Energy Systems Modeling: Principles and Applications [1st ed.] 978-981-13-6220-0;978-981-13-6221-7

Table of contents :
Front Matter ....Pages i-xi
The Systems Approach and Energy Models (Hooman Farzaneh)....Pages 1-15
Analytical Approaches in Energy Modeling (Hooman Farzaneh)....Pages 17-43
Energy Demand Models (Hooman Farzaneh)....Pages 45-80
Energy Supply Models (Hooman Farzaneh)....Pages 81-105
Climate Change Multiple Impact Assessment Models (Hooman Farzaneh)....Pages 107-129
Optimal Control of Energy Systems (Hooman Farzaneh)....Pages 131-147
Back Matter ....Pages 149-168

Citation preview

Hooman Farzaneh

Energy Systems Modeling Principles and Applications

Energy Systems Modeling

Hooman Farzaneh

Energy Systems Modeling Principles and Applications

Hooman Farzaneh Interdisciplinary Graduate School of Engineering Sciences Kyushu University Fukuoka, Japan

ISBN 978-981-13-6220-0 ISBN 978-981-13-6221-7 https://doi.org/10.1007/978-981-13-6221-7

(eBook)

© Springer Nature Singapore Pte Ltd. 2019 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To my dear mother, for her central role in my life and the many sacrifices she has made for me.

Preface

The consumption and production of energy around the world play a significant role in several sustainability problems, such as climate change and the depletion of resources. So far, world energy use has been dominated by energy consumption in industrialized countries. However, that situation is rapidly changing. Industrialization, the improvement of living standards, and population growth are leading to rapid increases in energy consumption in developing countries, with considerable impacts on global sustainability issues. Policymakers in these countries need meaningful tools and analysis methods to consider the impact of policy measures. Policy impact analysis is the most challenging task. How to appropriately reflect existing policy instruments and their interactions? How do we simplify the essence of future policies for policy scenarios? Energy systems modeling can provide policymakers with tools to explore and understand possible future changes in the energy system. In recent years, the total number of energy models made available has grown tremendously, partly because of the expansion in computational science. The variations among these models are many and mind-boggling. It all begs the following question: “Which model is most suited for a particular purpose or situation?” This book aims to answer this fundamental question through serving as an introductory reference for those studying the application of models to energy systems. The book opens with a taxonomy of energy models and treatment of descriptive and analytical models. It aims to provide the reader with a foundation in the basic principles underlying the energy models and to set those principles in the context of energy system studies. Therefore, the book will first provide an overview of different ways of classification. These can be helpful in cases where model developers themselves are looking for a convenient modeling approach. The book then provides insights for the reader into the varied applications of different energy models to answer complex questions, including those relating to specific aspects of energy policy measures dealing with issues of supply and demand. Case studies are provided in all of the chapters as examples of how existing models fit the classification methods outlined in this book.

vii

viii

Preface

The motivation for this book is to bridge the gap in the existing literature. While numerous books have been published in the area of energy modeling over the past decade, few source books are devoted to the basic principles and fundamentals of energy systems modeling. As a teacher of this course at research universities, I have yet to identify a single textbook that adequately covers both the principles and applications of energy systems modeling. With increasing interest in energy policy planning, particularly in developing countries, the role of the well-prepared sourcebook, and the early training thereof, is becoming increasingly important. This book, I believe, neatly fills this gap and will be of great value to those educating both the young researchers of tomorrow and the older model developers of today! The book is accessible enough for those who want a general understanding of the quantitative models used in the energy sectors. As conceived, the book consists of chapters devoted to various principles and applications, taking the reader from the basic principles involved, reviewing existing molding approaches, through to understanding the state-of-the-art processes of energy production and consumption through modeling and validation/illustration via different case studies. With its in-depth mathematical foundation, this book serves as a comprehensive collection of work on modeling energy systems and processes, taking the inexperienced graduate student from the basics through to a high-level understanding of the modeling processes in question, as well as providing professionals and academic researchers in the field of energy planning with an up-to-date reference covering the latest works. The content of this book is based on experiences achieved from more than 10 years of research and teaching at different universities and institutions in both Iran and Japan. I have designed this book to be a thorough review of the state-of-theart energy systems modeling and their applications. As such, in writing this, the readership I have had in mind includes graduate students, academics, and professional model developers. The book would also be a suitable text for a course on energy system analysis for graduate students in the field of energy and environmental engineering. I would like to express my gratitude to my colleagues, in particular, Dr. Eric Zusman, Prof. Hideaki Ohgaki, Prof. Keiichi Ishihara, Prof. Benjamin McLellan, Prof. Zhang Qi, Prof. Yadollah Saboohi and Dr. Mahendra Sethi, for their muchvalued input. I am deeply grateful to my literary editor, Dr. Elizabeth Webeck, for her excellent support and assistance in the editing and proofreading of this book. I wish to thank my supportive wife, whose love and guidance are with me in whatever I pursue and my wonderful son, who provides unending inspiration. I acknowledge with gratitude the research support I have received during past 3 years from the Japan Society for the Promotion of Science (Grant-in-aid for the Scientific Research (C), project No: 16K00650) and the Asia-Pacific Network for Global Change Research (Ref. CRRP2017-07SY-Farzaneh). Fukuoka, Japan November, 2018

Hooman Farzaneh

Contents

1

The Systems Approach and Energy Models . . . . . . . . . . . . . . . . . . . 1.1 What Is Energy Modeling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Modeling Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Methodological Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Mathematical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Spatial Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Classification of Energy Models . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 6 8 9 12 12 15

2

Analytical Approaches in Energy Modeling . . . . . . . . . . . . . . . . . . 2.1 Econometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Statement of Theory and Hypothesis . . . . . . . . . . . . . . . 2.1.2 Specification of the Econometric Model to Test the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Using Economic Theory in the Econometric Models . . . 2.2 Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Steps in a Simulation Modeling . . . . . . . . . . . . . . . . . . 2.3.2 Solution of the Simulation Problem . . . . . . . . . . . . . . . . 2.4 Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Mixed Integer Programming . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

17 17 18

. . . . . . . . . . . . .

18 20 21 21 27 29 30 31 33 33 38 41 42

ix

x

3

4

Contents

Energy Demand Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Objectives of the Methodology . . . . . . . . . . . . . . . . . . . . 3.1.2 Energy Demand Calculations . . . . . . . . . . . . . . . . . . . . . 3.1.3 Application of the Energy Demand Simulation Model to the Yokohama Smart City Project . . . . . . . . . . . . . . . . 3.2 Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Estimation of Energy Demand in the Middle East Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Optimization Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Optimal Fuel Consumption in a Gas Turbine Power Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Optimal Energy Demand in a Cement Factory . . . . . . . . . 3.3.3 Optimal Energy Demand in a Building . . . . . . . . . . . . . . 3.3.4 Minimum Energy of a Compression Unit . . . . . . . . . . . . 3.3.5 Thermal Efficiency Maximization in a Heat Recovery System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 47 51 58 59 61 65 66 72 74 75 80

Energy Supply Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1 Theory of Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Applying the Theory of Production to the Energy Supply System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Mathematical Formulation of the Energy Supply Model . . . . . . . 87 4.3.1 Objective Function of the Model . . . . . . . . . . . . . . . . . . . 87 4.3.2 Demand Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.3 Capacity Constraint on Production . . . . . . . . . . . . . . . . . 89 4.3.4 Availability of Resources . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.5 Speed of Resource Extraction . . . . . . . . . . . . . . . . . . . . . 89 4.3.6 Bound on Emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4 CO2 Zero-Emission Energy System in the Middle East Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.1 One Additional Concept to the Supply Model: “Peak Oil” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4.2 Long-Term Projection of Energy Demand and Supply . . . 92 4.4.3 Main Macroeconomic Assumptions . . . . . . . . . . . . . . . . . 92 4.4.4 CO2 Zero-Emission Scenario . . . . . . . . . . . . . . . . . . . . . 94 4.4.5 Availability of Renewable Energy Resources in the Middle East Region . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Optimal Power Generation from an On-site Power System . . . . . . 99 4.5.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5.2 Optimal Configuration of the Power System . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Contents

xi

5

Climate Change Multiple Impact Assessment Models . . . . . . . . . . . . 5.1 What Is the Co-benefits Approach? . . . . . . . . . . . . . . . . . . . . . . 5.2 The A-S-I Modeling Framework . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Public Health Co-benefit Assessment . . . . . . . . . . . . . . . . . . . . . 5.4 Assessing the Climate Co-benefits in the Transport Sector . . . . . . 5.4.1 Baseline Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 The Plan Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Assessing the Climate Co-benefits in the Power Sector . . . . . . . . 5.5.1 Scenario Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 109 109 112 115 116 118 123 125 128

6

Optimal Control of Energy Systems . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 What Is Optimal Control Theory? . . . . . . . . . . . . . . . . . . . . . . . 6.2 Optimal Fuel Consumption in a Passenger Vehicle . . . . . . . . . . . 6.3 Optimal Power Control Strategy in a Hybrid Renewable Energy System (HRES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Case of Fukushima Prefecture in Japan . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 133

Appendix A (GAMS Codes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power Company Problem (Simplex Method) . . . . . . . . . . . . . . . . . . . . Quadratic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Turbine Energy Demand Optimization Model . . . . . . . . . . . . . . . . Cement Factory Energy Demand Optimization Model (Cost Minimization) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Work of Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Building Energy Demand Optimization Model . . . . . . . . . . . . . . . . . . . Thermal Efficiency Maximization in a Heat Recovery System . . . . . . . . Middle East Energy Supply Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136 138 143 146 149 149 150 150 151 152 154 155 156 159

Chapter 1

The Systems Approach and Energy Models

Contents 1.1 What Is Energy Modeling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 The Modeling Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 The Methodological Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 The Mathematical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 The Spatial Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 Classification of Energy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1

What Is Energy Modeling?

To understand energy modeling, we need to consider a system. This system could be a building, like a house or an office block. Of course, it is vitally important that this building is structurally sound. It is also important that this building is energy efficient and comfortable for those who will inhabit and use the building: it should not be too hot or too cold. In fact, it is expected that the temperature inside the building will be just right at all times. Until very recently, it was only after buildings were built that the task of ensuring the temperature was just right was undertaken. Calculating the optimal way to adjust the temperature by hand was a daunting and somewhat costly procedure. Energy modeling allows us to save both resources and time. It allows us to take all the variables that play a role in the energy considerations of the building into account, including the weather, the people, and the utility rates, for example. Energy modeling allows us to represent, analyze, make predictions, and provide insight into real systems. In the case of buildings, it helps us to choose between different designs and materials. By adjusting variables, we can check their impact on the energy requirement. Say we want to change the lighting, and it turns out the lighting we want is less efficient and generates more heat. This means more ventilation will be required. The adjustments required can be provided by the energy model. Because © Springer Nature Singapore Pte Ltd. 2019 H. Farzaneh, Energy Systems Modeling, https://doi.org/10.1007/978-981-13-6221-7_1

1

2

1 The Systems Approach and Energy Models

the immediate consequences of changing variables can be seen, smart decisions can be made at the design stage to ensure the building is both efficient and comfortable. Anything that gives off heat or breathes, including lighting, weather, electrical devices, and human beings, can be included in the energy model. We can even do a lifecycle analysis to determine energy usage and to estimate utility bills over time. There are a few critical questions in energy modeling. What kind of model is needed to study a specific energy system, like that of a building? What are the structural characteristics of such a model? To answer these questions, we need some background knowledge about the process of modeling. The first step is to identify the real problem and to establish a clear objective for creating a model. The next step is to determine what is already known about the issue. Another list should also be drawn up including all of the information that can be assumed based on what is already known. The next step is to determine the physical principles that govern the desired model and the equations that are needed to obtain the results. Once the planning phase has been completed, it is possible to create a model. To make sure that the model is capable of obtaining the desired relationships among the data that we are trying to accomplish, our data needs to be inputted for a trial run. It is essential to verify the validity of the results provided by the model before we decide to use the model for any specific purpose. To determine the validity of the model after applying our data, we need to ask the following questions. Does the model provide the results we expected? Do the results make sense? Are the results repeatable? To make our model useful for further applications, we need to consider how it could be improved. Are there any other variables which should be taken into consideration? Are there any restrictions, and, if so, is there a way to overcome them? The key at this point is to tinker with the model to find the best way to improve it before using it again. Once the ways to improve the model have been identified, the changes should be made, and the model should be rigorously tested again to prove its validity. These procedures are shown in Fig. 1.1. In large-scale energy systems like a regional energy market, energy models help to provide an understanding of the quantitative relationships between different parts of the energy system and between different time periods, under various assumptions. In the following sections, we will discuss the main purposes and structural characteristics of the different energy models available. The scale and potential applications of these models vary considerably. After reflecting on the purpose of energy modeling, it is time to choose the model. A comprehensive list of considerations was published by Van Beeck (1999). Among them are the following: 1. What is the main focus of different models in analyzing an energy system? What questions can be answered by using the models? 2. What are the underlying hypotheses and assumptions used in the development of each model? 3. What kind of methodology is applied in each model? 4. Which mathematical approach is utilized, and what algorithm is applied to find solutions in the methodology of the model?

1.1 What Is Energy Modeling?

3

Fig. 1.1 The process of energy modeling

5. What geographical areas are addressed by the model (i.e., global, regional, local, sectoral)? 6. What is the time scale of the model? Short, medium, or long term? It is essential to bear in mind that a model is a representation of reality, but is not reality itself. Models are developed to enable discussions on the specific problems which underlie the main purpose they were designed to deal with. Behind all energy models, there are both general purposes and specific purposes. The general purposes of the energy model are reflected by how the model addresses the future. The general purposes of three typical energy models, as established by Hourcade et al. (1996), are these. Forecasting Models To anticipate the future challenges which may be faced with an energy system, by extrapolating historical trends to analyze the short-term impact of certain actions, such as economic behavior and general growth patterns. Backcasting Models To construct visions of various desired future outcomes for an energy system based on a backward approach, identifying policies and programs that will connect that specified future to the present. This type of modeling undertakes the challenge of discussing the future from the direction opposite to that of the forecasting models. Scenario Analysis Models To explore the future pathways for an energy system based on a comparison between a limited number of desired future scenarios with a reference scenario (i.e., a baseline). The intervention scenarios provide projections for a wide range of the factors which drive the energy scenario. These include

4

1 The Systems Approach and Energy Models

production, consumption, trade, prices, investments, technology mixes, and many others. The specific purposes reflect the detailed aspects addressed by an energy model, such as patterns in the way energy is generated and consumed in different sectors (Farzaneh et al. 2016). Consider the following. Demand-Side Models These consist of a broad range of methodologies which focus on determining the final energy consumption in the entire economy or a particular sector, such as the buildings (residential, industrial, and commercial), industrial energy use, and the transportation system. The overall methodological focus of this cluster of energy system models is to consider the demand side endogenously, and the supply-side issues are not considered at all. These models rely on bottom-up simulation techniques to estimate energy demand in energy systems to study their design and renovation or the way they are affected by certain behavioral and technical changes. Supply-Side Models Mostly focused on energy supply technologies, with a particular focus on renewable energy systems, fossil-based power plants, oil and gas industries, etc. They are characterized by a limited spatial scale and generally consider a single piece of technology using a simulation technique or experimental work to perform the analysis, including the design and performance of the system. The models may, therefore, be characterized as calculating supply-side parameters related to technology design or, in some cases, the operation of such technologies. Models involving optimization techniques have been proposed to predict the functional relationship between system performance and the critical system parameters through the estimation of the optimal design of supply technologies. Integrated Models The methodological foundations of demand and supply models are typically developed by focusing on specific aspects of energy use. Consequently, a significant challenge encountered by these types of models is the lack of integration between different subsectors on both the demand and supply sides of the energy sector. This challenge is addressed by the integrated models which look at the full set of processes within an energy system. Integrated models of energy systems can present a complete perspective to achieve overall reductions in energy consumption and are better at analyzing the direct and indirect effects of policies. In this category, agent-based simulation models are widely used. These models are especially designed to investigate the reliability of energy generation and supply networks in energy systems by a greater reliance on aggregate accounting and rule-based approaches than the behavioral profiles of consumers. In these models, the agents are characterized by bounded rationality and can learn, adapt, and reproduce. Other models include assessment models which represent studies of the entire economy and how local and global policy decisions might shape energy performance. It is possible to test the impact of various scenarios by decomposing trends in energy consumption and generation into socioeconomic, technological, and demographic developments. State-of-the-art integrated models are implemented as bottom-up technology assessment models which use cost-effectiveness (minimum total cost of the system), seeking the optimal combination of supply technologies through

1.2 The Model Structure

5

using such optimization techniques as linear or mixed integer programming to satisfy the exogenously specified energy demand. Differences in the essential features of integrated models of energy systems imply a difference in the potential for estimating such determinant factors as the demographic, economic, and technological parameters. The biggest challenge posed by these models is to create an interface to simultaneously study the interaction between supply and demand vectors. This can be done by developing equilibrium-seeking feedback processes to provide a sophisticated and robust modeling system. Despite the diversity of practices among the models, many of the challenges they face are common to them all. Some of them can be addressed as the complexity of the modeling techniques and data availability or resolution. Another problem is to create an analytical framework that explicitly captures all aspects of energy systems.

1.2

The Model Structure

For each type of energy model, decisions need to be made on how to accurately reflect all the assumptions involved in the modeling development process. An energy modeler inevitably has to deal with unknown or missing information while developing the model. In such cases, the requirement for the missing data has to be met by making certain assumptions. These assumptions made by the modeler are embedded in the approach. The assumptions are an important feature of the model and need to be considered carefully to ensure an energy system can be completely simulated: 1. Definition of the endogenous and exogenous variables: An endogenous variable is one that is explained by a model. Endogenous variables have values that are determined by other variables within the model equations and mathematical formation. These “other” variables are called exogenous variables. An exogenous variable is a variable that is not affected by other variables in the system. They are taken as a “given” in the model. Let’s suppose a power plant generates a certain amount of electricity. The amount of electricity is the endogenous variable and is dependent on any number of other variables: these may include the weather, technology, or the price of fuel, for example. Because the amount of electricity is entirely reliant on the other factors in the system, it is purely endogenous. Identifying the exogenous variables and the endogenous variables can be challenging. Using the example of the power plant we used before, something might cause the amount of electricity generated to rise. For example, adding a new technology might increase power output. To decide if this new variable is exogenous, the modeler has to determine whether the increase in the production would cause the new variables to change. However, a variable like “weather” is exogenous as a rise in power output would not affect the weather. But, what about “Price?” The price of electricity certainly isn’t affected by one small power plant’s output, but what if this was a major power company that suddenly increased its production and saturated the market? Price, in this case, would be

6

1 The Systems Approach and Energy Models

partially an endogenous variable and may be considered to be exogenous in certain scenarios. 2. Description of the non-energy use: Primary energy can either be used as process energy or as a feedstock which is converted to a product. Non-energy use includes energy products used as raw materials in the different sectors which are not consumed as fuel or transformed into another fuel. In our power plant example, non-energy would include the use of other petroleum products, such as lubricants. Another example is the natural gas used as a raw material in the petrochemical industry. The suitable models for measuring energy-related polices include a detailed description of non-energy use. 3. Description of the energy end-uses: The end-uses are basically the categories used by the modeler to break down the overall energy use. A detailed description of energy end-uses in the model would enable the modeler to assess the impact of technological change on the efficiency of an overall system. 4. Description of the technical aspects: To ensure that the expected changes in structure, policy, and technology are taken into account, a wide range of technologies should be integrated into the model. It is important that each specific energy service is represented. In our simple example of the power plant, the thermal energy generated from fuel combustion is utilized to heat water. When water is heated, it turns into steam and spins a steam turbine which drives an electrical generator. A number of technologies are involved here: the steam boilers, the turbines, the pumps, and the generators, for example. The link between the different technologies through the process of energy conversion needs to be reflected in detail in the model to enable the technological potential for fuel substitution and new supply technologies to be accurately assessed. 5. Projection of the demographic and macroeconomic parameters: This clarifies how the various scenario parameters affect the result. The main parameters considered in these categories are population, economic growth, price, and income. These parameters are exogenously defined.

1.3

The Modeling Paradigm

From the analytical perspective, the difference between different methodologies used in the energy models is in how they represent the interactions between the energy sector and its surroundings, including the society, economy, and environment. The top-down and bottom-up approaches are the two major differences between methodologies used to examine the relationship between energy consumption and other economic activities such as manufacturing, mining, etc.

1.3 The Modeling Paradigm

7

It is widely accepted that the terms “top-down” and “bottom-up” are synonymous with the terms “aggregated” and “disaggregated” when discussing models (Cao et al. 2008). The top-down approach focuses on the big picture and how the consumption of energy carriers (i.e., electricity, coal, oil, and gas) can be estimated by the application of macroeconomic theory and econometric techniques to historical data on such aggregate economic indices as capital, price, and income. It endogenizes behavioral relationships and assumes there are no discontinuities in historical trends. So, the modeler believes that if the economic sector is doing well, chances are, the subsectors in that sector will also do well. In simple terms, the top-down approach selects various sectors or industries and tries to achieve a balance between energy supply and demand. It measures the externalities of energy systems. For example, if the price of oil rises, the top-down analysis might focus on buying stocks in oil companies. Conversely, for companies that use large quantities of oil to make their product, a top-down model might consider how rising oil prices might hurt company profits. As such, this approach is simple and not data-intensive. However, some critics complain that the majority of top-down methodologies present an aggregate estimation of energy demand and that has limits in its ability to accurately analyze the sectors and to represent the complexity of the energy sector. They give a pessimistic estimate of the best performance by reflecting the technologies already adopted by the economic sector. On the other hand, bottom-up models depend on the selection of individual sectors. There is less focus on market conditions, macroeconomic indicators, and industry fundamentals in the bottom-up models. They concentrate on technology options in various sectors and are mostly used in the planning of cost-effectiveness, in assessing technology strategies, in simulating demand-side energy-saving measures, and in considering the efficiency opportunities on the supply-side. A bottomup model allows for a detailed description of technologies and targets specific technologies that perform well against the economic backdrop. Because this method uses disaggregated data to explore specific purposes, it is data-intensive. The bottom-up models lack feedback regarding economic growth (e.g., GDP and income) and the behavioral responses of market actors and tend to assume that interactions between the energy sector and other sectors are negligible (Fig. 1.2).

Fig. 1.2 Top-down vs. bottom-up in electricity demand estimation

8

1.4

1 The Systems Approach and Energy Models

The Methodological Approach

In the following part, an overview of commonly used methodological approaches in developing energy models will be presented: 1. Simulation models: Simulation models provide information about the possible behaviors of the energy system being modeled (Vepa 2013). Simulation models tend to be descriptive rather than prescriptive. They tell us how a system works under given conditions but not how to arrange the system to optimize its performance. Simulation models can be classified along three dimensions: deterministic versus stochastic, static versus dynamic, and discrete versus continuous. Deterministic simulations are entirely defined by the model, and rerunning a simulation does not change the outcome. Deterministic simulations are typically used when the system being modeled is too complicated to analyze analytically. Stochastic simulations include randomness. That means the different runs of the same simulation with the same initial conditions can generate different results. In a discrete model, there are innumerable relevant variables, and they are all represented by integers. In a continuous model, on the other hand, there are a set number of variables. For example, when analyzing the energy flow in a power plant, we might choose to model each piece of equipment: in such a case, a discrete model would be the best option. Another option would be to treat energy as a flow where differential equations can describe changes in the flow. In such a case, a continuous model would be the most suitable option. 2. Optimization models: Optimization problems often involve the words “maximize” or “minimize.” Optimization is also useful when there are limits or constraints on the resources required or boundaries restricting the possible solutions. Optimization models can be applied to a huge variety of situations and problems in the energy systems. Here are some examples: – Controlling the air/fuel ratio in a boiler throughout the combustion process to minimize fuel consumption – Selecting the best combination of technologies and energy resources to minimize the total cost of the power sector based on the predicted demand profile – Selecting the best set of stocks to invest in the renewable energy market to maximize returns based on anticipated performance 3. Macroeconomic models: The identification of the economic objectives, analyzing problems, and evaluating policies are central to macroeconomics. There are at least four macroeconomic objectives in developing energy models: – To achieve stable and sustainable growth in global energy demand – To have a stable price level in the energy market with no significant inflation and no deflation

1.5 The Mathematical Approach

9

– To have a balance between the energy sector and the rest of the economic sectors – To have an equitable distribution of energy recourses Economic problems arise when the objectives are not met in the energy sector. Some of the possible issues facing the energy sector include excessive energy consumption growth, an energy supply deficit, and extreme energy inequality. The macroeconomic models are used to explain the possible causes of the economic problems in the energy sector. For example, is inflation caused by excessive domestic aggregate energy demand? If so, which component is responsible for this? Alternatively, is it caused by an increase in global energy prices? Could it be caused by several factors acting at once? Is the cause predictable or the result of an unexpected shock? The macroeconomic models can identify the major costs and consequences associated with a problem in the energy sector and enable us to choose the most suitable policy to tackle the problem. They can also be used to identify the conflicts associated with the selected policies. Most policy objectives conflict with at least two other ones. For example, when using monetary policy to control inflationary pressure, aggregate demand may contract, which has a detrimental effect on growth and employment. In this category, the input/output approach is used to describe transactions among economic sectors and assist in the analysis of energyeconomy interactions. Macroeconomic models are often developed to explore purposes, using assumed parameters and scenarios which do not necessarily have to reflect reality (Hall and Henry 1988). These models cannot represent specific technologies and require a relatively high level of expertise. 4. Econometric models: Econometrics is the application of statistical methods to economic data and is described as the branch of economics that aims to give empirical content to economic relations. It is the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation. The basic tool for econometrics is the multiple linear regression model. This model is widely used in energy modeling. Generally, the main purpose of the application of econometric models in the energy sector is to predict the future consumption of energy commodities by sector under different policy frameworks as accurately as possible, using measured parameters. In econometric models, energy demand is driven by three factors: price, economic activity, and energy efficiency trends. They rely on aggregated historical data that have been measured and collected in the past to predict the short- or medium-term future of energy consumption.

1.5

The Mathematical Approach

Mathematical models are used to give a formalized representation of the energy system to permit computer-based operations that provide insight on the different energy policies possible. Commonly applied techniques in energy modeling include:

10

1 The Systems Approach and Energy Models

1. Mathematical programming Mathematical programming includes linear programming (LP), nonlinear programming, (NLP) and mixed integer programming (MIP). The purpose of mathematical programming is to optimize (maximize or minimize) a function subject to a set of constraints (Sinha 2006). These constraints tend to be the restrictions on resource availability or other technical, economic, or even institutional bounds. An example is petroleum product transport: the task is to find the cheapest option or combination of options among pipeline rails, tankers, road tanker river barges, and coastal tank ships. By applying mathematical programming, the cheapest selection among all possible combinations will be provided to satisfy optimality criteria, which is the minimum total cost of transportation. In linear programming, all relationships, including object function and constraints, are expressed in fully linear terms. The simplex technique is used for almost all linear programming models. It should be noted at this point that, in the real world, an energy system represents nonlinear behavior. Nonlinear optimization deals with nonlinear functions. Typical methods of solving these problems are first-order methods, such as gradient and conjugate gradient descent, and second-order methods (and approximations thereof), such as Newton’s method and Levenberg-Marquardt (Luenberger and Yinyu 2008). An example of the application of NLP in the energy sector is the maximization of the household utility function: a complex function of many determinants including time, technology, quality and quantity of demand, environment, comfort, and cost. The major difficulty lies in certifying the optimality conditions and finding the best local solution under the modeling assumptions. Mixed integer programming (MIP) is an extension of linear programming and allows for greater detail in formulating technical relations in energy modeling. Generally, it involves the optimization of a linear objective function, subject to linear equality and inequality constraints. This means that some or all of the variables are required to be an integer (Renata et al. 2015). MIP can be used when addressing uncontrollable, semi-controllable, and controllable variables in energy modeling. For example, a single-machine batch scheduling under time-of-use electricity prices to minimize the total energy cost. 2. Dynamic programming To put it simply, dynamic programming is a recursive approach which uses previous knowledge to construct a new solution dynamically (Puterman 1978). An example in energy planning is to find the optimal storage operation in the network and coupling storage and network constraints with the randomness of renewable generation. Dynamic programming is extremely useful in proving analytical and structural results for specific power system engineering problems. For example, it is well-suited to finding the optimal charge/discharge scheduling of a given battery to ensure the minimum electricity bill while reducing stress on the battery and prolonging battery life.

1.5 The Mathematical Approach

11

3. Genetic algorithm The genetic algorithm (GA) is a method for solving both constrained and unconstrained optimization problems. A comprehensive discussion of the GA is given by Goldberg (1989). We can apply the genetic algorithm to solve a variety of optimization problems in the energy sector that are not well-suited for standard optimization algorithms, including problems in which the objective function is discontinuous, is not differentiable, is stochastic, or is highly nonlinear. The genetic algorithm can address problems of mixed integer programming, where some components are restricted to be integer-valued. The genetic algorithm uses three main types of rules at each step to create the next generation from the current population: – It uses a set of the coded parameters that are to be optimized. – The optimal value is obtained from a group of possible solutions. – It uses probabilistic rules to generate new solutions instead of deterministic rules. GA is widely accepted in the different disciplines of energy applications in buildings including environmental parameters, renewable energy systems, naturally ventilated buildings, energy consumption and conservation, and HVAC systems. However, due to the random nature of GA, finding a unique and reproducible result is not an easy task. 4. Artificial Neural Networks (ANNs) Neural networks are algorithms based on the way the brain works. They can build predictive models by learning the patterns in historical data. Neural networks are made up of small interconnected processing elements referred to as nodes. Each node processes a small part of the task. Neural networks also consist of input and output layers, as well as a hidden layer comprised of units that transform the input which can be used by the output layer. A connection weight multiplies data to the input layer. In the simplest case, these products are merely summed, fed through a transfer function to generate a result, and then passed to the activation function (Hassoun 2003). In such a structure, some of the nodes interface with the real world to receive input (Fig. 1.3). The feedforward neural network is the basic type of neural net. In this system, information is delivered from input to output in one direction. A more widely used form of network is the recurrent neural network, in which data can flow in multiple directions. ANN models may be used as an alternative method in engineering

Fig. 1.3 Overall schematic of an ANN model

12

1 The Systems Approach and Energy Models

Fig. 1.4 PEST complexity in energy modeling

analysis and predictions. They can be applied to the energy sector to predict energy consumption, using economic and demographic variables.

1.6

The Spatial Perspective

Energy models include the regional difference in availability of resources and technologies, the price of energy carriers, policy frameworks, and demand patterns. Common geographic coverage is global, regional, national, local, or limited to certain states and districts. While a spatial perspective remains vital for energy modeling, the greater geographic resolution provided by these models increases model run time since the model needs highly aggregated data. An energy model can also focus on one unit like a factory or include a fleet or a society. However, according to the political, economic, social, and technical (PEST) analysis, the degree complexity which is reflected by the model increases at the society level (Fig. 1.4).

1.7

Classification of Energy Models

In Table 1.1, you will find a comparative overview of existing energy system models. The models are characterized according to their features, including their specific and general purposes, the modeling paradigm, methodological approach, mathematical approach, spatial perspective, and time horizon.

International Atomic Energy Agency

International Energy Agency

Brookhaven National Laboratory, USA University of Stuttgart, Germany IIASA, Austria

ENPEP

MARKAL

MARKALMACRO MESAP

DOE, USA

International Energy Agency

PNNL, USA

CEDRL, Canada

MICROMELODIE NEMS

WEPS

MiniCAM

RETScreen

CEA, France

Stockholm Environmental Institute

LEAP

MESSAGE

Developer European Commission

Model EFOM

Table 1.1 Classification of energy models

Energy demand Energy demand Energy demand Energy demand Energy supply Bottom-up

Top-down

Top-down

Top-down

Top-down

Bottom-up

Hybrid

Integrated Energy supply

Hybrid

Bottom-up

Hybrid

Hybrid

Modeling paradigm Bottom-up

Energy supply Integrated

Integrated

Purpose Energy supply Integrated

Optimization

Optimization/ simulation Simulation

Econometric

Macroeconomic

Macroeconomic/ optimization Econometrics/ simulation Optimization

Optimization

Macroeconomic/economic equilibrium

Accounting/ Simulation

Methodological approach Optimization

Mathematical programming Dynamic programming Mathematical programming

Regression

Regression/time series

Mathematical programming Dynamic programming Regression/mathematical programming Mathematical programming

Regression/mathematical programming

Mathematical approach Mathematical programming Analytical programming

Local, national Local, national

Global

National

National

Local, national Local, national Local, national Local, national

Local, national, global Local, national

Spatial perspective National

(continued)

Medium, long term Medium, long term Long term

Short, medium,, long term Medium, long term Medium, long term Medium, long term Short, medium, long term Medium, long term Medium term

Time horizon Medium, long term Medium, long term

1.7 Classification of Energy Models 13

European Commission

National Technical University of Athens, NTUA International Atomic Energy Agency

EnergyPLAN

POLES

PRIMES

MAED

Developer Open-source research community incl. UCL, USA Aalborg University

Model OSeMOSYS

Table 1.1 (continued)

Energy demand

Integrated

Integrated

Purpose Energy demand Energy supply

Top-down

Hybrid

Hybrid

Bottom-up

Modeling paradigm Bottom-up

Accounting/ simulation

Agent based

Optimization

Optimization

Methodological approach Optimization

Mathematical programming Dynamic programming Analytical programming

Mathematical approach Mathematical programming Mathematical programming

Local, national Local, national

Spatial perspective Local, national National, state, regional Global Medium to long term Medium to long term

Long term

Short time

Time horizon Short time

14 1 The Systems Approach and Energy Models

References

15

References Cao J, Mun S Ho, Jorgenson DW (2008) Co-benefits of greenhouse gas mitigation policies in China, Environment for Development, Discussion Paper Series, EfD DP 08-10, China Farzaneh H, Doll CNH, Puppim de Oliveira JA (2016) An integrated supply-demand model for the optimization of energy flow in the urban energy system. J Clean Prod 114:269–285 Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning., 978-0201157673. Addison-Wesley Professional, Boston, MA Hall SG, Henry SGB (1988) Macroeconomic modelling (Contributions to economic analysis), 978-0444704290. Elsevier Science Hassoun MH (2003) Fundamentals of artificial neural networks, 978-0262514675. MIT Press Hourcade JC et al (1996) Estimating the cost of mitigating greenhouse gases. In: Bruce JP, Lee H, Haites EF (eds) Climate change 1995: economic and social dimensions of climate change. Contribution of working group III to the second assessment report of the IPCC. University Press, Cambridge, pp 263–296 Luenberger D, Yinyu Y (2008) Linear and nonlinear programming. 978-1-4419-4504-4, Springer US Puterman ML (1978) Dynamic programming and its applications, 9781483258942. Academic Press Renata M, Włodzimierz O, Speranza MG (2015) Linear and mixed integer programming for portfolio optimization. Springer International Publishing, Switzerland Sinha SM (2006) Mathematical programming theory and methods., 9780080535937. Elsevier Science, New Delhi Van Beeck N (1999) Classification of energy models. Tilburg University & Eindhoven University of Technology, Tilburg Vepa R (2013) Dynamic modeling, simulation and control of energy generation., 978-1-4471-53993. Springer-Verlag, London

Chapter 2

Analytical Approaches in Energy Modeling

Contents 2.1 Econometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Statement of Theory and Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Specification of the Econometric Model to Test the Theory . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Using Economic Theory in the Econometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Steps in a Simulation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Solution of the Simulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Mixed Integer Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

17 18 18 20 21 21 27 29 30 31 33 33 38 41 42

Econometric Models

The word econometrics is derived from two Greek words, “Oukovouia” which means economy, and “Uetpov,” which means measure. Thus, econometrics means the measure of things in economics such as economic systems, markets, etc. “It is the results of a particular outlook on the role of economics, consists of the application of mathematical statistics to economic data to lend empirical support to the models constructed by mathematical economics and to obtain numerical results” (Tintner 1968). Econometrics may be defined as “the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation related by appropriate methods of an interface” (Samuelson 1983). The field of econometrics involves three major subjects: economic theory, mathematical economics, and statistics. Because all three contribute equally to the field of econometrics, the demand for this field of study is extremely high, and it is considered highly viable. © Springer Nature Singapore Pte Ltd. 2019 H. Farzaneh, Energy Systems Modeling, https://doi.org/10.1007/978-981-13-6221-7_2

17

18

2

Analytical Approaches in Energy Modeling

Fig. 2.1 The productivity of the cement factory as a function of the main factors

The steps for making an econometrics model are outlined in this section.

2.1.1

Statement of Theory and Hypothesis

The first step is the selection of a hypothesis or a theory which shows the cause and effect relationship between variables in the model. For example, since capital investment, raw material, and labor are the factors that affect the productivity of a cement factory, they all affect cement products (Fig. 2.1.) It is a matter that needs to be analyzed. In this example: • The product is the dependent variable. • Capital, material, and labor are independent variables.

2.1.2

Specification of the Econometric Model to Test the Theory

Once we have set our goals, we can develop an economic model. An econometrical analysis is carried out to derive economic relations or conduct a quantitative analysis. In other words, it is essential to have an economic model for our hypothesis or situation. In our example, this economic model shows that the product is a function of capital investment, material, and labor. In order to test this econometric model, we need to: • Represent it mathematically. • Take the other factors that affect the product into account. There could be many other factors, such as technology, energy, etc.

2.1 Econometric Models

19

• Introduce random error or disturbance which is another variable factor that can impact product. In general, an econometric model links an observable dependent variable Y (e.g., product) to observable explanatory variables X1, ..., Xi (i.e., capital, raw material, labor, etc.) and an unobservable variable ε (the error term):
 Y ¼ f X 1 ; : . . . ; X i ; β1 ; . . . ; β j ; ε

ð2:1Þ

The correlation between independent variables can be estimated through finding correct values for β as the coefficient of the econometric model. This is the simplest linear econometric model: Y ¼ β1 X 1 þ β 2 X 2 þ    þ βi X i þ βj þ ε

ð2:2Þ

j¼iþ1

ð2:3Þ

Data are required to obtain values for coefficients of the econometric model. However, there are many considerations that need to be taken into consideration at this step. Observed data collected by observing the real world are preferred over experimental data to determine near-correct results. Selecting the right data type depends on the size and type of the stated problem and also the time constraint. We need to specify the function f in Eq. (2.1) that can hold for all possible values of Y, X1, . . ., Xi and ε. Consider the linear regression model as follows: Y ¼ β 1 X 1 þ β2 þ ε

ð2:4Þ

Let Y be the demand for cement products and X1 be the price of cement. From the law of demand, we know that the demand Y declines when the price X1 rises. Therefore, β1 must be negative. This means that: If X 1 > β2 =β1 ) β1 X 1 þ β2 < 0

ð2:5Þ

This is impossible because demand cannot be negative. We could think of fixing this problem by specifying the demand model as: ln ðY Þ ¼ β1 X 1 þ β2 þ ε

ð2:6Þ

Y ¼ expðβ1 X 1 þ β2 ÞexpðεÞ

ð2:7Þ

According to the law of demand, β1 is negative. This model predicts that even if the price X1 becomes zero, the demand Y cannot exceed an upper limit and will be bounded. While this may happen in some special markets, it is not the case in the cement market. This model can be specified as follows:

20

2

Analytical Approaches in Energy Modeling

ln ðY Þ ¼ β1 ln ðX 1 Þ þ β2 þ ε β

Y ¼ expðβ1 ln ðX 1 Þ þ β2 ÞexpðεÞ ¼ X 11 expðβ2 ÞexpðεÞ

ð2:8Þ and

β1 < 0

ð2:9Þ

Therefore, demand Y increases toward infinity if the price X1 decreases toward zero and vice versa. This is what we expect from the law of demand.

2.1.3

Using Economic Theory in the Econometric Models

The general example is an application of production theory and microeconomic theory (Ramsey 1980). Using the cement factory example, where the production in this factory is governed by a constant elasticity of substitution (CES) production function as follows (Arrow et al. 1961): P ¼ γ fαLρ þ ð1  αÞK ρ g1=ρ expðεÞ

0 < α < 1,γ > 0,ρ  1

ð2:10Þ

where P is product, L is labor, K is capital, and ε is an error term. The question is this: What is the optimal cement production rate? We use microeconomic theory to answer this question. The profit function can be given as the difference between total revenue and total cost (Bierens 2017): ΠðL, KÞ ¼ pP  sL  cK ¼ pγfαLρ þ ð1  αÞK ρ g1=ρ expðεÞ  sL  cK

ð2:11Þ

where p is the market price of cement. s and c refer to the wage rate and unit cost of capital. In the short turn, capital is fixed. If the cement market is perfectly competitive, the price of cement is also fixed, and the wage rate is given for individual factories. Therefore, by taking the derivative of the total profit equation, setting it equal to zero and solving it for L, the maximal profit can be estimated as follows: ∂ΠðL, KÞ ∂pγfαLρ þ ð1  αÞK ρ g1=ρ expðεÞ ¼ s¼0 ∂L ∂L

ð2:12Þ

hence  ρ ϕ  ϕ γ s P¼L expðρϕεÞ p α ϕ¼

1 1þρ

ð2:13Þ ð2:14Þ

2.2 Statistical Models

21

Taking logs, Eq. (2.13) can be written as: ln ðPÞ ¼ ln ðLÞ þ ϕ ln ðγ ρ =αÞ þ ϕ ln ðs=pÞ þ ρϕε ¼ β0 þ β1 ln ðLÞ þ β2 ln ðs=pÞ þ E

ð2:15Þ

where β0 ¼ φln(γ ρ/α), β1 ¼ 1, β2 ¼ φ and E ¼ ρφε. The above model can be estimated by a least squares regression.

2.2

Statistical Models

As was discussed in Chap. 1, a regression analysis is a statistical tool for the investigation of relationships between variables. Regression techniques have long been the cornerstone of econometrics and have become essential for policymakers.

2.2.1

Regression

Consider we collect data on energy demand and production. Let P denote cement production, and let E denote that energy consumption. These data are plotted in a two-dimensional scatter diagram like that shown in Fig. 2.2. Each point in the below diagram represents the energy consumption in the sample. The diagram indeed suggests that higher values of P tend to yield higher values of E, but the relationship is not perfect and does not help to obtain an accurate prediction about P. Using a linear system like Eq. (2.4), the hypothesized relationship between P and E may be written as: P ¼ β0 þ β1 E þ ε

ð2:16Þ

where β0 and β1 are constant amounts and ε is the “noise” term reflecting other factors that influence production.

Fig. 2.2 Representation of a two-dimensional diagram

22

2

Analytical Approaches in Energy Modeling

Where is the location of this line on the scatter diagram? Here, the nature of the noise term plays a key rule. For the noise term with a large negative number, the true (observed) value of P will always be less than its value on the line. It means that the proposed line lies above most of the data points. Likewise, the line lies below the data points for a positive noise term. If the noise term is assumed to be zero, the line lies roughly in the middle of the data points. To pick the best of all possible lines, we will define an estimated error for each point as the vertical distance between the value of P along the estimated line and the true value of P for the same observation. As can be observed from this figure, each possible line that passes through the points will cause a different set of estimated errors. We use a regression analysis to find the one which the sum of the squares of the estimated errors is at a minimum. This is called “the minimum sum of squared errors” criterion (Draper and Smith 1998). If we consider Eq. 2.16, then the minimum sum of squared errors will be: X

e2i ¼

X

ðPi  β0  β1 E i Þ2

ð2:17Þ

i

Therefore, ∂


P

e2i

 ¼ 2

X

ðPi  β0  β1 E i Þ ¼ 0 ∂β0
P 2  X ∂ ei ¼ 2 E i ðPi  β0  β1 E i Þ ¼ 0 ∂β1

ð2:18Þ ð2:19Þ

Respectively, we have   n P  Ei

 P    P  β 0    E P i i      P 2 ¼ P E i  β 1   E i P i 

ð2:20Þ

If we assume that   n Δ ¼  P Ei

P  X 2 X  P E2i  ¼ n E 2i  E i 6¼ 0  Ei

ð2:21Þ

We can find coefficients with the Cramer’s rule:       β 0  1  P E 2  P E i  P P i   ¼  Pi   P  β  Δ   Ei n  E i P i  1 P P P n E i Pi  Ei Pi β1 ¼ P P n E 2i  ð E i Þ2 If we consider P ¼

P

Pi and E ¼ n

P

Ei , then n

ð2:22Þ ð2:23Þ

2.2 Statistical Models

23


 Pi  P E i  E 2 P
Ei  E

P
β1 ¼

ð2:24Þ

β0 ¼ P  β1 E

ð2:25Þ

P y xi β1 ¼ P i 2 xi

ð2:26Þ

yi ¼ Pi  P

ð2:27Þ

xi ¼ Ei  E

ð2:28Þ

Or, it can be rewritten as

By applying this model to the sample data represented in Table 2.1, the value of β0 and β1 can be estimated. Therefore, for the above data, β1 ¼ 1.4746, β0 ¼ 3.3057, and, finally, P ¼ 3.3057 + 1.4746E, as shown in Fig. 2.3. In a real cement production process, production is affected by a variety of factors in addition to energy consumption, which were aggregated into the noise term in the simple regression model above. The multiple regression method enables us to quantify the impact of these additional factors on production (dependent variable), simultaneously. Let M denote the raw material (ton) and then add it to the data previously collected, as shown in Table 2.2. Using the multiple regression method, we can modify Eq. (2.16) as represented below: P ¼ β 0 þ β1 E þ β 2 M þ ε

ð2:29Þ

β2 is expected to be positive. We can no longer think of using a two-dimensional diagram, since we have three explanatory variables in Eq. (2.29). Therefore, the Table 2.1 Sample data for the simple regression example

n 1 2 3 4 5 6 7 8 9 10 11 12

P 1 3 5 7 9 12 15 22 30 33 36 42

E 3 4 6 8 9 13 15 16 18 22 27 33

xi 11.39 10.89 8.39 6.59 5.19 1.39 0.11 1.61 3.31 7.61 12.61 18.61

yi 16.92 14.92 12.92 10.92 8.92 5.92 2.92 4.08 12.08 15.08 18.08 24.08

yixi 192.71 162.47 108.39 71.96 46.29 8.23 0.32 6.57 39.98 114.76 228.00 448.15

xi2 129.77 118.63 70.42 43.45 26.95 1.94 0.01 2.59 10.95 57.89 158.97 346.27

24

2

Analytical Approaches in Energy Modeling

Fig. 2.3 Simple regression example Table 2.2 Sample data, including the raw material

n 1 2 3 4 5 6 7 8 9 10 11 12

P 1 3 5 7 9 12 15 22 30 33 36 42

E 3 4 6 8 9 13 15 16 18 22 27 33

M 3 8 13 20 25 34 42 70 96 106 118 123

parameters β0, β1, and β2 are estimated in a hyperplane, to minimize the sum of squared errors between the observed value of P and the estimated plane. β0 represents the intercept of that plane with the P-axis, where E and M are zero. β1 and β2 imply the slope of that plane in the E and M dimensions. The geographical demonstration of this equation can be shown using a hyperplane in a 3D surface graph, as shown in Fig. 2.4. Using the Analysis ToolPak in Excel software, the hyperplane can be represented by the following linear equation: P ¼ 0:536 þ 0:360E þ 0:2416M

ð2:30Þ

We need to test the coefficients of determination, which show the extent of the effect of an independent variable on the dependent variable in our estimation. Here are the essential validation tests used in the regression modeling (Carlberg 2016):

2.2 Statistical Models

25

Fig. 2.4 Demonstration of the hyperplane in the 3D surface graph

R-squared The coefficient of determination. It tells us how many points fall on the regression line: P


2 Y i  Y i R ¼1P
2 i Yi  Yi 2

i

ð2:31Þ

where Yi and Y i refer to the observed and estimated values of the dependent variable (P), respectively. The R-squared (R2) measures the ratio of the sum of squared errors of our regression model to the sum of squared errors of our baseline model. In our example, the value of R2 is estimated at about 0.9973, which shows the success of the regression in predicting the values of the dependent variable within the sample. Standard Error It measures the statistical reliability of the coefficient estimates— the more significant the standard errors, the more statistical noise in the estimates. The standard error of the regression is a summary measure based on the estimated variance of the residuals. The standard error of the regression is computed as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  P
 2 i Yi  Yi s¼ nk1

ð2:32Þ

n and k represent the number of data points and number of independent variables, respectively. The standard error is estimated at about 0.7367, which indicates good regression in a model. We also need to test the validity of our model by posing the following question: Is there at least one independent variable linearly related to the dependent variable?

26

2

Analytical Approaches in Energy Modeling

Fig. 2.5 Excel report on multiple regression analysis

To answer this question, the following hypothesis needs to be tested: H : β1 ¼ β2 ¼ . . . ¼ βi ¼ 0 (null hypothesis). H1: At least one β is not equal to zero. 0

Therefore, if at least one β is not equal to zero, the model is valid. We usually use the F statistic in combination with the p-value (significance F) when deciding to support or reject the null hypothesis. The F statistic is calculated using the ratio of the mean square regression (MS regression) to the mean square residual (MS residual). df refers to the degree of freedom. Large F results from a large MS. Then, much of the variation in y is explained by the regression model, and the null hypothesis should be rejected. This shows the model is valid. Figure 2.5 shows the model results provided by the Analysis ToolPak in Excel. In the above figure, the SS regression is the variation explained by the regression line; SS residual is the variation of the dependent variable that is not explained. The significance F is comparing with a level of confidence of 5% to indicate rejection of the null hypothesis. As can be observed from the results, with F value about 2036.03 and significance F value about 1.11074  1012 which is less than 0.05, the null hypothesizes must be rejected. p-value This reflects the probability that we will mistakenly reject the null hypothesis. • A small p-value (typically 0.05) indicates strong evidence against the null hypothesis, so we reject the null hypothesis. • A large p-value (>0.05) indicates weak evidence against the null hypothesis, so we fail to reject the null hypothesis. • p-Values very close to the cutoff (0.05) are considered to be marginal (could go either way). t-Statistics The t-statistic is used to test how extreme a statistical estimate is. It can be computed by subtracting the hypothesized value from the statistical estimate and then dividing by the estimated standard error. It may be used to test the hypothesis that the true value of the coefficient is zero: t sat ¼

Value of the coefficient Standard error of the coefficient

Big value for t sat shows the null hypothesizes can be rejected.

ð2:33Þ

2.2 Statistical Models

2.2.2

27

Time Series Analysis

A sequence of observations ordered over time “T” can represent a time series, where T is a finite set of equidistant points in time. There are two types of time series analysis approaches: Kinetic model: the observations are defined as a function of time xt ¼ f(t). Dynamical model: the observations are a function of their own past or the history of other observed variables xt ¼ f(xt1, xt2, xt3, . . .). A time series analysis accounts for the fact that data points taken over time may have an internal structure. An example is as follows: • • • •

m: represents a long-term uniform change in the time series k: represents a periodic change in the time series s: represents a cyclic change in the time series u: the residual component

Equation (2.34) represents the simplest form that the four components add up to the time series (Shumway and Stoffer 2017): xt ¼ mðt Þ þ kðt Þ þ sðt Þ þ uðt Þ

ð2:34Þ

One can estimate the parameters of the functions m, k, and s with regression methods, making some assumptions about the period of the trade cycle component. The residual component u (t) is the regression residual. It is also possible to eliminate the residual component by adopting a smoothing procedure such as the moving averages method. In the following example, we will discuss the application of the moving average and regression methods in estimating the time series of electricity demand in a factory, using the sample data reported in Table 2.3. Using the minimum sum of squared errors technique, we have Eð t Þ ¼ where β0 ¼ xt  β1 t and β1 ¼

Xn t¼1

ðxt  β0  β1 t Þ2 ¼ 0

xt t  xt t
2 and therefore: t 2  t n
 P xt t  t

mðt Þ ¼ x  t¼1 n
P t¼1

n
 P xt t  t

 t¼1 n
2 :t þ P 2 :t t  t t  t

ð2:35Þ

t¼1

Therefore m(t) ¼ 41.525 – 0.2566t. Using this equation, the difference between observed and estimated values is reported in Table 2.4.

28

2

Table 2.3 Sample data for the time series of electricity demand

2003

2004

2005

2006

Table 2.4 Difference between the observed and estimated value

2003

2004

2005

2006

Analytical Approaches in Energy Modeling

Quarter I Quarter II Quarter III Quarter IV Quarter I Quarter II Quarter III Quarter IV Quarter I Quarter II Quarter III Quarter IV Quarter I Quarter II Quarter III Quarter IV

Quarter I Quarter II Quarter III Quarter IV Quarter I Quarter II Quarter III Quarter IV Quarter I Quarter II Quarter III Quarter IV Quarter I Quarter II Quarter III Quarter IV

Time sector (t) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Xt (Observed) 61.2 47 34.1 33 38.5 36.5 38.5 36.9 32.4 33.3 32.6 31.3 40.5 42 42.7 49

Xt (Trend) 41.27 41.01 40.76 40.50 40.24 39.99 39.73 39.47 39.22 38.96 38.70 38.45 38.19 37.93 37.68 37.42

Xt (kW) 61.20 47.00 34.10 33.00 38.50 36.50 38.50 36.90 32.40 33.30 32.60 31.30 40.50 42.00 42.70 49.00

Diff 19.93 5.99 6.66 7.50 1.74 3.49 1.23 2.57 6.82 5.66 6.10 7.15 2.31 4.07 5.02 11.58

To calculate the S component below equations can be used: SðtÞ ¼ Sj  B 4 P

B¼

ð2:36Þ

Sj

1

4

ð2:37Þ

2.3 Simulation Method

29

Table 2.5 The calculated values of Sj in different quarters in each year

Table 2.6 The values of S(t)

2003 2004 2005 2006 Sum Ave Sj

Quarter I 19.93 1.74 6.82 2.31 13.69 3.4213

Quarter II 5.99 3.49 5.66 4.07 0.91 0.2279

S(Quarter I) S(Quarter II) S(Quarter III) S(Quarter IV)

Quarter III 6.66 1.23 6.10 5.02 8.96 2.2404

Quarter IV 7.50 2.57 7.15 11.58 5.64 1.4088

3.4213 0.2279 2.2404 1.4088

where Sj, Sj , and B represent the difference between the trend and observed values in each of the seasons, the average of Sj in each quarter and the average of Sj during whole period, respectively. These values are shown in Tables 2.5 and 2.6. In this example, B ¼ 0. Using Eq. (2.34), the time series of electricity demand can be given by the following correlations:

2.3

xðt, Quarter IÞ ¼ 44:946  0:2566t

ð2:38Þ

xðt, Quarter IIÞ ¼ 41:763  0:2566t

ð2:39Þ

xðt, Quarter IIIÞ ¼ 39:284  0:2566t

ð2:40Þ

xðt, Quarter IVÞ ¼ 40:116  0:2566t

ð2:41Þ

Simulation Method

Simulation attempts to build an experimental device called a model which provides information about the possible behaviors of the actual system being modeled. Simulations can be explained in this simple way: problem-solving. Should we add a new boiler to our power plant? Will we receive a positive return on our investment? Simulations can answer these and many other questions because they provide a virtual environment to learn and experiment with a process. Simulations are a place where data is easy to come by and where mistakes result in improvements instead of ending careers. Simulations are based on the definition of a system. A system can be defined as a set of interacting components or entities operating together to achieve a common goal or objective. For example, we may consider a manufacturing system with its machine canters, inventories, conveyor

30

2

Analytical Approaches in Energy Modeling

belts, production schedule, and items produced. We use the simulation models to measure and estimate the performance of the system. The result will be used to improve operation or to provide useful information about possible failures.

2.3.1

Steps in a Simulation Modeling

1. Statement of the problem The first step in simulation modeling is to prepare a concise description of the current problem and the desired goal. This goal should provide an overall vision for the process in a system. 2. Model development Conceptualizing the problem is the most important part of the model development which deals with the identification of the essential features of a problem and the formulation of the assumptions that describe the system. The required data to construct the model is a function of the complexity of the model. The better the quality and accuracy of the input data, the higher the accuracy of the estimation provided by the model. 3. Model translation Most models try to simulate real-world systems. Since this requires a great amount of effort in terms of data collection and computation, the model must be entered into a computer-recognizable format. 4. Verification Verification refers to the process of ensuring that the model is free from logical errors. In other words, the model should do exactly what it is intended to do. 5. Validation of results Validating the results is an important step in the simulation. Some of the overall goals of this step are as follows: • Ensure whether the model is useful for a set of searches performed to satisfy the requirements of dealing with a specific problem. • Ensure that the simulation produces sufficient results within a reasonable time frame to assess and evaluate whether the initial set of searches needs to be modified. • Determine whether the results of the model allow for a comparison between alternative search methodologies. Figure 2.6 represents the process of making a simulation modeling.

2.3 Simulation Method

31

Fig. 2.6 Steps in a simulation study (Enhanced version of original adopted from Banks et al. 2009)

2.3.2

Solution of the Simulation Problem

After formulating our problem, we need to use our mathematical skills to solve it. In almost every case, we are faced with a simultaneously coherent mathematical system of equations like those depicted below: f 1 ð x1 ; x2 ; . . . ; xn Þ ¼ 0 f 2 ð x1 ; x2 ; . . . ; xn Þ ¼ 0 ⋮ f n ð x1 ; x2 ; . . . ; xn Þ ¼ 0 The Taylor polynomial solutions in the above systems with two state variables (x1, x2) can be expressed as follows: f 1 ð x1 ; x2 Þ ¼ 0

ð2:42Þ

f 2 ð x1 ; x2 Þ ¼ 0

kþ1 kþ1 
k k  ∂f 1 x1k ; x2k
kþ1  f 1 x1 ; x2 ¼ f 1 x1 ; x2 þ x1  x1k ∂x1
k k
 ∂f 1 x1 ; x2 þ xkþ1  x2k þ    2 ∂x2

ð2:43Þ

ð2:44Þ

32

2

Analytical Approaches in Energy Modeling


  ∂f 2 x1k ; x2k
kþ1 x1  x1k ¼ þ ∂x1
k k  ∂f 2 x1 ; x2
kþ1 þ x2  x2k þ    ∂x2

 where, Δx1k ¼ xkþ1  x1k and Δx2k ¼ xkþ1  x2k 1 2


kþ1 f 2 xkþ1 1 ; x2






f 2 x1k ; x2k






 ∂f 1 x1k ; x2k ∂f 1 x1k ; x2k k Δx1 þ Δx2k ¼ f 1 x1k ; x2k ∂x1 ∂x2
k k
k k
 ∂f 2 x1 ; x2 ∂f 2 x1 ; x2 Δx1k þ Δx2k ¼ f 2 x1k ; x2k ∂x1 ∂x2

ð2:45Þ

ð2:46Þ ð2:47Þ

The Newton algorithm is the most common iterative algorithm which can be used to solve the above equations: J Δx ¼ f ðÞ 2

∂f 1 ðx1k , x2k , . . . , xnk Þ ∂f 1 ðx1k , x2k , . . . , xnk Þ 6 ∂x1 ∂x2 6 6 k k k k 6 ∂f 2 ðx1 , x2 , . . . , xn Þ ∂f 2 ðx1 , x2k , . . . , xnk Þ 6 J ¼6 ∂x1 ∂x2 6 6 6 ⋮ ⋮ 6 4 ∂f ðx k , x k , . . . , x k Þ ∂f ðx k , x k , . . . , x k Þ n 1 2 n 1 2 n n ∂x1 ∂x2

ð2:48Þ 3 ∂f 1 ðx1k , x2k , . . . , xnk Þ 7 ∂xn 7 7 ∂f 2 ðx1k , x2k , . . . , xnk Þ 7 7 ... 7 ∂xn 7 7 7 ⋱ ⋮ 7 k k k 5 ∂f n ðx1 , x2 , . . . , xn Þ  ∂xn ð2:49Þ ...

And 2

3 Δx1k 6 Δx k 7 2 7 Δx ¼ 6 4 ⋮ 5 Δxnk

0

f ð xÞ ¼

 dðf ðxÞÞ dy Δy ¼ lim dx dx Δx!0 Δx

ð2:50Þ

ð2:51Þ

So, to proceed with the above algorithm, we need to first do the following: 1. 2. 3. 4. 5. 6.

Set initial values (‘0’ denote initial values). Substitute the x’s into (*). Solve for Δx. If a result is obtained, stop the calculation process. If not, go to 5. ¼ xik þ Δxik . xkþ1 i k xi ¼ xkþ1 , and go back to 2. i

2.4 Optimization Method

2.4

33

Optimization Method

In mathematics, the term “optimization” refers to the strategy used to maximize or minimize a real function by systematically choosing the values of variables from an allowed set. Based on this definition, for a given function f: f: A R from some set A to the real numbers, we seek to find an element x0 in A such that f(x0)  f(x) for all x in A (“minimization”) or such that f(x0)  f(x) for all x in A (“maximization”). Problems formulated using this technique in the fields of engineering may refer to the technique as energy optimization. In a more advanced formulation, the objective function, f(x), to be minimized or maximized, might be subject to constraints in the form of: Equality constraints, Hi(x) ¼ 0 (i ¼ 1, . . ., m) Inequality constraints, Gj(x)  0 ( j ¼ m þ 1, . . ., n) And/or parameter bounds, xl, xu A general problem (GP) description is stated as: Min f ðxÞ

x 2 Rn

ð2:52Þ

Subject to:

2.4.1

H i ðxÞ ¼ 0 i ¼ 1, . . . , m

ð2:53Þ

G j ðxÞ  0 j ¼ m þ 1, . . . , n

ð2:54Þ

xl < x < xu

ð2:55Þ

Linear Programming

Linear programming (LP) problems involve the optimization of a linear objective function, subject to linear equality and inequality constraints. Linear programs are usually expressed in canonical form, as follows: max CT x subject to : Ax  b x0 where x represents the vector of variables (to be determined), C and b are vectors of (known) coefficients, and A is a (known) matrix of coefficients.

34

2

Table 2.7 Power company problem

Analytical Approaches in Energy Modeling Solar S1 S2 S

E1 E2 Requirement

Wind W1 W2 W

Fossil F1 F2 F

Price P1 P2

1. Power company problem Suppose that we are living in a rural area with no access to the electric grid. Electricity can be generated from three different sources: solar energy (S), wind energy (W), and fossil fuel (F) subject to two different tariffs (P1 and P2) by a local power company. The data is summarized in Table 2.7. S1, S2, W1, W2, F1, and F2 show the share of solar energy, wind energy, and fossil fuel in each option (E1 and E2), respectively. Being environmentally conscious, we have decided to meet our electricity demand with the at least certain quantity of solar energy (S) and wind energy (W) and a limited quantity of fossil energy (F). So, we wish to optimize our purchase by finding the least expensive combination of E1 and E2. This can be expressed as a linear programming problem in the following way: Min P1 E 1 þ P2 E2

ð2:56Þ

S1 E 1 þ S2 E 2  S

ð2:57Þ

W 1 E1 þ W 2 E2  W

ð2:58Þ

F 1 E1 þ F 2 E2  F

ð2:59Þ

Subject to

E1  0 and E2  0. Which in matrix form becomes: Min ½P1 P2 



E1 E2

subject to3 2 2 3 S1 S2 S 4 W 1 W 2 5 E 1  4 W 5, E 1  0 E2 E2 F1 F2 F

ð2:60Þ

ð2:61Þ

The above linear constraints (Eqs. (2.57, 2.58, and 2.59)) produce a convex polyhedron, which is called the feasible area and includes the possible values for E1, E2.

2.4 Optimization Method

35

2. Duality Every linear programming problem can be converted into a dual problem, which provides an upper bound to the optimal value of the primal problem. We can express the primal problem as follows: Max C T x

ð2:62Þ

Ax  b x  0

ð2:63Þ

Subject to

The corresponding dual problem is Min bT y

ð2:64Þ

AT y  c y  0

ð2:65Þ

Subject to

where y is used instead of x as variable vector. We now adopt the perspective of the owner of the power company who supplies electricity with the combination of different technologies of solar, wind, and fossil. The owner now solves the following optimization problem: How can I set the prices for different generation technologies (C1, C2, and C3) so that the customers will buy energy from me and so that I will maximize my revenue?

The customer will only purchase energy from the owner if the electricity price is below P1 for E1 and below P2 for E2. These restrictions impose the dual problem: Max C 1 S þ C2 W þ C 3 F

ð2:66Þ

S1 C 1 þ W 1 C2 þ F 1 C 3  P1

ð2:67Þ

S2 C 1 þ W 2 C2 þ F 2 C 3  P2

ð2:68Þ

Subject to

And C1, C2, C3  0. As we can see from the above example, because the dual problem deals with economic values, it is possible to determine the unit pricing scheme which will maximize the total revenue. The primal problem, on the other hand, seeks the best combination of the physical quantities. The minimum value of the total cost is equal to the maximum value of the total revenue.

36

2

Analytical Approaches in Energy Modeling

To replace the inequalities with the equalities in the constraints and convert above problem into its augmented form, nonnegative slack variables should be added to Eqs. (2.67) and (2.68): Max C 1 S þ C2 W þ C 3 F

ð2:69Þ

S1 C 1 þ W 1 C2 þ F 1 C 3 þ Y 1 ¼ P1

ð2:70Þ

S2 C 1 þ W 2 C2 þ F 2 C 3 þ Y 2 ¼ P2

ð2:71Þ

E1 ,E2 ,Y 1 ,Y 2  0

ð2:72Þ

where Y1 and Y2 are the (nonnegative) slack variables. 3. Simplex method The simplex method is used for solving linear programming problems. It was invented by George Dantzig in 1974. This method tests adjacent vertices of the feasible set in sequence so that at each new vertex, the objective function improves or is unchanged. This method needs the creation of multiple tables consisting of all the coefficients of the decision variables of the original problem and the slack, surplus, and artificial variables. In general, the operation in standard simplex includes the following steps: selecting an entering variable X(n), selecting a leaving variable X(m), and updating the tableau X(mn). Consider the following example: Max Z ¼ cx

ð2:73Þ

Ax  b

ð2:74Þ

x0

ð2:75Þ

First, try to convert above system to its standard format as follows:

x ½A I  xs

¼

b

ð2:76Þ

where, in the above equation, xs is the added stack variable. The basic variables matrix for this problem can be represented as follows: 2

3 xB1 6 : 7 6 7 7 xB ¼ 6 6 : 7 AND 4 : 5 xBn

2

3 B11 : . . . . . . . . . B1n 6: : : : 7 6 7 7 B¼6 : : : : 6 7 4: : : : 5 Bn1 : . . . . . . . . . Bnn

2.4 Optimization Method

37

In the beginning B ¼ I. At any iteration nonbasic variables are set to zero. Therefore, BxB ¼ b and xB ¼ B1b. At any iteration, the vector b and, therefore, the inverse matrix B, can be calculated. The objective function then can be rewritten as Z ¼ cBxB, where cB is the coefficient of basic variables The solving steps are as follows: 1. The input variables xj are determined in a vector format of Pj. 2. Y ¼ cBB1. 3. zj – cj ¼ YPj – cj. For all nonbasic variables, the largest negative value is chosen. The variable with the largest negative value is selected as the input variable. If all values are positive, the loop stops. 4. The leaving variable, xr, and its vector, Pr, are selected based on the minimum ratio rule.   θ ¼ Min ðxB Þk =α j k ; α j k > 0 , α j ¼ B1Pj, and xB ¼ B1b k

5. The next basic variable is determined. 6. Go to step (1). Figure 2.7 shows the application of the simplex method to a simple example. Nowadays, computer science and software programming help us to solve much larger problems very quickly. GAMS (General Algebraic Modeling System) is one of the leading molding systems which is used worldwide for solving optimization problems (Rosenthal 2007). A numerical modeling of the power problem, using GAMS software, is given in Appendix A. In Table 2.8, the required numerical values used in this problem are listed.

Fig. 2.7 Numerical example of the simplex method

38

2

Analytical Approaches in Energy Modeling

Table 2.8 Numerical value for the power company problem E1 E2 Requirement (kWh)

Solar 80% 10% 3000

Wind 10% 40% 2000

Fossil 10% 50% 5000

Price (cent/kWh) 20 10

By using the above code in GAMS software, we find that the optimal combination of the two options is (E1 ¼ 2.56 MWh and E2 ¼ 9.49 MWh), and the monthly electricity bill is $121.80.

2.4.2

Nonlinear Programming

Nonlinear programing is the process of solving a constrained and unconstrained optimization problem, where the objective function or some of the constraints are nonlinear. Gradient methods are generally used for the unconstrained optimization when the function F(x) is continuous in its first derivative in a neighborhood of a point a. Therefore, the reduction in F(x) is the most rapid when one moves from a in the direction of the negative gradient of F at a, ∇F(a), if: b ¼ a  α∇F ðaÞ

ð2:77Þ

And for a small positive step size, α, which is allowed to change at every iteration, we have F(a)  F(b). Therefore, we can start with an initial guess x0 which satisfies a local minimum of F, through the sequence x0, x1, x2, . . . such that xk þ 1 ¼ xk + αkβk, where βk ¼ ∇f(xk). Let’s have a look at the following example: f ðx1 ; x2 Þ ¼ 10x21 þ 2x22 þ 4x1 x2  14x1  6x2 þ 15    k
k k 20x1k  4x2k þ 14 β1 k β ¼ ∇f x1 ; x2 ¼ ¼ 4x2k  4x1k þ 6 β2k

ð2:78Þ ð2:79Þ

We need to solve αk ¼ argminαh(α) ¼ f(xk + αβk):
 hðαÞ ¼ f xk þ αβk
k 2
2 ¼ 10 x1 þ αβ1k þ 2 x2k þ αβ2k

 þ4 x1k þ αβ1k x2k þ αβ2k
k 
 14 x1 þ αβ1k  6 x2k þ αβ2k þ 15

ð2:80Þ

2.4 Optimization Method

39

And this is a simple quadratic function of the scalar α. It is minimized at the following: αk ¼


k 2
k 2 β1 þ β 2
k 2
2  20 β1 þ 4 β2k þ 8β1k β2k

ð2:81Þ


when using the steepest descent algorithm to minimize f (x) starting from x11 ; x12 ¼ ð0; 5Þ and using a tolerance of ε ¼ 106. For a convex quadratic function, the contours of the function values will be shaped like ellipsoids, and the gradient vector at any point x will be perpendicular to the contour line passing through x. The 30 final results are x30 1 ¼ 0.5 and x2 ¼ 1. f(0.5, 1) ¼ 8.5. Quadratic programming (QP) is another solving method which is often used for linearly constrained optimizations with a quadratic objective function. The general quadratic program can be written as follows: 1 Min f ðxÞ ¼ cx þ xT Qx 2

ð2:82Þ

Ax  b

ð2:83Þ

x0

ð2:84Þ

Subject to

where c and Q describe the coefficients of the linear terms and quadratic terms, respectively, and Q is an (nxn) symmetric matrix. We assume that a feasible solution exists, that the constraint region is bounded, and that Q is positive definite to guarantee convexity. The Lagrangian function for the quadratic program can be represented as follows: 1 Lðx; μÞ ¼ cx þ xT Qx þ μðAx  bÞ 2

ð2:85Þ

where μ is an m-dimensional row vector. We now apply the general first-order conditions which are necessary for a global minimum when Q is positive definite. These conditions are called the Karush-KuhnTucker conditions and are given below (Jensen and Bard 2002): ∂L  0 j ¼ 1, . . . ,n ∂x j ∂L  0 i ¼ 1, . . . ,m ∂μi

c þ xT Q þ μA  0

ð2:86Þ

Ax  b  0

ð2:87Þ

40

2

xj

Analytical Approaches in Energy Modeling


 xT cT þ Qx þ AT μ ¼ 0

∂L ¼ 0 j ¼ 1, . . . ,n ∂x j

μi gi ðxÞ ¼ 0 i ¼ 1, . . . ,m

μðAx  bÞ ¼ 0

ð2:88Þ ð2:89Þ

xj  0

j ¼ 1, . . . , n

ð2:90Þ

μi  0

i ¼ 1, . . . , m

ð2:91Þ

We add nonnegative surplus and slack variables v and y to the inequalities (2.86) and (2.87) and then re-arrange above equations to introduce the following linear programming system: cT þ Qx þ AT μT  y ¼ 0

ð2:92Þ

Ax  b þ v ¼ 0

ð2:93Þ

μv ¼ 0

ð2:94Þ

yT x ¼ 0

ð2:95Þ

And x, μ, y, v  0 The simplex algorithm can be used to solve the above system. Consider the following example: Min f ðxÞ ¼ 5x1  8x2 þ x21 þ 8x22

ð2:96Þ

x1 þ x2  2

ð2:97Þ

x1  3

ð2:98Þ

x1  0

ð2:99Þ

x2  0

ð2:100Þ

The data and variable definitions are given below.

5 C ¼ 8 T



2 Q¼ 0

0 16



1 A¼ 1

1 0

2 b¼ 3

The linear constraints (2.92) and (2.93) take the following form: 2x1 þ μ1 þ μ2  y1 ¼ 5

ð2:101Þ

16x2 þ μ1  y2 ¼ 8

ð2:102Þ

x1 þ x2 þ v1 ¼ 2

ð2:103Þ

x1 þ v2 ¼ 3

ð2:104Þ

2.4 Optimization Method

41

We can add artificial variables to each constraint and minimize their sum as the object function of the problem. Min δ1 þ δ2 þ δ3 þ δ4

ð2:105Þ

2x1 þ μ1 þ μ2  y1 þ δ1 ¼ 5

ð2:106Þ

16x2 þ μ1  y2 þ δ2 ¼ 8

ð2:107Þ

x1 þ x2 þ v1 þ δ3 ¼ 2

ð2:108Þ

x1 þ v2 þ δ 4 ¼ 3

ð2:109Þ

and x1, x2, y1, y2, μ1, μ2, v2, v2  0 The optimal solution to the above problem is (x1, x2, y1, y2, μ1, μ2, v2, v2) ¼ (1.61, 0.38, 0, 0, 1.78, 0, 0, 1.39), which satisfies the slackness conditions (2.94) and (2.95).

2.4.3

Mixed Integer Programming

An integer programming problem is an optimization problem with integer variables. If all the decision variables are required to be integers, then this problem is called a pure integer programming problem. If only some of the decision variables are required to be integers, then this problem is called a mixed integer programming problem. Moreover, if all the design rebels must be 0 or 1, then this problem is called a 0–1 integer programming problem or binary integer programming problem. Here, we will focus on solving integer programming problems, using the branch and bound method first proposed by Land and Doig (1960). The branch and bound algorithm splits the original problem into two branches of subproblems and solves them through the following steps: Step 1 Linear programming (LP) relaxation: We ignore all integer constraints on variables and solve the linear programming problem. If the optimal solution satisfies the integer constraints, we have found the optimal solution to the integer problem (IP). Otherwise, we need to initialize the lower bound on the optimal objective value of the IP problem and proceed with step 2. Step 2 Branching: We select an LP problem among the LP problems and choose a variable, xi, which is integer-restricted in the IP problem but has a non-integer value in the optimal solution of the selected LP problem. Therefore, we will have the following two new LP problems by adding two constraints on xi: xi  [xi] and xi  [xi] þ 1. Step 3 Bounding: We solve these two LP problems from step 2 and calculate the objective values for each of them. Step 4 Redetection: An LP problem may be eliminated from the calculation process if the problem is deemed infeasible.

42

2

Analytical Approaches in Energy Modeling

Fig. 2.8 Solving integer programming problems, using the branch and bound method

The candidate solution is the optimal solution to the IP problem, if there are no LP problems that can be branched out. Otherwise, we return to Step 2 and perform another iteration. The branch and bound method is applied to solve the mixed integer programming problem in the following way: Max

z ¼ 4x1  2x2 þ 7x3  x4

ð2:110Þ

Subject to: x1 þ 5x3  10

ð2:111Þ

x1 þ x2  x3  1

ð2:112Þ

6x1  5x2  0

ð2:113Þ

x1 þ 2x3  2x4  3

ð2:114Þ

x1 ,x2 ,x3 ,x4  0

ð2:115Þ

And x1, x2, and x3 are integer variables. The solution of the above problem is shown in Fig. 2.8.

References Arrow KJ, Chenery HB, Minhas BS, Solow RM (1961) Capital-labor substitution and economic efficiency. Rev Econ Stat 43(3):225–250, The MIT Press Banks J, Carson JS, Nelson BL, Nicol DM (2009) Discrete-event system simulation, 5th edn. Prentice Hall-Pearson Education Inc, Upper Saddle River Bierens HJ (2017) Econometric model specification. World Scientific, Singapore Carlberg C (2016) Regression analysis Microsoft Excel, ISBN-10: 0789756552, Que Publishing

References

43

Dantzig G (1974) Linear programming and extensions, ISBN-10: 0691059136, Princeton University Press Draper NR, Smith H (1998) Applied regression analysis, 3rd edn, ISBN 0–471–17082-8, John Wiley Jensen PA, Bard JF (2002) Operations research models and methods, 978–0471380047, John Wiley & Sons Land AH, Doig AG (1960) An automatic method for solving discrete programming problems. Econometrica 28:497–520 Ramsey JB (1980) Evaluation of econometric models, 978–0–12-416550-2, Academic Press Rosenthal RE (2007) GAMS — a user’s guide. GAMS Development Corporation, Washington, DC Samuelson PA (1983) Foundations of economic analysis, Enlarged edition, 978–0674313033, Harvard Economic Studies Shumway RH, Stoffer DS (2017) Time series analysis and its applications. Springer. https://doi.org/ 10.1007/978-3-319-52452-8 Tintner G (1968) Methodology of mathematical economics and econometrics, Volume 2, Issue 6, University of Chicago Press

Chapter 3

Energy Demand Models

Contents 3.1 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Objectives of the Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Energy Demand Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Application of the Energy Demand Simulation Model to the Yokohama Smart City Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Estimation of Energy Demand in the Middle East Region . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Optimization Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Optimal Fuel Consumption in a Gas Turbine Power Plant . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Optimal Energy Demand in a Cement Factory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Optimal Energy Demand in a Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Minimum Energy of a Compression Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Thermal Efficiency Maximization in a Heat Recovery System . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

45 46 47 51 58 59 61 65 66 72 74 75 80

Simulation Models

Simulation models are designed for the specific purpose of evaluating the energy demand of a country or world region in the medium and long term. They are found in the scenario approach. “A scenario is viewed as a consistent description of a possible long-term development pattern of a country, characterized mainly regarding the long-term direction of governmental socioeconomic policy” (Chateau and Lapillonne 1982). Using this approach, the planner can make assumptions about the possible evolution of social, economic, and technological development patterns of a country. These assumptions allow for future projections for a long-term scenario based on current trends and governmental objectives. In summary, the simulation methodology comprises the following sequence of operations (IAEA 2006):

© Springer Nature Singapore Pte Ltd. 2019 H. Farzaneh, Energy Systems Modeling, https://doi.org/10.1007/978-981-13-6221-7_3

45

46

3 Energy Demand Models

Fig. 3.1 Main inputs and outputs of the energy demand simulation model (Adopted from IAEA 2006)

1. Disaggregating the total energy demand of the country or region into a large number of end-use categories in a coherent manner 2. Identifying the social, economic, and technological parameters which affect each end-use category of the energy demand 3. Establishing, in mathematical terms, the relationships which relate to energy demand and the factors affecting this demand 4. Developing (consistent) scenarios for social, economic, and technological development for the given country 5. Evaluating the energy demand resulting from each scenario 6. Selecting, from among all the possible scenarios proposed, the “most probable” patterns of development for the country Figure 3.1 shows the main inputs and outputs of an energy demand simulation model.

3.1.1

Objectives of the Methodology

The scenarios can be subdivided into two categories: 1. The structural changes in the energy demand: This is done by utilizing a detailed analysis of the social, economic, and technological characteristics of the given country. 2. The evolution of the potential markets of each form of final energy: This deals with the technological factors which affect the calculated energy demand. For example, the calculated energy demand is impacted by the efficiency and

3.1 Simulation Models

47

penetration potential of each alternative energy form and is also affected by such new technology as the smart grid. The bottom-up simulation models include an analysis and aggregation of the energy system, and they allow the performance function of the energy system to be identified and estimated. This is possible because the whole energy system is segregated into incremental elements such as end-user, final energy, energy conversion, and energy resources. When various energy forms, i.e., electricity, fossil fuels, etc., are competing for a given end-use category of energy demand, this demand is calculated first in terms of useful energy and then converted into final energy. In the process of doing this, market penetration and the efficiency of each alternative energy source and also the use of new technologies are taken into account. The demand for fossil fuels is therefore broken down in terms of coal, gas, or oil, and then the substitution of fossil fuels by alternative “new” energy forms (i.e., solar, district heat, etc.) is estimated. In completing this step, we have to consider the importance of the structural changes in the energy system that these energy forms may introduce in the future. Because these substitutions will be determined by policy decisions, they must be taken into account at the stage of formulating and writing the scenarios of development.

3.1.2

Energy Demand Calculations

The simulation model calculates the energy demand as a function of a scenario of possible development. This scenario covers two types of scenario elements (IAEA 2006): • The socioeconomic-related scenarios, which describe the social and economic evolutions of the country such as changes in GDP, income, population, etc. • The technology-related scenarios which relate to the technological changes, such as changes in the penetration of an alternative energy technology into its potential markets The energy demand is first calculated in terms of useful energy. Then, by considering the efficiency of each conversion technology and the penetration of each alternative energy carrier into the given market, this useful energy is then converted to final energy.

3.1.2.1

Industry Sector Energy Demand Calculation

This sector includes agriculture, mining, construction, and manufacturing industries. The energy demand of each economic subsector is estimated as a function of the level of economic activity of the subsector such as its value-added and the energy intensity of each energy form. The energy intensities (i.e., the consumption of

48

3 Energy Demand Models

electricity per unit of value added) are characteristics of each subsector and depend on the industrial process used. The following equation represents the demand of energy in the industry sector as a function of the value added and energy intensity: EInd ¼

XX τ

Q  SI τ j  EI τ j

ð3:1Þ

j

where Q is the value added ($), SI is the industry share in whole sector value added, and EI is the energy intensity in each industrial sector (PJ/$). τ and j refer to the technology type and subsector, respectively.

3.1.2.2

Transportation Sector Energy Demand Calculation

In the transport sector, the final energy is defined as a function of the transport demand (passenger-kilometers or ton-kilometers). The transport demand is determined from the average distances traveled by the people and goods within large cities or between cities. The modal share of the particular type of transportation (i.e., bus, passenger vehicle or train, etc.) reflects the transport policy of the region and therefore must be considered as the main scenario parameters. The energy demand in the transport sector can be estimated by the following formula: E Tran ¼

XX v

PKM  SRvf  EI vf

ð3:2Þ

f

And PKM ¼

XX v

V vf U vf Dvf Ovf

ð3:3Þ

f

where PKM is the total passenger-kilometers, SR is the Modal share (%), EI is the energy intensity (Liter/km), V is the number of each vehicle in use, U is the utilization rate (%), D is the average annual distance traveled (km), O is the occupancy rate (person/vehicle), and v and f refer to the vehicle type and technology type, respectively. The growth of vehicle stock (in use) can be related to a certain macroeconomic parameter (per capita GDP), using a sigmoidal shape function, which assumes a “Sshaped curve” for the long-term trend in vehicle growth. In this model, the growth of the vehicle in use is estimated by using the Gompertz function, as follows (Dargay and Gately 1999): V t ¼ Veδe

μGDPt

ð3:4Þ

3.1 Simulation Models

49

V indicates the saturation level of vehicle ownership. δ and μ can be estimated through a regression analysis, using the historical trends of vehicle ownership. The growth in per capita mobility over time typically follows a sigmoid or S-shaped curve. The following formula represents the logistic function which has been used in the simulation model (Farzaneh et al. 2016): 

PKM Capita

 ¼ t

α 1 þ γexpðβt Þ

ð3:5Þ

where α is the saturation level which is calculated on the basis of time budget and average speed of mobility in cities. γ and β are positive coefficients.

3.1.2.3

Household Sector Energy Demand Calculation

The demographic nature characterizes the energy demand in the household sector. The primary determining factors in estimating the energy demand in this sector are the population and the number of dwellings. The calculations for the household sector represent the impact of the living conditions of the population and the type of residence on the potential markets for the alternative energy forms of final energy. The following equations represent the extension of the above definition of energy demand in the residential and commercial sectors: For the residential sector ERes ¼

X X i

ND  DWSij  DWZ ij  EI ij



ð3:6Þ

j

And ND ¼

Pop  α θ

ð3:7Þ

For the commercial sector ECom ¼

X X k

FA  CSkj  EI kj



ð3:8Þ

j

where ND is the number of dwellings, DWS is the share of each dwelling type (%), DWZ is the size of each dwelling type (sqm), EI is the energy intensity (kWh/sqm), FA is the total floor area (sqm), and CS is the share of each building type in the total floor area (%), respectively. α indicates the share of population type, and θ refers to the average household size (person/household). i, j, and k refer to the dwelling type (i.e., single room, double room, etc.), type of technology (i.e., heating, cooling, cooking, lighting, etc.), and type of building (i.e., school, mall, shop, office, etc.).

50

3 Energy Demand Models

3.1.2.4

Service Sector Energy Demand Calculation

The energy consumption in the service sector is characterized by the economic level of activity such as labor force and value added in this sector. The major end-use categories in the service sector include space heating, cooking, lighting, air conditioning, motive power for motors, etc. Space heating  U RSHi ¼ ND  DWSi  DWZi  DSHi  DSHEFi  HDD  24

h Day

 ð3:9Þ

Hot water U RHWi ¼ ND  DWSi  DHWi  DHWEFi

ð3:10Þ

Cooling U RCOOLi ¼ ND  DWSi  DCCi  DWZi  DCOOLEFi  CDD   h  24 Day

ð3:11Þ

Cooking U RCOOKi ¼ ND  DWSi  DCOi  DCOOKEF

ð3:12Þ

U RELECi ¼ ND  DWSi  DSEi  DELECEF

ð3:13Þ

U SMOTFi ¼ GDP  SGDPi  MOTFINTi

ð3:14Þ

Electricity uses

Motor fuels

where DSHEF is the specific heating load rate by dwelling type (kWh/sqm. C1. h1), HDD is the heating degree day (C.Day), DWZ is the size of each dwelling type (Sqm/cap), DSH is the fraction of floor area that is actually heated by dwelling type (%), DHW is the share of total dwellings with hot water facilities (%), DHWEF is the specific energy consumption for water heating per

3.1 Simulation Models

51

person (kWh/cap), DCC is the fraction of floor area that is actually cooled by dwelling type, CDD is the cooling degree day (C.Day), DCOOLEF is the specific cooling load rate by dwelling type (kWh/sqm. C1. h1), DCO is the share of total dwellings with indoor cooking facilities (%), DCOOKEF is the specific energy consumption for cooking in dwellings (kWh/cap), DSE is the fraction of total dwellings that are electrified (%), DELECEF is the electricity consumption rate (kWh/cap), SGDP is the share of service sector in total GDP (%), and MOTFINT is the fuel consumption intensity in the service subsector (kWh/$), respectively.

3.1.3

Application of the Energy Demand Simulation Model to the Yokohama Smart City Project

In this example, city-level data from Yokohama, Japan, is taken to use in a simulation modeling technique to calculate the energy consumption in different subsectors in this city (Farzaneh et al. 2014). The simulation will also detail the role of executive policy in supporting the reduction in energy consumption in the city of Yokohama. This city lies on partially reclaimed land with an area of 242 ha and a population of 3.7 million. Yokohama is officially Japan’s largest incorporated city. Yokohama has a humid subtropical climate with hot and humid summers and chilly, but relatively mild, winters. Temperatures in winter rarely drop below freezing, while it can get quite warm in summer due to the effect of humidity. The city’s profile is shown in Table 3.1. This city is well known for its utilization of renewable energy, including wind and solar energy. A wind power generator capable of producing approximately 4 MW of energy is in use, and solar photovoltaic cells capable of generating about 19 MW of energy have been installed to help meet the electricity demand of the city. Besides this, about 219 MW of total electricity consumption is supplied by hydropower. As the largest city in Japan, Yokohama aims to build a “Next Generation Energy Infrastructure and Social System”: that is, Yokohama is making an earnest attempt to maximize CO2 reduction with innovative, cutting-edge techniques. In the near future, Yokohama will add to its efforts thus far by intensively introducing PV and HEMS (a Home Energy Management System). If we consider Yokohama City as a typical example, approximately 40% of the total energy is consumed in the residential and commercial sectors. This energy use was associated with over 7 million tons of CO2 equivalent (CO2e) emissions in 2010.

3.1.3.1

Baseline Scenario

This example takes demographic changes and economic activities in Yokohama into account. Among all the parameters which have an impact on energy consumption, these two parameters are the most fundamental. When calculations for the residential

52

3 Energy Demand Models

Table 3.1 Yokohama City profile

Item Total population (million) Number of households GDP (billion US $) Maximum ambient temp (0C) Maximum solar irradiation (MJ/sqm)

3.7 1.2 11.2 32 15.7

Table 3.2 Dwelling types and commercial subsectors in Yokohama Dwelling groups Under 29 sqm Between 30 and 49 sqm Between 50 and 69 sqm Between 70 and 99 sqm Between 100 and 149 sqm Over 150 sqm Commercial subsectors Hotel (Western style) Hotel (Japanese style) Office and bank Shop Theater and film Hospital Office Mall Department store Bank Theater and entertainment Others Primary school Junior high school

Share in total sector (%) 3 8 17 32 30 10 Share in total sector (%) 2 3 2 19 2 2 17 20 10 4 2 2 8 7

Average area (sqm) 28 40 60 85 125 200

sector are performed, the living conditions of the population are taken into account, that is, the place of residence (city local climate conditions) and the type of residence (dwelling mode and size). Tables 3.2, 3.3, 3.4, 3.5, and 3.6 show the underlying assumptions which are used to set the baseline scenario.

3.1.3.2

Energy Balance in the Baseline Scenario

The first law of thermodynamics is a statement of material balance—neither mass nor energy can be created or destroyed—it can only be transformed. This indicates the overall balance of energy at all times. The energy balance is then designed to illustrate the general energy flow (production to end-user) of the energy system. The

3.1 Simulation Models

53

Table 3.3 Fraction of floor area covered by the heating and cooling system in the residential sector in Yokohama

Under 29 sqm Between 30 and 49 sqm Between 50 and 69 sqm Between 70 and 99 sqm Between 100 and 149 sqm Over 150 sqm

Fraction of floor area which should be heated (%) 50 50

Fraction of floor area which should be cooled (%) 2.5 2.5

50

2.5

50

2.5

50

5

80

5

Table 3.4 Installed power capacity and annual electricity generation in Yokohama

Hydropower PV Biomass Wind Waste to electricity Fossil thermal power plant Power plant (coal) Power plant (oil) Power plant (LNG/LPG) Nuclear Total

Installed capacity (MW) 219 19 74 4 80

Annual operation (h/y) 7000 2000 7500 1600 7500

Maximum generation (GWh/y) 1533 38 555 6.4 600

31 265 639

7500 7500 7500

232.5 1987.5 4792.5

405 1735

7500

3037.5 12782.4

energy balance table has four main building blocks: the supply-side information (resources), conversion details, final distribution, and the demand-side information. The supply-side information captures the domestic supply of energy products (electricity and heat) through production. Energy production provides the quantities of energy domestically produced in a city. The conversion section of energy accounting captures the conversion of primary energies into secondary energies either through physical or chemical changes. Usually, the inputs used in the transformation process are given a negative sign, while the outputs are given a positive sign. The commonly used conversion process for an energy system is electricity generation. However, as with supply information, conversion is also a city-specific

54

3 Energy Demand Models

Table 3.5 Energy intensities used in the calculation of energy demand in the residential sector

Under 29 sqm Between 30 and 49 sqm Between 50 and 69 sqm Between 70 and 99 sqm Between 100 and 149 sqm Over 150 sqm

SHR (Wh/sqm/C/h) 3 3

SCR (kWh/dw/y) 1500 1500

SHW (kWh/cap/y) 350 350

SHC (kWh/dw/y) 930 930

SEC (kWh/dw/y) 900 900

3

2000

350

930

900

3

2500

350

930

900

3

3500

350

930

900

4

4000

350

930

900

SHR specific heat loss rate for heating, SCR specific cooling rate, SHW specific heat rate for water heating, SHC specific heat rate for cooking, SEC specific electricity consumption, cap person, dw dwelling, sqm square meter

Table 3.6 Energy intensities used in the calculation of energy demand in the commercial sector (kWh/sqm/y) Hotel (Western style) Hotel (Japanese style) Office and bank Shop Theater and film Hospital Office Mall Department store Bank Theater and entertainment Others Primary school Junior high school

Space heating 49.88 48.72 27.84 17.40 35.96 75.40 27.84 17.40 16.24 27.84 35.96 27.84 27.84 29.00

Cooling 85.84 30.16 35.96 76.56 47.56 6.50 35.96 76.56 102.08 35.96 47.56 35.96 3.48 3.48

Lighting 177.48 126.44 133.40 265.64 116.00 149.64 133.40 265.64 243.60 133.40 116.00 133.40 19.72 19.72

Other 247.08 258.68 9.28 35.96 26.68 185.60 9.28 35.96 56.84 9.28 26.68 9.28 24.36 12.76

section of the energy account and typically varies from one city to the next. The conversion section also captures information on the energy used by the end-users and the transmission and distribution losses. Both of these elements carry a negative sign as they represent a reduction in energy flows as energy is used by consumers. The final level is the demand side. In terms of accounting balance, this is the residual amount available for domestic consumption from primary supplies after accounting for conversion and other transmission losses. Generally, net supply is

3.1 Simulation Models

55

calculated from the supply side, while the net demand is calculated from the demand side, and these two figures should match, thus ensuring the correctness of the accounting. However, it is quite rare that the two items are the same. The statistical difference term is used as the balancing item. Its sign indicates whether the supply-side total is higher (thus requiring a deduction of any balancing amount) or lower (thus requiring some balancing amount) than the demand-side total. The energy balance is organized in four sections: supply, conversion, final distribution, and use. Depending on the requirements and purpose of the analysis, it is possible to gain insight into these areas. For example, the primary energy requirement indicates the total energy requirement of the city to meet final demand. The trend of the primary energy requirement of a city shows how the internal aggregate demand has changed over time. Similarly, the conversion section of the energy balance provides information on energy conversion efficiency and how the technical efficiency of aggregate conversion has changed over the study period. These can easily be analyzed from energy balance tables. Final consumption data can be used to analyze the evolution of useful energy demand of the city by fuel type and by sector of use. Such analyses provide a better understanding of the demand pattern of each sector and energy source. In this example, the energy balance shows how fossil fuel has been distributed in the energy system in Yokohama. As can be seen from Table 3.7, approximately 1.248 Mtoe of the 2.471 Mtoe provided by fossil fuels was consumed in the power sector to provide the majority of the city’s electricity demand. Some of the important indices which can be calculated from the energy balance table are the following: 3:538 1. Primary energy factor (PEF) ¼ TPE TFE ¼ 2:265 ¼ 1.56. This means that the use of about 1.56 unit of primary energy is required to produce one unit of useful energy in Yokohama City. 2. Herfindahl-Hirschman index (HHI) is calculated by squaring the share of each energy resources (Si) competing in an energy market and then summing the resulting numbers:

HHI ¼

N X

s2i

ð3:15Þ

i¼1

By collecting the value of each Si from the energy balance table, HHI can be calculated as follows: HHI ¼ ð2:471=3:538Þ2 þ ð0:652=3:538Þ2 þ ð0:016=3:538Þ2 þ ð0:002=3:538Þ2 þð0:118=3:538Þ2 þ ð0:155=3:538Þ2 þ ð0:124=3:538Þ2 ¼ 0:526:

Fossil fuels 2.471 0.000 2.471 1.248 0.000 1.223 0.544 0.679 0.000

Nuclear 0.652 0.000 0.652 0.652 0.000 0.000 0.000 0.000 0.000

TPE total primary energy, TFE total final energy

Levels Import Production TPE Electricity Network loss TFE Residential Commercial Sum

Solar 0.000 0.016 0.016 0.016 0.000 0.000 0.000 0.000 0.000

Table 3.7 Energy balance for Yokohama City (unit, Mtoe) Wind 0.000 0.002 0.002 0.002 0.000 0.000 0.000 0.000 0.000

Biomass 0.000 0.118 0.118 0.118 0.000 0.000 0.000 0.000 0.000

Geothermal 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Hydro 0.000 0.155 0.155 0.155 0.000 0.000 0.000 0.000 0.000

Waste 0.000 0.124 0.124 0.124 0.000 0.000 0.000 0.000 0.000

Electricity 0.000 0.000 0.000 1.097 0.055 1.042 0.494 0.548 0.000

Sum 3.123 0.415 3.538 1.217 0.055 2.265 1.038 1.227 0.000

56 3 Energy Demand Models

3.1 Simulation Models

57

As a rule of thumb: • • • •

HHI below 0.01 indicates a highly competitive industry. HHI below 0.15 indicates an unconcentrated industry. HHI between 0.15 and 0.25 indicates moderate concentration. HHI above 0.25 indicates high concentration.

The energy system in Yokohama City is highly concentrated on fossil fuel, with an oil dependency about 69.8% (2.471/3.538).

3.1.3.3

Scenario I: Alternative Energy

According to the Yokohama Smart City Project (YSCP), a next-generation energy infrastructure which enables the large-scale introduction of renewable energy is in the works in order to reduce CO2 emissions in this city. The target for the end of FY2014 was the introduction of renewable energy with a total capacity of about 27 MW of medium- and large-sized PV systems. This should make the percentage of power generated by residential PV systems more than 5% of the final energy consumption by households in the demonstration area. In addition to this, an offshore wind turbine capable of producing about 8 MW of energy will accelerate the increase in renewable energy equipment in the area. The other promising opportunity includes solar water heater technology to supply hot water demand for 200,000 dwellings through detecting upper-atmospheric weather conditions and the potential of solar irradiation in this city. Estimating the amount of electricity and heat generation in this scenario can be carried out using the following equations:     h Electricity ðGWhÞ ¼ New Added CapðGWÞ  OF  LF yr

ð3:16Þ

Heat  SWH ðGWhÞ       MJ sqm day ¼ RAD  NDWSWH ðMillionÞ  B0  OFSWH  YSWH ð%Þ sqm:day Dwelling yr =3:6

ð3:17Þ where, in the above equations, OF is the operation factor, LF is the load factor, RAD is the monthly mean solar radiation, NDW indicates the number of dwellings, B0 is the area of solar collector per dwelling, and Y is the thermal efficiency of the solar water heater, respectively. SWH indicates the solar water heater.

3.1.3.4

Scenario II: Energy Efficiency

Accelerating progress to make energy use in the residential and commercial sectors more efficient is indispensable. There is significant scope for adopting more efficient

58

3 Energy Demand Models

technologies in these sectors. The energy efficiency scenario can be defined as the introduction of end-user technologies which are useful in energy demand reduction through the provision of following techniques: – Wall-mounted-occupancy-sensor for lighting (WMOSL) which detects movements of people and automatically turns lights “on” and “off” – With LED lighting – Compact fluorescent lighting (CFL) The basic assumptions in this scenario are represented in Table 3.8. The amount of the energy saving can be calculated, using the following equation:   WF kWh=y   ElecSave ¼ Total Area Coverage ðMillion sqmÞ  WA sqm

ð3:18Þ

Table 3.9 shows the energy saving potential by the combined (alternative energy + energy efficiency) scenario that was considered in this case study. The results demonstrate that energy saving through the implementation of the combined scenario accounted for 2.3% of the total fossil fuel and nuclear power used in Yokohama by promoting the city’s energy performance in both the electricity supply system and at the end-user level. Table 3.10 shows the energy balance table which was recalculated, considering the implementation of the combined scenario in Yokohama.

3.2

Statistical Models

As was discussed in Chap. 2, the purpose of statistical models is to describe and predict the future projections of energy demand in an energy system. To achieve this, we need to collect data in an attempt to describe the characteristics of the projections. In the statistical models, the energy demand is usually estimated with a high-level aggregation of data such as the total and sectoral energy needs. Therefore, the development of energy demand is explained by a macroeconomic variable, and the functional relationship is then estimated by a regression analysis of historical Table 3.8 Basic assumptions in the energy efficiency scenario

WMOSL LED CFL

Coverage area 70% of total floor area in the commercial sector 50% of total floor area in the commercial sector 90% of total floor area in the residential sector

WF saving per each measure per year 45 kWh/y

WA minor lighting coverage area 30 sqm

50 kWh/y

10 sqm

37 kWh/y

10 sqm

3.2 Statistical Models

59

Table 3.9 Comparison between the bassline scenario (before) and the combined scenarios (after)

Unit: Mtoe Fossil fuels Nuclear Solar Wind Biomass Hydro Waste (Elec) Total

Before 2.471 0.652 0.016 0.002 0.118 0.155 0.114 3.538

After 2.371 0.614 0.066 0.006 0.118 0.155 0.114 3.455

Difference 0.099 0.038 0.050 0.004 0.000 0.000 0.000 0.083

observed data. GDP (gross domestic product) is one of the most important macroeconomic variables which represent the market value of the final services produced in a certain period of time in a country. It may be calculated by three different methods as total value added, consumption, and income: GDP ¼ CP þ CG þ IP þ IG þ JSD þ EX  IM

ð3:19Þ

where CP and CG represent the share of private and governmental sections in total GDP. IP and IG are the total investment of private and governmental sections, respectively. JSD is the total wealth storage. EX and IM represent the total imports and exports. The total final energy demand can be estimated using the following formula:  FE ¼ FE 0

GDP GDP0

α 

X X0

β 

Y Y0

γ

ð3:20Þ

In Eq. 3.20, subscript 0 refers to the base year. X and Y are the exogenous determinants of energy demand (e.g., energy price, population, income, etc.). α, β and γ are constant elasticities.

3.2.1

Estimation of Energy Demand in the Middle East Region

The Middle East currently produces 31% and 15% of the total global consumption of crude oil and natural gas, respectively. On the demand side, these countries have traditionally consumed a significant share of their own crude oil and natural gas production, and this share of domestic consumption is increasing sharply, raising questions over the future of crude oil exports from the Middle East. Due to the growing population, higher standards of living, accelerated growth of energyintensive industries, and highly subsidized energy prices, demand for Middle Eastern oil and gas has almost doubled.

Levels Import Production TPE Electricity Network loss TFE Residential Commercial Sum

Fossil fuels 2.371 0.000 2.371 1.175 0.000 1.196 0.517 0.679 0.000

Nuclear 0.614 0.000 0.614 0.614 0.000 0.000 0.000 0.000 0.000

Solar 0.000 0.066 0.066 0.039 0.000 0.027 0.027 0.000 0.000

Wind 0.000 0.006 0.006 0.006 0.000 0.000 0.000 0.000 0.000

Biomass 0.000 0.118 0.118 0.118 0.000 0.000 0.000 0.000 0.000

Table 3.10 Energy balance for Yokohama City after intervention (unit, Mtoe) Geothermal 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Hydro 0.000 0.155 0.155 0.155 0.000 0.000 0.000 0.000 0.000

Waste 0.000 0.124 0.124 0.124 0.000 0.000 0.000 0.000 0.000

Electricity 0.000 0.000 0.000 1.053 0.052 1.001 0.462 0.539 0.000

Sum 2.986 0.469 3.455 1.179 0.052 2.224 1.006 1.218 0.000

60 3 Energy Demand Models

3.3 Optimization Models

61

The following statistical model estimates the energy demand in the Middle East region: E ¼ GDPα Popβ T δ EIρ

ð3:21Þ

where E, GDP, Pop, T, and EI indicate the total energy demand, per capita GDP, population, total trade, and energy intensity, respectively. The model projects the macroeconomics and demographic data to estimate energy demand based on oil products, natural gas, and electricity consumption in the region. Taking the natural logarithms of Eq. (3.21) and adding a random error term produce the following equation: lnE ¼ αlnGDP þ βlnPop þ δlnT þ ρlnEI þ U

ð3:22Þ

where U is the random error term. The dataset contains annual observation over the period 1980–2010. The source of energy demand and intensity data is the BP statistical report for 2010 (BP 2012), while GDP and population are collated from the International Monetary Fund (IMF) and the UN data bank, respectively (IMF 2012; UN 2012). Table 3.11 shows the collated data. By extending a standard OLS regression to find the coefficients in Eq. (3.22), the following results can be obtained (Table 3.12). Therefore, Eq. (3.22) can be rewritten as follows: E ¼ GDP0:64 Pop0:79 T 0:03 EI0:44

ð3:23Þ

For every 1% increase in the population and per capita GDP, energy consumption increased by 0.79% and 0.64%, respectively. Next, for every 1% increase in energy intensity, energy consumption increased by as much as 0.44%. The results reveal that population increase results in more production and consumption activities, which, in turn, raises energy consumption and emissions. The growth in energy intensity has contributed to the higher energy consumption in the Middle East region. Using Eq. 3.23, Fig. 3.2 shows the future projection of energy demand in the Middle East region, assuming an annual average growth rate by 7%, 1.4%, 6.2%, and 0.3% for the per capita GDP, population, trade, and energy intensity, respectively.

3.3

Optimization Models

In a complex system, thousands of units of energy-consuming equipment are in constant operation, and the most critical information for useful energy demand modeling is to identify the following:

62

3 Energy Demand Models

Table 3.11 Collected data for the case of the Middle East region

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Energy demand (Mtoe) 133 142 154 168 189 201 214 223 233 247 259 268 282 296 321 336 349 366 380 392 407 429 448 469 501 540 565 587 634 654 689

Table 3.12 Results of the statistical model

Per capita GDP ($) 3348.46 3583.86 3682.20 3781.86 3903.09 3980.34 3874.18 3870.70 3829.55 4004.07 4188.20 4472.68 4569.62 4657.13 4762.53 4856.22 5133.73 5345.51 5505.40 5577.84 5871.03 6048.60 6242.00 6694.69 6777.50 7257.45 7737.06 8245.22 8662.20 8685.54 8909.24

POP (1000) 83061.00 86476.40 89891.80 93307.20 96722.60 100138.00 103696.00 107254.00 110812.00 114370.00 117928.00 121168.60 124409.20 127649.80 130890.40 134131.00 137308.40 140485.80 143663.20 146840.60 150018.00 153632.20 157246.40 160860.60 164474.80 168089.00 172677.60 177266.20 181854.80 186443.40 191032.00

R2 ¼ 0.999 F ¼ 935721.9 Coefficients Ln GDP 0.637 Ln Pop 0.785 Ln EI 0.438 Ln T 0.029

Trade (Billion $) 212.20 205.40 161.60 125.40 115.50 102.20 71.40 87.30 87.30 111.50 138.40 121.60 133.40 129.40 135.40 151.00 182.70 187.30 144.50 182.30 268.00 239.80 247.80 302.30 400.80 541.20 659.50 766.20 1023.40 708.20 896.40

EI (Mtoe/ Billion $) 0.13 0.13 0.16 0.17 0.19 0.21 0.23 0.25 0.27 0.25 0.25 0.25 0.25 0.23 0.28 0.29 0.28 0.29 0.29 0.27 0.27 0.27 0.28 0.28 0.28 0.29 0.28 0.26 0.27 0.29 0.29

Standard error 0.082 0.053 0.014 0.017

t stat 7.804 14.804 30.860 1.762

3.3 Optimization Models

63

Fig. 3.2 Projection of energy demand in the Middle East region

• • • •

Which equipment or units consume very high energy compared to the standards? What are the reasons for such high-energy consumption? How can the situation be improved upon? What impact does this have on the operating cost?

Models for optimizing energy consumption are developed to reply to the above questions. This type of model is based on the minimization of energy consumption, using the concept of the system integration. The task of system integration in this type of model is to explain the structure and the parameters that both influence the system and interact with the environment. The interactions are represented in the form of relations. The relations are described as the structure of the system. Each energy system consists of subsystems (control volumes), and therefore, the system integration theory is developed to study the relationship between each subsystem and whole system with its environment and the impact of these interactions on the amount of energy consumed by the system. Utilization of system integration becomes possible when the whole system is segregated into subsystems (control volumes). The level of segregation is usually determined by the ability to introduce the fundamental laws of physics, engineering, and economics. The Eulerian approach to each control volume can be represented as follow: Z dx ∂x ¼ þ ρuxdA dt ∂t CV

ð3:24Þ

CS

where x, ρ, u, and A refer to the state variable of the system, density of material following through control volume, the velocity of material following through the control volume, and surface of control volume, respectively. Operation of the control volume may change from level to level. This phenomenon may be due to changes in energy or mass flow variables which represent the state of a control volume.

64

3 Energy Demand Models

Equation (3.24) represents the change in the state of the control volume and should be written as the general form of fundamental laws of physics and technical rules: First law of thermodynamics  X   X  0 0 dEc:v V i2 V o2 ¼ Qc:v þ þ gzi  þ gzo  W c:v m i hi þ mo ho þ dt 2 2 o i ð3:25Þ Mass quantity law X dmc:v X ¼ mi  mo dt o i

ð3:26Þ

In the above equations, Ecv is the state variable of energy flow in the control volume; 0

Qcv is the absorbed heat from environment; m is the state variable of mass flow in the control volume; h is the specific enthalpy of flow; V is the velocity of flow; Z is the height of passage of flow from the ground; and Wcv is the shaft work done on the environment by the control volume. Implementation of system integration in a defined system may be represented by the help of the set of system Eqs. (3.25) and (3.26) for all control volumes of a system, as shown in Fig. 3.3. Therefore, the problem can be formed as an optimization problem through the minimization of total energy consumption, subject to satisfying Eqs. (3.25) and (3.26) and other constraints due to the availability of resources, technology, etc.

System Boundary

Input

Technology

CV(1)

Technology

Technology

CV(2)

Fig. 3.3 The concept of system integration approach

Technology

CV(n)

Output

3.3 Optimization Models

3.3.1

65

Optimal Fuel Consumption in a Gas Turbine Power Plant

Consider a simple open gas turbine cycle with a total installed capacity about 100 kW. (Fig. 3.4). The gas turbine is fed by two fuel types (1 and 2): the quality and cost of these two types of fuel are different. What is the least expensive combination of fuel types 1 and 2 in this system? Assuming an idealized Bryton cycle, the problem can be formulated as the minimization of the total cost of fuel consumption, as follows: Min TC ¼ m31 pNG1 þ m32 pNG2 8 Compressor Control Volume > > < m1  m2 ¼ 0 m hT 1 , P1 þ E 6  m2 hT2 2 , P2  E 7  E cw ¼ 0 > > : 1 1 E CW  E6 ð1  ηC Þ ¼ 0

8 Combustion Control Volume > > < m3  m31  m32 ¼ 0 m þ m  m4 ¼ 0 > > : 2 T 2 , P32 m2 h2 þ m3 hT3 3 , P3 þ m31 Ecc1 þ m32 E cc2  m4 hT4 4 , P4 ¼ 0 8 Turbine Control Volume > > > > < m4  m5 ¼ 0 m4 hT4 4 , P4  m5 hT5 5 , P5  E 6  E TW ¼ 0 ¼ 0 > > ð1  η T Þ > > E TW  E 6 : ηT 8 < Stack Control Volume m5  m8 ¼ 0 : m5 hT5 5 , P5  m8 hT8 8 , P8  ESW ¼ 0 8 < Generator Control Volume E  EU  EGW ¼ 0 : 7 EGW  E 7 ð1  ηG Þ ¼ 0

ð3:27Þ ð3:28Þ

ð3:29Þ

ð3:30Þ

ð3:31Þ

ð3:32Þ

The input data used in above model is reported in Table 3.13. TC is the total cost of fuel consumption. The revised simplex method can be used to find the optimal solution of the above linear system. The results are shown in Table 3.14. Based on the model result, fuel type 2 is selected as the best option. Although the price of fuel type 2 is higher than fuel type 1, its higher quality will result in lowering the rate of fuel consumption in the proposed power cycle. The energy balance of the gas turbine system based on the optimal results is shown in Fig. 3.5.

66

3 Energy Demand Models

Fig. 3.4 Gas turbine schema

What If the Price of Fuel Type 2 Increases? The result of the sensitivity analysis on the price of fuel type 2 is shown in Table 3.15. As can be observed from the table, fuel type 1 will be selected once the price of fuel type 2 exceeds 76 $/t.

3.3.2

Optimal Energy Demand in a Cement Factory

A simple process diagram of the cement production is shown in Fig. 3.6. In this simple example, we try to find the optimal rate of energy consumption which results in minimum total operation cost of the system. To achieve this, an optimization model can be developed by introducing the following constraints: 1. Demand inequality M 5 ηL  M 6

ð3:33Þ

8 M 1 ηC  M 2 ¼ 0 > > < M 2 ηR  M 3 ¼ 0 M > 3 ηK  M 4 ¼ 0 > : M 4 ηB  M 5 ¼ 0

ð3:34Þ

2. Mass balance

3.3 Optimization Models

67

Table 3.13 Input data for the illustrative example gas turbine EU PNG1 PNG2 ηC ηT ηG h1 h2 h3 h4 h5 h8 Ecc1 Ecc2

Electric power (100 kW) Fuel price1 Fuel price2 Mechanical efficiency of compressor Mechanical efficiency of turbine Mechanical efficiency of generator Enthalpy of the flow stream 1 Enthalpy of the flow stream 2 Enthalpy of the flow stream 3 Enthalpy of the flow stream 4 Enthalpy of the flow stream 5 Enthalpy of the flow stream 8 Net calorific value of fuel1 Net calorific value of fuel2

Table 3.14 Optimal value of the key variables

TC m1 m2 m31 m32 m4 m5 m8 E6 E7

Fig. 3.5 Gas turbine optimal energy balance

MJ/h $/t $/t % % % MJ/t MJ/t MJ/t MJ/t MJ/t MJ/t MJ/t MJ/t

$/h t/h t/h t/h t/h t/h t/h t/h MJ/h MJ/h

360 63 70 0.8 0.8 0.95 25 142 33 1080 400 300 46,000 56,000

1.390 1.164 1.164 0.00 0.020 1.184 1.184 1.184 643.881 378.947

68

3 Energy Demand Models

Table 3.15 Sensitivity analysis on price of fuel type 2 PNG2 TC m31 m32

70 1.390 0 0.020

71 1.410 0 0.020

72 1.430 0 0.020

73 1.450 0 0.020

74 1.470 0 0.020

75 1.490 0 0.020

76 1.510 0 0.020

77 1.522 0.024 0

Fig. 3.6 A simple representation of the cement production

3. Resource availability M1  Ht þ Y t

ð3:35Þ

where, in the above equations, M, η, H, and Y refer to the mass flow rate, efficiency, historical capacity at the starting point, and the new capacity developed during the time period, respectively. Subscripts L, C, K, R, and B indicate the loading, crushing, kiln furnace, and raw ball mill and cement ball mill. t refers to the time period. 4. Crusher electrical energy consumption The bond correlation is used to calculate electrical energy consumption based on product size and the Grindability index as follows (Peray 1979): 16 W¼ Gbp0:82

rffiffiffiffiffiffiffiffi P 100

ð3:36Þ

where W is the electricity demand, Gbp refers to the Grindability index, and P is the product size (μm). For a typical cement factory, the electricity consumption of the crusher varies from 1.5 to 1.7 kWh/t.

3.3 Optimization Models

69

5. Ball mill electrical energy consumption The ball mill is designed for grinding of clinker, gypsum, and dry or moist additives to any type of cement, using the electric power which is represented by the following formula (Peray 1979): N ¼ C  G  Di  n

ð3:37Þ

where N is the electricity load. C is the dynamic coefficient which can be calculated as follows: C¼

2πY sin α 6:12Di

ð3:38Þ

G is the weight of material (ton); Di is the mill effective inner diameter (m); α is the angle of repose; Y is the distance from the center mill to the powder gravity (m); and 42:5 ffiffiffiffi). finally, n is the critical mill speed which is measured from 0.7 to 0.75 rpm (n0 ¼ p Di For a typical cement factory (Table 3.16) 6. Rotary kiln energy consumption Energy consumption in a dry rotary kiln with the preheater is divided into two categories: Thermal energy (heat): 2.9–3.2 GJ/t Electrical energy: 20–26 kWh/t 7. Loading electrical energy consumption Loading belongs to the low energy intensity section of the factory. Its electrical energy consumption is around 5 kWh/t. 8. Labor Requirement Labor distribution used in this example is given in Table 3.17. And finally, the plant factor is considered to be about 7000 h/yr. 9. Objective function The total operation cost of the system can be defined as the total costs spent on raw material, energy, and wages:

Table 3.16 Technical parameter to calculate the critical mill speed

Raw material ball mill Cement mill

Di 2.93 2.71

n 16.8 23.2

c 0.44 0.91

70

3 Energy Demand Models

Table 3.17 Labor distribution used in the cement factory example Unit Crusher Raw material ball mill Rotary kiln Cement mill Loading

Labor (person/ton/h) 2 1 2.4 1 4

C t ¼ M Rt r t þ

X τ

Eτt eτt þ

X τ

Average wage ($/person.h) 20 20 20 20 20

Lτt wτt þ M f P f

ð3:39Þ

Ct is the annual operation cost; MRt is the total raw material consumed in a year; rt is the unit price of 1 ton raw material consumed in a year; Eτt is the energy consumption; eτt is the unit price of electricity; Lτt is the number of labors; and wτt represents the wage of labors working at different sections of the factory. Mf and Pf are fuel consumption and fuel price. τ refers to different technologies, like crushing, etc. Table 3.18 shows the underlying assumptions considered in this example. Using above technical information, the problem can be formulated as follows: Min C ¼ ð0:1M  1Þ þ  0:04ðE C þ E R þ E K þ EB þ E L Þ þ 63M f þ 7000  20 ðLC þ LR þ LK þ LB þ LL Þ Subject to Demand inequality 0:9M 5  1:0  106

Mass Balances 0:6M 1  M 2 ¼ 0 0:9M 2  M 3 ¼ 0 0:9M 3  M 4 ¼ 0 0:9M 4  M 5 ¼ 0

Resources Bound     M 1  5  106 þ 2  106

3.3 Optimization Models Table 3.18 Basic assumptions in the cement factory example

71

Annual cement production Plant factor Raw mat. price Fuel price Electricity price NG heating value Historical capacity New capacity

1 million tons 7000 h/year 0.1 $/t 63 $/t 0.04 $/kWh 46 GJ/t 5 million tons 2 million tons

Energy Balances 1:7M 1  EC ¼ 0 22M 2  ER ¼ 0 for ðD ¼ 2:93; n ¼ 16:8; c ¼ 0:44Þ 26M 3  EK ¼ 0 57M 4  EB ¼ 0 for ðD ¼ 2:71; n ¼ 23:2; c ¼ 0:91Þ 5M5  E L ¼ 0 46M f  3:2M 3 ¼ 0

Labor Requirement LC  2M 1 =7000 ¼ 0 LR  M 2 =7000 ¼ 0 LK  2:4M 3 =7000 ¼ 0 LB  M 4 =7000 ¼ 0 LL  4M 5 =7000 ¼ 0 By using the linear programming approach, the optimal solution can be found in Table 3.19. Figure 3.7 shows the optimal energy and mass balances for this example. We can also adopt the perspective of profit maximization by introducing a new object function for this problem: ( Max

P t ¼ M 5 pc 

M Rt r t þ

X τ

Eτt eτt þ

X τ

) Lτt wτt þ M ft Pft

where pc is the cement price in the market. So, we want to find the optimal rate of energy consumption which results in maximum total profit from the cement production. The results of both cost minimization and profit maximization problems are given in Table 3.20. It can be observed from the above table that the cost minimization problem will result in minimum rate of cement production, whereas the profit maximization problem will result in maximum rate of raw material consumption.

72

3 Energy Demand Models

Table 3.19 Optimal results obtained for the cement factory example

Annual operational cost Fuel intensity Raw material intensity Electricity intensity

61.2 M$ 0.09 t/t 2.5 t/t 150 kWh/t

Fig. 3.7 Optimal mass and energy balances in the cement factory example Table 3.20 Comparison of the optimal results obtained from the cost minimization and the profit maximization

Production (million tons) Total fuel consumption (1000 ton) Total electricity consumption (TWh) Total raw material consumption (million tons)

3.3.3

Cost minimization problem 1 95.5 149.4 2.5

Profit maximization problem 2.7 at pc ¼ 60$/ton 263.0 411.8 7

Optimal Energy Demand in a Building

In this example, a simple energy demand model for a building is presented. The full energy balance equation for a steady-state building energy model looks like this: QSolar þ QHeater þ QP ¼ QVent þ QLoss

ð3:40Þ

where QSolar, QHeater, QVent, QLoss, and QP are the solar heat gain by the building, thermal heating load provided by heater system, ventilation heat loss, heat loss from walls and windows, and metabolic heat gain, respectively. The above calculation is carried out to estimate the required fuel consumption for heating to maintain the required indoor climate conditions in the building. Solar heat gain is the amount of heat which is absorbed and passes through the windows, as follows (Clarke 2001):

3.3 Optimization Models

73

QSolar ¼ AG :SC:ESM:SF

ð3:41Þ

The heat loss due to ventilation is a function of the minimum airflow rate of a heated space required for hygienic reasons, Vinf, (Ashrae 2016): QVent ¼ 0:33:V inf ΔT

ð3:42Þ

And finally, the heat loss from walls, ceiling, floor, and windows can be calculated using the equation below: Qloss ¼ Aw U w ΔT

ð3:43Þ

In the above equations, AG is the glazing area of the windows (m2); SC is the shading coefficient of the glazing; ESM is the external shading multiplier, and SF is the solar factor (Wm2); ΔT is the indoor and outdoor temperature difference; Aw is the area represented by the walls, ceiling, floor, and windows (m2); and Uw is the U-values (Wm2 K1), respectively. The main decision variables in the building energy demand model are defined as the indoor temperature and the minimum airflow rate which indicates the thermal comfort condition in the building. The thermal conform can be satisfied by applying the following conditions: T iL  T i  T iU

ð3:44Þ

V inf  V inf , L

ð3:45Þ

Subscripts L and U refer to the lower and upper limits. The model of optimal energy demand in the building can then be written as follows: Min m f p f s:t 8 QHeater þ QSolar þ QP  QVent  QLoss  0 > > > > > QSolar  AWin :SC:ESM:SF ¼ 0 > > > > > QVent  0:34:V inf ðT i  T o Þ ¼ 0 > >  > < Q   A U Wall Wall þ AWin U Win þ ADoor U Door þ ACeiling U Ceiling þ AFloor U Floor ðT i  T o Þ ¼ 0 loss p ¼ 0 > Qp  n p Q > > > > > m f NCV  QHeater ¼ 0 > > > > > T iL  T i  T iU > > : V inf  V inf , L

The basic assumptions considered to solve the above linear system are given in Table 3.21. The ground plan of the building is shown in Fig. 3.8. The optimal results are given in Table 3.22. As can be observed from Table 3.22, the use of the fuel would be about 0.087 m3/h to maintain the required indoor climate conditions. The optimal

74

3 Energy Demand Models

Table 3.21 Basic assumption in the building example Building plan Wall Door Window #1 Window #2 Ceiling Floor Shading coefficient Solar factor External shading multiplier Indoor temp (lower limit) Indoor temp (upper limit) The minimum airflow rate Outdoor temp. Dwellers Average body extracted heat Fuel (NG) hating value Fuel price

Area (m2) 45 1 2 2 24 24 0.7 190W m2 1 20  C 27  C 20 m3/h 6 C 4 people 20 W/person 10 kWh/m3 1.5 $/m3

U-value (W m2 K-1) 1.6 1.3 0.45 0.45 0.3 0.7

Fig. 3.8 Ground plan of the building

conditions are satisfied when the indoor temperature and airflow rate are set at the lowest allowance level.

3.3.4

Minimum Energy of a Compression Unit

In this example, a two-stage reciprocating compressor is considered. The objective is to determine the minimum electrical energy required to pressurize air to a specific value above the ambient pressure, as shown in Fig. 3.9.

3.3 Optimization Models Table 3.22 Optimal results obtained for the building example

75

Fuel cost Fuel consumption Heat loss from walls and windows Solar gain Ventilation heat loss Metabolic heat gain Thermal heating load demand Optimum indoor temp. Optimum airflow rate

0.13 $/h 0.087 m3/h 1387.4 W 532 W 92.4 W 80 W 867.8 W 20  C 20 m3/h

Fig. 3.9 Two-stage reciprocating compressor with intercooler

If the gas is assumed to be ideal, the minimum theoretical work per mass of this compressor can be represented as follows:

Min

γRT 1 W¼ γ1

"  γ1 #  ðγ1 P2 ð γ Þ P3 γ Þ þ 2 P1 P2

ð3:46Þ

where R and γ represent the gas constant and capacity ratio, respectively. The above equation represents a nonlinear system. Suppose the following given data: T1 ¼ 298 K P1 ¼ 100 kPa P3 ¼ 4000 kPa γ ¼ 1.4 and R ¼ 8.314 J/mol. K Application of the gradient method to minimize W as a function of P2, starting with an initial value of 100 for P2 yields: P2 ¼ 632.45 kPa and W ¼ 12.02 KJ/mole.

3.3.5

Thermal Efficiency Maximization in a Heat Recovery System

This example shows how the optimization technique can be applied to a more complicated energy system. In this case, the system is the open indirect cycle of a

76

3 Energy Demand Models

External Heat Source 8

9

Heat Exchanger

0 4 G

C1

5 3

2 Expander

C2 1

10

7

Fig. 3.10 The open indirect cycle of gas turbine used for heat recovery from a nuclear reactor (Adopted from Farzaneh 2010)

gas turbine. Here, instead of minimizing the energy consumption of the system, we try to maximize the overall thermal efficiency of the system, which is a key operating parameter in thermal energy systems. The open indirect cycle of the gas turbine is almost the same as a simple Brayton cycle, but it can be used for the purpose of heat recovery in different industries. It is especially useful for recovering energy in nuclear power plants (see Fig. 3.10). In the above figure, fresh air is compressed through two stages of compression as a working fluid in the cycle (0–2). Compressed air absorbs the thermal energy of the coolant of the nuclear reactor in the heat exchanger (2–3). The compressed air is expanded through the turbine, and electricity is generated in the generator (3–6). Total thermal efficiency in the Brayton cycle may be defined as the ratio of net output power to total input or absorbed heat, as follows: ηth ¼

W net Qadd

ð3:47Þ

W net ¼ m0 Cp fðT 3  T 5 Þ  ðT 2  T 4 Þ  ðT 1  T 0 Þg

ð3:48Þ

Qadd ¼ m0 Cp ðT 3  T 2 Þ

ð3:49Þ

Therefore ηth ¼

ðT 3  T 5 Þ  ðT 2  T 4 Þ  ðT 1  T 0 Þ ðT 3  T 2 Þ

ð3:50Þ

3.3 Optimization Models

77

Wnet is the network produced by the cycle, and Qadd is the thermal energy absorbed by the compressed air in the heat exchanger. For the compression stage, we have T1  T0 ¼

T0 ðCc1  1Þ ηc1

ð3:51Þ

T2  T4 ¼

T2 ðCc2  1Þ ηc2

ð3:52Þ

And γ

γ

ðP0  βÞCc1 γ1  ðΔPint þ βÞCc2 γ1 Ccr ¼ P0

ð3:53Þ

where ηc, Cc, ΔPint, Ccr, β, and rc refer to the efficiency of the compressor, the dimensionless compression number, the total pressure drop in the intercooler, the total dimensionless compression number, the pressure drop in compressor, and compression ratio, respectively. When compressed flow enters the turbine, the pressure of that air decreases. It is clear that the outlet pressure of the turbine should never reach atmospheric pressure. The dimensionless parameter is introduced to control this ratio: L¼

P5 P0

ð3:54Þ γ1

C t ¼ ð r t LÞ γ  γ1  γ1 P3 γ P0 γ x¼ ¼ P2 P5   1 T 3  T 5 ¼ T 3 ηt 1  C t :x

ð3:55Þ ð3:56Þ ð3:57Þ

ηt is the turbine efficiency. Ct and x are dimensionless expansion number and dimensionless pressure ratio. According to the principles in the design of the gas turbine, the maximum power which may be obtained from the turbine is strictly affected by the turbine inlet temperature (T3). This parameter is affected by the amount of recoverable heat through the use of the heat exchanger. The dimensionless cycle temperature ratio (t) represents the ratio of maximum turbine inlet temperature to ambient temperature:

78

3 Energy Demand Models

t¼

T3 T0

ð3:58Þ

T 3 ¼ ð1  εH ÞT 2 þ εH T 8

ð3:59Þ

T 4 ¼ ð1  εint ÞT 1 þ εint T 7   Cc2  1 T2 ¼ T4 1 þ ηc2   Cc2  1 t ¼ ð1  εH Þð1  εint ÞCc1 1 þ ηc2     Cc2  1 T 7 T8 þ ð1  εH Þεint 1 þ þ εH ηc2 T0 T0

ð3:60Þ ð3:61Þ

ð3:62Þ

In the above equations, εH and εint refer to the recuperator and intercooler effectiveness, respectively. The total net output power can be determined from the following equation: W net ¼ m0 C p T 0 DSW       1 Cc2  1 Cc2  1 T 7 DSW ¼ tηt 1   εint  ð1  εint ÞCc1 C t :x ηc2 ηc2 T0

ð3:63Þ ð3:64Þ

where DSW is defined as dimensionless-specific work of cycle. Finally, the thermal efficiency of the open indirect cycle of the gas turbine may be represented using the following equation:





 tηt 1  C1t :x  ð1  εint ÞCc1 Ccη2 1  εint Ccη2 1 TT 70  Ccη1 1 h

c2i h c2  i c1 ηth ¼ t  ð1  εint ÞCc1 1 þ Ccη2 1  εint 1 þ Ccη2 1 TT 70 c2

ð3:65Þ

c2

This equation reveals that dimensionless numbers, such as the cycle temperature ratio, the total compression ratio, and the cycle pressure ratio, together with the efficiency of compressors, the effectiveness coefficients of the intercooler, and the recuperator, influence the thermal efficiency of the open indirect cycle of the gas turbine by using it as a heat recovery system. The above model can be formulated as a constrained optimization problem with a nonlinear objective function and constraints. In this example, the decision variables can be categorized as follows: 1. Cc and x are defined as independent variables. 2. t, Ct, ηc, and ηt are defined as first-order dependent variables. 3. Temperatures, pressures, and the flow rate of the working fluid are defined as second-order dependent variables.

3.3 Optimization Models

79

The limitations of Cc and x are included according to the design condition: 1:1 < Cc < 2 x 2.5). Small hydropower is playing an essential role in the competitive power supply sector in Delhi. It can

126

5 Climate Change Multiple Impact Assessment Models

Table 5.13 Policy options and assumptions for scenario generation in Delhi Scenario Alternative energy usage (Shift) End-user efficiency improvement (Improve)

Policy option 1) Introducing 50 MW solar PV 2) Increasing the installed capacity of hydro up to 100 MW 3) Increasing the installed capacity of waste-electricity to 46 MW 1) Replacing regular lighting system with compact fluorescent lighting for 3 million Delhi households 2) Improving COP of air conditioning to 2.7

provide sustainable energy services, based on the use of routinely available, indigenous resources, and provide better solutions to the long-standing energy problems encountered by the power supply system. Solid waste management remains one of the most neglected sectors in Delhi. On average, 80% of the municipal solid waste generated is collected: 90% of the collected solid waste is disposed of in landfills, and the remainder is composted. At a total of 5500 tons per day, the contribution of municipal solid waste to the total CH4 emission is estimated at approximately 80%. The implementation of technologies like incinerators would facilitate electricity production on the supply side and reduce the emission of GHG and other air pollutants in this city. Table 5.12 shows the potential of the calculated co-benefits of the implementation of the policy intervention scenarios considered in this survey (Table 5.14). Figure 5.14 shows the potential reduction of GHG emissions by each scenario in Table 5.13. Figure 5.15 shows the calculated MAC (marginal abatement cost) based on the midterm payback period by considering the following definition:
MAC

$ tCO2

 ¼

Total capital investment  Present value of the project ð5:9Þ Comulative GHG saving over the life of the project

The project total capital investment is estimated at 1500 million USD. It should be noted that the cost values derived in this survey are based on theoretical ideas which provide indicative potential value: these may differ from the actual field measurement value due to the number of factors which influence the capital and operational costs of different electricity generation technologies. Figure 5.15 also clearly indicates that improving energy efficiency at the end-user levels results in a rapid, cost-effective decrease in emissions. The cheapest strategy for managing the electrical load and reducing emissions in the domestic sector is making improvements to the lighting. The results show that the co-benefit policies which promote low-carbon communities have a GHG emission reduction potential of about 3.67 Mt/y. The proposed intervention scenarios help to motivate both the power supply system and the energy consumers to participate in reducing the peak demand in the system. The power supply system in Delhi also needs to be reformed through the deployment of renewable energy sources (mainly hydropower) and by introducing a system to generate electricity from municipal waste.

5.5 Assessing the Climate Co-benefits in the Power Sector Table 5.14 Potential reduction in GHG emissions and air pollutants (kt/y)

GHG CO NOx SO2 PM10 PM 2.5

Baseline 15658.3 29.00 46.91 123.90 5.74 1.80

127 After intervention 11982.7 19.68 32.61 84.74 3.96 1.22

Difference 3675.6 9.32 14.30 39.16 1.78 0.58

Fig. 5.14 The potential reduction of GHG emissions for the selected scenarios

Fig. 5.15 Midterm marginal cost abatement curve for policy intervention scenarios in 2030 (based on the average electricity price of about 6.5 Rs/kWh and the annual discount rate, at about 5.5%)

128

5 Climate Change Multiple Impact Assessment Models

References Doll CNH, Balaban O (2013) A methodology for evaluating environmental co-benefits in the transport sector: application to the Delhi metro. J Clean Prod 58:61–73 EEA, European Environmental Agency, EMEP/EEA air pollutant emission inventory guidebook (2013) Technical guidance to prepare national emission inventories, Group 7 road transport, 19 Apr 2016, ISBN 978–92–9213-403-7, Copenhagen, Denmark. http://www.eea.europa.eu/ publications/EMEPCORINAIR/page016.html Farzaneh H (2017a) Multiple benefits assessment of the clean energy development in Asian cities. Energy Procedia 136:8–14 Farzaneh H (2017b) Development of a bottom-up technology assessment model for assessing the low carbon energy scenarios in the urban system. Energy Procedia 107(2017):321–326 Farzaneh H, Doll CNH (2014) Guidebook for the Co-benefits evaluation tool for the urban energy system, The United Nations University, Institute for the advanced study of sustainability. http:// tools.ias.unu.edu/sites/default/files/manual/Energy_Evaluation_Tool_Guidebook.pdf. Accessed 01 July 2017 Farzaneh H, Puppim de Oliveira JA, Doll CNH, Suwa A, Dashti M (2014a) The co-benefits of energy efficiency policy to manage the electric load in Delhi, ECSEE 2014, Oral Presentation, UK, Brighton Farzaneh H, Suwa A, Doll CNH, Puppim de Oliveira JA (2014b) Developing a tool to analyze climate Co-benefits of the urban energy system. Procedia Environ Sci 20:97–105 Fischedick M, Roy J, Abdel-Aziz A, Acquaye A, Allwood JM, Ceron J-P, Geng Y, Kheshgi H, Lanza A, Perczyk D, Price L, Santalla E, Sheinbaum C, Tanaka K (2014) Industry. In: Edenhofer O, Pichs-Madruga R, Sokona Y, Farahani E, Kadner S, Seyboth K, Adler A, Baum I, Brunner S, Eickemeier P, Kriemann B, Savolainen J, Schlömer S, von Stechow C, Zwickel T, Minx JC (eds) Climate change 2014: mitigation of climate change. Contribution of working group III to the fifth assessment report of the intergovernmental panel on climate change. Cambridge University Press, Cambridge, UK/New York Gesellschaft fur D (2019) Sustainable urban transport: Avoid-Shift-Improve (A-S-I), division 44 water, energy, transport, international Zusammenarbeit(GIZ) Gmbh.. https://www.sutp.org/ files/contents/documents/resources/E_Fact-Sheets-and-Policy-Briefs/SUTP_GIZ_FS_AvoidShift-Improve_EN.pdf. Accessed 01 Jan 2019 Heather A (2013) An integrated approach to public transport, Tehran Islamic Republic of Iran, Case study for global report on Human settlements 2013. Available from http://www.unhabitat.org/ grhs/2013. Accessed 01 Apr 2017 Hyslop A (2006) Co-benefits of municipal climate change mitigation strategies, Hamilton, University of Waterloo, Canada. http://citeseerx.ist.psu.edu/viewdoc/download?doi¼10.1.1.488.268& rep¼rep1&type¼pdf. Accessed 01 Apr 2017 International Energy Agency, IEA (2014) World energy outlook 2014, ISBN: 978–92–6420805–6, France Kakouei A, Vatani A, Kamal Bin Idris A (2012) An estimation of traffic related CO2 emissions from motor vehicles in the capital city of Iran. Iranian J Environ Health Sci Eng 9(1):13 Karimzadegan H, Rahmatian M, Farhud DD, Yunesian H (2008) Economic valuation of air pollution health impacts in the Tehran Area, Iran. Iranian J Publ Health 37(1):20–30 Ministry of the Environment Japan (MOEJ) (2009) Co-benefits approach – development needsoriented efforts to address climate change and CDM. Overseas Environmental Cooperation Center (OECC), Japan Ostro B (2004) Outdoor air pollution, assessing the environmental burden of disease at national and local levels, World Health Organization Protection of the Human Environment, ISBN 9241591263, Geneva Saboohi Y, Farzaneh H (2009) Model for developing eco-driving strategy of a passenger vehicle. Appl Energy 86(10):1925–1932

References

129

Schipper L, Marie-Liliu M (1999) Flexing the link between transport greenhouse gas emissions: a path for the World Bank. International Energy Agency, Paris. http://documents.worldbank.org/ curated/en/826921468766156728/pdf/multi-page.pdf Accessed 01 April 2017 SCI, Statistical Center of Iran (2013) Iran statistical yearbook 2011, Presidency wise for strategic planning and supervision, Tehran, Iran. http://istmat.info/files/uploads/44179/iran_statistical_ yearbook_2010-2011_1389.pdf. Accessed 01 Apr 2017 TAQCC, Tehran Air Quality Control Center (2013) http://air.tehran.ir/Default.aspx?tabid=191. Accessed 01 April 2017 TAQCC, Tehran Air Quality Control Center (2016) http://air.tehran.ir/Default.aspx?tabid¼191. Accessed 01 Apr 2017 TSYB, Tehran Statistical Yearbook 2011–2012 (in Persian). http://www.tehran.ir/portals/0/ flipbook/tehran-statistical-yearbook-2011-2012/index.html. Accessed 01 Apr 2017 TTTOD, Tehran Traffic Transportation Organization and Deputy, Deputy of planning and studies (2013) An overview of Tehran transportation master plan (revised in 2013), Tehran, Iran. http:// www.iran.uitp.org/sites/default/files/documents/MasterPlanEnglishBook-old_Compressed.pdf. Accessed 01 Apr 2017 UN, Department of Economic and Social Affairs (2014) World urbanization prospects. ISBN 97892-1-151517-6, NY, US,. https://esa.un.org/unpd/wup/Publications/Files/WUP2014-High lights.pdf. Accessed 01 April 2017 UNU-DESA, United Nations Department of Economic and Social Affairs. http://www.un.org/en/ development/desa/population/theme/sdg/index.shtml. Accessed 01 Apr 2017

Chapter 6

Optimal Control of Energy Systems

Contents 6.1 What Is Optimal Control Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Optimal Fuel Consumption in a Passenger Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Optimal Power Control Strategy in a Hybrid Renewable Energy System (HRES) . . . . . 6.3.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Case of Fukushima Prefecture in Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

131 133 136 138 143 146

What Is Optimal Control Theory?

Consider you are driving a car from your home to your office and you want to reach your destination in the least possible time. Here, the system includes both the car and the route between your home and office. The question is, how should you press the accelerator pedal and change the gears in order to minimize the total traveling time? The optimality criterion can be defined as the minimization of the total traveling time, through controlling the speed and shifting the gears, subject to satisfying limited numbers of constraints such as availability of the fuel and traffic conditions. This problem formulation usually represents a controlled dynamic system which can be explained by the optimal control theory (Hocking 1991). Optimal control theory explains the dynamic systems in which the evolution trajectory can be rectified incessantly in time by choosing the value of a control parameter, like the accelerator pedal or the gears in the car example. A controlled dynamical system is usually governed by an ordinary differential equation of the form


x_ ðt Þ ¼ f ðxðt ÞÞ x ð 0Þ ¼ x 0

t>0

ð6:1Þ

The initial point is given as x0 2 Rn. f represents the dynamical evolution of the state of the system. In a controlled dynamics problem, f depends on some control parameters like ρ belonging to a set A  Rn, so that the corresponding controlled dynamics can be expressed as follows: © Springer Nature Singapore Pte Ltd. 2019 H. Farzaneh, Energy Systems Modeling, https://doi.org/10.1007/978-981-13-6221-7_6

131

132

6 Optimal Control of Energy Systems

Fig. 6.1 The evolution trajectory in the controlled system dynamic




x_ ðt Þ ¼ f ðxðt Þ; ρÞ x ð 0Þ ¼ x 0

t>0

ð6:2Þ

and 8 a1 > > > > a > 2 > < : ρðt Þ ¼ : > > > > : > > : an

0  t  t1 t1  t  t2 ð6:3Þ t n1  t  t n

Figure 6.1 illustrates the resulting evolution. The solution x() of Eq. 6.2 depends upon ρ() and the initial condition   xðÞ ¼ x ; ρðÞ; x0

ð6:4Þ

To find the best control for the system, a specific payoff criterion may be introduced as follows: ZT P½ρðÞ≔

Rðxðt Þ; ρðt ÞÞdt þ QðxðT ÞÞ

ð6:5Þ

0

where R is the running payoff and Q is the terminal payoff at a given terminal time T > 0. The optimal control problem is defined to find a control ρ*() which maximizes the payoff. In other words, we want

6.2 Optimal Fuel Consumption in a Passenger Vehicle

133

P½ρ ðÞ  P½ρðÞ

ð6:6Þ

For all controls ρ() E A, such a control ρ*() is called optimal.

6.2

Optimal Fuel Consumption in a Passenger Vehicle

Let’s use our previous car traveling example to answer this question: “How should the driver press the accelerator pedal and shifts gears in order to travel with least fuel consumption in a given traffic system?” We can formulate the above question in a form of an optimal control problem, using Eq. (6.2):


  m_ f ðt Þ ¼ f m f ðt Þ; PT ðt Þ m f ð 0Þ ¼ m f 0

t>0

ð6:7Þ

where mf refers to the fuel consumption rate in the passenger vehicle. The traction power of the vehicle, PT, is then the control variable which controls the level of energy flow (fuel mass flow) as a state variable of the model (Farzaneh and Saboohi 2007). The traction power can be defined as a function of the traction force, FT, and the speed of the vehicle, vt. Since FTt ¼ f(vt)


mf_ ðt Þ ¼ f ðmf ðt Þ; vðt ÞÞ mf ð0Þ ¼ mf 0

t>0

ð6:8Þ

The relationship between speed and the selection of the gear ratio may be identified through the basic dynamic equation of movement of the vehicle (Saboohi and Farzaneh 2009): KM

dv ¼ FT  FR dt η τi FT ¼ T r

ð6:9Þ ð6:10Þ

E Tt ¼ F T: xt:

ð6:11Þ

ETt NCV

ð6:12Þ

mft ¼

where M, FT, FR, and i are the mass of the vehicle, the traction force, the motion resistance forces, and the gear ratio, respectively. K is the weight factor. ETt is the traction energy (load), and NCV is the calorific heat value of the fuel.

134

6 Optimal Control of Energy Systems

The maximum speed of the vehicle can then be obtained through solving the equilibrium relationship based on the main traffic characteristics with the help of the following set of equations (Saboohi and Farzaneh 2008; Wu 2002): Fluid traffic θob  θco "

"

V T ¼ v0  ðv0  vc0 Þ

θob θco

n1 ## ð6:13Þ

Traffic jam θob  θjg   3600 1 1 VT ¼  τgo θob θmax

ð6:14Þ

Transient traffic θjg  θob  θco V T ¼ ðV T Fluid ζ Þ þ ðV T Jam ð1  ζ ÞÞ ζ¼

θco  θ jg θbo  θ jg

ð6:15Þ ð6:16Þ

where θob, θco, n, tco, tgo, θjg, and θmax are defined as the observed vehicle density at a defined period in time estimated from statistical reports, the vehicle density of the particular traffic state, the number of traffic lanes, the mean net time headway, the mean net gap, the vehicle density of the jam state in go position, and the maximum vehicle density. VT is the possible maximum speed in a specific state of the traffic flow. vt  V T

ð6:17Þ

Maintaining the optimal speed and gearing in a certain traffic system can be identified as the driving strategy which leads to the least fuel consumption. The search for the optimal combination of the state and control variables of the vehicle system to minimize the fuel consumption is addressed by the following optimality criterion: ZT Min F ¼

m f ðvÞdt 0

With the following initial and boundary conditions:

ð6:18Þ

6.2 Optimal Fuel Consumption in a Passenger Vehicle

135

x ð 0Þ ¼ x 0

vð0Þ ¼ v0

ð6:19Þ

xð T Þ ¼ x T

vðT Þ ¼ vT

ð6:20Þ

The application of the above model is demonstrated with the help of the traffic system represented in Fig. 6.2. The optimal results obtained in an intense combined traffic flow are described in Fig. 6.3.

Fig. 6.2 Given traffic conditions for the car traveling example (car net weight, 1.2 ton; drag coefficient, 0.3)

Fig. 6.3 Optimal fuel efficiency and speed profile in the car traveling example

136

6.3

6 Optimal Control of Energy Systems

Optimal Power Control Strategy in a Hybrid Renewable Energy System (HRES)

The hybrid renewable energy systems (HRES) are stand-alone power supply systems which combine two or more renewable technologies which operate with higher efficiency than each individual system. If the production of energy during extended periods is not guaranteed by the proposed HRES, energy storage requirements should be considered to bridge the lean times. A possible solution consists of adding a hydrogen storage chain, including a proton-exchange membrane fuel cell (PEM FC), a hydrogen producer, and a hydrogen storage tank. For the production of hydrogen, diverse methods and sources are available and are currently in use. Electrolysis, steam reforming of coal and natural gas, fermentation, and thermochemical and photochemical water splitting are among those methods which can be employed for hydrogen production in an HRES configuration. Water electrolysis using photovoltaic electricity is the most practical for the HRES purposes. Hydrogen can also be generated from waste biomass via thermochemical or biochemical conversion routes. Gasification is one of the popular options which can provide a broad range of products. However, one of the major problems in the gasification process is the formation of tar and ash that can conduct deposition, sintering, slagging, fouling, and agglomeration. The supercritical water gasification (SCWG) process is an alternative to the conventional gasification which uses wet waste, wet biomass, and aqueous-sludge destruction. This process uses water over its critical point (22 MPa and 374 C) as the gasifying agent (instead of air or steam) which heavily benefits the decomposition of the biomass allowing to achieve very high gasification ratios and hydrogen volumetric ratios. The progress in processing and reactor design of the SCWG dates back to the late 1970s. However, a major challenge to the development of the SCWG process is the high cost of the feedstock which can be tackled trough using low-cost feedstock like biowaste, including municipal sewage sludge which is often heavily moisture-laden. The combination of SCWG of biomass with fuel cell hybrid was studied in past investigations by many scholars. In this example, a technology assessment model is applied to quantify the power generated from a HRES based on the combination of hydrogen generation from biomass gasification (SCWG) and solar electrolysis coupled to a photovoltaic system and a fuel cell for providing heat and power. Because the electric power, heat, and hydrogen generated by the system cannot be affected by the weather, they can be used to form a composite energy system capable of all-weather operation. The methodology is based on the simultaneous optimization of the power and heat management through detailed estimation of the size of the components of the proposed HRES in order to obtain a balance between the load requirements and the power supply. Figure 6.4 shows the conceptual design for the proposed HRES. Mixed municipal sewage sludge is macerated, homogenized to a moisture content of up to 70 wt%.

6.3 Optimal Power Control Strategy in a Hybrid Renewable Energy System (HRES)

137

Fig. 6.4 The proposed HRES

The slurry feed is then pumped to the system operating pressure of approximately 3400 psig (Stream 2) and preheated in two stages—first, in a heat recovery heat exchanger (Stream 3) and second, in a gas-fired trim heater. The thermal energy of the hot flue gas from the fuel cell can be used to supply steam for the heat recovery heat exchanger. Having been preheated to the gasification temperature of 600 C, the process stream enters the SWC gasifier (Stream 4). The high-temperature syngas stream exiting the gasifier (Stream 5) is then cooled to the near-ambient temperature in a water-cooled heat exchanger. A portion of steam produced by this heat exchanger is used to preheat the slurry feed (Stream 10), with the remainder of steam available for export or other in-plant uses. Subsequent to cooling, but still at essentially full pressure, the syngas stream enters a gas-liquid separator (Stream 6). The liquid phase, comprised of water and a significant fraction of the CO2, exits from the bottom of the separator. The high-pressure gas stream exiting the separator is fed to a pressure swing adsorption (PSA) separator after being depressurized to 500 psig (Stream 7). The PSA separator removes all contaminants down to a parts per million level while recovering about 99.9% of the H2 at essentially feeding pressure (Stream 9). The gas-fired trim heater is fed by the off-gas from the PSA unit. Hydrogen from the PSA unit is subsequently mixed with the hydrogen produced from the electrolyzer which can be stored in a storage tank and fed to the fuel cell. The electric power, generated by the system, can meet external loads through the electric power output control device. In comparison with other stand-alone or hybrid renewable systems, the main advantages of the proposed HRES can be listed as follows: – The surplus electric power which is generated by solar photovoltaic can be used through the generation of hydrogen from water electrolysis.

138

6 Optimal Control of Energy Systems

– The hydrogen can be converted to electric power in the fuel cell which will be used during critical periods, when the power generated by solar photovoltaic is insufficient to meet the load. – Electric power which is needed for the pretreatment devices, mixer, and pumps can be provided by the HRES, itself. – The exhaust gas from the fuel cell can also be recovered to produce utilities. – The off-gas produced from PSA can be used as the fuel source for the gas-fired trim heater. – The extra hydrogen generated by the HRES can also be sold to be used for other purposes (i.E., fueling the fuel cell or hydrogen engines, etc.)

6.3.1

Model Development

6.3.1.1

SCWG Unit

The assumed basic expected reaction kinetics (using glucose as the model compound) are represented by the reaction listed below (Antal et al. 1993): 2C 6 H 12 O6 þ 10H 2 O $ 11CO2 þ CH 4 þ 20H 2

ð6:21Þ

In the above reaction, supercritical water behaves like a “non-ideal” gas which interacts and reacts with solute molecules at a faster rate than pure liquids or gases. The lower density of water at the supercritical condition results in a low relative dielectric constant of water which favors free radical reactions. This important property of supercritical water enables SCWG to quickly destroy the wet biomass and organic aqueous wastes while efficiently producing H2 and methane-rich gases. The properties of water in supercritical condition are reported in Table 6.1. In this example, the total Gibbs energy minimization method has been applied to determine the species and their amounts of the above reaction in the equilibrium state at a specific temperature and pressure. The pressure and temperature of supercritical water gasification systems result in a non-ideal thermodynamic behavior for the gaseous compounds. The total Gibbs free energy of the gas and aqueous phase supercritical region is given by the following equations: (

) X  fi G¼ ni μ0, i þ RTln f 0, i i Gaseous ( )

X 0 þ n j g j þ RTlnx j þ RTlnγ j j

ð6:22Þ Aqueous

where x is the mole fraction, n is the mole amount, f is the fugacity of components, γ is the molarity-based activity coefficient, and g0 is the standard molar Gibbs energy

6.3 Optimal Power Control Strategy in a Hybrid Renewable Energy System (HRES) Table 6.1 Properties of water in supercritical condition

Temperature ( C) Pressure (MPa) Density, ρ (g cm-3) Dielectric constant, ε (F m-1)

Normal 25 0.1 1 78.5

139

Supercritical 400 400 25 50 0.17 0.58 5.9 10.5

of components. μ0 and f0 are the chemical potential and fugacity under standard pressure. i and j represent the components in their gaseous and aqueous phases, respectively. To calculate the fugacity coefficients, the Peng-Robinson equation of state has been developed, using parameters derived from critical properties of substances, and is defined as pffiffiffi A Z þ ð1 þ 2ÞB pffiffiffi lnf i ¼ Z  1  lnðZ  BÞ  pffiffiffi ln 2 2B Z þ ð1  2ÞB

ð6:23Þ

    Z 3  ð1  BÞZ 2 þ A  3B2  2B Z  AB  B2  B3 ¼ 0

ð6:24Þ

and

where

A¼

0:45724

R2 T 2c Pc

qffiffiffiffiffiffi 2  1 þ K 1  TTc P R2 T 2

B¼

c 0:0778 RT Pc P

RT

K ¼ 0:37464 þ 1:5422ω  0:26992ω2

ð6:25Þ ð6:26Þ ð6:27Þ

Z is the compressibility factor (Z ¼ pv/RT). The subscript c refers to the critical point, and ω is the acentric factor of the substance. Solving Eq. (6. 24) results in three roots. The maximum of the real roots refers to the gas phase, and the minimum refers to the liquid phase in the two-phase region. An optimization problem with the constraints of mole and charge balance and nonnegativity of the number of species is conducted wherein the total Gibbs energy in Eq. (6.22) is minimized with respect to ncp (c ¼ 1,2,..., nc; p ¼ 1,2,. . ., np) subject to the following constraints: The material balance of component c nc ¼

np X

ncp ðc ¼ 1; 2; . . . ; nc Þ

p¼1

The nonnegative mole number of component c in phase p

ð6:28Þ

140

6 Optimal Control of Energy Systems

  0  ncp  nc c ¼ 1; 2; . . . ; nc ; p ¼ 1; 2; . . . ; np

ð6:29Þ

where the equilibrium state consists of “np” phases and each phase “p” consists of (n1, p, n2, p, n3p,. . ., nnc, p). The optimum values, ncp (c ¼ 1, 2, . . ., nc; p ¼ 1, 2, . . ., np), are the mole numbers of the equilibrium state.

6.3.1.2

Fuel Cell Unit

The maximum voltage, E, of the fuel cell is calculated with Gibbs free energy: E¼

ΔG 2F

ð6:30Þ

In the above equation, n is the number of moles of hydrogen reacted. F refers to the Faraday constant, which is about 96,485 (C/mole). The output power of the fuel cell can be calculated as follows: V C ¼ E:ηmax

ð6:31Þ

2:F:m_ H2 3600MW H2

ð6:32Þ

PFCOut ¼ I Total :V C

ð6:33Þ

I Total ¼

where MWH2 represents the molecular weight of hydrogen. PFC  Out and ηmax refer to the fuel cell electric power and maximum efficiency of hydrogen conversion, respectively.

6.3.1.3

Electrolyzer Unit

The hydrogen production flow rate from the electrolyzer, ṁH2  Elz(kg/h), can be defined as a function of the total electrical power consumption and the efficiency of the electrolyzer: m_ H2Elz ¼

E Elz :ηElz HHV

ð6:34Þ

In this study, the heating value of the produced hydrogen, HHV, is equal to 39.4 kWh/kg. The efficiency of a hydrogen production system may vary between 65% and 85% (Dufo-Lopez and Bernal-Agustin 2008).

6.3 Optimal Power Control Strategy in a Hybrid Renewable Energy System (HRES)

6.3.1.4

141

Solar Cell

The total solar incident received by the tilted surface of a photovoltaic cell can be formulated as follows:      ∅ ∅ þ ρðcosx þ CÞ sin 2 GT ¼ Gb, n ð cos ϕ cosx þ sinθ sinx cos ðφ  λÞ Þ þ C cos 2 2 2

ð6:35Þ where GT is the total solar radiation on a tilted surface. Gb, n is the direct normal irradiance on a surface perpendicular to the sun’s rays. ∅, x, and θ are the tilt angle, the zenith angle, and the angle between the tilted surface and the solar rays, respectively. φ and λ stand for sun azimuth and plate azimuth angle, respectively. C is the diffuse portion constant for calculating diffuse radiation, and ρ is the reflection index. The sun zenith and azimuth angles can be calculated using local data on latitude and the solar angle. The power generated by the panels can be estimated as a function of the solar radiation using the following formula: PPV Out ¼

  PRef : GT 1 þ kc T Cell  T Ref GRef

ð6:36Þ

where kc is the temperature coefficient of the PV panel and PRef refers to the rated power under the reference conditions. GRef is 1000 (W/m2), TRef is 25 C, and TCell is the cell temperature which is affected by ambient conditions, such as wind speed, solar irradiation, and temperature.

6.3.1.5

Hydrogen Storage

The hydrogen level of the storage tank at time step t can be estimated by considering a non-steady-state mass balance around the hydrogen storage tank, as follows: dm_ H2 m_ H2FC ¼ m_ H2SCWG þ m_ H2Elz  dt ηst

ð6:37Þ

where ṁH2  SCWG, ṁH2  Elz, and ṁH2  FC are the inlet mass flow rate of hydrogen from the SCWG, the inlet mass flow rate of hydrogen from the electrolyzer, and the hydrogen consumption of fuel cell, respectively. ηst refers to the storage efficiency which represents the pumping and leakage losses. In this study it is assumed to be about 95%.

142

6 Optimal Control of Energy Systems

The optimal control model for this example is described by the following system: Min TC ¼ ACC þ OC

ð6:38Þ

where ACC ¼ CRF t

X

Cλt FC λt

ð6:39Þ

E λmdht VC λmdht

ð6:40Þ

λ

OC ¼

XXXX λ

m

d

h

where TC, ACC, OC, CFR, FC, VC, C, and E represent the total cost of the system, the annualized capital cost, the operational cost, the capital recovery factor, the fixed cost, the variable cost, the rated capacity of a specific technology, and the power generated from a specific technology. λ denotes special technology such as the solar PV or fuel cell. m, d, and h are the respective months, days, and hours. The total cost of the system is minimized, while the demand of electricity is met by the power generated from the HRES: Load mdht 

XXXX λ

m

d

Eλmdht

ð6:41Þ

h

In this example, the level of hydrogen in the hydrogen tank (Eq. 6.37) is defined as the control variable of the optimal control problem. The control strategy used to manage the amount of power and hydrogen generated by the system is depicted in Fig. 6.5.

Fig. 6.5 The control strategy of power and hydrogen in the proposed HRES

6.3 Optimal Power Control Strategy in a Hybrid Renewable Energy System (HRES)

143

Fig. 6.6 Hourly typical rural household load profile (kWh/h)

6.3.2

Case of Fukushima Prefecture in Japan

The above model was used to verify the advantage of the proposed HRES in terms of energy supply and power generation in a subject district around Shinchi Station in Fukushima prefecture. This region has been transitioning to the “reconstruction and rebirth” phase from the “recovery” phase after the Great East Japan Earthquake. The selected area includes ten detached houses with a daily peak electric power load about 14 kW and a total annual electricity consumption of 43,700 kWh. Figure 6.6 shows the hourly load profile of the typical household in the selected area. The optimal configuration of the proposed system was found in order to support the annual electricity demand of the designated household area. Details of the hourly solar radiation were used to estimate the total power generation of the system. Figure 6.7 illustrates the hourly solar radiation available in the selected area. The detailed hourly simulation of the electricity generation by the system for a period of 3 days in cold and hot seasons is shown in Fig. 6.8. The annual electricity generation and consumption by the proposed HRES are presented in Table 6.2. According to the results of the model, about 12,279 kWh can be generated by the PV system: this can support less than 40% of the total annual electricity demand. The rest of the electricity for consumption should be provided by the fuel cell. The hydrogen required for the fuel cell can be provided from both electrolyzer and SCWG sources. The total annual hydrogen production is estimated at about 2680 kg, using the annual wet biomass consumption of 98 tons (see Fig. 6.9). Possible source materials include wastewater, kitchen waste, and organic waste.

144

6 Optimal Control of Energy Systems

Fig. 6.7 Hourly solar irradiation in Fukushima prefecture in 2016

Fig. 6.8 Hourly electric power and hydrogen generation from the proposed HRES

The total hydrogen generated by the electrolyzer and SCWG was estimated at about 74 kg and 2606 kg by the model, respectively. Figure 6.10 shows the monthly amount of the biomass feedstock, based on hydrogen consumption by the fuel cell system. The total capital cost of the HRES was estimated on the basis of the detailed estimation of the component sizes and capacities which is reported in Table 6.3.

6.3 Optimal Power Control Strategy in a Hybrid Renewable Energy System (HRES) Table 6.2 Annual electricity generation/consumption

KWh Jan Feb Mar Apr May Jun July Aug Sep Oct Nov Dec Total

Demand 5633 4773 4850 3464 2391 2298 3277 4055 2834 2279 3298 4488 43,640

PV 872 874 1103 1314 1351 1225 1148 1035 1016 1035 612 695 12,279

Fuel cell 4910 4087 4128 2581 1595 1488 2530 3326 2041 1729 2818 3935 35,169

145

Electrolyzer 149 187 401 485 555 430 415 306 223 381 132 143 3808

Fig. 6.9 Hydrogen generation from the proposed HRES

Based on all of the aforementioned calculation results, the total annual cost of such an HRES was estimated around USD 350,000. The levelized cost of energy (LCOE) of the system, including electricity, was estimated at about 0.92 $/kWh which is much higher than the average electricity price in Japan. As one of the most important advantages of the proposed HRES, the required thermal energy in the gas-fired trim heater and heat exchanger can be provided by using the fuel cell exhaust gas. Therefore, if the use of thermal energy is included, the LCOE supplied by the HRES will be reduced to 0.49 $/kWh. Because hydrogen generation from SCWG is still at the developmental stage, the cost of the necessary equipment is still quite high. Therefore, in order to reduce the cost of energy supply, it is necessary to optimize the capacity of different components in relation to the external loading demand. The results reveal that the cost of SCWG equals a large portion of the capital cost of the HRES in this study. However,

146

6 Optimal Control of Energy Systems

Fig. 6.10 Biomass feedstock annual consumption Table 6.3 Cost of the proposed HRES

Solar PV Fuel cell Electrolyzer H2 storageþ comp SCWG Invertor Total

Capacity 14 kW 14 kW 11 kW 9 kg 235 kg/ day 20 kW

Unit cost ($) 4300 5000 2000 1500

Capital cost ($) 60,200 70,000 22,000 13,500

Life time (Year) 25 5 10 25

Annualized cost($) 3867 18,406 3826 793

650

152,750

25

8869

1200

24,000 342,450

15

4104 39,865

the cost analysis showed that the future cost reduction potential in the photovoltaic manufacturing will result in a reduction in the total cost of the HRES and a subsequent reduction in the cost of energy. The technology for hydrogen generation from SCWG is still improving and developing. There is much room here for reducing costs. The application of the HRES has the potential to foster technological development for hydrogen generation from biomass gasification and subsequently lower the cost of related equipment.

References Antal MJ, Manarungson S, Mok WS (1993) Hydrogen production by steam reforming glucose in supercritical water. In: Bridgewater AV (ed) Advances in thermochemical biomass conversion. Blackie Academic and Professional, London, pp 1367–1377

References

147

Dufo-Lopez R, Bernal-Agustin JL (2008) Multi-objective design of PV– wind– diesel– hydrogen– battery systems. Renew Energy 33:2559–2572 Farzaneh H, Saboohi Y (2007) Evaluation of the optimal performance of passenger vehicle by integrated energy-environment-economic modeling. Int J Environ Sci Technol 4:103–109 Hocking L (1991) Optimal control: an introduction to the theory with applications. Oxford University Press, Oxford Saboohi Y, Farzaneh H (2008) Model for optimizing energy efficiency through controlling speed and gear ratio. Energy Effic J 1:37–45 Saboohi Y, Farzaneh H (2009) Model for developing eco-driving strategy of a passenger vehicle. Appl Energy 86(10):1925–1932 Wu N (2002) A new approach for modeling of fundamental diagrams. J Transp Res Part A 36:867–884

Appendix A (GAMS Codes)

Power Company Problem (Simplex Method)

* Input Parameters scalar S1 /0.8/; scalar S2 /0.1/; scalar W1 /0.1/; scalar W2/0.4/; scalar F1/0.1/; scalar F2/0.5/; scalar P1/20/; scalar P2/10/; scalar S/3000/; scalar W/2000/; scalar F/5000/; * Variables FREE VARIABLE TC objective function (Total purchase cost of electricity); POSITIVE VARIABLES E1 Option 1, E2 Option 1; * Equations EQUATIONS FT objective function, no1,no2, no3; FT.. TC=e=(P1*E1)+(P2*E2); no1.. (S1*E1)+(S2*E2)=g=S; no2.. (W1*E1)+(W2*E2)=g=W; no3.. (F1*E1)+(F2*E2)=g=F;

© Springer Nature Singapore Pte Ltd. 2019 H. Farzaneh, Energy Systems Modeling, https://doi.org/10.1007/978-981-13-6221-7

149

150

Appendix A (GAMS Codes)

*Solve control MODEL control powercompany/ALL/; SOLVE control USING LP Minimization TC;

Quadratic Problem

FREE VARIABLE Z objective function; POSITIVE VARIABLES x1, x2, S1, S2; EQUATIONS Target Objective function, no1, no2; Target.. Z=E=(-5*x1)-(8*x2)+(x1**2)+(8*(x2**2)); no1.. x1+x2=l=2; no2.. x1=l=3; MODEL qdp/ALL/; SOLVE qdp USING NLP Minimization Z;

Integer Problem

FREE VARIABLE Z objective function; POSITIVE VARIABLES S1, S2, S3, S4, x4;

Appendix A (GAMS Codes) integer VARIABLES integer VARIABLES integer VARIABLES

151 x1; x2; x3;

EQUATIONS Target Objective function, no1, no2, no3, no4; Target.. Z=E= (4*x1)-(2*x2)+(7*x3)-x4; no1.. x1+(5*x3)+S1=e=10; no2.. x1+x2-x3+S2=e=1; no3.. (6*x1)-(5*x2)+S3=e=0; no4.. (-x1)+(2*x3)-(2*x4)+S4=e=3; MODEL IP/ALL/; SOLVE IP USING MIP Maximization Z;

Gas Turbine Energy Demand Optimization Model

*------Defined Units * Energy Unit (MJ) * Specific Enthalpy (MJ/t) * Mass Unit (ton) scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar

EU /360/; pch4 /63/; pch41 /70/; YG /0.95/; YC /0.8/; Yt/0.8/; h5/400/; h8/300/; h4/1080/; h2/142/; h1/25/; h3/33/; Ecc1/46000/; Ecc2/56000/;

FREE VARIABLE TC objective function(Total operation cost Of System); POSITIVE VARIABLES m1 Inlet Air mass flow, m2 Compressed Air,

152

Appendix A (GAMS Codes) m3 m31 m32 m4 m5 m8 E6 E7 Ecw Esw Etw EE Egw

Total Fuel, Fuel 1, Fuel 2, Exhaust from combustion, Exhaust from turbine, Exhaust from stack, Oulet shaft energy from turbine, Inlet shaftenergy to generator, Waste energy from compressor, Waste energy from stack, Waste energy from turbine, Waste energy from stack, Waste energy from generator;

EQUATIONS FT objective function, no1,no2,no3, no4, no5, no6, no7, no8, no9, no10, no11, no12,no13; ******* Objective function FT.. TC=e=(m31*pch4)+(m32*pch41); ******** Generator Equation no1.. E7-EU-Egw=e=0; no2.. Egw=e=E7*(1-YG); ******** Stack Equation no3.. m5-m8=e=0; no4.. (m5*h5)-(m8*h8)-Esw=e=0; ******** Turbine Equation no5.. m4-m5=e=0; no6.. (m4*h4)-(m5*h5)-E6-Etw=e=0; no7.. Etw-(E6*(1-Yt)/Yt)=e=0; ******** Combustion Chamber Equation no8.. m2+m3-m4=e=0; no9.. (m2*h2)+(m3*h3)+(m31*Ecc1)+(m32*Ecc2)-(m4*h4)=e=0; ******* Compressoe Equation no10.. m1-m2=e=0; no11.. (m1*h1)+E6-(m2*h2)-E7-Ecw=e=0; no12.. Ecw=e=E6*(1-YC); no13.. m3=e= m31+m32; MODEL gt/ALL/; SOLVE gt USING LP MINIMIZATION TC;;

Cement Factory Energy Demand Optimization Model (Cost Minimization)

*------Defined Units * Energy Unit (kWh)

Appendix A (GAMS Codes)

153

* Net Calorific value (GJ/t) * Mass and capacity Unit (ton) * Price Unit($/ton) scalar scalar scalar scalar scalar scalar scalar scalar

m6 /1000000/; pf /7000/; pr /0.1/; png /63/; pe/0.04/; ncv/46/; hc/5000000/; nc/2000000/;

FREE VARIABLE TP objective function (Total operation cost of System); POSITIVE VARIABLES m1 Inlet mass to crusher, m2 Inlet mass to raw material ball mill, m3 Inlet mass to rotary kiln, m4 Inlet mass to cement mill, m5 Inlet mass to loading, Ec Elec of crusher, Er Elec of raw ball mill, Ek Elec of rotary kiln, Eb Elec of cement mill, El Elec of loading, Lc Labor of crusher, Lr Labor of raw ball mill, Lk Labor of rotary kiln, Lb Labor of cement mill, Ll Labor of loading, ET Total electricity consumption, mf Fuel consumption, TL Total wage; EQUATIONS FT objective function, no1,no2,no3,no4, no5, no6, no7,no8,no9, no10,no11,no12,no13,no14, no15,no16,no17,no18; ******* Objective function FT.. TP=e=(((pr*m1)+(pe*(Ec+Er+Ek+Eb+El))+(mf*png)+((20*Lc) +(20*Lr)+(20*Lk)+(20*Lb)+(11*Ll)*7000))); ********* Demand inequality no1.. (0.9*m5)-m6=g=0; * ******** Mass Balances no2.. (0.6*m1)-m2=e=0; no3.. (0.9*m2)-m3=e=0; no4.. (0.9*m3)-m4=e=0; no5.. (0.9*m4)-m5=e=0; ******** Resources Limitation no6.. m1-hc-nc=l=0; ******** Energy Balances no7.. (1.7*m1)-Ec=e=0; no8.. (22*m2)-Er=e=0;

154

Appendix A (GAMS Codes)

no9.. (26*m3)-Ek=e=0; no10.. (57*m4)-Eb=e=0; no11.. (5*m5)-El=e=0; no12.. (ncv*mf)-(3.2*m3)=e=0; ******* Labor requirement no13.. (Lc-(2*m1/7000))=e=0; no14.. (Lr-(1*m2/7000))=e=0; no15.. (Lk-(2.4*m3/7000))=e=0; no16.. (Lb-(1*m4/7000))=e=0; no17.. (Ll-(4*m5/7000))=e=0; no18.. Et=e=Ec+Er+Ek+Eb+El; MODEL control cement/ALL/; SOLVE control USING LP Minimization TP;

Minimum Work of Compression

scalar scalar scalar scalar scalar

T1 /298/; R /8.314/; k /1.4/; P1 /100/; P3 /4000/;

FREE VARIABLE W objective function(Total operation cost Of System); POSITIVE VARIABLES P2

pressure at point 2;

P2.l=100; EQUATIONS FT objective function; *--------------------------------------------------------------------******* Objective function FT.. W=e=((k*R*T1)/(k-1))*(((P2/P1)**((k-1)/k))+((P3/P2)** ((k-1)/k))-2); MODEL control house/ALL/; SOLVE control USING NLP MINIMIZATION W;

Appendix A (GAMS Codes)

155

Building Energy Demand Optimization Model

Building Energy Demand Optimization Model *------Defined Units * Energy Unit (kWh) * Net Calorific value (GJ/m3) * Mass and capacity Unit (m3) * Price Uint($/m3) scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar scalar

Wind1 /2/; Wind2 /2/; SC /0.7/; pf /0.04/; To /6/; ESM/1/; np/4/; SF/64/; D1/2/; Wall/45/; Door/1/; Ceiling/24/; Floor/24/; UWall/1.6/; UDoor/1.3/; UWind1/0.45/; UWind2/0.45/; UCeilling/0.3/; UFloor/0.7/; Qpp/0.3/; ncv/36/;

FREE VARIABLE TC objective function (Total operation cost of System); POSITIVE VARIABLES QHeater, Qsolar, QP, QVent, QLoss, Vinf, Ti, Vf; EQUATIONS FT objective function, no1, no2, no3, no4, no5, no6, no7, no8, no9; ******* Objective function FT.. TC=e=(pf*vf);

156

Appendix A (GAMS Codes)

******* Constraints no1.. QHeater+Qsolar+QP-QVent-QLoss=g=0; no2.. Qsolar-((Wind1+Wind2)*(SC*ESM*SF))=e=0; no3.. QVent-0.34*Vinf*(Ti-To)=e=0; no4.. QLoss-(((Wind1*UWind1)+(Wind2*UWind2)+(Wall*UWall) +(Door*UDoor)+(Ceiling*UCeilling)+(Floor*UFloor))*(Ti-To))=e=0; no5.. Qp-np*Qpp=e=0; no6.. (vf*ncv)-(QHeater*3.6)=e=0; no7.. Ti=g=20; no8.. Ti=l=27; no9.. Vinf=g=20; MODEL control building/ALL/; SOLVE control USING NLP MINIMIZATION TC;

Thermal Efficiency Maximization in a Heat Recovery System

SCALAR T6 /1073/; FREE VARIABLE TY objective function (Thermal Yield Of System); POSITIVE VARIABLES Yc Stage Yield Compressor, Yt Stage Yield Turbine, Cc OVER COMPRESS Ratio, Ct Over Press ratio OF Turbine, t Over Temp OF Cycle, x Dimless Over Press Drop, T1 COMPREDDOR(1) OUTLET TEMP(K), T2 COMPRESSOR(2) OUTLET TEMP(K), T3 TURBINE INLET TEMP(K), T4 COMPRESSOR(2) INLET TEMP, T5 TURBINE OUTLET TEMP, P1 COMPRESSOR(1) OUTLET PRESS(bar), P2 COMPRESSOR(2) OUTLET PRESS(bar), P3 TURBINE INLET PRESS(bar), P4 COMPRESSOR(2) INLET PRESS(bar), P5 TURBINE OUTLET PRESS(bar), rc COMPRESSOR PRESS RATIO, L TURBINE DIMLESS PRESS DROP, cct Overal comp press ratio, rt TURBINE PRESS RATIO;

Appendix A (GAMS Codes) *___ *___

157

INITIAL VALUES OF SOME VARIABLES BASED ON EXPRIMENTAL WORKS Cc.L=1.60; x.L=.97; Yc.L=.75; Ct.L=1.50; Yt.L=.85; t.L=3.3; rt.L=3.3; T4.l=300; EQUATIONS ST objective function, NO1 stage yield compressor equation, NO2 over press ratio of turbine equation, NO3 stage yield turbine equation, NO4 over temp ratio of cycle equation, NO5 OVER COMPRESS RATIO LOWER LIMITATION NON-EQUATION, NO6 OVER COMPRESS RATIO UPPER LIMITATION NON-EQUATION, NO7 DIMLESS OVER PRESS DROP LOWER LIMITATION NON-EQUATION, NO8 DIMLESS OVER PRESS DROP UPPER LIMITATION NON-EQUATION, NO9 OVER PRESS RATIO OF TURBINE LOWER LIMITATION NON-EQUATION, NO10 OVER PRESS RATIO OF TURBINE UPPER LIMITATION NON-EQUATION, NO11 STAGE YIELD COMPRESSOR UPPER LIMITATION NON-EQUATION, NO12 STAGE YIELD TURBINE UPPER LIMITATION NON-EQUATION, NO13 OVER TEMP RATIO OF CYCLE LOWER LIMITATION NON-EQUATION, NO14 OVER TEMP RATIO OF CYCLE UPPER LIMITATION NON-EQUATION, NO15 COMP(1) OUTLET TEMP EQUATION, NO16 TUTBINE INLET TEMP EQUATION, No17 COMP(2) INLET TEMP EQUATION, NO18 COMP(2) OUTLET TEMP EQUATION, NO19 TURBINE OUTLET TEMP EQUATION, NO20 COMP(1) OUTLET PRESS EQUATION, NO21 COMP(2) OUTLET PRESS EQUATION, NO22 TURBINE INLET PRESS EQUATION, NO23 COMP(2) INLET PRESS EQUATION, NO24 TURBINE OUTLET PRESS EQUATION, NO25 COMP PRESS RATIO EQUATION, NO26 TURB PRESS RATIO EQUATION, NO27 TURB DIMLESS PRESS EQUATION, No28 T1 lower limit non-equation, NO29 T2 lower limit non-equatioN, NO30 T3 lower limit non-equation, NO31 T3 upper limit non-equation, NO32 T4 lower limit non-equation, NO33 T5 upper limit non-equation, no34 p1 lower limit non-equation, NO35 p2 lower limit non-equation, NO36 p3 lower limit non-equation, NO37 p4 lower limit non-equation, NO38 P5 lower limit non-equation, NO39 rc lower limit non-equation, NO40 rt lower limit non-equation, NO41 CCT EQUATION;

158

Appendix A (GAMS Codes) ST..

TY=E=(t*Yt*((1-(1/(ct*x))))-((0.1*cc/Yc)*(cc-1)) -((0.9/

Yc) *(cc-1))-((cc-1)/Yc))/(t-(0.1*cc*(1+((cc-1)/Yc))) -(0.9*(1+((cc-1)/Yc)))); NO1.. Yc=E=(Cc-1.00)/((Cc**1.2)-1.00); NO3.. Yt=E=(((Ct*x)**1.91)-(Ct*x))/ (((Ct*x)**1.91)-((Ct*x)**0.91)); NO2.. Ct=E=x*Cct; NO4.. t=E=(0.01*cc*(1+((cc-1)/Yc)))+(0.09*(1+((cc-1)/yc))) +(0.9*(T6/298)); NO5.. Cc=G=1.10; NO6.. Cc=L=2; NO7.. x=G=0.95; NO8.. x=L=.99; NO9.. Ct=G=1.10; NO10.. Ct=L=2.0; NO11.. YC=L=1.00; NO12.. Yt=L=1.00; NO13.. t=G=3.20; NO14.. t=L=3.6; NO15.. T1=E=298*(1+((CC-1)/Yc)); NO16.. T3=E=t*298; NO17.. T4=E=(0.1*T1)+268.2; NO18.. T2=E=T4*(1+((CC-1)/Yc)); NO19.. T5=E=T3/(rt**0.26); NO20.. P1=E=CC**3.5; NO21.. P2=E=(0.98*(cc**7))-(0.04*(cc**3.5)); NO22.. P3=E=rt*P5; NO23.. P4=E=(0.98*rc)-0.02; NO24.. P5=E=L; NO25.. rc=E=cc*3.5; NO26.. rt=E=((CT*X)**3.5); NO27.. L=E=((1/X)**3.5); NO28.. T1=G=298; No29.. T2=G=298; NO30.. T3=G=298; NO31.. T3=L=1073; NO32.. T4=G=298; NO33.. T5=G=298; NO34.. P1=G=1.00; NO35.. P2=G=1.00; NO36.. P3=G=1.00; NO37.. P4=G=1.00; NO38.. P5=G=1.00; NO39.. rc=G=1.0; NO40.. rt=G=1.10; NO41.. CCT=E=P2**0.285; MODEL GTHR/ALL/; SOLVE GTHR USING NLP MAXIMIZING TY;

Appendix A (GAMS Codes)

159

Middle East Energy Supply Model

*################### Model Characteristics ################################## * Midin.xls is the database of the model which is called using an external link. * Energy Carriers *c | Crude Oil *o | Processed Oil *g | Natural Gas *r | Renewable Energy *e | Electricity *p | Oil Products *d | Distillate Oil *j | Jet Fuel *k | Kerosene *l | LPG *q | Gasoline *z | Other oil products *f | Fuel oil *-----------------------------------* System Levels *U | Final End User Energy Demand *T | Energy Transport *X | Conversion (Secondary Energy) *A | Processing (Primary Energy) *R | Energy Resources *------------------------------------* Regions Indicators *u | USA *c | Canada *s | S&C America+Mexico *m | Middle East *e | Europe *f | Africa *a | Asia & Pacific *w | World *-----------------------------------* Technology Indicators *o | Oil Fields *g | Gas Fields *r | Renewable Sources *i | Import Energy *u | Upstream Surface Facilities *k | Oil Refinery *n | Gas Refinery *p | Power Plant *l | Domestic Transport *q | Electricity Transmission

160

Appendix A (GAMS Codes)

*s | International Transport (Ships,) *------------------------------------– *#################### Scalar (Imput Prameters) ################################ sets t/2010,2015,2020,2025,2030,2035,2040,2045,2050,2055,2060, 2065,2070,2075,2080,2085,2090,2095,2100/; sets t/2010/; sets c/Iran,Iraq,Kuwait,Oman,Qatar,SaudiArabia, UAE,Bahrain, Others/; ****************************** Utility Input Data****************************** parameter Udmd(c,t), Ujmj(c,t),Ukmk(c,t),Ulml(c,t), Uqmq(c,t), Uzmz (c,t),Ufmf(c,t); $onUNDF $CALL GDXXRW Midin.xls par=Udmd rng=OP!a2:ax11 Rdim=1 Cdim=1 trace=0 par=Ujmj rng=OP!a14:ax23 Rdim=1 Cdim=1 trace=0 par=Ukmk rng=OP!a26: ax35 Rdim=1 Cdim=1 trace=0 par=Ulml rng=OP!a39:ax48 Rdim=1 Cdim=1 trace=0 par=Uqmq rng=OP!a52:ax61 Rdim=1 Cdim=1 trace=0 par=Uzmz rng=OP!a65:ax74 Rdim=1 Cdim=1 trace=0 par=Ufmf rng=OP!a78:ax87 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD Udmd $LOAD Ujmj $LOAD Ukmk $LOAD Ulml $LOAD Uqmq $LOAD Uzmz $LOAD Ufmf $GDXIN parameter Ugmg(c,t),Ueme(c,t),Uouo(c,t), Ugug(c,t), Upup(c,t),Uoco (c,t),Ugcg(c,t), Upcp(c,t),Uoso(c,t),Ugsg(c,t), Upsp(c,t), Uoeo(c,t), Ugeg(c,t),Upep (c,t),Uoao(c,t), Ugag(c,t),Upap(c,t),Uofo(c,t),Ugfg(c,t), Upfp(c,t) ; $onUNDF $CALL GDXXRW Midin.xls par=Ugmg rng=Demand!a14:ax23 Rdim=1 Cdim=1 trace=0 par=Ueme rng=Demand!a27:ax36 Rdim=1 Cdim=1 trace=0 par=Uouo rng=Demand!a39:ax48 Rdim=1 Cdim=1 trace=0 par=Ugug rng=Demand!a52: ax61 Rdim=1 Cdim=1 trace=0 par=Upup rng=Demand!a65:ax74 Rdim=1 Cdim=1 trace=0 par=Uoco rng=Demand!a78:ax87 Rdim=1 Cdim=1 trace=0 par=Ugcg rng=Demand!a91:ax100 Rdim=1 Cdim=1 trace=0 par=Upcp rng=Demand!a104: ax113 Rdim=1 Cdim=1 trace=0 par= Uoso rng=Demand!a117:ax126 Rdim=1 Cdim=1 trace=0 par= Ugsg rng=Demand!a130:ax139 Rdim=1 Cdim=1 trace=0 par= Upsp rng=Demand!a144:ax153 Rdim=1 Cdim=1 trace=0 par= Uoeo rng=Demand!a158:ax167 Rdim=1 Cdim=1 trace=0 par= Ugeg rng=Demand! a172:ax181 Rdim=1 Cdim=1 trace=0 par= Upep rng=Demand!a186:ax195 Rdim=1 Cdim=1 trace=0 par= Uoao rng=Demand!a200:ax209 Rdim=1 Cdim=1 trace=0 par= Ugag rng=Demand!a214:ax223 Rdim=1 Cdim=1 trace=0 par= Upap rng=Demand!a228:ax237 Rdim=1 Cdim=1 trace=0 par= Uofo rng=Demand!

Appendix A (GAMS Codes)

161

a242:ax251 Rdim=1 Cdim=1 trace=0 par= Ugfg rng=Demand!a255:ax264 Rdim=1 Cdim=1 trace=0 par= Upfp rng=Demand!a268:ax278 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD Ugmg $LOAD Ueme $LOAD Uouo $LOAD Ugug $LOAD Upup $LOAD Uoco $LOAD Ugcg $LOAD Upcp $LOAD Uoso $LOAD Ugsg $LOAD Upsp $LOAD Uoeo $LOAD Ugeg $LOAD Upep $LOAD Uoao $LOAD Ugag $LOAD Upap $LOAD Uofo $LOAD Ugfg $LOAD Upfp $GDXIN ****************************** Variable cost of operation********************** parameter CVTplp(c,t),CVTglg(c,t),CVTeqe(c,t),CVToso(c,t),CVTpsp (c,t),CVTgsg(c,t),CVXppe(c,t) CVXgpe(c,t), CVXrpe(c,t),CVAokp(c,t),CVAgng(c,t),CVAcuo(c,t) ; $onUNDF $CALL GDXXRW Midin.xls par= CVTplp rng=VCost!a2:ax11 Rdim=1 Cdim=1 trace=0 par=CVTglg rng=VCost!a14:ax23 Rdim=1 Cdim=1 trace=0 par=CVTeqe rng=VCost!a26:ax35 Rdim=1 Cdim=1 trace=0 par=CVToso rng=VCost!a38:ax47 Rdim=1 Cdim=1 trace=0 par=CVTpsp rng=VCost!a50: ax59 Rdim=1 Cdim=1 trace=0 par=CVTgsg rng=VCost!a62:ax71 Rdim=1 Cdim=1 trace=0 par=CVXppe rng=VCost!a75:ax84 Rdim=1 Cdim=1 trace=0 par=CVXgpe rng=VCost!a88:ax97 Rdim=1 Cdim=1 trace=0 par=CVXrpe rng=VCost!a100:ax109 Rdim=1 Cdim=1 trace=0 par=CVAokp rng=VCost! a112:ax121 Rdim=1 Cdim=1 trace=0 par=CVAgng rng=VCost!a124:ax133 Rdim=1 Cdim=1 trace=0 par=CVAcuo rng=VCost!a136:ax145 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD CVTplp $LOAD CVTglg $LOAD CVTeqe $LOAD CVToso $LOAD CVTpsp $LOAD CVTgsg $LOAD CVXppe $LOAD CVXgpe $LOAD CVXrpe

162

Appendix A (GAMS Codes)

$LOAD CVAokp $LOAD CVAgng $LOAD CVAcuo $GDXIN ****************************** Fixed Capital Investment ********************** parameter CFTplp(c,t),CFTglg(c,t),CFTeqe(c,t),CFToso(c,t),CFTpsp (c,t),CFTgsg(c,t),CFXppe(c,t) CFXgpe(c,t), CFXrpe(c,t),CFAokp(c,t),CFAgng(c,t),CFAcuo(c,t) ; $onUNDF $CALL GDXXRW Midin.xls par= CFTplp rng=FCost!a2:ax11 Rdim=1 Cdim=1 trace=0 par=CFTglg rng=FCost!a14:ax23 Rdim=1 Cdim=1 trace=0 par=CFTeqe rng=FCost!a26:ax35 Rdim=1 Cdim=1 trace=0 par=CFToso rng=FCost!a38:ax47 Rdim=1 Cdim=1 trace=0 par=CFTpsp rng=FCost!a50: ax59 Rdim=1 Cdim=1 trace=0 par=CFTgsg rng=FCost!a62:ax71 Rdim=1 Cdim=1 trace=0 par=CFXppe rng=FCost!a75:ax84 Rdim=1 Cdim=1 trace=0 par=CFXgpe rng=FCost!a88:ax97 Rdim=1 Cdim=1 trace=0 par=CFXrpe rng=FCost!a100:ax109 Rdim=1 Cdim=1 trace=0 par=CFAokp rng=FCost! a112:ax121 Rdim=1 Cdim=1 trace=0 par=CFAgng rng=FCost!a124:ax133 Rdim=1 Cdim=1 trace=0 par=CFAcuo rng=FCost!a136:ax145 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD CFTplp $LOAD CFTglg $LOAD CFTeqe $LOAD CFToso $LOAD CFTpsp $LOAD CFTgsg $LOAD CFXppe $LOAD CFXgpe $LOAD CFXrpe $LOAD CFAokp $LOAD CFAgng $LOAD CFAcuo $GDXIN ****************************** Energy Carriers Prices ($/tonne) *************** parameter pXoo(c,t),pXgo(c,t),pXpo(c,t),pRgi(c,t),pRpi(c,t),Au(c, t),pRei(c,t) ; $onUNDF $CALL GDXXRW Midin.xls par=pXoo rng=RPrice!a2:ax11 Rdim=1 Cdim=1 trace=0 par=pXgo rng=RPrice!a15:ax24 Rdim=1 Cdim=1 trace=0 par=pXpo rng=RPrice!a28:ax37 Rdim=1 Cdim=1 trace=0 par=pRgi rng=RPrice!a41: ax50 Rdim=1 Cdim=1 trace=0 par=pRpi rng=RPrice!a54:ax62 Rdim=1 Cdim=1 trace=0 par=Au rng=RPrice!a67:ax76 Rdim=1 Cdim=1 trace=0 par=pRei rng=RPrice!a79:ax88 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD pXoo $LOAD pXgo $LOAD pXpo $LOAD pRgi

Appendix A (GAMS Codes)

163

$LOAD pRpi $LOAD Au $LOAD pRei $GDXIN ****************************** Efficiency ********************* parameter YTplp(c,t),YTglg(c,t),YTeqe(c,t),YToso(c,t),YTpsp(c,t), YTgsg(c,t),YXppe(c,t) YXgpe(c,t), YXrpe(c,t),YAokp(c,t),YAgng(c,t),YAcuo(c,t) ; $onUNDF $CALL GDXXRW Midin.xls par= YTplp rng=Y!a2:ax11 Rdim=1 Cdim=1 trace=0 par=YTglg rng=Y!a14:ax23 Rdim=1 Cdim=1 trace=0 par=YTeqe rng=Y!a26: ax35 Rdim=1 Cdim=1 trace=0 par=YToso rng=Y!a38:ax47 Rdim=1 Cdim=1 trace=0 par=YTpsp rng=Y!a50:ax59 Rdim=1 Cdim=1 trace=0 par=YTgsg rng=Y!a62:ax71 Rdim=1 Cdim=1 trace=0 par=YXppe rng=Y!a75:ax84 Rdim=1 Cdim=1 trace=0 par=YXgpe rng=Y!a88:ax97 Rdim=1 Cdim=1 trace=0 par=YXrpe rng=Y!a100:ax109 Rdim=1 Cdim=1 trace=0 par=YAokp rng=Y! a112:ax121 Rdim=1 Cdim=1 trace=0 par=YAgng rng=Y!a124:ax133 Rdim=1 Cdim=1 trace=0 par=YAcuo rng=Y!a136:ax145 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD YTplp $LOAD YTglg $LOAD YTeqe $LOAD YToso $LOAD YTpsp $LOAD YTgsg $LOAD YXppe $LOAD YXgpe $LOAD YXrpe $LOAD YAokp $LOAD YAgng $LOAD YAcuo $GDXIN parameter YAdkp(c,t),YAjkp(c,t),YAkkp(c,t),YAlkp(c,t),YAqkp(c,t), YAzkp(c,t),YAfkp(c,t); $onUNDF $CALL GDXXRW Midin.xls par=YAdkp rng=YM!a2:ax11 Rdim=1 Cdim=1 trace=0 par=Yajkp rng=YM!a14:ax23 Rdim=1 Cdim=1 trace=0 par=YAkkp rng=YM!a26: ax35 Rdim=1 Cdim=1 trace=0 par=YAlkp rng=YM!a39:ax48 Rdim=1 Cdim=1 trace=0 par=YAqkp rng=YM!a52:ax61 Rdim=1 Cdim=1 trace=0 par=YAzkp rng=YM!a65:ax74 Rdim=1 Cdim=1 trace=0 par=YAfkp rng=YM!a78:ax87 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD YAdkp $LOAD YAjkp $LOAD YAkkp $LOAD YAlkp $LOAD YAqkp $LOAD YAzkp $LOAD YAfkp $GDXIN

164

Appendix A (GAMS Codes)

**********************Resources Availability (Mtoe) ********************* parameter Rgg(c,t),Rrr(c,t),Accumoil(c,t),Rref(c,t) ; $onUNDF $CALL GDXXRW Midin.xls par=Rgg rng=Res!a2:ax11 Rdim=1 Cdim=1 trace=0 par=Rrr rng=Res!a14:ax23 Rdim=1 Cdim=1 trace=0 par=Accumoil rng=Res! a26:ax35 Rdim=1 Cdim=1 trace=0 par=Rref rng=Res!a38:ax47 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD Rgg $LOAD Rrr $LOAD Accumoil $LOAD Rref $GDXIN **********************Capcity ********************* parameter HXfpe(c,t),HXrpe(c,t),HAokp(c,t),HAgng(c,t) ; $onUNDF $CALL GDXXRW Midin.xls par=HXfpe rng=Cap!a2:ax11 Rdim=1 Cdim=1 trace=0 par=HXrpe rng=Cap!a16:ax25 Rdim=1 Cdim=1 trace=0 par=HAokp rng=Cap! a30:ax39 Rdim=1 Cdim=1 trace=0 par=HAgng rng=Cap!a44:ax53 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD HXfpe $LOAD HXrpe $LOAD HAokp $LOAD HAgng $GDXIN **********************Plant Factor ********************* parameter PFXgpe(c,t),PFXppe(c,t),PFXrpe(c,t),PFAokp(c,t) ; $onUNDF $CALL GDXXRW Midin.xls par=PFXgpe rng=PF!a2:ax11 Rdim=1 Cdim=1 trace=0 par=PFXppe rng=PF!a16:ax25 Rdim=1 Cdim=1 trace=0 par=PFXrpe rng=PF! a30:ax39 Rdim=1 Cdim=1 trace=0 par=PFAokp rng=PF!a44:ax53 Rdim=1 Cdim=1 trace=0 $GDXIN Midin.gdx $LOAD PFXgpe $LOAD PFXppe $LOAD PFXrpe $LOAD PFAokp $GDXIN *######################## POSITIVE VARIABLES ################################## positive VARIABLES CFT(c,t),CFX(c,t), CFA(c,t), CVT(c,t), CVX(c,t),CVA(c,t), TRC(c,t), Uowo(c,t), Ugwg(c,t), Upwp(c,t), Tplp(c,t), Tglg(c,t), Teqe(c,t), Toso(c,t), Tpsp(c,t),Tgsg(c,t), Xppe(c,t), Xgpe(c,t), Xrpe(c,t), Aokp(c,t), Agng(c,t), Acuo(c,t), Rpi(c,t), Rgi(c,t), Roo(c,t),Alfa(c,t), Oilsupply(c,t), Gassupply(c,t), Poweroilprod(c,t), TherPower(c,t),

Appendix A (GAMS Codes)

165

RenPower(c,t), TXR(c,t) Tdmd(c,t),Tjmj(c,t),Tlml(c,t),Tkmk(c,t),Tqmq(c,t),Tzmz(c,t),Tfmf (c,t),Tpsp(c,t),Addd(c,t), Ajdj(c,t), Akdk(c,t), Aldl(c,t), Aqdq(c,t) ,Azdz(c,t), Afdf(c,t),Rei(c,t),UT(c,t), AAA(c,t),Aokpi(c, t),Aokpx(c,t),Rdi(c,t),Rji(c,t), Rki(c,t),Rqi(c,t),Rzi(c,t), Rli(c,t),Rfi(c,t),An(c,t),AGG(c,t),Adom (c,t), Exo(c,t),Exg(c,t); variable Baoil(c,t); *####################### Objective Function ################################### free variable TC; *########################## EQUATIONS ####################################### EQUATION no1(c,t), no2(c,t), no3(c,t), no4(c,t), no5(c,t), no6(c,t), no7(c,t), no8(c,t), no9(c,t), no10(c,t), no11(c,t), no12(c,t), no13(c,t), no14(c,t), no15(c,t), no16(c,t), no17(c,t),no18(c,t) no19(c,t), no20(c,t), no21(c,t), no22(c,t), no23(c,t), no24(c,t), no25(c,t), no26(c,t),no27(c,t), no28(c,t), no29(c,t), no30(c,t), no31(c,t),no32(c,t), no34(c,t), no35(c,t), no37(c,t), no38(c,t),no39(c,t), no40(c,t), no41(c,t), no42(c,t), no43(c,t), no44(c,t), no45(c,t), no46(c,t), no49(c,t), no50(c,t), no51(c,t),no52(c,t), no53(c,t),no54(c,t), no55 (c,t),no56(c,t), no57(c,t), no58(c,t), no59(c,t),no60(c,t),no61(c,t), FT; *############################### Model Main ################################### ************** Cost function items ******************************************** ****Transportation Cost no1(c,t).. CVT(c,t)=e=(Tplp(c,t)*YTplp(c,t)*CVTplp(c,t))+(Tglg(c, t)*YTglg(c,t)*CVTglg(c,t))+(Teqe(c,t)*YTeqe(c,t)*CVTeqe(c,t)) +(Toso(c,t)*YToso(c,t)*CVToso(c,t))+(Tpsp(c,t)*YTpsp(c,t) *CVTpsp(c,t))+(Tgsg(c,t)*YTgsg(c,t)*CVTgsg(c,t)); no2(c,t).. CFT(c,t)=e=(Tplp(c,t)*YTplp(c,t)*CFTplp(c,t))+(Tglg(c, t)*YTglg(c,t)*CFTglg(c,t))+(Teqe(c,t)*YTeqe(c,t)*CFTeqe(c,t)) +(Toso(c,t)*YToso(c,t)*CFToso(c,t))+(Tpsp(c,t)*YTpsp(c,t) *CFTpsp(c,t))+(Tgsg(c,t)*YTgsg(c,t)*CFTgsg(c,t)); ****Conversion Cost (M$/Mtoe) no3(c,t).. CVX(c,t)=e=(11.6*1000)*(Xppe(c,t)*YXppe(c,t)*CVXppe(c, t))+(Xgpe(c,t)*YXgpe(c,t)*CVXgpe(c,t)) +(Xrpe(c,t)*YXrpe(c,t)*CVXrpe(c,t)); no4(c,t).. CFX(c,t)=e=((PFXppe(c,t)*11.6*CFXppe(c,t))/PFXppe(c,t))

166

Appendix A (GAMS Codes)

+((PFXgpe(c,t)*11.6*CFXgpe(c,t))/PFXgpe(c,t)) +((PFXrpe(c,t)*11.6*CFXrpe(c,t))/PFXrpe(c,t)); ****Processing Cost no5(c,t).. CVA(c,t)=e=(Aokp(c,t)*YAokp(c,t)*CVAokp(c,t))+(Agng(c, t)*YAgng(c,t)*CVAgng(c,t)) +(Acuo(c,t)*YAcuo(c,t)*CVAcuo(c,t)); no6(c,t).. CFA(c,t)=e=(((HAokp(c,t))*PFAokp(c,t)/(24*7.33*1000)) *CFAokp(c,t)) +((HAgng(c,t))*CFAgng(c,t)) +(Acuo(c,t)*YAcuo(c,t)*CFAcuo(c,t)); ****Resources Cost no7(c,t).. TRC(c,t)=e=(Rei(c,t)*pRei(c,t)*1000000000*11.6)+(Rgi(c, t)*pRgi(c,t))+(Rpi(c,t)*pRpi(c,t)); ****Export Income no8(c,t).. TXR(c,t)=e=( Uowo(c,t)*pXoo(c,t))+(Upwp(c,t)*pXpo(c, t))+( Ugwg(c,t)*pXgo(c,t)); *************** End user constraints ****************************************** **** Middle East oil product demand-transport no9(c,t).. (Tdmd(c,t)*YTplp(c,t))-Udmd(c,t)=e=0; no10(c,t).. (Tjmj(c,t)*YTplp(c,t))-Ujmj(c,t)=e=0; no11(c,t).. (Tkmk(c,t)*YTplp(c,t))-Ukmk(c,t)=e=0; no12(c,t).. (Tlml(c,t)*YTplp(c,t))-Ulml(c,t)=e=0; no13(c,t).. (Tqmq(c,t)*YTplp(c,t))-Uqmq(c,t)=e=0; no14(c,t).. (Tzmz(c,t)*YTplp(c,t))-Uzmz(c,t)=e=0; no15(c,t).. (Tfmf(c,t)*YTplp(c,t))-Ufmf(c,t)=e=0; no16(c,t).. UT(c,t)=e=Udmd(c,t)+Ujmj(c,t)+ Ukmk(c,t)+Ulml(c,t)+ Uqmq(c,t)+ Uzmz(c,t)+ Ufmf(c,t); no17(c,t).. Tplp(c,t)-Tdmd(c,t)-Tjmj(c,t)-Tkmk(c,t)-Tlml(c,t)Tqmq(c,t)-Tzmz(c,t)- Tfmf(c,t)=e=0; **** Middle East gas and elec demand ********************************** no18(c,t).. (Tglg(c,t)*YTglg(c,t))+(Rgi(c,t))-Ugmg(c,t)=e=0; no19(c,t).. (Teqe(c,t)*YTeqe(c,t))+(Rei(c,t))-Ueme(c,t)=e=0; **** World oil product demand ***************************************** no20(c,t).. (Toso(c,t)*YToso(c,t))-Uowo(c,t)=e=0; no21(c,t).. (Tpsp(c,t)*YTpsp(c,t))-Upwp(c,t)=e=0; no22(c,t).. (Tgsg(c,t)*YTgsg(c,t))-Ugwg(c,t)=e=0; no23(c,t).. Uowo(c,t)-Uouo(c,t)-Uoco(c,t)-Uoso(c,t)-Uoeo(c,t)Uoao(c,t)-Uofo(c,t)=e=0; no24(c,t).. Upwp(c,t)-Upup(c,t)-Upcp(c,t)-Upsp(c,t)-Upep(c,t)Upap(c,t)-Upfp(c,t)=e=0; no25(c,t).. Ugwg(c,t)-Ugug(c,t)-Ugcg(c,t)-Ugsg(c,t)-Ugeg(c,t)Ugag(c,t)-Ugfg(c,t)=e=0;

Appendix A (GAMS Codes)

167

*************** Conversion Constraints **************************************** no26(c,t).. (Xppe(c,t)*YXppe(c,t))+(Xgpe(c,t)*YXgpe(c,t))+(Xrpe (c,t)*YXrpe(c,t))-Teqe(c,t)=e=0; no27(c,t).. (Xppe(c,t)*YXppe(c,t))-(((HXfpe(c,t))*PFXppe(c,t))/ 11.6)=l=0; no28(c,t).. (Xgpe(c,t)*YXgpe(c,t))-(((HXfpe(c,t))*PFXgpe(c,t))/ 11.6)=l=0; no29(c,t).. (Xrpe(c,t)*YXrpe(c,t))-((((HXrpe(c,t))*PFXrpe(c,t)))/ 11.6)=l=0; *************** Processing Constraints **************************************** ***** Middle East Oil product for Oil Rfinery no30(c,t).. (Adom(c,t)*YAdkp(c,t))-Addd(c,t)=e=0; no31(c,t).. (Adom(c,t)*YAjkp(c,t))-Ajdj(c,t)=e=0; no32(c,t).. (Adom(c,t)*YAkkp(c,t))-Akdk(c,t)=e=0; no33(c,t).. (Adom(c,t)*YAlkp(c,t))-Aldl(c,t)=e=0; no34(c,t).. (Adom(c,t)*YAqkp(c,t))-Aqdq(c,t)=e=0; no35(c,t).. (Adom(c,t)*YAzkp(c,t))-Azdz(c,t)=e=0; no36(c,t).. (Adom(c,t)*YAfkp(c,t))-Afdf(c,t)=e=0; ******* Middle East oil producttransport************************************* no37(c,t).. Rdi(c,t)+Addd(c,t)-Tdmd(c,t)=e=0; no38(c,t).. Rji(c,t)+Ajdj(c,t)-Tjmj(c,t)=e=0; no39(c,t).. Rki(c,t)+Akdk(c,t)-Tkmk(c,t)=e=0; no40(c,t).. Rli(c,t)+Aldl(c,t)-Tlml(c,t)=e=0; no41(c,t).. Rqi(c,t)+Aqdq(c,t)-Tqmq(c,t)=e=0; no42(c,t).. Rzi(c,t)+Azdz(c,t)-Tzmz(c,t)=e=0; no43(c,t).. Rfi(c,t)+Afdf(c,t)-Tfmf(c,t)=e=0; no44(c,t).. Rpi(c,t)-Rdi(c,t)-Rji(c,t)-Rki(c,t)-Rli(c,t)- Rqi(c, t)-Rzi(c,t) -Rfi(c,t)=e=0; no45(c,t).. (Aokpx(c,t)*YAokp(c,t))-Tpsp(c,t)=e=0; no46(c,t).. An(c,t)-Addd(c,t)-Ajdj(c,t)-Akdk(c,t)-Aldl(c,t)-Aqdq (c,t)-Azdz(c,t)-Afdf(c,t)=e=0; no47(c,t).. Aokp(c,t)=e= An(c,t)+ Aokpx(c,t); no48(c,t).. Adom(c,t)=e=AAA(c,t)- Aokpx(c,t); ********* Middle East Oil and Gas and Upstream Capacity************************* no49(c,t).. (Agng(c,t)*YAgng(c,t))-Tglg(c,t)-Tgsg(c,t)=e=0; no50(c,t).. (Agng(c,t)*YAgng(c,t))-(HAgng(c,t))=l=0; no51(c,t).. (Acuo(c,t)*YAcuo(c,t))-Aokp(c,t)-Toso(c,t)=e=0; *************** Recourse Constraints ***************************************** no52(c,t).. Agng(c,t)-Rgg(c,t)-Exg(c,t)=l=0; no53(c,t).. Xrpe(c,t)-Rrr(c,t)=l=0; no54(c,t).. Acuo(c,t)=l= Rref(c,t)+Exo(c,t);

168

Appendix A (GAMS Codes)

*-------------------------Supply Energy--------------------------------------no55(c,t).. Oilsupply(c,t)=e=Acuo(c,t); no56(c,t).. Gassupply(c,t)=e=Agng(c,t); no57(c,t).. TherPower(c,t)=e=(Xppe(c,t)*YXppe(c,t))+(Xgpe(c,t) *YXgpe(c,t)); no58(c,t).. RenPower(c,t)=e= Xrpe(c,t)*YXrpe(c,t); no59(c,t).. AAA(c,t)=e=((HAokp(c,t))*PFAokp(c,t)/(24*7.33*1000)); no60(c,t).. AGG(c,t)=e=(((HXfpe(c,t))*PFXgpe(c,t))/11.6); no61(c,t).. Baoil(c,t)=e=UT(c,t)+ Upwp(c,t)+Uowo(c,t)-Oilsupply(c, t)-Rpi(c,t); ************* Objective function ********************************************* FT.. TC=e=sum((c,t),(Au(c,t)*(CFT(c,t)+CFX(c,t)+CFA(c,t))) +(CVT(c,t)+CVX(c,t)+CVA(c,t)+TRC(c,t)-TXR(c,t))); MODEL Middleeast/ALL/; SOLVE Middleeast USING LP MINIMIZATION TC;