Advanced Methods for Processing and Visualizing the Renewable Energy: A New Perspective from Signal to Image Recognition 9789811586057, 9789811586064

Table of contents :
Preface
Contents
Editors and Contributors
Rational Quartic Spline Interpolation and Its Application in Signal Processing
1 Introduction
2 Construction of Rational Quartic Spline Interpolant
2.1 Rational Quartic Spline (RQS)
2.2 Derivative Estimation
2.3 Derivative Estimation
2.4 Convergence Analysis of RQS
3 Data Interpolation Using RQS
4 Discussion
5 Applications in Signal Processing
6 Conclusion
References
A Controller for Natural Gas Fuel Dispenser with Multi-Level-Pressure Banks
1 Introduction
2 NGV Dispenser Model
3 Time-Optimal Control Model of NGV Dispenser
3.1 Parameter Identification
3.2 Development of Switching Time Equation
3.3 Designing Governing Equations
3.4 Implementation of Pontryagin’s Minimum Principle
3.5 Designing Forced Trajectory
3.6 Derivation of Optimal Switching and Total Minimum Time
3.7 Modeling of Forced Trajectory Using MATLAB/Simulink
3.8 Simulation Example
4 Results and Discussion
4.1 Performance of Refueling by Varying Initial Pressure Inside Receiver Tank
4.2 Performance of Refueling Using Multi-Level-Pressure Storage Banks
5 Conclusion
References
Power Performance Analysis of Solar Tracking System in UTP
1 Introduction
2 Related Literature Review
3 Project Flow and Methodology
4 Data Collection
5 Results and Discussions
6 Conclusion
References
Artificial Neural Network Modeling of Nanoparticles Assisted Enhanced Oil Recovery
1 Introduction
2 Mathematical Modeling of Nanoparticles Flow in the Reservoir
3 Data Collection
4 ANN Model Development
5 Results and Discussion
6 Conclusion
Appendix: ANN Matlab Code
References
Viable Options and Opportunities for Energy Saving in a Distribution System Towards Sustainability: Taylor’s University as the Case
1 Introduction
2 PV Installation Option
3 Orientations of the Placements of the PV
4 Energy Flow Analysis
5 Economic Analysis
6 Conclusion
References
C1 Surface Interpolation Using Quartic Rational Triangular Patches
1 Introduction
2 Rational Quartic Triangular Patches
3 C1 Continuity Between Two Patches
4 Error Calculation
5 Numerical Results and Discussion
6 Conclusion
References
Construction and Application of Septic B-Spline Tensor Product Scheme
1 Introduction
1.1 Literature Review
2 Preliminaries
2.1 Properties of the Scheme
3 Construction and Analysis of Septic B-Spline Tensor Product Scheme
3.1 Preliminaries
3.2 Construction of Septic B-Spline Tensor Product Scheme
3.3 Analysis of Septic B-Spline Tensor Product Scheme
4 Numerical Examples
5 Conclusion
References
Bayes Meets Tikhonov: Understanding Uncertainty Within Gaussian Framework for Seismic Inversion
1 Introduction
2 Regularisation Method of A. N. Tikhonov
3 Bayesian-Tikhonov Formulation for Linear Inverse Problems
4 The Maximum a Posteriori (MAP) Point
5 A Generalised Formulation
6 The a Posteriori Covariance Matrix
7 The Resolution Matrix
8 The Optimal Low-Rank Approximation of a Posteriori Covariance Matrix
9 Numerical Examples
10 Conclusions
References
Index

Citation preview

Studies in Systems, Decision and Control 320

Samsul Ariffin Abdul Karim Nordin Saad Ramani Kannan   Editors

Advanced Methods for Processing and Visualizing the Renewable Energy A New Perspective from Signal to Image Recognition

Studies in Systems, Decision and Control Volume 320

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink.

More information about this series at http://www.springer.com/series/13304

Samsul Ariffin Abdul Karim Nordin Saad Ramani Kannan •

•

Editors

Advanced Methods for Processing and Visualizing the Renewable Energy A New Perspective from Signal to Image Recognition

123

Editors Samsul Ariffin Abdul Karim Fundamental and Applied Sciences Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System Universiti Teknologi PETRONAS Seri Iskandar, Selangor, Malaysia

Nordin Saad Electrical and Electronic Engineering Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System Universiti Teknologi PETRONAS Seri Iskandar, Selangor, Malaysia

Ramani Kannan Electrical and Electronic Engineering Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System Universiti Teknologi PETRONAS Seri Iskandar, Selangor, Malaysia

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-15-8605-7 ISBN 978-981-15-8606-4 (eBook) https://doi.org/10.1007/978-981-15-8606-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book is a collection of the works that have been conducting by researchers at Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS (UTP) as well as contribution from Taylor’s University, Malaysia, and Seismic Modeling and Inversion Group, King Abdullah University of Science and Technology (KAUST), Saudi Arabia. There are eight chapters contributed in this book. Data interpolation is essential in geometric modelling and computer graphics (CG) epecially to model some complex shapes or surfaces. There is a need for the industry of geometric modelling to make changes in the shapes of the curves/surfaces without relying on new data, and this topic has been well-described in this book. A refuelling algorithm using time-optimal control (TOC) for optimal switching of natural gas vehicle (NGV) refuelling with multi-level-pressure storage banks is presented. Fundamental issues in the development of simulation model for an NGV refuelling system and methodology for optimization of the switching are discussed. In the recent years, renewable energy has been a lot in talks, and the most booming renewable energy currently is the solar energy. This book presents the design of prototype and analysis of performance and effectiveness of a single axis and a fixed axis solar tracker with an active closed-loop system. The development of artificial neural network and deep learning algorithms in the last decade has provided a crucial development to solve complex mathematical modelling problems. The current state-of-the-art literature on implementation of artificial neural network and deep learning algorithms in simulation reservoirs for enhanced oil recovery applications has been well-discussed in this book. A chapter has been allocated for a typical energy audit program profound for energy-saving perspective and another chapter for the viability of a PV system through technical, environmental and financial aspects, based on PV sizing and cost analysis. Apart from these, a tensor product of septic B-spline subdivision scheme has been proposed and analysed. In this chapter, the sound connection between the Bayesian approach and the Tikhonov regularization within the Gaussian framework has been demonstrated. The editors would like to express their gratitude to all the contributing authors for their great efforts and full dedication in preparing the manuscripts for the book. We would like to thank all reviewers for reviewing all v

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Preface

manuscripts and providing very constructive feedback. The first editor is fully supported by Universiti Teknologi PETRONAS (UTP) and the Ministry of Education, Malaysia, through a research grant FRGS/1/2018/STG06/UTP/03/ 1015MA0-020 (New rational quartic spline for image refinement) and YUTP: 0153AA-H24 (Spline Triangulation for Spatial Interpolation of Geophysical Data). The reader from this book will know about current trend of technologies, and this book could be a good reference for the postgraduate studies and researches. Any feedback can be directed to the first editor. Seri Iskandar, Malaysia July 2020

Samsul Ariffin Abdul Karim Nordin Saad Ramani Kannan

Contents

Rational Quartic Spline Interpolation and Its Application in Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noor Adilla Harim, Samsul Ariffin Abdul Karim, Mahmod Othman, Abdul Ghaffar, and Kottakkaran Sooppy Nisar A Controller for Natural Gas Fuel Dispenser with Multi-Level-Pressure Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nordin Saad and Mahidzal Dahari Power Performance Analysis of Solar Tracking System in UTP . . . . . . Ramani Kannan and Ishwerjeet Singh Inderjeet Singh Artificial Neural Network Modeling of Nanoparticles Assisted Enhanced Oil Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sayed Ameenuddin Irfan and Afza Shafie Viable Options and Opportunities for Energy Saving in a Distribution System Towards Sustainability: Taylor’s University as the Case . . . . . . Chockalingam Aravind Vaithilingam, Reynato Andal Gamboa, and Yen Ling Lai C1 Surface Interpolation Using Quartic Rational Triangular Patches . . . Nur Nabilah Che Draman, Samsul Ariffin Abdul Karim, and Ishak Hashim

1

25 47

59

77

89

Construction and Application of Septic B-Spline Tensor Product Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Mudassar Iqbal, Samsul Ariffin Abdul Karim, and Muhammad Sarfraz Bayes Meets Tikhonov: Understanding Uncertainty Within Gaussian Framework for Seismic Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Muhammad Izzatullah, Daniel Peter, Sergey Kabanikhin, and Maxim Shishlenin Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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Editors and Contributors

About the Editors Samsul Ariffin Abdul Karim is a senior lecturer at Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS (UTP), Malaysia. He has been in the department for more than eleven years. He obtained his B.App.Sc., M.Sc. and Ph.D. in Computational Mathematics and Computer Aided Geometric Design (CAGD) from Universiti Sains Malaysia (USM). He had 20 years of experience using Mathematica and MATLAB software for teaching and research activities. His research interests include curves and surfaces designing, geometric modelling and wavelets applications in image compression and statistics. He has published more than 120 papers in journal and conferences as well as seven books including two research monographs and three Edited Conferences Volume and 25 book chapters. He is the recipient of Effective Education Delivery Award and Publication Award (Journal and Conference Paper), UTP Quality Day 2010, 2011 and 2012, respectively. He was Certified WOLFRAM Technology Associate, Mathematica Student Level. He has published four books with Springer.

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Editors and Contributors

Nordin Saad is currently an Associate Professor in the Department of Electrical and Electronics Engineering, Universiti Teknologi PETRONAS, Malaysia. He obtained a B.Sc. degree in Electrical Engineering from Kansas State University, USA; a M.Sc. in Power Electronics Engineering from Loughborough University, UK; and a Ph.D. in Control and Systems Engineering from The University of Sheffield, UK. He was the Head, Department of Electrical and Electronics Engineering at the current university from 1998 to 2000, having joined the university in 1997. He was appointed as Cluster leader for the Industrial Automation and Control from 2005 to 2012, and subsequently Co-Cluster leader for the Power control and Instrumentation from 2013 to 2014. Currently, he is leading the Centre for Smart Grid Energy Research. There is a total of seven Ph.D.s’ and four M.Sc.s’ (by research) that had graduated under his supervision. He had authored and co-authored two books and about 130 journals, transactions, book chapters and technical papers and received a number of best paper awards at international conferences for the research conducted by his team. His research work encompasses some of the issues that include computer/embedded systems/PLC control of processes, instrumentation and control of process plant facilities, condition monitoring and diagnostic of machines, and power electronic converters for high power transmissions, electrical drives control and smart electrical energy systems. He is a Chartered Engineer registered under the Engineering Council, UK; a senior member of the Institute of Electrical and Electronics Engineers (SMIEEE), the USA; and a member of the Institute of Measurement and Control (MInstMC), UK.

Editors and Contributors

xi

Dr. Ramani Kannan is a Senior Lecturer in Universiti Teknologi PETRONAS, Malaysia. He received his B.E. degree from Bharathiar University, India. Later, completed his M.E. and Ph.D. in Power Electronics and Drives from Anna University respectively. He holds more than 115 publications in reputed international and national journals and conferences. He is an active senior member in IEEE(USA), and members of IE(I), IET(UK), ISTE(I) and Institute of Advanced Engineering and Science. He is recognized with many awards, including “Career Award for Young Teacher” from AICTE India, 2012; “Young Scientist Award” in power electronics and Drives, 2015; “Highest Research publication Award” 2017. Award for Outstanding Performance, Service and Dedication 2019 at UTP, Malaysia; and “Outstanding Researcher Award” in UTP Q Day 2019, Best Presenter Award, IEEE CENCON 2019 international conference at Indonesia. EDITOR–Books Sustaining Electrical Power Resources through Energy Optimization and Future Engineering Springer Nature Singapore Pte Ltd 2018., and Practical Examples of energy optimization models Springer Nature Singapore Pte Ltd 2019. He has completed 5 funded projects and 7 research projects in progress. The grants are FRGS, ASEAN-India, YUTP, KETTHA and STIRF. He is the Editor-in-Chief for the Journal of Asian Scientific Research (2011–2018) and Regional Editor for International Journal of Computer Aided Engineering and Technology, Inderscience Publisher, UK, from 2015. He is an Associate Editor in IEEE Access journal since 2018. He is servicing many guest editors such as Elsevier journal, Inderscience, IGI Global and IJPAM. His research interest involves in power electronics, inverters, modelling of induction motor and optimization techniques.

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Editors and Contributors

Contributors Samsul Ariffin Abdul Karim Fundamental and Applied Sciences Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Mahidzal Dahari Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia Nur Nabilah Che Draman Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Reynato Andal Gamboa Batangas State University, Batangas City, Philippines Abdul Ghaffar Informetrics Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Noor Adilla Harim Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Ishak Hashim Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, UKM Bangi, Selangor, Malaysia Mudassar Iqbal Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Sayed Ameenuddin Irfan Shale Gas Research Group (SGRG), Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Muhammad Izzatullah Seismic Modeling and Inversion Group, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Sergey Kabanikhin Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia; Novosibirsk State University, Novosibirsk, Russia; Sobolev Institute of Mathematics, Novosibirsk, Russia; Mathematical Center in Akademgorodok, Novosibirsk, Russia Ramani Kannan Electrical and Electronics Engineering Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia Yen Ling Lai Faculty of Innovation and Technology, Taylor’s University, Subang Jaya, Malaysia Kottakkaran Sooppy Nisar Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser, Kingdom of Saudi Arabia

Editors and Contributors

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Mahmod Othman Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Daniel Peter Seismic Modeling and Inversion Group, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Nordin Saad Electrical and Electronics Engineering Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Muhammad Sarfraz Department of Information Science, College of Life Sciences, Kuwait University Sabah AlSalem University City, Safat, Kuwait Afza Shafie Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Maxim Shishlenin Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia; Novosibirsk State University, Novosibirsk, Russia; Sobolev Institute of Mathematics, Novosibirsk, Russia; Mathematical Center in Akademgorodok, Novosibirsk, Russia Ishwerjeet Singh Inderjeet Singh Electrical and Electronics Engineering Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia Chockalingam Aravind Vaithilingam Faculty of Innovation and Technology, Taylor’s University, Subang Jaya, Malaysia

Rational Quartic Spline Interpolation and Its Application in Signal Processing Noor Adilla Harim, Samsul Ariffin Abdul Karim, Mahmod Othman, Abdul Ghaffar, and Kottakkaran Sooppy Nisar

Abstract Data interpolation is essential in geometric modelling and computer graphics (CG) specially to model some complex shapes or surfaces. There is a need for the industry of geometric modelling to make changes in the shapes of the curves/surfaces without relying on new data. To achieve this, we proposed a new rational quartic spline (RQS) scheme with three free parameters. We derive the convergence analysis based on Peano-Kernel theorem. Furthermore, the proposed scheme used for shape control and error analysis by manipulating the values of the free parameters. We have calculated the absolute error, Root Mean Square Error (RMSE) and higher coefficient of determination (R2 ) as error measurement. It observed that the proposed scheme gives higher accuracy in term of smaller error as compared to some existing schemes. This showed that the data interpolation using new RQS scheme with three parameters gives better results as compared to existing N. A. Harim · M. Othman Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar Perak Darul Ridzuan 32610, Malaysia e-mail: [email protected] M. Othman e-mail: [email protected] S. A. A. Karim (B) Fundamental and Applied Sciences Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, Seri Iskandar, Perak Darul Ridzuan 32610, Malaysia e-mail: [email protected] A. Ghaffar Informetrics Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam e-mail: [email protected] K. S. Nisar Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser 11991, Kingdom of Saudi Arabia e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4_1

1

2

N. A. Harim et al.

quartic polynomial. Furthermore, an application in signal processing shows that the proposed RQS is highly accurate to increase the discrete-time signal sampling. Keywords Rational quartic spline · Data interpolation · Error analysis · Signal processing

1 Introduction Data interpolation is a field related to geometric modelling and computer graphics (CG) as well as classic differential geometries. In this study, we followed procedures for interpolating curves that could improve the closing shape of the interpolated data by manipulating the values of shape control parameters. The important tool of CG is data interpolation. As an example, if the assumed data is positive, then the subsequent interpolation curves and surfaces must be positive everywhere. Many methods can used to interpolate the data sets such as cubic spline interpolation and rational spline. Here, we give a brief review of data interpolation. Fritsch and Carlson [1] have given the concept that the cubic Hermite spline can be used for the shape preservation i.e. monotonicity and convexity preservation. For shape preservation, their method needed the adjustment of the first derivation of the monotonicity of the interpolating curve that can established on some interval. Brodlie and Butt [2, 3] have extended the works of Fritsch and Carlson [1] by constructing piecewise continuous cubic Hermite interpolant that preserves the positivity and monotonicity. Other works in shape preserving interpolation are well described in [1, 4, 5, 9, 15]. Wang and Tan [6] have constructed the new RQS scheme with two shape control parameters and they have derived the sufficient conditions for shape preservation of the RQS scheme. Wang and Tan [7] have also made extension in the univariate RQS scheme of [6] to the bivariate scheme; and obtained the necessary condition for the monotonicity preservation of the monotone surface data. The key disadvantage of their work is that the first order partial derivative still required to be improved if the non-monotonicity occurred in x- or y-dimension. Han [5] has introduced a new class of piecewise RQS interpolation scheme with one shape control parameter. Wang et al. [8] have introduced the weighted RQS scheme by using the work of Wang and Tan [6], which based on the RQS scheme of Deng et al. [9]. Liu et al. [10] have established the new RQS scheme based on the functional values. Zhu [11] has constructed new RQS scheme with four parameters, however his scheme is too complicated, and the derivation is difficult to understand. Therefore, to address this drawback we improved the RQS scheme of Wang and Tan [6] by increasing the degree of the denominator from one (linear) to two (quadratic) with three tension parameters. This will enable us to have more degree of freedom to interpolate the curves. This study is an extension to our previous work in Harim et al. [12, 13]. Three main features of the proposed rational quartic spline for data interpolation are summarized below:

Rational Quartic Spline Interpolation and Its Application …

3

1. The proposed scheme has three parameters meanwhile Wang and Tan [6] only has two parameters. By having more parameters, more accurate interpolating curve is obtained. 2. Error estimation shows that the proposed scheme gives higher accuracy in term of smaller error compared with quartic polynomial, Wang and Tan [6] and Zhu [11] scheme. 3. The proposed scheme is suitable for signal processing application. The rest of this chapter is organized as follows. Section 2 is dedicated to the construction of RQS scheme. In Sect. 3, we discuss data interpolation for 2D data using the proposed scheme including convergence analysis, as well as the Results and Discussion (including the comparison with existing schemes and default quartic polynomial using error analysis). The summary is given in the final section.

2 Construction of Rational Quartic Spline Interpolant In this section, we discussed the construction of the RQS scheme with three parameters and its analysis by employing the values of shape control parameter.

2.1 Rational Quartic Spline (RQS) Given scalar data set {(x I , f i ), i = 1, 2, . . . n} where x1 < x2 < · · · < xn and the first derivative d i , at the respective point. Then the RQS scheme with three shape control parameters αi > 0, βi > 0 and yi ≥ 0 on the interval is given by [12]: S(x) =

P1 (θ) Qi (θ)

(1)

where Pi (θ ) = (1 − θ )4 αi f i + (1 − θ )3 θ Ai + (1 − θ )2 θ 2 Bi + (1 − θ )θ 3 Ci + θ 4 βi f i+1 Q i (θ ) = αi (1 − θ )2 + γi (1 − θ )θ + βi θ 2 fi i with h i = xi+1 − xi , i = fi+1h− and θ = x−x . hi i 1 The RQS scheme in Eq. (1) gives the C continuity at the knots and satisfy the following conditions:

S(xi ) = f i S(xi+1 ) = f i+1 S (1) (xi ) = di S (1) (xi+1 ) = di+1

(2)

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N. A. Harim et al.

From Eq. (2) and after some derivations, the unknown Ai , Bi and Ci are given as: Ai = (2αi + γi ) f i + αi h i di Bi = (αi + γi ) f i+1 + (βi + γi ) f i Ci = (2βi + γi ) f i+1 − βi h i di+1

(3)

2.2 Derivative Estimation Suppose a 2D set of data {(xi , f i ), i = 1, 2, . . . , n} is considered that x1 < x2 < · · · < xn The first derivatives, d i at the point xi , i = 1, 2, . . . , n. Let h i = xi+1 − xi fi and i = fi+1h− . i At the end points of x1 and xn . 

 h1 d1 = 1 + (1 − 2 ) h1 + h2   h n−1 dn = n−1 + (n−1 − n−2 ) h n−1 + h n−2

(4) (5)

At the other point, xi = 2, 3, . . . , n − 1 the value of di are given as  di =

h i−1 i + h i i−1 h i−1 + h i

 (6)

2.3 Derivative Estimation Based on Eq. (1), some observations can be made as follows: (a) The RQS scheme in Eq. (1) is reduced to the standard quartic polynomial spline when αi = 1, βi = 1, γi = 2 as shown in Eq. (7). (b) S(x) = (1 − θ )4 f i + (3 f i + h i di )(1 − θ )3 θ + (3 f i + 3 f i+1 )(1 − θ )2 θ 2 + (3 f i+1 − h i di+1 )(1 − θ )θ 3 + θ 4 f i+1

(7)

(c) The RQS scheme defined in Eq. (1) converges to a straight line when γi → ∞ or αi , βi → 0 such that lim s(x) = lim s(x) = (1 − θ ) f i + θ f i+1

γi→∞

αi,βi →0

(8)

Rational Quartic Spline Interpolation and Its Application …

5

Table 1 Data from [14] i

1

2

3

4

5

6

xi

1

2

4

5

7

8

7 9

fi

24.616

2.4616

41.027

4.1027

57.437

5.7428

0.5744

(d) The RQS scheme in Eq. (1) is reduced to the rational spline of the form quartic/quadratic when αi > 0, βi > 0 and γi ≤ −2. There will be some vertical asymptotes or poles at θ = 0.5. For instance, when αi = βi = 1 and γi = −2, the proposed RQS defined in Eq. (1) can be written as s(x) =

      (1 − θ)4 f i + h i di (1 − θ)3 θ + − f i+1 − f i (1 − θ)2 θ 2 + −h i di+1 (1 − θ)θ 3 + θ 4 f i+1 2 (1 − 2θ)

(9)

Similar argument applied when αi < 0, or βi < 0. There are also vertical asymptotes in proposed RQS as in observation (9). For this observation, the poles at √ 5−1 θ= 2 . s(x) =

      (1 − θ)4 f i + 3 f i + h i di (1 − θ)3 θ + 2 f i+1 (1 − θ)2 θ 2 + 2 f i+1 + h i di+‘ (1 − θ)θ 3 + θ 4 f i+1 2 1−θ −θ

(10)

To show the example of shape control and geometric properties of the proposed RQS, we used the data of Sarfraz et al. [14] (Table 1) and found the effect of free shape control parameters of the RQS scheme defined in Eq. (1). Figure 1 represents the interpolating curve obtained by the proposed scheme using the parametric values given in Table 2. Figures 1a–c express the role of free shape control parameters when the RQS scheme is applied on discrete data points. Figure 1c produces smoother curve as compared to Fig. 1b, d converges the curve to a straight-line segment when the values αi and βi are the same. Figure 1e represents the interpolating curve when the values of parameters are negative. It can see clearly that in Fig. 1e, there are vertical asymptotes. With shape control analysis and geometric properties, we have restricted the free parameters to αi , βi > 0 and γi ≥ 0.

2.4 Convergence Analysis of RQS In this section, we will derive the error analysis of the proposed RQS by using Peano-Kernel Theorem. The main result is given in Theorem 1.   Theorem 1 Let f(x) ∈ C3 xI , xi+1 , i = 1, 2, . . . , n − 1. Since all parameters value satisfy αi , βi > 0 and γi ≥ 0 then | f (x) − S(x)| ≤



h i3 (3) h i   f i − di , f i+1 f + − di+1 96 4

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N. A. Harim et al.

(b)

(a)

(d) (c)

(e) Fig. 1 Shape control analysis

Rational Quartic Spline Interpolation and Its Application …

7

Table 2 RMSE value and R2 for non-polynomial fitting Figure 1

Value αi

Value βi

Value γi

Observation

(a)

1

1

1 (black)

–

1

1

0.5 (red)

–

1

1

2 (orange)

Default quartic polynomial

0.1 (black)

1

1

–

0.01 (orange)

1

1

–

3 (red)

1

1

1

0.2 (black)

1

1

0.02 (orange)

1

1

3 (red)

1

0.01 (black)

0.01 (black)

1

Converge to a straight line

0.001 (orange)

0.001 (orange)

1

Converge to a straight line

0.1 (red)

0.1 (red)

1

Converge to a straight line

1

1

−2 (black)

–

−3 (orange)

1

1

–

1

−3 (red)

1

–

(b)

(c)

(d)

(e)

 1 hi αi (−2i − di+1 + di ) + βi (−2i − di+1 − di ) + √ 2 αi βi + γi 27 1 1 + |γi (i − di )| + |γi (−2i + di+1 + di )| 16 27 Proof Since the interpolant is local, it is sufficient to consider on the interval   x , i = 1, 2, Let H3 (x) = (1 − θ)2 f i + 2 f i + h i f d i (1 − θ )2 θ + , x i i+1  . . . , n −1.  2 2 f i+1 − h i f d i+1 (1 − θ )θ + θ 2 f i+1 and setting f i = f  (x), i = 1, 2, . . . , n − 1,      then H3∗ (x) = (1 − θ)2 f i + 2 f i + h i f i (1 − θ)2 θ + 2 f i+1 − h i f i+1 (1 − θ )θ 2 + 2 θ f i+1 Whereas, after some simplification, the RQS defined in Eq. (1) can be written as h i (1 − θ)2 θ 2  αi (−2i − di+1 + di )((1 − θ )) Q i (θ )  +βi (−2i − di+1 − di )θ + γi (−2i + di+1 + di )θ + γi (i − di )

S(x) = H3 (x) +

Since H3∗ (x) is a cubic Hermite interpolant to f (x), then for f (x) ∈ C 3 [a, b], 3 f (x) − H ∗ (x) ≤ h i f (3) . 3 96

Meanwhile,

8

N. A. Harim et al.



 f (x) − H ∗ (x) ≤ h i f  − di , f  − di+1 3  i+1  4√ i Q i (θ ) ≥ (1 − θ )θ 2 αi βi + γi 1 1 1 , θ 2 (1 − θ ) ≤ 27 and (1 − θ)2 θ 2 ≤ 16 . By combining all these and (1 − θ)2 θ ≤ 27 terms, we will obtain the result stated in Theorem 1   Theorem 2 Let f(x) ∈ C4 xi , xi+1 , i = 1, 2, . . . , n − 1. Since all parameters value satisfy αI , βi > 0 and γi ≥ 0, then



h i4 (4) h i   f i − di , f i+1 + f − di+1 384 4  hi 1 αi (−2i − di+1 + di ) + βi (−2i − di+1 − di ) + √ 2 αi βi + γi 27 1 1 + |γi (i − di )| + |γi (−2i + di+1 + di )| . 16 27

| f (x) − S(x)| ≤

3 Data Interpolation Using RQS This section focuses on the example of the data interpolation by applying the RQS scheme with the existing schemes. As a validation, we have calculated error measurements for each data sets. Throughout this section, let f(x) represents true function for each example, p(x) is considered as quartic polynomial when αi , βi = 0 and γi = 2, w(x) is the Wang and Tan [6] scheme, z(x) is considered as Zhu [11] scheme and s(x) represents the proposed scheme with suitable parametric value. Formulas for absolute error, RMSE and R2 are described as: (a) Absolute Error Ae = |va − ve |

(11)

where va Approximation value ve Exact value from true function. (b) Root Mean Square Error (RMSE)

 RMSE = (c) Coefficient of determination (R2 )

n n=1 (ve

n

− va )2

(12)

Rational Quartic Spline Interpolation and Its Application …

9

   xy − x y R =   2  2   2  2  n x − n y − x y n

2



(13)

where n Number of samples x The value of data x y The value of data y. In order to choose the best possible parameters value that will give smaller absolute error and RMSE as well as higher R2 , we have implemented the following algorithm: Algorithm I Input: Number of points {(xi , f i )i = 1, 2, . . ., n}. Step 1: Plot the true function. Step 2: Plot the interpolation curve for Wang and Tan [6], Zhu [11] and the proposed scheme by selected value of parameters. Step 3: Manipulate the suitable value of parameters for the proposed scheme. Step 4: Calculate error analysis by using Eqs. (11), (12) and (13). Output: The smooth interpolating curves, the smallest error for absolute error, RMSE and R2 . The proposed Algorithm 1 is used in the numerical simulations. The simulations have been carried out several times, and the indicated parameters as listed Tables below are the best in numerical sense. MATLAB 2019 version installed on Intel® Core™ i5-8250U @ CPU 1.8 GHz is used to produce all graphical and numerical results. Example 1 Consider the function f (x) = e x , x ∈ [0, 4]. This is data taken from Duan et al. [15] and Table 3 shows the data points with first derivative values.   Example 2 The f (x) = cos π2x for x = {0, 0.25, 0.5, 0.75, 1} function of

  xπ  f (x) = cos 2 , x = 0, 0, 25, 0, 5, 0, 75, 1 is used. Table 8 shows the data points and values of derivatives using AMM by [4]. √ , k ∈ [0, 4] Example 3 The function of f (x) = − 1 − (x − 1) + 23 where, x = k+1 3 is used from Duan et al. [15]. Table 13 shows the sampling point of the function √ f (x) = − 1 − (x − 1) + 23 .

Table 3 Data from [15] i

0

1

2

3

xi

0

1

2

3

4 4

fi

1

2.71823

7.3891

20.0855

54.5982

di

0.2420

3.1945

8.6836

23.6045

45.4207

10

N. A. Harim et al.

4 Discussion In this section, the data interpolation given in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17) and in Figs. 1, 2, 3 and 4 by applying RQS scheme with three shape control parameters with the well-known methods of [4, 7, 10, 13, 15] are discussed. From these tables and figures of the general characteristics of the schemes, we can make the following observations: • In Example 1, we considered the function f (x) = e x , x ∈ [0, 4] (Duan et al. [15]). Figure 2a indicates the exact value for the original function and Fig. 2b represents the quartic polynomial when αi = 1, βi = 1, γi = 2. Figure 2c shows the result of Wang and Tan [10] scheme when αi = 1, βi = 2. Figure 2d indicates the Zhu [11] scheme with αi = 1, βi = 1, γi = 0.1, ei = 1. Meanwhile, Fig. 2e, f indicate the RQS scheme with αi = 1, βi = 1, γi = 0.1 and αi = 1, βi = 1, γi = 0.2, respectively. Table 4 shows the functional values while Table 5 indicates the absolute error obtained by using the different choices of proposed and existing schemes. Table 6 gives absolute error achieved from Eq. (9) by selecting three Table 4 Functional values for Example 1 x

f (x)

p(x)

w(x)

z(x)

s(x)

0.0

1.000000

1.000000

1.00000

1.000000

1.000000

0.2

1.221403

1.263704

1.23504

1.246712

1.228783

0.4

1.491825

1.586853

1.51743

1.557906

1.502644

0.6

1.822119

1.916987

1.84387

1.908855

1.821443

0.8

2.225541

2.267628

2.23386

2.288981

2.220894

1.0

2.718282

2.718282

2.71828

2.718282

2.718282

1.2

3.320117

3.435104

3.35719

3.374489

3.340178

1.4

4.055200

4.313514

4.12481

4.180823

4.084611

1.6

4.953032

5.210911

5.01216

5.107533

4.951196

1.8

6.049647

6.164053

6.07225

6.161674

6.037016

2.0

7.389056

7.389056

7.38906

7.389056

7.389056

2.2

9.025013

9.337582

9.12578

9.119782

9.079544

2.4

11.02317

11.72534

11.2123

11.21358

11.10312

2.6

13.46373

14.16472

13.6244

13.68388

13.45874

2.8

16.44464

16.75563

16.5060

16.60236

16.41031

3.0

20.08553

20.08553

20.0855

20.08553

20.08553

3.2

24.532530

25.382180

24.80643

24.666106

24.680761

3.4

29.964100

31.872798

30.47844

30.209091

30.181420

3.6

36.598234

38.503715

37.03516

36.871505

36.584667

3.8

44.701184

45.546535

44.86820

44.893181

44.607851

4.0

54.598150

54.598150

54.59815

54.598150

54.598150

Rational Quartic Spline Interpolation and Its Application …

11

Table 5 Values of absolute error for Example 1 X

p(x)

w(x)

z(x)

s(x)

0.00000

0.000000

0.00000

0.000000

0.000000

0.20000

0.042302

0.01364

0.025310

0.007380

0.40000

0.095028

0.02561

0.066081

0.010820

0.60000

0.094868

0.02175

0.086737

0.000675

0.80000

0.042088

0.00832

0.063440

0.004647

1.00000

0.000000

0.00000

0.000000

0.000000

1.20000

0.114988

0.03707

0.054372

0.020061

1.40000

0.258314

0.06961

0.125623

0.029411

1.60000

0.257879

0.05913

0.154501

0.001836

1.80000

0.114406

0.02260

0.112027

0.012631

2.00000

0.000000

0.00000

0.000000

0.000000

2.20000

0.312569

0.10076

0.094769

0.054531

2.40000

0.702171

0.18922

0.190405

0.079948

2.60000

0.700987

0.16074

0.220148

0.004991

2.80000

0.310987

0.06144

0.157720

0.034335

3.00000

0.000000

0.00000

0.000000

0.000000

3.20000

0.849649

0.27390

0.133576

0.148230

3.40000

1.908698

0.51434

0.244991

0.217320

3.60000

1.905480

0.43693

0.273270

0.013567

3.80000

0.845351

0.16701

0.191997

0.093333

4.00000

0.000000

0.00000

0.000000

0.000000

Table 6 Maximum absolute error Scheme

Value of parameter

Maximum error

αi

βi

γi

ei

Quartic polynomial

p(x)

1

1

2

–

1.905480

Wang and Tan [6]

w(x)

1

2

–

–

0.514342

Zhu [11]

z(x)

1

1

0.1

1

0.27327

The proposed rational quartic spline

s1(x)

1

1

0.1

–

0.217320

s2(x)

1

1

0.2

–

0.374045

s3(x)

1

1

0.01

–

0.187580

values of γi . Meanwhile Table 7 shows error analysis for RMSE and R2 by using Eqs. (10) and (11) respectively. It is clear from Tables 6 and 7 that the quartic polynomial gives better values for the maximum absolute error, RMSE and R2 . It is observed that the proposed scheme gives better results as compared to [6, 11].

12

N. A. Harim et al.

Table 7 Error analysis Scheme

Error analysis RMSE

R2

Quartic polynomial

p(x)

0.69249

0.80154

Wang and Tan [6]

w(x)

0.17532

0.79625

Zhu [11]

z(x)

0.13446

0.79591

The proposed rational quartic spline

s1(x)

0.06557

0.79488

s2(x)

0.10688

0.79553

s3(x)

0.06396

0.79423

Table 8 Data for Example 2 i

0

1

2

3

4

xi

0

0.2500

0.5000

0.7500

1.000

fi

1

0.9239

0.7071

0.3827

0

di

−0.0232

−0.5858

−1.0824

−1.4142

−1.6473

Table 9 Functional values for Example 2 x

f (x)

p(x)

w(x)

z(x)

s(x)

0.00000

1.00000

1.00000

1.00000

1.00000

1.00000

0.05000

0.99691

0.99304

0.99475

0.99557

0.99565

0.10000

0.98768

0.97897

0.98430

0.98385

0.98427

0.15000

0.97236

0.96365

0.96976

0.96847

0.96890

0.20000

0.95105

0.94718

0.95011

0.94965

0.94973

0.25000

0.92387

0.92388

0.92388

0.92388

0.92388

0.30000

0.89100

0.88772

0.89772

0.88994

0.88999

0.35000

0.85264

0.84525

0.86308

0.84951

0.84980

0.40000

0.80901

0.80163

0.81699

0.80572

0.80602

0.45000

0.76040

0.75712

0.76327

0.75916

0.75922

0.50000

0.70710

0.70710

0.70710

0.70711

0.70710

0.55000

0.64944

0.64725

0.66402

0.64882

0.64884

0.60000

0.58778

0.58285

0.61047

0.58584

0.58595

0.65000

0.52249

0.51756

0.53984

0.52030

0.52043

0.70000

0.4539

0.45179

0.46022

0.45310

0.45313

0.75000

0.38268

0.38268

0.38268

0.38268

0.38268

0.80000

0.30901

0.30824

0.32922

0.30891

0.30890

0.85000

0.23344

0.23171

0.26492

0.23289

0.23289

0.90000

0.15643

0.15470

0.18050

0.15561

0.15561

0.95000

0.07845

0.07769

0.08711

0.07805

0.07805

1.00000

0.00000

0.00000

0.00000

0.00000

0.00000

Rational Quartic Spline Interpolation and Its Application …

13

Table 10 Values of absolute error for Example 2 x

p(x)

w(x)

z(x)

s(x)

0.00000

0.00000

0.00000

0.00000

0.00000

0.050000

0.003872

0.00216

0.00134

0.00126

0.100000

0.008712

0.00338

0.00384

0.00341

0.150000

0.008712

0.00260

0.00390

0.00346

0.200000

0.003872

0.00094

0.00141

0.00132

0.250000

0.000000

0.00000

0.00000

0.00000

0.30000

0.00328

0.00671

0.00106

0.00101

0.35000

0.00738

0.01044

0.00313

0.00283

0.40000

0.00738

0.00797

0.00330

0.00298

0.45000

0.00328

0.00286

0.00124

0.00117

0.50000

0.00000

0.00000

0.00000

0.00000

0.55000

0.00219

0.01457

0.00062

0.00060

0.60000

0.00493

0.02268

0.00195

0.00183

0.65000

0.00493

0.01734

0.00219

0.00205

0.70000

0.00219

0.00623

0.00089

0.00085

0.75000

0.00000

0.00000

0.00000

0.00000

0.80000

0.00077

0.02020

0.00011

0.00010

0.85000

0.00173

0.03147

0.00055

0.00055

0.90000

0.00173

0.02406

0.00083

0.00081

0.95000

0.00077

0.00865

0.00041

0.00040

1.00000

0.00000

0.00000

0.00000

0.00000

Table 11 Maximum absolute error Scheme

Value of parameter

Maximum error

αi

βi

γi

ei

Quartic polynomial

p(x)

1

1

2

–

0.008712

Wang and Tan [6]

w(x)

1

5

–

–

0.03147

Zhu [11]

z(x)

1

1

0.5

1

0.00389

The proposed rational quartic spline

s1(x)

1

1

0.01

–

0.00026

s2(x)

1

1

0.1

–

0.00089

s3(x)

1

1

0.2

–

0.00161

s4(x)

1

1

0.5

–

0.00346

14

N. A. Harim et al.

Table 12 Error analysis Scheme

Error analysis RMSE

R2

Quartic polynomial

p(x)

0.00424

0.95495

Wang and Tan [6]

w(x)

0.01266

0.95000

Zhu [11]

z(x)

0.00179

0.95467

The proposed rational quartic spline

s1(x)

0.00015

0.95449

s2(x)

0.00042

0.95453

s3(x)

0.00075

0.95457

s4(x)

0.00163

0.95467

Table 13 Sampling point for Example 3 i

1

2

3

4

5

xi fi

6

7

8

9

10

0.33

0.500

0.666

0.8333

1

1.1667

1.333

1.5

1.667

0.33

0.754

0.634

0.557

0.5140

0.5

0.5140

0.557

0.6340

0.754

0.754

−0.855

−0.592

−0.360

−0.1716

0.0

0.1716

0.360

0.5924

0.855

−0.855

Table 14 Functional values for Example 3 X

f (x)

p(x)

w(x)

z(x)

s(x)

0.33333

0.75464

0.75464

0.75464

0.75464

0.75464

0.40000

0.70000

0.70433

0.73714

0.70220

0.69961

0.46666

0.65409

0.66391

0.69930

0.65833

0.65405

0.53333

0.61557

0.62543

0.65142

0.61868

0.61642

0.60000

0.58348

0.58789

0.60152

0.58449

0.58404

0.66666

0.55719

0.55719

0.55719

0.55719

0.55719

0.73333

0.53621

0.53918

0.54655

0.53767

0.53630

0.80000

0.52020

0.52688

0.55721

0.52265

0.52064

0.86666

0.50893

0.51561

0.56377

0.51055

0.50951

0.93333

0.50222

0.50519

0.54963

0.50271

0.50247

1.00000

0.50000

0.50000

0.50000

0.50000

0.50000

1.06666

0.50222

0.50519

0.49380

0.50360

0.50247

1.13333

0.50893

0.51561

0.54295

0.51096

0.50951

1.20000

0.52020

0.52688

0.59288

0.52138

0.52064

1.26666

0.53621

0.53918

0.60816

0.53651

0.53630

1.33333

0.55719

0.55719

0.55719

0.55719

0.55719

Rational Quartic Spline Interpolation and Its Application …

15

Table 15 Values of absolute error for Example 3 x

p(x)

w(x)

z(x)

s(x)

0.33333

0.00000

0.00000

0.00000

0.00000

0.40000

0.00433

0.03714

0.00220

0.00039

0.46666

0.00981

0.04521

0.00423

0.00004

0.53333

0.00986

0.03585

0.00312

0.00086

0.60000

0.00440

0.01803

0.00101

0.00056

0.66666

0.00000

0.00000

0.00000

0.00000

0.73333

0.00296

0.01034

0.00146

0.00009

0.80000

0.00667

0.03701

0.00245

0.00044

0.86666

0.00668

0.05484

0.00163

0.00058

0.93333

0.00297

0.04741

0.00048

0.00024

1.00000

0.00000

0.00000

0.00000

0.00000

1.06666

0.00297

0.00842

0.00138

0.00024

1.13333

0.00668

0.03402

0.00203

0.00058

1.20000

0.00667

0.07268

0.00118

0.00044

1.26666

0.00296

0.07195

0.00030

0.00009

1.33333

0.00000

0.00000

0.00000

0.00000

Table 16 Maximum absolute error for Example 3 Scheme

Value of parameter

Maximum error

αi

βi

γi

ei

Quartic polynomial

p(x)

1

1

2

–

0.00986

Wang and Tan [6]

w(x)

1

5

–

–

0.07268

Zhu [11]

z(x)

1

1

5

1

0.00423

Quartic spline

s1(x)

1

1

0.1

–

0.00057

s2(x)

1

1

0.05

–

0.00086

s3(x)

1

1

0.5

–

0.00382

s4(x)

1

1

5

–

0.01473

• In  Example 2,  we considered another function f (x) =  

cos π2x f (x) = cos xπ , x = 0.0.25.0.5.0.75.1 ([4]) and Table 8 shows z the data points and values of derivatives by using Arithmetic Mean Method (AMM). Figure 3a shows the interpolating curves for the original function. Figure 3b gives the interpolating curves by using quartic polynomial. Meanwhile, Fig. 3c shows the result of Wang and Tan [6] scheme at αi = 1, βi = 5 and Fig. 3d represents the interpolating obtained by Zhu [11] scheme. Lastly, Fig. 3e, f show the interpolating curves of the proposed scheme αi = 1, βi = 1, γi = 0.5 and αi = 1, βi = 1, γi = 0.3 respectively. Table 9 shows the original function and Table 10 indicates the absolute error by using the value of γi = 2 for the

16

N. A. Harim et al.

Table 17 Error analysis Scheme

Error analysis RMSE

R2

Quartic polynomial

p(x)

0.00005

0.605078

Wang and Tan [6]

w(x)

0.001466

0.489507

Zhu [11]

z(x)

0.00015

0.600007

The proposed rational quartic spline

s1(x)

0.00000

1

s2(x)

0.00000

1

s3(x)

0.00000

1

s4(x)

0.00000

1

quartic polynomial, the scheme of Wang and Tan [6] at γi = 5, the scheme of Zhu [11] at γi = 0.5 and the proposed schemes when the other parameters are fixed to 1, respectively. Table 11 shows the maximum value for absolute error for each scheme. It can observed that Wang and Tan [10] have the highest maximum value of absolute relative error followed by a quartic polynomial, Zhu [7] and the proposed scheme. For the proposed scheme s1(x) has the smallest absolute error as compared with some other existing schemes. Table 12 shows the values of RMSE and R2 for each scheme. √ • In Example 3, we use the function f (x) = − 1 − (x − 1) + 23 where, x = k+1 , k ∈ [0, 4] (Duan et al. [15]). Table 13 shows the data points. Figure 4a 3 shows the interpolating curve of original function. Figure 4b represents the quartic polynomial at αi = 1, βi = 1, γi = 2. Figure 4c shows the Wang and Tan [10] scheme at αi = 1, βi = 5 and Fig. 4d shows the Zhu [7] scheme, respectively. Figure 4e, f show the results obtained from the proposed scheme with parameters value αi = 1, βi = 1, γi = 0.05, αi = 1, βi = 1, γi = 0.1 respectively. Table 14 shows the original function and Table 15 gives the values of absolute error for all schemes. Table 16 shows the summarization maximum values for Example 3. From Table 16, the proposed scheme, s4(x) gives the smallest value for absolute error as compared with the other schemes by αi = 1, βi = 1, γi = 20 respectively. Table 17 obtained the values of RMSE and R2 for each scheme. Clearly, for this example, the proposed RQS outperform all established schemes. From Table 17 by using our proposed RQS the value of RMSE is zero and R2 is equal to 1. Graphically we also can see that the interpolating curves by using the proposed RQS scheme is similar with actual function. This shows that, RQS scheme is capable to produce smaller error, higher R2 as well as visually pleasing interpolating curves. • From Tables 16 and 17, it can be concluded that when the value of γi is the highest, the smallest error can be obtained. This shows that, parameter γi plays an important role as tension parameter since it can used to control the quality of the interpolating curve (and surface). The error estimation shows that the proposed scheme gives higher accuracy with smaller error as compared to quartic polynomial, Wang and Tan [6] and Zhu [11]

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17

50

y axis

40

30

20

10

0 0

1

x

2

3

4

axis

(a) True Function

(b) Quartic Polynomial

(c) Wang and Tan [12]

(d) Zhu [15]

(e) The proposed scheme

(f) The proposed scheme

Fig. 2 Interpolating curve for Example 1

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50

y axis

40

30

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10

0 0

1

2

x axis

3

4

(a) True Function

(b) Quartic Polynomial

(c) Wang and Tan [10]

(d) Zhu [13]

(e) The proposed scheme

(f) The proposed scheme

Fig. 3 Interpolating curves obtained by using the function example

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(a) True Function

(b) Quartic Polynomial

(c) Wang and Tan [10]

(d) Zhu [13]

(e) The proposed scheme

(f) The proposed scheme

Fig. 4 Interpolating curves obtained by using the function Example 3

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schemes. Based on numerical error analysis, we conclude that, the proposed RQS is better than existing rational quartic splines interpolation with smallest error.

5 Applications in Signal Processing Our final example is an application of the proposed RQS to interpolate discrete time signals obtained in signal processing. We consider the discrete-time signals discussed in Hussain et al. [16]. Let the discrete-time signal is defined as x[n] = A cos(ω0 n + ϕ), 5 ≤ n ≤ 20

(14)

where A is an amplitude, ω0 is an angular frequency (in radian per sample) and ϕ is the phase radian of x[n]. We choose A = 1, ω0 = 0.3 and ϕ = 1 [14]. Figure 5 shows the sampling of discrete-time signal i.e. stem by equally sampling 16 discrete-time signals. Figure 6 shows the example of interpolating curves by using the proposed RQS with parameters αi = 1, βi = 1, γi = 0.1 (black color) and true signal (red color). It can clearly seen that, the proposed RQS scheme produces identical curve as the original signal. The proposed RQS scheme give the maximum error as 8.7254e−04, RMSE as 2.1226e−07 (MSE is 4.5054e−14) and R2 is 1. Meanwhile Hussain et al. [16] has maximum error 2.74e−02. Based on maximum error, the proposed RQS scheme is better than Hussain et al. [16]. In fact, since the proposed RQS scheme has the coefficient of determination R2 = 1 and RMSE is almost zero. This is highly accurate compared with cubic spline interpolation of Hussain et al. [16]. We conclude that, the proposed RQS is suitable for this type of Fig. 5 Stem for discrete-time signal

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Fig. 6 Interpolating curves: the proposed RQS (black) and true signal (red)

data interpolation. Besides that, the absolute error for RQS scheme is confirm with error bound given in Theorem 1. Meanwhile Fig. 7 shows the interpolating curves with 8 equally sampling discretetime signals by using the proposed RQS with parameters αi = 1, βi = 1, γi = 0.1 (black color) and the true signal (red color). From this figure, the proposed RQS still give a very good result even though we only use 8 sampling. We obtain maximum error is 8.2e−03, RMSE is 1.5057e−05, MSE is 2.2673e−10 and R is equal to 1. This shows that, the proposed RQS scheme is significant in signal processing application. Fig. 7 Interpolating curves: the proposed RQS (black) and true signal (red)

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6 Conclusion In this study, a new RQS scheme (quartic/quadratic) with three shape control parameters constructed for curve interpolation. From the error analysis, we observed that the proposed scheme is more suitable for all data sets compared with some existing schemes. It is also obvious from the Examples 1, 2 and 3 that the absolute error, RMSE and R2 respectively, obtained from the proposed scheme are more accurate as compared to some existing methods. Error estimation shows that the proposed scheme gives high accuracy with smaller error as compared to quartic polynomial, quartic spline, Wang and Tan [6] and Zhu [11] schemes. Furthermore, our proposed RQS scheme is easy to use compared with Zhu [11] scheme. Furthermore, we have applied the proposed RQS scheme to interpolate discrete-time signal. From the result, the proposed RQS scheme is better than Hussain et al. [16]. Based on all examples, the proposed RQS is highly accurate for data interpolation. For future studies, we will be more focusing on shape-preserving interpolation by applying the RQS scheme. Application of RQS in image interpolation such as image rotation and image upscaling are also promising ideas. Acknowledgements The authors would like to acknowledge Universiti Teknologi PETRONAS (UTP) and Ministry of Education (MOE), Malaysia for the financial support received in the form of a research grant: FRGS/1/2018/STG06/UTP/03/1/015MA0-020.

References 1. Fritsch, F.N., Carlson, R.E.: Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17, 238–246 (1980) 2. Brodlie, K.W., Butt, S.: Preserving convexity using piecewise cubic interpolation. Comput. Graph. 15(1), 15–23 (1991) 3. Butt, S., Brodlie, K.W.: Preserving positivity using piecewise cubic interpolation. Comput. Graph. 17(1), 55–64 (1993) 4. Delbourgo, R., Gregory, J.A.: The determination of derivative parameters for a monotonic rational quadratic interpolant. IMA J. Numer. Anal. 5, 397–406 (1985) 5. Han, X.: Convexity-preserving piecewise rational quartic interpolation. SIAM J. Numer. Anal. 46(2), 920–929 (2008) 6. Wang, Q., Tan, J.: Rational quartic spline involving shape parameters. J. Inf. Comput. Sci. 1(1), 127–130 (2004) 7. Wang, Q., Tan, J.: Shape preserving piecewise rational biquartic surface. J. Inf. Comput. Sci. 3, 295–302 (2006) 8. Wang, Y., Tan, J., Li, Z., Bai, T.: Weighted rational quartic spline interpolation. J. Inf. Comput. Sci. 10(9), 2651–2658 (2013) 9. Deng, S., Fang, K., Xie, J., Chen, F.: Rational biquartic interpolating surface based on function values. In: International Conference on Technologies for E-Learning and Digital Entertainment, pp. 773–780. Springer, Berlin, Heidelberg (2008) 10. Liu, Z., Xiao, K., Liu, X., Jiang, P.: Local point control of a new rational quartic interpolating spline. In: 2016 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH), pp. 1–7. IEEE, July 2016

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11. Zhu, Y.: C2 positivity-preserving rational interpolation splines in one and two dimensions. Appl. Math. Comput. 316, 186–204 (2018) 12. Harim, A., Karim, S.A.A., Othman. M., Saaban, A.: High accuracy data interpolation using rational quartic spline with three parameters. Int. J. Sci. Technol. Res. 8, 1219–27432 (2019) 13. Harim, A., Karim, S.A.A., Othman. M., Saaban, A.: Data interpolation using C2 rational quartic spline. Int. J. Sci. Technol. Res., 1219–27433 (2019) 14. Sarfraz, M., Hussain, M.Z., Hussain, M.: Shape-preserving curve interpolation. Int. J. Comput. Math. 89(1), 35–53 (2012) 15. Duan, Q., Djidjeli, K., Price, W.G., Twizell, E.H.: Constrained control and approximation properties of a rational interpolating curve. Inf. Sci. 152, 181–194 (2003) 16. Hussain, M.Z., Irshad, M., Sarfraz, M., Zafar, N.: Interpolation of discrete time signals using cubic spline function. In: Proceeding of 19th International Conference on Information Visualization. Barcelona, Spain, 22–24 July 2015

A Controller for Natural Gas Fuel Dispenser with Multi-Level-Pressure Banks Nordin Saad and Mahidzal Dahari

Abstract A refueling algorithm using time-optimal control (TOC) for optimal switching of Natural gas vehicle (NGV) refueling with multi-level-pressure storage banks is presented. Fundamental issues in the development of simulation model for an NGV refueling system and methodology for optimization of the switching is discussed. Performance measures to evaluate and compare the proposed algorithm with a commercial NGV refueling algorithm are based on two criteria: refueling time and total mass stored, and are analyzed based on two experiments: the performance of refueling with fixed-level-pressure banks, i.e., a series of banks of same pressure and multi-level-pressure banks, i.e., a series of banks of increasing pressure. Results revealed the viability of TOC refueling algorithm when used in NGV refueling using multi-level-pressure banks. Keywords Natural gas · Fuel dispenser · NGV refueling · Optimal switching time · Multi-level pressure banks A refuelling algorithm using time-optimal control (TOC) for optimal switching of Natural gas vehicle (NGV) refuelling with multi-level-pressure storage banks is presented. Fundamental issues in the development of simulation model for an NGV refuelling system and methodology for optimization of the switching is discussed. Performance measures to evaluate and compare the proposed algorithm with a commercial NGV refuelling algorithm are based on two criteria, namely, the refuelling time and the total mass stored. The two criteria are analysed based on two N. Saad (B) Electrical and Electronics Engineering Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] M. Dahari Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Jalan Universiti, 50603 Kuala Lumpur, Malaysia e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4_2

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experiments, firstly, the performance of refuelling with fixed-level-pressure banks, i.e., a series of banks of same pressure and secondly, multi-level-pressure banks, i.e., a series of banks of increasing pressure. Results revealed the viability of the proposed TOC refuelling algorithm when used in NGV refuelling using multi-level-pressure banks.

1 Introduction Compressed natural gas (CNG) is natural gas that has been compressed for storage aboard natural gas vehicle (NGV), a vehicle in which its engine is fuelled by natural gas. Although natural gas is cheap and well suited to motoring uses, NGVs have not achieved widespread diffusion [1]. A limited availability of natural gas pipelines in strategic areas has been identified as a main problem that impeded the development of more refuelling facilities, and has contributed to the congestions at NGV dispenser stations. A strategy that has been proposed to reduce the congestion at the dispenser stations was to upgrade the dispensing facilities within the dispenser station where permissible [2, 3]. In an NGV dispenser station, the natural gas is transferred into vehicle tank by pressure differential. If only the differential storage pressure is used to transfer the gas, then the flow of gas would cease when the storage pressure and vehicle tank pressure has been equalized. A paper by Radhakrishnan et al. [4] has pointed out that during refuelling there is a limit of mass flow rate known as ‘choking’ of the flow. At this condition the dispenser pipe exit pressure is much higher than the NGV tank pressure and the resulting abrupt pressure drop is a major cause of energy loss. Since most of the NGVs that come in for refuelling will have its tank at a very low pressure once the tank connected to the dispenser maintained at 3600 psig, sonic flow would take place in the refuelling hose and there will be pressure discontinuity at the tip of the dispenser. This irreversible energy loss contributed considerably to the filling rate of an NGV tank. In [5], the authors continued the work of [4] who suggested that an alternative solution to reduce the time and energy losses during refuelling is by introducing an optimum filling schedule using a series of banks with increasing pressures, or multi-level-pressure storage banks. Here, the problem lies in finding a control algorithm to automatically and optimally switch the refuelling banks from lower to higher pressure bank sequentially as the vehicle storage pressure increases. The use of time-optimal control (TOC) methodology in the area of time-optimal switching problems have been reported by many authors including Kuo et al. [6], Chang and Chow [7], Park and Cho [8], Danbury [9], Moon et al. [10], and Zolfaghari et al. [5]. One of the pioneering works dealing with the time-optimal switching problem is dated back to Kuo et al. [6]. A controller with phase position feedback is designed and built for the minimum time control of a motor. Experimental results showed that the controlled motor is capable of travelling a prescribed number of steps in near minimum time by the injection of one or two single pulses and the proper adjustment of the lead angle of the feedback sensors. Chang and Chow [7] proposed

A Controller for Natural Gas Fuel Dispenser …

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a TOC of switchable series or shunt capacitors requiring only a single switching for damping power system swings resulting from large disturbances. The strategy is useful for a switchable series capacitor with a high compensation rating relative to the transmission system. It is illustrated for a single-machine infinite-bus system and an interconnected system. The used of TOC for nuclear reactor power coarse and fine control stages has been addressed in Park and Cho [8]. During the coarse control stage, the maximum control effort or time-optimal is used to direct the system toward the switching boundary set near the desired power level. At this boundary, the controller is switched to the fine control stage in which an adaptive proportionalintegral-feedforward (PIF) controller is used to compensate for any unprecedented reactivity feedback effects. This fine control is also introduced to obtain a constructive method for determining the (adaptive) feedback gains against the sampling effect. The feedforward control term is included to suppress the over or undershoot. In another work, using TOC Danbury [9] developed a nonlinear control law based on a switching function for a servomechanism. The switching function depends on the characteristics of the actuator and on the load. In practical servomechanisms, the friction load may be unknown and be varied in a slow manner. In such cases the optimal switching function is unknown prior to the start of each move. A controller is developed which estimates the optimal switching function during each move, thus enabling near TOC to be achieved. Moon et al. [10] designed a fourth-order model of a crane-load system and investigated the use of time-optimal control scheme with input constraints for transferring the crane load with swing angle regulation using a rule-based approach inspired by the manual crane operation. It is shown that the TOC problem becomes the determination of the switching times. Based on the heuristics obtained, a set of expert rules is generated for the adjustment of the switching times. A simple iteration method for obtaining the optimal switching times is presented which provides an alternative solution when there is no analytical expression for a solution. In [5], a formulation of a time-optimal control to transfer a given initial state to a specified final state in a minimum time is presented. The focus of this chapter will be to describe the development of switching time optimization for NGV dispenser system using time-optimal control (TOC) method, similar to the approach used in [6–10], to achieve optimal time and energy consumption [4]. The work involves the development of a simulation model via MATLAB/Simulink to represent an NGV refuelling system for used as a tool for evaluating and re-designing the optimal switching time to achieve optimal performance. An NGV dispenser test rig that has been developed in this university for the study of its metering system is used to investigate the effectiveness of the proposed TOC refuelling algorithm, specifically on two most important refuelling scenarios [11]. The paper is organized as follows: Sect. 2 discusses the NGV dispenser model, and the switching time researched. Section 3 summarizes the design and development of mathematical model and switching time-optimal design. Illustrative examples of the simulation and experimental design, and the test model and procedures used to demonstrate the applicability of the approach are presented in Sect. 4. Section 5 gives the conclusions of this paper.

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2 NGV Dispenser Model Figure 1 shows a simplified model for process modelling study to represent an existing NGV refuelling system. It consists of a reservoir, a straight horizontal pipeline and a receiver [4]. The reservoir’s pressure is assumed to be constant at 3600 psig and temperature at 30 °C. The pipeline is horizontally straight with length of 5 m and diameter of 0.0125 m. The cross-sectional area of the pipe is constant. Friction effect along the pipeline is taken into account since the flow is irreversible. It is assumed that NGV with an empty tank 14.7 psig is being filled. The NGV tank varies from the initial value of 14.7–3000 psig when fully filled. These sets of conditions are chosen as the extreme values. The volume of the NGV tank at receiver is 0.055 m3 , the standard value used in most countries. The initial temperature of the tank is assumed to be 30 °C and temperature rise is expected due to gas compression effects. The flow of natural gas between the banks and the receiver is modelled as a Fanno flow system. It is an adiabatic frictional flow and also a flow of a compressible fluid with frictional pressure drop. Flow through a straight pipe of constant cross section is adiabatic when heat transfer through pipe wall is negligible. The initial Mach number, Mo was assumed to be zero since the banks cross sectional area is very large compared to that of the pipe. With the banks at high pressure and the receiver tank at low pressure the flow rate can be determined using the knowledge that the Mach number becomes 1 at the pipe exit since for a long pipeline with a very low exit pressure, the speed of the gas may reach the sonic velocity. It is not possible, however, for the gas to pass through the sonic barrier from the direction of either subsonic or supersonic flow. If the gas enters the pipe at a Mach number less than 1, the Mach number will increase but will not become greater than 1. If an attempt is made to maintain the constant discharge pressure and lengthening the pipe to force the gas to change from subsonic to supersonic flow (or vice versa), there is a maximum mass flow limit, referred to as ‘choking’ of the flow that occurs when Mach number is equal to one. At this condition it was found that

Fig. 1 Conceptual model of the dispenser

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29

the pipe exit pressure is much higher than the tank pressure and the resulting abrupt pressure drop is a major cause of the energy loss.

3 Time-Optimal Control Model of NGV Dispenser The main objective of using optimal control in a control system is to determine a control signal to satisfy some physical constraints and at the same time intensify (maximize or minimize) a chosen performance criterion (performance index). The interest is to find the optimal control, u*(t), (‘*’ indicates optimal condition) that will drive the system from initial state to final states, x(t) with some constraints on controls, u(t) and at the same time intensifying the given performance index indicated by, J*. The statement of boundary conditions for the physical constraints on the controls and states variables, are constrained as Eq. (1), where, ‘+’ and ‘−’ indicate the maximum and minimum values these variables can attain. U− ≤ u(t) ≤ U+ and X− ≤ x(t) ≤ X+

(1)

For NGV switching system using multi-level-pressure storage banks, the formulation of time-optimal control problem requires a mathematical model, generally in state variables form and a specification of the performance index i.e., minimum switching time. The sequence of steps taken to solve the problem is as follows. A mathematical model for NGV refuelling system is first developed. The Pontryagin’s Minimum Principle has extensively been used to design time optimal control [12– 17]. The Pontryagin’s Minimum Principle is used to verify that the optimal control is of the appropriate form, i.e., in this case a ‘Bang-bang’. Next, the optimal forced trajectory is described and following this, the optimal switching time is derived. In the next step, the forced trajectories are simulated using MATLAB/Simulink before its implementation model is tested on an experimental NGV test rig. The necessary tests can be devised to simulate the various refuelling scenarios and detail study of the switching control algorithm’s performance. Finally, the performance of the TOC refuelling algorithm is examined and compared to a commercial refuelling algorithm.

3.1 Parameter Identification Mass flow rate is chosen as the unit for measuring natural gas of the NGV refueling system due to the fact that mass is constant at any pressure and temperature. It is important to point out here that although other methods may be applied to this switching problem, e.g., using gas flow dynamics, however, the approach used in this work is very related to the concept used in the measurement of the gas mass flow rate using a coriolis flow meter. Newton’s 2nd Law of Motion is applied to determine the resultant force between the NGV dispenser storage banks and the vehicle tank,

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as this will be the basis to calculate the switching time needed to switch from one storage bank to another. Figure 2 and Table 1 show the conceptual model of NGV refueling system with multi-level-pressure banks to implement the TOC refueling algorithm with mass and mass flow rate as the measurement variables. From the Newton’s 2nd Law of Motion, it is known that F = ma, where F is the force, m is the mass, and a is the acceleration. Pressure from the storage bank and NGV vehicle provide the force, Fstorage and Fngv , respectively. Note that there are two forces attracting the load, one at the front (force from the NGV vehicle Fngv ) and the other one at the rear of the load (force from the storage bank Fstorage ). Higher pressure banks at the rear of the load propels the load forward (in a positive direction) whilst higher pressure banks at the front of the load propels it backwards (in a negative direction). The total force in a pipeline when both pressures have equalized is

Fig. 2 Implementation of TOC in NGV refueling with multi-level pressure banks

Table 1 Values and definitions of parameters and variables for Fig. 2

Symbol

Description

Units

m1

Mass at storage bank 1

kg

mc

Mass at vehicle cylinder

kg

m¨ coriolis

Rate of mass flow rate

kg s−2

P1

Pressure at storage bank 1

kPa

Pc

Pressure at vehicle cylinder

kPa

A

Area of pipeline

m2

A Controller for Natural Gas Fuel Dispenser …

Fstorage + (−Fngv ) = F = 0

31

(2)

At the stage of mass equilibrium, the resultant force inside the pipeline between the NGV tank and storage bank is equal. However, to achieve such equilibrium, longer time is required due to declining pressure differential which drives the system towards equilibrium. In the case of the NGV refueling with multi-level pressure banks, there are several banks of natural gas at different pressure level. The refueling starts from lower pressure bank and switches over to higher pressure bank wisely as the NGV storage tank’s pressure approaches the banks’ pressure. Therefore, the objective is to determine the right instance for the switch-over to take place. The time of switching can be solved using a forced trajectory developed based on Pontryagin Minimum Principle. From Fig. 2 and notations from Table 1, the governing equations of the system is m1 − mc (m¨ coriolis ) = A(P1 − Pc ) Aρ

(3)

The equation is used as a basis to determine the optimal switching time in the development of NGV refueling algorithm with multi-level pressure banks. The following section explains the first step of the optimal switching design.

3.2 Development of Switching Time Equation Detail explanation on how the modeling and simulation via TOC is conducted can be found in [17–19]. The flow of a mass in natural gas pipeline can be considered as a simple motion of an inertial load in a frictionless pipeline environment as illustrated in Fig. 3.

Fig. 3 The motion of mass in a pipeline with force and initial state of mass and mass flow rate, (x1 , x2 )

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3.3 Designing Governing Equations The motion is described by m y¨ (t) = f (t), where m is the mass of natural gas, y(t) be the output (or angular displacement) and f (t) is the external force applied to = ms1 2 . The the system. The transfer function G(s) of the system is G(s) = y(s) f (s)

control is defined as u(t). Let u(t) = fm(t) , then equation y¨ (t) = u(t). Define the state variables, x1 (t) and x2 (t), as x1 (t) = y(t) = output and x2 (t) = y˙ (t) = output rate, respectively. In the case of motion of mass for natural gas in a pipeline, the output and output rate are the mass and mass flow rate. The system in state-space representation is obtained as        0 1 x1 0 x˙1 = + [u] x˙2 0 0 x2 1 x˙1 (t) = x2 (t) x˙2 (t) = u(t)

(4)

The system can be written in the form x(t) ˙ = f (x(t)) + g(x(t))u(t)

(5)

The control u(t) is assumed to be constrained in magnitude by the relation |u(t)| ≤ U (t) ∀t ∈ [t0 , t1 ]

(6)

where U − ≤ u(t) ≤ U + → |u(t)| ≤ U U + (t) = +1

U − (t) = −1

The magnitude constraint is the result of physical limitations on the amount of mass flow rate.

3.4 Implementation of Pontryagin’s Minimum Principle The mathematical model of NGV refueling system allows the objective of the problem to be specified precisely. When NGV tank is empty, pressure from storage bank is higher than the pressure inside the NGV tank. As a result, the force will push the mass towards the NGV tank. Let t1 be the time taken for the mass to move from storage bank to NGV tank. Consequently, when the mass inside the NGV tank

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increases, the pressure will start to increase and the force will push the mass towards the storage bank. Let t2 be the time taken for the mass to move back from NGV tank to the dispenser storage bank. If the storage bank is assumed to be the location of origin, the objective can be defined as to find the control input u(t) that minimizes the transition time to the origin. To be admissible each u(t) must satisfy the three conditions (a) the system is completely controllable, (b) the magnitude of the control u(t) is constrained as shown in Eq. (6), and (c) the initial state is x(t0 ) and the final (target) state is 0, see [17, 18] for details. For minimum time system, the performance index (PI) is t f J=

1 · dt = t f − t0

(7)

t0

where t0 is fixed and t f is free (t0 is the initial time and t f is the final time). From the system (4) and the PI (7), the Hamiltonian is H (x(t), p(t), u(t)) = 1 + p1 (t)x2 (t) + p2 (t)u(t)

(8)

Applying the Pontryagin Principle, the Hamiltonian is minimized to   H x(t), p(t), u ∗ (t) ≤ H (x(t), p(t), u(t)) =

min H (x(t), p(t), u(t)) |u| ≤ 1

(9)

Solving the Hamiltonian, gives 1 + p1 (t)x2 (t) + p2 (t)u ∗ (t) ≤ 1 + p1 (t)x2 (t) + p2 (t)u(t)

(10)

This leads to p2 (t)u ∗ (t) ≤ p2 (t)u(t) =

min { p2 (t), u(t)} |u| ≤ 1

(11)

If p2 (t) > 0 i.e., (positive) the optimal control must be the smallest admissible control value u(t) = −1. On the other hand, if p2 (t) < 0 i.e., (negative) the optimal control must be the largest admissible control value u(t) = +1. In a situation, if p2 (t) = 0, the optimal control u(t) is indeterminate. Therefore, the TOC is given by expression u(t) = −sign{ p2 (t)} =  = ±1. As has been mentioned earlier, this type of control action is also termed as “Bang-bang” since it is either at its maximum value or its minimum value. As will be discussed in the following section, the transitions between the extremes are referred as switches.

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3.5 Designing Forced Trajectory Consider the co state equations along with the Hamiltonian (6) given as p˙ 1 (t) = − p˙ 2 (t) = −

dH =0 d x1 (t)

dH = − p1 (t) d x2 (t)

(12)

Let π1 and π2 , the initial values of the co state variables given as π1 = p1 (0) and π2 = p2 (0). Solving Eq. (19), gives p1 (t) = π1 = constant and p2 (t) = π2 − π1 t. It can be shown that p2 (t) − t is a straight line in the p2 (t) plane; the four possible shapes of p2 (t) and the corresponding shapes of the H -minimal control u(t) = −sign{ p2 (t)} are shown in Fig. 4. The ‘shapes’ of p2 (t) shown in Fig. 4 indicates that the optimal control is a piecewise constant and could switched, at most once because the problem is normal. There exist four control sequences i.e., {+1}, {−1}, {+1, −1}, {−1, +1}, that could satisfy the time-optimal control problem, see Eq. (8). However, it is important to note that a control sequences like for example {+1, −1, +1} is not an optimal control sequence. The reason is the control sequence {+1, −1, +1} requires two switching which was in violation of the time-optimal theorem, see [17–19]. From Fig. 4, the optimal control for this second order system (double integral) can be seen as a piecewise constant and could switch only once. In order to arrive at closed-loop realization of the optimal control, the phase (x1 (t), x2 (t)) plane (state) trajectories need to be found. Fig. 4 The four possible ‘shapes’ of p2 (t) = π2 − π1 t and the corresponding controls u(t) = −sgn{π2 − π1 t}

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Since, over a finite interval of time, the time-optimal control constant u(t) =  = ±1, (6) using u(t) =  = const with the initial conditions x1 (0) ∼ = ξ1 , and x2 (0) ∼ = ξ2 to obtain the relations x2 (t) = ξ2 + t, and x1 (t) = ξ1 + ξ2 t + 21 t 2 . Eliminate t, gives 1 1 x1 (t) = ξ1 − ξ22 + x2 (t)2 2 2

(13)

2 where t = (x2 (t) − ξ2 ) = x2 (t)−ξ .  Equation (13) is the equation of the trajectory in the (x1 x2 ) plane originating at (ξ1 , ξ2 ) and due to the action of the control u(t) = . If u = +1, then t = x2 (t) − ξ2 , or x1 (t) = C1 + 21 x2 (t)2 , and if u = −1, then t = ξ2 − x2 (t), or x1 (t) = C1 − 21 x2 (t)2 , where C1 = ξ1 − 21 ξ22 and C2 = ξ1 + 21 ξ22 are constant. Figure 5 shows a family of parabolas in (x1 , x2 ) plane. In Fig. 5, it is shown that the forced (phase plane) trajectories in the state plane, the solid trajectories are generated by u = 1 and the dashed trajectories are generated by u = −1. For the control problem, the objective is to drive any initial state to (0, 0), that is, to the origin of the state plane. At t = t f , the state variables are assigned as x1 (t = t f ) = 0; x2 (t = t f ) = 0. Rewrite this for any initial state using x1 (t) = ξ1 , x2 (t) = ξ2 ,  = u, and solved for x 1 (t), gives x1 (t) = 21 ux22 (t). The problem can be restated as to find the time-optimal control sequence to drive the system from any initial state (x1 (t), x2 (t)) to the origin (0, 0) in a minimum time. In Fig. 5, it can be seen that there are two curves labeled γ+ and γ− which transfer any initial state (x1 (t), x2 (t)) to the origin (0, 0). The γ+ curve is the locus of all (initial) points (x1 (t), x2 (t)) which can betransferred to the final point (0, 0) by the control u = +1, and is given as γ+ = (x1 , x2 ): x1 = 21 x22 , x2 ≤ 0 . Also it can

Fig. 5 The forced (phase plane) trajectories in the state plane

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N. Saad and M. Dahari

be shown that, the γ− curve is the locus of all (initial) points (x1 (t), x2 (t)) which can be transferred to the final point(0, 0) by the control u = −1, and is given as γ− = (x1 , x2 ): x1 = − 21 x22 , x2 ≥ 0 . The complete switch curve, or the γ curve, is defined as the union of the partial switch curves γ+ and γ− . That is

1 |x | γ = (x1 , x2 ): x1 = − x2 2 2 = γ+ ∪ γ −

(14)

The switch curve γ is as shown in Fig. 6. The regions are defined by applying the control u = +1 or u = −1. Let R+ be the region of thepoints to the left of the switch curve γ such that R+ = (x1 , x2 ): x1 < − 21 x2 |x2 | . Similarly,  let R− be the region of the points to the right of the switch curve γ such that R− = (x1 , x2 ): x1 > − 21 x2 |x2 | . The various trajectory generated by the possible control sequences is shown in Fig. 7. Figure 7 shows four possible control sequences which drive the system from any initial condition to the origin. Briefly, if the system is initially anywhere (point a) on the γ+ curve, the optimal control is u = +1 to drive the system to origin in minimum time t f , and also if the system at rest anywhere (point b) on the γ− curve, the optimal control is u = −1 to drive the system to origin in minimum time t f . Similarly, if the system is initially anywhere (point c) in the R+ region, the optimal control sequence is u = {+1, −1} to drive the system to origin in minimum time t f , and also if the system is initially anywhere (point d) in the R− region, the optimal control sequence is u = {−1, +1} to drive the system to origin in minimum time t f . Fig. 6 Switch curve

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Fig. 7 Various trajectories generated by four possible control sequences

Suppose the system is initially at point d, where the control u = +1 and the optimal control sequence u = {−1, +1} were used, the system can also be drove to origin. However, if say a control sequence {+1, −1, +1} is used, which was not a member of the optimal control sequence, then the time t f taken is higher than the corresponding time t f taken for the system with control sequence {−1, +1}. The time-optimal control u as a function of the state [x1 , x2 ] is given by u∗ = u ∗ (x1 , x2 ) = +1 for (x1 , x2 ) ∈ γ+ ∪ R+ u∗ = u ∗ (x1 , x2 ) = −1 for (x1 , x2 ) ∈ γ− ∪ R−

(15)

Alternatively, if z = x1 + 21 x2 |x2 | is defined, then if z > 0, u∗ = −1 and z < 0, u∗ = +1

(16)

3.6 Derivation of Optimal Switching and Total Minimum Time The optimal switching and total minimum time can now be solved. For convenience let the initial point (ξ1 , ξ2 ) is at regional R− . Therefore  = u = −1, and x 1 = ξ1 +

 1 2 ξ2 − x22 2

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This intersect γ+ at a point (x1 , x2 ) where x1 is positive and x2 is negative that satisfies ξ1 +

 1 1 2 ξ2 − x22 = x22 2 2 1 x22 = ξ1 + ξ22 2

(17)

Since x2 < 0 at all points of γ+ , the switch occurs at the point where, x2 = − ξ1 + 21 ξ22 . The equation for the intersection in region γ+ is x1 = 1 ξ + 41 ξ22 . 2 1

1 2 x . 2 2

Therefore x1 =

 The time from start to switch with  = u = −1 is t1 = ξ2 + ξ1 + 21 ξ22 . With similar reasoning, the time from  switch to start with  = u = +1 is

t2 = x2 − ξ2 . With x2 = 0, the time is t2 = ξ1 + 21 ξ22 . Therefore, the minimum total time taken for the flow to be equalized at the point of origin is 1 T = t1 + t2 = ξ2 + 2 ξ1 + ξ22 2

(18)

The time taken for the system starting at any position in state space and ending at the origin can be calculated using the set of Eq. (13) for each portion of the trajectory. Finally, it can be shown that the minimum time t f for the system starting from (x1 , x2 ) and arriving at (0, 0) is given by  ⎧ ⎪ x + 4x1 + 2x22 If x1 > − 21 x2 |x2 | ⎪ ⎨ 2  t f = −x + −4x + 2x 2 If x < − 1 x |x | 2 1 1 2 ⎪ 2 2 2 ⎪ ⎩ |x2 | If x1 = − 21 x2 |x2 |

(19)

3.7 Modeling of Forced Trajectory Using MATLAB/Simulink The scheme of TOC law is shown in Fig. 8. Note that the closed-loop (feedback) optimal controller nonlinear control u is a nonlinear function of x1 and x2 , although the system is linear. The state variables x1 (output) and x2 (output rate) are measured at each instant of time. The signal x2 is introduced to the nonlinearity (function generator), whose output is 21 x2 |x2 |. The signal z(t) is given as, z(t) = x1 (t) + 21 x2 (t)|x2 (t)|, and after a sign reversal, a relay can be controlled. Basically, the relay is the engineering realization of the signum

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39

Fig. 8 Closed-loop implementation of time-optimal control (TOC) law

operation. The relay output u ∗(t) is the time-optimal control. If the system is initially at (x1 , x2 ) ∈ R− , then x1 > − 21 x2 |x2 |, this means z > 0 and hence the output of the relay is u = −1. On the other hand, if the system is initially at (x1 , x2 ) ∈ R+ , then x1 < − 21 x2 |x2 |, this means z < 0 and hence the output of the relay is u = +1. A basic difficulty occurs when the state (x1 (t), x2 (t)) ∈ γ , then z(t) > 0. In this case, the input signal to the relay is zero: however, since a relay is a physical element with small (but nonzero) inertia, the relay will not switch precisely at the γ curve but some distance away. Hence, by the time the relay actually switches, the state is no longer on the γ curve and the input to the relay is nonzero. The implementation of time-optimal control model is obtained by using abs and signum function block as shown in Fig. 9. The model is tested with four different initial conditions of output, x1 , and output rate, x2 , to simulate the (x1 , x2 ) phaseplane trajectories belonging to γ+ , γ− , R+ and R− , and compare that with Eq. (18) to valid t1 and t2 . The following section shows examples of forced trajectory simulated on the MATLAB/Simulink model.

3.8 Simulation Example The equations of switching are used to validate its performance with range of signals measured by a coriolis flow meter in the NGV test rig used in this study. Practically, the value of mass and mass flow rate measured by the flow meter cannot be negative, and this is only true for range of 0 ≤ x1 (t) ≤ x1 (t)+ , and 0 ≤ x2 (t) ≤ x2 (t)+ . Therefore, the input initial states for the actual system are only valid at region R− .

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Fig. 9 Simulink implementation of TOC

An example to illustrate the application of the simulation model needed in solving an optimal switching for an NGV refueling is as follows. The values are scaled down 20 times so that the trajectory can be fully shown on the Simulink model. Notably, initial state of 20 kg/min and 0.365 kg are normalized to 1 kg/min and 0.018 kg, respectively. More precisely, Eqs. (18) and (19) are used to calculate the time for switching (bank to receiver tank), time for switching to origin (receiver tank to bank) and total time i.e., t1 = 1.72 s, t2 = 0.72 s and t f =2.44 s, respectively. The total time is validated using Eq. (19), where, t f = x2 + 4x1 + 2x22 if x1 > − 21 x2 |x2 |. Hence, the actual time for initial state of (20, 0.365) is t1 = 34.4 s, t2 = 14.4 s and t f = 48.8 s.

4 Results and Discussion This section will examine and analyses several scenarios in the refueling experiments conducted on an NGV test rig. The investigation will be focusing on two areas: performance of refueling using two different control algorithms, namely the TOC and the commercial refueling, first on fixed-level-pressure storage bank, and secondly on multi-level-pressure storage bank, respectively. Each storage bank consists of three different banks. Results of experiments are based on two criterions, i.e., refueling time and total mass of natural gas stored. Two experiments were conducted to evaluate the performance of TOC and commercial refueling, firstly, the performance of refueling when storage pressures are set to 3600 psig whilst receiver tank is varied from 20 to 2000 psig, and secondly the performance of refueling when storage pressures are set to different increasing pressures while receiver tank is initially at 20 psig.

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4.1 Performance of Refueling by Varying Initial Pressure Inside Receiver Tank Initial conditions of the storage pressure banks are fixed to 3600 psig, whilst initial pressure inside receiver tank varies from 20 to 2000 psig. Performance of refueling is measured based on refueling time and total mass stored. Tables 2, 3, 4 and 5 show results based on two performance measures i.e., refueling time and total mass stored. As illustrated in Tables 2 and 3, for initial pressures ranges from 20 to 100 psig, the TOC refueling time would be faster than the commercial refueling with average difference of 6.60 s and average total mass loss of 0.04 kg, respectively. However, as illustrated in Tables 4 and 5, for initial pressures ranges from 200 to 2000 psig, TOC refueling time would be slower than the commercial refueling with average difference of 6.17 s but total mass stored would increase with average values of 0.47 kg, respectively. The results verify that at lower initial pressure ranges i.e., 20–100 psig, the TOC refueling is better in term of refueling time, whilst at higher initial pressure ranges Table 2 Analysis of comparison for total time taken between commercial refueling and TOC refueling when initial pressure inside receiver tank varies from 20 to 100 psig Initial pressure of receiver tank (psig)

Total time taken by commercial refueling (s)

Total time taken by TOC (s)

Difference commercial refueling to TOC (s)

Percentage of improvement (%)

20

66.00

51.00

+15.00

+22.73

40

59.00

51.00

+8.00

+13.56

60

58.00

50.00

+8.00

+13.79

80

52.00

51.00

+1.00

+1.92

100

51.00

50.00

+1.00

+1.96

Average

57.20

50.60

+6.60

+10.79

Table 3 Analysis of comparison for total mass stored between commercial refueling and TOC refueling when initial pressure inside receiver tank varies from 20 to 100 psig Initial pressure of Total mass stored Total mass stored Difference Percentage of receiver tank by commercial by TOC (kg) commercial improvement (%) (psig) refueling (kg) refueling to TOC (kg) 20

8.76

8.58

−0.18

−2.05

40

8.96

8.53

−0.43

−4.80

60

8.61

8.47

−0.14

−1.63

80

8.14

8.57

+0.43

+5.28

100

8.11

8.25

+0.14

+1.73

Average

8.52

8.48

−0.04

−0.29

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Table 4 Analysis of comparison for total time taken between commercial refueling and TOC refueling when initial pressure inside receiver tank varied from 200 to 2000 psig Initial pressure of receiver tank (psig)

Total time taken by commercial refueling (s)

Total time taken by TOC (s)

Difference commercial refueling to TOC (s)

Percentage of improvement (%)

200

50.00

55.00

−5.00

−10.00

400

46.00

47.00

−1.00

−2.17

500

45.00

45.00

0.00

0.00

1000

40.00

41.00

−1.00

−2.50

1500

24.00

34.00

−10.00

−41.67

2000

8.00

28.00

−20.00

−250.00

35.50

41.70

−6.17

−51.06

Average

Table 5 Analysis of comparison for total mass stored between commercial refueling and TOC refueling when initial pressure inside receiver tank varies from 200 to 2000 psig Initial pressure of Total mass stored Total mass stored Difference Percentage of receiver tank by commercial by TOC (kg) commercial improvement (%) (psig) refueling (kg) refueling to TOC (kg) 200

7.89

8.49

+0.60

+7.60

400

7.37

7.60

+0.23

+3.12

500

7.04

7.31

+0.27

+3.84

1000

5.79

5.80

+0.01

+0.17

1500

3.77

4.19

+0.42

+11.14

2000

1.41

2.72

+1.31

+92.90

Average

5.55

6.02

+0.47

+19.80

i.e., 200–2000 psig, the commercial refueling gives a better performance in term of total mass stored. The next section will address the performance of both refueling techniques using multi-level-pressure storage banks.

4.2 Performance of Refueling Using Multi-Level-Pressure Storage Banks In this experiment, storage pressures of the multi-level-pressure banks at the low, medium, and high banks are set to 290–2000 psig, 1450–3000 psig and 3000– 3600 psig, respectively, whilst the initial pressure at receiver tank is set initially at 20 psig. Tables 6 and 7 show results of refueling time and total mass stored of

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43

Table 6 Analysis of comparison for total time taken between commercial refueling and TOC refueling using different storage pressures with receiver tank initially at 20 psig Pressure at low bank, medium bank and high bank (psig)

Total time taken by commercial refueling (s)

Total time taken by TOC (s)

Difference commercial refueling to TOC (s)

Percentage of improvement (%)

290–1450–3600

66.00

1000–2000–3000

72.00

38.00

28.00

+42.42

42.00

30.00

+41.67

2000–3000–3600 Average

66.00

48.00

18.00

+27.27

68.00

42.67

25.33

+37.12

Table 7 Analysis of comparison for total mass stored between commercial refueling and TOC refueling using different storage pressures with receiver tank initially at 20 psig Pressure at low bank, medium bank and high bank (psig)

Total mass stored Total mass stored Difference Percentage of by commercial by TOC (kg) commercial improvement (%) refueling (kg) refueling to TOC (kg)

290–1450–3600

4.87

3.16

1.71

−35.11

1000–2000–3000 5.41

4.71

0.70

−12.94

2000–3000–3600 7.27

6.82

0.45

−6.19

Average

4.90

0.95

−18.08

5.85

the commercial refueling and TOC refueling when storage banks set to the different pressure values, whilst initial pressure inside receiver tank is set initially at 20 psig. Interestingly, the results show that TOC refueling time is much faster than the commercial refueling with average difference of 25.33 s and total mass loss of only 0.95 kg. Even though the results in Table 7 indicate that the commercial refueling has performance measure better the TOC in term of the total mass stored, however the performance of TOC in this respect is still within acceptable range. The results verify the viability of TOC as compared to the commercial refueling when used as a switching controller in NGV refueling using multi-level-pressure storage banks. Notably, the total mass loss although small, are due to only one performance index of optimal control theory has been considered in this work i.e., time-optimal. A few other performance indices should be considered such as fuel-optimal and energy-optimal where the objectives would be to determine the optimal total mass and optimal energy utilized with respect to time. It is worth to mention here that these performance indices are related to each other, hence, the choice of performance index for analysis in this work. Because it is not the scope of this research to analyse in detail on other performances, the discussion is limited only to time-optimal control. Nevertheless, the approach for other performances would be similar to what that has been presented.

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5 Conclusion The aim of this work has been to design and implement switching time strategies for an NGV refueling that would provide a minimum switching time and a faster filling rate for NGV refueling using multi-level-pressure storage banks. It discusses fundamental issues in the development of simulation model for an NGV refueling system and methodology for optimizing of the switching and has covered the following issues: development of mathematical model, switching time-optimal design and experimental design. In particular, it has been organized to answer questions such as, what optimization method is required to produce an optimized switching time for storage banks; how to implement the method and build a simulation model for optimization purpose; how to specify problems and scenarios to be analyzed (the relevant experimental design), and how to extract useful information for comparisons. The detailed comparisons of three criteria (the amount of time for refueling with a multi-level pressure banks, the filling time, and the filling capacity) for an NGV refueling system were given. The TOC algorithm, which has not been reported elsewhere to be used in an NGV refueling dispenser has shown to produce promising result is achieving faster refueling time with minimal loss of natural gas stored in receiver tank. As existing method applied in NGV refueling system is not able to reduce NGV congestion problem, this work demonstrated steps taken in analyzing and developing an optimized switching strategies for an NGV refueling system using the well establish technique of optimal control theory and validating the results on actual multi-level-pressure banks NGV test rig. This work has contributed to an improved switching time, and show that up to some extends the congestion at NGV dispenser stations could be reduced.

References 1. Di Pascoli, S., Femia, A., Luzzati, T.: Natural gas, cars and environment. A (relatively) ‘clean’ and cheap fuel looking for users. Ecol. Econ. 38(2), 179–189 (2001) 2. Oester, U.: Refueling stations for the new century. In: 7th International Conference and Exhibition on Natural Gas Vehicles. URL: https://www.iangv.org. 5 June 2003 3. Thomas, G., Goulding, J.W., Munteanu, C.: Measurement, Approval and Verification of CNG Dispensers. KT11 Report, p. 10. URL: https://www.nwml.gov.uk/legis/refs/kt11.pdf. 7 July 2004 4. Radhakrishnan, V.R. , Hisam, N.A., Mutalib, M.I.A., Abdullah, M.N.: Mathematical model of the refueling system of a compressed NGV. In: Proceedings of ANGVA 2004 Conference, Buenos Aires, Argentina, Oct 2004 5. Zolfaghari, Z., Baradarannia, M., Hashemzadeh, F., Ghaemi, S.: Time optimal control and switching curve analysis for Caputo fractional systems. In: 2016 International Conference on Soft Computing & Machine Intelligence, pp. 229–233 (2016) 6. Kuo, B., Yackel, R., Singh, G.: Time-optimal control of a stepping motor. IEEE Trans. Autom. Control 14(6), 747–749 (1969) 7. Chang, J., Chow, J.H.: Time-optimal control of power systems requiring multiple switchings of series capacitors. IEEE Trans. Power Syst. 13(2), 367–373 (1998)

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8. Park, M.G., Cho, N.Z.: Time-optimal control of nuclear reactor power with adaptive proportional-integral-feed forward gains. IEEE Trans. Nucl. Sci. 40(3), 266–270 (1993) 9. Danbury, R.N.: Near time-optimal control of nonlinear servomechanisms. IEEE Proc. Control Theory Appl. 141(3), 145–153 (1994) 10. Moon, M.S., Van Landingham, H.F., Beliveau, Y.J.: Expert rule based time-optimal control of crane loads. In: Proceedings of the 1996 IEEE on Control Application, pp. 602–607, Sept 1996 11. Dahari, M.: Switching time optimization via time optimal control for natural gas vehicle refueling. PhD thesis, Department of Electrical and Electronics Engineering, Universiti Teknologi PETRONAS (2012) 12. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelideze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience Publishers, Inc., New York (1962) 13. Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. Wiley, New York (1966) 14. Hermes, H., Lasalle, J.P.: Functional Analysis and Time Optimal Control. Academic Press (1969) 15. Kirk, D.E.: Optimal Control Theory: An Introduction. Prentice Hall (1970) 16. Sage, A.P., White, C.C.: Optimal Systems Control. Prentice-Hall, Englewood Cliffs, NJ (1977) 17. Macki, J., Strauss, A.: Introduction to Optimal Control Theory. Springer-Verlag, New York (1982) 18. Naidu, D.S.: Optimal Control Systems. CRC Press, Boca Raton, FL (2003) 19. Athans, M., Falb, P.: Optimal Control: An Introduction to the Theory and Its Applications. McGraw Hill, New York, NY (1966)

Power Performance Analysis of Solar Tracking System in UTP Ramani Kannan and Ishwerjeet Singh Inderjeet Singh

Abstract In the recent years, Renewable energy has been a lot in talks, the most booming renewable energy currently is the Solar energy. This is because solar energy is a simple, cheap and easy method to generate electricity. In Malaysia, if you have installed solar panel the government will provide a certain compensation, as well as buy excessive solar power which is used to supply power to homes or buildings around it. This study presents the design of prototype and analysis of performance and effectiveness of a single axis and a fixed axis solar tracker with an active closed loop system utilizing Arduino as a microcontroller. Furthermore, the ultimate aim of this study is to test the performance and effectiveness of the two types of tracking system, developing and testing the prototype for the two systems. This study can be divided into two phases, which is the software and hardware development. In hardware development for the single axis tracker, two light dependent resistor (LDR) is being used in order to detect light rays from the sunlight. The highest light intensity which is detected by the LDR will determine the position of the sun. Hence, upon determining the position of sunlight, the LDR will send information on the highest light intensity produced to the linear actuator in order to direct the solar photovoltaic panel to maintain perpendicular to the light intensity which will maximize the output produced. Based on conventional research, the single axis solar tracking system is shown to have the higher effectiveness compared to the fixed axis solar tracker. Keywords Solar tracker · Renewable energy · Linear actuator · Light intensity

R. Kannan (B) Electrical and Electronics Engineering Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] I. S. I. Singh Electrical and Electronics Engineering Department, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4_3

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R. Kannan and I. S. I. Singh

In the recent years, Renewable energy has been a lot in talks, the most booming renewable energy currently is the Solar energy. This is because solar energy is a simple, cheap and easy method to generate electricity. This study presents the design of prototype and analysis of performance and effectiveness of a single axis and a fixed axis solar tracker with an active closed loop system utilizing Arduino as a microcontroller. Furthermore, the aim of this study is to test the performance and effectiveness of the tracking system, developing and testing the prototype for this system. This study can be divided into two phases, which is the software and hardware development. In hardware development for the single axis tracker, two light dependent resistor (LDR) is being used in order to detect light rays from the sunlight. The highest light intensity which is detected by the LDR will determine the position of the sun. Hence, upon determining the position of sunlight, the LDR will send information on the highest light intensity to the linear actuator in order to direct the solar photovoltaic panel to maintain perpendicular with the light intensity which will maximize the output produced. The single and fixed axis solar tracking system will determine the amount of power can be produced in a day at Universiti Teknologi PETRONAS (UTP). Based on conventional research, the single axis solar tracking system is shown to have the higher effectiveness compared to the fixed axis solar tracker.

1 Introduction Electricity or power is the most important need for human being. To be able to carry out our daily lifestyle without any obstacles, electrical energy plays a drastic role. Electrical energy can be obtained from many sources for example Wind energy, Fossil Fuels, Hydro energy, Solar energy and Nuclear & Coal energy. There are two types of energy which is renewable and non-renewable energy. Renewable energy is energy that can be used repeatedly without it replenishing or naturally replaced where else non-renewable energy is energy that can only be used once and cannot be reused many times, this will cause non-renewable energy to extinct one day if used excessively. Solar energy is among the most under-rated renewable energy because its demands had been extremely increasing over the last few years. Therefore, many researches and development on the effectiveness and performance of the solar energy is being carried out. This brings me to my research to analyze the performance and effectiveness of single axis and fixed axis solar tracking system. The main purpose of using a solar tracker is to monitors and track the movement of the sunlight throughout the day and directs the solar photovoltaic panel towards the highest light intensity so that maximum energy can be generated from the solar photovoltaic panel. Solar trackers increase the energy produced by solar panel by 21–31%. The main objective of the present studies is: (a) To evaluate the effectiveness and performance of a fixed and single axis solar tracking system.

Power Performance Analysis of Solar Tracking System in UTP

49

(b) To develop a prototype of a single axis solar tracking system utilizing an active closed loop system (c) To maximize the power per unit area and generate power throughout the day

2 Related Literature Review From [1], there are two different types of tracking driving method which is the Active system and the passive system, In order to design a solar tracking system there are certain parameters that needs to be taken into consideration which includes, light intensity, Angle of incidence, tilt angle, solar azimuth angle, declination angle, inclination angle, orientation angle, zenith angle, latitude, and the elevation angle. If the solar tacking system fulfils all the parameters, it will produce the maximum energy from the solar tracker. Passive tracking is been used way before active tracking system was introduced. Zomeworks Track Racks is the company that manufactures the passive tracking system and supplies it around the world [1]. The advantages of using passive tracking system is that it has a low cost and can help increase the power output and efficiency by more than 20%. Furthermore, Active tracking system uses the electrical, electronic and mechanical engineering concept. Active tracking system is made by using DC Motors, controllers etc. There are four different types of Active Solar tracking system which is the open and closed loop driver system, Microprocessor system, sensor driven system and lastly the intelligent driven system [1]. Active Solar Tracking system can be broken into open loop system and closed loop system. Open loop system uses microcontroller to ease the movement of the dc motors or the linear actuator used. It moves by predicting the position of the sun using a computer-controlled algorithms, GPS or simple timing system [2]. Open loop solar tracking system do not have a feedback sensor so it cannot adjust its output results based on updated requirement by the solar tracking system [2]. In Ref. [3], A study was conducted in Mossoro, Brazil to analyze the performance of a single axis and a fixed axis solar tracker. The author used an active closed loop system as a driving method for the tracking system. It used two LDR which will detect light ray from the sunlight using partial shaded method. How the system works is simple, if one LDR is being shaded by a cover, the microcontroller will detect the difference between the light intensity between both the LDR. This will cause the microcontroller to send signal to the DC Motor to move the solar photovoltaic panel to the position to where both the LDR has the same light intensity readings. The testing was done from 13 July 2014 until 20 July 2014. The results show that fixed solar tracker manage to capture 0.1463 kWh where else single axis solar tracking system manage to capture 2.631 kWh. The results were taken at different condition which is sunny day, cloudy day and scattered clouds day. Single Axis solar tracking system produced 20% more energy compared to fix solar tracking system and has an efficiency of 11%.

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Fig. 1 Block diagram of open loop solar tracking system

In Ref. [4], An experiment related to the concept of solar tracking system was done to determine how the position of the sun is being tracked by the orientation and inclination of the sun. The experiment was done to maximize the total energy that is collection from the sun. A simulation was done to compare with the actual results. The results show that solar tracking system can generate 40% more energy than fixed axis solar tracker. In [5], an experiment was conducted between single axis and fixed axis tracking system using two different concept which is the normal tracking system and the daily adjustment strategy system. Both the systems have one thing in common which is to achieve maximum power output. Normal tracking system consists of an active closed loop tracking system which will require feedback information in order to function. This experiment was focused on minimizing the tracking error. However, daily adjustment strategy is conducted using an active system with an open loop system. The system employed in this experiment uses a time-based adjustment to determine the sunlight position. The results from this experiment shows that normal tracking system strategy has 23.6% energy captured while daily adjustment strategy has 31.8% energy captured. This shows that daily adjustment can produce more energy and power compared to normal tracking system. In Conclusion, comparison between fixed axis and single axis tracker was done. This project was proposed in order to determine a suitable method to maximize the output power generation and energy produced using a solar tracking system. It can also be used as a guide on how the solar tracking system functions and the limitation of the system. Comparison on which method is more suitable and cost saving was analyzed. There is 30 weeks for this project to achieve its objective and start generating cleaner energy which can be a main source of energy one day which will help the county to turn into a green and renewable county (Figs. 1 and 2).

3 Project Flow and Methodology In this study a single axis will be designed and assembled using an active closed loop driving method. The main component of this project is the Arduino Uno Microcontroller which acts as the brain of this study. The Arduino will control the movement of the linear actuator upon receiving information from the Light Dependent Resistor (LDR) which will determine the highest light intensity spot using a partial shading

Power Performance Analysis of Solar Tracking System in UTP

51

Fig. 2 Block diagram of closed loop solar tracking system

method and sends the input of the position from the Arduino to move the linear actuator to that spot. In the Single axis solar tracking system, there is only one linear actuator which will control the movement of the solar photovoltaic panel from east to west. The LDR arrangement will be using the partial shading concept, LDR 2 will be in the shaded region therefore LDR 1 will have higher light intensity, the Arduino will send impulse to the linear actuator when the reading of LDR 2 is lesser than LDR 1 which mean that the position of the sun producing the highest light intensity is at LDR 1, when the reading of LDR 2 is equal LDR 1 the system will start adjusting and looking for different position at where LDR 2 is lesser than LDR 1 [6]. During night, the LDR will detect the low light intensity it will switch off the system (Fig. 3). The arrangement of the LDR sensor plays a crucial role, If the arrangement of the LDR sensors is not done correctly, it will cause disturbance for the LDR to detect the position with the highest light intensity and send the information to the Arduino so that the solar panel can move in that particular location. Figure 4 shows the concept of partial shading method and the arrangement of LDR.

4 Data Collection The data from the solar photovoltaic panel is collected manually using a multimeter at every 30 min interval from 8:00 am up to 6:00 pm. The data collected is the voltage across a 15.3  power Resistance which is the load which drains out the power from the solar panel. The formula used to calculate the current and power is as follows (Figs. 5, 6 and 7): (a) Current (A) = Voltage (V)/Resistance (R) (b) Power (W) = Voltage2 (V)/Resistance (R)

5 Results and Discussions From the results obtained from this project, it can be observed that single axis solar tracking system produces more voltage and current in the morning which results in

52

Fig. 3 General flow chart for single axis tracking system

Fig. 4 Arrangement of LDR using partial shading method

R. Kannan and I. S. I. Singh

Power Performance Analysis of Solar Tracking System in UTP

Fig. 5 Solar tracker facing the sun during sun rise

Fig. 6 Solar tracker facing the sun at noon

53

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R. Kannan and I. S. I. Singh

Fig. 7 Solar tracker facing the sun during sun set

high power generated. At noon, the current and voltage produced by the single axis solar tracker and the fixed solar panel is the same this is because both the position of solar panel is perpendicular towards the sunlight. In the evening, single axis produces more voltage and current compared to fix solar panel because the sunlight faces the solar panel of the single axis directly. There is a drop in voltage at 1:00 pm which is caused the interruption of clouds blocking the sunlight which caused the current and power generated to drop (Figs. 8, 9 and 10; Tables 1 and 2). Voltage vs Time 18 16

Voltage (V)

14 12 10 8 6 4 2 0

8

8.3

9

9.3 10 10.3 11 11.3 12 12.3 1 1.3 Time (hour) Fixed Axis

2

2.3

3

Single Axis

Fig. 8 Voltage versus time graph for single axis and fixed solar tracker

3.3

4

4.3

5

5.3

6

Power Performance Analysis of Solar Tracking System in UTP

55

Current Vs Time 1.2 1

Current (A)

0.8 0.6 0.4

6

4.3

5

4

5.3

4

3.3

4.3

3

3.3

2

2.3

1.3

1

12.3

12

11.3

11

10.3

10

9.3

9

8

0

8.3

0.2

5

5.3

6

Time (hour) Single Axis

Fixed Axis

Fig. 9 Current versus time graph for single axis and fixed solar tracker

Power Vs Time 18

Power (Wa)

16 14 12 10 8 6 4 2 0

8

8.3

9

9.3 10 10.3 11 11.3 12 12.3 1

1.3

2

2.3

3

Time (hour) Fixed Axis

Single Axis

Fig. 10 Power versus time graph for single axis and fixed solar tracker

6 Conclusion Solar energy is an unlimited source of energy, Sunlight which obtained from the sun is free. The implementation of this Single will help to maximize the total energy produced daily and the total power produced daily. In Malaysia, if solar panel were installed in every building and house in the country it will help generate 1.4 times more power that is generated currently. This system is built with a very small cost

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Table 1 Data collected for fixed axis solar tracking system Time (h)

Resistance of load ()

Current (A)

8.00 am

Voltage (V) 3.521

15.3

0.230131

Power (W) 0.81029

8.30 am

4.432

15.3

0.289673

1.283832

9.00 am

5.63

15.3

0.367974

2.071693

9.30 am

7.55

15.3

0.493464

3.725654

10.00 am

10.95

15.3

0.715686

10.30 am

12.85

15.3

0.839869

10.79232

11.00 am

14.02

15.3

0.91634

12.84708

11.30 am

14.58

15.3

0.952941

13.89388

12.00 pm

15.29

15.3

0.999346

15.28001

12.30 pm

15.56

15.3

1.016993

15.82442

1.00 pm

14.65

15.3

0.957516

14.02761

1.30 pm

15.4

15.3

1.006536

15.50065

2.00 pm

14.96

15.3

0.977778

14.62756

2.30 pm

14.43

15.3

0.943137

13.60947

3.00 pm

14.2

15.3

0.928105

13.17908

3.30 pm

13.01

15.3

0.850327

11.06275

4.00 pm

11.98

15.3

0.783007

9.380418

4.30 pm

10.32

15.3

0.67451

6.960941

5.00 pm

8.79

15.3

0.57451

5.049941

5.30 pm

6.52

15.3

0.426144

2.778458

6.00 pm

6.13

15.3

0.400654

2.456007

7.836765

which will allow many people who tend to invent in solar tracking system can do so as they will be maximizing the power output and there won’t be any losses made. In the coming future, the usage of solar power will produce a clean and green environment and by utilizing the tracking system it will help maximize the power produced. By using solar power as a source of energy it will also help to reduce global warming and the greenhouse effect as non-renewable energy will be cut down.

Power Performance Analysis of Solar Tracking System in UTP

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Table 2 Data collected for single axis solar tracking system Time (h)

Resistance of load ()

Current (A)

Power (W)

8.00 am

Voltage (V) 4.865

15.3

0.317973856

1.546943

8.30 am

6.83

15.3

0.446405229

3.048948

9.00 am

8.9

15.3

0.581699346

5.177124

9.30 am

11.02

15.3

0.720261438

7.937281

10.00 am

13.13

15.3

0.858169935

11.26777

10.30 am

13.91

15.3

0.909150327

12.64628

11.00 am

14.4

15.3

0.941176471

13.55294

11.30 am

14.8

15.3

0.967320261

14.31634

12.00 pm

15.5

15.3

1.013071895

15.70261

12.30 pm

15.61

15.3

1.020261438

15.92628

1.00 pm

14.8

15.3

0.967320261

14.31634

1.30 pm

15.42

15.3

1.007843137

15.54094

2.00 pm

15.08

15.3

0.985620915

14.86316

2.30 pm

14.67

15.3

0.958823529

14.06594

3.00 pm

14.34

15.3

0.937254902

13.44024

3.30 pm

13.87

15.3

0.906535948

12.57365

4.00 pm

12.52

15.3

0.818300654

10.24512

4.30 pm

11.52

15.3

0.752941176

8.673882

5.00 pm

10.53

15.3

0.688235294

7.247118

5.30 pm

8.87

15.3

0.579738562

5.142281

6.00 pm

7.84

15.3

0.512418301

4.017359

References 1. Al-Rousan, N., Mat, A.I., Khairunaz, M., Desa, M.: Advances in solar photovoltaic tracking systems: a review. Sci. Renew. (2018) 2. Sidek, M.H.M., Azis, N., Hasan, W.Z.W., Kadir, M.Z.A.A., Shafie, S., Radzi, M.A.M.: Automated positioning single-axis solar tracking system with precision elevation and azimuth angle control. Sci. Renew. (2017) 3. Vieira, R.G., Guerra, F.K.O.M.V., Vale, M.R.B.G., Araújo, M.M.: Comparative performance analysis between static solar panels and single-axis tracking system on a hot climate region near to the equator. Sci. Renew. (2016) 4. Singh, S.P., Srikant, K., Jairaj, K.S.: Performance comparison and cost analysis of single axis tracking and fixed tilt PV systems. Eng. Sci. Technol. ICCTEST (2017) 5. Eke, R., Senturk, A.: Performance comparison of a single-axis sun tracking versus fixed PV system. Sol. Energy 86(9), 2665–2672 (2012) 6. Ahmed, J., Salam, Z.: An improved method to predict the position of maximum power point during partial shading for PV arrays. IEEE Trans. Ind. Inf. 11(6), 1378–1387 (2015) 7. https://www.instructables.com/id/Stepper-Motor-Arduino-Solar-Tracker-EV/

Artificial Neural Network Modeling of Nanoparticles Assisted Enhanced Oil Recovery Sayed Ameenuddin Irfan and Afza Shafie

Abstract The development of artificial neural network and deep learning algorithms in the last decade has provided a crucial development to solve complex mathematical modeling problems. The application of artificial neural network and deep learning algorithms to solve the complex flow problems arises in the reservoir simulation. The reservoir simulation is a complex phenomenon that requires understanding of the complex fluid flows phenomenon’s and solving nonlinear partial differential equations numerically. Artificial neural network and deep learning algorithm helps to simulate the flow phenomenon by reducing the numerical error and also useful in reducing the approximations in the mathematical model. This chapter deals with current state of the art literature on implementation of artificial neural network and deep learning algorithms in simulation reservoirs for enhanced oil recovery applications. Keywords Artificial neural network · Deep learning · Fluid flow · Reservoir simulation and enhanced oil recovery

1 Introduction The understanding of the reservoir characteristics is an essential issue for improving the oil recovery efficiency and reducing the damage to the reservoir. One of the significant components of reservoir characterization is the in-depth understanding of fluid flow in the reservoir, which could affect the permeability and porosity of the S. A. Irfan Shale Gas Research Group (SGRG), Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] A. Shafie (B) Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4_4

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reservoir and reduces the oil recovery efficiency. The fluids in the reservoir consists of crude oil, water and natural gas can be defined as the fluids, rocks and fine particles with the size range of 1–100 μm present in the sandstone formation [1–4]. Before exploration of oil from the reservoir, the fines attain the equilibrium in the reservoir due to gravitational force, electrostatic force, friction between reservoir components, and chemical reactions. After the oil extraction takes place from the reservoir, water injection is started in the secondary and tertiary oil recovery. The change in reservoir temperature, mechanical and electrostatic force, and chemical interaction starts the detachment of fines [5–9]. Fine migration is a challenging aspect in reservoir mechanics and need to be studied carefully for the improved oil recovery efficiency, since the fine migration decreases the reservoir permeability by blocking the pores and reduces the oil recovery percentage. The fine migration also affects machines because fine particles resulting from the production well, may damage pipeline and machinery, leading to difficulties in implementation of enhanced oil recovery techniques. One of the techniques to reduce this migration in the reservoir, is to disturb the fine particles in the equilibrium, but this will be very difficult when water flooding takes place in secondary and enhanced oil recovery stages [10]. One possible method is using specialize organic and inorganic acids as a clay agent to reduce the movement of the fine particles in large clay reservoirs. Gravel packs of different acids have been utilized to reduce the quantity of free fine particles near the wellbore region. Specialized polymers have also been used as surface changing agents to reduce the fines particle movement. A recent study has also been conducted on using nanoparticles as a fine reducing agent. Researchers [11–14] have used coated nanoparticles to reduce the formation of fines. A case study has been implemented in the offshore wells. The nanoparticles are attached to surface because of high van der Waals and electrostatic forces in prop-pant stages of fracture pumping treatment; and when the fine particles travel through this region, the particles get absorbed to the nanoparticles which lead to reduced movement of fine particles in the porous media [15–17]. Experimental studies have been carried out on nanoparticles as enhanced oil recovery agent and improvement in recovery efficiency [2, 18–20]. In addition, numerical studies carried out for nanoparticles transport in the porous media has given an insight into how the nanoparticles behave in the porous media and helps to use it for fine particle reduction application. Mathematical modeling helps to understand the effects of the parameters on movement of fine particles and their behaviour after the injection of nanoparticles. The fine particle transport is studied by using the empirical modeling method to develop a microscopic model. Experimental study has also been combined with a mathematical model for studying the fine particle movement in a reservoir size setup. In addition, a compositional model for nanoparticles transport and fine migration in the porous media was also developed to consider these four components; the flow of water or oil, salts, nanoparticles and fine particles. The model gives better understanding of the geochemical phenomenon and spatial distribution of fine particle and nanoparticles movement in the porous media [21–25].

Artificial Neural Network Modeling of Nanoparticles Assisted …

61

The current study focused on the development of a simplified model for the nanoparticles flow in the porous media and the effect of fine migration on the adsorption of nanoparticles in the porous media. Experimental analysis has shown that there are many factors influencing fine migration, such as low salinity, reservoir temperature, pH, rock, and fluid porosity. In many cases, injection/production at a high rate would cause fines migration in reservoirs. The injection/production rate should be lower than the critical rate in order to prevent damage to reservoirs. After injecting nanofluid such as SiO2 , into the porous medium, the nanoparticles can be absorbed in the pore surfaces and increase the critical rate and reduce the rate of fines migration. Fines separation from pore surface is mainly controlled by rolling mechanism. Glass beads coated with SiO2 nanoparticles increases the surface roughness and accordingly controls fines migration. This is the primary mechanism how SiO2 nanofluids control the fines migration. In addition, the concentration of the nanofluid will affect the performance of the nanoparticles in controlling fines migration. The best concentration is at which the roughness of the pore surface will be maximized, and fines migration distance will be minimized in the porous medium. This chapter focus on the development of a simplified model for the nanoparticles flow in the porous media and utilized the data generated from mathematical model to develop a ANN model for reservoir simulation.

2 Mathematical Modeling of Nanoparticles Flow in the Reservoir Nanoparticle flow can be represented mathematically by considering the Brownian diffusion equations relating the loss term consists of detachment rate and attachment rate of nanoparticles with the change in velocity, salinity and temperature in the reservoir. A mathematical model is developed by Yuan et al. [5, 16, 26, 27] by considering the retention factors and forces experienced by the fine particle. The mathematical model deals with two cases. In the first case, the nanoparticles are inserted along with the brine in the reservoir. In the second case, the reservoir is filled with nanoparticles then the brine fluids are injected into the reservoir. For case one, the expression for calculation of nanoparticles that is going to interact with fine particles and settle at the surface is presented in the model. Analytical solution is carried out to solve the model. Langmuir adsorption isotherm is applied by [26, 27] to calculate the amount of nanoparticles absorbed at the surface by fine particles. The transport of nanofluid in two-phase flow (oil and water) leads to changes in the wettability and permeability of the reservoir. In order to study these changes, [1] developed a mathematical model for the transport of nanoparticles in porous media by considering the following assumptions: (1) the flow is one-dimensional under isothermal conditions, and the rock and fluids are supposed to be incompressible; (2)

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the porous medium is heterogeneous; (3) the oil and water flow in porous media follow Darcy’s law and the gravity force is neglected; (4) the nanoparticles are discretized into n-sized intervals; and (5) the viscosity and density of the fluids are constant, and oil and water are Newtonian fluids. Firstly, the model considers the flow of fluids in the porous media using Darcy’s law using the following Eq. (1) [28, 29]: ∂ K l ∂ Pl ∂C N P ∂φ Sl + + =0 ∂t ∂ x μl ∂ x ∂t

(1)

where x is the distance of particle in the reservoir, in the reservoir, t is time, ϕ is the porosity of the reservoir medium, K l is the effective permeability, and S l , μl , and Pl are the saturation, viscosity, and porosity of phase l (oil or gas), respectively. Material balance approach is used to represent the nanoparticles transport in the porous media. The material balance equation for concentration of nanoparticles flow in the porous media is given by [23, 24]. The material balance equation is given below. u(x, t) ∂C ∂ 2C ∂C ∂C N P + −D 2 + =0 ∂t φ ∂x ∂x ∂t

(2)

C: nanoparticles concentration t: time, u(X,T ): in-situ velocity, φ: porosity, D: dispersion coefficient, x: distance, R: loss term. The first term of Eq. (2) represents the storage of nanoparticles in the porous media and after filling the porous media, it will then flow to the different part of the reservoir. The next term in the given equation is the convection of nanoparticles which gives the concentration of nanoparticles at a different time. The third term explains about the dispersion of nanoparticles. Nanoparticles can experience dispersion because of Brownian motion. The last term represents the loss of the nanoparticles when in contact with fines particles and combined with the fines. Fluid flow in porous media describes the energy involves in the nanoparticles flow; it is shown as Eq. 3 ∂2 P φμc ∂ P − =0 k ∂t ∂x2

(3)

P: pressure, μ: carrier fluid viscosity, c: total comprehensibility, k: permeability of porous media. The first term describes the storage condition of porous media. Pressure will change over time as there is a “lag” condition in porous media. The second term describes the convection term of pressure. There will be pressure variation along the reservoir caused by the flowing fluid. To get the prediction of concentration along the specific distance and time, Eqs. 2 and 3 need to be solved simultaneously using proper boundaries and initial conditions which are listed below. Injection condition at inner boundary (Neumann boundary): QμB ∂P =− ∂ x x=0 kA

Artificial Neural Network Modeling of Nanoparticles Assisted …

63

Constant outlet pressure at outer boundary (Dirichlet boundary): Px=L = Patmospheric Constant injected concentration at inner boundary (Dirichlet boundary): C x=0 =in j No concentration flux condition at outer boundary (Neumann boundary): ∂C =0 ∂ x x=L Initial pressure at all points in the core is confining pressure: P(x, 0) = Pcon f ining Initial concentration at all points in the core is zero: C(x, 0) = 0 First, Eq. 1 is solved with the above initial and boundary conditions, and then the fine particle concentration is solved using Eq. 2, which leads to the solution of Eq. 1. Using the solution of Eq. 1, the concentration of nanoparticles is calculated using the Eq. 2 with the same initial and boundary conditions.

3 Data Collection The data for ANN model is gained from solving the mathematical model explained in the previous section by Eqs. 2 and 3. The model is solved using the finite difference method. The solution obtained is utilized for development of ANN model for reservoir simulation. The Nanoparticles concentration versus core length at different time is shown in Fig. 1. Some of the data points utilized in the development of ANN model is presented in Table 1.

4 ANN Model Development The neural network structure comprises of the architecture of network and number of hidden neurons and layers. The multilayer perceptron (MLP) structure was the

64

S. A. Irfan and A. Shafie 1 0.9 0.8

Concentration

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.05

0.1

0.15

0.2

0.25

0.3

Reservoir length

Fig. 1 Nanoparticles concentration versus core length for various values of different time

Table 1 Nanoparticle concentration data utilized in the ANN model development Nanoparticle concentration

Reservoir length

0.020147

0.852113

0.044417

0.901408

0.203263

0.152582

Time 39.8 79.59 119.4

most commonly used prediction model in ANN architecture. The MLP model used in this study has a hidden layer with a single output shown in Fig. 2.

Fig. 2 MLP schematic diagram with inputs values, weights and hidden neurons

Artificial Neural Network Modeling of Nanoparticles Assisted …

65

The input vector x and output vector y consists of information of input and output layers. It can be defined as [30–32]    y(k) = f y(k − 1), y(k − 2), . . . , y k − n y , u(k − 1), u(k − 2), . . . , u(k − n u ))

(4)

Here, the subsequent output value y(k) is dependent on the past outputs as well as past input values u(k). In addition, the delay-line taps of the input and output are represented by ny and nu respectively. In this study, different vector inputs ranging from x 0 to x ny+nu were used to determine the value of x. These vector inputs were noted every hour as shown in Table 2. The output y comprised of a single output vector. The nanoparticles’ concentration in addition to pressure, space, and time were carried by each vector. In this study, the single hidden layer MLP model was used. The non-linear neurons constructed the structure of MLP’s hidden structure. These neurons are represented by the hyperbolic tangent function, that is, f (x) = tanh(x), that is attributed as a non-linear function or activation function. This neural network defined by a non-linear mapping of any given input x to an output y having (ny + nu + 1) input neurons, h hidden neurons and only one output neuron is presented by the relation: Table 2 Input and output vectors of the proposed neural network model Input variable

Output variable

k

Description Hour (a time step between each input values)

x0

1

Bias

x1

u(k − 1)

Vectors of previous values of concentration of nanoparticles at time k − 1

x2

u(k − 2)

Vectors of previous values of concentration of nanoparticles at time k − 2

x nu

u(k − nu )

.. .

.. .

Vectors of previous values of concentration of nanoparticles at time k − nu .. .

x nu+1

y(k − 1)

Vectors of previous values of pressure, reservoir length and time at time k – 1

x nu+2

y(k − 2)

.. .

.. .

Vectors of previous values of pressure, reservoir length and time at time k − 2 .. .

x nu+ny

Y (k − ny )

Vectors of previous values of pressure, reservoir length and time at time k − ny

y(k)

[IT(k); IH(k)]

Vectors of previous values of pressure, reservoir length and time at time k

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S. A. Irfan and A. Shafie

⎡ ⎞⎤ ⎛ n y +n u +1 h   ⎣w j × f ⎝ y = y(x, w) = w ji xi ⎠⎦ j=0

(5)

i=0

The parameters of the given neural network model are characterized by weights (w) and biases (b) connecting the different layers. The weight represented by vector W describes the non-linear mapping. The values of the parameters W and b are calculated during the training stage. The learning, then testing and finally generalization are the three fundamental steps to acquire the optimal ANN model. The training dataset having N inputs and N . Table 1 described variable output were used in learning phase such asD = {xi , ti }i=1   x each having sample n y + n u + 1 as input vector. The variable t, likewise called the objective variable, is the comparing estimation of the concentration of nanoparticles. This stage comprise of altering w in order to limit the error function J, which is generally the addition of errors square between the test yield t i and the ANN model yield, yi = y(xi ; w) : 1 1 2 {yi − ti }2 = e , 2 i=1 2 i=1 N

J(w) =

N

(6)

Levenberg–Marquardt is used as learning algorithm due to its fast convergence with least mean square error (MSE). The testing and generalization were second and third phases of the ANN. Dataset of testing and generalization phases were used during this phase. In this study, the mean square error (MSE) and the correlation coefficient (R) were utilised to assess the performance of the model. MSE and R can be assessed as:  e2  i , (7) MSE = N   − 2  N  i=1 (yi − t ) R = ± (8) N − 2 ) (t − t i=1 i

5 Results and Discussion The result of ANN model obtained is presented in this section. In Table 3, the results of nanoparticle concentration ANN model is presented together with accuracy obtained at different hidden neurons. The model has been also varied with different data initialization points. The data has been divided into three different categories and has been pointed out as M1, M2, and M3, the first 70% of data used for training has

Artificial Neural Network Modeling of Nanoparticles Assisted …

67

Table 3 Assessing performances of nanoparticle concentration for different test initialization parameters M1 Hidden neurons

Mean squared error for test

5

0.09654

10

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been labelled as M1, and 16–75% of data had been utilized as model M2 and last 70% of the data utilized for training had been categorized as M3. The mean square error of the model is presented in Figs. 3 and 4 and the best validation of the model is achieved with 0.004. The ANN model is compared with R squared method for training, testing, validation, and overall with R squared of 0.99845, 0.99037,0.99622, and 0.9992 respectively. The mean square error of the model is presented in Figs. 5 and 6 and the best validation of the model is achieved with 0.00949. The ANN model is compared with Best Validation Performance is 4.6293e-24 at epoch 31 105 Train Validation Test

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Best Validation Performance is 4.7142e-25 at epoch 11 Train Validation Test

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R squared method for training, testing, validation, and overall with R squared of 1, 0.9968,0.9165, and 0.9745 respectively.

6 Conclusion In the current study, a neural network-based intelligent prediction of nanoparticle concentration was made for a mathematical model-based data obtained for nanoparticle flow in the porous media. The developed ANN model has shown the good results with r squared value more than 0.9 and it easy to utilize the developed model to evaluate the concentration of nanoparticles for different input conditions.

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References 1. Ju, B., Fan, T.: Wettability alteration and its effects on production in water flooding. Pet. Sci. Technol. 30(16), 1692–1703 (2012) 2. Ju, B., Fan, T.: Experimental study and mathematical model of nanoparticle transport in porous media. Powder Technol. 192(2), 195–202 (2009) 3. El-Amin, M.F., Sun, S., Salama, A.: Modeling and simulation of nanoparticle transport in multiphase flows in porous media: CO2 sequestration. Fluid Dyn. Simul. 1–10 (2012) 4. El-Amin, M.F., Saad, A.M., Sun, S., Salama, A.: Numerical simulation of magnetic nanoparticles injection into two-phase flow in a porous medium. Procedia Comput. Sci. 108, 2260–2264 (2017) 5. Yuan, B., Moghanloo, R.G.: Analytical modeling nanoparticles-fines reactive transport in porous media saturated with mobile immiscible fluids. AIChE J. 65(10) (2019) 6. Habibi, A., Ahmadi, M., Pourafshary, P., Ayatollahi, S., Al-Wahaibi, Y.: Reduction of fines migration by nanofluids injection: an experimental study. SPE J. 18(02), 309–318 (2013) 7. You, Z., Yang, Y., Badalyan, A., Bedrikovetsky, P., Hand, M.: Mathematical modelling of fines migration in geothermal reservoirs. Geothermics 59, 123–133 (2016) 8. You, Z., Badalyan, A., Yang, Y., Bedrikovetsky, P., Hand, M.: Fines migration in geothermal reservoirs: laboratory and mathematical modelling. Geothermics 77, 344–367 (2019) 9. Hasannejad, R., Pourafshary, P., Vatani, A., Sameni, A.: Application of silica nanofluid to control initiation of fines migration. Pet. Explor. Dev. 44(5), 850–859 (2017) 10. Irfan, S.A., Shafie, A., Yahya, N., Zainuddin, N.: Mathematical modeling and simulation of nanoparticle-assisted enhanced oil recovery—a review. Energies 12(8), 1575 (2019) 11. Yang, Y., Bedrikovetsky, P.: Exact solutions for nonlinear high retention-concentration fines migration. Transp. Porous Media 119(2), 351–372 (2017) 12. Dehghan Monfared, A., Ghazanfari, M.H., Jamialahmadi, M., Helalizadeh, A.: Adsorption of silica nanoparticles onto calcite: equilibrium, kinetic, thermodynamic and DLVO analysis. Chem. Eng. J. 281, 334–344 13. Al-Sarihi, A., Zeinijahromi, A., Genolet, L., Behr, A., Kowollik, P., Bedrikovetsky, P.: Effects of fines migration on residual oil during low-salinity waterflooding. Energy Fuels 32(8), 8296– 8309 (2018) 14. Mansouri, M., Nakhaee, A., Pourafshary, P.: Effect of SiO2 nanoparticles on fines stabilization during low salinity water flooding in sandstones. J. Pet. Sci. Eng. 174, 637–648 (2019) 15. Assef, Y., Arab, D., Pourafshary, P.: Application of nanofluid to control fines migration to improve the performance of low salinity water flooding and alkaline flooding. J. Pet. Sci. Eng. 124, 331–340 (2014) 16. Yuan, B., Moghanloo, R.G., Wang, W.: Using nanofluids to control fines migration for oil recovery: nanofluids co-injection or nanofluids pre-flush? A comprehensive answer. Fuel 215, 474–483 (2018) 17. Bedrikovetsky, P., Caruso, N.: Analytical model for fines migration during water injection. Transp. Porous Media 101(2), 161–189 (2014) 18. Murphy, M.J.: Experimental Analysis of Electrostatic and Hydrodynamic Forces Affecting Nanoparticle Retention in Porous Media. The University of Texas at Austin (2012) 19. Sepehri, M., Moradi, B., Emamzadeh, A., Mohammadi, A.H: Experimental study and numerical modeling for enhancing oil recovery from carbonate reservoirs by nanoparticle flooding. Oil Gas Sci. Technol. Rev. d’IFP Energies Nouv. 74, 5 (2019) 20. Adil, M., Lee, K., Mohd Zaid, H., Ahmad Latiff, N.R., Alnarabiji, M.S.: Experimental study on electromagnetic-assisted ZnO nanofluid flooding for enhanced oil recovery (EOR). PLoS One 13(2), e0193518 (2018) 21. El-Amin, M.F., Salama, A., Sun, S.: Modeling and simulation of nanoparticles transport in a two-phase flow in porous media. In: SPE International Oilfield Nanotechnology Conference, pp. 1–9 (2012) 22. Tunio, S.Q., Tunio, A.H., Ghirano, N.A., El Adawy, Z.M.: Comparison of different enhanced oil recovery techniques for better oil productivity (2011)

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Viable Options and Opportunities for Energy Saving in a Distribution System Towards Sustainability: Taylor’s University as the Case Chockalingam Aravind Vaithilingam, Reynato Andal Gamboa, and Yen Ling Lai Abstract A typical energy audit program profound for energy saving perspective. However, there is always room for improvement for the electrical system in further optimizing energy savings. One great option considering the terrain of Malaysia solar technology is a promising solution. However, the investment to the return on the economic is more of critical to be sustainable. Viability of the PV system through technical, environmental and financial aspects, based on PV sizing and cost analysis, and performed. The design requirement for a PV system that were considered in this study such as cost, Return of Investment (ROI), payback period, available space, suitable location for installation, power sizing were conducted. Upon the implementation of the system for the chosen landscape (Taylor’s University Distribution system), it is estimated that 197,028 kWh/year of energy, RM72,569.14/year of cost saving making it a strategy towards sustainability. Keywords Optimization · Sustainability · PV systems · Economic and environmental · Sizing of PV systems · Cost effectiveness

1 Introduction In terms of the greenhouse gas (GHG) emission, Malaysia is said to be high among the other developing countries in the Southeast Asia. Research have been done by Shahid [1] and found out that the per capita GHG emission of Malaysia is around 3.5 C. A. Vaithilingam (B) · Y. L. Lai Faculty of Innovation and Technology, Taylor’s University, 1, Jalan Taylor’s, 47500 Subang Jaya, Malaysia e-mail: [email protected]; [email protected] Y. L. Lai e-mail: [email protected] R. A. Gamboa Batangas State University, 4200 Batangas City, Philippines e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4_5

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times the values of Indonesia and 1.6 times the value of Thailand. They are emitted as a form of air pollutants and are produced from burning of fossil fuels [2]. Air pollution like this is undoubtedly detrimental to the public health and damageable to the environment. Aside from that, Malaysia is still highly dependent on nonrenewable energy (NRE) sources such as the fossil fuels and natural gas [3]. However, adoption of NRE is still difficult due to the issue of their pricing, environmental issue, and limitation of resource. Imagine as there are more populations in this world, the building demand increases, and the energy demand increases at the same time. If the NRE to conventional sources were to be exploited, however depletion is unavoidable over time. On the other side, renewable energy (RE) serves as alternative energy source. There are currently few types of RE in Malaysia, for instance, hydropower, biomass, biogas, geothermal, and solar shown in Table 1. Solar power is produced by converting the solar energy to electrical energy. It has high potential in providing a cleaner and greener environment with zero GHG emission. Dealing with RE like solar can be an advantage since Malaysia is located in the equatorial region. This strategic location results in a high solar irradiation of at least 6 h sunlight every day even though with the covering of the clouds [4]. Other than that, because there is no seasonal change in this country due to the geographical factor, the solar irradiation is constant as compared to other countries that have four seasons. This can be proven as there are emerging new large-scale solar projects (LSS) and the residential or commercial rooftop PV installation. In this work we have taken the case of Taylor’s University Lakeside Campus (TULC) which is served by 4 distribution transformers with a total maximum demand of 36,732 kW. We analyse in detail the energy trend, the option and opportunity to become solar powerhouse of the future is presented in detailed way. As the subject need to be dealt fairly, energy consumption of TULC in each month of 2017 are analyzed in detail. Based on the energy bill, the average total electricity consumption is 999.21 kWh per month, starting from January to December. The load curve of the transformers in TULC based on energy audit data which is the maximum active Table 1 Annual power generation (MWh) of commissioned RE installations Year Biogas

2019

2709.10

Biogas Biomass (Landfill/agri waste)

Biomass (solid waste)

Small hydro

Geothermal

48,945.00

51,183.77

2018 15,679.17 226,106.42

225,468.05

2017 16,319.90 198,985.22

247,542.95 19,303.86 74,831.27 431,938.79 0.00

2016 17,143.13

70,486.00

151,385.22 36,751.74 49,026.53 319,560.53 0.00

2015 16,988.66

41,122.39

192,372.22 18,090.07 55,406.38 264,185.56 0.00

2014 19,772.25

31,844.44

226,196.38

2013 12,962.68

9804.10

209,407.59 11,144.25 79,081.75

2012

7465.40

101,309.87

98.11

Source SEDA, Malaysia [7]

0.24 13,692.41

Solar PV

71,958.21 0.00

4102.42 65,377.23 463,456.20 0.00

4347.83 67,567.90 184,647.78 0.00 3234.52 25,629.78

50,933.63 0.00 5321.15 0.00

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1400 1200 1000 800 600 400 200 0 12:00 12:45 01:30 02:15 03:00 03:45 04:30 05:15 06:00 06:45 07:30 08:15 09:00 09:45 10:30 11:15 12:00 12:45 13:30 14:15 15:00 15:45 16:30 17:15 18:00 18:45 19:30 20:15 21:00 21:45 22:30 23:15

Energy Consumption (kW)

Load Curve of Taylor's University 1600

Time Transformer 1, T1

Transformer 2, T2

Transformer 4, T4

Total Transformer

Transformer 3, T3

Fig. 1 Load curve of four feeder transformers in TULC

power of each transformer have been plotted as shown in Fig. 1. During the period of 12:00 a.m. till 7:25 a.m., the load consumption is the lowest. The load consumption starts to increase gradually at around 7:30 a.m. when the university starts the day and decrease slowly from 8:30 p.m. when there is lesser academic activities. The peak hour of total electricity consumed for the transformers is roughly around 1:30 p.m. In this university compound, it comprises of 6 blocks in total, which are known as the commercial Syopz Mall block, Block A, Block B, Block C, Block D, and Block E. There are 4 transformers in total for the components of the buildings. The highest proportion of the total energy consumed is usually by the chiller, the commercial block as well as the hospitality block. Based on the monthly electricity bill of TULC, the average maximum demand is 3061 kW out of the 12 months in Year 2017. The utmost concerned is that, TULC need to spend the amount of RM476,612.10 per month on average of Year 2017 and RM494,342.43 per month just for the electricity bill as reference to the latest electricity bill in October 2019.

2 PV Installation Option The site location is Taylor’s University Lakeside Campus Block A building with the latitude and longitude of 3.064846, 101.617144 respectively or 3°03 53.5 N 101°37 01.7 E that is obtained from Google Map as shown in Fig. 2. The capacity of the physical roof space of the building of Block A is obtained by using Helioscope software. Different panel rating has different physical dimension and that affect the space available to place the panels. As for the roof condition, the

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Fig. 2 Site location

roof plan is obtained from the chargeman of TULC so that the results that have been analysed through Helioscope software is more practical. In industrial practice, access to the roof is important to check out the roof condition. Sometimes, it even involves the usage of drone for backup purpose of the data collection. There are few useful information that have been extracted from the interview with the chargeman as well as the roof plan provided. The height of the building of Block A is approximately 15 m. The type of roof is metal deck and its tilt is 5°. On the rooftop, there is only one area that is suitable for the application of PV system. When designing a new project of PV system, usually we always focus on the nameplate capacity, however there are few factors such as temperature, shading, mismatch, and other performance drivers can vastly influence the performance of the system in reality. This software is commonly used in the industry sectors to predict the performance of the PV system with the computation of assumption of losses. Through few sets of variants that is run as the simulation of different setting of parameter, optimizations and parameter analysis can be performed. PV array sizing is performed by defining the choice of the PV modules in the library and the specification of losses. Loss diagram through system simulation by hourly steps is performed. Figure 3a how the sizing of the array according to their number of modules and strings were performed. It shows the performance ratio of 1.30 and array nominal power of 156 kWp. Figure 3b, the orientation and titling of solar panel is as shown. The field structure is a fixed, titled plane of 5° and azimuth angle of 180°. The orientation part of the angle is facing towards South. The angle of the panels is fixed for the inclination of tilt angle. The optimization is according to yearly irradiation yield.

3 Orientations of the Placements of the PV After choosing the panel brand, the rating of the panel is to be considered. There are few models from Risen Energy such as 72 cells with 340 and 345 W, 144 cells with

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(a) Array Design Settings

(b) Orientation Settings Fig. 3 Array design, orientation and tilting of solar panel

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345, 350 and 355 W. Based on the usual operation, it is recommended to choose the possibly highest power that can fit in to the design. Thus, the panel chosen is Risen Energy RSM 144-6-355P [5] due to its availability in Malaysia reliable local supplier and it has high efficiency of 17.80%, long product warranty period of 12 years. The inverter chosen is Sungrow SG-60KTL due to its availability in the Malaysia local supplier and it provide enough power to the system. Other than choosing module, there are also various inverter brands such as ABB, Huawei, Sungrow and so on. The inverter is one of the crucial components in a PV system which convert the direct current (DC) output of a PV panel into alternating current (AC) that can be fed into the electrical grid. It brings a major impact on the performance of system because it has inverter efficiency losses which is considered as one of the larger efficiency losses in the system. The inverter chosen is Sungrow SG-60KTL [6] due to its availability in the Malaysia local supplier and it provide enough power to the system. Upon grasping all the necessary information, the design and layout of the solar system are done by using Helioscope software. Based on the layout, with the total area of 1693.0 m2 , 440 modules of solar panels are placed on the site location rooftop. The type of racking in the field segment is fixed tilt racking. Frame size is 10 by 2, row spacing of 1.5 m, frame spacing of 0.5 m, and setback of 0.5 m. As an output, 156.2 kWp of capacity based on rooftop is designed. The arrangement of the modules are 20 panels in series and 22 strings in parallel with a landscape orientation or namely horizontal orientation as in Fig. 4a with the electrical design, it is important to understand about the DC wiring zone of the system, which is the

(a) DC Wiring Zone

(c) Southwestern Angle Fig. 4 Analysis using numerical tools

(b) Shading Heatmap

(d) Southeastern Angle

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arrangement of the panels connection to the inverters. Each of the inverters are equally connected to 20 strings of panels. The shade modelling is assessed as in Fig. 4b, and it is shown that there are no modules that are shaded upon installation. This analysis helps to determine the feasibility of the project when there is any shading involved. Optimization can be done when the shaded modules are removed, or layout are relocated so that there is minimal shading losses by minimal effect of shade on the power production. Figure 4c shows the southwestern angle which is 225° shows the isometry view of the site layout design and Fig. 4d shows southeastern angle which is 135° shows the isometry view of the site layout design. Figure 5 shows the estimated monthly production for the system according to the modelling in the software. March has the highest production which is 19,035.8 kWh whereas June has the lowest production which is 15,513.0 kWh. Estimated performance monthly as shown in Fig. 6 can be analysed using the PV syst software. In normalized performance index, the array nominal installed power at STC with the global irradiance in outdoor conditions of 1000 W/m2 [7] which is obtained from the PV-module manufacturer. Collection loss (Lc ) is the

Fig. 5 Monthly production by Helioscope

(a) Per installed kWp Production Fig. 6 Production capacity and performance ratio

(b) Performance Ratio

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array losses which include the module quality, mismatch and soiling, as well as other inefficiencies. System loss (Ls ) is the inverter losses in the grid connected system. System Yield (Yf ) is the daily useful energy in the system. Performance ratio (PR) is a crucial parameter of a PV system, particularly for the investor’s focus. As an outcome of using the PV systs software, the performance ratio is 79.3%, which meant that there are 20.7% energy generated through the PV panels is lost in the system losses. The energy produced is 197,253 kWh/year with a specific production of 1263 kWh/kWp/year.

4 Energy Flow Analysis The PV array loss factor such as the array soiling losses, wiring ohmic loss, module quality losses, module mismatch losses as shown in Fig. 7. One of the detailed losses set is called soiling loss, it is due to rain, dust and bird droppings and other environmental conditions on the PV modules. Another loss is the module quality loss due to manufacture’s technical specifications and +0.5% is considered an estimated quality gain. Incidence Angle Modification (IAM) losses known as reflection loss as there is some portion of light that is reflected by the module surface when the sunlight strikes a module from a shallow angle. Light Induced Degradation (LID) is the degradation of the modules at the first exposure to light. Based on the general industry experience, Tier-1 modules commonly has the LID of 0.8–1.5% [8]. Inverter loss during operation is the loss of power of the inverter converting its power from DC to AC. AC ohmic losses is the loss in the cable from the inverter to the substation. Mismatch losses [9] also need to be considered in the system performance analysis as it is caused by the uneven environmental conditions or manufacturing processes. First, different irradiance level across the array which is caused by the difference in soiling rates of the modules, bird droppings or cloud cover on that part of the array. The irradiation variance of this project is 5.0%. There is also temperature difference in the module of the same array due to the wind difference across it which the modules at the perimeter of the array have lower temperature. However, in Helioscope, the temperature is modeled with a constant temperature in overall with temperature spread is 4.0°. Other than that, during the manufacturing process, minor differences occur from one unit to the consequent, thus it is being binned to minimize the mismatch losses and sold according to specific binning tolerance with minimum module tolerance is −2.5% and the maximum module tolerance is 2.5% [10].

5 Economic Analysis Economic analysis is conducted in various methods such as estimating the ROI which show the efficiency of an investment. The payback period is also estimated to aid in

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Fig. 7 Loss diagram over the whole year

the estimation of the time taken to cover the initial cost of investment. Upon obtaining the simulated system report from PV systs, financial analysis can be completed. With the system price and tariff rate, the payback period is calculated. The impact of cost savings and energy savings were calculated and emphasized as shown in Table 2. Based on the calculation, payback period for this project is around 7 years. With the designed system size of 156.00 kWp and quoted commercial price per kWp which is RM3500/kWp, the system price is RM546,000.00. With the annual specific yield obtained from the PV systs results based on the system size, the annual solar generation from the system is 197,028 kWh. Based on the Risen Energy warranty certificate, the end-of-first-year rated minimum output power of polycrystalline solar PV modules is lesser than 97.5% [6] and at the end of each year after the first year,

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Table 2 Payback period Information

Value

Unit

System size

156.00

kWp

Price per kWp

3500.00

MYR/kWp

System price

546,000.00

MYR

Annual specific yield

1263.00

kWh/kWp

Annual solar generation

197,028

kWh

Y1 system performance

100.00%

p.a

Y2 system performance

97.50%

p.a

Annual sub. system degradation

0.70%

p.a

TNB tariff rate

0.3650

MYR/kWh

ICPT rate

0.0255

MYR/kWh

Project payback period

7.22

Years

the power output is not be decreased by more than 0.70% every year. With the estimated system performance over 25 years, the total cost savings after implementing this system is RM1,814,236.00. With the accumulate net savings over the 25 years, including the Operation and Maintenance (O&M) cost the ROI chart is plotted to summarize the annualized the profitability over costs as in Fig. 8.

ACCUMMULATE NET SAVINGS (MYR)

Return of Investment (ROI) Chart - Cash Model 2,500,000 2,000,000 1,500,000 1,000,000 500,000 0 0

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Fig. 8 Return of investment chart

Accumulate Annual Return (MYR)

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6 Conclusion As the manuscript is documented, few analyses are conducted to assess how the PV system impact the overall performance of the TULC distribution system. Technical feasibility of PV system in TULC is proven with the design and evaluation of the PV array sizing, layout design, load curve analysis, and other aspects. Helioscope software were used to configure the layout of the PV system. The PV array sizing is completed using PV systs software with the loss diagram shown. As an outcome, 197,028 kWh/year of energy, RM72,569.14/year of cost and 136,893.59 kgCO2 /year of CO2 is saved in applying PV system for optimizing the energy consumption of TULC.

References 1. Shahid, S., Minhans, A., Puan, O.C.: Assessment of greenhouse gas emission reduction measures in transportation sector of Malaysia. J. Teknol. 70(4), 1–8 (2014) 2. Sharvini, S.R., Noor, Z.Z., Chong, C.S., Stringer, L.C., Yusuf, R.O.: Energy consumption trends and their linkages with renewable energy policies in East and Southeast Asian countries: Challenges and opportunities. Sustain. Environ. Res. 28(6), 257–266 (2018) 3. Bekhet, H.A., Harun, N.H.: Role of non-renewable energy for sustainable electricity generation in Malaysia. World Acad. Sci. Eng. Technol. Int. J. Environ. Ecol. Eng. 100005282 10(9), 1–10 (2016) 4. MET Malaysia. MetMalaysia: Iklim Malaysia, Malaysia Climate Change & Green House Effect Climate Malaysia (Online). Available: https://www.met.gov.my/pendidikan/iklim/iklimmala ysia. Accessed: 18 Aug. 2019 5. Risen Solar Technology. Risen RSM-6-335-355P Datasheet (Online). Available: https://www. enfsolar.com/pv/panel-datasheet/crystalline/40019. Accessed 23 Nov. 2019 6. Sungrow, SG60KTL String Inverter Datasheet (Online). Available: https://www.sungro wpower.com/en/products/pv-inverter/string-inverter/sg60ktl/technical-data. Accessed 23 Nov. 2019 7. Gupta, P., Kumar, R.K., Pachauri, Y.: Chauhan, Effect of environmental conditions on single and double diode PV system: a comparative study. Int. J. Renew. Energy Res. 4, 849–858 (Jan.) 8. Sopori, et al.: Understanding light-induced degradation of c-Si solar cells. In: Conference Record of the IEEE Photovoltaic Specialists Conference, NREL/CP-5200-54200 Jun. 2012. Available: https://www.nrel.gov/docs/fy12osti/54200.pdf 9. Folsom Labs (Online). Available: https://www.folsomlabs.com/modeling. Accessed 19 Nov. 2019 10. Field, H., Gabor, A.M.: Cell binning method analysis to minimize mismatch losses and performance variation in Si-based modules. In: Conference Record of the Twenty-Ninth IEEE Photovoltaic Specialists Conference, 2002. New Orleans, pp. 418–421 (2002)

C1 Surface Interpolation Using Quartic Rational Triangular Patches Nur Nabilah Che Draman, Samsul Ariffin Abdul Karim, and Ishak Hashim

Abstract A version of quartic rational triangular patches is used to C1 construct a surface comprising two composite triangles. Like previous scheme such as cubic Ball and cubic Bézier, they have no free parameter that can be adjusted for surface interpolant. Thus, our proposed method has three free shape parameters, α, β, and γ . This study shows the comparison of the three methods—quartic rational, cubic Ball and cubic Bézier triangular patches. To validate the performances, the error measurement used are root mean square errors (RMSE), maximum error and Central Processing Unit (CPU) times. Besides that, we also made a comparison between C1 and G1 continuity. Based on the results, the proposed scheme is better than the three previous schemes in terms of a smaller value of RMSE, and maximum error. Meanwhile, G1 continuity gives less computation rather than C1 . Keywords Rational quartic triangular patches · C 1 continuity · G 1 continuity · Error · Computation

N. N. C. Draman Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] S. A. A. Karim (B) Fundamental and Applied Sciences Department, Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] I. Hashim Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4_6

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1 Introduction Interpolation problems arise in many areas which require reconstruction of smooth surface from irregular data set. The main objective of scattered data interpolation is to produce a surface which attains some degree of smoothness between two adjacent triangles. Many researchers have studied scattered data interpolation by using C1 continuity condition. For example, Goodman and Said [1] have employed the cubic Bézier to construct a triangle interpolant which interpolates partial derivative at the three vertices of each triangle. The befitting Bézier control points can be determined from the given data and consequently adjacent patches joined together with C1 continuity. Nevertheless, the inner control points in the patch replaced by three different points because of the implementation of three local schemes. Besides that, Piah et al. [2] presented C1 interpolant to scattered data which is interpolant positive everywhere. Sufficient conditions are derived on Bézier points to guarantee surface of cubic Bézier triangular patches is perpetually positive. It is an extension of a sufficient condition of Foley and Opitz [3]. Luo and Peng [4] described the C1 rational spline as a piecewise rational convex combination of three cubic Bézier triangular patches that share the same boundary Bézier ordinates. Besides, Ong and Wong [5] explained a global C1 scattered data interpolation scheme subject to constant lower and upper bounds. Moreover, Beatson and Ziegler [6] discussed the monotonicity preserving using C1 quadratic spline for monotone data arranged over triangular elements obtained by subdividing each mesh rectangle into a grid of sixteen triangles. Renka [7] proposed the Fortran implementation for convexity preserving of scattered data interpolation. Meanwhile, a new method to preserve the monotonicity multivariable scattered data has been developed by Beliakov [8]. However, this scheme is only applicable to preserve the shape of monotone data from a function that does not fall under Lipschitz continuous functions definitions. A weakness of this scheme is that it involves a quadratic programming problem that needs the proper choices of some initial values. In addition, Han and Schumaker [9] discussed about a monotone interpolation problem for gridded and scattered data using C1 cubic spline defined on triangulations. Saaban et al. [10] proposed positivity preserving scheme by using quartic triangular Bézier patches. The surface is constructed using three convex combination of quartic triangular Bézier patches and is tested to the real data that are collected distribution of rainfall at various stations in West Peninsular of Malaysia. Goodman et al. [11] proposed a method of derivative that used a convex combination of all derivatives on triangular patches. This scheme is more accurate to the existing method of least-squares minimization but with less computation. Meanwhile, Hussain et al. [12] proposed a C1 convex surface interpolation to preserve data arranged in triangular grid. This scheme is more flexible and involves more relaxed constraints. Karim et al. [13] have discussed scattered data interpolation to visualize and predict rainfall data in Peninsular of Malaysia. All the existing schemes applied in scattered data interpolation do not have any free parameters for shape preserving. The proposed method by Zhu et al. [14] has

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advance characteristics with three shape parameters α, β, and γ . The main objective of this study is to derive sufficient conditions for C1 continuity for two triangle patches. In addition, we investigate the influence of the shape parameters for smaller RMSE, maximum error and CPU time (in seconds).

2 Rational Quartic Triangular Patches Zhu et al. [14] proposed quartic rational triangular patches with three shape parameters. Besides, the proposed scheme can reduce to cubic Ball when α = β = γ = 0. The quartic rational Said-Ball is defined as 

T (U, V, W ) =

Hi,3 j,k (u, v, w)Ni, j,k

(1)

i+ j+k=3

with three shape parameters α, β, γ where Hi,3 j,k (u, v, w) is the basis function given in (2) and Ni, j,k are the control points as shown in Fig. 1. The quartic rational basis functions are given as follows: u2 ; 1 + α(1 − u) u 2 v[2 + α + 2α(1 − u)] 3 H2,1,0 ; (u, v, w) = 1 + α(1 − u) uv 2 [2 + β + 2β(1 − v)] 3 H1,2,0 ; (u, v, w) = 1 + β(1 − v) v2 3 H0,3,0 ; (u, v, w) = 1 + α(1 − v)

3 H3,0,0 (u, v, w) =

Fig. 1 Control points of triangle T

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v 2 w[2 + β + 2β(1 − w)] ; 1 + β(1 − w)   vw 2 2 + γ + 2γ (1 − w) 3 H0,1,2 ; (u, v, w) = 1 + γ (1 − w) w2 3 H0,0,3 ; (u, v, w) = 1 + α(1 − w) a 2 w[2 + α + 2α(1 − w)] 3 H2,0,1 ; (u, v, w) = 1 + α(1 − w)   aw 2 2 + γ + 2γ (1 − w) 3 H1,0,2 ; (u, v, w) = 1 + γ (1 − w) 3 H1,1,1 (u, v, w) = 6uvw

3 H0,2,1 (u, v, w) =

(2)

Figure 2 shows the changes on the patches without changing the control points. Let V1 , V2 and V3 indicate the vertices of the triangle T and u, v, w as the barycentric coordinates such that any points of the triangle can be defined as below: V = uV1 + vV2 + wV3 , u + v + w = 1

(3)

Figure 3 shows a triangle T with respective barycentric coordinate.

3 C1 Continuity Between Two Patches In this section, the C1 continuity condition is discussed in order to create a smooth surface as much as possible. Figure 4 shows two adjacent triangles of rational quartic triangular patches with a common edge, V2 V3 = W3 W2 . The triangle patch will attach with another patch across all shared edges and certain smooth conditions will satisfied. The vertices g0 = h 0 and g3 = h 3 are the ordinates of vertices, meanwhile g6 , g4 , h 6 , h 4 , g0 = h 0 , g3 = h 3 and h 4 , can be obtained by it gradients and data points while g5 and h 5 quartic ordinates are to be determined. Assume that the control points for both triangles are arranged as shown in Fig. 4. Therefore, the sufficient conditions for C1 continuity along a common boundary can be written as h 4 = ug4 + vg0 + wg1

(4)

h 5 = ug5 + vg1 + wg2

(5)

h 6 = ug6 + vg2 + wg3

(6)

C1 Surface Interpolation Using Quartic Rational Triangular …

Basis

Basis

Basis

Basis

Basis

Basis

Basis

Basis

Basis

Basis

93

Fig. 2 One patch of rational quartic triangular patches with shape parameter (α = 0, β = 1, γ = 0)

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Fig. 3 Triangle T

Fig. 4 Control points of two adjacent triangles of quartic rational triangular patches for triangle T and R

where W1 = uV1 + vV2 + wV3 and u + v + w = 1. Observed that, (4) and (6) are automatically fulfilled since h 4 , g4 , g1 can be determined at V2 and h 6 , g6 , g2 can be determined at V3 respectively. Following Goodman and Said [1], we obtain h 5 by using the inward normal direction to the edge that shows in Fig. 5. Let h 5 be normal derivatives as shows Fig. 5. Fig. 5 Notation of triangle T

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4 Error Calculation Three types of errors measurement are calculated in this study such as which RMSE, and max error. They are defined as follows:  RMSE =

SS E n

(7)

Maximum error = max|Z i − Z o |

(8)

where n is the degree of freedom. ei is the error deviation between actual values Z e and the estimates value Z 0 that defined as ei = Z e − Z 0 . Z 0 is the mean of data. SSE =

n 

(Z i − Z O )2 =

n 

i=1

(Z i − Z O )2 =

i=1

SSR =

n  

Ze − Z 0

n 

ei2

(9)

i=1

2

(10)

i=1

SST = SSR + SSE

(11)

CPU time or known as CPU execution time is the time between the start and the end of execution of a given program. In this paper, we used MATLAB 2016 version installed on Intel (R) Core (TM) i3-5005U CPU @ 2.00 GHz.

5 Numerical Results and Discussion We test the scheme by using two different test functions with 10 control points. We also compare C1 and G1 continuity to determine RMSE maximum errors and CPU times using three methods such as cubic Ball, cubic Bézier and quartic rational triangular patches. The test function is stated as below  64−81((x−0.5)2 +(y−0.5)2 ) 1. F(x, y) =  9−0.5 2    − 0.5) + (y − 0.5)2 /3 2. G(x, y) = e − 81 4 (x Table 1 shows 10 control points used to test our scheme with actual value when using F(x, y) and G(x, y).

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Table 1 10 Control points

x

y

F(x, y)

G(x, y)

0

1

3.86E−02

1.34E−05

0

0.67

2.15E−01

1.18E−03 1.18E−03

0

0.33

2.15E−01

0

0

3.86E−02

1.34E−05

0.33

0.33

3.56E−01

1.03E−01 1.03E−01

0.67

0.67

3.56E−01

1

1

3.86E−02

1.34E−05

0.67

1

1.79.E−01

1.02E−01

0.33

1

1.79.E−01

1.02E−01

0.33

0.67

3.56E−01

1.03E−01

Figure 6 shows the surface of the true function and surface interpolant of F(x, y), while Fig. 7 shows the true surface and surface interpolant of G(x, y) by using the proposed scheme. Table 2 summaries the errors for all three schemes. Tables 2 and 3 show the errors measurement using C1 and G1 continuity for three different methods. In Table 2, the proposed scheme for F(x, y) has produced the desired result of 0.0726, 0.5614 and 0.2294 respectively. Since the proposed method has free parameter of α, β, and γ , the best result is obtained when the parameters value is α = β = γ = 4. G1 has also gives an outstanding result when using α = 2.5, β = 2, and γ = 2.5, resulting a reading on RMSE, max error and CPU times are 0.0692, 0.2490 and 0.2229 respectively. Table 3, the values of three types of errors are shown when using G1 continuity with two types of test functions. From F(x, y), the proposed scheme gives values 0.0836, 0.1538 and 0.1837 respectively for RMSE, max error and CPU times when

(a) True function for F (x, y)

(b) Surfaceinterpolant for F (x, y) function using the proposed method.

Fig. 6 Surface interpolant for F(x, y) function using the proposed method

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(a) True function for G (x, y)

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(b) Surface interpolant for G (x, y) function using the proposed method.

Fig. 7 The true function and surface interpolant for G(x, y)

Table 2 The result of C1 continuity of F(x, y) and G(x, y) with three different method C1

Function

Method

Shape parameter α

β

γ

RMSE

Max error

CPU time

F(x, y)

Cubic Ball

0

0

0

0.1320

0.1914

0.2451

0.0891

0.1157

0.2347

Quartic rational

4

4

4

0.0726

0.1107

0.2294

Cubic Ball

0

0

0

0.0706

0.2728

0.2533

0.0692

0.2507

0.2280

2.5

2

2.5

0.0692

0.2490

0.2229

Cubic Bézier G(x, y)

Cubic Bézier Quartic rational

Table 3 The result of G1 continuity of F(x, y) and G(x, y) with three different methods G1

Function

Method

Shape parameter α

β

γ

RMSE

Max error

CPU time

F(x, y)

Cubic Ball

0

0

0

0.1338

0.1914

0.1768

0.0929

0.1565

0.1460

Quartic rational

3

3

3

0.0836

0.1538

0.1424

Cubic Ball

0

0

0

0.0720

0.2728

0.1837

0.0731

0.2507

0.1633

4

3.8

4

0.0757

0.2444

0.1436

Cubic Bézier G(x, y)

Cubic Bézier Quartic rational

using α = β = γ = 3. This proven that our proposed method is indeed better compared to cubic Ball and cubic Bézier. For G(x, y), the results obtain surface interpolation for G1 give slightly difference. The propose scheme give higher RMSE which is 0.0757 but maximum error and CPU times are smallest than two other schemes which is 0.2444 and 0.1436.

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Table 4 The comparison between C1 and G1 continuity for the proposed scheme G1

Shape parameter α C1 G1

β

γ

RMSE

Max error

CPU time

F(x, y)

4

4

4

0.0726

0.1107

0.2294

G(x, y)

2.5

2

2.5

0.0692

0.2490

0.2229

F(x, y)

0

0

0

0.0720

0.2728

0.1837

G(x, y)

3

3

3

0.0836

0.1538

0.1424

Based on error analysis that has been made, we conclude that C1 continuity gave a smaller RMSE and maximum error compared to G1 continuity. Unfortunately, CPU times for G1 is less computation rather than C1 continuity. Table 4 shows a comparison of values of those variables for the quartic rational triangular patches. Remark In Zhu et al. [14], the authors have discussed the construction of the rational quartic spline and they derive the conditions for G1 . In this paper, we have utilized their scheme but we have constricted C1 composite triangular patches. As can be seen in Sect. 3, our derivation is different than [11].

6 Conclusion This chapter discussed quartic rational triangular patches with three shape parameters that has been proposed by Zhu et al. [14]. This research using C1 continuity conditions to combine two adjacent triangle patches. The numerical result has been done by using 10 control points and test two different test functions to determine which method can produce smaller RMSE, maximum error and CPU time. Besides that, a comparison between C1 and G1 has also been made. The results show our scheme can generate better interpolating surfaces by smaller RMSE, maximum error as CPU time compared to cubic Ball and cubic Bézier triangular patches. Acknowledgements This research was fully supported by Universiti Teknologi PTERONAS (UTP) and Ministry of Education, Malaysia through FRGS/1/2018/STG06/UTP/03/1015MA0020 and Universiti Teknologi Petronas (UTP) through a research grant YUTP: 0153AA-H24 (Spline Triangulation for Spatial Interpolation of Geophysical Data). The first author is supported through Graduate Research Assistant (GRA) Scheme.

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References 1. Goodman, T.N.T., Said, H.B.: A triangular interpolant suitable for scattered data interpolation. Commun. Appl. Numer. Methods 7(6), 479–485 (1991) 2. Piah, A.R.M., Goodman, T.N.T., Unsworth, K.: Positivity-preserving scattered data interpolation. Lect. Notes Comput. Sci. (LNCS) 3604, 336–349 (2005) 3. Foley, T.A., Opitz, K.: Hybrid cubic Bézier triangle patches. In: Mathematical Methods in Computer Aided Geometric Design II, pp. 275–286. Academic Press (1992) 4. Luo, Z., Peng, X.: A C1 -rational spline in range restricted interpolation of scattered data. J. Comput. Appl. Math. 194(2), 255–266 (2006) 5. Ong, B.H., Wong, H.C.: Positivity preserving scattered data interpolation scheme. In: Fontanella, F., Jetter, K., Laurent, P.J. (eds.) Proceedings of Advanced Topics in Multivariate Approximation, pp. 259–274. World Scientific Publishing Company (1996) 6. Beatson, R.K., Ziegler, Z.: Monotonicity preserving surface interpolation. SIAM J. Numer. Anal. 22(2), 401–411 (1985) 7. Renka, R.J.: Interpolation of scattered data with a convexity preserving surface. ACM Trans. Math. Softw. 30(2), 200–211 (2004) 8. Beliakov, G.: Monotonicity preserving approximation of multivariate scattered data. BIT 45(4), 653–677 (2005) 9. Han, L., Schumaker, L.L.: Fitting monotone surfaces to scattered data using C1 piecewise cubics. SIAM J. Numer. Anal. 34(1), 569–585 (1997) 10. Saaban, A., Majid, A.A., Piah, A.R.M.: Visualization of rainfall data distribution using quintic triangular Bézier patches. Bull. Malays. Math. Sci. Soc. 32(2), 137–150 (2009) 11. Goodman, T.N.T., Said, H.B., Chang, L.H.T.: Local derivative estimation for scattered data interpolation. Appl. Math. Comput. 68(1), 41–50 (1995) 12. Hussain, M., Majid, A.A., Hussain, M.Z.: Convexity-preserving Bernstein–Bézier quartic scheme. Egypt. Inform. J. 15(2), 89–95 (2014) 13. Karim, S.A.A., Saaban, A., Hasan, M.K., Sulaiman, J., Hashim, I.: Interpolation using cubic B`ezier triangular patches. Int. J. Adv. Sci. Eng. Inf. Technol. 8(4–2), 1746–1752 (2018) 14. Zhu, Y., Han, X., Liu, S.: Quartic rational Said-Ball-like basis with tension shape parameters and its application. J. Appl. Math. (2014)

Construction and Application of Septic B-Spline Tensor Product Scheme Mudassar Iqbal, Samsul Ariffin Abdul Karim, and Muhammad Sarfraz

Abstract The subdivision schemes have already considered as a fundamental research part of CAGD for the aim of curve shaping. A lot of work has been done in this famous research area of CAGD. Due to the high demand for curve and surface modeling in the industry, subdivision schemes show a very efficient and fast enough method to produce the smooth curves in this regard. Various properties and their applications have also introduced in this significant research field. Subdivision algorithms have this significant because of its simplicity, flexibility and easy to apply. In this paper, we propose and analyze a tensor product of Septic B-spline subdivision scheme. Some essential features of proposed scheme have discussed here like continuity, polynomial generation and holder regularity. Some results of the scheme have been shown using surface modeling with the help of computer programming. Keywords B-spline · Limit stencil · Tensor product · Subdivision scheme · Continuity and Laurent polynomial

M. Iqbal Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] S. A. A. Karim (B) Fundamental and Applied Sciences Department and Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] M. Sarfraz Department of Information Science, College of Life Sciences, Kuwait University Sabah AlSalem University City, BShadadiya, P.O. Box 5969, 13060 Safat, Kuwait e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4_7

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1 Introduction A subdivision scheme, when recursively applied, shows consistency and a wellorganized iterative algorithm to produce a limiting curve or surface from discrete data set of initial control polygon or mesh by subdividing it according to some refined rules. These days subdivision schemes took a new direction due to various applications and generalizations. These subdivision algorithms are useful for the creation of smooth curves in fields such as CAD/CAM, CAGD and Computer Graphics. There are various types of subdivision schemes that have already been introduced such as stationary or non-stationary, uniform or non-uniform, binary, ternary, Quaternary and interpolating or approximating, etc. Now here, we are going to discuss interpolating and approximating subdivision schemes. The interpolating subdivision schemes interpolate all initial points and newly formed points are plugged as linear combinations of old points. Recursively applying the process leads to smooth limiting curve. Approximating subdivision schemes creates curves that approximate the control polygons. The limit surfaces or curves of an approximating subdivision scheme do not pass through the control points of control polygon or control mesh. The polygon frequently shrinks towards the final limit curves with the levels of alterations increasing. While, the interpolating subdivisions schemes are more attractive than approximating subdivisions schemes due to their interpolating property. In interpolating subdivision schemes all points in the control polygon are located on the limit curve, which helps and simplifies the graphic techniques and engineering designs.

1.1 Literature Review Rham [1] was the first scholar who started work on subdivision schemes. He constructed a scheme which generated a function with the first derivative. Chaikin [2] also used subdivision to design a curve. Subdivision schemes gained importance when people generalized the tensor product in an arbitrary topology. This idea was presented by Doo and Sabin [3]. Catmull and Clark [4] used subdivision schemes to surface design and to control meshes in an arbitrary topology. In the mid 80s, Dyn et al. [5] formed a four point interpolating subdivision scheme for curve design. In the late part of 80’s, Deslauriers and Dubuc [6] defined a symmetric iterative interpolation process which have the properties come from an associated function F. Dyn et al. [7] generalized the scheme of Dubuc and Deslauriers, known as butterfly scheme and concluded that the scheme is C 1 continuous in a certain range of parameter. In 1995, Dyn and Levin [8] introduced the analysis of Hermite-Type subdivision schemes for surface designs. In 1994, Cai [9] used 4-point scheme with non-uniform control points to calculate convergence and error estimation. He also showed that curves and surfaces generated from 4-point schemes give better results. In 1998, Stam [10] showed that surfaces and its derivatives can be described in terms of Eigen basis functions. In 2002, Hassan et al. [11, 12] worked on arity and number of control

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points. In 2005, Mustafa and Liu [14] worked on the scheme of Bajaj with the new parameter which controlled the shape of models and gave them more flexibility to design a model over the soft and rough mesh network. Beccari et al. [15] produced an interpolating subdivision scheme which produced conic curves shapes. Siddiqui and Ahmad [16] presented 6-point subdivision scheme and claimed that his schemes can give better smoothness. In 2007, Hormann and Sabin [17] produced a family of subdivision scheme and calculated that how support, holder’s regularity, precision set and degree of polynomial spanned limit curve. Khan and Mustafa [18], in 2008, calculated six-point interpolating subdivision scheme for complex Eigen values. In 2009, Khan and Mustafa [19] worked on four-point approximating scheme with one shape parameter and showed that their scheme had the higher smoothness and small support size as compared to other 4-point schemes. Mustafa et al. [20] also worked on m-point approximating scheme and showed that his scheme has higher smoothness as compared to other schemes. Zheng et al. [21] used B-spline to construct a 2n-ary subdivision scheme. Mustafa and Rehman [22], in 2010, constructed (2b + 4)-point, n-ary interpolating and approximating schemes. Siddiqi and Rehan [23] worked on a 4-point binary scheme which generated the family of curves. They also produced a corner cutting scheme which generated a curve of C 1 continuity. During 2011, Mustafa et al. [24] worked on odd point ternary approximating schemes and developed a formula to generalize them. In 2013, Mustafa et al. [25] worked on odd point ternary families of approximating subdivision schemes, in which they showed that their schemes have high smoothness, they also worked on subdivision regularization, in which they proposed that unified frame work can work well for both over-fitting preventation and noise removal in subdivision as well as regularization. In 2013, Ghaffar et al. [26] designed a three point tensor product scheme and showed some applications of given scheme. Mustafa et al. [27] present a family of (2n − 1)point binary approximating subdivision schemes with free parameters for describing curves. Almost all existing odd-point binary symmetric approximating schemes belong to this family of schemes. Again in 2013, Mustafa et al. [28] generalized unified families of interpolating subdivision schemes of 2n point and 2n − 1 point p-ary which generated Lagrange polynomial for n ≥ 2 and p ≥ 3. In 2014, Mustafa and Bari [29] worked on the new family of non-stationary odd point ternary interpolating subdivision schemes by using Lagrange identities. In 2014, Mustafa et al. [30] worked on univariate dual and primal subdivision schemes. They started their work to derive two point binary schemes and extend it for constructing of the family of binary univariate subdivision schemes.In 2015, Siddiqui et al. [31] proposed non-stationary binary 3-point and 4-point schemes using the hyperbolic function as a basis. In 2015, Siddiqui et al. [32] presented a new binary 6-point non-stationary interpolating subdivision scheme in hyperbolic form. In 2015, Rehan and Siddiqi [33] discussed the continuity a new class of 3-point ternary schemes. Also, they generate limiting curves using proposed schemes. Again in 2015, Rehan and Siddiqi [34] discussed a combined 6-point interpolating and approximating scheme using tension parameter in their paper. Again, Mustafa et al. [35] presented a general algorithm to generate a new class of binary approximating subdivision schemes and also given the derivation of some family members. In 2016, Hameed and Mustafa [36] constructed and analyzed

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binary subdivision schemes using Lane-Riesenfeld algorithm for curves and surfaces with Chaikin scheme. In 2016, Ghaffar and Mustafa [37] proposed three different algorithms for approximating subdivision scheme with application in curve modeling. In 2016, Rehan and Sabri [38] proposed a 4-point ternary scheme which creates C 0 interpolating and C 1 , C 2 , C 3 approximating limiting curves. In 2016, Siddiqui et al. [39] developed 4-point and 5-point binary subdivision schemes using hyperbolic B-spline basis functions. In 2016, Siddiqui et al. [40] worked on the hyperbolic form of ternary subdivision schemes using hyperbolic B-spline basis Function. In 2016, Salam et al. [41] discussed two non-stationary types of Chaikin’s, perturbation subdivision scheme, presented in Dyn et al. (2004), described with tension parameter w. In 2017, Cheng and Zhou [42] explained the necessary conditions of subdivision schemes with finite masks. During 2017, Akram et al. [43] discussed the properties of the binary four point interpolating non stationary subdivision scheme [15]. In 2018, Manan et al. [44] focused on an algorithm to solve 3rd order boundary value problem using 8-point approximating scheme. It concludes the results with stability and convergence that is evaluated with the illustration of numerical example. In 2019, Kanwal et al. [45] formulated a numerical approximating collocation algorithm that is based on binary 6-point approximating subdivision scheme to generate the curves. It is examined that the scheme is generating more smooth continuous solutions of the problems. Numerical example was given to illustrate the algorithm with its graphically representation. During mid 2019, Ghaffar et al. [46] presented a new class of 2q-point non-stationary subdivision schemes to generate the curves. Also in 2019, Ghaffar et al. [47] constructed a tensor product of nine-tic b-spline approximating subdivision scheme to generate the smooth curves and surfaces. Numerical examples were given to demonstrate the surface representations.

2 Preliminaries Here, we are going to discuss a special type of subdivision scheme called Septic Bspline subdivision scheme of degree 7 with different properties. First, we introduce the Septic B-spline scheme, then analyze the scheme by discussing its important properties like: continuity, hölder exponent, polynomial generation and reproduction and limiting curve produced by the Septic B-spline subdivision scheme. We defined it as:  k+1 8 k 56 k 56 k 8 k ηu−1 + 128 ηu + 128 ηu+1 + 128 ηu+2 , η2u = 128 (2.1) k+1 1 k 28 k 70 k 28 k 1 k η2u+1 = 128 ηu−1 + 128 ηu + 128 ηu+1 + 128 ηu+2 + 128 ηu+3 , The Laurent polynomial of this scheme is ξ(x) =

 x−4  8 x + 8 x7 + 28 x6 + 56 x5 + 70 x4 + 56 x3 + 28 x2 + 8 x + 1 . 128 (2.2)

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The mask of scheme (2.1) is   1 8 28 56 70 56 28 8 1 ξ = ..., 0, 0, , , , , , , , , , 0, 0, ... , 128 128 128 128 128 128 128 128 128 8 56 56 8 1 + 128 + 128 + 128 ] and [ 128 + The odd and even stencil of the scheme are [ 128 70 28 1 + 128 + 128 ] respectively. 128 Since both sum of coefficient of even and odd stencil equal to one i.e.



8 56 56 8 + + + 128 128 128 128

28 128

+

 = 1,

(2.3)

and 

1 28 70 28 1 + + + + 128 128 128 128 128

 = 1,

(2.4)

For convergence of above proposed scheme, the necessary conditions are satisfied. From (2.2), the Laurent polynomial ξ(x) can be written as ξ(x) = (1 + x) (1+x) , 128x4 7

where b(x) =

(1+x)7 . 128x4

(2.5)

(2.6)

Put x = 1 in b(x), we get b(1) = 1. The condition is equal to ξ(−1) = 0, ξ(1) = 2, from this we have ξ(x) = (1 + x)b(x), and b(1) = 1.

2.1 Properties of the Scheme 2.1.1

Smoothness of the Scheme

Theorem 2.1 Consider the scheme (2.1) Sξ be convergent if and only if the scheme Sd is contractive, then for contractiveness dn  < 1 for some n > 0 with dn  = n n n n maxu |dv−2 l | : 0 ≤ v < 2 , where du are the coefficients of the scheme Sd with u n symbol d n (x) = d (x)d (x2 )...d (x2 −1 ). Theorem 2.2 If Sξ converges, then the limit curves can denotes by ξ(x) = ( 1+x )q d (x). 2 Sd be the scheme for the qth divided differences.

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Proof Laurent polynomial of proposed scheme (2.1) can be elaborated as: ξ(x) =

(1 + x)8 128

where   ξ(x) = x8 + 8 x7 + 28 x6 + 56 x5 + 70 x4 + 56 x3 + 28 x2 + 8 x + 1 /128. In order to prove C 0 continuity of the subdivision scheme Sξ related to ξ(x), we have to show the convergence of d1 (x). To see this we generate another scheme Sη1 related to η1 (x) collected from d1 (x) as:  ξ(x) =

1+x d1 (x) 2

where   d1 (x) = x7 + 7 x6 + 21 x5 + 35 x4 + 35 x3 + 21 x2 + 7 x + 1 /64. The scheme Sη1 is contractive. For this we have,    



 1 + 21 + 35 + 7   7 + 35 + 21 + 1  1 ,  , ≤ Sη1 ∞ = max     2 64 64                



 1   21   35   7   7   35   21   1  1 ≤ Sη1 ∞ = max   +   +   +   ,   +   +   +   , 2 64 64 64 64 64 64 64 64



Sη

1

∞

=

                 1   21   35   7   7   35   21   1  1 max   +   +   +   ,   +   +   +   < 1. 2 64 64 64 64 64 64 64 64

So scheme Sη1 is contractive, Sd1 is convergent and Sξ is C 0 continuous. Similarly by applying this method we have C 7 smoothness level.

2.1.2



Holder Exponent

Here, we will find holder continuity of the Septic B-spline. l Theorem 2.3 Consider the scheme (2.1) Sξ with symbol ξ(x) = 1+x d (x) pro2 d m  duces limit curves with Holder continuity r ≥ l − log2 m for some m. Proof The symbol of the scheme (2.1) is:

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 x−4  8 x + 8 x7 + 28 x6 + 56 x5 + 70 x4 + 56 x3 + 28 x2 + 8 x + 1 . 128  1+x 8 d (x) = 2

ξ(x) =

where d (x) =

2 . x4

Here l = 8 and d = [2], this implies that r ≥ 8 − log2 (2) = 7. 2.1.3



Polynomial Generation and Reproduction

In this section, we will talk about the degree of polynomial generation and polynomial reproduction of Septic B-spline scheme. Theorem 2.4 The degree of polynomial generation of scheme (2.1) is 7. Proof Since the Laurent polynomial ξ(x) of the scheme (2.1) is ξ(x) = (1 + x)(7+1) d (x), where d (x) =

1 {1 + x} , 128x4

then 9 is the degree of polynomial generation.  ξ(x) = (1 + x)

7+1

1 128x4

,

this shows that degree of polynomial generation is 7.



Theorem 2.5 The polynomial reproduction of scheme (2.1) is primal parametrization. Proof For any scheme that generates linear functions with symbol ξ(x) = (1 + x)8 d (x) Let τ =

a (x) 2

attach the data ηuv to parameter tuv = −τ +

u+τ 2v

(2.7)

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then the scheme also reproduces linear functions. The Laurent polynomial of the scheme (2.1) is:  x−4  8 x + 8 x7 + 28 x6 + 56 x5 + 70 x4 + 56 x3 + 28 x2 + 8 x + 1 , 128  1+x 8 d (x). = 2

ξ(x) =

ξ  (x) = {−4x−4 − 24x−4 − 56x−3 − 56x−2 + 56 + 56x + 24x2 + 4x3 }/128. After putting x = 1, we get ξ  (1) = {−4 − 24 − 56 − 56 + 56 + 56 + 24 + 4}/128 = 0. So the value of τ =

ξ  (1) 2

=

0 2

= 0, putting the value of τ in above equation(2.7): tuv = −τ + tuv = 0 +

u+0 2v

u+τ , 2u

=

u . 2v

So the Septic B-spline scheme generates linear reproduction w.r.t primal parametrization. This scheme is primal parametrization. 

3 Construction and Analysis of Septic B-Spline Tensor Product Scheme 3.1 Preliminaries In this section, we construct septic B-Spline tensor product subdivision scheme. We analyze the scheme by reviewing the continuity of the scheme and limiting behavior of the curve generated by septic B-spline tensor product subdivision scheme. The Laurent polynomial of tensor product scheme can be acquired by the accompanying principle: ξ(x) = ξ(x1 , x2 ) = ξ(x1 )ξ(x2 ),

(3.1)

where ξ(x1 ) and ξ(x2 ) are the Laurent polynomials of univariate schemes. A general compact form of binary subdivision scheme S which maps a polygon k k+1 }u,v∈Z to a refined polygon ηk+1 = {ηu,v }u,v∈Z is defined by ηk = {ηu,v

Construction and Application of Septic B-Spline …

ηδk+1 =



109

ξδ−2γ ηγk , δ ∈ Zs .

(3.2)

γ ∈Zs

where S = 1 for curve and S = 2 for surface. In case of univariate subdivision schemes the two rules ( for u is even and odd) given below as: Let u = 2m, then  k+1 = ξ2m−2v ηuk . η2m v∈Z

To get second rule, we assume u = 2m + 1, then k+1 η2l+1 =



ξ(2m+1)−2v ηuk .

v∈Z

Thus one rule is based on the even coefficients of the mask, and the other on the odd coefficients. In the case of bivariate subdivision scheme there are four rules depending on the parity of each component in the multi-index u = (u1 , u2 ). Writing all the multi-indices by components, we have four rules k+1 = η2u 1 ,2u2



k+1 ξ2l1 ,2l2 ηuk1 −l1 ,u2 −l2 , η2u = 1 +1,2u2

l1 ,l2 ∈Z k+1 η2u = 1 ,2u2 +1



 l1 ,l2 ∈Z

k+1 ξ2l1 ,2l2 +1 ηuk1 −l1 ,u2 −l2 , η2u = 1 +1,2u2 +1

l1 ,l2 ∈Z

ξ2l1 +1,2l2 ηuk1 −l1 ,u2 −l2 ,



ξ2l1 +1,2l2 +1 ηuk1 −l1 ,u2 −l2 .

l1 ,l2 ∈Z

A necessary condition for uniform convergence of scheme (3.2) is given in the following theorem.  Theorem 3.1 [26] Let ξ(x) = ξ(x1 , x2 ) = u,v ξu,v x1u x2v be the symbol or Laurent polynomial of bivariate subdivision scheme S, which is defined on quad-meshes. Then a necessary condition for the convergence of S is: 

ξδ−2γ = 1, δ ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}.

(3.3)

γ ∈Z2

This implies that: ξ(1, 1) = 4, ξ(−1, 1) = ξ(1, −1) = ξ(−1, −1) = 0.

(3.4)

ξ(x) = (1 + x2 )b(x) Theorem 3.2 [26] Suppose the schemes with symbols ξ [1] (x) = 1+x 1 ξ(x) = (1 + x )b(x), are both contractive, namely and ξ [2] (x) = 1+x 1 2

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lim (Sξ [1] )k η0 = 0,

k→∞

lim (Sξ [2] )k η0 = 0,

k→∞

for any initial data η0 then the scheme Sξ with the symbol: ξ(x) = (1 + x1 )(1 + x2 )b(x), x = (x1 , x2 ), is convergent. Conversely, if Sξ is convergent then Sξ [1] and Sξ [2] are contractive. Remark 3.1 Thus convergence is checked in this case by checking the contractivity of two subdivision schemes Sξ1 , Sξ2 . If b(x1 , x2 ) = b(x2 , x1 ), which is typical for schemes having the symmetry of the square grid, then ξ(x1 , x2 ) = ξ(x2 , x1 ), and the contractivity of only one scheme has to be checked. Theorem 3.3 [26] Let ξ(x1 , x2 ) = (1 + x1 )n (1 + x2 )n b(x).

(3.5)

If the schemes with the masks ξu,v (x1 , x2 ) =

2u+v ξ(x1 , x2 ) , u, v = 0, ..., n (1 + x1 )u (1 + x2 )v

(3.6)

are convergent, then Sξ generate C n function. Remark 3.2 For C n continuity of Sξ , we have to show that the subdivision schemes Su,v , corresponding to masks ξu,v (x1 , x2 ) foru, v = 0, 1, ..., n are convergent and it [1] [2] and Su,v corresponding to the is equivalent to checking whether schemes Su,v ξu,v (x1 ,x2 ) ξu,v (x1 ,x2 ) [1] [2] masks ξu,v (x1 , x2 ) = 1+x1 and ξu,v (x1 , x2 ) = 1+x2 are contractive, which is











[1] L

[2] L

equivalent to checking whether 21 Su,v

< 1 and 21 Su,v

< 1, for some ∞ ∞ integer L > 0. Since there are four rules for computing the values at next refinement level, we define the norm as: ⎧ ⎫



⎨ ⎬    

1 [k]

[k] [k] [k] [k]

S = 1 max |ξ2s,2t |, |ξ2s+1,2t |, |ξ2s,2t+1 |, |ξ2s+1,2t+1 | ,

2 u,v

⎩ ⎭ 2 ∞ s,t∈Z

s,t∈Z

s,t∈Z

s,t∈Z

(3.7) where k = 1, 2.

3.2 Construction of Septic B-Spline Tensor Product Scheme Consider the proposed Septic B-spline subdivision scheme (2.1) 

k+1 8 k 56 k 56 k 8 k = 128 ηu−1 + 128 ηu + 128 ηu+1 + 128 ηu+2 , η2u k+1 1 k 28 k 70 k 28 k ηu+2 + η2u+1 = 128 ηu−1 + 128 ηu + 128 ηu+1 + 128

1 k η , 128 u+3

(3.8)

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111

and its mask is:   1 8 28 56 70 56 28 8 1 ξ = ..., 0, 0, , , , , , , , , , 0, 0, ... , 128 128 128 128 128 128 128 128 128 and its Laurent polynomial is given as: ξ(x) =

 x−4  8 x + 8 x7 + 28 x6 + 56 x5 + 70 x4 + 56 x3 + 28 x2 + 8 x + 1 . 128

This implies that  x1−4  8 x1 + 8 x17 + 28 x16 + 56 x15 + 70 x14 + 56 x13 + 28 x12 + 8 x1 + 1 , 128  x−4  ξ(x2 ) = 2 x28 + 8 x27 + 28 x26 + 56 x25 + 70 x24 + 56 x23 + 28 x22 + 8 x2 + 1 , 128 ξ(x1 ) =

Since ξ(x1 , x2 ) = ξ(x1 )ξ(x2 ), then we have the following Laurent polynomial of Septic B-spline scheme Sξ is:ξ(x1 , x2 ) =

x1−4 x2−4 (x1 8 x2 8 + 8 x1 8 x2 7 + 8 x1 7 x2 8 + 28 x1 8 x2 6 + 64 x1 7 x2 7 + 28 x1 6 x2 8 + 56 x1 8 x2 5 + 224 x1 7 x2 6 16,384 +224 x1 6 x2 7 + 56 x1 5 x2 8 + 70 x1 8 x2 4 + 448 x1 7 x2 5 + 784 x1 6 x2 6 + 448 x1 5 x2 7 + 70 x1 4 x2 8 + 56 x1 8 x2 3 +560 x1 7 x2 4 + 1568 x1 6 x2 5 + 1568 x1 5 x2 6 + 560 x1 4 x2 7 + 56 x1 3 x2 8 + 28 x1 8 x2 2 + 448 x1 7 x2 3 +1960 x1 6 x2 4 + 3136 x1 5 x2 5 + 1960 x1 4 x2 6 + 448 x1 3 x2 7 + 28 x1 2 x2 8 + 8 x1 8 x2 + 224 x1 7 x2 2 +1568 x1 6 x2 3 + 3920 x1 5 x2 4 + 3920 x1 4 x2 5 + 1568 x1 3 x2 6 + 224 x1 2 x2 7 + 8 x1 x2 8 + x1 8 + 64 x1 7 x2 +784 x1 6 x2 2 + 3136 x1 5 x2 3 + 4900 x1 4 x2 4 + 3136 x1 3 x2 5 + 784 x1 2 x2 6 + 64 x1 x2 7 + x2 8 + 8 x1 7 +224 x1 6 x2 + 1568 x1 5 x2 2 + 3920 x1 4 x2 3 + 3920 x1 3 x2 4 + 1568 x1 2 x2 5 + 224 x1 x2 6 + 8 x2 7 + 28 x1 6 +448 x1 5 x2 + 1960 x1 4 x2 2 + 3136 x1 3 x2 3 + 1960 x1 2 x2 4 + 448 x1 x2 5 + 28 x2 6 + 56 x1 5 + 560 x1 4 x2 +1568 x1 3 x2 2 + 1568 x1 2 x2 3 + 560 x1 x2 4 + 56 x2 5 + 70 x1 4 + 448 x1 3 x2 + 784 x1 2 x2 2 + 448 x1 x2 3 +70 x2 4 + 56 x1 3 + 224 x1 2 x2 + 224 x1 x2 2 + 56 x2 3 + 28 x1 2 + 64 x1 x2 + 28 x2 2 + 8 x1 + 8 x2 + 1).

=

x1−4 x2−4 (1 + x1 )8 (1 + x2 )8 . 16,384

From (2.1), we suggest the following Septic B-spline tensor product scheme: 64 448 k 448 k 64 + + + ηk η η ηk 16,384 u−1,v−1 16,384 u,v−1 16,384 u+1,v−1 16,384 u+2,v−1 448 k 3136 k 3136 k 448 k + + + + η η η η 16,384 u−1,v 16,384 u,v 16,384 u+1,v 16,384 u+2,v 448 k 3136 k 3136 k 448 k + + + + η η η η 16,384 u−1,v+1 16,384 u,v+1 16,384 u+1,v+1 16,384 u+2,v+1 64 448 k 448 k 64 + + + + , ηk η η ηk 16,384 u−1,v+2 16,384 u,v+2 16,384 u+1,v+2 16,384 u+2,v+2 8 56 56 8 k+1 = + + + η2u+1,2v ηk ηk ηk ηk 16,384 u−1,v−1 16,384 u,v−1 16,384 u+1,v−1 16,384 u+2,v−1 224 k 1568 k 1568 k 224 k + + + + η η η η 16,384 u−1,v 16,384 u,v 16,384 u+1,v 16,384 u+2,v k+1 = η2u,2v

(3.9)

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560 k 3920 k 3920 k 560 k + + + η η η η 16,384 u−1,v+1 16,384 u,v+1 16,384 u+1,v+1 16,384 u+2,v+1 224 k 1568 k 1568 k 224 k + + + + η η η η 16,384 u−1,v+2 16,384 u,v+2 16,384 u+1,v+2 16,384 u+2,v+2 8 56 56 8 + + + + , ηk ηk ηk ηk 16,384 u−1,v+3 16,384 u,v+3 16,384 u+1,v+3 16,384 u+2,v+3 8 224 560 224 8 k+1 = + + + + ηk ηk ηk ηk ηk η2u,2v+1 16,384 u−1,v−1 16,384 u−1,v 16,384 u−1,v+1 16,384 u−1,v+2 16,384 u−1,v+3 56 1568 k 3920 k 1568 k 56 + + + + + ηk η η η ηk 16,384 u,v−1 16,384 u,v 16,384 u,v+1 16,384 u,v+2 16,384 u,v+3 56 1568 k 3920 k 1568 k 56 + + + + + ηk η η η ηk 16,384 u+1,v−1 16,384 u+1,v 16,384 u+1,v+1 16,384 u+1,v+2 16,384 u+1,v+3 8 224 k 560 k 224 k 8 + + + + + , ηk η η η ηk 16,384 u+2,v−1 16,384 u+2,v 16,384 u+2,v+1 16,384 u+2,v+2 16,384 u+2,v+3 1 28 70 28 1 k+1 η2u+1,2v+1 = + + + + ηk ηk ηk ηk ηk 16,384 u−1,v−1 16,384 u,v−1 16,384 u+1,v−1 16,384 u+2,v−1 16,384 u+3,v−1 28 784 k 1960 k 784 k 28 + + + + + ηk η η η ηk 16,384 u−1,v 16,384 u,v 16,384 u+1,v 16,384 u+2,v 16,384 u+3,v 70 1960 k 4900 k 1960 k 70 + + + + + ηk η η η ηk 16,384 u−1,v+1 16,384 u,v+1 16,384 u+1,v+1 16,384 u+2,v+1 16,384 u+3,v+1 28 784 k 1960 k 784 k 28 + + + + + , ηk η η η ηk 16,384 u−1,v+2 16,384 u,v+2 16,384 u+1,v+2 16,384 u+2,v+2 16,384 u+3,v+2 1 28 70 28 1 + + + + + ηk ηk ηk ηk ηk 16,384 u−1,v+3 16,384 u,v+3 16,384 u+1,v+3 16,384 u+2,v+3 16,384 u+3,v+3 +

(3.10)

3.3 Analysis of Septic B-Spline Tensor Product Scheme In this section, we present C 7 continuous Septic B-Spline tensor product scheme. To check the continuity of the Septic B-Spline tensor product scheme(3.9), we apply similar analysis tools to those in the case. From (3.5) for u = 0, v = 0 and then from (3.6), we get ξ0,0 (x1 , x2 ) = ξ(x1 , x2 ) =

x1−4 x2−4 (1 + x1 )8 (1 + x2 )8 16,384

This implies ξ0,0 (x1 , x2 ) =

x1−4 x2−4 16,384

(x1 8 x2 8 + 8 x1 8 x2 7 + 8 x1 7 x2 8 + 28 x1 8 x2 6 + 64 x1 7 x2 7 + 28 x1 6 x2 8 + 56 x1 8 x2 5

+224 x1 7 x2 6 + 224 x1 6 x2 7 + 56 x1 5 x2 8 + 70 x1 8 x2 4 + 448 x1 7 x2 5 + 784 x1 6 x2 6 + 448 x1 5 x2 7 +70 x1 4 x2 8 + 56 x1 8 x2 3 + 560 x1 7 x2 4 + 1568 x1 6 x2 5 + 1568 x1 5 x2 6 + 560 x1 4 x2 7 + 56 x1 3 x2 8 +28 x1 8 x2 2 + 448 x1 7 x2 3 + 1960 x1 6 x2 4 + 3136 x1 5 x2 5 + 1960 x1 4 x2 6 + 448 x1 3 x2 7 + 28 x1 2 x2 8 +8 x1 8 x2 + 224 x1 7 x2 2 + 1568 x1 6 x2 3 + 3920 x1 5 x2 4 + 3920 x1 4 x2 5 + 1568 x1 3 x2 6 + 224 x1 2 x2 7 +8 x1 x2 8 + x1 8 + 64 x1 7 x2 + 784 x1 6 x2 2 + 3136 x1 5 x2 3 + 4900 x1 4 x2 4 + 3136 x1 3 x2 5 + 784 x1 2 x2 6 +64 x1 x2 7 + x2 8 + 8 x1 7 + 224 x1 6 x2 + 1568 x1 5 x2 2 + 3920 x1 4 x2 3 + 3920 x1 3 x2 4 + 1568 x1 2 x2 5 +224 x1 x2 6 + 8 x2 7 + 28 x1 6 + 448 x1 5 x2 + 1960 x1 4 x2 2 + 3136 x1 3 x2 3 + 1960 x1 2 x2 4 + 448 x1 x2 5 +28 x2 6 + 56 x1 5 + 560 x1 4 x2 + 1568 x1 3 x2 2 + 1568 x1 2 x2 3 + 560 x1 x2 4 + 56 x2 5 + 70 x1 4 +448 x1 3 x2 + 784 x1 2 x2 2 + 448 x1 x2 3 + 70 x2 4 + 56 x1 3 + 224 x1 2 x2 + 224 x1 x2 2 + 56 x2 3 +28 x1 2 + 64 x1 x2 + 28 x2 2 + 8 x1 + 8 x2 + 1).

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[1] [2] [1] If S0,0 and S0,0 are subdivision schemes corresponding to the masks ξ0,0 (x1 , x2 ) and [2] ξ0,0 (x1 , x2 ) respectively, then [1] ξ0,0 (x1 , x2 ) =

ξ0,0 (x1 , x2 ) x−4 x−4 = 1 2 (1 + x1 )7 (1 + x2 )8 1 + x1 16,384

[2] ξ0,0 (x1 , x2 ) =

ξ0,0 (x1 , x2 ) x−4 x−4 = 1 2 (1 + x1 )8 (1 + x2 )7 1 + x2 16,384

and

This implies [1]

ξ0,0 (x1 , x2 ) =

x1−4 x2−4 16,384

{x1 7 x2 8 + 8 x1 7 x2 7 + 7 x1 6 x2 8 + 28 x1 7 x2 6 + 56 x1 6 x2 7 + 21 x1 5 x2 8 + 56 x1 7 x2 5

+196 x1 6 x2 6 + 168 x1 5 x2 7 + 35 x1 4 x2 8 + 70 x1 7 x2 4 + 392 x1 6 x2 5 + 588 x1 5 x2 6 + 280 x1 4 x2 7 +35 x1 3 x2 8 + 56 x1 7 x2 3 + 490 x1 6 x2 4 + 1176 x1 5 x2 5 + 980 x1 4 x2 6 + 280 x1 3 x2 7 + 21 x1 2 x2 8 +28 x1 7 x2 2 + 392 x1 6 x2 3 + 1470 x1 5 x2 4 + 1960 x1 4 x2 5 + 980 x1 3 x2 6 + 168 x1 2 x2 7 + 7 x1 x2 8 +8 x1 7 x2 + 196 x1 6 x2 2 + 1176 x1 5 x2 3 + 2450 x1 4 x2 4 + 1960 x1 3 x2 5 + 588 x1 2 x2 6 + 56 x1 x2 7 +x2 8 + x1 7 + 56 x1 6 x2 + 588 x1 5 x2 2 + 1960 x1 4 x2 3 + 2450 x1 3 x2 4 + 1176 x1 2 x2 5 + 196 x1 x2 6 +8 x2 7 + 7 x1 6 + 168 x1 5 x2 + 980 x1 4 x2 2 + 1960 x1 3 x2 3 + 1470 x1 2 x2 4 + 392 x1 x2 5 + 28 x2 6 +21 x1 5 + 280 x1 4 x2 + 980 x1 3 x2 2 + 1176 x1 2 x2 3 + 490 x1 x2 4 + 56 x2 5 + 35 x1 4 + 280 x1 3 x2 +588 x1 2 x2 2 + 392 x1 x2 3 + 70 x2 4 + 35 x1 3 + 168 x1 2 x2 + 196 x1 x2 2 + 56 x2 3 + 21 x1 2 +56 x1 x2 + 28 x2 2 + 7 x1 + 8 x2 + 1},

and [2]

ξ0,0 (x1 , x2 ) =

x1−4 x2−4 16,384

{x1 8 x2 7 + 7 x1 8 x2 6 + 8 x1 7 x2 7 + 21 x1 8 x2 5 + 56 x1 7 x2 6 + 28 x1 6 x2 7 + 35 x1 8 x2 4

+168 x1 7 x2 5 + 196 x1 6 x2 6 + 56 x1 5 x2 7 + 35 x1 8 x2 3 + 280 x1 7 x2 4 + 588 x1 6 x2 5 + 392 x1 5 x2 6 +70 x1 4 x2 7 + 21 x1 8 x2 2 + 280 x1 7 x2 3 + 980 x1 6 x2 4 + 1176 x1 5 x2 5 + 490 x1 4 x2 6 + 56 x1 3 x2 7 +7 x1 8 x2 + 168 x1 7 x2 2 + 980 x1 6 x2 3 + 1960 x1 5 x2 4 + 1470 x1 4 x2 5 + 392 x1 3 x2 6 + 28 x1 2 x2 7 +x1 8 + 56 x1 7 x2 + 588 x1 6 x2 2 + 1960 x1 5 x2 3 + 2450 x1 4 x2 4 + 1176 x1 3 x2 5 + 196 x1 2 x2 6 +8 x1 x2 7 + 8 x1 7 + 196 x1 6 x2 + 1176 x1 5 x2 2 + 2450 x1 4 x2 3 + 1960 x1 3 x2 4 + 588 x1 2 x2 5 +56 x1 x2 6 + x2 7 + 28 x1 6 + 392 x1 5 x2 + 1470 x1 4 x2 2 + 1960 x1 3 x2 3 + 980 x1 2 x2 4 + 168 x1 x2 5 +7 x2 6 + 56 x1 5 + 490 x1 4 x2 + 1176 x1 3 x2 2 + 980 x1 2 x2 3 + 280 x1 x2 4 + 21 x2 5 + 70 x1 4 +392 x1 3 x2 + 588 x1 2 x2 2 + 280 x1 x2 3 + 35 x2 4 + 56 x1 3 + 196 x1 2 x2 + 168 x1 x2 2 + 35 x2 3 +28 x1 2 + 56 x1 x2 + 21 x2 2 + 8 x1 + 7 x2 + 1}.

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By utilizing (3.7), we get





1 [1]



S = 1 max  1

2 0,0

 16,384 (7 + 35 + 196 + 21 + 980 + 490 + 1 + 588 + 2450 + 196 2 ∞ +28 + 1470 + 980 + 7 + 70 + 588 + 35 + 28 + 21 + 1)| ,   1   16,384 (1 + 21 + 28 + 35 + 588 + 70 + 7 + 980 + 1470 + 28 +196 + 2450 + 588 + 1 + 490 + 980 + 21 + 196 + 35 + 7)| ,   1   16,384 (56 + 280 + 392 + 168 + 1960 + 8 + 1176 + 1960 +56 + 56 + 1176 + 392 + 280 + 56 + 168 + 8)| ,   1   16,384 (8 + 168 + 56 + 280 + 1176 + 56 + 56 + 1960 + 1176 + 8 + 392 +1960 + 168 + 392 + 280 + 56)|}   1 1 1 1 1 = max , , , < 1, 2 2 2 2 2

(3.11)





1 [2]



S = 1 max  1

2 0,0

 16,384 (7 + 35 + 196 + 21 + 980 + 490 + 1 + 588 + 2450 + 196 2 ∞ +28 + 1470 + 980 + 7 + 70 + 588 + 35 + 28 + 21 + 1)| ,   1   16,384 (1 + 21 + 28 + 35 + 588 + 70 + 7 + 980 + 1470 + 28 +196 + 2450 + 588 + 1 + 490 + 980 + 21 + 196 + 35 + 7)| ,   1   16,384 (56 + 280 + 392 + 168 + 1960 + 8 + 1176 + 1960 +56 + 56 + 1176 + 392 + 280 + 56 + 168 + 8)| ,   1   16,384 (8 + 168 + 56 + 280 + 1176 + 56 + 56 + 1960 + 1176 + 8 + 392 +1960 + 168 + 392 + 280 + 56)|}   1 1 1 1 1 = max < 1, , , , 2 2 2 2 2

(3.12)

Let S0,0 be the subdivision scheme corresponding to the Laurent polynomial, ξ0,0 (x1 , x2 ), then by using (3.7), we have from (3.11) and (3.12), we see that [k] ; k = 1, 2, & u, v = 0 are contractive, so by the theorem 3.2, the subdiviSu,v sion schemes Su,v , corresponding to masks ξu,v (x1 , x2 ) for u, v = 0 are convergent. Hence by theorem 3.3, proposed scheme Sξ is C 0 continuous. And so on. By applying above procedure repeatedly, it is obvious that the proposed scheme has C 7 continuity. (See appendix A [47])

4 Numerical Examples Here, performance of our Septic B spline tensor product scheme is shown. The refinement algorithm of Septic B spline tensor product scheme involves computing a new vertex related to each pair(vertex, face) of the original mesh. The new vertices are computed as weighted averages of the points corresponding to each face of the original mesh. For this case, these weights (around a face) are:

Construction and Application of Septic B-Spline …

(a) Initial Polygon

115

(b) Limit Curve

Fig. 1 Shows initial polygon and its corresponding limit curve

1 8 28 56 64 70 224 448 560 784 1568 1960 { 16,384 , 16,384 , 16,384 , 16,384 , 16,384 , 16,384 , 16,384 , 16,384 , 16,384 , 16,384 , 16,384 , 16,384 , 3136 3920 , }. 16,384 16,384

The newly generated vertices are then joined to form the faces of the refined control mesh. Below here we also discussed the visual performance of the Septic B-spline subdivision scheme. In order to present the achievements of the scheme (3.10), we discuss continuity and shape of limit curves. • Figure 1a Shows the initial polygon of alphabet W and (b) shows smoothness of limit curve at level 3. • Figure 2a Shows the initial polygon of an Arrow and (b) shows its smoothness of limit curve at level 3. • Figure 3a Shows the initial polygon of a saw and (b) shows its smoothness of limit curve at level 3. • Figure 4a Shows the initial polygon of symbol  and (b) shows its smoothness of limit curve at level 3. • Figure 5a Shows the initial polygon of a Star and (b) shows its smoothness of limit curve at level 3. The following method is used to generate refined polygon from the control polygon. Each vertex point is computed in the proportion of [8: 56: 56: 8:]/128 and each edge point is computed in the proportion of [1: 28: 70: 28: 1:]/128. A face point is obtained as the centroid of every mesh of the given control polygon and vertex point is obtained as the normal of a vertex mesh in the control polygon. The new points are then show association with each other. There will be two edges along each side of each vertex in the previous mesh. These pairs are linked and form quadrilaterals over the old edges. Inside every control polygon, there will be the same number of new vertices. These are related to each other inside the control polygon. Finally, around every old vertex there is another vertex in the adjoining corner of each old polygon.

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(a) Initial Polygon

(b) Limit Curve

Fig. 2 Shows initial polygon and its corresponding limit curve

(a) Initial Polygon

(b) Limit Curve

Fig. 3 Shows initial polygon and its corresponding limit curve.

(a) Initial Polygon

(b) Limit Curve

Fig. 4 Shows initial polygon and its corresponding limit curve

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(a) Initial Polygon

117

(b) Limit curve

Fig. 5 Shows initial polygon and its corresponding limit curve.

These are used to form another polygon with the same number of edges. The new mesh will generate quadrilaterals for each edge in the old vertex, by making a little n-sided polygon. For each n-sided polygon, it will produce a n-sided polygon for each n-valence (Valence are the number of edges that touches the vertex). Using first iteration of our scheme all vertices will have a valence of four, so resulting applications will generate quadrilaterals for the vertices. To make smooth model the scheme is applied repeatedly. Figure 1 shows that each face of the block has been partitioned into four quadrilaterals. The vertex points, the corners of the block, have been “squeezed internally”. We have applied our new scheme as a plugin in our modeling and animation, by using MAPLE and MATLAB softwares. We are able to obtain the required model after 5th subdivision level of the proposed scheme. In Fig. 1a–f, models of visual performance of our proposed scheme are displayed. It is clearly seen that proposed scheme shows smoothness in generating different curves (Fig. 6).

5 Conclusion Here, we have offered Septic B-spline approximating subdivision scheme using symbolical form of Laurent polynomial method to determine the smoothness of our scheme. Our scheme gives good results for modeling of curves and surfaces. It produces continuous models as shown in figures. We have discussed some well known properties to improve the validity of our scheme.

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1

1

0.5

0.5

0

0

-0.5

-0.5

-1 4

-1 4 2 0 -2 -4

-4

-2

0

2

4

2 0 -2 -4

-4

-2

(a) At k=0

(b) At k=1

(c) At k=2

(d) At k=3

(e) At k=4

(f) At k=5

Fig. 6 a Initial polygon while b–f are results upto 5th subdivision levels.

0

2

4

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References 1. De Rham, G.: Un peu de mathématiques à propos d’une courbe plane. Elem. der Mathe. 2, 73–76 (1947) 2. Chaikin, G.M.: An algorithm for high-speed curve generation. Comput. Graph. Image Process. 3(4), 346–349 (1974) 3. Doo, D., Sabin, M.: Behaviour of recursive division surfaces near extraordinary points. Comput. Aided Des. 10(6), 356–360 (1978) 4. Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10(6), 350–355 (1978) 5. Dyn, N., Levin, D., Gregory, J.A.: A 4-point interpolatory subdivision scheme for curve design. Comput. Aided Geometr. Des. 4(4), 257–268 (1987) 6. Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. In: Constructive Approximation (pp. 49–68). Springer, Boston (1989) 7. Dyn, N., Levine, D., Gregory, J.A.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graph. (TOG) 9(2), 160–169 (1990) 8. Dyn, N., Levin, D.: Analysis of Hermite-type subdivision schemes. Ser. Approxim. Decomposit. 6, 117–124 (1995) 9. Cai, Z.: Convergence, error estimation and some properties of four-point interpolation subdivision scheme. Comput. Aided Geometr. Des. 12(5), 459–468 (1995) 10. Stam, J.: Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, pp. 395–404. ACM (1998) 11. Hassan, M., Dodgson, N.A.: Ternary and three-point univariate subdivision schemes (No. UCAM-CL-TR-520). University of Cambridge, Computer Laboratory, pp. 1–15 (2001) 12. Hassan, M.F., Ivrissimitzis, I.P., Dodgson, N.A., Sabin, M.A.: An interpolating 4-point C 2 ternary stationary subdivision scheme. Comput. Aided Geometr. Des. 19(1), 1–18 (2002) 13. Dyn, N., Floater, M.S., Hormann, K.: A C 2 four-point subdivision scheme with fourth order accuracy and its extensions. Mathematical methods for curves and surfaces. Tromsø, 99. 145– 156 (2004) 14. Mustafa, G., Xuefeng, L.: A subdivision scheme for volumetric models. Appl. Math. J. Chin. Univ. 20(2), 213–224 (2005) 15. Beccari, C., Casciola, G., Romani, L.: An interpolating 4-point C 2 ternary non-stationary subdivision scheme with tension control. Comput. Aided Geometr. Des. 24(4), 210–219 (2007) 16. Siddiqi, S.S., Ahmad, N.: A C 6 approximating subdivision scheme. Appl. Math. Lett. 21(7), 722–728 (2008) 17. Hormann, K., Sabin, M.A.: A family of subdivision schemes with cubic precision. Comput. Aided Geometr. Des. 25(1), 41–52 (2008) 18. Faheem, K., Mustafa, G.: Ternary six-point interpolating subdivision scheme. Lobachevskii J. Math. 29(3), 153–163 (2008) 19. Mustafa, G., Khan, F.: A new 4-point quaternary approximating subdivision scheme. In: Abstract and Applied Analysis, Vol. 2009, pp. 1–14. Hindawi (2009) 20. Mustafa, G., Khan, F., Ghaffar, A.: The m-point approximating subdivision scheme. Lobachevskii J. Math. 30(2), 138–145 (2009) 21. Zheng, H., Hu, M., Peng, G.: Ternary even symmetric 2n-point subdivision. In: International Conference on Computational Intelligence and Software Engineering, 2009. CiSE 2009, pp. 1–4. IEEE (2009) 22. Mustafa, G., Rehman, N.A.: The mask of (2b + 4)-point n-ary subdivision scheme. Computing 90(1–2), 1–14 (2010) 23. Siddiqi, S.S., Rehan, K.: Modified form of binary and ternary 3-point subdivision schemes. Appl. Math. Comput. 216(3), 970–982 (2010) 24. Mustafa, G., Ghaffar, A., Khan, F.: The odd-point ternary approximating schemes. Am. J. Comput. Math. 1(02), 111–118 (2011)

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25. Mustafa, G., Ghaffar, A., Aslam, M.: A subdivision-regularization framework for preventing over fitting of data by a model. Appl. Appl. Math. Int. J. (AAM) 8(1), 178–190 (2013) 26. Ghaffar, A., Mustafa, G., Qin, K.: Construction and application of 3-point tensor product scheme. Appl. Math. 4(3), 477–485 (2013) 27. Mustafa, G., Ghaffar, A., Bari, M.: (2n − 1)-point binary approximating scheme. In: Eighth International Conference on Digital Information Management (ICDIM 2013), pp. 363–368. IEEE (2013) 28. Mustafa, G., Ashraf, P., Deng, J.: Generalized and unified families of interpolating subdivision schemes. Numer. Math. Theor. Methods Appl. 7(2), 193–213 (2014) 29. Mustafa, G., Bari, M.: A new class of odd-point ternary non-stationary schemes. Br. J. Math. Comput. Sci. 4(1), 133–152 (2014) 30. Mustafa, G., Ashraf, P., Aslam, M.: Binary univariate dual and primal subdivision schemes. SeMA J. 65(1), 23–35 (2014) 31. Siddiqi, S.S., us Salam, W., Rehan, K., : Binary 3-point and 4-point non-stationary subdivision schemes using hyperbolic function. Appl. Mathe. Comput. 258, 120–129 (2015) 32. Siddiqi, S.S., us Salam, W., Rehan, K.: A new non-stationary binary 6-point subdivision scheme. Appl. Math. Comput. 268, 1227–1239 (2015) 33. Rehan, K., Siddiqi, S.S.: A family of ternary subdivision schemes for curves. Appl. Math. Comput. 270, 114–123 (2015) 34. Rehan, K., Siddiqi, S.S.: A combined binary 6-point subdivision scheme. Appl. Math. Comput. 270, 130–135 (2015) 35. Mustafa, G., Ashraf, P., Saba, N.: A new class of binary approximating subdivision schemes. J. Teknol. 78(4–4), 65–72 (2016) 36. Hameed, R., Mustafa, G.: Construction and analysis of binary subdivision schemes for curves and surfaces originated from Chaikin points. International Journal of Analysis, vol. 2016. Article ID 1092476, 1–15 (2016) 37. Ghaffar, A., Mustafa, G.: An Alternative Method for Constructing Subdivision Algorithm. Sci. Int. 28(5), 5011–5015 (2016) 38. Rehan, K., Sabri, M.A.: A combined ternary 4-point subdivision scheme. Appl. Math. Comput. 276, 278–283 (2016) 39. Siddiqi, S.S., us Salam, W., and Rehan, K., : Construction of binary four and five point nonstationary subdivision schemes from hyperbolic B-splines. Appl. Math. Comput. 280, 30–38 (2016) 40. Siddiqi, S.S., us Salam, W., Rehan, K.: Hyperbolic forms of ternary non-stationary subdivision schemes originated from hyperbolic B-splines. J. Comput. Appl. Math. 301, 16–27 (2016) 41. Us Salam, W., Siddiqi, S.S., Rehan, K.: Chaikin ’s, perturbation subdivision scheme in nonstationary forms. Alexandria Eng. J. 55(3), 2855–2862 (2016) 42. Cheng, L., Zhou, X.: Necessary conditions for the convergence of subdivision schemes with finite masks. Appl. Math. Comput. 303, 34–41 (2017) 43. Akram, G., Bibi, K., Rehan, K., Siddiqi, S.S.: Shape preservation of 4-point interpolating non-stationary subdivision scheme. J. Comput. Appl. Math. 319, 480–492 (2017) 44. Manan, S.A., Ghaffar, A., Rizwan, M., Rahman, G., Kanwal, G.: A subdivision approach to the approximate solution of 3rd order boundary value problem. Commun. Math. Appl. 9(4), 499–512 (2018) 45. Kanwal, G., Ghaffar, A., Hafeezullah, M.M., Manan, S.A., Rizwan, M., Rahman, G.: Numerical solution of 2-point boundary value problem by subdivision scheme. Commun. Math. Appl. 10(2), 1–11 (2019) 46. Ghaffar, A., Bari, M., Ullah, Z., Iqbal, M., Nisar, K.S., Baleanu, D.: A new class of 2q-point nonstationary subdivision schemes and their applications. Mathematics 7(7), 639 (2019) 47. Ghaffar, A., Iqbal, M., Bari, M., Muhammad Hussain, S., Manzoor, R., Sooppy Nisar, K., Baleanu, D.: Construction and application of nine-Tic B-spline tensor product SS. Mathematics 7(8), 675 (2019)

Bayes Meets Tikhonov: Understanding Uncertainty Within Gaussian Framework for Seismic Inversion Muhammad Izzatullah, Daniel Peter, Sergey Kabanikhin, and Maxim Shishlenin

Abstract In this chapter, we demonstrate the sound connection between the Bayesian approach and the Tikhonov regularisation within Gaussian framework. We provide a thorough uncertainty analysis to answer the following two fundamental questions: (1) How well is the estimate determined by a posteriori PDF, i.e. by the combination of observed data and a priori information? (2) What are the respective contributions of observed data and a priori information? To support the proposed methodology, we demonstrate it through numerical applications in seismic inversions. Keywords Bayesian framework · Inverse problems · Seismic inversion In this chapter, we demonstrate the sound connection between the Bayesian approach and the Tikhonov regularisation within Gaussian framework. We provide an uncertainty analysis framework to answer the following two fundamental questions: (1) How well is the estimate determined by the a posteriori PDF, i.e. by the combination of observed data and a priori information? (2) What are the M. Izzatullah (B) · D. Peter Seismic Modeling and Inversion Group, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia e-mail: [email protected] D. Peter e-mail: [email protected] S. Kabanikhin · M. Shishlenin Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 630090 Novosibirsk, Russia e-mail: [email protected] M. Shishlenin e-mail: [email protected] Novosibirsk State University, 630090 Novosibirsk, Russia Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia Mathematical Center in Akademgorodok, 630090 Novosibirsk, Russia © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4_8

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respective contributions of observed data and a priori information? To support the proposed methodology, we demonstrate it through numerical application in linear inverse problem for the seismic travel-time tomography.

1 Introduction The objective of seismic inversion is to provide information about the internal structure of the Earth from observed data. In most geophysical applications, seismic inversion is introduced as an iterative, local optimisation problem that attempts to minimise the least-squares residuals between observed and synthetic data. Mathematically, the inverse problem is ill-posed [1–7], leading to a non-uniqueness of the solutions or unstability of the solutions. It remains challenging to solve inverse problems practically due to limitations in data acquisition, measurement uncertainties and the non-uniqueness of the solution. Several advancements in geophysics have been proposed to tackle these challenges [8–12]. Despite these rapid advancements, the research on uncertainty analysis for the seismic inversion solutions is progressing slowly due to limitations of computational power and algorithms advancement. Uncertainty analysis conventionally performs within the Bayesian framework. Within the geophysics community, uncertainty analysis commonly refers to as the resolution analysis [13–15]. Backus and Gilbert [16, 17] developed the concept of resolution for linear geophysical inverse problems, and Kabanikhin and Shishlenin [44] developed the similar concept from [3] and [39] for solving coefficient inverse problems. Although this concept is not an original concept for uncertainty analysis within the Bayesian framework, we can extend its definition to accommodate the framework. Hence, the resolution defines a balance measurement between the information from the observed data in comparison with the a priori information in determining the parameter estimates [18–22]. One should notice that by this definition, parameters can be well resolved by the entire state of information while being poorly resolved from the observed data. In practice, the observed seismic data typically are finite and corrupted by noise. The main consequence of these features is that the observed data may be informative, relative to the a priori information, only on a low-dimensional subspace of the entire parameter space [8, 9, 23–25]. Focusing back in the Bayesian framework, firstly the parameters of interest are estimated by combining the information from observed data with a priori information on the parameters. The information from observed data reflects in the likelihood function, and the a priori information on the parameters imbeds in the a priori probability density function (PDF). The product of the two determines the a posteriori PDF which is the solution to the inverse problem within the Bayesian framework. The uncertainty analysis can be performed through sampling the a posteriori PDF with the family of Monte Carlo methods [20, 26, 27]. Through the accepted samples, we can infer the parameter fields and conduct the uncertainty analysis. However, for a large-scale inverse problem, e.g. seismic inversion, the family of Monte Carlo methods are infeasible and impractical. As an alternative, the most trivial approach

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to inspecting the a posteriori PDF through the whole of parameter space, its maximum is taken as a point estimate of the parameters of interest. This approach is known as the Laplace approximation in the statistical literature [28–30]. It is based on a second-order Taylor approximation of the log posterior around the maximum a posteriori point (MAP) estimate, which results in a Gaussian approximation to the posterior. However, such a point estimate contains a limited amount of information; hence, it only infers uncertainty about a point instead of the whole parameter space. Despite the fact, the Laplace approximation approach has strong ties with the field of optimisation in estimating the MAP. The problems of estimating the MAP can be simplified into minimising a misfit function of the negative logarithm of a posteriori PDF. The negative logarithm of a posteriori PDF resembles the general Tikhonov regularisation misfit function [31–44]. This simplification enables us to infer the uncertainty at the MAP point. Based on our extended definition of the concept of resolution, the uncertainty analysis problem is essentially one of information reduction. This information reduction reflects in the standard deviations or variances of the parameters contains in the a posteriori covariance matrix. These can be interpreted as overall uncertainty bounds in the directions along which the parameters are poorly resolved. Although this is the most convenient to analyse the uncertainty, note that this a posteriori covariance matrix is evaluated at the MAP point; thus, the form of information is limited. In this chapter, we demonstrate the sound connection between the Bayesian approach and the Tikhonov regularisation within Gaussian framework. We provide an uncertainty analysis framework to answer the following two fundamental questions: (1) How well is the estimate determined by the a posteriori PDF, i.e. by the combination of observed data and a priori information? (2) What are the respective contributions of observed data and a priori information? To support the proposed methodology, we demonstrate it through numerical application in linear inverse problem for the seismic travel-time tomography.

2 Regularisation Method of A. N. Tikhonov We start this chapter by introducing the Tikhonov regularisation technique. From the perspective of inverse and ill-posed problems, we can pose the problems as Am = d , where m represents the model parameter, and d is the observed data. We define M and D as metric spaces of model parameter and data, respectively. A is an operator A : M → M, which maps m from model parameter space M onto data space D. ˜ ⊂M In many ill-posed problems, Am = d has a class of possible solutions M which is not compact, but due to the data d contains measurement errors, the output ˜ lies beyond the existence class. For constructing the approximate solution A(M) solutions to such problems, A. N. Tikhonov (1963, 1964) proposed a regularisation technique. Let me to be the exact solution of an ill-posed problem Am = d for some d ∈ D.

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Definition (Regularising Family) The family of operators {Rα }α>0 is called regularising for the problem Am = d if 1. for any α > 0 operator Rα : D → M continuous, 2. for any ε > 0, there is α∗ > 0 such that for all α ∈ (0, α∗ ) ρM (Rα d , me ) < ε, other word, lim Rα d = me .

α→+0

(1)

If the right side of the equation Am = d is given approximately and the error of the original data ρD (dδ , d ) ≤ δ is known, then the regularising family {Rα }α>0 allows not only to build an approximate solution mαδ = Rα dδ , but also evaluate the evasion of the approximate solution mαδ from the exact me . In fact, by virtue of the triangle inequalities, ρM (mαδ , me ) ≤ ρM (mαδ , Rα d ) + ρM (Mα d , me ).

(2)

When α → +0, the second term on the right-hand side (2) tends to zero. Due to the ill-posedness of the problem, to estimate the first term for α → +0, δ → +0 is a complex problem. However, this can be solved by taking into account the a priori and/or a posteriori information about the exact solution. Consider the case where M, D are Banach spaces as an example. Let A : M → D as a linear, completely continuous operator, and Rα as a linear operator for all α > 0. Assume that for d ∈ D there is a single solution me and instead of d , given its approximation dδ ∈ D such that d − dδ  ≤ δ.

(3)

We estimate the norm of the difference between the exact solution me and the regularised one mαδ = Rα dδ by me − mαδ  ≤ me − Rα d  + Rα d − Rα dδ .

(4)

Denote me − Rα d  = γ (me , α). By virtue of the property (1) of the regularising family, the first term on the right-hand side of (4) tends to zero when α → 0, i.e., lim γ (me , α) = 0. From the linearity of Rα and the condition (3) it follows that

α→0

Rα d − Rα dδ  ≤ Rα  δ. Recall that the norm of the operator A : M → D, where M and D are Banach spaces, defined by the formula

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Am . m∈M m

A = sup

m=0

Because the problem is ill-posed, the norm Rα  cannot be uniformly bounded, otherwise lim Rα = A−1 , and the problem Am = d would be classically wellα→+0

posed. However, if α and δ tend to zero consistently, then the right-hand side of the received estimate me − mαδ  ≤ γ (me , α) + Rα δ

(5)

approach zero. In fact, let’s denote ω(me , δ) = inf {γ (me , α) + Rα δ} and we show α>0

that

lim ω(me , δ) = 0.

δ→0

Let ε > 0 be an arbitrary number. Because lim γ (me , α) = 0, then there is such α→0

an α0 (ε) that for all α ∈ (0, α0 (ε))

γ (me , α) < ε/2. Denote μ0 (ε) = (0, δ0 (ε))

inf

α∈(0,α0 (ε))

Rα  and take δ0 (ε) = ε/(2μ0 (ε)). Then for δ ∈

inf {Rα  δ} ≤ δ

α>0

inf

{Rα } ≤ ε/2.

α∈(0,α0 (ε))

Therefore, for an arbitrary ε > 0, we can find α0 (ε) and δ0 (ε) such that for all α ∈ (0, α0 (ε)) and δ ∈ (0, δ0 (ε)) me − mαδ  < ε. We have defined the regularisation family, and now let us introduce the Tikhonov functional, a well-known approach in regularisation technique, which can be described as T (m, dδ , α) = Am − dδ  + α(m). For specific operators A and regularisation families {Rα }α>0 , the relationship between the regularisation parameter α and the level errors δ in the observed data can be obtained explicitly. To construct a regularisation family, we need to choose a sequence of αn where αn → 0 as n goes to infinity. This sequence is called the regularisation sequence. Consider an example, let D = M, where M is a separable Hilbert space. Let A be a linear operator, a completely continuous self-adjoint positive, and {ϕn }, {λn } are the corresponding sequences of eigenfunctions and eigenvalues of operator A (λk+1 ≤ λk , k ∈ N). The eigenvalues of the operator A can be obtained analytically for the

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ill-posed continuation problem for Helmholtz equation [43]. The exact solution me of the equation Am = d is can be represented in the form me =

∞  dk ϕk , λk

dk = d , ϕk .

(6)

k=1

the solution exists and belongs to a Hilbert space M, the series (6) and Since ∞ (dk /λk )2 converge. But, for dδ ∈ D, if there is no condition that satisfies d − k=1 ∞ dδ  < δ for the solutions of the equation Am = dδ , then the series (dδk /λk )2 , k=1 dδk = dδ , ϕk , diverges [6, 35–44]. Consider a sequence of operators {Rn } Rn dδ =

n  dδk k=1

λk

ϕk ,

(7)

and we show that it has all the properties of a regularising families at n → ∞. The Rn operators are obviously continuous, Rn  = 1/λn and for all m ∈ M lim Rn Am = lim

n→∞

n→∞

n 

mk ϕk = m,

mk = m, ϕk .

k=1

But then for the exact solution (6) of the equation Am = d and a regularised solution (7) mδn = Rn dδ can be fairly approximate by me − mδn  ≤ me − Rn d  + Rn d − Rn dδ  ≤ me − Rn Ame  + Rn (d − dδ ) ≤

∞  k=n+1

∞ 

m2k + Rn δ =

k=n+1

m2k +

δ . λn

Therefore, for any ε > 0 , we may first select number n0 such that ∞ 

m2k < ε/2,

k=n0 +1

and then select δ > 0 from the condition δ < λn ε/2. Then, for all n > n0 and δ ∈ (0, λn ε/2), we obtained a completed inequality me − mδn  < ε. Now, we introduce the sequential approximation method. Let all the conditions of the previous example be met and, additionally, λ1 < 1. Let’s define the sequence {mn } by the rule mn+1 = mn − Amn + d , m0 = d , n = 0, 1, 2, . . . . We show that if there is an exact solution to the equation Am = d , me ∈ M, then

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lim mn = me .

n→∞

Indeed, mn =

n n   (E − A)k d = (E − A)k Ame . k=0

k=0

Consider now the regularising family, which is built on the basis of minimizing the Tikhonov functional T (m, d , α) = Am − d 2 + α(m), where (m) is continuous non-negative convex functional (stabilizing functional— Tikhonov and Arsenin [1]), satisfying certain conditions. Consider the simplest example of the Tikhonov functional T (m, d , α) = Am − d 2 + αm2 , α > 0.

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Theorem 1 Let M and D be separable Hilbert spaces, A—linear completely continuous operator. Then for all d ∈  and α > 0 functional T (m, d , α) reaches its lower edge on a single element mα . The proof is presented, for example, in the following book [5, 35–42]. We will discuss only some of its stages. First, the regularisation term enforces the nonnegativity of the functional in M . There is an element m0 , where its minimum T0 = inf T (m, d , α) = T (m0 , d , α) ≥ 0 is reached. It is clear that if T0 = 0, then m0 m∈M

will be the solution of the equation Am = d . To prove the uniqueness of m0 , we may write the increment of the functional as T (m + δm, d , α) − T (m, d , α) = 2 Am − d , Aδm + 2α m, δd  + Aδm, Aδm + α δm, δm = 2 A∗ Am − A∗ d + αm, δm + Aδm, Aδm + α δm, δm. Here A∗ is the conjugate operator to A. From the last inequality, by taking into account the approximation of | Aδm, Aδm + α δm, δm ≤ (A2  + α)δm2 , it follows that for any m ∈ M, the functional T (m, d , α) has a gradient of the form T (m, d , α) = 2(A∗ Am − A∗ d + αm). Therefore, m0 is the solution of the equation αm + A∗ Am = A∗ d .

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By virtue of the linearity of the operator A, it is sufficient to prove that homogeneous equation (10) αm + A∗ Am = 0 has only the trivial solution to guarantee the uniqueness of solution in Eq. (9). We can show this through contradiction. That is, assume that there is a non-zero element m ¯ ∈ ¯ = 0. But, then the equality α m ¯ + A∗ Am, ¯ αm ¯ + A∗ Am ¯ = M such that α m ¯ + A∗ Am 0 can be rewritten in the form ¯ m ¯ + 2α Am, ¯ Am ¯ + A∗ Am, ¯ A∗ Am ¯ = 0. α 2 m, This is a contradiction, because in the left part of the latter inequalities the first term is strictly positive, and the other two are non-negative. Thus, we show the uniqueness of solution in Eq. (9). For an approximated observed data dδ ∈ D which satisfies the condition d − dδ  ≤ δ, there is an approximate solution mαδ where its converge to the exact solution me of the equation Am = d as α and δ parameters go to zero, respectively. In fact, there exists and unique minimum of the functional T (m, dδ , α) at the point mαδ , which described by the Theorem 1. Theorem 2 ([5]) Let the conditions of the Theorem 1 satisfied. Suppose that for some d ∈ D, there is a single solution me the equation Am = d . Denote {dδ }δ>0 as a family of approximated data, each element of which satisfies the condition d − dδ  < δ. Then, if δ is chosen tends to zero, the regularisation parameter α = α(δ) is selected so that limδ→0 α(δ) = 0 and limδ→0 δ 2 /α(δ) = 0. Then, the element mα(δ),δ , at which the minimum regularising value is reached by the functional T (m, dδ , α), tends to the exact solution me of the equation Am = d , i.e. limδ→0 mα(δ),δ − me  = 0.

3 Bayesian-Tikhonov Formulation for Linear Inverse Problems In this section, we introduce the problem formulation for linear inverse problems in the Bayesian framework. Next, we demonstrate the sound connection between the Bayesian approach and the Tikhonov regularisation, as explained in the previous section. Here, we introduce the derivation of our problem formulation, and interested readers may refer to [27–30] for more details problem formulation in the Bayesian framework. Consider a linear system of equations with additive Gaussian noise in the observed data, d = Am + e (11) m ∈ Rn , d , e ∈ Rm , A ∈ Rm×n ,

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and let the parameters m and the observed data d be the random variables. We assume that m and e are independent and identical Gaussian distribution with zero means, m ∼ N (0, γ 2 Cm ), e ∼ N (0, σ 2 I ).

(12)

The covariance of the noise, σ 2 I indicates that each component of D is contaminated by independent and identically distributed random noise. Now, we define the a priori density of the form  πprior (m) = exp

 1 T −1 − m Cm m , 2γ 2

(13)

and the likelihood density by assuming that the noise covariance is known is  πlike (d |m) = exp

−

 1 2 . ||d − Am|| 2σ 2

(14)

By Bayes theorem, we can define our a posterior density by the product of the likelihood and the a priori densities. The a posteriori density defines as πpost (m|d ) ∝ πlike (d |m)πprior (m)   1 T −1 1 2 ||d − Am|| − m Cm m ∝ exp − 2σ 2 2γ 2 = exp(−V (m|d )), where V (m|d ) =

1 T −1 1 ||d − Am||2 + m Cm m. 2σ 2 2γ 2

(15)

(16)

Based on the Gaussian likelihood and a priori densities, the a posteriori density πpost (m|d ) is also Gaussian. This condition is only valid for the linear case, and for the non-linear case, this might not hold. Next, we demonstrate the sound connection between the Bayesian approach and the Tikhonov regularisation. Consider the matrix Cm is symmetric positive definite, so is its inverse; thus, it admits Cholesky factorisation of the form Cm−1 = LT L.

(17)

With this notation, we can reformulate our V (m|d ) into the Tikhonov regularisation functional, defines as T (m) = 2σ 2 V (m|d ) = ||d − Am||2 + λ2 ||Lm||2 , λ =

σ , γ

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where λ is the ratio of the noise and the a priori variances. It also acts as a parameter which balancing the information contributions between the likelihood and the a priori densities. The Tikhonov regularisation functional describes in Eq. (18), plays a crucial role in the classical regularisation theory. From the Tikhonov regularisation perspectives, the first term controls the closeness of the solution to the observed data and the second one acts as a penalty. The regularisation matrix L is typically selected so that large ||Lm|| corresponds to an undesirable feature of the solution [31–44]. In order to provide a balance between these two terms, the choice of the λ is a non-trivial challenge in Tikhonov regularisation. In general, Tikhonov regularisation solution is the outcome of a balance between fitting the data and eliminating unwanted features. As we have observed, the Tikhonov regularisation fits well into the Bayesian framework, since its interpretation parallels to the concept of likelihood and a priori densities.

4 The Maximum a Posteriori (MAP) Point In the previous section, we have demonstrated the sound connection between the Bayesian approach and the Tikhonov regularisation. In this section, we discuss the MAP point estimation of the a posteriori density. The MAP estimate is the point in the parameter space that maximises the a posteriori probability density function (PDF). The problems of estimating the MAP can be simplified into minimising a misfit function of the negative logarithm of the a posteriori PDF. The negative logarithm of a posteriori PDF resembles the general Tikhonov regularisation misfit function, as shown in Eq. (18) in the previous section. The Maximum A Posteriori (MAP) estimator defines as, mMAP = arg max πpost (m|d ) m

= arg min − log πpost (m|d ) m

(19)

= arg min V (m|d ). m

In this particular case, the MAP estimator is better describes as mMAP = arg min(||d − Am||2 + λ2 ||Lm||2 ). m

(20)

5 A Generalised Formulation To prepare ourselves for the uncertainty analysis, introducing a generalised problem formulation is essential. This generalised problem formulation should fit a balance measurement between the information from the observed data in comparison with

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the a priori information in determining the parameter estimates. The objective here is to express the parameters in the dimensionless unit so that it is scaling invariant under a linear transformation as proposed by Albert Tarantola [18–20]. By doing so, our uncertainty analysis is consistent with the units. In this section, we generalise the formulation of the negative logarithm of a posteriori PDF in Eq. (18). We let the parameter λ equals unity, and statistically normalise the parameters with the linear transformation so that the transformed parameters have the identity matrix as a priori covariance matrix. This approach is similar in spirit to the standard Tikhonov regularisation functional [31–44], and the concept of prior-preconditioned data misfit Hessian as proposed in [8–10, 23]. Consider the following formulation of the negative logarithm of a posteriori PDF T (m) = − log πpost (m|d ) 1 1 ¯ 2Cprior , = ||d − Am||2CD + ||m − m|| 2 2

(21)

where ||x||C = xT C −1 x. The data noise and a priori covariance matrices are presented by CD and Cprior , respectively. At ∇T (m) = 0, we can determine the MAP estimator through the following derivations: −1 (m − m) ¯ ∇T (m) = AT CD−1 (Am − d ) + Cprior

(22)

= 0.

Consider that CD and Cprior are invertible symmetric positive definite matrices, and they admit the Cholesky factorisation in the form of 1/2

1/2

1/2

1/2

CD = CD CD , Cprior = Cprior Cprior .

(23)

Now we proceed with −1 −1 )m = AT CD−1 d + Cprior m ¯ (AT CD−1 A + Cprior 1/2

−1/2

(Cprior AT CD −1/2

((CD =

1/2

−1/2

CD

−1/2

−1 ACprior + I )Cprior m = Cprior AT CD−1 d + Cprior Cprior m ¯

−1/2

ACprior )T (CD

1/2

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−1/2

ACprior ) + I )Cprior m

1/2 −1/2 −1/2 (Cprior AT CD )CD d

+

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1/2

(24)

−1/2 Cprior m ¯

˜¯ ˜ = d˜ + m, (A A˜ + I )m ˜T

1/2 −1/2 −1/2 ACprior and m ˜ = Cprior m. We define d˜ = A˜ T CD d as the weighted −1/2 noise-normalized data, and m ¯˜ = Cprior m ¯ as the weighted a priori mean. As we have observed in Eq. (24), our problem formulation has reduced into the standard Tikhonov regularisation functional through prior-preconditioned data misfit Hessian approach. In this formulation, we incorporate an a priori mean m ¯ and thus introduces bias into the MAP estimate towards this m. ¯ Although such an a priori mean −1/2

where A˜ = CD

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m ¯ can have a dramatic effect on the MAP estimate, for simplicity of formulation onwards, we consider it equals zero. We consider the SVD approach as a means to estimate the MAP point. We will introduce the concept of filter factors [31, 33–42] as a balance measurement between the information from the observed data in comparison with the a priori information in determining the parameter estimates. Through this approach, we can understand these aspects in more detail and can assist us in our uncertainty analysis. Consider implementing SVD on matrix A˜ = U V T , we can rewrite the formulation in Eq. (24) and define our MAP estimator as follows: (A˜ T A˜ + I )m ˜ = d˜ (V U T U V T + I )m ˜ = d˜

(25)

(V 2 V T + I )m ˜ = d˜ V ( 2 + I )V T m ˜ = d˜ , and we can define our MAP estimator for this generalised formulation as m ˜ MAP = V ( 2 + I )−1 V T d˜ n  v T d˜ fi i 2 vi . = σi i=1

(26)

σ2

i is the filter factor The singular values of A˜ T A˜ denotes by σi2 . The function fi = σ 2 +1 i which acts as the balance measurement between the information from the observed data in comparison with the a priori information in determining the parameter estimates. The filter factor satisfy the following conditions:

 σi2 1, σi2  1, fi = 2 ≈ σi + 1 0, σi2  1.

(27)

We observe that for singular values σi2 larger than 1, the filter factors are close to one. The corresponding SVD components define the well-resolved or certainty directions concerning the observed data in comparison with the a priori information. On the other hand, for singular values much smaller than 1, the filter factors are small. The corresponding SVD components define the poor-resolved or uncertainty directions. In these directions, a priori information over-influence the a posteriori PDF than the likelihood PDF does [22, 25, 32, 35–42]. We note that the optimal balance measurement between the information from the observed data in comparison with the a priori information is at σk2 = 1, where k is the index of optimal truncation of singular values. By quoting Albert Tarantola [18–20], the high certainty directions in the model space are that have a maximum ratio between the a posteriori information

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and the a priori. We will return to this concept of filter factor in later sections since it is essential to the analysis a posteriori covariance and resolution matrices.

6 The a Posteriori Covariance Matrix In the previous section, we introduced the standard and the generalised problem formulation in the Bayes-Tikhonov framework. In the next two sections, we discuss the a posteriori covariance and resolution matrices for analysis of uncertainties. In this work, we focus on the framework within Gaussian assumptions based on the Laplace approximation, which apply to both linear and non-linear cases. The Laplace approximation takes on a second-order Taylor approximation of the log posterior around the MAP estimate, which results in a Gaussian approximation to the posterior [28–30]. Although this is the most convenient to analyse the uncertainty, note that this a posteriori covariance matrix is evaluated at the MAP point; thus, the form of information is limited. For the non-linear case, i.e. non-linear forward model, an exact computation of the a posteriori covariance matrix would involve a full computation of the a posteriori PDF through the whole parameter space. However, this is too computationally demanding in practice even for the most trivial cases. By using Laplace approximation, we can linearise our non-linear forward model about the MAP point resulting the a posteriori PDF looks like a Gaussian distribution with mean mMAP and covariance matrix Cpost = H (mMAP )−1 , where H (mMAP ) denotes the Hessian at MAP point [9, 10, 20, 22]. Following [8–10, 23], assuming that the forward model A(m) is differentiable, we linearise the right-hand side of Eq. (11) around mMAP to obtain d ≈ A(mMAP ) +

∂A(m) (m − mMAP ) + e, ∂m

(28)

where ∂A(m) is the derivative of A(m) evaluated at mMAP . Consequently, this gives the ∂m approximation of the negative logarithm of a posteriori PDF in Eq. (21) around the mMAP 1 −1 (m − mMAP ), (29) T (m) ≈ T (mMAP ) + (m − mMAP )T Cpost 2 thus we have 

 1 T −1 πpost (m|d ) ≈ πpost (mMAP |d ) exp (m − mMAP ) Cpost (m − mMAP ) . 2

(30)

Note that this linearisation is a reasonable approximation when the non-linear forward model is nearly linear or mildly non-linear, i.e., the limits of data noise are

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small and with an abundance of observed data. For the linear case, the Gaussian assumptions describe in Eqs. (29) and (30) are exact. As we discussed in Sect. 4, we should consider a balance measurement between the information from the observed data in comparison with the a priori information in determining the parameter estimates. We statistically normalise the parameters through the linear transformation so that the transformed parameters have the identity matrix as a priori covariance matrix. Consequently, we express the parameters in the dimensionless unit so that it is scaling invariant under a linear transformation—this approach address in details in [9, 10, 20–23] and theoretically supported by [25, 32]. Returning to our generalised formulation, our MAP point for the general case (after transformation) define as 1/2

˜ MAP mMAP = Cprior m = Cprior V ( 2 + I )−1 V T d˜ 1/2

(31)

= Cprior V ( 2 + I )−1 V T Cprior AT CD−1 d 1/2

1/2

= Cpost AT CD−1 d , and the a posteriori covariance is then given by −1 −1 Cpost = (AT CD−1 A + Cprior ) 1/2

1/2

−1/2

= Cprior (Cprior AT CD

−1/2

CD

ACprior + I )−1 Cprior 1/2

1/2

= Cprior (A˜ T A˜ + I )−1 Cprior 1/2

1/2

= Cprior (V 2 V T + I )−1 Cprior 1/2

1/2

= Cprior V ( 2 + I )−1 V T Cprior 1/2

1/2

(32)

= V˜ ( 2 + I )−1 V˜ T n  1 v˜i v˜iT , = 2 σ + 1 i i=1 where V˜ = Cprior V . To further understand the balance between the information from the observed data in comparison with the a priori information in a posteriori covariance matrix, we consider two extreme situations as described in [18, 19, 22, 32] as follows: 1/2

1. The observed data entirely determines the solutions, i.e. the likelihood function entirely determines the shape of the a posteriori PDF. In this case, have our a posteriori PDF in the form of Cpost = (AT CD−1 A)−1 .

(33)

Let’s perform SVD analysis on Eq. (33). Consider CD = γ 2 I and A = U V T ,

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Cpost = (AT CD−1 A)−1  −1 1 T T = (V U )(U V ) γ2 −1  1 2 T = V

V γ2 n  γ2 T = vi v . σ2 i i=1 i

(34)

In Eq. (34), the singular vectors direction are pointing towards where the observed data have an enormous influence in an absolute sense, i.e., not relative to or in the absence of the a priori information. The singular values σi2 which comes from the forward modelling are inversely proportional to the noise level γ 2 in the observed data. For the observed data with a significantly low noise level, the parameters are excellently well-resolved with low variance in the a posteriori covariance matrix. However, this is not the case in practice, since the observed data is commonly noisy with a significant level of noise. On the other hand, if the noise level is enormously high, we are unable to resolve the parameters well enough, even with perfect forward modelling. We 2 can evaluate this based on the ratio of noise level to the singular values γσ 2 . The i a posteriori covariance matrix will remain high in values which reflects high uncertainty. 2. The a priori information entirely determines the solutions, i.e. the shape of the a priori PDF reflects the shape of a posteriori PDF. In this case, it reflects that the uncertainty after inversion is as significant as before inversion, i.e., there is almost no significant reduction in the variance, and there is no significant information gain. Consider Eq. (32) as a basis for the analysis in this case, and we restate the equation below, Cpost = V˜ ( 2 + I )−1 V˜ T n  (35) 1 v˜i v˜iT , = 2 σ +1 i=1 i The singular values σi2 in Eq. (35) are inversely proportional to the noise level in the observed data (Kindly refer to Sect. 4). If we consider that in the extreme of the noise level going to zero, the a posteriori covariance matrix will remain when there are singular values equals zero. There will always remain a term Cpost =



v˜i v˜iT ,

(36)

i∈O

where O denotes the set of singular vectors correspond to σi2 = 0. The singular vectors v˜i along these directions are purely pointing towards a priori information,

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i.e., the observed data does not provide any meaningful information, no matter how low the noise level is. On the other hand, in the absence of a priori information, when the term 1 in the denominator is absent, the a posteriori covariance matrix becomes very large when some of the singular values are very small as addressed in the previous case. Based on the two extreme situations described above, the a posteriori covariance matrix should take into account the interactions between the information from the observed data and the a priori information. Accounting for such interaction is a crucial feature of the optimal covariance approximation which will discuss in a later section.

7 The Resolution Matrix In this section, we discuss the concept of the resolution matrix. As we mentioned earlier, this concept is not an original concept for uncertainty analysis within the Bayesian framework; however, we can extend its definition to accommodate the framework. We define the resolution as a balance measurement between the information from the observed data in comparison with the a priori information in determining the parameter estimates [18–22]. Conventionally, the resolution matrix R is defined through the linear relation between the estimated and exact model [16, 17]. Here, we will formulate the resolution matrix in the Bayesian framework by incorporating the a priori information. Assume dexact as exact data in the absence of uncertainty from an artificial exact model parameters mexact , and we define the relationship between exact data and exact model parameters as, (37) dexact = Amexact . The resolution matrix R is defined through the following linear relation, ¯ = R(mexact − m), ¯ mMAP − m

(38)

−1 −1 T −1 ) A CD A. We need to stress that this where m ¯ = mprior and R = (AT CD−1 A + Cprior concept of the resolution matrix is only valid within the Laplace approximation, i.e., when the exact model parameters mexact are close enough to the maximum a posteriori point mMAP . In practice, neither we know the exact model parameters or the actual resolution matrix that maps the exact model parameters on the estimated ones. However, we can justify this through a posteriori PDF where the exact model parameters are close to the maximum. Hence, if estimated model parameters have a higher probability of being close to the exact one, it should replace the maximum as the point estimate.

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The relationship between the resolution matrix and the a posteriori covariance matrix can be derived within Gaussian framework by manipulating Eq. (32) [13, 20, 22], and we get (39) R = I − Cpost . The relationship stated above is for the statistically normalised model parameters on the a priori information, and we have addressed this in Sect. 4. The essential features of the resolution matrix are the deviations from the identity matrix. Based on the relationship above, it clearly stated that it is the deviations by the a posteriori covariance matrix. To further understand this concept and to relate with a posteriori covariance matrix, consider the following extreme situations: 1. For solutions determined entirely by observed data, we have the a posteriori covariance matrix significantly smaller than the a priori covariance matrix, Cpost < Cprior , hence R = I , i.e., we have a full resolution from the observed data. 2. For solutions determined entirely by a priori information, we have the a posteriori covariance matrix equals (or without a significant variance reduction) to the a priori covariance matrix, Cpost = Cprior hence R = 0, i.e., there is no resolution from the observed data at all. Based on these extreme situations, the resolution matrix is a measure of the balance between the information from the observed data in comparison with the a priori information in determining the parameter estimates, as we defined earlier. In other words, the resolution matrix informs the deviations of the ratio of a posteriori to a priori variance from unity. The relationship in Eq. (39) can contribute more insights through SVD analysis. By combining Eq. (32) with Eq. (39), we get R=I−

n 

σ2 i=1 i

1 v˜i v˜ T , +1 i

(40)

where the low singular values σi2 determine the deviations of the resolution matrix R from the ideal shape I . Following [20], the interpretation of the traces of matrices I , R, and I − R are as follows: • trace(I ) = n, the total number of model parameters,  σi2 • trace(R) = i=1 σ 2 +1 , the number of model parameters determined by the i observed data,  • trace(I − R) = i=1 σ 21+1 , the number of model parameters determined by the a i priori information. As we observe, the singular values of R is exactly the filter factor stated in Sect. 4, Eq. (27). The filter factor is a tool to assist us in understanding which singular components in the information from the observed data in comparison with the a priori

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information contributing in determining the parameter estimates. Consequently, it provides the balance measure we need in approximating the optimal a posteriori covariance matrix which will be discussed in the next section.

8 The Optimal Low-Rank Approximation of a Posteriori Covariance Matrix In the previous sections, we introduce the concepts of a posteriori covariance and resolution matrices. We also describe the relationship between these matrices and the filter factor as in the Tikhonov regularisation. The a posteriori covariance matrix only informative in specific directions due to the influence of noise in the observed data and the a priori information, however. In this section, we propose the optimal lowrank approximation of a posteriori covariance matrix. We describe the optimality by the a posteriori covariance matrix should take into account the interactions between the information from the observed data and the a priori information to determine the boundary between information and noise. We propose a threshold criterion based on the filter factor in approximating the optimal low-rank a posteriori covariance matrix. To obtain the optimal low-rank approximation of a posteriori covariance matrix, firstly we need to identify the boundary between information provided by measurements in the observed data with respect to the prior information and extract the information effectively. For this purpose, SVD analysis is the best approach. Consider implementing SVD on matrix A˜ = U V T as in Eq. (25) and we restate the equation below: ˜ = d˜ (A˜ T A˜ + I )m (V U T U V T + I )m ˜ = d˜ (V 2 V T + I )m ˜ = d˜ 2 T V ( + I )V m ˜ = d˜ . In the equation above, V ( 2 + I )V T is a linear mapping from the model space m ˜ to data space d˜ . The column of V in A˜ spans the model space m, ˜ and the column of U spans the data space d˜ . If the singular values are zero or significantly small, the column of V constructs the null-space, or the model space which models cannot be determined by the observed data. However, if the singular values are significantly large, the corresponding column of U constructs the information space where the observed data spanning this space provide valuable information for the model. The singular values provide us a hint about the boundary between information provided by measurements in the observed data with respect to the prior information. Ideally, there is a clear distinction in the singular values spectrum to identify this boundary. Unfortunately, there is a smooth spectrum of singular values due to noise corruption and computational round-off error. This fact indicates the complexity of the deter-

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mination of the boundary between information provided by measurements in the observed data with respect to the prior information. To facilitate the determination of the boundary between information provided by measurements in the observed data with respect to the prior information from the singular values spectrum, we can relate this to the concept of filter factor which introduced in Sect. 4. Consider a filter factor as in Eq. (27), yet now we generalise its conditions into,  σi2 1, σi2  λ, ≈ fi (λ) = 2 (41) σi + λ 0, σi2  λ, where λ resembles the ratio of the noise and the a priori variances. In Tikhonov regularisation, λ is the regularisation parameter which needs to be estimated [31– 42]. The filter factor above measures how well-resolved or certain the solution concerning the observed data in comparison with the a priori information, which depends on the value of λ. Thus, the role of λ is essential for the optimality of low-rank approximation as it acts as the threshold in truncating the spectrum of singular values of the inverse of a posteriori covariance matrix. If the truncation of the singular spectrum occurs above the threshold value λ, some of the information contains within the observed data might not be included, we lose the valuable information. However, if the truncation occurs below the threshold value λ, the influence of noise and a priori information is visible in the solution. Thus, it leads to the reduction in the resolution and increases the variance in the solution. There are many approaches to estimate the value of lambda; however, it is beyond the scope of this chapter. The readers may kindly refer to [31–44] for more details upon regularisation parameters estimation. We demonstrate the optimality concept further in the numerical examples.

9 Numerical Examples To further understand the concepts introduced in the previous sections, we demonstrate them in the following numerical examples. In this section, we consider the linear seismic travel-time tomography problem as in [34]. The numerical examples simulate geophysical problems where one records the travel time of seismic waves between sub-surface sources and receivers located either at or below the surface. Our first task is to determine the maximum a posteriori model (MAP) of sub-surface attenuation, or slowness, in the domain represented as m, the model parameters by minimising the negative log posterior as in Eq. (24). In accomplishing this task, we minimise the negative log posterior with conjugate gradient algorithm [35]. We consider a square domain of size [0, 100] × [0, 100] with 75 equally spaced sources located at the right side of the domain, and 100 equally spaced receivers located on the top of the surface and the left side of the domain. The true slowness model is illustrated in Fig. 1a. The dominating frequency of the propagating wave is

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set to 10 Hz. We consider Gaussian prior with covariance matrix Cprior = (λLT L)−1 , where λ equals unity and L is the Laplacian matrix. Then, we approximate the MAP by minimising the negative log posterior with 100 iterations of the conjugate gradient algorithm starting from initial slowness model illustrated in Fig. 1b. The results are illustrated in Fig. 2. In Fig. 2, we show the approximation of the MAP of the slowness model, together with the model error norms and residual norms plots. In Fig. 2b, by knowing the true slowness model, we can compute the relative model error norms at each iteration through me − m(k) 2 (k) = . rmodel me 2 As we can observe, the model error norms start to stabilise at the 70-th iteration, which indicates we reached the optimal MAP model, and the iteration can be terminated. However, in real-life applications of inverse problems, the true model is unknown, and the computation of model error norms are impossible. Thus, we need to rely upon other relative norms to guarantee our convergence. Typically, the relative residual norms between the observed data dobs and the synthetic data dsyn = Am [as in Eq. (11)] are being considered. The data and the normal equation relative residual norms can be computed by rdata =

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not depending on the model parameter, and this simplifies the computation for quantifying the uncertainties. For this task, we compute the posterior covariance based on Eq. (32), then truncate its singular spectrum based on the optimal criterion given by Eq. (41). For this specific example, the optimal truncation point is at singular value equals unity. In Fig. 3, we plot the singular spectrum for the prior covariance, the Hessian, the posterior covariance, and the normalised data misfit Hessian [A˜ T A˜ as in Eq. (24)], respectively. We could observe the fast decaying spectrum from all the singular values plots; hence this informs us that the data are informative only on a low-dimensional subspace. Figure 4 (top row) shows the convergence of the approximate posterior variance for different value of singular value truncation. The first column in Fig. 4 represents the prior variance and its approximation of the mean model without information from the observed data. For this formally 10,000-dimensional problem, a good approximation of the posterior variance is achieved with optimal truncation at the 437-th ˜ hence the data are singular values of the the normalised data misfit Hessian (A˜ T A); informative only on a low-dimensional subspace. The quality of the covariance matrix approximation is also reflected in the structure of mean model approximation from the approximate posterior covariance (bottom row). All three of these approximated mean model are significantly close to the MAP model computed by minimising the negative logarithm of the posterior distribution. Already with truncation at singular value approximately unity (at the 437-th singular values), the features of the approximate posterior variance and mean model match those of the MAP posterior variance

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and mean model. In applications, agreement in this variance and mean model are important. Based on this numerical example, we observed that the optimal truncation point for obtaining the low-rank approximation of a posteriori covariance can be determined by the filter factor (Eq. (41)) introduced in Sect. 8.

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10 Conclusions This chapter has presented and demonstrated the sound connection between the Bayesian approach and the Tikhonov regularisation within Gaussian framework for linear inverse problems. We discussed the Tikhonov regularisation technique from a functional point of view. By connection between Bayesian approach and Tikhonov regularisation technique, we provided an uncertainty analysis framework to answer the following two fundamental questions: (1) How well is the estimate determined by the a posteriori PDF, i.e. by the combination of observed data and a priori information? (2) What are the respective contributions of observed data and a priori information? We also discussed the concept of the resolution matrix and introduced the optimal low-rank approximation of a posteriori covariance matrix for the Bayesian solution of linear inverse problems. We demonstrated it through the seismic travel-time tomography example, with Gaussian prior and noise distributions defined on finitedimensional spaces. In a typical large-scale inverse problem, e.g. seismic inversions, observations may be informative relative to the prior only on a low dimensional subspace of parameter space. An important task that needs to be done in the future is to generalise this framework for inverse problems with nonlinear forward modelling, i.e., nonlinear inverse problems.

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Index

A Active closed loop system, 47 Alternating current, 82 Approximating, 102 Artificial neural network, 59

B Bayesian, 121, 122, 128–130, 136, 143 Bayesian framework, 122, 128, 136 Bayesian-Tikhonov, 128 B-spline, 101, 103, 104, 106–108, 110–112, 115, 117

C C1 continuity, 89–92, 95–98 Central Processing Unit (CPU), 89 Characteristics, 10 Computation, 89, 90, 98 Connected, 83 Continuity, 3, 112 Contractive, 114 Convergent, 114 Convex combination, 90 Coriolis flow meter, 39 Cost analysis, 77 Cost effectiveness, 85 Cubic Ball, 89 Cubic Bézier, 89 Cubic spline, 2

D Data interpolation, 1–3, 8, 10, 21, 22

Deep learning, 59 Derivative di , 3 Discrete time signals, 20

E Economic analysis, 84 Economic and environmental, 77 Electricity consumption, 78 Energy audit, 77 Energy saving, 77 Error, 89, 91, 95–98 Error analysis, 1, 3, 5, 9, 11, 12, 14, 16, 20, 22

F Fanno flow, 28 Fixed axis, 48 Fluid flow, 59, 62

G Gaussian, 121, 123, 128, 129, 133, 134, 137, 140, 143 G1 continuity, 89, 95–98 Geometric modelling, 2 Geophysics, 122

H Hermite, 2 Holder continuity, 106

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim et al. (eds.), Advanced Methods for Processing and Visualizing the Renewable Energy, Studies in Systems, Decision and Control 320, https://doi.org/10.1007/978-981-15-8606-4

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148 I Impulse, 51 Interpolating, 102 Interpolation, 2 Inverse problems, 122, 123, 128, 140, 143 Irregular, 90

L Laurent polynomial, 117 Least-squares minimization, 90 Light dependent resistor, 47 Light intensity, 47–51 Limit stencil, 105 Linear, 124, 125, 127–129, 131, 133, 134, 136, 138 Linear actuator, 47–51 Linear inverse problems, 122, 123, 128, 140, 143 Linear seismic, 139 Load curve, 78 Local schemes, 90 Lower, 90

M Mach number, 28 Mass flow rate, 26 Mathematical modeling, 59–61 Maximum error, 89 Mesh, 114 Minimum output, 85 Minimum switching time, 29 Mismatch losses, 84 Model, 59–70 Multi-level-pressure banks, 25, 26, 30, 31, 42, 44

N Nanofluids, 61 Nanoparticles, 59–67, 69 Natural gas, 25–29, 31, 32, 44 NGV dispenser, 26 NGV refueling, 25, 27, 29–32, 40, 43, 44 NGV tank, 28

O Observations, 4 Oil and gas, 59, 62 Optimal switching time, 27, 29, 31 Optimization, 80, 83

Index P Panel, 80 Parameters, 3 Payback period, 77 Poles, 5 Polygon, 102 Polynomial generation, 107 Polynomial reproduction, 107 Porous media, 60–62, 69 Primal parametrization, 107 PV modules, 84 PV sizing, 77 PV systems, 77, 80, 82, 84, 87

Q Quartic rational, 89

R Rational quartic spline, 1–3, 11–14, 16, 20 Rational quartic triangular patches, 91–93 Rational spline, 2 Reflection loss, 84 Regularisation, 123, 125, 127, 128, 130, 139 Renewable energy, 47, 48, 56, 78 Reservoir modeling, 61, 63 Reservoir simulation, 59, 61 Reservoir simulation and enhanced oil recovery, 59 Resolution, 122, 123, 133, 136, 137, 139, 143 Return of Investment, 77 Root Mean Square Errors (RMSE), 89

S Seismic, 122, 123, 139, 143 Seismic inversion, 121, 122, 143 Sensor, 51 Septic, 104 Shape, 3 Shape preserving, 2 Signal processing, 2, 3, 20, 21 Simulation, 59, 61, 63 Single axis, 48 Sizing of PV systems, 77, 80, 87 Smallest, 16 Smooth surface, 90 Solar energy, 48 Solar photovoltaic panel, 48, 51 Solar technology, 77 Solar tracker, 47–50, 53–55 Solar tracking system, 48

Index Stencil, 105 Subdivision scheme, 101–104, 106, 108– 110, 113, 114, 117 Surface modeling, 101 Sustainability, 77 System, 85 T Temperature, 84 Tensor product, 101–104, 108, 110–112, 114 Tikhonov, 123, 125, 127, 139 Tikhonov regularisation, 121, 123, 128–131, 138, 143 Time-optimal control, 26

149 Tomography, 143 Track the movement of the sunlight, 48

U Uncertainty, 121–123, 130–133, 135, 136, 143 Upper bounds, 90

V Vertex, 115 Vertical asymptotes, 5 Voltage, 51