Metaheuristic Optimization Algorithms in Civil Engineering: New Applications (Studies in Computational Intelligence, 900) 303045472X, 9783030454722

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Studies in Computational Intelligence 900

Ali Kaveh Armin Dadras Eslamlou

Metaheuristic Optimization Algorithms in Civil Engineering: New Applications

Studies in Computational Intelligence Volume 900

Series Editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Computational Intelligence” (SCI) publishes new developments and advances in the various areas of computational intelligence—quickly and with a high quality. The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life sciences, as well as the methodologies behind them. The series contains monographs, lecture notes and edited volumes in computational intelligence spanning the areas of neural networks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, self-organizing systems, soft computing, fuzzy systems, and hybrid intelligent systems. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution, which enable both wide and rapid dissemination of research output. The books of this series are submitted to indexing to Web of Science, EI-Compendex, DBLP, SCOPUS, Google Scholar and Springerlink.

More information about this series at http://www.springer.com/series/7092

Ali Kaveh Armin Dadras Eslamlou •

Metaheuristic Optimization Algorithms in Civil Engineering: New Applications

123

Ali Kaveh Department of Civil Engineering School of Civil Engineering Iran University of Science and Technology Tehran, Iran

Armin Dadras Eslamlou Department of Civil Engineering School of Civil Engineering Iran University of Science and Technology Tehran, Iran

ISSN 1860-949X ISSN 1860-9503 (electronic) Studies in Computational Intelligence ISBN 978-3-030-45472-2 ISBN 978-3-030-45473-9 (eBook) https://doi.org/10.1007/978-3-030-45473-9 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Advances in civil engineering technology require greater accuracy, efficiency, and speed in the analysis and design of the corresponding systems. It is therefore not surprising that new methods have been developed for the optimal design of real-life systems and models with complex configurations and a large number of elements. This book can be considered as an application of metaheuristic algorithms to some important optimization problems in civil engineering. This book is addressed to those scientists and engineers, and their students, who wish to explore the potential of newly developed metaheuristics by some practical problems. The concepts presented in this book are not only applicable to civil engineering problems but can equally be used for optimizing the problems involved in mechanical and electrical engineering. The authors and colleagues have been involved in various developments and applications of various metaheuristic algorithms to structural optimization in the last two decades. This book contains part of this research suitable for various aspects of optimization in civil engineering. The book is likely to be of interest to civil, mechanical, and electrical engineers who use optimization methods for design, as well as to those students and researchers in structural optimization who will find it to be necessary professional reading. In Chap. 1, a short introduction is provided for the goals and contents of this book. Chapter 2 employs several metaheuristic algorithms for stacking sequence optimization of laminated plates to maximize the buckling capacity. Chapter 3 involves the rigidity of jointed composite castellated beams in their optimal designs. In this chapter, particle swarm optimization, colliding bodies optimization, and enhanced colliding bodies optimization algorithms are used for the optimization of semi-rigid jointed beams. Chapter 4 presents the optimal design of steel curved roof frames with its roof being part of a circular arc. In the objective function, different factors affecting the weight of frames are considered, and it is optimized by different metaheuristic algorithms. v

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Chapter 5 studies the steel pitched roof frames with gable roofs. In this chapter, design optimization of the steel member sections is performed by different apex heights and tapered lengths for steel pitched roof frames. Chapter 6 introduces a two-stage optimal sensor placement method for modal identification of structures. At the first stage, using a graph-theoretic technique, the structure is partitioned into equal substructures. At the second stage, a predefined number of triaxial sensors are allocated to the substructures through optimization. Chapter 7 presents an adaptive node moving refinement in the discrete least squares meshless method using the charged system search for optimum analysis of elasticity problems. Chapter 8 employs a hybrid version of the non-dominated sorting genetic algorithm with differential evolution operators to solve the performance-based multi-objective optimization problem. Chapter 9 applies three well-known metaheuristic algorithms, including colliding bodies optimization, enhanced colliding bodies optimization, and particle swarm optimization for size and performance optimization of steel plate shear wall systems. An example of such a system is a lateral load resisting system that contains an infill plate attached to the surrounding beams and columns. Chapter 10 utilizes the collided bodies optimization for the optimization of large-scale offshore wind turbine supporting structures. Here, the OC4 reference jacket is considered as the case study. Both the ultimate limit state and frequency constraints are considered, and the aerodynamic, hydrodynamic, and wave loads under extreme weather conditions are exerted. Chapter 11 presents the application of the colliding bodies optimization algorithm for analysis, design, and optimization of water distribution systems (WDSs). The design and cost optimization of WDS are performed simultaneously with the analysis process using a new objective function to satisfy the analysis criteria, design constraints, and cost optimization. Chapter 12 optimizes the location of the tower crane that has an important effect on material transportation costs. The appropriate location of tower cranes for material supply and engineering demands is a combinatorial optimization problem that is difficult to resolve. In this chapter, the performance of the particle swarm optimization, colliding bodies optimization, enhanced colliding bodies optimization, vibrating particles system, and enhanced vibrating particles system are compared in terms of their effectiveness in resolving a practical tower crane layout problem. Chapter 13 proposes a framework for the optimization of building components with sustainability aspects in the BIM environment as one of the successful tools in the architecture, engineering, and construction industry. Chapter 14 develops the multi-objective version of two metaheuristics, CBO and ECBO and applies for construction site layout planning. Besides, the data envelopment analysis is utilized for calculating the efficiency of optimal Pareto front layouts that can help decision-makers to select the final layout among the non-dominated candidates.

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Chapter 15 tailors a comprehensive framework that merges MCDM techniques with metaheuristic multi-objective optimization techniques. The most proper schedule for appliances is selected by creating a trade-off among various optimization criteria. A multi-objective ant lion optimizer is applied and tested on a smart home case study to detect all the Pareto solutions. Tehran, Iran March 2020

Ali Kaveh Armin Dadras Eslamlou

Acknowledgements

We would like to take this opportunity to acknowledge a deep sense of gratitude to a number of colleagues and friends who in different ways have helped in the preparation of this book. Our special thanks are due to Thomas Ditzinger, the Editorial Director of the Applied Sciences of Springer, for his constructive comments, editing, and unfailing kindness in the course of the preparation of this book. My sincere appreciation is extended to our Springer colleagues Anja Seibold and Viradasarani Natarajan for publishing the present book. We would like to thank our colleagues for using our joint papers and for their help in various stages of writing this book: Dr. M. Khanzadi, Dr. S. R. H. Vaez, Dr. P. Hosseini, Dr. H. Arzani, Dr. K. Laknejadi, Dr. B. Alinejad, Dr. M. H. Ghafari, Mr. N. Geran Malek, Mr. B. Ahmadi, Mr. M. Dehghan, Mr. Y. Vazirinia, Mr. M. Rastegar Moghaddam, Mr. M. Bakhtyari, Mr. S. Sabeti, Mr. M. Farhadmanesh, and Mr. F. Shokohi. We would like to thank the publishers who permitted some of our papers to be utilized in the preparation of this book, consisting of Springer, John Wiley and Sons, and Elsevier. Every effort has been made to render the book error-free. However, the authors would appreciate any remaining errors being brought to his attention through their e-mail addresses: [email protected] (Ali Kaveh) and [email protected]. ac.ir (Armin Dadras Eslamlou).

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Contents

1

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Engineering Design and Optimization . . . . . . . . . . . . . 1.2 Application of Metaheuristic Optimization Algorithms in Civil Engineering . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of the Present Book . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Optimum Stacking Sequence Design of Composite Laminates for Maximum Buckling Load Capacity . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 JAYA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Grey Wolf Optimizer . . . . . . . . . . . . . . . . . . . . . 2.4.3 Colliding Bodies Optimization . . . . . . . . . . . . . . 2.4.4 Salp Swarm Algorithm . . . . . . . . . . . . . . . . . . . . 2.4.5 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Quantum-Inspired Evolutionary Algorithm . . . . . . 2.5 Anti-optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Golden Section Search (GSS) . . . . . . . . . . . . . . . 2.6 Numerical Results for Deterministic Loading . . . . . . . . . . 2.6.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6 Case 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Numerical Results for Uncertain Loading . . . . . . . . . . . . . . 2.7.1 A Comparison of the Effect of Different Materials . 2.7.2 An Investigation on the Effect of Aspect Ratio . . . 2.7.3 An Investigation on the Effect of Loading Domain . 2.7.4 A Comparison Among the Performance of the Different Optimization Algorithms . . . . . . . . . . . . . 2.8 Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Optimum Design of Castellated Beams with Composite Action and Semi-rigid Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Design of Castellated Beams . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Flexural Capacity . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Shear Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Web Post-buckling . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Design of Composite Beams . . . . . . . . . . . . . . . . . . . . . . . 3.4 Semi-rigid Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Semi-rigid Composite Castellated Beam . . . . . . . . . . . . . . . 3.5.1 Deflection of Semi-rigid Composite Castellated Beam . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 The Vibration of Semi-rigid Composite Castellated Beam . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 CBO and ECBO . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Penalty Function . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Optimal Design of Steel Curved Roof Frames by Enhanced Vibrating Particles System Algorithm . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Curved Roof Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Design Constraints . . . . . . . . . . . . . . . . . . . . . .

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Structural Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Loading Combinations . . . . . . . . . . . . . . . . . . 4.4.2 The Dead and Collateral Loads (D) . . . . . . . . . 4.4.3 The Live Load (L) . . . . . . . . . . . . . . . . . . . . . 4.4.4 The Balanced and Unbalanced Snow Loads (S) 4.4.5 The Seismic Load (E) . . . . . . . . . . . . . . . . . . . 4.4.6 The Wind Loads (W) . . . . . . . . . . . . . . . . . . . 4.5 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Vibrating Particles System . . . . . . . . . . . . . . . 4.5.2 Enhanced Vibrating Particles System . . . . . . . . 4.5.3 Gray Wolf Optimizer . . . . . . . . . . . . . . . . . . . 4.5.4 Enhanced Colliding Bodies Optimization . . . . . 4.5.5 Salp Swarm Algorithm . . . . . . . . . . . . . . . . . . 4.5.6 Grasshopper Optimization Algorithm . . . . . . . . 4.5.7 Harmony Search . . . . . . . . . . . . . . . . . . . . . . . 4.6 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Geometry and Sizing Optimization of Steel Pitched Roof Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Objective Function . . . . . . . . . . . . . . . . . 5.2.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Loading . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Structural Analysis . . . . . . . . . . . . . . . . . 5.2.5 Strength Design Criteria . . . . . . . . . . . . . 5.2.6 Displacement Criteria . . . . . . . . . . . . . . . 5.2.7 Penalty Function . . . . . . . . . . . . . . . . . . 5.3 Optimization Algorithms . . . . . . . . . . . . . . . . . . . 5.3.1 Simulated Annealing Optimization . . . . . 5.3.2 Particle Swarm Optimization . . . . . . . . . . 5.3.3 Artificial Bee Colony . . . . . . . . . . . . . . . 5.3.4 Whale Optimization Algorithm . . . . . . . . 5.3.5 Grey Wolf Optimizer . . . . . . . . . . . . . . . 5.3.6 Invasive Weed Optimization . . . . . . . . . . 5.3.7 Harmony Search . . . . . . . . . . . . . . . . . . . 5.3.8 Colliding Bodies Optimization . . . . . . . . 5.3.9 Enhanced Colliding Bodies Optimization .

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Examples . . . . . . . . . . . . . . 5.4.1 Example 1 . . . . . . . 5.4.2 Example 2 . . . . . . . 5.5 Discussions and Conclusion . References . . . . . . . . . . . . . . . . . .

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Two-Stage Optimal Sensor Placement Using Graph-Theory and Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Sensor Placement Criterions . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Modal Assurance Criterion . . . . . . . . . . . . . . . . . 6.2.2 Visualization of Mode Shapes . . . . . . . . . . . . . . . 6.3 Partitioning Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Preliminaries from Graph Theory . . . . . . . . . . . . 6.3.2 k-Means Method . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Spectral Partitioning . . . . . . . . . . . . . . . . . . . . . . 6.4 Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Steps of the QEA . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 The Dynamical Quantum-Inspired Evolutionary Algorithm (DQEA) . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Proposed Two-Stage Approach . . . . . . . . . . . . . . . . . 6.5.1 Stage 1 (Structural Partitioning) . . . . . . . . . . . . . 6.5.2 Stage 2 (Optimization of Sensor Placement) . . . . . 6.6 Numerical Results and Discussions . . . . . . . . . . . . . . . . . 6.6.1 Benchmark Model . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Performance of the Methods on TMAC Criterion . 6.6.3 Assessing the Mode Shape Visualization Criterion 6.7 Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Charged System Search Algorithm for Adaptive Node Moving Refinement in Discrete Least-Squares Meshless Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Discrete Least Squares Meshless (DLSM) . . . . . . . . . . . 7.2.1 Moving Least Squares Shape Functions . . . . . . 7.2.2 Discrete Least-Squares Meshless Method . . . . . 7.3 Charged System Search . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Error Indicator and Adaptive Refinement . . . . . . . . . . . 7.5 The Link Between the CSS and Adaptivity . . . . . . . . . . 7.5.1 Objective Function . . . . . . . . . . . . . . . . . . . . . 7.5.2 Selected Parameters . . . . . . . . . . . . . . . . . . . .

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Numerical Examples . . . . . . . . . . . . . . . . . . . 7.6.1 Infinite Plate with a Circular Hole . . . 7.6.2 A Cantilever Beam Under End Load . 7.7 Discussions and Conclusion . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Optimal Seismic Design of Steel Plate Shear Walls Using CBO and ECBO Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Different Techniques for Simulating Steel Plate Shear Walls 9.2.1 Strip Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Pratt Truss Model . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Truss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Partial Strip Model . . . . . . . . . . . . . . . . . . . . . . . . 9.2.5 Multi-angle Model . . . . . . . . . . . . . . . . . . . . . . . . 9.2.6 Modified Strip Model . . . . . . . . . . . . . . . . . . . . . . 9.2.7 Cyclic Strip Model . . . . . . . . . . . . . . . . . . . . . . . . 9.2.8 Orthotropic Membrane Model . . . . . . . . . . . . . . . . 9.3 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Requirements for Low Seismic Design . . . . . . . . . 9.3.2 Requirements for High Seismic Design . . . . . . . . .

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Performance-Based Multi-objective Optimization of Large Steel Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Employed Multi-objective Optimization Algorithm . . . . 8.2.1 NSGA-II-DE . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 GA Operators . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Constraint Handling . . . . . . . . . . . . . . . . . . . . 8.3 Seismic Optimum Design Procedure . . . . . . . . . . . . . . . 8.3.1 Loading and Constraints for Optimum Seismic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Nonlinear Static Analysis (Pushover Analysis) . 8.3.3 Lifetime Seismic Damage Cost . . . . . . . . . . . . 8.4 Meta-modeling for Predicting the Response . . . . . . . . . 8.4.1 Approximation Model Selection and Training . 8.4.2 Model Management . . . . . . . . . . . . . . . . . . . . 8.5 The Proposed Framework . . . . . . . . . . . . . . . . . . . . . . 8.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 2D Example . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 3D Example . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.4

CBO and ECBO Algorithms . . . . . . . . . . . . . . . . . 9.4.1 Colliding Bodies Optimization (CBO) . . . . 9.4.2 Enhanced Colliding Bodies Optimization . . 9.5 Structural Optimization . . . . . . . . . . . . . . . . . . . . . 9.5.1 Optimization Formulation . . . . . . . . . . . . . 9.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Low Seismic Design Example . . . . . . . . . . 9.6.2 High Seismic Design Example . . . . . . . . . 9.6.3 Performance-Based Design Optimization of 9.6.4 Optimum Design of 6- to 12-Story SPSW . 9.7 Discussions and Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Colliding Bodies Optimization Algorithm for Structural Optimization of Offshore Wind Turbines with Frequency Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Configuration of the OC4 Reference Jacket . . . . . . . . 10.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Loading Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Wave Loading . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Wind Loading . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Load Combinations . . . . . . . . . . . . . . . . . . . 10.5 The Structural Optimization Problem . . . . . . . . . . . . . 10.5.1 Design Variables . . . . . . . . . . . . . . . . . . . . . 10.5.2 Cost Function . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Colliding Bodies Optimization Algorithm . . . 10.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Hydrodynamic Loading . . . . . . . . . . . . . . . . 10.6.2 Aerodynamic Loading . . . . . . . . . . . . . . . . . 10.6.3 Final Results . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Colliding Bodies Optimization for Analysis and Design of Water Distribution Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Water Distribution Network Optimization Problem . . . . . . . . 11.2.1 Analysis Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Design Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Colliding Bodies Optimization Algorithm . . . . . . . . . . . 11.3.1 Collision Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 The CBO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 11.4 A New Algorithm for Analysis and Design of the Water Distribution Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11.5 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 A Two-Loop Network . . . . . . . . . . . . . . . 11.5.2 Hanoi Water Distribution Network . . . . . . 11.5.3 The Go Yang Water Distribution Network . 11.6 Discussions and Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Optimization of Tower Crane Location and Material Quantity Between Supply and Demand Points . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Colliding Bodies Optimization . . . . . . . . . . . . . . . 12.3.2 Enhanced Colliding Bodies Optimization . . . . . . . . 12.3.3 Vibrating Particles System . . . . . . . . . . . . . . . . . . 12.3.4 Enhanced Vibrating Particles System . . . . . . . . . . . 12.3.5 Encoding of Solutions . . . . . . . . . . . . . . . . . . . . . 12.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Results and Discussion on Single Tower Crane Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Results and Discussion for the Multi-tower Crane Layout Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.3 Discussions and Conclusion . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 Optimization of Building Components with Sustainability Aspects in BIM Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Proposed Framework to Opt Desired and Optimum Selection for Building Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Initial Preparation Phase . . . . . . . . . . . . . . . . . . . . . 13.2.2 Optimization Phase . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Efficiency Evaluation Phase . . . . . . . . . . . . . . . . . . 13.2.4 Multi-attributes Decision Making Phase . . . . . . . . . . 13.3 Methods Used in the Proposed Framework . . . . . . . . . . . . . . 13.3.1 Enhanced Non-dominated Sorting Colliding Bodies Optimization (ENSCBO) . . . . . . . . . . . . . . . . . . . . 13.3.2 Data Envelopment Analysis (DEA) . . . . . . . . . . . . . 13.3.3 The Compromise Ranking Method VIKOR . . . . . . . 13.4 Implementation of a Case Study and the Corresponding Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14 Multi-objective Optimization of Construction Site Layout 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Optimization Metaheuristic Algorithms . . . . . 14.2.2 Data Envelopment Analysis . . . . . . . . . . . . . 14.3 Case Study and Discussion of Results . . . . . . . . . . . . 14.3.1 Description of the Case Study . . . . . . . . . . . . 14.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 Multi-objective Electrical Energy Scheduling in Smart Homes Using Ant Lion Optimizer and Evidential Reasoning . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Preparing Required Information About Appliances Scheduling Operation . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Multi-objective Optimization (MOO) . . . . . . . . . . . 15.2.3 Multi-criteria Decision Making (Shannon’s Entropy) . . . . . . . . . . . . . . . . . . . . . . . 15.2.4 Evidential Reasoning . . . . . . . . . . . . . . . . . . . . . . 15.3 The Multi-objective Home Appliance Scheduling Problem . 15.3.1 Objective Functions . . . . . . . . . . . . . . . . . . . . . . . 15.4 Implementation of the Proposed System . . . . . . . . . . . . . . . 15.4.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Parameter Configuration . . . . . . . . . . . . . . . . . . . . 15.4.3 Pareto Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.4 Determining the Weights . . . . . . . . . . . . . . . . . . . 15.4.5 Ranking Solutions . . . . . . . . . . . . . . . . . . . . . . . . 15.4.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

1.1 Engineering Design and Optimization The archaeological records demonstrate the beginning of the technological development of behaviorally modern humans (Homo sapiens) from more than thirty thousand years ago, which has been confirmed by traces of bone awls, eyed needles, fire-making drills, heated shelters and spear-throwers [1]. When the ancient human was constructing the primary technological innovations such as spear-thrower, probably she/he was intuitively thinking about the specifications of the thrower by which the spear could achieve greater velocity by imparting less energy. Thus, the engineering design that provides the most desired features with the fewest negatives is not a new thing. Although engineering has existed since ancient times, the word engineer dates back to less than 700 years ago. The industrial revolution was a pivotal turning point in the definition of engineering, which was based on Newtonian mechanics and mathematics. In this era, the major branches of engineering, such as civil engineering and mechanical engineering, were established. First, the civil engineers were responsible for the design and construction of buildings, bridges, lighthouses, canals and harbors. However, in the contemporary era, numerous responsibilities and multidisciplinary professions devolved into civil engineers. For example, their responsibilities comprise a spectrum of various disciplines ranging from structural engineering, earthquake engineering, coastal and offshore engineering to construction management, transportation and environmental engineering. The growing engineering sciences and mechanized production made it easy to construct structures and consume the resources much faster than before. After decades of low prices for commodities and fuels, due to the growing global population and unprecedented demand for buildings, the extraction from non-renewable resources has increased drastically. Nowadays the resource depletion has become a serious problem so that the foreseeable future construction mineral prices are likely to be

© Springer Nature Switzerland AG 2020 A. Kaveh and A. Dadras Eslamlou, Metaheuristic Optimization Algorithms in Civil Engineering: New Applications, Studies in Computational Intelligence 900, https://doi.org/10.1007/978-3-030-45473-9_1

1

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1 Introduction

high. Besides the economic perspective, this issue makes adverse impacts on the environment by generating hazardous wastes and air emissions. These impacts beyond the near-term can manifest itself negatively on the life quality of the next generations. Therefore, we have to change course if we are going to preserve the earth for them. To confront this alarming challenge for the environment and economy, new policies have to move toward emphasizing sustainability aspects and resource efficiency. Technical changes such as innovational design and efficient production are effective solutions for this formidable trouble. To be more specific, the design and production schemes that are employed by engineers must be switched from classical methods to optimal ones in order to moderate resource consumption. During recent decades, various modern and classic optimization methods have been developed regarding the diverse type of problems. Classic methods say linear programming, utilizes differential calculations and mathematics for finding the optimal solution [2]. Although these techniques provide the exact optimal solution for some problems with specific conditions, they are not appropriate for many of the large and realistic problems. On contrary, the stochastic methods, such as metaheuristics, do not guarantee the optimality of obtained solutions, but they are capable of finding good solutions for a wide variety of problems with a feasible computing time [3]. Metaheuristic algorithms are nature-inspired, gradient-free, iterative and often population-based methods that start searching from a population of initial random solutions. They gradually improve the quality of solutions employing searching strategies, so-called exploration and exploitation. In engineering, we usually deal with problems such as, maximizations or minimization decisions and designs with conditions for which the latter techniques are preferable [4]. Metaheuristic algorithms are not restricted to specific problems and can be applied to almost all optimization problems. In these methods, the constraints can easily be implemented through different techniques such as penalty technique or repairing strategies. These capabilities have made them suitable to the majority of applications in industrial projects, engineering design and advanced sciences [5].

1.2 Application of Metaheuristic Optimization Algorithms in Civil Engineering In the present book, we attempted to consider eclectic works in the field of optimization in civil engineering that are selected from the outcomes of our research group in recent years. This book provides a diverse application of metaheuristics in structural engineering, earthquake engineering, hydraulics and construction management. It not only covers some prevalent problems that civil engineers continuously will face but also deals with some modern applications such as optimal sensor placement for modal identification, optimal design of offshore turbines, BIM and smart-home energy scheduling.

1.2 Application of Metaheuristic Optimization Algorithms …

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It can be a useful guideline for both researchers and practitioners to get familiar with diverse applications and get insight into algorithms. In most cases, inspiring discussions and suggestions are proposed in such a way that can lead to new researches and solutions. Moreover, the problems are clearly stated and can be implemented by the readers.

1.3 Organization of the Present Book The remaining chapters of this book are organized as follows: Chapter 2 employs several metaheuristic algorithms for stacking sequence optimization of laminated plates to maximize the buckling capacity. Two loading types are considered as optimization problems: deterministic and probabilistic loading. Moreover, different cases with various panel aspect ratios, number of layers and materials are examined to provide the optimal fiber orientations. To account for the uncertainty in loading, the Golden Section Search (GSS) is applied for finding a robust design based on the worst-case scenario. In this chapter, the results are investigated from different perspectives, and sensitivity analyses are performed [6, 7]. Chapter 3 involves the rigidity of jointed composite castellated beams in their optimal designs. In this chapter, Particle Swarm Optimization (PSO), Colliding Bodies Optimization (CBO), and Enhanced Colliding Bodies Optimization (ECBO) algorithms are used for the optimization of semi-rigid jointed beams. Several variables consisting of profile section, cutting depth, cutting angle, holes spacing, the number of filled end holes of the castellated beams and rigidity of connection are considered as the optimization variables. Constraints include the construction, moment, shear, deflection and vibration limitations. Effect of partial fixity and commercial cutting shape of a castellated beam for a practical range of beam spans and loading types are studied through numerical examples. The efficiency of the meta-heuristic algorithms is also compared [8]. Chapter 4 presents the optimal design of steel curved roof frames with its roof being part of a circular arc. In the objective function, different factors affecting the weight of frames are considered and it is optimized by basic and Enhanced Vibrating Particle System (VPS and EVPS), Grey Wolf Optimizer (GWO), ECBO, Salp Swarm Algorithm (SSA), Grasshopper optimization algorithm (GOA) and Harmony Search (HS) algorithm. Pitched roof steel frames are used in a wide range of span lengths that are usually considered as low-rise buildings. For example, they are used in the construction of single-story factories, warehouses, gymnasiums, hypermarkets, and hangars. The discussed results can provide invaluable information for designers to decide about the prospective design while fulfilling the displacement, buckling and stability constraints [9]. Chapter 5 studies the steel pitched roof frames with gable roofs. In this chapter, design optimization of the steel member sections is performed by different apex heights, and tapered lengths for steel pitched roof frames. The effective variable definition has helped to reduce the variable domain and ignore the unwanted part of

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1 Introduction

the bounds before starting the optimization process. Nine meta-heuristic algorithms are used for optimization whose performances are compared for this type of frame structure. The suitable number of optimization algorithms provides reliable results and ensures the reader about the optimum apex height, roof angle and tapered ratio [10]. Chapter 6 introduces a two-stage Optimal Sensor Placement (OSP) method for modal identification of structures. At the first stage, using a graph-theoretic technique, the structure is partitioned into equal substructures. At the second stage, a predefined number of triaxial sensors are allocated to the substructures through optimization. In this chapter, a dynamic version of the Quantum-Inspired Evolutionary algorithm (QEA) is introduced and applied to this optimization problem. Furthermore, the graph theory resulted in the reduction of the search space and improved the mode shape visualization [11]. Chapter 7 presents an adaptive node moving refinement in the Discrete Least Squares Meshless (DLSM) method using the Charged System Search (CSS) for optimum analysis of elasticity problems. The CSS Physics-inspired algorithm obtained suitable locations of the nodes, increasing the accuracy of the analysis method. To demonstrate the effectiveness of the proposed method, some benchmark examples with available analytical solutions are examined [12]. Chapter 8 employs a hybrid version of the non-dominated sorting genetic algorithm (NSGA-II) with differential evolution (DE) operators to solve the performancebased multi-objective optimization problem. In this problem, the initial and life cycle cost of large-scale structures both are considered and a specific meta-model is utilized for reducing the number of fitness function evaluations. Moreover, the required computational time for pushover analysis is decreased by a simple numerical method. The obtained results for the application of the framework demonstrate its capability in solving the present complex multi-objective optimization problem [13]. Chapter 9 applies three well-known metaheuristic algorithms, including CBO, ECBO, and PSO for size and performance optimization of steel plate shear wall (SPSW) systems. An SPSW is a lateral load resisting system that contains an infill plate attached to the surrounding beams and columns. The results derived in this chapter not only can be employed in the design of new buildings but also can be utilized for the retrofitting of existing buildings [14]. Chapter 10 utilizes the Collided Bodies Optimization (CBO) for the optimization of large-scale offshore wind turbine supporting structures. Here, the OC4 reference jacket is considered as the case study. Both the Ultimate Limit State (ULS) and frequency constraints are considered and the aerodynamic, hydrodynamic and wave loads under extreme weather conditions are exerted. The results show a noticeable weight and cost reduction comparing the optimized design with the initial values [15]. Chapter 11 presents the application of the CBO for analysis, design, and optimization of Water Distribution Systems (WDSs). The design and cost optimization of WDS is performed simultaneously with the analysis process using a new objective function to satisfy the analysis criteria, design constraints, and cost optimization. Several practical examples of WDSs are selected to demonstrate the efficiency of the

1.3 Organization of the Present Book

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presented method. The comparison of obtained results signifies the efficiency of the CBO method in terms of construction cost and computational time of the analysis [16]. Chapter 12 optimizes the location of the tower crane that has an important effect on material transportation costs. The appropriate location of tower cranes for material supply and engineering demands is a combinatorial optimization problem that is difficult to resolve. In this chapter, the performance of the PSO, CBO, ECBO, Vibrating Particles System (VPS), and Enhanced Vibrating Particles System (EVPS) are compared in terms of their effectiveness in resolving a practical Tower Crane Layout (TCL) problem [17]. Chapter 13 proposes a framework for the optimization of building components with sustainability aspects in the BIM environment as one of the successful tools in the Architecture, Engineering, and Construction (AEC) industry. This approach allows the project stakeholders to choose the optimal combination for their building components with the least human interference in the selection of suitable materials from a wide range of candidates. In this chapter, Enhanced Non-Dominated Sorting Colliding Bodies Optimization (ENSCBO) algorithm as a multi-objective version of the recently developed meta-heuristic algorithm (ECBO) is employed for reducing cost, increasing energy saving, applying recyclable materials and localization [18]. Chapter 14 develops the multi-objective version of two meta-heuristics, CBO and ECBO and applies for Construction Site Layout Planning (CSLP). Besides, the Data Envelopment Analysis (DEA) is utilized for calculating the efficiency of optimal Pareto front layouts that can help decision-makers to select the final layout among the non-dominated candidates. The optimal site layout has several positive impacts on the financial issue, quality of construction, productivity, security, safety and environmental effects [19]. Chapter 15 tailors a comprehensive framework that merges MCDM techniques with metaheuristic multi-objective optimization techniques. The most proper schedule for appliances is selected by creating a trade-off among various optimization criteria. A Multi-Objective Ant Lion Optimizer (MOALO) is applied and tested on a smart home case study to detect all the Pareto solutions. A benchmark instance of the appliance scheduling is solved by the proposed methodology, and Shannon’s entropy technique is utilized to find the objectives’ corresponding weights. Afterward, the acquired Pareto optimal solutions are ranked based on the Evidential Reasoning (ER) method, and the optimal solution is selected. By inspecting the efficiency of every solution considering multiple criteria such as unsafety, electricity cost, delay, Peak Average Ratio (PAR), and CO2 emission, the proposed approach confirms its effectiveness in smart appliance scheduling [20].

6

1 Introduction

References 1. Hoffecker, J. F. (2005). Innovation and technological knowledge in the Upper Paleolithic of Northern Eurasia. Evolutionary Anthropology: Issues, News, and Reviews, 14(5), 186–198. https://doi.org/10.1002/evan.20066. 2. Snyman, J. A. (2005). Practical mathematical optimization. Cham: Springer. 3. Kaveh, A. (2017) Advances in metaheuristic algorithms for optimal design of structures (2nd ed.). Cham: Springer. 4. Yang, X. (2010). Engineering optimization. Engineering optimization (pp. 15–28). NJ, USA: Wiley. 5. Kaveh, A. (2017). Applications of metaheuristic optimization algorithms in civil engineering. Cham: Springer. 6. Kaveh, A., Dadras, A., & Geran Malek, N. (2019). Optimum stacking sequence design of composite laminates for maximum buckling load capacity using parameter-less optimization algorithms. Engineering with Computers, 35(3), 813–832. https://doi.org/10.1007/s00366018-0634-2. 7. Kaveh, A., Dadras, A., & Geran Malek, N. (2019). Robust design optimization of laminated plates under uncertain bounded buckling loads. Structural and Multidisciplinary Optimization, 59(3), 877–891. https://doi.org/10.1007/s00158-018-2106-0. 8. Kaveh, A., & Ghafari, M. H. (2018). Optimum design of castellated beams: Effect of composite action and semi-rigid connections. Scientia Iranica, 25(1), 162–173. 9. Kaveh, A., Vaez, S. R. H., Hosseini, P., & Bakhtyari, M. (2019). Optimal design of steel curved roof frames by enhanced vibrating particles system algorithm. Periodica Polytechnica Civil Engineering, 63(4), 947–960. 10. Kaveh, A., & Ghafari, M. H. (2019). Geometry and sizing optimization of steel pitched roof frames with tapered members using nine metaheuristics. Iranian Journal of Science and Technology, Transactions of Civil Engineering, 43(1), 1–8. https://doi.org/10.1007/s40996-0180132-1. 11. Kaveh, A., & Dadras Eslamlou, A. (2019). An efficient two-stage method for optimal sensor placement using graph-theoretical partitioning and evolutionary algorithms. Structural Control and Health Monitoring, 26(4), e2325. https://doi.org/10.1002/stc.2325. 12. Arzani, H., Kaveh, A., & Dehghan, M. (2014). Adaptive node moving refinement in discrete least squares meshless method using charged system search. Scientia Iranica, 21(5), 1529– 1538. 13. Kaveh, A., Laknejadi, K., & Alinejad, B. (2012). Performance-based multi-objective optimization of large steel structures. Acta Mechanica, 223(2), 355–369. https://doi.org/10.1007/ s00707-011-0564-1. 14. Kaveh, A., & Farhadmanesh, M. (2019). Optimal seismic design of steel plate shear walls using metaheuristic algorithms. Periodica Polytechnica Civil Engineering, 63(1), 1–17. 15. Kaveh, A., & Sabeti, S. (2018). Structural optimization of jacket supporting structures for offshore wind turbines using colliding bodies optimization algorithm. The Structural Design of Tall and Special Buildings, 27(13), e1494. https://doi.org/10.1002/tal.1494. 16. Kaveh, A., Shokohi, F., & Ahmadi, B. (2014). Analysis and design of water distribution systems via colliding bodies optimization. International Journal of Optimization in Civil Engineering, 4(2), 165–185. 17. Kaveh, A., & Vazirinia, Y. (2017). Tower cranes and supply points locating problem using CBO, ECBO, and VPS. International Journal of Optimization in Civil Engineering, 7(3), 393–411. 18. Khanzadi, M., Kaveh, A., Moghaddam, M. R., & Pourbagheri, S. M. (2019). Optimization of building components with sustainability aspects in BIM environment. Periodica Polytechnica Civil Engineering, 63(1), 93–103. https://doi.org/10.3311/PPci.12551.

References

7

19. Kaveh, A., Rastegar Moghaddam, M., & Khanzadi, M. (2018). Efficient multi-objective optimization algorithms for construction site layout problem. Scientia Iranica, 25(4), 2051–2062. https://doi.org/10.24200/sci.2017.4216. 20. Kaveh, A., & Vazirinia, Y. (2020). Smart-home electrical energy scheduling system using multi-objective antlion optimizer and evidential reasoning. Scientia Iranica, 27(1), 177–201. https://doi.org/10.24200/sci.2019.53783.3412.

Chapter 2

Optimum Stacking Sequence Design of Composite Laminates for Maximum Buckling Load Capacity

2.1 Introduction In this chapter, metaheuristic algorithms are applied for maximizing the buckling capacity of laminated plates. Two loading types are considered as optimization problems: deterministic and uncertain loaded composite laminates. Furthermore, different cases with various panel aspect ratios, number of layers and materials are examined to provide the optimal configurations [1]. To account for the uncertainty in loading, the anti-optimization approach is employed. Golden Section Search (GSS) is applied for finding a robust design based on worst-case biaxial compressive loading [2]. The results are investigated from different perspectives and sensitivity analyses are performed. In the past few decades, significant researches have been conducted to study the design problem of composite structures. Comprehensive reviews on the classification and comparison of optimization methods for the problem of lay-up selection of the laminated composite structures have been presented. Among the various classifications described in these reviews, four of them are outlined as follows: (1) an approach proposed by Venkataraman and Haftka [3], in which the procedure was categorized into two single laminate and stiffened plate design; (2) Abrate [4], in which the problems were classified according to their objective functions; (3) Fang and Springer [5], that used four categories as designing approaches, i.e. analytical approaches, enumeration procedures, heuristic methods, and nonlinear programming and; (4) Setoodeh et al. [6] that considered two approaches for the design of composite structures, namely constant stiffness with the aim of obtaining optimum stacking sequence for material distribution and variable stiffness over the domain of structure. The hybrid laminate comprises the laminated composites with more than one type of constituent as the matrix and reinforcement phases. The development of hybrid laminates led to improvements in terms of principal features, for example, improvements in performance, flexibility, weight, and cost. Several studies can be found on the optimal design of hybrid laminated composites in the literature. Huang © Springer Nature Switzerland AG 2020 A. Kaveh and A. Dadras Eslamlou, Metaheuristic Optimization Algorithms in Civil Engineering: New Applications, Studies in Computational Intelligence 900, https://doi.org/10.1007/978-3-030-45473-9_2

9

10

2 Optimum Stacking Sequence Design of Composite Laminates …

and Haftka [7] optimized the fiber orientations near a hole in a single layer of a multilayered composite laminate for increased strength using gradient-based and Genetic Algorithms. Lee et al. [8] incorporated variable critical load cases (bending, shear and torsion) within the design optimization problem of hybrid (fiber–metal) composite structures (HCS). Kalantari et al. [9] investigated the design problem of carbon-glass/epoxy hybrid composite laminates in the presence of three different uncertainty sources, including uncertainties in lamina thickness, fiber orientation, and matrix voids. In their study, the conflicting objectives were considered as material cost and density, where minimum flexural strength was chosen as a constraint. Adali et al. [10] compared the optimal stacking configuration of constant thickness symmetric hybrid laminates for both robust and deterministic buckling load cases. They observed that the stacking sequence design for a deterministic load case considerably differs from that of a robust laminate designed by taking the load uncertainties into account. In the past, the stacking sequence optimization problems were solved by classical gradient-based methods. The deficiencies of these methods in dealing with non-convex spaces and discrete variables led to their limited success. During recent decades, different metaheuristic algorithms have been established as promising and effective tools for dealing with optimal design problems. It should be noted that one of the challenges in the implementation of evolutionary algorithms is the difficulty in the determination of algorithm-specific parameters, such as crossover and mutation rates of GA. Finding the ideal parameters is a time-consuming task, and an inappropriate parameter setting may disturb the performance of the algorithms. For instance, de Almeida [11] demonstrated that the harmony memory size (HMS), harmony memory consideration rate (HMCR) and pitch adjusting rate (PAR) have an apparent effect on the reliability of the Harmony Search Algorithm (HSA) for optimizing the laminated composites. From this point of view, the parameter-less algorithms are at an advantage, which need only the general parameters. In this chapter, we will focus on the parameter less algorithms. At the first step the JAYA, Grey Wolf Optimizer (GWO), Salp Swarm Algorithm (SSA), Colliding Bodies Optimizer (CBO) and two versions of the Genetic Algorithm (GA) are utilized for optimization of laminates with deterministic loads. At the next step, an improved version of the Quantum-inspired Evolutionary Algorithm (QEA) is applied for finding robust configurations. The Golden-Section Search (GSS) method is employed as the anti-optimizer for comparison of the results with those of Adali et al. [10]. The remainder of this chapter is organized as follows: In the next section, the theoretical framework of buckling calculations is presented and in Sect. 2.3 the optimization problem is stated. In Sects. 2.4 and 2.5 the optimization and anti-optimization algorithms are respectively explained. The numerical results are reported in Sects. 2.6 and 2.7. Finally, the concluding remarks are provided in Sect. 2.8.

2.2 Theoretical Framework

11

2.2 Theoretical Framework Based on the classical laminated plate theory (CLPT), the governing equation of buckling for a symmetric N layer laminate can be expressed as: ∂ 4w ∂ 4w ∂ 4w ∂ 4w ∂ 4w + 4D + 2D + 4D + D + 2(D ) 16 12 66 26 22 ∂x4 ∂ x 3∂ y ∂ x 2∂ y2 ∂ x∂ y 3 ∂ y4  2 2  ∂ w ∂ w (2.1) + λ Nx 2 + N y 2 = 0 ∂x ∂y

D11

In Eq. (2.1), W is the deflection in the z-direction, and h is the total thickness of the laminate. In addition, Di j stands for the bending stiffness coefficients, and can be expressed by Eq. (2.2) h

2 Di j =

N    Q i j z 2 dz =

zk+1

k=1 z k

− h2

N  1  (k)  3 (k)   Q i j z 2 dz = Q i j z k+1 − z k3 3 K =1

(2.2)

(k)

where Q i j represents the transformed reduced stiffness of this layer, which is expressed by: T  (k) Q i j = [T ]−1 [Q] [T ]−1

(2.3)

The coefficients of the reduced stiffness matrix ([Q]) can be stated as the following equation: ⎤ Q 11 Q 12 0 [Q] = ⎣ Q 12 Q 22 0 ⎦; 0 0 Q 66 E1 υ12 E 2 υ21 E 1 , Q 12 = = , Q 11 = 1 − υ12 υ21 1 − υ12 υ21 1 − υ12 υ21 E2 Q 22 = , Q 66 = G 12 1 − υ12 υ21 ⎡

(2.4)

In the above equation, E 1 , E 2 and G 12 are longitudinal Young’s modulus, transverse Young’s modulus, and shear modulus, respectively and υ12 and υ21 are Poisson’s ratios. Furthermore, the transformation matrix [T ] can be calculated by the following formula: ⎤ ⎡ 2 S 2 2C S C (2.5) [T ] = ⎣ S 2 C 2 −2C S ⎦; C = cos(θ ), S = sin(θ ) −C S C S C 2 − S 2

12

2 Optimum Stacking Sequence Design of Composite Laminates …

For simply supported edges, the boundary conditions are stated as: w = 0,

Mx = 0 at x = 0, a

(2.6)

w = 0,

M y = 0 at y = 0, b

(2.7)

Considering the governing equation expressed in Eq. (2.1) and the boundary conditions in Eqs. (2.6) and (2.7), the following value for the critical buckling load multiplier λcb of biaxial compression case is obtained [12, 13]: λb (m, n) = π

2

m 4 D11 + 2(D12 + D66 )(r mn)2 + (r n)4 D22 (am)2 N x + (ran)2 N y

(2.8)

where the length and width of the plate are denoted by a and b, respectively. r = a/b is the aspect ratio, and bending stiffness matrix coefficients are indicated by Di j . Different mode shapes can be acquired by inserting corresponding values of m and n associated with the transverse displacement patterns. Here, the smallest of λb (1, 1), λb (1, 2), λb (2, 1) and λb (2, 2) is taken as the critical buckling load. The results given by Nemeth [14] indicate that when the following constraints are satisfied, D16 and D26 terms are insignificant δ=

D16

≤ 0.2,

(2.9)

D26

− 41 ≤ 0.2.

(2.10)

− 1 3 D11 D22 4

γ =

3 D11 D22

In computations, the satisfaction of the above constraints is ensured. Therefore, the mentioned terms do not arise in Eq. (2.8).

2.3 Problem Statement Studying the performance of laminated composites under buckling load, as one of the most frequent causes of failure in laminates, is fairly crucial. In Fig. 2.1a, the schematic illustration of a simply supported composite plate, subject to an in-plane compressive load, is demonstrated. The optimization problem can be defined as the maximization of the critical buckling load factor λcb as the objective function, which can be formulated as follows: 4

m D11 + 2(D12 + D66 )(r mn)2 + (r n)4 D22 (2.11) Maximize: λcb (θ ) = π 2 (am)2 N x + (ran)2 N y

2.3 Problem Statement

13

Fig. 2.1 Schematic configuration of a multilayered sandwich panel: a geometry of the laminated plate and applied loads, b sequence of plies

where θ is the vector of design variables, containing the fiber ply angles of laminates. Besides, λcb can be found using combinations of m and n that yield the lowest buckling load. For deterministic optimization, 64-layer two-ply stacks with 02 , ±45 and 902 degrees are replaced by integers 1, 2 and 3, respectively. In the binary implementation for discrete ply angles of robust optimization, 16-layer stacks with θ = 0◦ , +45◦ , −45◦ , and 90◦ are encoded to binary numbers 00, 01, 10, and 11, respectively. Additionally, the representation of solutions is in the order of coming from the outermost lamina to the innermost one. As shown in Fig. 2.1b, in the case of hybrid carbon–glass/epoxy laminate, glass–epoxy layers are assumed as the inner plies of the plate, and carbon–epoxy layers are the outer plies. The mechanical properties of these materials are provided in Table 2.1. In many conditions, precise probabilistic data on uncertainty in buckling loads of laminated composites are not available a priori, however, these can be bounded to a defined set. These bounded domains which are denoted by U p ( p = 1, 2, . . . , ∞), can be formulated as: U p = {(N x , N y )|N x ≥ 0, N y ≥ 0, N xp + N yp ≤ 1}

(2.12)

Based on the value of exponent p, distinct shapes for uncertain domains are obtained. As illustrated in Fig. 2.2, p = 1 and p = 2 stand for triangular and circular domains, respectively. For a given p, the minimum value of the buckling load factor Table 2.1 Material properties of carbon–glass/epoxy laminates E1

E2

Carbon–epoxy (T300/280)

181

10.3

Glass–epoxy (Scotch-ply 1002)

38.6

8.27

G 12

υ12

7.17

0.28

4.14

0.26

14

2 Optimum Stacking Sequence Design of Composite Laminates …

Fig. 2.2 Domains of uncertainty: a triangular, b circular

is determined by solving the following anti-optimization problem:   λ θ K ; N ∗ = min λ(θ K ; N ) N ∫ Up

(2.13)

  In the level of anti-optimization, N = N x , N y ∈ U p are considered as the design variables corresponding to the lowest buckling capacity (worst case).

2.4 Optimization Algorithms The details of the utilized meta-heuristics are presented in the following sections. Except for GA, other algorithms are parameter-less, which only require the definition of population size and the number of iterations. To avoid the complexities of parameter tuning which is one of the priorities of the present chapter, the population size and number of iterations are set as 50 and 100, respectively.

2.4.1 JAYA Algorithm JAYA, a global search population-based algorithm was proposed by Rao [15]. The simplicity and high efficiency of this method were evaluated by implementing various constrained and unconstrained benchmark problems in [15]. The concept of this method is exploring the best solution while avoiding the worst one, hence it is named as a Sanskrit word meaning victory, i.e. JAYA. One of the significant features of this algorithm is that it does not need the algorithm-specific parameters. Consequently, the population size and number of iterations are the only two required standard controlling parameters.

2.4 Optimization Algorithms

15

Fig. 2.3 A schematic of position updating in JAYA algorithm

The solution updating of JAYA algorithm is performed by the following equation:       X j,k,i = X j,k,i + r1, j,i X j,best,i −  X j,k,i  − r2, j,i X j,wor st,i −  X j,k,i 

(2.14)

In the above equation, X j,k,i is the updated value of the X j,k,i . Here, r1, j,i and r2, j,i are two random numbers ranged from 0 to 1, during the ith iteration, in order to ensure the better exploration in the search space. The number of design variables and population size are indicated by ‘m’ ( j = 1, 2, . . . , m), and ‘n’ (k = 1, 2, . . . , n), respectively. Moreover, X j,best,i , and X j,wor st,i denote the  solu best and worst   X X − tions inthe ith iteration. In Eq. (2.14), the terms “r 1, j,i j,best,i j,k,i ” and   “−r2, j,i X j,wor st,i −  X j,k,i  ”, respectively represent that the solution attempts to move towards the best solution and avoid the worst one. The presentation of Eq. (2.14) in 2D space is illustrated in Fig. 2.3. At the end of each if the value of the objective function improves,   iteration, the updated position X j,k,i participates in the next iteration. The aforementioned procedure continues until the termination conditions are met. The maximum number of iterations is considered as the termination condition. A flowchart of the JAYA algorithm is presented in Fig. 2.4. The pseudo-code of JAYA algorithm is as follows: Randomly initialize the population Calculate the fitness of each search agent Sort Population Find the best and worst solutions while (i 0 within the support domain. W x − x j  = 0 outside the support domain. W x − x j  monotonically decreases from the point of interest x. W x − x j is sufficiently smooth, especially on the boundary  j . Here the cubic spline weight function is employed for better performance in the meshless method as follows: ⎧2 for d¯ ≤ 21 ⎨ 3 − 4d¯2 + 4d¯ 3   4 4 ¯3 2 ¯ ¯ ¯ W x − x j = W (d) = 3 − 4d + 4d − 3 d for 21 < d¯ ≤ 1 (7.3) ⎩ 0 for d¯ > 1   where d¯ = x − x j /dw and dw is the size of the influence domain of point x j . Minimization of Eq. (7.2) with respect to the coefficient α(x) leads to φ(x) = P T (x) A−1 (x)B(x)φ h

(7.4)

where A(x) =

n        w x − xj P xj PT xj

(7.5)

j=1

and, B(x) = [w(x − x1 ) P(x1 ), w(x − x2 ) P(x2 ), . . . , w(x − x n ) P(x n )]

(7.6)

Equation (7.4) can be written in the following compact form: φ(x) =

n 

NiT (x)φi (x) = N T (x)φ h

(7.7)

i=1

Leading to the definition of MLS shape function defined as: N T (x) = P T (x) A−1 (x)B(x)

(7.8)

where N T (x) contains the shape functions of nodes at the point x, which are called moving least squares (MLS) shape functions.

142

7 The Charged System Search Algorithm for Adaptive Node …

7.2.2 Discrete Least-Squares Meshless Method Consider the following partial differential equation: ⎧ ⎨ L(φ) + f = 0 in  B(φ) − t¯ = 0 in t ⎩ φ − φ¯ = 0 in u

(7.9)

where L and B are partial differential operators; φ¯ and t¯ are vectors of prescribed displacements and tractions on the Dirichlet and Neumann boundaries, respectively;  is the considered domain; u and t are the displacement and traction boundaries, respectively; f is the vector of external force or source term on the problem domain. Suppose the value of the estimating function φ in a point such as xk is denoted as follows: φ(xk ) =

m 

Ni (xk )φi

(7.10)

i=1

According to the discretization of the problem domain and its boundaries using Eq. (7.4) the residual of the partial differential equation at the point xk is defined as follow: R (xk ) = L(φ(xk )) + f (xk ), k = 1, . . . , M

(7.11)

The residual of Neumann boundary condition at the point xk on the Neumann boundary can also be presented as Eq. (7.12): Rt (xk ) = B(φ(xk )) − t¯(xk ), k = 1, . . . , Mt

(7.12)

Finally, the residual of Dirichlet boundary condition at the point xk on the boundary can be written as: ¯ k ), k = 1, . . . , Mu Ru (xk ) = φ − φ(x

(7.13)

where, Md is the number of internal points, Mt is the number of points on the Neumann boundary, Mu is the number of points on the Dirichlet boundary, and M is the total number of points. A penalty approach can now be used to form the least-squares functional of the residuals defined as: M Mt Mu d   1  2 2 2 J= R (xk ) + α . Rt (xk ) + β . Ru (xk ) 2 k=1  k=1 k=1

(7.14)

7.2 Discrete Least Squares Meshless (DLSM)

143

where, α and β are the penalty coefficients for the Importance of Neumann and Dirichlet boundary conditions, respectively. Minimization of the functional with respect to nodal parameters (φi , i = 1, 2, . . . , n) leads to the following system of equations: Kφ = F

(7.15)

where Ki j =

M 

[L(N )]kT [L(N )]k + α

k=1

Fi = −

Mt 

[B(N )]kT [B(N )]k + β

k=1

Mu 

NkT Nk

Mt Mu M    [N ]kT φ¯ k [L(N )]kT [L(N )] f k + α [B(N )]kT t¯k + β k=1

(7.16)

k=1

k=1

(7.17)

k=1

The stiffness matrix K in Eq. (7.16) is square, symmetric, and positive definite. Therefore, the final system of equations can be solved directly via efficient solvers.

7.3 Charged System Search The charged system search is based on electrostatics and Newtonian mechanics laws [3]. The Coulomb and Gauss laws provide the magnitude of the electric field at a point inside and outside a charged insulating solid sphere, respectively, as follows:

kq

e i

Ei j =

r a3 i j ke qi 2 ri j

i f ri j < a i f ri j ≥ a

(7.18)

where ke is a constant known as the Coulomb constant, ri j is the separation of the center of a sphere and the selected point; qi is the magnitude of the charge; and a is the radius of the charged sphere. Using the principle of superposition, the resulting electric force due to N charged spheres is equal to:    N  ri − r j qi qi i 1 = 1, i 2 = 0 ⇔ ri j < a F j = ke q j (7.19) r . i + .i , i j 1 2 2 3 i 1 = 0, i 2 = 1 ⇔ ri j ≥ a a r − r r i j ij i=1 Also, according to Newtonian mechanics, we have: r = rnew − rold

(7.20)

rnew − rold t

(7.21)

V =

144

7 The Charged System Search Algorithm for Adaptive Node …

a=

Vnew − Vold t

(7.22)

where rold and rnew are the initial and final position of a particle, respectively, v is the velocity of the particle; and a is the acceleration of the particle. Combining the above equations and using Newton’s second law, the displacement of any object as a function of time is obtained as: rnew =

1F . t 2 + vold . t + rold 2m

(7.23)

Inspired by the above electrostatics and Newtonian mechanics laws, the pseudocode of the CSS algorithm is presented as follows. Level 1: Initialization Step 1. Initialization. Initialize the parameters of the CSS algorithm. Initialize an array of charged particles (CPs) with random positions. The initial velocities of the CPs are taken as zero. Each CP has a charge of magnitude q defined considering the quality of its solution as qi =

f it(i) − f itwor st , i = 1, 2, . . . , N f itbest − f itwor st

(7.24)

where f itbest and f itwor st are the best and the worst fitness of all the particles; f it (i) represents the fitness of agent i. The separation distance ri j between two charged particles is defined as ri j = 

Xi − X j  X i + X j /2 − X best + ε

(7.25)

where X i and X j are the positions of the ith and jth CPs, respectively, X best is the position of the best current CP, and ε is a small positive number to avoid singularities. Step 2. CP ranking. Evaluate the values of the fitness function for the CPs, compare with each other and sort them in increasing order. Step 3. CM creation. Store the number of the first CPs equal to charged memory size (CMS) and their related values of the fitness functions in the charged memory (CM). Level 2: Search Step 1. Attracting force determination. Determine the probability of moving each CP toward the others considering the following probability function:

pi j =

1 0

f it (i)− f itbest f it (i)− f it ( j)

> rand ∨ f it ( j) > f it (i) else

(7.26)

7.3 Charged System Search

145

Furthermore, calculate the attracting force vector for each CP as follows: ⎧   N j = 1, 2, . . . , N ⎨    qi qi X Fj = q j r .i + .i − X p i = 1, i 2 = 0 ⇔ ri j < a ij 1 2 ij i j ⎩ 1 a3 ri2j i,i= j i 1 = 0, i 2 = 1 ⇔ ri j ≥ a (7.27) where F j is the resultant force affecting the jth CP. Step 2. Solution construction. Move each CP to the new position and find its velocity using the following equations: X j,new = rand j1 .ka .

Fj .t 2 + rand j2 .kv .V j,old .t + X j,old mj V j,new =

X j,new − X j,old t

(7.28) (7.29)

where rand j1 and rand j2 are two random numbers uniformly distributed in the range (0,1). m j is the mass of the CPs, which is equal to q j . The mass concept may be useful for developing a multi-objective CSS. t is the time step, and it is set to 1. ka is the acceleration coefficient; kv is the velocity coefficient to control the influence of the previous velocity. Here, ka and kv are taken as 0.5. Step 3. CP position correction. If each CP exits from the allowable search space, the algorithm corrects its position using the HS-based handling approach as described for the HPSACO algorithm [4]. Step 4. CP ranking. Evaluate and compare the values of the fitness function for the new CPs, and sort them in increasing order. Step 5. CM updating. If some new CP vectors are better than the worst ones in the CM, in terms of their objective function values, include the better vectors in the CM and exclude the worst ones from the CM. Level 3: Controlling the terminating criterion Repeat the search level steps until a terminating criterion is met.

7.4 Error Indicator and Adaptive Refinement Adaptivity is an important tool for the efficiency and effectiveness of any numerical method. Any adaptive procedure is formed of two main parts, error estimation and mesh refinement. To estimate errors, different methods have been introduced and applied in the numerical methods which can be separated into two categories: methods based on residual of the differential equations governing the problem and

146

7 The Charged System Search Algorithm for Adaptive Node …

those based the restoration of the error which consider the error as the gradient of the solution. In this chapter, error estimation based on the least-squares function that is formed from weighted residuals is utilized; the error for each point is defined as follows:   1 2 R (xk ) + α . Rt2 (xk ) + β . Ru2 (xk ) ek = (7.30) 2  where ek is the error of any point in the domain or its boundaries. The advantage of this choice is the availability of the error estimator in the process of the main simulation. Different methods for refinement and achieving more accurate solutions after identifying the error distribution can be used. There are three general methods of refinement in finite element, mesh moving (r-method), mesh enrichment (p-method) and p-refinement, whereby higher order of shape function is used. In mesh moving methods, the number of the nodes is constant, but the location of the nodes is altering according to the achieved errors. In FEM, the connectivity of the nodes may be disturbed, and some of the elements can overlap or have zero areas. In the mesh enrichment process as the most common method in FEM, the initial mesh remains, and some new elements are added to the domains with a higher amount of errors, or a new mesh is created based on the error distribution. In the increasing process of the order of shape functions, the order increases with higher errors in the domain. This process requires the employing of hierarchical shape functions, which is associated with complications. It is clear that the most appropriate adaptivity is mesh moving because of the constant number of the nodes; therefore, its computational costs are less than the other methods. The use of this method involves some complexity in FEM due to the deformation of the element after removing them. In meshless methods and especially in the DLSM method, there is no element scheme, and the solution is sensitive to neither the distance between nodes and nor the method of exposure, hence the use of the node moving method is easily possible.

7.5 The Link Between the CSS and Adaptivity For the CSS algorithm, each node is used as a CP of zero radiuses and m j is defined for masses. For each node, the error is formed from weighted residuals that are available using Eq. (7.30). The charge of each node is defined as the normalized error as following: qj =

e j − emin emax − emin

(7.31)

The distance between any two nodes, j and i is defined by L P norm, and in particular, for planar problem L 2 norm is used as follow:

7.5 The Link Between the CSS and Adaptivity

ri j =

 (xi − x j )2 + (yi − y j )2

147

(7.32)

Considering that the error reduction in meshless methods requires the densities of the nodes in the accumulation zones of the errors. Therefore, as seen in Eq. (7.30), only nodes with less error are allowed to move toward the nodes with higher errors  pi j =

1 qi > q j 0 else

(7.33)

For each node, the force vectors in two directions are calculated according to the following equations:  qi x i − x j F jx = k e q j pi j cos(θ ) ri2j xi − x j i=1,i= j   N  qi yi − y j F jy = ke q j pi j sin(θ ) 2 yi − y j r ij i=1,i= j N 



(7.34)

(7.35)

Here θ is the angle between the horizontal direction and the line connecting the two nodes. According to the formulation of the CSS, new positions of the nodes in the two-dimensional case is obtained similarly.

7.5.1 Objective Function In the DLSM method, the value of residuals represents the scope to which the numerical solution satisfies the governing differential equation and its boundary condition. Also, it should be mentioned that the least-squares functional defined as the squares residual can be considered as a measure of the error of the numerical solution. The method is especially efficient since the least-squares calculations are already available from the solution procedure. Hence, the objective function to be minimized is taken as the normalized sum of squared residuals of all points in the domain. The search level of the CSS is continued as long as the improvement in the reduction of the objective function is possible. The algorithms are coded in compact visual FORTRAN, and the systems of linear equations are solved via an efficient solver.

7.5.2 Selected Parameters In the first examples, the nodes in the arc edge and corner nodes have been fixed. The nodes on the horizontal boundaries can only move horizontally, and the vertical

148

7 The Charged System Search Algorithm for Adaptive Node …

movement is restricted. t is set to 1. ka and kv both are taken as 0.5. For all nodes, m j is selected as constant. When it takes an arbitrary value such as 5 the algorithm will need almost 10 iterations but the results are from one iteration in taking 20 for the best value of m j . The objective function is taken as the ratio of the least-squares functional of the residuals to the total number of points which is available in the process of the main simulation as follows:  Minimize

T otal E Estimate

=

Md k=1

2 R (xk ) + α .

 Mt

Rt2 (xk ) + β . M

k=1

 Mu k=1

Ru2 (xk )



(7.36) Here Md is the number of internal points, Mt is the number of points on the Neumann boundary, Mu is the number of points on the Dirichlet boundary, and M is the total number of points. In the second example, the corner nodes are fixed and boundary nodes can only move on their directions. Similar to the first example,t is set to 1; ka and kv both are taken as 0.5. The value of m j which leads to the best results in one iteration is taken as 52.

7.6 Numerical Examples In this section, two examples of planar elasticity are presented whose analytical solutions are available. The comparison of the initial solution and the solution after refinement by the CSS with the analytical solution shows the efficiency of the presented method. The first example considers an infinite plate with a circular hole subject to uniaxial traction, and the second example is a cantilever beam under end load.

7.6.1 Infinite Plate with a Circular Hole The first example considers an infinite plate with a circular hole under a uniaxial load at infinity and with boundary conditions, as shown in Figs. 7.1 and 7.2. The exact solutions of this problem can be defined as: For the stresses     3a 4 a2 3 (7.37) cos(2θ ) + cos(4θ ) + 2 cos(4θ ) σx = t 1 − 2 r 2 2r    2 3a 4 a 1 σ y = −t 2 (7.38) cos(2θ) − cos(4θ ) + 2 cos(4θ ) r 2 2r

7.6 Numerical Examples

149

Fig. 7.1 An infinite plate with a circular hole under a uniaxial load P

Fig. 7.2 Boundary conditions of the plate

   3a 4 a2 1 (7.39) cos(2θ ) + sin(4θ ) − 2 sin(4θ ) τx y = −t 2 r 2 2r     a2 k−1 t a4 ur = r + cos(2θ ) + [1 + (1 + k) cos(2θ )] − 3 cos(2θ ) 4G 2 r r (7.40)  2 4 a t a uθ = (7.41) (1 − k) − r − 3 sin(2θ ) 4G r r 

In the above equations, G is the shear modulus and k = (3 − ν)/(1 + ν) where ν representing the Poisson’s ratio. Due to symmetry, only the upper right square quadrant of the plate is simulated (see Fig. 7.2). The edge length of the square is 5a, where a is the radius of the circular hole. The Dirichlet boundary condition is imposed on the left and bottom boundaries and the tractions are applied to the top and right edges. The problem is solved under plane stress conditions. The process begins with

150

7 The Charged System Search Algorithm for Adaptive Node …

the simulation of the problem on two distributions of 95 nodes referred to as initial and refined distributions, which are shown in Figs. 7.3 and 7.4, respectively. The error distribution is estimated based on the numerical solution obtained by Eq. (7.14). Fig. 7.3 The initial distribution of the nodes

Fig. 7.4 The refined distribution by the CSS

7.6 Numerical Examples

151

After processing the CSS algorithm, with a uniform distribution of the nodes, the adapted nodes are more concentrated around the curved edge where the numerical errors are much higher due to stress concentration. A comparison of the horizontal displacement u x of the hole and the normal stress σx along x = 0, with the analytical solution of the problem, a tremendous improvement can be seen in the accuracy of the DLSM method, as shown in Figs. 7.5 and 7.6. Stress tensor for the initial solution, the solution of the refined distribution of the nodes by CSS and the real stress tensor for analytical solution are shown in Fig. 7.7. Fig. 7.5 Normal stress σx along x = 0

Fig. 7.6 The horizontal displacement u x of hole

152

7 The Charged System Search Algorithm for Adaptive Node …

Fig. 7.7 Contours of the normal stress σx . a Initial solution, b refined solution, c exact solution

Fig. 7.8 A cantilever beam under a point load at the end

7.6.2 A Cantilever Beam Under End Load As the second example, the problem of a cantilever beam under a point load at the end is considered that is shown in Fig. 7.8. For this problem, the exact stresses and displacements in the plane stress are provided by following equations σx = −

v=

(7.46)

σy = 0

(7.47)

 P 2 c − y2 2I

(7.48)

  Py  3x(2L − x) + (2 + υ) y 2 − c2 6E I

(7.49)

τx y = u=−

P(L − x)y I

 Py  2 x (3L − x) + 3υ(L − x)y 2 + (4 + 5υ)c2 x 6E I

(7.50)

where E is the elastic modulus and υ presents Poisson’s ratio. The moment of inertia I = 2c3 /3 is considered for a beam with a rectangular cross-section and unit thickness. The problem is solved using the DLSM method under plane stress conditions.

7.6 Numerical Examples

153

Fig. 7.9 Initial nodal distribution

The distribution of 125 nodes is used for DSLM, as shown in Fig. 7.9. The errors are calculated, and the nodal locations are adaptively altered using the CSS algorithm, as shown in Fig. 7.10. The problem is then solved again with the refined nodal configuration, and the results are compared to those obtained with the initial configuration and the exact analytical results. Figures 7.11 and 7.12 compare the vertical displacement u y along the upper surface of the beam and the normal stress σx along the upper surface of the beam obtained by the initial and adapted distribution with the analytical solutions. It can be seen that the result obtained by the adapted nodal distribution is virtually exact indicating the effectiveness of the CSS algorithm in the DLSM method.

Fig. 7.10 Refined nodal distribution by the CSS

Fig. 7.11 The vertical displacement u y along upper surface of the beam

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7 The Charged System Search Algorithm for Adaptive Node …

Fig. 7.12 Normal stress σx along upper surface of the beam

For large-scale and complex problems, the use of metaheuristic approaches becomes time-consuming; however, for such a problem, one can decompose it into its components, and after solving each component, the solution of the main problem can be obtained using the methods provided in Ref. [5].

7.7 Discussions and Conclusion Though the DLSM method has been developed for achieving more accurate solutions, the refinement process is inevitable for some problems. The process of refinement proposed in this chapter makes it possible to achieve more accurate responses without increasing the number of points or imposing computational costs. Considering the points in the discretized domain as charged particles in the CSS algorithm and using the normalized errors in the DLSM process as the charges of particles, an accurate model is developed. In meshless methods, different techniques have introduced, which occasionally imposes a considerable computational cost. However, since the CSS explores the most suitable locations of the nodes in one or two iterations, its computational cost is much less than the existing algorithms.

References 1. Firoozjaee, A. R., & Afshar, M. H. (2009). Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations. Engineering Analysis with Boundary Elements, 33(1), 83–92. https://doi.org/10.1016/j.enganabound.2008.03.004.

References

155

2. Arzani, H., Kaveh, A., & Dehghan, M. (2014). Adaptive node moving refinement in discrete least squares meshless method using charged system search. Scientia Iranica, 21(5), 1529–1538. 3. Kaveh, A., & Talatahari, S. (2010). A novel heuristic optimization method: Charged system search. Acta Mechanica, 213(3), 267–289. https://doi.org/10.1007/s00707-009-0270-4. 4. Kaveh, A., & Talatahari, S. (2009). Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures. Computers & Structures, 87(5), 267–283. https://doi.org/10.1016/j.compstruc.2009.01.003. 5. Kaveh, A. (2013). Optimal analysis of structures by concepts of symmetry and regularity. Wien: Springer.

Chapter 8

Performance-Based Multi-objective Optimization of Large Steel Structures

8.1 Introduction In recent years, the importance of economic considerations in the field of structures has motivated many researchers to propose new methods for minimizing the initial and life cycle cost of the structures subjected to seismic loading. In this chapter, a new framework is presented to solve the performance-based multi-objective optimization problem considering the initial and life cycle cost of large-scale structures. In order to solve this problem, the non-dominated sorting genetic algorithm (NSGA-II) using differential evolution (DE) operators is employed to solve the optimization problem while a specific meta-model is utilized for reducing the number of fitness function evaluations. The required computational time for pushover analysis is decreased by a simple numerical method. The constraints of the optimization problem are based on the FEMA codes. The presented results for the application of the proposed framework demonstrate its capability in solving the present complex multi-objective optimization problem [1]. Economically speaking, the costs of a structure are not only dependent on its initial operation costs but also the secondary costs such as maintenance, damage, and repair expenses. The later costs can have a major effect on the entire expected cost of a structure in its lifetime and should be considered in decision making, and it is clear that if more initial money is spent for the construction of a building, naturally, its life cycle cost will be lower. In the literature, the entire expected life cost of the structure is called Life Cycle Cost. Seismic damage costs caused by severe earthquakes have a direct relation with the behavior of the structure against such earthquakes. The maximum inter-story drift ratio of a structure can be an appropriate indicator of the damage levels since it nearly indicates the presented rotations and displacements in the elements of the structure. Wen and Kang [2] have proposed a relationship using the exceedance probability of the maximum inter-story drift from the predefined drift levels to evaluate the seismic damage cost. In this way, a pushover analysis is needed to obtain the maximum © Springer Nature Switzerland AG 2020 A. Kaveh and A. Dadras Eslamlou, Metaheuristic Optimization Algorithms in Civil Engineering: New Applications, Studies in Computational Intelligence 900, https://doi.org/10.1007/978-3-030-45473-9_8

157

158

8 Performance-Based Multi-objective Optimization …

inter-story drifts. In recent decades, the performance-based design of structures as a design philosophy has been implemented in many works, and many studies on this concept have been carried out. Various guidelines such as AISC (2002), ATC-40 (1996), and NEHRP guidelines [FEMA-273 (1997) and FEMA-356 (2000)] among many others have been introduced for the analysis and design of new structures as well as evaluation and rehabilitation of the existing ones. A performance-based design depends on the performance levels considered for the structure, and the design procedure is accomplished in such a manner that the respective defined demand and capacity of the structure are balanced. Consequently, a nonlinear analysis should be carried out. The purpose of the nonlinear static analysis known as the pushover analysis is to specify the structural performance in terms of the deformation and strength capacity of that structure. For different performance levels of the structure, different damage states have been defined and therefore, the evaluation of the life cycle cost has become feasible. Because of the economic importance of this subject, in recent years many studies have been conducted on the estimation of the real life cycle cost of a structure. Wen and Shinozuka [3] investigated cost-effectiveness in active control of structures. Frangopol et al. [4] presented a lifetime optimization methodology for planning the repair of structures against the deterioration. Bucher and Frangopol [5] developed a concept to optimize the maintenance strategies based on the lifetime cost of deteriorating structures. Wen and Kang [2] introduced a design criterion for minimum building life-cycle costs. The issue of optimum seismic design of structures, considering the initial cost and life cycle cost, has been investigated in [6] as a multi-objective optimization problem. The obtained Pareto front by the optimization algorithm provides invaluable information about the strategies that investors can spend their money most suitably and reliably. Due to the complexity of the present optimization problem, evolutionary algorithms are employed. In recent years, many different forms of metaheuristics and evolutionary algorithms have been proposed in the literature of evolutionary optimization. However, the major problem in utilizing every evolutionary algorithm is the need to perform a large number of fitness function evaluations. This problem is involved to a great extent in our specific optimization problem because each fitness function evaluation contains a pushover analysis, which takes a long time even if advanced computers are employed. Consequently, the required computational time for this problem, especially for large structures, may exceed several hundred hours, and this high computational time may convert this solution algorithm to an illogical one. In recent years, many different remedies have been proposed to tackle the abovementioned problem. The application of such methods in this specific optimization problem can cause a considerable reduction in computational time. This chapter aims to propose a computational framework for the optimum seismic design of large-scale structures with an acceptable computational time. To achieve this, first, the initial and seismic damage costs are considered as two different objectives of the optimization problem. Then a modified version of NSGA-II [7, 8] is employed for solving the problem. Additionally, the procedure of pushover analysis is modified to reduce the necessary computational time and finally, a meta-model is

8.1 Introduction

159

utilized in order to reduce the required number of fitness function evaluations. The results of the solved examples demonstrate the power of the proposed framework. It is shown that by the use of this framework, the present complex multi-objective optimization problem can be solved using an acceptable computational time.

8.2 Employed Multi-objective Optimization Algorithm In the literature of evolutionary algorithms for multi-objective optimization, a wide range of different algorithms have been proposed by researchers. In this chapter, the utilized algorithm for solving multi-objective optimization is NSGA-II [7]. The wide application of this algorithm in engineering problems proves its great ability in covering Pareto front and solving the multi-objective optimization problems. The main features of this algorithm can be summarized in this way that comparing the qualities of different solutions in NSGA-II is based on their non-domination ranks and crowded distances, which can be obtained by a fast sorting algorithm proposed in [7]. The lower the non-domination rank of a solution is, the better it is. If two solutions have the same non-domination rank, NSGA-II prefers the solution with a larger crowded distance.

8.2.1 NSGA-II-DE NSGA-II-DE used in the experimental studies is the same as NSGA-II-SBX in [8], except that it replaces the SBX operator in NSGA-II-SBX by the DE operator. NSGAII-DE maintains a population Pt of size N at generation t and generates Pt+1 from Pt in the following way: Step (1) Do the following independently N times to generate new solutions. Step 1.1: Select three solutions x1 , x2 and x3 from Pt by using binary tournament selection. Step 1.2: Generate a solution x  from x1 , x2 and x3 by the crossover operator, and then perform a mutation operator on x  to produce a new solution xnew . Step 1.3: If an element of xnew is out of the boundary of the acceptable domain, its value is reset to be the value on the nearest boundary. Step (2) Combine all the new solutions generated in Step 1 with all the solutions in Pt , and form a combined population of size 2 N. Select the N best solutions from the combined population to constitute Pt+1 . Several variants of NSGA-II with DE have been proposed for dealing with rotated multi-objective problems (rotated MOPs) or MOPs with nonlinear variable linkages. There is no big difference between these variants. None of them employs mutation

160

8 Performance-Based Multi-objective Optimization …

after DE operators. The experimental studies of [8] show that the polynomial mutation operator does slightly improve the algorithm performance, particularly, on MOPs with complicated Pareto fronts.

8.2.2 GA Operators For the DE operator used in Step 1.2, each element x j in x  is generated as follows: x j =



Fix(x1 + F × (x2 − x3 )) with probability C R, with probability 1 − C R x1

(8.1)

where CR and F are two control parameters. The polynomial mutation in Step 1.2 is obtained as follows:  xnew =

  Fix x j + σk × (bk − ak ) with probability pm , x j

with probability 1 − pm

(8.2)

with  σk =

i f rand < 0.5, (2 × rand)1/η+1 − 1 1 − (2 − 2 × rand)1/η+1 otherwise

(8.3)

where rand is a uniform random number from the interval [0,1], the distribution index η and the mutation rate pm are two control parameters. ak and bk are the lower and upper bounds of the kth decision variable, respectively. In this chapter, CR and F values are assumed to be 1 and 0.5, respectively. When a design variable violates the allowable bounds of variables, it is equated to the nearest boundary value of the allowable interval.

8.2.3 Constraint Handling In order to handle the given constraints, a relatively simple scheme is adopted. Whenever two individuals are compared, first, they are checked for constraint violation. If both are feasible, non-dominance is directly applied to decide the winner. If one is feasible and the other is infeasible, the feasible dominates. If both are infeasible, then the one with the lowest amount of constraint violation dominates the other. This is the approach that has been utilized in [7] to handle the constraints.

8.3 Seismic Optimum Design Procedure

161

8.3 Seismic Optimum Design Procedure 8.3.1 Loading and Constraints for Optimum Seismic Design In the optimization procedure, the steps include considering an arbitrary design and then performing pushover analysis of that structure against gravity and seismic loads to assess the structural performance. According to the available provisions, the design of a structure should fulfill some initial constraints. Thus, before entering the analysis phase for an elected design, some simple constraints should be checked. One of the constraints is the strength ratio of the beam to column that is known as the strong column–weak beam design philosophy and the other constraint that should be considered before the analysis phase is that the selected sections for two consecutive columns in height shall be in such an order that the lower column section has greater or equal strength compared to the upper column. Besides these constraints, all AISC checks must be satisfied for the gravity loads to perform the pushover analysis. According to the ASCE-7, the gravity load combinations are as following: Q G = 1.2 Q D + 1.6 Q L

(8.4)

Once the expressed constraints are satisfied, the pushover analysis is performed. In this analysis the component gravity load combinations are as follows according to FEMA-273 [9]: Q G = 1.1 (Q D + Q L )

(8.5)

Q G = 0.9 Q D

(8.6)

Among the above combinations, whichever that produces the most unfavorable effect in the structure is considered. In pushover analysis, the seismic loads are applied incrementally while the structure constantly loaded under gravity loads.

8.3.2 Nonlinear Static Analysis (Pushover Analysis) The purpose of the nonlinear static analysis is to specify the structural performance within the deformation and strength capacity of the structure. Under pushover analysis, a structural model is pushed to a target displacement considering inelastic behavior for its material, and deformations to determine the internal forces. The structural model is pushed with the monotonically increasing loads or displacements until reaching the predefined target displacement or collapse state. The development of the pushover analysis procedure is based on the assumption that the structural

162

8 Performance-Based Multi-objective Optimization …

response is related to the first mode of an equivalent single degree of freedom structural system. In this analysis, the combinations for gravity loads follow the Eq. (8.5) following FEMA-273. The target displacement represents the maximum displacement probably to be experienced during the occurrence of the design earthquake. It is obtained from the following relationship [9]: δt = C0 C1 C2 C3 Sa

Te2 g 4π 2

(8.7)

where, C0 is the modification factor relating the spectral displacement and likely roof displacement. Its value depends on the number of stories. C1 is a modification factor relating the expected maximum inelastic displacements to the displacements calculated for a linear elastic response. C2 is a modification factor to represent the effect of hysteresis shape on the maximum displacement response. C3 is a modification factor to represent the increased displacements due to dynamic P −  effects. Te is the effective fundamental period of the building that is calculated using the force-displacement relationship in pushover analysis. From FEMA-273 we have:  Te = Ti

Ki Ke

(8.8)

where Ti is the fundamental elastic period in seconds, and K i and K e are elastic and effective lateral stiffnesses of the building, respectively. The relation between base shear and displacement of target point that is nonlinear is replaced with a bilinear graph to estimate K e and Vy , which are corresponding to the effective stiffness and the yield strength of the building respectively. In the pushover curve, the conversion process to a bilinear curve is performed in such a manner that the area above and below the curve is balanced approximately. Additionally, the bilinear graph should coincide with the main pushover curve at 0.6Vy which means the effective lateral stiffness is taken as the secant stiffness, estimated at a base shear equal to 60% of the yield strength [9]. The main factor which affects the required computational time for constructing the pushover curve is the number of steps of pushing the structure. However, by reducing the number of steps, the accuracy of the obtained pushover curve decreases, and additionally, the distance between computed target displacement and the nearest point of the pushover to target displacement increases. Since the structure’s internal forces and inter-story drifts should be checked in the state corresponding to target displacement, the large distance leads to this case that the structure might be checked for the forces and inter-story drifts which relate to a displacement that is much larger or smaller than the computed target displacement. In this case, the optimization procedure may accept an infeasible solution or reject other feasible solutions. In this chapter, in order to reduce the number of steps and alleviate the mentioned problem, after performing the pushover analysis with a few numbers of pushing steps, the pushover curve is interpolated linearly, as seen in Fig. 8.1. Additionally,

8.3 Seismic Optimum Design Procedure

163

Fig. 8.1 The obtained P −  curve by the pushover analysis and interpolation process

the interpolation procedure is repeated by the internal forces and inter-story drifts. The employed procedure reduces the required computational time considerably and prevents all the mentioned problems. After obtaining the pushover curve, the target displacement is accomplished by a numerical algorithm as follows: Target displacement determination algorithm { Set δt (i+1) := max (i.e., as initial assumption set target displacement to be the final displacement) of point (1) on the pushover curve) Set δt (i) := d1 (displacement  While δt (i) − δt (i+1) /δt (i) > 0.005 Calculate E, the area under the pushover curve from origin to δt (i+1) . Set  E := I n f For k = 2: number of points with displacement smaller than δt (i+1) Set ds := displacement of point (k) on the pushover curve Set vs := Base shear corresponding to the point (k) Set vy := vs/0.6 and dy := vy.ds/vs Compute the area under the constructed bilinear curve and set it as E 0 If abs(E − E 0 )/E <  E Set Vy := vy and Dy := dy EndIf EndIf Compute Te by the Eq. (8.8) Compute δt (i+1) by the Eq. (8.7) Set δt (i) := (δt (i) + δt (i+1) )/2 Set i := i+1 Endwhile } For pushover analysis, the considered constraints are related to the maximum inter-story drift ratio, because it is a good gauge of both structural and nonstructural

164

8 Performance-Based Multi-objective Optimization …

damage states as discussed in Fragiadakis et al. [10]. Here, three hazard levels are considered. The levels correspond to earthquakes with 50, 10, and 2% probability of exceedance in 50 years. Consequently 0.7%, 2.5% and 5% allowable maximum inter-story drift ratios are followed for each limit-state, respectively.

8.3.3 Lifetime Seismic Damage Cost For a structure that is designed against probable earthquakes, ignoring the maintenance cost, the whole expected life cost of the structure is the total of initial cost and seismic damage costs. The lifetime seismic damage cost consists of the cost of damage or repair, loss of contents, injuries, and human fatality, as well as other economic loss caused by structural damage. This cost is known as the life cycle cost of the structure, Wen and Kang [2]. For the purpose of our work in this chapter, the life cycle cost of a structure is calculated similarly to Wen and Kang [2] by:  ν 1 − e−λt · Ci Pi λ i=1 N

Cseismic (T, X ) =

(8.9)

where T is the service lifetime of a new structure or remaining life of a retrofitted structure; N is the number of limit-states considered; ν is the annual occurrence rate of major earthquakes; λ is the annual monetary discount rate that will be constant and equal to 5%; Ci is the retrofitting cost of ith damage state violation; Pi is the probability of ith damage state violation given the earthquake occurrence; the cost function of the ith damage state violation could be defined as expressed in Table 8.1 (from Wen and Kang [2]). In the equation above, it is assumed that after a significant earthquake occurrence, the damaged structure will be immediately retrofitted to its initial state. The probability of ith damage state violation can be evaluated from the following equation: Table 8.1 Damage states, Drift ratio limits and corresponding costs Performance level

Damage state

Drift ratio limits (%)

Cost (% of initial cost)

I

None

 < 0.2

0

II

Slight

0.2 <  < 0.5

0.5

III

Light

0.5 <  < 0.7

5

IV

Moderate

0.7 <  < 1.5

20

V

Heavy

1.5 <  < 2.5

45

VI

Major

2.5 <  < 5.0

80

VII

Destroyed

5.0 < 

100

8.3 Seismic Optimum Design Procedure

165

Pi = Pi ( > i ) − Pi+1 ( > i+1 )

(8.10)

Considering the Poisson process for modeling major earthquakes, the exceedance probability over a period (0, t) can be given by: Pt ( > i ) = 1 − e−ν Pt (>i )t

(8.11)

in which,  is the drift ratio and Pi ( > i ) is the exceedance probability which is obtained as: Pi ( > i ) = −

1 [ln(1 − Pt ( > i ))] ν.t

(8.12)

The damage states listed in Table 8.1 are reached when the corresponding maximum inter-story drift values have exceeded from the prescribed limitations listed in column 3 of that table. P i ( > i ) as the annual exceedance probability of the ith damage state can be derived from the following relationship based on the work of Papadrakakis et al. [11]: P i ( > i ) = α e−β i

(8.13)

The preceding expression in a logarithmic format is linear with an intercept to α and the slope −β. For earthquakes with 2, 10, and 50% probability of exceedance in 50 years, P i that is the annual probability of exceedance, which can be calculated using the Poisson’s model. The probability of exceedance p in t years is as the following:

p=

−1 t

 ln (1 − p)

(8.14)

In Eq. (8.13) i is the maximum inter-story drift that is obtained from the pushover analysis. Subsequently, the parameters α and β can be obtained by the best fit of the known pairs P i and i .

8.4 Meta-modeling for Predicting the Response In the present optimization problem, fitness function evaluation is the most timeconsuming part of the solution algorithm. If all of the required fitness function evaluations are performed by pushover analysis, it may need many days, weeks, or even months to complete the solving process. Consequently, the prohibitive required computational time for solving large scale structures converts this process to an illogical algorithm. The solution to this problem is the use of computationally efficient approximations of the fitness function.

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8 Performance-Based Multi-objective Optimization …

So far, several models have been used for fitness function approximation. The most popular ones are polynomials (often known as response surface methodology), the kriging model, most popular in design and analysis of computer experiments, the feedforward neural networks, including multi-layer perceptron and radial-basisfunction (RBF) networks and the support vector machines. In the engineering problems, due to the lack of data and the high dimensionality of the input space, it is very difficult to obtain a perfect global functional approximation of the original fitness function. To tackle this problem, two main measures can be taken. Firstly, the quality of the approximate model should be improved as much as possible, given a limited number of data. Several aspects are important to improve the model quality, such as the selection of the model, use of the active data sampling and weighting (both on-line and off-line), selection of training method and selection of error measures. Secondly, the approximate model should be used together with the original fitness function. In most cases, the original fitness function is available, although it is computationally expensive. Therefore, it is very important to use the original fitness function efficiently. This is known as model management in conventional optimization. Now, these two concerns are reviewed in the specific multi-objective optimization problem.

8.4.1 Approximation Model Selection and Training In [12], a comparative study is performed by the authors among RBF networks, Cokriging, and Kriging models. The obtained results demonstrate that with respect to the model precision and the required computational time, RBF networks perform better than the other models. Neural networks are adaptive statistical models that can be trained and utilized for predicting the response of a function. The required input data for training the neural network contains the exact response of the considered function to some initial solutions in addition to some model tuning parameters. In this chapter, a radial basis function (RBF) network is constructed and trained for predicting the response of the considered solution in the optimization process. In the application of RBF networks to our optimization problem, three major concerns should be replied: 1. The selection of the input data should be made in a way that firstly it can present the considered structure properly, and secondly, the trained RBF network by these input data can predict the response of the structure with acceptable precision. 2. The selection of the output data should be made in a way that it can cover both the response of the structure and its constraint violation. 3. Since this problem is a multi-objective optimization, the solutions on the Pareto front may differ significantly from each other. Consequently, the trained RBF network with these widely ranged input data can have low precision in estimating the response or even may generate completely wrong answers. As much as the

8.4 Meta-modeling for Predicting the Response

167

input data of the RBF network are similar, its estimation of the response of the function to an arbitrary point (in the same range as data utilized for training the network) is accurate. There are many choices for the representative input data for training the RBF network. For example, the input data can be the section numbers (or section area) considered for the structural members. While, in large structures, this is the number of member groups (design variables) that makes the training process complicated and time-consuming. Additionally, the section numbers of the structural members cannot be an acceptable representative of the structural behavior under lateral loads. The considered data should be capable of representing the stiffness of the structure. It is well known that the natural frequencies are fundamental parameters representing the behavior of structures, and because the mass of the structure is approximately constant in the optimization process of a specific structure, the natural frequencies of the structure can be a good representative of its stiffness. On the other hand, the natural frequencies of a structure can be determined by a linear analysis in a truly short time. These properties of natural frequencies make them a proper choice for the input data of RBF networks in our problem. Similarly, for selecting the output data for RBF networks, different problems should be covered. First of all, it should be noted that the first objective of the optimization problem, the weight of the structure, does not need any structural analysis. Two other major parameters are the life cycle cost factor (CLC factor), the life cycle cost coefficient, and the factor representing the amount of constraint violation. In this chapter, these two parameters are considered as the output parameters of the RBF networks. In order to solve the third concern in our multi-objective optimization problem and also to improve the quality of the approximate model, in this chapter a concept of nearest data selection is utilized. In this method, all of the solutions which are evaluated by the original fitness function, are stored in an archive. Moreover, in the process of fitness approximation, when the optimization process generates a new solution, first its m nearest neighbors (existing in the archive) in the natural frequency space is determined, then a new RBF network is constructed and trained by these accurate solutions. Finally, the trained RBF network is used for estimating the fitness of the given solution. It should be stated that the value of M should be determined in a way that first, the trained RBF should be able to estimate the response accurately, second, the RBF network should not be overtrained. Here the distance between two different solutions is defined as follows: d = Ti − T j

(8.15)

where, in this equation . is the norm of the vector T, and T is the vector of k natural frequencies. More details of this procedure are illustrated in Fig. 8.2. In this chapter, m and k are equated to 15 and 5 respectively.

168

8 Performance-Based Multi-objective Optimization …

Fig. 8.2 The natural frequency space and the procedure of the data selection for training an RBF network and use it for estimating the fitness of the newly generated solution

8.4.2 Model Management The application of approximation models to evolutionary optimization processes is not as straightforward as one may expect. One essential point is that it is very difficult to construct an approximate model that is globally correct due to the high dimensionality, ill distribution, and a limited number of training samples. There are three major concerns in using approximate models for fitness approximation. Firstly, it should be ensured that the evolutionary algorithm converges to the global optimum or a nearoptimum of the original fitness function. Secondly, the computational cost should be reduced as much as possible. Thirdly, in the process of evolutionary optimization, the range of the solutions may change significantly, and the model trained by the initial data may converge to a false solution. Therefore, it is quite essential in most cases that the approximate model is used together with the original fitness function. The issue of incorporation of these two functions in the process of evolutionary computation is regarded as model management. In the literature, the incorporation of the approximate model and the original fitness function is performed in three different ways: 1. The initial constructed approximate model is assumed to be of high-fidelity, and therefore, the original fitness function is not used in evolutionary computation. 2. The importance of using both the approximate model and the original function for fitness evaluation has been recognized. There are generally two approaches, one is solution-based, and the other is generation-based. By solution-based management, it is meant that in each generation, some of the solutions use the approximate model and others the original function for fitness evaluation. While in generation-based management, in some specified generations, all of the solutions are evaluated by the original fitness function.

8.4 Meta-modeling for Predicting the Response

169

3. Other more complicated methods, in which the probability of evaluating a solution by the original fitness function depends on the fidelity of the approximate model. The decision about the model management should be made based on the properties of the problem we face. In our problem, because of its features, i.e., multi-objective optimization problem, we utilize a combined method. The employed method can be summarized in this way: 1. The first 100 acceptable generated solutions are evaluated by the original fitness function (pushover analysis). The obtained results of these solutions are stored in an archive (as mentioned in Sect. 8.2.1). 2. In each generation of the optimization process, 1/50 of the solutions are evaluated by the original fitness function and the others by the approximate models. In this way gradually some solutions of the new regions of the search space are added to the archive. 3. After every 50 generations of the optimization process, in one generation, all of the generated solutions are evaluated by the original fitness function. In this way, it is ensured that the evolutionary computation converges to the main global optimum of the search space. In the mentioned method, all of the solutions evaluated by the original fitness function are added to the archive, and the approximate models utilize the provided data in this archive for training the RBF networks (for additional details of this process refer to Sect. 8.2.1). The numerical procedure of the implemented algorithm is proposed in Sect. 8.4.1.

8.5 The Proposed Framework In the main algorithm, all of the introduced components are incorporated in a simple framework. The proposed framework makes it possible to perform performancebased multi-objective optimization of the large and complex structures. In this problem, all constraints are categorized into three groups: 1. Initial constraints: The constraints of this group are fulfilled by modifying the given solution. These constraints are (1) the Beam-column strength ratio, which should be less than 1.2 in each joint. In this chapter, this condition is checked in each joint, and if it is not fulfilled the cross-section number of the columns connecting to the joint is increased one number and then it is checked again. This procedure continues until all joints fulfill this constraint. (2) Lower columns should have the same or larger cross-section number than the upper columns. This constraint is checked from the last story and gradually modifies the crosssection of columns in order to fulfill this constraint. (3) The structure should be checked for gravity loads. If the stress ratio of each member of the structure

170

8 Performance-Based Multi-objective Optimization …

is more than 1, its cross-section number is increased by one. This procedure continues until all members fulfill the constraints. 2. Final constraints: This group contains the checking of the drift ratio limit and member forces limit as mentioned in Sect. 8.3.2. For this group, the constraint violation is reported by a factor that guides the optimization process, as mentioned in Sect. 8.2.3. The main procedure can be summarized as follows: Main procedure { (a) (b) (c) (d)

Select all of the input parameters Initialize the population randomly Evaluate all members of the population NSGA-II-DE generates a new generation of the population by the use of the provided data of previous generations. (e) Evaluate all members of the population (f) Stop the procedure if stopping criteria is fulfilled; otherwise go to step d.

} Additionally, the evaluation of the generated population is done as follows: Evaluate { (a) Prepare the input data for model analysis (b) Check three initial constraints (c) If model management conditions are not fulfilled then (c-1) Perform the modal analysis. (c-2) Select 15 nearest solutions in the natural frequency space of the archive and train an RBF network by these input data. (c-3) Estimate the fitness function of the newly generated solution by the trained RBF. Else {Perform pushover analysis} (c-4) Perform the modal analysis. Prepare the input data for pushover analysis {Distribution of the forces, maximum required displacement of the roof, modified gravity loads for this specific design} (c-5) Perform pushover analysis by the provided input data. (c-4) Linearly interpolate the obtained P − curve with 10 points in each segment (c-6) Find the target displacements corresponding to 2, 10 and 50% probability of exceedance. (c-7) Compute the life cycle cost factor (CLC factor) accurately. (c-8) Check the final constraint violation in points corresponding to the computed target displacements.

8.5 The Proposed Framework

171

(c-9) Store the obtained data to the archive (natural frequencies, fitness values, the factor representing the amount of constraint violation). (d) End }

8.6 Numerical Results In this section, the structural analysis is conducted by the combination of Matlab and Opensees software. First, the required data for analyzing the structure is provided by Matlab, then by the use of these data, Opensees performs the analysis. As mentioned above, two different models are used in this chapter for analyzing the given structure. In the first part, a linear model of the structure is constructed by the use of the “elasticBeamColumn” element while the joint masses are computed by the Matlab and are given as input data to Opensees. Then a linear static and modal analysis are done by the Opensees. In the second part of the analysis, a nonlinear model of the structure is constructed by the use of fiber elements (nonlinearBeamColumn element) while the other required data, such as the distribution of the lateral loads, are provided by the Matlab. Finally, Opensees performs static nonlinear analysis (pushover). In both of the examples, the cross-sections are W-shape available from the manuals of the American Institute of Steel Construction [13]. In this chapter, the available cross sections are 127 W-shape sections consist of a database of 23 W40, 22 W36, 13 W33, 15 W27, 18 W21, and 36 W14 sections. Each design variable, beam or column or brace sections, is free to select each of these sections.

8.6.1 2D Example In this part, an example considered in [10] is examined to illustrate the efficiency of the proposed algorithm. This structure is a 2D five-bay, ten-story moment-resisting steel frame shown in Fig. 8.3. The FE model of the frame consists of 110 members and in-total 180 degrees of freedom. The columns and the beams of the structure have I-shaped cross-sections and are grouped into 13 sets, each corresponding to an independent design variable. Its geometric characteristics and the group members are also shown in Fig. 8.3. The modulus of elasticity is equal to 210 GPa and the yield stress is σ y = 235 MPa. The constitutive relation is bilinear with a strain hardening ratio of 0.01. The frame is assumed to have rigid connections and fixed supports. The permanent load is taken as DL = 2.66 kN/m2 , and the live load is taken as LL = 1.5 kN/m2 . The gravity loads are contributed from an effective area of 250 m2 . The static pushover analysis is performed to estimate structural performance in accordance with FEMA-273. The seismic inputs could be obtained from the target

172

8 Performance-Based Multi-objective Optimization …

Fig. 8.3 Geometry and member grouping of the ten-story steel frame in example 1

5% damped smooth elastic response spectra for the hazard levels associated with 50-year exceedance probabilities for the Los Angeles area that is available on the ground shaking hazard maps of guidelines. The type of soil profile assumed to be D, and a damping ratio of 2% has been considered for the structure. A first mode-based lateral load distribution is adopted. The lateral loads are applied in 10 displacement-controlled increments until the target displacement of the 2/50 earthquake is reached. By the use of the proposed framework, this optimization problem is solved, and the obtained Pareto front curves are shown in Fig. 8.4 for the life-cycle cost against the initial material weight (total weight, including dead and live loads). Because life cycle cost is defined as a ratio of initial cost, both values are based on kN. In this example, a population of 100 individuals is employed for the optimization process, and the main algorithm performs 500 generations. The total computing time required for solving the multi-objective optimization problem by the proposed framework was approximately 3.2 h on a Pentium 4, 2.83 MHz. It is seen that this algorithm needs 50,000 fitness function evaluations and without the use of employed meta-model, the solution process requires about 208 h.

8.6 Numerical Results

173

Fig. 8.4 Obtained Pareto front in five different runs of the proposed framework for 10 story frame

Because of the stochastic nature of the solution algorithm, this problem is solved for 5 times. To compare the behavior of the different designs of the Pareto front curve, two characteristic designs were selected. These designs are the extreme points that correspond to the single-objective optimum designs where the initial material weight and the life-cycle cost were the objective functions, respectively. The properties of these two designs are listed in Table 8.2. It should be mentioned that in the deigns obtained for this structure, the ratio of the life cycle cost to its initial cost changes between 0.013 and 8.03.

8.6.2 3D Example The second design example is a 20-story braced steel frame structure consisting of 416 joints and 1040 members. Figure 8.5 shows 3D, plan, and facade views of this structure. The considered moment frame is stiffened against lateral forces by bracing members located in the corner bays on each side of the building. The columns in each story are collected in three member groups as corner columns, inner columns, and outer columns, whereas beams are divided into two groups as inner beams and outer beams. The corner columns are grouped together as having the same section over two adjacent stories, as are inner columns, outer columns, inner beams, outer beams, and braces. This results in a total of 60 distinct member groups, i.e. 60 design variables. The gravity loads acting on floor slabs cover dead (DL) and live (LL) loads. All the floors including the roof, are subjected to a design dead load of 2.88 kN/m2 and a design live load of 2.39 kN/m2 . In distributing the gravity loads, it is assumed that all

Member sections

W40x327

W40x149

Design B (Minimum initial cost)

1

Design A (Minimum life cycle cost)

Group No.

W21x62

W36x441

2

W33x130

W36x529

3

W14x22

W40x264

4

W21x48

W33x221

5

W40x593

W14x730

6

Table 8.2 Section properties of two selected design of the ten-story 2D structure 7

W40x593

W14x730

8

W40x431

W14x730

9

W40x324

W14x665

10

W40x593

W36x800

11

W40x593

W36x800

12

W40x593

W36x800

13

W40x593

W36x487

174 8 Performance-Based Multi-objective Optimization …

8.6 Numerical Results

175

Fig. 8.5 Geometry of the 1040-member steel space frame, a 3D view, b facade view, c plan view

loads are distributed uniformly between all beams while the contribution of the inner beams is twice as much as the outer beams. All the other parameters are assumed the same as the previous example. It should be noted that in this example the lateral earthquake load is applied to structure just in the x-direction. Similar to the previous example a static pushover analysis is performed. The seismic input data also derived in a similar manner. Similar to the previous example, the Pareto front curve is obtained by the proposed framework and illustrated in Fig. 8.6 for the life-cycle cost against the initial material weight. In this example, a population of 100 individuals is employed for the optimization process, and the main algorithms perform 500 generations. The total computing time required for solving the multi-objective optimization problem was approximately 35 h on a Pentium 4, 2.83 MHz. It is seen that this algorithm requires 50,000 fitness function evaluations and without the use of the employed meta-model, the solution process would have required about 835 h. In order to demonstrate the ability of the proposed framework, similarly, this problem is solved for 5 times and the results of each run are presented in Table 8.3. Same as the previous example, two characteristic designs were selected. The properties of these two designs are listed in Table 8.3. In this example, in all of the designs obtained for this structure, the ratio of life cycle cost to its initial cost changes between 0.016 and 8.22. It should be mentioned that in these two examples, each design variable is allowed to select one section number of the 127 sections provided in the section

176

8 Performance-Based Multi-objective Optimization …

Fig. 8.6 Obtained Pareto front in five different runs of the proposed framework for 20-story 3Dframe

database. Consequently, any initial information about each design variable can help the optimization algorithm to find better solutions in much less time.

8.7 Discussions and Conclusion In this chapter, a new framework for the multi-objective performance-based optimal design of large-scale steel structures is proposed. Considering the initial and life cycle cost of the structures as two objectives of the optimization problem and obtaining the Pareto front of all the possible optimal designs, provides invaluable economic information which helps investors or insurance companies make the best decisions. This problem is more involved in large-scale structures. It should be noted that here, we have tried to consider all the mentioned constraints included in guidelines and practical matters in a way that the results can be useful for all practicing engineers in real-life projects. In the proposed framework, a modified NSGA-II is employed for multi-objective optimization. Additionally, two simple numerical procedures are utilized for computing the target displacement and reducing the number of points in figuring out of the pushover curve that leads to an enormous reduction in the required computational time for each pushover analysis. In the end, a specific meta-model is used for predicting the fitness value and reducing the required number of pushover analysis. The proposed methodology is based on the FEMA provisions. This methodology has been applied for the design of a ten-story steel moment-resisting frame and the other twenty-story 3D steel braced frame structures. It is demonstrated that by the

Design A (Minimum life cycle cost)

W36X529 W36X800 57 W36X302

Section

Group No.

Section

41

Group No. 49

W40X593

Section

Group No.

33

Group No.

Section

W36X800

Section

17

Group No. W40X327

W27X161

Section

25

9

Group No.

Group No.

W36X302

Section

Section

1

Group No.

W40X277

58

W40X593

50

W40X593

42

W40X593

34

W36X800

26

W27X161

18

W40X593

10

W40X593

2

W40X397

59

W40X249

51

W40X431

43

W36X800

35

W40X593

27

W40X593

19

W40X593

11

W36X652

3

Table 8.3 Section properties of two selected design of the twenty-story 3D structure 4

W36X247

60

W36X135

52

W40X593

44

W40X593

36

W36X529

28

W27X94

20

W40X593

12

W40X593

5

W33X387

53

W40X593

45

W40X593

37

W27X307

29

W40X593

21

W40X593

13

W40X235

6

W36X441

54

W36X800

46

W36X800

38

W40X593

30

W36X800

22

W40X372

14

W27X161

7

W27X217

55

W40X593

47

W40X503

39

W36X529

31

W36X800

23

W36X231

15

W40X593

8

(continued)

W36X652

56

W36X529

48

W40X593

40

W36X529

32

W40X593

24

W40X593

16

W14X34

8.7 Discussions and Conclusion 177

Design B (Minimum initial cost)

Table 8.3 (continued)

W40X397 W40X324 57 W14X730

Section

Group No.

Section

41

Group No. 49

W40X593

Section

Group No.

33

Group No.

Section

W36X800

Section

17

Group No. W14X53

W14X132

Section

25

9

Group No.

Group No.

W14x193

Section

Section

1

Group No.

W14X550

58

W40X278

50

W14X82

42

W40X503

34

W40X593

26

W14X22

18

W14X22

10

W33x118

2

W27X102

59

W14X22

51

W40X593

43

W40X593

35

W40X593

27

W14X74

19

W14X342

11

W14X34

3

W21X132

60

W21X201

52

W40X392

44

W40X593

36

W36X800

28

W14X22

20

W33X318

12

W14X22

4

5

W40X503

53

W14X82

45

W40X503

37

W40X593

29

W40X593

21

W14X22

13

W40X149

6

W40X215

54

W40X593

46

W40X397

38

W40X593

30

W40X593

22

W14X22

14

W14X22

7

W14X159

55

W40X331

47

W40X593

39

W36X800

31

W36X441

23

W14X22

15

W27X235

8

W33X221

56

W14X82

48

W40X397

40

W40X593

32

W40X593

24

W14X22

16

W14X22

178 8 Performance-Based Multi-objective Optimization …

8.7 Discussions and Conclusion

179

use of the proposed framework solving the proposed problem can be accomplished in too large structures within an acceptable amount of computational time.

References 1. Kaveh, A., Laknejadi, K., & Alinejad, B. (2012). Performance-based multi-objective optimization of large steel structures. Acta Mechanica, 223(2), 355–369. https://doi.org/10.1007/ s00707-011-0564-1. 2. Wen, Y. K., & Kang, Y. J. (2001). Minimum building life-cycle cost design criteria. I: Methodology. Journal of Structural Engineering, 127(3), 330–337. https://doi.org/10.1061/(asce)07339445(2001)127:3(330). 3. Wen, Y. K., & Shinozuka, M. (1998). Cost-effectiveness in active structural control. Engineering Structures, 20(3), 216–221. https://doi.org/10.1016/S0141-0296(97)00080-1. 4. Frangopol, D. M., Lin, K.-Y., & Estes, A. C. (1997). Life-cycle cost design of deteriorating structures. Journal of Structural Engineering, 123(10), 1390–1401. https://doi.org/10.1061/ (ASCE)0733-9445(1997)123:10(1390). 5. Bucher, C., & Frangopol, D. M. (2006). Optimization of lifetime maintenance strategies for deteriorating structures considering probabilities of violating safety, condition, and cost thresholds. Probabilistic Engineering Mechanics, 21(1), 1–8. https://doi.org/10.1016/j.probengmech. 2005.06.002. 6. Liu, M., Burns, S. A., & Wen, Y. K. (2003). Optimal seismic design of steel frame buildings based on life cycle cost considerations. Earthquake Engineering and Structural Dynamics, 32(9), 1313–1332. https://doi.org/10.1002/eqe.273. 7. Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182–197. https://doi.org/10.1109/4235.996017. 8. Li, H., & Zhang, Q. (2009). Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Transactions on Evolutionary Computation, 13(2), 284–302. https://doi.org/10.1109/TEVC.2008.925798. 9. Council, B. S. S. (1997). NEHRP guidelines for the seismic rehabilitation of buildings. FEMA273, Federal Emergency Management Agency, Washington, DC, 2-12. 10. Fragiadakis, M., Lagaros, N. D., & Papadrakakis, M. (2006). Performance-based multiobjective optimum design of steel structures considering life-cycle cost. Structural and Multidisciplinary Optimization, 32(1), 1. https://doi.org/10.1007/s00158-006-0009-y. 11. Lagaros, N. D., & Papadrakakis, M. (2007). Robust seismic design optimization of steel structures. Structural and Multidisciplinary Optimization, 33(6), 457–469. https://doi.org/10.1007/ s00158-006-0047-5. 12. Kok Sung, W., & Ray, T. (2004). Performance of kriging and cokriging based surrogate models within the unified framework for surrogate assisted optimization. In: Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753) (Vol. 1572, , pp. 1577– 1585), 19–23 June 2004. 13. American Institute of Steel Construction. (2002). Seismic provisions for structural steel buildings. American Institute of Steel Construction.

Chapter 9

Optimal Seismic Design of Steel Plate Shear Walls Using CBO and ECBO Algorithms

9.1 Introduction In this chapter, three well-known metaheuristic algorithms including Colliding Bodies Optimization (CBO), Enhanced Colliding Bodies Optimization (ECBO), and Particle Swarm Optimization (PSO) are employed for size and performance optimization of steel plate shear wall (SPSW) systems [1]. Recently, the number of high-rise residential and commercial buildings is significantly increased and engineers tend to use more tall and slender structures. In these buildings, the effect of lateral loads such as wind loads and seismic forces is substantial so that special attention is required for their design. There are different lateral load resisting systems, and the use of steel plate shear walls is one of the efficient systems with high energy dissipation. An SPSW is a lateral load resisting system that contains an infill plate attached to the surrounding beams and columns. This system acts as a cantilever wall in the total height of the building. They show high initial stiffness in a very ductile manner and absorb a considerable amount of energy when subjected to high seismic loads. The steel shear walls not only can be employed in the design of new buildings but also can be utilized for the retrofitting of existing buildings. In 1980, engineers were utilizing SPSWs but they were reinforcing them with several stiffeners in both directions for preventing buckling and using their yield capacity. This approach had economic problems because of the cost of materials and construction. Nevertheless, quite a few experiments proved that unstiffened thin steel plate shear walls have high ductility and strength after the buckling in compression. SPSW’s buckling strength in pressure depends on the slenderness of the plate— the ratio of the length and width to thickness. These ratios are usually high for conventional buildings, and moreover, the erection of these walls is not quite flat due to manufacturing errors. Consequently, the buckling strength of the SPSWs in compression is very low. The generated principal compressive stresses are more than the compressive strength in the plate when seismic loads are exerted to the © Springer Nature Switzerland AG 2020 A. Kaveh and A. Dadras Eslamlou, Metaheuristic Optimization Algorithms in Civil Engineering: New Applications, Studies in Computational Intelligence 900, https://doi.org/10.1007/978-3-030-45473-9_9

181

182

9 Optimal Seismic Design of Steel Plate Shear Walls …

walls; hence, they buckle in compression directions and form fold lines as well as tensile stresses perpendicular to the directions. The lateral loads are transferred by these principal diagonal tension stresses through the web plate and this behavior is defined as the post-buckling tension field action. For the first time, post-buckling properties such as stiffness and strength of the steel plate shear walls were recognized by Thorburn et al. [2] and were confirmed by experimental analyses of Timler and Kulak [3]. Gholizadeh and Shahrezai [4] employed the bat algorithm to optimize the placement of SPSWs and minimized the weight of 2D frames. The beam and column sections, as well as web plate thicknesses, were defined as design variables in 2D frames. In this research, the orthotropic membrane model was used for simulating steel plate shear walls and linear static analysis was used in the design process. Sadeghi Eshkevari et al. [5] proposed a new form of couple elements named as coupling panels in SPSWs and investigated their performance on a 9-story steel frame. In order to optimize the performance of the steel frame and minimize the story drifts differences, a non-metaheuristic algorithm was employed. In this study, the thicknesses of web plates were defined as design variables. Furthermore, the nonlinear static analysis, modal pushover analysis, and time history analysis were three types of analysis employed in their research. Performance-based optimization on steel moment-resisting frames was also investigated by Tehranizadeh and Moshref [6], and frequency-based design optimization of plates was performed by Armand [7]. The remainder of the chapter is organized as follows. Section 9.2 provides some of the modeling methods for steel plate shear walls. In Sect. 9.3, a brief compilation of the design requirements for SPW and SPSW systems subjected to low and high seismic loads is presented in order to use as design constraints of Sect. 9.5. Section 9.4 includes a brief explanation of the CBO and ECBO algorithms. The optimization examples including low and high seismic design, a comparison between moment frame and SPW in optimal form, performance optimization of the SPSW, base shear sensitivity analysis on optimal high seismic design of SPSW, and size optimization of the 6- to 12-story SPSWs are studied in Sect. 9.6. Finally, the last section discusses the results and concludes the chapter.

9.2 Different Techniques for Simulating Steel Plate Shear Walls In order to determine the stiffness and strength of SPSW systems, there are various techniques. In this chapter, some of these models for different purposes like specifying forces in the elements and calculating the lateral displacement of the system are described.

9.2 Different Techniques for Simulating Steel Plate Shear Walls

183

Fig. 9.1 Schematic of a typical strip model

9.2.1 Strip Models In this method, the infill plate in the panel is replaced with a set of parallel steel bar members which can resist only tension stresses and are inclined to the diagonal direction (Fig. 9.1). It is recommended that a minimum of 10 strips be employed for simulating the effect of resulted forces on elements of the frame with good accuracy. The horizontal distance of the strips on the beam for n strips is calculated as x = 1/n[L + h . tan(α)]

(9.1)

where L is the width of the panel and h is its height. The cross-sectional area of the equivalent strip is calculated as follows As =

[L . cos(α) + h . sin(α)]tw n

(9.2)

In this chapter, the strip model is employed for modeling SPSW in low seismic load cases and for performance-based optimization subject to high seismic loads.

9.2.2 Pratt Truss Model Thorburn et al. [2] proposed the Pratt Truss Model (equivalent brace model) in which the steel plate at each story is replaced by only one diagonal brace—which resists tension stresses—to simplify the process of modeling a steel plate shear wall for preliminary design. The area of the brace is given by:

184

9 Optimal Seismic Design of Steel Plate Shear Walls …

Fig. 9.2 Schematic of a truss model

A=

tw . L . sin2 (2α) 2 sin(ϕ) . sin(2ϕ)

(9.3)

where ϕ is the angle between the brace and the column.

9.2.3 Truss Model The aim of this approach is to approximate the stiffness of the steel plate shear wall employing a truss element as depicted in Fig. 9.2. The horizontal member is assumed to be rigid, and the area of vertical and the diagonal members are calculated as follows Aver = Ad =

Im 0.5(L + dc )2    L 3d I 2 βm

2.6h(L + dc )2

(9.4) (9.5)

where Im , βm , and other variables are defined in Topkaya and Atasoy [8].

9.2.4 Partial Strip Model This model is inspired by the current strip model with one difference that tension strips are utilized in the medial area near the diagonal line of the SPSW panel. This technique is based on the idea of the effective length of the web plate in tension. Because the corner area of the panel is in the compression range and its strength and stiffness are not considered [9].

9.2 Different Techniques for Simulating Steel Plate Shear Walls

185

Fig. 9.3 Schematic of a multi-angle model

9.2.5 Multi-angle Model In each panel of this model which was developed by Rezai et al. [9], 5 multi-angle strips are utilized instead of infill plates. Figure 9.3 illustrates the configuration of the strips in the panel. This configuration is selected to simulate the behavior of the steel plate shear walls with good accuracy. Because the angle of tension field in the panel corner is near vertical and in the middle of the panel is near horizontal. The proposed method employed the effective width concept to explain incomplete tension-field action and is able to approximate the initial stiffness of the SPSW system. However, the ultimate strength of the system is not simulated in a very precise manner.

9.2.6 Modified Strip Model In order to approximate the real behavior of the SPSW system, in this model the compression limit of the post-buckling strength is not considered. The truss elements that passed the yield limit and reached their ultimate stress must be eliminated during the analysis. Considering the design codes which recommend elimination of the compression strength of the steel plate shear wall, this model is not recommended for design purposes.

186

9 Optimal Seismic Design of Steel Plate Shear Walls …

Fig. 9.4 Schematic of an orthotropic membrane model

9.2.7 Cyclic Strip Model In this model, the strips in both directions are utilized to predict the hysteretic cyclic behavior of the SPSW. In each cycle just tension strips resist and the compression strips do not resist. This model is very useful for analyses that need the cyclic behavior of the system like dynamic analyses.

9.2.8 Orthotropic Membrane Model Orthotropic elements can be used to model the steel plate shear wall. These elements distinguish between tension and compression resistance of the steel plate. To this end, the local axis of the membrane elements must be set to match the angle of tension stress as depicted in Fig. 9.4. In this direction the complete modulus of elasticity is assigned; however, the zero stiffness is assigned to the perpendicular direction (compressive strength). It is recommended that the in-plane shear stiffness of the membrane elements be assumed as zero and at least four divisions in each direction are employed to achieve good accuracy.

9.3 Design Requirements The steel structures, depending on the construction site, are designed for low seismic or high seismic demands. The differences in these modes are on the ductility of the steel plates. The low seismic design requirements are for limited ductility of the web

9.3 Design Requirements

187

plate and the high seismic design requirements are for high ductility of the web plate with special provisions of AISC 341. In the low seismic design mode, the columns are defined as vertical boundary elements and beams defined as horizontal boundary elements. The boundary elements should remain elastic, while the formation of the plastic hinges in two ends of the horizontal boundary elements is allowed for high seismic design mode.

9.3.1 Requirements for Low Seismic Design In this section, design constraints for a low seismic mode of structures that are in sites with a response modification factor equal or less than 3 are stated. The high seismic design should also fulfill these requirements. As AISC 360 does not include the requirements for SPSWs, some general constraints from AISC 341 are used for low seismic design. There are two analysis approaches in low seismic states: using internal forces directly resulted from analysis, or utilizing the forces calculated with the assumption of the uniform distribution of the average tension stresses in the steel plates. In this chapter, the former approach is employed. Design constraints are as the following limitations: a. The allowable shear strength is considered as: φVn = (φ)0.42Fy L c f tw sin(2α)

(9.6)

The angle of the yielding web is defined as: tan4 α =

1 + t2wA.Lc   3 1 + tw . h A1b + 360Ih c . L

(9.7)

where Fy is the infill panel yield stress, L c f is the clear distance between vertical boundary elements’ flanges, tw is the thickness of the infill plate, φ is the resistance factor (φ = 0.9), L is the bay width, h is the story height, Ic is the moment of inertia of the vertical boundary element, and Ac and Ab are the cross-sectional area of the vertical and horizontal boundary elements, respectively. b. Stiffness constraint of the vertical boundary elements is limited as: Ic ≥ 0.00307

tw h 4 L

(9.8)

c. Stiffness constraint of the horizontal boundary elements is limited as: Ib ≥ 0.003

(tw )L 4 h

(9.9)

188

9 Optimal Seismic Design of Steel Plate Shear Walls …

where tw is the difference between the web plate thicknesses above and below the beam. d. The strength constraints of the boundary elements are defined as: Pr Mr Pr < 0.2 : + ≤ 1.0 Pc 2Pc Mc   Pr Pr 8 Mr if ≥ 0.2 : + ≤ 1.0 Pc Pc 9 Mc if

(9.10) (9.11)

where Pr and Pc are the required and available axial—compressive and tensile— strength, respectively. Mr and Mc are the required and available flexural strength, respectively. The compressive and tensile strengths are defined based on nominal strengths as Eqs. (9.12) and (9.13). for compressive elements: Pc = φc Pn(compr ession) , φc = 0.9 for tensile elements: Pc = φt Pn(tension) , φt = 0.9

(9.12) (9.13)

e. The inter-story displacement is constrained as:   Cd ≤ 0.02 i = δe(i) − δe(i−1) Ie h si

(9.14)

where δe(i) is the deflection at ith story, Cd is the deflection amplification factor, Ie is the importance factor, and h si is the story height below story i.

9.3.2 Requirements for High Seismic Design In this section, additional design constraints for high seismic mode are presented. These constraints are enforced for the structures that are in the sites with a response modification factor greater than 3. Due to the high ductility of this system, a combined plastic and linear analysis are recommended to determine the internal forces in the boundary elements. Since the web plates are not designed to carry the gravity loads, it is preferred that the failure in web plates happen prior to failure in boundary elements. As shown in Fig. 9.5, two mechanisms are considered for the collapse of the SPSW system: first, the complete uniform yielding mechanism in the height of the structure and second, the local mechanism in one story (i.e. soft story). To ensure that the former mechanism is dominated, it is essential to avoid using extra thickness in the web plate of panels. Because if unnecessary thicknesses are used in the web

9.3 Design Requirements

189

Fig. 9.5 Schematics of SPSW collapse mechanisms

plates, inappropriate relative displacement happens which leads to undesirable soft story mechanisms. Also, this can be a compelling reason for designing this system in an optimal form. In soft story mechanism, plastic hinges form in two ends of the columns and just in one story. It is an undesirable collapse mechanism since only one story reaches its ultimate capacity. On the other hand, using the capacity of all stories in uniform

190

9 Optimal Seismic Design of Steel Plate Shear Walls …

yielding mechanism due to the formation of plastic hinges at two ends of beams makes it a desirable collapse mechanism. Some design requirements in this section such as Strong-Column Weak-Beam constraint in connections of beams to columns lead to the formation of this suitable collapse mechanism. These design constraints are formulated in the following. a. The strong-column weak-beam constraint is fulfilled with the following limitations: ∗ M pc > 1.0 (9.15) M pb   M ∗pc = Fy − Pu /Py Z (9.16) M pb = M pr + Vu sh

(9.17)

sh = 1/2(db + dc )

(9.18)

M pr = 1.1R y Fy Z R B S

(9.19)

∗ columns’ plastic moment strengths at a connection where M pc is the sum of (reduced due to axial force), M pb is the sum of beam plastic moment strengths at a connection, Pu is the axial force in the columns, Py is equal to A g Fy , db and dc are the beam and column depths, Vu is the shear force in the beam at the location of the formed hinge, M pr is the beam plastic moment strength in the absence of axial force, R y is the ratio of the expected yield stress to the specified minimum yield stress (Fy ) and Z R B S is the plastic section modulus of the reduced beam section (=2/3Z x ). All these forces and dimensions are illustrated in Fig. 9.6. b. Compactness constraint for wings of the W-shaped sections is defined as

bf E ≤ 0.3 2t f Fy

(9.20)

and for webs of the W-shaped sections: 1 h f or : Ca ≤ : ≤ 3.14 8 tw



E [1 − 1.54Ca ] Fy

E 1 h ≤ 1.12 f or : Ca > : [2.23 − Ca ] 8 tw Fy

(9.21)

(9.22)

9.3 Design Requirements

191

Fig. 9.6 Forces at column centerline resulted from beam plastic hinge

Ca =

Pu φb Py

(9.23)

where Ca is the axial force ratio, and E is the modulus of elasticity of steel elements.

9.4 CBO and ECBO Algorithms The colliding bodies optimization (CBO) algorithm introduced by Kaveh and Mahdavi [10] is based on the idea of a collision between two objects in one dimension. In this method, CBs (colliding objects) collide with each other until they reach an optimum location. The momentum physics is used to calculate the corresponding quantities in the CBO algorithm. In this section, first, the steps of the CBO algorithm are described; then, an enhanced version of CBO is elaborated which was proposed by Kaveh and Ilchi Ghazaan [11].

9.4.1 Colliding Bodies Optimization (CBO) In order to have a good insight into the algorithm, physical laws used in the process of the algorithm are described here.

192

9.4.1.1

9 Optimal Seismic Design of Steel Plate Shear Walls …

Collision Laws

When two objects collide together, the velocity of the objects before and after the collision is derived based on the laws of momentum and energy. In an isolated system, the total kinetic energy and the total momentum of the objects are conserved; this can be expressed as the following equations: m 1 v1 + m 2 v2 = m 1 v1 + m 2 v2

(9.24)

1 1 1 1 m 1 v12 + m 2 v22 = m 1 v1 2 + m 2 v2 2 + Q 2 2 2 2

(9.25)

where v1 and v2 are the velocity of the objects before the collision, v1 and v2 are the velocity of the objects after the collision, m 1 and m 2 are the mass of pair colliding objects, and Q is the loss of energy as a consequence of the collision. Finally, the velocity of the objects after the collision can be obtained using Eqs. (9.26) and (9.27): v1 =

(m 1 − εm 2 )v1 + (m 2 + εm 2 )v2 m1 + m2

(9.26)

v2 =

(m 2 − εm 1 )v2 + (m 1 + εm 1 )v1 m1 + m2

(9.27)

where ε is the coefficient of restitution which can be defined as follows:    v − v   2 1 ε= |v2 − v1 |

(9.28)

Two types of collision can be considered: 1. A perfectly elastic collision: Q = 0 & ε = 1 2. An inelastic collision: Q = 0 & ε ≤ 1.

9.4.1.2

CBO Algorithm

In CBO, each CB is a solution vector that includes a number of variables. Objects are divided into two equal groups named as stationary and moving bodies. Improvement of the position of moving bodies and alteration of stationary body positions are two main purposes for the classification of the objects. The moving bodies hit to stationary bodies and the positions are updated. The main steps of CBO can be described as follows: Level 1: Initialization Step 1: A random initialization for CBs first positions is employed in the search space:

9.4 CBO and ECBO Algorithms

193

xi0 = xmin + rand(xmax − xmin ), i = 1, 2, . . . , n

(9.29)

where xi0 is the initial value of the ith CB vector, xmin and xmax are the allowable boundary value of the variable vectors, rand is a decimal number in the interval [0,1] and n is the total population of the CBs. Level 2: Searching Step 1: Evaluation of the value of the objective function for each CB and sorting them in ascending order. Step 2: Defining groups of the stationary (good agents) and moving objects like a pair of colliding bodies in the virtual impact. Stationary CBs are the ones in the lower half of CBs after the sorting, and similarly moving CBs are in the upper half. Step 3: Calculating the values of the mass and velocity of the CBs before the collision by Eqs. (9.30)–(9.32): m k = n

1 f it (k)

1 i=1 f it (i)

, k = 1, 2, . . . , n

(9.30)

The velocity of stationary bodies before the collision is zero as: vi = 0, i = 1, . . . ,

n 2

(9.31)

The velocity of moving bodies before the collision is calculated according to Eq. (9.32): vi = xi − xi− n2 , i =

n + 1, . . . , n 2

(9.32)

where xi and xi− n2 are the position of the ith CB and its pair in the previous group, respectively. Step 4: Calculating the velocity of CBs after the collision: The velocity of stationary bodies after the collision is calculated as: vi

=

  m i+ n2 + εm i+ n2 vi+ n2 m i + m i+ n2

, i = 1, . . . ,

n 2

(9.33)

where vi+ n2 and vi are the velocity of the ith stationary bodies before and after the collision, m i and m i+ n2 are the mass of the ith CB and its pair in the next group, respectively. The velocity of moving bodies after the collision is: vi

=

  m i + εm i− n2 vi mi + m

i− n2

, i=

n + 1, . . . , n 2

(9.34)

194

9 Optimal Seismic Design of Steel Plate Shear Walls …

where vi and vi are the velocity of the ith moving CB before and after the collision, m i and m i− n2 are the mass of the ith CB and its pair in the previous group, respectively. Also, ε can be defined as: ε =1−

iter itermax

(9.35)

where itermax is the maximum number of iteration and iter is the number of the current iteration. Step 5: Updating the CBs’ positions: The position of stationary bodies is updated as: xinew = xi + rand ◦ vi , i = 1, . . . ,

n 2

(9.36)

where xinew and xi are the new position and old position of the ith stationary bodies, respectively. The position of moving bodies is updated as: xinew = xi− n2 + rand ◦ vi , i =

n + 1, . . . , n 2

(9.37)

where xinew and xi− n2 are the new position of the ith moving bodies and old position of its pair in the previous group, respectively. Level 3: Continue previous levels until a terminating criterion is satisfied.

9.4.2 Enhanced Colliding Bodies Optimization In the ECBO algorithm, a number of best CBs from previous iterations are replaced by the current worst CBs using a parameter named colliding memory (CM); consequently, the convergence rate increases. In addition, another parameter named pro is utilized in order to escape the objects from local optima in a probabilistic manner. The main steps of the ECBO can be described as follows: Level 1: Initialization Step 1: A random initialization for CBs first positions is employed in the search space [using Eq. (9.29)]. Level 2: Searching Step 1: Evaluating the value of the objective function for each CB in order to calculate the value of its mass [using Eq. (9.30)]. Step 2: Using the colliding memory (CM) to substitute a number of old best CBs by the current worst CBs. Step 3: Sorting the objective function in ascending order and defining groups of the stationary (good agents) and moving objects as pairs of colliding bodies.

9.4 CBO and ECBO Algorithms

195

Step 4: Calculate the velocity of the colliding bodies (moving and stationary objects) before and after the collision [using Eqs. (9.31)–(9.34)]. Step 5: Update the CBs positions [using Eqs. (9.36) and (9.37)]. Step 6: Use the pro to escape from local optimal answers. To this end, a number within interval (0, 1) is chosen randomly and is compared with the value of pro, which is a predefined number between 0 and 1; if this random number is lower than pro, one random variable of the colliding body is selected and is changed to an available random value in the search space. This action is performed for all CBs. Level 3: Continue previous level steps until a terminating criterion is satisfied.

9.5 Structural Optimization 9.5.1 Optimization Formulation In this work, the objective function is defined as the minimization of the weight and standard deviation of the story drifts in the SPSW system while stiffness, strength, and displacement constraints must be satisfied. The optimization problem is formulated in the following:

Find {X } = x1 , x2 , . . . , xng To minimize

W ({X }) =

nm 

ρi Vi

(9.38)

i=1

  N 1   2 dri f ti − dri f t Std({X }) =  N i=1  g j ({X }) ≤ 0, j = 1, 2, . . . , nc Subjected to : ximin ≤ xi ≤ ximax

(9.39)

where {X } is the vector of design variables, ng is the number of design variables, W ({X }) is the weight of structure, nm is the number of elements in the structure, ρi and Vi are the material density and volume of the ith member, respectively; Std({X }) denotes the standard deviation of the story drifts, N is the number of stories, dri f ti and dri f t are the story drift in the ith story and the average value of story drifts, respectively; ximin and ximax are the lower bound and upper bound of the design variables, g j ({X }) represents design constraints, and nc is the number of design constraints.

196

9 Optimal Seismic Design of Steel Plate Shear Walls …

The penalty approach is employed to handle the constraints of numerical examples. The penalty function multiplied by objective function constitutes a new objective function that is used in optimization algorithms. The penalty function is formulated as follows: Penalt y({X }) = (1 + ε1 . υ)

ε2

υ=

nc 



max 0, g j ({X })

(9.40)

j=1

where υ is the total violations of the constraints, ε1 is a constant considered for tuning the exploration rate of the search space which is set to one, and the ε2 constant is considered for setting exploitation rate of the search space which is changed from 1.5 to 6 during the optimization process.

9.6 Numerical Examples 9.6.1 Low Seismic Design Example The optimal design of a 2D frame braced by steel plate shear walls is shown in Fig. 9.7. This building is in the zone of low seismicity (with response modification coefficient equal to 3) which is considered to examine the algorithms and to investigate the optimal configuration of the SPSW system. This example is proposed in

Fig. 9.7 Schematic of a typical floor plan and SPW elevation

9.6 Numerical Examples Table 9.1 Earthquake load acting on the 9-story SPW

197 Forces and Shears in each SPW Level

Frame force (kips)

Frame shear (kips)

Roof

105

105

9th floor

81.4

186

8th floor

70.9

257

7th floor

60.4

317

6th floor

50.0

367

5th floor

39.6

407

4th floor

29.7

437

3rd floor

20.1

457

2nd floor

11.1

468

AISC Design Guide No. 20 (Steel Plate Shear Walls) and is used in this study to have an authentic numerical example in terms of structural geometry, loading, as well as design constraints. Total weight of the building is 20,700 kips and is assumed to be located in Chicago; ASTM A36 (F y = 36 ksi, F u = 58 ksi) and ASTM A992 (F y = 50 ksi, F u = 65 ksi) are used for web plate and boundary elements material, respectively; the steel has a modulus of elasticity equal to E = 29,000 ksi. The W-shaped sections introduced in AISC instructions are the sections bank for frame members, and 13 thicknesses (0.0625, 0.0673, 0.0747, 0.1046, 0.125, 0.1345, 0.1875, 0.250, 0.3125, 0.375, 0.4375, 0.500, and 0.625 in.) are used for web plate thicknesses in frame panels. Beam-to-column connections are rigid according to AISC 341 requirements for SPW systems, and a strut beam is embedded in the middle of the first story panel. A second-order P- analysis is employed in order to involve the secondary effects of axial loads in frame elements. SAP 2000 software is utilized for simulation and analysis. Also, the optimization process is programmed in MATLAB. Base shear in this building is distributed vertically based on Eqs. (9.41) and (9.42) and the exponent k is 1.12. Table 9.1 presents the calculated earthquake loads in all levels of the frame for each SPSW. Fx = Cvx V

(9.41)

wx h k Cvx = n x k i=1 wi h i

(9.42)

Low seismic optimum design of a 9-story SPW and a comparison with optimum design of a 9-story moment frame in the same condition, are investigated. Columns in each story are categorized in one group, while such a category is not considered for beams. Because their moment inertia (stiffness) constraint may require heavy sections in some stories, which are not required in the other ones.

198

9 Optimal Seismic Design of Steel Plate Shear Walls …

In CBO, ECBO, and PSO, the population of n = 30 agents is used for the design problems. In ECBO, the size of colliding memory is taken as 5, and the pro parameter increases linearly from 0.3 to 0.5 during the optimization process. The first column in Table 9.2 belongs to the AISC Design Guide example and the second column is the optimal design found by ECBO that is 38% lighter than the AISC Design example. In the second design, like Design Guide example only W14 sections are used for vertical boundary elements (VBEs); additionally, stiffness constraint for horizontal boundary elements (HBEs) is eliminated for having more categorized beam sections. On the other hand, there is no extra consideration in the rest of the columns in Table 9.2. Therefore, all W-shaped sections for VBEs and stiffness constraints for HBEs are considered. The third, fourth, and fifth columns in Table 9.2 are for optimal design obtained by ECBO, CBO, and PSO, respectively. The best weight is for the ECBO algorithm that is 43% lighter than the AISC Design Guide example that contains the most use of steel plates in comparison with other methods. As is seen in Fig. 9.8, both the best answer and the best average answer are obtained by ECBO with the fastest convergence rate. This shows the high ability of the ECBO algorithm for finding the optimum design of such a complex structure with complicated constraints. It is worth mentioning that the best convergence curve obtained by ECBO is near to its average answer, which elucidates the reliable performance of this algorithm for structural problems. It can be seen that the PSO is not appropriate for this complex structural problem since it is severely trapped in a local optimal solution. The force, stiffness, and displacement constraints for the optimum design obtained by ECBO are transformed into the ratio of existing values to allowable value. Figure 9.9 depicts the correspondence between the strength constraint for combined compression, flexure and stiffness constraints for VBEs and HBEs, respectively. It is shown that strength is the dominant constraint for columns; however, stiffness requirement for beams in 3rd, 5th, 6th, and 9th stories exceeds the required strength. Finally, Fig. 9.10 illustrates the shear force and nominal shear strength in web plates which are very close together. This implies that the structure reaches almost its full capacity. Drift to allowable value ratio of the optimum design is demonstrated in Fig. 9.11. Comparison between moment frame and SPSW system The last column in Table 9.2 demonstrates the optimal design of the moment frame similar to the 9-story SPW. The optimization process, in this case, is performed only by the ECBO algorithm. Although all the available capacity of the structure is nearly utilized in the optimal moment frame in Fig. 9.12, the weight of its optimum solution is 25% heavier than the optimum structure of the SPW system. This indicates the good performance of the steel plate shear walls in seismic loads due to the high ability of the steel plates in energy dissipation.

9.6 Numerical Examples

199

Table 9.2 Comparative results of the 9-story SPW and result of the moment frame Level

Low seismic SPW

Moment frame Optimal design

AISC design example

ECBO*

ECBO

CBO

PSO

ECBO

Roof

W27X94

W12X26

W30X99

W30X99

W36X182

W24X55

9th floor

W24X84

W6X9

W14X22

W14X22

W30X99

W27X84

8th floor

W24X84

W12X26

W27X84

W30X90

W27X84

W30X116

7th floor

W24X84

W16X26

W24X55

W24X55

W33X141

W36X135

6th floor

W24X84

W12X26

W18X40

W21X44

W44X248

W40X149

5th floor

W24X84

W12X26

W18X35

W21X44

W27X84

W40X149

4th floor

W24X84

W12X26

W30X90

W30X90

W33X130

W44X198

3rd floor

W24X84

W12X26

W16X31

W16X26

W44X224

W44X198

2nd floor

W24X84

W30X99

W16X26

W30X99

W44X224

W24X76

(Strut)

1st floor

W10X45

W14X22

W14X22

W14X22

W14X22

W40X167

Column sections

9th floor

W14X132

W14X48

W18X40

W21X50

W30X90

W21X50

8th floor

W14X132

W14X53

W16X45

W21X57

W27X94

W27X84

7th floor

W14X233

W14X74

W24X68

W24X68

W27X114

W30X108

6th floor

W14X233

W14X99

W18X86

W30X90

W27X114

W30X132

5th floor

W14X233

W14X99

W27X114

W24X104

W36X160

W36X160

4th floor

W14X233

W14X132

W14X120

W21X147

W24X207

W44X198

3rd floor

W14X370

W14X159

W27X146

W27X161

W36X210

W40X215

2nd floor

W14X370

W14X176

W27X178

W40X199

W44X285

W36X230

1st floor

W14X370

W14X370

W36X260

W40X297

W40X297

W40X328

Beam sections

(continued)

200

9 Optimal Seismic Design of Steel Plate Shear Walls …

Table 9.2 (continued) Level

Low seismic SPW

Moment frame Optimal design

AISC design example

ECBO*

ECBO

CBO

PSO

9th floor

0.0625

0.0625

0.0625

0.0625

0.0747

8th floor

0.0625

0.0673

0.0625

0.0625

0.1345

7th floor

0.1046

0.1046

0.1046

0.1046

0.1345

6th floor

0.1046

0.125

0.125

0.125

0.1875

5th floor

0.125

0.125

0.1345

0.1345

0.25

4th floor

0.1345

0.1875

0.1345

0.1345

0.25

3rd floor

0.1875

0.1875

0.1872

0.1875

0.25

2nd floor

0.1875

0.1875

0.1875

0.1875

0.375

1st floor

0.1875

0.3125

0.1875

0.25

0.625

88,208.6

54,403.1

49,661.1

54,608.5

86,544.1

62,691.7

Ave. weight (lb)

55,883.4

52,408.8

60,766.1

166,591.7

64,774.4

No. of analyses

19,800

7200

7260

510

8490

tw (in)

Weight (lb)

ECBO

*Just W14 sections are used for columns (does not include required HBE stiffness) Wall/column (%)

37.50

34.60

31.80

41.20

Wall/structure (%)

24.30

21.00

19.50

20.90

9.6.2 High Seismic Design Example In this section, we are going to optimize a high seismic design of a 9-story special steel plate shear wall for a minimum weight and obtain a uniform stance for the relative displacement of the stories. The plan of a 2D frame inhibited by special steel plate shear walls is presented in Fig. 9.13. This building is located in a zone of high seismicity in San Francisco (with a response modification coefficient equal to 7). The ductile detail requirements are considered to verify the mentioned algorithms and to investigate the optimal form of the SPSW system.

9.6 Numerical Examples

201

Fig. 9.8 Convergence curves of the 9-story SPW; best and average CBO, ECBO and PSO

Fig. 9.9 Strength and stiffness constraints for the 9-story SPW

Since in high ductility design all web plates need to be strictly braced, unlike low seismic design examples, there is no strut beam in the first story panel, but a beam on the foundation is embedded for inhibiting the first story web plate. All of the optimization and analysis processes, based on the Combined Plastic and Linear Analysis Approach, are coded in MATLAB platform. The laterally unbraced length for each beam is taken as the entire length of the beam. One-third of the span length is used for in-plane and out-of-plane slenderness calculations. Also, the out-of-plane

202

9 Optimal Seismic Design of Steel Plate Shear Walls …

Fig. 9.10 Strength constraint for the 9-story SPW

Fig. 9.11 Displacement constraint for the 9-story SPW

Fig. 9.12 Strength and displacement constraints for the moment frame

9.6 Numerical Examples

203

Fig. 9.13 Schematic of a typical floor plan and SPSW elevation

effective length factors of the members are equal to 1, and the in-plane factor is conservatively specified as a unit for a frame with side-sway inhibition. All columns are considered non-braced in their length. In order to find the best configuration of the uniform drift in high seismic optimal design, SAP 2000 is utilized for simulation and analysis. Table 9.3 provides the calculated earthquake loads for each SPSW in all stories of the frame. In CBO, ECBO, and PSO, the population of n = 50 agents is used for optimization. In ECBO, the size of colliding memory is taken as 10, and the parameter pro increases linearly from 0.3 to 0.5 during the optimization process. Table 9.3 Earthquake load acting on the 9-story SPSW

Forces and shears in each SPSW Level

Frame force (kips)

Frame shear (kips)

Roof

197

197

9th floor

152

349

8th floor

133

482

7th floor

113

595

6th floor

93.5

689

5th floor

74.0

763

4th floor

55.4

818

3rd floor

37.5

855

2nd floor

20.8

876

204

9 Optimal Seismic Design of Steel Plate Shear Walls …

Table 9.4 Comparative results of the 9-story SPSW Level

High seismic SPSW Optimal design

Beam sections

Column sections

Uniform drift state

AISC design example

ECBO*

ECBO

CBO

ECBO

Roof

W30X108

W30X108

W24X76

W12X111

W24X76

9th floor

W27X94

W18X40

W14X22

W12X19

W14X22

8th floor

W27X94

W30X90

W30X99

W33X130

W30X99

7th floor

W27X94

W21X44

W36X135

W18X40

W36X135

6th floor

W30X108

W18X35

W16X36

W27X84

W16X36

5th floor

W27X94

W21X50

W27X84

W24X55

W27X84

4th floor

W30X116

W21X44

W30X90

W40X149

W30X90

3rd floor

W27X94

W30X90

W27X84

W24X55

W27X84

2nd floor

W27X94

W24X68

W21X57

W30X99

W21X57

1st floor

W30X108

W30X90

W44X198

W44X198

W44X198

9th floor

W14X283

W14X283

W30X99

W30X124

W30X99

8th floor

W14X283

W14X120

W24X76

W30X99

W24X76

7th floor

W14X283

W14X257

W36X160

W40X215

W36X160

6th floor

W14X398

W14X233

W40X215

W30X173

W40X215

5th floor

W14X398

W14X257

W40X192

W33X241

W40X192

4th floor

W14X665

W14X311

W40X268

W40X268

W40X268

3rd floor

W14X665

W14X342

W40X324

W33X387

W40X324

2nd floor

W14X665

W14X500

W33X387

W33X354

W33X387 (continued)

9.6 Numerical Examples

205

Table 9.4 (continued) Level

High seismic SPSW Optimal design

tw (in)

Weight (lb)

Uniform drift state

AISC design example

ECBO*

ECBO

CBO

ECBO

1st floor

W14X665

W14X605

W33X515

W30X527

W33X515

9th floor

0.0673

0.0673

0.03

0.03

0.6

8th floor

0.1046

0.0747

0.03

0.03

0.57

7th floor

0.125

0.125

0.085

0.11

0.38

6th floor

0.1345

0.125

0.11

0.11

0.22

5th floor

0.1875

0.125

0.11

0.11

0.41

4th floor

0.1875

0.1345

0.15

0.12

0.06

3rd floor

0.25

0.1345

0.155

0.14

0.175

2nd floor

0.25

0.1875

0.1775

0.145

0.04

1st floor

0.25

0.1875

0.1775

0.19

0.03

150,667.8

97,749.6

104,320.3

86,207

90,923

Ave. weight (lb)

106,288.3

89,520.1

92,620.9

No. of analyses

25,500

45,900

37,860

Wall/column (%)

12.70

12.80

11.60

Wall/structure (%)

9.90

9.40

8.60

*Just W14 sections are used for columns

The first column in Table 9.4 belongs to the high seismic design example of AISC Design Guide No. 20. The second column in Table 9.4 is the optimal structure found by ECBO which is 30% lighter than the Design Guide example. In the optimum design, like Design Guide example only W14 sections are used for vertical boundary elements (VBEs). In the rest of the columns in Table 9.4, all W-shaped sections for

206

9 Optimal Seismic Design of Steel Plate Shear Walls …

Fig. 9.14 Convergence curves of the 9-story SPSW; best and average CBO, ECBO and PSO

VBEs can be assigned. The third and fourth columns in Table 9.4 are for optimal design obtained by ECBO and CBO, respectively. The best weight is for the ECBO algorithm that has 42% lighter weight in comparison with the AISC Design Guide example. It also contains steel plates more than the optimal design obtained by CBO. Both the best answer and the best average answer—illustrated in Fig. 9.14— are obtained by ECBO with a faster convergence rate. This proves the high ability of ECBO in finding the optimal answer for a complex structure (SPSW) with even more complicated constraints than SPW systems. Like the low seismic section, the best answer obtained by ECBO and its average answers is close together. These indicate that the performance of the ECBO algorithm is quite good for SPSW design problems, the same as it was for the SPW system. In Fig. 9.14, since PSO is trapped in local optima, it is demonstrated that it is obviously an impotent algorithm for finding optimum solutions for this complex structural problem. Figure 9.15 indicates that strength constraint is dominant in VBEs for high seismic optimum design and similarly the stiffness requirement for beams exceeds the required strength in some stories (5th, 6th, and 10th stories). Strong-Column WeakBeam is the last constraint that is applied to prevent the soft-story collapse mechanism. Figure 9.16 shows that this constraint does not govern in any of the connections because the optimal design of the SPSW system provides a design that tends to collapse in the uniform yielding mechanism.

9.6 Numerical Examples

207

Fig. 9.15 Strength and stiffness constraints for the 9-story SPSW

Fig. 9.16 Strong-Column Weak-Beam constraint for the 9-story SPSW

9.6.3 Performance-Based Design Optimization of SPSW In this section, uniformity of the story drifts is considered as the structural performance which is mostly based on displacement. The standard deviation of the story drifts in the optimized design of the SPSW is defined as the objective function. In this example, frame elements are those obtained for minimum weight objective optimization (third column in Table 9.4), and web plate thicknesses are the design variables. The story drifts of the 9-story SPSW system are obtained by linear static analysis as shown in Fig. 9.17. They belong to before and after the performance optimization. Sensitivity Analysis of the Base Shear Optimization of the SPSW design subjected to the five base shear loading—with 0.8, 1, 1.2, 1.4, and 1.6 times of the calculated base shear—are investigated in this

208

9 Optimal Seismic Design of Steel Plate Shear Walls …

Fig. 9.17 Comparison of the drift for the 9-story SPSW between uniform state and optimal design

section. Figure 9.18 and Table 9.5 provide the convergence histories and comparative results of the 9-story SPSW for different base shears, respectively. It is noteworthy that the consumption of the building materials in different structural elements needs significant heed which is paid in this work by sensitivity analysis of the base shears. In this regard, the amount of consumed material in columns and web plates which are resisting elements against seismic lateral loads are demonstrated in Fig. 9.19. It can be seen that the percentage of the wall usage in different base shears is almost fixed and near 8 or 9% of the structure. The little decline of the wall usage with base shear increase can be explained as follows. This may happen because boundary elements would leave their elastic range if thicker plates were employed—as a consequence of the web plate yielding forces exerted on boundary elements. According to Fig. 9.20,

Fig. 9.18 Convergence curves of the 9-story SPSW subjected to different base shears

9.6 Numerical Examples

209

Table 9.5 Comparative results of the 9-story SPSW subjected to different base shears Level

High seismic SPSW Optimal design using ECBO

Beam sections

Column sections

AISC design example

0.8 base shear

1.0 base shear

1.2 base shear

1.4 base shear

1.6 base shear

Roof

W30X108

W27X94

W24X76

W40X149

W30X99

W30X108

9th floor

W27X94

W12X22

W14X22

W24X62

W24X55

W30X108

8th floor

W27X94

W16X31

W30X99

W30X90

W36X150

W30X90

7th floor

W27X94

W21X50

W36X135

W40X149

W24X68

W33X118

6th floor

W30X108

W21X50

W16X36

W21X44

W30X90

W30X99

5th floor

W27X94

W30X116

W27X84

W30X116

W30X90

W36X135

4th floor

W30X116

W27X84

W30X90

W21X44

W36X160

W24X62

3rd floor

W27X94

W30X90

W27X84

W27X84

W30X99

W44X198

2nd floor

W27X94

W24X68

W21X57

W30X90

W44X248

W40X249

1st floor

W30X108

W36X135

W44X198

W40X268

W40X397

W40X531

9th floor

W14X283

W33X118

W30X99

W40X249

W33X130

W40X149

8th floor

W14X283

W16X67

W24X76

W40X167

W27X114

W30X173

7th floor

W14X283

W18X86

W36X160

W40X199

W36X230

W40X192

6th floor

W14X398

W21X111

W40X215

W40X297

W40X199

W40X268

5th floor

W14X398

W36X135

W40X192

W40X268

W40X244

W40X298

4th floor

W14X665

W40X221

W40X268

W33X354

W40X297

W36X393

3rd floor

W14X665

W36X260

W40X324

W33X354

W33X468

W36X393

2nd floor

W14X665

W33X318

W33X387

W40X436

W33X468

W36X588

1st floor

W14X665

W33X468

W33X515

W36X588

W36X798

W36X848 (continued)

210

9 Optimal Seismic Design of Steel Plate Shear Walls …

Table 9.5 (continued) Level

High seismic SPSW Optimal design using ECBO

tw (in)

Weight (lb)

AISC design example

0.8 base shear

1.0 base shear

1.2 base shear

1.4 base shear

1.6 base shear

9th floor

0.0673

0.03

0.03

0.095

0.03

0.05

8th floor

0.1046

0.03

0.03

0.11

0.04

0.095

7th floor

0.125

0.035

0.085

0.12

0.12

0.13

6th floor

0.1345

0.05

0.11

0.125

0.13

0.155

5th floor

0.1875

0.06

0.11

0.125

0.13

0.155

4th floor

0.1875

0.125

0.15

0.13

0.14

0.17

3rd floor

0.25

0.155

0.155

0.13

0.21

0.175

2nd floor

0.25

0.165

0.1775

0.165

0.21

0.22

1st floor

0.25

0.175

0.1775

0.18

0.23

0.23

150,667.8

70,483.3

86,207

108,931.4

118,183.9

132,641.9

Ave. weight (lb)

73,059.7

89,520.1

110,213.8

120,878.2

136,091.3

No. of analyses

48,200

45,900

38,600

36,700

49,850

Wall/column (%)

13.30

12.80

11.20

11.10

10.90

Wall/structure (%)

9.60

9.40

8.40

7.90

7.80

Fig. 9.19 Weight ratio comparison among optimal answers for different base shears

9.6 Numerical Examples

211

Fig. 9.20 Weight comparison among optimal answers for different base shears

the total weight of the frame for the SPSW system increases very close to a small slope line with the increase of the base shear. In comparison between SPW and moment frame. It is asserted that the steel shear wall system has a lighter optimum structure. In a similar manner, it is predicted that an increment of the structural weight versus augmentation of the base shear would be a line with an exponential slope if other conventional lateral load resisting systems like moment frames and braced frames were applied.

9.6.4 Optimum Design of 6- to 12-Story SPSW To investigate the steel plate performance with the change of the structural height, seven structures, including 6, 7, 8, 9, 10, 11, and 12-story SPSW shown in Fig. 9.21 are studied in their optimum design form. It is assumed that the base shears of the

Fig. 9.21 Schematic of 6- to 12-story SPSWs

212

9 Optimal Seismic Design of Steel Plate Shear Walls …

frames are directly proportional to the number of stories. Thus, the amount of base shear for n-story SPSW is taken as n/9 times of the calculated base shear for 9-story SPSW (=876 kips) and is distributed vertically based on Eqs. (9.41) and (9.42). Table 9.6 presents the comparative results of the structures. Figure 9.22 shows that the percentage of wall usage in SPSW system versus the story number declines from about 12 to 6%. The decline of the wall usage may be owing to web plate yielding forces exerted on boundary elements if thicker plates are used. Because heavier columns are required in lower stories of the higher SPSWs due to the remarkable increase in the axial load. According Fig. 9.23, the total weight of the SPSW frame increases near exponential with an increment of the story number. It is expected that drastic changes in the total weight of the building in optimum design would happen if other conventional lateral load resisting systems like moment frames and braced frames were considered. All of the results show the high ability of the SPSW systems in energy dissipation for high and low seismic loads.

9.7 Discussions and Conclusion In this chapter, the seismic design optimization of the steel plate shear wall is performed by metaheuristic algorithms. The weight of the structure is defined as its cost. Hence, minimizing the weight of the structures and enhancing the performance of them are the objective functions of the problems. The most imperative discussions and conclusions can be summarized as follow: • The ECBO algorithm’s excellent performance is demonstrated in all optimum design examples of the steel plate shear wall system. • The CBO algorithm in seismic optimum design of the steel plate shear wall system shows an acceptable performance. This algorithm does not have complicated parameters to be tuned for such sophisticated problems with complex constraints. • The weak performance of the PSO algorithm shows that the new improved PSO algorithms, such as DPSO, are a better substitute for PSO. • The results indicate the high capability of the SPSWs in energy dissipation even in their minimum weight form which is also recommended by AISC Design Guide No. 20. This guideline recommends avoiding overdesign of the thicknesses of the steel plates. • Domination of the uniform yielding mechanism in optimum design of steel plate shear walls is obtained by metaheuristic algorithms, which is resulted from meeting the Strong-Column Weak-Beam constraint in a decisive manner. • SPSW structures have less weight and superior performance in optimum design form in comparison with the moment frame system. • The performance-based optimized design of SPSW demonstrates that thicker web plates are required in top panels of the frame if the minimum weight and drift uniformity objective functions are defined separately.

Column sections

Beam sections

W24X104

12-story

W36X182

W27X146

W14X283

8th floor

W24X76

W36X135 W33X141

W14X283

9th floor

W30X99

W44X198

W30X99

10th floor W40X192

W27X84

W21X57

W21X50

W40X149

W40X436

W36X160

W44X248

W40X244

W40X480

W40X192

W40X215

W40X215

W27X84

W21X101

W16X67

(continued)

W40X268

W40X215

W30X173

W14X61

W30X90

W24X55

W27X84

W30X90

W30X99

W40X192

W24x68

W27X84

W30X90

W18X35

W30X90

W30X99 W30X90

W33X130

1st floor

W16X36

W18X40

W24X62

W27X84

W44X198 W36X135

W21X44

W30X108

W30X108

2nd floor

W14X34

W24X55

W18X35

W18X35 W30X90

W30X90 W30X90

11th floor

W27X94

3rd floor

W33X118

W36X135 W16X36

W27X84

W21X50

12th floor

W30X116

W27X94

4th floor

W30X90

W27X94

W21X44

W21X50

5th floor

W24X84 W21X44

W30X108

6th floor

W12X19

W27X94

7th floor

W21X83

W27X94

8th floor

W30X99

W24X76 W21X44

W27X94

9th floor W21X50

W24X84

11-story

W8X18

W30X90

10th floor W14X22

W24X76

10-story

W33X118

W33X118

9-story

W21X44

W24X84

8-story

W8X21

W27X84

7-story

Optimal design 6-story

11th floor

W30X108

AISC design example

SPSW

High seismic

12th floor

Roof

Level

Table 9.6 Comparative results of the 6- to 12-story SPSW

9.7 Discussions and Conclusion 213

tw (in)

Table 9.6 (continued)

0.04

0.1875

0.055

0.11

0.105

0.095

5th floor

0.03 0.045

0.1345

6th floor

0.04

0.125

7th floor

0.085 0.11

0.11

0.135

0.12

0.12

0.095

0.1046

8th floor

0.03

0.055 0.085

0.0673

9th floor 0.08

W36X650

10th floor 0.03

W33X515

W33X424

W33X468

W40X328

W36X280

W40X192

W36X798

W36X650

W33X515

W40X297

W40X268

W40X268

W36X848

W36X650

W36X588

W33X424

W33X387

W40X328

W40X268

12-story

0.1

0.065

0.045

0.045

0.03

(continued)

0.135

0.13

0.13

0.13

0.11

0.11

0.03

W40X436

W33X387

W40X324

W40X268

W40X192

W40X199 W40X215

11-story

0.105

W36X359

W40X328

W40X244

W40X215

W40X192

W36X160 W40X215

10-story

0.045

W40X268

W36X230

W40X215

W40X215

W24X104

W33X141 W30X173

9-story

0.03

W14X665

1st floor

W40X192

W30X108 W30X99

8-story

11th floor

W14X665

2nd floor

W30X173

W36X170

W24X76

W33X118

7-story

Optimal design 6-story

12th floor

W14X665

W14X665

W14X398

5th floor

3rd floor

W14X398

4th floor

W14X283

6th floor

AISC design example

SPSW

High seismic

7th floor

Level

214 9 Optimal Seismic Design of Steel Plate Shear Walls …

0.25

0.25

2nd floor

1st floor 39,509.9

0.11

0.11

0.11

0.095

40,172 15.10 10.90

No. of analyses

Wall/column (%)

Wall/structure (%)

9.80

13.70

18,750

54,854.4

53,331.2

0.13

0.13

0.13

0.115

7-story

Optimal design 6-story

41,100.9

150,667.8

0.1875

0.25

3rd floor

AISC design example

SPSW

High seismic

4th floor

Level

Ave. weight (lb)

Weight (lb)

Table 9.6 (continued)

10.30

13.50

30,300

74,342

71,089.9

0.145

0.145

0.115

0.11

8-story

9.40

12.80

45,900

89,520.1

86,207

0.1775

0.1775

0.155

0.15

9-story

9.20

12.10

29,500

112,331

110,652

0.21

0.16

0.16

0.145

10-story

6.40

9.00

46,602

136,286

132,468

0.23

0.23

0.19

0.125

11-story

6.70

9.00

36,140

170,856

166,815

0.185

0.15

0.145

0.14

12-story

9.7 Discussions and Conclusion 215

216

9 Optimal Seismic Design of Steel Plate Shear Walls …

Fig. 9.22 Weight ratio comparison among optimal answers for different base shears

Fig. 9.23 Weight comparison among optimal answers for different base shears

• Sensitivity analysis of the base shear for 9-story SPSW shows that the percentage of the wall usage is almost constant for different base shears in this particular example. • In optimum SPSW system, although wall to column weight percentage decreases with the increment of the number of stories, making use of web plates with least possible thicknesses is still vital for high performance of the system subjected to seismic loads.

References 1. Kaveh, A., & Farhadmanesh, M. (2019). Optimal seismic design of steel plate shear walls using metaheuristic algorithms. Periodica Polytechnica Civil Engineering, 63(1), 1–17. 2. Thorburn, L. J., Montgomery, C., & Kulak, G. L. (1983). Analysis of steel plate shear walls. 3. Timler, P. A., & Kulak, G. L. (1983). Experimental study of steel plate shear walls. 4. Gholizadeh, S., & Shahrezaei, A. M. (2015). Optimal placement of steel plate shear walls for steel frames by bat algorithm. The Structural Design of Tall and Special Buildings, 24(1), 1–18. 5. Sadeghi Eshkevari, S., Dolatshahi, K. M., & Mofid, M. (2017). Optimized design procedure for coupling panels in steel plate shear walls. The Structural Design of Tall and Special Buildings, 26(1), e1301.

References

217

6. Tehranizadeh, M., & Moshref, A. (2011). Performance-based optimization of steel moment resisting frames. Scientia Iranica, 18(2), 198–204. 7. Armand, J.-L. (1971). Minimum-mass design of a plate-like structure for specified fundamental frequency. AIAA Journal, 9(9), 1739–1745. 8. Topkaya, C., & Atasoy, M. (2009). Lateral stiffness of steel plate shear wall systems. ThinWalled Structures, 47(8–9), 827–835. 9. Rezai, M., Ventura, C. E., & Prion, H. (2004). Simplified and detailed finite element models of steel plate shear walls. In Proceedings of 13th World Conference on Earthquake Engineering. 10. Kaveh, A., & Mahdavi, V. R. (2014). Colliding bodies optimization: a novel meta-heuristic method. Computers & Structures, 139, 18–27. 11. Kaveh, A., & Ilchi Ghazaan, M. (2014). Enhanced colliding bodies optimization for design problems with continuous and discrete variables. Advances in Engineering Software, 77, 66–75.

Chapter 10

Colliding Bodies Optimization Algorithm for Structural Optimization of Offshore Wind Turbines with Frequency Constraints

10.1 Introduction Considering the size and dimension of offshore wind turbines, the optimization of such structures is a tedious task. Nonetheless, in this chapter, a meta-heuristic algorithm named Colliding Bodies Optimization (CBO) is employed when investigating the optimal design of jacket supporting structures for offshore wind turbines. The OC4 reference jacket is considered as the case study, validating the outcomes of this algorithm. The structural optimization is performed when both the Ultimate Limit State (ULS) and frequency constraints being considered. During the optimization process, a noticeable weight reduction is observed, while all the constraints mentioned above being fulfilled [1]. Considering the increase in the population of the world, and remarkable abatement in fossil-fuel resources, renewable energies, more specifically wind energy, has engrossed researchers’ attention within the past few decades. When it comes to renewable energies, offshore wind energy has been always considered as a suitable option. The existence of myriad appropriate locations for the installment of wind turbines in the marine environment further justifies this popularity. In addition, since noise and visual pollutions are no longer an issue in the offshore wind industry, this industry has become more popular as the environmental issues are being broadened. Generally, bottom-fixed and floating supporting structures are common structural options in the offshore wind industry. Monopiles—amongst all the existing supporting structures—are the predominant structural system, mainly due to its tangible simplicity in both manufacturing and design processes. However, when moving to deeper areas as an attempt to search for higher wind potentials, these supporting structures are not as formerly applicable, since they cannot stand the harsher environment of such regions. In such an environment, the frame supporting structures—for instance, jacket supporting structures—are of more efficiency. Additionally, these supporting structures could bear the weight of larger wind turbines, which is assumed as an appreciable advantage; thus, frame-supporting structures have become an important © Springer Nature Switzerland AG 2020 A. Kaveh and A. Dadras Eslamlou, Metaheuristic Optimization Algorithms in Civil Engineering: New Applications, Studies in Computational Intelligence 900, https://doi.org/10.1007/978-3-030-45473-9_10

219

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10 Colliding Bodies Optimization Algorithm for Structural …

part of the offshore renewable industry in the past few years. Tripod and jacket supporting structures, which have been already deployed in the oil and gas industry, are generally recognized as the best choices in the offshore wind turbine industry, mainly due to the features mentioned above. Considering the importance of structural optimization in offshore wind turbine supporting structures due to their size and dimension, this task has been tackled by several researchers, as follows: Using a zeroth-order search algorithm, the optimal design of monopile offshore wind turbine structures is explored by Uys et al. [2]. Chen and Yang et al. deployed the Particle Swarm Optimization (PSO) algorithm in searching the optimal design of both the shape and size of the lattice partition in a wind turbine tower with lattice-tubular hybrid supporting structure [3]. Thiry et al. utilized the Genetic Algorithm (GA) to explore the optimal design of a monopile offshore wind turbine while both FLS, ULS and frequency constraints being considered [4]. Considering ULS conditions, Long et al. investigated the characteristics of tripod and jacket supporting structures [5]. Long and Moe further expanded their results considering FLS conditions in compliance with design standards [6]. Zwick et al. then presented a full-height lattice offshore wind turbine—a new concept in the industry [7]. Furthermore, utilizing an iterative optimization approach, its optimal design under both FLS and ULS constraints are then investigated. Zwick and Muskulus presented a method for simply assessing fatigue load based on statistical regression models [8]. Oest et al. investigated the optimal design of jacket supporting structures when considering fatigue and Ultimate Limit State constraints [9]. In their research, FLS, ULS and frequency constraints are taken into consideration as the main structural constraints. Kaveh and Sabeti explored the optimal design of the OC4 reference jacket when wind and wave loads are considered in-plane, embedding the whole structure in 2D space [10]. This research utilizes meta-heuristic algorithms in the investigation of the optimal design of jacket supporting structures. Many such algorithms have recently been developed considering natural phenomena—the collision between bodies and the free vibration of a system. They all share simplicity in implementation and less timeconsumption in comparison to the other algorithms as their conspicuous advantages. Colliding Bodies Optimization (CBO)—a recently developed algorithm based on the physic laws governed the collision between bodies by Kaveh and Mahdavi—is the utilized algorithm in this research. Its simple formulation and parameter independence are the main merits over most of the other meta-heuristic algorithms [11]. Synoptically, this research aims to employ metaheuristic algorithms, more specifically Colliding Bodies Optimization, to explore the optimal design of a jacket supporting structure. Therefore, utilizing Finite Element Method principles, the whole structure is modeled in MATLAB. The algorithm is then employed, exploring to find the lightest structural members that satisfy the considered constraints—Ultimate Limit States and frequency constraints. The efficiency of the utilized algorithm is then investigated employing a case study. The OC4 reference jacket, which bears the weight of an NREL 5 MW reference wind turbine, is the considered example. Finally,

10.1 Introduction

221

the outcomes of this research are compared with the original structure, validating the efficiency of the utilized algorithm.

10.2 Configuration of the OC4 Reference Jacket As mentioned, in today’s offshore wind industry, searching for offshore locations having stronger wind potentials has resulted in the utilization of frame supporting structures. In fact, these structures comprise two different sections: the lattice section and the tower. In this chapter, the main goal is to obtain the optimal design of the lattice section. This research is performed based on the OC4 reference jacket characteristics (Fig. 10.1). Locating at K13 deep-water site in the North Sea, the mean water level in this site is considered 50 m above the seabed. The well-known 5 MW horizontal axis NREL wind turbine is the utilized turbine in this wind turbine, having 3 m/s and 25 m/s as its cut-in and cut-out wind speeds, respectively. In this wind turbine, the rotor weighs 110,000 kg while the nacelle mass is approximately 240,000 kg, having a mass of 350,000 kg in aggregate. The 5 MW NREL wind turbine is surmounted on a 68 m long tower. It approximately weighs 218,000 kg when the weight of its equipment being ignored. Utilizing a transition piece made of concrete weighing 660,000 kg, Fig. 10.1 The OC4 reference jacket supporting structure

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10 Colliding Bodies Optimization Algorithm for Structural …

the tower and supporting structure are connected. The weight of supporting structure, which is made of several hollow circular members, is approximately 673,718 kg.

10.3 Finite Element Model In this chapter, the Finite Element Method is utilized in simulating and analyzing the design example. Each member, comprising two nodes, is modelled as a 3D frame element, each of which has six degrees of freedom. The whole structure is then modeled in MATLAB using the abovementioned approach. The cross-sectional properties are considered constant throughout the length of each member. In jacket supporting structures, the role of transition piece in the integrity of the structure is irrefutable; therefore, the transition piece is simulated using four elements, each of which is assigned one-fourth of the total weight of the original transition piece. Additionally, to control the dynamic behavior of the structure, two frequency constraints are considered; hereupon, the frequency of the structure must be calculated. Thoroughly modeling the structure, eight extra elements are considered to take the structural role of the tower into consideration. The weight of the Rotor Nacelle Assembly (RNA) is assigned, as a lumped mass on the top of the tower. The mass matrix of each element is then calculated using a consistent mass matrix. Assessing both stiffness and mass matrices, the frequencies of the structure could be easily calculated using an eigenvalue analysis.

10.4 Loading Conditions Offshore structures are, generally, subject to different loading cases—wave, wind, and currents actions (Fig. 10.2). Thus, accurately evaluating the loading conditions when designing offshore wind turbine structures are of utmost importance. Environmental loads in this chapter are assessed based on the DNV standard [12]. Many load cases must be taken into consideration when designing offshore wind turbines, such as regular power production, extreme weather conditions, and shut down. However, the extreme weather condition is the mode taken into account. In this condition, it is presumed that the turbine, due to encountering the extreme values of environmental phenomena such as wave and wind, is stopped, and both ULS and frequency constraints are taken into consideration. Generally, the loads applied to offshore wind turbines can be categorized as either permanent or environmental load cases. The weight of both structural and non-structural elements, which are constant in any arbitrary period, are considered in the former, while wind and wave actions are considered in the latter group. Despite permanent loads, environmental load cases are a function of metocean data—such as wave height, and wind velocity—resulting in their difference from site to site. In this chapter, wind and wave actions are considered as load cases, which are briefly described here.

10.4 Loading Conditions

223

Fig. 10.2 Aero-hydro dynamic loads applying on an offshore wind system

10.4.1 Wave Loading Morrison equation has been widely utilized in the calculation of wave actions on slender structures. However, its usage is restricted to the cases where the diameter of the structure is noticeably smaller than the wavelength. Based on this equation, hydrodynamic load on a unit length of a slender structure comprises of two different terms: drag and inertia, which are written as follows: d F = d Fm + d Fd =

Cm πρ D 2 Cd ρ D |u w |u w dz u˙ w dz + 4 2

(10.1)

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10 Colliding Bodies Optimization Algorithm for Structural …

where: d Fm d Fm Cm Cd D ρ uw u˙ w

Inertia force (N/m) Drag force (N/m) Inertia coefficient Drag coefficient Element diameter (m) Mass density of seawater (kg/m3 ) Horizontal velocity of water particle (m/s) Horizontal acceleration of water particle (m2 /s).

In the abovementioned equation, drag and inertia coefficients are functions of Keulegan-Carpenter number, relative roughness and Reynolds number, which are considered 0.7 and 2, respectively. In order to assess hydrodynamic load on oblique members, some geometrical manipulation are employed to find the normal velocity and acceleration of water particles to the axis of each inclined member; then, the normal wave force to the axis of element is calculated using Morrison equation.

10.4.2 Wind Loading 10.4.2.1

Wind Force on Tower

Based on DNV standard, the effect of wind on the tower of offshore wind turbines could be calculated as follows (DNV 2014): F=

1 × ρa × C S × S × U 2 2

(10.2)

where: ρa CS S U

Air density (kg/m3 ) Shape coefficient Projected area of the member normal to the direction of the force (m2 ) Wind velocity (m/s).

In this chapter, the shape coefficient is considered 0.15. In order to calculate the wind load, the required parameters are calculated as follows [12]: C = 5.73 × 10−2 ×



1 + 0.15 × U0

IU = 0.06 × (1 + 0.043 × U0 ) ×

 z −0.22 h

(10.3) (10.4)

10.4 Loading Conditions

225



  z   T × 1 − 0.41 × IU × ln U (T, z) = U0 × 1 + C × ln h T0

(10.5)

where: U0 h T0 T < T0 z

10.4.2.2

1 h wind mean speed at 10 m height (m/s) 10 m 3600 s Desired time (s) Desired height from still water level (m).

Wind Force on Rotor and Nacelle Assembly

In order to accurately calculate the wind effect on the Rotor and Nacelle Assembly (RNA), a comprehensive study regarding 3D aero-servo-elastic analysis is required. However, since such data is not readily accessible, in this chapter, the effect of wind on the RNA is calculated using a scaling relationship [13]. Utilizing this relationship, the wind loads imposed on any arbitrary wind turbine could be easily calculated based on a known wind turbine as follows: T1 = T2 M1 = M2

 

R1 R2

2

R1 R2

(10.6) 3 (10.7)

where: R1 /R2 The ratio of rotor diameters T Aerodynamic thrust M Aerodynamic moment To use the relationship mentioned above, some restrictions must be controlled. For instance, the Tip Speed Ratio (TSR) must be constant between the actual and scaled wind turbines. Additionally, all basic characteristics of wind turbines, such as the number of blades, the utilized material in the construction of blades, and the utilized airfoil must be identical. Finally, both wind turbines must be geometrically similar to the greatest extent. Therefore, aerodynamic thrust and moment for the 5 MW NREL wind turbine are readily determined using the aforementioned methodology, which results in an acceptable approximation for the initial steps of designing offshore wind turbines.

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10.4.3 Load Combinations Based on DNV 2014, following load combinations are considered when evaluating ULS constraints in offshore wind turbines [12]: First Load Combination dead load (containing self-weight of the whole structure including the tower, supporting structure, the weight of wind turbine multiplied by 1.25) + wind load (consisting of wind load on the tower, supporting structure, and turbine multiplied by 0.7) + wave load (multiplied by 0.7). Second Load Combination dead load (containing self-weight of the whole structure including the tower, supporting structure, the weight of wind turbine multiplied by 1) + wind load (consisting of wind load on the tower, supporting structure, and turbine multiplied by 1.35) + wave load (multiplied by 1.35).

10.5 The Structural Optimization Problem A typical structural optimization problem can be stated as follows: Find X = [x1 , x2 , x3 , x4 , . . . , xn ] To minimize Mer (X ) = f (X ) × f penalty (X ) Subject to gi (X ) ≤ 0, i = 1, 2, . . . , m xi min ≤ xi ≤ xi max

(10.8)

In the abovementioned formulas, X is the vector of design variables with n unknowns, gi is the ith constraint from m inequality constraints. A well-known penalty approach is employed for taking constraint handling into account which is expressed as follows:  f penalty (X ) = 1 + ε1

m i=1

ε2 max(0, gi (X ))

(10.9)

10.5 The Structural Optimization Problem

227

In the abovementioned penalty function, Mer (X ) is the merit function, f (x) is the cost function and ε1 and ε2 are the parameters controlling the balance between exploration and exploitation rates within the search space in the algorithm, which are taken 1 and 3, respectively.

10.5.1 Design Variables The design variables of this optimization problem are the diameter and thickness of each member in the jacket supporting structure. Supporting structure members are categorized into ten different groups (Fig. 10.3); thus, perceivably, the design variable vector of this problem consists of 20 entities. X = [D1 , D2 , . . . , D10 , t1 , t2 , . . . , t10 ]

10.5.1.1

(10.10)

Design Constraints

The optimal design of jacket supporting structures is performed when considering both ULS and frequency constraints. Due to the sensitiveness of offshore structures, more specifically offshore wind turbines, to the dynamic excitements, limiting the dynamic behavior of the structures should be considered so that the occurrence of undesired phenomena, such as dynamic resonance, be prevented. To do so, the first and second frequencies of the structure are calculated and restricted to the soft-stiff range shown in Fig. 10.4. The lower and upper bounds of this region, in this design example, are taken as 0.22 and 0.31 Hz, respectively [9]. In addition, ULS constraint (Buckling Failure) is considered based on Eurocode 3. All elements, except the elements of the tower and transition piece, are included when assessing buckling failure of the supporting structural elements. The two following constraints must be fulfilled in all the aforementioned elements [14]: Be =

My,ED Mz,ED NED + kyy + kyz χy NRK /γM1 χLT My,RK /γM1 Mz,RK /γM1

(10.11)

Ge =

My,ED Mz,ED NED + kzy + kzz χz NRK /γM1 χLT My,RK /γM1 Mz,RK /γM1

(10.12)

In the abovementioned formulas, NED , My,ED , and Mz,ED are design compression force and maximum moments about the local y-y and z-z axis, respectively. NRK , My,RK , and Mz,RK are the resistance force and moments of the critical cross-section, respectively. Furthermore, γM1 is a partial safety factor for global stability, which is considered 1.2 in this chapter. χLT is the reduction factor taking lateral-torsional

228 Fig. 10.3 Design grouping (Gray elements not being optimized)

10 Colliding Bodies Optimization Algorithm for Structural …

10.5 The Structural Optimization Problem

229

Fig. 10.4 The frequency spectrum of the dynamic loads

buckling into account. However, since the utilized elements are circular hollow members, which are not susceptible to such failure, this coefficient is considered as 1. kyy , kyz , kzy , and kzz are interaction factors, which are calculated based on Eurocode 3 [14]. χy , and χz are reduction factors taking flexural buckling into account, which are calculated as follows:   1  ,1 (10.13) χ y = χz = min Φ + Φ 2 − λ¯ 2     (10.14) Φ = 0.5 1 + α λ¯ − 0.2 + λ¯ 2  λ¯ =

A fy Ncr

(10.15)

where λ¯ is non-dimensional slenderness, Ncr is the Euler critical force, and f y is the yield stress of the utilized material. Additionally, in order to make sure that local instability will not take place, according to Eurocode 3, the ratio of diameter over thickness in all elements is restricted to 59.4.

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10 Colliding Bodies Optimization Algorithm for Structural …

10.5.2 Cost Function In this optimization problem, the cost function equals the supporting structure weight, which is written as follows: f (X ) =

n i=1

ρgVi =

n i=1

ρg Ai L i =

n

ρg(π Di ti L i )

(10.16)

i=1

10.5.3 Colliding Bodies Optimization Algorithm Colliding Bodies Optimization (CBO) has been recently developed based on momentum and energy conservation laws in the one-dimensional collision between bodies. This algorithm is a multi-agent algorithm, which contains a number of Colliding Body (CB) with determined mass and velocity [11]. CBs move toward new positions after collisions, having new velocities in correspondence to old velocities, masses and coefficient of restitution. Randomly selecting the agents within the search space, the algorithm is initialized. Then, in accordance with the values of the cost function, in ascending order, the agents are sorted. Afterward, CBs are divided into two equal categories named stationary and moving groups. Good agents, which are defined in accordance with the objective function, are considered as stationary agents, whose velocities before the collision is considered zero. Before the collision, moving category members move toward stationary agents, having the better and worse CBs collided together. This results in improving moving CBs positions while forcing stationary CBs toward better locations. In this algorithm, the velocity of CBs before the collision is considered as the value of change in the body position. The velocities of CBs are defined in Eq. (10.17) vi = 0, i = 1, 2, . . . , n vi = xi − xi−n i = n + 1, n + 2, . . . , 2n

(10.17)

Next, considering momentum and energy conservation laws, the velocities of each body after the collision are calculated as follows. (m i+n + εm i+n )vi+n i = 1, 2, . . . , n m i + m i+n (m i − εm i−n )vi vi = i = n + 1, n + 2, . . . , 2n m i + m i+n vi =

(10.18)

where vi and vi are the velocities of the ith CB before and after the collision, respectively. Additionally, the mass of each CB is determined as follows:

10.5 The Structural Optimization Problem

231

1 fit(k)

m k = n

1 i=1 fit(i)

k = 1, 2, . . . , 2n

(10.19)

In the abovementioned formula, fit(i) is the value of the objective function for the ith agent. Perceivably, larger and smaller masses are carried by better and worse CBs, respectively. In CBO algorithm, the coefficient of restitution (ε) is utilized so that the rate of exploration and exploitation could be controlled during optimization. This ratio is defined as follows: ε =1−

iter itermax

(10.20)

where iter and itermax are the actual iteration number and the maximum number of iterations, respectively. New position of CBs, after the collision, could be calculated as follows: xinew = xi + rand ◦ vi i = 1, 2, . . . , n xinew = xi−n + rand ◦ vi i = n + 1, n + 2, . . . , 2n

(10.21)

Reaching a pre-defined evaluation number such as the maximum number of iterations, the optimization process is terminated.

10.6 Results This chapter is aimed to exhibit how meta-heuristic algorithms can be utilized in the structural optimization of jacket supporting structures for offshore wind turbines. The OC4 reference jacket is chosen as a case study. The structure is modeled in MATLAB using Finite Element Method principles, and its optimal design is then investigated. Wind and wave effects on the structure are assessed based on DNV standard as the environmental loads. These actions are quantified based on the characteristics indicated in the Table 10.1. The structural properties of the utilized steel in the construction of the jacket supporting structure are ( f y = 355 MPa, E = 2×105 MPa, ρ = 7885 kg/m3 ). The mass density of seawater is considered (ρ = 1025 kg/m3 ) while this number for air when assessing wind load is taken (ρ = 1.225 kg/m3 ). Table 10.1 Simplified load cases used in the case study

Wave

Wind

Significant wave height (m)

9.4

Wave period (s)

13.7

Water depth (m)

50

1 h mean wind speed at hub height (m/s)

42.73

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10 Colliding Bodies Optimization Algorithm for Structural …

Table 10.2 Aerodynamic forces in the structure

Total force (kN)

696.96

Total moment (kN m)

74.30

10.6.1 Hydrodynamic Loading When calculating hydrodynamic load, it is assumed that both drag and inertia terms of Morrison equation simultaneously take place. They both, in addition, are calculated in the phase angle equal to zero. Overall, using the Morrison equation, the wave load in both ends of each member is calculated and then exerted on the member as a uniformly distributed load by averaging the hydrodynamic loads acting on the start and end nodes.

10.6.2 Aerodynamic Loading The same process as the hydrodynamic load is carried out for wind load acting on both towers and supporting structure elements, which means that the wind load is considered as a uniformly distributed load on each element. As mentioned, the thrust and aerodynamic moments acting on the RNA in the stopped mode are approximately calculated as follows (Table 10.2).

10.6.3 Final Results In this chapter, the optimal design of the OC4 reference jacket supporting structure is investigated using the Colliding Bodies Optimization algorithm. As mentioned, this optimization problem deals with 20 design variables—diameters and thicknesses of supporting structure elements—which are further categorized in ten design groups. Thirty colliding bodies in 500 iterations are employed to explore the optimal design of the problem. Indicating the accuracy of the algorithms, the outcomes of this research are compared with those of [9]. The thickness and diameter of each element of the structure are chosen from 0.01 m to 0.1 m and from 0.1 m to 5 m, respectively. The outcomes of the optimization process are mentioned in Table 10.3. The weight of supporting structure during the optimization process in each iteration and its corresponding penalized cost function is depicted in Figs. 10.5 and 10.6, respectively. As noticed, none of the constraints is violated during the optimization.

10.7 Discussions and Conclusion Table 10.3 Optimum design variables using CBO algorithm

233

Design variable

Original supporting structure

Oest et al. [9]

Proposed method

D1 (m)

0.8

0.5034

0.4226

D2 (m)

1.2

0.9266

1.4420

D3 (m)

0.8

0.5941

0.5449

D4 (m)

1.2

0.9266

1.4395

D5 (m)

0.8

0.5795

0.5294

D6 (m)

1.2

0.7854

1.4004

D7 (m)

0.8

0.5801

0.4970

D8 (m)

1.2

0.7546

0.6001

D9 (m)

0.8

0.5680

0.3928

D10 (m)

1.2

0.9661

0.2605

t1 (m)

0.020

0.0126

0.0113

t2 (m)

0.050

0.0315

0.0244

t3 (m)

0.020

0.0149

0.0114

t4 (m)

0.035

0.0223

0.0687

t5 (m)

0.020

0.0145

0.0104

t6 (m)

0.035

0.0220

0.0242

t7 (m)

0.020

0.0145

0.0126

t8 (m)

0.035

0.0255

0.0304

t9 (m)

0.020

0.0154

0.0163

t10 (m)

0.040

0.0256

0.0452

First frequency (Hz)

0.2262

Second frequency (Hz)

0.2748

10.7 Discussions and Conclusion The structural optimization of the offshore structures, more specifically offshore wind turbines, is of the most complex engineering tasks. The outstanding dependency between the intensity of environmental load cases and utilized cross-sections in members further aggravates this difficulty. Thus, the optimal design of jacket supporting structures is investigated using meta-heuristic algorithms—Colliding Bodies Optimization (CBO) algorithm. The OC4 reference jacket is considered as the design example in this research. The structure is modeled in MATLAB, and its optimal design is then performed while ULS and frequency constraints being considered. The wind effect on the Rotor Nacelle Assembly (RNA) is calculated using a scaling approximation suggested by Manwell et al. [13]. That utilized, the aerodynamic loads of a known wind turbine are readily converted to the desired case. Additionally, hydrodynamic load on the supporting structure elements is quantified based on the Morrison equation. Since the structure comprises

234

10 Colliding Bodies Optimization Algorithm for Structural …

Fig. 10.5 Supporting structure weight during the optimization process

Fig. 10.6 Convergence curve during the optimization process

10.7 Discussions and Conclusion

235

of both horizontal and oblique members, using some geometrical manipulations, the normal water particle kinematics to the member axis, including velocity and acceleration, are firstly assessed. Employing the Morrison equation, the normal wave load to the member axis of each element is then calculated. Afterward, the CBO algorithm is deployed, attempting to explore the optimal design of the jacket supporting structure under extreme weather conditions. Comparing the outcomes of this research with the initial values, noticeable weight reduction is observed while fulfilling all constraints.

References 1. Kaveh, A., & Sabeti, S. (2018). Structural optimization of jacket supporting structures for offshore wind turbines using colliding bodies optimization algorithm. The Structural Design of Tall and Special Buildings, 27(13), e1494. https://doi.org/10.1002/tal.1494. 2. Uys, P. E., Farkas, J., Jármai, K., & van Tonder, F. (2007). Optimisation of a steel tower for a wind turbine structure. Engineering Structures, 29(7), 1337–1342. https://doi.org/10.1016/j. engstruct.2006.08.011. 3. Chen, J., Yang, R., Ma, R., & Li, J. (2016). Design optimization of wind turbine tower with lattice-tubular hybrid structure using particle swarm algorithm. The Structural Design of Tall and Special Buildings, 25(15), 743–758. https://doi.org/10.1002/tal.1281. 4. Thiry, A., Rigo, P., Buldgen, L., Raboni, G., & Bair, F. (2011). Optimization of monopile offshore wind structures. In W. F. Carlos Guedes Soares (Ed.), Advances in marine structures. London: CRC Press. 5. Long, H., Moe, G., & Fischer, T. (2011). Lattice towers for bottom-fixed offshore wind turbines in the ultimate limit state: Variation of some geometric parameters. Journal of Offshore Mechanics and Arctic Engineering, 134(2). https://doi.org/10.1115/1.4004634. 6. Long, H., & Moe, G. (2012). Preliminary design of bottom-fixed lattice offshore wind turbine towers in the fatigue limit state by the frequency domain method. Journal of Offshore Mechanics and Arctic Engineering, 134(3). https://doi.org/10.1115/1.4005200. 7. Zwick, D., Muskulus, M., & Moe, G. (2012). Iterative optimization approach for the design of full-height lattice towers for offshore wind turbines. Energy Procedia, 24, 297–304. https:// doi.org/10.1016/j.egypro.2012.06.112. 8. Zwick, D., & Muskulus, M. (2016). Simplified fatigue load assessment in offshore wind turbine structural analysis. Wind Energy, 19(2), 265–278. https://doi.org/10.1002/we.1831. 9. Oest, J., Sørensen, R., T. Overgaard, L. C., & Lund, E. (2017). Structural optimization with fatigue and ultimate limit constraints of jacket structures for large offshore wind turbines. Structural and Multidisciplinary Optimization, 55(3), 779–793. https://doi.org/10.1007/s00158016-1527-x. 10. Kaveh, A., & Sabeti, S. (2018). Optimal design of jacket supporting structures for offshore wind turbines using CBO and ECBO algorithms. Periodica Polytechnica Civil Engineering, 62(3), 545–554. https://doi.org/10.3311/PPci.11651. 11. Kaveh, A., Mahdavi, V. R. (2015). Colliding bodies optimization: Extensions and applications. Berlin: Springer. 12. DNV, G. (2014) DNV-OS-J101–Design of offshore wind turbine structures. Oslo: DNV GL. 13. Manwell, J. F., McGowan, J. G. & Rogers, A. L. (2010). Wind turbine design and testing. Wind energy explained (pp. 311–357). Hoboken, NJ: Wiley. 14. CEN, E. (2010). 3: Design of steel structures-Part 1-1: General rules and rules for buildings. Brussels: CEN.

Chapter 11

Colliding Bodies Optimization for Analysis and Design of Water Distribution Systems

11.1 Introduction This chapter describes the application of the Colliding Bodies Optimization algorithm (CBO) for simultaneous analysis, design, and optimization of Water Distribution Systems (WDSs). In this method, analysis is carried out using CBO, which is a population-based search approach imitating nature’s ongoing search for better solutions. Also, design and cost optimization of WDSs are performed simultaneously with the analysis process using a new objective function in order to satisfy the analysis criteria, design constraints, and cost optimization. Several practical examples of WDSs are selected to demonstrate the efficiency of the presented algorithm. The comparison of obtained results signifies the efficiency of the CBO method in reducing the WDSs construction cost and computational time of the analysis [1]. Nowadays, due to the huge extension in size and dimension of the structures, there has been a great increase in the weight and cost of construction materials used for structures. Therefore, it is not surprising that much attention is being paid by engineers to the optimal design of structures, which leads to a significant decrease in their costs. One of the most imperative fields in which the optimization and resource management needs special consideration is the water distribution system. The water distribution system is an essential infrastructure, which consists of hydraulic components such as pipes, valves, reservoirs, and pumps. This system supplies water in the highly capitalized societies in the desired quantity for consumers in a reliable form. This configuration usually is simplified by a graph layout that has a number of nodes denoting the places in the urban area, lines denoting the pipes, and other features such as reservoir and pumps. The construction and maintenance of the water distribution system pipelines to supply water can cost millions of dollars every year. Due to the high costs associated with the construction of water distribution systems (WDSs), much research has been dedicated to the development of methods to minimize the capital costs associated with such infrastructure. © Springer Nature Switzerland AG 2020 A. Kaveh and A. Dadras Eslamlou, Metaheuristic Optimization Algorithms in Civil Engineering: New Applications, Studies in Computational Intelligence 900, https://doi.org/10.1007/978-3-030-45473-9_11

237

238

11 Colliding Bodies Optimization for Analysis and Design …

Traditionally, water distribution system design is based on trial-and-error methods employing the experience. However, in the light of the optimization of cost and profits, designing the best layout of the water supply system, counting the best selection of water demands and pipe length and diameter within the millions of possible configurations, attracted a large amount of literature during the last decades. The majority of the literature has focused on cost, though; some others deal with other aspects of designing, such as reliability. The nonlinear nature of equations involved in the water distribution system, conservation of mass and energy (hydraulic head loss) equations, made this field of engineering as a challenging scope. The research in optimization has attracted many researchers focusing on various programming methods such as linear and non-linear programming. Alperovits and Shamir [2] reduced the complexity of an original nonlinear problem by solving a series of linear sub-problems. In this method, a linear programming problem is solved for given flow distribution, and then a search is conducted in the space of the flow variables. This method was followed, and other methods developed, examples of which are Goulter et al. [3], Kessler and Shamir [4], and Fujiwara and Kang [5] who used the two-phase decomposition method. Metaheuristic methods such as Genetic Algorithm [6], Ant colony optimization [7], the Shuffled Frog-Leaping Algorithm [8] were also utilized in several optimization approaches for water distribution networks. Geem [9], who developed harmony search (HS) and particle-swarm harmony search (PSHS) and Eusuff and Lansey [9], who proposed the SFLA model, also employed their techniques for water distribution system optimization. Tolson et al. [10] developed a hybrid discrete-dynamically dimensioned search (HD-DDS) algorithm to perform the optimal design of the water distribution system. One of the new meta-heuristic methods that recently developed by Kaveh and Mahdavi [11] is the Colliding Bodies optimization method (CBO). In this chapter, the CBO algorithm is used as a metaheuristic optimization algorithm together with a classic analyzer such as the Newton-Raphson approach. In classic methods, pipe demands are often calculated using indirect methods, and pre-selected pipe sizes are utilized. However, here, the pipe sizes and demands are considered as the optimization variables leading to simultaneous analysis, design, and optimization.

11.2 Water Distribution Network Optimization Problem The water distribution network optimization problem is defined as the selection of the most desirable configuration of the circulation network, considering the allowable pipe diameter and water demand in each point while satisfying various possible objectives such as network reliability, redundancy, and water quality. The most common and favorable objective function of the water distribution system is considered as minimizing the network arrangement cost, by suitable selection of pipe diameters and lengths. This cost can be expressed as

11.2 Water Distribution Network Optimization Problem

C=

N 

f (Di , L i )

239

(11.1)

i=1

where f (Di , L i ) is the cost of ith pipe, with diameter Di and length L i , and N is the number of pipes in the network configuration. In engineering problems, usually, two phases should be performed to achieve a goal, analysis, and design. In the water distribution systems problem, as a complex system of pipes, the goal is defined as searching for the length and diameters of the pipes and obtaining the required water demands at certain points of the network.

11.2.1 Analysis Phase In the analysis phase, the goal is to achieve a suitable water distribution for the postulated configuration of pipe lengths and diameters among an infinite number of scenarios. This goal is achieved in the light of the fact that our proposed distribution should satisfy the continuity equation in each node, and satisfy the hydraulic head loss principle in the system loops. In other words, only a few distributions can assure the continuity equation in each node, and through these distributions, only one distribution can satisfy the hydraulic head loss equations. Continuity equation or mass conservation at each node is given by 

Q in −



Q out = Q e

(11.2)

where Q in is the volumetric flow rate to the node, Q out is the flow rate out of the node, and Q e is the external inflow rate to the node. Each loop is defined as a series of pipe configurations, where the differences between the head losses of the two end nodes of its pipes should be summed in order to find the head loss of the entire loop. For the conservation of energy, this sum should be equal to zero. If a loop has other features such as pumps, its energy interactions should also be added to the conservation equation formula as 

hf −



Ep = 0

(11.3)

where h f is the hydraulic head loss calculated by the Hazen-Williams or DarcyWeisbach formulae and E p is the energy added to water at the loop by a pump. The above equation is also known as the hydraulic head loss equation. For the analysis of a water distribution system, fundamental principles of water systems are used. The principle of water branching has an interesting analogy with characteristics of an electric circuit when the rate of the flow corresponds to the electric current, and the head loss corresponds to the drop in potential. The hydraulic head loss, between two nodes i and j, can be expressed by Hazen-Williams formula as:

240

11 Colliding Bodies Optimization for Analysis and Design …

hf = ω

L C α Dβ

Qα

(11.4)

where ω is a numerical conversion constant; α is a coefficient equal to 1.85, and β is coefficient equal to 4.87. Based on the analogy between the electric circuits and the pipe branching, when two pipes are in the form of series, the head loss in this series configuration will equal to the sum of head losses of the constituting pipes (determined by Eq. 11.4), and the flow is equal to the flow rate of each pipe. h t = ω

La β α C a Da

Qα + ω

Lb β C α Db

Qt = Qa = Qb

(11.5) (11.6)

where a and b denote the pipe a and pipe b which are used in the series configuration of a pipe network. Now considering the fact that each network may include a combination of parallel and series arrangement of branching pipes, the formulation of the water distribution network is obvious. However, a network configuration has other features such as loops and reservoirs, which should be carefully dealt with, and as a result, other equations should be set to achieve the best supply system.

11.2.2 Design Phase In the design phase of the water distribution system, the pipe diameters satisfying the water demand in each node and the place of the urban area should be determined. As previously mentioned, in this section, the third imperative requirement of the water distribution system design should be set. This requirement is the minimum pressure, which imposes some limitations in order to prevent system failure. Thus, during the network configuration assortment, the pressure in each point should be checked. For each node in the network, the minimum pressure constraint is given in the following form: H j ≥ H jmin ; j = 1, . . . , M

(11.7)

where H j , H jmin and M denote the pressure head at node j, the minimum required pressure at node j, and the number of nodes in the network, respectively. Other requirements such as reliability, minimum and maximum limit of the velocity and the maximum pressure should be satisfied in the design phase. To attain the network that satisfies the water requirement, conservation of mass and energy equations in each node and loop should be coupled and solved. These equations can be arranged in the following form:

11.2 Water Distribution Network Optimization Problem

241



 Q H × qp − =0 N ull (M, 1)

(11.8)

where Q is the demand in each node, and N ull(M, 1) is a M × 1 zero vector with M being the number of loops. This zero vector indicates that in each loop, the summation of the pipe’s head loss should be zero, as the conservation of energy implies. It can be seen that N demands node (N conservation of mass equation for each node) and M loop energy conservation equation, construct the above form of equations. q p denotes the flow rate of each node. The matrix H consists of two essential parts. The first part corresponds to the equation of the conservation of mass consisting of some positive and negative 1, indicating the input and output flow rate of each node. Besides, there are some 0 entries which obviously denote the pipes that are not relevant to the considered node in that equation whose flow rate is considered in q matrix in the same row. The second part of H, corresponds to M loops, an contains some positive and negative coefficients. The signs are determined according to the flow direction in each pipe, being assumed at the first step of the analysis (conservation of mass) and the postulated direction of the loops. These coefficients are determined using the Hazen-Williams formula. As previously mentioned, the primary directions assigned to the pipes may not satisfy the conservation of energy equation, and the correct directions are decided in the design process. As an example, Fig. 11.1 depicts a fundamental simple WDS example whose satisfaction equations can be presented as follows:  ±ql = −Q K k = 1, 2, . . . , 5 (11.9) l→k

Fig. 11.1 An example of simple fundamental WDSs

242

11 Colliding Bodies Optimization for Analysis and Design …



±Al |ql |n−1 ql = 0.0 m = 1, 2

(11.10)

l→m

⎤⎡ ⎤ ⎡ ⎤ −Q 2 q1 1 −1 −1 0 0 0 ⎢0 ⎥⎢ q ⎥ ⎢ −Q ⎥ 1 0 −1 −1 0 3⎥ ⎢ ⎥⎢ 2 ⎥ ⎢ ⎢ ⎥⎢ ⎥ ⎢ ⎥ 0 1 1 0 −1 ⎢0 ⎥⎢ q3 ⎥ ⎢ −Q 4 ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎢0 ⎥⎢ q4 ⎥ ⎢ −Q 5 ⎥ 0 0 0 1 1 ⎢ ⎥⎢ ⎥ ⎢ ⎥ n−1 n−1 n−1 ⎣ 0 A2 |q2 | ⎦⎣ q5 ⎦ ⎣ 0.0 ⎦ −A3 |q3 | A4 |q4 | 0 0 q6 0 0 0 −A4 |q4 |n−1 A5 |q5 |n−1 −A6 |q6 |n−1 0.0 (11.11) ⎡

where A = ω C αLDβ . As an example, in the first four rows of the H matrix (corresponding to four nodes where the water is being used), the first part of H is presented. In the first row of the matrix, the entry for pipe number 1 is positive since the direction of the flow in this pipe has an input role to the point. While the pipes 2 and 3 play the output role. As an illustration of the second part, bearing in mind the first loop, the direction of pipe numbers 2 and 4 are negative. In the second part of the matrix, considering the loop 1, the direction of the pipes 2 and 4 are the same as the direction of loop 1, thus have positive signs. While the pipe number 3 acts in the reverse direction of the loop. Finally, it should be mentioned that, in this chapter, similar to that of the Fujiwara and Kang [5], to achieve a feasible design, the configuration of series pipes that have the standard diameters are used. For example, if the program chooses the pipes with the diameter 38 inch for the system, which corresponds to neither the standard 30 inch nor to the 40 inch pipes, the later subroutine would change the pipe to two series pipes. One of the pipes would have a diameter equal to 30 inch, and the other will be 40 inch. This exchange should be made such that the sum of the lengths of two pipes is equal to the length of the primary pipe. Because these two pipes should have the same demand as that of the primary pipe, and the total hydraulic head loss of these two pipes should be equal to the primary pipe.

11.3 The Colliding Bodies Optimization Algorithm Many meta-heuristic approaches are inspired by solutions that nature seems to have chosen for hard problems. The collision is a natural occurrence, which happens between objects, bodies, cars, etc. The CBO algorithm is one of the populationbased meta-heuristic search methods that developed based on this phenomenon. In this method, each agent (CB) is considered as a colliding body with mass m and mimics the collision between two objects in one-dimension, in which one object collides with other object, and they move toward minimum energy level [11].

11.3 The Colliding Bodies Optimization Algorithm

243

11.3.1 Collision Laws In physics, collisions between bodies are governed by (i) laws of momentum and (ii) laws of energy. When a collision occurs in an isolated system, Fig. 11.2, the total momentum and energy of the system of objects is conserved. The conservation of the total momentum requires the total momentum before the collision to be the same as the total momentum after the collision, and can be expressed as: m 1 v1 + m 2 v2 = m 1 v1 + m 2 v2

(11.12)

Likewise, the conservation of the total kinetic energy is expressed by 1 1 1 1 m 1 v12 + m 2 v22 = m 1 v1 2 + m 2 v2 2 + Q 2 2 2 2

(11.13)

where v1 is the initial velocity of the first object before impact, v2 is the initial velocity of the second object before impact, v1 is the final velocity of the first object after impact, v2 is the final velocity of the second object after impact, m 1 is the mass of the first object, m 2 is the mass of the second object, and Q is the loss of kinetic energy due to impact. The velocity after a one-dimensional collision can be obtained as: v1 =

(m 1 − εm 2 )v1 + (m 2 + εm 2 )v2 m1 + m2

(11.14)

v2 =

(m 2 − εm 1 )v2 + (m 1 + εm 1 )v1 m1 + m2

(11.15)

where ε is the coefficient of restitution (COR) of two colliding bodies, defined as the ratio of the relative velocity of separation to the relative velocity of approaching:    v − v  v 2 1 ε= = |v2 − v1 | v

(11.16)

According to the coefficient of restitution, two special cases of collision can be considered as:

Fig. 11.2 The collision between two bodies. a Before the collision. b After the collision

244

11 Colliding Bodies Optimization for Analysis and Design …

A perfectly elastic collision is defined as the one in which there is no loss of kinetic energy in the collision (Q = 0 and ε = 1). In reality, any macroscopic collision between objects will convert some kinetic energy to internal energy and other forms of energy. In this case, after the collision the velocity of separation is high. An inelastic collision is the one in which part of the kinetic is changed to some other form of energy in the collision. Momentum is conserved in inelastic collisions (as it is for elastic collision), but one cannot track the kinetic energy through the collision since some of it is converted to other forms of energy. In this case, the coefficient of restitution does not equal to one (Q = 0 and ε ≤ 1). Here, after the collision the velocity of separation is low. For most of the real objects, ε is between 0 and 1.

11.3.2 The CBO Algorithm In the CBO algorithm, each solution candidate X i is considered as a colliding body (CB). The massed objects are composed of two main equal groups, i.e., stationary and moving objects, where the moving objects move to follow stationary objects, and a collision occurs between pairs of objects. This collision is performed for two purposes: (i) to improve the positions of moving objects; (ii) to push stationary objects towards better positions. After the collision, the new positions of the colliding bodies are updated based on the new velocity by using the collision laws, as discussed in Sect. 11.3.1. The pseudo-code for the CBO algorithm can be summarized as follows: Step 1: Initialization. The initial positions of CBs are determined randomly in the search space: xi0 = xmin + rand.(xmax − xmin ) i = 1, 2, . . . , n

(11.17)

where xi0 determines the initial value vector of the ith CB. xmin and xmax are the minimum and the maximum allowable values vectors of variables; rand is a random number in the interval (0, 1), and n is the number of CBs. Step 2: Determination of the body mass for each CB. The magnitude of the body mass for each CB is defined as: m k = n

1 f it (k)

1 i=1 f it (i)

, k = 1, 2, . . . , n

(11.18)

where f it (i) represents the objective function value of the agent i; n is the population size. Obviously, a CB with good values exerts a larger mass than the bad ones. Also, for maximizing the objective function the term f it1(i) is replaced by f it (i). Step 3: Arrangement of the CBs. The arrangement of the CBs objective function values is performed in ascending order (Fig. 11.3a). The sorted CBs are equally

11.3 The Colliding Bodies Optimization Algorithm

245

divided into two groups: • The lower half of CBs (stationary CBs); These CBs are good agents that are stationary, and the velocity of these bodies before the collision is zero. Thus: vi = 0 i = 1, 2, . . . ,

n 2

(11.19)

• The upper half of CBs (moving CBs): These CBs move toward the lower half. Then, according to Fig. 11.3b, the better and worse CBs, i.e., agents with upper fitness value of each group will collide together. The change of the body position represents the velocity of these bodies before the collision as: vi = xi − xi− n2 i =

n + 1, . . . , n 2

(11.20)

where vi and xi are the velocity and position vector of the ith CB in this group, respectively, xi− n2 is the ith CB’s pair position in the previous group.

Fig. 11.3 a The sorted CBs in increasing order. b The pairs of objects for the collision

246

11 Colliding Bodies Optimization for Analysis and Design …

Step 4: Calculation of the new position of the CBs. After the collision, the velocity of bodies in each group is evaluated using Eqs. (11.14) and (11.15) and the velocities before the collision. The velocity of each moving CB after the collision is: vi =

(m i − εm i− n2 )vi m i + m i− n2

i=

n + 1, . . . , n 2

(11.21)

where vi and vi are the velocity of the ith moving CB before and after the collision, respectively; m i is the mass of the ith CB; m i− n2 is the mass of ith CB’s pair. Also, the velocity of each stationary CB after the collision is: vi =

(m i+ n2 + εm i+ n2 )vi+ n2 mi + m

i+ n2

i = 1, . . . ,

n 2

(11.22)

where vi+ n2 and vi are the velocity of the ith moving CB pair before and the ith stationary CB after the collision, respectively; m i is the mass of the ith CB; m i+ n2 is mass of the ith moving CB pair. As mentioned previously, ε is the coefficient of restitution (COR), and for most of the real objects, its value is between 0 and 1. It is defined as the ratio of the separation velocity of two agents after the collision to the approaching velocity of two agents before the collision. In the CBO algorithm, this index is used to control the exploration and exploitation rate. For this goal, the COR decreases linearly from unit to zero. Thus, ε is defined as: ε =1−

iter itermax

(11.23)

where iter is the actual iteration number and itermax is the maximum number of iterations, with COR being equal to the unit and zero representing the global and local search, respectively. New positions of CBs are obtained using the generated velocities after the collision in the position of stationary CBs. The new positions of each moving CB are: xinew = xi− n2 + rand ◦ vi i =

n + 1, . . . , n 2

(11.24)

where xinew and vi are the new position and the velocity after the collision of the ith moving CB, respectively, xi− n2 is the old position of the ith stationary CB pair. Also, the new positions of stationary CBs are obtained by: xinew = xi + rand ◦ vi i = 1, . . . ,

n 2

(11.25)

where xinew , xi and vi are the new position, old position, and the velocity after the collision of the ith stationary CB, respectively. rand is a random vector uniformly

11.3 The Colliding Bodies Optimization Algorithm

247

distributed in the range (−1,1) and the sign “◦” denotes an element-by-element multiplication. Step 5: Termination criterion control. Steps 2–4 are repeated until a termination criterion is satisfied. It should be noted that a body’s status (stationary or moving body) and its numbering are changed in two subsequent iterations.

11.4 A New Algorithm for Analysis and Design of the Water Distribution Networks As explained in Sect. 13.2, the matrix H, known as the stability matrix of the network, cannot be solved by a direct method. Thus, this matrix is solved utilizing different indirect approaches such as the Newton-Raphson method. Classic methods that use the above-mentioned indirect approaches perform the analysis and design steps separately, which requires a considerable amount of computational time. However, in the presented method, analysis, design, and optimization steps are performed simultaneously. In order to analyze a network, we have to find a set of pipe demands that satisfy the Eq. (11.8) mentioned in Sect. 13.2. In the present approach, the analysis phase is performed using the CBO algorithm by searching a vector of the pipe demands that satisfy the above equation. The lefthand side of this equation is a zero vector and should be changed to a scalar. The best is to find its norm. When the norm of a vector equals to zero, then all the arrays of the vector equal to zeros. Considering the norm of the above matrix as the analysis constraints can be a reliable technique. Then simultaneous with the design, the analysis phase will be performed by considering the following objective function as the optimization objective function: f (qp , D) =

L  i=1

+

  Q li × cos t (Di ) × 1 + nor m H × qp = Null(M, 1)

L 

gi (qi , Di )

(11.26)

i=1

Figure 11.4 shows the schematic procedure of designing and analysis of a water distribution system using the CBO algorithm, which is used in this chapter.

11.5 Design Examples In order to assure that this method is reliable and efficient in this field, three famous networks are selected from literature, which was studied by many other researchers.

248

11 Colliding Bodies Optimization for Analysis and Design …

Fig. 11.4 Flowchart of the present method

The following sections explain the comparative study of cost optimization of the water distribution system for these networks.

11.5 Design Examples

249

Fig. 11.5 Two-loop water distribution network

11.5.1 A Two-Loop Network The two-loop network, shown in Fig. 11.5, was first introduced by Alperovits and Shamir [2] for the implementation of linear programming to acquire the least cost solution, considering the weight of the pipes. This basic configuration was employed by different authors for comparison of their results for the optimal design of the water distribution system as a simple illustrative network. This network consists of 8 pipes, seven nodes, and two loops. The network is fed by gravity from a constant reservoir, which has 210 m fixed head. The length of all the pipes is assumed to be 1000 m with a Hazen-Williams coefficient (C) is considered as 130. Allowed pipe diameter and corresponding costs are available in Table 6 [9]. The minimum head limitation in each pipe is set to 30 m above ground level. Here, ω = 10.5088 is employed for the Hazen-Williams formulation as Savic and Walters [12]. Table 11.1 compares the results obtained using the CBO algorithm with those obtained by other methods. Also, Table 11.2 shows the corresponding nodal heads obtained in this chapter. As can be seen, in all nodes, the minimum nodal head requirement is satisfied.

11.5.2 Hanoi Water Distribution Network The Hanoi network is a real water circulation network that formerly studied by Fujiwara and Kang [14] in Vietnam (Fig. 11.6). This medium-size network includes 32 nodes, 34 pipes, 3 loops, and 1 gravity reservoir with a 100 m fixed head for

250

11 Colliding Bodies Optimization for Analysis and Design …

Table 11.1 Comparison of the pipe diameters for the two-loop network Pipe number

Alperovits and Shamir [2]

Goulter et al. [3]

Kessler and Shamir [4]

Kaveh et al. [13]

Present work (CBO)

Pipe length (m)

Pipe diameter (in)

Pipe length (m)

Pipe diameter (in)

1

20 18

20 18

18

L1 = 595.52 L2 = 404.48

D1 = 18 D2 = 16

L1 = 987.85 L2 = 12.15

D1 = 18 D2 = 16

2

8 6

10

12 10

602.78 397.22

10 8

74.8 925.2

12 10

3

18

16

16

94.36 905.64

20 18

998.25 1.75

16 14

4

8 6

6 4

3 2

582.75 417.25

8 6

981.93 18.07

3 2

5

16

16 14

16 14

806.91 193.09

16 14

934.62 65.38

16 14

6

12 10

12 10

12 10

174.46 825.54

10 8

996.85 3.15

10 8

7

6

10 8

10 8

934.91 65.09

8 6

751.14 248.86

10 8

2 1

996.25 3.75

2 1

8

6 4

2 1

3 2

978.63 21.37

Cost ($)

497,525

435,015

417,500

432,358

Table 11.2 Optimal pressure heads for two-loop network

415,070

Pipe number

Min pressure req. (m)

Pressure

1

–

–

2

30

53.21

3

30

30.80

4

30

43.38

5

30

30.87

6

30

30.08

7

30

30.02

its feeding. Like the previous example, the Hazen-Williamz coefficient C = 130 is employed for network water distribution equations. The tolerable of the pipe diameters, which indicates the difference in upper limitation diameter with the two-loop network, is displayed in Fig. 11.6. The water required in this network is much higher than the accustomed demands for other ones, so for satisfying these demands, the maximum velocity limitation is set to 7 m/s. As shown in Table 11.3, the CBO algorithm achieved good results in comparison to most of the previous researches. The obtained results show that CBO has produced significant improvement in the total cost of the network, and it is one of the best solutions for this network.

11.5 Design Examples

251

Fig. 11.6 Hanoi water distribution network

11.5.3 The Go Yang Water Distribution Network Kim et al. [15] originally presented the GoYang network in South Korea, as shown in Fig. 11.7. The system information such as elevations and water demand in each node are given in Table 11.4. As the table and figure show, the system consists of 30 pipes, 22 nodes, and 9 loops, and is fed by a pump (4.52 kW) from a 71 m fixed head reservoir. Pipe length and their designed diameters are presented in Table 11.5, considering that the Hazen- Williams coefficient C is taken as 100, and 8 commercial pipe diameters that presented in Table 11.6 are used for this network. The minimum head limitation is assumed to be 15 m above the ground level. Table 11.4 shows the corresponding node pressure obtained using the CBO method. It can be observed that the minimum pressure limitation is satisfied in all nodes of the network. Also, Table 11.5 compares the selected diameters obtained using CBO with those obtained using other methods. It is apparent from Table 11.5 that the CBO algorithm gives better results than other methods, and the corresponding cost obtained by this algorithm is equal to 176,946,211 Won(≈$176,946), while the original cost was 179,428,600 Won(≈$179,429).

252

11 Colliding Bodies Optimization for Analysis and Design …

Table 11.3 Comparison of the pipe diameters and the total cost for the hanoi network Pipe number

Pipe Length (m)

Fujiwara

Savic and Walters

1

100

40

40

2

1350

40

3

900

4

Harmony

Kaveh et al. [13]

Present work (CBO)

Pipe length (m)

Pipe diameter (in)

Pipe length (m)

Pipe diameter (in)

40

L1 = 99.96 L2 = 0.04

D1 = 40 D2 = 30

L1 = 99.70 L2 = 0.30

D1 = 40 D2 = 30

40

40

1349.75 0.25

40 30

1347.10 2.90

40 30

40

40

40

852.17 47.82

40 30

853 47

40 30

1150

40

40

40

1084.35 65.65

40 30

1084.40 65.60

40 30

5

1450

40

40

40

1299.37 150.62

40 30

1299.70 150.30

40 30

6

450

40

40

40

360.93 89.06

40 30

361.10 88.90

40 30

7

850

38.16

40

40

496.46 353.53

40 30

496.90 353.10

40 30

8

850

36.74

40

40

399.38 450.61

40 30

397.50 452.50

40 30

9

800

35.33

40

40

224.15 575.85

40 30

789.40 10.60

40 30

10

950

29.13

30

30

258.49 691.51

30 24

950

30

11

1200

26.45

24

24

1002.79 197.2

24 20

1200

24

12

3500

23.25

24

24

338.32 3161.68

24 20

1016.90 2483.10

30 24

13

800

19.57

20

20

684.30 115.70

20 16

800

20

14

500

15.62

16

16

402.94 97.06

16 12

447.60 52.40

16 12

15

550

12.00

12

12

6.99 543.01

16 12

550

12

16

2730

22.50

12

12

2687.58 42.42

20 16

2730

16

17

1750

25.24

16

16

1480.29 269.70

24 20

1750

20

18

800

29.01

20

20

475.23 324.77

30 24

800

24

19

400

29.28

20

20

246.80 153.20

30 24

15.30 384.70

24 20

20

2200

38.58

40

40

1573.23 626.77

40 30

1578.90 621.10

40 30 (continued)

11.6 Discussions and Conclusion

253

Table 11.3 (continued) Pipe number

Pipe Length (m)

Fujiwara

Savic and Walters

21

1500

17.36

20

22

500

12.65

23

2650

24

Harmony

Kaveh et al. [13]

Present work (CBO)

Pipe length (m)

Pipe diameter (in)

Pipe length (m)

Pipe diameter (in)

20

272.62 1227.38

20 16

1500

20

12

12

2.82 497.18

16 12

500

16

32.59

40

40

2529.05 120.95

30 24

2534.90 115.10

30 24

1230

22.06

30

30

1112.98 117.02

20 16

1111.40 118.60

20 16

25

1300

18.34

30

30

223.13 1076.87

20 16

222.30 1077.70

20 16

26

850

12.00

20

20

6.01 843.99

16 12

850

20

27

300

22.27

12

12

299.62 0.38

20 16

300

12

28

750

24.57

12

12

484.67 265.33

24 20

704.50 45.50

16 12

29

1500

21.29

16

16

1258.09 241.91

20 16

1500

12

30

2000

19.34

16

12

848.55 1151.45

20 16

834.80 1165.20

20 16

31

1600

16.52

12

12

1309.85 290.15

16 12

1600

16

32

150

12.00

12

16

0.28 149.72

16 12

150

12

33

860

12.00

16

16

4.40 855.60

16 12

860

20

34

950

22.43

20

24

888.35 61.65

20 16

950

30

Cost ($)

–

6,320,000

6,073,000

6,056,000

5,562,343

5,741,900

11.6 Discussions and Conclusion In this chapter, the CBO algorithm is applied to the least-cost design of WDSs. One of the most important features of this method is the simultaneous analysis, design, and optimization requiring less computational time. While the analysis and optimal design of WDSs are performed in two separate phases in the existing methods (some use software such as Epanet 2, and some others employ different optimization methods). Also, the new algorithm, so-called CBO, utilizes simple formulation, and it requires no parameter selection. For water distribution network problems, tuning of parameters is a really hard task, and appropriate parameter values are very difficult

254

11 Colliding Bodies Optimization for Analysis and Design …

Fig. 11.7 GoYang water distribution network Table 11.4 Nodal data and the computational results for the GoYang network Pipe number

Water demand (cmd)

Ground level (m)

Pressure (original) (m)

Pressure (NLP) (m)

Pressure (HS) (m)

Pressure (CBO) (m)

1

−2550.0

71.0

15.61

15.61

15.61

15.61

2

153.0

56.4

28.91

28.91

24.91

28.18

3

70.5

53.8

31.18

31.15

26.32

27.58

4

58.5

54.9

29.53

29.1

24.11

26.31

5

75.0

56.0

28.16

27.47

22.78

24.92

6

67.5

57.0

26.91

25.44

20.67

23.36

7

63.0

53.9

30.46

30.75

25.34

27.18

8

48.0

54.5

29.80

29.48

24.41

26.17

9

42.0

57.9

26.05

24.48

20.01

20.16

10

30.0

62.1

21.50

20.17

15.43

15.16

11

42.0

62.8

20.92

19.79

15.06

15.18

12

37.5

58.6

24.34

22.95

18.16

20.50

13

37.5

59.3

23.54

22.07

17.38

17.67

14

63.0

59.8

21.43

20.84

15.27

16.0

15

445.5

59.2

21.59

20.78

15.42

16.54

16

108.0

53.6

31.06

30.65

25.88

26.8

17

79.5

54.8

29.05

28.97

24.29

24.7

18

55.5

55.1

28.76

28.87

23.99

24.18

19

118.5

54.2

29.49

29.14

24.89

27.54

20

124.5

54.5

28.80

27.96

24.43

27.2

21

31.5

62.9

21.06

20.18

16.89

20.04

22

799.5

61.8

21.47

20.07

17.21

20.28

11.6 Discussions and Conclusion

255

Table 11.5 Comparison of the pipe diameters for the GoYang network Pipe number

Pipe length (m)

Diameter (original) (mm)

Diameter (NLP) (mm)

Diameter (HS) (mm)

Length (CBO) (mm)

Diameter (CBO) (mm)

1

165.0

200

200

150

L1 = 134.62 L2 = 30.38

D1 = 200 D2 = 150

2

124.0

200

200

150

108.54 15.46

125 100

3

118.0

150

125

125

0.15 117.85

125 100

4

81.0

150

125

150

15.21 65.79

100 80

5

134.0

150

100

100

120.91 13.09

100 80

6

135.0

100

100

100

113.74 21.26

100 80

7

202.0

80

80

80

202.0

80

8

135.0

100

80

80

135.0

80

9

170.0

80

80

80

170.0

80

10

113.0

80

80

80

113.0

80

11

335.0

80

80

80

335.0

80

12

115.0

80

80

80

115.0

80

13

345.0

80

80

80

345.0

80

14

114.0

80

80

80

114.0

80

15

103.0

100

80

80

103.0

80

16

261.0

80

80

80

261.0

80

17

72.0

80

80

80

72.0

80

18

373.0

80

100

80

373.0

80

19

98.0

80

125

80

98.0

80

20

110.0

80

80

80

110.0

80

21

98.0

80

80

80

98.0

80

22

246.0

80

80

80

10.96 235.04

100 80

23

174.0

80

80

80

174.0

80

24

102.0

80

80

80

55.62 46.38

100 80

25

92.0

80

80

80

40.28 51.72

100 80

26

100.0

80

80

80

100.0

80 (continued)

256

11 Colliding Bodies Optimization for Analysis and Design …

Table 11.5 (continued) Pipe number

Pipe length (m)

Diameter (original) (mm)

Diameter (NLP) (mm)

Diameter (HS) (mm)

Length (CBO) (mm)

Diameter (CBO) (mm)

27

130.0

80

80

80

130.0

80

28

90.0

80

80

80

18.79 71.21

100 80

29

185.0

80

100

80

185.0

80

30

90.0

80

80

80

90.0

80

Cost (Won)

–

179,428,600

179,142,700

177,135,800

176,946,211

Table 11.6 Candidate pipe diameters Network

Candidate diameter

Corresponding cost

Two-loop

{1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24} in inches

{2, 5, 8, 11, 16, 23, 32, 50, 60, 90, 130, 170, 300, 550} in dollar/meter

Hanoi

{12, 16, 20, 24, 30, 40} in inches

{45.726,70.4,98.378, 129.333, 180.748, 278.28} in dollar/meter

GoYang

{80, 100, 125, 150, 200, 250, 300, 350} in millimeters

{37,890; 38,933; 40,563; 42,554; 47,624; 54,125; 62,109; 71,524} in won/meter

to select, while this algorithm does not involve any internal parameter. This feature of CBO is a definite strength. In order to show that this method is reliable and capable in this field of engineering, three famous WDSs are selected from literature, which are studied by many other researchers. It is observed that optimization results obtained by the colliding bodies optimization method have less cost in comparison with more results obtained using other common methods. Therefore, this method is a reliable approach for the optimal design of water distribution networks.

References 1. Kaveh, A., Shokohi, F., & Ahmadi, B. (2014). Analysis and design of water distribution systems via colliding bodies optimization. International Journal of Optimization in Civil Engineering, 4(2), 165–185. 2. Alperovits, E., & Shamir, U. (1977). Design of optimal water distribution systems. Water Resources Research, 13(6), 885–900. https://doi.org/10.1029/WR013i006p00885. 3. Goulter, I. C., Lussier, B. M., & Morgan, D. R. (1986). Implications of head loss path choice in the optimization of water distribution networks. Water Resources Research, 22(5), 819–822. https://doi.org/10.1029/WR022i005p00819. 4. Kessler, A., & Shamir, U. (1989). Analysis of the linear programming gradient method for optimal design of water supply networks. Water Resources Research, 25(7), 1469–1480. https:// doi.org/10.1029/WR025i007p01469.

References

257

5. Fujiwara, O., & Khang, D. B. (1990). A two-phase decomposition method for optimal design of looped water distribution networks. Water Resources Research, 26(4), 539–549. https://doi. org/10.1029/WR026i004p00539. 6. Wu, Z. Y., Boulos, P. F., Orr, C. H., & Ro, J. J. (2001). Using genetic algorithms to rehabilitate distribution systems. Journal AWWA, 93(11), 74–85. https://doi.org/10.1002/j.1551-8833. 2001.tb09335.x. 7. Maier, H. R., Simpson, A. R., Zecchin, A. C., Foong, W. K., Phang, K. Y., Seah, H. Y., et al. (2003). Ant colony optimization for design of water distribution systems. Journal of Water Resources Planning and Management, 129(3), 200–209. https://doi.org/10.1061/(ASCE)07339496(2003)129:3(200). 8. Eusuff Muzaffar, M., & Lansey Kevin, E. (2003). Optimization of water distribution network design using the shuffled frog leaping algorithm. Journal of Water Resources Planning and Management, 129(3), 210–225. https://doi.org/10.1061/(ASCE)0733-9496(2003)129:3(210). 9. Geem, Z. W. (2006). Optimal cost design of water distribution networks using harmony search. Engineering Optimization, 38(3), 259–277. https://doi.org/10.1080/03052150500467430. 10. Tolson, B. A., Asadzadeh, M., Maier, H. R., Zecchin, A. (2009) Hybrid discrete dynamically dimensioned search (HD-DDS) algorithm for water distribution system design optimization. Water Resources Research, 45(12). https://doi.org/10.1029/2008wr007673. 11. Kaveh, A., Mahdavi, V. R. (2015) Colliding bodies optimization: Extensions and applications. New York: Springer. 12. Savic Dragan, A., & Walters Godfrey, A. (1997). Genetic algorithms for least-cost design of water distribution networks. Journal of Water Resources Planning and Management, 123(2), 67–77. https://doi.org/10.1061/(ASCE)0733-9496(1997)123:2(67). 13. Kaveh, A., Ahmadi, B., Shokohi, F., & Bohlooli, N. (2013). Simultaneous analysis, design and optimization of water distribution systems using supervised charged system search. International Journal of Optimization in Civil Engineering, 3(1), 37–55. 14. Sodeyama, H., Sunakoda, K., Fujitani, H., Soda, S., Iwata, N., & Hata, K. (2003). Dynamic tests and simulation of magneto—rheological dampers. Computer-Aided Civil and Infrastructure Engineering, 18(1), 45–57. 15. Kim, J., Kim, T., Kim, J., & Yoon, Y. (1994). A study on the pipe network system design using non-linear programming. Journal of Korea Water Resources Association, 27(4), 59–67.

Chapter 12

Optimization of Tower Crane Location and Material Quantity Between Supply and Demand Points

12.1 Introduction In the construction projects, the location of the tower crane as expensive equipment has an important effect on material transportation costs. The appropriate location of tower cranes for material supply and engineering demands is a combinatorial optimization problem within the tower crane layout problem that is difficult to resolve. Due to the construction site conditions, there are several tower crane location optimization models. Meta-heuristics are popular and useful techniques to resolve these complex optimization problems. In this chapter, the performance of the Particle Swarm Optimization (PSO) and four newly developed meta-heuristic algorithms Colliding Bodies Optimization (CBO), Enhanced Colliding Bodies Optimization (ECBO), Vibrating Particles System (VPS), and Enhanced Vibrating Particles System (EVPS) are compared in terms of their effectiveness in resolving a practical Tower Crane Layout (TCL) problem. In recent decades, many researchers have tried to provide the best method for solving the Construction Engineering Optimization Problems (CEOPs). Construction site layout problems (CSLPs) are the most interesting CEOPs, because they brought the consideration of layout aesthetics and usability qualities into the facility design process. Many building construction projects utilize tower cranes for transporting heavy construction materials. Material transportation is one of the major activities in the building construction industry, lifting and hoisting heavy materials by cranes in construction sites are usual tasks that need precise planning. Every construction project requires enough space for temporary facilities to perform construction activities safely and efficiently. Planning construction site spaces for safe and efficient working status is a complex and multi-disciplinary task as it involves accounting for a wide range of scenarios. CSLPs are known as largescale optimization problems. There are two methods to solve large size problems,

© Springer Nature Switzerland AG 2020 A. Kaveh and A. Dadras Eslamlou, Metaheuristic Optimization Algorithms in Civil Engineering: New Applications, Studies in Computational Intelligence 900, https://doi.org/10.1007/978-3-030-45473-9_12

259

260

12 Optimization of Tower Crane Location and Material Quantity …

the meta-heuristics and the exact methods with the global search for smaller search sized problems. For example, Li and Love [1] developed a construction site-level facility layout problem for allocating a set of predetermined facilities into a set of locations, while satisfying the layout constraints and requirements. They applied the genetic algorithm to solve the CSLP by assuming that the predetermined locations are rectangular and are large enough to accommodate the largest facility. Gharaie et al. [2] resolved their model by Ant Colony Optimization. Kaveh et al. [3] applied Colliding Bodies Optimization (CBO) and its enhanced version (ECBO). The tower crane is a major facility in the transportation of materials, especially heavy prefabricated units such as steel beams, ready-mixed concrete, prefabricated elements, and large-panel formworks. TCLP tries to find the best position of tower cranes and supply points in a building construction site for supplying all requests in a minimum time, has been raised since about twenty years ago. Zhang et al. [4] developed an analytical model by considering the travel time of tower crane hooks and adopting a Monte Carlo simulation to optimize the tower crane location. However, their considered tower crane was a single crane, and the effect of the location of supply points on lifting requirements and travel was neglected. An artificial neural network model by Tam et al. [5] was applied for predicting tower crane operations. They employed a genetic algorithm model to optimize the crane and supply points layout. The case study used by Tam et al. was subsequently used in several kinds of research to compare the effectiveness of other optimization methods. For example, mixed-integer linear programming (MILP) used by Huang et al. [6] to optimize the crane and supply points location showed that their method reduced the travel time of the hook by 7% compared to the results obtained from the previous genetic algorithms. Kaveh and Vazirinia [7] compared the performance of CBO, ECBO, and VPS in this model and discussed the results. A particle bee algorithm (PBA) with two cases was developed by Lien and Cheng [8] to optimize the tower crane layout and material quantity between supply and demand points. Herdani [9] developed an Evolutionary Big-Bang Big-Crunch EBB-BC and applied it for optimizing this model. In this chapter, four newly developed meta-heuristic algorithms called Colliding Bodies Optimization (CBO) [10], Enhanced Colliding Bodies Optimization (ECBO) [11], Vibrating Particles System (VPS) [12], and Enhanced Vibrating Particles System (EVPS) [13] are used to optimize the tower crane layout and material quantity between supply and demand points. Solving real-life problems by meta-heuristic algorithms has become an interesting topic in recent years. Many meta-heuristics with different mechanisms and characteristics are developed and applied to a wide range of optimization problems. The main objective of these optimization methods is to efficiently explore the search space in order to find global or near-global solutions. Since these algorithms are not problem-specific and do not require derivatives of the objective function, they have received increasing attention from both academia and industry. Meta-heuristic methods are global optimization methods that attempt to reproduce natural phenomena

12.1 Introduction

261

humans social or physical phenomena. Exploitation and exploration are two important characteristics of meta-heuristic optimization methods. Exploitation serves to search around the current best solutions and select the best possible points, and exploration allows the optimizer to explore the search space more efficiently, often by randomization. In the next section, the tower crane layout with material quantity supply and demand optimization is described. A brief explanation of the optimization algorithms presented in Sect. 12.3. Numerical examples are studied in Sect. 12.4. In Sect. 12.5, the results are discussed and the conclusions are derived.

12.2 Problem Statement Many types of research have worked on the locating and transporting time of a tower crane. A mathematical model for determining the most suitable tower crane location was developed by Choi and Harris [14], Zhang et al. [4] developed the Monte Carlo simulation approach to optimize tower crane location; Tam et al. employed an artificial neural network model for predicting tower crane operations and a genetic algorithm model for site facility layout [5]. Huang et al. [6] developed mixed-integer linear programming (MILP) to optimize the crane and supply locations. However, Tam and Huang considered only operation time cost of material operation flow and ignored other important cost factors such as rent, labour and tower crane setup. Lien and Chuang [8] developed a Particle Bee Algorithm (PBA) to solve a TCL model that more practically reflect the actual conditions on a construction site and also considered the rent, labour, and tower crane setup cost. Herdani [9] applied PSO, BBBC, and EBB-BC meta-heuristic algorithms to solve this PBA model and discussed the results. In this chapter, the performance of PSO and four newly developed metaheuristic algorithms CBO, ECBO, VPS, and EVPS are compared in terms of their effectiveness in resolving a practical TCLP. The meta-heuristics are used to optimize the location of the tower crane. Also, these methods are used to optimize the operating distance and frequency between demand and supply points in terms of total operating costs based on the material requirements at the points. Travel distance between the supply and demand points can be calculated by Eqs. (12.1)–(12.3) referring to Figs. 12.1 and 12.2.    y y ρ D kj = (D xj − Crkx )2 + (D j − Crk )2

(12.1)

   y y ρ Sik = (Six − Crkx )2 + (Si − Crk )2

(12.2)

L=

 y y (Six − D xj )2 + (Si − D j )2

(12.3)

262

12 Optimization of Tower Crane Location and Material Quantity …

Fig. 12.1 Radial and tangent movements of the hook

Fig. 12.2 Vertical movement of the hook

Hook movement time is an important parameter to evaluate the total time of material transportation using a tower crane. This parameter has been split up into horizontal and vertical paths to reflect the operating costs by giving an appropriate cost-time factor. Corresponding movement paths along different directions can be seen from Figs. 12.1 and 12.2.

12.2 Problem Statement

263

A continuous type parameter ψ indicates the degree of coordination of the hook movement in radial and tangential directions which depend on the controlling skills of a tower crane operator. The times for horizontal and vertical hook movements can be calculated from Eqs. (12.4)–(12.7).       ρ D kj − ρ Sik 

Ta =

Va

 2  k 2 ⎤ 2 k + ρ D + ρ Si L j 1 ⎥ ⎢   Tω = arccos⎣   ⎦, [0 < arccos(θ) < π] k k Vω 2 ∗ ρ D j ∗ ρ Si ⎡

Th = max{Ta + Tω } + ψ ∗ min{Ta + Tω }

Tv =

   z  D j − Siz  Vh

(12.4)

(12.5)

(12.6)

(12.7)

The total travel time of tower crane at location k between supply point i and demand point j, Ti,k j , can be calculated using Eq. (12.8) by specifying the continuous type parameter β for the degree of coordination of hook movement in horizontal and vertical planes. Ti,k j = max + β · min{Th + Tv }

(12.8)

The objective function of the TCLP is required to satisfy two requirements: (1) The function must be high only for those solutions with a high design preference and (2) The function must be high only for those solutions that satisfy the layout constraints. The objective function of this problem presents as follow: minT C =

I  J K   k

i

R=

Tki,j ∗ Q i, j ∗ CUk + R + S + L

(12.9)

j K 

Mk ∗ (int(DYk /30))

(12.10)

I Sk + M Sk ∗ M STk + RS Ik

(12.11)

k

S=

K  k

L=

K  k

LCk ∗ L Ak ∗ DYk

(12.12)

264

12 Optimization of Tower Crane Location and Material Quantity …

where TC is total cost; K is the number of potential cranes; I is the number of supply points; J is the number of demand points; Q i, j is the quantity of material flow from Si to D j ; CUk is the cost of material flow from Si to D j per unit quantity and unit time by kth crane; R is total rent cost; S is total setup cost; L is total labor cost; Mk is rent cost per month of kth crane; DYk denotes the days of renting tower crane/labor work of kth crane; I Sk stand for tower crane initial setup cost; M Sk is tower crane modified setup cost by kth crane; M STk is modified setup times by kth crane; RS Ik is disassembly cost; LCk is labor cost per person by kth crane; L Ak is labor amount linked with kth crane. Subject to: If actual supply capacities (i) > limit supply capacities (i) then T C = T C + 40, 000

(12.13)

If actual demand capacities (i) < limit demand capacities(i) then T C = T C + 40, 000

(12.14)

It is noteworthy that the objective function limit of actual supply capacities should be smaller than or equal to the limit supply capacities. Besides, the objective function limit of actual demand capacities should be equal to limit demand capacities. The constraints will add a penalty when the objective function breaks the above rules.

12.3 Optimization Algorithms Particle Swarm Optimization (PSO) is a well-known algorithm that has been presented in many papers and books, hence for brevity, and it is not repeated here.

12.3.1 Colliding Bodies Optimization An efficient algorithm, inspired by the momentum, and energy rules of the physics, named Colliding Bodies Optimization, was developed by Kaveh and Mahdavi [10]. CBO does not depend on any internal parameter, and also it is extremely simple to implement and use. In this method, one body collides with another body, and they move to a lower cost.Each solution candidate “X” at CBO, contains several variables i.e., Xi = Xi, j and is considered as a colliding body (CB). The bodies with masses being assigned are divided into two main equal groups, i.e., stationary and moving bodies (Fig. 12.3), where the moving bodies move to stationary bodies and a collision occurs between the body pairs. The goal of this process is: (i) to improve the locations of moving bodies and (ii) to push stationary bodies toward the better locations. After the collision, new locations of colliding bodies are updated based on the new velocity by using the collision rules. The main procedure of the CBO is described as follow:

12.3 Optimization Algorithms

265

Fig. 12.3 Pairs of CBs for collision

Step 1: The initial positions of colliding bodies are determined with random initialization of a population of individuals in the search space as Eq. (12.15): xio = xmin + rand × (xmax − xmin ), i = 1, 2, 3, . . . , n

(12.15)

where, xio determines the initial value vector of the ith colliding body. xmax and xmin are the minimum and the maximum allowable value vectors of variables, respectively; rand is a random number within the interval [0, 1], and n is the number of colliding bodies. Step 2: The magnitude of the body mass for each colliding body is defined as: mk =

n  (1/fit(i))/fit(k)

(12.16)

i=1

where fit(i) represents the objective function value of the ith colliding body; n is the population size. It seems that a colliding body with good values makes a larger mass than the bad ones. Also, for maximization, the objective function fit(i) will be replaced by 1/fit(i). Step 3: Then, colliding bodies’ objective function values are arranged in ascending order. The sorted colliding bodies are divided into two equal groups as the following: • The lower half of the CBs (stationary CBs); These CBs are good agents which are stationary and the velocity of these bodies before the collision is zero. Thus: vi = 0, i = 1, 2, 3, . . . ,

n 2

(12.17)

• The upper half of CBs (moving CBs): These CBs move toward the lower half. Then, according to Fig. 12.3, the better and worse CBs, i.e. agents with upper fitness value, of each group will collide together. The change of the body position represents the velocity of these bodies before the collision as: vi = xi − xi− n2 , i =

n + 1, . . . , n 2

(12.18)

266

12 Optimization of Tower Crane Location and Material Quantity …

where, vi and xi are the velocity and position vector of the ith CB in this group, respectively; xi− n2 is the ith CB pair position of xi in the previous group. Step 4: After the collision, the velocities of the CBs in each group are evaluated as: • Stationary CBs: vi

  m i+ n2 − εm i+ n2 × vi+ n2 n   = , i = 1, 2, 3, . . . , ; 2 m i + m i+ n2

(12.19)

• Moving CBs: vi

  m i − εm i− n2 × vi n n n   , i = + 1, + 2, + 3, . . . , n; = n 2 2 2 m i + m i− 2

(12.20)

where ε is the Coefficient of Restitution (COR) calculated for the two colliding bodies, defined as: ε =1−

iter itermax

(12.21)

with iter and itermax being the current iteration number and the total number of iterations for the optimization process, respectively. New positions of CBs of each group are updated using the generated velocities after the collision in the position of stationary CBs, as follows: • Moving CB: xinew = xi− n2 + rand◦ vi , i =

n n + 1, + 2, . . . , n; 2 2

(12.22)

where, xinew and vi are a new position and the velocity after the collision of the ith moving CB, respectively; xi− n2 is the old position of the ith stationary CB pair. • Stationary CB: n xinew = xi + rand◦ vi , i = 1, 2, 3, . . . , ; 2

(12.23)



where, xinew , xi and vi are the new positions, previous positions and the velocity after the collision of the ith CB, respectively. rand is a random vector uniformly distributed in the range of (−1, 1) and the sign ‘◦’ denotes an element-by-element multiplication. Step 6: The process is repeated from step 2 until one termination criterion is satisfied. Termination criterion is the predefined maximum number of iterations. After getting the near-global optimal solution, it is recorded to generate the output. The pseudo of CBO is shown in Fig. 12.4.

12.3 Optimization Algorithms

267

Pseudo Code of the Colliding Bodies Optimization Initial location is created randomly by Eq. (12.15) The value of the objective function is evaluated and masses are defined by Eq. (12.16) While stop criteria is not attained (like max iteration) for each CBs Calculating the velocity of Stationary and moving CBs before collision according Eqs. (12.17) and (12.18) Calculate CBs velocity after collision according by Eqs. (12.19) and (12.20) Update CBs position according Eqs. (12.22) and (12.23) End for End while End Fig. 12.4 Pseudo code of the CBO

12.3.2 Enhanced Colliding Bodies Optimization In order to improve CBO for faster and more reliable solutions, Enhanced Colliding Bodies Optimization (ECBO) was developed employing memory capacity to save a number of historically best CBs and also utilizes a mechanism to escape from local optima [11]. The pseudo of ECBO is shown in Fig. 12.5 and the involved steps are given as follows: Step 1: Initialization Initial positions of all CBs are determined randomly in an m-dimensional search space by Eq. (12.15). Step 2: Defining mass The value of mass for each CB is evaluated according to Eq. (12.16). Step 3: Saving Pseudo Code of Enhanced Colliding Bodies Optimization Initial location is created randomly by Eq. (12.15) The value of objective function is evaluated and masses are defined by Eq. (12.16) While stop criteria is not attained (like max iteration) for each CBs Calculate Stationary and moving CBs velocity before collision according Eqs. (12.17) and (12.18) Calculate CBs velocity after collision according by Eqs. (12.19) and (12.20) Update CBs position according Eqs. (12.22) and (12.23) If rand i < Pro One dimension of the ith CB is selected randomly and regenerate by Eq. (12.24) End if End for End while End

Fig. 12.5 Pseudo code of the ECBO

268

12 Optimization of Tower Crane Location and Material Quantity …

Considering a memory saving some historically best CB vectors and their related mass and objective function values can improve the algorithm performance without increasing the computational cost. For that purpose, a Colliding Memory (CM) is utilized to save a number of best-so-far solutions. Therefore, in this step, the solution vectors saved in CM are added to the population, and the same number of current worst CBs are removed. Finally, CBs are sorted according to their masses in descending order. Step 4: Creating groups CBs are divided into two equal groups: (i) stationary group and (ii) moving group. The pairs of CBs are defined according to Fig. 12.3. Step 5: Criteria before the collision The velocity of stationary bodies before the collision is zero (Eq. 12.17). Moving objects move toward stationary objects and their velocities before collision are calculated by Eq. (12.18). Step 6: Criteria after the collision The velocities of stationary and moving bodies are calculated using Eqs. (12.19) and (12.20), respectively. Step 7: Updating CBs The new position of each CB is calculated by Eqs. (12.22) and (12.23). Step 8: Escape from local optima Metaheuristic algorithms should be capable to escape from the trap when agents get close to a local optimum. In ECBO, a parameter like Pro within (0, 1) is introduced and it is specified whether a component of each CB must be changed or not. For each colliding body, Pro is compared with r n i (i = 1, 2, . . . , n) which is a random number uniformly distributed within (0, 1). If r n i < Pr o, one dimension of the ith CB is selected randomly and its value is regenerated as follows:   xi j = x j,min + rand × x j,max − x j,min , i = 1, 2, . . . , n;

(12.24)

where xi j is the jth variable of the ith CB. x j,min and x j,max are the lower and upper bounds of the jth variable, respectively. To protect the structures of CBs, only one dimension is changed. This mechanism provides opportunities for the CBs to move all over the search space for diverse exploration. Step 9: Terminating condition check The optimization process is terminated after a fixed number of iterations. If this criterion is not satisfied go to Step 2 for a new round of iteration.

12.3.3 Vibrating Particles System The VPS is a population-based algorithm which simulates a free vibration of single degree of freedom systems with viscous damping [12]. Similar to other multi-agent methods, VPS has a number of individuals (or particles) consisting of the variables

12.3 Optimization Algorithms

269

of the problem. In the VPS each solution candidate is defined as “X”, and contains j a number of variables (i.e., X i = {X i }) and is considered as a particle. Particles are damped based on three equilibrium positions with different weights, and during each iteration, the particle position is updated by learning from (i) the historically best position of the entire population (HB), (ii) a good particle (GP), and (iii) a bad particle (BP). The solution candidates gradually approach their equilibrium positions that are achieved from the current population and historically best position in order to have a proper balance between diversification and intensification. The main procedure of this algorithm is defined as: Step 1: Initialization Initial locations of particles are created randomly in an n-dimensional search space, by Eq. (12.25): j

xi = xmin + rand × (xmax − xmin ), i = 1, 2, 3, . . . , n;

(12.25)

j

where, xi is the jth variable of the particle i. xmax and xmin are respectively the minimum and the maximum allowable values vectors of variables. rand is a random number within the interval (0, 1), and n is the number of particles. Step 2: Evaluation of candidate solutions The objective function value is calculated for each particle. Step 3: Updating the particle positions In order to select the GP and BP for each candidate solution, the current population is sorted according to their objective function values in increasing order, and then GP and BP are chosen randomly from the first and second half, respectively. According to the above concepts, the particle’s position is updated by the following equation:       j xi = ω1 . D.A. rand1 + HB j + ω2 . D.A. rand2 + GP j + ω3 . D.A. rand3 + BP j (12.26) j

where xi is the jth variable of the particle i. ω1 , ω2 , ω3 , are three parameters to measure the relative importance of HB, GP and BP, respectively (ω1 +ω2 +ω3 = 1). rand1 , rand2 , and rand3 are random numbers uniformly distributed in the range of (0, 1). The parameter A is defined as:          j j j + ω1 · GP j − xi + ω1 · BP j − xi A = ω1 · HB j − xi

(12.27)

Parameter D is a descending function based on the number of iterations:  D=

iter itermax

−α (12.28)

In order to have a fast convergence in the VPS, the effect of BP is sometimes considered in updating the position formula. Therefore, for each particle, a parameter

270

12 Optimization of Tower Crane Location and Material Quantity …

like p within (0, 1) is defined, and it is compared with rand [a random number uniformly distributed in the range of (0, 1)] and if p < rand, then ω = 0 and ω2 = 1 − ω1 . Three essential concepts consisting of self-adaptation, cooperation, and competition are considered in this algorithm. Particles move towards HB so the selfadaptation is provided. Any particle has the chance to affect the new position of the other one, so the particles are cooperating. Because of the p parameter, the influence of GP (good particle) is more than that of BP (bad particle), and therefore the competition is provided. Step 4: Handling the side constraints There is a possibility of boundary violation when a particle moves to its new position. In the proposed algorithm, for handling boundary constraints a harmony searchbased approach is used. In this technique, there is a possibility like harmony memory considering rate (HMCR) that specifies whether the violating component must be changed with the corresponding component of the historically best position of a random particle or it should be determined randomly in the search space. Moreover, if the component of a historically best position is selected, there is a possibility like Pitch Adjusting Rate (PAR) that specifies whether this value should be exchanged with the neighbouring value or not. Step 5: Terminating condition check Steps 2 through 4 are repeated until a termination criterion is fulfilled. Any terminating condition can be considered, and in this chapter, the optimization process is terminated after a fixed number of iterations. The pseudocode of the VPS is shown in Fig. 12.6. Pseudo code of Vibrating Particles System (VPS) Initialize algorithm parameters Create initial positions randomly by Eq. (12.25) Evaluate the values of objective function and store HB While maximum iterations is not fulfilled for each particle The GP and BP are chosen if P 0.5 study), and r (t) = is a stochastic function where rand 0 i f rand ≤ 0.5 is a random number produced with uniform distribution in the range of [0, 1]. Then random walks should be normalized by Eq. (15.10) to prevent ants from overshooting and also keeping random walks in the boundaries of the search space:    X it − ai × dit − cit + cit = (bi − ai ) 

X it

Step 6.

(15.10)

cit = Antlion tj + ct

(15.11)

dit = Antlion tj + d t

(15.12)

where cit presents the minimum of all variables for the ith ant, dit is the maximum of all variables for the ith ant, and Antlion tj shows the position of the selected jth antlion at tth iteration. Update the ant’s position using Eq. (15.13). Antit =

R tA × R tE 2

(15.13)

15.2 Methodology

339

where Antit shows the position of the ith ant at the tth iteration, R tA presents the random walk around the antlion chosen using the roulette wheel at tth iteration, and R tE is the random walk around the elite at tth iteration. Step 7. If every ant has been traversed, then go to step 8; otherwise, go to step 2. Step 8. Calculate the objective values of all ants. Step 9. Update the archive. Step 10. If the archive is full, eliminate some solutions by roulette wheel and Eq. (15.14) from the archive to accommodate the new solutions. Pi =

Ni c

(15.14)

Step 11. Check whether the termination condition is met. If the ending condition is met, then go to Step 12; otherwise, go to step 2. Step 12. Output the Pareto-optimal solutions. Pareto optimal solutions can be obtained and ranked in various ways using metaheuristic algorithms. MOALO uses an archive to store Pareto optimal solutions, and convergence of MOALO algorithm inherits from the ALO algorithm. Once a solution is selected from the archive, the ALO algorithm is used in order to improve its quality. Nonetheless, finding the Pareto optimal solutions set by a great variety is a challenging task. To overcome this challenge, The MOPSO based leader selection and archive maintenance strategies are employed. Of particular importance is providing a limit for the archive, and increasing the distribution solutions that should be selected from the archive. Solutions’ distribution in the archive is measured by the niching technique in which the proximity of every solution is checked upon a prearranged radius. Then, the number of solutions in the proximity is counted and considered as the distribution measure. To improve the distribution of solutions in the archive, two mechanisms identical to those in MOPSO are considered. First, the solutions with the minimum inhabited vicinity are picked as antlions. Equation (15.1) is then used to define the probability of picking a solution from the archive. The flowchart of the MOALO algorithm is shown in Fig. 15.1.

15.2.3 Multi-criteria Decision Making (Shannon’s Entropy) Several methods can be utilized to discover the objectives’ normalized weights, such as Shannon’s entropy technique analytical hierarchy process (AHP), ordered weighted averaging (OWA), and simple additive weighted (SAW) approach. In this chapter, the attributes’ weighting is based on crude values of optimal Pareto solutions because, according to the above-mentioned methods, users’ decisions might be insufficient and might result in a partial judgment on weights. To evaluate the relative weights, Shannon’s entropy method declares the corresponding importance weights of the attributes based on the differentiation amongst data. Thus, Shannon’s entropy can present a more reliable measure for the corresponding weights of the objectives in

340

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

the loss of the users’ preferences. Shannon’s entropy acts as a measure for the degree of uncertainty in information formulated in terms of probability theory. Shannon’s entropy as a measure of uncertainty is associated with the information source. The information can be easily defined as objective values. The uncertainty in information is addressed by Shannon’s entropy utilizing the theory of probability. The inherent hypothesis is that a lower probability of an event shows its higher chance for providing more information by its happening, i.e., an objective with a biased distribution offers more relative importance in comparison to a sharply peaked one. Shannon’s entropy parameter (E j ) of the jth objective is formulated as follows: n Ej = −

i=1

Pi j ln Pi j , wher e i ∈ {1, 2, . . . , n}and j ∈ {1, 2, . . . , m} (15.15) ln n

Pi j = n

fi j

i=1

fi j

, wher e i ∈ {1, 2, . . . , n}and j ∈ {1, 2, . . . , m}

  m

1 − Ej  , wher e ω j = m  ωj = 1 j=1 1 − E j j=1

(15.16) (15.17)

where f i j indicates the jth objective function of the ith solution, and Pi j is the jth linear normalized objective of ith solution that is utilized to calculate the value of E j for the jth objective; n and m are the number of solutions and number of objectives, respectively. Finally, ω j shows the corresponding relative normalized weight of the jth solution which is determined by Eqs. (15.15) and (15.16).

15.2.4 Evidential Reasoning The decision making is widely used in different sides of engineering, and several approaches have been proposed and employed in dealing with MCDM problems, e.g., additive utility function approaches, outranking approaches, and Evidential Reasoning (ER). Using MCDM approaches makes users’ preference criteria controllable and more efficient, and the data may be easily transferred to the controller. Hence, the daily repetitive and time-consuming procedure of review and action of the schedule can be reduced. The evidential reasoning is a comprehensive approach for integrated investigation of the MCDM problems under various uncertainty types, like ignorance and fuzziness jointly. The ER approach comprises all parts of the MCDM framework, employing the belief matrices and the belief structures. The ER information aggregation methodology contains a rule-or-utility-based information transformation procedure concerning different information types in both natures of quantitative and qualitative under the required circumstances of utility and value equality. In multi-objective optimization, where objectives are frequently conflicting, the Pareto solutions might be so copious, and it could be time-consuming to ultimately

15.2 Methodology

341

select a single compromising solution. The output of multi-objective optimization algorithms is a set of non-dominated solutions. Every non-dominated solution meets the scheduling objectives to some extent, which requires the utilization of the MCDM methods to pick the most suitable non-dominated solution. The MCDM problems handle the procedure of ordering solutions by considering various criteria. Therefore, taking multiple attributes into consideration, non-dominated solutions can be ordered by using the ER approach, which is able to present more efficient and practical appliance scheduling alternatives. The ER approach includes the following steps [6]: (1) Identification and analysis of the multiple assessment criteria using a comprehensive study from engineering judgments or expert interviews with regard to the weight assigned to each criterion. This step collects and models various kinds of supporting attributes such as qualitative, quantitative, precise numbers, fuzziness, uncertainties, comparison numbers, and belief structures concerning criteria weights and utility by a belief decision matrix. Precise numbers show single or exact values without any uncertainty, whereas interval numbers denote estimates in ranges, and belief structures indicate an evaluation as a distribution (for instance, unsafety of a specific alternative is “Good” to a belief degree of 71% and, at the same time, it can be evaluated to be “Moderate” to a degree of belief of 29%; such an evaluation can be represented as {(Good, 0.71), (Moderate, 0.29)} and is referred as a belief structure). When the assessor is not sufficiently sure on the assessment because of a lack of knowledge or evidence, the sum of probability is unequal to one (incomplete assessment). Furthermore, in the state where no data is available to assess an alternative’s performance on a criterion, the total belief degree is assumed to be zero in belief structure. (2) Transformation of different types of assessment degrees into a general framework of judgment by unifying the belief structures and employing rule-andutility-based information transformation procedures. Thus, they can be consistently compared and aggregated. Belief structures should be translated during this step; for instance, ‘Very Bad’ and ‘Very Good’ respectively indicate zero and one, and remaining grades may/may not be uniformly distributed. (3) Employing the ER formulation and algorithm to agglomerate the assessing information on multiple criteria types to attain the overall assessment of each alternative. (4) Generation of utility scores or utility intervals in the state of the lack of information. The utility-based ranking is able to assess the overall performance of every alternative respecting all aspects (criteria). Moreover it jointly uses a systematic-rational prioritizing framework which presents the best schedule for smart home appliances, and one which satisfies all the users’ favorites. Because the ultimately selected solution is a trade-off among users’ preferences. There are various techniques for calculating the weights of every criterion, such as pairwise matching, defined by users, and Shannon’s entropy. Since the weights denote the relative importance, normalizing the values is more useful in comparison to using absolute values and can be calculated by the following formulas:

342

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

Wj ωi = L i=1

Wj

, wher e i = {1, 2, . . . , L}

S . L . 0 ≤ ωi ≤ 1, wher e

L

ωi = 1

(15.18)

(15.19)

i=1

The linguistic phrases like ‘worst’, ‘good’, and others are known as grades where the entire set of them is denoted by H = {Hn , n = 1, 2, . . . , N }. The analytical format of the ER procedure is able to determine the combined beliefs degree βn of the nth grade, where n ∈ {1, 2, . . . , N } and β H denotes the assessment of incompleteness for the entire set of H. Unlike the recursive evidential reasoning, its analytical format necessitates no iteration to assess several attributes, hence, presenting more flexibility for optimization and assessment, it is preferred. Following formulations are presented to calculate βn and β H : L

L − i=1 1 − ωi Nj=1 βi, j

 βn = L N N N L L n=1 i=1 1 − ωi j=1, j =n βi, j − (N − 1) i=1 1 − ωi j=1 βi, j − i=1 (1 − ωi ) i=1 1 − ωi

N

j=1, j =n βi, j

L

N



L

(15.20)

i=1 1 − ωi j=1 βi, j − i=1 (1 − ωi )

 L L N N L n=1 i=1 1 − ωi j=1, j =n βi, j − (N − 1) i=1 1 − ωi j=1 βi, j − i=1 (1 − ωi )

βH = N

(15.21) where βi, j is the degree of belief of the ith primary criterion for the jth grade, and N is the number of grades in set H. Combined belief degrees and incomplete assessment (βn and β H ) need to be translated into a single utility score for ranking the alternatives. Therefore, it is essential to produce numerical values corresponding to the belief structures: u max =

N −1

βn u(Hn ) + (βn + β H )u(Hn )

(15.22)

n=1

u min = (β1 + β H )u(H1 ) +

N

βn u(Hn )

(15.23)

n=2

u ave =

u max + u min 2

(15.24)

The minimum, maximum, and average values of utility scores are denoted by u max , u min , and u ave , respectively, and u(Hn ) is a function showing the nth grade’s utility score. For instance, if n = 6, and all of the grades are equally ranged in the range of [0, 1], then u(Hn ) = {0, 0.2, 0.4, 0.6, 0.8, 1}. According to Eqs. (15.7) and (15.8), if there is no incomplete assessment (bH = 0), all three states of minimum,

15.2 Methodology

343

maximum, and average values of utility scores are equal and can be calculated by the following formulation: u max = u min = u ave =

N

βn . u(Hn )

(15.25)

n=1

15.3 The Multi-objective Home Appliance Scheduling Problem The main model of this chapter is derived from the multi-objective demand-side scheduling (MODSS) problem, which is presented by Du et al. and Smart-home appliance scheduling problem which is presented by Sou et al. [5]. Du et al. [4] considered the objectives of Operational Unsafety of Appliances, Electricity Cost of Household, and Operational Delay of Appliances, Peak to Average Ratio has a common formula, and the CO2 emission is derived from Sou et al. Objective functions and formulations of the model are presented in the following:

15.3.1 Objective Functions 15.3.1.1

Operational Unsafety of Appliances

The users’ awake and at-home statuses for controlling the appliances’ operations are considered as operational unsafety of the appliances. This objective function decreases the appliance operation in not at-home or sleeping conditions of users, and the unsafety time rate quantifies this situation. The minimization formulation for appliances’ operational unsafety ( f 1 (x)) for a home with n time-adjustable appliances was presented by Du et al. [4] as follows: min f 1 (x) x

n

(15.26)

ρaU T Ra (X a )

(15.27)

γa − Sa (X a ) γa

(15.28)

Sa (X a , t). M(t). N (t)

(15.29)

f 1 (x) =

a=1

U T Ra (X a ) = Sa (X a ) =

T

t=1

344

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

 Sa (X a , t) =  M(t) = 

  1, t ∈ Xa , X a + γa − 1  0, t ∈ H X a , X a + γa − 1

1, i f : user s ar e at home 0, i f : user s ar e away

1, i f : user s ar e awake 0, i f : user s ar e awsleep   H = {1, 2, . . . , T }, X a ∈ αa , βa γa + 1 N (t) =

(15.30) (15.31) (15.32) (15.33)

X = {X 1 , X 2 , . . . , X a , . . . , X n }

(15.34)

  X a ∈ αa , βa − γa + 1 , wher e a = {1, 2, . . . , n}

(15.35)

subjected to:

Within this formulation, unsafety time rate of appliance a is denoted by U T Ra , and X a is the starting time slot of the appliance’s operation. ρa is the unsafety parameter, and with the higher value of ρa , the cost of operational unsafety will be higher. The set X = {X 1 , X 2 , . . . , X a , . . . , X n } indicates the appliances’ starting time slots. Sa is the number of time slots that users are awake and at-home during the appliance a is in operation and determined by the appliance’s operation status Sa (X a , t) with the information  status M(t) and awake status N (t)  of users’ at-home in a day. The phrase t ∈ H X a , X a + γa − 1 demonstrates that t pertains to H = {1, 2, . . . , T } excluding the range. Within this formulation, appliance a unsafety time rate is denoted by U T Ra , and X a is the appliance’s operation starting time slot. ρa is the unsafety parameter. The higher the value of ρa , the higher the cost of operational unsafety will be. The set X = {X 1 , X 2 , . . . , X a , . . . , X n } indicates the appliances’ starting time slots. Sa represents the number of time slots when users are awake and at-home while the appliance a is in operation and is determined using the appliance’s M(t) and operation status Sa (X a , t) with the information of users’ at-home status  awake status N (t) in a day. The phrase t ∈ H X a , X a + γa − 1 demonstrates  that t pertains to H = {1, 2, . . . , T } excluding the range X a , X a + γa − 1 and T = 120 is the boundary of scheduling that intimates the number of time slots forward that the  schedule of energy consumption is made for time-adjustable appliances. X a ∈ αa , βa γa + 1 because the operation should start ahead of the deadline with at least the length of the operation time. After determining the start time slots of appliances, the unsafety time rates (UTR) of appliances are calculated by Eqs. (15.28– 15.33), and the operational unsafety is calculated by UTR. The UTR is a ratio between the unsafe operation time slots (the time slots when users are asleep or away, but the appliance is in operation) and the operation length. Consider that various appliances may have an identical U T R and ρa is presented for distinguishing the operational unsafety of appliances and that both the U T R and ρa jointly define the appliance’s operational unsafety by ρaU T Ra . The unsafety time rate concept is depicted in Fig. 15.2.

15.3 The Multi-objective Home Appliance Scheduling Problem

γ a − sa AAAAAA

345

sa

AA

γa AA Appliances in operation Users are at home and awake 0

2

4

6

8

10

12

14

16

18

20

22

24

Fig. 15.2 Representation of the unsafety time rate concept

The users’ at-home status M(t) and awake status N (t) are separately determined by users as various users have various at-home status and awake status. Based on the users’ predefined at-home status and awake status, the appliances’ operational unsafety is calculated by Eqs. (15.26–15.35). The operational unsafety of the same energy consumption schedule differs under various statuses of users.

15.3.1.2

Electricity Cost of Household

Pa shows the power of appliance a. By considering each 1 h as 5 identical time slots with the fixed P5a energy consumption during each time slot, the energy consumption schedule of appliance a is calculated by [5].    eat eat = P5a , t ∈ X a , X a + γa − 1 , Ea = eat = 0, t ∈ H X a , X a + γa − 1 ,  ⎩ H = {1, 2, . . . , T }, X a ∈ αa , βa γa + 1 ⎧ ⎨

(15.36)

where eat denotes the energy consumption of appliance a during time slot t. According to the appliances’ energy consumption, and the day-ahead real-time electricity price, the minimization formulation of electricity cost is: min f 2 (x) x

f 2 (x) =

T

pr ct .lt (x)

(15.37)

(15.38)

t=1

lt (x) =

n

eat

(15.39)

a=1

X = {X 1 , X 2 , . . . , X a , . . . , X n }

(15.40)

346

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

subjected to   X a ∈ αa , βa − γa + 1 , wher e a = {1, 2, . . . , n}

(15.41)

where pr ct is the real-time electricity price at time slot t, and lt is the total energy consumption of all the time-adjustable appliances during time slot t, and it is obtainable after determining the start time slots of all appliances X and the energy consumption of every appliance is scheduled by Eq. (15.36).

15.3.1.3

Operational Delay of Appliances

Figure 15.3 illustrates that the appliance’s operational delay is the delay time from αa , the earliest start time of the operation, and the longest delay occurs when the appliance reaches the deadline to complete its operation, i.e., the appliance starts at the time slot βa − γa + 1. For a home including n time-adjustable appliances, the operational delay minimization is formulated as follows [4]: min f 3 (x)

(15.42)

x

f 3 (x) =

n

σaDT Ra (X a )

(15.43)

a=1

xa − αa βa − γa + 1 − αa

(15.44)

X = {X 1 , X 2 , . . . , X a , . . . , X n }

(15.45)

DT Ra (X a ) =

subjected to βa − γ a + 1

αa

x a − αa AAAAA

αa

βa

βa − γ a + 1 − α a AAAAAAAAAAA xa

γa AA Appliance in operation

βa

βa − γ a + 1 AAAAAAAA

γa AA Appliance in operation

Fig. 15.3 Representation of the delay time rate concept

15.3 The Multi-objective Home Appliance Scheduling Problem

  X a ∈ αa , βa − γa + 1 , wher e a = {1, 2, . . . , n}

347

(15.46)

where DT Ra indicates the delay time rate of appliance a. σa > 1 shows the delay parameter of appliance a. The higher the value of σa , the higher the cost of operational delay will be.

15.3.1.4

Peak to Average Ratio (PAR)

Supporting the stability of the entire electricity network is an important issue that is obtainable by minimizing PAR. PAR is the rate of maximum daily power demand per average daily power demand. Minimization formulation of PAR is presented as follows: min f 4 (x)

(15.47)

maxx lt (x) f 4 (x) = T t=1 l t (x)

(15.48)

x

lt (x) =

n

eat

(15.49)

a=1

X = {X 1 , X 2 , . . . , X a , . . . , X n }

(15.50)

  X a ∈ αa , βa − γa + 1 , wher e a = {1, 2, . . . , n}

(15.51)

Subjected to

15.3.1.5

CO2 Emission

During day and night, various renewable and nonrenewable sources with different CO2 footprints are used for electricity generation, which means it causes a dynamic CO2 emission footprint during hours of a day. By shifting appliances’ operation to the hours of the day with low CO2 emissions, it will result in reducing the total CO2 emissions of households. The minimization formulation of total CO2 emission is presented as follows: min f 5 (x) x

f 5 (x) =

T

t=1

lt (x).Ct

(15.52)

(15.53)

348

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

lt (x) =

n

eat

(15.54)

a=1

X = {X 1 , X 2 , . . . , X a , . . . , X n }

(15.55)

  X a ∈ αa , βa − γa + 1 , wher e a = {1, 2, . . . , n}

(15.56)

Subjected to

where Ct is the carbon emission in time-slot t.

15.4 Implementation of the Proposed System 15.4.1 Numerical Example In order to verify and show the effectiveness of the proposed appliance scheduling system and unify the ER approach with MOALO for solving SHEMS, a benchmark of smart home, first proposed by Zhao et al. [7], is adopted and the relationships within the different objectives are investigated. Du et al. altered the data to consider the safety of appliances. Eight common appliances are recognized, and some of them are used more than once during the day, and corresponding parameters are illustrated in Table 15.1. Table 15.1 Parameters of appliances

Appliance Rice

cookera

OTI

LOT

Power (kW)

1–40

2

0.5

Rice cookerb

56–65

2

0.5

Rice cookerc

71–90

2

0.5

Water heater

86–105

3

1.5

Dishwasher

101–120

2

0.6

Washing machine

1–60

5

0.38

Electric kettlea

1–40

1

1.5

Electric kettleb

81–90

1

1.5

Clothes dryer

71–90

5

0.8

Oven

71–90

3

1.9

Electric radiatora

56–65

5

1.8

Electric radiatorb

81–110

20

1.8

*a, *b and *c indicate that appliance * is used three times within various OTIs in a day

15.4 Implementation of the Proposed System

349

Du et al. [4] considered the at-home and awake statuses of users, as shown in Figs. 15.4 and 15.5. The electricity price data and CO2 footprint data are shown in Figs. 15.6 and 15.7, respectively [8]. Both the unsafety (ρa ) and delay (σa ) parameters are considered to be 2. It is worth mentioning that the users’ at-home status and awake status in Figs. 15.4 and 15.5 are demonstrated to show how the users’ statuses are considered in the appliances’ operational unsafety. The users individually define the at-home status and awake status. Regarding the effect of parameter setting on the performance of metaheuristic algorithms, the Taguchi method is hired. Before calibration of the applied algorithms, some preliminary tests are run to find proper parameter levels. To achieve more accurate, also better-sustained results for the proposed algorithm, the population number and iteration parameters are configured.

users’ at-home status

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour Fig. 15.4 Users’ at-home status

users’ awake status

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour Fig. 15.5 Users’ awake status

350

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

electrisity price (cents/kWh)

10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour Fig. 15.6 Electricity price data

CO2 footprint (g/(kWh)

80 70 60 50 40 30 20 10 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour Fig. 15.7 The CO2 footprint data

15.4.2 Parameter Configuration Due to the effect of parameter configuration on the performance of metaheuristics, the Taguchi method of design-of-experiment (DOE) is performed to the configuration of the MOALO parameters. The MOALO contains two key parameters: the maximum number of iterations (T ), and the number of antlions (n). Before calibration of the used algorithms, some preliminary tests are run to find appropriate parameter levels, and five levels are considered for each of the two parameters. For each run, the maximum number of iterations is set to Max I t = {1000, 2000, 3000, 4000, 5000}, and the number of ants is set as n = {100, 150, 200, 250, 300}. The performance indicators of multi-objective algorithms differ from those of single-objective algorithms, so that in the single-objective case, the optimal solution has the global optimum of a particular objective function, whereas, in the multi-objective case, there may not exist a unique solution that is optimal in terms of all objective functions. Hence, a different method is required for comparing the performance of each test of the algorithm. Therefore, the relative performance metrics (e.g., Spread of Non-dominance Solutions (SNS), Diversification Matrix (DM), Mean Ideal Distance (MID), The rate of achievement

15.4 Implementation of the Proposed System

351

Main Effects Plot for SN ratios Data Means

5.75

Mean of SN ratios

nAnt

MaxIt

5.50 5.25 5.00 4.75 4.50 1

2

3

4

5

1

2

3

4

5

Signal-to-noise: Larger is better

Fig. 15.8 Diagram of mean effect of the SN ratio

to two objectives simultaneously (RAS), and Quality Metric (QM)) are employed to analyze the results quantitatively. For a thorough explanation of this method and its enhanced version, the reader may refer to Jolay et al. [9]. The trend of each parameter for different sets is illustrated in Fig. 15.8. Taguchi is a well-known method of designof-experiment (DOE) and comprehensive explanations about it are presented in [9]. As demonstrated in Fig. 15.8, the second level of iteration numbers and the fourth level of population numbers have better performance. So, in this case, the population number is set to be 2000, and the population number is selected as 250.

15.4.3 Pareto Selection Table 15.4 presents the acquired Pareto solutions of benchmark instance of SHEMS and the utility scores for the solution of each objective. Figure 15.9 shows the parallel coordinates plot of the Pareto solutions. This data can be used for prospective research studies. As can be seen from Fig. 15.10, except for cost and unsafety, and CO2 emission and unsafety objective pairs which have no clear relationship, the relationship between all other objective pairs is oppositional; hence a Pareto diagram can be plotted for them.

352

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

Fig. 15.9 Parallel coordinates plot of the Pareto solutions

15.4.4 Determining the Weights Shannon’s entropy technique determined the weights for all objectives (see Sect. 15.2.3). The PAR objective possessed the highest value of normalized weight by 41.65%, which for the delay, unsafety, cost, and CO2 emission are equal to 36.65%, 16.49%, 4.91%, and 0.3%, respectively. Table 15.2 shows that the PAR objective forces a higher impact on the associated uncertainty on acquired results, and possesses the higher weight. It is clear that these weights will be modified daily by changing the day-ahead CO2 footprint and day-ahead electricity price. This is the advantage of this weighting method to the previous methods.

15.4 Implementation of the Proposed System

353

80

23 21

75

19

Cost

Delay

70

17

65 15 60

55 12

13

14

16

18

11 12

20

14

Unsafety 9.0

755

8.5

750

8.0

745

CO2 emission

7.5

PAR

7.0 6.5 6.0 5.5

18

20

730 725

4.5

715 16

20

735

720

14

18

740

5.0

4.0 12

16

Unsafety

18

710 12

20

14

Unsafety

16

Unsafety

23

9.0 8.5

21

8.0 7.5 7.0

PAR

Delay

19 17

6.5 6.0

15

5.5 5.0

13

4.5 11 58

63

68

Cost

73

4.0 58

63

68

Cost

Fig. 15.10 Obtained pareto optimal solutions shown for each pair of objectives

73

15 Multi-objective Electrical Energy Scheduling in Smart Homes … 755

9.0

750

8.5

745

8.0 7.5

740

7.0

735

PAR

CO2 emission

354

730

6.5 6.0

725

5.5

720

5.0

715

4.5

710 58

63

68

4.0 12

73

17

755

755

750

750

745

745

740

740

735 730 725

735 730 725

720

720

715

715

710 12

17

22

Delay

CO2 emission

CO2 emission

Cost

22

710 4.0

5.0

6.0

Delay

7.0

8.0

9.0

PAR

Fig. 15.10 (continued)

Table 15.2 The objectives’ normalized weights by Shannon’s entropy Objectives

Unsafety

Cost

Delay

PAR

CO2 emission

Normalized weights

0.1649

0.0491

0.3665

0.4165

0.0030

15.4.5 Ranking Solutions The acquired Pareto solutions now should be ranked based on their overall performance, indicating the satisfactory degree of every alternative by taking into account all the criteria simultaneously. As seen in Fig. 15.11, in order to assess the overall performance of each solution, a hierarchical structure is required to associate the appliances’ unsafety, electricity cost, appliances’ delay, PAR, and CO2 emission attributes with their related normalized weights to the indicator of the overall performance. The belief structure of each attribute is determined, as demonstrated in Fig. 15.12. The x-axis shows the grades (‘Worst’, ‘Poor’, ‘Moderate’, ‘Good’, and ‘Best’) and the corresponding values of the attribute. For example, consider

15.4 Implementation of the Proposed System

355

Unsafety Objective (0.1649) Cost Objective (0.0491) Delay Objective (0.3665)

Overall Performance (1.0)

PAR Objective (0.4165) CO2 emission Objective (0.0030)

Fig. 15.11 Hierarchical structure of objective weights 93.22%

64.16%

39.74%

100% 8.2577

Worst

12.8923 18.9562

19.3677 Unsafety 75.4792 $ Cost 22.6271 Delay 8.2577 PAR CO2 emission 750.9595 g

722.0227

Poor 17.5895 71.3715 $ 20.0001 7.2475 741.5271 g

Moderate 15.8114 67.2638 $ 17.3731 6.2372 732.0947 g

Good 14.0332 63.1560 $ 14.7461 5.2270 722.6622 g

60.9583

Best 12.2550 59.0483 $ 12.1191 4.2167 713.2298 g

Fig. 15.12 Transfer of each attribute to the belief structure

the 40th alternative. The delay of 18.9562 lies between ‘Worst’ and ‘Poor’ grades with the delay values of 22.6271 and 20.0001, respectively. So, this attribute of 40th alternative belongs to the grade of ‘Worst’ with 60% belief, and with 40% belief degree belongs to ‘Poor’ grade. The identical procedure is performed for all other attributes. The acquired belief structures of all attributes are presented in Table 15.3. Figure 15.13 shows the overall performance utility assessment of all solutions. After that, the solutions are sorted by their acquired overall performance utility scores and Table 15.3 The attributes Belief structures of 40th alternative Grades

Worst (%)

Good (%)

Best (%)

Unsafety

35.84

64.16

Cost

46.50

53.50

93.22

6.78

Delay

60.26

PAR

100

CO2 emission

Poor (%)

Moderate

39.74

356

15 Multi-objective Electrical Energy Scheduling in Smart Homes … 1

Overall utility score (%)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190 197 204 211 218 225 232 239

0.1

Solution Number

Fig. 15.13 Overall performance utility assessment of all solutions

the first one is recommended to the users. In this case, the 6th Pareto solution is selected as the best solution because of having the highest utility score (i.e. 79.52%) among all the Pareto solutions, which means this appliance scheduling alternative satisfies the overall performance by concurrently regarding all objectives. However, the overall performance might be insufficient for the final decision in selecting the best solution schedule, because every schedule requires to be reviewed for its weakness and strength points to be identified considering utility scores of every objective. As can be observed, the 6th alternative has the best utility score for the PAR objective, which is equal to 100% and other objectives of this alternative have a good level of utility scores. The ER presents an obvious sense of each alternative’s performance for every single criterion for users and enables them to investigate schedules quickly.

15.4.6 Discussions Among the Pareto solutions, the 6th, 40th, 99th, 137th, 181st, and 230th alternatives (Pareto solutions) were chosen to show the procedure of overall assessment. Figure 15.14 shows the corresponding utility scores of every objective, along with the overall performance of every alternative. Furthermore, the 181st alternative possesses a 21% overall utility score, which is quite low to be selected and does not present satisfactory performance in points of the unsafety objective. By careful investigation of the best solutions, the users are able to select the most suitable scheduling alternative. Evidential reasoning expedites the investigation of the overall performance of scheduling alternatives by providing more pieces of information about the performance of each scheduling alternative concerning each objective. The users’ presumptions on the performance of each alternative provide more confidence for users to perform their preferred appliance schedule, and therefore, more efficient energy control in the smart home. As remarked before, the ER approach presents resultant informative data showing the weaknesses of each scheduling alternative

15.4 Implementation of the Proposed System

357

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Alternative 6

Alternative 40

Alternative 99

Alternative 137

Alternative 181

Alternative 230

Overall Performance

80%

28%

58%

29%

21%

48%

Unsafety

56%

91%

86%

81%

68%

55%

Cost

60%

88%

95%

90%

86%

73%

Delay

66%

35%

43%

16%

24%

25%

PAR

100%

0%

61%

22%

0%

61%

CO2 emission

59%

77%

57%

92%

87%

95%

Fig. 15.14 The utility scores of the 6th, 40th, 99th, 137th, 181st, and 230th alternatives concerning each objective

at any desirable level. This study divides the overall performance into five grades including ‘worst grade’, ‘poor grade’, ‘Moderate grade’, ‘good grade’, and ‘best grade’, which are spaced equally within the range of [0, 1]. Then, these grades can be utilized to show the combined degree of belief βn (Fig. 15.15). Accordingly, the 6th solution’s overall performance is believed to pertain to the ‘best’ grade by a degree equal to 79.52%. As well, the 6th alternative has no belief degree on the ‘worst’ grade, in comparison to the 40th, 137th, and 181st alternatives with belief degrees for the ‘worst’ grade of 46.05%, 18.10%, and 47.58%, respectively. Consequently, the users decide to select the 6th alternative as the best one. If the first alternative was not suitable, the users could start investigating other alternatives. In this case, there is no incomplete assessment, so the incompleteness assessment is considered to be zero, β H = 0. Poor

Moderate

Good

Best

degrees of belief (

)

Worst 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00%

Alternative 6

Alternative 40

Alternative 99

Alternative 137

Alternative 181

Worst

0.00%

46.05%

0.00%

18.10%

47.58%

Alternative 230 0.00%

Poor

0.00%

22.49%

9.23%

67.60%

35.98%

34.72%

Moderate

26.10%

14.83%

56.48%

0.00%

3.42%

38.89%

Good

29.70%

6.42%

26.57%

9.86%

11.48%

26.24%

Best

44.20%

10.20%

7.72%

4.44%

1.54%

0.14%

Fig. 15.15 Combined belief degrees (βn ) for the 6th, 40th, 99th, 137th, 181st, and 230th alternatives concerning the overall performance

358

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

Evidential reasoning is highly effective in determining the performance of each alternative, and it empowers the users to make a practical and transparent decision on the best scheduling alternative. Employing the ER procedure in smart home appliance scheduling presents a more efficient electrical energy control system, and provides more confidence for users on their decision because the users have a transparent knowledge of the performance of each scheduling alternative. In some cases, an inappropriate alternative (from the users’ point of view) may own the highest value of the utility score. In such cases, the next one can be investigated. Concerning the purpose of the trade-off making, a solution that builds a balance among unsafety, cost, delay, PAR, and CO2 emission is preferred more. In comparison to the other solutions, the 6th solution provides a suitable trade-off among the objectives because the objectives are concurrently satisfied at an acceptable degree. The best schedule is obtained as well as the relative objectives are shown in Fig. 15.16 and if the user recognizes that the schedule cannot satisfy their preference in terms of unsafety or delay of appliances, they can change the relative ρa or σa of the appliance. 1

Appliance

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Operational Unsafety Operational Delay 1.4142 1.0000

Rice cooker1 Rice cooker2

1.4142

1.4142

Rice cooker3 Water heater Dishwasher Washing machine

2.0000

1.3094

0.7071 1.3195 2.0000

1.0850 1.0801 1.7411

Electric kettle1

2.0000

1.3055

Electric kettle2 Clothes dryer Oven

0.7071

1.3608

2.0000 0.5359

1.2030 1.0000

0.5743

1.1487

0.6926

2.0000

Electric radiator

1

Electric radiator 2

Time slot

operation time interval

scheduled operation time

3

Load

2.5

2 1.5 1 0.5

Electricity cost (cents)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

16

17

18

19

20

21

22

23

24

16

17

18

19

20

21

22

23

24

Time of Day (Hour) 3 2.5 2 1.5 1 0.5 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

CO2 Emission (g/kW)

Time of Day (Hour) 30 25

20 15 10

5 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Time of Day (Hour)

Fig. 15.16 The best solution acquired by this research

15.5 Conclusion

359

15.5 Conclusion In this chapter, the appliances’ unsafety, electricity cost, appliances’ delay, Peak to Average Ratio, and CO2 emission are considered as the objectives of a smart home appliance scheduling model and jointly optimized. Moreover, a comprehensive MCDM framework for ranking acquired Pareto solutions is tailored and evaluated for making a trade-off between objectives. A multi-objective Ant Lion optimizer (MOALO) is tailored to solve SHEMS and is applied to a benchmark appliance scheduling instance from the literature. After that, an ER approach is adopted to rank the optimal Pareto solutions employing the weights acquired from Shannon’s entropy. The proposed ER approach helps users to identify the efficiency of every alternative by a panoramic view concerning all the attributes and consequently selecting one of the Pareto solutions with higher assurance. A comprehensive framework to amalgamate MCDM techniques into an evolutionary multi-objective optimization is employed to facilitate the process of making a trade-off among objectives. The proposed approach in employing the MCDM technique can select the most efficient appliance schedule. The evidential reasoning approach represents a transparent and comprehensive sense of the efficiency of the alternatives, and the users can find the strengths and weaknesses of each alternative. The implementation of the developed SHEMS will need easy access to the data of day-ahead electricity cost and CO2 emission. According to the multi-user nature of smart homes, integrating the developed system with game theory concepts may be beneficial. Testing several metaheuristics for solving this problem will also be an interesting topic.

Appendix See Table 15.4.

15.2226

13.9364

15.3974

15.0665

15.2570

13.1506

15.3970

15.1382

13

14

15

16

17

18

19

15.2293

8

12

14.9617

7

17.0116

15.3649

6

11

12.2550

5

14.7989

13.6921

4

14.8493

15.2293

3

10

14.6435

9

17.1417

2

61.8578

70.1594

71.4902

63.9868

63.4514

63.5847

71.4542

63.5840

63.9736

63.0491

64.2737

62.5355

64.3873

65.6434

60.2432

62.2955

71.4569

71.4542

63.4492

17.1284

12.5776

12.3977

15.4740

15.5510

15.0823

12.6965

15.6671

15.5519

14.5018

14.9529

14.1747

14.2542

15.6477

22.6271

21.7105

12.4987

12.3762

17.8280

5.7980

5.7980

5.7980

4.7438

5.2709

5.7980

5.7980

5.7980

4.7438

5.7980

5.2709

5.7980

5.7980

4.2167

6.8521

5.7980

5.7980

5.7980

5.7980

718.4045

745.2823

745.3243

726.9526

735.3550

720.0563

746.8006

719.6484

730.9634

725.4890

734.7895

733.7811

738.1254

728.5132

715.6594

714.9864

746.6918

746.8006

714.1940

CO2 emission

57

71

77

75

71

65

74

63

72

69

73

69

69

80

32

46

72

74

52

59

56

87

58

60

56

76

58

33

64

64

58

62

56

100

80

58

66

31

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

1

#

Table 15.4 All the acquired pareto solutions for the smart home appliance scheduling benchmark

83

32

24

70

73

72

24

72

70

76

68

79

68

60

93

80

24

24

73

Cost (%)

52

96

97

68

67

72

95

66

67

77

73

80

80

66

0

9

96

98

46

Delay (%)

61

61

61

87

74

61

61

61

87

61

74

61

61

100

35

61

61

61

61

PAR (%)

(continued)

86

15

15

64

41

82

11

83

53

68

43

46

34

59

94

95

11

11

97

CO2 emission (%)

360 15 Multi-objective Electrical Energy Scheduling in Smart Homes …

CO2 emission

13.1063

16.9856

13.0207

15.7953

14.7750

15.3871

13.2207

14.3759

15.7184

17.7241

15.3974

14.9617

14.8214

16.5320

13.2907

13.3738

13.2768

13.5136

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

61.6662

64.1006

63.7130

63.6540

63.5697

62.9751

62.0403

63.0814

62.9968

62.3380

62.9120

62.2586

63.1638

62.6034

65.1334

59.9607

63.7382

63.3554

17.8799

17.0175

17.5972

17.6197

17.0100

18.0663

15.4956

14.9029

17.0731

16.4147

15.2081

20.9829

15.5615

16.7423

13.6117

22.0699

13.7257

21.3051

7.3792

5.7980

6.8521

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

7.3792

5.7980

5.7980

725.0781

719.1450

716.8148

716.8579

717.1116

717.8952

725.2368

724.7709

718.7073

718.0575

731.2745

719.8696

727.7293

723.3095

732.1948

721.0676

731.6613

716.9784

40

61

47

59

55

55

65

66

54

59

67

49

63

60

68

26

67

48

82

86

84

85

40

64

62

56

23

51

70

86

56

65

50

89

33

88

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

20

#

Table 15.4 (continued)

84

69

72

72

72

76

82

75

76

80

76

80

75

78

63

94

71

74

Cost (%)

45

53

48

48

53

43

68

74

53

59

71

16

67

56

86

5

85

13

Delay (%)

22

61

35

61

61

61

61

61

61

61

61

61

61

61

61

22

61

61

PAR (%)

(continued)

69

84

90

90

90

88

68

69

85

87

52

82

62

73

50

79

51

90

CO2 emission (%)

Appendix 361

CO2 emission

13.5136

14.3358

12.8923

14.5812

14.2213

14.6813

14.2213

14.2213

13.4135

13.9742

15.3923

14.6363

14.7812

14.5136

14.3136

14.1336

13.8252

13.7279

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

60.0275

60.0048

59.0726

59.5317

59.4906

60.6823

59.5679

60.5147

61.8152

61.6166

61.4180

61.5587

61.8451

60.4876

60.7093

60.9583

61.0696

61.5659

21.9785

21.9048

21.1977

18.9923

18.7327

18.2502

19.8617

18.7253

18.2966

18.5765

18.8964

18.8636

18.0378

18.6142

18.2616

18.9562

18.9242

18.3079

7.3792

7.3792

7.3792

5.7980

7.3792

7.3792

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

7.3792

8.2577

6.8521

5.7980

723.1382

720.9801

721.3806

722.9183

723.3209

723.3192

721.1451

722.8884

721.8166

721.4916

721.1667

720.6617

723.8403

726.1011

726.5390

722.0227

722.5242

723.5365

25

25

27

54

35

37

51

52

57

56

54

54

56

55

37

28

41

57

79

78

74

71

68

64

67

56

76

84

72

72

66

72

67

91

71

82

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

38

#

Table 15.4 (continued)

94

94

100

97

97

90

97

91

83

84

86

85

83

91

90

88

88

85

Cost (%)

6

7

14

35

37

42

26

37

41

39

36

36

44

38

42

35

35

41

Delay (%)

22

22

22

61

22

22

61

61

61

61

61

61

61

61

22

0

35

61

PAR (%)

(continued)

74

79

78

74

73

73

79

74

77

78

79

80

72

66

65

77

75

73

CO2 emission (%)

362 15 Multi-objective Electrical Energy Scheduling in Smart Homes …

CO2 emission

14.1336

14.0505

14.3136

13.9279

13.9279

14.3136

14.4110

14.6161

14.4282

15.2777

14.2213

14.1336

14.1336

14.1336

12.8408

13.9645

13.0207

16.6578

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

61.4601

59.4680

62.4396

59.7234

59.6788

59.5069

60.1509

59.3630

61.3393

62.4868

62.2988

59.4418

59.9450

59.3325

59.9384

59.1240

59.1530

59.5729

17.5023

21.9931

19.5292

22.2827

21.5244

21.5452

20.4066

20.5508

19.0967

19.9747

19.3456

20.1251

20.5745

21.1022

20.6726

20.4415

19.6956

19.3477

5.9736

8.2577

5.7980

6.8521

7.3792

7.3792

5.7980

8.2577

5.9736

5.7980

5.7980

7.3792

7.3792

5.7980

7.3792

8.2577

5.7980

8.2577

730.0047

718.3342

719.9357

719.2443

719.0413

718.4550

721.2573

720.8116

720.3384

719.6376

719.4611

720.8947

716.8616

723.8097

723.9357

722.8441

722.2769

721.0120

52

14

53

33

26

26

50

19

49

50

52

30

29

48

29

19

52

25

38

89

76

92

74

74

74

72

58

69

67

70

71

76

76

71

75

74

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

56

#

Table 15.4 (continued)

85

97

79

96

96

97

93

98

86

79

80

98

95

98

95

100

99

97

Cost (%)

49

6

29

3

10

10

21

20

34

25

31

24

20

15

19

21

28

31

Delay (%)

57

0

61

35

22

22

61

0

57

61

61

22

22

61

22

0

61

0

PAR (%)

(continued)

56

86

82

84

85

86

79

80

81

83

83

80

90

72

72

75

76

79

CO2 emission (%)

Appendix 363

CO2 emission

16.2847

16.2847

15.1550

15.7036

15.2151

16.5080

15.5877

13.5136

17.5776

17.5776

13.5136

13.2207

13.2207

13.6764

14.0505

14.0505

14.2254

13.4883

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

62.5930

61.9653

60.5958

60.3698

60.4028

61.7535

61.1238

63.1664

67.5448

65.9606

62.7928

60.5352

62.3312

62.2222

62.8599

62.6959

66.5647

66.7463

17.3593

16.6060

19.3735

19.3792

17.1293

17.2806

18.0882

17.0801

14.8774

15.2584

17.2595

16.8389

16.2180

16.2553

15.9773

16.4547

14.8513

14.7403

5.7980

5.7980

5.7980

7.3792

5.7980

5.7980

5.7980

8.2577

5.7980

5.7980

8.2577

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

733.5048

737.1763

726.1201

726.0636

733.9633

735.2347

726.9711

735.6544

737.0504

734.6493

733.0318

730.8753

733.2398

733.4252

733.3305

733.6621

737.7221

737.1302

59

62

53

33

61

60

58

33

62

61

33

58

59

61

61

60

64

64

83

72

75

75

80

86

86

82

25

25

82

53

40

58

52

59

43

43

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

74

#

Table 15.4 (continued)

78

82

91

92

92

84

87

75

48

58

77

91

80

81

77

78

54

53

Cost (%)

50

57

31

31

52

51

43

53

74

70

51

55

61

61

63

59

74

75

Delay (%)

61

61

61

22

61

61

61

0

61

61

0

61

61

61

61

61

61

61

PAR (%)

(continued)

46

37

66

66

45

42

64

41

37

43

48

53

47

46

47

46

35

37

CO2 emission (%)

364 15 Multi-objective Electrical Energy Scheduling in Smart Homes …

CO2 emission

19.3677

13.2207

13.6484

14.5441

14.5136

13.3555

15.0524

13.2207

14.5136

13.2207

13.2207

14.5136

14.5441

14.5441

13.6484

14.5136

14.8065

14.5136

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

59.4036

60.0030

59.1566

60.2172

61.4770

60.3401

59.7548

59.7321

60.3757

59.0930

59.8390

60.6945

60.0092

59.1926

60.3545

61.1524

61.8930

65.7170

18.4209

17.8386

19.3477

17.2796

16.6261

16.6917

18.4267

18.1703

17.9799

19.1686

18.1093

18.7427

18.1028

19.4110

16.8152

17.1547

18.0217

14.9940

7.3792

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

7.3792

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

8.2577

5.7980

728.1273

731.3030

726.1765

732.2641

734.8168

732.3622

728.5667

730.2261

729.4188

727.6424

729.2846

724.4386

727.4578

725.2773

732.2314

735.3783

728.4742

734.3737

36

56

53

60

61

61

55

58

40

53

58

53

58

52

60

60

31

59

68

64

68

80

68

68

68

86

86

68

86

61

85

68

68

80

86

0

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

92

#

Table 15.4 (continued)

98

94

99

93

85

92

96

96

92

100

95

90

94

99

92

87

83

59

Cost (%)

40

46

31

51

57

56

40

42

44

33

43

37

43

31

55

52

44

73

Delay (%)

22

61

61

61

61

61

61

61

22

61

61

61

61

61

61

61

0

61

PAR (%)

(continued)

61

52

66

50

43

49

59

55

57

62

57

70

62

68

50

41

60

44

CO2 emission (%)

Appendix 365

CO2 emission

14.5136

15.5183

13.6484

14.9413

13.3555

14.5136

15.2293

13.5136

13.5136

16.5222

15.2293

15.2293

13.6484

13.6484

14.5136

14.9996

14.5692

14.7320

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

62.7145

63.0770

62.7050

59.5796

61.1046

61.6413

59.9566

60.1073

59.8713

60.4125

60.3321

61.1173

59.7321

59.9869

60.0796

60.0652

59.5558

59.2375

15.9001

16.0686

15.8599

18.6534

17.2761

17.2025

16.7661

16.5709

16.8378

17.6816

17.9319

16.4521

18.0644

18.0087

17.8576

17.6483

17.6075

18.6406

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

7.3792

7.3792

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

734.6931

735.2087

735.0816

726.5798

734.8627

734.9969

733.3702

732.9533

732.8086

731.5180

729.5955

735.3054

730.9709

728.2227

729.7118

730.7418

731.4037

730.6881

63

63

63

55

60

60

59

60

56

41

40

60

56

58

56

59

55

55

65

67

61

68

80

80

58

58

40

82

82

58

68

85

62

80

54

68

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

110

#

Table 15.4 (continued)

78

75

78

97

87

84

94

94

95

92

92

87

96

94

94

94

97

99

Cost (%)

64

62

64

38

51

52

56

58

55

47

45

59

43

44

45

47

48

38

Delay (%)

61

61

61

61

61

61

61

61

61

22

22

61

61

61

61

61

61

61

PAR (%)

(continued)

43

42

42

65

43

42

47

48

48

52

57

41

53

60

56

54

52

54

CO2 emission (%)

366 15 Multi-objective Electrical Energy Scheduling in Smart Homes …

CO2 emission

14.5136

17.8452

17.5270

14.6562

14.3653

14.4361

14.5854

14.6562

14.2205

13.6355

15.4883

15.0741

14.4889

15.1923

14.7817

15.7413

15.3270

15.1062

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

60.7093

61.0949

61.8527

60.0392

60.1108

59.9507

60.5304

60.1020

60.6206

74.0222

74.2133

74.0150

75.4792

75.2809

74.2133

70.3729

70.2607

59.0483

18.9790

19.1113

19.7089

20.3422

20.2116

20.5186

18.9947

18.7093

20.9640

12.2218

12.1191

12.1653

12.1973

12.1832

12.1999

13.3445

13.2550

18.9812

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

7.3792

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

721.7106

721.7035

718.7407

716.7063

717.4020

716.6109

725.3321

726.3938

716.0881

748.8843

748.8843

749.1795

750.6642

750.9595

748.8843

743.2157

742.9204

728.9206

52

51

48

49

48

49

52

52

29

75

74

74

74

74

74

66

66

54

60

57

51

64

59

69

60

55

81

72

66

67

69

70

66

26

21

68

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

128

#

Table 15.4 (continued)

90

88

83

94

94

95

91

94

90

9

8

9

0

1

8

31

32

100

Cost (%)

35

33

28

22

23

20

35

37

16

99

100

100

99

99

99

88

89

35

Delay (%)

61

61

61

61

61

61

61

61

22

61

61

61

61

61

61

61

61

61

PAR (%)

(continued)

78

78

85

91

89

91

68

65

92

6

6

5

1

0

6

21

21

58

CO2 emission (%)

Appendix 367

CO2 emission

14.3136

14.1336

14.3136

13.8408

14.0252

14.5142

14.8741

14.7817

15.6065

14.3555

14.2207

14.2207

15.6065

15.2207

15.2207

15.0207

15.6065

15.0207

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

59.2967

59.4557

59.2639

59.8044

59.8368

60.5588

60.8284

60.9932

61.2191

60.8887

60.5681

60.3285

60.7141

60.1744

60.3456

60.2350

60.2739

61.9817

19.5771

19.2238

19.2798

20.7963

20.8187

18.6509

20.9273

20.5122

20.4670

19.5327

19.2197

19.2865

19.7164

19.7064

20.2310

20.6913

20.8586

20.1714

7.3792

7.3792

8.2577

8.2577

8.2577

8.2577

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

7.3792

5.7980

7.3792

7.3792

7.3792

725.1592

726.0846

725.6042

719.5635

719.4964

725.2547

717.0842

717.0914

717.1479

718.0555

719.5987

719.6764

719.6693

722.6284

722.9539

716.0953

716.2160

715.7793

31

32

23

16

16

25

48

49

49

49

52

52

51

32

51

29

28

30

61

53

61

58

58

53

72

72

70

53

64

63

68

75

78

71

74

71

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

146

#

Table 15.4 (continued)

98

98

99

95

95

91

89

88

87

89

91

92

90

93

92

93

93

82

Cost (%)

29

32

32

17

17

38

16

20

21

29

32

32

28

28

23

18

17

23

Delay (%)

22

22

0

0

0

0

61

61

61

61

61

61

61

22

61

22

22

22

PAR (%)

(continued)

68

66

67

83

83

68

90

90

90

87

83

83

83

75

74

92

92

93

CO2 emission (%)

368 15 Multi-objective Electrical Energy Scheduling in Smart Homes …

CO2 emission

13.2207

13.3555

14.7812

14.5136

14.7812

14.0505

14.4265

14.7812

14.7812

14.5136

13.1181

14.5136

14.5136

14.5136

12.9284

13.1181

14.9413

14.5136

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

61.3615

61.2790

62.0484

61.5858

60.4055

61.9126

62.4492

62.4439

61.8159

61.8159

61.8483

60.9960

60.9606

61.2792

61.3297

62.0419

61.1973

61.0038

20.0766

18.0720

20.2701

20.5625

20.3002

19.9932

19.8929

20.3365

19.6197

19.4957

19.5491

18.4767

18.6518

19.7701

19.8860

19.4133

20.4932

20.6531

8.2577

5.7980

8.2577

8.2577

7.3792

8.2577

8.2577

7.3792

8.2577

7.3792

7.3792

7.3792

7.3792

5.7980

5.7980

7.3792

5.9736

7.3792

718.1755

727.6208

717.1133

716.8914

718.6664

717.1323

717.2666

717.2122

719.1609

719.1609

719.0938

726.8657

725.8881

719.0266

718.7829

719.2174

716.9006

716.7770

21

55

22

21

29

21

22

31

23

31

31

36

36

50

51

32

48

30

68

62

88

91

68

68

68

88

68

64

64

69

75

64

68

64

85

86

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

164

#

Table 15.4 (continued)

86

86

82

85

92

83

79

79

83

83

83

88

88

86

86

82

87

88

Cost (%)

24

43

22

20

22

25

26

22

29

30

29

39

38

27

26

31

20

19

Delay (%)

0

61

0

0

22

0

0

22

0

22

22

22

22

61

61

22

57

22

PAR (%)

(continued)

87

62

90

90

86

90

89

89

84

84

84

64

66

85

85

84

90

91

CO2 emission (%)

Appendix 369

CO2 emission

14.5457

14.9413

14.5457

13.4821

13.4821

14.5136

14.8065

14.5136

14.8065

14.5136

14.5136

15.0631

15.2136

12.9284

12.8408

14.6786

14.3604

12.7025

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

61.0248

61.2371

61.2371

62.5973

62.5168

61.3981

61.1721

60.7659

62.0538

61.9225

61.8182

62.6212

61.7713

62.4377

62.2730

60.5910

61.3064

60.5910

20.9104

20.6775

20.6427

20.3016

20.0203

17.9809

18.0313

19.5954

19.8999

19.7583

20.3324

19.5930

20.0256

20.4962

21.0476

18.4202

18.1093

18.5135

7.3792

7.3792

7.3792

7.3792

7.3792

5.7980

5.7980

7.3792

8.2577

8.2577

7.3792

8.2577

7.3792

5.7980

6.8521

5.7980

5.7980

5.7980

715.2125

715.4032

715.4032

717.0914

718.1382

727.3948

727.3383

722.7390

718.4546

719.0797

718.6522

718.8785

717.0970

716.3709

716.3638

726.0632

727.3524

726.0632

30

28

28

32

32

55

55

32

22

22

29

22

30

50

34

55

55

55

94

70

66

92

91

58

61

68

68

64

68

64

68

83

83

68

62

68

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

182

#

Table 15.4 (continued)

88

87

87

78

79

86

87

90

82

83

83

78

83

79

80

91

86

91

Cost (%)

16

19

19

22

25

44

44

29

26

27

22

29

25

20

15

40

43

39

Delay (%)

22

22

22

22

22

61

61

22

0

0

22

0

22

61

35

61

61

61

PAR (%)

(continued)

95

94

94

90

87

62

63

75

86

84

86

85

90

92

92

66

63

66

CO2 emission (%)

370 15 Multi-objective Electrical Energy Scheduling in Smart Homes …

CO2 emission

14.5812

13.9954

16.9143

17.0170

17.0170

17.0170

16.6350

16.4350

17.4317

17.1099

16.6988

16.6988

17.3099

17.5241

17.0170

16.8488

16.8488

16.4509

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

62.7894

63.6916

63.5465

63.4589

63.5670

63.6615

63.0341

62.7554

63.4934

63.5465

63.0007

62.9733

63.3123

63.0336

63.7985

63.8164

60.7175

61.2508

19.5575

16.9903

17.0416

17.3143

17.4538

17.3631

15.9150

16.0713

17.5999

17.3139

19.6884

19.6836

16.1666

16.2494

16.2233

16.4717

20.9286

20.6619

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

7.3792

717.7272

716.8232

717.1675

716.7173

718.0009

715.9986

727.6929

727.2426

715.8325

718.2905

716.4735

716.7419

723.1146

722.6644

720.5454

720.0298

715.6962

715.2690

47

55

55

53

52

53

60

59

52

53

47

46

58

58

58

57

48

28

41

35

35

33

26

29

38

38

32

27

41

38

33

33

33

34

76

67

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

200

#

Table 15.4 (continued)

77

72

73

73

72

72

76

77

73

73

76

76

74

76

71

71

90

87

Cost (%)

29

54

53

51

49

50

64

62

48

51

28

28

61

61

61

59

16

19

Delay (%)

61

61

61

61

61

61

61

61

61

61

61

61

61

61

61

61

61

22

PAR (%)

(continued)

88

90

90

91

87

93

62

63

93

87

91

91

74

75

81

82

93

95

CO2 emission (%)

Appendix 371

CO2 emission

16.0367

16.8037

16.6406

16.7872

16.3290

16.0108

15.9026

15.0473

16.7246

16.0170

16.1168

15.6102

15.4350

16.8488

15.5108

16.8170

16.8170

16.8170

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

62.5599

62.8481

62.4509

63.8872

63.3073

63.4394

63.4531

63.4053

62.8571

63.4485

63.4531

63.2819

63.0908

63.0908

63.0548

62.7620

63.8872

62.8235

16.9742

17.0022

17.0874

19.6028

16.2750

19.9664

19.9504

19.8602

17.4775

16.0845

20.0844

19.7192

19.6325

19.5748

19.4907

19.4548

19.5354

19.6282

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

717.7043

717.4218

717.8897

715.4207

726.9442

715.2620

715.1278

715.6859

716.9415

724.5290

715.1278

716.1252

716.1252

716.1252

715.9776

717.9956

715.4207

717.3033

55

55

55

49

58

48

48

47

54

59

48

48

48

47

46

47

46

48

36

36

36

54

35

55

53

46

47

37

61

49

47

43

36

38

36

47

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

218

#

Table 15.4 (continued)

79

77

79

71

74

73

73

73

77

73

73

74

75

75

76

77

71

77

Cost (%)

54

54

53

29

60

25

25

26

49

62

24

28

28

29

30

30

29

29

Delay (%)

61

61

61

61

61

61

61

61

61

61

61

61

61

61

61

61

61

61

PAR (%)

(continued)

88

89

88

94

64

95

95

93

90

70

95

92

92

92

93

87

94

89

CO2 emission (%)

372 15 Multi-objective Electrical Energy Scheduling in Smart Homes …

CO2 emission

17.1135

17.5241

17.2059

16.7246

17.6215

17.7241

17.3099

15.7184

17.2072

16.8255

237

238

239

240

241

242

243

244

245

63.7301

63.6505

62.8775

63.2076

63.6313

63.5533

63.4580

63.7836

63.7836

63.7973

18.4221

17.9371

19.9174

17.4845

17.4408

17.4930

16.0320

18.1527

18.0555

18.2044

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

5.7980

717.6390

713.4983

716.7736

715.9986

716.1046

718.1351

724.7762

713.3640

713.3640

713.2298

50

51

48

53

52

52

59

50

50

50

36

30

51

29

23

25

37

30

26

32

Unsafety (%)

Overall Performance (%)

PAR

Utility scores in terms of

Delay

Unsafety

Cost

Objectives

236

#

Table 15.4 (continued)

72

72

77

75

72

73

73

71

71

71

Cost (%)

40

45

26

49

49

49

63

43

44

42

Delay (%)

61

61

61

61

61

61

61

61

61

61

PAR (%)

88

99

91

93

92

87

69

100

100

100

CO2 emission (%)

Appendix 373

374

15 Multi-objective Electrical Energy Scheduling in Smart Homes …

References 1. Kaveh, A., & Vazirinia, Y. (2020). Smart-home electrical energy scheduling system using multiobjective antlion optimizer and evidential reasoning. Scientia Iranica, 27(1), 177–201. https:// doi.org/10.24200/sci.2019.53783.3412. 2. Mirjalili, S., Jangir, P., & Saremi, S. (2017). Multi-objective ant lion optimizer: A multi-objective optimization algorithm for solving engineering problems. Applied Intelligence, 46(1), 79–95. https://doi.org/10.1007/s10489-016-0825-8. 3. Belton, V., & Stewart, T. (2002). Multiple criteria decision analysis: An integrated approach. Springer Science & Business Media. 4. Du, Y. F., Jiang, L., Li, Y. Z., Counsell, J., & Smith, J. S. (2016). Multi-objective demand side scheduling considering the operational safety of appliances. Applied Energy, 179, 864–874. https://doi.org/10.1016/j.apenergy.2016.07.016. 5. Sou, K. C., Weimer, J., Sandberg, H., & Johansson, K. H. (2011). Scheduling smart home appliances using mixed integer linear programming. In 2011 50th IEEE Conference on Decision and Control and European Control Conference, 12–15 Dec 2011, pp. 5144–5149. 6. Bazargan-Lari, M. R. (2014). An evidential reasoning approach to optimal monitoring of drinking water distribution systems for detecting deliberate contamination events. Journal of Cleaner Production, 78, 1–14. https://doi.org/10.1016/j.jclepro.2014.04.061. 7. Zhao, Z., Lee, W. C., Shin, Y., & Song, K. (2013). An optimal power scheduling method for demand response in home energy management system. IEEE Transactions on Smart Grid, 4(3), 1391–1400. https://doi.org/10.1109/TSG.2013.2251018. 8. Kristinsdóttir, A. R., Stoll, P., Nilsson, A., & Brandt, N. (2013). Description of climate impact calculation methods of the CO2 e signal for the active house project. In KTH Royal Institute of Technology. 9. Jolai, F., Asefi, H., Rabiee, M., & Ramezani, P. (2013). Bi-objective simulated annealing approaches for no-wait two-stage flexible flow shop scheduling problem. Scientia Iranica, 20(3), 861–872. https://doi.org/10.1016/j.scient.2012.10.044.