Intelligent Renewable Energy Systems E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
INTELLIGENT RENEWABLE ENERGY SYSTEMS
This collection of papers on artificial intelligence and other methods for improving renewable energy systems, written by industry experts, is a reflection of the state of the art, a must-have for engineers, maintenance personnel, students, and anyone else wanting to stay abreast with current energy systems concepts and technology.
Renewable energy is one of the most important subjects being studied, researched, and advanced in today’s world. From a macro level, like the stabilization of the entire world’s economy, to the micro level, like how you are going to heat or cool your home tonight, energy, specifically renewable energy, is on the forefront of the discussion.
This book illustrates modelling, simulation, design and control of renewable energy systems employed with recent artificial intelligence (AI) and optimization techniques for performance enhancement. Current renewable energy sources have less power conversion efficiency because of its intermittent and fluctuating behavior. Therefore, in this regard, the recent AI and optimization techniques are able to deal with data ambiguity, noise, imprecision, and nonlinear behavior of renewable energy sources more efficiently compared to classical soft computing techniques.
This book provides an extensive analysis of recent state of the art AI and optimization techniques applied to green energy systems. Subsequently, researchers, industry persons, undergraduate and graduate students involved in green energy will greatly benefit from this comprehensive volume, a must-have for any library.
Audience
Engineers, scientists, managers, researchers, students, and other professionals working in the field of renewable energy.
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Veröffentlichungsjahr: 2021
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Table of Contents
Cover
Title Page
Copyright
Preface
1 Optimization Algorithm for Renewable Energy Integration
1.1 Introduction
1.2 Mixed Discrete SPBO
1.3 Problem Formulation
1.4 Comparison of the SPBO Algorithm in Terms of CEC-2005 Benchmark Functions
1.5 Optimum Placement of RDG and Shunt Capacitor to the Distribution Network
1.6 Conclusions
References
2 Chaotic PSO for PV System Modelling
2.1 Introduction
2.2 Proposed Method
2.3 Results and Discussions
2.4 Conclusions
References
3 Application of Artificial Intelligence and Machine Learning Techniques in Island Detection in a Smart Grid
3.1 Introduction
3.2 Islanding in Power System
3.3 Island Detection Methods
3.4 Application of Machine Learning and Artificial Intelligence Algorithms in Island Detection Methods
3.5 Conclusion
References
4 Intelligent Control Technique for Reduction of Converter Generated EMI in DG Environment
4.1 Introduction
4.2 Grid Connected Solar PV System
4.3 Control Strategies for Grid Connected Solar PV System
4.4 Electromagnetic Interference
4.5 Intelligent Controller for Grid Connected Solar PV System
4.6 Results and Discussion
4.7 Conclusion
References
5 A Review of Algorithms for Control and Optimization for Energy Management of Hybrid Renewable Energy Systems
5.1 Introduction
5.2 Optimization and Control of HRES
5.3 Optimization Techniques/Algorithms
5.4 Use of GA In Solar Power Forecasting
5.5 PV Power Forecasting
5.6 Advantages
5.7 Disadvantages
5.8 Conclusion
Appendix A: List of Abbreviations
References
6 Integration of RES with MPPT by SVPWM Scheme
6.1 Introduction
6.2 Multilevel Inverter Topologies
6.3 Multilevel Inverter Modulation Techniques
6.4 Grid Integration of Renewable Energy Sources (RES)
6.5 Simulation Results
6.6 Conclusion
References
7 Energy Management of Standalone Hybrid Wind-PV System
7.1 Introduction
7.2 Hybrid Renewable Energy System Configuration & Modeling
7.3 PV System Modeling
7.4 Wind System Modeling
7.5 Modeling of Batteries
7.6 Energy Management Controller
7.7 Simulation Results and Discussion
7.8 Conclusion
References
8 Optimization Technique Based Distribution Network Planning Incorporating Intermittent Renewable Energy Sources
8.1 Introduction
8.2 Load and WTDG Modeling
8.3 Objective Functions
8.4 Mathematical Formulation Based on Fuzzy Logic
8.5 Solution Algorithm
8.6 Simulation Results and Analysis
8.7 Conclusion
References
9 User Interactive GUI for Integrated Design of PV Systems
9.1 Introduction
9.2 PV System Design
9.3 Economic Considerations
9.4 PV System Standards
9.5 Design of GUI
9.6 Results
9.7 Discussions
9.8 Conclusion and Future Scope
9.9 Acknowledgement
References
10 Situational Awareness of Micro-Grid Using Micro-PMU and Learning Vector Quantization Algorithm
10.1 Introduction
10.2 Micro Grid
10.3 Phasor Measurement Unit and Micro PMU
10.4 Situational Awareness: Perception, Comprehension and Prediction
10.5 Conclusion
References
11 AI and ML for the Smart Grid
11.1 Introduction
11.2 AI Techniques
11.3 Machine Learning (ML)
11.4 Home Energy Management System (HEMS)
11.5 Load Forecasting (LF) in Smart Grid
11.6 Adaptive Protection (AP)
11.7 Energy Trading in Smart Grid
11.8 AI Based Smart Energy Meter (AI-SEM)
References
12 Energy Loss Allocation in Distribution Systems with Distributed Generations
12.1 Introduction
12.2 Load Modelling
12.3 Mathematical Model
12.4 Solution Algorithm
12.5 Results and Discussion
12.6 Conclusion
References
13 Enhancement of Transient Response of Statcom and VSC Based HVDC with GA and PSO Based Controllers
13.1 Introduction
13.2 Design of Genetic Algorithm Based Controller for STATCOM
13.3 Design of Particle Swarm Optimization Based Controller for STATCOM
13.4 Design of Genetic Algorithm Based Type-1 Controller for VSCHVDC
13.5 Conclusion
References
14 Short Term Load Forecasting for CPP Using ANN
14.1 Introduction
14.2 Working of Combined Cycle Power Plant
14.3 Implementation of ANN for Captive Power Plant
14.4 Training and Testing Results
14.5 Conclusion
14.6 Acknowlegdement
References
15 Real-Time EVCS Scheduling Scheme by Using GA
15.1 Introduction
15.2 EV Charging Station Modeling
15.3 Real Time System Modeling for EVCS
15.4 Results and Discussion
15.5 Conclusion
References
About the Editors
Index
End User License Agreement
List of Tables
Chapter 1
Table 1.1 CEC 2005 benchmark function [67].
Table 1.2 Parameters of different algorithms for benchmark functions.
Table 1.3 Optimization result of CEC-2005 benchmark functions.
Table 1.4 Variation of load demand (pu) and solar power generation (pu) with loa...
Table 1.5 Cost and lifetime of different DGs.
Table 1.6 Optimum placement of RDGs and shunt capacitors to the 33-bus distribut...
Table 1.7 Optimum placement of RDGs and shunt capacitors to the 69-bus distribut...
Chapter 2
Table 2.1 List of one-dimensional chaotic maps.
Table 2.2 Unimodal and multimodal test problems and their details.
Table 2.3 Results of the test problems with the proposed methods.
Table 2.4 Non-parametric test outcomes (part-1).
Table 2.5 Non-parametric test outcomes (part-2).
Table 2.6 Datasheet values of PV modules at STC conditions.
Table 2.7 Estimated parameters of the three-diode model for Kyocera (KC200GT mul...
Table 2.8 Optimal parameter identification of the three-diode model for Canadian...
Table 2.9 Normalized statistical analysisof the error function for the three-dio...
Table 2.10 Normalized results of the sum of square error (SSE) for the three-dio...
Chapter 3
Table 3.1 Comparison of machine learning and artificial intelligence based islan...
Chapter 4
Table 4.1 Rule base for fuzzy logic controller.
Chapter 5
Table 5.1 Application of Genetic Algorithms for Hybrid Renewable Energy system o...
Chapter 6
Table 6.1 Pulse pattern of H-bridge.
Table 6.2 Specifications of SunPower SPR-305E-WHT-D module at STC.
Table 6.3 Simulation parameters.
Table 6.4 Harmonic distortion of line voltage w.r.t M.
Chapter 7
Table 7.1 Electrical characteristics data of sun power SPR-305E-WHT-U module.
Table 7.2 Parameter of wind turbine model.
Chapter 8
Table 8.1 Values of exponential parameters for commercial, industrial and reside...
Table 8.2 Load demand, candidate buses to access, and year of connection of the ...
Table 8.3 Economic and environmental data.
Table 8.4 Alternatives for substation and feeder upgradation.
Table 8.5 Cost of upgrading feeders (×10
3
USD).
Table 8.6 Different planning cases.
Table 8.7 Stage-wise investment plans for different planning cases.
Table 8.8 Obtained values of objectives for different planning cases.
Table 8.9 Detail analysis of economic performances of planning cases 1−3.
Table 8.10 Detail analysis of Economic performances of planning cases 1−3.
Table 8.11 Comparison of different optimization techniques.
Table 8.12 WSRT based comparison between different algorithms.
Chapter 9
Table 9.1 Cost estimate of PV system.
Table 9.2 Market survey data for PV system design.
Table 9.3 Module cost estimate.
Table 9.4 Standards for PV installations.
Table 9.5 Panel efficiencies.
Table 9.6 Load description of a household consumer.
Table 9.7 Lumped load data for a household.
Table 9.8 Load consumption pattern.
Chapter 10
Table 10.1 Fault class corresponding to fault type.
Table 10.2 Training data set with known fault class.
Table 10.3 Initial weight vector.
Table 10.4 Updated weight vector after training.
Table 10.5 Euclidean distance test data.
Chapter 11
Table 11.1 Conventional grid versus smart grid.
Chapter 12
Table 12.1 Consumers type and their connected nodes.
Table 12.2 DG data at different nodes.
Table 12.3 Allocated losses (kW) at 4 AM with different load models in scenario ...
Table 12.4 Allocated losses (kW) at 11 AM with different load models in scenario...
Table 12.5 Node-wise allocated energy losses (kWh) without DGs for entire day on...
Table 12.6 Allocated losses (kW) at 4 AM with different load models in scenario ...
Table 12.7 Allocated losses (kW) at 11 AM with different load models in scenario...
Table 12.8 Node-wise allocated energy losses (kWh) with DGs for entire day on di...
Table 12.9 Summary of total allocated energy losses (kWh) ahead of a day.
Chapter 13
Table 13.1 Eigenvalues of 2-level STATCOM with PI controller.
Table 13.2 Eigenvalues of 2-level STATCOM with nonlinear feedback for suboptimal...
Table 13.3 Parameters used for optimization with genetic algorithm.
Table 13.4 Eigenvalues of 2-level STATCOM with nonlinear feedback for suboptimal...
Table 13.5 Eigenvalues of 3-level STATCOM with optimal controller parameters bas...
Table 13.6 Parameters used for optimization with PSO.
Table 13.7 Eigenvalues of 2-level STATCOM with nonlinear feedback for optimal co...
Table 13.8 Operating combinations of VSC HVDC controllers.
Table 13.9 Eigenvalues of the combined system in case-1 to case-4.
Table 13.10 Eigenvalues of the combined system in case-5 to case-8.
Chapter 14
Table 14.1 Gas turbine specifications.
Table 14.2 Comparison between the actual load and predicted load.
Chapter 15
Table 15.1 Car specification.
Table 15.2 Nomenclature table.
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Begin Reading
About the Editors
Index
Also of Interest
End User License Agreement
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Scrivener Publishing100 Cummings Center, Suite 541JBeverly, MA 01915-6106
Artificial Intelligence and Soft Computing for Industrial Transformation
Series Editor: Dr S. Balamurugan ([email protected])
Scope: Artificial Intelligence and Soft Computing Techniques play an impeccable role in industrial transformation. The topics to be covered in this book series include Artificial Intelligence, Machine Learning, Deep Learning, Neural Networks, Fuzzy Logic, Genetic Algorithms, Particle Swarm Optimization, Evolutionary Algorithms, Nature Inspired Algorithms, Simulated Annealing, Metaheuristics, Cuckoo Search, Firefly Optimization, Bio-inspired Algorithms, Ant Colony Optimization, Heuristic Search Techniques, Reinforcement Learning, Inductive Learning, Statistical Learning, Supervised and Unsupervised Learning, Association Learning and Clustering, Reasoning, Support Vector Machine, Differential Evolution Algorithms, Expert Systems, Neuro Fuzzy Hybrid Systems, Genetic Neuro Hybrid Systems, Genetic Fuzzy Hybrid Systems and other Hybridized Soft Computing Techniques and their applications for Industrial Transformation. The book series is aimed to provide comprehensive handbooks and reference books for the benefit of scientists, research scholars, students and industry professional working towards next generation industrial transformation.
Publishers at ScrivenerMartin Scrivener ([email protected])Phillip Carmical ([email protected])
Intelligent Renewable Energy Systems
Edited by
Neeraj Priyadarshi
Akash Kumar Bhoi
Sanjeevikumar Padmanaban
S. Balamurugan
and
Jens Bo Holm-Nielsen
This edition first published 2022 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA
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Library of Congress Cataloging-in-Publication Data
ISBN 978-1-119-78627-6
Cover image: Pixabay.com
Cover design by Russell Richardson
Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines
Printed in the USA
10 9 8 7 6 5 4 3 2 1
Preface
This book presents intelligent renewable energy systems integrating artificial intelligence techniques and optimization algorithms. The first chapter describes placement of distributed generation (DG) sources including renewable distributed generation (RDGs) such as biomass, solar PV, and shunt capacitor has been considered for the study purpose. The second chapter develops a new approach to chaotic particle swarm optimization (CPSO) technique. In the third chapter, comprehensive reviews of different artificial intelligence and machine learning techniques have been explicated. To bring out its advantages over other methods used in island detection, the traditional methods are first explained and then compared with artificial intelligence and machine learning island detection techniques. The performance of the intelligent controller is found to be good under steady conditions for grid connected photovoltaic systems and has been discussed in chapter four. Chapter five explains various uses of Genetic Algorithms (GA) and Solar PV forecasting are described; further, many stimulated algorithms which have been used in optimization, controlling, and methods of supervising of power for renewable energy analysis, which include hybrid power generation strategies are discussed. Chapter six presents the integration of 100 kW solar PV source to the 25 kV AC grid by using generalized r-s based SVPWM algorithm. Chapter seven aims to discuss the idea of hybrid system configuration, dynamic modeling, energy management, and control strategies. A multi-stage planning framework is proposed in chapter eight to divide the planning period into several stages so that investments can be made in each stage as per the requirements. A unique and a novel GUI is presented to design the entire solar PV systems has been discussed in Chapter nine. Chapter ten addresses micro-grid situational awareness using micro PMU. Role of AI & ML in smart grid entities such as Home Energy Management System (HEMS), Energy Trading, Adaptive Protection, Load Forecasting and Smart Energy Meter are presented in Chapter eleven. Chapter twelve presents a new method for energy loss allocation in radial distribution network (RDN) with distributed generationin the context of deregulated power system. Chapter thirteen presents the optimization of controller parameters for FACTS and VSC based HVDC. Chapter fourteen describes Short Term load forecasting for a Captive Power Plant Using Artificial Neural Network. Chapter fifteen defines Real-time EV Charging Station Scheduling Scheme by using Global Aggregator.
Neeraj PriyadarshiAkash Kumar BhoiSanjeevikumar PadmanabanS. BalamuruganJens Bo Holm-NielsenEditors
1Optimization Algorithm for Renewable Energy Integration
Bikash Das1, SoumyabrataBarik2*, Debapriya Das3 and V. Mukherjee4
1Department of Electrical Engineering, Govt. College of Engineering and Textile Technology, Berhampore, West Bengal, India2Department of Electrical and Electronics Engineering, Birla Institute of Technology and Science Pilani, K. K. Birla Goa Campus, Goa, India3Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India4Department of Electrical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, India
*Corresponding author: [email protected]
Abstract
With the development of society, the electrical power demand is increasing day by day. To overcome the increasing load demand, renewable energy resources play an important role. The common examples of renewable energy resources are solar photovoltaic (PV), wind energy, biomass, fuel-cell, etc. Due to the various benefits of the renewable energy, the incorporation of renewable energy resources into the distribution network becomes an important topic in the field of the modern power system. The incorporation of renewable energy resources may reduce the network loss, improve voltage profile, and improve the reliability of the network. In this current research work, optimum placements of renewable distributed generations (RDGs) (viz. biomass and solar PV) and shunt capacitors have been highlighted. For the optimization of the locations and the sizes of the RDGs and the shunt capacitors, a multi-objective optimization problem is considered in this book chapter in presence of various equality and inequality constraints. The multi-objective optimization problem is solved using a novel mixed-discrete student psychology-based optimization algorithm, where the key inspiration comes from the behaviour of a student in a class to be the best one and the performance of the student is measured in terms of the grades/marks he/she scored in the examination and the efficacy of the proposed method is analyzed and compared with different other optimization methods available in the literature. The multi-objective DG and capacitor placement is formulated with reduction of active power loss, improvement of voltage profile, and reduction of annual effective installation cost. The placement of RDGs and shunt capacitors with the novel proposed method is implemented on two different distribution networks in this book chapter.
Keywords: Renewable energy integration, shunt capacitors, distributed generation, mixed discrete student psychology-based optimization algorithm, distribution networks
1.1 Introduction
In order to satisfy the increasing electricity load demand, electrical power generation needs to be scheduled properly [1–25]. Electrical power sources can be classified into two categories named as non-renewable and renewable sources. Non-renewable sources mainly include fossil fuels [26–45]. To generate electrical power from fossil-fuels, the fossil-fuels need to be burned. But the combustion of fossil-fuel causes pollution which affects the atmosphere. On the other hand, renewable energy resources cause zero or very little pollution. The main drawback of renewable energy resources is that the extraction of energy is dependent on nature [46–55]. In spite of having the disadvantages, the renewable energy resources are gaining more and more interest in the extraction of electrical power and to satisfy the increasing load demand.
To get better benefits, the placement of distributed generation (DG) to the distribution network needs proper strategy and planning [56–71]. Improper placement of DG may lead to increase in network loss, as well as may cause instability to the network. DG injects power into the distribution network. Based on the power injection, the DG sources can be classified into three categories viz.:
a) Unity power factor (UPF) DG,
b) Lagging power factor (LPF) DG and
c) Reactive power DG.
UPF DG injects active power only to the network whereas LPF DG injects both active and reactive power to the distribution network. On the other hand, reactive power DG generates only reactive power. An example of UPF DG is solar photovoltaic cells. Biomass and wind turbines can be considered as an example of LPF DG. The shunt capacitor injects reactive power to the network and it can be said as the reactive power DG.
Proper incorporation of renewable distributed generation (RDG) may reduce the network power loss, improve the voltage profile, improve the voltage stability index (VSI), improve reliability, etc. Due to the various benefits of the incorporation of distributed generation to the distribution network, various researchers have considered this topic as their research interest. The literature review reveals that for the placement of DG to the distribution network, various researchers have considered different approaches to optimize the location and the size of the DG sources. The approaches include analytical, classical optimization methods as well as the metaheuristic optimization algorithm. In order to reduce the active power loss of the distribution network, Acharya et al. [1] have proposed an analytical approach to optimize the size and location of the DG source. Gozel and Hocaoglu [2] have proposed another analytical expression to determine the optimum size and location of DG. This approach is based on the current injection method with an objective to reduce the network power loss. Wang and Nehair [3] have proposed an analytical approach to optimize the size of UPF DG [3]. Wang and Nehair have considered different types of load demands of the distribution network [3]. On the other hand, Hung et al. [4] have proposed an analytical expression to optimize the location and size of LPF DG which is capable of supplying both active and reactive power to the distribution network. It may be observed that most of the researchers have developed analytical expressions to determine the optimum size of the DG in order to reduce the network loss. Aman et al. [5] have proposed an analytical approach for optimum placement of DG considering active power loss reduction and improvement of voltage profile of the network. They have considered a voltage sensitivity analysis approach based on the power stability index to determine the size of DG using a stepwise iterative approach. Some researchers [6] have also considered classical optimization methods to optimize the size of DG. At that same time, an analytical approach to determine the optimum location of DG in the distribution network may also be seen in [7]. Analytical methods can be implemented easily and take less computation time. But the direct formulation of complex problems using the analytical method is quite difficult. The application of an analytical approach, to solve complex problems, may lead to inaccurate solutions due to the assumptions made during the problem formulation process.
To overcome the problem, some of the researchers have adopted the linear and nonlinear programming approach for optimizing the DG size [8]. A considerable number of researchers have applied the metaheuristic optimization algorithm for the DG placement problem. Different nature-inspired algorithms like genetic algorithm (GA) [9], tabu search [41], particle swarm optimization (PSO) [42, 43], combined GA-PSO [44], artificial bee colony (ABC) algorithm [45], harmony search algorithm [46, 47], differential evolution (DE) [48], teaching-learning based optimization (TLBO) [49] may be found in the literature. The application of different hybrid optimization algorithms to solve the DG placement problem may also be noticed in the literature [50−53]. In [54], the authors proposed a multi-objective GA-based approach for the optimal positioning of multiple types of DG to reduce investment costs, cost due to annual energy loss and increase the system reliability. Many of the researchers have taken into account the VSI as the primary objective function while optimizing the DG size [55, 56]. Various objective functions have been considered by different researchers to determine the size and the locations of the DG. In [57], considering the probabilistic behavior of renewable resources, the authors attempted to solve the DG optimization problem. Singh and Goswami [58] have considered nodal pricing methodology for optimizing the DG locations and capacity. They have also studied the economical aspect of DG incorporation into the distribution network. At that same time, in [59], the authors have studied the power penetration by the DG sources considering the average hourly load demand. Some other research works on the optimum DG placement problem may also be noticed in the literature [60−65].
In this current book chapter, placement of DG sources including RDGs (such as biomass, solar PV) and shunt capacitor has been considered for the study purpose. The study has been performed by considering a multi-objective function that includes reduction of active power loss, the betterment of voltage profile, and minimization of effective annual installation cost. To optimize the locations of considered DG sources, a novel optimizing technique named mixed-discrete student psychology-based optimization (SPBO) algorithm is used. The proposed algorithm is inspired by the natural behaviour of the students to be the best student in the class. The criteria to be the best student is to perform well in the examination and the student needs to give more effort to be the one. The study has been carried out considering the hourly average load demand of the distribution network for a day. In this book chapter, the proposed method is tested on two distribution networks namely 33-bus and 69-bus distribution networks.
The remaining book chapter is categorized as follows. In the next section, a new algorithm named mixed-discrete SPBO is presented. The problem formulation is discussed in Section 1.3. Section 1.4 presents the optimum DG placement in the distribution networks using the proposed mixed-discrete SPBO algorithm. Finally, the conclusions are drawn in Section 1.5.
1.2 Mixed Discrete SPBO
1.2.1 SPBO Algorithm
SPBO algorithm has been proposed by Das et al. [66]. Similar to other meta-heuristic methodology, SPBO also uses a set of populations. The population considered in SPBO is analogous to a group of students present in the class. In general, the performance of a student is analyzed based on the marks/grade obtained by the student in the examination. The student with the maximum marks/grade is considered as the best one in the class and the student is awarded accordingly. It is also very clear that in the test the students will aim to achieve the highest marks/grade for which they need to provide more effort in their study. The overall grade is the final score of the students in the examination. The overall grade of a student depends on the cumulative effort given by the student in each subject offered to them. The students enhance their performance in the examination by paying more attention and effort in their studies. SPBO is based on the psychology of the students who are trying to improve their performance in the examination as well as trying to get the highest marks/grade in the examination.
For the improvement of the overall performance, the students need to enhance their performance in each subject which are offered to them. However, the improvement of the students’ performance is not the same for all the students. Improvement of the students’ performance depends upon a few factors like the capability and efficiency of the student as well as how the student gets interested in the subject. If a student is interested in a subject, he/she will be more involved in that subject. As a result, the performance of the student will be improved in that subject. As the overall grades/marks depend on each subject, improvement in any subject will improve the student’s overall performance. To improve performance, some students use to give similar or better kind of effort as given by the best student while some use to give more effort than that of the average student. If a student is less interested in a subject, then he/she will try to give the subject an average effort to improve his/her overall performance in the exam. The effort of a student can’t be measured directly. Marks obtained by the student in a subject are the outcome of her/his effort given to that subject. So, it may be considered, in general, that the effort given by the student is equivalent to the marks/grade obtained. Considering all these facts and the psychology of the students, the students can be divided into four different categories (such as the best student, good student, average student, and the students trying to improve randomly) [66].
(a) Best student: The best student is to be considered as the student who scored the overall highest marks/grade in the examination. It is an obvious fact that the best student will always try to secure his/her position in the class. For securing his/her position, the best student should have to provide more effort towards each subject than the same given by the rest of the students in the class. Therefore, it can be concluded that the best student will always provide more effort than the same given by other students in the class. Mathematically, the student of this category can be visualized with the help of (Equation 1.1) [66]
where, Xbest indicates the marks/grades obtained by the best student, Xj indicates the marks of the student, randomly selected, in the same subject, and rand generates a random number between 0 and 1. In (Equation 1.1), the D parameter randomly takes the value either 1 or 2. Marks obtained in the subjects by the best student may be increased or maybe decreased depending upon the student and the subjects. However, the primary objective of the best student in the class is to secure his/her position by continuously scoring the highest overall marks in the examination.
(b) Good student: If a student is interested in a subject, then he or she may strive to make more effort to enhance his or her performance in that subject. As a result, their overall performance in the examination will be improved. The students of this kind can be classified as subject wise good student. The effort given by all the students in this category may not be the same because the psychologies of the students are different. Some students try to give a similar or even better effort than the best student, where as; some students try to improve their performance by giving more effort after considering the effort given by the best student as well as the average effort given by the students of the class. This group of students may be further divided into two categories. The first category of this group of students is those who try to give similar or even better effort as given by the best student. This category of students gives effort to the subjects in which they are interested. So, their performance in that subject, as well as their overall performance, gets improved. Improvement of these kinds of students in a subject can be explained using (Equation 1. 2a) [66]. The other category of good students is those students who give effort considering the effort given by the best student as well as try to give more effort than the average effort of the class so that their performance in that subject, as well as overall performance, gets improved. Mathematically, the student of this category can be represented as (Equation 1. 2b) [66].
Here, Xnewi is the improved performance of the ith student in that subject; Xbest and Xi are the marks/grade obtained in that subject by the best student and the ith student, respectively; mean represents the average marks obtained in the class in that particular subject and rand variable produces a random number between 0 and 1.
(c) Average student: In order to increase their overall success in the test, students with less interest in a subject aim to offer an average effort. The students of this category can be named as subject-wise average student. While giving average effort in a particular subject, students will try to improve their overall performance by paying more effort to the other subjects which are offered to them. The performance of this category of students may be explained with the help of (Equation 1. 3) [66]
where, Xi indicates the marks/grade of the ith student of the class in a subject, mean indicates the average marks obtained in that particular subject, and rand generates a random number between 0 and 1.
(d) Students who try to improve randomly: Except the three categories of students discussed above, there are certain students who strive to enhance their results without recognising the effort offered by rest of the students in the class. The students of these kinds always try to improve their performance randomly to some extent up to their limitations, depending upon their interest in the subject. Improvement of the students’ performance based on this concept may be represented by (Equation 1. 4) [66]
where min and max are the two variables that indicate the minimum and the maximum marks of the subject, respectively, and rand generates a random number between 0 and 1.
The process of getting interested in a subject for different students is not deterministic. It depends on the students’ psychology. So, it may be said that the selection of different categories of students is a random process.
Incorporating the psychology of the aforementioned four categories of students, the SPBO algorithm can be visualized easily using the flowchart given in Figure 1.1. Each (student) population consists of various variables analogous to the subjects offered to the students. The students try to give the subjects effort so that their overall performance in the exam is improved. The fitness function is selected as the overall marks/grades obtained by the students. The effort given by the student is appreciated if her/his performance gets improved. Similarly, if the fitness function improves, a variable change is accepted. And, finally, the performance of the best student will be considered as the best solution or optimum solution. The performance of different optimization algorithms depends upon their own parameters. But SPBO has no such tuneable parameter.
1.2.2 Performance of SPBO for Solving Benchmark Functions
In order to evaluate the performance of SPBO for optimizing the benchmark functions, ten of the CEC-2005 benchmark functions are considered. The ten different CEC-2005 benchmark functions among twenty-five are presented in Table 1.1. To compare the performance of SPBO with other optimization methods, the results obtained by using SPBO are compared to those obtained by using the different optimization methods namely PSO, TLBO, CS, and SOS. Twenty-five individual runs are performed for each of the functions and for each of the algorithms.
The performance of the algorithms selected is evaluated on the basis of the optimal result obtained and on the basis of convergence mobility. For the purposes of analysis, the algorithms are found to converge when the gap between the optimal function result and the result obtained crosses below 1×10-5. The results obtained below 1×10-5 are considered as equal to zero. The parameters of PSO, TLBO, CS and SOS are considered according to the dimension of the benchmark function. But SPBO does not have any parameter and the size of the population needs not vary according to the dimension of the benchmark functions. With the increase of dimension of the functions, the size of the population of the proposed SPBO needs not to be increased. That’s why the population size of SPBO for all the considered benchmark functions is considered to be constant. It is considered as 20 for the proposed SPBO. In order to have a fair comparison of the performance of all the algorithms, the analysis is done based on the number of fitness function evaluations (NFFE) taken to converge.
Figure 1.1 Flowchart of the SPBO algorithm.
Table 1.1 CEC 2005 benchmark function [67].
Problem
Type of the function
Name of the functions
F(x*)
Initial range
Bounds
Dimension (
D
)
F1
Unimodal
Shifted Sphere Function
-450
[-100,100]
D
[-100,100]
D
30
F2
Unimodal
Shifted Schwefel’s Problem 1.2
-450
[-100,100]
D
[-100,100]
D
30
F3
Unimodal
Shifted Rotated High Conditioned Elliptic Function
-450
[-100,100]
D
[-100,100]
D
30
F4
Unimodal
Shifted Schwefel’s Problem 1.2 with Noise in Fitness
-450
[-100,100]
D
[-100,100]
D
30
F5
Unimodal
Schwefel’s Problems 2.6 with Global Optimum on Bounds
-310
[-100,100]
D
[-100, 100]
D
30
F6
Basic multimodal
Shifted Rosenbrock’s Function
390
[-100, 100]
D
[-100, 100]
D
30
F7
Basic multimodal
Shifted Rotated Griewank’s Function without Bounds
-180
[0, 600]
D
[0, 600]
D
30
F8
Basic multimodal
Shifted Rotated Ackley’s Function with Global Optimum on Bounds
-140
[-32, 32]
D
[-32, 32]
D
30
F9
Basic multimodal
Shifted Rastrigin’s Function
-330
[-5, 5]
D
[-5, 5]
D
30
F10
Basic multimodal
Shifted Rotated Rastrigin’s Function
-330
[-5, 5]
D
[-5, 5]
D
30
1.2.3 Mixed Discrete SPBO
In general, the optimization algorithm is used to optimize (minimize) the objective function by obtaining the optimum value of the variable vector X. To optimize the variable vector X, consist of some continuous and discrete variables, a mixed discrete version of SPBO may be used. For an n-dimensional problem that includes continuous and discrete variables, the variable vector may be represented as in (Equation 1.5).
where [Xcont] and [Xdisc] are the continuous and discrete variable vectors, respectively. For the scenario of m continuous variables and remaining (n-m) discrete variables, the [Xcont] and [Xdisc] maybe expressed as in (Equation 1.6) and (Equation 1.7), respectively.
As mentioned earlier the mixed discrete SPBO is capable to handle both the continuous and discrete variables. In the mixed discrete SPBO, the continuous variables are updated as the conventional SPBO. The updating process of the continuous variables is the same as discussed in the previous section using four categories of students (namely best student, good student, average student, and students who want to improve randomly) with the help of (Equations 1.1–1.4). For the discrete variables, the discretization may be done with the help of the nearest vertex approach (NVA). The NVA method is normally based on finding out the Euclidean norm in the design space. The discrete variables may be expressed in terms of a hypercube, which are represented by the sets of ordered pair and can be represented as in (Equation 1.8)
where, are the lower and upper limits of the discrete variables, in the hypercube. In the hypercube, the lower and upper limit can be defined by floor and ceiling functions as presented in (Equation 1.9) and (Equation 1.10), respectively.
where, ℤ+ is the set of integers. In the hypercube H, the closest vertex of the discrete variables can be determined using the NPV method as expressed in (Equation 1.11).
where, is the discrete version of the continuous variable Xij. To restrict the variables within the boundary of minimum and maximum limit the same procedure may be used as used in the conventional SPBO algorithm.
1.3 Problem Formulation
1.3.1 Objective Functions
The selection of the locations and the proper sizes of the RDGs (biomass and solar PV) and shunt capacitors in the distribution networks depend on the selection of the proper objective functions. Improper placement of the RDGs and shunt capacitors leads to the maloperation of the distribution networks which includes the increment in active power loss, poor voltage profile, and huge installation cost. In this book chapter, the active power loss, voltage deviation, and the effective annual installation cost of RDGs and shunt capacitors are considered as the main objective functions. The multi-objective function is converted to the single objective function by using the weighted sum approach, where the weights for the objective functions are selected based on their preferences. The objective function with the weights is stated in (Equation 1.12).
(a) Power Loss Index (PLI)
PLI is the ratio of the active power losses after and before the placement of RDGs and the shunt capacitors. PLI is defined as in (Equation 1.13).
The are the active power loss after and before the placement of RDGs and shunt capacitors, respectively, at the time instant t.
The active power loss at any time instant t can be defined as in (Equation 1.14)
where NBr is the total number of branches, Zi is the branch impedance and |Iij| is the magnitude of the branch current, connected between bus-i and bus-j, defined as in (Equation 1.15).
After the RDGs and shunt capacitors connected to the distribution networks, the branch current magnitude |Iij| is modified by (Equation 1.16).
Here, are the voltage magnitude and the load angle after inserting the DGs and shunt capacitors.
(b) Voltage Deviation Index (VDI)
Improvement in the voltage profile in the distribution network is measured by voltage deviation from the nominal voltage of 1.0 pu. The voltage deviation index is measured as the ratio of the voltage deviations (VDs) after and before the placement of the RDGs and shunt capacitors which are defined for the time instant t as
where VDDG is defined as
Here, are the voltage of ith bus after and before placement of DG to the distribution network. The value of voltage deviation near zero indicates an improved voltage profile.
(c) Installation Cost Index (ICI)
Total installation cost includes the cost of the RDGs and the shunt capacitors. Lifetime (LT) of the RDGs and the shunt capacitors are used to obtain the annual installation cost. The ICI is defined as
Here, CostDG(t) is the effective annual installation cost of the DGs and shunt capacitors at time instant t, and can be defined as
and the installation cost, considering the maximum sizes of DGs and shunt capacitors, is defined as
In this book chapter, equal preference is given to each objective function in the weighted sum approach. Therefore, the weights for the three objective functions are
1.3.2 Technical Constraints Considered
The minimization of the weighted average of the three objective functions is subjected to the fulfilment of certain equality and inequality constraints as stated below [68],
(a) Equality Constraints
At any time instant t, the active and the reactive power balance for the distribution network should be maintained by the equation as follows
This equality constraint is fulfilled by the load flow solution of the distribution networks. Putility, Qutility are the active and reactive power taken from the utility, respectively, PDG, QDG are the active and reactive power supplied by the RDGs, in order, and shunt capacitors, and is the percentage load level expressed in per unit.
(b) Inequality Constraints
(i) The bus voltages of the distribution networks at any time instant must be within the minimum (
V
min
) and maximum (
V
max
) voltage level as in
The placement of RDGs and the shunt capacitors may increase the bus voltage beyond the limit at low voltage level and decrease the bus voltage at high load level. Therefore, bus voltages are kept within the prescribed limit by introducing the penalty function defined as
(ii) The active and reactive power generation of the DGs should be within the specified limit given as
where are the minimum and the maximum limit of the active power injection of the DGs and are the minimum and the maximum limit of the reactive power injection of the DGs which are to be maintained at any time instant. These inequalities are maintained by introducing the penalty function as below
(iii) The total active and reactive power injections by the RDGs and shunt capacitors at any time instant should be less than the total active and reactive power load demand of the distribution network at the same instant. This inequality constraint is defined as
These inequality constraints are considered by introducing the two penalty functions as
(iv) Branch currents, after the placement of DGs and shunt capacitors, should be within the maximum allowable level to maintain the feeder thermal limit. The branch current limit should be
This limit is considered in the objective function with the help of a penalty function described as
After considering all the penalty functions, the objective function, to be minimized, is
The values of the penalty factors τfeed are to be selected as very high value for a minimization problem. In this book chapter, the penalty values are selected as 1016 [68]. If any of the inequality constraints violates then the corresponding penalty function adds a very high value to the objective function.
1.4 Comparison of the SPBO Algorithm in Terms of CEC-2005 Benchmark Functions
For the comparison purpose, different algorithms (viz. PSO [69], TLBO [70], CS, and SOS) are considered in this book chapter and the parameters for the different algorithms considered are tabulated in Table 1.2 as below. The study has been carried out to analyze the performance of SPBO to solve CEC-2005 benchmark functions [67, 71]. The analysis has been done considering 30-dimensional problems. The comparison performance study of SPBO with PSO, TLBO, CS, SOS has been shown in Table 1.3. It may be noticed from Table 1.3
