 [https:// courses.cs.washington.edu/courses/cse576/book/ch11.pdf]. 7. Beong In Yun, Iterative Methods for Solving Nonlinear Equations with Finitely Many Roots in an Interval, 2012. 8. C. Dong, Math. Numer. Sinica, A Basic Theorem of Constructing an Iterative Formula of the Higher Order for Computing Multiple Roots of an Equation, 1982.

8

Finding the Surface Area and Volume of the Hyperspheres Using Simple Calculus Shantanu, Ritu Sahni, and Manoj Sahni

CONTENTS 8.1 Introduction................................................................................................... 125 8.2 Emergence of String Theory......................................................................... 126 8.3 Idea Behind String Theory............................................................................ 126 8.4 Multi-Dimensions or Hyper Dimensions – The Need................................... 126 8.5 Our Own Notion and Some Misconceptions about Volume and Surface Area........................................................................................... 127 8.6 Volume of Hypersphere................................................................................. 128 8.7 Surface Area of Hypersphere........................................................................ 128 8.8 Verification For Surface Area and Volume of Hyperspheres With Results For Various Dimensions.......................................................... 128 8.9 Conclusion..................................................................................................... 129 References............................................................................................................... 130

8.1 INTRODUCTION We live in a wonderfully complex universe, which is quite intriguing; hence we are curious about nature. We’ve asked ourselves, “Why are we here?” several times. Where did we and the world come from? And the most important one – what is the world made of? It’s our privilege to live in a time when enormous progress has been made toward finding some of the answers. String theory (see [1–8]) is our most recent attempt to answer the question of our world’s existence. We know that the current model of what is the world made of shows that the ordinary matter is made up of atoms, which are in turn made of three basic components – electrons, protons and neutrons, where electrons whirls around the nucleus, which is then further made up of neutrons and protons. The electron is a truly fundamental particle, while neutrons and protons are made of smaller particles, known as quarks. Quarks are, as far as we know, truly elementary. Our current knowledge about the sub-atomic composition of the universe is summarized in what is known as the standard model of particle physics. It describes DOI: 10.1201/9781003159124-8

125

126

Applied Mathematical Modeling and Analysis in Renewable Energy

both the fundamental building blocks the world is made up of and the forces through which these blocks interact. There are 12 basic building blocks, six of them are quarks and they go by interesting names of up, down, charm, strange, bottom and top. A proton, for instance, is made up of two up quarks and one down quark. The other six are leptons; these include electrons and their heavier siblings, the muons and the tau, as well as neutrinos. There are four fundamental forces in the universe: gravity, electromagnetism, weak nuclear force and strong nuclear force, and each of them are produced by fundamental particles that act as the carrier of force. The most familiar of them is photon, a particle of light which is the mediator of electromagnetic forces; this means that, for instance, a magnet attracts another magnet because both objects exchange photons (to be specific here, virtual). The graviton is the particle associated with gravity. The strong force is carried by eight particles known as gluons. Finally, the weak force is transmitted by three particles: W + , W − and Z.

8.2  EMERGENCE OF STRING THEORY The behavior of all the above-mentioned particles and forces is described with precision by standard model, with notable exception: gravity because it is very difficult to describe it microscopically and this has been the most important problem in theoretical physics to formulate a quantum theory of gravity. In the last few decades, string theory has emerged as the most promising candidate for a microscopic theory of gravity, and it somehow attempts to provide a complete, unified and consistent description of fundamental structure of our universe, and for this reason, sometimes it is also called the theory of everything (see for instance [3–4]).

8.3  IDEA BEHIND STRING THEORY The essential idea behind string theory [7] is that all the different ‘fundamental’ particles of standard model are really just different manifestations of one basic object: a string. How can that be? Well, we would picture an electron as a point. But if string theory is true, then an extremely powerful microscope could find that they are not actually a point but a tiny loop of string inside it. The frequency of oscillation is, in brief, what makes it different. If it oscillates in certain ways, we have electrons, and in other way, we can have a photon or quark, which is the idea behind it. So, if string theory is correct, the entire world is made up of strings!

8.4  MULTI-DIMENSIONS OR HYPER DIMENSIONS – THE NEED It’s not so hard to construct higher dimensional worlds using Einstein’s equation, but the question is then – why bother? It’s because physicists dream of a unified theory – a single mathematical framework in which all fundamental forces and units of matter can be described together in a manner that is internally consistent with consistent future observations (for instance, one can see [9, 10]). And it turns out that having

Finding the Surface Area and Volume of the Hyperspheres

127

extra dimensions of space makes it possible to build candidates for such a theory. Then, the question arises – how to have extra dimensions? There are possibly two ways of having extra dimensions: 1. Roll up the extra dimensions into some very tiny but nonetheless interesting spaces of their own; it is called Kaluza-Klein compactifications. 2. Make the extra dimensions into some really big, but constrain all the matter and gravity to propagate in a three-dimensional subspace called branes.

8.5 OUR OWN NOTION AND SOME MISCONCEPTIONS ABOUT VOLUME AND SURFACE AREA The misconception about volume in our mind is that it is just an analogy used to describe three-dimensional shapes, and surface area is viewed as the area of a surface if we flatten the surface and consider its shape to be two-dimensional. But as per our view, volume is the region occupied by that shape in our space and surface area is just the outline or bounded region for that shape. Similarly, whenever we usually talk about dimensions, we often encounter questions like where is the fourth perpendicular line just like we have other threes in Cartesian coordinate systems. Mathematically, it can still make some sense, but what about the physical nature of it? In simple language, dimensions are just the minimum number of data required to locate something in space (not considering time as a dimension just for now), which ‘can be’ more than three if we get to know about the three spatial dimensions. For now, we have made an attempt to find out volume and surface area of one of the family of shapes from hyper dimensions. Here, we are considering hyperspheres only. Hypersphere (see Fig. 8.1) is an analogy used to describe higher dimensional spheres. For formulating their surface area and volumes, we have opted

FIGURE 8.1  Hypersphere.

128

Applied Mathematical Modeling and Analysis in Renewable Energy

many methods, which include calculus, mathematical reasoning and own observations.

8.6  VOLUME OF HYPERSPHERE Let us consider a hypersphere with its center at O of nth dimension. With our notion th  of understanding, we can break it into its components of ( n − 1) dimension. Let the th  th  radius of that n hypersphere be R, and let the hypersphere of ( n − 1) dimension be at height y from the origin with radius x. Now, in right-angled ∆OAB, with ∠B = 90° and using Pythagoras theorem, R 2 = x 2 + y 2   ⇒ x = R 2 − y 2 (8.1)

the volume of ( n − 1) r = x = R2 − y2 .

th 



dimension hypersphere is given as = Vn −1 ( r ), where

Therefore, in right  ∆OAB,  y = R sin θ (8.2)

By equations (8.1) and (8.2), we have x = R cos θ .

Thus, 

dy = R cos θ   ⇒  dy = R cos θ dθ (8.3) dθ

∫

Therefore, Vn ( r ) = 2 Vn −1 ( r ) dy

(8.4)

In equation (8.4), it is multiplied by 2 because of symmetry. Therefore, by equations (8.3) and (8.4), Vn ( r ) = 2 ∫ Vn −1 ( r ) R cos θ dθ  , where r = R cos θ

8.7  SURFACE AREA OF HYPERSPHERE As per our understanding, if we differentiate the equation of volume with respect to radius, we can get the surface area, or in the better words, the bounded region because as far as our observations and knowledge are concerned in geometry, if by integrating a curve, we can get the region bounded in it, then by differentiating it again, we can get the curve again, i.e., the bounded region for that hypersphere. −1 ( R ) Thus, An ( r ) = dVndR .

8.8 VERIFICATION FOR SURFACE AREA AND VOLUME OF HYPERSPHERES WITH RESULTS FOR VARIOUS DIMENSIONS Case 1: For Two-Dimensional

π 2

∫

V2 ( R ) = 2 V1 ( r ) R cos θ dθ = π R 2   =  Volume 0

129

Finding the Surface Area and Volume of the Hyperspheres

TABLE 8.1 Volume and Surface Area Calculated for Various Multi-Dimensions Dimension 3-D 4-D 5-D

Volume 4 π R3 3

Surface Area

1 2 4 π R 2 8 2 5 π R 15

2π 2 R3

4π R 2

8 2 4 π R 3

6-D

1 3 6 π R 6

π 3 R5

7-D

16 3 7 π R 105

16 3 6 π R 15

For bounded region,

A2 ( R ) =

d (V2 ( r )) d (π R 2 ) = = 2π R   =  Surface area dR dR

Case 2: For Three-Dimensional π 2



∫

V3 ( R ) = 2 V2 ( r ) R cos θ dθ = 0

4 π R3   =  Volume 3

For bounded region,

A3 ( R ) =

d (V3 ( r )) d ( 43 π R3 ) = = 4π R 2   =  Surface area dR dR

Similarly, if we go on for further dimensions in the same way, we can have the following results shown in Table 8.1. It is seen from Table 8.1, that the surface area of nth dimensions is directly proportional to R n−1 and volume is directly proportional to R n.

8.9 CONCLUSION This chapter intends to find the surface area and volume of a hypersphere using calculus in a simpler manner. The general formula for surface area and volume is derived for multi-dimensions. In Table 8.1, the calculated formula is shown for volume and surface area for higher dimensions. In this way, the surface area and volume are calculated for any dimensions using simple calculus.

130

Applied Mathematical Modeling and Analysis in Renewable Energy

REFERENCES

1. Michael B. Green, John H. Schwarz, Edward Witten, Superstring Theory, Volume-I and II, Cambridge University Press, 2012. 2. Joseph Polchinski, String Theory: Volume 1, An Introduction to the Bosonic String (Cambridge Monographs on Mathematical Physics), Cambridge University Press, 2005. 3. John R. Gribbin, The search for superstrings, symmetry, and the theory of everything, Little, Brown, c1998. p. 212. 4. Steve Adams, A theory of everything. New scientist, v. 161, Feb. 20, 1999: 1–4. 5. Sunil Muhki, The theory of strings: An introduction. Current science, v. 77, Dec. 25, 1999: 1624–1634. 6. Michael J. Duff, The theory formerly known as strings. Scientific American, v. 278, Feb. 1998: 64–69. 7. Hans Christian Von Baeyer, World on a string. The sciences, v. 39, Sept./Oct. 1999: 10–13. 8. Brian Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, W. W. Norton, c1999. p. 448. 9. Brian Greene, The heart of matter. Natural history, v. 109, Feb. 2000: 80–83. 10. Michio Kaku, Hyperspace: A Scientific Odyssey through Parallel Universes, Time Warps, and the Tenth Dimension, Oxford University Press, 1994. p. 359.

Part III Mathematical Modeling in Renewable Energy

9

Analysis of RSM Method for Optimization of Ultrasound-Assisted KOH Catalyzed Biodiesel Production from Waste Cotton-Seed Cooking Oil Suvik Oza, Pravin Kodgire, and Surendra Singh Kachhwaha

CONTENTS 9.1 Introduction�������������������������������������������������������������������������������������������������� 133 9.2 Material and Methodology............................................................................ 135 9.2.1 Raw Materials.................................................................................... 135 9.2.2 Design of Experiment (DOE)............................................................ 135 9.2.3 Ultrasound-Assisted Transesterification Reaction............................. 136 9.2.4 Response Surface Methodology (RSM)............................................ 136 9.3 Results and Discussion.................................................................................. 138 9.3.1 RSM Statistical Analysis................................................................... 138 9.3.2 Effect of Different Parameters on Biodiesel Yield............................ 143 9.3.3 Optimization of the Process Parameters........................................... 144 9.3.4 Comparison of RSM Method of BBD and CCD Approaches........... 146 9.4 Conclusion..................................................................................................... 147 Acknowledgments................................................................................................... 147 References............................................................................................................... 147

9.1 INTRODUCTION The worldwide energy frameworks are profoundly subject to petroleum products. With rapid industrial and cosmopolitan development, the absolute global energy utilization is anticipated to enhance by 28% by the year 2040 [1]. Today, the most predominant assets for worldwide energy requirements are coal, flammable gas, and unrefined petroleum [1, 2]. These regular fuel sources are ceaselessly draining and non-sustainable. In this way, scientists are centered on the issues to discover a DOI: 10.1201/9781003159124-9

133

134

Applied Mathematical Modeling and Analysis in Renewable Energy

substitute source. For this, biodiesel helps in the insurance of climate because of its non-harmful, sustainable, and biodegradable nature as it produces fewer sulfur emanations and ozone-depleting substances. Biodiesel is easy to utilize and safe as compared to fuel diesel [3]. Biodiesel is produced using renewable natural plantderivatives such as vegetable oils that might be either edible or non-edible oils, animal flabs, and algae, as well as using waste vegetable oil from houses and restaurants. Biodiesel produced by the transesterification of triglycerides contain a 10% built-in oxygen fraction that supports complete combustion [2, 3]. For biodiesel production, the main aspect is to choose the best, easily available, and low-cost raw material. The second-generation feedstocks have resolved the problems associated within the first-generation biodiesel. As the biodiesel is made from edible oil, it will create food security problem [2]. The second-generation biodiesel products from non-edible oil plant sources are not utilized for human consumption. The Government of India took up a countrywide biodiesel mission program in 2003 to produce biodiesel from second-generation feedstock using nonagricultural land; however, this mission couldn’t succeed. Until this point of time, India has accomplished just 0.12% biodiesel blending [4]. It projected that 85 million liters of biodiesel is expected to be blended in 2019 as compared to 83 million liters blended a year ago. India’s National Bio-fuel Policy 2018 had specified a biodiesel blending objective of 5% by 2030 [5]. As per estimates, for India, waste cooking oil (WCO) can be seen as a conservative and cost-effective option, being a raw material with ease of availability [3]. The worldwide WCO market size was esteemed at $6,041.2 million (42,288.4 crores in India) for the year 2018. Producing biodiesel from WCO may save 21% of fossil energy as compared with unrefined petroleum (where 96% of saved energy would have been used for manufacture the petrodiesel) [6]. The investigations indicated that the utilization of mechanical stirring (MS) technique for biodiesel production requires a higher reaction time, a high amount of process variables, and larger activation energy alongside extended post-processing times. However, the use of process intensification (PI), especially ultrasonication (US) methods, is energy-efficient mostly due to low processing time, reduced steps for purification of product, and absence of the mass transfer limitation. Various PI-based biodiesel production techniques include membrane reactor, hydrodynamic cavitation (HC), US, microwave irradiation (MW), and the combination of US and MW. Among the above methods, US has an amazing process of interest [3, 7, 8]. There are some beneficial outcomes of US, which include production, development, and implosive breakdown of bubbles because of constant development and recompression in a fluid medium [3, 9]. During biodiesel production, US irradiation causes bubble cavitation close to the phase boundary between the methanol and triglyceride phase, thereby delivering an enormous number of micro-bubbles. A few bubbles remain steady following the above cycle, while others get exposed to brutal breakdown after growing to a specific bubble size [9]. Enormous energy is passed upon the incredibly unbalanced breakdown of cavitation bubbles that ranges around 20–40 kHz. In such emulsions, the interfacial area and mass exchange among the alcohol and oil phase increase. The homogeneous movement of reactants moves the equilibrium in a forward way by simultaneously giving physical and chemical influences in transesterification [3].

135

Analysis of RSM Method

Further, US gives physical blending to boost activation energy needed to initiate transesterification, which enhances the reaction rate. KOH catalyst is favored because it is readily available and has nominal cost. KOH catalyzed reaction has less process time, less utilization of catalyst, and is broadly utilized because of its ability to deliver the greatest yield under moderate temperature condition [10]. For optimization of process, researchers used various methods, i.e., response surface methodology (RSM), artificial neural network (ANN), and extreme learning machine (ELM); however, the mostly used method is RSM. The RSM method includes techniques, viz. box- Behnken design (BBD) and central composite design (CCD), which are widely used for optimization. The analysis of variance (ANOVA) is commonly adopted to develop a mathematical model through RSM techniques such as BBD and CCD. ANOVA is a statistical technique which is utilized to examine the impact of various components/terms (liner impact (xi)/quadratic (xi2)/interaction impact (xixj) terms) existing in the mathematical model. The Fisher test (t-test) in hypothesis testing can be used for the significance of each term. For the most part, 95% of the certainty stretch is chosen, and it might differ depending on the exactness needed in the forecast utilizing the mathematical model. Many researchers have utilized RSM and ANOVA methods to obtain results from examination. In the present chapter, the goal is to compare analyses obtained through the RSM coupled box-Behnken and central composite methods. Besides, it is furthermore critical to analyze the efficiency to reduce the resource accomplished through successful usage of these RSM techniques which can improve the viability of the process.

9.2  MATERIAL AND METHODOLOGY 9.2.1 Raw Materials WCCO is taken from neighborhood restaurants in Ahmedabad, Gujarat. Prior to its use in experimentation, this oil was filtered to remove the suspended food particles. Methyl alcohol (99% pure, A.R. grade) and KOH pellets (99% purity grade) were purchased from M/s Fisher Scientific, India.

9.2.2 Design of Experiment (DOE) The biodiesel production from WCCO using KOH catalyst condition was optimized using RSM methods. Table 9.1 provides the process parameters (independent variables) and the range of level for experimentation. TABLE 9.1 Independent Variables and Range of Levels Used for Experimentation Process Variables Methyl alcohol/oil molar ratio KOH wt% Temperature ºC

Levels for Variables –1 0 4.5 6 0.3 0.5 40 50

1 7.5 0.7 60

136

Applied Mathematical Modeling and Analysis in Renewable Energy

RSM-based BBD and CCD were used for studying the effects of process parameters on biodiesel yield. In the BBD approach, total 15 exponential runs were created, while in the CCD method, a total of 20 experimental runs were created. A steady reaction time of 10 min and range of three levels were chosen based on preliminary performed experimentation.

9.2.3 Ultrasound-Assisted Transesterification Reaction Programmable sonicator set up model no: VCX 500; M/s sonic vibracell, USA was utilized for the biodiesel production. Sonicator can vary power up to 500 W (amplitude range of 20–100%; for experimental runs, 50% amplitude is used) with a fixed frequency of 20 kHz. Three necks borosil rector (250 mL capacity) was utilized as a reaction vessel with a 50 mL oil reactant. Post reaction with a catalyst of KOH, the formation of two layers of the reaction mass was noticed. The top layer comprises unsaturated fatty acid methyl ester (FAME) with a trance of unconverted oil, and the lower thick brown-colored layer is of glycerol. Three-time washing of biodiesel with distilled (DI) water is done to remove trances of KOH, and is then dried to eliminate trances of dampness.

9.2.4 Response Surface Methodology (RSM) RSM is a collection of statistical mathematical methods beneficial for evolving, refining, and optimizing processes. The objective of the watchful design of experimentation is to enhance a response that is influenced by numerous self-governing parameters. Based on the controlled value of independent variables, the output is obtained from a well-designed regression analysis [11, 12]. The design of experiment (DOE) and RSM are widely adopted in the optimization of biodiesel production processes. DOE technique is valuable for acquiring the most extreme data from a negligible number of all-around arranged investigations by shifting at the same time all the cycle factors, while the RSM is an assortment of numerical and measurable apparatuses for building an experimental model associating result with the powerful process factors [13]. RSM has two models based on linearity or polynomial behavior: the first-order model and the second-order model [11, 13]. The regression analysis was performed on the experimental yield data to assess the response of biodiesel yield as y function (refer to equation 9.1) fitted for a quadratic polynomial equation, given by: n



y = B0 +

∑ i =1

n

Bi xi +

∑ i =1

n

Bii xi2 +

i −1

∑∑ B x x + ε (9.1) ij i

j

i

i =1 j =1

Where y is the predicted response value, Bi and Bij are regression coefficients obtained and can represent the linear, second-order, and interaction effect of x1, x2, x3, … while n is the number of independent variables and ε is the random error [12]. BBD and CCD are the types of design of RSM. BBD uses fewer design points than CCD; this makes it an expensive method to run with the same number of factors.

137

Analysis of RSM Method

The BBD has treatment combinations at the mid-level of the edge of the experiment space, and requires at least three continuous factors [14]. The BBD also ensures that all factors are not set to their high levels at the same time. BBD has less number of experiments compared to CCD. For the development of BBD, the number of experiments (N) is obtained by expression N = 2n (n – 1) + m; where n is the number of experiments and m denotes the number of center points [15]. Both ‘n’ and ‘m’ has a value of 3. Thus, BBD has 15 experimental runs for three levels and three factors as shown in Table 9.2. CCD technique is a full or partial factorial design method with the center point amplified with a gathering of the axial point that grants evaluation of non-linearity in the predicted model. For the developments of CCD, the number of experiment (N) is calculated by N = 2n + 2n + m, where n is the number of independent variables and m is the number of replicated central point [12, 16]. For the current system, n = 3 and m = 6. This forms a set of eight cube points, six axial points, and six repeated center data; thus total 20 experiments run (refer to Table 9.3), which are analyzed in a randomized order, where axial point, α =1.682. By means of Design-Expert® software (edition 11, stat-Ease Inc., USA), experimental yield data of both BBD (15 experimental runs) and CCD (20 experimental runs) were used as input to examine the optimized biodiesel. ANOVA was carried out on the experiment yield values to determine the coefficient of the secondorder polynomial model. Its results gave a relationship among the variations caused by experimental values and asserted suitability of the predicted model. This was

TABLE 9.2 RSM-Based DOE Runs for BBD Method and Their Respective Biodiesel Yield Values Run Order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Methyl Alcohol: Oil Molar Proportion (A) 6.00 6.00 4.50 4.50 7.50 6.00 6.00 4.50 7.50 6.00 4.50 7.50 6.00 6.00 7.50

KOH wt% (B) 0.70 0.50 0.30 0.70 0.50 0.30 0.50 0.50 0.70 0.70 0.50 0.50 0.30 0.50 0.30

Temperature °C (C) 60.00 50.00 50.00 50.00 60.00 40.00 50.00 60.00 50.00 40.00 40.00 40.00 60.00 50.00 50.00

Experiment Yield % 79.30 96.48 84.29 72.85 91.67 81.98 98.00 83.26 82.38 88.42 89.25 81.54 91.69 97.23 81.54

Predicted Yield % 79.11 97.20 84.45 74.15 91.94 82.09 97.20 82.03 82.14 87.34 88.90 82.70 92.69 97.20 82.70

138

Applied Mathematical Modeling and Analysis in Renewable Energy

TABLE 9.3 RSM-Based DOE Runs for CCD Method and Their Respective Biodiesel Yield Values Run Order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Methyl Alcohol: Oil Molar Proportion (A) 6.00 6.00 7.50 3.48 6.00 7.50 4.50 6.00 6.00 6.00 7.50 6.00 6.00 4.50 7.50 4.50 6.00 6.00 4.50 8.52

KOH wt% (B) 0.50 0.50 0.30 0.50 0.50 0.70 0.70 0.50 0.50 0.50 0.30 0.50 0.16 0.30 0.70 0.70 0.84 0.50 0.30 0.50

Temperature °C (C) 50.00 50.00 60.00 50.00 66.82 40.00 40.00 33.18 50.00 50.00 40.00 50.00 50.00 60.00 60.00 60.00 50.00 50.00 40.00 50.00

Experiment Yield % 97.23 97.20 86.64 73.17 90.10 79.36 79.43 88.11 98.00 97.20 67.98 97.20 75.17 82.86 79.19 63.14 68.16 96.48 80.32 76.29

Predicted Yield % 97.23 97.23 86.65 73.18 90.12 79.37 79.44 88.11 97.23 97.23 67.99 97.23 75.18 82.87 79.20 63.15 68.17 97.23 80.33 76.30

verified by analyzing the values of sum of the square root (SS), mean square (MS), p-value, F-test, and ‘lack of fit’ (LOF) test. The evolution of model was done by the coefficient of determination (R2), and the accuracy of the obtained quadratic polynomial model was also evaluated. The model is whether significant or not was checked by F-test.

9.3  RESULTS AND DISCUSSION 9.3.1 RSM Statistical Analysis The statistical optimization of the process parameters was performed using the Design-Expert® software (edition 11; Stat-Ease, Inc., USA) to maximize the biodiesel yield as per the process mentioned in section 9.4. Here, a second-order quadratic regression surface model was found to be the best fit for both data sets as per the DOE software. As mentioned earlier, for 15 factorial run points with three factors for BBD model (refer Table 9.2), and 20 factorial points in the CCD method (refer Table 9.3). The predicted regression equations are

Analysis of RSM Method

139

mentioned here for BBD method (refer to equation 9.2) and CCD method (refer to equation 9.3). 1. Regression Equation for BBD YBBD = −106.916 + 24.444 A + 271.621B + 2.486C + 10.233 AB + 0.269 AC − 2.354 BC − 3.531 A2 − 225.677 B 2 − 0.028C 2

(9.2)

2. Regression Equation for CCD YCCD = −107.196 + 24.476 A + 271.774 B + 2.492C + 10.225 AB + 0.269 AC − 2.354 BC − 3.534 A2 − 225.859 B 2 − 0.029C 2

(9.3)

Here A, B, and C are the encoded forms of methyl alcohol: oil molar ratio, KOH wt%, and temperature, respectively. AB, AC, and BC are the interaction effects parameters, and A2, B2, and C2 are the quadratic effects of the process variables. Both BBD and CCD methods predicted the maximum yield of 98% (refer to Tables 9.2 and 9.3). Table 9.4 shows the details of ANOVA parameters for BBD and CCD methods. As per Table 9.4, the F-values of 40.98 for BBD model and 2289.74 for CCD model were obtained with a corresponding p-value of less than 0.05 for both the models, indicating that they are significant [17]. The ‘lack of fit’ F-values for the BBD and CCD are founded 5.14 (refer Table 9.4 (a)) and 5.16 E-7 (refer Table 9.4 (b)), respectively, moreover the P-value for both model are greater than 0.05 also it’s justified that the lack of fit terms is no significant for this experiment. This observation overall designates that the investigational data fitted satisfactorily with the quadratic model. The higher value of the coefficient of determination (R2) indicates the accuracy and developed the model is the best fit with the predicted model data and additional, terms adjusted coefficient of determination (Adj.R2) was assessed which indicates the model fitness. For the Box-Behnken design model, the R2 and Adj.R2 values were obtained 98.66% and 96.25%, respectively, also both term’s differences are less than 0.4. When, in the CCD model, the R2 and Adj.R2 values were obtained 99.95% and 99.91% respectively and both terms between differences in very less. The adequate precision is 19.90 and 141.67 for BBD and CCD models, respectively, which is greater than 4. As per the analysis of the developed quadratic model, as seen in Table 9.4, the significant model terms found in BBD model are B, AB, AC, BC, A2, B2, and C2, and the linear terms of A and C are insignificant in the model, while A, B, C, AB, AC, BC, A2, B2, and C2 are the significant terms in CCD-based model equation. The R2 value is higher in CCD model as compared to BBD model. Figure 9.1 shows a plot of predicted vs. actual biodiesel yield for (a) BBD model and (b) CCD model. The biodiesel yield values are stereotypically distributed along with the reference straight line of 45°. It is indicate that the developed regression model is best fit in the model and the obtained the response is correct.

140

TABLE 9.4 Result of the Analysis of Variance (ANOVA) for the Biodiesel Production Using (a) BBD and (b) CCD Method (a) Box-Behnken Design (BBD) F-value 40.98 3.48 17.03 1.39 18.75 32.31 44.08 115.89 149.63 15.04 5.14

p-value 0.0004 0.1212 0.0091 0.2913 0.0075 0.0023 0.0012 0.0001 < 0.0001 0.0117 0.1673

Sum of df Squares Mean square 9 2384.91 264.99 1 11.75 11.75 1 59.35 59.35 1 4.79 4.79 1 75.28 75.28 1 129.93 129.93 1 177.28 177.28 1 911 911 1 1176.25 1176.25 1 118.57 118.57 5 5.97E-06 1.19E-06 5 1.16 0.2315 10 1.16 0.1157 19 2386.06 R2 = 99.95%; Adj.R2 = 99.91%; Adequate Precision = 141.67

F-value 2289.74 101.52 512.82 41.38 650.45 1122.69 1531.89 7871.82 10163.8 1024.57 5.16E-06

p-value < 0.0001 < 0.0001 < 0.0001 < 0.0001 < 0.0001 < 0.0001 < 0.0001 < 0.0001 < 0.0001 < 0.0001 1

Applied Mathematical Modeling and Analysis in Renewable Energy

Source of Sum of Variation df Squares Mean square Model 9 741.62 82.4 A 1 6.99 6.99 B 1 34.24 34.24 C 1 2.8 2.8 AB 1 37.7 37.7 AC 1 64.96 64.96 BC 1 88.64 88.64 A2 1 233.05 233.05 B2 1 300.88 300.88 C2 1 30.25 30.25 Lack of Fit 3 8.9 2.97 Pure Error 2 1.16 0.5776 Residual 5 10.05 2.01 Total 14 751.67 R2 = 98.66%; Adj.R2 = 96.25%; Adequate Precision = 19.90

(b) Central Composite Design (CCD)

Analysis of RSM Method

141

FIGURE 9.1  Predicted vs. actual data plot of biodiesel yield for (a) BBD and (b) CCD methods.

Figure 9.2 shows the plots of normal percentage probability vs. residual data of biodiesel yield for (a) BBD and (b) CCD methods. It exhibits a good connection between externally studentized residuals and their normal percentage probability rank position data. In Fig. 9.2(b), mostly all the data points are arranged in straight line as the externally studentized residual data are almost zero (i.e., error is 0.04). Figure 9.3(a) and (b) are response plots of externally studentized residuals versus plotted against the predicted data of yield for BBD model and CCD model, respectively. In these plots, all the data points are randomly scattered to the reference line. It generally indicates that the regression model shows amazing adequacy of the biodiesel process. Figure 9.4 (a) and (b) shows the perturbation plots of process variables affecting biodiesel yield for both BBD and CCD methods. Perturbation graphs aid in

FIGURE 9.2  Normal probability vs. residual plots of biodiesel yield for (a) BBD and (b) CCD methods.

142

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 9.3  Externally studentized residual vs. predicted biodiesel yield for (a) BBD and (b) CCD methods.

understanding the affectability of individual factor and their variations while keeping the excess elements consistent. The nature of factor which has a more inclined slope when contrasted with that of a planar one indicates that it affects the biodiesel yield significantly [8]. In BBD model (refer to Fig. 9.4(a)), process parameter A has the perturbation effect influence (exhibiting steep slope) between lower level (–1) and intermediary level (0), followed by the process variables B and C. Among intermediary and higher levels, parameter B presented overwhelming effects trailed by variables A and C. A comparable approach exemplified in Fig. 9.4 for CCD method, process parameter A has sticking effect (i.e., steepest slope) between lower level (–1) and middle level (0), followed by process variables B and C. Amongst intermediary and higher level, variable B displays governing effect trailed by parameters A and C.

FIGURE 9.4  Perturbation plots for process variables for (a) BBD and (b) CCD methods.

Analysis of RSM Method

143

9.3.2 Effect of Different Parameters on Biodiesel Yield The analysis of effect of singular process variable on WCCO-biodiesel yield offers detailed understanding; however, it is incapable of giving an all-encompassing perspective concerning the variation in yield. To discover the impact among demonstrated variables, 3-D surfaces along with 2-D contour plot need to be inspected. Figure 9.5(a) exhibits an interaction plot of WCCO-biodiesel yield for molar proportion and catalyst (KOH) wt% for BBD (@ keeping temperature constant at the intermediary level). The 90%-yield profile zone exists for a catalyst amount between 0.37% and 0.65%, and molar proportion in the range of 5.2–6.9 at a fixed temperature of 50 ºC. The residual region parts in the biodiesel yield decrease with an increase in the loading of the molar proportion and KOH amount. Figure 9.5(b) shows an interaction plot of WCCO-biodiesel yield concerning molar proportion and catalyst (KOH) wt% for CCD (@ keeping temperature constant at the intermediary level). Yield profile region more than 90% exists for a catalyst (KOH) wt% in the range of 0.37%–0.65% and molar proportion in the range of 4.6–7.1 at a constant temperature of 50 °C. Figure 9.6 (a) presents the conjoint effect of process temperature and molar ratio for BBD. A yield profile region higher than 95% exists for a molar ratio in the range of 5.3–7.1, and temperature in the range of 42 ºC–60 ºC. Figure 9.6(b) displays the interaction effect of process temperature and molar proportion for CCD. Yield profile region more than 90% exists for a molar proportion in the range of 0.47–7.5, and temperature in the range of 40 ºC–60 ºC. Figure 9.7(a) shows the interaction effect of reaction temperature and catalyst (KOH) amount for BBD (molar ratio kept constant at the intermediate level). The realistic region for maximum yield (≥ 90%) is 0.38%–0.64% range of the catalyst wt % and 40 ºC–60 ºC temperature range. Figure 9.7(b) shows the interaction effect of process temperature and catalyst (KOH) amount for CCD (molar ratio kept constant at the intermediate level). The realistic region for higher yield (≥ 90%) is 0.30%–0.63% range of the KOH wt% and

FIGURE 9.5  3-D surface plot and contour plot of interaction effect (molar ratio and catalyst (KOH) amount) on biodiesel yield for (a) BBD and (b) CCD methods.

144

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 9.6  3-D surface plot and contour plot of interaction effect (molar ratio and temperature) on biodiesel yield for (a) BBD and (b) CCD methods.

40 ºC–60 ºC temperature range. It is found that biodiesel yield increases with the increase in the operating factors until a specific value beyond which the biodiesel yield declines. The decline in the biodiesel yield is due to reversible response of transesterification reaction [3].

9.3.3 Optimization of the Process Parameters The RSM mathematical optimization procedure was used for investigating the ideal response conditions within the scope of the control factors by contemplating the standard blunder (StdErr) that occurred in the model [2]. Figure 9.8 provides the optimal reaction condition for BBD and CCD methods. The best quality capacity was applied to acquaint a powerful strategy with acquiring optimum response conditions. As

FIGURE 9.7  3-D surface plot and contour plot of interaction effect (catalyst (KOH) amount and temperature) on biodiesel yield for (a) BBD and (b) CCD methods.

Analysis of RSM Method

145

FIGURE 9.8  The optimal reaction condition for (a) BBD and (b) CCD methods.

indicated by the breaking point basis for biodiesel yield optimization, the secondorder polynomial equation, i.e., equations (9.1) and (9.2) were applied to optimize the operating process conditions by utilizing mathematical RSM optimization in BBD and CCD approaches. By utilizing the Design-Expert® software, various arrangements of various optimal operating factors and corresponding biodiesel yields were created. While using BBD, the optimal solution of parameters were found to be 6.160, 0.464%, 53.274 ºC, respectively, for methyl alcohol: oil molar ratio, KOH wt%, and temperature; and corresponding to this optimal condition, the predicted yield of 97.57% was observed, which is shown in Fig. 9.8(a). For this optimal condition, the value of desirability function was 0.911, which was nearest to 1. A comparison of the predicted yield (97.57%) with the experimental yield (98%) showed that the error was less than 1%. Similarly, for CCD method, the optimal solution of parameters were found to be 6.160, 0.463%, 53.264 ºC, respectively, for methyl alcohol: oil molar ratio, KOH wt%, and temperature;

146

Applied Mathematical Modeling and Analysis in Renewable Energy

TABLE 9.5 Comparison of Optimum Condition Obtained from BBD and CCD Methods Optimum Condition Molar ratio Catalyst amount (w/w %) Temperature Predicated yield

BBD 6.160:1 0.464 53.274 97.572

CCD 6.159:1 0.463 53.264 97.555

Error 0.001 0.001 0.01 0.017

% Error 0.02% 0.22% 0.02% 0.02%

and corresponding to this optimal condition, the predicted yield of 97.56% was observed as shown in Fig. 9.8(b). The desirability got for optimal condition was around 0.934. The error in predicted yield (97.56%) with the experimental yield (98%) was less than 1%. Table 9.5 shows the optimum conditions for both BBD and CCD, and also shows the percentage error between both approaches. The percentage error of predicated yield between BBD and CCD is less than 1%. The optimum conditions give the best yield and will be beneficial for biodiesel production.

9.3.4  Comparison of RSM Method of BBD and CCD Approaches In this study, both analytical tools of RSM, i.e., BBD and CCD approaches have given the outstanding outputs for optimization of process variables of transesterification process. Examination of the outcomes indicated that the results obtained from both BBD and CCD models of RSM statistical procedures were in agreement with one another. However, the sequence of important parameters is not the same in both approaches. In BBD, the sequence of parameters are BB, AA, BC, AC, AB, B, CC, A, and C, while in CCD, the sequence is BB, AA, BC, AC, CC, AB, B, A, and C. In both methods, the regression equation in terms of mathematical indicator is the same as shown in equations (9.2) and (9.3). The R2 value is higher in CCD (99.95%) as compared to that in BBD (98.66%). The higher values of R2 terms indicate that the optimization of process parameters is accurate. It means that the applied CCD method has given good outcomes compared to BBD. According to the predicted vs. actual graph the CCD model is give good response compare to the BBD model because of all the data points are very closely fit to the reference line. Its means that difference between the actual and predicted yield value is very less. The desirability value of 1 is indicated the optimization of the process gives accurate outcomes in the process. Here, the desirability function values obtained are 0.934 for the BBD method and 0.911 for the CCD method. RSM indicates quantifiable centrality of all potential blends of interaction and quadratic term parameters dependent on a 95% certainty interval, and this can help with perceiving the future course of an ideal reaction. Furthermore, the alluring work in RSM can, without a very remarkable stretch, choose the ideal operating process condition inside the extent of levels of components [2, 17].

Analysis of RSM Method

147

9.4 CONCLUSION In this chapter, the process parameters were optimized using BBD and CCD method of the RSM technique. Both BBD and CCD agreed that KOH wt% amount was the most influencing factor in the biodiesel production from WCCO catalyzed by KOH. The optimum WCCO yield of 98% was reached at a methyl alcohol/oil molar ratio of 6:1, KOH amount of 0.50% w/w, and process temperature of 50 ºC for BBD and CCD. These two techniques have given a decent expectation of the biodiesel yield with respect to molar proportion, catalyst amount, and temperature. According to this study, CCD has higher R2 values as compared to the other method of BBD, and also, CCD gives better outcomes of the other optimal parameters as compared to the outcomes of the parameters of BBD method.

ACKNOWLEDGMENTS The authors desire to recognize Gujarat Energy Development Agency (GEDA), Gujarat, India, for supporting this research work, and the Centre for Biofuels and Bioenergy Studies (CBBS) in Pandit Deendayal Energy University (PDEU) for providing laboratory facilities for this project.

REFERENCES

1. S. Ardabili, A. Mosavi, and A. R. Várkonyi-Kóczy, “Systematic review of deep learning and machine learning models in biofuels research,” Lect. Notes Networks Syst., vol. 101, pp. 19–32, August 2020, doi: 10.1007/978-3-030-36841-8_2. 2. Y. H. Tan, M. O. Abdullah, C. Nolasco-Hipolito, and N. S. Ahmad Zauzi, “Application of RSM and Taguchi methods for optimizing the transesterification of waste cooking oil catalyzed by solid ostrich and chicken-eggshell derived CaO,” Renew. Energy, vol. 114, no. PB, pp. 437–447, 2017, doi: 10.1016/j.renene.2017.07.024. 3. A. Sharma, P. Kodgire, and S. S. Kachhwaha, “Investigation of ultrasound-assisted KOH and CaO catalyzed transesterification for biodiesel production from waste cotton-seed cooking oil: Process optimization and conversion rate evaluation,” J. Clean. Prod., vol. 259, p. 120982, 2020, doi: 10.1016/j.jclepro.2020.120982. 4. S. X. Tan, S. Lim, H. C. Ong, and Y. L. Pang, “State of the art review on development of ultrasound-assisted catalytic transesterification process for biodiesel production,” Fuel, vol. 235, pp. 886–907, July 2019, doi: 10.1016/j.fuel.2018.08.021. 5. M. V. Rodionova, R. S. Poudyal, I. Tiwari, R. A. Voloshin, S. K. Zharmukhamedov, H. G. Nam, B. K. Zayadan, B. D. Bruce, H. J. M. Hou, S. I. Allakhverdiev, “Biofuel production: Challenges and opportunities,” Int. J. Hydrogen Energy, vol. 42, no. 12, pp. 8450–8461, 2017, doi: 10.1016/j.ijhydene.2016.11.125. 6. M. G. Kulkarni and A. K. Dalai, “Waste cooking oil – An economical source for biodiesel: A review,” Ind. Eng. Chem. Res., vol. 45, no. 9, pp. 2901–2913, 2006, doi: 10.1021/ie0510526. 7. M. Aghbashlo, S. Hosseinpour, M. Tabatabaei, and M. Mojarab Soufiyan, “Multiobjective exergetic and technical optimization of a piezoelectric ultrasonic reactor applied to synthesize biodiesel from waste cooking oil (WCO) using soft computing techniques,” Fuel, vol. 235, July 2018, pp. 100–112, 2019, doi: 10.1016/ j.fuel.2018.07.095.

148













Applied Mathematical Modeling and Analysis in Renewable Energy

8. M. Agarwal, G. Chauhan, S. P. Chaurasia, and K. Singh, “Study of catalytic behavior of KOH as homogeneous and heterogeneous catalyst for biodiesel production,” J. Taiwan Inst. Chem. Eng., vol. 43, no. 1, pp. 89–94, 2012, doi: 10.1016/j.jtice.2011.06.003. 9. P. A. Oliveira, R. M. Baesso, G. C. Moraes, A. V. Alvarenga, and R. P. B. Costa-Félix, “Ultrasound Methods for Biodiesel Production and Analysis,” Biofuels – State of Development, Intech Open, vol. 7, 2018, doi: 10.5772/intechopen.74303. 10. R. V. Quah, Y. H. Tan, N. M. Mubarak, M. Khalid, E. C. Abdullah, and C. NolascoHipolito, “An overview of biodiesel production using recyclable biomass and nonbiomass derived magnetic catalysts,” J. Environ. Chem. Eng., vol. 7, no. 4, p. 103219, 2019, doi: 10.1016/j.jece.2019.103219. 11. E. Ambrosio, Diego L. Lucca, Maicon H. B. Garcia, Maísa T. F. de Souza, Thábata K. F. de S. Freitas, Renata P. de Souza, Jesuí V. Visentainer, and Juliana C. Garcia, “Optimization of photocatalytic degradation of biodiesel using TiO2/H2O2 by experimental design,” Sci. Total Environ., vol. 581–582, pp. 1–9, 2017, doi: 10.1016/ j.scitotenv.2016.11.177. 12. A. I. Khuri, “A general overview of response surface methodology,” Biometrics Biostat. Int. J., vol. 5, no. 3, pp. 87–93, 2017, doi: 10.15406/bbij.2017.05.00133. 13. R. K. Elango, K. Sathiasivan, C. Muthukumaran, V. Thangavelu, M. Rajesh, and K. Tamilarasan, “Transesterification of castor oil for biodiesel production: Process optimization and characterization,” Microchem. J., vol. 145, pp. 1162–1168, 2019, doi: 10.1016/j.microc.2018.12.039. 14. S. L. C. Ferreira, R. E. Brunsb, H. S. Ferreira a, G. D. Matos a, J. M. David a, G. C. Brand~ao a, E. G. P. da Silva a, L. A. Portugal a, P. S. dos Reis c,a, A. S. Souza a, and W. N. L. dos Santos, “Box-Behnken design: An alternative for the optimization of analytical methods,” Anal. Chim. Acta, vol. 597, no. 2, pp. 179–186, 2007, doi: 10.1016/ j.aca.2007.07.011. 15. E. Soria-Figueroa, V. Y. Mena-Cervantes, M. García-Solares, R. HernándezAltamirano, and J. Vazquez-Arenas, “Statistical optimization of biodiesel production from waste cooking oil using CaO as catalyst in a Robinson-Mahoney type reactor,” Fuel, vol. 282, no. 186, p. 118853, 2020, doi: 10.1016/j.fuel.2020.118853. 16. M. Tripathi, A. Bhatnagar, N. M. Mubarak, J. N. Sahu, and P. Ganesan, “RSM optimization of microwave pyrolysis parameters to produce OPS char with high yield and large BET surface area,” Fuel, vol. 277, December 2019, p. 118184, 2020, doi: 10.1016/ j.fuel.2020.118184. 17. V. B. Veljković, A. V. Veličković, J. M. Avramović, and O. S. Stamenković, “Modeling of biodiesel production: Performance comparison of Box–Behnken, face central composite and full factorial design,” Chinese J. Chem. Eng., vol. 27, no. 7, pp. 1690–1698, 2019, doi: 10.1016/j.cjche.2018.08.002.

10

Energy Data Analysis of an Educational Institution in India Chandana Sasidharan and Aman Aggarwal

CONTENTS 10.1 Introduction................................................................................................. 149 10.2 Literature Review........................................................................................ 150 10.3 Developing an Energy Consumption Baseline............................................ 151 10.3.1 Total Energy Consumption Baseline............................................ 151 10.3.2 Energy Baselines for All the Load Centers.................................. 152 10.3.3 Identifying the Major Load Centers............................................. 154 10.4 Analysis of Energy Bill............................................................................... 156 10.5 Savings From Balancing the System........................................................... 158 10.6 Savings From Air Conditioning.................................................................. 159 10.7 Conclusion................................................................................................... 159 Acknowledgments................................................................................................... 160 References............................................................................................................... 160

10.1 INTRODUCTION Over the past decade, the energy consumption in India’s buildings sector grew at a rate of 8% as per the energy demand per year [1]. In the next decade, the growth in India would be double the global average [2]. Educational institutions have been an interesting example of building’s electricity usage as energy consumption trends are dependent on the occupancy pattern [3]. Savings in energy bill is beneficial for educational institutions as energy is typically the second-highest reported expense [4]. Sustainable energy savings can be achieved by focusing on energy conversation measures. The National Buildings Energy Efficiency Program in India aims to minimize the buildings’ energy consumption and contribute to the grid demand reduction. Analysis of the energy performance of buildings is critical to identify the strategies which can reduce electricity consumption such as retrofitting. Energy audit is the first step to identify the target area for retrofitting such as lighting and cooling. Typically, energy audits are performed with a minimum period for measurement of building energy data and using historical data available in the form of energy bills and operator logbooks. The short-term data and long-term historical data are analyzed to understand the energy consumption pattern. With smart metering, it is possible to increase the ease of data collection and improve the quality of energy DOI: 10.1201/9781003159124-10

149

150

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 10.1  Block diagram showing major smart metering points.

data analysis. In this chapter, the electricity consumption data of TERI School of Advanced Studies (TERI SAS) recorded by smart meters and the historical energy consumption data are studied. The smart meters have been installed in the outgoing circuits of all the major loads in TERI SAS, as shown in Fig. 10.1. The study combines smart meter data, historical data, and ground surveys to undertake energy consumption analysis. The goal is to achieve an energy signature for each load center. Potential savings from modifications to the distribution system by strategies such as balancing and time delaying the air conditioning (AC) operations are also discussed in detail.

10.2  LITERATURE REVIEW Modeling energy consumption in buildings is a complicated exercise. The spatial and temporal resolution of data is an essential factor when studying the energy demand of buildings. Modeling the systems is an important option to study the energy consumption patterns of institutional buildings. Black-box models are ideal in this case, as the rationale behind the interrelationship between the data gathered can be determined [5]. The most crucial step for building energy simulations is data collection, and smart metering is convenient in data collection. Availability of extensive data can help in identifying the energy consumption pattern of a building. The method allows developing monitoring strategies to detect anomalies and take appropriate remedial steps by identifying the reasons [6]. Smart energy meters with advanced metering infrastructure allows real-time monitoring of load centers in buildings. Monitoring of energy data at major load centers helps in better modeling and analysis. This opens a pathway for the development of customized energy saving advice to customers. Better decision-making is possible by applying tools, which will help in visualizing and benchmarking electricity use [7].

Energy Data Analysis of an Educational Institution in India

151

The initial step in audits is to establish the baseline of energy consumption, which would serve as the benchmark to vet any corrective action [8]. A study for building energy in Portugal identified that focusing on lighting systems can result in significant energy saving [9]. Another study on educational buildings in Romania suggests a method to identify the focus areas by studying energy performance [10]. In another study undertaken in Argentina for school buildings, the key indicators used for energy efficiency are the energy consumption per unit area, energy used for heating, and cost of energy [11]. Internet of Things (IoT)-based energy analysis for a group of buildings showed that it is possible to inculcate a behavioral change toward energy efficiency by raising awareness among energy users in educational institutions [12]. Researchers have found that data is the key to encourage decision-makers to tap into the available potential for energy improvements in educational buildings [13]. Energy audit of a Russian educational institution was able to identify specific energy improvement opportunities based on the benchmarked class of energy efficiency [14]. Researchers have also highlighted that there is no significant difference between non-LEED certified buildings and certified buildings [15]. Another study on Finnish educational buildings has found that there is no clear correlation between the age of a building and electricity use [16]. To the best of the knowledge of authors, there are hardly any studies focusing on India’s educational institutions, which is a sizeable share of buildings in India. The chapter intends to develop energy signature for institution and major load centers, which can be used to develop building energy models. The chapter also looks into the possible investment opportunities to improve energy efficiency in the building.

10.3  DEVELOPING AN ENERGY CONSUMPTION BASELINE 10.3.1 Total Energy Consumption Baseline The base year considered for the energy consumption analysis of TERI SAS is 2016. M/s Zenatix installed smart meters at major load centers on the campus in late 2015. The consumption trend generated from the smart meter data for the year 2016 is presented in Fig. 10.2. The monthly average electricity consumption is around 1700 kWh; the recorded minimum and maximum daily consumptions reported are 614.5 kWh and 3068.5 kWh, respectively. A study of the consumption trend in the base year in Fig. 10.2 shows that the monthly consumption for the first quarter of the year is around 1000 kWh in a day. The consumption doubles in May and increases further for the next two summer months. In July, the consumption decreases marginally, and in August, the consumption increases again. For the remainder of the year, the energy consumption decreases to 1000 kWh per day. The monthly averages for benchmarking are available in Fig. 10.3. The maximum consumption is recorded in summer and, in particular, in May, the daily average increases to 2425 kWh. The consumption remains above 2300 kWh from June to September. Compared with the cooling load in summer, heating load in winter months is not high. Hence, it is not surprising that the energy use in the winter months is half of the summer average.

152

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 10.2  Energy consumption trend for the base year.

10.3.2 Energy Baselines for All the Load Centers The next step in the study is to develop energy baselines for all the load centers. Four medium voltage panels operate at 415 V designed to cater to the lighting and plug-in load requirements. The energy baseline for all these panels is shown in Fig. 10.4. Out of the four panels, MVP 2 and MVP 2-1 supply power to the academic and administration buildings in the campus. These panels are designed to share the load in 1:2 ratio. The overall consumption pattern of TERI SAS can be correlated with the consumption pattern of MVP 2-1. The major noticeable difference between the energy signatures is that there is a constant base load on the MVP 2-1 year around. MVP 2 has less number of AC loads than MVP 2-1; hence it has a distinct energy signature. The energy pattern for the panel supplying hostel block is also

FIGURE 10.3  Monthly average consumption in the base year.

Energy Data Analysis of an Educational Institution in India

FIGURE 10.4  Consumption trend of MVPs.

153

154

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 10.5  Consumption trend of AC panels.

distinct as the major loads occur in winter due to additional water heating loads. The cafeteria block maintains a low consumption, which can be correlated with low occupancy. The peak in October and November 2015 (refer to Fig. 10.5) was due to construction activities in the cafeteria block. The consumption trend of the five AC supply panels shown in Fig. 10.5 reinforces the inference that AC constitutes major load of the campus. The consumption trend of the panels can be correlated to the overall energy consumption trend. All of them have peak energy consumption in summer with a noticeable fall in July. The AC panel for hostel operates only during the March–October time period. An interesting observation is that the consumption of that did not really fall to zero in winter. This shows that there are some loads that were connected to the AC panels catering to heating and other loads. Also, the AC panel of the cafeteria block appears as an anomaly, and the ground survey showed that the panel was supplying power to loads other than AC year-round. The energy consumption pattern for remaining feeders at 415 V level shown in Fig. 10.6 shows that that the mobile towers installed on the roof of TERI SAS form about 10% of the energy consumption. The office block panel supplies a commercial area within the campus, but this space was unoccupied during the study. The load on the AC panel of the office block was due to mobile tower. Similarly, the load on the air washer unit panel is also due to mobile tower. The mobile towers alone add a constant load of 7.2 kW, which translated to 63 MWh annually. The common service panel provides supply to the construction and maintenance activity that happens in the basement. The sump motor pumps water from the sump in basement to the water tanks of roof.

10.3.3 Identifying the Major Load Centers The study identifies the major load centers in the institution. As highlighted in Fig. 10.7, the Medium Voltage Panel (MVP 2-1), which caters to the bulk of the load

Energy Data Analysis of an Educational Institution in India

155

FIGURE 10.6  Consumption trend for other panels.

in the academic block and the administration block, is the major load center. The annual energy consumption for MVP 2-1 is 257 MWh, which constitutes approximately 40% of the total energy consumption of TERI SAS. This panel supplies power to a significant number of cassette-type AC units. These AC units have been installed recently in classrooms, laboratories, and staff rooms where the original centralized AC system is not catering. The AC units are the major downstream loads that are responsible for the lion’s share of energy consumption of this panel. The six medium voltage panels supplying power to the centralized AC system are the remaining major load centers. The air handling unit (AHU) panel supplying power to the AC system in the biotechnology lab and the hostel block records the next highest annual consumption of 68.4 MWh and 60 kWh, respectively. These are closely followed by the AHU systems for the lecture hall and academic block. The top five load centers altogether account for more than 75% of the annual energy

FIGURE 10.7  Load center-wise energy consumption.

156

Applied Mathematical Modeling and Analysis in Renewable Energy

consumption. However, when the consumptions of the second, third, fourth, and fifth major load centers are put together, the total is only 220 MWh. This is because the load centers supply both individual AC units apart from the centralized AC systems.

10.4  ANALYSIS OF ENERGY BILL TERI SAS receives power at 11 kV voltage, which is then stepped down to 415 by a 1600 kVA transformer installed within the campus. The meters used for energy billing are installed upstream of the transformer at the 11 kV level. The smart meters are installed in the low tension (LT)/voltage panel at 415 V. A comparison of the meter readings is done as part of energy bill analysis for the recorded maximum demand, power factor, and billed energy. The maximum demand and the power factor recorded in the bill by the electricity distribution company matches the 15-minute interval data recorded by the smart meters. However, there is a substantial difference between the energy data recorded by the upstream and downstream meters. Hence, further detailed investigation is performed. On analyzing the difference between the high tension (HT)/voltage and LT side meters (Fig. 10.8), it is observed that the difference is not constant. The difference between the energy meter readings varies in proportion to the average daily energy consumption of the month. The smart meter infrastructure confirms that there are chances of measurement errors in the measurement of low-cost meters. Accounting for the error, it is observed that the trend lines of average consumption follow each other. As the HT side and LT side are interconnected via a transformer, supplementary investigation is undertaken to understand transformer losses. The transformer’s loading is studied for the highest and lowest consumption using smart meter data (Fig. 10.9). Maximum loading happens in summer due to AC load. The highest energy consumption happened in May 2017, and the highest load drawn was less than 12% of the transformer rated capacity. A contributing factor for the low transformer load is solar power generated from the rooftop plant. The minimum loading, which happened on February 17, was 2% of the rated transformer capacity.

FIGURE 10.8  The plot between average consumption and difference in energy readings.

Energy Data Analysis of an Educational Institution in India

157

FIGURE 10.9  Maximum and minimum transformer loading.

The consumption in winter months is less as the AC system is off, and the rooftop solar is performing well. The transformer is heavily oversized for the TERI SAS. Hence, the no-load losses of the transformer should be a factor of consideration for energy efficiency improvement. A curve fitting (Fig. 10.10) between the LT side smart meter and HT side meter readings shows a good linear fit with an R2 value of 0.99. The intercept of line accounts for no-load losses, which are to the tune of 1525 kWh in a month. This is not a small number, as annually, this accounts for an expenditure of INR 1.53 lacs

FIGURE 10.10  Curve fitting between HT and LT side data.

158

Applied Mathematical Modeling and Analysis in Renewable Energy

in a year at the energy rate of INR 8.4 per unit. The contracted kilo volt ampere (KVA) demand for TERI SAS has been brought down to 680 KVA from 1100 KVA after installation of solar rooftop. The existing transformer has a rated capacity for more than double the sanctioned demand. The study recommends performing financial feasibility analysis for transformer replacement of lower capacity and energyefficient transformer.

10.5  SAVINGS FROM BALANCING THE SYSTEM The study has also looked into the unbalances present in the system. A significant unbalance is observed for the Y phase current (Fig. 10.11) compared to the other phases. It is interesting to note that this presents a complex mathematical challenge as the unbalance in Y is counterbalanced by the unbalance in R phase. The B phase current is very close to the resultant value. The voltage unbalance of the system is quite low, less than 2% in most cases. The unbalance in the power factor is also observed in Y phase, where generally, during mid-day, a dip in power factor is observed. The ground survey revealed this could be correlated with the starting of the sump motor. It should be noted that TERI has a capacitor bank for power factor correction. However, all the capacitors are connected in three-phase. The remedial action to correct the power factor is to perform load balancing. In case the unbalance in the Y phase continues, it is possible to add an additional single phase capacitor for Y phase if the unbalance still prevails after the load balancing exercise is performed. The unbalance in current translates to a significant unbalance in Y phase with respect to power also. Averaging the unbalance present and if load balancing is performed, a saving of 3.4 lacs can be achieved annually.

FIGURE 10.11  Unbalance in power system.

Energy Data Analysis of an Educational Institution in India

159

FIGURE 10.12  Savings in air conditioning.

10.6  SAVINGS FROM AIR CONDITIONING The load of TERI SAS increases as early as 7 AM (Fig. 10.12) though the classes begin only at 8:30 AM. Staff and students start to arrive on the campus by 8 AM. There exists a practice to switch on the AC systems as early as 7 AM in most areas. If the time of switching on of AC units is shifted to 8 AM or 8:30 AM, the savings possible are to the tune of INR 2.4–3.2 lacs in a year. Similarly, ensuring all AC is off at 5 PM helps to save INR 0.7 lacs a year. A simple behavioral change can lead to energy savings and energy bill reduction. Though temperature sensors are available in the reception area, it is not linked to the control of refrigeration system. During the ground survey, the electrical maintenance team also pointed out that many of the centralized AC systems are obsolete, and their temperature cut-off function does not work properly. For the unit AC systems, as a large number of cassette AC systems get switched on by a single remote control inside a classroom, the AC provided is more than required. The students also observed that the classrooms are either too cold or uncomfortable. An increase in one degree of cooling temperature can save 3–5% of the total consumption as per the ACEEE study. For TERI SAS, the estimated Air Conditioning (AC) energy consumption for a year is 340 MWh. If 5% of the energy is saved, the energy expenditure savings are to the tune of INR 1.27 lacs per year. Currently, though the setting of cut-off temperature is possible, the effectiveness of the cut-off is questionable. The long-term solution is to provide a smart monitoring and control system for the AC system in TERI SAS.

10.7 CONCLUSION The chapter includes the details of a study, which demonstrates that smart, metering data can be effectively combined with historical data. The energy baselines developed during the study for educational institutes can help develop tools in benchmarking and detect anomalies in the system for preventive and corrective actions

160

Applied Mathematical Modeling and Analysis in Renewable Energy

in the future. With the help of a ground survey, it is possible to identify the major load centers. The study finds that AC is the primary load of the educational institution. The study also identifies three types of important interventions for energy efficiency: behavioral efficiency improvement, no investment system re-configuration, and an equipment replacement opportunity. The study highlights that savings can be achieved by controlling the on/off time and temperature setting. The study helps in identifying the underlying oversizing problem of the transformer, and recommends replacement with an efficient model of the right capacity. The study also throws light on the savings from balancing the system. However, to avail the best possible energy savings from a real-time monitoring system, the control element also needs to be developed. A smart energy management system that can automatically generate energy savings advice and perform demandside management can be developed using the study findings. This requires modeling of the building energy patterns, which is identified as a future research area. The major challenge experienced with the metering data is frequent data loss due to loss of connectivity. The foolproof design of smart monitoring systems should include data loggers for storage of data to prevent data loss.

ACKNOWLEDGMENTS The authors would like to acknowledge the support received from the management of TERI School of Advanced Studies. The authors also thank M/s Zenatix for the smart metering data. They also convey their sincerest gratitude to faculty guide Mr. Sapan Thapar.

REFERENCES

1. International Energy Agency: India Energy Outlook. World Energy Outlook special report. (2015). 2. US Energy Information Administration: International Energy Outlook. (2017). 3. Allab, Y., Pellegrino, M., Guo, X., Nefzaoui E., and Kindinis, A., Energy and comfort assessment in educational building: Case study in a French university campus. Energy and Buildings 143, 202–219 (2017). 4. Gul, M.S., and Patidar, S., Understanding the energy consumption and occupancy of a multi-purpose academic building. Energy and Buildings 87, 155–165 (2015). 5. Yang, J., Santamouris, M., Lee, S.E., and Deb, C.; Energy performance model development and occupancy number identification of institutional buildings. Energy and Buildings 123, 192–204 (2016). 6. Belussi, L., Danza, L., Salamone, F., Meroni, I., Galli, S., and Svaldi, S.D., Integrated smart system for energy audit: Methodology and application. In: Proceedings of 50th AiCARR conference on Beyond NZEB buildings, Matero, Italy (2017). 7. Kimura, O., Komatsu, H., Nishio, K.I., and Mukai, T., A prototype tool for automatically generating energy-saving advice based on smart meter data. Energy Efficiency, 1–18 (2018). 8. Dall, O.G., Green Energy Audit of Buildings, Springer-Verlag, London (2015). 9. Soares, N. Dias Pereira, L., Ferreira, J., Conceição, P., and Pereira da Silva, P., Energy efficiency of higher education buildings: A case study. International Journal of Sustainability in Higher Education 16(5), 669–691 (2015).

Energy Data Analysis of an Educational Institution in India

161

10. Zanni, D., Righi, A., Dalla, T., and Peron, F., The energy improvement of school buildings: Analysis and proposals for action. Energy Procedia 82, 526–532 (2015). 11. Filippın, C., Benchmarking the energy efficiency and greenhouse gases emissions of school buildings in Central Argentina.  Building and Environment  35(5), 407–414 (2000). 12. Amaxilatis, D., Akrivopoulos, O., Mylonas, G., and Chatzigiannakis, I.; An IoTbased solution for monitoring a fleet of educational buildings focusing on energy efficiency. Sensors 17(10), 2296. (2017). 13. Erhorn, H., Mroz, T., Mørck, O., Schmidt, F., Schoff, L., and Thomsen, K.E., The Energy Concept Adviser—A tool to improve energy efficiency in educational buildings. Energy and Buildings 40(4), 419–428 (2008). 14. Vatin, N., Petrichenko, M., Nemova, D., Staritcyna, A., and Tarasova, D., Renovation of educational buildings to increase energy efficiency. Applied Mechanics and Materials 633, 1023–1028 (2014). 15. Agdas, D., Srinivasan, R.S., Frost, K., and Masters, F.J., Energy use assessment of educational buildings: Toward a campus-wide sustainable energy policy. Sustainable Cities and Society 17, 15–21 (2015). 16. Sekki, T., Airaksinen, M., and Saari, A., Measured energy consumption of educational buildings in a Finnish city. Energy and Buildings 87, 105–115 (2015).

11

Factors to Consider A Review of Smart Grid Implementation in India Atmiya Patel, Vipul N. Rajput, Kartik S. Pandya, and Dipayan Guha

CONTENTS 11.1 Introduction................................................................................................. 164 11.2 Power Grid of India..................................................................................... 165 11.2.1 Introduction to the Indian Grid..................................................... 165 11.2.2 Current State of Indian Grid......................................................... 165 11.2.3 Key Issues of Indian Grid............................................................. 166 11.2.3.1 Insufficient Fuel Supply............................................... 166 11.2.3.2 Pricing.......................................................................... 166 11.2.3.3 Infrastructure............................................................... 167 11.2.3.4 Investment/Financial Problem..................................... 167 11.2.3.5 Theft of Power.............................................................. 167 11.2.3.6 Underperformance....................................................... 167 11.2.3.7 Slippage in Generation................................................. 167 11.2.3.8 Equipment Shortage..................................................... 167 11.2.3.9 Land Acquisition and Environment Clearance..................................................................... 167 11.2.3.10 Skilled Manpower Shortage......................................... 167 11.2.3.11 Approval and License.................................................. 168 11.3 Smart Grid Growth in India........................................................................ 168 11.3.1 Smart Grid Technology................................................................. 169 11.3.1.1 Advanced Meter Infrastructure.................................... 169 11.3.1.2 Demand Response........................................................ 169 11.3.1.3 Power Quality Measurement........................................ 169 11.3.1.4 Outage Management.................................................... 170 11.3.1.5 Renewable Integrations/Microgrids............................. 170 11.3.2 Policies, Standards, and Regulations............................................ 170 11.3.3 Smart Grid Projects in India......................................................... 170 11.4 Factors Influencing the Implementation of the Smart Grid in India........................................................................................................ 170 11.4.1 Enablers for the Implementation of Smart Grid........................... 172 11.4.2 Barriers to the Implementation of a Smart Grid........................... 172 DOI: 10.1201/9781003159124-11

163

164

Applied Mathematical Modeling and Analysis in Renewable Energy

11.5 Discussion................................................................................................... 173 11.5.1 Discussion on Enablers................................................................. 174 11.5.1.1 Measuring and Sensing................................................ 174 11.5.1.2 Advance Components.................................................. 174 11.5.1.3 Advanced Control Methods......................................... 175 11.5.2 Discussion on Barriers.................................................................. 175 11.5.2.1 Policy and Regulation.................................................. 175 11.5.2.2 Maturity of Technology and Risk of Delivery............. 176 11.5.2.3 Business Scenario........................................................ 176 11.5.2.4 Lack of Awareness....................................................... 176 11.5.2.5 Abilities and Learning................................................. 176 11.5.2.6 Cybersecurity and Data Privacy.................................. 177 11.5.2.7 Power Theft.................................................................. 177 11.6 Conclusion................................................................................................... 177 References............................................................................................................... 178

11.1 INTRODUCTION A smart grid can be considered as a modern electricity system that can proficiently assimilate all users’ behavior and activities associated with it (Cecati et al. 2010). The necessity of improving the power sector’s consistency and efficacy has led to the origin of smart grid. The depletion of fossil fuels and environmental awareness has led to the increase in the potential interest in renewable energy, which is also one of the pioneers in the smart grid’s growth. As discussed in Telang, Bedekar, and Wakde (2020), a smart meter is a smart grid apparatus that facilitates the vice versa information from the grid to load. The modernization of the electric control network is integral to these endeavors. Several merits can be achieved by making the grid smarter, including improved quality and reliability of power, two-way communication of information, improved grid control and security, self-healing characteristic, consumer choice availability, and smooth integration of renewable energy sources. For availing these concrete benefits, smart grid development has increased drastically in the past few years worldwide. A step-by-step investigation of the smart grid transmission advancement is portrayed under four principles: intelligent and powerful segments, smart control and security focus, smart transmission systems, and brilliant substations (Li et al. 2010; Ourahou et al. 2020). Smart systems are created all the more rapidly because of the progress in correspondence developments and engaging the bidirectional flow of data (Usman and Shami 2013). The main objectives of smart gird are: (1) usage of ongoing valuing and charging, (2) integration of sustainable power source assets, (3) accommodation of plug-in hybrid electric vehicles (PHEV) and plug-in electric vehicles (PEV), (4) two-way data stream among utility and shopper, (5) minimizing generation cost and lessening the emission of greenhouse gases (GHG) and other gases, and (6) advancement of vitality creation (Usman and Shami 2013; Wang et al. 2012; Rihan 2019; Gungor et al. 2011; Dileep 2020). With these broad objectives, the advancement of various policies and pilot projects for smart gird has been started worldwide. India is one of those countries where

A Review of Smart Grid Implementation in India

165

more emphasis has been laid on smart grid development. In India, the reasons for smart grid promotion are high distribution losses, electricity theft in a distribution operation system, long and complex distribution system, etc. On the other hand, several factors can affect the implementation of the Indian smart grid. The chapter’s main objective is to review the level of deployment of the smart grid in India. Additionally, various enablers and barriers related to the smart grid employment in India are identified and discussed.

11.2  POWER GRID OF INDIA In this section, the Indian grid is introduced briefly and key challenges to it are discussed.

11.2.1 Introduction to the Indian Grid Power is the key factor for industrialization, urbanization, economic development, and personal satisfaction in the public’s eyes (Roy et al. 2005). India is the fourthlargest customer and third largest producer of electrical power over the world, along with a critical installed capacity of power reaching 364.96 GW as of November 30, 2019 (Central Electricity Authority 2019). More importantly, the administration targets the limit’s expansion by about 100 GW under the 13th Five Year Plan of 2017–2022 (Tiewsoh, Jirásek, and Sivek 2019). Indian power sector is electrically demarcated into five regions: Northern, Western, Eastern, Southern, and North Eastern (Bhatt and Jani 2019). As mentioned in Kumar (2019), in December 2013, India’s grid became a single synchronized grid operating at the same frequency. By the 2003 Act, every region has a regional load dispatch center (RLDC), which is the mainframe for confirming the concerned region’s power networks joined operations. Furthermore, on a national level, the National Load Despatch Centre (NLDC) was developed to achieve the optimal scheduling and dispatching of electric power within a region. The RLDCs with state load dispatch centers (SLDCs) execute the real-time grid function and electrical energy dispatch in the desired area through the grid’s safe and reasonable operations. All actions are initiated based on the grid standards specified by the Central Electricity Authority (CEA) and grid codes given by the Central Electricity Regulatory Commission (CERC). With effect since January 3, 2017, Power System Operation Corporation Limited (POSOCO) has been introduced in India as a selfregulating government organization that runs the NLDC and RLDCs. It safeguards autonomous system operations and delivers level-playing-field to all stakeholders.

11.2.2  Current State of Indian Grid India’s power sector mainly depends on fossil fuels, and specifically coal, which delivered around three-fourths of all power in 2017–2018. As a preventive measure, the government is enforcing awareness about sustainable sources of power. The Government of India’s 2018 National Electricity Plan expressed that the nation will not require the additional non-inexhaustible generation plants until 2027 by

166

Applied Mathematical Modeling and Analysis in Renewable Energy

employing 50,025 MW generation plants based on coal under development and achieving an additional 275,000 MW to introduce sustainable power capacity (Conti 2006; Rekha 2019; Yang et al. 2019). India has surplus power generation and a satisfactory framework for providing power to all. The Government of India proposed a plan as “Power for All” with specific electrical energy supply objectives to the country’s general populations up to 2019 (Power Grid Corporation of India Limited 2019). This planning ensured a reliable and constant electrical supply to all by upgrading the significant structure. It is a joint and coordinated work of India’s Government that can share subsidizing and generally make financial growth (National Smart Grid Mission 2018).

11.2.3 Key Issues of Indian Grid The difficulties of India’s capacity division are apparent. It needs monstrous speculation to extend its age limit generously and, in the long run, to make the part proficient and industrially feasible. The accompanying segment examines the key issues that should be attended (Colak et al. 2016) by the Indian government with extraordinary criticalness to guarantee the power improvement. As the Indian power part expands the age and transmission limits, critical difficulties lie ahead, bringing about authentic underperformance (Kappagantu and Daniel 2018; Jadhav and Dharme 2012; Akpojedje, Ogujor, and Idode 2019). The fundamental issues are illustrated in Fig. 11.1. 11.2.3.1  Insufficient Fuel Supply A lack of coal and gaseous petrol supplies to control age is a problem that needs to be addressed in India’s capacity segment. A low coal transport structure has declined this issue. India’s imposing business models, such as coal India Ltd and statecontrolled coal India, are compelled by crude mining systems and are infested with burglary and defilement. The bringing in of gas is additionally troublesome on account of estimating and framework issues. 11.2.3.2 Pricing The main problem with assessing is its powerlessness to send a suitable banner to the suppliers and the customers to affect expected lead changes. The current hardened evaluating framework limits the choice of methodology instruments for demand side management (DSM).

FIGURE 11.1  Key issues of Indian grid.

A Review of Smart Grid Implementation in India

167

11.2.3.3 Infrastructure Foundation here broadly implies physical workplaces of transmission and appointment frameworks, and the necessary equipment and organizations to control associations. 11.2.3.4  Investment/Financial Problem The significant test for actualizing the Indian smart grid is the accessibility of assets. Extraordinary hypotheses are essential to remember the real objective of only a relationship among the clients and the smart gird. 11.2.3.5  Theft of Power In India, financial loss due to theft of power might be around $16 billion yearly. Power system operational losses and transmission losses are increasing rapidly. Populist expert free power measures equally drain the power organizations. Some power companies continue to bleed and lead to bankruptcy due to this factor. 11.2.3.6 Underperformance India has verifiably neglected to satisfy its capacity segment focused on an exciting advantage and with massive open doors ahead. The facility division keeps on being influenced by the deficiency both on the age and, therefore, the transmission side. 11.2.3.7  Slippage in Generation The vital purposes behind slippage of power are slow domestic work progress, poor geology, flash flood, divisional aggravation, law and order issue, shortage of human resources, and troublesome site conditions. 11.2.3.8  Equipment Shortage Although, inadequacy has mainly been seen within the middle sections of boilers, turbines, and generators, there has been the non-appearance of a sufficient supply of balance of plant outfit. Aside from this, hardware development is deficient too. 11.2.3.9  Land Acquisition and Environment Clearance The land acquisition in India is the most challenging process. Land for power plant construction and suitable atmosphere are not easily provided. Government approval, and rules and regulations are very complex and long time processes. The new bill associated with the land acquisition has continued to face political resistances continuously. 11.2.3.10  Skilled Manpower Shortage Manpower is an essential requirement in a power plant. But due to some reasons manpower does not like this work. Besides, other significant issues at power plants are lack of skill and lack of information. In project management, engineering, estimating, surveying, and contract management, the educational system frequently fails to produce the requisite number of specialists.

168

Applied Mathematical Modeling and Analysis in Renewable Energy

11.2.3.11  Approval and License There is no prerequisite for a license to install a new power plant. It is done through a straightforward offering process and power obtainment is allowed. Working up the deficient foundation, the permanent nonexistence of fuel, and fundamental monetary shortcoming of state-possessed power organizations are few of the Indian power area’s fundamental difficulties. To tackle these issues, the National Smart Grid Mission in 2015 by the Ministry of Power has decided to plan and monitor the execution of policies associated with India’s smart grid actions.

11.3  SMART GRID GROWTH IN INDIA The vision for smart grid implementation in Indian states: “Change the power sector of India into a safe, adaptive, sustainable and intelligently enabled ecosystem that offers reliable and quality electricity for all with dynamic participation of stakeholders” (National Smart Grid Mission 2018). India’s government plans for “Access, Availability, and Reasonableness of Quality Power for all.” India Smart Grid Forum and India Smart Grid Task Forum under the Ministry of Power have authorized 14 pilot projects of the smart grid all over the nation. The Indian government will subsidize half of the expense of the endeavor as permit and remaining cost must be afforded by the utilities of individual state (India Brand Equity Foundation 2018; Central Electricity Authority 2019). Another vision of the smart grid is to lead innovation, trendy expressions like energy preservation and outflow degeneration, environment friendly power vitality, manageable improvement, well-being factor, degeneration of transmission and distribution hardships, and the ideal use of benefits. Because India is attempting to meet its power requests, both as far as electric energy and peak load, the smart grid can accomplish the absence of control and progression of the country’s electrical network status. A “powerful grid” is an impression of renovating the situation of the country’s electric power framework by the meeting of data and functional innovation connected to the network, enabling a maintainable alternative to the clients and redesigned security, unwavering quality, and productivity to utilities. The smart grid technology permits electrical energy to be generated, transmitted, dispatched, and used more effectively and skillfully. The smart grid has several constructive characteristics that provide advantage to the customers. These are as follows. • Improved energy management: Tracking and managing the usage of power by real-time monitoring as along with a chance for reducing the energy consumption. • Integration of web portal as well as mobile applications for in-house display. • Automatic managing of power outage and restoration. • Choices of energy usage from different companies based on the online pricing mechanism.

A Review of Smart Grid Implementation in India

169

FIGURE 11.2  Smart grid technology.

11.3.1 Smart Grid Technology The smart grid advances include the organization of Information Communication Technology (ICT) and Information Technology (IT) foundation. A portion of the functionalities/innovative headways received for the Indian situation is illustrated in Fig. 11.2. 11.3.1.1  Advanced Meter Infrastructure Advanced meter infrastructure (AMI) controls the two-way communication from customers to control centers. The main objective of this system is smart metering by giving errorless data and information about customers’ usage units. AMI can identify power supply network problems, load profiling, and energy auditing (Ghosal and Conti 2019). Moreover, AMI provides operational, financial, and security benefits to customers. 11.3.1.2  Demand Response Demand response (DR) system improves the electric grid efficiently, balancing overall consumer demand and reducing the overall energy cost. DR collects different types of information such as load forecast, SCADA system, meter data management system, etc. (Doolla et al. 2016). Moreover, other benefits of the system are additional investment options, reduction of aggregate transmission and commercial (AT&C) losses, reduction of peak load demand, overgeneration load shifting, and fast ramping. 11.3.1.3  Power Quality Measurement Power quality measurement (PQM) is an essential term in power generation, transmission, and distribution. PQM controls voltage flickering and harmonics as well as lowers losses and enhances consumer loyalty. PQM will incorporate voltage control, stack adjusting, and so on.

170

Applied Mathematical Modeling and Analysis in Renewable Energy

11.3.1.4  Outage Management Outage management system (OMS) provides customer satisfaction and improves availability and reliability. OMS controls the scheduled and unscheduled operation and manages the outage distribution such as distribution transformers (DTs), high-tension/low-tension feeders (HTs/LTs). 11.3.1.5  Renewable Integrations/Microgrids A microgrid works parallel to the electric grid. Microgrid generation resources embody the microturbine, solar, wind, and other different energy sources (National Smart Grid Mission 2018). Microgrids offer the electrical power to the critical load during islanding and enhance the quality of power, efficiency, and reliability through mixing various energy sources optimally.

11.3.2 Policies, Standards, and Regulations This section discusses different strategies, standards, and guidelines that Indian government made for effective smart grid implementation. Table 11.1 classifies different plans of the Indian government for the smooth implementation of the smart grid (India Smart Grid Forum 2013).

11.3.3 Smart Grid Projects in India The financial procedure of a developing country like India depends on the obligation and nature of the electrical power supply (Redmon and Gentz 1981). To assess the essential focal points and to spot fitting advancements/models of the smart grid, the Ministry of Power initiated 14 pilot projects (Kappagantu 2015). Ministry of Power allotted some smart grid-based ventures in various states. These allotted projects are: (1) Chamundeshwari Electricity Supply Corporation Ltd., Mysore, (2) Uttar Haryana Bijli Vitran Nigam, (3) Himachal Pradesh State Electricity Board, (4) Smart Grid Knowledge Center, Manesar, (5) Electricity Department, Government of Puducherry, (6) West Bengal State Electricity Distribution Company Ltd., (7) Tripura State Electricity Corporation Ltd., (8) Assam Power Distribution Company Ltd., (9) Uttar Gujarat Vij Company Ltd. (UGVCL), (10) Punjab State Power Corporation Ltd., (11) IIT Kanpur Smart City Pilot, (12) Chandigarh Electricity Department, (13) Amravati, Maharashtra State Electricity Distribution Company Ltd. (MSEDCL), and (14) Congress Nagar, (MSEDCL), (National Smart Grid Mission 2018). Amongst the projects (1) – (9) are awarded, and (10) UGVCL, Gujarat is under the awarded project, whereas remaining projects are under process by National Smart Grid Mission.

11.4 FACTORS INFLUENCING THE IMPLEMENTATION OF THE SMART GRID IN INDIA The enablers mainly support the fast, smooth, and effective implementation of the smart grid, whereas barriers cause the smart grid’s slow growth. The enablers and barriers are explained in detail in the following subsections (Bayindir et al. 2016).

A Review of Smart Grid Implementation in India

171

TABLE 11.1 Various Plans of Smart Grid Deployment in India Agenda Policies and Tariffs

12th Plan (2012–2017) • Deployment of some appropriate arrangement tariffs • PV cell and smart metering: less assessments and endowment

Empower Access and Availability of Quality Power for All

• Electrification of all households by 2017 • Accessibility of 24 hours of intensity for significant urban areas; 22 hours for all urban areas just as give Life Line (8 hours likewise as night top) to any or all or any by 2017 • Get smart grid pilots, full smart grid rollout into primary urban areas • Development of WAMS • Improvement of little networks in 1000 towns/mechanical parks/business center points • Enablement of “Prosumers” in peak zones • Framework for advanced meter infrastructure (AMI) starts for all customers with stack >20 KW or reliable with sorted out objective domains of utilities

Smart Grid Rollouts likewise as Automation, Microgrids and Different Improvements

13th Plan (2017–2022) • Choice of a power supplier for clients in metros and peak urban zones • Obligatory demand response programs for more prominent segments of buyers • 24 hours supply in all urban areas • Provide at least 12 hours to any or all customers by 2022

14th Plan (2022–2027) • Selection opportunity of a power supplier for any client

• Enter SG rollout every single urban region • Positioning of WAMS at all substations and framework associated age components • Development of little matrices altogether 10,000 towns/modern parks/business center points • Enablement of “Prosumers” in metros and major city regions • Nationally, AMI takes off for buyers with three-stage associations

• SG rollout nationwide • Development of little networks in 20,000 towns/mechanical parks/business centers • Active participation of “Prosumers” • Nationwide AMI rollout for all customers

• Constant and quality 24 × 7 power offer to all or any classes of shoppers across the nation

172

Applied Mathematical Modeling and Analysis in Renewable Energy

11.4.1 Enablers for the Implementation of Smart Grid The smart grid technologies’ practical implementation can be achieved over an extended period, counting sequential layers of operation and competence over current systems and equipment. The enablers are key factors that support the building of a smart grid and can be defined by broader characteristics. These key enablers are listed and discussed in Table 11.2 (Faheem et al. 2018; Appasani, Maddikara, and Mohanta 2019).

11.4.2 Barriers to the Implementation of a Smart Grid Barriers to implementing the alleged smart grid in India area unit are great as people have obstructed electrification improvement ever since market relief was initially declared nearly 20 years ago. The issues that distress India’s capacity segment are numerous and genuine and extremely outstanding to approach producers, industry

TABLE 11.2 Identified Enablers for Smart Grid Deployment in India Enablers Increase in electricity demand and provide shortfall

Measurement and sensing

Increase in cybersecurities Communications

Reliability Efficiency Conventional energy combination Grid improvement Advanced components Advanced control methods

Remarks India is a developing country, and the financial growth of a developing country depends on electricity demand and supply. India introduces a smart grid very fast for long-term benefits to fight the electricity crisis and supply shortfall. Measurement and sensing are the most important part of the smart grid. Hence, advanced meter infrastructure bolster checking, control, protection, and basic leadership capacities are introduced. The considerable securities are introduced using Internet of Things (IoT) to protect smart meters and smart devices. The smart grid is created on a bidirectional information path, and the IoT is used to make it faster to lessen the network’s multifaceted nature. The smart grid can enhance blackout administration execution by reacting quicker to restoration hardware. The smart grid can be able to enhance stack features and decrease framework losses. Green energy is an essential part of the smart grid and electricity generation; CO2 reeducation and capacity increase. A smart grid improves flexibility and the robustness of the grid also reduces the risk of losses. A smart grid integrates advanced power electronic devices, advanced computing and smart technique, smart metering etc. Some advanced microelectronics manage the smart grid for better control and stability and improved power system.

A Review of Smart Grid Implementation in India

173

TABLE 11.3 Identified Barriers to Smart Grid Deployment in India Barriers Policy and regulations

Maturity of technology and risk of delivery

Business scenario

Lack of awareness

Poor economic condition of services

Skills and knowledge Cybersecurity and data privacy Power theft

Remarks In a few cases, utilities do not take the route as a business model would for smart grid as their square measure regulative and approach hindrances in situ either create invert motivating forces or neglect constructive impetuses of saving time of individual area speculation. Technologies have vital technological risks associated with them as a result agreement standards haven’t emerged. Additionally, there are a few tests exclusively of goliath scale usage for every 50,000 premises. Thus, they keep it up being essential conveyance changes evaluated into the estimates. The suppliers and utility managers of the regional unit are receptive to the role played by significant matrices. They are generally unfit to make the business case for a profitable frame. There is an absence of mindfulness among partners regarding smart grid’s part during a sectionalized low-carbon future. In India, the debt-burdened utilities find the situations tough for taking a position as smart grid initiatives. A deficiency is anticipated in vital skills that will be needed by the designer to build smart grids. In smart grids, significant issues are information leakage and hacking. In India, power theft is a significant issue. There can be huge losses between the distribution system and transmission system operations, and the smart grid system is greatly affected.

specialists, and, for sure, the general population. Mainly eight barriers are holding back the smart grid’s employment; however, none of them is impossible to remove. The overriding challenge might be a restrictive context that adjusts to current business and wants people’s broader environmental objectives (Kappagantu and Daniel 2018; Asaad et al. 2019). These key barriers are listed and discussed in Table 11.3.

11.5 DISCUSSION This section discusses the substantial enablers and barriers to smart grid implementation in India.

174

Applied Mathematical Modeling and Analysis in Renewable Energy

11.5.1 Discussion on Enablers The smart grid can go about as a spine foundation to adjust novel plans of action like the smarter town, electric vehicles, smart systems except for a huge amount of adaptable and efficient vitality structure and tariff structures (Kabalci and Kabalci 2019). Some major enablers are discussed in the following sections. 11.5.1.1  Measuring and Sensing Detecting and estimation framework can be the foundation of a smart grid. The smart meters and associated AMI should be set up to help perception, control, security, and unraveling capacities. High goals ongoing estimations can be adjusted to ideally utilize available assets, maintain a strategic distance from clogs, help showcase tasks, and make request viewpoint administration potential. 11.5.1.2  Advance Components Advance components in smart grid implementation increase the grid values and effectiveness, including advanced use of power electronic, superconducting devices, advanced generation technologies, advanced storage technologies, composite conductors, and grid-friendly appliances. 11.5.1.2.1  Advance Use of Power Electronics For effective voltage regulation, sensitive power adjustment, and control exchange over long separations, FACTS devices may be used. The back-to-back HVDC system can be used in a long-distance transmission line for improving the system stability. 11.5.1.2.2  Superconducting Apparatus Superconducting wires are used for an incredibly realistic and effective exchange in intensity and an improvement in the standard of intensity. High-temperature superconducting devices can be considered a good alternative for transferring massive power for future power system frameworks. Some of these are superconducting magnetic energy storage (SMES), superconducting synchronous condensers, blame current limiters, and high-productivity engines and generators. 11.5.1.2.3  Advanced Generation Technologies The advanced generation technologies involve changes in traditional power generation, improvement in distribution generation resources (generally in 3 kW to 10 MW extend), and exploration of advanced power electronics and electrical materials. 11.5.1.2.4  Advanced Storage Technologies A smart grid is an innovative and new technology through storage capacity increase. Some new battery innovations are highly efficient, store large energy capacity, and are economical. These batteries are described as follows. 11.5.1.2.5  Sodium Sulfur (NaS) Battery This battery is exceptionally efficient (~89%) with lower monetary cost, massive back up for energy stockpiling, and peak saving.

A Review of Smart Grid Implementation in India

175

11.5.1.2.6  Vanadium Redox Flow Battery This battery has a large capacity of energy storage and modular design. It can improve the stability of the grid. 11.5.1.2.7 Ultracapacitors These store energy like a battery and release it rapidly like an electrical gadget. 11.5.1.2.8  Superconducting Magnetic Energy Storage The main advantages of SMES are low loss and fast response. The SMES can be used for power quality improvement and stability enhancement. 11.5.1.2.9  Composite Conductors and Grid-Friendly Appliances The benefits of composite conductors are high temperature operation, increased amperage, and reduced sag. To conserve peak load, the appliances such as heaters, air conditioners, washing machines, dryers, etc. can automatically turn on or off. This will help customers to participate effectively in making the grid smarter. 11.5.1.3  Advanced Control Methods The main three categories of the advance control methods are discussed as follows. 11.5.1.3.1  Improve Control of the Bulky Generation Advanced microelectronics devices can be used for better control of the generators and also improve the generators’ transient strength. 11.5.1.3.2 Monitor and Control of Power System Stability with Real-Time Mode This strategy includes control of voltage and frequency, self-mending of the grid following an unsettling influence. 11.5.1.3.3  Innovative Control of Distribution Systems The control of the distribution system can be enhanced by considering microgrids and load sharing.

11.5.2 Discussion on Barriers There are mainly eight barriers that affect the implementation of the smart grid in India. These are: (1) policy and regulation, (2) business scenario, (3) maturity of technology and risk of delivery, (4) lack of awareness, (5) skills and knowledge, (6) cybersecurity and data privacy, (7) poor economic health of services, and (8) power theft (Appasani, Maddikara, and Mohanta 2019; Kappagantu and Daniel 2018; Kaushal 2011). 11.5.2.1  Policy and Regulation A recent approach and administrative systems are, for the most part, intended to deal with the current power systems. To some degree, the present model has energized

176

Applied Mathematical Modeling and Analysis in Renewable Energy

rivalry in the generation and given intensity, which otherwise can’t be advertised for clean vitality supplies. For the most part, the government sets approach while the controllers screen the usage to shield the clients and avoid advertising abuse. 11.5.2.2  Maturity of Technology and Risk of Delivery Development can be one of the major constituents of the smart grid that controls a wide variety of instruments, programming, and correspondence propels. At times, innovations are all-around growth, for instance, in joint advancements in a few zones. Nevertheless, it is still at a too introductory period of progress and is yet to be produced at an extraordinary level. On the equipment aspect, the fast advancement of innovation can be seen by the merchants around the world. The real difficulties are to beat the mixture of the entire equipment framework and deal with a high volume of information on the product framework and information administration aspects. With various programming framework, providers restore different information designs. Moreover, there is a high demand for cutting-edge information models. The expansion of information puts pressure on the information administration outline that is more relatively like the broadcast communication business than industry utility. 11.5.2.3  Business Scenario Most of the business models bring about negative business cases, weakened by two fundamental difficulties: (1) High initial and running costs, and (2) capital and working expenses. They fuse tremendous settled expenses related to sorting out the interminable correspondences. Equipment costs do not cause imperative developments in economies of scale, and the reconciliation has a massive conveyance and incorporation dangers. Advantages are stressed by the administrative structure while computing the points of interest. Associations tend to be preservationists in what they can assemble as cash preferences to the investors. 11.5.2.4  Lack of Awareness An ordinary client’s level of understanding, in terms of knowing how the control is conveyed to their homes, is usually very low. Therefore, it is desirable that before implementing a smart grid, an ordinary client should be made mindful of what smart grids are? In this regard, clients should be mindful of their vitality utilization design at home and offices. Methodology creators and controllers should be clear about the long-haul possibilities of keen frameworks. The utilities have to concentrate on the general limits of the smart grid instead of the execution of smart meters. They have to think about a comprehensive view. 11.5.2.5  Abilities and Learning As the utilities can transfer toward smart grids, it may be an essential substitution limit sets to cross any obstruction. They need to grow new abilities in the investigation, information administration and assisting choices. To address this issue, a unit of architects and directors can be prepared to deal with the changes. These changes require venture and cash from organizations and private players to help in preparing the plans that may empower the structure directors and models for tomorrow. To bring such a modification, the utilities should be expected with unwieldy concern.

A Review of Smart Grid Implementation in India

177

The changes that occur due to the progress will have to be dealt with to maintain a strategic distance from workers’ overloading. 11.5.2.6  Cybersecurity and Data Privacy Along with the progress of a similar computerized power framework comes the related security and information administration test. Advanced systems are more prone to pernicious assaults from the programming framework. Security turns out to be a vital issue to act instantly. Moreover, issues of the attack on protection and security by utilizing private information emerge. The information accumulated from the utilization data may offer important information on buyers’ conduct and inclinations. This potential information may well be mishandled if rectify conventions and safety efforts are not implemented. Thus, these two issues cannot be handled in a standard straightforward way. It can create a negative impression on clients’ recognition and will end up being an obstruction for appropriation. 11.5.2.7  Power Theft The metering, billing, and receiving charges are not effective in developing countries such as India. Subsequently, organized theft is a shockingly authentic concern. It is nothing but a surprise to see that AT&C losses are relatively high in such countries. To deal with such a profound established issue, just specific and authoritative estimates won’t be sufficient. Social mindfulness is also required to deal with such a far-reaching issue. Social exercises to change people’s outlook who are engaged in such corrupt practices are required alongside the execution of law and rules. From the Smart Grid Pilot Project of Puducherry, it has been seen that a 10–12% enhancement for the losses of AT&C can be achieved.

11.6 CONCLUSION In this chapter, various Indian grid issues are extensively elaborated, which are the primary sources of driving India’s smart grid execution. Also, various smart grid technologies, policies, and regulations for smart grid implementation in India are discussed with the status of various smart grid projects. After that, various factors affecting India’s smart grid deployment are identified and divided into two categories: enablers and barriers. The remarks and suggestions based on these factors are summarized as follows: • The main enablers, such as advanced components, measuring and sensing devices, and advanced control methods, can be helpful for the fast smart grid deployment. In a smart grid, the advanced metering and other advanced components can improve the grid efficiency and generation capacity. The advance control methods enhance the controlling of losses, grid’s self-monitoring system following unscheduled influence, better control of frequency and voltage, load schedule planning and control DSM. • The main identified barriers are power theft, cybersecurities, technology maturity and delivery risk, and lack of awareness. Among these barriers, power theft is a major problem in the Indian grid, i.e., yearly $16 billion

178

Applied Mathematical Modeling and Analysis in Renewable Energy

power losses. However, it can be recovered by using smart grid control methods. Other barriers are technology maturity, delivery risk, lack of awareness about cybersecurity and data privacy. Cybersecurity and information protection are substantial obstructions in smart networks. However, they can be overcome in smart grids by utilizing IoT high-evaluation protections. Furthermore, IoT secures the smart grid against the hacking and losses of generation. It provides security mainly in the application of data confidentiality, users’ privacy with the interaction of utility, control of access of the data received integrity of data, and authentication.

REFERENCES Akpojedje, France Onoabedje, Emmanuel A Ogujor, and Michael O Idode. 2019. A survey of smart grid systems on electric power distribution network and its impact on reliability. Journal of Advances in Science and Engineering 2 (1): 37–52. doi:10.37121/jase.v2i1.44. Appasani, Bhargav, Jaya Bharata Reddy Maddikara, and Dusmanta Kumar Mohanta. 2019. Standards and communication systems in smart grid. In Smart Grids and Their Communication Systems, Springer. Asaad, Mohammad, Furkan Ahmad, Mohammad Saad Alam, and Mohammad Sarfraz. 2019. Smart grid and Indian experience: A review. Resources Policy: 101499. doi:10.1016/ j.resourpol.2019.101499. Bayindir, Ramazan, Ilhami Colak, Gianluca Fulli, and Kenan Demirtas. 2016. Smart grid technologies and applications. Renewable and Sustainable Energy Reviews 66: 499–516. doi:10.1016/j.rser.2016.08.002. Bhatt, Jignesh G, and Omkar K Jani. 2019. Smart development of Ahmedabad-Gandhinagar twin city metropolitan region, Gujarat, India. In Smart Metropolitan Regional Development, Springer. Cecati, Carlo, Geev Mokryani, Antonio Piccolo, and Pierluigi Siano. 2010. An overview on the smart grid concept. Paper read at IECON 2010-36th Annual Conference on IEEE Industrial Electronics Society, at Glendale, AZ, USA. Central Electricity Authority. All India installed capacity (in MW) of power stations. December 2019. November, 2019 [cited 15 December 2019]. Available from http:// www.cea.nic.in/reports/monthly/installedcapacity/2019/installed_capacity-11.pdf. Central Electricity Authority. Executive summary on power sector report for month of October, 2019. 2019 [cited 22 November 2019]. Available from http://www.cea.nic.in/ reports/monthly/executivesummary/2019/exe_summary-10.pdf. Colak, Ilhami, Seref Sagiroglu, Gianluca Fulli, Mehmet Yesilbudak, and Catalin-Felix Covrig. 2016. A survey on the critical issues in smart grid technologies. Renewable and Sustainable Energy Reviews 54:396–405. doi:10.1016/j.rser.2015.10.036. Conti, Juan Pablo. 2006. Let the grid do the thinking [intelligent networks]. Power Engineer 20 (2): 34–37. doi:10.1049/pe:20060207. Dileep, G. 2020. A survey on smart grid technologies and applications. Renewable Energy 146:2589–2625. doi:10.1016/j.renene.2019.08.092. Doolla Suryanarayan, Singh Amit, Banerjee Rangan. 2016. Demand Response in India: Technology Assessment, M&V Approach and Framework for DR Implementation. New Delhi, India: Shakti Sustainable Energy Foundation. Faheem, Muhammad, Syed Bilal Hussain Shah, Rizwan Aslam Butt, et al. 2018. Smart grid communication and information technologies in the perspective of Industry 4.0: Opportunities and challenges. Computer Science Review 30: 1–30. doi:10.1016/j. cosrev.2018.08.001.

A Review of Smart Grid Implementation in India

179

Ghosal, Amrita, and Mauro Conti. 2019. Key management systems for smart grid advanced metering infrastructure: A survey. IEEE Communications Surveys and Tutorials 21 (3): 2831–2848. doi:10.1109/COMST.2019.2907650. Gungor, Vehbi C, Dilan Sahin, Taskin Kocak, et al. 2011. Smart grid technologies: Communication technologies and standards. IEEE Transactions on Industrial informatics 7 (4): 529–539. doi:10.1109/TII.2011.2166794. India Brand Equity Foundation. India brand Equity Foundation 2018 [cited 28 April 2018]. Available from https://www.ibef.org/download/Power-February-20181.pdf. India Smart Grid Forum. Smart grid vision and roadmap for India. 2013  [cited 22 April 2018]. Available from www.indiasmartgrid.org/reports/Smart%20Grid%20Vision%20 and%20Roadmap%20for%20India.pdf. Jadhav, Ganesh N, and Anjali A Dharme. 2012. Technical challenges for development of smart grid in India. Paper read at IEEE-International Conference on Advances In Engineering, Science And Management (ICAESM-2012), at Nagapattinam, Tamil Nadu, India. Kabalci, Ersan, and Yasin Kabalci. 2019. Introduction to smart grid architecture. In Smart Grids and Their Communication Systems, Springer. Kappagantu, Ramakrishna, and S Arul Daniel. 2018. Challenges and issues of smart grid implementation: A case of Indian scenario. Journal of Electrical Systems and Information Technology 5 (3): 453–467. doi:10.1016/j.jesit.2018.01.002. Kappagantu, Ramakrishnan. 2015. Smart grid implementation in India: A case study of Puducherry Pilot Project, India. Paper read at 2015 International Conference on Energy Economics and Environment (ICEEE), at Noida, India. Kaushal, R. Challenges of implementing smart grids in India 2011  [cited 25 April 2018]. Available from https://www.greatlakes.edu.in/gurgaon/sites/default/files/SMAR_ GRID_CHALLENGES.pdf. Kumar, Ankit. 2019. Beyond technical smartness: Rethinking the development and implementation of sociotechnical smart grids in India. Energy Research and Social Science 49: 158–168. doi:10.1016/j.erss.2018.10.026. Li, Fangxing, Wei Qiao, Hongbin Sun, et al. 2010. Smart transmission grid: Vision and framework. IEEE Transactions on Smart Grid 1 (2): 168–177. doi:10.1109/ TSG.2010.2053726. National Smart Grid Mission. SG Projects Status, December 2019. National Smart Grid Mission 2018 [cited 28 December 2019]. Available from https://www.nsgm.gov.in/en/ sg-status. National Smart Grid Mission. SG technologies. National Smart Grid Mission 2018 [cited 20 December 2018]. Available from https://www.nsgm.gov.in/en/content/ sg-technologies. Ourahou, Meriem, Wiam Ayrir, Bennacer el Hassouni, and Ali Haddi. 2020. Review on smart grid control and reliability in presence of renewable energies: challenges and prospects. Mathematics and Computers in Simulation 167: 19–31. doi:10.1016/j. matcom.2018.11.009. Power Grid Corporation of India Limited. One nation one grid. 2019 [cited 20 January, 2019]. Available from www.powergridindia.com/one-nation-one-grid. Redmon, John R, and Charles H Gentz. 1981. Affect of distribution automation and control on future system configuration. IEEE Transactions on Power Apparatus and Systems PAS-100 (4):1923–1931.doi:10.1109/TPAS.1981.316536. Rekha, Santhi. 2019. Role of smart grid in power sector and challenges for its implementation: A review on Indian scenario. Journal of Renewable Energy and Smart Grid Technology 14 (1):77–86. Rihan, Mohammad. 2019. Applications and requirements of smart grid. In Smart Grids and Their Communication Systems, Springer.

180

Applied Mathematical Modeling and Analysis in Renewable Energy

Roy, Anjan, Srikrishna A Khaparde, Polagani Pentayya, S Usha, and AR Abhyankar. 2005. Operating experience of regional interconnections in India. Paper read at IEEE Power Engineering Society General Meeting, 2005, at San Francisco, CA, USA. Telang, Aparna S, PP Bedekar, and SD Wakde. 2020. towards smart energy technology by integrating smart communication techniques. In Techno-Societal 2018: Springer. Tiewsoh, Lari Shanlang, Jakub Jirásek, and Martin Sivek. 2019. Electricity generation in India: Present state, future outlook and policy implications. Energies 12 (7):1361. doi:10.3390/en12071361. Usman, Ahmad, and Sajjad Haider Shami. 2013. Evolution of communication technologies for smart grid applications. Renewable and Sustainable Energy Reviews 19: 191–199. doi:10.1016/j.rser.2012.11.002. Wang, Zhu, Lingfeng Wang, Anastasios I Dounis, and Rui Yang. 2012. Integration of plugin hybrid electric vehicles into energy and comfort management for smart building. Energy and Buildings 47:260–266. doi:10.1016/j.enbuild.2011.11.048. Yang, Benqiang, Shuli Liu, Mark Gaterell, and Yang Wang. 2019. Smart metering and systems for low-energy households: challenges, issues and benefits. Advances in Building Energy Research 13 (1): 80–100. doi:10.1080/17512549.2017.1354782.

12

Integration and Modeling of SmallScale Pumped Storage Jyoti Gupta and Arun Kumar

CONTENTS 12.1 Introduction................................................................................................. 181 12.1.1 Study Area.................................................................................... 183 12.1.2 Power Demand and Supply Power................................................ 185 12.2 Modeling Pumped Storage As Battery....................................................... 187 12.3 Solar-Hydro Hybrid System........................................................................ 189 12.4 Simulation Result........................................................................................ 189 12.5 Conclusion................................................................................................... 190 References............................................................................................................... 190

12.1 INTRODUCTION The renewable sources of energy, i.e., wind energy and solar energy, are intermittent in nature. In order to limit the impact of intermittence nature in power generation, storage of energy is required. The most common form of current electricity storage is pumped storage in the electricity sector, which is a method of long-term energy storage relative to other sources. It turns low-cost off-peak power into highvalue on-peak power. It is, therefore, important to hybridize the energy generated by renewable sources with storage such as pumped storage. Energy storage system can be broadly classified as large-scale and small-scale technologies, as shown in Table 12.1. Hybrid power system is a small-scale stand-alone system that generates electricity from more than one renewable energy sources with their respective storage devices (Lukuyu and Cardell 2014). Rapid advancement in the field of pumped storage was made on a large scale (MW). This large-scale pumped storage can be scaled down for its utilization at small scale in remote area. Since the remote areas are far from the main transmission line, therefore in such a situation, small-scale pumped storage, as an energy storage device, is the most feasible and economic solution in off-grid mode for hybrid system comprising solar and hydro power generation (in present study). The driving elements of the hybrid system, as shown in Fig. 12.1, are: (i) solar photovoltaic (PV) power plant, (ii) small hydro power (SHP) station, and (iii) pumped storage plant (PSP). DOI: 10.1201/9781003159124-12

181

182

Applied Mathematical Modeling and Analysis in Renewable Energy

TABLE 12.1 Energy Storage Technologies Technology Large scale Small scale

Storage System Pumped storage Compressed air energy storage Battery Flywheel Capacitors

Challenges a. High environmental impact b. Geographically limited –

PSP is used as an emergency power supply at the peak demand time. During offpeak demand, it converts the low-price excess electricity into high value via pumping water from lower to higher reservoir. Thus, it smoothens the peak to valley differences in supply by peak shaving and valley filling. Thermal, large hydro, natural gas, and nuclear power plant are used to fulfill base load requirement, whereas oil and pumped storage plants well satisfy the fluctuating demand (Jog 1989). Katsaprakakis et al. (2008) found that PSP installation in an isolated area is not always economical. It will be beneficial when specific cost of energy production and annual diesel oil

FIGURE 12.1  Hybrid power system.

Integration and Modeling of Small-Scale Pumped Storage

183

consumption are very high. Connolly et al. (2010) developed a program to examine the user-provided terrain and determine if the PSP system is feasible on it. Stoppato et al. (2014) did the particle swarm optimization on hybrid system of a small village composed by PV-system, pump as turbine, batteries, and diesel set. The proposed reservoir for PSP serves the purpose of irrigation as well as electricity generation during peak demand. Dhillon et al. (2014) reviewed the basics of wind energy, and suggested the integration of wind energy system with PSP system to smoothen the electricity supply. The small-scale pumped storage system integrated with the PV system was studied to meet the requirements of a few kilowatts (Rehman et al. 2015). A case study on small pumped storage facilities in the Czech Republic was presented by Soukal et al. (2011). Due to environmental concerns, large-scale construction of a pumped storage was no longer easy. Hence, the concept of small-scale pumped storage was proposed, which can be accomplished by using man-made tanks of large capacity. The capability to store huge amount of energy to satisfy variable demand directs toward the pumped storage for hybrid power system. The objective of this chapter is to analyze the demand and supply of the selected remote area, and based on that model, the small-scale pumped storage system is integrated with hydro and solar power. For evaluating the performance of hybrid power system, hybrid optimization model for electric renewables (HOMER) is used. The software is developed by National Renewable Energy Laboratory, in the US. HOMER software performs energy balance, and based on the existing internal models, it calculates unit cost of energy of hybrid system. It is possible to model the hybrid system containing wind turbines, diesel sets, batteries, solar photovoltaic modules, hydro power plants, and batteries. But it does not have internal model for pumped storage (reversible hydroelectric power plants). To account model for pumped storage plant in HOMER, storage batteries are used, which are discussed in detail in section 12.2.

12.1.1 Study Area Most of the states are connected with central grid system for continuous supply of electricity, while hilly region requires setting up a localized generation plant for the power demand satisfaction. In rural and remote areas, the renewable energy sources such as solar, wind, hydro, and biomass are available in sufficient quantity to satisfy their local demands. In the present chapter, remote rural site of Uttarkashi district has been selected, which includes three villages, viz. Harsil, Mukhuba, and Dharali, where good sunshine during winter and good rainfall during monsoon occur. The general information of the study area has been shown in Fig. 12.2. A small hydro plant of capacity 2 × 100 kW is situated in Harsil in Uttarkashi Gangotri Road. The layout map of Harsil SHP is shown in Fig. 12.3. The already existing SHP at Harsil generated the power by diverting small fraction of water from Kakoragad stream. The capacity of the power channel supports more water to divert, which can increase the capacity of the existing SHP plant by 100 kW.

184

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 12.2  Site location in map. (Administrative Atlas-Uttarakhand 2011 http://www. censusindia.gov.in/2011census/maps/atlas/Uttrakhand.html.)

FIGURE 12.3  Harsil hydro power plant layout. (Google Earth.)

Integration and Modeling of Small-Scale Pumped Storage

185

12.1.2 Power Demand and Supply Power The area is facing difficulties in supplying electricity due to limited generation through small hydro. Therefore, to estimate the power demand for Harsil, Mukhuba, and Dharali villages, primary survey has been conducted. The power requirement is categorized as domestic and commercial load. The area has a large number of apple orchards and due to non-availability of storage facilities, fruits are sold at very cheap rates. Thus, to increase the financial returns on apple, setting-up of cold storage of 50 tons capacity is proposed in Mukhuba, which contributes to rise in the total electricity demand of the area in future. The location of apple orchard is shown in Fig. 12.4. To draw the load profile of cold storage, the following assumptions have been taken: 1. The insulating material polyurethane foam (PUF) is equally applied to floor, ceiling, and side walls of thickness 100 mm. 2. Ambient temperature is taken as a relatively hottest day during the month of May with normal temperature 17.7°C and warmest temperature 23.7°C. 3. Apples are at ambient temperature initially. The basic equation used for drawing the load profile is as under:

Qe =

kA ( Ta − Ts ) (12.1) L



Ql =

mc p ( Ta − Ts ) (12.2) t



Qt = Qe + Ql (12.3)

FIGURE 12.4  Location of apple orchard. (Google Earth.)

186

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 12.5  Load profile of cold storage.

Where Qe is the amount of heat penetrated inside the cold storage (kW), Ql is the amount of heat loss by apples (kW), Qt is the total amount of heat (kW) which is required to remove from the cold storage to keep the apples fresh for long time, k is the thermal conductivity of the PUF, A and L is the cross-sectional area and thickness of the wall, respectively, Ta is the ambient temperature, Ts is the temperature inside the cold storage which is required to maintain, m is the mass of the apple, and c p is the heat capacity. The load profile of the cold storage is shown in Fig. 12.5. The overall hourly load profile for the day containing both the current and future demand put into HOMER software which is simulated further in seasonal profile and yearly profile, as shown in Fig. 12.6. Due to availability of flat land, a solar panel of 160 kW capacity can be installed at the site location in Harsil. To estimate the potential, solar radiation data has been taken from Nation Renewable Energy

FIGURE 12.6  Total load profile.

Integration and Modeling of Small-Scale Pumped Storage

187

FIGURE 12.7  Solar power output.

Laboratory (NREL 2011). Figure 12.7 shows the annual average hourly global solar radiation, plotted against hours of the day.

12.2  MODELING PUMPED STORAGE AS BATTERY HOMER software does not facilitate the selection of pumped storage as direct resource component. Therefore, it is modeled as an electrical storage of certain capacity and efficiency. This section elaborates the procedure and consideration made in the modeling of pumped storage as battery. To establish the relationship between battery and pumped storage some basic equations of physics are used. The consideration made are: 1. Round trip efficiency of battery is reduced to 65% to represent a pumped storage system (Fausto et al. 2016) which signifies that there are some losses during energy storage. 2. The nominal voltage is selected as 100 V. 3. Minimum state of charge must be set at 0% (Sinha and Chandel 2014), which signifies that the volume of the reservoir is evacuated during generation. 4. For a given life time, life cycle cost (LCC) replacement and maintenance cost is calculated and entered. 5. Float life of the battery representing reservoir must be related to the planning, operation, and other considerations. 6. While modeling of equivalent battery, losses due to evaporation and infiltration in the reservoir are neglected.

188

Applied Mathematical Modeling and Analysis in Renewable Energy

HOMER calculates the hydro power output from the basic governing equation of power generation given by equation (12.4), where water is falling from a height of H ( m ) with a flow rate of Q ( m 3 / s ) and conversion efficiency of η , which on combination calculates the generated power in kilowatt as:

P = 9.81 ⋅ η ⋅ Q ⋅ H (12.4)

If υ(m3) is the volume of the upper reservoir which is emptying at mean flow rate Q ( m 3 / s ), the duration of generating hours t g is given as:

( )

tg =



υ (12.5) Q ⋅ 3600

The energy produced in kW-h or combining equation (12.4) and equation (12.5),

Eg =

9.81 ⋅ η ⋅ H ⋅ υ (12.6) 3600

The amount of energy stored by battery is referred to as its power capacity. The power capacity of the battery C is defined in Ampere-hour (A-h). Since the voltage is fixed due to its internal chemistry, therefore watt-hour (W-h) capacity can be calculated by multiplying with nominal voltage (V ) which is selected as 100 V, suggested in HOMER. But any other value of the nominal voltage can be selected. Thus, the total storage energy of the battery can be expressed in kW-h as: EB =



C ⋅V (12.7) 1000

Based on the data shown in Table 12.2, the specification of the battery can be calculated to represent the small-scale PSP as equivalent battery. Using equations (12.4–12.7), the relation between PSP and equivalent battery is evaluated and shown in Table 12.3. The capacity of the equivalent battery is obtained as 49000 A-h with round trip efficiency of 65% that is able to supply 2275 A of current for 14 hours. During pumping hours, the reservoir is filled in 10 hours with flow rate of 0.42 m3/s for which the equivalent battery corresponds to maximum charge rate of 0.065 A/A-h. If the maximum capacity of the battery is set as 49 A-h, the total TABLE 12.2 Parameters of Pumped Hydro Parameters Reservoir size Available head Flow rate (turbine) Overall efficiency Flow rate (pump)

Symbol Vol (m3) H (m) m3/s Η m3/s

Value 15120 148 0.3 80% 0.42

189

Integration and Modeling of Small-Scale Pumped Storage

TABLE 12.3 Specification of Battery Equivalent to Pumped Hydro Q (m3/s) 0.3 0.42

Operating Hours 14 10

E g (kWh) 4900 4900

P(kW) 350 350

C (A-h) 49000 49000

I (A) 2275 3185

number of batteries required to store 350 kW of power with maximum charge rate of 0.065 is 1000.

12.3  SOLAR-HYDRO HYBRID SYSTEM Following considerations are made to model the hydro power plant to integrate with pumped storage: 1. The hydroelectric run-of-river system must be the only equipment on the DC bus, which ensures battery as reservoir to simulate with water flow. 2. The head losses in pipe must be set at zero as all the losses are included in efficiency. 3. The residual flow must be subtracted from the total flow before entering the flow data in HOMER. 4. The cost data entered is from the existing site. 5. The discharge data are entered on monthly basis. To fulfill the basic load demand, the optimization result shows that the nominal power rating of the solar PV panel is 160 kW. The derating factor is assumed to be taken as 80% and life time 20 years. To convert AC to DC, a rectification unit is used, called converter. The capacity of converter is defined as the maximum amount of AC power delivered by inverting the DC power. In order to facilitate the integrated PSP system, the converter must fulfill the following conditions: 1. The efficiency is set at 100% as all losses are already considered in hydro inputs. 2. The inverter must be operating with AC power source in parallel. 3. The rectifier capacity must be zero. This enables that the excess electricity would be used in pumping operation.

12.4  SIMULATION RESULT The unit cost of energy from hydro and solar is Rs. 6.37 and Rs. 8.03, respectively. The simulation results of the hybrid power plant are shown in Fig. 12.8. Based on the results obtained, following observation can be made: a. The cost of energy (COE) of hybrid power plant is 12.31 INR/kWh. b. 100% electricity production is from renewable sources.

190

Applied Mathematical Modeling and Analysis in Renewable Energy

FIGURE 12.8  Simulation result.

12.5 CONCLUSION To meet the load requirements of remote area, small-scale pumped storage is integrated with renewable energy sources such as solar and hydro. To evaluate the performance of this, hybrid system HOMOR software is used. As the HOMER software does not include inbuilt model of pumped storage, therefore it is modeled as equivalent battery. This paper explains the method and consideration made to represent PSP system as an equivalent battery. To represent the pumped storage of 350 kW, a single battery of 49000 A-h capacity or 1000 battery of 49 A-h capacity with maximum charge rate of 0.065 A/A-h is selected. The cost of energy of hybrid power plant is estimated using HOMER software as 12.31 INR/kWh.

REFERENCES Lukuyu, J. M., and Cardell, J. B. 2014. Hybrid power system options for off-grid rural electrification in Northern Kenya. Smart Grid Renewable Energy 5(5):89–106. Jog, M. G. 1989. Hydro Electric Pumped Storage Plants. Katsaprakakis, D. A., Christakis, D. G., Zervos, A., Dimitris P., and Voutsinas, S. 2008. Pumped storage systems introduction in isolated power production systems. Renewable Energy 33:467–490. Connolly, D., MacLaughlin, S., and Leahy, M. 2010. Development of a computer program to locate potential sites for pumped hydroelectric energy storage. Energy 35:375–381. Stoppato, A., Cavazzini, G., Ardizzon, G., and Rossetti, A. 2014. A PSO (particle swarm optimization)-based model for the optimal management of a small PV (photovoltaic)pump hydro energy storage in a rural dry area. Energy 76:168–174. Dhillon, J., Kumar, A., and Singal, S. K. 2014. Optimization methods applied for WindPSP operation and scheduling under deregulated market: A review. Renewable and Sustainable Energy Reviews 30: 682–700. Rehman, S., Al-Hadhrami, L. M., and Alam, Md. M. 2015. Pumped hydro energy storage system: a technological review. Renewable and Sustainable Energy Reviews 44:586–598. Soukal, J., Sigma W. U., Jager, R., and Pochyly, F. 2011. Small pumped-storage plants and their role in the Czech grid. Hydropower and Dams 18:1–4. Administrative Atlas-Uttarakhand. 2011 http://www.censusindia.gov.in/2011census/maps/ atlas/Uttrakhand.html (Accessed on November 4, 2015).

Integration and Modeling of Small-Scale Pumped Storage

191

Nation Renewable Energy Laboratory (NREL). 2011. http://www.nrel.gov/rredc/solar_data. htm (Accessed on December 1, 2015). Fausto, A., Canales, A. B., and Carlos, A. B. M. 2016. Modeling a hydropower plant with reservoir with the micro power optimization model (HOMER). International Journal of Sustainable Energy 40:145–156. Sinha, S., and Chandel, S. S. 2014. Review of software tools for hybrid renewable energy systems. Renewable and Sustainable Energy Reviews 32:192–205.

Index Note: Page numbers followed by f and t refer to figures and tables, respectively. 1/10 approximation algorithm, 54 1/2-approximate EFX, 54 1/2-approximate MMA1, 54 2/3-approximate maximin fair allocation, 54 1.45-approximation algorithm, 54 3D solid, toroid or sphere, 117–120 3-D surface plot and contour plot, 113–120 16 layered architecture (VGG16), 29f 33-bus radial system, 67 1993 milestone National Polyp Study, 22

A ACEEE study, 159 Adenoma detection rate (ADR), 22 Adenomatous polyps, 22 Administrative Atlas-Uttarakhand 2011, 184f Advanced components, 172t, 174 control methods, 172t, 174 meter infrastructure (AMI), 169 Advancement of vitality creation, 164 Aggregated TCLs, 60 AI-based detection, 21–33 Air handling unit (AHU) panel, 155 Algorithms based on ordinal ranking, 54 Algorithms based on valuations/cardinality, 54 Allocation of a divisible/ indivisible goods, 48, 53 Analysis of energy bill, 156 of variance (ANOVA), 135, 140t Anatomy of the gastrointestinal system, 22f Ancillary services, 58 Annotation, 25, 26 Annual energy consumption, 155 Anticipated spike, 59 Appell’s symbol, 81 Approval and License, 168 Approximation algorithms, 52 Artificial neural network (ANN), 23 Automated examination, 23 Automation of DR programs, 59 Auxiliary Equations, 6 Average consumption, 152f, 156f

B Backward time central space (BTCS) method, 8

Barriers to the Implementation of a Smart Grid, 172 Benchmarking electricity, 150 Beta-integral, 86 Bi-criteria approximation algorithm, 55 Bidirectional flow of data, 164 Biodiesel blending, 134 production techniques, 134 Black-box models, 150 BMX allocations, 54 Borda maximin (BMX), 53 Box-Behnken design (BBD) method, 135, 139, 140t Brilliant substations, 164

C Cake-cutting protocols, 50 Catalyst of KOH, 136 Celiac disease classification, 24 Central composite design (CCD), 135, 140t Central Electricity Authority (CEA) 2019, 165 Central Electricity Regulatory Commission (CERC), 165 Central microgrid controller, 63 Cleat compressibility, 4, 20 permeability, 4, 7 CNN architecture for polyp detection, 23 Coal Bed Methane Reservoir, 3 Coal formation heterogeneous, 4 isotropic, 4 Coefficient of determination (R2), 138 Colonoscopy, 22–23, 25, 26t Colorectal cancer, 21, 25f malignant, 22 tumors, 22 Colored Red Green Blue (RGB) images, 26 Combinatorial identities, 90, 94 Composite conductors, 175 Companion matrix, 88, 89 Computer-aided diagnosis, 25 Concentric circles, 113, 114f Controllable loads (deferrable and interruptible loads), 58

193

194 Control system technologies, 59 Conventional colonoscopy, 23 energy combination, 172t machine learning, 24f Convolutional layers, 27 neural network, 23 Cost minimization Supply chain model, 36 CVC-Clinic DB, 25, 26f Cost of load shedding, 63, 65 Curve from negative infinity, 101 Curve fitting, 157f Customer-rewards scheme, 61 Cut-and-choose protocol, 50 Cybersecurity, 177, 178

D Data privacy, 177 Dataset CVC-Clinic DB, 25 Deep Learning Neural Network, 23 Deferrable load electric vehicles, washers, and dryers, 58–59 Demand elasticity parameters, 36 Demand response (DR) controller, 57–72 Demand-side management (DSM), 166 Design-Expert® software, 137 Design of experiment (DOE), 135 Desmos, 101, 122 Desorption pressure, 7 Deterministic market demand, 36 Dewatered coal, 4 Diffusive, Parabolic Equation, 7 Discrete and bounded envy-free protocol, 52 Discussion on enablers, 174 Divide-and-choose protocol, 47 Division of the whole cake, 51 DOE software, 138 Double series relations, 81 Drained distance, 5f DR device, 67 DR objectives with respect to timescale, 59

E Educational Institution in India, 149 Economic Order Quantity (EOQ), 36 Edible oil, 134 Egalitarian social welfare, 48, 50 Electric control network, 164 Electricity theft, 165 Empirical fluid properties, 7 Empirical relations stress-dependent cleat permeability, 7 stress-dependent cleat porosity, 7

Index Enablers for Smart Grid Deployment in India, 172t Endoscopic images vision challenge, 25–26 Endoscopist, 22 End-user comfort, 59, 63 Energy audit of a Russian educational institution, 151 consumption baseline, 151 supply, 57, 166 Energy data analysis, 149–159 Energy storage technologies, 182t Environmental and economic drawbacks, 57 Envy-ensuring, 54 Envy-free (EF) up to one good (EF1), 48 up to least valued good (EFX), 48 allocation, 48 α-approximately, 52 Exponential curve, 102f Extended forms of inversion pairs of Gessel and Stanton, 90 Externally studentized residual vs. predicted biodiesel, 142f Extreme learning machine (ELM), 135

F Fair allocation of items, 47–55 Finite difference, 19 First-generation biodiesel, 134 Fisher test, 135 Floor space constraints, 42 Fluid flow stages, 3–4 relative permeability values, 7 saturations, 7 Fokker-Planck equations, 59 Food security problem, 134 Fossil fuels, 164, 165 F-test, 138 Fuzzy environment, 36

G Gas saturation equation, 13 Gastrointestinal polyps, 21, 29 General class of polynomials, 79–99 General Lagrange inversion pair, 79 Generalized hypergeometric function, 81 Genetic algorithm, 36, 54 Gessel and Stanton, 79 Global minima, 119, 121 Greenhouse gases (GHG), 164 Grid-friendly appliances, 174 Google Earth, 184f, 185f

195

Index H

K

Harsil in Uttarkashi Gangotri road, 183 Hessian matrix, 40 Hierarchical Demand Response Controller, 57–72 relationship, 61 High electricity prices, 59 High-level optimizer (HLO), 61 High tension/voltage side meters, 156 Homogeneous movement of reactants, 134 Horizontal stress, 7 Hybrid optimization algorithm, 36 Hybrid optimization model for electric renewable (HOMER), 183 Hybrid power system, 181, 182f, 183 Hydrodynamic cavitation (HC), 134 Hydro power generation, 181 Hyper dimensions, 126 Hypergeometric function, 81, 86–88 Hypergeometric form of extended Racah polynomial, 98 Hypersphere, 127f

Kaluza-Klein compactifications, 127 Kalman filter, 59 Kakoragad stream, 183 Key issues of Indian grid, 166–168 KOH amount, 143

I ImageNet database, 24 Image segmentation, 25 Independent squeezing, 106 Indian grid, 165–168 Indivisible or non-shareable items, 53 Industrialization, 165 Information Communication Technology (ICT), 169 Information Technology (IT), 169 Insufficient fuel supply, 166 Instant price discounts policy, 36 Integral of the Jacobi polynomial, 98 Integral representation, 86 Intelligent and powerful segments, 164 Intermittent VRE sources, 58 Internet of Things (IoT), 151 Interruptible loads, 59 Inverse series relations, 81 Inversion pair of extended Bessel polynomial, 94 extended Hahn polynomial, 93 extended Jacobi polynomial, 93 extended Legendre polynomial, 94 extended Laguerre polynomial, 94 extended Racah polynomial, 98 extended Wilson polynomial, 93 Inverter-based microgrids, 61 Investment/financial problem, 167 Iterative method, 19, 121

J Jacobian matrix, 64

L Lack of fit (LOF) test, 138 Lagrange’s inversion formula, 79 Lagrangian multiplier, 39 Land acquisition, 167 Large-scale loads, 59 Layered architecture of VGG16, VGG19, 29f Layout map, 183, 184f LEED certified buildings, 151 Life cycle cost (LCC), 187 Linearly squeezed exponential curve, 103f Load-following DR controller, 60 Load profile equation, 185 Load profile of cold storage, 185, 186f Loads (industrial, commercial, residential), 58 Load shedding strategies, 61 Loss of generation, 59 Low-level controller (LLC), 61 Low tension/voltage (LT) panel, 156

M Major load centers, 154 Major smart metering points, 150f Map of Uttarkashi district of Uttarakhand, 184f Mass continuity equation, 7 Mass transfer limitation, 134 Mathematical program, 62, 64 Maturity of technology, 173t, 175–176 Max pooling layers, 29 –30 Maximin aware (MMA), 53 Maximin share (MMS), 53 Material balance, 7 Mean square (MS), 138 Mean values of the physical characteristics, 67t Measuring and sensing, 174 Mechanical stirring (MS), 134 Medium voltage panel (MVP 2-1), 152, 154 Membrane reactor, 134 Methyl alcohol: oil ratio (molar ratio), 135t, 137t, 138t, 139 Microgrids grid-connected mode, 58 islanded mode, 58 optimization, 63 reliability, 58 Microscopic theory of gravity, 126 Microwave irradiation (MW), 134

196 Minima, maxima function, 121 Mixed-integer linear programming, 62 MMX allocation, 54 Mobile tower, 154 Modified IEEE 33-bus radial system, 67 Modeling TCLs, 59 Monochrome polyp masks, 26 M/s sonic vibracell, USA, 136 M/s Zenatix, 151 Multi-dimensions, 126, 129t

N National Bio-fuel Policy 2018, 134 National Buildings Energy Efficiency Program, 149 National Electricity Plan, 165 National Load Despatch Centre (NLDC), 165 National Renewable Energy Laboratory, 183 National Smart Grid Mission 2018, 166 Nash social welfare, 48 Neocognitron, 23 Net load prediction, 60–62 Non-commutative forms, 79 Non-contiguous pieces, 48 Non-linear mixed inventory problem, 36 partial differential equation, 7 Normal distributions, 66 Normal probability vs. residual plots of biodiesel, 141f

O Optimality criteria, 40 Optimal reaction condition, 144, 145f Optimal selling price, 40 Optimization at a single decision time step, 62 of the process parameters, 144 of biodiesel production processes, 136 Optimizer, 61, 62f Ordinal/cardinal preferences, 53 Outage management system (OMS), 170

P Parameters of pumped hydro, 188t Pareto-efficient allocation, 54 Pareto optimality, 48 Peak-load reduction, 61 Penetration level of VRE, 58 Permeability and porosity relation, 15 Permeability model, 4 Perturbation plots, 141, 142f

Index Physical and chemical influences in transesterification, 134 Plug-in electric vehicles (PEV), 164 hybrid electric vehicles (PHEV), 164 Pochhamer symbol, 81 Polynomial time algorithm, 54 Polypshyperplasias, 25 metastasis stage, 25 Polyurethane foam (PUF), 185 Porosity-permeability cubic relation, 7 Power demand, 183, 185 electronics, 174 grid, 57, 165 quality measurement (PQM), 169 system flexibility, 58 outages, 168 Power Grid Corporation of India Limited 2019, 166 Power System Operation Corporation Limited (POSOCO), 165 Price break policy, 36 sensitive demand, 35 Process intensification (PI), 134 Process temperature, 143 Profit maximization inventory model, 35 Programmable sonicator set up model, 136 Proportional solution, 50 Pseudo polynomial algorithm, 54 Pump as turbine, 183 Pumped storage as battery, 187 Pumped storage plant (PSP), 181 Purification of product, 134 P-value, 138 PV array output, 65 PV system, 183 Pythagoras theorem, 128 Python, 122

Q q-analogs, 79 q-Lagrange’s inversion pair, 79 Quadratic polynomial equations, 136 Quantum theory of gravity, 126

R Radially squeezing, 106 Ramp-up event in the net load, 67 Raw materials, 135 Reaction temperature, 143 Regional load dispatch center (RLDC), 165 Regression analysis, 136

197

Index Regression equation, 139 Reliability of power, 164 Remote area, 181, 183 Renewable energy sources, 57, 164, 181, 183, 190 Renewable integrations/microgrids, 170 Reservoir pressure, 3 Residential loads (critical and controllable), 58 Residential space heaters, 67 Resolution of data (Spatial, Temporal), 150 Reversible hydroelectric power plants, 183 Right angle triangle, 112f Risk of delivery, 176 Round-robin algorithm, 54 Round trip efficiency, 187 Response surface methodology (RSM) method, 136 RSM models, 136 RSM statistical analysis, 138

S Sample temperature profiles, 67 Seasonal and annual profile, 186 Second-generation feedstocks, 134 Second-order quadratic regression surface model, 138 Security focus, 164 Selfridge-Conway Protocol, 50 Sensitivity table, 41t Sequential representation of CNN architecture, 28f Shrinkage stress, 7 Simulation results, 66 Single-phase water flow, 7 Site location in map, 184f Skilled manpower shortage, 167 Slack bus, 67 variables, 63, 68 Slippage in generation, 167 Small hydro power (SHP) station, 181 Small-scale pumped storage, 181 Smart grid deployment in India, 171t growth in India, 168 implementation, 164 projects in India, 170 technology, 169 Smart transmission systems, 164 Sodium sulfur (NaS) battery, 174 Solar energy, 181 Solar-hydro hybrid system, 189 Solar photovoltaic (PV) power plant, 181 Solar power output, 187f Special cases of integrals, 97 inverse series relation, 90

Specification of battery, 189t Squeezed grid, 105f Squeezing graphs, 101 Squeezing in infinite space, 103 limited/definite space, 103 Squeezing the cartesian coordinates, 103 Stackelberg-Nash equilibrium game approach, 36 Standards and regulations, 170 Stakeholders, 168 State load dispatch centers (SLDCs), 165 State queuing, 59 Sticking effect, 142 Stochastic nature, 60 String theory, 126 Sum of the square root (SS), 138 Sump motor pumps, 154 Superconducting apparatus, 174 Superconducting magnetic energy storage (SMES), 174 Supercritical carbon dioxide (sCO2), 7 Supermarket problem, 40 Supervised learning algorithms, 23 Supply-demand balance, 57 Surface area of the hyperspheres, 128, 129t Surveillance research, 22f Sustainable energy infrastructure, 57 power source assets, 164 Swelling and shrinkage stresses, 7

T Tariff structures, 174 TCL-based DR modeling and control, 59 Temperature dead-band of TCLs, 62 dynamics of TCLs, 62 TERI School of Advanced Studies (TERI SAS), 150 Theft of power, 167 Theory of everything, 126 Thermostatically controlled loads (TCLs), 59 Theta-form differential equation, 81 Three-dimensional coal formation, 5f Three necks borosil rector, 136 Toroidal surface, 117f, 118f, 119f Total absolute revenue, 36 Total load profile, 186f Transesterification reaction, 136 triglycerides, 134 Transfer learning, 24 Transformer’s loading, 156 Tri-diagonal matrix algorithm (TDMA) approach, 8 Two-level controller, 61 Two-way communication of information, 164

198 Types of valuations (Additive, Binary additive, Subadditive (SA), Submodular (S)), 53

U Ultracapacitors, 175 Ultrasonication (US) methods, 134 Ultrasound-assisted KOH catalyzed biodiesel production, 133 Unsupervised learning algorithms, 23 Utilitarian social welfare, 50 Unbalance in power system, 158f Unsaturated fatty acid methyl ester, 136 Urbanization, 165

V Vanadium redox flow battery, 175 Variable coefficients, 6 Variable generation tracking, 69f VCX 500, 136 Vendor’s total revenue, 37 VGG implantation, 30 Visual pattern recognition mechanism, 23

Index Voltage feedback DR controllers, 61 Voltage magnitude at all buses, 71f Voltage regulation, 174 Volume of the hypersphere, 129, 129t VRE curtailment, 63

W Waste cotton-seed cooking oil (WCCO), 133 Waste/used cooking oil (WCO/UCO), 134 Water-gas saturation relation, 12 Water flow through cleats, 8 Water saturation equation, 13 Weighted moving average (WMA) predictor, 64 Wind energy, 181 World Health Organization, 21 Worldwide energy requirements: Coal, flammable gas, unrefined petroleum, 133 Worldwide UCO market, 134

Z ZIP model of the load, 65 Zenatix installed smart meters, 151