Optimization in Renewable Energy Systems. Recent Perspectives [1st Edition] 9780081012093, 9780081010419

Table of contents :
Content:
Front-matter,Copyright,List of ContributorsEntitled to full textChapter 1 - Introduction to Renewable Energy Systems, Pages 1-26
Chapter 2 - Introduction to Optimization, Pages 27-74
Chapter 3 - Optimal Procurement of Contingency and Load Following Reserves by Demand Side Resources Under Wind-Power Generation Uncertainty, Pages 75-116
Chapter 4 - Optimum Bidding of Renewable Energy System Owners in Electricity Markets, Pages 117-158
Chapter 5 - Impacts of Accurate Renewable Power Forecasting on Optimum Operation of Power System, Pages 159-175
Chapter 6 - Optimum Transmission System Expansion Offshore Considering Renewable Energy Sources, Pages 177-231
Chapter 7 - Optimum Sizing and Siting of Renewable-Energy-based DG Units in Distribution Systems, Pages 233-277
Chapter 8 - Optimum Design of Small-Scale Stand-Alone Hybrid Renewable Energy Systems, Pages 279-306
Index, Pages 307-313

Citation preview

Optimization in Renewable Energy Systems

Optimization in Renewable Energy Systems Recent Perspectives

Edited by

Ozan Erdinc¸, PhD Yildiz Technical University, Istanbul, Turkey

Butterworth-Heinemann is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright r 2017 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-101041-9 For Information on all Butterworth-Heinemann publications visit our website at https://www.elsevier.com/books-and-journals

Publishing Director: Joe Hayton Senior Editorial Project Manager: Kattie Washington Production Project Manager: Kiruthika Govindaraju Designer: Mark Rogers Typeset by MPS Limited, Chennai, India

List of Contributors Amirsaman Arabali LCG Consulting, Los Altos, CA, United States James B. Bassett University of Washington, Bothell, WA, United States Jose´ L. Bernal-Agustı´n University of Zaragoza, Zaragoza, Spain Abebe W. Bizuayehu C-MAST, University of Beira Interior, Covilha˜, Portugal Joa˜o P.S. Catala˜o ´ C-MAST, University of Beira Interior, Covilha˜, Portugal; INESC-ID, Instituto Superior Tecnico, University of Lisbon, Lisbon, Portugal; INESC TEC and Faculty of Engineering of the University of Porto, Porto, Portugal Ting Dai Siemens, Greater Minneapolis-St. Paul, MI, United States Rodolfo Dufo-Lo´pez University of Zaragoza, Zaragoza, Spain Yavuz Eren Yildiz Technical University, Istanbul, Turkey Desta Z. Fitiwi C-MAST, University of Beira Interior, Covilha˜, Portugal Mahmoud Ghofrani University of Washington, Bothell, WA, United States Madeleine Gibescu Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands ˙Ibrahim B. Ku¨c¸u¨kdemiral Yildiz Technical University, Istanbul, Turkey; Glasgow Caledonian University, Glasgow, Turkey Juan M. Lujano-Rojas ´ C-MAST, University of Beira Interior, Covilha, Portugal; INESC-ID, Instituto Superior Tecnico, University of Lisbon, Lisbon, Portugal Moein Moeini-Aghtaei Sharif University of Technology, Tehran, Iran Gerardo J. Oso´rio C-MAST, University of Beira Interior, Covilha, Portugal Nikolaos G. Paterakis Eindhoven University of Technology, Eindhoven, The Netherlands

xi

xii

List of Contributors

My Pham University of Washington, Bothell, WA, United States ´ Sergio F. Santos C-MAST, University of Beira Interior, Covilha˜, Portugal Miadreza Shafie-khah DII, University of Salerno, Salerno, Italy ˘ Akın Ta¸scıkaraoglu Yildiz Technical University, Istanbul, Turkey Shahab S. Torbaghan Vlaamse Instelling voor Technologisch Onderzoek (VITO), EnergyVille, Poort Genk, Belgium ˙Ilker U¨stoglu ˘ Yildiz Technical University, Istanbul, Turkey

CHAPTER

INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

1

´ Sergio F. Santos1, Desta Z. Fitiwi1, Miadreza Shafie-khah2, Abebe W. Bizuayehu1 and Joa˜o P.S. Catala˜o1,3,4 1

C-MAST, University of Beira Interior, Covilha˜, Portugal 2DII, University of Salerno, Salerno, Italy ´ INESC-ID, Instituto Superior Tecnico, University of Lisbon, Lisbon, Portugal 4INESC TEC and Faculty of Engineering of the University of Porto, Porto, Portugal

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1.1 AN OVERVIEW OF RENEWABLE ENERGY SYSTEMS All societies need energy services to satisfy their needs (such as cooking, lighting, heating, communications, etc.) and to support productive services. In order to secure sustainable development, the delivery of energy services needs to be safe and cause low environmental impacts [1 3]. Social sustainability and economic development require security and easy access to energy resources, which are indispensable to promote sustainable energy and essential services. This means applying different strategies at different levels to revamp economic development. To be environmentally benign, energy services should provoke low environmental impacts, including greenhouse gas (GHG) emissions. According to the study in Ref. [4], fossil fuels are still the main primary energy sources. A major revolution is required in how energy is produced and used in order to preserve a sustainable economy capable of providing the required public services (both in the developed and developing countries), and laying effective support mechanisms to climate change mitigation and adaptation efforts [5]. A major concern in both the developed and developing countries, including emerging economies, is that without having abundant and accessible energy sources, it is not possible to maintain the current paradigm in the medium and long term, from an economic point of view. In accordance with the International Energy Agency (IEA) reference scenario, the primary global energy consumption will grow between 40% and 50% until 2030, at an annual average rate of 1.6%. Without a major paradigm shift in energy policies throughout the world, fossil fuels are still expected to cover about 83% of the increase in demand [4]. The reasons for this strong growth are essentially two: The continuous increase in world population and the economic convergence between the developed and developing countries, especially with emerging economies such as India and China that are leading the economic recovery from the recent global economic crisis, and becoming the major consumers of nonrenewable energy sources. This change must be answered with structural measures, such as by putting a real monetary value to energy. Some of the promising solutions are accelerating renewable energy integration, Optimization in Renewable Energy Systems. DOI: http://dx.doi.org/10.1016/B978-0-08-101041-9.00001-6 Copyright © 2017 Elsevier Ltd. All rights reserved.

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CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

promoting energy efficiency, and supporting transport systems modernization. This can be achieved by promoting more transparent markets to flourish and creating an enabling environment for competition in all sectors of the economy and energy production [6]. The sustainability of energy systems is now an important factor for socioeconomic development. Sustainability depends on three major components (as schematically demonstrated in Fig. 1.1): (1) The security of access to energy, (2) the accessibility of services, and (3) environmental compatibility. Changing the energy scenario presents itself as a huge challenge whose solution ultimately depends on the political will of governing bodies to make the necessary investments on a global scale. In the medium and long-term horizon, investment decisions will affect the cost and the environmental impacts of infrastructures. Most likely, the energy supply will be the main factor of possible models for future development at global, regional, and national levels.

FIGURE 1.1 Sustainability in the electricity sector. Adapted from Ref. [23].

1.1 AN OVERVIEW OF RENEWABLE ENERGY SYSTEMS

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1.1.1 CLIMATE CHANGE GHG emissions associated with energy services are the major causes of climate change. The report in Ref. [4] indicates, “most of the observed increase in global average temperature since the mid20th century is very likely due to the observed increase in anthropogenic greenhouse gas concentrations”. The carbon dioxide (CO2) concentrations have grown continuously to about 390 ppm of CO2 in 2010, a 39% increase since preindustrial levels [7]. The global average temperature increased by 0.76 C (from 0.57 C to 0.95 C) between 1850 and 1899. And between 2001 and 2015, the warming trend has increased significantly. Note that forest abatement, fires, and the release of non-CO2 gases from industry, trade, and agriculture also contribute to global warming [7]. Moreover, all indicators show that there will be a significant increase in demand for primary energy during the 21st century [7]. The emission rates are also expected to substantially exceed the natural removal rates, causing a continuous increase in GHG concentrations in the atmosphere, and consequently, the rise in average global temperature. The Cancun agreement [8] appeals to reduce GHG emissions and limit the global average temperature rise below 2 C, taking the preindustrial value as a reference. It has recently been agreed on to a target level of 1.5 C in the average temperature rise. Historically, the developed countries are the main contributors to global CO2 emissions, and continue to have the highest total history of per capita emissions [8]. In recent years, GHG emissions in most developing countries have been increasing, currently, covering more than half of the total emissions. For instance, the total annual emissions in China surpassed that of the USA in recent years [8]. However, the latest climate conference, COP-21 (UN climate conference) in Paris [9], brings hope for the fight against climate change where, for the first time, representatives of almost every country in the world convened together in an effort to reduce emissions and counter the effects of global warming. The Paris Agreement, which will take effect after 2020, underscores the fact that the participation of all nations—Not just rich countries—Is crucial to combat climate change. On the whole, 195 member countries of the UN Climate Convention and the European Union have ratified the document [9]. The long-term goal of the agreement is to keep global warming “well below 2 C.” This is the point in which scientists argue that the planet is doomed to a future of no return, leading to devastating effects such as rising sea levels, extreme weather events (droughts, storms, and floods), and lack of water and food.

1.1.2 RENEWABLE ENERGY TREND An increase in an overall world trend in the awareness of climate change and the need for mitigation efforts is bringing forth huge increase in the deployment of renewable energy in comparison to fossil fuel energy sources. The landmark that signals the dawning of this renewable age goes hand in hand with the degree of advancement in technologies and a higher degree of renewable energy system (RES) penetration, which is being achieved around the world. Furthermore, there are several driving factors for these remarkable growths among which are favorable government support policy and increasing competitiveness in costs. After several decades of efforts in research and continuous development in RES, the yearly growth in the capacity of these plants is becoming greater than the total investment capacity added in power plants based on coal, natural gas, and oil all combined together [10]. Nowadays, RESs have reached a significant level of share in energy supply options, becoming one of the prominent global alternative power supply sources. This trend will continue

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increasing at faster rates as long as the world’s desire for industrial scale clean energy sources is on the higher side [4]. The latest global trends in renewable energy investment status reports indicate that renewables represented a 58.5% of net additions to global power capacity in 2014, with significant growth in all regions, which represents an estimated 27.7% of the world’s power generating capacity, enough to supply an estimated 22.8% of global electricity. Wind, solar, and biomass power generations reached an estimated 9.1% of the world’s electricity in 2014, up from 8.5% in 2013. According to renewables status report [11], the overall cost-cutting achieved to date helped to ensure such a strong momentum in 2014, reaching an investment boom up to 29% in solar, and 11% in wind technologies, and geothermal managing to raise 23%. Further cuts in the cost of generation for both solar and wind look to be on the cards in 2015 [11]. The report on global renewable energy 2015 [11] also indicates the continued growth of RES participation in parallel proportion with the energy consumption and the falling oil prices. In addition, issues related to the untapped RES potentials indicate that it still requires a growing effort in pursuing innovative approaches to increase its participation in order to guarantee a clean energy future. Concerning the regional expansion of RES utilization, such growth scheme is not limited to the industrialized regions, but also an increasing number of developing countries are even becoming important manufacturers and installers of this fashionable energy source. Another essential growth trend currently being observed, which is worth mentioning here, is the diversity of applications of the renewable sources. The use of renewables is no more limited to the power generation only, but its use is expanding in heat related and transpiration applications. In this regard, several supporting technologies like heat supply and storage systems are helping flourish the deployment of these important energy resources across many countries. Also, a significant contribution to the world transport sector is being promoted with an increased share in the use of ethanol and biodiesel in combination with fossil fuels. In relation to the job creation opportunities, renewable energy employment continues expanding, which according to International Renewable Energy Agency (IRENA) [12], in 2014; an estimated 7.7 million people are working directly or indirectly in this sector. Also, concerning government policies, the number of countries, states, and provinces which adopted renewable policies and targets tripled since 2004. Regarding investment mechanisms, innovative approaches have been introduced like in the case of Asian investment banks, representing a new investment vehicles for renewable energy projects such as green bonds, yield companies, and crowd funding which have attached new classes of capital providers and are helping to reduce the cost of capital for financing renewable energy projects [11]. As a result, the investment flow in renewables has outpaced fossil fuels for five consecutive years in all regions. According to the global status report [11], at present, there is no systematic linkage between the so called renewable energy twin pillars: The renewable energy sources and energy efficiency, in technical as well as policy wise.

1.2 TYPES OF RENEWABLE ENERGY SYSTEMS 1.2.1 INTRODUCTION A major change in the energy sector between 2014 and 2015 has been the rapid fall of oil prices, as well as natural gas and coal but not so drastically. After an extensive period of stable high oil

1.2 TYPES OF RENEWABLE ENERGY SYSTEMS

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Table 1.1 Renewable Electric Power Capacity (Top Regions/Countries in 2013) [8] World

EU-28

Brics

China

Technology Wind power Solar energy PV Solar energy CSP Geothermal Hydropower Bioenergy Ocean power Total renewable power capacity

United States

Germany

Spain

Italy

India

34 36 B0 B0 5.6 8.1 0 78

23 5.6 2.3 0 17.1 1 B0 32

8.6 17.6 B0 0.9 18.3 4 0 31

20 2.2 0.1 0 44 4.4 0 27

(GW) 318 139 3.4 12 1000 88 0.5 560

117 80 2.3 1 124 35 0.2 235

115 21 0.1 0.1 437 24 B0 162

91 19.9 B0 B0 260 6.2 B0 118

61 12.1 0.9 3.4 78 15.8 B0 93

price, it has been falling from more than $100 until the middle of 2014 to a level below $50 at the beginning of 2015 [12]. In 2016, further fall is observed, and as it stands now the price of oil oscillates around 30 $/barrel. Renewable technologies are becoming increasingly competitive in a number of countries but government support is still needed to enhance the development of these schemes in many other countries. The capacity increase of base renewable generation is estimated to be 128 GW in 2014 (Table 1.1) [8], out of which 37% is related to wind, nearly a third to solar energy and more than one quarter to hydropower [8] (see in Fig. 1.2). The growth in installed wind power capacity has been developed mainly onshore, but offshore wind development has also shown substantial. China continues to have the largest wind power market with a 20 GW installed capacity. Germany stands second by installing more than 5 GW of wind power, whereas the wind capacity added in the United States was at a very low level in 2013 and 2014, standing at almost 5 GW [13]. The solar photovoltaic (PV) was greatly expanded in Asia, especially in China and Japan. In Japan, the expansion is supported by generous feed-in tariffs. The low price of oil proves to be a challenge for other forms of renewable energy, including biofuels for transport and renewable-based heating system, as the latter directly competes with natural gas-based heating (whose price is still, in many cases, linked to the oil price). Although biofuels face challenges stemming from lower oil prices, some other developments served to improve their prospects. For instance, to overcome the current dark prospects of biofuels in Brazil, the government increased the ethanol rate of mixture from 25 to 27% and from 5 in biodiesel to 7%, and increased gasoline taxes, whereas Argentina and Indonesia have increased their biofuels mandates [8]. A long period of low oil prices could result in neglecting the promotion of energy efficiency and instead returning to wasteful consumption. However, there is no evidence to date that this has occurred [10].

1.2.2 WIND POWER Wind power has been used for thousands of years in a variety of applications. Wind energy can be transformed into mechanical energy or electricity. But wind power remained in the background in

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CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

FIGURE 1.2 (A) Average annual growth rates of renewables (2008 13); (B) global electricity production (2013). Adapted from Ref. [8].

detriment of other fuels for various technical, social, and economic reasons. It was the oil crisis in the 1970s, which led to a renewed interest in wind power technology especially for electricity generation connected to the grid, to pump water, and to provide power in remote areas. The technical potential of wind power to serve the energy needs is immense. Although wind resource varies around the globe, there is enough potential in most regions to support high levels of wind power generation. Wind resources are not a barrier to the global expansion of this technology in the coming decades. New wind power technologies have contributed to significant advances in wind power penetration. In a general perspective, the global wind power capacity has been increasing [14], smoothly from 2000 to 2006, and in a more accentuated way from 2007 to 2013 as shown in Fig. 1.3. More than 51 GW of wind power were added to the power systems, representing a 44% increase compared with 2013, making an overall contribution of approximately 370 GW to the energy production mix, as shown in Table 1.2. The top 10 countries accounted for 84% of the installed capacity in the world at the end of 2013, but there are dynamic and emerging markets in most regions [8]. In continental terms, Asia is the one that has successively grown in recent years and holds half of the new capacity added, followed by the European Union in Europe (23% in 2014, compared

1.2 TYPES OF RENEWABLE ENERGY SYSTEMS

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FIGURE 1.3 Wind power total world capacity (2000 13). Adapted from Ref. [14].

Table 1.2 Wind Power Global Capacity and Additions [8] Country

Total End 2012 (GW)

Added 2013 (GW)

Total End 2013 (GW)

China United States Germany Spain India United Kingdom Italy France Canada Denmark Rest of world World total

60.8/75.3 60.0 31.3 22.8 18.4 8.6 8.1 7.6 6.3 4.2 41 283

14.1/16.1 1.1 3.2/3.6 0.2 1.7 1.9 0.4 0.6 1.6 0.7 7 35

75.5/91.4 61.1 34.3/34.7 23 20.2 10.5 8.6 8.3 7.8 4.8 48 318

with about 32% in 2013) and North America, which has grown by 13% in 2014, an 8% less compared with 2013. From Table 1.2, it can be seen that only China accommodates 45% of the new wind added globally, followed by Germany, the United States, and India. Other countries in the top 10 are Canada, the United Kingdom, Sweden, France, and Denmark. Growth in some of the major markets was driven by uncertainty about future policy changes.

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CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

Table 1.3 Wind Power Generation Technologies [8] Technology

Typical Characteristics

Capital Costs (USD/kW)

Wind onshore

• • • • • • • • • •

• • • • • • •

Wind offshore • Wind • Onshore • Small-scale

Turbine size: 1.5 3.5 MW Capacity factor: 25 40% Turbine size: 1.5 7.5 MW Capacity factor: 35 45% Turbine size: up to 100 kW Average: 0.85 kW (global) 0.5 kW (China) 1.4 kW (United States) 4.7 kW (United Kingdom)

925 1470 (China and India) 1500 1950 (elsewhere) 4500 5500 (global) 2250 6250 (OCDE) 2300 10,000(United States) 1900 (China) 5870 (United Kingdom)

The significant wind power growth is due to the continuous technological advances and relative maturity, supporting mechanisms and incentive packages, favorable policy, continuously falling capital costs among others. Table 1.3 [8] shows the main wind power generation technologies, their technical characteristics and associated costs. The three main turbine types are classified by their sizes and deployment site (onshore or offshore). For each technology, two types of capital costs are shown in US dollars per kW and the typical costs of energy production in US cents per kWh.

1.2.3 SOLAR ENERGY 1.2.3.1 Solar photovoltaic Solar energy is the main and largely inexhaustible source of energy for most countries [15]. In recent years, the deployment of PV has been breaking records year after year. After nearly a stagnated period, it has steadily grown to be one of the leading technologies in terms of installed capacity. More than 39 GW has been added in 2013, bringing the total installed capacity to over 139 GW in this technology by 2013. There has been a geographical shift of the biggest installers, led by China, Japan, and the United States, and Asia is becoming the largest solar PV market worldwide instead of Europe. China have witnessed higher growth than Europe, and other promising markets such as the United States and others have experienced an extremely slow growth [16]. In 2013, nine countries added more than 1.0 GW of solar PV to their networks, and new facilities continue to appear as can be seen in Table 1.4. At the end of 2013, at least 10 GW of total capacity was added in five countries instead of two in 2012. The leaders of solar energy per capita were Germany, Italy, Greece, Czech Republic, and Australia [8]. The Asia added 22.7 GW at the end of 2013, bringing the total amount of solar PV in operation to almost 42 GW. China had almost a one-third share of the global installed capacity. Apart from Asia, about 16.7 GW were added around the world, mainly in the European Union (about 10.4 GW) and North America (5.4 GW) in which the United States led became the third largest market in 2013. The solar PV technologies can be divided into two main types depending on where they are placed, rooftop or ground-mounted. Each has a set of characteristics. However, it can be said that there are three transverse characteristics of both types of technologies: The peak capacity, the

1.2 TYPES OF RENEWABLE ENERGY SYSTEMS

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Table 1.4 Solar PV Global Capacity and Additions [8] Country

Total End 2012 (GW)

Added 2013 (GW)

Total End 2013 (GW)

Germany China Italy Japan United States Spain France United Kingdom Australia Belgium Rest of world World total

32.6 7.0 16.4 6.6 7.2 5.4 4.0 1.8 2.4 2.7 13.8 100

3.3 12.9 1.5 6.9 4.8 0.2 0.6 1.5 0.8 0.2 6.5 39

35.9 19.9 17.6 13.6 12.1 5.6 4.6 3.3 3.3 3.0 20.2 139

Table 1.5 Solar Energy Technologies [8] Technology

Typical Characteristics

Capital Costs (USD/kW)

Solar PV (rooftop)

Peak capacity: • 3 5 kW (residential); • 100 kW (commercial); • 500 kW (industrial) Capacity factor: • 10 25% (fixed tilt)

Solar PV (groundmounted; utility-scale)

• Peak capacity: 2.5 250 MW • Capacity factor: 10 25% (fixed tilt) • Conversion efficiency: 10 30%

Residential costs: • 2200 (Germany); • 3500 700 (USA) • 4260 (Japan); • 2150 (China); • 3380 (Australia); • 2400 3000 (Italy) Commercial costs: 3800 (United States); • 2900 3800 (Japan) • 1200 1950 (typical global); • As much as 3800 including Japan. • Averages: 2000 (United States); 1710 (China); 1450 (Germany); 1510 (India)



LCOE—Levelized cost of energy.

capacity factor, and the conversion efficiency (as depicted in Table 1.5 [8]). These technologies are also distinguished based on where/how they are being deployed: Residential, commercial, and industrial consumer in particular with respect to peak capacity. Note that the receptivity of each type of technology varies greatly from one geographical area to another mainly due to the differences in energy cost of each of these areas, and often the incentives/compensation for the adherence to these technologies [15].

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1.2.3.2 Concentrated solar power The concentrated solar power (CSP) is a market that is so far small, but it is growing mainly thanks to the increased efficiency levels in places with direct sunlight and low humidity. This technology continues to spread to new markets with significant projects already completed in late 2013 in Australia, Italy, and the United States and progress in Chile, Namibia, Portugal, Saudi Arabia, among others [8]. The biggest market is China with 50 MW. More than 165 MW were added in systems operating in over 20 countries led by China and the United States, as can be seen in Table 1.6 [8]. The solar thermal power market continues to grow after the record in 2012. In 2013, the overall capacity grew by 36%, more than 3.4 GW, with Spain and the United States being the major markets [15]. A summary of the main CSP technologies can be found in Table 1.7 [8], which shows the main characteristics as well as the cost of these technologies by country or area. In the typical characteristics, the main types of CSP, the size of the plants and the capacity factor are shown.

Table 1.6 Concentrating Solar Thermal Power (CSP) [8] Country

Total End 2012 (MW)

Added 2013 (MW)

Total End 2013 (MW)

Spain United States United Arab Emirates India Algeria Egypt Morocco Australia China Thailand World total

1950 507 0 0 25 20 20 12 0 5 2540

350 375 100 50 0 0 0 0 10 0 885

2300 882 100 50 25 20 20 12 10 5 3425

Table 1.7 Solar Energy Technologies [8] Technology

Typical Characteristics

Capital Costs (USD/kW)

Concentrating solar thermal power (CSP)

• Types: parabolic, trough, tower, dis • Plant size: • 50 250 MW (trough); • 20 250 MW (tower); • 10 100 MW (Fresnel) • Capacity factor: • 20 40% (so storage); • 35 75% (with storage)

• • • • • •



LCOE—Levelized cost of energy.

Trough, no storage: 4000 7300 (OCDE) 3100 4050 (not OCDE) Trough, 6 h storage: 7100 9800 Tower: 5600 (United States without storage) • 9000 (United States with storage)

1.2 TYPES OF RENEWABLE ENERGY SYSTEMS

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1.2.4 GEOTHERMAL POWER Geothermal energy can be used efficiently in the development of networks whether they are connected or not and is especially useful in rural electrification schemes and direct applications such as district heating, cooking, bathing and industrial processes, and so on [17]. Geothermal resources provide energy in electrical form and heating/direct cooling. Global electricity generation from geothermal sources is estimated to be just under half of the total geothermal production of 76 TW h, with the remaining 91 TW h accounting for direct use [17]. From Table 1.8, it can be seen that, in 2013, the estimated generation capacity added was at least 530 MW, bringing the total global capacity to 12 GW, with an estimated annual generation of 76 TW h [8]. The countries that added more production capacity in 2013 were New Zealand, Turkey, the United States, Kenya, Mexico, Philippines, Germany, Italy, and Australia. By the end of 2013, the countries with the largest installed generation capacity were the United States with 3.4 GW, the Philippines with 1.9 GW, Indonesia with 1.3 GW, Mexico with 1.0 GW, Italy with 0.9 GW, New Zealand with 0.9 GW, Iceland with 0.7 GW, and Japan with 0.5 GW. This resource can be classified into two categories, which are as follows [18]: •

•

High temperature (T . 150 C): This resource is usually associated with areas of volcanic activity, seismic, or magma. At these temperatures, it is possible to use the geothermal resource for power generation purpose; Low temperature (T , 100 C) generally results from the meteoric rise of water circulation in faults and fractures as well as resident water inside porous rocks at deep underground.

The geothermal energy conversion process involves energy transfer by convection, transforming the heat produced and contained inside the earth into a useful energy in the form of electricity or other forms. The energy can also be extracted using the water injection technology from the surface in hot rock formations. Table 1.9 summarizes some general characteristics of geothermal technologies.

Table 1.8 Geothermal Power Global Capacity and Additions [8] Top Countries by Total Capacity

Net Added 2013 (MW)

Total End 2013 (MW)

United States Philippines Indonesia Mexico New Zealand Top countries by new additions New Zealand Turkey United States Kenya Philippines Mexico World total

507 0 0 25 20 (MW) 196 112 84 36 20 10 465

375 100 50 0 0 (MW) 0.9 0.3 3.4 0.2 1.9 1.0 12

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Table 1.9 Geothermal Power Technology [8] Technology

Typical Characteristics

Capital Costs (USD/kW)

Geothermal power

• Plant size: 1 100 MW • Capacity factor: 60 90%

• Condensing flash: 1900 3800 • Binary: 2250 2200

Table 1.10 Hydro Power Global Capacity and Additions [8] Top Countries by Total Capacity

Net Added 2013 (GW)

Total End 2013 (GW)

China Brazil United States Canada Russia India Top countries by new additions China Turkey Brazil Vietnam India Russia World total

28.7 1.5 0.2 0.5 0.7 0.8 (GW) 28.7 2.9 1.5 1.3 0.8 0.7 40

260 86 78 76 47 44 (GW) 260 22 86 14 44 47 1000

1.2.5 HYDRO POWER The production of hydroelectricity is mainly through hydroelectric plants, which are associated with large-capacity or medium-capacity dams, forming a reservoir of water by interrupting the flow of water. Also, this energy has been exploited by applying the so-called small hydro plants, which consist of the construction of small reservoirs or dams that divert part of rivers for an unleveled location (where the turbines are installed), thereby producing electricity. The production of hydroelectricity is the most efficient and one of the least polluting processes. Many of the effects are reversible, and nature with the human contribution, ultimately find a new balance. The capacity of global hydropower production in 2013 increased by 4% (approximately 40 GW), which varies every year according to the metrological conditions of the places where they are located, was estimated at 3750 TW h in 2013. The countries with the highest production capacity are China (260 GW/905 TW h), Brazil (85.7 GW/415 TW h), the United States (78.4 GW/ 269 TW h), Canada (76.2 GW/388 TW h), Russia (46.7 GW/174.7 TW h), India, and Norway, which together have 62% of global installed capacity (Table 1.10) [4].

1.2 TYPES OF RENEWABLE ENERGY SYSTEMS

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Table 1.11 Hydropower Power Technologies [8] Technology

Typical Characteristics

Capital Costs (USD/kW)

Hydropower (grid-based)

• • • • • •

• Projects . 300 MW: 1000 2250 • Projects 20 300 MW: 750 2500 • Projects , 20 MW: 750 4000

Hydropower (off-grid/rural)

Plant size: 1 MW 18,000 MW Plant type: Reservoir, run-of-river Capacity factor: 30 60% Plant size:0.1 1000 kW Plant type: run-of-river, hydrokinetic, diurnal storage

1175 6000

It is estimated that a pumped storage capacity in the order of 2 GW was added in 2013, bringing the global hydropower to 135 140 GW. The country that installed more hydropower capacity in 2013 was China. Other countries with significant installed hydropower capacity were Turkey, Brazil, Vietnam, India, and Russia. Hydropower is the most developed technology among the renewables, reaching levels of optimality when coupled with the wind. The summary of the main technological characteristics can be found in Table 1.11, including the typical technology cost for each type. It can be emphasized that, among the presented technologies, it is the one that has the lowest typical cost.

1.2.6 BIOENERGY Bioenergy is the designation for the energy obtained from biomass. There are three forms of fundamental energy: Heat energy, mechanical energy, and electricity, all of which can be obtained from biomass sources. The systems that produce mechanical energy as combustion engines or turbines of direct and indirect combustion are coupled to electrical generators, which convert mechanical energy into electrical energy. The conversion of mechanical energy to electrical energy generates heat approximately two-third to one-third of the generated electricity, which demonstrates the increased economic efficiency of cogeneration (simultaneous production of heat and electricity) in stationary applications. The biogas from landfills, recycling of agricultural wastes, and other organic wastes can be used in stationary power plants for energy production [19]. Bioenergy has shown steady growth rates in the last years and it is expected to keep on this path in the future. In European Union, the consumption of biomass energy is projected to increase by at least 33 Mtoe by 2020, as shown in Fig. 1.4 [46]. The electricity generation in the European Union from solid biomass in 2014 was approximately 81.6 TW h. The five top producers were the United States followed by Germany, Finland, the United Kingdom, Sweden, and Poland having a total production in Europe of 63% [20]. The estimated growth of bioenergy market in 2014 was 5 GW, bringing the total capacity worldwide to approximately 93 GW. Country wise, the biogeneration leaders are the United States with 69.1 TW h, Germany with 49.1 TW h, China with 41.6 TW h, Brazil with 32.9 TW h, and Japan with 30.2 TW h.

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CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

FIGURE 1.4 Final energy consumption for bioenergy in European Union. Adapted from Ref. [46].

Table 1.12 Ocean Power Technologies [8] Technology

Typical Characteristics

Capital Costs (USD/kW)

Ocean power (tidal range)

Plant size: ,1 to .250 MW Capacity factor: 23 29%

5.290 5870

1.2.7 OCEAN POWER Ocean power refers to any energy harnessed from the ocean waves, tidal range (up and down), tidal currents, ocean currents, temperature gradients, and salinity gradients [21]. The development of this emerging sector can contribute to the achievement set targets for renewable energy integration, reduction of GHGs and their harmful effects, and simultaneously boost economic growth through innovation and new job creation [22]. At the end of 2013, the capacity of the ocean energy was about 530 MW, most of this coming from the tidal power category [4]. Majority of ocean power projects currently in operation generate power from ocean tides. Among these is the Sihwa plant in Korea, completed in 2011 with a capacity of 254 MW, the central Rance in France with a capacity of 240 MW, the central Annapolis Nova Scotia, Canada with a capacity of 3.9 MW, and Jiangxia in China with the capacity of 3.9 MW. Other projects are smaller, and many of them are precommercial demonstration projects with a remarkable concentration of wave and tidal development facilities (in the order of 11 MW) in the United Kingdom [8]. Some of the tidal technology characteristics are summarized in Table 1.12.

1.3 ECONOMIC ASPECTS OF RENEWABLE ENERGY SYSTEMS

15

1.3 ECONOMIC ASPECTS OF RENEWABLE ENERGY SYSTEMS 1.3.1 INTRODUCTION Installation cost, net annual energy production, and value of energy are the three main economic factors to make a decision about employing RESs. The value of energy is equal to the electricity price or tariff for RESs located on the supply side; while it can be equal to the retail price for demand side renewable resources (the systems that use energy on site). The reason for using retail price in studies of RESs on the demand side is that the purchased energy from the grid by the consumer is displaced by the on-site generation. In order to investigate economic feasibility of renewable resources, they have to enter in a competition with other available energy resources and technologies. Fossil fuel prices have had considerable variations over the past 10 years, and there is an uncertainty about these prices in future. Considering carbon emission is also another factor that has affected the fossil fuel cost [23]. In order to concentrate on the fuel cost and its uncertainty in more details, it should be noted that a cost between $0.5/gallon and $1.0/gallon is added to the gasoline cost in the United States that is related to the military expenditures just to ensure the oil flow from the Middle East [24]. In order to improve the penetration level of renewable energy resources in the power system, the installation costs should be returned during a rational period. This would be obtained by producing sufficient power at an appropriate price. In the cases that RESs are installed in places where there is no connection to the power system network, the price of electricity would be high, because it would be obtained by a cost competition with other available energy carriers. On this basis, the electricity price of RESs is associated with the range of prices of the energy carriers. There are many factors that induce uncertainty in the future cost of energy (COE) carriers. These factors are mostly related to the level of dependency on imported energy carriers, policies on emission reduction, as well as policies on developing renewable energy resources [25]. The price of all energy carriers is strongly associated with the price of oil that has been difficult to forecast due to many factors such as political aspects [26]. Fluctuations of oil price in the past few years prove this claim that prediction of energy prices has become more complex. For example, in some reports, the peak time of oil price was forecasted to occur in 2007, whereas other reports forecasted it to happen in 2015 and even 2040 [27]. Note that, as the oil price is also related to the demand growth, a wide range of its fluctuations must be considered for each time and geographic area [28]. It should be noted that, energy economics is highly dependent on incentive- and penalty-driven policies. On this basis, it is very difficult to impose life-cycle costs without considering the impacts of emission and government supports that motivate investors for investing in RESs [29]. According to the aforementioned description, the regulatory supports for RESs have been driving the world market.

1.3.2 AFFECTING FACTORS Many factors affect investors to invest in RESs. Incentives are a key element in choosing the renewable systems, as both type and size of RESs are determined by investors based on the differences between market incentives. Another affecting factor is the cost of land that has an important

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CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

impact on type and number of RESs. In order to have the optimal rate of return, RESs should have the highest amount of availability to ensure they can produce an appropriate level of energy. On this basis, they should have the capability to be operated as much time as possible. To this end, the reliability of the network and consequently the unavailability to transfer generated power due to network failures should be estimated. It should be noted that, if the RES generates when the demand peak occurs, the income of the system is augmented due to the increase in energy price. Owners of on-site renewable resources can also benefit more when the generated energy by the systems is required by the on-site demand. For example, the wind power generator produces electricity at nights when the space heating systems are highly required during winter. RESs are able to generate power to supply the on-site demand, or to inject to the grid. The amount of energy that is consumed by on-site demand is replaced with the supplied energy by the grid. On this basis, if the amount of generation is less than the on-site demand, using RESs can reduce the net load. On the contrary, if the on-site consumption is less than the generated energy, the surplus is injected into the grid at a price/tariff based on an agreement with the utility. Externalities are also an important factor for making a decision about renewable energy resources. This is due to the fact that in life-cycle cost analysis emission and CO2 costs should be considered [30,31]. There is a wide range of rates of externalities that depends on the rules and regulations in various countries. As power producers do not like paying the cost of externalities, in the United States, there are litigations by all sides to decrease the externality rates based on the reason that there is no reliable evidence to prove CO2 emission is harmful to society. In European countries, there are various costs for CO2 emission that provide a better base for renewable energy resources by making them more cost competitive [31].

1.3.3 LIFE-CYCLE COSTS The life-cycle cost is a method that analyzes the total cost of the system by considering the expenditures during the system life and salvage value [13]. By using a life-cycle cost analysis different investment options can be compared. Moreover, the most economic design for the system can be achieved. There are some other options that the RESs must compete with such as small-scale diesel generators and electrical energy storage systems. In this case, the effective factors are the initial cost, the electricity price, and the required infrastructures [32,33]. It is noteworthy that the lifecycle cost is also useful to compare different plans even if the RES is the only choice. Furthermore, this analysis is employed to determine if a hybrid renewable-based system can be the most economic plan. The life-cycle cost analysis enables to investigate impacts of employing several components with various reliability and lifetimes. Discount rate indicates how much increase or decrease in finance happens over the time. Note that, using the inaccurate amount of discount rate for calculating life-cycle cost can cause to unrealistic solutions. Although most of the RESs are economical, in order to select the best plan between all available options, the life-cycle cost analysis is the best method [32,33]. The financial assessment can be carried out over an annual base to calculate economic indices such as payback period and cash flow. It should be mentioned that, annualized COE considered for RESs should be compared with that for other resources. In other words, the annualized COE should not be directly compared with

1.3 ECONOMIC ASPECTS OF RENEWABLE ENERGY SYSTEMS

17

the current COE, as it is not sensible. The mentioned costs of energy create an appropriate base to compare different plans considering alternative resources and to select the best resource of energy. There are some calculating tools to analyze renewable energy projects and assess the life-cycle cost, and even emission considerations [34]. These calculations prove that the current RESs are economical.

1.3.4 ECONOMIC TREND OF RENEWABLE ENERGY SYSTEMS RESs are strongly promoted by policy makers because they are a key element for economic development. The renewable systems are able to compete with current thermal power plants [26]. They can also improve the economy by job creation, as more than 100 jobs for installing a wind power plant and more than 10 jobs for its operation are required for each 100 MW project [24]. There are some attempts to reduce the property tax of wind power plants in order to better motivate the investor to invest in the renewable system and consequently the economic development [35]. Wind power generation is one of the most economic RESs, as its COE is about $70/MW h that can compete with thermal units. Trend studies have shown that before 2003, COE of wind power plants was higher than fuel fossil units; however, in the period of 2003 to 2009, it approximately equaled to the thermal units [24]. In 2009, the COE of wind farms dropped to the levels lower than conventional power. The studies on various resources indicate that annualized COE of wind power plants can even compete with the one of combined cycle gas turbines, although the fuel price of these thermal units is low [36]. In some countries such as the United States, conventional power plants benefit from that the fuel costs are not taxed, whereas the RESs do not have the cost of fuel at all. On this basis, the main issue of renewable resources is the high-investment cost, which causes people to prefer paying for the fuels. It should be noted that, in the case of on-site RESs, the small-scale wind power generation cannot compete with the retail prices [37]. In the case of supply side, another barrier to integration of RESs is the capacity of transmission network that may cause the power curtailment in order to ensure the system security [38]. As most of the renewable energies such as wind, solar, and geothermal resources are far from the load centers, they can impose an extra cost on the transmission system [39]. Although the future of energy is uncertain and ambiguous, and every prediction can be risky, as oil price forecast has been a challenge, the trend of renewable energies is almost evident [28]. On this basis, it can be expected that in future distributed RESs will have more penetration and even some new distributed electricity markets will help these new resources [37]. In addition, with developments of the high-voltage transmission network, large farms of RESs will be installed much far from the load centers [38]. In near future, renewable energy resources can better compete with other energy alternatives just due to the carbon cost. Even other air emission costs such as NOx and SOx will motivate people to invest in systems without fuel and emission costs. This can cause that renewable energies, particularly wind power, become the most economic resource of producing electricity [32,33]. It should be noted that one part of the installation cost of RESs has been supplied by the income resulted from carbon trading. The future of energy without RES developments would be an unsolvable problem due to the growth of environmental concerns. In order to avoid this problem and to

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CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

provide a sustainable energy, policy makers should put more weight on the renewable energy as well as conservation and energy efficiency.

1.4 OPPORTUNITIES AND CHALLENGES OF RENEWABLE ENERGY SYSTEMS INTEGRATION 1.4.1 RENEWABLE ENERGY SYSTEMS INTEGRATION OPPORTUNITIES Most of the electric energy consumed comes from nonrenewable energy sources (mainly, fossil fuels). This has led to a series of questions from energy dependence concerns to climate change issues, which are some of the major drivers of RES integrations in many power systems across the world. It is now widely recognized that integrating RESs in power systems brings about a lot of economic, environmental, societal, and technical benefits to all stakeholders. These are some of the reasons behind the rapid growth of RES integrations in many power systems across the world, as indicated in a 2015 report by the IEA. The report further shows that, in 2013, an approximately 19.1% of global electric energy consumption came from RESs, most of which was from hydropower [4]. Among the wide-range benefits of RESs is their significant contribution to the GHGs which are leading to not only climate change and its dire consequences but also series environmental and health problems. Most RES technologies (wind and solar PV, for instance) have very low carbon footprints, making them very suitable to mitigate climate change and reduce its consequences. Hence, integrating RESs in power systems partly replaces polluting (conventional) power generation sources, resulting in a “cleaner” energy mix that is, one with lower emission levels. RES integration also has an undisputable positive impact on the social and economic development of nations. It is widely understood that the three socioeconomic indicators, per capita income, per capita energy use and economic growth, are highly correlated with each other. Economic growth can be considered, for instance, as the main driver for energy consumption. Therefore, RESs can spur economic growth and create job opportunities. Because of their distributed nature, RES integration can also be integrated into a national policy (especially, in the developing countries) to foster rural development. At present, the RES business currently employs an estimated 7.7 million people throughout the world [4]. RESs also play an important role in energy access. At present, more than 1.2 billion people do not have access to electricity globally (85% of which are in rural areas where RESs are abundant) [4]. Exploiting the potential of RESs should be at the forefront to address this societal problem. Energy security concern is also one of the main drivers of RES integration. Current electric energy production scheme is dominated by conventional generation sources, which use fossil fuels whose prices are subject to significant volatilities. In addition to these volatilities, geopolitical availability of fossil fuels is also becoming a concern for many countries. The combination of all these can have significant impacts on the energy supply security. Because of this, generation of electricity locally using RESs can significantly contribute to the energy security of nations. As a result, this can reduce the heavy dependence on fossil fuels for power generation. In addition to the benefits briefly explained above, RESs can bring technical benefits such as improved system stability and voltage profiles, reduced losses and electricity prices, and so on.

1.4 OPPORTUNITIES AND CHALLENGES OF RESs INTEGRATION

19

FIGURE 1.5 Development and future trend of generation capacity, demand and wholesale electricity market price in Central Europe from 2010. Adapted from Ref. [47].

The combination of conventional generation capacity with the renewable generation capacity will be able to address the continuous increase in demand, in opposition to the scenario of conventional generation capacity only, which according to forecasts would not be able to meet the demand, Fig. 1.5 [47].

1.4.2 RENEWABLE ENERGY SYSTEMS INTEGRATION CHALLENGES AND BARRIERS Despite the robust growth of integration RES in many power systems, there are still certain challenges and barriers that impede the smooth integration of RESs. These challenges and barriers can be broadly classified into two categories: Technical and nontechnical. The nontechnical category includes challenges and barriers related to capital costs, market and economic issues, information and public awareness, sociocultural matters, the conflict between stakeholders, regulation, and policy. The variable COE production by RESs is very small (close to zero); however, they are generally capital intensive. Even if the capital costs are declining for most RES technologies, their levelized costs of energy are yet to be competitive with that of conventional energy sources. This can make investing in RESs less attractive for potential investors. However, this is likely to change as RES costs continue to fall while that of conventional energy sources become more expensive amid resource depletion and policies to internalize external costs such as environmental pollution costs. Market and economic barriers exist when there is a lack of clearly designed economic and financial instruments to support RES integration efforts. For instance, whenever there is market failure associated with internalizing the cost of environmental pollution, it is very difficult to expect a lot of investments coming in from RES developers. Information gap and lack of public awareness on RESs and their benefits can also significantly hinder RES integration. Moreover, sociocultural issues such as conflicting land use requirements can sometimes lead to contentious issues regarding RES development, directly affecting the level of penetration. The conflict between stakeholders is

20

CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

FIGURE 1.6 Main reasons in the EU 27 for issue “Lack of communication.” Adapted from Ref. [48].

another barrier, more specifically the lack of communication, as shown in Fig. 1.6 [48]. The challenges and barriers related to the regulatory and policy issues emanate from the structure of energy industries and existing technical regulations, the level of support for technology transfer and R&D, and so on. Technical limitations (barriers), on the other hand, are related to the nature of the resources and the power systems. Some of the most common RESs depend on primary energy sources such as wind speed, solar radiation and wave, which are subject to high-level variability and unpredictability (the latter also known as uncertainty), resulting in considerable grid integration challenges [40]. This is because the uncontrollable variability and uncertainty of such sources introduce a lot of technical problems in the system, making the real-time operation of the system very challenging. The intermittent nature of power production from these resources also dramatically affects the reliability of energy supply. Moreover, such sources are found geographically dispersed across a vast

1.4 OPPORTUNITIES AND CHALLENGES OF RESs INTEGRATION

21

area, and their availability is site specific. Unlike conventional power sources, these energy sources cannot be transported to areas close to demand centers. This means harnessing these resources requires the higher need for network investments than conventional ones. In addition, RESs based on the aforementioned primary energy sources are characterized by low capacity factors (i.e., low energy production per MW installed) compared to conventional power sources. In other words, the spatial energy intensity of RESs based on these sources is very low. This means that, for the same amount MW installed, the size of land required for such RESs is several times higher than that of the conventional one, which can sometimes be problematic during integration efforts because this creates fierce competition with other land use claims or requirements [41].

1.4.3 ALLEVIATING THE CHALLENGES AND BARRIERS Most of the challenges and barriers explained before have proven solutions that happen to be overlooked in many systems [42]. In general, these are summarized as follows: •

•

•

•

• •

• •

• •

Market and economic barriers are often fixed by streamlining appropriate market and economic signals related to carbon taxes, emission trading schemes, finances, and incentive mechanisms as well as enhancing public support for R&D and creating a conducive environment for RES development. All this can have a considerable positive impact on the level of RES integration. Setting energy standards, continuous information campaigns, and technical training about RESs and their benefits can enhance public knowledge and awareness, which can in the end have supportive roles in RES development. Creating an enabling environment for R&D, improving technical regulations, scaling up international support for technology transfer, liberalizing energy industries, providing incentive packages to RES developers, designing appropriate policies of RESs and conventional energy sources, and others minimize the regulatory and policy barriers to developing RESs. Coordinating investments of RESs based on variable generation resources such as wind and solar power with large-scale energy storage systems, demand side management participation and grid expansion can significantly increase the level of RES integration. Enhancing operation and the flexibility of conventional power generation sources can also be very useful to scale up RES integration. Designing an efficient wholesale market such as dynamic retail pricing and developing coordinated operation and planning tools (such as joint network and generation investment planning models) can have a positive role in RES integration. For full utilization of RESs, the coordination between distribution system operators and transmission system operators is also vital. Ensuring regional interconnections via regional cooperation and increasing the level of participation of all stakeholders (including RES generators) in voltage control, provision of reserves, reactive power support, and others are significantly helpful for the stated purpose. It is also important to improve prediction tools, monitoring and control protocols that can help efficient utilization of the RESs. Using smart-grid technologies and concepts are also expected to facilitate smooth integrations of large-scale RESs because these are equipped with advanced control and management tools to counterbalance the intermittent nature of most RES energy productions.

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CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

1.4.4 CURRENT TRENDS AND FUTURE PROSPECTS During the past decades, the level of global RES integration has been steadily growing. This has been against a number of odds such as the recent global financial crisis, the dramatically falling fuel prices and the slowdown of increasing global electricity consumption that have been thought to decelerate or stall this trend [11]. In general, there is a general consensus globally that RESs will cover a significant amount of electricity consumption in the years to come. The high uncertainty of RESs can be partially solved by the introduction of a bigger operational flexibility, as shown in Fig. 1.7 [49], through coordinated participation of various stakeholders. The recent developments in the 2015 Paris climate conference (COP-21), overall trends in international policy on RESs, energy dependence concerns, the falling capital costs of several matured RES technologies, and other technoeconomic factors are all favorably expected to further accelerate the level RES integrated into power systems.

FIGURE 1.7 Sources of operational flexibility in power systems. Adapted from Ref. [49].

1.6 CONCLUSIONS

23

1.5 THE NEED FOR OPTIMIZATION TOOLS TO SOLVE PROBLEMS RELATED TO RENEWABLE ENERGY SYSTEMS The power system operation must be conducted in order to optimize the production cost and to ensure the security of supply. The classic problem aims to optimize the provided power by each group of service generators, leading to an overall minimum cost of power production while satisfying the load and system security. The availability of powerful calculation methods and efficient algorithms for optimization has made it possible to obtain “optimal” solution of these problems for large electrical systems [43]. The need for optimization tools is indispensable in power system operation and planning in the presence of RESs. This is because of the increased variability and uncertainty introduced to the electric system as a result of integrating variable energy sources [44]. In addition, the demand variability over time and the uncertainty related to unexpected interruptions of generators (or other system components) all suggest the need for efficient optimization tools. Uncertainty is introduced to the system as a result of the increasing level of intermittent integration and uncontrollable generation (like wind or solar energy) because of the limited predictability. Unexpected changes in outputs usually lead to the need for higher levels of reserves for the efficient and reliable operation of the power supply [44,45]. Therefore, when a large amount of intermittent energy is used to feed a conventional electrical network, scheduling has to be taken into account by placing controllable energy sources as a backup to cover variations and unplanned injections of renewable energy. Operators need to know how much energy a renewable plant can deliver in order to gain the best price for each megawatt hour [44]. Therefore, the need to develop optimization tools to solve RES-related problems is crucial to the sustainable electrical system development, so that it operates in a more efficient and reliable way while respecting all operational constraints minimizing energy costs for end users. This is further discussed in subsequent chapters of this book.

1.6 CONCLUSIONS The potential of RESs is colossal because in principle they can meet several times the world demand. RESs such as wind, biomass, hydro, and geothermal can provide sustainable energy services based on available resources in all parts of the world. The transition to renewable-energybased power systems tends to increase, while their costs continuously decline as gas and oil prices continue to oscillate. In the last half century, the demand for wind and solar systems has been continuously increasing, experiencing a reduction in capital costs and generated electricity costs. There have been continuous performance improvement and R&D undergoing in the sector in the past decades. As a result, the prices of renewable energy and fossil fuels, as well as social and environmental costs are to diverge in opposite directions. Economic and political mechanisms must support the wide spread of sustainable markets for the rapid development of RES. At this point, it is clear that the present and future growth will occur mainly in renewable energy and in some natural-gas-based systems, and not common sources like coal or oil. The progress of RESs can increase diversity in the electricity markets, contributing to obtain long-term sustainable energy, helping reduce local

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CHAPTER 1 INTRODUCTION TO RENEWABLE ENERGY SYSTEMS

and global GHG emissions and promote attractive trade options to meet specific energy needs, particularly in the developing countries and rural areas helping to create new opportunities.

ACKNOWLEDGMENTS This work was supported by FEDER funds through COMPETE 2020 and by Portuguese funds through FCT, under Projects FCOMP-01-0124-FEDER-020282 (Ref. PTDC/EEA-EEL/118519/2010), POCI-01-0145-FEDER-016434, POCI-01-0145-FEDER-006961, UID/EEA/50014/2013, UID/CEC/50021/2013, and UID/EMS/00151/2013. Also, the research leading to these results has received funding from the EU Seventh Framework Programme FP7/2007-2013 under grant agreement no. 309048. Moreover, S´ergio Santos gratefully acknowledges the UBI/Santander Totta doctoral incentive grant in the Engineering Faculty.

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[41] GEA, Global energy assessment—toward a sustainable future. Cambridge University Press, Cambridge, UK and New York, NY, USA and the International Institute for Applied Systems Analysis, Laxenburg, Austria, 2012. [42] Fraunhofer IWES (2015): The European power system in 2030: flexibility challenges and integration benefits. An analysis with a focus on the pentalateral energy forum region. Analysis on behalf of Agora Energiewende. [43] J. Pedro Suceno Paiva, 2013 Redes de Energia El´ectrica—Uma An´alise Sist´emica, IST Press. Lisboa. [44] Bazmi AA, Zahedi G. Sustainable energy systems: Role of optimization modeling techniques in power generation and supply—a review. Renewable Sustainable Energy Rev 2011;15(no. 8):3480 500. [45] Bhattacharyya SC, Timilsina GR. Energy demand models for policy formulation: a comparative study of energy demand models. World Bank Policy Res Work Pap Ser 2009 vol. 4866, pp. 144. [46] European Biomass and Association (AEBIOM), 2015 AEBIOM-statistical-report-2015: European bioenergy outlook, European Biomass Association, Brussels. [47] Auer, H., Burgholzer, B., Opportunities, challenges and risks for RES-E deployment in a fully integrated European electricity market, Report D2.1, Market4RES, 2015. Available in: ,http://market4res.eu/wpcontent/ uploads/D2.1_Market4RES_20150217_Final.pdf.. [48] E.B. Zane, R. Bru¨ckmann, and D. Bauknecht, Integration of electricity from renewables to the electricity grid and to the electricity market—RES—integration, Final Report, European Commission Berlin, 2012. [49] Ulbig, A., Anderson, G., Analyzing operational flexibility of power systems. In: Proceedings of power systems computation conference (PSCC), Wrocław, Poland, USB flash drive, Aug 18 22, 2014.

CHAPTER

INTRODUCTION TO OPTIMIZATION

2

˘ 1 Yavuz Eren1, ˙Ibrahim B. Ku¨c¸u¨kdemiral1,2 and ˙Ilker U¨stoglu 1

Yildiz Technical University, Istanbul, Turkey 2Glasgow Caledonian University, Glasgow, United Kingdom

2.1 INTRODUCTION The existence of optimization methods can be traced back to the days of Newton, Lagrange, and Cauchy. Newton and Leibnitz made invaluable contributions to the literature of calculus which allowed the development of differential calculus methods for optimization. However, the minimization of some functions using calculus of variations has been established by Bernoulli, Euler, Lagrange, and Weierstrass. Then, Lagrange introduced the method of adding unknown multipliers to the minimization problem, which led him to cope with constrained optimization problems. For the solution of optimization problems, Cauchy introduced the first application of the steepest descent method to solve unconstrained optimization problems. By the middle of the 20th century, the high-speed digital computers made implementation of the complex optimization procedures possible and stimulated further research on newer methods. Spectacular advances followed, producing a massive literature on optimization techniques. This advancement also resulted in the emergence of several well-defined new areas in optimization theory. Some of the major developments in the area of numerical methods of unconstrained optimization can be outlined as follows. The work on optimization theory was born by the development of the simplex method by Dantzig in 1947 for linear programming (LP) problems [1]. Then, the principle of optimality was presented by Bellman for dynamic programming problems in 1957 [2]. Work by Kuhn and Tucker in 1951 on the necessary and sufficient conditions for the optimal solution of programming problems laid the foundation for later research in non-LP (NLP) [3]. But the most valuable contributions to NLP were made by Zoutendijk and Rosen during the early 1960s [4,5]. Again in the same year, Duffin, Peterson, and Zener developed geometric programming, and Gomory did pioneering work in integer programming, which is one of the most exciting and rapidly developing areas of optimization [6,7]. The reason for this is that most real world applications fall under this category of problems. Also, Dantzig, Charnes, Cooper and, Symons developed stochastic programming techniques and solved problems by assuming design parameters to be independent and normally distributed [8
10]. In the last decade, simulated annealing (SA), genetic algorithms (GAs), and neural network methods were introduced to represent a new class of mathematical programming. Between them, SA is analogous to the physical process of annealing of metals and glass. The GAs are search techniques based on the mechanics of natural selection and natural genetics and finally neural network methods are based on solving the problem using the computing power of a network of interconnected “neuron” processors. Optimization in Renewable Energy Systems. DOI: http://dx.doi.org/10.1016/B978-0-08-101041-9.00002-8 Copyright © 2017 Elsevier Ltd. All rights reserved.

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CHAPTER 2 INTRODUCTION TO OPTIMIZATION

To indicate the widespread scope of the subject, some major applications in different engineering disciplines can be listed as follows: Design of civil engineering structures such as frames, foundations, bridges, towers, chimneys, and dams for minimum cost; design of minimum weight structures for earth quake, wind, and other types of random loading; design of water resources systems for obtaining maximum benefit; design of aircraft and aerospace structure for minimum weight; finding the optimal trajectories of space vehicles; design of pumps, turbines and heat transfer equipment for maximum efficiency; optimum design of electrical machinery such as motors, generators, and transformers; optimum design of electrical networks; finding out the optimal energy production and distribution strategy; optimum design of control systems; optimum design of chemical processing equipments and plants; optimal selection of a site for an industry; optimal planning of maintenance and replacement of equipment to reduce operating costs; allocation of resources or services among several activities to maximize the benefit; controlling the waiting and idle times in production lines to reduce the cost of production; designing the shortest route to be taken by a salesperson to visit various cities in a single tour and optimal production planning, controlling, and scheduling.

2.2 FORMULATION OF OPTIMIZATION PROBLEM In the development of an optimization problem, maybe the most important and the crucial thing is to produce the optimization model of the physical problem, and it requires a very good understanding of the physical process itself together with some expertise. The modeling and building of an optimization problem require the following steps: • • • • •

Data collection, Problem definition and formulation, Model development, Model validation and evaluation of performance, and Model application and interpretation.

Between these steps, the data collection is the most time-consuming step, but it is the fundamental basis of the model-building process. The availability and accuracy of data can have considerable effect on the accuracy of the model and on the ability to evaluate the model. Problem definition and formulation step involves the identification of the decision variables, formulation of the model objective(s), and the formulation of the model constraints. In this step, one must identify the important elements that the problem consists of. Then, the determination of the number of independent variables, the number of equations required to describe the system, and the number of unknown parameters are performed. After this, one needs to evaluate the structure and complexity of the model and finally in the last step, the accuracy of the model is selected. Model development includes the mathematical description, parameter estimation, and software development. Note that the model development phase is an iterative process and it may require returning back to the model definition and formulation phase several times. During the model validation and evaluation of the performance, one checks the performance of the model as a whole. The performance of the model

2.2 FORMULATION OF OPTIMIZATION PROBLEM

29

is to be evaluated using standard performance measures such as root mean squared error or some other metrics. A sensitivity analysis should also be performed in this step to test the model inputs and parameters. This phase also is an iterative process and may require returning to the model definition and formulation phase. Finally, model application and interpretation include the use of the model in the particular area of the solution and the translation of the results into operating instructions issued in understandable form to the individuals who will administer the recommended system.

2.2.1 GENERAL MODELS OF OPTIMIZATION PROBLEMS In mathematics, the term optimization, or mathematical programming, refer to the study of problems in which one tries to minimize or maximize a real function by systematically choosing the values of real or integer variables within an allowed set. Generally, the mathematical programming (optimization) model is a problem of the form c% 9 min f0 ðxÞ subject to fi ðxÞ # 0; i 5 1; . . . ; m x

(2.1)

where xARn is the vector of decision variables, f0 : Rn -R is the objective (cost/penalty) function, fi : Rn -R; i 5 1; . . . ; m stand for the constraints and finally c% is the optimal cost. Decision variables are essential. If there are no variables, we cannot define the objective function and the problem constraints. In many practical problems, one cannot choose the design variable arbitrarily. They have to satisfy certain specified functional and other requirements. A set of constraints, fi ; i 5 1; . . . ; m are those which allow the unknowns to take on certain values but exclude others. However, constraints are not essential in an optimization problem and optimization problems can be defined without any constraints as well. The term programming does not refer to computer coding. It is used instead of scheduling or planning. The term “subject to” often replaced by “:” On the other hand, the set O 5 fxARn : fi ðxÞ # 0; i 5 1; . . . ; mg is called the feasible set. Then, it is obvious that x% AO which satisfies c%5f0 ðx% Þ is called the optimal solution of the optimization problem. Here, a set of unknowns or variables control the value of the objective function. Optimization problems can be classified based on the type of constraints, nature of design variables, physical structure of the problem, nature of the equations involved, permissible values of the decision variables, deterministic nature of the variables, and the number of objective functions. If the optimization problem is subject to one or more constraints, then it is called a constrained optimization problem; otherwise, it is called unconstrained optimization problem. The field of unconstrained optimization is quite a large one, for which a lot of algorithms and software algorithms are available. Constraints are restrictions that must be satisfied to produce an acceptable design. Note that constraints can be broadly classified into two subclasses, behavioral/functional constraints and geometric/side constraints. The former one represents limitations on the behavior performance of the system, whereas the latter one represents physical limitations on design variables such as availability, fabricability, and transportability. The classification of optimization problems can be made on the basis of the nature of the design variables as well. If the design variables are static, then we call these kinds of problems as static optimization problems. In contrary, if the decision variables

30

CHAPTER 2 INTRODUCTION TO OPTIMIZATION

are the functions of some independent variable such as time, then we call these problems as dynamic optimization problems. Based on the physical structure, we can also classify optimization as optimal control (OC) and non-OC problems. An OC problem is a mathematical programming problem involving a number of stages, where each stage evolves from the preceding stage in a prescribed manner. It is defined by two types of variables: The control or design and state variables. The control variables define the system and controls how one stage evolves into the next and the state variables describe the behavior or status of the system at any stage. The problem is to find a set of control variables such that the total objective function (also known as the performance index, PI) over all stages is minimized, subject to a set of constraints on the control and state variables. For OC problems, one of the constraints fi ; i 6¼ 0 is in the form of _ 5 h½xðtÞ; uðtÞ xðtÞ

(2.2)

where x_ denotes the time-derivative of the variable x, and u stands for the control variable. Different from the non-OC problems, the objective function in OC problems is in the form of min f0 ðx; u; tÞ x;u

(2.3)

where t stands for the independent variable, time. An objective function expresses the main aim of the model, which is either to be minimized or maximized. For example, in a manufacturing process, the aim may be to maximize the profit or minimize the cost. In comparing the data prescribed by a user-defined model with the observed data, the aim is minimizing the total deviation of the predictions based on the model from the observed data. In designing a bridge pier, the goal is to maximize the strength and minimize size. Based on the nature of equations for the objective function and the constraints, optimization problems can be classified as linear, nonlinear, geometric, and quadratic programming problems. For example, depending on the nature of f0 : Rn -R, we have different optimization models. For 2 instance, if f0 5 minx :Ax2b:2 where AARm 3 n , bARm , and :U:2 refers to the Euclidean norm, this problem is called least-squares problem and arises in many situations for example in statistical estimation problems such as linear regression. If the problem is stated as min cT x : aTi x # bi ; i 5 1; . . . ; m x

(2.4)

where cARn , ai ARn , bi ARn , it is called LP problem, which was introduced by Dantzig in the 1940s in the context of logistical problems arising in military operations. This model of computation is perhaps the most widely used optimization problem today. If any of the functions among the objectives and constraint functions is nonlinear, the problem is called a NLP problem. This is the most general form of a programming problem and all other problems can be considered as special cases of the NLP problem. If the objective function and the constraint functions are expressed as polynomials of decision variables x such as gðxÞ 5 c1 xa111 xa221 ?xann1 1 c2 xa112 xa222 ?xann2 1 ? 1 cm xa11m xa22m ?xannm

(2.5)

then this type of optimization problem is known as geometric programming problem. Between the NLP problems, maybe the most well-behaved problem is known as quadratic programming problem. A quadratic programming has a quadratic objective function and linear

2.2 FORMULATION OF OPTIMIZATION PROBLEM

31

constraints and is concave (for maximization problems). It can be solved by suitably modifying the LP techniques. In a quadratic programming problem, the objective function can be represented as f0 ðxÞ 5 d 1

n X i51

qi xi 1

n X n X

Qij xi xj

(2.6)

i51 j51

where d, qi , and Qij are some constants with appropriate dimensions. We can also classify the optimization problems in terms of permissible values of the decision variables. Under this classification, objective functions can be classified as integer and real-valued programming problems. If some or all of the design variables of an optimization problem are restricted to take only integer (or discrete) values, the problem is called an integer programming problem. In contrast to the integer programming problem, a real-valued problem is that in which it is sought to minimize or maximize a real function by systematically choosing the values of real variables from within an allowed set. When the allowed set contains only real values, it is called a real-valued programming problem. Sometimes, we can classify optimization problems on the basis of deterministic nature of the decision variables. Under this classification, if the problem is a deterministic problem, the system will produce the same output for a same input. If the optimization problem involves some probabilistic (nondeterministic or stochastic) variables, then we classify these kinds of optimization problems as stochastic optimization problems. Finally, we can classify the optimization problems depending on the number of objective functions used in the problem. If there is only a single objective function in the problem, then we call this kind of problems as single-objective programming problem. On the other hand, if there are more than one objective functions, then we call this problem as a multi-objective programming problem.

2.2.2 CONVEX OPTIMIZATION Generally speaking, each of the LP, least squares, quadratic programming, geometric programming problems can be transformed into a convex optimization problem via a change of variables. Note that if f0 is a convex function and D is a convex region then the optimization problem is classified as a convex optimization problem. The convexity property can make optimization in some sense “easier” than the general as if a local minimum exits, it is guaranteed that this minimum is also the global minimum of the optimization problem. Convex optimization problems can be solved by some modern methods such as subgradient projection and interior point methods or by some old methods such as cutting plane methods, ellipsoid methods, and subgradient methods [11]. Definition: Consider a given set S : Rn -R, we say that S is convex set, if for every x; yARn such that λx 1 ð1 2 λÞyAS

for all λAð0; 1Þ. Otherwise, the set is called concave set (Fig. 2.1).

(2.7)

32

CHAPTER 2 INTRODUCTION TO OPTIMIZATION

FIGURE 2.1 (A) Convex set and (B) Concave set.

FIGURE 2.2 Convex function.

Definition: Consider the function f : S-R, where S is a convex set. For every x; yAS and for all λAð0; 1Þ, we say f is convex function if the following inequality satisfies.   f λx 1 ð1 2 λÞy # λf ðxÞ 1 ð1 2 λÞf ðyÞ

(2.8)

Otherwise, it is called concave function. Fig. 2.2 clearly illustrates that a convex function does not have a local minimum, and any local minima is also a global minimum.

2.2.3 BASIC CONCEPTS AND MODELS OF LINEAR OPTIMIZATION In a general manner, a LP model involves an n-dimensional linear cost function and requires conditions to seek the best possible solution to minimize it. Assume that, the solution vector of this problem is x 5 ðx1 ; . . . ; xn Þ subject to a set of linear equality and inequality constraints. In this case, general formulation of a minimization-based LP model is as follows: minimize cx

ai x # bi ; iAf1; . . . ; mg

subject to

kj # xj # lj ; j 5 1; . . . ; n

ar x 5 er ; rAf1; . . . ; tg xj Aθj

(2.9)

2.2 FORMULATION OF OPTIMIZATION PROBLEM

33

where ai is the n-dimensional row vector, bi and er are scalars. Upper and lower bounds on decision variable xj can be defined by kj and lj , respectively and any decision variable in the n-dimensional set can be constrained by nonnegative or nonpositive limits as given in (2.9). In many real world problems, those bounds are required to be chosen as nonnegative, but in fact, those bounds can be chosen as 7 N. In this case, xj is called an unrestricted variable. When the variable can be freely assigned to any value within the bounds, this model is called continuous variable optimization model. Any vector x% satisfying all the conditions is defined as feasible solution. A feasible solution % x that satisfies cx # cx% for all possible solution x is called optimum solution. Then, the value of cx% is called optimal cost. It is possible to define a maximization problem by seeking the decision variable x that makes the cx maximum. But, this is not necessary because the minimization problem can be transformed to maximization problem by just replacing the cost function cx by minimization of 2 cx and the constraints can be converted to maximization problem by multiplying the both sides of the associated (in)equality by 21. Note that, constraint (2.9) also captures the equality constraints by choosing the same upper and lower bounds. Moreover, this constraint can be decomposed to present upper or lower constraints as kj # xj or xj # lj , respectively. In a similar fashion with cost function, these constraints can be adapted to maximization problem by multiplying the both sides by 2 1. Considering the above discussion, we can state the LP model in a more compact model by using the inequality constraints. For this aim, assume that there exists m constraints that are indexed by i 5 1; . . . ; m and let A be m 3 n matrix formed by 1 3 n dimensioned constraint vectors ai such that 2

a1

3

6 7 A 5 4 ^ 5:

(2.10)

am

This formulation allows us to consider inequalities as component wise, such as, ith component of Ai is directly associated with the ith component of vector bi . So, by the merit of this formulation, constraints can be expressed compactly as Ai x $ bi . Then, the LP model can be constructed such that minimize cx subject to Ax $ b

(2.11)

x $ 0:

This formulation can be also represented in standard form such that minimize cx subject to Ax 5 b

(2.12)

x $ 0:

In this formulation, we try to synthesize vector b by using a nonnegative amount x of each resource vector A as minimizing the total cost of cx [11]. Note that, both of the formulations are equivalent as the feasible solution has the same cost. Therefore, any LP problem in general formulation can be reduced to a standard form counterpart.

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CHAPTER 2 INTRODUCTION TO OPTIMIZATION

For this aim, we encounter two basic problems, which are elimination of free variable and elimination of inequality constraints. For the first problem, an unrestricted variable x should be replaced by x1 2 x2 and so that x1 $ 0 and x2 $ 0 are new variables. Hence, by considering the fact that any real number can be stated as the differences of two positive numbers, we can add the free variable to the problem formulation. For the latter problem, let us consider such a constraint Ax # b:

(2.13)

To state this constraint in the standard form, we can introduce a new slack variable s $ 0 such that Ax 1 s 5 b s $ 0:

(2.14)

In a similar manner, the constraints as Ax $ b can be reduced to the standard form as subtracting a slack variable s such that Ax 2 s 5 b s $ 0:

(2.15)

Sometimes, it is much more convenient to represent a LP in canonical form. In canonical form, the objective function is always optimized as defining every constraint is a # constraint and all variables are nonnegative. The advantage of representing all constraint set as less-than-or-equal constraints is that we can write the entire constraint set using matrix notation and therefore we can adapt efficient numerical methods to solve these kind of problems. Hence, a maximization-based LP in canonical form can always be represented as maxfcxjx $ 0; Ax # bg

(2.16)

where vectors are compared component wise. That is, x # y if and only if xi # yi for all entries i. However, not all LP problems have solutions. A linear program is not solvable if the problem has inconsistent set of constraints. This is only possible if there is no valid assignment to entries of x, which satisfies all constraints simultaneously. Such LP problems are called infeasible. On the other hand, in some cases although there is a feasible solution to the problem, there still not be an optimal solution to (2.11). That is the objective function cx does not have a minimum for a minimization problem or a maximum for a maximization problem. These kind of linear programs are called unbounded. Infeasibility and unboundedness are the only things that can go wrong in a LP problem. Any LP problem that is neither unbounded nor infeasible has at least one optimum solution.

2.2.3.1 Duality Assume, we have a linear maximization program in canonical form as in (2.16) and suppose that we think a candidate solution x% is an optimal solution to this problem. How can we prove that this solution is optimal? Assume that the maximum profit to this problem is z. Then, cx # z

(2.17)

for all feasible x values. Now take the constraint inequality Ax # b and multiply both sides of the inequality by a row vector y from the left which yields to yAx # yb where y $ 0. Then, if we can

2.2 FORMULATION OF OPTIMIZATION PROBLEM

35

make yA 5 c while making yb as small as possible, then we can reach to the maximum profit value z. Therefore, the dual of our linear maximization problem in canonical form can be defined as   min ybjy $ 0; yA 5 c :

(2.18)

The duality theorem says that the solution to the dual and the solution to the original or with well-known name primal problem match exactly. That is       max cxx $ 0; Ax # b 5 min yby $ 0; yA 5 c :

(2.19)

The duality theorem allows us to find out whether a feasible solution is optimal or not. Whenever cx 5 yb is achieved for a specific x and a specific y, then we can conclude that the solution is optimal. This relationship is the basis of a family of LP algorithms known as primal-dual algorithms. This algorithm iterates between primal and dual problem by improving the solution in each step until both sides of (2.19) are equal. There are many algorithms for solving linear programs. Between them, some are practical and some run in polynomial time in the worst case. However, there is a trade-off between these two properties. That is, a practical algorithm may not converge in polynomial time whereas a guaranteed polynomial time algorithm is not generally practical to be used in high dimensional problems. Generally, there are two fundamental methods for solving linear programs. They are: Simplex method and Ellipsoid method. Simplex method is recommended when the order of the problem is small, whereas an ellipsoid method is recommended when the order of the problem is big. For instance, for a typical problem the cost of a simplex method is generally around Oðn2 mÞ, and its worst case cost is problematic. The simplex method was developed by Dantzig in 1951 [12]. The fundamental idea behind the simplex method is that the set of points that satisfy the constraints of a feasible linear program forms an n-dimensional polytope which is a convex body. The body is surrounded by the constraints of the form Ax # b, each of which cuts the space in half along the hyperplane fxjAx 5 bg. Then, a solution to the LP problem is the corner of this polytope that is farthest from the origin in the c direction for a maximization problem and nearest to the origin for a minimization problem where cx is the objective function. To find the optimal corner, the algorithm starts from one arbitrary corner and walks through edges until it cannot go any further. The current location in each step is represented by a set of n constraints of the form Ax 5 b. Note that this is called basis and uniquely identifies some corner as n equations with n unknowns can be uniquely solved. Moving from one corner to the next involves sweeping a new constraint for one of the constraints in the basis, which is known as pivoting.

2.2.3.2 The Applied Base Technique to Solve Linear Optimization Problems: Simplex Method A linear program with n decision variables and m constraints can be defined as max

z 5 cx

s:t:

Ax 5 b x$0

where c 5 ½ c1

c2

c3 ? cn , x 5 ½x1 x2 ?xn  and T

(2.20)

36

CHAPTER 2 INTRODUCTION TO OPTIMIZATION

2

a11

6 6 a21 A56 6 ^ 4 am1  b 5 b1

a12

?

a22

?

a1n

3

7 a2n 7 7; ^ & ^ 7 5 am2 ? amn T b2 ? bm :

As some artificial decision variables are basic and some are not, we can always decompose x as  xT 5 xTB xTN where the subscripts B and N denote basic and nonbasic variables, respectively. Then, the linear program can be alternatively written as max

z 5 cB xB 1 cN xN

s:t:

BxB 1 NxN 5 b

(2.21)

x$0

 where A and c has been partitioned to its columns accordingly as A 5 B For instance, consider the LP example max

z 5 2x1 1 3x2

s:t:

x1 1 x2 # 50

  N and c 5 cB

 cN .

(2.22)

2x1 1 x2 # 30 x1 ; x2 $ 0:

This linear program can be transformed into standard form by introducing new variables e1 and e2 such as max

z 5 2x1 1 3x2

s:t:

x1 1 x2 1 e1 5 50

(2.23)

2x1 1 x2 1 e2 5 30

 T  Then, xB 5 e1 e2 , xN 5 x1

1 1 finally N 5 . 2 1 Solving for xB gives

x2

T

x1 ; x2 ; e1 ; e2 $ 0:

 , cN 5 2

  3 , cB 5 0

  0 , b 5 50

xB 5 B21 ðb 2 NxN Þ 5 B21 b 2 B21 NxN:

30

T

, B 5 I2 3 2 and

(2.24)

So, the linear program can be rewritten as max

z 5 cB B21 b 2 ðcB B21 N 2 cN ÞxN

s:t:

xB 1 B21 NxN 5 B21 b x $ 0:

Now putting everything in tableau form leads to: z

xB

xN

Right hand side

1 0

0

cB B21 N 2 cN B21 N

cB B21 b B21 b

Im 3 m

(2.25)

2.3 SOLUTION TECHNIQUES FOR UNCONSTRAINED OPTIMIZATION

37

Note that taking xN 5 0 leads to a solution known as basic feasible solution. It is well known that every basic solution is an extreme point on the polyhedral solution set. This corresponds to the last block row of the tableau. This matrix representation (or tableau representation) contains all of the information we need to execute the simplex algorithm. An entering variable is chosen from among the columns containing the reduced costs [right hand side (RHS)] and matrix B21 N. Naturally, a column with a negative reduced cost is chosen. We then chose a leaving variable by performing the minimum ratio test on the chosen column and the RHS column. We pivot on the element at the entering column and leaving row and this transforms the tableau into a new tableau that represents the new basic feasible solution. So, the general algorithm can be summarized as follows: Simplex algorithm 1. Convert the linear program to standard form. 2. Obtain an initial basic feasible solution (if possible). 3. Determine whether the basic feasible solution is optimal. If yes, stop. 4. If the current basic feasible solution is not optimal, then determine which nonbasic variable (zero valued variable) should become basic (become nonzero) and which basic variable (nonzero valued variable) should become nonbasic (go to zero) to make the objective function value better. 5. Determine whether the problem is unbounded. If yes, stop. 6. If the problem doesn’t seem to be unbounded at this stage, find a new basic feasible solution from the old basic feasible solution. Go back to step 3. The details of the algorithm and modifications for special cases like degeneracy, unbounded case can be found in Ref. [13]. If the feasible region of linear problem has no degenerate extreme points, then the simplex algorithm will terminate in a finite number of steps with an optimal solution to the LP problem [14].

2.3 SOLUTION TECHNIQUES FOR UNCONSTRAINED OPTIMIZATION The field of unconstrained optimization is quite a large and prominent one, for which a lot of algorithms and software are available. Although these methods are seldom used in applications, as in the real world the decision variables are subject to some constraints, the methods of unconstrained optimization are useful to solve more general problems where efficient local unconstrained optimization techniques are required for the solution of constrained and global optimization problems. If the optimization is at a stage where no constraints are active then determining a search direction and travel distance for minimizing/maximizing the objective function involves an unconstrained minimization/maximization algorithm. Recall that, any optimization algorithm starts by an initial point x0 and carries out a series of iterations to reach the optimal point, xopt . Let us consider, the problem of minimizing the function h:Rn -R and suppose that rh and r2 h exist and are continuous where r stands for gradient function. Hence, the solution of this generic optimization problem can be described in the following way [15].

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CHAPTER 2 INTRODUCTION TO OPTIMIZATION

Generic algorithm for an unconstrained optimization problem n 1. Start with  a npoint x0 AR and  set k 5 0: 2. If xk A xAR : rhðxÞ 5 0 then stop. 3. Find a direction of search dk ARn . 4. Find a step αk ARn along dk : 5. Compute xk11 5 xk 1 αk dk ; set k 5 k 1 1and go to step 2. The choice of αk affects both convergence and the speed of convergence of the algorithm. Computing the step αk along a given direction dk is the most important part of the so-called line search algorithms which is known as one of the basic iterative approaches to find a local minimum of an objective function. The general line search algorithm begins with an initial point, finds a descent search direction, determines the suitable step length, and checks the termination criteria. At the ith iteration the next point is given by the sum of the previous point and the direction (di ) multiplied by the step length (αi ) in that direction, thus xi11 5 xi 1 αi di . Furthermore, the next point must be such that the objective function value at that point should be less than or equal to the previous point hðxi 1 αi di Þ # hðxi Þ. Next, some pure line search algorithms such as Nelder
Mead, Hooke and Jeeves, Fibonacci, and Golden Section are explained in detail.

2.3.1 NELDER
MEAD METHOD The Nelder
Mead method is one of the most well-known gradient-free nonlinear optimization algorithms that rely on difference vectors for exploring the objective function landscape where the simplex can shrink as well as expand to adapt to the objective function surface. The method proposed by John Nelder and Roger Mead is used to find the minimum/maximum of an objective function that uses a polytope (the so-called simplex) of n 1 1 vertices in an n-dimensional space [16]. In two dimensions, a simplex is a triangle, in three dimensions it is a tetrahedron, but not necessarily the regular tetrahedron and so on. The basic idea behind this method is that the worst vertex of the simplex at ith iteration is replaced by one of the reflection, expansion, or inside/outside contraction points. The simplex adapts itself to the local surface, moving downhill at inclined landscapes, changing direction when a valley is encountered, and contracting in the neighborhood of the minimum. Recall that, this technique requires only function evaluations, not derivatives. The minimization algorithm of the function hðxÞ where xARn and the test points or vertices are given as x1 ; x2 ; . . .; xn11 is explained as follows: Nelder
Mead algorithm: 1. Order the values at the vertices such that hðx1 Þ # hðx2 Þ # ? # hðxn11 Þ. 2. Calculate the centroid xG of all vertices except xn11 . 3. Compute the reflected point by xR 5 xG 1 αðxG 2 xn11 Þ with α 5 1 being the reflection coefficient. If hðx1 Þ # hðxR Þ # hðxn Þ then replace xn11 , the worst point, with the reflected point and go to the first step. 4. If hðxR Þ # hðx1 Þ then find the expanded point by xE 5 xG 1 γðxR 2 xG Þ, where γ 5 2 is the standard value for expansion coefficient. If hðxE Þ # hðxR Þ then find a new simplex by replacing xn11 with xE and go to the first step. Else find a new simplex by replacing xn11 with xR and go to the first step. Finally, if the reflected point is not better than the second worst point then continue with step 5.

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5. If hðxR Þ $ hðxn Þ obtain the contracted point by xC 5 xG 1 ρðxn11 2 xG Þ. If hðxC Þ , hðxn11 Þ then find a new simplex by replacing xn11 with xC and go to step 1. Else go to step 6. Here, the standard value for ρ is 0:5. 6. For all but the best vertex (jAf2; :::; n 1 1g) compute n new vertices by xj 5 x1 1 σðxj 2 x1 Þ with σ 5 0:5 and go to the first step. An approach for finding the minimum of a function hðxÞ in a given interval is to evaluate the function many times and search for a local minimum. To reduce the quantity of function evaluations, it is critical to have a good strategy for finding where is to be evaluated. In this manner, two effective sectioning strategies are the Golden Section and Fibonacci searches.

2.3.2 GOLDEN SECTION SEARCH The golden section is a line segment divided into two parts. Let the point C be positioned on the line segment AB such that AC2 5 BC  AB, that is, the ratio of the segment’s entire length to the longer length is equal to the ratio of the longer length to the shorter length, then the segment has been divided into the Golden ratio and denoted by τ [17]. Golden section search algorithm 1. Given initial interval ½a1 ; b1  and tolerance δ. Recall that τ 5 0:618. Find x11 5 b1 2 τðb1 2 a1 Þ and x12 5 a1 1 τðb1 2 a1 Þ. 2. Set j 5 1. j j11 2.1 If hðxj2 Þ . hðxj1 Þ then aj11 5 aj , bj11 5 xj2 , xj11 2 5 x1 , x1 5 bj11 2 τðbj11 2 aj11 Þ. j j j j11 j j11  2.2 If hðx2 Þ# hðx1 Þ then aj11 5 x1 , bj11 5 bj , x1 5 x2 , x2 5 aj11 1 τðbj11 2 aj11 Þ. 3. If bj11 2 aj11  # δ then stop, else set j 5 j 1 1 and go to step 2.1. In this method, the ratio for the reduction of intervals at each iteration remains constant. In Fibonacci method this ratio is not constant and the search is based on the sequence of Fibonacci numbers.

2.3.3 FIBONACCI SEARCH Starting with F0 5 1 and F1 5 1, the Fibonacci numbers Fn11 5 Fn 1 Fn21 for n 5 1; 2; . . .. Here, the so called reduction each iteration such that it is given as the ratio of two consecutive unimodal function in ½a; b, that is hðxÞ is decreasing for xopt # x # b [18].

are defined by the equation factor τ is allowed to change at Fibonacci numbers. Let hðxÞ be a a # x # xopt and increasing for

Fibonacci search algorithm 1. Set k 5 0, δ . 0 and n such that Fn11 $ ðb0 2 a0 Þ=δ. Find τ 5 Fn =Fn11 , λ 5 b 2 τðb 2 aÞ, γ 5 a 1 τðb 2 aÞ, ha 5 hðaÞ, hb 5 hðbÞ, hλ 5 hðλÞ, hγ 5 hðγÞ. 2. If hλ . hγ go to step 3, else go to step 4. 3. If b 2 λ # δ stop and the solution is γ, else set a 5 λ, λ 5 γ, hλ 5 hγ find γ 5 a 1 τðb 2 aÞ and hγ 5 hðγÞ. Go to step 5. 4. If γ 2 a # δ stop and the solution is λ, else set b 5 γ, γ 5 λ, hγ 5 hλ find λ 5 b 2 τðb 2 aÞ and hλ 5 hðλÞ. Go to step 5. 5. Set k 5 k 1 1 and τ 5 Fn2k =Fn2k11 . Go to step 2.

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On the other hand, for specifying special functions for desired direction through the optimal point, Hooke and Jeeves’ method which is explained in the following section, can be utilized.

2.3.4 HOOKE AND JEEVES’ METHOD The method of Hooke and Jeeves’ is a globally convergent method having a pattern search which guesses the shape of a function to locate a downhill direction. Hooke and Jeeves’ method consist of a sequence of exploratory moves about a base point. The exploratory moves by considering the local behavior of the function to locate the direction of current stepping valleys. The pattern moves by employing the data achieved in the exploration to step along the valleys. If an exploratory move leads to a decrease of the value of hðxÞ (xARn ), it is called a success and is followed by the socalled “pattern moves”; otherwise, it is a failure [19]. Hooke and Jeeves’ algorithm (Exploratory moves) 1. Choose a base point p1 and a step length s1 . 2. Find hðp1 1 s1 u1 Þ, where u1 5 ð1; 0; 0; :::; 0ÞT . If hðp1 1 s1 u1 Þ , hðp1 Þ replace p1 by p1 1 s1 u1 , else find hðp1 2 s1 u1 Þ and if hðp1 2 s1 u1 Þ , hðp1 Þ replace p1 by p1 2 s1 u1 . Otherwise hold p1 . 3. Apply step 2 for the second variable in hðxÞ with step length s2 and u2 5 ð0; 1; 0; :::; 0ÞT . Repeat the procedure to each variable in turn, finally arriving at a new base point p2 . 4. If p2 5 p1 , halve each of sj ’s and go to step 1. End if the length is reduced to the prescribed tolerance. 5. If p2 6¼ p1 perform the pattern moves as defined as follows: Hooke and Jeeves’ algorithm (Pattern moves) 1. Move from p2 to q1 5 2p2 2 p1 and start over with a new sequence of exploratory moves about p1 . 2. If the minimum value achieved is less than hðp2 Þ then a new base point p3 has been reached. Go to step 1 and increase the counter by 1. Otherwise, quit pattern move procedure and continue with a new sequence of exploratory moves. If the objective function hðxÞ is smooth enough, it is possible to accelerate the convergence by curve fitting, where hðxÞ is approximated by a polynomial such that the coefficients of the polynomial are chosen in order to fit it to the derivatives, if they are computable.

2.3.5 GRADIENT DESCENT METHOD A new family of methods for unconstrained minimization problem is gradient descent (ascent), also known as steepest descent (ascent) which is a first-order iterative optimization algorithm where the local minimum (maximum) of the objective function hðxÞ is determined using gradient descent (ascent), such that the steps proportional to the negative (positive) of the gradient of the function at the current point is taken [11]. The algorithm of steepest descent is given as follows: Gradient descent algorithm 1. Given x0 , set k 5 0. 2. Calculate dk 5 2 rhðxk Þ. If dk 5 0, then stop. 3. Solve for the step size αk , minα hðxk 1 αk dk Þ. 4. Set xk11 5 xk 1 αk dk , k 5 k 1 1 and go to the first step.

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It is obvious that due to its simplicity each such lower dimensional or scalar minimization/maximization problems can be used as a part of the full problem. For example, coordinate descent algorithms rely on this principle.

2.3.6 COORDINATE DESCENT METHOD Coordinate descent methods are also iterative methods where each iterate is determined by fixing some components of xARn at their current values. The full optimization problem is reduced to a sequence of simple optimization problems where the minimization or maximization is done with respect to the remaining components. Let us again consider the unconstrained minimization problem minx hðxÞ where xARn -R is continuous [20]. Coordinate descent algorithm: 1. Set k 5 0 and choose x0 ARn . 2. Choose coordinate jk Af1; 2; . . . ; ng 3. xk11 5 xk 2 αk ½rhðxk Þjk ejk , and k 5 k 1 1, where αk and ejk being the step-length and coordinate direction, respectively. 4. If termination test is satisfied then stop, else go to step 2.

2.4 SOLUTION TECHNIQUES FOR CONSTRAINED OPTIMIZATION AND RELEVANT APPLICATIONS TO RENEWABLE ENERGY SYSTEMS The structure of the optimization problem in electric power systems including renewable energy units is generally in constrained form of optimization. Such problems can be categorized as power system scheduling with renewable resources, market operation under renewable energy based uncertainty, home energy management systems, and demand response strategies with household size renewable energy units, sizing and sitting of renewable-based-distributed generation systems within distribution system, sizing of renewable energy units in stand-alone or grid-connected mode of operation, optimal design and operation of wind turbines, wind farm optimal layout, optimal operation of fuel cell powered power systems, finding the maximum power tracking point of photovoltaic systems, and others. Thus in this regards, this subsection represents the constrained optimization problem solution techniques.

2.4.1 GENETIC ALGORITHM The GA is a kind of optimization method based on simulation of the biological processes. As the GA establishes analogy with the biological systems, GA uses the terms of chromosome, selection, crossover, and mutation [21]. Chromosomes represent the candidate solution of the optimization problem and include n-bit (locus) length arrays depending on the size of the solution. Every bit of the chromosomes has binary coded structure so that, chromosomes are sequence of the 0s and 1s depending to the extent of n. On the other hand, selection, crossover, and mutation operators have probabilistic models to mimic the functions in the biological processes and manipulate the candidate solutions to increase the convergent rate of the optimization problem [22].

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Main characteristic feature of the GA is that it is a random process due to the inspiration in developing from biological systems. Hence, GA is a good option for applying to nonstandard optimization problems which have discontinuous, stochastic and nonlinear constraints, or nonlinear objective function [23]. Moreover, this technique has a property for setting the randomization level, so that, this makes the algorithm much more efficient than the random search methods [24]. Besides, GA has advantages compared with the other standard techniques which are involving a large number of arguments of an objective function that can be coded on chromosomes, achieving a population of candidate solution points at each iteration randomly, and selecting the next population by using the efficient probability tools. GA involves five basic elements which are random population generator, fitness function and the execution operators of selection, crossover, and mutation [25]. Random population generator specifies the chromosomes between the lower and upper bound of the search space of the candidate solution set of the problem. These initial sets are also known as first parents set to produce the children for the next generation. After being constructed the initial set, each of the chromosomes is tested by fitness function to see how well it solves the problem. As it is understood from the functionality of the fitness function, then GA moves on the next step and selection operator selects the chromosome pairs at random to reproduce the next generation. Next, the crossover operator steps in and chooses randomly a specific locus interval for each pair of chromosomes to exchange the subsequences in this interval of chromosomes. In this way, two new offsprings come into existence from every pair for the next generation. Consequently, the algorithm terminates when the termination criterion is satisfied. Fig. 2.3 renders the general GA concept which is applied to many optimization problems in the literature. On the other hand, there have been some research activities to offer advanced version of GA. For this aim, some points should be reconsidered to present more convergent and faster GA. Synthesis problem of GA formulation requires analyzing deeply the factors of the effects of selecting of the initial population, probability models of crossover operator, and mutating operator. Selection of initial population depends on probability models and by imposing better probabilistic model makes the algorithm faster. Similarly, crossover operator also utilizes the probability theory. But, compared to the initial population generator this operator has more complex structure. Such as, in the reproducing phase of new offspring, it specifies the dimensions of bits to exchange between two elements of the pairs. So, this process needs to construct complex probability models to present more convergent algorithm. On the other hand, mutating operator focuses on a single bit on the chromosomes to check if, whether the result is local or global optimal point. So that, this tool of GA makes this approach very attractive. This phenomenon is very important for optimization problems and GA inherently provides this to check the feasibility of the solution. Then, presenting new mathematical approaches to choose the bits to mutate enables to provide faster GAs. As discussed above, GA provides many advantages as mimicking the biological processes. On the other hand, this formulation also put extra burden on coding complexity which makes this technique relatively harder than the other competitive heuristic methods [25]. Optimal sizing problem of renewable energy systems is one of the most popular problems that is dealing with multi-constrained OC techniques. For example, Shahirinia et al. considered the optimal sizing problem of stand-alone hybrid power source systems formed with photovoltaic array, wind turbine, diesel generator, and battery [26]. All the economic constraints of the related power source components were imposed on the GA formulation and optimal sizing of the sources

2.4 SOLUTION TECHNIQUES FOR CONSTRAINED OPTIMIZATION

FIGURE 2.3 Flowchart of GA optimization technique.

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have been calculated by minimizing the electrification costs. At the same perspective, Koutroulis et al. and Senjyu et al. formulated GA scheme for optimal sizing the components of photovoltaic
wind power generation units for ensuring medium term operation with minimal capital and maintenance costs [27,28]. With the similar aim, Al-Shamma’a and Addoweesh deal with the sizing problem of hybrid power source that supplies northern part of Saudi Arabia by using GA optimization approach [29]. They presented a detailed analysis considering not only the economical and environmental constraints, but also technical and renewable energy fraction perspectives. After testing the nine different hybrid configurations, they concluded feasible three options of hybrid composition in terms of economical
environmental conditions, various limitation interactions on wind source and solar energy density. Besides these of the technical and environmental issues, hybrid powered systems should fulfill the power reliability concerns. Another real sizing application of solar
wind-battery hybrid system is realized by Bilal et al. [30]. They implemented electrification problem of an isolated site with the mentioned hybrid energy system by employing GA technique under the objectives of minimization of regular cost and the losses for the supplying power. Yang et al. presented an optimal sizing problem of power supplying hybrid system composed of photovoltaic
wind and battery modules for telecommunication relay station [31]. To solve the problem, they recommended a methodology depending on GA concept considering the constraints of module number and dimensions and also they adopted an approach to show the correlation between power reliability and configuration of the hybrid system. Another trendy problem for hybrid renewable energy system is placement of wind turbines for obtaining maximum production capacity. Grady et al. addressed this problem by employing GA method for different wind conditions as aiming to limit the number of wind turbines [32]. Productive operation of hybrid system depends on a key requirement of utilizing the energy more than the enough for the load demands in the system. Dufo-Lo´pez et al. dealt with this problem for the hybrid systems and developed a GA based control strategy to make a decision to evaluate the spare energy as charging the batteries or to produce hydrogen depending on the energy using pattern of the related hybrid system [33]. Gonzalez et al. also studied on the layout problem of wind farms with more attaching importance to economical returns of the investment [34]. They exploited GA to solve the optimal wind farm configuration problem by considering the technical requirements, initial investments, and net cash flow for wind farm life span.

2.4.2 PARTICLE SWARM OPTIMIZATION Particle swarm optimization (PSO) technique is proposed by Eberhart and Kennedy [35] inspired by the swarming act of the bird, fish, and insect groups as searching for food. Every individual in the group is called “particle”, and this technique generally simulates the social behavior of the animal groups focusing on position changing of the particles. As reaching the goal, positioning patterns of the particles and whole group enable to define an algorithm based on swarming methodology. For example, let us consider a flock of birds attempting to find source of food. When one gets closer position to the source than the rest, chirps loudly and the others go toward to that bird. In this attempt of keeping up the closest one, all the group adjust their velocities depending on their own position. And, this process depending on the variation of position and velocity happens iteratively until any of the members reaches the food source in the searching area.

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From broader perspective, there is a hypothesis that swarm behaviors of the animal groups can be categorized in two behaviors: Exploratory and exploiting. Exploratory behavior defines the attempts for enlarging search-space, besides exploiting behavior describes the actions taken by the particles for reaching the possible optimal solution in the space [36,37]. In the search space, every particle which is also defined as solution has fitness function to determine the velocities through the goal and each particle is represented by a vector to calculate the next movement of the particle. So that, fitness function takes a candidate solution vector as argument and produces a real value. Then, the output of the function is compared with the other particles output in the search space. In this way, every particle (solution) contains four parameters which are current position in the search space, current fitness value, current velocity, and memory contains the best position ever in the flock. In each iteration, the velocity and position of the particles are updated as considering the current velocity, best position of the particles taken so far and global best position which indicates the position of the particles that is closest to the target. So that, by including the population of the flock as the counter i and dimension the search space as the counter n, position and velocity set can be given as follows [38,39]: xi 5 ½ xi;1 vi 5 ½ vi;1

vi;2

xi;2

? xi;n 

? vi;n ; i 5 f1; 2; . . . ; mg

(2.26)

The particles obey the simple rules, but there is no centralized control regulating how the particles behave [38]. So that, following equations are calculated in every iteration counted by t to update velocity (vi ) and position (xi ) of the ith particle, respectively [40].     vi ðt 1 1Þ 5 ωvi ðtÞ 1 c1 r1 pj ðtÞ 2 xj ðtÞ 1 c2 r2 pg ðtÞ 2 xj ðtÞ xi ðt 1 1Þ 5 xi ðtÞ 1 vi ðt 1 1Þ

(2.27)

where ω is inertia weight, pi ðtÞ, and pg ðtÞ denote the best position vector of the ith particle ever found and whole swarm, respectively. c1 (c2 ) is cognitive (social) accelerative coefficient, and r1 (r2 ) is the independent random uniformly distributed variable taken the values in the range of [0,1]. Considering (2.27), PSO technique encompasses three main factors: Inertia, memory, and cooperation. These factors characterize the swarming behavior of the group and supplied by the two main cognitive process which are own experience of each particle and communication with the other particles. Thus, particles gather and share information to increase their quality of fitness function [41]. As seen in Fig. 2.4, PSO algorithm starts with defining the constraints of the problem and related parameters. Then, the fitness function is evaluated for each of the particles to measure the optimality of the current results. Then, interacting with the other particles the velocity and position values are modified according to (2.27), respectively. In this way, the process is iterated until satisfying the termination criterion. Termination criteria of the algorithm can be imposed as achieving the minimum distance between the current position and target or reaching the minimum number of iteration. Besides that, alternative stopping criteria can be added for the case of no-target is found in the limit of the predetermined iterations. For discussion of the pros and cons of the PSO technique, first, it should be mentioned that PSO algorithm proposes two version of solutions which are global and local. Local version of the solution comprises a limited space which is called “working space” and focuses on the optimal solution. From this point of view, this version is slower than the global one and more reliable for

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FIGURE 2.4 Flowchart of PSO technique.

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converging the optimal solution. Besides, global version of the algorithm focuses on outer space to find the solution space which includes the optimal solution. So that, this version of the technique is faster than the local one but convergence rate is less reliable than the local one [35,36]. The combination of the both versions of the techniques give rise to more robust solution and it avoids getting trapped to local optimal solution. Moreover, to overcome this handicap, PSO technique can be hybridized with other techniques such that simulating annealing or tabu search (TS) [42]. In parallel with above discussion, PSO technique has similarities with GA. Such as, both technique start with a group of random generated initial set and have values of fitness function to evaluate the convergence rate of the population. As GA uses genetic operators to change the convergence rates, PSO utilize velocity equation. Indeed, modeling and control of the swarm patterns of the flock the velocity data is defined on Cartesian coordinates for each bird. So, it should be noted that PSO is more convenient for the problems having less than three parameters, but GA is convenient for more than three parameters. On the other hand, GA algorithm requires to share information between each of the chromosomes, but in PSO, all the group follow the particle which has the global best position value to reach the target. From this point of view, it can be commented that PSO has simpler tools to implement in software environment and needs less computational time and allocated memory for reaching the optimal point when compared to GA. On the other hand, PSO technique proposes less reliable solution than GA [25,35]. In literature, there have been enormous number of papers adopting PSO technique for presenting solutions to the problems of the system including alternative energy sources. For looking at sizing and configuration problems, Ardakani et al. dealt with the optimal sizing problem of wind
photovoltaic
battery hybrid power system by taking into account of the technical and economic constraints such as system reliability and maintenance costs, respectively [43]. Sizing of the components was determined by PSO-based optimization algorithm, and the result was implemented as real-time application. Another hybrid configuration consisting of thermal, wind, and solar units for the optimal generation problem proposed by Chakraborty et al., and they suggested a solution model by utilization of PSO methodology [44]. Lo´pez et al. focused on biological waste management system and realized optimal location the components of biomass-based power plant as considering the constraints of supply area and power generation limit [45]. In another investigation, Wang and Singh employed PSO procedure for optimal operation setup of hybrid power system built by wind, solar, and storage sources for reaching the most favorable criteria of the cost, emission, and reliability [46]. Parallel to the above discussion, Sharafi and ELMekkawy searched similar optimization problem for a more multi-source hybrid configuration which is incorporated by wind
solar
diesel generator
batteries
fuel cellelectrolyzer, to minimize total cost of the system, unmet load and emissions [47]. Zhao et al. suggested PSO technique established on the objectives of investment cost and reliability of the system as providing for formulation of the optimal configuration problem of wind
photovoltaic powered hybrid system [48]. Kaviani et al. highlighted the case of any component outage in the operating of hybrid system with the units of wind
PV
fuel cell and they proposed an optimal solution model exploiting the PSO methodology as also considering the above mentioned factors [49]. Moreover, Dehghan et al., Wang and Singh, Pourmousavi et al. conducted research as considering the constraints of reliability and investment of hybrid power systems, in the Refs. [50
52], respectively. For penetration of photovoltaic power system into the grid, Kornekalis found a PSO-based solution to the problem of needed optimal numbers of photovoltaic module as satisfying the maximization conditions of economical and environmental during the operational period of the system [53].

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As the solar energy depended power systems need to be operated on maximum power point (MPP) tracking mode, Miyatake et al. presented MPP tracking algorithm based on PSO approach [54]. According to the results after the testing in shading and unshading cases, it can be commented that the output of the algorithm ensures higher overall efficiency. In a similar fashion, to maximize the capturing energy from the wind power systems for fixed and unfixed speed turbines for variable wind velocity was subjected to the RES researches. Kongnam and Nuchprayoon addressed PSObased optimization solution to this problem as considering stochastic wind speed distribution model to actuate the proposed system [55]. For supporting the electrification of remote area, photovoltaic panels, wind turbine, and fuel cells can be utilized to charge the batteries or producing hydrogen to store the tanks for using in sunless periods. For this aim, Avril et al. studied on this scheme for stand-alone photovoltaic system to fulfill the comfort demands of the consumers and minimization expectation of the setup cost under stable operation and they presented PSO solution to overcome these techno-economic requirements for short and medium terms [56]. Hakimi et al. considered similar problem with the configuration of wind unit, fuel cell, electrolyzer, anaerobic reactor, and hydrogen tank [57]. Instead of utilizing photovoltaic panels, wind energy and waste materials provided energy for generating pure hydrogen. Also, Safari et al. evaluate the idea to store the unused energy as hydrogen for later using from another perspective [58]. They suggested fuzzy logic controller for conducting hybrid power system as using PSO algorithm to optimize membership functions and very significant results have been reached in terms of longer life of the batteries, getting better state of charge value up to 6.18% and decreasing the operating time of fuel cell, and others. Recently, sizing, management, scheduling, and operation solutions of grid and smart grids contain RES that have been addressed in many studies. For example, Chung et al. developed parameter tuning method utilizing PSO technique to maintain reliable and high-power quality even for harsh conditions for island operation microgrid [59]. Moghaddam et al. concentrated on the problem of multiobjective operational planning of wind
solar power system for dispatching the generation energy through the microgrid [60]. The problem formulated within the joint scope of fuzzy self-adaptation and PSO concepts with the objectives of total emission reduction and increasing the utilization rate of the RESs and the results compared with the other evolutionary optimization techniques. Yao et al. dealt with the economic dispatch model of wind power subject to uncertainty in power forecast and the constraint of carbon tax and proposed quantum affected PSO [61]. The model was tested related IEEE benchmark system for dispatch concept, then compared to the results with other methods to reveal the advantageous of the devoted approach. Lee provided a feasible PSO algorithm for scheduling operation of hydroelectric power system which is hybridized with wind turbine [62]. As having great effect for cost decreasing strategies of energy consumption, demand side management is also very trendy issue. Gudi et al. focused on this problem and established a demand side management tool by using PSO [63]. A case study realized for changeable household situations to test the merits of the strategy regarding minimizing the electricity consumption and regulating the performance deficits. Similarly, for meeting the isolated load demand, S´anchez et al. utilized PSO method for optimal sizing problem of the wind-photovoltaic-fuel cell power system [64]. On the other hand, Pedrasa et al. concentrated on interruptible loads in the demand side and presented a PSO-based approach as dividing the swarm into subswarms for minimizing the total requirements of uninterruptible loads and achieving near optimal solution for computational time [65].

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2.4.3 SIMULATING ANNEALING SA algorithm is one of the most preferred heuristic methods for solving the optimization problems. Kirkpatrick et al. introduced SA by inspiring the annealing procedure of the metal working [66]. Annealing procedure defines the optimal molecular arrangements of metal particles where the potential energy of the mass is minimized and refers cooling the metals gradually after subjected to high heat. In general manner, SA algorithm adopts an iterative movement according to the variable temperature parameter which imitates the annealing transaction of the metals [67]. A simple optimization algorithm compares iteratively the outputs of the objective functions running with current and neighboring point in the domain so that, if the neighboring point generates better result than the current one, then it is saved as base solution for the next iteration. Otherwise, the algorithm terminates the procedure without searching the wider domain for better results. So that, the algorithm is prone to be getting trapped in local minima or maxima. Instead, SA algorithm proposes an effective solution to this problem as incorporating two iterative loops which are the cooling procedure for the annealing process and Metropolis criterion [68,69]. Basic idea behind the Metropolis criterion is to be executed randomly to extra search the neighborhood of the candidate solution to avoid being trapped to local extreme points. To be more precise, let us consider the minimization problem and define an objective function f ðxi Þ corresponding to the argument set of xi 5 fx1 ; x2 ; ?; xn g with nAR. Therefore, if f ðxi11 Þ , f ðxi Þ, then take xi11 as a new candidate extreme point to check. Otherwise, define w 5 exp [2f f ðxi11 Þ 2 f ðxi Þg=Tc  where Tc is the current temperature parameter and generate a random number s, such that 0 , s , 1 [70,71]. Then, if the relation of w . s is true, you also accept the xi11 as a new candidate, else reject and go back to previous step and generate another s. Hence, Metropolis criterion allows for the motion of the current step to a certain extent even the objective function’s trajectory is getting convergent through the potential local minimum point. On the other hand, Metropolis algorithm proposes solution for a constant temperature. So that, for larger values of Tc , the algorithm requires wider search area. Therefore, this raises the probability for reaching the global minimum. Besides that, as the iterative motion of algorithm is initialized randomly, it may skips the global minima or adopts non precision approach through the extreme point. On the contrary, for smaller values of Tc , it requires smaller search area and this case may give rise of being trapped in local minima. For removing this handicap, SA offers an iterative solution as incorporating nested loops for changing the temperature parameter and the solution point. The motion of SA algorithm begins with a larger value of temperature to execute the iteration for inner loop and immediately, the point leads the best objective function in the current inner loop is assigned as new candidate solution. Then, the outer loop is run by incrementing the temperature and updating the starting point. This iterative process continues until reaching the lowest limit of temperature or realizing the predetermined number of iterations. The summary of this procedure is illustrated in Fig. 2.5. Note that, it is very critical to choose the temperature changing steps in the outer loop and randomly step sizes of inner loop. If one chooses Tc very large, then w . r is satisfied at any iteration so that the algorithm will be terminated most likely at a random minimum in the domain of the function to be optimized. Besides, if one chooses Tc very small the condition w . s never be satisfied and the motion will be terminated at the first minima. Moreover, selecting rating of temperature change has a crucial effect on the quality of the solution [72]. For preventing of those handicaps, SA algorithm deals with the optimization problem with large value of temperature and

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FIGURE 2.5 Flowchart of SA technique.

travels a large number of points in the domain of the objective function. For the further steps, it decreases the temperature gradually and narrows the search sector by adopting the local minimum point at the previous inner loop. By this way, it eliminates the local minimum points from the search space, and also, it converges to the global minimum point sensitively. Consequently, it can be commented that SA is very preferable technique among the other heuristic approaches as providing practical randomness into the search to avoid the local extreme points [73]. On the other hand, SA contains a trade-off between computational time and the sensitivity of the solution.

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Considering the literature related with renewable energy technologies (RETs), SA algorithm have been generally appealed for optimal sizing and operation problems. For example, Askarzadeh dealt with optimal sizing problem of photovoltaic/wind hybrid system and presented simple and efficient discrete optimization methodology as combining the chaotic, harmony, and SA approaches [74]. In a similar manner, Katsigiannis et al. considered the hybrid configuration of SA and Tabu search method for sizing problem of hybrid power systems comprises wind, photovoltaic, diesel
biodiesel generator, fuel cell, and battery sources [75]. They tested the system for many configurations and scenarios and concluded that the provided solution gives more sensitive results than the single use of those techniques. Penetration of the renewable energy sources (RESs) into the grid have introduced new technological concept such as smart grid, smart home, zero emission vehicles, and others. In this manner, Sousa et al. proposed a SA-based OC strategy for maintaining the energy sources of smart grid in terms of the generation, storage, and direction phases of the system energy [76]. They obtained favorable results with regard to execution time and compatible management of distributed energy sources. In a parallel optimization aim, Gandomkar et al. presented a hybrid algorithm incorporating of SA and GA techniques for allocation of the sources in the distribution network [77]. In a similar fashion, Velik and Nicolay offered modified SA algorithm for the testing the microgrid in terms of trading profit and utilization rate of the photovoltaic system embedded into the microgrid [78]. Another important aspect of renewable energy research is optimal integrating problem of wind energy systems. In this regard, Chen developed a wind
thermal coordination algorithm adopted SA methodology to regulate integrating energy capacity into the main power system [79]. By the merit of algorithm, they clarified optimal scheduling problem for the isolated wind power source.

2.4.4 ANT COLONY OPTIMIZATION Ant colony optimization (ACO), which is developed by Dorigo et al., is also a population-based heuristic approach like PSO technique [80,81]. The motivation of this technique is based on the behavior of the ant colonies for finding food. Each member of the colony makes an effort to find a commonly used path through the food source. In this regard, the ants secrete signaling pheromone to mark their own path for the source. This is the main characteristic of this approach and the follower ants prefer to go on the path with stronger pheromone so that, the change on the density of the pheromone level specifies the selecting probability of each paths. This technique is developed by observing the food searching efforts of ant clusters. The ant colony follows an organized and smart technique for approaching the food source. The details of the technique is modeled with mathematical tools, and then the approach is transformed into an optimization problem framework to utilize for engineering problems [82,83], such as the search area is defined as graph and the agents (ants) are described as moving point on this graph. As the agents move on the graph, a simulation version of pheromone secreting model is realized with a stochastic approach to mark the most popular paths through the source [84]. Each ant starts to move from randomly selected points on the graph. The connection line from the starting point to the target describe a path and each path is categorized with pheromone level and correlating heuristic value so that, higher of these parameters for a path gives rise to higher the probability an ant prefers this shorter path through the source. The rest of the ants use the pheromone deposited on the path for searching more promising direction through the food target. Then, this

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iterative procedure goes on until each of the ants finishes their travel for the food and pheromone level is updated on each paths visited by the ants. Consequently, each ant provides a solution and, at least, one path among the solutions should fulfill the termination criterion to finish the all procedure. As the main characteristic of this technique depends on the pheromone level on the route through the food source, the higher the depositing pheromone load, the higher optimality a solution is categorized. Fig. 2.6 depicts the general flowchart of ACO technique which is summarized below. Advantages and disadvantages of ACO show similarity with PSO technique, so that the position of the ants are defined on coordinate planes and this technique is also suitable for the problem with

FIGURE 2.6 Flowchart of ACO technique.

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parameters lower than three [25]. Moreover, as each ant in the colony makes an effort for following the most feasible path, ACO algorithm is very compatible with parallel implemented solutions which gives the algorithm responsive character. As increasing global environmental concerns and the needs for more electrical power, alternative energy sources have been penetrated into the grid and ACO technique have been utilized many research activities subjected to these microgrid systems. For controlling of these microgrids and dispatching the energy in the whole system, Colson et al. proposed a supervisory controller based on ACO technique by considering the constraints of fuel availability, economical issues, and environmental problems [85]. They concluded that, the proposed strategy promises an important step for synthesizing heuristic-based autonomous operation strategy for microgrids which provides more sustainable and less operational costs and emissions. Besides that, Pothiya et al. also considered ACO heuristic model for finding more economic dispatch model and concluded that the obtained solution gives a result with higher quality and less computational time compared to the other similar heuristic methods [86]. Another microgrid solution for the above mentioned problem is to utilize wind and solar energy. In this manner, Ero˘glu and Sec¸kiner improved an ACO algorithm for the aim of maximization the expected output power of wind energy source by considering the constraints of turbine locations, wind directions, and some assumption raised in formulation of the problem [87]. The authors tested the algorithm under three different scenarios built by the various cases of the constraints and also compared it with other competitive techniques to reveal the merits of the adopted strategy. Sizing of hybrid configuration of renewable sources is another current issue. For example, Kefayat et al., utilized ACO and artificial bee colony (ABC) technique for optimal sizing and location of a distributed energy system which is composed of fuel cell, wind, and gasoline sources. The adopted strategy is applied to this hybrid system to satisfy the minimizing the objectives of the power loses, emissions rates, cost of produced energy, and the instability of the voltage. The results obtained expose the effectiveness of the solution techniques compared with the similar evolutionary optimization methods [88]. Real-time application of wind energy based electricity plant considered in [89]. The authors presented a heuristic hybrid optimization solution compromising ACO and PSO techniques for testing the wind-based output power with changeable weather conditions and consequently they obtained a robust forecasting model for the related systems. By considering the solar energy systems under the variable insulation condition, it is extremely important to operate the PV arrays near to MPP in the characteristics of power-voltage. For this aim, Jiang et al. provided a MPP tracking scheme based on ACO technique and they demonstrated the feasibility of this scheme by testing the PV system under different irradiance shading conditions [90], so they concluded that the presenting algorithm ensures for tracking the global MPP in the characteristic as the PV system was operating various shading cases.

2.4.5 TABU SEARCH TS which is known as the first metaheuristic local search method in the literature, was proposed by Glover [91]. The philosophy behind the TS is to avoid cycling action which means to return recently visiting potential solution point and is based on the ground of keeping the visited points in a tabu and then disallow the future solution to repeat the same solution in the list. The term tabu means to prevent the repeating search for previously visited solution points in the search area. Moreover, it also

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covers the banned solutions kept in TS memory accordingly user-defined set of rules, namely TS algorithm records the previously visited points for a certain period, then the solutions marked as tabu if they have been visited in a certain period or violated by a pre-defined rule [92]. TS approach provides an efficient algorithm for classification of the optimization problem regarding with local and global search and it embodies the element of search space and the neighboring definition structure of visited points which can be classified as combinations of local search equipped with certainterm memories [93]. As bearing the characteristic of local search, TS methods provides promising solution feedback to the optimization problem and enables to look for adjacent solution to get better (improved) point. From this point of view, it is faster than GA for searching the global optima, but gives poor performance as comparing to GA for the aspects of solution quality. On the other hand, as characterized as metaheuristic, it yields very similar results when it is operated with similar parameters. The flowchart of TS is illustrated in Fig. 2.7. As a first step, special and required constraints of the problem, searching domain for seeking the solution of the problem, allocation tabu list memory,

FIGURE 2.7 Flowchart of TS technique.

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and objective function should be defined. Then, the candidate solution is brought forward equipped with possible neighboring solutions and immediately this solution is checked for the measures in terms of local or global necessities and also tabu list. Neighboring solutions denote the transformed local solution points in the search space and contains attractive and unattractive solutions relative to the current local solution [93]. In other words, by looking these prospective solutions, it can be decided to move through better solutions or remain stopped. As moving away from the local optimal solution, tabus prevent to revisit previously stopping points, thus, the TS algorithm converges through the global optima without tracing back to the search. For checking the quality of the candidate solution, termination criteria stepped in for finishing or moving the next iteration. Termination criteria can be determined as a fixed number of iteration or fixed amount of computational time. Besides that, this criterion can be made more specialized as considering the advancement in objective function [93]. Finally, depending on the termination criterion, the process is finished or the tabu list is updated for the next iteration. Updating procedure is realized by exchanging the newest tabu with oldest one or enlarged the memory of tabu list. It should be noted that all the solutions can be overlapped on the tabu list, but this may requires considerable number of storage modules and also, this approach gives rises extra burden of computational during running of the TS algorithm. Instead, tabus are recorded in a volatile memory of the search and the memory is allocated for the commonly used tabus relating the last transformation process on the current solution [93]. For preventing cycling or revisiting the suboptimal points in the search area the length of the tabu list are usually instructured as variable format [91,94,95]. In the literature, TS algorithm has been addressed in many studies related with RESs. For example, Gandomkar et al. studied optimal allocation of dispersed power sources in the grid for ensuring to minimize distribution loses for the demand side perspective. For this aim, they proposed a hybrid algorithm composed of the techniques of GA and TS and demonstrated the effectiveness of the algorithm by comparing GA [96]. In a similar perspective, Nara et al. utilized TS algorithm for optimal placement of the RESs (fuel cell, photovoltaic, wind etc.) to minimize the operating losses of distributed system. According to the results, it is shown that TS method has better performance than SA for the considered problem [97]. Golshan and Arefifar dealt with the grid configuration problem in terms of the mutual relationship between optimal distributed generation sources and reactive power sources. Because of this planning problem formulation yields a combinatorial form, the solution algorithm was based on TS technique and the performance of the algorithm was tested on 33-bus and 69-bus radial distribution system to examine the fitness of the parameters and control variables [98]. In the same fashion, Tan et al. illustrated the feasible utilization of combinational forms of GA
TS and Fuzzy logic-TS approaches for optimal placement of the distributed generation systems and increasing the sharing of renewable power system in the system [99]. Moreover, Katsigiannis et al. investigated for electrification problem of remote areas by using the renewable sources and formulated a combinatorial optimization problem by using the TS, SA, and combination of those techniques, respectively [75]. Under the many number of renewable based scenarios, they demonstrated the effectiveness of the TS-SA hybrid technique compared to the sole performance of TS and SA methods with regard to the notions of convergence and quality of the results obtained. Moreover, Katsigiannis and Georgilikas implemented a TS approach for the optimal sizing problem of small-scale isolated power system equipped with conventional and renewable sources and the authors showed that the proposed technique is feasible in terms of computational time and solution quality [100].

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By the reason of photovoltaic and wind power systems have increasing trend for using in power supply side, some inevitable problems for keeping the supply-demand balancing have emerged from the operational nature of those sources. Hence, those problems give rise to severe fluctuation in power distribution. For elimination of the fluctuation in supply, Tanaka et al. addressed a TSbased optimal methodology for operating of smart home concept [101]. The method provided manipulated the activation and deactivation process of the controllable loads such as battery and heat pump to minimize the power fluctuating problem of the system. Moreover, the approach enhanced the productivity of the system which is combined with PV generator, battery, solar collector, and heat pump in cost of energy. As fluctuating in wind power has considerable influence for stability of the power system equipped with large scale of WT, prediction algorithms come into prominence for sizing of such power systems. For this aim, Ramirez-Rosado et al. presented a fuzzy logic approach based on TS method. By the merit of TS, they formulated a multi-objective optimal planning technique for sizing and location problems occurred in power distribution systems [102]. In the same fashion, Novoa and Jin addressed a stochastic approach for modeling the wind characteristics and load variation to optimize the production capacity of the power system with regard to energy reliability and cost challenges [103]. As the bearing characteristic of local search, TS has a powerful feature to combine sophisticated constraints of real-life problems into the optimization formulation. On the other hand, as a common problem to synthesize better version of optimization technique based on metaheuristic approach, it is crucial to define probability tools to determine certain or near-certain neighboring structure for the local optima point [93]. Moreover, this notion indicates the excluding capability of the algorithm from local optima and helps to update tabu list accurately. In parallel with this consideration, Kalinli and Karaboga utilized TS methodology in the training process of neural network technique and also they demonstrated the advantageous of TS for nonlinear problems looked for optimal solution in on-line mode [104].

2.4.6 FIREFLY ALGORITHM Firefly algorithm is classified as swarm intelligent, metaheuristic and nature-inspired, and it is developed by Yang in 2008 by animating the characteristic behaviors of fireflies [105]. In fact, the population of fireflies show characteristic luminary flashing activities to function as attracting the partners, communication, and risk warning for predators [106]. As inspiring from those activities, Yang formulated this method under the assumptions of all fireflies are unisexual such that all fireflies has attracting potential for each other and the attractiveness is directly proportionate to the brightness level of individuals [107]. Hence, the brighter fireflies attract to the less brighter ones to move toward to them, besides that in the case of no fireflies brighter than a certain firefly then it moves randomly. In the formulation of firefly algorithm, the objective function is associated with flashing light characteristics of the firefly population. Considering the physical principle of the light intensity, it is inversely quadratic proportional to the square of the area, so that this principle enables to define fitting function for the distance between any two fireflies. For the optimization of fitting function, the individuals are forced to systematic or random moves in the population [106]. In this way, it is ensured that all the fireflies move toward to more attractive ones which have brighter flashing until

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the population converge to brightest one. Within this procedure, firefly algorithm is executed by three parameters which are attractiveness, randomization, and absorption. Attractiveness parameter is based on light intensity between two fireflies and defined with exponential functions. When this parameter is set to zero, then it happens to the random walk corresponding to the randomization parameter which is determined by Gaussian distribution principle as generating the number from the [0,1] interval [106]. On the other hand, absorption parameters affect to the value of attractiveness parameters as changing from zero to infinity. And, for the case of converging to the infinity, the move of fireflies appears as random walk [106]. The procedure of the firefly optimization technique is briefly rendered in Fig. 2.8. Utilization of firefly algorithm to solve the problems emerging from the setup and operation phases of RETs has been gained momentum recently. Considering the performance of this algorithm compared with well-known PSO and GA algorithm, it is claimed that firefly algorithm deals with the formulation of multi-modal functions more efficiently and has more success factor than those techniques [106,107]. Despite that the firefly algorithm method has been emerged recently; it has been referenced to significant number of researches. Considering the problem for photovoltaic systems in terms of optimization framework, MPP tracking on power-voltage characteristic is very familiar problem. Moreover, this problem is getting more complicated under impeding conditions for direct sunshine due to observed multiple local maxima points on the curve. In this respect, Shi et al. proposed modified firefly algorithm for the related problem under shading conditions [108]. They concluded that the proposed algorithm enabled to converge the global maximum point efficiently and quickly compared with the conventional techniques. Another optimization problem for solar system is the prediction of global solar radiation by using the meteorological data of sunshine. For this issue, Olatomiwa et al. addressed a method developed as hybridization of support vector machines and firefly algorithm to predict the monthly solar radiation [109]. The effectiveness of the method was validated as comparing with artificial neural networks and genetic programming models. In the similar manner, as considering the detrimental effect of variable weather conditions to produce electricity from solar system, Sulaiman IS et al. presented grid connected photovoltaic power systems. They proposed artificial neural network approach to model the output voltage of the adopted system and utilized firefly algorithm to search for optimal neuron numbers and other performance criterions in the training process [110]. In this context, firefly algorithm has been utilized to solve the problems confronted in smart grid and distributed generation applications. For example, Farhoodnea et al. dealt with the problem of the optimal placement and sizing of power conditioners used for smart grid applications [111]. They formulated dynamic firefly algorithm-based solution for the related problem. Moreover, the proposed multi-objective method helped to improve the overall performance of the smart grid as eliminating the factors causing decrease in performance [111]. To show the effectiveness of the method, the results were compared with the stationary firefly algorithm, hybrid GA, and dynamic PSO techniques and it was proved the superiority of the proposed method for the optimal sizing and location of the power conditions in smart grid applications. In another study, for minimizing the total power loss of the grid and improving the quality of the voltage form, Sulaiman MH et al. proposed firefly algorithm to solve the optimal allocation problem of distributed generation units [112]. They tested the effectiveness of the adopted method as comparing the GA and concluded that the performance of the both technique was nearly same. Another good aspect of smart grid

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FIGURE 2.8 Flowchart of firefly algorithm technique.

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concept is to provide potential to monitor the status of electrical networks. Besides that, in most cases, monitoring of the system status is a challenging problem, and it necessitates rigorous techniques to extract the related information from the system. Khorshidi et al. examined this complex problem and presented a hybrid approach as combining the weighted least squares and firefly algorithm methods for the state estimation problem of smart distribution system contains RESs [113]. Saxena and Ganguli addressed firefly-based solution for economic load dispatching problem in terms of the fuel cost of power generator [114]. They also considered the power generator compromised with solar and wind power sources and tested the proposed approach for different scenarios. In addition to the operative constraints, reliability factors should be incorporated in the construction of renewable power systems. In this perspective, Chandrasekaran et al. proposed firefly algorithm to maintain system reliability for solar integrated power plant [115]. On the other hand, the investigation on optimal setup and operation problems of wind-energybased RETs has been associated with firefly algorithm approach. Hendrawati et al. focused on optimal placement problem of turbines along the wind farms to achieve the optimal objectives of the setup cost and output power [116]. For this aim, they applied and tested firefly algorithm for the cases generated by the status of wind speed and direction to find the most feasible turbine number. As focusing the similar problem, Deb et al. developed firefly-based method for wind farm to regulate the output power and to find the optimal location of turbines [117]. As the power curve characterization constructed by correlating speed and output power of the wind turbine enables to observe the performance of the wind power unit, Karthik et al. applied firefly algorithm to obtain parametric modeling of those curves [118]. In this way, they estimated the utilization rate of the output power just for using the historical data of related turbines. Mekhamer et al. dealt with the challenging problem to balance the inversely proportional effects of power demand increase and power quality factors in hybrid distribution system [119]. They utilized firefly algorithm technique to overcome the mentioned problem and reported the related results for optimal design of the hybrid power system. As providing the economic and environmental advantages, combined heating and power microgrids which consisted of RESs was also addressed in the literature. Xiaopeng et al. studied on optimal dispatch model of the mentioned grid under consideration of the stochastic behaviors of the renewable energy systems and formulated an optimization problem based on leapfrog firefly algorithm method [120].

2.4.7 ARTIFICIAL BEE COLONY ALGORITHM ACO technique introduced by Dorigo et al. as inspiring of intelligence behavior of honey bees for searching the food source [80,81]. Depending on this metaheuristic approach, ABC algorithm is presented by Karaboga [121]. In this technique, depending on their tasks, bees are classified into three main groups as employed, onlooker, and scout. Scout bees are responsible for finding food source, besides, employed and onlooker bees are in charge of exploitation job in the source. For reviewing the tasking of the bees, scout bees search the area for food sources and mark the sources randomly without considering the richness for food content. Then, employed bees perform their tasks as choosing the some sources which have rich content of nectar. Depending on the abundance of nectar, they specify their dancing style as well as adjusting their speed to attract the onlooker bees which are waiting in the hive. Immediately, onlooker bees decide to tend toward the sources

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with their experiences by the dancing activities of the employed bees. When the food source is depleted, employed bees become scout bees as charging with exploring duty for new food sources. Establishing the analogy with the activities in a bee colony through optimization framework, every food source can be considered as a solution for the optimization problem. Hence, the richness of the food content for the source specifies the optimal quality of the solution as expressed by objective function. For more details for ABC readers are addressed to Refs. [121,122]. ABC algorithm is fast converging and proper for conducting the multi-objective complex problems [123]. On the other hand, ABC algorithm provides some easiness for implementation of the optimization technique as it does not require the more effort to tune the control parameters compared with GA [124]. Because of those advantages, ABC algorithm has been preferred to overcome the problems related with RES research. For example, Nasiraghdam and Jadid addressed photovoltaic/wind turbine/fuel cell power system to reconfigure the main power (grid) system as taking the hybrid system a share in the side of the grid [123]. They addressed a multi-objective ABC optimization scheme with respect to minimal power losses, energy production cost, deviation of the voltage for realization of the grid power system and they also tested the effectiveness of the proposed scheme on 33-bus distribution system. Another study for optimal sizing of renewable hybrid energy system into the grid is concluded by El-Zonkoly as considering ABC algorithm to solve the problems of design and allocation for PV-diesel hybrid energy system [125]. Besides of the design related problems, maximization of the operational profit of microgrid also emerged as an optimization problem for RES. In this context, Malakar et al. investigated the most economical conditions for the operation of small scale hybrid system composed of wind farm and pumped storage unit [126]. For this aim, they proposed ABC algorithm as taking into consideration of the water storage level and uncertainties emerging at wind-power production and supplying loads. After solving the optimization problem, they obtained strategies for optimal operation periods of pump storage unit according to the low pricing periods. Smart grid technology is another important application area for optimization framework as increasing the ratio of RES solution into the supplying side of the power sources. Corresponding to those conducted researches, development of smart grid technologies have gained momentum as the development of new operational strategies for better utilization of renewable energy systems. For considering the end users in smart grid methodology, they should be classified in terms of energy consumption patterns for economical energy dispatching strategies. In this context, Marzband et al. introduced ABC optimization algorithm for offering proper load types in a multi period scale to the end users for economical running of the power systems [127]. Moreover, they experimentally validated the performance of proposed solution under nominal and uncertain conditions and they examined the potential of the strategy in terms of the factors of extendibility, reliability, and flexibility [127]. As parallel with realization the growing potential and development of computational technologies, scheduling concept on smart grid is widely studied by the researchers. For example, Govardhan et al. considered the distributed energy recourses of distributed generation, demand response, and the vehicles plugged to the grid and developed an scheduling algorithm by utilization of ABC optimization approach for reaching remarkable reduction in the total energy cost [128]. In the same perspective, El-Zonkoly studied on optimally allocation problem of PV, wind, diesel units, and parking lots of plug-in-hybrid electrical vehicles [129]. Formulation of the problem is realized by methodology of ABC algorithm in terms of the setup and operational constraints. They

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operated the algorithm for optimal allocation and sizing problem of the considered units and proved that the resulting power system ensured cost reduction in the main grid system. For maximization of output power of photovoltaic systems, Oshaba et al. proposed a proportional-integral controller design for using ABC algorithm to control the system composed of PV panel and water pump [130]. They also compared the proposed optimization task with GA to prove the effectiveness under the disturbances. Moreover, shading patterns arising from atmospheric conditions influence negatively the output power of PV unit. For considering this problem, Sundareswaran et al. presented a ABC-based optimization algorithm to maximization the output power under the shading conditions and also demonstrated this strategy as improving the energy saving comparing the alternative approaches [131]. As wind energy is considered as one of the best clean energy technologies, researchers study on to ensure optimality conditions for wind-based energy systems. In this regard, Jadhav and Roy provided an optimal design approach by using ABC technique for wind
thermal hybrid power system as being considered the random nature of wind power [132]. They tested their method in the way of dispatching cost and emission for IEEE 30-bus test system and validated that their method is better solution accuracy and convergence rates as comparing the some other related techniques.

2.4.8 ARTIFICIAL IMMUNE SYSTEM Artificial immune system (AIS) algorithm is presented as an optimization technique by inspiring from the biological operation process of immune system [133
135]. Operation of this system based upon the existing of the antigens which are known as disease causing elements and the immune cells has adaptive capability to recognize and destroy them [136]. For catching and destroying the antigens, immune cells carry receptor molecules which are also defined as antibodies. Excessive increasing of antigens activates the immune system and antibody molecules are produced by immune cells to eliminate the antigens infected to the tissues. As the annihilation process depending on the affinity threshold between the antibodies and antigens, immune cells multiply rapidly and expose to hyper mutation. Mutation rate that is also defined as receptor editing, has inverse proportionality in degree of affinity between defending antibodies and antigens, such as the antibodies with lower antigenic affinity are hyper mutated and vice versa. By this way, similar to the mutation operator of GA, hypermutation provides to become diversification of immune cells to resist the various antigens. Moreover, the immune cells with nonadaptive antibodies for the antigens are disposed of the cell population. Thus, an efficient population of immune cells is ensured for the future infections. As observing immune system process, antigen population can be defined as the solutions in the search space [25]. Then, defining the initial antibody population, the optimization progress starts. In every iteration, antigen population changes gradually as applying the mutation and receptor editing operators. By this way, the fitness value of the population is maximized as elimination of the infeasible solution which corresponds to lower affinity in the immune process. AIS algorithm is terminated when the stopping criteria satisfied. Similar to the special operators of other fundamental optimization algorithms, mutation operator provides a fast search for local optima, on the other hand, the receptor editing operator helps to

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escape from local optima through global optima. From this point of view, AIS algorithm is getting popular especially for sizing problems of RES-related power systems. Addressing the AIS-based solution for the problems of optimal sizing of the energy suppliers and optimal energy management strategies in a smart grid concept have been recently appeared in the literature. For example, Ramachandran et al. formulated a hybrid algorithm advanced with both of the immune system and PSO techniques for profitable operation of smart grid in the energy market [137]. They aimed to provide an agent-based structure for the auction process to satisfy the profitability expectations of the seller and buyer and they demonstrated the effectiveness of their approach as simulating the related distributed energy sources under different energy market scenarios. Basu investigated dynamical economical dispatch problem of power generators with uncertain load demands and proposed AIS technique for optimal solution for this problem [138]. Although it does not include a direct solution for RES technologies, the results and formulation of the problem allows for utilization for the problems of RES-based power system application. In the context of scheduling problem for hybrid system, Lakshmi et al. considered the power system of wind and thermal energy to optimal operation under the constraints for satisfying minimum start/stop status of the power generators [139]. They demonstrated the effectiveness of their optimal solution formulated with the technique of AIS on a generation company with 10 thermal units and 2 wind farms. In a similar fashion, planning and sizing of the distributed network energy system which includes fuel cell and nonrenewable power sources studied by Soroudi et al. as adopting AIS-based method to optimize the cost and emission factors of the power system [140]. Moreover, they tested and compared the solution with the results of other competitive methods to reveal the dominance of the adopted method.

2.4.9 GAME THEORY Game theory optimization is a method of using mathematical tools to find the most feasible strategy to minimize one’s maximum losses or maximize one’s minimum winnings in a game structure such as war, business competition, and others [141]. Games are formulated with the basic elements of players, strategies, and utility function. Players or decision makers establish completion or form cooperation with other players and chooses best advantageous strategy [141]. Strategies are known by the players and can be categorized as behavioral and chance strategies. Besides, the utility function is also known by the players and has an optimal pay off depending on the preferred strategies [142]. Despite that, there are lots of game types, most of the familiar types are cooperate/noncooperate and zero-sum/nonzero sum games. The more details for game theory concept [141
143] can be referred. In the literature, game theory approach has been utilized in the studies of renewable-energy based technologies. For example, Mei et al. defined cooperative and noncooperative models of grid connected hybrid power system in terms of the payoffs of the units in the system [144]. Nguyen et al. defined cooperative game theory-based agent approach for the optimum participation of renewable power system actors (system operators, producers, and consumers) within the power system operation [145]. Considering the smart grid concept, Saad et al. formulated a coalition game between microgrids powered by solar and wind energy as transferring the needed energy of each other and they

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improved the reasonable energy efficiency in the distribution line [146]. In a similar manner, the energy trading within the smart grid structure through distributed storage units based on a noncooperative game theory optimization approach studied by Wang et al. [147]. Through the same perspective, incentive-based strategies for smart grid have been popular in the energy market. Adopting the game theory approach, Mohsenian-Rad et al. proposed scheduling algorithm to find the optimal energy consumption for the subscribers [148]. In a similar fashion, Atzeni et al. provided a two side game theoretical optimization approach between end users and system operator for the demand side management concept [148]. Another demand side management approach similar to [149] was also realized by Maharjan et al. [150]. For further reading of this concept, one can refer to [151]. For optimization of the energy production in wind farm, Marden et al. reported a model free control strategy based on game theory approach as considering the aerodynamic interaction among the turbines [152]. In this context, for the aggregation of the wind energy and developing fair sharing mechanism, Baeyens et al. addressed coalition game theory for analyzing the benefits of forming coalition between the wind power producers [153]. A combined evaluation of electric vehicle charging, local renewable power generation, and energy prices considering physical power system constraints through a game theoretical analysis was given by Lee et al. [154]. In another study, Zhang et al. realized the scheduling of electric vehicle charging at multiple charging stations considering local renewable energy availability and electric vehicle arrival time and power-price-based uncertainties [155].

2.4.10 SIMPLEX TECHNIQUE SPECIFICALLY FOR MIXED INTEGER LINEAR PROGRAMMING CONCEPT The simplex technique previously described in Section 2.2.3.2 can be used for models in mixed integer linear programming (MILP) context that has been widely used in the literature for different power system problems. Let us consider the LP model in (2.4). As the characteristic of this model, objective function and constraints are in linear form. Besides, according to the type of problem, some variables can be defined as real-valued (fractional values) and some others as integer valued. In this case, this optimization formulation is called MILP. MILP model has a wide range of application area for optimizing the complex systems developed in renewable energy concept. For detailed review of MILP technique one can refer to [156]. Regarding MILP technique applied to different areas of electric power system, there are several literature examples. Among them, Eren and Gorgun developed a methodology for the optimization of the components of fuel cell battery hybrid vehicle using MILP approach [157]. Erdinc conducted the demand response strategies for smart household by adopting the MILP technique and analyzed the technical and economical effects of distribution generation units and energy storage under constraints of dynamic pricing and peak power limiting [158]. Paterakis et al. formulated a novel optimal operating strategy based on MILP framework for home energy management as interacting the economic and technical aspects of bi-directional power flow [159]. In the same manner, Paterakis et al. proposed another home energy management structure for day-ahead scheduling of the household appliances as modeling them in MILP concept and they tested the overall model with reasonable low time steps to prove the computational performance of the methodology [160].

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Paterakis et al. presented a method for the problem of reconfiguration of distributed network adopting the MILP technique and tested the formulation on a distribution system [161]. On the other hand, regarding the off-grid hybrid power systems, Chen et al. developed a MILP-based approach for allocation of the power units and also to meet the demanded power from the appliances [162].

2.4.11 OPTIMAL CONTROL TECHNIQUE It is well-known that H2 and HN-based controller design such as linear quadratic regulator (LQR) design or linear quadratic Gaussian (LQG) type controller design problems can be formulated as a convex problem and can be efficiently solved by convex optimization techniques. This formulation can be obtained by a slight modification of the Lyapunov stability theorem. Assume that a linear time-invariant system is governed by x_ 5 Ax n

(2.28)

n3n

where xAR , AAR . It is well known that (2.28) is asymptotically stable for every initial condition xð0Þ 6¼ 0 if and only if there exist a symmetrical positive-definite matrix P such that VðxÞ 5 xT Px . 0 ’x 6¼ 0

(2.29)

and V_ 5 xT ðAP 1 PAÞx , 0

’x 6¼ 0

(2.30)

Since we are interested in an LQR problem, we need to find out a control law of the form u 5 Kx

(2.31)

that minimizes the performance index N ð

Jðx; uÞ 5

ðxT Qx 1 uT RuÞdt

(2.32)

0

along the system trajectory (2.28). Here, Q 5 QT $ 0 and R 5 RT . 0. In view of (2.28) and (2.31), (2.32) can be rewritten as N ð

JðxÞ 5

ðxT Qx 1 xT K T RKxÞdt:

(2.33)

0

Also, by using the TraceðUÞ operator and its useful rotation and linearity properties, the integrand in (2.33) can be manipulated to obtain N ð

JðxÞ 5

2 3 N ð   Trace ðQ 1 K T RKÞxxT dt 5 Trace4ðQ 1 K T RKÞ xxT dt5:

0

Hence, defining P9

ÐN 0

(2.34)

0

xxT dt, (2.34) can be written as

  JðxÞ 5 Trace ðQ 1 K T RKÞP :

(2.35)

2.4 SOLUTION TECHNIQUES FOR CONSTRAINED OPTIMIZATION

65

On the other hand, replacing A by A 1 BK and assuming that the closed-loop system is stable in the sense of Lyapunov, then in view of (2.28) and (2.31), Lyapunov stability criteria (2.30) can be rewritten as ðA 1 BKÞP 1 PðA1BKÞT 1 x0 xT0 , 0:

(2.36)

However, (2.36) is nonconvex therefore introducing a new variable Y 5 KP, (2.36) can be converted to a convex constraint such as AP 1 BY 1 PAT 1 Y T BT 1 x0 xT0 , 0:

(2.37)

On the other hand, the objective function (2.35) can rewritten as  JðxÞ 5 Trace QP 1 R1=2 YP21 YR1=2 :

(2.38)

Note that (2.37) is a homogeneous function of Y and P. That is if (2.37) is satisfied for a specific values of Y % and P% , then for any r . 0, rY % and rP% will also satisfy the same inequality. Hence, (2.37) can be rewritten as [11,163] AP 1 BY 1 PAT 1 Y T BT 1 I , 0:



Also, by introducing an auxiliary term X, the nonlinear term Trace R replaced by Trace ðX Þ subject to X . R1=2 YP21 YR1=2:

1=2

21

YP YR

1=2



(2.39)

can be (2.40)

Hence, using well-known Schur formula [164], and in view of (2.39) and (2.40), the complete convex optimization technique for LQR problem can be described as min Trace ðQPÞ 1 Trace ðXÞ

(2.41)

AP 1 BY 1 PAT 1 Y T BT 1 I , 0; " # X R1=2 Y . 0; P . 0: Y T R1=2 P

(2.42)

P;Y;X

subject to

If one can find a solution to the convex optimization problem (2.41), the optimal LQR gain can be obtained as K 5 YP21 . Note that, in the formulation of LQR type optimal controller, whole state of the related system is assumed as available for control. On the other hand, most of the application can be exposed to the measurement noise and it is needed to estimate the states. In this case, first, the states are reconstructed by adding the Gaussian noise to model the disturbances, then the estimated states are used for feedback controller. This type of controllers is defined as LQG controller which is combined with Kalman filter and LQR. For more details, one can refer to Refs. [164,165]. Considering the practices of OC technique for the RETs, the literature does not contain rich publication content. Besides that, some remarkable studies have been carried out. For example, Pukrushpan et al. provided a model based analysis and design formulation for a fuel processing system to regulate the hydrogen supplied to the fuel cell and to balance the temperature during the

66

CHAPTER 2 INTRODUCTION TO OPTIMIZATION

peak power demands from the fuel cell [166]. For realization of the analysis and design requirements, they addressed LQR
LQG controller technique to achieve good response under wide range of operating conditions. Considering the studies on wind-based RETs, Boukhezzar and Siguerdidjane investigated energy conversion systems for wind energy as focusing to capture optimal wind energy [167]. By this aim, they synthesized LQG controller based on linear and nonlinear model of the wind source. In a similar fashion, Kalbat used the LQG technique to increase the capturing wind energy and to reduce the operational costs of the wind power system [168]. In another study, Imran et al. discussed the disturbance rejection problem of wind power system and presented a pitch regulation method based on LQG [169]. Regarding the context of power grid applications, Huerta et al. studied on control methodology of voltage-source converter coupled to the power grid [170]. They proposed LQG controller for the current control of the related converter and highlighted some useful criteria for choosing the controller parameters. Liu et al. presented decentralized LQG controller for parallel power converters in a power network [171]. By this approach, they aimed to drive the power system through stable operating point and also damping the disturbance effect by a cooperative action.

2.5 CONCLUSION AND DISCUSSION Optimization concept has always found an area of use for different power system problems and together with the increasing RES penetration, the operation of the electric power system needs to be analyzed in deeper in optimization framework. Traditional optimization techniques and modern heuristic techniques have been addressed greatly in the literature to solve the problems related with the design and operation process engineering systems. On the other hand, most of the optimization problem in the framework of RETs involves nonlinear objective functions and multi constraints. For this reason, intelligent-based methods have been applied to the mentioned problems to satisfy the expected requirements of economic feasibility and technical reliability by the engineering systems. Most of the intelligent methods have been developed by the intelligence behind of collective behaviors of the animal colonies or the intelligence mechanism of the biological or physical processes. Moreover, the admirable metaphor of those intelligent events is to allow the new adaptation to formulate the challenging problems in the scope of RETs so that, as we search to understand the basic behavior characteristics of the creatures or the complex processes in the nature, it leads us to formulate engineering problems as much as appropriate the nature of them. In this respect, penetration rate of the optimization algorithms into the modeling, controlling, and planning problems of RETs have been an increasing tendency. The main factor behind this increasing has been based to eliminate the disadvantages of those technologies by using of the novel optimization and control techniques. Moreover, as the developments of the computer technology, the traditional optimization algorithms allow for some unsolvable problems become solvable. Hence, some complex hybrid configurations of RETs have been formulated as optimization framework and utilization trend of those technologies have been also accelerated.

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CHAPTER

OPTIMAL PROCUREMENT OF CONTINGENCY AND LOAD FOLLOWING RESERVES BY DEMAND SIDE RESOURCES UNDER WIND-POWER GENERATION UNCERTAINTY

3 Nikolaos G. Paterakis

Eindhoven University of Technology, Eindhoven, The Netherlands

3.1 INTRODUCTION The qualification and quantification of the appropriate ancillary services (ASs) in order to ensure the secure operation of the power system and the provision of uninterrupted and quality service to the consumers play a primordial role in the short-term operations of the System Operator (SO). More specifically, it should be guaranteed that sufficient capacity is kept in order to allow for corrective actions with the purpose of facing energy imbalances. Such imbalances may occur due to a generating unit outage or because of the failure of a transmission line. These events, which are commonly referred to as contingencies, constitute a severe jeopardy for the operation of the power system and should be tackled through the deployment of reserves to ensure the satisfaction of system constraints. Another source of imbalances is the deviation of the intrahour load demand from its forecasted value. In addition to that the large-scale penetration of renewable energy sources (RES), especially of wind-power generation, has resulted in an increased need for procuring reserves in order to accommodate the volatility in the power output of such resources. Different SOs across the world utilize different definitions and procurement procedures as regards reserves [1,2]. Furthermore, apart from the more traditional reserve services, a new type of AS was recently introduced by Midcontinent Independent SO [3] and California Independent SO [4], namely the flexible ramping products. These services are designed to increase the robustness of the procurement of load-following reserves under uncertainty and are especially related to major ramping events caused by solar and wind-power resources. It is nowadays a common ground that demand-side resources may be also deployed in order to provide system services, presenting significant potential technical and economic benefits, especially in the presence of high levels of RES penetration in the generation mix. The question this chapter deals with is whether demand response (DR) could facilitate the system operation when apart from Optimization in Renewable Energy Systems. DOI: http://dx.doi.org/10.1016/B978-0-08-101041-9.00003-X Copyright © 2017 Elsevier Ltd. All rights reserved.

75

76

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system contingencies and intrahour load deviations, the SO must also confront the uncertainty in the production of wind farms. Providing AS in a market framework primarily involves the solution of the unit commitment problem. Many different solution approaches have been proposed so far. A significant number of recent studies employ meta-heuristic techniques such as genetic and evolutionary algorithms [5
7], particle swarm optimization [8], tabu search and simulated annealing [9], as well as hybrid approaches [10]. Artificial intelligence methods such as fuzzy and expert systems [11], as well as neural networks [12] have been also used. Among the first methods applied to the solution of the unit commitment problem were the ones based on priority list [13] and mathematical programming. For instance, Lagrangian relaxation [14] is applied in Ref. [15] to a transient stability-constrained network structure. The Lagrangian relaxation method and its improved versions are also employed in Refs. [16,17]. The combination of Lagrangian relaxation with mixed-integer nonlinear programming is applied in Ref. [18]. Dynamic programming has been also extensively applied to the solution unit commitment problem in the past [19]. Mixed integer linear programming (MILP) approach may be currently considered as the state-of-the-art for the unit commitment problem solution, being almost exclusively employed in modern centralized market clearing engines and having been extensively studied in the literature [20
22]. A detailed discussion regarding the solution approaches of the unit commitment problem can be found in Ref. [23], while a recent review covering stochastic unit commitment is presented in Ref. [24]. Numerous technical studies that propose market designs in order to procure reserve services also exist. In Refs. [25,26], a stochastic security-constrained market-clearing procedure is developed, in which line and generator failures are taken into account through a set of preselected contingencies, while reserves are determined using the expected load not served as a penalty metric. A model to evaluate the economic impact of reserve provision under high wind-power generation penetration based on stochastic programming is provided in Ref. [27]. Moreover, the two-stage stochastic programming model presented in Ref. [28] considers the participation of DR providers in meeting the system constraints. A day-ahead market structure in which DR participation in contingency reserves is considered on the basis of bidding an offer curve representing the cost of rendering load curtailments available is also presented in Ref. [29]. A multi-agent based market model was proposed by Jafari et al. [30]. The day-ahead and several intraday markets, as well as a spot real-time market for energy and operating reserves are considered. However, in Ref. [30] involuntary load shedding is the only considered form of demand side participation. In Ref. [31], the operation of the power system in the presence of wind-power generation was analyzed on the basis of stochastic contingency analysis, neglecting the participation of DR resources. In Ref. [32], a switching operation between two separate energy markets named “conventional energy market” and “green energy market” was proposed, using a stochastic programming framework for profit maximization, without investigating the contribution of demand-side resources. Similar studies neglecting the contribution of DR to reserve procurement for managing uncertainties were conducted in Refs. [33
35]. It is worth mentioning though that the aforementioned studies made use of approaches to achieve computational efficiency. The computational efficiency of solving the problem of unit commitment under uncertainty was also the subject of Ref. [36]. Uncertainties stemming from wind-power generation and load demand were confronted by optimally scheduling hourly reserves using a security-constrained unit commitment in Ref. [37]. The dual decomposition algorithm was applied to solve the two-stage stochastic programming unit commitment problem in Ref. [38]. Jin et al. [39] modeled demand as a linear function of price in order

3.2 MATHEMATICAL MODEL

77

to investigate the contribution of DR to reserves under significant wind-power generation penetration. Load uncertainty and generation unavailability were considered in Ref. [40] without accounting for RES uncertainty. Apart from the stochastic-programming-based literature referred above, there exist studies considering different uncertainty modeling frameworks such as probabilistic [41,42], rolling stochastic [43], and Monte Carlo criteria [44]. In this chapter a two-stage stochastic-programming-based joint energy and reserve marketclearing model within a MILP framework is developed. The uncertainty that is introduced by the penetration of wind-power generation, load deviations, and system contingencies is considered, so that economic optimality can be achieved. In order to guarantee the secure operation of the system, reserve services are procured from both generation and demand side resources which are represented by load serving entities (LSE). Two different time scales are introduced. The first stage of the model stands for the day-ahead market operation and is cleared on an hourly basis, while the second stage considers possible instances of uncertainty with a subhourly granularity.

3.2 MATHEMATICAL MODEL 3.2.1 OVERVIEW AND MODELING ASSUMPTIONS The overview of the proposed model is portrayed in Fig. 3.1. The model is fed with a set of plausible wind-power generation scenarios and the technical and economic characteristics of the demand and generation side resources that might be employed in order to provide reserve services. The SO collects all the relevant information, as well as other system parameters (e.g., inelastic load forecast) and clears the market in a centralized fashion by solving the two-stage stochastic programming model. The model consists of two stages: The first stage represents the day-ahead market and involves variables and constraints that are not dependent on any specific scenario realization, while the second stage represents a possible instance of the day-ahead market in terms of uncertainty

FIGURE 3.1 Overview of the market clearing model.

78

CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

realization and comprises variables and constraints which are dependent on each individual scenario and its probability of occurrence. The first stage of the problem is cleared considering an hourly granularity, while the second stage is cleared using intrahour intervals. The time granularity of the second stage may be adjusted to the required time interval. In this chapter, demand may belong to one of three categories. The first category of demand comprises the so-called inelastic loads. The consumption of the loads that fall in this category has to be fully satisfied under normal operating conditions and cannot be altered. Nonetheless, as a last resort and under a very high penalty, the SO may activate involuntary load shedding in order to satisfy the system constraints. Another type of loads is represented by LSE that contribute to load following reserves (LSE of type 1). Their consumption can be altered within prespecified limits in order to respond to wind-power generation fluctuations and intrahour load deviations. Finally, LSE, which are eligible for providing contingency reserves (LSE of type 2), stand for loads the consumption of which can be modified given that several constraints are satisfied. For example, it may be the case that there is a limited number of times that these resources may be called to respond and that every call has a specific maximum duration. More complex constraints (e.g., minimum time between two calls) and contract types can be easily integrated within the proposed methodology. Two different types of reserve services are discerned. The first type is load following reserves that might be procured from both generating units and LSE that are committed to provide this service. More specifically, with the purpose of balancing intrahour and wind-power generation deviations, generating units provide synchronized up and down, as well as nonspinning reserves. The LSE of type 1 can provide this type of reserve by continuously adjusting their consumption upward or downward. The participation of the demand side in load following reserves is compensated at the utility value of the demand on the top of a cost paid for instructing them to be on stand-by. In the case of a generating unit or a transmission line failure, a second type of reserves, namely contingency reserves, is activated. The energy deficit is counterbalanced by available synchronized and nonsynchronized units, or LSE that opt to provide this service. The LSE of type 2 that provide this service are compensated both at a value related to the period of the day they are called to provide this service, as well as for being on stand-by. In order to facilitate the mathematical modeling of the problem, several assumptions are adopted. First of all, the only source of uncertainty taken into consideration is related to the wind production, since it is considered that the transmission line and unit contingencies on trial are perfectly known. The network constraints are considered in terms of a lossless linear representation, while active power losses may be included in the formulation as explained in Ref. [27]. Furthermore, the power output of units that become unavailable due to a contingency are instantly set to zero. Similarly, the active capacity of a transmission lines that trip is considered to be zero. Due to the short length of the horizon, it is assumed that units that fail are unavailable thereafter; though, it is possible that a line becomes available again within the examined horizon. Without loss of generality, the participation of wind-power producers is considered to be promoted by the SO. This means that as regards the market clearing procedure, wind energy is considered to be free of cost. In practice, it could be paid a regulated tariff out of the day-ahead market on the basis of the energy actually produced [27]. Regarding the modeling of the demand side resources, since in practice DR can respond in a few minutes [45], ramping constraints are not enforced for the LSE. Finally, only loads that are not subject to any resource offering scheme may incur involuntary load shedding.

3.2 MATHEMATICAL MODEL

79

3.2.2 OBJECTIVE FUNCTION EC 5

X

(

t1 AT 1

1

1

X

X j2 AJ2

f AF i

CjDN;LSE1 ULSE1DN j1 ;t1 1 ;t1

πs

sAS

1

iAI

X j1 AJ1

" # X X  R;DN R;DN R;UP R;UP R;NS R;NS 1 1 Ci;f ;t1 Ubi;f ;t1 1 SUCi;t1 1 SDCi;t1 1 Ci;t1 URi;t1 1 Ci;t1 URi;t1 1 Ci;t1 URi;t1

X t2 AT2

(

"

X X iAI

0 LSE2 λLSE2 j2 ;t2 Uψj2 ;t2 ;s

f AF i

1

1 CjUP;LSE1 ULSE1UP j1 ;t1 1 ;t1



1 #

X j2 AJ2

CjDN;LSE2 ULSE2DN;con j2 ;t1 2 ;t1

1 CjUP;LSE2 ULSE2UP;con j2 ;t1 2 ;t1

) 

   X 0 u d C 0i; f ;t2 Uri;Gf ;t2 ;s 1 CAi;t2 ;s 1 λLSE1 j1 ;t2 U LSE1j1 ;t2 ;s 2 LSE1j1 ;t2 ;s

X wAW

j1 AJ1

Vspill w;t2 UΔT2 USw;t2 ;s ΔT1

! 1

X rAR

!9 = LOL Vr;t shed 2 UΔT2 ULr;t2 ;s ; ΔT1 (3.1)

The objective function (3.1) stands for the minimization of the total expected cost (EC) of the system operation. The first line of (3.1) expresses the costs associated with energy provided from the generating units, the startup and shutdown costs, and the commitment of the units to provide reserves. The second and third lines represent the costs of scheduling reserves from the LSE of type 1 and type 2, respectively. The rest of the objective function is scenario dependent, as indicated by the summation over the scenario index. Furthermore, in the second stage the intrahour intervals are taken into account, since a different set of time intervals is considered. The fourth line of the objective function takes into consideration the cost of changing the status of the generating units and the cost of actually deploying reserves from the generators. Similarly, the fifth line considers the cost of deploying reserves from the LSE of type 1. The sixth line stands for the cost of calling LSE of type 2 to provide contingency reserves. Finally, the last line takes into account the wind spillage cost and the EC of the energy not served to the inelastic loads. 0

Ci;f ;t2 5 0

λLSE1 j1 ;t2 5 0

λLSE2 j2 ;t2 5

Ci;f ;t1 UΔT2 ’i; f ; t2 AT2in ; t1 ΔT1

(3.2)

λLSE1 j1 ;t1 UΔT2 ’i; f ; t2 AT2in ; t1 ΔT1

(3.3)

λLSE2 j2 ;t1 UΔT2 ’i; f ; t2 AT2in ; t1 ΔT1

(3.4)

Eqs. (3.2)
(3.4) are required in order to adjust the units of the marginal cost of the generating units and the utilities of the LSE. The cost unit is h/MW h which is suitable for the first stage of the problem in which the duration of the time interval is 1 h; however, in the second stage of the problem, intrahour intervals (min) are considered, and therefore, the cost units should be appropriately adjusted.

80

CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

3.2.3 CONSTRAINTS 3.2.3.1 First stage constraints This section presents the first stage constraints of the optimization problem. These constraints involve only decision variables that do not depend on any specific scenario. Furthermore, the time dependence of variables refers to the time interval utilized in the first stage (i.e., hourly in this study) that is denoted by t1 AT1 .

3.2.3.1.1 Generator output limits Psch i; t1 5

X

bi;f ;t1 ’i; t1

(3.5)

f AF i

0 # bi; f ;t1 # Bi; f ;t1 ’i; f ; t1

(3.6)

DN min 1 Psch i; t1 2 Ri;t1 $ Pi Uui;t1 ’i; t1

(3.7)

UP max 1 Psch Uui;t1 ’i; t1 i; t1 1 Ri; t1 # Pi

(3.8)

The generator bidding function is considered to be monotonically increasing and is approximated using a step-wise linear function as in Ref. [46]. This is enforced by (3.5) and (3.6). An example of a marginal cost function for a unit that offers its available energy in three blocks is illustrated in Fig. 3.2. Constraints (3.7) and (3.8) limit the output power of a generating unit, taking also into account the hourly scheduled up and down reserve margins.

3.2.3.1.2 Generator minimum up and down time constraints t1 X τ5t1 2UT1i 11

FIGURE 3.2 Example of a step-wise linear bidding function.

y1i;τ # u1i;t1 ’i; t1

(3.9)

3.2 MATHEMATICAL MODEL

t1 X

z1i;τ # 1 2 u1i;t1 ’i; t1

81

(3.10)

τ5t1 2DT1i 11

Constraint (3.9) forces a unit to remain committed for at least UT1i hours once a startup decision is made ðy1i;t1 5 1Þ, while (3.10) forces a unit to remain offline for at least DT1i hours once a shutdown decision is realized ðz1i;t1 5 1Þ.

3.2.3.1.3 Unit commitment constraints y1i;t1 2 z1i;t1 5 u1i;t1 2 u1i;ðt1 21Þ ’i; t1

(3.11)

y1i;t1 1 z1i;t1 # 1 ’i; t1

(3.12)

Eq. (3.11) enforces the startup and shutdown status change logic. The logical requirement that a unit cannot start up and shut down simultaneously during the same period is modeled using (3.12). Note that, these constraints indicate only the hour for which a startup or shutdown decision is taken but not the exact subhourly interval in which the startup or shutdown decision will actually occur.

3.2.3.1.4 Startup and shutdown costs SUC1i;t1 $ SUCi Uy1i;t1 ’i; t1

(3.13)

SDC1i;t1 $ SDCi Uz1i;t1 ’i; t1

(3.14)

The cost that incurred when an off-line unit receives a command by the SO to start up or when an online unit is commanded to shut down is considered through constraints (3.13) and (3.14), respectively.

3.2.3.1.5 Ramp-up and ramp-down limits sch Psch i;t1 2 Pi;ðt1 21Þ # ΔT1 URUi ’i; t1

(3.15)

sch Psch i;ðt1 21Þ 2 Pi;t1 # ΔT1 URDi ’i; t1

(3.16)

In order to consider the effect of the ramp rates that limit the changes in the output of the generating units, constraints (3.15) and (3.16) are enforced. ΔT1 is the time length of the optimization interval of the first stage in minutes, for example, ΔT1 5 60 min in the case of hourly granularity.

3.2.3.1.6 Generation side reserve scheduling S 1 0 # RUP i;t1 # T URUi Uui;t1 ’i; t1

(3.17)

S 1 0 # RDN i;t1 # T URDi Uui;t1 ’i; t1

(3.18)

  NS 1 ’i; t1 0 # RNS i;t1 # T URUi U 1 2 ui;t1

(3.19)

Constraints (3.17)
(3.19) impose limits on the procurement of reserves from the conventional generating units. Up and down spinning reserves and nonspinning reserves are defined by (3.17)
(3.19), respectively. Note that T S and T NS denote the time in minutes during which the reserves should be

82

CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

fully deployed. The deployment time for each reserve type is defined by the rules that hold for each system. Note also that the aforementioned constraints are responsible for scheduling the total amount of reserve that is needed to cover all the imbalances considered in this study, that is, wind and load fluctuations, as well as contingencies. UP;load RUP 1 RUP;wind 1 RUP;con ’i; t1 i;t1 i;t1 i;t1 5 Ri;t1

(3.20)

DN;load 1 RDN;wind 1 RDN;con ’i; t1 RDN i;t1 i;t1 i;t1 5 Ri;t1

(3.21)

NS;load 1 RNS;wind 1 RNS;con ’i; t1 RNS i;t1 i;t1 i;t1 5 Ri;t1

(3.22)

Up spinning reserves, down spinning reserves and nonspinning reserves are scheduled in order to maintain the system balance during the actual operation of the power system that is disturbed due to positive or negative elastic or inelastic load deviations, wind ramp-ups and -downs, as well as contingency events. Up spinning reserves imply the increase in a synchronized unit’s power output, while down spinning reserves stand for the opposite. Nonspinning reserves are provided by nonsynchronized units as stated by (3.19). Eqs. (3.20)
(3.22) decompose the unit’s total scheduled up, down, or nonspinning reserves to different services that correspond to the different factors that can trigger the need of such reserves. The decomposition of reserves from the generation side is displayed in Fig. 3.3.

FIGURE 3.3 Reserve scheduling from generating units.

3.2 MATHEMATICAL MODEL

83

FIGURE 3.4 Load and reserve scheduling from LSE of type 1.

3.2.3.1.7 Wind-power scheduling WP;max 0 # PWP;S ’w; t1 w;t1 # Pw

(3.23)

Typically, the wind-power generation scheduled in the day-ahead market is considered equal to its forecasted value. However, in this study, it is considered that the SO schedules the optimal amount of wind at each period t1 according to the technoeconomic optimization within the limits imposed by (3.23). Several studies consider that the upper bound of wind-power scheduling in the day-ahead market is N. However, in this study, the upper limit is considered equal to the installed capacity of each wind farm.

3.2.3.1.8 Load serving entities It was stated before that the demand side can also contribute in reserves. In this study, two types of LSEs that are able to provide different reserve services are considered. First, the LSE of type 1 can provide up and down load following reserves in order to balance the wind fluctuations and the intrahour load deviations. Second, the LSE of type 2 may provide up and down reserve in order to confront contingencies. The two types of LSE are graphically illustrated in Figs. 3.4 and 3.5 in which the basic parameters of these loads are identified. sch max LSE1min j1 ;t1 # LSE1j1 ;t1 # LSE1j1 ;t1 ’j1 ; t1

(3.24)

sch min 0 # LSE1UP j1 ;t1 # LSE1j1 ;t1 2 LSE1j1 ;t1 ’j1 ; t1

(3.25)

UP;load 1 LSE1UP;wind ’j1 ; t1 LSE1UP j1 ;t1 j1 ;t1 5 LSE1j1 ;t1

(3.26)

84

CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

FIGURE 3.5 Load and reserve scheduling from LSE of type 2.

max sch 0 # LSE1DN j1 ;t1 # LSE1j1 ;t1 2 LSE1j1 ;t1 ’j1 ; t1

(3.27)

DN;load LSE1DN 1 LSE1DN;wind ’j1 ; t1 j1 ;t1 j1 ;t1 5 LSE1j1 ;t1

(3.28)

X t1 AT1

req LSE1sch j1 ;t1 $ Ej1 ’j1

(3.29)

According to (3.24), the load may be scheduled within an upper and lower limit around its nominal value that defines its flexibility. The amount of up reserves that may be scheduled during a period t1 are between zero and the margin that is defined by the difference between the scheduled and the minimum allowed load as stated in (3.25). These reserves are further decomposed into a component related to a reduction in order to balance wind fluctuations and a component that is related to balancing an intrahour deviation of the load as stated by (3.26). Similarly, the amount of down reserve that may be scheduled in each period is between zero and the capacity that is defined by the difference between the maximum allowed and the scheduled load, a fact that is stated by (3.27). The decomposition of down reserves in its components is realized by (3.28). Finally, in order to ensure that the LSE of type 1 energy needs are fulfilled during the horizon, despite the fact that it may be scheduled for partial curtailment in several periods, the energy requirement constraint (3.29) is enforced. sch max LSE2min j2 ;t1 # LSE2j2 ;t1 # LSE2j2 ;t1 ’j2 ; t1

(3.30)

min 0 # LSE2UP;con # LSE2sch j2 ;t1 j2 ;t1 2 LSE2j2 ;t1 ’j2 ; t1

(3.31)

3.2 MATHEMATICAL MODEL

sch 0 # LSE2DN;con # LSE2max j2 ;t1 j2 ;t1 2 LSE2j2 ;t1 ’j2 ; t1

85

(3.32)

Similar to LSEs of type 1, the load of LSEs of type 2 may be scheduled within an upper and lower limit around its nominal value. This is enforced by (3.30). The up and down reserves that are scheduled by the LSEs of type 2 in order to confront system contingencies are defined by (3.31) and (3.32), respectively. This type of load is not subject to an energy requirement constraint due to the fact that it is paid to be curtailed for a prespecified number of periods.

3.2.3.1.9 Day-ahead market power balance X iAI

PSi;t1 1

X

PWP;S w;t1 5

wAW

X rAR

D1r;t1 1

X j1 AJ1

LSE1sch j1 ;t1 1

X j2 AJ2

LSE2sch j2 ;t1 ’t1

(3.33)

Eq. (3.33) enforces the market power balance. In other words, it states that the total generation of the conventional units and the total production of the wind farms must be equal to the demand of the inelastic load and the demand of the LSE of the two types at any given time interval t1 . Although any scheme could be implemented within the proposed formulation, it is common not to enforce network constraints in the first stage of the problem [27].

3.2.3.2 Second-stage constraints This section presents the second-stage constraints of the optimization problem. These constraints involve only decision variables that do depend on a specific scenario realization. Furthermore, the time dependence of variables refers to the time interval utilized in the second stage (i.e., subhourly intervals, e.g., 15 min) that is denoted by t2 AT2 .

3.2.3.2.1 Generating units Constraints (3.34)
(3.43) are related to the operation of the generation side in the light of each individual scenario outcome. min 2 PG i;t2 ;s $ Pi Uui;t2 ;s ’i; t2 ; s

PG i;t2 ;s

# Pmax Uu2i;t2 ;s i

’i; t2 ; s

(3.34) (3.35)

The minimum and maximum generation limits are also enforced in the second stage of the problem through (3.34) and (3.35). G PG i;t2 ;s 2 Pi;ðt2 21Þ;s # ΔT2 URUi ’i; t2 ; s   G PG i;ðt2 21Þ;s 2 Pi;t2 ;s # ΔT2 URDi 1 N1 U 1 2 UCi;t2 ’i; t2 ; s

(3.36) (3.37)

As stated before, a ΔT2 -minute time interval is adopted in the second stage of the model constraints (3.36) and (3.37), which hold to limit the ramp-up and down of the units. As the ramp-up and -down rates of the generators are given in MW/min, the power output of a unit can change by its ramp-up or down rate multiplied by ΔT2 in each scenario. Note that constraint (3.37) is relaxed when the unit i fails by using a sufficiently large value for the constant N1 .

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

t2 X

y2i;τ;s # u2i;t2 ;s ’i; t2 ; s

(3.38)

z2i;τ;s # 1 2 u2i;t2 ;s ’i; t2 ; s

(3.39)

UT2 τ5t2 2ΔTi 11 2

t2 X DT2 τ5t2 2ΔTi 11 2

In the second stage of the problem, the minimum up and down times of the generating units are given in minutes. Thus, in (3.38) and (3.39) these times are divided by the duration of each interval ΔT2 in order to express the minimum up and down times in a number of intervals. Evidently, UT2i and DT2i must be integer multiples of ΔT2 . y2i;t2 ;s 1 z2i;t2 ;s # 1 ’i; t2 ; s

(3.40)

y2i;t2 ;s 2 z2i;t2 ;s 5 u2i;t2 ;s 2 y2i;ðt2 21Þ;s ’i; t2 ; s

(3.41)

Similarly to (3.11) and (3.12), constraints (3.40) and (3.41) ensure that the logic of unit commitment is preserved. SUC2i;t2 ;s $ SUCi Uy2i;t2 ;s ’i; t2 ; s

(3.42)

SDC2i;t2 ;s $ SDCi Uz2i;t2 ;s ’i; t2 ; s

(3.43)

The startup and shutdown costs of the generators incurred in the second stage are considered using (3.42) and (3.43).

3.2.3.2.2 Wind spillage limits 0 # Sw;t2 ;s # PWP w;t2 ;s ’w; t2 ; s

(3.44)

For technical and economic reasons the SO may decide to spill a portion of available wind production to facilitate the operation of the power system. This is enforced by (3.44).

3.2.3.2.3 Involuntary-load shedding limits 2 0 # Lshed r;t2 ;s # Dr;t2 ’w; t2 ; s

(3.45)

As a last resort the SO may decide to shed a portion of the inelastic demand. This requirement is enforced by constraint (3.45).

3.2.3.2.4 Energy requirement constraint for LSE of type 1 X LSE1ac j1 ;t2 ;s $ Ejreq ’j1 ; s 1 ΔT2 t2 AT2

(3.46)

Constraint (3.46) enforces the energy requirement constraint for the LSE of type 1 in each scenario. The division with the duration of the time interval ΔT2 is required in order to appropriately match the units of energy and power.

3.2 MATHEMATICAL MODEL

87

3.2.3.2.5 Reserve deployment from LSE of type 2 Eqs. (3.47)
(3.55) enforce several constraints related to the deployment of reserves from the LSE of type 2. u LSE2u;con j2 ;t2 ;s # N2 Uυj2 ;t2 ;s ’j2 ; t2 ; s

(3.47)

# N2 UυDN LSE2jd;con j2 ;t2 ;s ’j2 ; t2 ; s 2 ;t2 ;s

(3.48)

u DN υLSE2 j2 ;t2 ;s 5 υj2 ;t2 ;s 1 υj2 ;t2 ;s ’j2 ; t2 ; s

(3.49)

υuj2 ;t2 ;s 1 υDN j2 ;t2 ;s # 1 ’j2 ; t2 ; s

(3.50)

Constraints (3.47)
(3.50) are used in order force the LSE of type 2, once called, to provide only up or down contingency reserves. More specifically, (3.47) and (3.48) determine the amount of up and down reserve that may be deployed. The right hand side of these inequalities involves the multiplication of a sufficiently large constant N2 with a binary variable that indicates whether an LSE of type 2 provides up or down reserve. If the LSE of type 2 is called in period t2 then u υLSE2 j2 ;t2 ;s 5 1. The call implies that either up or down reserves are provided (either υj2 ;t2 ;s 5 1 or υDN j2 ;t2 ;s 5 1). These states are mutually exclusive, a fact that is expressed by (3.49) and (3.50). LSE2 LSE2 LSE2 ψLSE2 j2 ;t2 ;s 2 ζ j2 ;t2 ;s 5 υj2 ;t2 ;s 2 υj2 ;ðt2 21Þ;s ’j2 ; t2 ; s

(3.51)

LSE2 υLSE2 j2 ;t2 ;s $ ψj2 ;t2 ;s ’j2 ; t2 ; s

(3.52)

LSE2 υLSE2 j2 ;t2 ;s $ ζ j2 ;ðt2 11Þ;s ’j2 ; t2 ; s

(3.53)

Constraints (3.51)
(3.53) enforce the deployment logic of this type of resource. X

t2 AT2

call ψLSE2 j2 ;t2 ;s # Nj2 ’j2 ; t2 ; s

t2 X

(3.54)

ψLSE2 j2 ;τ;s $ υj2 ;t2 ;s ’j2 ; t2 ; s

(3.55)

T dur j τ5t2 2ΔT2 11 2

The deployment of demand side resources to provide reserve services may be subject to several rules, for example, maximum number of calls, duration of a call, and many more. Eq. (3.54) limits the maximum number of times each LSE of type 2 can be utilized to procure contingency reserves during the scheduling horizon. Finally, (3.55) constrains the maximum duration of each call to last at most Tjdur periods. 2

3.2.3.2.6 Network constraints P

  P P WP PG fb;t2 ;s i;t2 ;s 1 wANnw Pi;t2 ;s 2 Sw;t2 ;s 1 nABnn b   P P P P ac ac j SE1 j LSE2 5 nABn fb;t2 ;s 1 rANnr D2r;t2 2 Lshed r;t2 ;s 1 j1 ;t2 ;s L 1 j2 ;t2 ;s ’b; ðn; nnÞABðn; nnÞ; t2 ; s j1 AN 1 j2 AN 2

iANni

b

n

n

(3.56)

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

fb;t2 ;s 5 Bb;n Uðδn;t2 ;s 2 δnn;t2 ;s ÞULCb;t2 ’b; ðn; nnÞABðn; nnÞ; t2 ; s

(3.57)

2 fbmax ULCb;t2 # fb;t2 ;s # fbmax ULCb;t2 ’b; t2 ; s

(3.58)

2 π # δn;t2 ;s # π ’n; t2 ; s

(3.59)

δn;t2 ;s 5 0’t2 ; s; if n  ref

(3.60)

In the second stage of the problem, the network constraints are taken into account using a lossless DC power flow formulation. More specifically, Eq. (3.56) stands for the power balance at each node of the system which states that the total power generated at each node by conventional units, the net production of wind farms plus the power injection from incoming transmission lines must equal the total net consumption of inelastic and elastic loads, as well as the power that is injected to outgoing transmission lines. The flow over a transmission line is defined by (3.57), while a power flow limit is set according to the maximum capacity of a transmission line by (3.58). In case of a transmission line failure, the active power flow through a transmission line is forced to zero. Finally, (3.59) and (3.60) state that the voltage angles must be bounded between 2 π and π and that at the slack bus the voltage angle must be specified, respectively.

3.2.3.3 Linking constraints The set of linking constraints bridges the day-ahead market decisions and the decisions made based on the outcome of each plausible scenario. As a result, the constraints pertaining this stage involve both scenario independent and scenario dependent decision variables. Linking constraints enforce the fact that reserves in the actual operation of the power system are no longer a stand-by capacity but are materialized as energy. To simplify the mathematical formulation presented below, the following should be noted: The equations that refer to reserve deployment by generating units hold only for units that are not under contingency. Furthermore, as long as there are no contingencies or wind/load deviations, the reserves provided by the demand side are also zero, and the relevant equations are not enforced. It should be also noted that the notation t2 AT2in means that t2 is a subhourly interval of the hour t1 that appears in the same equation.

3.2.3.3.1 Additional cost due to change of commitment status of units CAi;t2 ;s 5

X t2 AT2

SUC2i;t2 ;s 2 SUC1i;t1 1

X t2 AT2

SDC2i;t2 ;s 2 SDC1i;t1 ’i; t2 AT2in ; t1 ; s

(3.61)

In the case of a difference occurring in the commitment status, a commitment scheduling change cost is charged through (3.61).

3.2.3.3.2 Generation side reserve deployment sch UP NS DN in PG i;t2 ;s 5 Pi;t1 1 ri;t2 ;s 1 ri;t2 ;s 2 ri;t2 ;s ’i; t2 AT2 ; t1 ; s

(3.62)

UP;load UP;wind UP;con UP 5 ri;t 1 ri;t 1 ri;t ’i; t2 ; s ri;t 2 ;s 2 ;s 2 ;s 2 ;s

(3.63)

3.2 MATHEMATICAL MODEL

89

DN;load DN;wind DN;con DN ri;t 5 ri;t 1 ri;t 1 ri;t ’i; t2 ; s 2 ;s 2 ;s 2 ;s 2 ;s

(3.64)

NS;load NS;wind NS;con NS 5 ri;t 1 ri;t 1 ri;t ’i; t2 ; s ri;t 2 ;s 2 ;s 2 ;s 2 ;s

(3.65)

UP in 0 # ri;t # RUP i;t1 ’i; t2 AT2 ; t1 ; s 2 ;s

(3.66)

DN in # RDN 0 # ri;t i;t1 ’i; t2 AT2 ; t1 ; s 2 ;s

(3.67)

NS in 0 # ri;t # RNS i;t1 ’i; t2 AT2 ; t1 ; s 2 ;s

(3.68)

The power output of a unit i in a scenario s in period t2 that is a subhourly interval of t1 is equal to the scheduled generation output during period t1 augmented by the deployment of up spinning and nonspinning reserves, minus the deployment of down spinning reserve as stated by (3.62). Furthermore, the deployed reserves are further decomposed into several components related to the factor that triggered their deployment. This is enforced by (3.63)
(3.65). Constraints (3.66)
(3.68) limit the deployment of the different types of reserves in period t2 by their scheduled amount in the corresponding hourly interval t1 . Therefore, the scheduled reserves of each type in each interval of the day-ahead market coincide with the maximum reserve deployment within that interval in the second stage of the problem. Finally, it should be noted that similarly to (3.66)
(3.68) that impose restrictions to the total amount of deployed reserves, each reserve component should be also constrained by its corresponding scheduled amount. UP NS DN ri;t 1 ri;t 2 ri;t 5 2 ;s 2 ;s 2 ;s

X

f AF i

G in ri;f ;t2 ;s ’i; t2 AT2 ; t1 ; s

(3.69)

G in ri;f ;t2 ;s # Bi;f 2 bi;f ;t1 ’i; f ; t2 AT2 ; t1 ; s

(3.70)

G in ri;f ;t2 ;s $ 2 bi;f ;t1 ’i; f ; t2 AT2 ; t1 ; s

(3.71)

In the second stage of the problem, generation side reserves are materialized as an energy alteration, and therefore, the cost increase or decrease that occurs is priced according to the marginal cost function of each generation. Constraints (3.69)
(3.71) are used in order to decompose the deployed reserves into the power blocks of the generation cost function.

3.2.3.3.3 Demand side reserve deployment sch u d in LSE1ac j1 ;t2 ;s 5 LSE1j1 ;t1 2 LSE1j1 ;t2 ;s 1 LSE1j1 ;t2 ;s ’j1 ; t2 AT2 ; t1 ; s

(3.72)

u;wind LSE1uj1 ;t2 ;s 5 LSE1u;load j1 ;t2 ;s 1 LSE1j1 ;t2 ;s ’j1 ; t2 ; s

(3.73)

d;wind LSE1dj1 ;t2 ;s 5 LSE1d;load j1 ;t2 ;s 1 LSE1j1 ;t2 ;s ’j1 ; t2 ; s

(3.74)

in 0 # LSE1uj1 ;t2 ;s # LSE1UP j1 ;t1 ’j1 ; t2 AT2 ; t1 ; s

(3.75)

in 0 # LSE1dj1 ;t2 ;s # LSE1DN j1 ;t1 ’j1 ; t2 AT2 ; t1 ; s

(3.76)

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

Constraint (3.72) adjusts the actual consumption of the LSE of type 1 according to the deployed reserves, while (3.73) and (3.74) decompose the up and down deployed reserves into their components. Constraints (3.75)
(3.76) limit the deployment of the different types of reserves in period t2 by their scheduled amount in the corresponding hourly interval t1 . Note that constraints which limit the deployment of the individual reserve components should be also enforced. u;con d;con sch in LSE2ac j2 ;t2 ;s 5 LSE2j2 ;t1 2 LSE2j2 ;t2 ;s 1 LSE2j2 ;t2 ;s ’j2 ; t2 AT2 ; t1 ; s

(3.77)

UP;con 0 # LSE2u;con ’j2 ; t2 AT2in ; t1 ; s j2 ;t2 ;s # LSE2j2 ;t1

(3.78)

DN;con ’j2 ; t2 AT2in ; t1 ; s 0 # LSE2d;con j2 ;t2 ;s # LSE2j2 ;t1

(3.79)

As in the case of the LSE of type 1, constraints (3.77)
(3.79) hold for the case of the LSE of type 2.

3.2.3.3.4 Load following reserves determination   WP;S PWP i;t2 ;s 2 Sw;t2 ;s 2 Pw;t1  P  DN;wind UP;wind NS;wind 2 ri;t 2 ri;t 5 iAI ri;t 2 ;s 2 ;s 2 ;s   P u;wind in 1 j1 AJ1 LSE1d;wind j1 ;t2 ;s 2 LSE1j1 ;t2 ;s ’i; j1 ; t2 AT2 ; t1 ; s

P

wAW

  1 D2r;t2 2 Lshed r;t2 ;s 2 Dr;t1  P  UP;load NS;load DN;load 1 ri;t 2 ri;t 5 iAI ri;t 2 ;s 2 ;s 2 ;s   P d;load in 1 j1 AJ1 LSE1u;load j1 ;t2 ;s 2 LSE1j1 ;t2 ;s ’i; j1 ; t2 AT2 ; t1 ; s

(3.80)

P

rAR

(3.81)

Eqs. (3.80) and (3.81) enforce the logic of deploying load following reserves. Constraint (3.80) states that if the net accepted wind in a subhourly interval t2 in a specific scenario is greater than the scheduled wind power during the corresponding hourly interval t1 in the day-ahead market, then down reserves should be deployed. This may be accomplished either by decreasing the power output of the generating units or by increasing the consumption of the LSE of type 1. The opposite holds when the wind deviation is negative. In order to procure reserves to balance the intrahour deviations of the load, (3.81) must be enforced. According to (3.81), when the load deviation is positive, then either the units should increase their production or the LSE of type 1 should decrease their consumption. The opposite holds if there is a negative load deviation. Note that in both cases, a combination of up and down reserves from the different resources is also possible, as long as the imbalances are covered.

3.3 CASE STUDIES 3.3.1 WIND-POWER PRODUCTION SCENARIO GENERATION The scenario generation technique adopted in this chapter is based on forecasting using time series models [47] utilizing the ECOTOOL MATLAB toolbox [48]. Historical data regarding the total

3.3 CASE STUDIES

91

production of the wind farms located in the island of Crete were collected from the database of the SiNGULAR project [49] for the years 2011 and 2012. The wind farms of the island have a total installed capacity of 176.5 MW. Wind-power generation scenarios are created for the randomly selected day 4/9/2012. First, in order to stabilize the variance of the time series, the logarithmic transformation is applied to the original data. Subsequently, an autoregressive integrated moving average (ARIMA) model is applied to the logarithmically transformed time series. The generic form of the ARIMA model is represented by Eq. (3.82). log ψt 5 c 1

1 ð12BÞ

d0

ð12Bs1 Þd1 . . .ð12Bsk Þdk

θq0 ðBÞ θq1 ðBs1 Þ θqk ðBsk Þ ... Et φp0 ðBÞ φp1 ðBs1 Þ φpk ðBsk Þ

(3.82)

where logψt stands for the observed time series; εt is Gaussian white noise with zero mean and constant variance; sj, j 5 0; . . . ; k are a set of seasonal periods, with s0 5 1; ð12Bsj Þdj ; j 5 0; . . . ; k are the k 1 1 differencing operators necessary to reduce the time series to mean stationarity; θqj ðBsj Þ and ϕqj ðBsj Þ; j 5 1; . . . ; k are invertible and stationary polynomials in the backshift operator B :Bl 5 yt2l of type θqj ðBsj Þ 5 ð1 1 θq1 Bsj 1 . . . 1 θqj Bqj sj Þ; c is a constant. To generate wind-power generation scenarios, several ARIMA models are fit to the observed time series. The rationale followed is that forecasting is performed for the requested number of periods for a specific day by considering different ranges of historical data. More specifically, starting from the first week in the past, a day is added to the historical time series and the forecasting is repeated, while a new ARIMA model is fit when a whole new week is added to the historical data range. Following this procedure and by considering historical data spanning from 20/6/2012 to 3/9/2012, an initial pool of 70 equiprobable scenarios is constructed. The computational performance of the stochastic programming models strongly depends on the size of the scenario set. In this respect, a scenario reduction technique based on the k-means clustering algorithm [50] is applied in order to reduce the number of scenarios by substituting the initial scenario set by an approximate representative set of nonequiprobable scenarios. Provided that historical data with the desired subhourly granularity are available, the methodology described above can be readily applied. Nevertheless, the data utilized were given for hourly intervals, and the scenario generation methodology was slightly altered. More specifically, scenarios with hourly resolution are first constructed and subsequently, it is considered that in each intrahour interval the wind-power production may randomly deviate up to 5% from the corresponding hourly value according to a normal distribution.

3.3.2 ILLUSTRATIVE EXAMPLE To demonstrate the application of the proposed model, the sample 6-bus system comprising four conventional generators, a wind farm with installed capacity 100 MW, one inelastic load, a LSE of type 1 and a LSE of type 2 shown in Fig. 3.6 is analyzed over a 6-h horizon, considering that the intrahour granularity is 10 min. The characteristics of the transmission system are presented in Table 3.1. The technical and economic data of the generators are presented in Tables 3.2 and 3.3, respectively. Spinning reserves must be fully available in 15 min, while the nonspinning reserves in 30 min. The cost of providing spinning and nonspinning reserves from the generating units is equal to 20% and 10% of their most expensive power block, respectively. Three wind-power generation scenarios (low, moderate, and high) are considered with probabilities of occurrence 54.29%, 30%,

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

FIGURE 3.6 Topology of the 6-bus system.

Table 3.1 Characteristics of the Transmission Lines (6-Bus System) Line No.

From Bus

To Bus

X (pu)

Flow Limit (MW)

1 2 3 4 5 6 7

1 1 2 2 3 4 5

2 4 3 4 6 5 6

0.170 0.258 0.037 0.197 0.018 0.037 0.140

410 200 500 250 500 250 230

Table 3.2 Technical Characteristics of the Generating Units (6-Bus System) Unit

U1

U2

U3

U4

Minimum capacity (MW) Maximum capacity (MW) Minimum up time (h) Minimum down time (h) Minimum up time (min) Minimum down time (min) Ramp up rate (MW/min) Ramp down rate (MW/min) Initial output (MW) Time committed/decommitted at the beginning of the scheduling horizon (h) Time committed/decommitted at the beginning of the scheduling horizon (min)

150 500 3 3 180 180 5 5 300 5

120 450 12 12 720 720 15 15 450 5

40 400 0 0 20 10 40 40 0 25

20 150 0 0 10 10 40 40 0 25

300

300

2 300

2 300

3.3 CASE STUDIES

93

Table 3.3 Economic Characteristics of the Generating Units (6-Bus System) Power Blocks (MW)

Marginal Costs (h/MW h)

Unit

B1

B2

B3

B4

B5

C1

C2

C3

C4

C5

Startup Cost (h)

U1 U2 U3 U4

250 150 200 50

120 110 80 50

60 90 60 30

50 60 40 10

20 40 20 10

5 9 20 22

5.5 10 20.5 24

6 10.5 21 25

6.5 11 22 26

7 12 23 28

30,000 25,000 2000 1000

Shutdown Cost (h) 5000 2000 1000 500

FIGURE 3.7 Wind-power generation scenarios (6-bus system).

and 15.71%, respectively. The three wind-power generation scenarios are presented in Fig. 3.7. Also, it is to be noted that the wind spillage cost and the involuntary load shedding costs are considered equal to 1000 h/MW h. Regarding the demand side resources, the LSE of type 1 offers continuous up and down load following reserves at a cost of 5 h/MW h. The LSE of type 2 may contribute to scheduled contingency reserves at a cost of 10 h/MW h. In addition, it is paid a fixed 40 h when called to provide reserve. Contingency reserves from the LSE of type 2 may be procured two times within the scheduling horizon, and the service should last for a maximum of 30 min. The nominal system load is presented in Table 3.4, while the intrahour inelastic load profile is provided in Table 3.5. Note that, it is considered that the demand of the LSE of both types is equal to their nominal load. The LSE of type 1 may provide up and down reserves by altering its load in both directions by 20%. The LSE of type 2 may provide only up contingency reserves by reducing its consumption up to 50%. In order to elaborate the reserve scheduling methodology, the following tests are performed: First the loads of LSE are considered inflexible and, therefore, cannot participate in reserve provision.

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

Table 3.4 Hourly Load (6-Bus System) Time

Inelastic Load (MW)

Nominal LSE1 Load (MW)

Nominal LSE2 Load (MW)

1 2 3 4 5 6

900 800 550 750 600 450

80 90 100 80 70 60

90 120 110 80 70 50

Table 3.5 IntraHour Load (6-Bus System) Time

Inelastic Load (MW)

Time

Inelastic Load (MW)

Time

Inelastic Load (MW)

1 1:10 1:20 1:30 1:40 1:50 2 2:10 2:20 2:30 2:40 2:50

822 851 965 918 952 901 847 858 698 840 875 640

3 3:10 3:20 3:30 3:40 3:50 4 4:10 4:20 4:30 4:40 4:50

503 561 502 532 497 580 821 615 790 704 780 785

5 5:10 5:20 5:30 5:40 5:50 6 6:10 6:20 6:30 6:40 6:50

564 601 648 627 562 575 450 450 430 450 440 450

First, the system is considered to be free of contingencies (case C1-A). Subsequently, two contingency scenarios are investigated: (1) The must-run unit 2 is considered to fail at 4:10 (case C1-B), and (2) the transmission line 2 (that connects buses 1 and 4) is considered to fail at 4:10 (case C1-C). It should be noted that owing to the small size of the test system, concurrent contingencies would lead to an infeasible optimization problem. Then, the same cases are studied considering also the participation of LSE (cases C2-A, C2-B, and C2-C). Results concerning period 4 of the day-ahead market and the intrahour interval 4:10 in which the contingencies are considered to occur are analyzed in detail. Table 3.6 presents the scheduled power output of the generating units and the scheduled reserve levels from generation and demand side resources for period 4 of the day-ahead market. In all the cases during period 4, the wind power scheduled coincides with the installed capacity of the wind farm (100 MW). In C1-A, the total upward reserves that are scheduled are 121.848 MW, while the

3.3 CASE STUDIES

95

Table 3.6 Scheduled Generator Output, Generation and Demand Side Reserves (MW) Unit 1

Unit 2

Unit 3

Unit 4

LSE 1

LSE 2

Scheduled output Spinning up reserve Spinning up reserve (load) Spinning up reserve (wind) Spinning up reserve (contingency) Spinning down reserve Spinning down reserve (load) Spinning down reserve (wind) Spinning down reserve (contingency) Scheduled output Spinning up reserve Spinning up reserve (load) Spinning up reserve (wind) Spinning up reserve (contingency) Spinning down reserve Spinning down reserve (load) Spinning down reserve (wind) Spinning down reserve (contingency) Scheduled output Nonspinning reserve Nonspinning reserve (load) Nonspinning reserve (wind) Nonspinning reserve (contingency) Scheduled output Nonspinning reserve Nonspinning reserve (load) Nonspinning reserve (wind) Nonspinning reserve (contingency) Up reserve (load) Up reserve (wind) Down reserve (load) Down reserve (wind) Up reserve (contingency) Down reserve (contingency) Total upward reserve Total downward reserve

C1-A

C1-B

C1-C

C2-A

C2-B

C2-C

500 0 0 0 0 50 50 0 0

454.205 0 0 0 0 135 135 0 0

500 0 0 0 0 300 45.666 0.467 253.866

500 0 0 0 0 50 50 0 0

407.428 0 0 0 0 119 119 0 0

500 0 0 0 0 300 40.899 0.467 258.634

310 43.435 38.105 5.330 0 90.701 90.701 0 0

355.795 32.440 29.034 3.376
0 0 0 0

310 8.697 0 8.697 0 151.877 140.658 11.219 0

310 43.435 38.105 5.330 0 90.701 90.701 0 0

402.572 26.822 26.092 0.730 0 0 0 0 0

310 4.017 0 4.017 0 156.558 145.339 11.219

0 78.413 32.895 45.518 0 0 0 0 0 0 0 0 0 0 0 0 121.848 140.701

0 378.786 31.583 41.502 305.701 0 150 95.353 4.553 50.094 0 0 0 0 0 0 561.226 135

0 370.852 71 45.985 253.866 0 0 0 0 0 0 0 0 0 0 0 379.549 451.877

0 78.413 32.895 45.518 0 0 0 0 0 0 0 3.333 10 0 0 0 125.181 150.701

0 367.176 13 41.604 312.572 0 142.005 84.908 7.097 50 16 0 16 0 40 0 592.003 135

0 366.518 101.899 45.985 218.634 0 0 0 0 0 0.557 0 3.341 0 40 0 411.092 459.899

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

total downward reserves are 140.701 MW. These reserves are exclusively used in order to balance intrahour deviations of the load demand and the wind-power generation uncertainty and should be sufficient to cover the highest intrahour deviations. The highest load increase is 71 MW and occurs in interval 4, while the maximum load decrease is 135 MW and occurs at period 4:10. Thus, the total up reserve scheduled to balance the load increase is 71 MW, while the down reserve scheduled for this purpose is 140.701 MW which exceeds the maximum load decrease. This implies that in order to cover this negative deviation in the consumption, both up and down reserves should be deployed. The maximum energy deficit that has to be balanced because of wind deviations is 49.431 MW in period 4 and 50.848 MW upward reserves are scheduled. In C2-A, the same amount of upward and downward reserves as in C1-A are scheduled. In addition to these reserves, 3.333 MW of upward reserves are scheduled to balance wind deviations, and 10 MW of down reserves are scheduled in order to accommodate changes in the inelastic load. In C1-B during period 4, the scheduled output power of unit 2 is 355.795 MW, and therefore, once the contingency occurs, this energy deficit has to be balanced by the other generating units and especially the off-line units 3 and 4 that provide nonspinning reserves. Furthermore, the load following reserves that would be normally provided by unit 2 must be replaced by other units. In C2-B in which LSE of type 2 are eligible resources to provide contingency reserve 40 MW are called by the SO and are active for 50 min. Furthermore, the maximum available of load following reserves that may be deployed from LSE of type 1 are scheduled (16 MW) in order to increase the ramping capability of the system. In C1-C and C2-C, the unit commitment status of the generating units is the same as in cases C1-A and C2-A. However, the power generated by unit 1 can be provided only through transmission line 1 that connects buses 1 and 2. As a result, the output of unit 1 which is scheduled to operate at its maximum capacity should be reduced, and therefore, 300 MW of down spinning reserve are scheduled. Moreover, LSE of type 2 are also employed in C2-A to provide contingency up demand side reserve and a small amount of load following reserve is scheduled by LSE of type 1 in C2-C. The reserve needs increase in comparison with the contingency free cases, however are less than in the case of unit failures. This implies that the impact of the considered transmission line contingency is less severe than the unit outage. Figs. 3.8 and 3.9 illustrate a specific instance of the actual operation of the system in period 4:10 in the moderate wind-power generation scenario, both neglecting and considering the contribution of the two different types of LSE. It may be noticed that when contingency is anticipated, the operation of the system is the same in this instance since the cost of scheduling load following reserves from the LSE of type 1 is higher than procuring reserves from the generation side. The difference between the scheduled wind-power generation and the moderate scenario is 41.604 MW, while the inelastic load deviation is negative and equal to 135 MW. Thus, the required net demand change that must be balanced by the generation side is a decrease of 93.396 MW which is implemented by deploying 44.299 MW down spinning reserve from unit 1, 90.701 MW down spinning reserve from unit 2, and 41.604 MW of nonspinning reserves from unit 3. In the case of the contingency of unit 2, in addition to the load following requirements, the deficit of 355.795 MW has to be covered. As a result, unit 4 is also contributing to nonspinning reserves. If the contribution of LSE of types 1 and 2 is considered, the consumption of the LSE of type 1 is increased by 16 MW, while the LSE of type 2 is curtailed by 40 MW. Finally, in the case of the transmission line 2 contingency case, the LSE of type 2 may also be curtailed by half in order to procure less reserves from the generation side.

3.3 CASE STUDIES

97

FIGURE 3.8 Analysis of period 4:10 in moderate scenario when contribution of LSEs is neglected. (A) without contingencies, (B) U2 fails at 4:10, and (C) transmission line 2 fails at 4:10. Red color (light gray in print versions): Generation and consumption scheduled in the day-ahead market. Green color (dark gray in print versions): Generation, consumption and active power flows in moderate scenario. All values are in MW.

3.3.3 APPLICATION ON A 24-BUS SYSTEM 3.3.3.1 Description of the case study In this section, the proposed methodology is tested on a modified version of the IEEE Reliability Test System for a 12-h horizon, using 15-min intervals in the second stage of the problem. Complete data regarding the technical and economic characteristics of the system may be found in Appendix A. The nuclear units at buses 18 and 21 and the hydro units at bus 22 are considered as must-run units. A wind farm is added to the generation mix and is located at bus 10. To account for the wind-power generation stochasticity, 10 nonequiprobable scenarios are generated for the total wind production according to the methodology described in Section 3.3.1. For the sake of simplicity, no intrahour load deviations are considered in this section. Note that the wind spillage cost and the involuntary load shedding cost are considered equal to 1000 h/MW h. All the generators

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FIGURE 3.9 Analysis of period 4:10 in moderate scenario when contribution of LSEs is considered. (A) without contingencies, (B) U2 fails at 4:10, and (C) transmission line 2 fails at 4:10. Red color (light gray in print versions): Generation and consumption scheduled in the day-ahead market. Green color (dark gray in print versions): Generation, consumption and active power flows in moderate scenario. All values are in MW.

except for the units at bus 22 (must-run at constant output) can participate in spinning up and down reserves that must be fully available in 15 min. The proposed formulation explicitly allows units that are off-line to be committed in the day-ahead to alter their status and provide nonspinning reserves. Nonspinning reserves must be fully deployed within 30 min and units 3, 4, 5, 6, and 7 are considered eligible for the provision of this service. The following cases are investigated: •

C1-A. The loads connected at buses 18 and 20 which stand for approximately 11.7% and 4.5% of the total system load, respectively, are considered to represent LSE of type 1. The only source of imbalances is the uncertain wind-power generation of the 200 MW wind farm. The cost of scheduling reserves from the LSE of type 1 is equal to 5 h/MW h, while the cost of deploying reserves is 50 h/MW h.

3.3 CASE STUDIES

•

•

•

•

•

99

C1-B. The cost of scheduling reserves from the LSE varies from 1 to 5 h/MW h, while the cost of deploying reserves receives a value equal to 10 times the reserve scheduling cost. The LSE of type 1 located at bus 20 is considered available only for reserve provision (cannot be rescheduled) with a flexibility of 20%. C2-A. Apart from the wind-power output uncertainty, a unit outage and a transmission line failure are considered. More specifically, the must-run unit 10 fails at 7:30 causing a deficit of 300 MW, while the transmission line 33 that connects buses 20 and 23 fails at 7:30, is repaired at period 9:15 and fails again at 11:30. The LSE of type 1 located at bus 20 is considered capable of providing load following reserve. C2-B. The LSE of type 2 located at bus 19 (6.4% of total system load) may provide up contingency reserve. The cost of scheduling contingency reserves from LSE of type 2 is 0.25 h/MW h, while the price paid by the SO in order to deploy reserve is 40 h/MW h. This type of reserve may be called at most two times and each call may last maximum 30 min. C2-C. The LSE of type 1 at bus 20 may provide load following reserve with an upward and downward flexibility of 30%, while the LSE of type 2 at bus 19 may provide upward contingency reserves with an upward flexibility of 50%. C3. The wind farm at bus 10 is considered to have different installed capacities while the LSE of type 1 and 2 have the same characteristics as in the C2-C.

3.3.3.2 Results and discussion Prior to delving into the analysis of the results concerning the aforementioned cases, it should be noted that due to the high wind spillage cost, no available wind energy spillage is noticed in any of the studied cases. Figs. 3.10 and 3.11 present the nominal load of the LSE of type 1 connected to buses 18 and 20, respectively. It may be noticed that in both cases, when a certain amount of flexibility is available for the LSE of type 1, its demand is rescheduled so that load is shifted from the relatively higher system loading periods (8
12) to the relatively low system loading periods (1
7). As a result, the day-ahead energy cost is expected to be reduced with the increase of the available flexible demand, a fact that is confirmed by the results portrayed in Fig. 3.12. It is interesting to notice that when the LSE of type 1 that is connected to bus 18 is considered, the decrease in the energy cost is more significant because of the larger amount of load reallocation. It is also important to investigate the effect of the contribution of the LSE of type 1 to reserves in order to balance the wind-power generation deviations on the cost of scheduled reserves from the generation side in the day-ahead market. The cost of scheduled day-ahead generation side reserves with respect to different levels of flexibility regarding the LSE of type 1 is illustrated in Fig. 3.13. The LSE of type 1 connected to bus 20 leads in a reduction in cost of generation side reserves as the flexibility increases from 0 to 20%. Note that for all the degrees of flexibility, the amount of reserves scheduled by LSE of type 1 is the same. The energy and reserve reduction costs are a consequence of the flexible demand rescheduling. Increasing the flexibility to 25% and 30% does not cause any further reduction in the generation side reserve cost. Considering that the load of bus 18 represents a LSE of type 1, the generation side reserve cost is reduced more because the amount of load that is rendered available to be rescheduled is larger. Although the amount of reserves scheduled by the LSE is the same as in the previous case, the load reallocation facilitates

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FIGURE 3.10 Scheduled load of LSE of type 1 connected at bus 18.

FIGURE 3.11 Scheduled load of LSE of type 1 connected at bus 20.

the wind-power integration and therefore, reduces the cost of reserve procurement by the generation side. It is noticeable that the cost of generation side reserves for 10% and 15% is the same, while it increases for 20%, 25%, and 30%. This increase is linked to the fact that the load reallocation leads to significant reduction in the conventional generation energy production cost on the expense of slightly increasing the generation side reserve cost.

3.3 CASE STUDIES

101

FIGURE 3.12 Energy cost for different values of LSE of type 1 flexibility (C1-A).

FIGURE 3.13 Reserve cost for different values of LSE of type 1 flexibility.

Another case that is examined is related to enforcing the requirement of the LSE of type 1 not being able to be rescheduled in the day-ahead market. Nevertheless, it may be scheduled to provide up and down reserves. This results in a constant day-ahead energy cost of 162,011 h regardless of the LSE of type 1 flexibility. Evidently, for the LSE of type 1 located at either bus 18 or 20, the minimum required flexibility of 5% yields the maximum possible reduction in the reserve cost.

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FIGURE 3.14 Generation scheduled reserve cost for different costs of LSE of type 1 reserve cost.

One determining factor for the utilization of LSE of type 1 as reserve providers is the cost at which their service is provided. In order to be an appealing alternative to the deployment of generation side resources, the cost of demand side reserves should be less than the cheapest reserve service offered by the generators, that is, 5 h/MW h from units 8 and 9. To demonstrate the importance of the demand side reserve offering cost, in C1-B a parametric analysis is performed. First, in Fig. 3.14, the generation side reserve cost versus the cost of LSE of type 1 reserve scheduling cost is depicted. Due to the fact that the load cannot be rescheduled with respect to its nominal value, the reserve cost reduction is purely the effect of scheduling more reserves from the LSE of type 1 with the reduction in the LSE of type 1 reserve scheduling cost. For instance, the nominal load and the deployed load in scenario 10 are displayed in Fig. 3.15. It may be noticed that for a LSE of type 1 scheduling cost of 1 h/MW h, the changes in the load pattern are substantial, while for a slight increase of the cost by 1 h/MW h, the reserves are significantly reduced. For higher costs, no reserves are deployed by the LSE of type 1 in scenario 10. Thus, it may be concluded that the sensitivity of scheduling reserves from the demand side is highly sensitive to the cost of this service. In cases C2-A, C2-B, and C2-C, a unit outage and a transmission line failure are considered. This implies that contingency reserves should be also scheduled in order to balance the energy production deficit and the network disturbances. In case C2-A, only the generation side may alter its production to provide contingency reserves. However, in cases C2-B and C2-C, in addition to the generation side, LSE of type 2 may also contribute to contingency reserves. In Fig. 3.16, the baseline consumption of the load connected to bus 19 that serves as an LSE of type 2 is illustrated together with the deployment of up contingency reserve considering two different degrees of flexibility. The maximum amount of load that may be curtailed is scheduled for deployment of contingency reserve in all scenarios. The SO calls two times the LSE of type 2 to provide contingency reserve for the maximum allowed duration (1 h). The first call is activated in period 11 and the second in period 11:30. These calls not only coincide with the second failure of the transmission line 33 but also with the highest system load periods that implies that the demand side provision of

3.3 CASE STUDIES

103

FIGURE 3.15 Scheduled load of LSE of type 1 and actual consumption in scenario 10.

FIGURE 3.16 Baseline load of LSE of type 2 and deployed contingency reserves.

contingency reserves is an alternative to providing contingency reserves from the already highly loaded units (generation side reserves would have a higher deployment cost). The day-ahead energy and reserve cost for the cases C2-A, C2-B, and C2-C are presented in Table 3.7. When the participation of the demand side resources is not considered, the contingencies cause an increase of 8110 h in the scheduled generation side reserves while maintaining the scheduled day-ahead energy cost. In C2-A as the flexibility of the LSE of type 1 increases, the day-ahead

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Table 3.7 Energy and Reserve Costs for Cases C2-A, C2-B and C2-C Case

Flexibility (%)

Energy Cost (h)

Reserve Cost (h)

LSE of Type 1 Reserve Cost (h)

LSE of Type 2 Reserve Cost (h)

C2-A

0 10 20 30 20 50 (30% and 50%)

162,011 161,657 161,366 161,105 162,011 162,011 161,105

10,770 10,682 10,647 10,629 10,627 10,434 10,271


8.185 8.185 8.185

8.185





8.150 20.375 20.375

C2-B C2-C

Table 3.8 Energy and Reserve Costs for Different Installed Capacity of Wind Farm (C3) Wind-Farm Capacity (MW) 200

500

800

Case Without LSE Nonschedulable LSE 1 load Schedulable LSE 1 load Without LSE Nonschedulable LSE 1 load Schedulable LSE 1 load Without LSE Nonschedulable LSE 1 load Schedulable LSE 1 load

Energy Cost (h)

Reserve Cost (h)

LSE of Type 1 Reserve Cost (h)

LSE of Type 2 Reserve Cost (h)

162,011 162,011

10,770 10,368


8.185


20.375

161,105

10,271

18.785

20.375

143,766 143,766

14,580 14,129


92.083


20.375

142,818

14,069

213.125

20.375

128,147 128,147

19,020 18,575


171.333


20.375

127,574

18,390

333.333

20.375

energy and reserve cost decrease as a result of optimally rescheduling its load demand. In C2-B in which the LSE of type 2 renders available its load to provide contingency reserve, higher cost reductions occur since its reserves address the source of imbalances that is responsible for the large amounts of day-ahead scheduled reserves. Finally, the greatest energy and reserve reduction costs are noticed in C2-C. The energy cost in this case coincides with the energy cost of C2-A with a flexibility of 30%, while the reserve cost is the lowest among the different cases. In the previous cases, the capacity of the wind farm was considered to be 200 MW. In order to investigate the effect that the demand side resources have on the energy and reserve costs with the increase in the installed capacity of the wind farm, case C3 is investigated. In this case, the wind farm is considered to have a progressively increasing capacity of 200, 500, and 800 MW, while the aforementioned contingencies are also taken into account. The relevant results are listed in Table 3.8. It may be noted that on the one hand, the energy cost when no flexible demand side

3.3 CASE STUDIES

105

Table 3.9 Computational Statistics (6-Bus System)

Equations Continuous variables Discrete variables Time (s)

Without Contingency/ Without LSE

Unit Contingency/ without LSE

Line Contingency/ without LSE

Without Contingency/ with LSE

Unit Contingency/ with LSE

Line Contingency/ with LSE

36,304 288,200

34,252 287,741

36,292 288,200

36,304 288,200

34,252 287,741

36,292 288,200

2890

2788

2890

2890

2788

2890

12.62

4.59

3.24

23.81

7.80

4.77

Table 3.10 Computational Statistics (24-Bus System) C2-C Equations Continuous variables Discrete variables Time (s)

566,648 2,364,079 32,028 562

resources are considered drops from 162,011 to 143,766 and 128,147 h for increasing the wind farm capacity to 500 and 800 MW due to integrating more free of cost wind energy in the dayahead market. On the other hand, the generation side reserve scheduling cost increases by 26.13% and 43.37% in these cases, respectively. If the LSE of type 1 is considered eligible for providing reserves, only (nonschedulable load) the energy cost does not change, while the generation side reserve cost is reduced. If the LSE of type 1 is considered schedulable, the energy cost is reduced together with the reserve cost. It is important to notice that with the increasing penetration of windpower generation, the LSE of type 1 offers more reserves, especially in the case in which the load may be optimally scheduled, while the total amount of power curtailment available from the LSE of type 2 is utilized in all the cases.

3.3.4 COMPUTATIONAL STATISTICS All the simulations were performed on a workstation with 256 GB of RAM memory, employing two 16-core Intel Xeon processors clocking at 3.10 GHz running on a 64-bit windows distribution. The maximum allowed relative optimality gap is set to 10
4%. Indicative results from the simulations presented in this chapter are presented in Tables 3.9 and 3.10. It may be noticed that the simulations on the 6-bus system are trivial from the perspective of the computational burden. On the other hand, the 24-bus system is characterized by an increased number of constraints and variables, especially discrete. As a result, the computational time required to solve these cases increases. Nevertheless, the computational time in all the cases is deemed acceptable.

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

3.4 CONCLUSIONS Contingencies are major events that cause energy deficits or disturbances in the active power flow through the transmission lines. Moreover, the variations of the wind-power generation and the intrahour load demand require reserves to be procured in order to maintain the balance between the generation and the demand. In this chapter, a two-stage stochastic joint energy and reserve market structure that incorporates two different types of demand side resources capable of providing reserve in order to confront imbalances caused by load demand and wind-power generation deviations, as well as system contingencies were presented. The proposed formulation was applied both on an example test system in order to explain its functionality and on a modified version of the IEEE Reliability Test System in order to obtain more generalized results. Based on the numerical findings, the following may be noticed: • •

•

•

The need in reserves increases when contingencies are anticipated, leading in scheduling nonspinning reserves from units that are scheduled to be off-line. The flexibility of the demand side is an important parameter; the higher the flexibility the more the day-ahead energy cost decreases due to shifting the load from the relatively high loading periods to relatively low demand periods. Apart from the energy cost, the cost of scheduling reserves from the generation side decreases when the demand side resources are considered, especially when the LSE renders its demand available to be optimally scheduled by the SO. It should be noted that the decrease in the reserve cost depends on the cost of utilizing demand side resources. The SO utilizes the contingency reserves offered by the demand side to their full extent.

Through the investigated test cases, it was rendered evident that the contribution of the two types of LSE to reserves bears economic benefits for the SO. Given that the services offered by the demand side may be procured at lower prices in comparison with the generation side reserves, they constitute an appealing alternative resource to confront power imbalances and transmission system disturbances. Especially, the contribution of the demand side resources was demonstrated to be more significant when higher levels of wind-power generation penetration are considered.

APPENDIX A : 24-BUS SYSTEM TEST SYSTEM DATA In this appendix, the data of the test systems used in this chapter are presented. The simulations are performed on a suitably modified version of the IEEE Reliability Test System [51] based on data presented in Refs. [52,53]. The topology and the characteristics of the transmission system can be found in Ref. [52]. Units are grouped by type and bus, which results in a reduction from 32 to 12 generating units, while the numbering used in Ref. [52] is adopted. The technical and economic data of the conventional generators are presented in Tables 3.11 and 3.12, respectively. Data concerning the system loading are presented in Table 3.13. Finally, the 10 wind-power generation scenarios that are used are displayed in Fig. 3.17, while their probability of occurrence is listed in Table 3.14.

Table 3.11 Technical Data of Conventional Generators

Unit

Maximum Output (MW)

Minimum Output (MW)

Minimum up time (h/min)

Minimum Down Time (h/min)

Ramp Up Rate (MW/min)

Ramp Down Rate (MW/min)

Initial Output (MW)

Periods Committed (h/min)

U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12

152 152 300 591 60 155 155 400 400 300 310 350

30.4 30.4 75 206.85 12 54.25 54.25 100 100 300 108.5 140

8 8 8 12 4 8 8 1 1 0 8 24

4 4 8 10 2 8 8 1 1 0 8 48

5 5 10 18 2 5.2 5.2 13.4 13.4 10 10.4 8

5 5 10 18 2 5.2 5.2 13.4 13.4 10 10.4 8

35 35 0 0 60 0 55 400 400 300 140 140

22 22 220 210 10 220 10 769 16 24 10 30

480 480 480 720 240 480 480 0 60 0 480 1440

240 240 480 600 120 480 480 60 60 0 480 2880

1320 1320 21200 2 600 600 21200 60 76140 960 1440 600 1800

Table 3.12 Economic Data of Conventional Generators Power Blocks (MW)

Marginal Costs (h/MW h)

Unit

B1

B2

B3

B4

C1

C2

C3

C4

Reserve Cost (h)

Startup Cost (h)

Shutdown Cost (h)

U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12

30.4 30.4 75 206.85 12 54.25 54.25 100 100 300 108.5 140

45.6 45.6 75 147.75 18 38.75 38.75 100 100 0 77.5 87.5

45.6 45.6 90 118.2 18 31 31 120 120 0 62 52.5

30.4 30.4 60 118.2 12 31 31 80 80 0 62 70

11.46 11.46 18.6 19.2 23.41 9.92 9.92 5.31 5.31 0 9.92 10.08

11.96 11.96 20.03 20.32 23.78 10.25 10.25 5.38 5.38 0 10.25 10.66

13.89 13.89 21.67 21.22 26.84 10.68 10.68 5.53 5.53 0 10.68 11.09

15.97 15.97 22.72 22.13 30.4 11.26 11.26 5.66 5.66 0 11.26 11.72

16 16 23 23 30 11 11 5 5 0 12 12

1430.4 1430.4 1725 3056.7 437 312 312 0 0 0 624 2298

1430.4 1430.4 1725 3056.7 437 312 312 0 0 0 624 2298

APPENDIX A : 24-BUS SYSTEM TEST SYSTEM DATA

Table 3.13 System Load Period

Inelastic System Load (MW)

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

1776 1670 1590 1563 1563 1590 1963 2281 2520 2546 2546 2520

FIGURE 3.17 10 Wind-power generation scenarios.

Load Bus

Percentage of System Load (%)

1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20

3.802 3.404 6.304 2.597 2.503 4.790 4.402 6 6.095 6.793 9.291 6.793 11.105 3.503 11.703 6.404 4.500

109

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

Table 3.14 Probability of Occurrence for Each Scenario Scenario

Probability (%)

s1 s2 s3 s4 s5 s6 s7 s8 s9 s10

10 4.28 14.28 2.85 20 5.71 17.14 1.42 14.28 10

NOMENCLATURE The main notation used in this chapter is listed below. Other symbols are defined where they first appear. Note that in order to state that a constraint holds “for every” element of a set, instead of for example, ’iAI, for the sake of brevity, ’i is used, unless strict notation is necessary to identify the domain of a constraint.

SETS AND INDICES bðBðn; nnÞÞ Bnb Bnn b f ðF i Þ iðIÞ j1 ðJ1 Þ j2 ðJ2 Þ nðNÞ Nnx rðRÞ sðSÞ t1 ðT1 Þ t2 ðT2 Þ wðWÞ

index (set) of transmission lines. set of sending nodes of transmission line b. set of receiving nodes of transmission line b. index (set) of steps of the marginal cost function of unit i. index (set) of conventional generating units. index (set) of LSE of type 1. index (set) of LSE of type 2. index (set) of nodes.   set of resources of type xA i; j1 ; j2 ; r; w connected to node n. index (set) of inelastic loads. index (set) of wind-power generation scenarios. index (set) of time intervals in the first stage of the problem. index (set) of time intervals in the second stage of the problem. index (set) of wind farms.

PARAMETERS Bi;f ;t1 Bb; n Ci; f ; t1

size of step f of unit i marginal cost function in period t1 (MW). susceptance of transmission line b (per unit). marginal cost of step f of unit i bidding function in period t1 (h/MW h).

Ci;R;DN t1

offer cost of down spinning reserve by generating unit i in period t1 (h/MW h).

CjDN;LSE1 1 ; t1

offer cost of down reserve by LSE of type 1 j1 in period t1 (h/MW h).

NOMENCLATURE

CjDN;LSE2 2 ;t1

offer cost of down reserve by LSE of type 2 j2 in period t1 (h/MW h).

R; NS Ci;t 1 R;UP Ci;t 1 CjUP;LSE1 1 ;t1 CjUP;LSE2 2 ;t1 D1r;t1 D2r;t2 DTi1 DTi2 Ejreq 1 fbmax

offer cost of nonspinning reserve by generating unit i in period t1 (h/MW h).

111

offer cost of up spinning reserve by generating unit i in period t1 (h/MW h). offer cost of up reserve by LSE of type 1 j1 in period t1 (h/MW h). offer cost of up reserve by LSE of type 2 j2 in period t1 (h/MW h). demand of inelastic load r in period t1 (MW). demand of inelastic load r in period t2 (MW). minimum down time of unit i (h). minimum down time of unit i (min). energy requirement of LSE of type 1 j1 (MW h).

LSE1max j1 ;t1

maximum capacity of transmission line b (MW). transmission line b contingency parameter—0 if transmission line b is under contingency in period t2 , else 1. maximum load of LSE of type 1 j1 in period t1 (MW).

LSE1min j1 ;t1

minimum load of LSE of type 1 j1 in period t1 (MW).

LCb;t2

LSE2max maximum load of LSE of type 2 j2 in period t1 (MW). j2 ;t1 LSE2min j2 ;t1

minimum load of LSE of type 2 j2 in period t1 (MW).

Njcall 2

maximum number of calls of LSE of type 2 j2 .

Pmax i Pmin i PWP w;t2 ;s PWP;max w

maximum power output of unit i (MW).

RDi

ramp down rate of unit i (MW/min).

RUi

ramp up rate of unit i (MW/min).

SDCi

shutdown cost of generating unit i (h).

SUCi

startup cost of generating unit i (h).

Tjdur 2

maximum duration of contingency reserve provision by LSE of type 2 j2 (min).

UCi;t2

unit i contingency parameter—0 if transmission line i is under contingency in period t2 , else 1.

UT1i UT2i LOL Vr;t 2 spill Vw;t 2

minimum up time of unit i (h).

ΔT1

duration of time interval in the first stage (min).

ΔT2

duration of time interval in the second stage (min).

λLSE1 j1 ;t1

utility of LSE of type 1 j1 in period t1 (h/MW h).

λLSE2 j2 ;t1

utility of LSE of type 2 j2 in period t1 (h/MW h).

πs

probability of occurrence of wind-power scenario s.

T

NS

TS

minimum power output of unit i (MW). power output of wind farm w in period t2 in scenario s (MW). maximum amount of wind that may be scheduled in the day-ahead market (MW).

minimum up time of unit i (min). cost of involuntary load shedding of inelastic load r in period t2 (h/MW h). wind spillage cost of wind farm w in period t2 (h/MW h).

nonspinning reserve deployment time (min). spinning reserve deployment time (min).

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CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

VARIABLES bi;f ;t1

power output scheduled from the f th block by unit i in period t1 (MW).

CAi;t2 ;s fb;t2 ;s

additional cost incurring due to a change in the commitment status of unit i in period t2 in scenario s (). active power flow through transmission line b in period t2 in scenario s (MW).

Lshed r;t2 ;s

load shed from inelastic load r in period t2 in scenario s (MW).

LSE1DN j1 ; t1

total down reserve scheduled from LSE of type 1 j1 in period t1 (MW).

LSE1jDN;load 1 ; t1 LSE1jDN;wind 1 ;t1 LSE1UP j1 ;t1 LSE1UP;load j1 ;t1 LSE1UP;wind j1 ;t1 LSE1ac j1 ;t2 ;s LSE1dj1 ;t2 ;s LSE1sch j1 ;t1 LSE1uj1 ;t2 ;s LSE1d;load j1 ;t2 ;s

down reserve scheduled to balance load deviations from LSE of type 1 j1 in period t1 (MW). down reserve scheduled to balance wind deviations from LSE of type 1 j1 in period t1 (MW). total up reserve scheduled from LSE of type 1 j1 in period t1 (MW). up reserve scheduled to balance load deviations from LSE of type 1 j1 in period t1 (MW). up reserve scheduled to balance wind deviations from LSE of type 1 j1 in period t1 (MW). actual consumption of LSE of type 1 j1 in period t2 in scenario s (MW). total down reserve deployed from LSE of type 1 j1 in period t2 in scenario s (MW). scheduled consumption of LSE of type 1 j1 in period t1 (MW). total up reserve deployed from LSE of type 1 j1 in period t2 in scenario s (MW).

LSE2jDN;con 2 ;t1

down reserve deployed to balance load deviations from LSE of type 1 j1 in period t2 in scenario s (MW). down reserve deployed to balance wind deviations from LSE of type 1 j1 in period t2 in scenario s (MW). up reserve deployed to balance load deviations from LSE of type 1 j1 in period t2 in scenario s (MW). up reserve deployed to balance wind deviations from LSE of type 1 j1 in period t2 in scenario s (MW). down reserve scheduled from LSE of type 2 j2 in period t1 (MW).

LSE2UP;con j2 ;t1

up reserve scheduled from LSE of type 2 j2 in period t1 (MW).

LSE2ac j2 ;t2 ;s LSE2d;con j2 ;t2 ;s LSE2sch j2 ;t1 LSE2u;con j2 ;t2 ;s PG i;t2 ; s Psch i;t1 PWP;S w; t1 RDN i;t1 RDN;con i;t1 RDN;load i;t1 RDN;wind i;t1 RNS i;t1 RNS;con i;t1

actual consumption of LSE of type 2 j2 in period t2 in scenario s (MW).

LSE1d;wind j1 ;t2 ;s LSE1u;load j1 ;t2 ;s LSE1u;wind j1 ;t2 ;s

down reserve deployed from LSE of type 2 j2 in period t2 in scenario s (MW). scheduled consumption of LSE of type 2 j2 in period t1 (MW). up reserve deployed from LSE of type 2 j2 in period t2 in scenario s (MW). actual power output of unit i in period t2 in scenario s (MW). power output scheduled for unit i in period t1 (MW). scheduled wind-power for wind farm w in period t1 (MW). total down spinning reserve scheduled from unit i in period t1 (MW). contingency down spinning reserve scheduled from unit i in period t1 (MW). down spinning reserve scheduled to balance load deviations from unit i in period t1 (MW). down spinning reserve scheduled to balance wind deviations from unit i in period t1 (MW). total nonspinning reserve scheduled from unit i in period t1 (MW). contingency nonspinning reserve scheduled from unit i in period t1 (MW).

NOMENCLATURE

RNS;load i; t1

nonspinning reserve scheduled to balance load deviations from unit i in period t1 (MW).

RNS;wind i; t1 RUP i; t1 RUP;con i; t1 RUP;load i; t1 RUP;wind i; t1 ri;Gf ;t2 ; s DN ri;t 2 ;s DN;con ri;t 2 ;s DN;load ri;t 2 ;s

nonspinning reserve scheduled to balance wind deviations from unit i in period t1 (MW).

113

total up spinning reserve scheduled from unit i in period t1 (MW). contingency up spinning reserve scheduled from unit i in period t1 (MW). up spinning reserve scheduled to balance load deviations from unit i in period t1 (MW). up spinning reserve scheduled to balance wind deviations from unit i in period t1 (MW). reserve deployed from the f -th block of unit i in period t2 in scenario s (MW). total down spinning reserve deployed from unit i in period t2 in scenario s (MW). contingency down spinning reserve deployed from unit i in period t2 in scenario s (MW).

NS ri;t 2 ;s

down spinning reserve deployed to balance load deviations from unit i in period t2 in scenario s (MW). down spinning reserve deployed to balance wind deviations from unit i in period t2 in scenario s (MW). total nonspinning reserve deployed from unit i in period t2 in scenario s (MW).

NS;con ri;t 2 ;s

contingency nonspinning reserve deployed from unit i in period t2 in scenario s (MW).

NS;wind ri;t 2 ;s UP ri;t 2 ;s

nonspinning reserve deployed to balance load deviations from unit i in period t2 in scenario s (MW). nonspinning reserve deployed to balance wind deviations from unit i in period t2 in scenario s (MW). total up spinning reserve deployed from unit i in period t2 in scenario s (MW).

UP;con ri;t 2 ;s

contingency up spinning reserve deployed from unit i in period t2 in scenario s (MW).

UP;load ri;t 2 ;s

SDC1i;t1

up spinning reserve deployed to balance load deviations from unit i in period t2 in scenario s (MW). up spinning reserve deployed to balance wind deviations from unit i in period t2 in scenario s (MW). shutdown cost of unit i in period t1 (h).

SDC2i;t2 ;s

shutdown cost of unit i in period t1 in scenario s (h).

SUC1i;t1

startup cost of unit i in period t1 (h).

SUC2i;t2 ;s

startup cost of unit i in period t1 in scenario s (h).

Sw;t2 ;s

wind spilled from wind farm w in period t2 in scenario s (MW).

u1i;t1 u2i;t2 ;s y1i;t1 y2i;t2 ;s z1i;t1 z2i;t2 ;s

binary variable—1 if unit i is committed in period t1 , else 0.

δn;t2 ;s ζ LSE2 j2 ;t2 ;s

binary variable—voltage angle of node n in period t2 in scenario s (rad). binary variable—1 if LSE of type 2 j2 stops providing contingency reserve in period t2 in scenario s, else 0. binary variable—1 if LSE of type 2 j2 is providing contingency reserve in period t2 in scenario s, else 0.

DN;wind ri;t 2 ;s

ðNS;loadÞ rði;t 2 ;sÞ

UP;wind ri;t 2 ;s

υLSE2 j2 ;t2 ;s

binary variable—1 if unit i is committed in period t2 in scenario s, else 0. binary variable—1 if unit i is starting up in period t1 , else 0. binary variable—1 if unit i is starting up in period t2 in scenario s, else 0. binary variable—1 if unit i is shutting down in period t1 , else 0. binary variable—1 if unit i is shutting down in period t2 in scenario s, else 0.

114

υDN j2 ;t2 ;s υuj2 ;t2 ;s ψLSE2 j2 ;t2 ;s

CHAPTER 3 OPTIMAL PROCUREMENT OF RESERVES

binary variable—1 if LSE of type 2 j2 is providing down contingency reserve in period t2 in scenario s, else 0. binary variable—1 if LSE of type 2 j2 is providing up contingency reserve in period t2 in scenario s, else 0. binary variable—1 if LSE of type 2 j2 is called to provide contingency reserve in period t2 in scenario s, else 0.

REFERENCES [1] Rebours Y, Kirschen D, Trotignon M, Rossignol S. A survey of frequency and voltage control ancillary services—Part I: technical features. IEEE Trans Power Syst 2007;22(1):350
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CHAPTER

OPTIMUM BIDDING OF RENEWABLE ENERGY SYSTEM OWNERS IN ELECTRICITY MARKETS

4 Ting Dai

Siemens, Greater Minneapolis-St. Paul, MI, United States

4.1 RENEWABLE ENERGY SYSTEM OWNERS IN ELECTRICITY MARKETS 4.1.1 ELECTRICITY MARKET OVERVIEW The electric power industry has evolved from a regulated operational structure to a competitive one in many countries around the world. In the United States, the Federal Energy Regulatory Commission established the foundation for developing competitive bulk electricity markets by enacting order 888 and 889 in 1996 to ensure nondiscriminatory open access to transmission services. Since then, the electricity industry has encountered numerous reconstruction activities, including increasing the number of market participants and the establishment of market operators as managers of a fair and secure market environment. Up to now, about two-thirds of the US electricity load is operated in the electricity markets. Based on the type of production to be traded, the electricity market can be categorized into the energy market, ancillary services market, financial transmission rights (FTR) market, and capacity market. 1. Energy market (also known as “the pool-based market”). Energy is bought and sold in the energy market through a two-settlement process: Day-ahead (DA) market and real-time (RT) market (also known as the balancing market). The DA market clears energy transactions each hour of the next operating day, whereas RT market transactions are performed just minutes before actual power deliveries. Energy is bought and sold in the RT market at RT spot prices to make up imbalances when system conditions change from the DA market. In some European countries, the energy market also includes an adjustment market which is similar to the DA market but is cleared closer to power delivery. The adjustment market is not discussed in this chapter. 2. Ancillary services market. In addition to the energy market, the ancillary services market ensures the reliability of the power system. Various types of ancillary services exist in the market today, such as regulation, contingency, and reserves. Other ancillary services, such as voltage control and black start services do not typically have auction-based markets. Optimization in Renewable Energy Systems. DOI: http://dx.doi.org/10.1016/B978-0-08-101041-9.00004-1 Copyright © 2017 Elsevier Ltd. All rights reserved.

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3. FTR market. In the FTR markets, forward auctions are held to allow market participants to submit their bids for purchase and sale of transmission rights. The transmission rights holders can obtain revenues or be charged penalties based on the locational differences in hourly energy prices. 4. Capacity market. The capacity market ensures long-term grid reliability by procuring the appropriate power capacity that is needed to meet growth predictions for future demand. This market creates long-term price signals and attracts promising bidders to invest in different resource supplies. The scope of this chapter includes only the US short-term energy markets which are DA market and RT market.

4.1.1.1 Electricity Market Participants and Operators The participants in an electricity market can be divided into three categories 1. Consumers. They are end users who buy energy or services (such as reserves) from the electricity market. A consumer can submit consumption bids to the electricity market with the goal of minimizing the cost of purchasing energy or services. 2. Retailers. They submit bids to purchase energy or services from the electricity market and sell them back to the consumers who don’t participate directly in the electricity market. 3. Energy Owners. They provide energy or services by submitting offers in the electricity market. Their intention is to maximize their total profits from selling energy or services. With so many participants in the electricity market, a neutral entity is required to establish sound market rules to guarantee the functioning of the economy and the processing of secure transactions. In the United States, independent system operators (ISOs) or regional transmission organizations (RTOs) are used to fulfill this role. The ISOs or RTOs administer transmission tariffs, maintain system security levels, coordinate and schedule maintenance, and play an important role in coordinating long-term planning. In the electricity market, the ISOs/RTOs play an important role as independent, unaffiliated market operators providing unbiased, open access to all market participants. The ISOs/RTOs also forecast electricity demand, schedule adequate reserve, run market clearing models, and send price signals to all of the market participants.

4.1.1.2 Time Frame of the Short-Term Electricity Market The time framework for the short-term electricity market is shown in Fig. 4.1. The DA market, as its name implies, refers to the market that operates 24 h prior to the RT market. It is used for scheduling the power production for the next day. The reserve market operates almost at the same time as the DA market. In most US markets, the DA market and reserve market are cooptimized by using a single clearing process, as determined by the ISOs/RTOs, from which the energy and reserve transactions are determined during each hour of the next operating day. Once the markets are cleared, the locational marginal prices (LMP), the reserve clearing prices, and the cleared energy volume of each participant are settled. The RT market is carried out just minutes before the actual power delivery by energy owners to ensure a RT balance between generation and demand. This is done by offsetting the differences between the RT operation and the corresponding energy program settled in the DA market. RT prices are calculated for the RT market based on the operating conditions.

4.1 RENEWABLE ENERGY SYSTEM OWNERS IN ELECTRICITY MARKETS

119

FIGURE 4.1 Time framework for clearing the short-term electricity market.

Price

Customer’s Bidding Curve Producer’s Offering Curve

λ

P

Power Quatity

FIGURE 4.2 Illustrations of an energy owner’s offering curve and a customer’s bidding curve.

4.1.1.3 Energy Owners in the Electricity Market In the short-term electricity market, the energy owners submit nondecreasing stepwise offering curves and the customers submit nonincreasing stepwise bidding curves indicating their willingness to sell and buy certain amounts of energy or reserve at certain prices, respectively. An illustration of an energy owner’s offering curve and a customer’s bidding curve is shown in Fig. 4.2. The ISOs/RTOs aggregate all of the bidding and offering curves, and determine the market clearing prices and power quantity committed from each energy owner and customer. As shown in Fig. 4.2, if the market clearing price is λ, the committed power of the energy owner is P. Failing to fulfill

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

their commitments brings penalties to the energy owners. The RT penalty mechanisms are not the same in the electricity markets of different countries. For example, in most European countries, two RT prices are determined separately in two different situations: a positive RT price for positive energy deviation (higher production or lower consumption than scheduled) and a negative RT price for negative energy deviation (lower production or higher consumption than scheduled).

4.1.2 IMPORTANCE OF INTEGRATING RENEWABLES INTO THE ELECTRICITY MARKET Before entering the electricity market, wind power is sold through long-term power purchase agreements (PPAs) which require the purchasers to buy all of the wind energy generated at a fixed price. However, after topping out at nearly $70/MW h for PPAs executed in 2009, the national average price of PPAs has shown a continuously declining trend [1]. In 2014, wind PPA prices fell to the lowest point at $23.5/MW h, which is almost competitive with wholesale prices in the electricity market. Also, since the availability of PPA contracts for wind remain in short supply, wind power energy owners can no longer obtain stable revenues by selling their power through PPAs. Studies show that wind energy integration costs are almost always below $12/MW h and often below $5/MW h—for wind power capacity penetrations of up to or even exceeding 40% of the peak load of the system in which the wind power is delivered. System operators continue to implement a range of methods to accommodate increased wind energy penetrations, including: centralized wind forecasting, treating wind as dispatchable, shorter scheduling intervals, and balancing areas consolidation and coordination. Today, around 23% of the wind power capacity in the United States is sold either through short-term contracts or directly into the electricity market. This percentage is expected to increase further according to the authors of the report [1]. They mentioned another important factor. The recent completion of transmission lines in Texas provides market access to a significant amount of newly installed wind capacity. As a consequence, wind energy owners are becoming more and more interested in trading their power in the electricity market instead of locking their revenues in the PPAs.

4.1.3 CURRENT MARKET RULES FOR RENEWABLE ENERGY SYSTEM OWNERS IN NORTH AMERICA Nowadays, wind and solar energy owners are permitted to participate in DA and RT market in North America but are not required to do so. For example, New England ISO has allowed renewable system owners to submit energy bid curve or self-schedule into the DA market (can include negative price bids). The proposal to require wind to submit bid curves into the DA market is under consideration. In most of the RT markets, renewable energy owners are not fully penalized for their deviation from DA schedule. Pennsylvanian
New-Jersey
Maryland interconnection (PJM) charges no penalty for deviation power less than 5% or 5 MW; Electric Reliability Council of Texas (ERCOT) charges wind a penalty when it is given an economic dispatch below its high dispatch limit (or capability) and is generating more than 10% above its basepoint. However, treating renewable energy owners the same way as the conventional power owners is a trend for the

4.1 RENEWABLE ENERGY SYSTEM OWNERS IN ELECTRICITY MARKETS

121

electricity markets. Table 4.1 provides the summary of the rules for integrating renewable energy in six North America electricity DA and RT markets: PJM, New York ISO (NYISO), New England ISO, southwest power pool (SPP), ERCOT, and California (CAISO). For more detailed information, please refer to Ref. [2]. In this chapter, the market bidding rules applied to conventional energy owners are assumed to be applied to the renewable energy owners. The renewable energy owners can submit multisegment bidding curves into the both DA and RT market. The deviation from DA schedule will be settled in RT price: if the actual production is less than the DA scheduled, the renewable energy owners will have to buy the deviated power at the RT price; otherwise, the renewable energy owners can bid the extra power into the RT market and get paid for the accepted power at the RT prices.

Table 4.1 Rules for Renewable Energy Owner to Participate in DA and RT Markets

ISO

PJM

NYISO

DA market

Allow to bid into DA market, but not required

Allow to bid into DA market, but not required

RT market

Deviations from DA schedules are settled at the RT priceDifferentials less than 5% or 5 MW incur no deviation charges

Required to submit bid curve into the RT market. Deviations from DA schedules are settled at the RT price

New England ISO

SPP

ERCOT

CAISO

Allow to bid into DA market, but not required

Allow to bid into DA market, but not required

Qualified renewable energy owners can participate in DA market

Deviations from DA schedules are settled at the RT price

Deviations from DA schedules are settled at the RT price. Can receive uplift credits if cleared in the DA market

Deviations from DA schedules are settled at the RT priceWind is charged a penalty when it is given an economic dispatch below its high dispatch limit (or capability) and is generating more than 10% above its basepoint

Allow to bid into DA market, but not required Can be dispatched down to less than forecasted output if LMP is less than the bid price Differences between the 5-min dispatch and the metered energy are considered uninstructed imbalance energy and are settled at 5-min market LMPs

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

Thus, the profit of the renewable energy owner π can be expressed as: π 5 λDA PDA 1 λRT ðPRT 2 PDA Þ

(4.1)

where λ and λ are DA and RT market clearing prices. P and P are power scheduled in DA and RT. As can be seen from Eq. (4.1), λDA and λRT are both uncertain variables. PDA and PRT are variables determined by the renewable energy owners but are also constrained by the actual power production which is also uncertain. In order to obtain the maximum revenue of the renewable energy owners, optimization models need to take consideration of these uncertain variables. DA

RT

DA

RT

4.2 PRICE-TAKER MODELS—MULTISTAGE STOCHASTIC PROGRAMMING APPROACH 4.2.1 MULTISTAGE STOCHASTIC PROGRAMMING APPROACH Stochastic programming is an approach for modeling optimization problems that involves uncertainties. In the short-term electricity market, decision makers have to make optimal decisions throughout a decision horizon involving several stages. For example, DA market decisions are first-stage decisions which are made before knowing the market clearing prices in the DA and RT markets. RT market decisions, on the other hand, are made after the clearing of the DA market but before knowing the market clearing prices in the RT market, which constitutes second-stage decisions. Thus, a two-stage stochastic programming model would be an appropriate tool for energy owners trading in both DA and RT markets. In stochastic programming, the stochastic process, such as electricity market prices and wind power production, can be represented using continuous or discrete random variables. In a best case scenario, stochastic programming problems with continuous random variables can only be solved in small or illustrative instances [3]. For this reason, scenario representation of random variables becomes indispensable in solving stochastic problems. The set of values used to model a random variable is usually arranged in a so-called “scenario tree,” as illustrated in Fig. 4.3. A scenario tree comprises a set of nodes and arcs. The node in the first-stage is called the root node. The nodes in the last stage are called leaves. A scenario is a path from the root to the leaf. A stage is a moment in the time line when the decisions are taken.

4.2.2 SCENARIO GENERATION AND REDUCTION METHOD Different techniques have been proposed in the literature to generate scenarios and build scenario trees. An easy way to generate a scenario for market prices is the sampling method, as proposed in Ref. [4]. Høyland and Wallace [5] proposed a moment matching method to generate a limited number of scenarios which satisfied certain properties. Pflug [6] described an optimal discretization method to find the reduced scenario set of the initial one that minimizes an error based on an objective function. Given that the computational burden of a stochastic programming problem increases rapidly with the number of scenarios, scenario reduction becomes necessary. The scenario reduction

4.2 MULTISTAGE STOCHASTIC PROGRAMMING APPROACH

123

Scenario1

Scenario2

Scenario3

Scenario4 1st Stage (root node)

2nd Stage

3rd Stage (leaves node)

FIGURE 4.3 A typical scenario tree.

techniques for two-stage stochastic programming were presented in Refs. [7,8]. The reduced number of scenarios that best retains the essential features of the original scenario set is chosen according to a probability distance. In this chapter, a procedure combining a path-based method and a scenario reduction technique is used to generate multistage scenario trees [9]. A seasonal autoregressive integrated moving average (ARIMA) model [10] is used to generate a large number of scenarios for uncertainties in the market. The general expression of seasonal ARIMA model can be expressed as follows: 

12

Pp

 Φj BjS ð12BÞd ð12Bs ÞD yt 5 P P jS ð1 2 qj51 θj Bj Þð1 2 Q j51 Θj B Þεt

j51

φj Bj



12

PP

j51

(4.2)

with p autoregressive parameter φ1 ; . . . ; φp , q moving average parameter θ1 ; . . . ; θp , a seasonal component of Pautoregressive parameter Φ1 ; . . . ; ΦP , Q moving average parameter Θ1 ; . . . ; Θp , and where yt and εt are the actual value and error at time t, respectively. B is the backshift operator, that is, Bj yt 5 yt2j . As εt is the forecast error which a random variable following normal distribution, a large number of scenarios can be generated by generating random value for εt . Then, a fast-forward scenario-reduction algorithm [9] was used to obtain a reduced scenario set with a sufficiently small number of scenarios from an iterative process. In each iteration, the scenario that minimized the Kantorovich distance between the reduced set and the original set is chosen from the set of unselected scenarios and included in the reduced set. The algorithm stops if either the required number of scenarios or a certain Kantorovich distance is attained. Fig. 4.4 shows an example of generating 10 DA LMP scenarios for each during one week by using PJM data [11].

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FIGURE 4.4 Example of reduced scenarios of DA LMP.

4.2.3 RISK MANAGEMENT Despite the numerous advantages of using stochastic programming to model the uncertainties associated with wind power trading, it only shows that the decision makers expect profits and neglect the probabilities of negative profits or losses. Thus, risk management constitutes an important part of the stochastic programming and helps the decision makers avoid undesirable outcomes. Different types of risk management models have been reported and analyzed, such as the variance, the shortfall probability, the VaR [12], and the conditional value at risk (CVaR) [13] models. For a given αAð0; 1Þ, VaRα is a measure computed as the maximum profit value such that the probability of the profit being lower than or equal to this value is lower than or equal to 1 2 α. VaRα 5 max⁡fxjpðB # xÞ # 1 2 αg

(4.3.1)

where p and B represent the probability and expected value of the profit, respectively, x is the decision variable of the maximization problem (4.3.1
4.3.5). For a given αAð0; 1Þ, the CVaRα represents the expected value not surpassing VaRα. CVaRα 5 EðBjB , VaRα Þ

(4.3.2)

A discrete formulation of CVaRα is the following: maxζ;ηω ζ 2

1 X pω ηω 12α ω

(4.3.3)

Subject to 2 Bω 1 ζ 2 ηω # 0

(4.3.4)

4.2 MULTISTAGE STOCHASTIC PROGRAMMING APPROACH

ηω $ 0

125

(4.3.5)

where pω and Bω represent the probability and expected profit of scenario ω, respectively, ζ is an auxillary variable used to compute CVaR, and ηω an auxillary variable used to compute CVaR in scenario ω. There is an additional difficulty in using VaR for stochastic problems in that it requires the use of binary variables for its modeling. Instead, the computation of CVaR does not require the use of binary variables; and it can be simply modeled by using linear constraints. The CVaR has been successfully implemented in various models considering uncertainties. Especially for wind energy owners, CVaR can help the decision makers to model the tradeoff between profit and risk.

4.2.4 BIDDING STRATEGY MODELS FOR PRICE-TAKER RENEWABLE ENERGY OWNER In this section, we will model the bidding strategy for the price-taker renewable energy owner using multistage stochastic programming. The mathematical formulation will be presented and a simple example will be shown.

4.2.4.1 Basic Mathematical Formation for the Price-Taker Renewable Energy Owner The basic mathematical formulation for the price-taker renewable energy owner in DA and RT market are expressed in the following mathematical problem (4.4): maxWtωD ;Δ1tω ;Δ2tω ;Δtω ;ζ;ηω πT 5

NΩ XNT  X  1  2 D r pr ω λD tω Wtω 1 λtω Δtω 2 Δtω t51 ω51

1 β w ðζ 2

1 XNΩ pr ω ηω Þ ω51 12α

(4.4.1)

Subject to D 0 # Wtω # W max ; ’t; ω

Δ1 tω

2 Δ2 tω

(4.4.2)

5 Δtω ; ’t; ω

(4.4.3)

ac D Δtω 5 Wtω 2 Wtω ; ’t; ω

(4.4.4)

0 # Δ1 tω

ac # Wtω ;

’t; ω

(4.4.5)

0 # Δ2 tω

D # Wtω ;

’t; ω

(4.4.6)

0



ζ2

D D D D 5 Wtω 0 ; ’t; ω; ω : λ Wtω tω 5 λtω0    0 D D D λD 2 Wtω 0 Wtω $ 0; ’t; ω; ω tω 2 λtω0

(4.4.7)

ηω $ 0; ’ω

(4.4.9)

XNT

ðλD W D t51 tω tω

1 λrtω Δ1 tω

2 λrtω Δ2 tω Þ # ηω ;

(4.4.8)

’ω

(4.4.10)

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2 D D where Wtω , Δ1 tω , Δtω ,Δtω , ζ, and ηω are the decision variables. Wtω represents the power offered by

2 the renewable energy owner in the DA market at time period t. Δ1 tω , Δtω , andΔtω represent the positive deviation, negative deviation, and total deviation of the renewable energy owner in the RT market at time period t. The objective function comprises two terms: (1) The expected profit of the renewable energy owner, which equals the revenue from the DA market plus the revenue from positive energy deviations in the RT market minus the cost for negative energy deviations in the RT market; and (2) the CVaR multiplied by a weighting factor β W, which controls the degree of risk aversion of the renewable energy owner. D In this model, the first-stage decision variable is the hourly bid of the energy volume Wtω of the renewable energy owner, whereas the second-stage variables are the RT energy deviations Δ1 tω and Δ2 of the renewable energy owner. Constraint (4.4.2) limits the amount of wind energy that can be tω traded in the DA market. Constraints (4.4.3)
(4.4.6) determine the total positive and negative energy deviations incurred by the wind energy owner per period and scenario. Constraint (4.4.7) constitutes the nonanticipativity conditions related to the decisions made in the first-stage. Constraint (4.4.8) enforces a nondecreasing offer curve. Constraints (4.4.9) and (4.4.10) are used to compute CVaR.

4.2.4.2 Examples In this section, we will present the example to demonstrate the effectiveness of the proposed model in the previous section. The historical data speed and market prices are obtained from PJM website [11]. The maximum wind generating capacity is set to be 300 MW. The ARIMA model is used to generate 5000 scenarios for wind power production, DA, respectively. Then, the scenario reduction method is applied to reduce the number of scenarios for each uncertainty to 5. Table 4.2 shows the information of these scenarios. The risk control parameter is set to be 0.2. By solving problem (4.4), the bidding strategy capacity into DA market for each scenario is W1D 5 127:43, W2D 5 126:80,W3D 5 252:42,W4D 5 268:17,W5D 5 300. Using these solved bidding capacities and their corresponding DA LMP, the readers can build the nondecreasing bidding curve shown in Fig. 4.5. The risk management control parameter can be used to control the risk associated with uncertainties. As we increase β W, the value of CVaR increases; but the expected profit decreases as shown in Figs. 4.6 and 4.7. This is as expected, the energy owner increases β W in order to reduce the trading risks, thus, the cost associated with risks increase and the expected profit decreases. Table 4.2 Scenario Information Scenario #

Wind Power Production (MW)

Probability

DA LMP ($/MW h)

Probability

1 2 3 4 5

109.52 150.20 205.53 251.98 289.61

0.302 0.2049 0.105 0.187 0.201

17.55 17.58 22.97 23.90 26.03

0.251 0.271 0.178 0.160 0.139

4.2 MULTISTAGE STOCHASTIC PROGRAMMING APPROACH

FIGURE 4.5 Bidding curve.

FIGURE 4.6 The impact β W of on expected profit.

FIGURE 4.7 The impact β W of on CVaR.

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4.2.5 MITIGATING THE TRADING RISK BY PURCHASING ADDITIONAL POWER FROM CONVENTIONAL ENERGY OWNER 4.2.5.1 Introduction To mitigate the trading risk associated with renewable energy trading, many approaches have been discussed and applied, such as combining renewable energy units with energy storage system, signing bilateral contract, and others. In this chapter, we will introduce the approach to mitigate the risk for bidding in the electricity market by buying additional amount of power from conventional energy owner and use it as “reserve” to hedge against the RT market penalties. In Ref. [14], the authors introduced a bilateral reserve settlement mechanism between renewable energy owners and conventional energy owners. The authors proposed several assumptions: (1) The bilateral reserve settlement introduced can be mixed with the system-wide reserve to provide standby power to cover the intermittency and uncertainty of renewable energy sources. The bilateral reserve is provided by conventional energy owners and consumed by renewable energy owners. (2) The bilateral reserve settlement among renewable and conventional energy owners can be seen as a new trading mechanism for a new type of reserve, adding to the existing system-wide reserve. The bilateral reserve settlement price and volumes of specific providers are cleared among the renewable and conventional energy owners who provide this new type of reserve. This new bilateral trading mechanism does not change current implementation of the system-wide reserve, regulation, and other auxiliary services for mitigating other uncertainties, for example, large load uncertainties, in the system. Both renewable and conventional energy owners try to compete with other energy owners and maximize their profit in the market. The bilateral reserve price and power amount settled between these two types of energy owners can be solved using game theory. The optimal bidding strategy of both energy owners can be modeled using stochastic programming introduced in the previous section.

4.2.5.2 Auction Game and Nash Equilibrium This section gives an introduction of game theory, Nash Equilibrium, and the application settle the bilateral reserve price and power amount between conventional and renewable energy owners. A game is a “formal representation of a situation in which a number of individuals interact in a setting of strategic independence [15].” There are four elements in a game: (1) The players, (2) the rules of the game, (3) the outcomes, and (4) the payoff and preferences (utility functions) of the players. A game can be either cooperative, where the players collaborate to achieve a common goal, or noncooperative, where they act on their own. In this section, different energy owners are the game players. Each player has the historical information of other players’ past actions. A strategy is a rule that tells the players which action(s) they should take. Assuming that the players are noncooperative, know the payoff functions of other players, and try to maximize their payoff functions while considering their rivals’ bidding strategies, the Nash equilibrium [15] occurs when no player has the incentive to change its offering/bidding strategy. In order to benefit from participating in the bilateral reserve market, renewable energy owners will not buy reserve from the bilateral reserve market if the reserve clearing price exceeds the RT price, whereas the conventional energy owners will not offer reserve in the bilateral reserve market

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129

if the RT price is lower than the DA price. This means that the offer price λRt for reserve at a certain time t can take any value between the DA price λD t and whichever has the higher value: the RT price λrt or the energy price cap ccap specified by the market operator. R r cap λD Þ t # λt # max⁡ðλt ; c

(4.5)

A renewable energy owner can bid reserve energy provided by a conventional energy owner at any value between zero and the predicted power imbalance during the market operation. 0 # WtR # Δ2 t

(4.6)

where WtR represents the power bought by the wind energy owner in the bilateral reserve market and Δ2 t is the negative deviation of wind power in time period t. Also, a conventional energy owner can offer reserve energy of its unit g to a wind energy owner at any value between zero and the maximum power output of the unit. 0 # PRgt # Pmax g

(4.7)

where PRgt represents the power offered by the conventional unit g in the bilateral reserve market at time period t and Pmax is the maximum power output of the conventional unit g. g Let γ i denote player i’s strategy and γ2i denote other players’ strategies. Player i’s total profit is πi, which includes the revenue from both energy and bilateral reserve markets and can be obtained by solving a two-stage stochastic programming problem. Let Гi be the set of continuous strategies of player i. For a continuous game, a strategy tuple γ i i51toI is a Nash equilibrium if the following equilibrium condition is satisfied [17] for all continuous strategies γ i s, where i 5 1,. . ., I:       $ πi γ i ; γ2i ; ’γ i AΓi ; ’i πi γ i ; γ2i

(4.8)

The continuous equilibriums are difficult to obtain because the payoff function π has no explicit formula. To simplify the solution process, the continuous strategy set of player i, Гi, is appropriately discretized into Ni choices; then the set of the resulting discrete strategies of player i can be written as Ωi 5 {γ i,n, n 5 1,. . ., Ni}. As player i has NG generating units and each unit can have Nig NG discrete strategies, Ni can be expressed as Ni 5 Lg51 Nig . Moreover, as there are total I players, a I

N

G game can then be formed with a total number of N 5 Li51 Lg51 Nig strategy tuples. The Nash solution can then be searched among the N strategy tuples. Similar to (3.4), is γ i i51toI , a Nash equilibrium for the matrix game if the following discrete equilibrium condition is satisfied.

      $ πi γ i;n ; γ2i ; ’γ i;n AΩi ; ’i πi γ i ; γ2i

(4.9)

4.2.5.3 Mathematical Model In this section, we will present the stochastic model to obtain the optimal bidding strategies for both conventional energy owners and renewable energy owners. 1. Conventional energy owner model The conventional energy owners, such as thermal and hydro power plants, can control their power outputs if no generator failure is considered. The problem (4.10) of obtaining the best

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

bidding strategy for conventional power energy owners is formulated as a two-stage stochastic program to maximize the profit of a thermal energy owner maxPDgtω ;Prgtω ;Pacgtω ;ζ;ηω ;ugt πT 5

NΩ X ω51

pr ω

XNT XNG h D r r R R λD tω Pgtω 1 λtω Pgtω 1 λt Pgt 2 t51 g51

  i r R Cg PD gtω 1 Pgtω 1 Pgt 2 max⁡ð0; StUPgt ðugt 2 ugðt21Þ ÞÞ 1 β T ðζ 2

1 XNΩ pr ω ηω Þ ω51 12α

(4.10.1)

Subject to: D r R max ugt Pmin g # Pgtω 1 Pgtω 1 Pgt # ugt Pg ; ’t; g; ω

(4.10.2)

D max ugt Pmin g # Pgtω # ugt Pg ; ’t; g; ω r max ugt Pmin g # Pgtω # ugt Pg ; ’t; g; ω D r R max Pac gtω Pgtω 1 Pgtω 1 Pgt # ugt Pg ; ’t; g; ω ac Pac gtω 2 Pgðt21Þω # RU g ; ’t; g; ω ac Pgðt21Þω 2 Pac gtω # RDg ; ’t; g; ω

(4.10.3)

Cg ðPÞ 5 ag 1 bg P 1 cg P   D  D 0 Pgtω 2 PD λtω 2 λD tω0 gtω0 $ 0; ’t; g; ω; ω 2

0

D D D PD gtω 5 Pgtω0 ; ’t; g; ω; ω :λtω 5 λtω0

ηω $ 0; ’ω h   D D r R R r D r R ζ2 d P 1 λ P 1 λ P 2 C P 1 P 1 P λ 2 t g tω tω t gtω gtω gt gtω gtω gt t51 g51   i max 0; StUPgt ugt 2 ugðt21Þ # ηω ; ’ω

(4.10.4) (4.10.5) (4.10.6) (4.10.7) (4.10.8) (4.10.9) (4.10.10) (4.10.11)

XNT XNG

(4.10.12)

r ac D r where PD gt ; Pgt ; Pgt ; ζ; η; ugt are the decision variables. Pgt and Pgt represent the power offered by the conventional unit g in the DA and RT market at time period t, respectively. Pac gt represents the total actual power output of the conventional unit g at time period t. ugt is the binary variable which represents the state of the conventional unit g in time period t (μ 5 1means ON and μ 5 0 means OFF). ζ and η are auxiliary variables used to compute the CVaR. r The random variables include λD t and λt which represent the DA and RT market prices at time period t, respectively. The variables, if augmented with a subscript ω, represent their realization in a scenario ω. max The constants and parameters are pr ω , Pmin g , Pg , RU g , RDg , ag , bg , cg , α, and β T , where min max pr ω is the probability of a scenario ω, Pg and Pg are the minimum and maximum power outputs of the conventional unit g, respectively, RU g and RDg are the ramp-up and ramp-down rates of the conventional unit g, respectively, ag , bg , and cg represent the thermal heat rate curve parameters of the conventional unit g, α is the per-unit confidence level, and β T is the risk aversion parameter of the conventional power energy owner. The objective function (4.10.1) comprises two terms: (1) The expected profit, which equals the revenues from the DA, RT, and bilateral reserve markets minus the production and start-up costs; and (2) the CVaR multiplied by a weighting factor β T, which allows the degree of risk aversion of the conventional power energy owner to be controlled.

4.2 MULTISTAGE STOCHASTIC PROGRAMMING APPROACH

131

In this model, the first-stage decision variable is the hourly bid of the energy volume PD gtω of the thermal units while the second-stage decision variable is the RT output Prgtω of the thermal units. Constraints (4.10.2)
(4.10.4) bind the maximum power capacity of each thermal unit. The actual total power generated by each thermal unit is expressed in constraint (4.10.5). Constraints (4.10.6) and (4.10.7) represent the ramp-up and ramp-down limits of each thermal unit, respectively. The production cost Cg of each thermal unit is expressed as a quadratic constraint (4.10.8). Constraint (4.10.9) enforces a nondecreasing offer curve. Constraint (4.10.10) constitutes the nonanticipativity conditions related to the decisions made in the firststage. Constraints (4.10.11) and (4.10.12) are used to compute CVaR. 2. Renewable energy owner model The model (4.11) to maximize the profit of a renewable energy owner is maxWtωD ;Δ1tω ;Δ2tω ;Δtω ;ζ;ηω πT 5

NΩ X ω51

pr ω

XNT  t51

 1 2 r r R R D λD tω Wtω 1 λtω ðΔtω 2 λtω Δtω 2 λt Wt 1 β w ðζ 2

1 XNΩ prω ηω Þ ω51 12α (4.11.1)

Subject to D 0 # Wtω # W max ; ’t; ω

Δ1 tω

2 Δ2 tω

5 Δtω ; ’t; ω

(4.11.3)

ac D 1 WtR 2 Wtω ; ’t; ω Δtω 5 Wtω

(4.11.4)

ac R 0 # Δ1 tω # Wtω 1 Wt ; ’t; ω

(4.11.5)

0 # Δ2 tω

D # Wtω ;

’t; ω

0

ζ2

(4.11.2)

(4.11.6)

D D D D 5 Wtω 0 ; ’t; ω; ω : λ Wtω tω 5 λtω0  D  D  0 D Wtω 2 Wtω $ 0; ’t; ω; ω 0 λtω 2 λD tω0

(4.11.7)

ηω $ 0; ’ω

(4.11.9)

XNT t51

1 2 D r r R R ðλD tω Wtω 1 λtω Δtω 2 λtω Δtω 2 λt Wt dt Þ # ηω ; ’ω

(4.11.8)

(4.11.10)

2 D D where Wtω , Δ1 tω , Δtω ,Δtω , ζ, and ηω are the decision variables. Wtω represents the power offered 2 by the renewable energy owner in the DA market at time period t. Δ1 tω , Δtω , andΔtω represent the positive deviation, negative deviation and total deviation of the renewable energy owner in the RT market at time period t. r ac The random variables include Wtac , λD t , and λt , where Wt is the actual renewable energy production at time period t. The variables, if augmented with a subscript ω, represent their realization in a scenario ω. The constants and parameters are pr ω ,W max , α, and β w , where W max is the installed capacity of the wind power energy owner and β w is the risk aversion parameter of the conventional power energy owner. The objective function (4.11.1) comprises two terms: (1) The expected profit, which equals the revenue from the DA market plus the revenue from positive energy deviations in the RT

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

market minus the cost for negative energy deviations in the RT market and the cost in the bilateral reserve market; and (2) the CVaR multiplied by a weighting factor β W, which controls the degree of risk aversion of the wind energy owner. Again in this model, the first-stage decision variable is the hourly bid of the energy volume D Wtω of the renewable energy owner, whereas the second-stage variables are the RT energy 2 deviations Δ1 tω and Δtω of the owner. Constraint (4.11.2) limits the amount of renewable energy that can be traded in the DA market. Constraints (4.11.3)
(4.11.6) determine the total positive
and negative energy deviations incurred by the renewable energy owner per period and scenario. Constraint (4.11.7) constitutes the nonanticipativity conditions related to the decisions made in the first-stage. Constraint (4.11.8) enforces a nondecreasing offer curve. Constraints (4.11.9) and (4.11.10) are used to compute CVaR. The two models of conventional and renewable energy owners are connected through the reserve volumes PRgt and WtR and the reserve clearing price λRt in the bilateral reserve settlement. The total reserve volume, that is, the sum of reserve volumes PRgt , of conventional energy owner should be equal to WtR . 3. Solving the matrix game The discretization of the strategy variables γ i s may cause loss or artificial creation of a Nash equilibrium [16]. To capture a possibly missing Nash solution, the standard discrete equilibrium condition (4.9) is loosened by ε to yield an approximate Nash equilibrium.       $ πi γ i;n ; γ2i 2 ε; ’γ i;n AΩi ; ’i πi γ i ; γ2i

(4.12)

The matrix payoffs are suppliers’ profits from both the energy and bilateral reserve markets. These profits are obtained by solving the two-stage constrained stochastic programming problems described in model (4.10) and (4.11) for all strategy tuples in the increasing order of bidding price and energy. After the payoffs for each strategy tuple are obtained, the strategy tuples are examined for the Nash equilibrium condition (4.12). Ref. [16] presented a method to find an appropriate value of ε for determining an approximate Nash equilibrium. The complete solution process for trading renewable energy in both energy market and the bilateral reserve settlement is depicted in a flow chart in Fig. 4.8, which consists of two parts: obtaining the matrix payoffs by stochastic programming on the left-hand side and solving for the Nash equilibrium on the right-hand side. The strategies contain two variables: the reserve energy volume PRgt or WtR that a conventional generating unit would like to offer or a renewable energy owner would like to buy, respectively, and the reserve clearing price λRt settled among conventional and renewable energy owners. These variables are discreated separately into a limited number of values within their limits defined by (4.5)
(4.7). A strategy tuple is then created by combining the values of these variables for all of the players in the game. For each strategy tuple, the stochastic optimization models (4.10) and (4.11) are executed for conventional and renewable energy owners, respectively, during which PRgt (or WtR ) and λRt are constant values in this strategy tuple. Then, the Nash equilibrium condition is examined for each strategy tuple solved to determine which strategy tuple yields the maximum profits for both conventional and renewable energy owners. Finally, the stochastic programming is executed for the best strategy tuple, that is, the Nash equilibrium, to obtain the optimal bidding curve. The parameter εmax is relatively small compared with the expected profit obtained from

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133

FIGURE 4.8 Flow chart for trading wind power using the proposed model.

the stochastic models, for example, 1% of the expected profit. If no solution is obtained from the search for the Nash equilibrium, wind energy owners will not bid reserve and conventional power energy owners will not offer reserve in the new bilateral reserve settlement.

4.2.5.4 Case Study Consider a game with two players: one thermal power plant with three units and one wind power plant. The installed capacities of the wind and thermal power plants are 100 and 140 MW, respectively. As there is only one thermal power energy owner in the market of providing the new type of bilateral reserve, the wind energy owner decides how much reserve it would like to buy from the thermal power energy owner and at what price. The thermal power energy owner decides

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

Table 4.3 Thermal Unit Data Unit Number min

P (MW) Pmax (MW) RU (MW/h) RD (MW/h) uIC (h) StUp ($) Fuel type a b c

1

2

3

0 50 50 50 0 0 Gas 0 80 0

5 45 15 15 0 88 Gas 85.509 70.85831 0.18819

5 45 15 15 0 88 Gas 82.342 68.23393 0.18122

the price of the reserved power and how much reserve it would like to sell. The thermal units’ operating characteristics are given in Table 4.3. The reserve price bid cap ccap is $1000/MW h in the reserve market. The ARIMA model is used to generate 5000 scenarios for wind power, DA price, and RT price predictions, respectively. Scenario reduction is then performed to reduce the scenarios of wind power, DA price, and RT price predictions to five each. Therefore, the final reduced scenario tree has 125 scenarios. The wind plant data is obtained from the National Renewable Energy Laboratory website [17]. The energy prices, including both the DA price and RT price, are obtained from the PJM website [11]. The risk aversion parameter is β T 5 β W 5 0.5. The confidence level is α 5 0.95. The maximum approximation parameter εmax (see Figure 4.8) is $10. For each thermal unit, three discrete reserve power volumes and three reserve clearing prices are generated. Assume that the reserve clearing price strategies are identical for the three units. A total of 81 strategy tuples are generated in this case study. The bidding curve for the wind energy owner is generated by solving the two-stage stochastic optimization problem (4.12) where the wind generation, RT price, and DA price are obtained through forecasting and scenario generation and reduction. The reserve price settled between wind and conventional power energy owners is obtained by using game theory. The expected profit is then calculated by applying the bidding curve obtained to the electricity market using real data obtained from the PJM market. 1. Case 1: RT price is lower than DA price A day is selected in which the RT price has a low standard deviation; and during some hours, the RT price is lower than the DA price. The RT and DA prices obtained from the PJM market and the reserve price settled between wind and conventional power energy owners obtained from the proposed model are shown in Fig. 4.9. The total expected profits to be gained by the wind energy owner from participating in the energy market only and from participating in both the energy and bilateral reserve markets are shown in Fig. 4.10. The increased profit for the wind energy owner from playing a game with the thermal power energy owner in the bilateral reserve market to buy reserve energy is shown in Fig. 4.11. The energy market bidding curves of the wind energy owner participating in the energy market only and in both the energy and bilateral reserve markets for the 7th and 24th hours are shown in Fig. 4.12.

4.2 MULTISTAGE STOCHASTIC PROGRAMMING APPROACH

FIGURE 4.9 Case 1: RT, DA, and settled reserve prices.

FIGURE 4.10 Case 1: expected profits of the wind power energy owner.

135

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

FIGURE 4.11 Case 1: increased profit of the wind power energy owner from participating in both the energy and bilateral reserve markets.

FIGURE 4.12 Case 1: energy market bidding curves of the wind energy owner generated for the 7th and 24th hour.

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137

In this case, when the RT price is lower than the DA price, the thermal units would rather sell power in the DA market than the bilateral reserve market. The reserve price is then set to zero as there is no transaction of reserve in the bilateral reserve market. During hours when the RT price is higher than the DA price, the reserve price is settled between the RT and DA price. Thus, the wind energy owner could buy cheaper energy from the bilateral reserve market to gain a higher profit, as shown in Figs. 4.9 and 4.10. Buying energy from the bilateral reserve market has changed the wind bidding curve, as illustrated in Fig. 4.12. Compared with the case where the wind energy owner only participates in the energy market, if the wind energy owner participates in both the energy and bilateral reserve markets, it will bid at a higher price for low capacities and then decline its price to bid at a lower price for high capacities in the energy market. Moreover, the maximum capacity and price that the wind energy owner wishes to bid in the energy market are higher than if it only participated in the energy market. These observations are expected as the wind energy owner tends to first bid a higher price to cover the cost it will spend to buy reserve power and then to make more profit by selling more power. 2. Case 2: RT price has a high mean values and high standard deviations In this case, a day is chosen in which the mean value and standard deviation of the RT price is high. This means that the RT price is more difficult to predict for market participants. The RT, DA, and settled reserve prices are shown in Fig. 4.13. The total expected profits of the wind energy owner to gain from participating in the energy market and from participating in both the energy and bilateral reserve markets are shown in Fig. 4.14. The increased profit of the wind energy owner from buying reserve energy is shown in Fig. 4.15. The energy market bidding curves of the wind energy owner in participating in the energy market only and in both the energy and bilateral reserve markets for the 18th and 19th hours are shown in Fig. 4.16.

FIGURE 4.13 Case 2: RT, DA, and settled reserve prices.

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FIGURE 4.14 Case 2: Expected profits of the wind power energy owner.

FIGURE 4.15 Case 2: Increase in profits of the wind power energy owner from participating in both the energy and bilateral reserve markets.

In this case, playing a game in the bilateral reserve market to buy reserve energy will not always benefit the wind power energy owner. In Fig. 4.13, the reserve price in the 19th hour is even higher than the RT price. Due to an inaccurate forecasting of the RT price, the wind energy owner buys expensive power from the bilateral reserve market, which results in a lower

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139

FIGURE 4.16 Case 2: Energy market bidding curves of the wind energy owner generated for the 18th and 19th hour.

(negative) expected profit in that hour compared with the case where the wind energy owner does not participate in the bilateral reserve market, as shown in Fig. 4.15. However, most of the time, the wind energy owner gains more profits from playing the game in the bilateral reserve market. The bidding curves in the 18th and 19th hours in Fig. 4.16 show the same features as in Fig. 4.12. The mean values of the increased profits of the wind power energy owner by playing the game in the bilateral reserve market are $22.5 in Case 1 and $244.8 in Case 2. A higher RT price results in more profit. The reserve price also has a tight correlation with the RT price and depends more on its fluctuations. A highly fluctuated RT price is more difficult to predict, which increases the risk of the wind energy owner losing money in the joint energy and bilateral reserve markets. 3. Impact of risk management In the previous cases, both β W and β T were chosen to be 0.5. To study the impact of risk aversion, the expected profit of the wind energy owner and the CVaR for different β W are calculated for Case 2, where β W changes from 0 to 0.9 with an interval of 0.1. The influence of β T of the thermal units is also considered. Three curves are plotted in Fig. 4.17 for β T to be 0.1, 0.5, and 0.9, respectively, where each curve represents the relationship among the expected profit of the wind energy owner, CVaR, and β W. As shown in Fig. 4.17, the three curves have the same trend. When β W increases, the value of CVaR increases, but the expected profit decreases. In the case of β T 5 0.5, the CVaR

140

CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

FIGURE 4.17 Expected profit and CVaR for the wind power energy owner for different β W and β T for Case 2.

increases by approximately 43%, whereas the expected profit decreases by only 1.7% when β W increases from 0 to 0.9. When β T increases but β W remains constant, both the expected profit and CVaR increase. This is expected as when β T increases, the thermal units are willing to take less risk by selling more reserve to the wind energy owner, which increases the expected profit of the wind energy owner. The impact of the risk aversion parameters for Cases 1 and 3 is similar to that of Case 2.

4.3 PRICE-MAKER MODELS—MPEC APPROACH 4.3.1 WHY CONSIDER WIND POWER ENERGY OWNERS AS PRICE-MAKERS? The previous section is based on the assumption that the renewable energy owner’s bidding strategies will not impact the market clearing prices. However, in real-world electricity market, this assumption is not always true. As the Lerner index [18] has stated that the extent to which the bidding prices exceed the marginal cost is a measurement of market power L5

P 2 MC P

(4.13)

where L is the Lerner index, P is the market clearing price, and MC is the marginal cost of this power energy owner.

4.3 PRICE-MAKER MODELS—MPEC APPROACH

141

As we know, the renewable energy which has very low marginal costs and thus has nonnegligible market power and should be considered as price-makers.

4.3.2 PRICE-MAKER MODELS—MPEC APPROACH 4.3.2.1 Introduction to MPEC The strategic bidding model for a price-maker energy owner in the electricity market is different from that for a price-taker energy owner, as the price-maker energy owner has the market power to influence the market clearing prices; and the market clearing price, on the other hand, can influence the clearing power quantity of the price-maker energy owner [23]. To address this dependency, the problem can be formulated as a bilevel model. The upper-level model maximizes the profit of the price-maker energy owner from the electricity market. The lower level model represents the market clearing process. The general formulation of a bilevel optimization problem (4.14) is the following: min f U ðx; yÞ

(4.14.1)

gU ðx; yÞ # 0

(4.14.2)

x:y

Subject to: h ðx; yÞ 5 0 U

(4.14.2)

yAargminy f f ðx; yÞ subject to g ðx; yÞ # 0; h ðx; yÞ 5 0g L

L

L

(4.14.2)

Under the assumption that Karush
Kuhn
Tucker (KKT) conditions are necessary and sufficient for optimality in the lower level problem, we can employ them to replace condition (4.14.2) and transfer the bilevel problem into a singlelevel one (4.15) as follows: minx:y;λ;μ f U ðx; yÞ

(4.15.1)

gU ðx; yÞ # 0

(4.15.2)

Subject to h ðx; yÞ 5 0 U

(4.15.3)

ry f ðx; yÞ 1 λ ry f ðx; yÞ 1 μ ry g ðx; yÞ 5 0

(4.15.4)

gL ðx; yÞ # 0

(4.15.5)

h ðx; yÞ 5 0

(4.15.6)

μ$0

(4.15.7)

L

T

L

T

L

L

μ g ðx; yÞ 5 0 T L

(4.15.8)

where λ and μ represent the dual variables associated to constraints h ðx; yÞ 5 0 and g ðx; yÞ # 0, respectively, in the lower level problem. The lower level problem can be replaced by its first-order optimality condition, such as the widely used KKT condition. Thus, the original bilevel optimization problem can be transformed to a singlelevel problem, which is called a mathematical program with equilibrium constraints L

L

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

(MPEC). The MPEC approach has been widely used to obtain optimal bidding strategies for conventional energy owners. Readers who are interested in learning the MPEC approach application can find detailed information in Refs. [19
22].

4.3.2.2 Multistage Stochastic MPEC The challenge to use the MPEC approach for modeling the bidding strategies of the price-maker renewable energy owners is the uncertainties involving in the modeling. Thus, the stochastic programming and MPEC approach will be combined together to model the bidding strategies of the price-maker renewable energy owner. 1. Uncertainties in the DA market The main uncertainty in the DA market clearing process comes from the load and renewable energy forecast errors. The load and renewable energy forecast errors are unintentional and depend on the forecasting models. These forecast errors are handled by up and down reserves, which are scheduled as functions of the standard deviations of the forecasted renewable power and the total forecasted demand of the system are as follows: D RU t 5 Rt 5 κ

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2 W Þ2 ðσD Þ 1 ðσ jt t

(4.16)

j

D where RU t and Rt are the total up and down reserve scheduled in a time period t, respectively. D W σt and σjt are the standard deviation of the total forecasted demand and forecasted renewable power of the renewable generating unit j in a time period t, respectively. κ is a parameter decided by the system operator and is selected to be 3 in this chapter. The reserve clearing process is assumed to be independent from the DA energy clearing and is modeled below:

minRUit ;RDit

X

RD D U λRU it Rit 1 λit Rit

(4.17.1)

i

Subject to Umax 0 # RU ; ’i; t it # Ri

0 # RD it X

# RDmax ; i

’i; t

(4.17.2) (4.17.3)

U RU it 5 Rt ; ’t

(4.17.4)

D RD it 5 Rt ; ’t

(4.17.5)

i

X i

D U D where RU it and Rit are the decision variables of the optimization problem (4.17). Rit and Rit represent the up and down reserve scheduled for the conventional energy owner i in a time period t, respectively. RUmax and RDmax are the parameters which represent the maximum up and i i down reserve that can be provided by the conventional energy owner i, respectively. 2. Uncertainties in the RT market In the RT market clearing process, the main uncertainties come from the intentional misscheduling of load and wind power production deviation. If the power purchasers predict that the RT price will be lower than the DA price, they may intentionally underschedule load in

4.3 PRICE-MAKER MODELS—MPEC APPROACH

143

the DA market and buy extra demand in the RT market at the RT price. Otherwise, if the power purchasers expect that the RT price will be higher than DA price, they may intentionally overschedule load and sell any extra energy procured in the DA market back to the system at the RT price. The behavior of these purchasers will increase the RT price volatility. For a wind power energy owner, the deviation of the actual production from that scheduled in the DA market will be balanced in the RT market. If the actual production is less than the DA scheduled, the wind energy owner will have to buy the deviated power at the RT price. Otherwise, the wind energy owner can bid the extra power into the RT market and get paid for the accepted power at the RT prices. Overall, the problem to determine the optimal bidding strategy for a strategic wind power energy owner is formulated as a bilevel stochastic optimization model. Fig. 4.18 shows the structure of the stochastic bilevel model, which consists of an upper
lower problem and two lower level problems. The upper-level problem maximizes the total profit obtained by the wind power energy owner in both DA and RT markets to determine its optimal bidding curves in both markets. The maximal profit and the resulting bidding curves depend on the cleared information (LMPs and scheduled energy) in the DA and RT markets, which will be obtained in the lower level problems. One lower level problem collects the bidding information of each energy owner and runs the DA market clearing process to generate the DA LMPs and energy schedule for each energy owner. The other lower level problem is carried out after the DA clearing to deal with the RT energy deviation. This RT market clearing process depends not only on the RT offering curve of each energy owner but also on the DA energy schedule. Both the RT LMPs and RT energy schedule will be announced after the clearing.

Upper-level problem Minimize: The negative profit of the wind power producer Determine: Day-ahead offering price of the wind power producer

Real-time offering price of the wind power producer

Lower-level Problem Day-ahead Market Cleaning Process Determine: Day-ahead LMP and cleared wind power

Day-ahead market scheduling

Lower-level Problem Real-time Market Clearing Process Determine: Real-time LMP and cleared wind power

FIGURE 4.18 The Structure of the proposed bilevel stochastic optimization model.

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

4.3.3 MATHEMATICAL FORMULATION AND CONVERSION 4.3.3.1 Bilevel Problem 1. Upper-level problem The upper level problem (4.18) maximizing the total profit of the wind power energy owner in the DA and RT market is shown below: minΞ

X jtω

"

πω

# X DA WD RT WR ð2 λðm:jAΨW Þ tω pbjtω Þ 2 λðm:jAΨW Þ tω Pjtω 2 βðζ 2 m

b

m

1 X πω ηω Þ 12α ω

(4.18.1)

Subject to WD CapD 0 # λWD ; ’j; b $ 2; t; ω ðb21Þjtω # λbjtω # λ

(4.18.2)

CapR 0 # λWR ; ’j; t; ω jtω # λ

(4.18.3)

ηω $ 0; ’ω

"

ζ 2 ηω #

#

X X WD RT WR ð2 λDA ðm:jAΨWm Þ tω pbjtω Þ 1 λðm:jAΨWm Þ tω Pjtω ; ’ω jt

(4.18.4) (4.18.5)

b

where m : wAΨW m denotes the bus where the wind power generating unit w is located; and WR ,λ , ζ, ηω , ΞD, ΞR } are the set of all decision variables of the problem (4.18), where Ξ 5 {λWD bjtω jtω ΞD and ΞR are the sets of all the decision variables in the lower level problems (4.19) and (4.20), respectively, and will be defined later. It should be noted that the total formulation for (4.18) consists of (4.18.1)
(4.18.5) and in addition the total formulations of (4.19) and (4.20). λWD bjt represents the offer price of block b of the wind generating unit j in a time period t in the DA market. λWR jt represents the offer price of the wind generating unit j in a time period t in the RT market.ζ and η are auxiliary variables used to compute the CVaR. These variables, if augmented with a subscript ω, represent their realization in a scenario ω. λCapD and λCapR are constants which represent the cap bidding price in the DA and RT markets, respectively. The objective function (4.18.1) minimizes the sum of two terms: (1) The negative expected profit of the wind power energy owner, which is the negative revenue obtained in the DA market plus the negative revenue from the RT market, and (2) the negative CVaRα multiplied by the weighting factor β. In (4.18.1), λDA and pWD bjtω are the variables determined in the ðm:jAΨW m Þtω RT lower level problem (4.19); while λðm:jAΨW Þtω and PWR are the variables determined in the lower jtω m level problem (4.20). The constraints (4.18.2) and (4.18.3) enforce the acceptable DA and RT bidding prices and nondecreasing DA bidding curves of the wind power energy owner. The constraints (4.18.4) and (4.18.5) are used to compute the CVaR, which is used in this paper as a measurement for risk. The CVaRα represents the expected profit associated with the (1 2 α) 3 100% worst scenarios. The weighting parameter β is set by the wind power energy owner to model the tradeoff between profit and risk. If the wind power energy owner is willing to gain less profit in order to bear less risk, it will select a higher value for β. 2. Lower level problem—DA market clearing The lower level problem (4.19) is formulated as follows to represent the DA market clearing process.

4.3 PRICE-MAKER MODELS—MPEC APPROACH

minΞD

X

CD λCD bitω pbitω 1

X

bi

WD λWD bjtω pbwtω 2

bj

X

LD λLD ldt pldtω

145

(4.19.1)

ld

Subject to X lðdAΨ D mÞ

pLD ldtω 2

X bðiAΨ Im Þ

pCD bitω 2

X bðjAΨ W mÞ

pWD bjtω 1

X

  D DA Bmn δD mtω 2 δntω 5 0:λmtω ; ’m; t; ω

(4.19.2)

nAΦm

CDmax Cmax 0 # pCD :μbitω ; μCmin bitω # pbit bitω ; ’b; i; t; ω

(4.19.3)

Wf Wmax Wmin 0 # pWD bjtω # Pjtb :μbjtω ; μbjtω ; b 5 1; ’j; t; ω

(4.19.4)

Wf Wf Wmax Wmin 0 # pWD bjtω # Pjtb 2 Pjtðb21Þ :μbjtω ; μbjtω ; ’b $ 2; j; t; ω

(4.19.5)

LDmax Lmax :μldtω ; μLmin 0 # pLD ldtω # pldt ldtω ; ’l; d; t; ω

(4.19.6)





D Dmin max Dmax jBmn δD mtω 2 δntω j # Cmn :β mntω ; β mntω ; ’m; nAΦm ; t; ω

(4.19.7)

D max Dmax Dmin δmin m # δmtω # δm :θmtω ; θmtω ; ’m; t; ω

(4.19.8)

D1 δD mtω 5 0:θtω ; m 5 1; ’t; ω

(4.19.9)

D D CD WD LD where PWf jtb is the bth element in the set WP(j,t); and Ξ 5 {pbitω ,pbjtω ,pldtω ,δmtω , DA Cmax Cmin Wmax Wmin Lmax Lmin Dmax Dmin Dmax Dmin D1 λmtω ,μbitω ,μbitω ,μbjtω ,μbjtω ,μldtω ,μldtω ,β mntω ,β mntω ,θmtω ,θmtω ,θtω } is the set of all decision variables of the problem (4.19). The decision variable set contains both primal and dual variables, where the dual variables are defined following the colon in each constraint. pCD bit and pWD bjt represent the power produced in block b by the conventional power energy owner i and the wind power energy owner j in a time period t in the DA market, respectively. pLD ldt is the power bought by demand d in block l in a time period t in the DA market. δD is voltage mt angle of bus m in a time period t in the DA market. λDA mt is the DA LMP at bus m in a time period t. λCD bitω is the random variable which represents the offer price of block b of the conventional power energy owner i in a time period t in the DA market. Other variables include pCDmax and bit CDmax pLDmax , where p is the capacity of block b of the conventional power energy owner i in a ldt bit time period t in the DA market and pLDmax is the capacity of block l of the demand d in a time ldt max max period t in the DA market. The constants are Bmn , Cmn , δmin m , and δm , where Bmn is the max imaginary part of the admittance of line m
n, Cmn is the transmission capacity of lines m
n min δmax m and δ m are the upper and lower limits of voltage magnitude of bus m, respectively. The objective function (4.19.1) minimizes the total cost of energy offered by both conventional and wind power energy owners minus the revenue from supplying demand in each period and each scenario. Eq. (4.19.2) enforces the DA power balance at each bus. The constraints (4.19.3)
(4.19.5) represent the limits of the power offered by the conventional and wind power units in each block. The constraint (4.19.6) represents the bounds of the demand in each block. The constraint (4.19.7) imposes the transmission capacity limits of each power line. The voltage angle limits of each bus are expressed in the constraint (4.19.8). The reference bus

146

CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

is selected in (4.19.9). Note that if the uncertainty of the bidding prices of the conventional 0 CD power energy owner i is not considered, then λCD bitω 5 λbitω0 , ’ω; ω . 3. Lower level problem—RT market clearing The lower level problem (4.20) modeling the RT market clearing process is formulated below: minΞR

X X X X CR2 CR2 WR U RU U LR ðλCR1 PCR1 Pitω Þ 1 λWR ðλRU λLR it itω 2 λit jtω Pjtω 1 it ritω 1 λit ritω 2 dtω Pdtω i

Subject to

j

X dAΨ D m

X

ðPLR dtω 1

pLD ldtω Þ 2

l

X bðjAΨ W mÞ

pWD bjtω

2

X

i

X bð

iAΨ Im

PWR jtω

X

X

CR2 U D ðPCR1 itω 2 Pitω 1 ritω 2 ritω Þ 2

iAΨ Im

  Bmn δRmtω 2 δRntω 5 0:λRT mtω ; ’m; t; ω

nAΦm

(4.20.3)

max2 max2 0 # PCR2 :μitω ; μmin2 itω # Pi itω ; ’i; t; ω X max CR1 2 pCD PCR1 itω # Pi bit :μitω ; ’i; t; ω

(4.20.4)

X

(4.20.5)

b CR2 pCD bit :μitω ; ’i; t; ω

b Wf WR pWD bjt # Pjtω # Pjtω 2

X

WRmax pWD ; μWRmin ; ’j; t; ω bjt :μjtω jtω

(4.20.6) (4.20.7)

b

LRf LRf LRmax LRmin 0 # PLR dtω # Pdtω ; : μdtω ; μdtω ; ’d; t; ω; if Pdtω $ 0 LRf LR LRmax LRmin PLRf dtω # Pdtω # 0; : μdtω ; μdtω ; ’d; t; ω; if Pdtω , 0 U Umax Umin 0 # rU itω # Rit :ϕitω ; ϕitω ; ’i; t; ω

jBmn

(4.20.2)

max1 max1 0 # PCR1 :μitω ; μmin1 itω # Pi itω ; ’i; t; ω

b

(

X

1

jAΨ W m

PCR2 itω # 2

Þ

pCD bitω 2

(4.20.1)

d



D Dmax Dmin 0 # rD itω # Rit :ϕitω ; ϕitω ; ’i; t; ω  max Rmax δRmtω 2 δRntω j # Cmn :β mntω ; β Rmin mntω ; ’m; nAΦm ; t; ω

δmin m

# δRmtω

Rmax Rmin # δmax mt :θmtω ; θmtω ;

’m; t; ω

δRmtω 5 0:θR1 tω ; m 5 1; ’t; ω

(4.20.8) (4.20.9) (4.20.10) (4.20.11) (4.20.12) (4.20.13)

where Ξ is the set of all primal and dual decision variables in the problem (4.20.1), where R RT CR2 WR LR U D max1 min1 max2 min2 CR1 CR2 ΞR 5 {PCR1 itω , Pitω , Pjtω , Pdtω , ritω , ritω , δmtω ,λmtω , μitω , μitω , μitω , μitω , μitω , μitω , Rmax Rmin Rmax Rmin WRmax WRmin LRmax LRmin Umax Umin Dmax Dmin μjtω , μjtω , μdtω , μdtω , ϕitω , ϕitω , ϕitω , ϕitω , β mntω , β mntω , θmtω , θmtω , θR1 tω }. CR2 PCR1 and P represent the increased and decreased power of the conventional power it it energy owner i in a time period t in the RT market, respectively. PWR is the rescheduled jt power of the wind generating unit j in a time period t in the RT market. PLR dt is accepted deviD ation of the demand d in a time period t in the RT market. rU and r represent up and down it it reserve deployed by the conventional power energy owner i in a time period t in the RT market, respectively. δRmt is voltage angle of bus m in a time period t in the RT market. λRT mt is the RT LMP at bus m in a time period t. R

4.3 PRICE-MAKER MODELS—MPEC APPROACH

147

LRf Wf The random variables include PWf jt and Pdt , where Pjt is the forecasted wind power production of the wind generating unit j in a time period t and PLRf dt is the forecasted deviation of demand d in a time period t. max1 The constants include Pmax1 ,Pmax2 , and Pmax and Pmax2 represent the i i i , where Pi i maximum increased and decreased power that can be provided by the conventional power energy owner i, respectively, and Pmax is the maximum power output of the conventional power i energy owner i. The objective function (4.20.1) minimizes the total cost of redispatching energy and deploying reserve minus the revenue from the deviation demand in each period and each scenario. Eq. (4.20.2) enforces the RT power balance at each bus. The constraints (4.20.3)
(4.20.6) limit the power of each conventional generator that can be sold into the RT market. The amount of wind power that can be sold in the RT market in each scenario is constrained by the forecasted wind power in the scenario, as described by (4.20.7). The bounds of the deviated demand in each scenario are represented by the constraint (4.20.8). The up and down reserves deployed are bounded by the scheduled up and down reserves in the DA market, respectively, as expressed in (4.20.9) and (4.20.10), respectively. The constraint (4.20.11) imposes the transmission capacity limits of each power line. The voltage angle limits of each bus are expressed in the constraint (4.20.12). The reference bus is selected in (4.20.13). The wind power trading in the RT market is different from the conventional power energy owners due to the uncertainty in the wind power production. The wind power energy owner may fail to fulfill the productions settled in the DA market and is forced to correct its negative deviation in the RT market. Meanwhile, the wind power energy owner can bid the extra power into the RT market if the deviation is positive. The rescheduled wind power is constrained by (4.20.8) for both cases. As the DA market is cleared prior to the RT market, the decision D CD WD LD variables RU it and Rit of the problem (4.18) and pbitω , pbjtω , and pldtω of the problem (4.19) are considered to be parameters in the problem (4.20).

4.3.3.2 Model Conversion The problem (4.19) is transferred by its KKT condition as follows where (4.21.5) is the same as (4.19.2): DA Cmax Cmin I λCD bitω 2 λmtω 1 μbitω 2 μbitω 5 0; ’b; iAΨ m ; t; ω

λWD bwtω X nAΦm

2 λDA mtω

1 μWmax bjtω

2 μWmin bjtω

5 0;

’b; jAΨ W m ; t; ω

DA Lmax Lmin D 2 λLD ldt 1 λmtω 1 μldtω 2 μldtω 5 0; ’l; dAΨ m ; t; ω   Dmax Dmax Dmin Dmin Dmax Dmin D1 DA Bmn λDA mtω 2 λntω 1 β mntω 2β nmtω 1β mntω 2β nmtω 1 θmtω 2 θmtω 1 θtω jm51 5 0; ’m; t; ω

X lðdAΨ D mÞ

pLD ldtω 2

X bðiAΨ Im Þ

pCD bitω 2

X bðjAΨ W mÞ

pWD bjtω 1

X

  D DA Bmn δD mtω 2 δntω 5 0:λmtω ; ’m; t; ω

(4.21.1) (4.21.2) (4.21.3) (4.21.4) (4.21.5)

nAΦm

Cmin 0 # pCD bitω \μbitω $ 0; ’b; i; t; ω  Cmax 0 # pCDmax 2 pCD bit bitω \μbitω $ 0; ’b; i; t; ω



Wmin 0 # pWD bjt \μbjtω

$ 0; ’b; w; t; ω

(4.21.6) (4.21.7) (4.21.8)

148

CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

  WD Wmax 0 # PWf jtb 2 pbjtω \μbjtω $ 0; b 5 1; ’j; t; ω   Wf WD Wmax 0 # PWf jtb 2 Pjtðb21Þ 2 pbjtω \μbjtω $ 0; ’b $ 2; j; t; ω

(4.21.9) (4.21.10)

Lmin 0 # pLD ldtω \μldtω $ 0; ’l; d; t; ω   Lmax 2 pLD 0 # pLDmax ldt ldtω \μldtω $ 0; ’l; d; t; ω  max   Dmin D 0 # Cmn 1 Bmn δD mtω 2 δntω \β mntω $ 0; ’m; nAΦm ; t; ω  max   Dmax D 0 # Cmn 2 Bmn δD mtω 2 δntω \β mntω $ 0; ’m; nAΦm ; t; ω

(4.21.11) (4.21.12) (4.21.13) (4.21.14)

D Dmin 0 # ðδmin m 1δmtω Þ\θmtω

$ 0; ’m; t; ω

(4.21.15)

D Dmax 0 # ðδmax m 2δmtω Þ\θmtω

$ 0; ’m; t; ω

(4.21.16)

where \ denotes complementarity operation. Similarly, the problem (4.20) is transferred by its KKT condition as follows where (4.22.8) is the same as (4.20.2): max1 CR1 I λCR1 2 λRT 2 μmin1 it mtω 1 μitω itω 1 μitω 5 0; ’iAΨ m ; t; ω

2 λCR2 it

1 λRT mtω

1 μmax2 2 μmin2 itω itω

1 μCR2 itω

5 0;

(4.22.1)

’iAΨ Im ; t; ω

(4.22.2)

RT WRmax 2 μWRmin 5 0; ’jAΨ W λWR jtω 2 λmtω 1 μjtω jtω m ; t; ω

X nAΦm



Bmn λRT mtω X dAΨ D m

2 λRT mtω

(4.22.4)

RT Dmax Dmin I λRD it 1 λmtω 1 ϕitω 2 ϕitω 5 0; ’iAΨ m ; t; ω

(4.22.5)

2 λLR dtω 2 λRT nt

ðPLR dtω 1

2

1 λRT mtω

pLD ldtω Þ 2

X

PWR jtω

X bð

1

jAΨ W m

Þ X

pCD bitω 2

iAΨ Im

2 ϕUmin itω

1 μLRmax itω

Rmax Rmin 1 β Rmax mntω 2β nmtω 1β mntω

X l

1 ϕUmax itω

(4.22.3)

5 0; ’iAΨ Im ; t; ω

λRU it

2 μLRmin itω

2 β Rmin nmtω X

Bmn δRmtω

5 0;

1 θRmax mtω

2

θRmin mtω

2 δRntω



5 0:λRT mtω ;

(4.22.6)

1 θR1 t jm51

CR2 U D ðPCR1 itω 2 Pitω 1 ritω 2 ritω Þ 2

iAΨ Im





’l; dAΨ D m; t

’m; t; ω

5 0; ’m; t; ω

X bð

jAΨ W m

Þ

(4.22.7)

pWD bjtω (4.22.8)

nAΦm min1 0 # PCR1 itω \μitω $ 0; ’i; t; ω

0 # ðPmax1 i

max1 2 PCR1 itω Þ\μitω

$ 0; ’i; t; ω

min2 0 # PCR2 itω \μitω $ 0; ’i; t; ω



0 # ðPmax2 i

2 0 # Pmax2 i 0#

X b

max2 2 PCR2 itω Þ\μitω

X



$ 0; ’i; t; ω

CR1 \μCR1 pCD bit 2 Pitω itω $ 0; ’i; t; ω

(4.22.9) (4.22.10) (4.22.11) (4.22.12) (4.22.13)

b

 CR2 pCD \μCR2 bit 2 Pitω itω $ 0; ’i; t; ω

(4.22.14)

4.3 PRICE-MAKER MODELS—MPEC APPROACH

0#

X

 WR WRmin pWD $ 0; ’j; t; ω bjt 1 Pjtω \μjtω

149

(4.22.15)

b

  X WR WRmax pWD $ 0; ’j; t; ω 0 # PWf bjt 2 Pjtω \μjtω jtb 2

(4.22.16)

b

(

LRmin $ 0; ’d; t; ω; if PLRf 0 # PLR dtω \μdtω dtω $ 0 LR LRmin $ 0; ’d; t; ω; if PLRf 0 # ðPLRf dtω 2 Pdtω Þ\μdtω dtω , 0

(

LR LRmax $ 0; ’d; t; ω; ifPLRf 0 # ðPLRf dtω 2 Pdtω Þ\μdtω dtω $ 0 LRmax $ 0; ’d; t; ω; ifPLRf 0 # PLR dtω \μdtω dtω , 0

(4.22.17)

(4.22.18)

Umin 0 # rU itω \ϕitω $ 0; ’i; t; ω

(4.22.19)

U Umax $ 0; ’i; t; ω 0 # ðRU it 2 ritω Þ\ϕitω

(4.22.20)

Dmin 0 # rD itω \ϕitω $ 0; ’i; t; ω

(4.22.21)

D Dmax 0 # ðRD it 2 ritω Þ\ϕitω $ 0; ’i; t; ω

(4.22.22)

 max   0 # Cmn 1 Bmn δRmtω 2 δRntω \β Rmin mnt $ 0; ’m; nAΦm ; t

(4.22.23)

 max   0 # Cmn 2 Bmn δRmtω 2 δRntω \β Rmax mnt $ 0; ’m; nAΦm ; t

(4.22.24)

R Rmin 0 # ðδmin m 1δmtω Þ\θmtω $ 0; ’m; t; ω

(4.22.25)

R Rmax 0 # ðδmax m 2δmtω Þ\θmtω $ 0; ’m; t; ω

(4.22.26)

4.3.3.3 Reform MPEC as MILP The MPEC problem having the objective function (4.19.1) and subject to the constraints (4.19.2)
(4.19.5), (4.21), and (4.22) is a nonlinear programming problem, where the nonlinearities come from the following three main sources. By linearizing the nonlinear terms, the MPEC problem is converted into a MILP problem, which can be solved effectively. P WD 1. The jb λDA ðm:jAΨ Wm Þtω pbjtω term in (4.19.1). According to (4.21.2), (4.21.3), (4.21.9), (4.21.10) and the strong duality theorem, this term can be linearized to (4.23). P

P WD WD Wmax Wmin WD λDA ðm:jAΨ Wm Þtω pbjtω 5 jb ðλbjtω 1μbjtω 2 μbjtω Þpbjtω P P CDmax Cmax P LD LD P CD 5 2 bi λCD μbitω 1 ld λldt pldtω 2 ld pLDmax μLmax bitω pbitω 2 ldt ldtω 2 bi pbit     P P Dmax Dmin max Dmax min Dmin max mðnAΦm Þ Cmn β mntω 1 β mntω 2 m δm θmtω 2 δm θmtω jb

(4.23)

150

2. The

CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

P j

WR λRT ðm:jAΨ W Þtω pjtω term in (4.20.1). m

Similar to (4.23), according to (4.22.3), (4.22.15), (4.22.16) and the strong duality theorem, this term can be linearized to (4.24). P j

WR λRT ðm:jAΨ W Þtω pjtω 5 m

P

WD WRmax j ðλjtω 1μjtω

2 μWRmin ÞpWR jtω jtω

P P P LR LR CR2 CR2 RD D U 5 2 i ðλCR1 PCR1 Pitω Þ 2 i ðλRU it it ritω 1 λit ritω Þ 1 itω 2 λit d λdtω Pdtω P P CR1  max P CD  2 i ðPmax1 μmax1 1 Pmax2 μmax2 Pi 2 b pbit itω itω Þ 2 i i i μitω 8P LRf LRmax  < d PLRf dtω μdtω ; if Pdtω $ 0 P  Umax U P CR2 P CD Dmax D ϕ 2 p 2 R 1 ϕ R 2 i μitω it it itω itω b bit i : P LRf LRmin LRf d Pdtω μdtω ; if Pdtω , 0  Rmax  P  max Rmax  P min Rmin max 2 mðnAΦm Þ Cmn β mntω 1 β Rmin mntω 2 m δm θmtω 2 δm θmtω

(4.24)

P CD CR2 There is still a nonlinear term ðμCR1 itω 2 μitω Þ b pbit in (4.24). It can be linearized using the method in [24]. 3. The MPEC model includes the nonlinear complementarity constraints (4.21.6)
(4.21.16) and (4.22.9)
(4.22.26). The complementarity constraint in the form of 0 # P\Q $ 0 can be replaced by the following formulation. P $ 0; Q $ 0; P # μM; Q # ð1 2 μÞM; μA0; 1

(4.25)

where M is a sufficiently large constant. Its value can be different for different complementarity constraints. Usually, M is set to be (dual variable 1 1) 3 100.

4.3.4 CASE STUDY The proposed model is tested using the IEEE Reliability Test System, which has 10 conventional power energy owners and one wind power energy owner. The detailed information for this test system can be found in Appendix. The total installed wind capacity is 1000 MW, which is approximately 23% of the total installed generation capacity in the system. In all case studies except for those explained specifically, all wind power generating units are located at bus 8 and the wind power energy owner is a strategic player in both the DA and the RT markets. The wind power data is obtained from the National Renewable Energy Laboratory website [17]. The historical data of the RT demand is obtained from the PJM market [11]. The parameters α and β in (4.18.1) are set to be 0.95 and 0, respectively. λCapD and λCapR are set to be $1000/MW h. In all of the case studies, the MILP problem is solved using Gurobi 5.5 in MATLAB [25]. The ARIMA model is used to generate 5000 scenarios of wind power and RT demand respectively. Then the scenario reduction method is applied to reduce the number of scenarios for the wind power and RT demand. Table 4.4 lists the information (values and probabilities)

4.3 PRICE-MAKER MODELS—MPEC APPROACH

151

Table 4.4 Scenario Information Wind Power Scenario (MW)

Probability

RT Demand Scenario

Probability

221.76 309.21 393.67 466.65 521.90 652.34 765.52 803.24

0.021 0.052 0.0917 0.213 0.101 0.391 0.128 0.0848

0.712 0.784 0.890 0.913 0.956 1.089 1.125 1.236

0.038 0.071 0.092 0.158 0.175 0.126 0.201 0.1390

FIGURE 4.19 DA and average RT LMPs at bus 8 and average selling price of the wind power energy owner for different wind power penetration levels.

of the resulting 8 wind power scenarios and 8 RT demand scenarios for a certain hour, where a RT demand scenario is described by the ratio of the forecasted RT demand to the DA demand. 1. The impact of wind power penetration level In this study, the installed wind power capacity is changed from 200 MW (5% penetration) to 1600 MW (33% penetration) with an increment of 200 MW. No capacity limits are imposed on the transmission constraints. The DA and average RT LMPs of the 64 scenarios at bus 8 and the average selling price of the wind power energy owner are shown in Fig. 4.19 for different wind power penetration levels.

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Due to the fact that wind power has no fuel cost, selling wind power into the market will lower the LMP. As shown in Fig. 4.19, both the DA LMP and the average RT LMP decrease as the penetration level of wind power increases. However, due to the effect of the uncertainties in the RT market, the RT LMP is more sensitive than the DA LMP to the changes of the demand and wind power supply. Therefore, the rate of decrease of the average RT LMP is much greater than that of the average DA LMP. When the penetration level is relatively low (5
15%), the average selling price of wind power increases with the increase of the wind power penetration level. At these penetration levels, as the RT LMP is much higher than the DA LMP, selling more wind power into the RT market will increase the average selling price. However, when the penetration level is relatively high (15
32%), the average selling price of wind power decreases with the increase of the wind penetration level. At these penetration levels, as the RT LMP decreases faster than the DA LMP and is even lower than the DA LMP when the penetration level becomes higher than 25%, selling the extra wind power into the RT market will decrease the average selling price. As can been seen more clearly from Fig. 4.20, the profit of the wind power energy owner in the DA market stays almost the same first, then increases dramatically, and finally stabilizes at a certain value when the penetration level increases. The cause of this result is explained below. When the penetration level of wind power is below 18%, the wind power energy owner does not have enough market power to influence the DA LMPs. Therefore, the DA LMP stays constant when the wind power penetration level changes within 18%. Moreover, when the wind power penetration level changes, the load demand in the DA market does not change. Therefore, the amounts of power bid into the DA market by the wind energy owner and the conventional generating units almost do not change. As a result, the profit of the wind energy owner in the DA market almost does not change. The results also show

FIGURE 4.20 DA, RT, and total profits of the wind power energy owner for different penetration levels of wind power.

4.3 PRICE-MAKER MODELS—MPEC APPROACH

153

that the wind power energy owner intends to sell more power into the RT market to gain more profit because the RT LMP is higher than the DA LMP. This, however, results in a decline of the RT price. As the wind power penetration level goes higher, the DA LMP decreases dramatically; but the profit of the wind energy owner obtained from the DA market still increases because more power is sold into the DA market. However, the system’s ability to consume wind power in the DA market is limited. As the wind power penetration level reaches 25%, further increasing the penetration level only results in a slight change in the DA LMP and a slight increase in the wind power sold into the DA market. Therefore, the DA profit curve is almost constant as the wind power penetration level goes beyond 25%. The profit in the RT market increases first until the penetration level reaches 18%. Beyond this penetration level, the RT profit decreases when the DA profit increases and then increases but is always below the DA profit when the penetration level exceeds 21%. As the RT price decreases to a certain level, the wind power energy owner will not sell all of its extra power into the RT market to prevent further decrease of the RT price and possible decrease of the total profit. As a result, some of the wind power generation capacity will be wasted as the wind power penetration level goes high. Therefore, it is important for the wind power energy owners to choose their installation capacities in a certain system. As Fig. 4.20 shows, the total profit of the wind energy owner increases with the wind penetration level. 2. The impact of transmission constraints In this study, the wind power capacity is 1000 MW. The proposed model is solved to obtain the optimal bidding strategy for the wind energy owner for four cases: (a) no capacity limits are imposed on any transmission lines; and a capacity limit of (b) 190 MW, (c) 100 MW, and (d) 30 MW is imposed on the transmission line 8
9. Fig. 4.21 compares the RT LMPs at bus

FIGURE 4.21 RT LMPs at bus 8 in Cases (a), (b) and (c) in different scenarios.

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CHAPTER 4 OPTIMUM BIDDING OF RENEWABLE ENERGY

Table 4.5 Profits of the Wind Energy Owner and DA LMP at Bus 8 in Different Cases Case

Profit in DA Market ($)

DA LMP at Bus 8 ($/MW h)

Profit in RT Market ($)

Total Profit ($)

a b c d

4012.71 3098.45 1796.23 1756.37

10.66 10.68 11.09 10.66

1674.7518 2813.6867 3470.8958 2151.8819

5687.4618 5912.1367 5267.1258 3908.2519

8 in different scenarios for Cases (a), (b), and (c). Case (d) is not plotted because the RT LMPs of Cases (c) and (d) are the same. The profits of the wind power energy owner in the DA and RT markets, the total profit of the wind energy owner, and the DA LMP at bus 8 for different cases are given in Table 4.5. When no capacity limits are imposed on the transmission lines, the power flow through line 8
9 is 75 MW in the DA market and maximally 292 MW in the RT market. As shown in Table 4.5, when the transmission capacity of line 8
9 is limited to 190 MW [Case (b)], the RT LMP in Scenario 64 is higher than that in Case (a). Although the amount of wind power sold into the DA market in Case (b) is lower than that in Case (a), the total profit in Case (b) is higher than that in Case (a) owing to the higher RT LMPs. Limiting the transmission capacity to 100 MW in Case (c) will further increase the RT LMP when compared to Case (b), as shown in Fig. 4.21. As a consequence, less amount of wind power will be sold in the DA market in Case (c) when compared with Cases (a) and (b). However, the increase of the RT LMP will not be able to compensate for the profit loss due to the decrease of the wind power sold in the DA market caused by the transmission congestion. In Case (d), the profit in the DA market increases slightly compared with Case (c) owing to the increase of the DA LMP. However, the total profit decreases dramatically compared with the previous three cases due to the small transmission capacity limit. Based on the four case studies, it can conclude that transmission congestion sometimes can be beneficial to wind power energy owners and can be used as a strategic mechanism to increase their profits. 3. The impact of risk management In this study, the installed wind power is 1000 MW. The proposed model is solved for different values of β in (3a). As shown in Fig. 4.22, the expected total profit of the wind power energy owner decreases and the CVaR increases with the increase of β. The wind bidding capacity in the DA market decreases when β increases, as plotted in Fig. 4.23. These results are expected. A higher β indicates that the wind power energy owner is willing to take less risk by selling less power into the DA market. As a consequence, the chance of the wind

4.3 PRICE-MAKER MODELS—MPEC APPROACH

155

FIGURE 4.22 The expected total profit and CVaR of the wind power energy owner for different values of β.

FIGURE 4.23 The wind power sold into the DA market for different values of β.

power energy owner to counter a negative deviation in the RT market decreases. The bidding curves of the wind power energy owner in the DA market for three selected values of β in Fig. 4.24 also show this trend: the bidding capacity decreases while the bidding price increases as β increases. Moreover, the expected total profit and bidding strategy are quite sensitive to the value of β when it is between 0 and 1.0, and its value should be selected carefully by the wind power energy owner. These results show the significance of risk management for the wind power energy owner to obtain the optimal bidding strategy.

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FIGURE 4.24 The DA bidding curves of the wind power energy owner for different values of β.

4.4 SUMMARY AND CONCLUSION This chapter starts with the introduction of the challenges and opportunities faced by the renewable energy owners participating in the electricity market. The main obstacle for renewable energy owners to compete with conventional power owners in the electricity market is the uncertainty in their production. However, as the PPA price goes down, the necessity for renewable energy owners to participate in the electricity market increases. Thus, appropriate models need to be built to help renewable energy owners benefit from electricity market. In this chapter, the stochastic programming model is introduced to maximize the profit of the renewable energy owners and handle the uncertainties, for example, electricity market prices, renewable energy production, and others faced by renewable energy owners. The risk management CVaR is applied to managing the trade risk associated with uncertainties. This chapter also considers two different types of renewable energy owners: Price-takers and price-makers. For the price-takers, their bidding strategies do not affect the market prices which can be considered as forecasted values in the stochastic model. Section 4.2.1 presented the basic stochastic programming model for this type of renewable energy owners and showed the readers how to build the bidding strategies. Section 4.2.2 introduced an approach to reduce the trading risks of the renewable energy owners by purchasing additional power from conventional energy owners. This approach has been analyzed using stochastic programming and game theory. Section 4.3 proposed the model to solve the bidding strategies for the price-maker renewable energy owners whose bidding strategies have nonnegligible impact on the market clearing prices. The optimal bidding strategy of the renewable energy owner has been generated by a bilevel stochastic optimization model, which has been converted into a singlelevel MILP problem using the duality theory and KKT condition to facilitate solving the model.

REFERENCES

157

FIGURE 4.25 24-Bus system.

4.5 APPENDIX: IEEE RELIABILITY TEST SYSTEM This appendix illustrates the 24-bus system that used to exam the model in Section 4.3. This 24-bus system is based on the single-area version of the IEEE Reliability Test System [26]. See Figure 4.25.

REFERENCES [1] Wiser R, Bolinger M. 2014 wind technologies market report, Lawrence Berkeley National Laboratory. [Online] Available at: ,http://energy.gov/sites/prod/files/2015/08/f25/2014-Wind-Technologies-MarketReport-8.7.pdf.; 2015

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[2] Porter K, Starr K, Mills A. Variable generation and electricity market [Online] Available at: ,http:// uvig.org/wp-content/uploads/2015/05/VGinmarketstableApr2015.pdf.; 2015. [3] Morales JM, Conejo AJ, Perez-Ruiz J. Short-term trading for a wind power energy owner. IEEE Trans Power Syst 2010;25:553
64. [4] Olsson M, So¨der L. Generation of regulating power price scenarios, Proceedings of the. IEEE PMAPS’04 Conference, Ames, USA, 2004. [5] Høyland K, Wallace SW. Generating scenario trees for multistage decision problems. Manage Sci 2001;47:295
307. [6] Pflug GC. Scenario tree generation for multiperiod financial optimization by optimal discretization. Math Program 2001;89:251
71. [7] Dupacov´a J, Gro¨we-Kuska N, Ro¨misch W. Scenario reduction in stochastic programming: an approach using probability metrics. Math Program (ser. A 2003;95:493
511. [8] Gro¨we-Kuska N, Heitsch H, Ro¨misch W. Scenario reduction and scenario tree construction for power management problems. Proceedings of the IEEE power tech conference, Bologna, Italy, 2003. [9] Morales JM, Pineda S, Conejo AJ, Carrion M. Scenario reduction for futures market trading in electricity markets. IEEE Trans Power Syst 2009;24:878
88. [10] Conejo AJ, Contreras J, Espı´nola R, Plazas MA. Forecasting electricity prices for a DA pool-based electric energy market. Int J Forecast 2005;21:435
62. [11] PJM market [Online]. Available at: ,http://www.pjm.com/.. [12] Jorion P. Value at risk. New York: McGraw-Hill; 1997. [13] Rockafellar RT, Uryasev S. Optimization of conditional value-at-risk. J. Risk 2000;2:21
4. [14] Dai T, Qiao W. Trading wind power in a competitive electricity market using stochastic programming and game theory. IEEE Trans Sustain Energy 2013;4:805
15. [15] Fudenberg D, Tirole J. Game theory. 5th ed Cambridge, MA: MIT Press; 1996. [16] dela Torre S, Contreras J, Conejo AJ. Finding multiperiod Nash equilibrium in pool-based electricity markets. IEEE Trans Power Syst 2004;19:643
51. [17] National Renewable Energy Laboratory (NREL) [Online]. Available at: ,http://www.nrel.gov/wind/ integrationdatasets/eastern/methodology.html.. [18] Lerner AP. The concept of monopoly and the measurement of monopoly power. Rev Econ Stud 1934;1:157
75. [19] de Lujan Latorre M, Granville S. The Stackelberg equilibrium applied to AC power systems—a noninterior point algorithm. IEEE Trans Power Syst 2003;18:611
18. [20] Hobbs BF, Metzler CB, Pang JS. Strategic gaming analysis for electric power systems: an MPEC approach. IEEE Trans Power Syst 2000;15:638
45. [21] Carlos R, Conejo AJ. Pool strategy of a producer with endogenous formation of locational marginal prices. IEEE Trans Power Syst 2009;24:1855
66. [22] Zhang G, Zhang GL, Gao Y, Lu J. Competitive strategic bidding optimization in electricity markets using bilevel programming and swarm techniques. IEEE Trans Ind Electron 2011;58:2138
46. [23] Dai T, Qiao W. Optimal bidding strategy of a strategic wind power energy owner in the short-term market. IEEE Trans Sustain Energy 2015;6:707
19. [24] Kazempour SJ, Conejo AJ, Ruiz C. Strategic generation investment considering futures and spot markets. IEEE Trans Power Syst 2012;27:1467
76. [25] Yin W, Gurobi Mex: A MATLAB interface for Gurobi. [Online]. Available at: ,http://www.convexoptimization.com/wikimization/index.php/Gurobi_Mex:_A_MATLAB_interface_for_Gurobi.. [26] Reliability Test System Task Force. The IEEE reliability test system—1996. IEEE Trans Power Syst 1999;14: 1010
1020.

CHAPTER

IMPACTS OF ACCURATE RENEWABLE POWER FORECASTING ON OPTIMUM OPERATION OF POWER SYSTEM

5 ˘ Akın Ta¸scıkaraoglu

Yildiz Technical University, Istanbul, Turkey

5.1 INTRODUCTION Maintaining the balance between energy supply and demand is of vital importance for reliable and secure operation of power systems [1,2]. In conventional power systems designed to generate electricity from high-capacity power plants (thermal, hydroelectric, nuclear, etc.) and to operate with unidirectional power flow, this balancing task could be accomplished by managing the output power of the plants considering the changes in load demand. Large penetration of renewable energy sources into power systems, however, causes an additional uncertainty that comes from generation side. In order to cope with this challenge, using the expected near-term values of renewables sources, that is, renewable power forecasts from wind and solar power plants, is considered as one of the most effective and feasible methods [3,4]. Renewable power forecasts significantly improve the controllability of power systems including high-capacity renewable sources. Therefore, the essential grid integration issues such as balance management, power generation scheduling, real-time dispatch, and determination of operating reserve requirements can be addressed by reducing the technical and economic risks of uncertainty in renewable power generation. Moreover, the forecasts play a key role in reducing the costs of renewable energy sources by minimizing curtailment operations and then increasing the revenue in electricity market operations [5,6]. Considering all these benefits of power forecasts, it can be concluded that high-quality renewable power forecasts are required by independent system operators, regional transmission operators, and wind power producers for the purpose of providing the optimum system operation decisions under the stochasticity and intermittency from renewable power sources. In this chapter, first, the basic definitions and explanations about renewable power forecasts are presented in Section 5.2. Then, the studies conducted for developing forecasts with higher performance, which are essential for optimum power system operations, are discussed in detail in Section 5.3. Section 5.4 summarizes the literature examples on the use of renewable power forecasts in power system operations. The last section includes a discussion that covers the existing challenges of renewable power forecasting and summarizes the most important remarks presented in this chapter. Optimization in Renewable Energy Systems. DOI: http://dx.doi.org/10.1016/B978-0-08-101041-9.00005-3 Copyright © 2017 Elsevier Ltd. All rights reserved.

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5.2 FUNDAMENTALS OF RENEWABLE POWER FORECASTING TECHNIQUES Forecasting tools presented in the literature are mostly developed for a certain type of an application area; however, it can be indicated that most of these methods can be also used for other areas with a reasonable performance. Nevertheless, for a higher forecasting performance required for optimum power system operations, different model structures and configurations for wind and solar forecasting are preferred in the literature as these two variables have different data characteristics. Generally speaking, it can be concluded that wind forecasting approaches are more complex and harder to model due to the stochastic nature of wind. Regarding the certain differences in the approaches adopted in wind and solar forecasts, two fields are discussed separately in the following subsections.

5.2.1 WIND SPEED AND POWER FORECASTS Wind energy is the fastest growing renewable energy source with a cumulative market growth of over 17% among the different renewable energy resources and expected to reach 2000 GW by 2030, supplying 17
19% of global electricity [7]. With the objective of supporting this growth, a large amount of effort has been devoted to the development of higher quality forecasting methods. The final target of wind forecasting methods is to acquire the expected power values from a wind turbine or a wind farm. However, two different approaches can be followed for reaching these values: (1) Forecasting power values directly, and (2) Forecasting wind speed and converting it into power forecasts using a power curve. In terms of the model structure, these two approaches use almost the same models for forecasting implementations. Under some certain conditions, such as a wind farm that includes turbines with different power ratings, using wind speed forecasts and then power curves for power forecasts can be a better option. A great number of wind speed/power forecasting approaches has been presented in the literature. In terms of their model structures, these approaches are generally divided into two classes in the literature: (1) Physical methods that use physical considerations such as orography and obstacles to estimate the wind output and (2) Statistical (data-driven) methods that use historical time series to forecast the near-term wind speed or power. It can be indicated that the physical models give generally better results for longer terms as they use numerical weather predictions (NWP), which provide the forecasts of weather characteristics by modeling the atmosphere mathematically. On the contrary, the data-driven models are generally used for relatively short terms for utilizing the high correlations between the past time series of wind speed/power at the same location. Among the data-driven models, machine learning-based methods such as artificial neural networks (ANNs) and support vector machines (SVM) are considered to be the most effective algorithms due to their capability of modeling nonlinear relationships between the time series. However, it is not possible to define a forecasting model as the model that has the highest accuracy compared to the other methods. Instead, each forecasting approach can provide reasonable results depending on various factors such as the forecasting horizon, data characteristics, and training data size. Considering the main advantages and disadvantages of the widely used wind forecasting models in the literature, a comparison between these approaches is given in Table 5.1 [3].

5.2 FUNDAMENTALS OF RENEWABLE POWER FORECASTING TECHNIQUES

161

Table 5.1 Brief Comparison of the Main Methods Used for Forecasting of Wind Speed and Power in the Literature Wind Speed/Power Forecasting Approach NWP models

Time series models (AR, ARMA, ARIMA, f-ARIMA, etc.) ANN-based models

Forecast Horizon

Modeling Efficiency

Effective in long terms, not suitable for short terms Suitable particularly for very short and short terms

Deficient in handling smaller scale phenomena

Effective for short and medium terms

SVM-based models

Effective for short and medium terms

Fuzzy logic models

Suitable for short terms

Bayesian networks

Effective in shorter terms

Kalman filter models

Suitable for short terms

Comparatively basic structure, difficult to model nonlinear problems No need to specify any mathematical model a priori, higher adaptability to online measurements High generalization performance, highaccuracy depending on the tuning of parameters appropriately Suitable for systems which are difficult to model exactly

Ability to handle missing observations and to avoid the overfitting of data, depend on the user’s expertise level Require previous knowledge about the system

Computational Efficiency Require large computational resource and time Very effective for low model orders Relatively highcomputational costs for large training data sets Complex optimization process and longer training time High complexity and a long process time in the case of many rules Require relatively more effort

Relatively high calculation times

Training Data Require a huge amount of data

Require a great deal of historic record Performance depends on the size of training data Require a large number of training data

Training data size may increase the accuracy Suitable for small training data sets

No requirement to store all historic data because of its recursive form

5.2.2 PV IRRADIANCE AND POWER FORECASTS Similar to the strategy adopted in wind forecasting literature, solar power forecasts are also realized in two different ways: (1) Using power measurements to forecast future power values, and (2) Forecasting the solar irradiance and converting these forecasts into power forecasts using a power conversion model. In the second group, it is also possible to take the temperature forecasts

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CHAPTER 5 IMPACTS OF ACCURATE RENEWABLE POWER FORECASTING

into account in conversion model in order to acquire higher accuracy forecasts. Furthermore, additional variables such as humidity and wind speed are also included in a number of literature studies with the same objective.

5.3 STATE-OF-THE-ART IN RENEWABLE POWER FORECASTING Together with the increasing rate of renewable energy in power grids, the accuracy of the forecasting algorithms has gained more importance. In addition to the conventional approaches, therefore, advanced algorithms have been discussed in the area of forecasting. First, advanced machine learning models have been employed and then various optimization algorithms have been applied to wind time series for higher accuracy forecasts. Recently, new ideas have put forward in the literature in order to improve forecasting performances. These efforts can be gathered into three groups: (1) Using different methods together to take advantage of the superiority of each model, (2) Incorporating data from different sites to exploit the spatial correlations among these sites, and (3) Using probabilistic forecasting methods to include the uncertainty information about the forecasts. These approaches are elucidated in the following subsections.

5.3.1 COMBINED RENEWABLE POWER FORECASTING METHODS The performance of renewable power forecasting methods generally depends on three different factors: (1) The length of forecast horizon (i.e., very short-term, short-term, medium term, and long-term), (2) The time resolution of forecasts (minutely, hourly, etc.), and (3) The size of training data. It is, therefore, possible to conclude that any single forecasting model cannot achieve the best forecasts for different cases and various approaches are generally tested for deciding the most suitable one for certain conditions. In order to deal with this problem and to provide reasonable results under varying conditions, combining two or more models has come into prominence particularly at the last decade. These combined approaches can mostly improve the forecasting performance compared to single methods by exploiting the unique advantages of single methods. In order to improve the results, several combination techniques have been used in the literature depending on many factors such as the required forecasting horizon, data availability, and required calculation time. Regarding these techniques adopted in the combination of various single models, the relevant studies presented in the literature of renewable energy forecasting can be gathered into four different groups: (1) Weighting-based combined approaches, (2) Data preprocessing-based combined approaches, (3) Parameter-optimization-based combined approaches, and (4) Data postprocessing-based combined approaches [3].

5.3.1.1 Weighting-based combined approaches Among the combined approaches presented in the renewable energy forecasting literature, the prevailing approach can be indicated as the weighting-based combined approaches. These approaches combine the forecasts of various models by assigning them a weighting factor taking the relative effectiveness of each model into account. In other words, a certain time series is forecasted with each model simultaneously and the forecasting results are evaluated by using different performance

5.3 STATE-OF-THE-ART IN RENEWABLE POWER FORECASTING

163

metrics. Regarding the performance measures, a weight coefficient is then calculated for each of the single models. Lastly, using the corresponding weight coefficients, the forecasts are merged. The general forecasting scheme of these approaches can be illustrated as in Fig. 5.1. The weighting-based combined forecasts are generally calculated as given in the following equation: F^ t1hjt 5

N X

wi  f^i;t1hjt

(5.1)

i51

where F^ t1hjt is the combined forecast for the time t 1 h carried out at time t, f^i;t1hjt is the forecast of ith model, wi is the weight coefficient for the ith model and N is the number of single models. The weight coefficients shown in Eq. (5.1) can be calculated using various methods and the models are generally called regarding these methods, such as equally-weighted combination model, variedweighted combination model, and variance
covariance combination model. It is noted that different formulations or methods can be used for combining the forecasting methods. The approaches combined with such a method have a large number of advantages. Maintaining always a reasonable accuracy for varying weather conditions by using different type of methods, such as using both an autoregressive (AR)-based model and a machine learning model, is one of the most important benefits of these approaches. Furthermore, the weighting-based models enable to use different amount and type of variables as inputs. For instance, only lagged wind speed time series can be applied to one of the models, whereas several meteorological parameters such as wind speed, wind direction, temperature, and pressure are used for the other model depending on its model structure and the contribution of the variables on the forecasting accuracy. It is to be noted that using more than two models might increase the forecasting accuracy; however, this could be impractical for real-time implementations due to the resulting high-computational costs.

FIGURE 5.1 Basic flowchart of the weighting-based combined forecasting approaches.

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5.3.1.2 Data preprocessing-based combined approaches This group of combined forecasting approaches consists of two different models; one for the forecasting of the desired variable and other one for a preprocessing on the available time series data. In these approaches, the main aim of data preprocessing is to decompose wind speed/power time series or solar irradiance/power time series into more regular subseries which are easier to forecast, and/or to filter out the irrelevant and redundant data in the data set. Both processes allow obtaining more predictable and informative data, enabling to increase accuracy and decrease calculation time. The flowchart of data preprocessing-based approaches can be denoted as in Fig. 5.2. In these approaches, the decomposition model generates a number of components with different frequencies and the forecast of each component is obtained with a forecasting model. These forecasting models can be the same for all components; however, using a different model or a model with different parameters generally provides better results as the time series with low and high frequencies can be modeled to a higher accuracy level using different methods. Among the data decomposition methods, wavelet transform (WT) has been the most widely preferred method for forecasting applications due to its efficiency in time
frequency analysis and relatively easy implementation. WT method basically decomposes time series into one low-frequency approximate level and a number of high-frequency detail levels. The number of detail levels is chosen considering the characteristics of the time series to be decomposed. Furthermore, the other parameters of WT such as the type of mother wavelet are also determined by analyzing the data. The decomposition level chosen and the other parameters in WT might considerably affect the forecasting performance and therefore they should be chosen with sufficient examination. In order to overcome this challenging problem, another decomposition method has been recently proposed by Huang et al. [8]. This method, namely empirical mode decomposition (EMD), decomposes the nonstationary time series into a series of oscillatory functions. The number of these functions, called intrinsic mode functions, depends on the considered time series itself and

FIGURE 5.2 Basic flowchart of the data preprocessing-based combined forecasting approaches.

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165

is hence not affected by the problems faced in parameter selection. Apart from WT and EMD, Fourier transform and blind source separation techniques are also used within the combined approaches with the objective of facilitating forecasting task with more stable components.

5.3.1.3 Parameter-optimization-based combined approaches In order to determine the parameters of a forecasting model, the trial-and-error methods, which analyze the effects of different model parameters on the model performance, are traditionally used in the literature. The unfavorable effects of these time-consuming and not very effective methods can be removed by employing various parameter optimization approaches for optimal model parameters. The basic model of these cascade structures is shown in Fig. 5.3. The prevailing method used in this group of combined approaches is genetic algorithms (GAs) due to its fast convergence to global optima and easy implementation. Other evolutionary algorithms such as evolutionary programming and differential evolution have been also used for selecting the optimal parameters to be used for a forecasting model. Besides, Kalman filter, which is a recursive algorithm used for state estimation, has been employed in several studies on combined renewable energy forecasting.

5.3.1.4 Data postprocessing-based combined approaches In the literature, a few studies have discussed the benefits of taking the residual errors into account in combined approaches. It has been shown that this approach improves the forecasts particularly when a considerable systematic error exists in the forecasts. The main reason of this error might be the over or underestimate of the related variable, and it is, therefore, possible to examine the characteristics of the error signal and then to account for its effect in the forecasting process, as denoted in Fig. 5.4. A brief evaluation on the main features of the four groups of the different combined approaches is shown in Table 5.2. The main advantages and disadvantages of the combined methods compared to others are also summarized in Table 5.2.

FIGURE 5.3 Basic flowchart of the parameter-optimization-based combined forecasting approaches.

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CHAPTER 5 IMPACTS OF ACCURATE RENEWABLE POWER FORECASTING

FIGURE 5.4 Basic flowchart of the data postprocessing-based combined forecasting approaches.

Table 5.2 Brief Evaluation of Combined Approaches Applied for Forecasting of Wind Speed and Power in the Literature Combined Wind Speed/Power Forecasting Approach

Strategy

Forecast Horizon

Forecast Accuracy

Modeling Efficiency

Computational Efficiency

Easy to implement and code, require an extra model for determining the weights Require a detailed mathematical knowledge on decomposition models

Highcomputational cost, adaptive to new data

A relatively basic structure, harder to code

Computationally intensive

Easy implementation

Computational time inefficiency

Weighting-based combined approaches

Assign weights regarding the model performances

Suitable for a wide range of forecast time

Different performances along the prediction horizon

Combined approaches including data preprocessing techniques

Decompose wind signal and forecast each component separately before an aggregation

Relatively longer horizons

Combined approaches including parameter selection and optimization techniques Combined approaches including error processing techniques

Tune the parameters of main model

More effective in shorter terms

Higher performance compared to others, robustness to wind fluctuations Relatively high-accuracy

Include the forecasts of residual error caused by model

Suitable for short terms

Effective in reducing systematic error

Require a huge computational resource, provide very slow response to new data

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167

5.3.2 SPATIOTEMPORAL RENEWABLE POWER FORECASTING METHODS Renewable energy forecasting methods provide valuable information about the expected changes in the energy to be generated in the near future. Combined approaches can improve these forecasts by taking advantage of the peculiarities of single forecasting models, as explained in the previous section. Both the mentioned single methods and the combined approaches are aiming at providing forecasts using historical time series collected from a certain point, such as a meteorological station, and a wind turbine or a solar panel. In order to further increase the forecasting accuracy, using information from areas close the exact location where the forecasts are performed has gained importance particularly at the last few years. A number of studies showed that in addition to temporal information, spatial dependency of wind and solar time series can also be utilized in the forecasts. Moreover, the increasing of the availability of data from different sources together with the emerging smart grid technologies, enabled especially by the development of advanced measurement and communication infrastructures, has paved the way for these studies. These methods, namely spatiotemporal forecasting methods, are based on the assumption that weather variables such as wind speed and direction, solar irradiance, and temperature tend to exhibit considerable correlations among the areas close to each other, possibly with time lags depending on several factors such as the distance and elevation difference between the sites. For an instance as given in Fig. 5.5, it is most likely that the wind speed profile of wind turbine 1 will be observed in the area of wind turbine 3 in a few hours due to both high speed and prevailing direction of wind at wind turbine 1. Similarly, the movements of the clouds from region 1 to regions 2 or 3 might cause to observe the lower solar irradiance values at these regions in a certain time. Considering the correlation among the data collected from the site of interest and other points in its vicinity, therefore, spatiotemporal methods can accomplish higher accuracy forecasts compared to the forecasts depending on only temporal data [9,10].

FIGURE 5.5 A map showing geographically dispersed wind and solar power plants subjected to different weather characteristics. The symbols on the left side of each power plant represent cloudiness, and the speed and direction of wind in the related area for a certain period.

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The most important consideration in these models is to deal with the problem of highcomputational costs caused by incorporating a large amount of data in the model structure. The studies on this topic therefore focus on addressing the trade-off between the amount of data used as input and the forecasting accuracy. In various studies, it is concluded that the selection of certain locations might improve the forecasting accuracy as well as decrease the computational burden. In order to determine these specific locations, several studies define a threshold distance based on the size of geographical areas, whereas the recent studies in the literature generally consider detailed correlation analyses that assign different weighting coefficients depending on the correlations between the target location and the other candidate locations. These coefficients are mostly based on the relative distance between the locations for wind speed and solar irradiance; the roughness of the area, wind profile, and prevailing wind direction for wind speed; and the speed of cloud movements for solar irradiance. As regards to wind speed/power forecasts, several studies present spatiotemporal forecasting methods including wind direction in the designing of the models for a higher forecasting accuracy. These studies consider one or mostly two dominant wind directions in the given area and the upstream locations are chosen based on this information. Various studies also incorporate wind direction as an exogenous input to the model instead of using it as a part of the model. Apart from wind direction, other meteorological quantities such as temperature and pressure are also used in spatiotemporal forecasting methods for higher performances. There are also a few studies considering geostrophic wind, which represents the wind affected by only air-pressure gradient force and Coriolis force, as an input for higher accuracies. Among the spatiotemporal forecasting methods used in renewable power forecasting, it can be indicated that a large part of the related literature focuses on the AR-based approaches due to their capability of including numerous data from different sources and easy implementation. As explained above, AR-based models are based on the assumption that output of a system can be obtained as a combination of its weighted previous values. This approach can be extended for multivariate systems for the purpose of incorporating data from different sources, as shown in the following equation: yvt

 

;s

5

V;S X n X

v51 s51

v;s yv;s t2i xi

(5.2)

i51

 

where yv ;s is the output variable to be forecasted for sample time t, variable v and source s. V and S represent the numbers of variables and sources, respectively. n is model order and xv;s i is regression coefficients. Eq. (5.2) can be shown in a format as in the following equation: 2

2





;s yvn11 6   6 yv ;s 6 n12 6 6 ^ 4  

;s yvn1M

3

2

7 6 7 6 7 6 756 7 6 5 4

y1;1 n

...

y1;1 1

y1;1 n11

&

^

^

&

^

y1;1 n1M21

...

y1;1 M

    ...    ... 

   . . .   . . . 

ynV;S

...

y1V;S

V;S yn11

&

^

^

&

^

yV;S n1M21

...

V;S yM

x1;1 1

6 6 ^ 36 6 6 x1;1 n 76 76 76 76 ^ 76 56 6 ^ 6 6 V;S 6 x 6 1 4 ^ xV;S n

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

(5.3)

5.3 STATE-OF-THE-ART IN RENEWABLE POWER FORECASTING

169

where M 1 n represents the number of data used in model training. In Eq. (5.3), the number of coefficients would be equal to N in which N :¼ nP and P 5 VxS (the total number of measurements of the variables from different sources). The objective in the training stage is the calculation of x coefficient vector that provides the best forecasts for the subsequent period. It is assumed in Eq. (5.3) that all the n order values are equal, that is, all variables from different locations have the same effect on forecasting performance. It is, however, obvious that the real case will most probably be different. Therefore a modified version of Eq. (5.3) can be used for a better modeling approach, as shown in the following equation: 2 2





yv ;s 6 nmax 11 6 v ;s 6 ynmax 12 6 6 6 ^ 4 v ;s ynmax 1M

3

2

7 6 7 6 7 6 756 7 6 7 6 5 4

y1;1 nmax

... ...

y1;1 nmax 2n1 11

y1;1 nmax 11

& &

^

^

& &

^

... ...

y1;1 nmax 2n1 1M

y1;1 nmax 1M21

    ...    ... 

   . . .   . . . 

yV;S nmax

...

ynV;S max 2nP 11

ynV;S max 11

&

^

^

&

^

...

yV;S nmax 2nP 1M

ynV;S max 1M21

6 6 36 6 6 76 76 76 76 76 76 56 6 6 6 6 4

x1;1 1

3

7 ^ 7 7 7 7 x1;1 n1 7 7 7 ^ 7 7 7 ^ 7 7 7 V;S 7 x1 7 ^ 5 xV;S nP (5.4)

P where ni is the order associated with ith location and N :¼ Pi51 ni . This structure enables to give importance to the data that are more related with the output variable. Considering their high capability of modeling complex dependences between different time series data, machine learning methods have been also used in spatiotemporal forecasts. Among them, ANN and SVM can be pointed out as the most effective ones for a high-forecasting accuracy. The other machine learning-based methods including Bayesian Network, Fuzzy Logic, Markov Chain and GA have been applied to various wind speed and irradiance data sets effectively in the recent studies in the literature. Apart from the machine learning methods, the contributions of other methods such as random forest, Kriging, decision trees and least absolute shrinkage and selection operator have been also investigated for spatiotemporal forecasting tasks.

5.3.3 PROBABILISTIC RENEWABLE POWER FORECASTING METHODS Conventional renewable energy forecasting methods produce a single forecasted mean value of the output power for each prediction time period. However, these forecasts, so-called deterministic forecasts (or point forecasts) cannot provide the exact information about future meteorological quantities and also their forecasting accuracies vary with time, resulting in a considerable uncertainty in the forecasts. Contrary to deterministic forecasting methods, the methods so-called probabilistic forecasting (or uncertainty forecasting) methods provide the probabilities of future values and also information on uncertainty, which is of great importance for power systems including a high portion of renewable energy. Particularly, with the help of stochastic optimization and appropriate bidding strategies, the quantitative uncertainty information about power forecasts can be

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CHAPTER 5 IMPACTS OF ACCURATE RENEWABLE POWER FORECASTING

effectively used in operational decisions related to trading future renewable energy production, unit commitment, and reserve management, resulting in a significant increase in penetration of renewable power and higher economic benefits. The uncertainty on the forecasts of wind and solar power generation mainly comes from the forecast of related highly variable meteorological quantity and its conversion into power values using power curves and conversion models. The wind or solar to power conversion techniques are obtained considering the ideal weather and terrain conditions, in other words, without taking into account the changes in wind speed and solar irradiance due especially to the impacts of other meteorological variables and landforms. Therefore, in addition to the wind speed and solar irradiance forecasting errors, these conversions cause an uncertainty in forecasts, which hinders the highaccuracy modeling of renewable energy generation. For the purpose of considering the uncertainty of renewable power generation, these forecasts should be carried out in a probabilistic framework. Hence, the uncertainty information can be expressed by various probability measures such as probability density functions (PDFs) and cumulative distribution functions (CDFs), and the other measures based on these functions such as quantiles (especially quantile regression and kernel density estimation) and intervals. In order to obtain the predictive distributions, two main techniques are used; parametric approaches which assume that predictive distribution follows a predefined shape (Gaussian, Beta, etc.) and nonparametric approaches in which predictive PDFs or CDFs are estimated at different points without depending on any assumption on the shape of distribution. As the distribution shape in parametric approaches relies on several parameters, the probabilistic approaches are relatively simpler and have lower computational costs. It is noted that when the assumption of power distribution shape is not appropriate, the effectiveness of these approaches may be decreased substantially. In these cases, the distribution-free nonparametric approaches might provide better results by considering more densities or quantiles, which is certainly more expensive in terms of computational costs.

5.4 THE LITERATURE ON THE CONTRIBUTION OF HIGH-ACCURACY RENEWABLE POWER FORECASTS IN OPTIMUM POWER SYSTEM OPERATION As previously explained in Section 5.1, the renewable energy forecasts are of great importance for the power system operations; however, in order to ensure the optimum system operation, the forecast quality must be very high during the time scale required by the related system operation. With this objective, different forecasting approaches are generally used for different power system operations depending on the time period required for the related system operation to be realized and maintained effectively. In other words, the most appropriate forecasting approach is mainly chosen regarding its purpose of use in power systems as each approach is effective in certain forecasting horizons and therefore suited to certain type of system and market activities. For instance, very short-term power forecasts (up to a few hours) are useful for active turbine controls, real-time power system operations, and regulation actions. Short-term forecasts (ranging from a few hours to a few days) are generally used for the planning of economic load dispatch, unit commitment, scheduling, regulation, competitive electricity trading in electricity markets, and guaranteeing the

5.4 HIGH-ACCURACY RENEWABLE POWER FORECASTS

171

security in power markets. The operations that require long-term information about the expected power from renewable sources, such as unit commitment, energy storage operations, and maintenance planning of renewable or conventional power plants, are carried out by using long-term forecasts (ranging from 2 to 7 days). All the forecasting approaches in different time scales help electric power systems operate in a more economic, reliable, and flexible way; however, in order to guarantee the optimum system operation, it can be indicated that the accuracy of power forecasts is the key factor. Any underforecast in renewable power values generally leads to considerable increments in the costs of power generating units, such as start-up and fixed costs, and overforecasts require the purchases of expensive peaking power to compensate the deficient power in the system. Therefore, a large number of studies is presented in the literature with the objective of optimally operating power systems together with the high-accuracy renewable power forecasts. Among these studies, Watson et al. employed an NWP model combined with a physical flow model and a statistical model for wind power forecasts, and evaluated the effectiveness of forecasts on the planning of online reserve capacities in the case of high-wind power penetration [11]. Benefits of storage systems and power output forecasts for allowing the scheduled generation of wind farms were analyzed in Ref. [12]. Wang et al. introduced a security-constrained unit commitment approach accounting for forecasted intermittent wind power generation with the objective of maintaining the security of power system operation [13]. Furthermore, Martin et al. investigated the effectiveness of different statistical models for solar irradiance forecasts and used these forecasts in combination with a solar thermal power plant for a better operation and improved scheduling of power production [14]. The effects of wind energy forecast errors on wind curtailment, related system costs, and economic dispatch in electricity systems with high-wind penetration were studied in Ref. [15]. The detailed information about the use of single forecasting methods for optimizing power system operations can be found in the comprehensive literature review studies, such as Refs. [6,16]. Together with the increasing penetration of wind and solar power plants in the power systems, the accuracy of power forecasts has gained more importance. Considering that the conventional forecasting models are not capable of providing the required accuracy criteria in these cases, therefore, advanced approaches have been introduced in the literature. First, using more than one method in forecasting, as explained in Section 5.3, has been proposed for essential power system operations. Mahoney et al. presented a wind forecasting system for optimized grid integration, which combines a number of different forecasting technologies and methods [17]. The model used weather model data from the National Centers for Environmental Prediction as input. A modified weather research and forecasting (WRF) model and also an ensemble system including both WRF and the Penn State-National Center for Atmospheric Research (NCAR) MM5 models were also integrated into the forecasting system. The forecasts obtained were then combined with dynamic integrated forecast system of NCAR. A combined model consisting of EMD, cascade-forward neural network and a linear model was proposed in Ref. [3] for the purpose of using the forecasts in an economic operation-based load dispatching strategy. The main objective of this strategy was to deal with the problem of stochastic wind and solar energy generation and to be able to participate in electricity markets with higher benefits. As the combination of forecasting methods is a relatively new research area, the studies on this area have been generally focused on the development of these models and their model performance assessment. Therefore, the studies dealing with the evaluation of

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combined models for system operations have been limited to a few studies given above. Further information about the structure and benefits of these models as well as their suitability to different system operations can be found in Ref. [18]. Smart grid technologies equipped with advanced measurement and communication technologies have enabled the collection of a great amount of data from geographically different sites. Therefore, the idea of exploiting data from different sources for the purpose of increasing the forecasting performance has been investigated in the literature, particularly in the last decade. The main motivation of these studies is that weather conditions propagate from one site to others in a certain time without losing their main characteristics due to the similarity of influencing factors on these conditions in adjacent sites, and therefore using all this information in the forecasts of a certain point might improve the forecasting accuracy. It can be indicated that the effects of data from one point on another point can be observed in a certain time depending mainly on the geographical conditions between the sites and the weather characteristics such as speed of the wind, prevailing wind direction, and the speed and direction of cloud motions. Among the studies based on spatiotemporal data, Xie et al. presented a spatiotemporal wind power forecasting approach for improved forecast quality and integrated the forecasts with a look-ahead economic dispatch framework for the purpose of reducing generation cost [19]. Space-time wind forecasts were used in a power system economic dispatch model with a modified regime-switching model in Ref. [20]. Using the proposed model, it was aimed to achieve economic benefits from the cost savings in both system-wide generation and ancillary service. Similar to the case for combined forecasting approaches, it can be also indicated for spatiotemporal approaches that only a few studies exist for the implementations of these methods in power system operations. The certainty of renewable power forecasts is as vital as the accuracy of the forecasts for the optimum operation of electric power systems. Making use of the information about the uncertainties related with the forecasts in power systems facilitates the exploitation of renewable power and enables the dealing with the complexity of decision-making problems such as stochastic unit commitment in these systems. These forecasts, called probabilistic renewable power forecasts, quantify the uncertainty associated with the conventional point forecasts and provide a distribution overforecasts instead of a single forecast. Therefore, it can be possible to evaluate the possible energy excess and deficiency problems caused by the under and overforecasts of renewable power and thus the expected balancing costs can be significantly reduced. Considering the mentioned benefits of probabilistic forecasts, the trend in renewable power forecasting approaches has been oriented to these forecasts, for both temporal and spatiotemporal methods. Methaprayoon et al. presented an ANN-based wind-power forecasting method and used these power forecasts in a unit commitment scheduling problem by considering the uncertainty of the forecasts with a probabilistic concept [21]. Pinson et al. used probabilistic wind generation forecasts within a methodology for achieving optimal bidding strategies for market participation and hence increasing the revenues [22]. Furthermore, Tuohy et al. examined the effects of stochastic nature of wind and load on the unit commitment and dispatch of power systems with high-wind power penetration [5]. Subsequently, they investigated the benefits of updated wind and load forecasts on the system planning. A reserve management tool using probabilistic wind power forecasts was proposed in Ref. [23] with the objective of setting the required operating reserve levels for daily and intraday markets. Furthermore, Makarov et al. studied on the uncertainties of balancing and ramping capabilities as well as ramping requirements and proposed a probabilistic approach including uncertainties caused

5.5 DISCUSSION AND CONCLUSION

173

by the errors in wind and solar generation forecasts [24]. Lowery and O’Malley investigated the effects of wind power error statistics on the performances of stochastic unit commitment and system operations for a system with high-wind power penetration [25]. An approach, which combines stochastic wind forecasts with a day-ahead unit commitment, was also presented in Ref. [26] in order to determine the optimal amount of spinning reserve required for a certain period because of wind power integration and hence to minimize total energy cost. In order to determine the dynamic operating reserve requirements, demand dispatch was combined with a probabilistic wind power forecasting method in Ref. [27]. Another method integrating a convolution process with Monte Carlo simulation method and taking into account wind power forecasting error was present in Ref. [28] in order to estimate the cost/benefit relationship used for the determination of optimal spinning reserve requirements. An extended literature survey on probabilistic forecasting methods and explanations about the model structures and advantages of these approaches can be found in Ref. [29].

5.5 DISCUSSION AND CONCLUSION Accurate renewable energy generation forecasts are one of key requirements in the optimum operation of power systems. Considering the high-level integration of stochastic renewable generation in power system, it is clear that even a small improvement in the outputs of conventional forecasting methods (i.e., single value forecasts) might provide a considerable technical and economic benefit for electric generation, transmission, and distribution systems. The efforts aimed to increase the forecasting accuracies without affecting the calculation times too much are investigated in this chapter of the book and these studies are classified into three groups: (1) Combined forecasting approaches, (2) Spatiotemporal forecasting approaches, and (3) Probabilistic forecasting approaches. The model structures and superiorities of each approach are elucidated in different subsections in Section 5.3, and their implementations on power systems in the literature are presented in Section 5.4 referring to the related studies. Regarding the remarks presented in the literature studies, it can be concluded that the conventional forecasting methods provide favorable results for relatively low-capacity renewable energy systems such as residential wind turbines and PV panels; however, their performances are not sufficient to satisfy the requirements of an effective operation in large-scale power systems including a high-capacity of renewable power. For such systems, using advanced forecasting algorithms consisting of different methods and/or accounting for various data collected from different sources located in a large area may improve the forecasting accuracies at different time and spatial resolutions, resulted in a more effective power scheduling and management. Including also the uncertainty information on the forecasts in power system management may provide an additional benefit for estimating the unexpected changes in wind and solar generations and taking an action accordingly. Despite the advancements in forecasting algorithms, there are still several challenges faced in the real-world applications, which prevents the exploitation of these approaches within power systems. First, a high number of data belonging to the various variables from different locations, particularly used in spatiotemporal forecasting methods, enables better modeling of the dynamics of renewable power generation; however, analyzing of such a large amount of data is not a

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straightforward task. Therefore, the “big data” technologies might be a promising topic in the context of renewable energy forecasting. Furthermore, the importance and use of temporal data at higher resolutions have been increasing recently, particularly for very short-term planning of renewable power management, which significantly complicates the examination of the data. Refining these data so that the most informative data is obtained can be pointed as another hot topic in renewable energy forecasting. Lastly, developing forecasting algorithms that integrate one group of forecasting approaches given above with another one, such as a probabilistic spatiotemporal forecasting method, can be considered as an effective and valuable method for improving the operation of power systems.

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93. [27] Botterud A, Zhou Z, Wang J, Sumaili J, Keko H, Mendes J, et al. Demand dispatch and probabilistic wind power forecasting in unit commitment and economic dispatch: a case study of Illinois. IEEE Trans Sustainable Energy 2013;4(1):250
61. [28] Lujano-Rojas JM, Oso´rio GJ, Matias JCO, Catala˜o JPS. A heuristic methodology to economic dispatch problem incorporating renewable power forecasting error and system reliability. Renewable Energy 2016;87:731
43. [29] Zhang Y, Wang J, Wang X. Review on probabilistic forecasting of wind power generation. Renewable Sustainable Energy Rev 2014;32:255
70.

CHAPTER

OPTIMUM TRANSMISSION SYSTEM EXPANSION OFFSHORE CONSIDERING RENEWABLE ENERGY SOURCES

6

Shahab S. Torbaghan1 and Madeleine Gibescu2 1

Vlaamse Instelling voor Technologisch Onderzoek (VITO), EnergyVille, Poort Genk, Belgium 2 Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands

6.1 INTRODUCTION The transmission system is the backbone of the electric power system. Its traditional role is to securely transport electrical energy from the remote large-scale generation power plants to distribution systems and industrial consumers. Nowadays, the transmission system also enables open access to electricity markets and promotes competition among producers. It facilitates the operation of electricity markets by providing capacity for conducting power trades between various price areas. The ultimate goal of the transmission system operator (TSO) is to maintain the reliability and security of supply at an affordable cost for the consumers. In recent years, electric power systems have been subjected to substantial changes, both on the generation side (e.g., large-scale integration of renewable energy sources (RES), energy storage systems), as well as the demand side (e.g., new distributed energy resources, electric heating, and transportation technologies). These changes present significant challenges to the operation of the power system. Next to that, the unbundling of the electricity production and transport sectors has introduced additional uncertainty in the planning processes of the TSOs, which must now align with anticipated long-term market developments. Transmission expansion planning (TEP) is the process of identifying the transmission network expansions that are necessary for dealing with these challenges and preparing the system for serving all current and future connected customers. Transmission is a capital-intensive investment for both large equipment and sophisticated control centers. Investment in transmission has a long lead-time. In addition, the transmission investment is in practice irreversible; thus, once an interconnector is built, it cannot be redeveloped. Moreover, the resale value of the installed asset is very low. Finally, transmission assets have a long lifetime ranging from 20 to 40 years. Therefore, decisions regarding locations, capacities and timing of transmission investments is a crucial step in any transmission expansion project and must be made with great caution. TEP algorithms traditionally followed a least-cost philosophy. The aim of the planners was to minimize the investment cost of transmission (sometimes also including generation) expansions to meet the future demand, subject to physical, and operational constraints. Optimization in Renewable Energy Systems. DOI: http://dx.doi.org/10.1016/B978-0-08-101041-9.00006-5 Copyright © 2017 Elsevier Ltd. All rights reserved.

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After deregulation and subsequent separation of generation and transmission assets, the transmission system continues to be a monopoly. Therefore, the operation and development of the transmission grid remain the responsibility of TSOs. With the introduction of competition in generation, the generation units are owned by different market parties that follow their own strategic interests. This change has imposed new challenges, both in planning and operation of the power system. In addition, large-scale integration of renewable energy sources (RES) has increased the number of uncertainties that the TSOs have to deal with. First, RES [including offshore wind (OW)] are often located remotely from load centers. Hence, new robust transmission infrastructure is a prerequisite for transferring power from the generation location to the load centers. Second, the power production levels of renewable resources are highly dependent on meteorological conditions. Thus, variable renewable generation, such as that from wind, wave, and solar sources, may consequently be difficult to predict over various time scales. Large penetrations of such energy generation technologies into the power system can lead to increase in the variability and uncertainty in the system’s generation patterns and drive a need for greater flexibility in market and system operations. Refs. [1
3] propose several solutions to reduce the impact of variability and the associated risks of RES. One possible solution at the transmission level is to extend the geographical distribution of the power system by building new transmission interconnectors1 between neighboring control areas. Since the traditional TEP algorithms are no longer viable in this highly competitive and uncertain environment, the methods have to be improved to account for the increased uncertainty induced by RES expansion and to address the needs of different stakeholders that are involved, for example, by minimizing total cost and maximizing profit of investors, while also maximizing the benefit to society.

6.2 CLASSIFICATION OF TEP FORMULATIONS It is the task of transmission planners to forecast the growth of load, generation fleet, and resultant power flows over short- and long-term horizons. In addition, they need to perform security checks to verify whether various technical limits will be violated. For those limits that are violated, planners propose a range of solutions to overcome problems and provide a cost-benefit analysis based on several measures, including the economic evaluation, socioeconomic feasibility, and technical performance [5]. This analysis relies on future development scenarios of load and generation of the power systems under study. System planners use future scenarios to determine the required reinforcements and the appropriate timing of building the assets. In general, TEP is a multiperiod, nonlinear, nonconvex, mixed-integer and large-scale optimization problem. It, therefore, has a high degree of complexity, as there are so many factors that have to be considered. TEP problems can be classified based on criteria related to regulatory structure, power system uncertainties, planning horizon, and solution methods [5
8]. The TEP problems are respectively 1 European Union electricity legislation defines an interconnector as: “a transmission line which crosses or spans a border between Member States and which connects the national transmission systems of the Member States.” [4].

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classified as regulated/deregulated, deterministic/nondeterministic, and static/dynamic. Depending on the mathematical properties of the formulation of the TEP problem, a heuristic, metaheuristic, or exact mathematical optimization method can be applied as the solution algorithm. A TEP problem can have features from all the above-mentioned categories in terms of problem formulation and solution method. For example, TEP problems to be discussed in this chapter are nondeterministic, static, or dynamic frameworks that are formulated for a deregulated environment. They are solved using mathematical optimization algorithms.

6.2.1 REGULATED VERSUS DEREGULATED In a regulated environment, the power system has a vertically integrated structure. The system operator owns all transmission and generation assets. Its aim is to provide electrical energy to consumers in the most economical manner, while maintaining a certain level of security and reliability. In such an environment, the transmission and generation expansion planning are conducted together centrally, with the purpose of minimizing the overall system cost. The separation of ownership of transmission and generation assets after deregulation complicates the planning process. The objective of long-term planning in the new environment is seen differently, depending on the perspective of the entity that is performing the assessment. The generation units seek to maximize their own profit. The TSO strives to maintain the reliability and security of the grid, at the same time providing nondiscriminatory access to electricity markets for all market players. The TSO also tries to find effective solutions that meet society’s needs and promote sustainability. In the deregulated power system, TSOs have limited information on the future development plans of the generation units. Therefore, uncertainty has become a major element in the decisionmaking process of TSOs, especially with respect to the choice of capacity. For transmission corridors, operation issues such as the acceptance level of congestion and the structure of the markets have become the main elements of long-term transmission planning. The situation is even more complicated, as the expansion decisions made by one entity will affect the benefits and so the decisions to be made by other partners. For example, building a new interconnector that connects the Netherlands to Denmark will induce significant changes in market prices, power flows, and the revenues of all the interconnectors that are connected to either system. As a result, after the establishment of the competitive environment the traditional TEP frameworks are no longer viable. It is, therefore, necessary to review both the ways, in which the problems are formulated and the algorithms are used to solve them [9
13]. Today, new expansion investments must be evaluated for their possible economic implications (on the market prices), as well as their social implications (on the behavior of transmission investors and, from there, on the competitive markets [14]). “Cooptimization” or “anticipatory transmission planning” are the computer-aided decision-support tools that consider generation dispatch and investment decisions on transmission capacities, congestion, and from there, the network investments. The anticipatory TEP enables transmission investors to evaluate how different network configurations will change investment and operational decisions to be made by generation investors. The cooptimization TEP can be an effective tool for regulators, as well as investors, to better understand various risks, benefits, and costs when assessing resource options, and to identify improved integrated solutions [15].

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6.2.2 CENTRALIZED VERSUS DECENTRALIZED DECISION-MAKING PROCESS Developing power system infrastructure is an expensive and time consuming process. It is normally conducted over the course of several years. From an economic perspective, the cash flow of a transmission expansion project consists of a large initial investment followed by transmission revenues and minimal operating and maintenance costs that occur every year over the lifetime of the project. Therefore, planning analysis generally compares alternatives that have different costs/ revenues at different points in time. In this section, the “time-value of money” concepts which are widely used in investment planning analysis are discussed. Financing the investment through external resources is a common practice, especially in power system planning projects. Therefore, it is more sensible to base the analysis on loan payments, rather than investment costs from a commercial perspective. However, when considering the system-wide, social perspective, direct investment costs are used instead in the cash flow calculations. Transmission infrastructure can exist as either a merchant system or as a regulated resource infrastructure. In the case of merchant interconnectors, the transmission investment is a profit-driven exercise. In this regard, the transmission revenue generated every hour would be considered as the source for investment recovery. In the deregulated electricity industry, two major paradigms of decision-making processes in transmission planning can be identified 1. Centralized: Decision-making, in which the planners’ objective is to minimize/maximize the system-wide social cost/social welfare under a central planning, and 2. Decentralized: Decision-making, with profit-maximizing objectives of multiple competing private investors (merchant transmission investors) that are in compliance with system operator’s guidelines. Regulated investment is a common practice in Europe. Under this scheme, TSOs are promoted to invest, build and operate the interconnector [16]. Their task includes the construction, maintenance and operation of the interconnectors [17]. The cost of new investments is normally financed through regulated transport tariffs. The TSOs must agree upon the rate and obtain the regulator’s approval for implementing the tariff. Therefore, the TSOs need to convince their respective regulators that the extra investment(s) in the new capacities are socially beneficial, so that they obtain their approval and may procure the investment. In addition to the regulated tariffs, a part of the investment may be recovered through transmission revenues the TSO receives from (implicitly) auctioning the interconnector capacities. According to Article 6(6) of Electricity Regulation (EC) No. 1228/2003, the TSO may utilize any revenues resulting from the allocation of the transmission interconnector only for the following purposes: • • •

Guaranteeing the availability of the allocated transmission capacity. Maintaining and/or increasing transmission investments. Reimbursing the cost of the regulatory authorities that are working toward modifying or approving methodologies that are used for calculating network tariffs.

Large price differences between electricity market areas create incentives for market parties to invest in new interconnectors to explore financial potentials and capture transmission revenues.

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These private interconnectors are referred to as “merchant interconnectors” and are considered as a commercial alternative to the regulated TSO investments. Merchant investment opens the way for profit-motivated investors to participate in electricity transmission projects (which has been so far considered a natural monopoly). Some argue that it is the only way to address the perceived problem of underinvestment in transmission system [18]. Moreover, in the absence of political willingness to increase the transmission tariffs for making new investments, merchant investment can be an option for transmission capacity expansion [19]. Merchant interconnections differ fundamentally from the regulated interconnectors in at least the three following points: • • •

Merchant investments are repaid through transmission revenues over the interconnector instead of the regulated transport tariffs and so involve higher risk for investors. In contrast to the regulated interconnectors, TSOs are not allowed to invest/participate in merchant interconnectors. To alleviate the investment risk, merchant interconnectors may be granted exemption from regulations such as: nondiscriminatory third-party access, restrictions on the use of transmission revenues, tariff regulation, and ownership unbundling provided that the exemption would not hamper market competition [20,21].

Regulated TEP is the state of practice. However, in recent years, decentralized merchant investment is penetrating some liberalized markets (e.g., Cross Sound Cable Interconnector [22], Path 15, TransBay cable and Green Line in the United States [23], Basslink in Australia [24], and Brit-Ned, Swe-Pol and Baltic Cable in Europe [25]). It is a move toward ending the transmission monopoly that prevailed for decades. For the moment, merchant transmission is viewed as a solution for large and risky investments, as would be the case for meshed offshore grids [23,25]. From an economic perspective, a major obstacle toward wider adoption of merchant transmission is the difficulty in determining the economic benefits and identifying the stakeholders, who may benefit from the facility. From a legal perspective, a good regulatory regime is required that provides opportunities for merchant investors to participate in transmission projects, when they are the most viable option. By 2015, the EU had granted authorization to few merchant projects, such as BritNed, Estlink, NorGer, France-Angleterre (IFA), 3 the East West Cables, and a new interconnector between Austria and Italy. Merchant transmission is a new practice in Europe. The introduction of the merchant transmission investment has created new challenges for the regulators, especially from an institutional level [25,26]. In this regard, the centralized (regulated monopoly) and decentralized (merchant transmission) network management approaches result in distinctly different expansion decisions and grid designs. Shrestha et al. [26] proved that in theory (under restrictive mathematical assumptions) that the centralized expansion approach results in socially optimal network capacity, which ensures maximum benefit to society (i.e., electricity consumers and producers). They also show that in contrast to the centralized approach, monopolistic merchant entrants under a decentralized system will always result in underinvestment in transmission capacities. However, enforcing proper competition in transmission investments or implementation of transmission revenue rights can make it plausible to achieve near socially optimal grid expansion, even under decentralized expansion [27].

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6.2.3 DETERMINISTIC VERSUS NONDETERMINISTIC METHODS There is a significant level of uncertainty involved in the future development of generation, transmission, demand, and electricity markets. The uncertainty stems from the nature of various technical, environmental, economic, and social/regulatory factors. In general uncertainty sources can be divided into two groups 1. Random sources (or aleatory uncertainties): The uncertainty can be represented statistically, using historical data, such as: load, variable renewable energy production, and others [28,29]. 2. Nonrandom sources (or epistemic uncertainties): Uncertainties that cannot be foreseen from previous history, such as changes in economic rules, regulatory regimes, and trends in public acceptance and technical developments of the power system. There are two types of transmission planning algorithms for dealing with the uncertainties: Deterministic and nondeterministic. In the classic deterministic approaches, planning is performed for a reduced number of operating states that represent the worst case conditions, for example, peak load or generation out of service [30
32]. The planner seeks to find the most essential reinforcements that are required to attain planning objectives (a certain economic goal and/or reliability level) and, at the same time, operate within the acceptable operating range for transmission equipment. The advantage of this approach is that it gives a unique set of investment decisions. The disadvantage of deterministic approaches is that uncertainties in various parameters, such as prices and load forecast, location of new power plants in the future and others, are not included in solving the problem [33]. Consequently, deterministic models are very likely to miss the whole picture, as the model looks at only a few snapshots of future scenarios and, therefore, their final results usually lack robustness. For example, by looking only at peak-load hours, the planning analysis fails to account for operating conditions of major interest that can happen during offpeak hours, in combination with high production from RES. An alternative approach is to use scenario planning. In scenario planning, several scenarios are defined that represent future regulatory and economic conditions. For each scenario, an optimal transmission configuration is developed using deterministic TEP or a production costing-based comparison of predefined plans. Then, the resulted designs are aggregated and those that are attractive for most cases are identified as robust solutions [34]. The purpose of nondeterministic planning methods is to better capture uncertainties associated with the analysis of the random and nonrandom factors in the future scenarios. This can be done by considering various operating snapshots (multitime period formulation), to which a probability of occurrence is assigned. Note that in many cases, the probability denotes a degree of importance of the operating states that are considered. The nondeterministic approaches include: Information gap decision theory [35], probabilistic (Monte Carlo simulations [36], point estimate [37], scenario-based modeling [38]), interval-based analysis [39], robust optimization [40], hybrid possibilistic-probabilistic approach (fuzzy-scenario [41,42], fuzzy Monte Carlo [43,44]), or combination of them [39,45].

6.2.4 STATIC VERSUS DYNAMIC METHODS Depending on the implementation horizon, TEP problems are classified into static and dynamic problems. In static TEP problems, the solution provides an optimal planning, assuming that system

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expansions are implemented instantly at a certain point in the future. The optimal design can include grid topology, transmission capacities, or merely a set of possible candidate reinforcements to attain a particular objective (e.g., minimize cost and maximize benefit) [6
48]. In reality, transmission network developments take place gradually in multiple development stages, because: (1) Building large infrastructures is costly and time consuming; (2) other parts of the power system and electricity markets develop in a gradual manner; (3) there can be disruptive delays that happen due to unforeseen technical and legal complications; and (4) the transmission development plan can adapt to aforementioned changing circumstances (closed loop rather than open loop design). Therefore, in addition to the optimal grid design, time restrictions should ideally be included in the analysis. Any static model will fail to provide information regarding the timing of the project development steps. In Section 4.3, a static framework for TEP in the North Sea is introduced. There are three planning horizons: Short term (1
5 years), medium term (up to 10 years), and long term (planning horizon longer than 10 years) [49]. Dynamic TEP (DTEP) takes into account temporal continuity of expansion projects. Note that unlike the models discussed in [7,50], the term “dynamic” does not merely refer to a series of statically built-up plans. The optimal plan includes development strategy and timing considerations, in addition to the sizing and placement of the assets. Although DTEP is computationally intensive [7], it usually leads to more economically efficient grid design and development strategy [51].

6.3 TRANSMISSION EXPANSION PLANNING IN EUROPE The construction of new infrastructures in Europe was initially driven by the need for increased cross-border power exchanges (PX) and the integration of the wholesale electricity markets. The liberalization of the power industry and emergence of electricity markets has changed the way of thinking about the operation of the system from a national to a regional or even European level. In the new environment, planners pursue solutions that facilitate cross-border power trades and encourage more efficient use of energy resources over all power systems. The European Union’s (EU’s) third internal energy market package is a good example. It was one of the major policy initiatives that aimed at “accelerating infrastructure investments, with the goal of ensuring the proper functioning of the EU electricity market” [20]. Today, the demand for integrating sustainable and renewable low-carbon energy resources has become an important supporting factor [20]. With the ambitious “20/20/20” targets, Europe aims at reducing CO2 emissions by 20% compared with 1990 levels, increasing the share of renewable sources in European energy systems to 20%, and increasing energy efficiency by 20%. It was the starting point for Europe’s transition to low carbon and sustainable energy supply. The 20/20/20 target is guided by the EU’s energy and climate change policies core objectives which are 1. Security of energy supply (by ensuring a reliable and uninterrupted supply of energy and electricity), 2. Competitiveness as electricity markets are restructured (by reducing the energy prices and increasing market efficiency), and 3. Sustainability (by limiting the footprint of energy production, transmission, and use on the environment).

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The development of new transmission networks both on and offshore, can improve the capability of the system to accommodate the variability and uncertainty in the power balance (i.e., due to the fluctuating and uncontrollable nature of wind power), while maintaining satisfactory levels of performance. So far, several projects have been launched in the EU2 to align Pan-European power grid development with the EU’s policy targets and move toward a Pan-European Supergrid. Building a transnational meshed grid and reinforcing existing onshore transmission systems may encounter either barriers or incentives in technical, economic, political, and regulatory domains. The technical issues have been well defined and addressed in literature [52
54]. From an economic viewpoint, developing a transnational transmission infrastructure requires a massive investment. They will have significant impact on the market operation of different countries. From a legal viewpoint, differences among heterogeneous national regulatory regimes, lack of legal certainty, and international cooperation, reduced social acceptance of projects, and lack of a long-term vision are factors that can hamper the development of cross-border offshore transmission and OW projects. They should be addressed adequately; otherwise, the development of a transnational grid may be suboptimal, not cost-efficient, or might even be prevented from coming into existence. To summarize, the European power system has become confronted with two major challenges in recent years: Growing share of RES and increasing share of cross-border power trades, while keeping the same level for security of supply. To meet these challenges, substantial grid reinforcements are required both on and offshore [55,56]. These developments are considered as the key priority in the national development plans of EU and several North Sea coastal states [56]. The North Sea offshore grid is identified as one of the six infrastructure priorities for the EU by the Second Strategic Energy Review and EU regulation No 347/2013 on guidelines for trans-European energy infrastructure [56]. OW is expected to provide a substantial contribution to the energy supply system, especially in North-Western Europe, due to greater technical maturity and decreasing cost. Particular attention has been focused on the North Sea, where the development of OW is already driving the construction of new connections from shore to sea. The total installed capacity of OW installations in Europe is expected to amount to 40 GW by 2020, could reach 150 GW by 2030, and even more by 2050 [57]. Special attention is focused on the North Sea, where there is a great potential for OW power plant development. In this regard, developing the transmission infrastructure in the North Sea is considered as a key priority. It is in fact, identified as 1 of the 12 “strategic energy infrastructure priority corridors” by European Regulation 347/2013, concerning guidelines for trans-European energy infrastructures. Large-scale integration of OW energy demands a secure and reliable network to transport the energy to the remote onshore load centers. In this regard, new transmission networks are needed to be developed both on and offshore. There are three main drivers for developing an offshore grid in the North Sea region. The first driver is to improve the security of supply by interconnecting the power 2

Several point-to-point HVDC projects have already been established [e.g., between The Netherlands and Norway (NorNed 1), The Netherlands and the United Kingdom (BritNed)] and many more are planned [e.g., between Germany and Norway (NorGer and NorD.Link), The Netherlands and Norway (NorNed 2), Denmark and The Netherlands (Cobra)]. There are also several projects for offshore wind connection to the shore that have recently been commissioned or construction started (mostly in Germany) such as BorWin 1, BorWin 2, SylWin 1, HelWin 1, and DolWin 1 projects.

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systems of different countries around the North Sea. This will help to bypass the bottlenecks of the onshore connections. The second driver is to enhance competition among European electricity markets by easing power trades and increasing possibilities for arbitrage and limiting the price spikes [58]. The production of RES is weather dependent. Large-scale integration of RES requires more flexible power systems. As the scope of weather patterns (e.g., wind availability and solar irradiation) is smaller than the size of Europe, RES resources are always available somewhere on the continent. As a result, the third driver for developing an offshore grid is to expand the geographical distribution of the system to increase the systems flexibility and create capacity for conducting cross-national power exchange, from which there is a surplus of renewable generation where there is a demand [59]. Significant expansion of transmission capacity can encounter technical, regulatory, social, and/or legal obstacles. From a technical viewpoint, proper accounting of physical flows using actual branch parameters, rather than transport models, are key to understanding the connections between transmission lines or cables, their cost, and benefits to different regions in conducting electricity trades. Inclusion of the correlation and location of actual injections can be obtained by using several periods, or even all hours of the year or several years, in order to achieve a grid design that is adequate, and yet not overbuilt [60]. From a financial standpoint, the economic fundamentals of consumption and generation also constrain realistic development. In addition, developing transmission infrastructure requires a massive investment and will have significant impact on the operation of different stakeholders and electricity markets of different countries. Therefore, appropriate choices regarding technology and line routing are necessary for proper cost optimization in actual implementation [61
63]. The pace of required development brings challenges of finance; the magnitude of capital expenditures associated with anticipated grid development has been evaluated as a strain on financial viability of the usual financiers of transmission projects, namely the TSOs of Europe [64]. Several national regulators are seeking possibilities to encourage private investment in grid projects [65]. The issue of investment is a crucial factor, because it could slow or threaten the feasibility of transmission developments, and yet it is not extensively studied. Issues of repaying investment costs and equitable distribution of costs and benefits have been examined, but only using a simple transport model to balance generator and investor benefits [66], or only considering a fixed grid [67]. Acknowledging that the grid grows in stepwise increases can be a valuable element of realism to inform industry and transmission companies [68]. For expansion planning, a formulation where the branch capacities are free to change is key to discovering alternate possibilities that can reward both society and investors. The main focus of this chapter is on TEP for a high voltage direct current (HVDC) grid in the North Sea region. The following section presents an overview of existing TEP formulation.

6.4 REVIEW OF TEP FORMULATIONS This section provides a market-based approach to solve a long-term TEP for meshed grids that connect large amount of RES to regional onshore markets. The findings of this part provide economic insight into the operation of a meshed AC as well as multiterminal HVDC grids. The proposed framework can support transmission

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system planners and private investors, as it determines the most economically efficient design to invest in. In classic benefit
cost economics, economic efficiency is measured through social welfare or surplus, the sum of economic surpluses across all market parties [69]. According to spot-pricing theory, the marginal-cost pricing leads to social welfare maximization [70]. Social welfare is defined as the sum of surpluses gained by all market participants (minus externality costs, if any) [31,69]. In theory, the maximum social welfare can only be attained under a perfectly competitive market [71]. Actual markets attempt to achieve a higher social welfare close to the maximum level. Therefore, the market clearing process is a social welfare maximization optimization problem that meets the physical constraints of the system (transmission constraints) and is called the optimal power flow (OPF) problem. The solution to the OPF gives equilibrium between the supply and demand in the whole region. The problem is that for capital intensive investment, such as for the transmission system, the marginal cost value can easily drop below the average cost3. In that case, marginal pricing can lead to underrecovery or deficit, which is an economic inefficiency. Perez-Arriaga et al. have previously shown that congestion rents can only contribute to a fraction of the total grid costs in practice [72]. Therefore, for transmission investments, a solution that only gives the highest social welfare is not necessarily the most economically sound, and provisions are required to recover the unallocated investments. Repaying transmission investment costs and equitable distribution of costs and benefits have been a challenging issue for TSOs4. Several methods have been developed to address the revenue reconciliation problem, such as neutral tax revenues or the so-called “second-best pricing schemes” (as they induce deviations from marginal cost pricing [75]), including average cost pricing, right of way pricing, fair rate of return regulation, welfare optimal breakeven point and peak-load pricing [74,76
78]. Peak-load pricing is based on the theory of long-run marginal cost. It is obtained by taking the investment capital cost explicitly into objective of the traditional welfare maximization problem. It was formalized by Crew et al. [77] to take the generation investment costs implicitly into account for welfare maximization. By adding a model of the AC transmission network, Lecinq and Ilic applied peak-load pricing for AC transmission pricing [74]. In what follows, static TEP (STEP) framework for both AC and DC systems are discussed. It highlights the basic concepts that are used when formulating the problem in the context of competitive electricity markets.

3

Average cost is defined as total cost divided by the quantity of the product, here, the amount of energy produced, consumed, or transferred. 4 The problem of economies of scale in infrastructure concerning the need for pricing the infrastructure, such that costs be recovered when marginal price is less than the average price, was first addressed by Frank Ramsey and M. Boiteux. Ramsey
Boiteux pricing was originally developed for naturally monopolistic public firms that seek social welfare maximization subject to a budget constraint [73]. When the marginal price is less than the average price, then the price charged by a natural monopoly cannot be equal to the marginal cost because this would generate losses. Ramsey
Boiteux prices can be used to satisfy the budget constraint [74] and are formally proven to present the secondbest optimal, as they deviate from marginal costs [74].

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6.4.1 FORMULATION OF TRANSMISSION EXPANSION PLANNING PROBLEM In this section, weighted STEP frameworks that take into account the probability of occurrence of various system states (ωt ). The formulation carries the advantage of including all operating states relevant to the design, but compresses their representation. Consider a power system with nðΩz Þ buses indexed i 5 1; 2; . . . ; nðΩz Þ. Ωz is the set of all buses. Each bus is assumed to be a price area of producers and consumers that participate in a zonal competitive market. For a hybrid AC and DC system, the STEP problem takes on the form maxΩ 5 φ 2 ψ

(6.1)

Subject to X

PGt g 2

gAGi

Pti 5

X

PDt d 5 Pti ; ’i; jAΩz ; ’tAΩO

(6.2)

dADi

X jAΩz

t fij;AC 1

X

t fij;DC ; ’iAΩz ; ’tAΩO

(6.3)

jAΩz

Pimin;t # Pit # Pimax;t ; ’iAΩz ; ’tAΩO

(6.4)

t t 5 Nij;AC  fij;cbl ; ’i; jAΩz ; ’tAΩO fij;AC AC

(6.5)

t t 5 Nij;DC  fij;cbl ; ’i; jAΩz ; ’tAΩO fij;DC DC

(6.6)

t;max t # fij;AC  wij;AC ; ’i; jAΩz ; ’tAΩO fij;cbl AC

(6.7)

t;max t # fij;DC  wij;DC ; ’i; jAΩz ; ’tAΩO fij;cbl DC

(6.8)

h

  2   t fij;cbl 2 gij;AC vti;AC 2 vti;AC  vtj;AC  cos δti 2 δtj 1 bij;AC vti;AC  vtj;AC  sin δti 2 δtj # AC   1 2 wij;AC  M; ’i; jAΩz ; ’tAΩO h i  2  t 2 gij;DC vti;DC 2 vti;DC  vtj;DC # 1 2 wij;DC  M; ’i; jAΩz ; ’tAΩO fij;cbl DC

(6.10)

Nij;AC $ 0; ’i; jAΩz

(6.11)

Nij;DC $ 0; ’i; jAΩz

(6.12)

0 # PtGg # Pmax;t Gg ; ’gAGi ; ’iAΩz ; ’tAΩO

(6.13)

0 # PtDd # Pmax;t Dd ; ’dADi ; ’iAΩz ; ’tAΩO

(6.14)

δti 5 0; ’tAΩO     t t max 2δmax ij 2 1 2 wij;AC  M # δi 2 δj # δij 1 1 2 wij;AC  M; ’iAΩz ; ’tAΩO

(6.15)

t max vmin i;DC # vi;DC # vi;DC ; ’iAΩz ; ’tAΩO

(6.17)

t max vmin i;AC # vi;AC # vi;AC ; ’iAΩz ; ’tAΩO

(6.18)

wij;DC Af0; 1g; ’i; jAΩz

(6.19)

wij;AC Af0; 1g; ’i; jAΩz

(6.20)

(6.9)

(6.16)

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

The first term in the objective function (6.1), φ5

X tAΩO

"

XX iAΩz dADi

PDt d



λtDd

2

XX

# PGt g



λtGg

 ωt

(6.21)

iAΩz gAGi

presents the social welfare over the whole planning horizon. The second term in the objective function, ψ5

   i 1X Xh 0 0 t;max t;max  Lij 1 wij;DC kijDC 1 kijDC  Nij;DC  fij;DC  Lij  nðΩO Þ (6.22) wij;AC kijAC 1 kijAC  Nij;AC  fij;AC 2 iAΩ jAΩ z

z

denotes the total investment cost of building AC and DC transmission infrastructures. Note that the investment cost includes both investment and installation costs as proposed in Ref. [79]. In the formulation above, Ωt is a set of all operating hours included in the planning horizon. ΩZ is the set of indexes of all price zones. Gi is the set of indexes (g) of all generating units in zone i. Likewise, Di is the set of all indexes (d) of the demands located in zone i. Eq. (6.2) enforces power balance at every operating scenario. Eq. (6.3) defines the power injection Pit of zone i into the rest of the system. Constraint (6.4) enforces power injection constraints, and Pimin;t # 0 # Pimax;t . Constraint (6.4) limits the combined power import/export of every country over AC and DC connections to certain range. Note that the value of Pimin;t ; Pimax;t might be determined based on security provisions that every price zone considers. In the context of this work, the AC interconnector connects zone i and j is assumed to compose t of a number Nij;AC of identical parallel AC cables fij;cbl , each with conductance gij;AC and suscepAC tance of bij;AC . In a similar manner, the DC interconnector connecting zone i and j is assumed to t composed of Nij;DC of identical parallel DC cables fij;cbl , each with conductance gij;dc . Eqs. (6.5) and DC (6.6) define the power flow over respectively AC and DC interconnectors connecting zone i and j. Eqs. (6.7) and (6.8) express the Kirchhoff’s Voltage Laws as disjunctive constraints for the candidate cables, for AC and DC technology, respectively. Note that wij;AC and wij;DC are integer variables and take a one value for integer that exist. For AC interconnectors that exist, wij;AC 51 constraint (6.7) limits the power flow of AC cables, to the maximum capacity. Likewise, for DC interconnectors that exist wij;DC 51 and (6.8) limits the power flow of DC cables to the maximum capacity. For solutions with near-zero Nij;AC and Nij;DC values, wij;AC 5wij;DC 50 and therefore the constraints become inactive. In a similar manner, for wij;AC 5 1 and constraint (6.9) becomes an equality constraint equivalent to  2     t fij;cblAC 2gij;AC vti;AC 2vti;AC  vtj;AC  cos δti 2 δtj 1 bij;AC vti;AC  vtj;AC  sin δti 2 δtj 5 0. For wij;DC 51,  2  t constraint (6.9) becomes an equality constraint equivalent to fij;cbl 2gij;DC vti;DC 2 vti;DC  vtj;DC 50. DC For wij;AC 5 wij;DC 5 0, Eqs. (6.9) and (6.10) do not constrain the power flow in the left-hand side since M is an adequately large positive number. Constraints (6.11) and (6.12) enforce the number of AC and DC cables respectively to be a positive real number. Eq. (6.13) states that each generator must produce below its capacity. Eq. (6.14) states that power consumed by each consumer must be a value between zero and its capacity. Eq. (6.15) sets the angle of the reference zone to zero. Constraint (6.16) enforces the fact that the angle difference between the two ends of a line cannot exceed a certain level. Note that vti is per-pole line-to-ground voltage of DC converters at each end of the bipolar interconnector. Due to operating limits, the DC voltage of the converters is bound between 0.9 p.u. and 1.1 p.u. As power flow of each DC cable is precisely controlled through voltage control of the converters,

6.4 REVIEW OF TEP FORMULATIONS

189

zonal voltages are considered independent decision variables. Therefore, Eq. (6.17) ensures that the voltage of the DC converters at the gate remains within acceptable operational range. Eqs. (6.18) and (6.19) define the integer variables wij;AC and wij;DC as discussed above. Only the steady state operating conditions are considered. Thus, the power system is assumed to be dynamically secure. That is, the DC converters are utilized with a robust control system (e.g., using voltage margin method [4] or voltage droop control [80]) which maintains transient stability of the system after a disturbance. The results of the optimization formulation (6.1)
(6.17) include an angle, a voltage (independent optimization variables), and an injection (dependent optimization variable) for each converter, which represent its steady-state operating point. Therefore, the optimization model determines the reference values of the voltage angles, DC-voltage as well as the supply/demand bids of the generation unites and consumers in each zone, for each operating state. In addition, the optimal solution includes grid topology and transmission capacities to meet the objective function. The optimization problem (6.1)
(6.17) is a multiperiod, nonlinear, nonconvex, mixed-integer and large-scale optimization problem. Therefore, it has a high degree of complexity and is computationally very intensive to solve.

6.4.2 SIMPLIFYING THE PROBLEM FORMULATION 6.4.2.1 AC Network The main objective of TEP is to expand existing power systems to enable them to create sufficient capacity for cross-border power exchange, to accommodate new types of renewable sources of energy and to serve growing demand in the future. TEP for AC networks has been investigated thoroughly in literature [45,81
83]. Early models were based on linear programming [84]. Proper accounting of physical flows using linearized DC approximation [70], allows Kirchhoff’s Voltage Laws (KVL) to be enforced with disjunctive constraints instead of nonlinear ones and, therefore, provided a key to understanding the connections between actual lines, their cost, and benefits to different regions [32,85]. In Refs. [86,87], the authors propose multistage TEP, but ignore the interactions of transmission and generation investments. Sauma et al. [88] propose a multistage game
theoric
based transmission and generation expansion planning framework that incorporates the effects of strategic interaction between generation and transmission, but such models are computationally burdensome, especially when applied to real-world problems. A number of studies [89,90] propose cooptimization of generation and transmission expansion based on mixed-integer formulation of flexible topology controls discussed in Refs. [91,92]. Using a large-scale wind generation and TEP model, the authors in Refs. [93,94] show that ignoring the interdependency between transmission and wind expansions can lead to a suboptimal solution. Physical limitations of the AC technology, especially for long distances and offshore applications (i.e., excessive reactive current drawn by the cable capacitances that induces excessive cable losses and demands reactive shunt compensation to control voltages and avoid overvoltages [95]), in addition to recent advances in the HVDC technology, have triggered interest in exploration of the application of HVDC technology for large-scale, long-distance transmission applications. There are two types of HVDC transmission systems • •

Current Source Converter (CSC) HVDC (or classical HVDC) and Voltage Source Converter (VSC) HVDC.

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

VSC has significant advantages over CSC, which makes this technology favorable for long distance and offshore cables. VSC HVDC technology utilizes better control of the electricity flow and direction. VSC is more compact compared with CSC and is easier to design. It enables “black start” capability and connection to weak power systems. Last but not least, it enables multiterminal configurations. Therefore, it is preferred technology for the development of a meshed grid design [96]. Previous studies [97,98] show that the DC transmission system can be considered as an alternative to AC technology in competitive electricity markets. Therefore, the rest of this chapter is focused on transmission planning problem using VSC-HVDC technology. In what follows a set of simplifying assumptions are proposed to simplify the STEP problem for VSC-HVDC technology to make the power flows linear, and allows for deriving the analytical solution of the problem. The analytical derivation of the solution is later shown to be useful for investigating the impact of different economic and policy conditions.

6.4.2.2 Continues Social Welfare As nonlinear, step-wise, aggregated supply
demand curves are more difficult to work with, we assume that aggregated supply
demand curves are linear function of power generation and consumptions. The linear supply and demand curves are presented in Fig. 6.1 and read respectively as follows: λtGg 5 atGg  PGt g 1 bGt g ; ’gAGi ; ’iAΩz ; ’tAΩO

(6.23)

λtDd 5 2atDd  PtDd 1 btDd ; ’dADi ; ’iAΩz ; ’tAΩO

(6.24)

where atGg and atDd are the slope of the supply and demand curves, btGg is the lowest price producers require before participating in any production activity, and btDd is the maximum price the consumers are willing to pay for the electricity. This assumption allows determination of the changes in social welfare (i.e., incremental social welfare) as a quadratic function of power injection of each zone in the rest of the system (see Fig. 6.2) and, therefore, makes the problem quadratic and so, easier to solve. It also enables Price

Price

PD

PD

PS

PS

Pi=PGi–PDi>0

ρi

ρi Pi=PGi–PDi< 0

quantity 0

PDi

PGi

(A) Net generation node

quantity 0

PGi

PDi

(B) Net consumption node

FIGURE 6.1 Linear supply and demand curve of a given market: (A) Net generation node and (B) net consumption node.

6.4 REVIEW OF TEP FORMULATIONS

191

ISCi(Pi)

b Pimin = – i 2ai

Pi 0

Pimax

FIGURE 6.2 and Pmax represent minimum and maximum power that region A typical incremental social cost curve. Pmin i i i can inject during operating hour t, with the sign convention such that negative injection means import (local load exceeds local generation).

implicit modeling of the operation of the onshore electricity markets and their responses to the power trades. Note that one could use more complex assumptions (i.e., nonlinear supply/demand curves). Then one would need to define social welfare explicitly as a function of total quantities demanded and supplied. This makes the objective function nonsmooth and more difficult to solve. Note that, for such nonsmooth problem, the Karush
Kuhn
Tucker (KKT) optimality conditions cannot be applied, and therefore, the analytical solution is not accessible.

6.4.2.3 Linear Transmission Investment Cost For the sake of simplicity, the cost of converters and interconnectors were represented as a linear function of the length and rated capacity of interconnectors [i.e., kijAC 5kijDC 50 in Eq. (6.22)]. All interconnectors were assumed to have been built using VSC-HVDC technology. In the absence of economies of scale, the cost of converters and interconnectors were assumed to be linearly dependent on length and rated capacity of the cables. However, high-voltage converters are extremely expensive equipment. Adding extra converters results in higher investment and operational costs; therefore, considering the cost of converters implicitly in the cost of interconnectors may induce an error in the cost calculations as investment cost of transmission infrastructure might be different from what is reported in this chapter. The formulation, resulted after applying the simplifications outlined above, is a continuous, nonlinear optimization formulation and allows for the consideration of multiple time periods in multiple development stages of the grid. The final results include optimal grid topology, transmission capacities, construction timing, and the resulting remuneration and distribution of the social welfare increase among the various onshore price zones.

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

6.4.3 STATIC TRANSMISSION EXPANSION PLANNING In this section, we present a weighted STEP framework that takes into account the probability of occurrence of various system states. This formulation carries the advantage of including all operating states relevant to the design, but compresses their representation5.

6.4.3.1 Assumptions We have assumed electricity markets are perfectly competitive. This implies that generators participate in the market with their short run marginal costs. For the sake of simplicity, a zonal market model is used in which the aggregated supply and demand bidding curves of each onshore zone are linear functions of the power generation/consumption of that zone. Only maximum and minimum import and export power constraints per zone were enforced. No intertemporal constraints on generation were considered. Only transmission investment decisions were considered. Intrazonal transmission constraints were neglected. In addition, the interconnector between zone i and j is assumed to be composed of a number Nij of identical parallel cables. In order to apply KKT conditions and make the analytical solution accessible, the problem was made continuous (as discussed in the previous section). Therefore, we assumed that Nij is continuous and differentiable variable. Any transmission operating cost component is neglected. The impact of discount, inflation and interest rates were not taken into account. Finally, a centralized decision-making entity (e.g., centralized offshore grid system operator) is considered that is regulated to maximize the social welfare and to provide nondiscriminatory transmission service.

6.4.3.2 Problem Formulation The simplified STEP problem for meshed VSC-HVDC grids that connect regional markets takes on the form   maxΩ vti ; Nij;DC 5 φ 2 ψ

(6.25)

Subject to  t t t max vmin i;DC # vi;DC # vi;DC εi ; γ i Þ; ’iAΩz ; ’tAΩO   Pmin # Pit # Pmax ; αti ; β ti ; ’iAΩz ; ’tAΩO i i

t fij;cbl DC



(6.27)

t t fij;DC 5 Nij;DC  fij;cbl ; ’i; jAΩz ; ’tAΩO DC

(6.28)

wij;DC Af0; 1g; ’i; jAΩz   t;max  wij;DC ; # fij;DC  wij;DC μtij ; ’i; jAΩz ; ’tAΩO

(6.29)

t fij;cbl 2gij;DC vti;DC DC

5

(6.26)

2

2 vti;DC  vtj;DC



  # 1 2 wij;DC  M; ’i; jAΩz ; ’tAΩO

(6.30) (6.31)

Due to the computational burden of TEP problems, the size of the input set has to be reduced to a limited number of representative operating states (or operating scenarios). Each scenario represents a group of operating hours from the original set that are similar to each other. The weights reflect the number of operating states each scenario represents. It also denotes the extent, to which every scenario contributes in the final grid design. Neglecting the weights results in a less realistic suboptimal grid design.

6.4 REVIEW OF TEP FORMULATIONS

  Nij;DC $ 0; ξ ij ; ’i; jAΩz ; X

t Pit 2 Ploss 5 0; ðπt Þ; ’i; jAΩz ; ’tAΩO

193

(6.32) (6.33)

iAΩz

P t where Pti 5 2  jAΩz fi;cbl . Constraints (6.26) and (6.27) enforce nodal voltage and power injection DC # 0 # Pmax . For interconnectors that exist, wij;DC 51 and constraints, respectively, and Pmin i i constraint (6.29) limits the power flow of each cable to its maximum capacity. For solutions with near-zero Nij;DC values, wij;DC 5 0, and therefore, the constraint becomes inactive. Eq. (6.29) defines wij;DC is an integer variable. For interconnector i 2 j that exists, Eq. (6.30) limits power flow of each cable to its maximum capacity. Eq. (6.31) expresses the Kirchhoff’s Voltage Laws as disjunctive constraints for the candidate lines. For wij;DC 5 1, (6.31) becomes an equality constraint  2  t 5gij;DC vti;DC 2 vti;DC  vtj;DC . For wij;DC 50, Eq. (6.31) does not constrain the equivalent to fij;cbl DC power flow in the left-hand side since M is an adequately large positive number6. In order to make the problem continuous, also to reduce the computational complexity, we have simplified the problem by assuming that transmission capacity can be added in small increments  2  t (i.e., wij;DC 51; ’i; jAΩz ). Then, fij;cbl 5gij;DC vti;DC 2 vti;DC  vtj;DC ; ’i; jAΩz and constraint DC (6.31) becomes:    2  t;max t fij;cbl 5gij;DC vti;DC 2 vti;DC  vtj;DC # fij;DC ; μtij ; ’i; jAΩz ; ’tAΩO DC

(6.34)

Constraint (6.34) limits power flow of each cable to its maximum capacity7. Note that for Nij;DC near-zero (i.e., meaning that there is no interconnector between zone i and j), constraint (6.34) unnecessarily enforces a power flow limit (as the corridor does not exist) in the final solution. As a result, applying the simplification yields an optimization problem formulation that is more constrained than the original problem. An analysis of the possible inefficiencies caused by this simplification is beyond the scope of this chapter and is a subject for future research. Eq. (6.32) enforces the number of cables to be a positive real number. Constraint (6.33) enforces the real power balance. Ptloss here represents the transmission power losses over all interconnectors and DC converters during hour t. It states that total power injected into the transmission grid equals total power withdrawn from the grid and transmission losses. We have assumed that the power system is dynamically secure and therefore have only considered the steady state operating conditions. That is, the DC converters are utilized with a robust control system (e.g., using voltage margin method [100] or voltage droop control [4]) which maintains transient stability of the system after a disturbance. The results of the optimization formulation (6.25)
(6.28) and (6.32)
(6.34) include a voltage (independent optimization variable) and an injection (dependent optimization variable) for each converter, which represent its steadystate operating point. Therefore, for each operating state, the reference values of the DC-voltage or active power of the DC converters have to be set to the optimal value determined by the model.

6

Similar approach have been used for representing the KVL constraints in literature [9,88]. Note that in a multiterminal HVDC grid, power flows according to the Kirchhoff’s laws and, unlike single HVDC circuits with just terminals at each end, cannot be independently controlled in each circuit [99]. 7

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

This is a nonlinear, nonconvex optimization problem. The Lagrangian (L) correspondingly reads as X X X  t;max     ℒ vti;DC ; Nij;DC 5 Ωðvti;DC ; Nij;DC Þ 1 μtij fij;DC 2 gij;DC ðvti;DC Þ2 2 vti;DC  vtj;DC  ωt  nðΩO Þ 1

XX iAΩz tAΩO

1

XX

iAΩz jAΩz tAΩO

αti

 max  Pi 2 Pti  ωt  nðΩO Þ

  t β ti Pti 2 Pmin  ω  nðΩO Þ i

iAΩz tAΩO

1

XX

(6.35)

t t γti  ðvmax i;DC 2 vi;DC Þ  ω  nðΩO Þ

iAΩz tAΩO

1

XX

t εti  ðvti;DC 2 vmin i;DC Þ  ω  nðΩO Þ

iAΩz tAΩO

1

XX

ξ ij  Nij;DC 1

iAΩz tAΩO

X tAΩO

πt 

X

Pti 2 Ptloss  ωt  nðΩO Þ

iAΩz

6.4.3.3 Analytical Solution The KKT optimality conditions state that any set of vti;DC , Nij;DC that satisfies the constraints (6.26)
(6.28) and (6.32)
(6.34) is a local solution to the problem (6.25) if, and only if, there exists a set of nonnegative Lagrangian multipliers such that 1. Multipliers associated with inactive constraints are zero, and also; 2. Derivatives of the Lagrangian with respect to the two decision variables are zero, that is, @Lðvti;DC ; Nij;DC Þ=@vti;DC 5 0 and @Lðvti;DC ; Nij;DC Þ=@Nij;DC 5 0; ’i; jAΩz ; ’tAΩO .

6.4.3.3.1 Full HVDC Power Flow The derivative of Lðvti;DC ; Nij;DC Þ with respect to vti;DC reads as X @Φðvti;DC ; Nij;DC Þ @Ptj @Φðvti;DC ; Nij;DC Þ @Pti  t 1  t t @Pi @Ptj @vi;DC jAZ @vi;DC X t;max t μtij  ð fij;DC 2 fij;cbl Þ  ωt  nðOÞ t X tAO @fij;cbl 1  t DC t @vi;DC @fij;cblDC jAZ X t;max t μtij  ð fij;DC 2 fji;cbl Þ  ωt  nðOÞ t X tAO @fij;cbl 1  t DC t @v @f i;DC ij;cblDC jAZ X X αtk  ðPmax 2 Ptk Þ  ωt  nðOÞ k @Pt 1 tAO kAZ  ti @vi @Pti XX t t αtk  ðPmax 2 P Þ  ω  nðOÞ k k X tAO kAZ @Ptj 1  t t @vi;DC @Pj jAZ X X t β tk  ðPtk 2 Pmin Þ  ω  nðOÞ k @Pt tAO kAZ 1  t i t @vi;DC @Pi XX t β tk  ðPtk 2 Pmin Þ  ω  nðOÞ k X tAO kAZ @Ptj 1  t 1 ðEti 2 γti Þ  ωt  nðOÞ t @vi;DC @Pj jAZ  X X  t t @ πt P 2 P  nðOÞ i loss iAZ 1

tAO

@vti;DC

50

(6.36)

6.4 REVIEW OF TEP FORMULATIONS

195

The partial derivations infer that changing the voltage of one zone induces a change in power flow of all interconnectors that are connected to this zone. Expanding @Lðvti;DC ; Nij;DC Þ=@vti;DC 50, and limiting the analysis to the case where voltage constraints are not binding (i.e., Eti 5 γ ti 50) one obtains: 2 X

X

  gij  Nij;DC  vtj;DC  ρtj 1 αtj 2 β tj 2 πt  ωt  nðOÞ 2 2

jAZ

    gij  Nij  2  vti;DC 2 vtj;DC  ρti 1 αti 2 β ti 2 πt  ωt  nðOÞ

jAZ

5

X

(6.37)

    gij  μtij  2  vti;DC 2 vtj;DC 2 μtji  vtj;DC  ωt  nðOÞ

jAZ

By multiplying vti;DC to both sides of (6.37), taking the sum over all zones at all times, after some manipulation one obtains: XXX

  t fij;cbl  ρtj 1 αtj 2 β tj 2 πt  ωt  nðOÞ DC

tAO iAZ jAZ

2

XXX

  t fij;cbl  ρti 1 αti 2 β ti 2 πt  ωt  nðOÞ DC

tAO iAZ jAZ

5

XXX

(6.38)

t fij;cbl  μtij  ωt  nðOÞ DC

tAO iAZ jAZ

μtij

where is Lagrangian multipliers associated with interconnector i 2 j and is strictly positive at line capacity, αti and β ti are Lagrangian multipliers associated with maximum and minimum power injections of each zone, respectively. Following the definition proposed by Schweppe et al. [70], the short  t run marginal cost of power injection of zone i at hour t is defined as: t t t ρi 5 @ ISCi Pi =@ Pi as the zonal price. Eq. (6.38) is the pricing mechanism. This formulation expresses the relation between hourly zonal prices, power flows of all interconnectors, and associated congestion revenues. An ideal pricing mechanism is one, in which the power injections are only a function of zonal price difference [101]. It states the amount of power to be exchanged between zones as a function of price and associated (shadow) congestion rent magnitudes. Due to power the power content  losses,  of a cable t t at the sending end is different from of it at the receiving end fij;cbl ¼ 6 f ; ’i; j; t . As a result, ji;cblDC DC the HVDC-driven pricing mechanism is not ideal and its interpretation is not consistent with the existing AC counterpart. In the next subsection, we show that this problem can be resolved by approximating the HVDC power flow.

6.4.3.3.2 Approximated HVDC Power Flow Fig. 6.3 presents a bipolar HVDC interconnector connecting zones i and j. The power flow over an HVDC interconnector at sending end i can be rewritten as follows:

h

   2  2 i Nij;DC  gij Nij;DC  gij  vti 2 vtj 1  ðvti 2vtj Þ2 fijt DC 5 Nij;DC  gij  ðvti Þ2 2 vti  vtj 5 2 2

(6.39)

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

FIGURE 6.3 Block diagram of bipolar HVDC system with earth return.

The first term on the right side of the equation expresses the power flow at the midpoint (Fijt m in t t Fig. 6.4) of the interconnector and the second term represents half of the line losses (Δfij;s 5Δ fij;r in Fig. 6.4). Losses typically amount to less than 5% of total power flows of a HVDC cable. Therefore, neglecting the second term of (6.39) introduces no significant error to the power flow calculation and yields the HVDC power flow as Fijt 





 Nij;DC  gij  uti 2 utj 2

(6.40)

where  uti 5 vti Þ2 :

Transmission losses are dependent on voltage magnitude and length of the cable: Losses increase, as length of the cable increases or operating voltage decreases. The highest transmission losses (and so approximation errors) are expected at the receiving end of the cable, when the system is operated at lowest operating voltage allowed (i.e., 0:9 p.u. at receiving end). Fig. 6.4 presents the monopole power content of a 2000 MW 500 kV bipolar conventional HVDC link with copper conductor of cross-section 1800 mm2 [99] and a length of 1800 km (the longest connection distance expected across the North Sea), when power flows from i to j. Approximation error at the receiving end amounts to no more than about to 2.12% of the rated power. Considering 1.1% losses for each of the two VSC converters, total approximation error amounts to less than 3.3% at each end. Note that all financial calculations carried out, hereafter, are based on approximated power flow (6.39)8.

8

In HVAC system, losses are estimated to account for 6
8% of the net generation. Congestions are estimated to account for 7
10%of the net generation. As HVDC technology presents approximately 25% lower losses compared to HVAC [101], one could argue that, for an HVDC system, congestion costs account for a larger proportion of total costs compared to losses. To simplify the problem in this chapter, an approximation of HVDC power flow was used, in which transmission losses are neglected.

6.4 REVIEW OF TEP FORMULATIONS

197

Power flow over a 2000 MW, +/– 500 KV bipolar conventional HVDC link with copper conductor of cross section of 1800 mm2 1005 Power flow over cable Power flow at the midpoint 1000 995

ΔFij,s

Power flow MW

990

Fij,m

985 980 975 970

ΔFij,r

965

ΔFij,s =ΔFij,r = (Vi –Vj).(Vi –Vj)/2

960 955

0

50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 Length of the cable from the sending end

FIGURE 6.4 Power flow over a 2000 MW, 1 /500 kV bipolar conventional HVDC link with, copper-conductor of cross-section of 1800 mm2.

6.4.3.4 Analytical Solution to the Optimization Problem Using the Approximated HVDC Power Flow By definition, power losses are neglected in the proposed linearly approximated HVDC power flow (i.e., Pt loss 5 0). Replacing the approximated HVDC power flow in Eq. (6.33) and expanding it gives nb X i51

Pti 5

nb X nb X i51 j51

Nij;DC  gij 

t

ui 2 utj 50 2

(6.41)

which is trivial. Therefore, the power balance constraint becomes trivial and should no longer be considered in the formulation of the problem. Consequently, the Lagrangian multiplier associated with the power balance constraint πti is not required and thus does not appear in the analytical solution.

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

Substituting the approximated power flow (6.41) in (6.25)
(6.34), also limiting the analysis to the case, in which voltage constraints are not binding, one derives the analytical solution to the problem, as follows: @ℒðuti ; Nij;DC Þ 50 @uti

XX gij  Nij;DC  ðρti 1 αti 2 β ti Þ  ωt  nðOÞ ðuti 2 utj Þ  2 iAZ jAZ XX gij 5 ðuti 2 utj Þ   μtij  ωt  nðOÞ 2 iAZ jAZ

.22 

(6.42)

The left side of Eq. (6.42) can be rewritten as 22 

XX



ðuti 2 utj Þ 

iAZ jAZ

52

XX

gij  Nij;DC 2

ðuti 2 utj Þ 

iAZ jAZ

1

XX

ðuti

2 utj Þ



iAZ jAZ

5

XX

ðuti 2 utj Þ 

iAZ jAZ

5

XX

Fijt 

h

gij  Nij;DC 2 gij  Nij;DC 2

gij  Nij;DC 2

   ρti 1 αti 2 β ti  ωt  nðOÞ

   ρti 1 αti 2 β ti  ωt  nðOÞ



  ρtj 1 αtj 2 β tj  ωt  nðOÞ

h   i   ρtj 1 αtj 2 β tj 2 ρti 1 αti 1 β ti  ωt  nðOÞ

(6.43)

    i ρtj 2 ρti 1 αtj 2 αti 2 β tj 2 β ti  ωt  nðOÞ

iAZ jAZ

52

X

  Pti  ρti 1 αti 2 β ti  ωt  nðOÞ

iAZ

The right side of Eq. (6.42) gives

 g XX

XX ij uti 2 utj   μtij  ωt  nðOÞ 5 Fijt  μtij  ωt  nðOÞ 2 iAZ jAZ iAZ jAZ

(6.44)

t where Fij;cbl is monopole power transmitted over every cable of interconnector i 2 j. KKT DC t;max t 5 fij;DC ), optimality conditions state that μtij is zero unless constraint (6.34) is binding (i.e., Fij;cbl DC thus:

XX iAZ jAZ

Fijt  μtij  ωt  nðOÞ 5

XX

t;max fij;DC  μtij  ωt  nðOÞ

(6.45)

iAZ jAZ

Substituting (6.43) and (6.45) in (6.42), then taking sum overall operating states gives: XXX

Fijt 

h

XX     i   ρtj 2 ρti 1 αtj 2 αti 2 β tj 2 β ti  ωt  nðOÞ 5 Pti  ρti 1 αti 2 β ti  ωt  nðOÞ

tAO iAZ jAZ

tAO iAZ

5

XXX

t;max fij;dc  μtij  ωt  nðOÞ

tAO iAZ jAZ

(6.46)

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199

Eq. (6.46) is the pricing mechanism for a meshed HVDC grid. The pricing mechanism inferred expresses the relation between amount of power to be exchanged, electricity prices in different zones and congestion revenues. The proposed linearized DC power flow results in a simplified pricing mechanism that is consistent with its AC counterpart.  Expanding @L uti ; Nij;DC =@Nij;DC 5 0, and then multiplying both sides by Nij;DC =2 gives X tAO

Fijt 



 0 t;max  kijDC  Nij;DC  Lij  nðOÞ 2  fij;DC      ρtj 2 ρti 1 αtj 2 αti 2 β tj 2 β ti  ωt  nðOÞ 5 2

(6.47)

Note that Nij;DC  ξij 5 Nji;DC  ξ ji 5 0 (KKT optimality conditions). Eq. (6.47) implies that for an optimal plan, the transmission revenue collected throughout the lifetime of the interconnector i 2 j will recover the investment cost of building it. Now taking sum of Eq. (6.47) over all zone pairs gives: XXX

Fijt 

h

    i ρtj 2 ρti 1 αtj 2 αti 2 β tj 2 β ti  ωt  nðOÞ 5 Ψ

(6.48)

tAO iAZ jAZ

Substituting (6.48) in (6.46), one encounters the condition that at the optimal grid design X

Rtλ 5

tAO

X

Rtμ 5

tAO

X

Rtρ 5 Ψ

(6.49)

tAO

where Ψ is the total investment cost, and the other terms comprise total nodal payment Rtλ at time t: Rtλ 5 2

X

  Pti  ρti 1 αti 2 β ti  ωt  nðOÞ

(6.50)

iAZ

Total congestion revenue Rtμ at time t Rtμ 5 2

XX

t;max μtij  fij;DC  ωt  nðOÞ

(6.51)

iAZ jAZ

and total transmission revenue Rtρ at that time Rtρ 5

XX

Fijt 

h

    i ρtj 2 ρti 1 αtj 2 αti 2 β tj 2 β ti  ωt  nðOÞ

(6.52)

iAZ jAZ

The associated grid design and operation pattern comprised by the solution uti and Nij;DC is one that establishes a balance such that the total transmission revenue generated from nodal price differences is precisely the amount paid by transmission costumers at all zones and also equals congestion revenues. Therefore, the grid design Nij;DC pays off its initial investment capital Ψ through the operation given by uti Note that the proof of investment recovery is valid in the case of bilateral transactions, if the TSO sets the transmission fee to the difference of the spot prices at the injection and reception zones [31].

6.4.3.5 Summary This section has proposed a multiple time-period, STEP framework applicable to HVDC meshed grids. The term “static” reveals the fact that the transmission capacities are considered to be built instantly and remain constant throughout the study period.

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

The analytical solution to the problem provides a pricing mechanism that expresses the relationship between the electricity price of different zones and the congestion charges associated with the interconnectors between them. The proposed approach computes the expansion plan, under which the investment capital will be fully paid-off through congestion revenues by the end of the chosen lifetime of the infrastructure. The proposed expansion plan could be used as a starting point for conducting further technical feasibility studies. The model gives a quantitative insight into market operation. Moreover, it enables the calculation of welfare distribution among the regions involved. The next section introduces an improved dynamic version of the TEP framework proposed here that considers transmission capacity reinforcements, which can vary over time.

6.4.4 DYNAMIC TRANSMISSION EXPANSION PLANNING In this section, we derive a dynamic optimization formulation for the market-based TEP approach to include time-varying nature of transmission reinforcements. In practical situations, transmission infrastructure reinforcements occur in multiple development stages because: (1) Building large transmission infrastructures is costly and takes place over a relatively long course of time, (2) other parts of the power system and electricity markets develop in a gradual manner, and (3) there can be disruptive delays that happen due to unforeseen technical and legal conflicts as discussed in Refs. [102,103]. We perceive that the delays in the implementation of large infrastructure projects can be related to technical (e.g., unavailability of DC breakers), economic (e.g., supply chain shortages), and legal conflicts (e.g., heterogeneous permitting criteria) [104
107]. Particularly for cross-border projects, the legal conflicts and the differences and uncertainty of national regimes results in long and complex permitting procedures and increases the risk for delays. Potential delays that result from the permitting or the public opposition are assumed to prevent 50% of commercially viable cross-border infrastructure projects from being implemented [106]. This is particularly true for the North Sea states, where the permitting regimes for offshore infrastructure are rather complex. Different authorities with different requirements and standards are involved. The authorization procedures vary between 2 years and 10 years, depending on the respective EU Member State [107]. Yet, there is a trend toward the establishment of a more coordinated approach to permitting and toward the introduction of one-stop shops (see also Refs. [108,109]). Therefore, in addition to the optimal grid design, a TEP framework must deliver an optimal expansion strategy. It contains information on project timing, considering the possible impact of different sources of delays. Any static model will fail to provide information regarding the timing of the project. Therefore, a dynamic approach is needed. The remainder of this section is organized as follows. Section 6.4.4.1 reviews the modeling assumptions considered in this section. Section 6.4.4.2 presents problem formulation and Section 6.4.4.3 presents the analytical solution to the problem.

6.4.4.1 Assumptions The electricity markets are assumed to be perfectly competitive. Similar to pervious section, a zonal market model is used, in which the aggregated supply and demand bidding curves of onshore zones are linear functions of the power generation/consumption. The model accounts for technical constraints,

6.4 REVIEW OF TEP FORMULATIONS

201

such as admissible voltage ranges, minimum/ maximum import of every price zone and maximum power flow limit over the offshore grid segment. Other intertemporal and interzonal constraints have been neglected. HVDC flows are approximated linearly, as proposed in Section 6.4.3. We have P assumed that every interconnector is built of a number of identical bipolar cables (Nijy 5 yk51 ΔNijk ), t;max t;max each with the capacity of 2  fij;DC . Here, 2  fij;DC (in MW) denotes the rated bipole capacity of the k identical cables, and ΔNij is the number of cables added to in a preceding year kð # yÞ. Here, Nijy represents the number of identical cables which the interconnector i 2 j consists of, by the end of year y. As for the previous section, Nij is assumed to be continuous and differentiable variable. All interconnectors are assumed to have been built using VSC-HVDC technology. In the absence of economies of scale, the cost of converters and interconnectors are assumed to be linearly dependent on length and rated capacity of the cables. We have assumed that the investment cost of each identical cable built in year k will be recovered in yrec ðkÞ 5 min½nðY Þ 2 k; 25 years, where 25 is the maximum investment recovery time. We also consider for an interconnector, nðY Þ is the last year in the study period Y, and Y is the set of operating years. Any transmission operating cost component is neglected. Finally, a centralized decision-making entity (e.g., centralized offshore grid system operator) is considered that is regulated to maximize the social welfare and to provide nondiscriminatory transmission service.

6.4.4.2 Problem Formulation Consider a power system with nðΩZ Þ nodes, where ΩZ 5 ΩZon , ΩZoff is the set of all on and offshore nodes. Each node represents a price zone within a perfectly competitive market. The whole study period is divided into nðΩY Þ operating years indexed yA1; 2; . . . ; nðΩY Þ. Every price zone experiences different operating conditions (i.e., market prices, wind speed) throughout its economic lifetime. Thus, each year is divided into nðΩOy Þ clustered operating states (indexed by tA1; 2; :::; nðΩOy Þ). Likewise, ΩOy and nðΩOy Þ denote respectively the set of operating states analyzed in year y and the associated number of members. Each Pclustered operating state has the temporal resolution of 1 h and probability of occurrence of ωt;y , tAΩOy ωt;y 5 1, yA ΩY where ωt;y represents the number of operating hours from the original space in year y that are represented by the clustered operating state t. Consequently, ωt;y indicates the significance of the operating state t among all operating states in year y. The contribution of zone i in the power flows of the rest of the system is reflected via its t;y power injection. Pt;y i . Pi is positive if the zone is a net power exporter, and negative otherwise. The annual aggregated incremental social welfare of all zones at all times is defined as in the previous section. The annually amortized aggregated transmission investment cost in year y reads as ψy 5

 1X X y t;max 2  fij;DC  Lij  keq;ij  nðΩOy Þ 2 iAΩ jAΩ Z

(6.53)

Z

y where Lij is the length of the cables (in km), and i; jAZ. keq;ij is the amortized cost of the asset base y keq;ij 5

y X k51

k k ΔNi;j  ρk  kinv;ij  β Base ði; j; k; yÞ;

(6.54)

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

where ρk 5 r  ðr11Þyrec ðkÞ =ðr11Þyrec ðkÞ 2 1 is the annual amortization rate in year k, r denotes the discount rate and, β Base ði; j; k; yÞ is a binary input variable and defined as  β Base ði; j; k; yÞ 5

1; 0;

if if

k # y # k 1 yrec ðkÞ otherwise

(6.55)

β Base ði; j; k; yÞ indicates whether the amortized investment cost of cable that was built in year k t;max k k  ΔNi;j ) should be considered in the investment calculation of year y. kinv;ij denotes (i.e., 2  fij;DC unit cost (in =MW km h) of transmission and converter stations for interconnector i 2 j in year k. y Thus, keq;ij is the amortized cost factor that represents the equivalent cost of transmission interconnectors and converter stations in year y. The DTEP problem is formulated as follows:  X  maxΩ uti ; ΔNi;k j 5

X ψy φy y 2 ð11r Þ ð11r Þy yAY yAY

(6.56)

Subject to   max t;y t;y εi ; γ i ; ’iAΩz ; ’tAΩOy ; ’yAΩY umin # ut;y i # ui i

(6.57)

 t;y  max y # Pt;y ; αt;y Pmin i # Pi i ; β i ; ’iAΩz ; ’tAΩO ; ’yAΩY i

(6.58)

 t;max

t;y 

t;y fij;cbl # fij;DC μij ; ’i; jAΩz ; ’tAΩOy ; ’yAΩY DC



y

y 0 # ΔNi;j # β Base ði; j; y; Y Þ  M; ξ yij ; ζ ij ; ’i; jAΩz ; ; ’yAΩY

 gij;DC  t;y y  Cui;DC 2 ut;y j;DC ; ’i; jAΩz ; ’tAΩO ; ’yAΩY 2 X t;y Pt;y 2  fij;DC ; ’i; jAΩz ; ’tAΩOy ; ’yAΩY i 5

t;y 5 fij;cbl DC

(6.59) (6.60) (6.61) (6.62)

jAΩz t;y t;y 5 2  fij;cbl  Nijy ; ’i; jAΩz ; ’tAΩOy ; ’yAΩY fij;DC DC

Nijy 5

y X

ΔNijk ; ’i; jAΩz ; ’tAΩOy ; ’yAΩY

(6.63) (6.64)

k51

t;y where Pt;y i denotes the total power injection of zone i, and fij;DC denotes the total monopole power transmitted over the interconnector i 2 j during the operating state t in year y. The objective function (6.56) represents the sum of incremental social welfare (φ) of all zones at all times minus the aggregated investment cost of transmission infrastructure (ψ). Note that all financial terms are discounted back to their net present value (NPV). Constraints (6.57) and (6.58) impose the voltage and power limits. ut;y i represents the squared voltage of DC converter station at zone i. WF;y 5 0, Pmax 5 Yt;y where Yt;y For offshore zones we have: Pmin i  Ki i denotes available wind power i i WF;y at offshore zone i during hour t and Ki is the installed wind capacity in year y. Constraint (6.60)

6.4 REVIEW OF TEP FORMULATIONS

203

enforces the upper and lower limits on number of cables and so the capacity of interconnectors to be built in every year. The investment decision variable β Base ði; j; y; Y Þ is a binary number, which is set to one if the model is allowed to make new investment in interconnector i 2 j in year y. M is an adequately large positive number. For β Base ði; j; y; Y Þ 5 1, the upper cap of constraint (6.60) implies that the capacity of the interconnector i 2 j can be increased as large as required. Constraint t;y (6.61) limits the power flow over each cable. fij;cbl denotes the linearly approximated power flow DC over a single cable that interconnector (i 2 j) is made of and gij;DC is the conductance of the cable. Note that constraint (6.61) should be relaxed if there no interconnector is built between (i 2 j). However, power flow constraint is simplified by assuming that transmission capacity can be added in small increments. As a result, constraint (6.61) is enforced between every two zones, even if they are not connected (i.e., no interconnector is built). This may lead to a suboptimal solution and should be researched further in a future study. One solution to this problem is to introduce a binary variable as explained before in previous sections.

6.4.4.3 Analytical Solution to the Problem Eqs. (6.56)
(6.64) represent the DTEP formulation, which is a nonlinear, nonconvex optimization problem. The problem can be solved by forming the Lagrangian dual problem and formulating the KKT conditions. The Lagrangian function takes the following form y ℒðvti ; Nij;DC Þ 5 Ωðut;y i ; ΔNij;DC Þ 1

 t;y t;y t;max X X X uij fij;DC 2 fij;cblDC  ωt;y  nðΩO Þ ð11rÞy

iAΩz jAΩz iAΩO

1

X X X αt;y ðPmax 2 Pt;y Þ  ωt;y  nðΩOy Þ i i i y ð11rÞ iAΩ yAΩ tAΩ y z

Y

O

X X X β t;y ðPt;y 2 Pmin Þ  ωt;y  nðΩOy Þ i i i 1 ð11rÞy iAΩ yAΩ tAΩ y z

Y

O

X X X γ t;y ðumax 2 ut;y Þ  ωt;y  nðΩOy Þ i i i 1 ð11rÞy iAΩ yAΩ tAΩ y z

1

(6.65)

O

X X X εt;y ðut;y 2 umin Þ  ωt;y  nðΩOy Þ i i i y ð11rÞ iAΩ yAΩ tAΩ y z

1

Y

Y

O

y X X X ξ t;y ij;DC  ΔNij;DC

ð11rÞy h i y y X X X ς ij;DC β Base ði; j; y; YÞ  M 2 ΔNij;DC

yAΩY iAΩz JAΩz

1

yAΩY iAΩz JAΩz

ð11rÞy

 y The KKT optimality conditions state that any combination of ut;y ; ΔN that satisfies i ij;DC (6.56)
(6.64) is a possible solution to the optimization problem (6.55) if and only if nonnegative y t;y1 1 Lagrangian multipliers exist such that @Lðut;y and i ; ΔNij;DC Þ=@ui 5 0 t;y1 y y @Lðui ; ΔNij;DC Þ=ΔNij;DC 5 0. Note that the Lagrangian multipliers associated with nonbinding constraints are zero.

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

y t;y1 t;y1 1 Expanding @Lðut;y i to the both sides, while also limiti ; ΔNij;DC Þ=@ui 5 0 and multiplying ui t;y ing the study to the case where voltage constraints are not binding so that Et;y i 5 γ i 5 0 gives

 y @L ut;y i ; ΔNij;DC

5 0.

1 @ut;y i

1 1 1 Rt;y Rt;y Rt;y μ ρ λ y1 5 y1 5 ð11r Þ ð11r Þ ð11r Þy1

where 1 Rt;y λ

5

XX

t;y1 fij;cbl DC



y1 X

! k ΔNij;DC

 t;y1 1  ωt;y1  nðΩOy1 Þ; ’t; y1  λt;y j 2 λi

(6.66)

(6.67)

k51

iAΩz jAΩz 1 Rt;y μ 5

XX

t;max t;y1 1 fij;DC  μt;y  nðΩOy1 Þ; ’t; y1 ij  ω

(6.68)

iAΩz jAΩz 1 Rt;y ρ 52

XX

1 1 Pt;y  λt;y  ωt;y1  nðΩOy1 Þ; ’t; y1 i i

(6.69)

iAΩz jAΩz

 t;y  t;y   t;y  t;y =@ Pi is the short-term marginal price of generation. αt;y where ρt;y i 5 @ ISCi Pi i and β i are the Lagrangian multipliers associated with the max/min power injection constraints (6.55). t;y t;y t;y λt;y i 5 ρi 1 αi 2 β i is the wholesale zonal market price of zone i during operating state t in year y. Eq. (6.66) is the pricing mechanism. It expresses the relation between nodal prices, power flows over every interconnector and the associated congestion charges. Taking the sum of Eq. (6.66), overall operating hours and years gives X X yAΩY tAΩO

X X Rt;y Rt;y μ λ 5 ð11rÞy ð11r Þy y yAΩ tAΩ y Y

(6.70)

O

t;y where Rt;y λ and Rμ are respectively transmission revenue and congestion revenue at time t in year y. Eq. (6.68) states that the total transmission revenue collected throughout the study period equals the total congestion revenue collected in the same period. y y1 Expanding @Lðut;y i ; ΔNij;DC Þ=@ΔNij;DC 5 0 gives y @ℒðut;y i ; ΔNij;DC Þ y1 @ΔNij;DC

50 t;y t;y t;y t;y1 X gij  ðut;y  nðΩOy1 Þ i 2 uj Þ  ðλj 2 λi Þ  ω y 2  ð11rÞ y5y1 tAΩOy hXnðΩ Þ i Y y1 t;max y 2  fij;DC  Lij  nðΩOy1 Þ  β ði; j; y; YÞ  ρ  k Base inv;ij y5y

.2  5

nðΩ XY Þ

(6.71)

1

2  ð11rÞy1

y1 Multiplying both sides with ΔNij;DC and taking the sum over all years gives

2

nðΩ XY Þ y1 51

5

nðΩ XY Þ y1 51

t;y t;y t;y t;y1 X gij  ðut;y  nðΩOy1 Þ i 2 uj Þ  ðλj 2 λi Þ  ω y 2  ð11r Þ y5y1 tAΩOy hXnðΩ Þ i Y y1 t;max y 2  fij;DC  Lij  nðΩOy1 Þ  β ði; j; y; y Þ  ρ  k 1 Base inv;ij y5y

y1 ΔNij;DC 

y1 ΔNij;DC

nðΩ XY Þ

1

2  ð11rÞy1

(6.72)

6.5 ALGORITHMS AND MODELS FOR SOLVING TEP

Considering

Pn i51

2

ð ai 3 bi Þ 3

nðΩ XY Þ

X gij 

Pn

ðut;y i

j5i cj

5

2 ut;y j Þ



5

j51 cj

ðλt;y j

y51 tAΩOy nðΩ XY Þ 2

Pn

t;max  fij;DC  Lij  nðΩOy Þ 

3

2 λt;y i Þ

hXy



Pj i51

Xy

y1

205

ðai 3 bi Þ, one rewrites Eq. (6.70) as

 y1  ωt;y  nðΩOy Þ ΔN ij;DC 51

2  ð11rÞy

y1 y1 ΔNij;DC  β Base ði; j; y1 ; yÞ  ρy1  kinv;ij y 51

i

(6.73)

1

2  ð11rÞy

y51

The left side of Eq. (6.73) represents the NPV of total transmission revenue generated by the bipole HVDC interconnector i 2 j throughout the study period. The right side equals the NPV of total investment cost of building the interconnector. Eq. (6.73) shows that interconnector i 2 j generates revenue since the first year of operation. Thus, Eq. (6.73) implies that the NPV of total transmission revenues interconnector i 2 j generates till the end of the study period [i.e., year nðΩY Þ] equals the NPV of total amortized investment cost of building it. Therefore, from a financial perspective, it shows that the model sets the capacities in such a way that the NPV of transmission revenues that will be collected throughout the lifetime of the interconnector equals the NPV of investment cost that is needed to build the interconnector and is expected to be recovered within the desired investment recovery period yrec ðkÞ. This results from assuming a linear investment cost and neglecting the quadratic HVDC losses [70]. Taking the sum over all zones, after some manipulations one obtains nX ðΩ Y Þ

X

y51 tAΩOy

Rt;y λ 5

nX ðΩ Y Þ y51

Ψy ð11r Þy

(6.74)

From Eqs. (6.70) and (6.74), it can be inferred that an optimal development plan determines the grid design such that the aggregated NPV of transmission revenues generated from price differences between different zones equals the NPV of investment cost of building the grid that would be recovered within the desired economic lifetime. Again, this is only true under assumption of linearity of expansion costs and absence of quadratic losses. Note that the proof of investment recovery holds true regardless of number of investors that are involved. However, in the case of multiple investors, an independent financial entity is required to collect the transmission revenues from the grid operators and distribute them appropriately among the investors. Under such conditions, the investment recovery of every cable of every interconnector will be completely fulfilled by the end of its economic lifetime yrec ðkÞ. In addition, the transmission revenues (which equals congestion revenues) that are generated afterwards [y . k 1 yrec ðkÞ] will contribute to paying off the investment cost of other cables.

6.5 ALGORITHMS AND MODELS FOR SOLVING TEP The goal in the TEP problem is to find the solution that gives the optimal objective function value and satisfies the constraints. In deregulated environments, the objective function can be to minimize the total cost, maximize the social welfare or optimize other economic metrics. Optimization constraints correspond to technical and economic limitations and/or reliability issues.

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The TEP problem is inherently a nonlinear, nonconvex and large-scale optimization problem. It is therefore a very complex problem that has attracted attention from both academia and industry. This section provides a brief classification of various optimization models and algorithms that are proposed to address the TEP problem. The most relevant methods for solving TEP problems are based on exact, heuristics, and metaheuristics optimization algorithms.

6.5.1 EXACT ALGORITHMS Exact mathematical optimization algorithms were traditionally used. One reason for using these models is, in contrary to heuristic algorithms, exact optimization methods can prove the optimality of the solution found. The most common exact methods include linear programing [84,110,111], nonlinear programing [83,112], dynamic programing [113], mixed-integer programing [110,114], bilevel programing [115,116], branch and bound techniques, [117,118], and decomposition techniques. Decomposition techniques decompose the optimization problem into several different subproblems using different techniques such as Bender’s decomposition [119] and hierarchical decomposition [120]. In the context of this chapter, nonlinear programing is used.

6.5.2 HEURISTIC & METAHEURISTIC ALGORITHMS Due to complexity of the TEP problems, finding an optimal solution using exact algorithms can be very challenging. To overcome this problem, several heuristic and metaheuristic algorithms have been developed which require less computational effort. The word heuristic means “based on experimentation, evaluation, or trial-and-error” methods9. As the name implies, they are experience-based techniques that are inspired by learning processes that exists in nature (e.g., evolution inspired algorithms, collective animal behaviors) or in industrial processes and phenomena. Heuristic models define a set of rules to be used in an iterative manner to find a solution to the TEP problem. The search process continues, until no better solution can be found. Heuristic models outperform the exact models, as they usually find a solution in relatively shorter time. However, contrary to exact models, they produce an approximate solution close to the optimal solution that cannot be proven to be optimal. In addition, these models are potentially more prone to get stuck in local minima or even diverge. Their final results depend on the parameter settings of the model [121
123]. In literature, the TEP problem is solved using heuristic models to perform sensitivity analysis on the required reinforcements with respect to increasing load supply capability or reducing energy not supplied [124] or reducing overloads [125]. The most widely used (meta)heuristic algorithms to select the required reinforcements in a transmission system are genetic algorithm [126,127], differential evolution (DE) [128], greedy randomized adaptive search procedure [129], ant colony optimization [130], particle swarm optimization (PSO) [131], artificial immune system [131], plant growth simulation algorithm [132], simulated annealing [133], tabu search [134], or a combination of the previous [135,136]. 9

http://dictionary.reference.com/.

6.6 NUMERICAL SIMULATION

207

6.6 NUMERICAL SIMULATION This section discusses some numerical results. It first provides data preparation (Section 6.6.1) and scenario development (Section 6.6.2). Section 6.6.3 presents the result of data clustering. Numerical results is presented in Section 6.6.4 and analyzed in Section 6.6.5.

6.6.1 ASSUMPTIONS We considered four North Sea coastal states, including Germany (DE), the Netherlands (NL), Norway (NO), and the United Kingdom (UK). Each EU member state/country was modeled with one onshore price zone and the respective offshore price zone(s). The offshore zones encompass the total installed OW capacity that every member state is envisaged to install by the end of the year under study. The information regarding current and projected OW capacities was gathered from the online database provided by the marine consultancy firm, 4C offshore [137]. A generic wind turbine was considered for all offshore locations with cut-in and cut-out speeds of 3 and 25 m/s, respectively. All simulations were hourly based with wind speed data from the year 1994, simulated at 120 masl with the spatial resolution of 9 km 3 9 km3. The equivalent speed to power conversion curve of each offshore zone was determined by taking into account the statistical properties of the wind patterns in a given offshore area, as discussed in Ref. [138]. Cost curve coefficients were determined hourly from aggregated supply
demand curves of onshore electricity markets (i.e., APX-UK [139], NordPoolSpot [140], and EPEXSpot [141]) from April 1, 2011 to March 30, 2012 (8760 data points). All interconnectors were assumed to be built of HVDC cables with rated power of 1000 MW per pole and voltage rating of 6 500 kV. We used the NorNed cable as the reference (with investment cost of 600 MAC, length of 580 km and a capacity of 700 MW [142,143]) to calculate the unit cost for building offshore transmission interconnectors and conk verter stations at kinv;ij 5 0.1687 k=ðMW km hÞ. For simplicity in illustrating the method, we have used a unique unit cost figure for building all interconnectors. We have assumed that the cost of reinforcing existing interconnectors is 20% less expensive than the cost of building a new one, due to technical and practical reasons. Finally, we have considered a discount rate of 5%. The proposed nonconvex optimization problem must be solved for every delay scenario separately.

6.6.2 DEFINING DEVELOPMENT STAGES Transmission planning is a multidisciplinary problem with the scope of several decades (e.g., 25
50 years). The large number of operating conditions that the system encounters every year makes the problem complicated and computationally expensive. To reduce the computational burden, I divided the study period into nðDs Þ development stages indexed ds A1; 2; . . . ; nðDs Þ. Each stage was assumed to be comprised several similar operating years. I have assumed that system changes (i.e., increasing OW capacities, that develop the onshore grid network and electricity markets) occur only at transitional years from one development stage to another (e.g., years ym ; yx ; yyrecð0Þ in Fig. 6.5). This assumption reduces our analysis to study nðDs Þ{nðΩY Þ as many operating years.

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FIGURE 6.5 The blank columns represent new transmission investments made in year 0 and yx . The hatch and black columns represent the annual recurring investments equivalent to the investment made in year y 5 0 and year y 5 yx , respectively. yðyrec Þ ð0Þ is the recovery time for the investment that is made in y 5 0. dsi denotes the ith development stage.

We have assumed that an interconnector may connect (1) an onshore zone to its OW farm (national shore-to-hub), (2) two onshore zones (cross-border shore-to-shore), (3) an onshore zone to an OW farm of another country (cross-border shore-to-hub), or (4) two OW farms (cross-border hub-to-hub)10. We defined three delay constraints as follows: • • •

yx : Number of years after the reference year during which expansion of cross-border connections is not allowed. yw : Number of years after the reference year that it takes every offshore zone to increase its installed capacity from 60% to 100%. ym : Number of years after the reference year that it takes every onshore market zone to gain its full capacity for wind feed-in.

The longest delays are expected to be observed on the development of the cross-border projects, especially those that connect OW farms. Therefore, we have assumed that cross-border shore-tohub and hub-to-hub connections will be built yx years after the reference year, whereas national shore-to-hub and cross-border shore-to-shore connections can be expanded any time during the study period, as per current trends in OW and cross-border trade. We have also assumed only 60% of the envisaged OW capacities, are installed in the first yw years. Table 6.1 presents the presumed installed wind capacities before and after year yw for every offshore zone. Only a limited volume of today’s total power trades are settled through the PX of the European markets (47% for DE, 35% for NL, 73% for NO, and 10% for the UK in 2012 [144]). We have assumed that by the end of the study period (yf ), all the power trades are settled in the PX. It implies that onshore markets will be highly liquid with stable prices and small sensitivity to the imports/exports. 10 We assume no changes in demand or fuel prices over time. This is, of course, a limitation in my modeling and can be considered as a direction for future studies.

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Table 6.1 Installed Offshore Wind Capacities before and after Year yw Zone

DE (GW)

NL (GW)

NO (GW)

UK-S (GW)

UK-N (GW)

Before yw After yw

10.12 16.87

2.88 4.8

1.61 2.68

4.5 7.5

7.08 11.8

Table 6.2 A Coefficients of the Quadratic Cost Curve of the Onshore Zones, before and after Year ym Zone

DE (h/MW h2)

NL (h/MW h2)

NO (h/MW h2)

UK (h/MW h2)

Before yw After yw

0.0016 0.001

0.0044 0.0023

0.0017 0.0014

0.0038 0.0007

We have also assumed that this development will take place gradually in a two-step manner, before and after year ym . We have considered the share of every zonal PX from the trades to be 50% above the current rate during the development stage 1 ({y: yA½1; ym }) and reaches 100%, as y 5 ym 1 1. Thus, for yA½ym11 ; yf , markets are considered to be completely liquid. The change is reflected in the empirical market data. That is, at each development stage, a larger volume of power trades will be cleared at the same clearing price of today. Accordingly, the quadratic cost curve coefficient of every country must be decreased proportionally to account for the share of the volume that will be settled in the PX. Table 6.2 presents the annual mean value of the cost curve coefficient a before and after the adjustment, at different development stages. For the purpose of this research, we have defined four development stages ds1 5 ½1; ym , ds2 5 ½ym11 ; yx , ds3 5 ½yx11 ; yyrec ð0Þ , ds4 5 ½yyrec ð0Þ11 ; yf  as presented in Fig. 6.5.

6.6.3 DATA CLUSTERING Due to the computational intensity of the optimization algorithm, it is intractable to solve the problem for one year data points with hourly resolution (i.e., 8760 h). Therefore, the number of representative operating hours must be selectively reduced using a clustering technique. It is not always easy to identify patterns of clusters for highdimensional data due to the curse of dimensionality. One way to make clusters more distinctive is to project the data from the original space to a lower dimension subspace using a feature transformation technique. We employed principal component analysis (PCA), which is a simple and effective statistical technique [145]. We used the K-means algorithm due to its simplicity and applicability to large scale data [146]. The connection between K-means clustering and PCA is explored in Ref. [147]. There, it is proven that clusters become more distinctive in the PCA-reduced subspace than in the original space. K-means sets the mean value of the observations allocated to every cluster, as the centroid of that cluster. As the clustering is carried out in the PCA-reduced subspace, it can be difficult to map the newly produced sample from the reduced space to the original space. To simplify this problem,

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Table 6.3 Annual Averaged Market Clearing Price (MCP) of the Onshore Zones (represented by cost curve coefficient “b”), before and after Clustering with K 5 50 Zone

DE (h/MW h)

NL (h/MW h)

NO (h/MW h)

UK (h/MW h)

Before yw After yw

49.37 50.98

52.64 53.37

45.15 46.38

41.63 37.09

Table 6.4 Wind Availability Coefficient for the Offshore Wind Zones before and after Clustering with K 5 50 Zone

DE (%)

NL (%)

NO (%)

UKN (%)

UKS (%)

Original Clustered

39.59 39.96

37.43 38.58

42.16 41.96

37.93 37.28

41.13 41.22

Table 6.5 Variance of Wind Availability Coefficient for the Offshore Wind Zones Before and after Clustering with K 5 50 Zone

DE (%)

NL (%)

NO (%)

UKN (%)

UKS (%)

Original Clustered

12.33 13.63

11.66 13.85

12.66 13.92

11.34 12.93

12.27 13.86

we redefine the centroid of every cluster to be the actual sample that has the shortest (Euclidean) distance  (calculated Tin the reduced subspace) from the mean value. The full set xfull 5 x1 ; x2 ; . . . ; x8760 contains 8760 operating hours. Each operating hour (xj ) comprise (4 3 2 5 )8 cost curve coefficients of the four onshore price zones and wind speed data of the five offshore zones. The full data set contains eight cost curve coefficients of the four onshore price zones and wind speed data of the five offshore zones. The PCA analysis shows that, out of the 13 features related to the different zones, the first three components in the PCA-reduced subspace contain 89% of the information in the data set. Using the PCA/ K-means  approach, we T reduced the complete year of data to 50 clustered operating states xcls 5 x01 ; x02 ; . . . ; x050 , where x0j are the operating hours selected from the full data set xfull . Tables 6.3 and 6.4 present the clustering results for K 5 50. It can be observed that the average price and available wind at different offshore locations in the clustered set are adequately close to those of the original set. The largest difference can be observed in the average market clearing price of the UK onshore zone. Table 6.5 presents the variance of the wind availability coefficient for the OW zones of the original and the clustered data sets. It can be seen that the variance of wind availability coefficient at different offshore locations in the clustered data set are adequately close to those of the original set.

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Table 6.6 The Sum (NPV) of Aggregated Social Welfare, the Total Revenue, the Net Economic Benefit of Onshore Zones, and the Total Interconnector Total Social Welfare

Total Revenue of Onshore Zones

Aggregated Net Economic Benefit of Onshore Zones

Total Investment Cost

Total Installed Interconnector Capacity

107.1 96.6 89.2 83.3 94.4 85 79.2

2 78.8 2 68.7 2 64 2 60.4 2 67.9 2 63.6 2 60.7

28.3 27.9 25.2 22.9 26.5 21.7 18.5

14 13 12.3 11.6 13.4 13.6 14.2

58 63.5 61.8 59.2 62.9 61.5 59.2

6.6.4 NUMERICAL RESULTS To better illustrate the sensitivity of the grid design to the delay constraints, we have defined three case studies. Case study A is the reference case, in which we have solved the planning problem using the STEP framework introduced in Section 6.4.3. We have used the results of the STEP module to evaluate the performance of the dynamic framework proposed in this chapter. In Case studies B and C, the TEP problem is solved using the proposed dynamic framework. We have assumed that the grid design evolves in a stepwise manner. We have introduced three different delay scenarios for each case study. Under Case study B, we have only accounted for delays associated with wind development. Therefore, moving from Scenarios 1 to 3, we have assumed that the delay time associated with OW integration (yw ) is extended in 5-year steps (i.e., yw 5 5, 10, and 15), whereas, in Case study C, in addition to the OW integration (yw ), we have considered the delay associated with the development of cross-border interconnectors. Thus, here we have assumed that the delay time associated with the OW integration (yw ), as well as with the cross-border connections (yx ) are extended in 5-year steps. Note that the delay time associated with the increase in the capacity of onshore zones for OW feed-in (i.e., ym ) is considered to occur at the end of the 5th year for all scenarios under the two case studies (i.e., ym 5 5). Table 6.6 indicates the total social welfare, total revenue, aggregated net economic benefit11, investment cost, and installed capacity for each case. Note that in all cases, the total revenue of all onshore zones is negative, as they are net importer of electrical energy. Also, note that longer delays lead to significant reduction in social welfare and net economic benefit of consumers and producers, with consequences that are discussed in detail below. It can be seen that the two types of delay scenarios can still produce a comparable, if slightly smaller net economic benefit compared to the base case if delays are small. However, the total amount of capacity in B1 and C1 is different and configured differently, due to factors discussed further in this section.

11 The net economic benefit is defined as sum of changes in revenues (defined as power injection quantity times the price for every onshore zone at every hour) plus the change in SW of every onshore zone at every hour.

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6.6.4.1 Case Study A: Instant Development of the Electricity Markets, Offshore Wind and Transmission Interconnectors Fig. 6.6 shows the grid design determined using under all case studies and scenarios. It can be seen that the grid design (straight 1 dashed lines) is highly meshed. All transmission capacities are presented under ‘Case A’ column in Table 6.7. The model sets large shore-to-hub radial connections and relatively smaller hub-to-hub connections. As no delay constraint is imposed, the grid design determined under Case A does not account for construction challenges that can affect the remuneration of different stakeholders and so the development of the grid. Yet, it can be useful for comparison and informative as it describes how the grid would evolve in a perfect world.

6.6.4.2 Case Study B: Gradual Market and Offshore Wind Development In all the three scenarios of Case study B, Table 6.7 indicates larger radial and cross-border hub-to-shore transmission capacities after year yw , when the total amount of installed OW

FIGURE 6.6 Geographical representation of the proposed expansions plan under different delay scenarios regarding wind integration and market developments in Case Studies A, B, & C. The straight connections are common under all cases. The dashed connections are chosen for both Cases A & B. The dotted connections are only chosen under Case Study C.

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Table 6.7 Interconnector Length and Capacities (GW) Determined under Case Study A as Reference Case and Under Case Study B for Different Wind Delay Scenarios Case B

Shore to hub

Hub to hub

dsi K4 K12 K17 K19 K23 K25 K26 Subtotal K28 K30 K32 K35 Subtotal

l (km)

Case A

ym 5 5

yw 5 5

ym 5 5

yw 5 10

ym 5 5

yw 5 15


167 51 433 198 305 279 90 46.1 254 297 101 295 12


13.2 7.6 1.7 5.1 0.1 4.5 13.9

1

3 13.6 7.3 1.6 6.5 0.3 4.7 14.9

1

4 13.5 7.5 1.3 6.2 0.5 4.4 14.3

1

4 13.1 7.6 0.8 6 0.5 4.1 13.7

0.7 3.8 4.4 3.1

8.8 4.9 0.5 5 0 2 4.1 48.9 0.7 1.8 2.1 2.9 14.5

0.7 4.4 4.7 4.7

8.8 4.9 0.3 5.1 0 2 4.1 47.7 0.9 1.7 2.2 2.9 14

0.9 4.1 4.5 4.5

8.8 4.9 0.2 5.2 0 2 4.1 45.8 1 1.7 2.2 2.9 13.3

1 3.9 4.1 4.3

Note that intermediate development stage(s) of every scenario is omitted due to lack of space. For each interconnector category (shore-to-hub, hub-to-hub), column ‘subtotal’ gives the total installed capacity built by the end of the study period.

capacity is boosted. No delay constraints were imposed on building cross-border hub-to-hub connections under this case study. Therefore, the model sets large hub-to-hub connections from the early development stages, which results in a grid topology similar to the one obtained under Case study A. It is already seen from Table 6.6 that in Scenario B1, social welfare is similar to the base case, but a larger total capacity has been built for the same final offshore fleet. This can be explained by early price differences present due to the delay of OW. Table 6.8 presents the annual weightedaveraged wholesale market clearing price of all price zones. These lead to relatively large nodal price differences of many interconnectors, for example K19, K26, and K35. These interconnectors experience a large capacity increase with K35 being the largest. It connects OW farm of Norway to the UK North. The attention of the reader is directed to relatively large nodal price difference between the two offshore zones at the end of ds1 , under Case study B1 (i.e., 35.29 2 30.75 5 4.54 =MW h). Comparing this amount with of the same value at the end of the study period (i.e., 0.69 =MW h) and with of Case study A (i.e., 1.51 =MW h), it can be concluded that first, under Case study B1, there is large potential for capacity increase between OW farms in the UK North and Norway. Second, the price difference is significantly reduced once the grid construction is completed. This shows that expansion of K35 is an economically efficient decision that induces a large increase in the aggregated social welfare of all zone (which is the objective of this framework).

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Table 6.8 Annual Averaged Wholesale Market price (h/MW h) of Different Price Zones Determined under Case Study A and Different Wind, Market and Transmission Delay Scenarios of Case Study B Case A

Development Stage DE NL NO UK OW-DE OW-NL OW-NO OW-UKS OW-UKN

Case B Scenario 1

Scenario 2

Scenario 3

ym 5 5, yw 5 5

ym 5 5, yw 5 10

ym 5 5, yw 5 15




ds1

ds2

ds3

ds1

ds2

ds3

ds4

ds1

ds2

ds3

ds4

38.3 37.4 41.3 38.6 36.3 36.8 39 35.7 37.5

37.2 32.2 38.3 32 34.8 31.5 35.3 30 30.7

37.7 37 39.8 38.1 36.1 36.5 37.9 35.7 37.2

37.7 37 39.8 38.1 36 36.5 37.9 35.7 37.2

37.2 32.2 38.3 32 34.8 31.5 35.3 30 30.7

42.4 42.3 42.5 42.7 41.2 41.9 41.1 41.4 42.1

37.7 36.8 40 38 35.9 36.2 37.7 35.2 37

37.7 36.8 40 38 35.9 36.2 37.7 35.2 37

37.2 32.2 38.3 32 34.7 31.5 35.3 30 30.7

42.4 42.2 42.5 42.6 41.2 41.8 40.9 41.3 42

37.7 36.6 40 37.8 35.6 35.9 37.5 34.6 36.6

37.7 36.6 40 37.8 35.6 35.9 37.5 34.5 36.6

The same discussion holds for K19 and K26, other connectors that contribute to the larger capacity of B1 compared to A. The larger capacity of B1 actually comes at a lower investment cost than Case study A. This is true also for B2 and B3, and is possible only because a lower capacity is assigned to long and large interconnectors that have strong influence on the investment cost. Table 6.7 shows the length of all interconnectors, including K17, the longest connection in grid designs of Case studies A and B. Moving from A to B1 and to B3, it can be seen that the model sets a smaller capacity to K17, as stronger delay constraints are imposed. This results in a relatively large reduction in the transmission investment cost. Interconnectors K23 and K28 are also relatively long and are instead increased, but their capacity starts significantly smaller than K17. The optimal distribution of capacity is across different interconnectors at different times, but apparently offers comparable power trades and therefore social welfare. Scenarios B1
B3 show how delay limits the possible exploitation of social welfare, and in the worst case, how “last minute” grid design may have relatively a same total capacity and similar or greater cost of investment, but with a poorer social welfare and net economic benefit to onshore zones (Table 6.6). Additional information on Scenarios B1
B3 is available from Table 6.9, which presents the aggregated NPV of the transmission revenues that will be collected and the investments that are made in each development stage in each scenario. By comparing the transmission revenues generated annually, it can be observed that for years before ym (i.e., during ds1 ) the transmission revenues generated for B1 are 30% smaller than for A. This difference can be associated with the smaller OW capacities and less market maturity, therefore, higher sensitivity of onshore market prices to

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Table 6.9 The Annual Transmission Revenues Collected by the end of Every Year under Case Study A and Each Development Stage of the Three Case A

Case B Scenario 1

Scenario 2

Scenario 3

ym 5 5, yw 5 5

ym 5 5, yw 5 10

ym 5 5, yw 5 15

Development Stage




ds1

ds2

ds3

ds1

ds2

ds3

ds4

ds1

ds2

ds3

ds4

Total nodal payments

0.99

0.71

0.897

0.915

0.71

0.5

0.97

0.97

0.71

0.51

1.08

1.08

wind imports during this period under Case study B1. For years after ym , however, the two numbers become close (Case study A1 is still slightly higher). This shows that the two grids perform similarly financially for years after ym . Going from B1 to B3 in Table 6.9, an increase in transmission revenue collected during ds2 can be noted. This is a consequence of price increase in the onshore markets, as can be seen in Table 6.8. The price increase itself is due to fact that, during this period, markets have become more mature and gained more capacity for wind feed-in. Note that due to the delay represented by yw , the OW capacities are not increased in this period. Thus, on one hand, markets become more liquid (and so less sensitive to power imports). Market liquidity is represented by “a” coefficient in the quadratic social cost curve. On the other hand, the new, less expensive offshore capacities are not integrated to the system yet. Thus, under case B2 and B3, onshore markets experience a price increase during ds2 . In practice, the price increase can happen when the onshore operators improve the flexibility of their control area by utilizing more expensive (but flexible) power production plants, such as gas-fired plants or grid-level energy storage units (in the future). Table 6.10 shows the aggregated investment cost made in each development stage of building the grid. By comparing the investments that are made, one observes that only 20% of total investment is made in years before ym (i.e., during ds1 ). This can also be associated with the smaller OW capacities and less market maturity, as explained above. Considering the magnitude of investments that are made in ds2 in the three scenarios under B, a decrease and then an increase in the investments can be seen. The investment decrease from B1 to B2 during ds2 occurs because the new less expensive offshore capacities are not yet integrated to the system. And, therefore, there is no need for making extra investments in this period. For a same reasoning, a large investment is made during ds3 , as more OW capacities are integrated into the system. Now looking at B3, it can be seen that almost a similar level of investments is made in ds2 and ds3 . This is because the investments that are made in the later years after yw are more expensive, as they should be recovered in fewer years. This effect is called the end effect [148]. As a result, delay of OW can challenge the financial viability of hub-to-hub interconnections: The less time to recover the investment, the more expensive the prospect. This is evident in the optimal solutions for B2 and B3 in Table 6.7; compared to B1, the longer the delay, the smaller capacities assigned for these connections.

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Table 6.10 The Aggregated (NPV) Transmission Revenues Collected by TSOs and the Transmission Investments Made in Each Development Stage under the Three Delay Scenarios of Case Study B Case B

Development Stage Total investment cost

ds1 2.4

Scenario 1

Scenario 2

Scenario 3

ym 5 5, yw 5 5

ym 5 5, yw 5 10

ym 5 5, yw 5 15

ds2 10.48

ds3 0

ds1 2.54

ds2 2.78

ds3 6.59

ds4 0

ds1 2.54

ds2 4.9

ds3 4.14

ds4 0

Note that no investment is assumed to be made in the last development stage (i.e., ds3 under B1 and ds4 under B2 & B3).

6.6.4.3 Case Study C: Gradual Market, Offshore Wind and Cross-border Transmission Development As outlined above, under Case study B, we have only accounted for delays associated with wind development. In the following, we present the results for Case C, in which, in addition to the OW integration (yw ), we have considered the delay associated with the development of cross-border interconnectors. Fig. 6.6 also presents the grid design determined under Case study C (Straight 1 dotted lines). Although strongly meshed, the grid topology determined under Case study C is significantly different from the two previous cases. The main reason for this difference is the extra delay constraint applied to developing cross-border hub-to-hub connections, which oblige the model to invest in other less economically efficient alternatives [e.g., shore-to shore connections (K3 and K16)12]. Table 6.11 presents the transmission capacities determined for every development stage under every scenario. Under each scenario, one sees an increase in the capacity of radial shore-to-hub (K4, K12, K19, K25, and K26) connections as the markets evolve and gain more capacity for wind feed-in (i.e., after ym ). As of year yx 5 yw (beginning of ds2 under Scenario 1 and ds3 under Scenarios 2 and 3), the model receives permission for expanding wind and cross-border interconnector capacities. This results in increasing the capacity of the existing radial shore-to-hub connections and setting large capacities for cross-border hub-to-hub connections (K28, K30, K32, and K35) to deliver the relatively cheaper OW energy of all offshore zones to the onshore consumers. In addition, moving from Scenarios 1 to 3, it can be seen that a longer delay time results in smaller national shore-tohub and hub-to-hub connections (K4, K19, K25, K26, K28, K30, K32, K35, except K12). The change in K3 and K16 can be attributed to the need for larger cross-border capacities for conducting power trades between onshore zones. Moving from C1 to C3, delays on the development of the hub-to-hub cross-border interconnectors become longer. This means that the recovery period for such investments becomes shorter. Therefore, the cost of building a similar interconnector at a later development stage becomes higher as yw becomes longer and so, they become too expensive to invest in. Therefore, it becomes more economically efficient to invest in relatively 12 Note that K3 and K16 are different from what they appear in the figure. These connections should pass through the shortest path in the middle of the North Sea. But due to lack of space are plotted as they are.

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Table 6.11 Interconnector Length and Capacities Determined under Case Study C for Different Wind, Market and Transmission Delay Scenarios Case C

Shore to shore

Shore to hub

Hub to hub

dsi K3 K16 Subtotal K4 K12 K19 K25 K26 Subtotal K28 K30 K32 K35 Subtotal

Scenario 1

Scenario 2

Scenario 3

l (km)

ym 5 5, yw 5 yx 5 5

ym 5 5, yw 5 yx 5 10

ym 5 5, yw 5 yx 5 15


531 567

1

1

1

167 51 198 279 90 254 297 101 295

0 3.3 3.3 9.4 2.8 1.5 2.9 15 47.7 0 0 0 0 12.3

3 0 3.3 14.5 7.2 6 5 15 1.5 4 4.3 2.5

1.3 4.4 5.7 10.1 2.9 1.5 3.6 8.4 45.8 0 0 0 0 10.2

4 1.3 4.4 14.2 7.3 5.1 4.9 14.3 1.3 3.3 4 1.6

3 4.9 8 9.8 2.9 1.5 3.8 9 30 0 0 0 0 7.9

4 3.1 4.9 13.8 7.4 4.3 4.3 13.4 1.1 2 3.7 1.1

Note that intermediate development stage(s) of every scenario is omitted, due to lack of space. For each interconnector category (shore-to-shore, shore-to-hub, hub-to-hub) column ‘subtotal’ gives the total installed capacity built by the end of the study period.

long and expensive shore-to-shore cross-border capacities, but from the early development stages. In addition, the higher cost of hub-to-hub cross-border connections justifies larger investments in shore-to-shore cross-border capacities, such as K3 and K16 moving from C1 to C3 in Table 6.11. A similar trend in transmission revenue collected and a similar decreasing trend in the capacity of interconnectors (again except for K12) and investments made in the last development stages (i.e., the end effect) can be observed, moving from C1 to C3. These effects and the exception are shown in Figures below. They can be reasoned in a similar manner as in Case study B, and so, are not further discussed here.

6.6.5 COMPARISON Consider the columns and dashed lines in Fig. 6.7. They exhibit a decreasing trend in the aggregated incremental social welfare, aggregated net economic benefit to producers and consumers of onshore zones, as delays become more severe. From left to right, both measures decrease as stronger delay constraints are imposed regarding the development of the OW integration and capacity of the onshore markets for wind feed-in (Scenarios 1 to 3 under Case study B) or regarding the capacity of OW, electricity markets and construction of cross-border connections (Scenarios 1 to 3 under Case study C).

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FIGURE 6.7 Total social welfare, aggregated net economic benefit and total investment cost of transmission infrastructure determined for different delay case studies and scenarios.

Comparing the results obtained under the Case studies B and C for a similar wind and market delay scenario, it can be seen that the length of the delay is the most influential factor on social welfare losses; a shorter delay time returns a higher social welfare increase. Although the length of the delay induces net economic benefit losses, one could see that the most influential factor on benefit losses is the severity of the delay (i.e., whether or not delay on cross-border connections are imposed). Therefore, in practice, the highest economic gain is achieved when OW capacities are developed at yw 5 5, and no delay is considered regarding development of the cross-border offshore connections. In Fig. 6.7, the social welfare increase declines from 107 B (under case A—30 years study period and 25 years investment recovery period) to 96.6BAC (under B1 with ym 5 yw 5 5) to 79.22 B (under C3 with ym 5 5, yw 5 yx 5 15). Thus, the cost of a 10-year delay in development of cross-border OW and transmission projects is 17.41 B (in terms of social welfare loss) to society. Comparing the worst case scenario with Case study A, the consequence of development delays are estimated at 107.10 2 79.2 5 27.9 B cost to society. Fig. 6.7 also shows that net economic benefit of onshore zones inclines from 28.3 B (Case study A) to 27.9 B (under B1 with ym 5 yw 5 5) to 18.5 B (under C3 with ym 5 5, yw 5 yx 5 15). Thus, the cost of a 10-year delay in developing cross-border OW and transmission projects is 9.4 B (in terms of net benefit loss) for the onshore zones. Looking at Tables 6.12 and 6.13, it appears that under all scenarios of the two case studies, the social welfare and net economic benefit of all countries increases after the grid is built. It may be noted that different countries are affected differently (i.e., unequally) by the delays. In this regard, the Netherlands has the largest social welfare loss (7 B), followed by the UK (6 B), Germany (4 B), and Norway (2 B), as the delay time grows longer. The unequal impact of delays on the onshore zones can be explained by their power import and market liquidity. Table 6.14 shows the annual weighted averaged power import of every onshore system in various development stage of Cases Studies A & B. These numbers present the power

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219

Table 6.12 Aggregated Social Welfare of Onshore Zones in Bh Scenario

DE

NL

NO

UK

Sum

B1 C1 B2 C2 B3 C3

31.51 31.1 29.19 29.32 27.34 28.87

17.76 15.74 16.63 12.9 15.7 10.79

14.93 13.94 13.7 13.79 12.75 13.47

32.43 33.58 29.67 29.26 27.54 26.09

96.63 94.38 89.2 85.27 83.33 79.22

Table 6.13 Aggregated Net Economic Benefit to Consumers and Producers of Onshore Zones in Bh Scenario

DE

NL

NO

UK

Sum

B1 C1 B2 C2 B3 C3

7.98 8.38 6.94 7.5 6.1 7.07

5.56 4.49 5 3.39 4.49 2.54

3.31 2.87 2.93 2.6 2.63 2.34

11.04 10.71 10.29 8.21 9.7 6.53

27.89 26.54 25.16 21.7 22.92 18.48

import when the grid is still at the early development stage (before ym ), and when their development is completed (after yw 5 yx ). The numbers show that the UK and the Netherlands have the largest power import increase, as their markets gain maturity and their grids become complete. This is due to the configuration of the input set to the model. The large increase in the power imports of 2 the UK can be attributed to extremely high liquidity [represented by a 5 0.0007 =MW h coefficient in Table 6.2] of this market, which enables larger power imports. Also, the large increase in power import of the Netherlands can be attributed to the large stand-alone market price of the Netherlands (53.37 =MW h Table 6.3), which incentives this country to import large amount of OW energy. The social welfare increase and net economic benefit to consumers and producers of each country is a function of its power injection. Thus, the social welfare of countries that have larger change in their power imports is subjected to more sever changes than others. As longer delay means longer periods of low power injection (i.e., ds1 and ds2 ) and shorter periods of large power injection (i.e., ds3 and ds4 ), the Netherlands, and the UK are affected the most by the delays. The UK also has the largest net economic benefit loss (4.51 B) followed by the Netherlands (3.02 B), Norway (0.97 B), and Germany (0.91 B). The large net economic benefit losses of the UK and the Netherlands can be attributed to relatively larger change in their revenue (Table 6.15) compared to their social welfare, going from B1 to C3.

Table 6.14 Annual Averaged Energy Export/Import (in TW h) of the Onshore Price Zones Determined Under Case Study A and Different Wind, Market and Transmission Delay Scenarios of Case Study B Case B

Case A

Scenario 1

Scenario 2

Scenario 3

ym 5 5, yw 5 5

ym 5 5, yw 5 10

ym 5 5, yw 5 15

Development Stage




ds1

ds2

ds3

ds1

ds2

ds3

ds4

ds1

ds2

ds3

ds4

DE NL NO UK

21.11 20.61 20.37 20.93

20.76 20.42 20.44 20.14

21.15 20.63 20.42 20.82

21.15 20.63 20.42 20.82

20.77 20.42 20.44 20.14

20.75 20.43 20.24 20.4

21.16 20.64 20.41 20.82

21.16 20.64 20.41 20.82

20.77 20.42 20.44 20.14

20.75 20.43 20.24 20.4

21.15 20.64 20.4 20.82

21.15 20.64 20.4 20.82

The minus sign denotes that onshore price zones are net importers.

6.6 NUMERICAL SIMULATION

221

Table 6.15 Aggregated Revenue of Onshore Zones in Bh Scenario

DE

NL

NO

UK

Sum

B1 C1 B2 C2 B3 C3

223.34 222.37 222.25 221.82 221.24 221.8

212.2 211.26 211.63 29.51 211.21 28.25

211.61 211.1 210.77 211.19 210.12 211.13

221.39 222.87 219.38 221.05 217.84 219.56

268.54 267.6 264.03 263.57 260.41 260.74

Moving from B1 to C3, it can be seen that the revenue and social welfare increase of Germany changes in almost similar magnitudes. This explains the small net economic benefit losses of Germany compared to UK and the Netherlands. A similar reasoning applies to the case of Norway. The unequal impact of delays on the onshore zones implies that the Netherlands and UK have the strongest incentive to avoid delays and the associated extra costs in multinational cross-border wind and transmission projects. It can further complicate resolving the legal conflicts that may arise between different countries.

6.6.6 DISCUSSION The numerical results provided here are based on a restrictive set of assumptions regarding the operation of the electricity markets. For instance, the proposed interzonal expansions inevitably induce new power flow patterns in the onshore power systems [5]. Subsequently, the onshore grid would also need to be expanded to a large extent before connecting the offshore grid. Thus, the transmission reinforcement that is presented here may require an even larger investment, which could conceivably lead to a different project planning and set of investment decisions. In this study, we have identified the total installed OW capacity of offshore zones, liquidity, and stand-alone prices of the onshore electricity markets and distance between the connections points as the dominant factors. They drive the decisions regarding the design of the grid, the power import/export of every zone and so the variations in their social welfare and net economic benefit at different stages of the grid development. The results presented in this chapter are based on empirical market data. It can be argued that once the grid is built, the onshore operators may change their strategy for participating in the markets. Then the aggregated supply/demand curves will be different from their use in this chapter. Accordingly, the cost curve coefficients determined based on supply
demand curves will be different. These effects could affect the final results. Therefore, one area for future research would be to investigate a clustering method that better preserves the correlations between the connection points over time and space.

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CHAPTER 6 OPTIMUM TRANSMISSION SYSTEM EXPANSION

6.7 CONCLUSIONS AND FUTURE RESEARCH 6.7.1 CONCLUSIONS The work presented in this chapter provides the theoretical background for TEP of meshed grids, both onshore and offshore. The analytical part lays the basis for investigating the impact of implementing different regulatory regimes on the development of such grids. The TEP is an ongoing research area. The TEP formulations discussed need to be improved to enable inclusion of the changes that are happening to power systems around the globe. More research is required to assess the economic impact of the integration of HVDC transmission systems into AC power systems. The static formulation discussed in Section 6.4.3 “sets the stage” for the dynamic framework described in Section 6.4.4, while several case studies were provided in Section 6.6. Considering nodal voltages as independent variables allows for accounting for technical realities, such as the loop flow effect in the proposed HVDC grid. It is shown, with no significant loss of accuracy, that linearly approximating the HVDC power flow simplifies the market mechanism and makes it consistent with its existing AC onshore grid counterpart. We start with STEP and then derive DTEP, a dynamic optimization formulation for our marketbased TEP framework, which allows the consideration of the time-varying nature of transmission reinforcements. We have utilized the model to investigate the impact of unanticipated delay constraints on the final grid development plan, on the development strategy and on the benefits of different stakeholders, on and offshore. The proposed formulation is a continuous, nonlinear optimization framework and allows for the consideration of multiple time periods in multiple development stages of the grid. The final results include optimal grid topology, transmission capacities, construction timing, and the resulting remuneration and distribution of the social welfare increase among the various onshore price zones. For both STEP and DTEP, the analytical solution to the optimization problem gives the pricing mechanism. In addition, the optimal plan is shown to allow investment recovery through congestion revenues as an implicit strict equality constraint. The proposed framework is driven by historical market-data in the form of hourly aggregated bidding curves and enables simulating the gradual development of the future offshore grid.

6.7.2 REFLECTION The analysis in this chapter strongly relied on the initial assumptions and selection of the input data. Thus, we reflect here on how the model design and assumptions made may have affected the conclusions of this chapter. All interconnectors were assumed to have been built using VSC-HVDC technology. In the absence of economies of scale, the cost of converters and interconnectors were assumed to be linearly dependent on length and rated capacity of the cables. However, high-voltage converters are extremely expensive equipment. Adding extra converters result in higher investment and operational costs. Therefore, considering the cost of converters implicitly in the cost of interconnectors may induce an error in the cost calculations as actual investment costs of transmission infrastructure might be different from what is reported in the numerical section of this chapter. In addition, to keep the problem linear, an approximation of the HVDC power flow was used, in

6.7 CONCLUSIONS AND FUTURE RESEARCH

223

which transmission and converter losses were ignored. HVDC transmission losses were shown to account for around 1% of net power trades. However, due to high frequency switching, VSCHVDC converter losses also account for a large portion of total losses. In effect, the cost of losses of converters was not included in the analysis. It could have distorted the results, as the final grid expansion investments may be larger than calculated here. In the optimization problem formulation, several operational constraints of power plants, such as ramp rates and start-up and shut-down decisions were ignored. Furthermore, while the simulation does reflect the limited capacity of the intermittent OW, the power system is assumed to be dynamically secure. Therefore, only the steady state operating conditions were considered. These limitations together underestimate the effect large-scale integration of intermittent OW will have on the future power system. Security constraints, as well as the cost of security, could be incorporated in the optimization problem; however, these are expected to change the design of the grid [149
152]. In the dynamic expansion part, we only consider nonrandom delay scenarios in the development of the grid, such that delays are provided as input to the model. An alternative would be to study random completion of the grid (i.e., a line is planned, but is randomly delayed or canceled). This could be an extension of the current framework, but would require stochastic programing and is beyond the scope of this chapter. Intrazonal expansions necessary to accommodate the new developments in wind energy and offshore grids were not considered in the scope of this work, even though the new proposed interzonal expansions presumably induce new power flows and congestion patterns in the onshore transmission systems [5]. Consequently, the onshore grid would need to be reinforced excessively before connecting the offshore grid. Therefore, the final grid expansion and reinforcement investments will be larger than calculated here and could conceivably lead to a different set of optimal investment decisions. The modeling of the electricity markets was abstracted from reality to reach an analytical solution and enable computationally feasible run times. We have assumed that the aggregated supply and demand bidding curves of each onshore zone are linear functions of the power generation/consumption of that zone. This was an important asset of the methodology, and, at the same time, a limitation. This assumption allows the determination of the changes in social welfare (i.e., incremental social welfare) as a quadratic function of power injection of each zone to the rest of the system and, therefore, makes the problem quadratic and thus, easier to solve. It also enables implicit modeling of the operation of the onshore electricity markets and their responses to the power trades. However, this approach has ignored the realistic situation, in which the aggregated supply and demand curves are step-functions (rather than linear functions) of power generation and consumption. In that case, social welfare in the objective function must be defined explicitly as a function of total quantities demanded and supplied. This makes the objective function nonsmooth and more difficult to solve. In such situations, other optimization algorithms (i.e., heuristic or metaheuristic) could be used to solve the problem. Note that, for such nonsmooth problems, the KKT optimality conditions cannot be applied and, therefore, the analytical solution will not be accessible. In this chapter, a zonal market model was considered with electricity markets that are perfectly competitive. The buyers and sellers were assumed to make their bids and offers in a double auction market with a single market price per zone. However, in the real-world market, parties are likely to engage in gaming to develop new strategies that maximize their own profit based on their learning from previous time periods and how the other player(s) and the market reacted [69]. It can be

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argued that once the grid is built, onshore power plant owners might need to develop new strategies for how to respond to the new market environment. For example, the surplus of conventional generators in the net importer countries would decrease drastically. Subsequently, the surplus of generation units in the net exporter countries (e.g., hydro storage capacities in Norway) will increase significantly. This will encourage owners of generation assets to adopt new bidding strategies and look for other opportunities. All these considerations mean that the aggregated supply/demand curves will be different in the future from those in 2011
12, which were used for the numerical results in this chapter. Accordingly, the cost curve coefficients determined based on the historical aggregated supply
demand curves will evolve over time and, therefore, a different optimal grid design might result. Therefore, the numerical results presented should be taken only as an illustration of the proposed STEP and DTEP methods, and not as a representation of the actual future of the North Sea grid.

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36.

CHAPTER

OPTIMUM SIZING AND SITING OF RENEWABLE-ENERGY-BASED DG UNITS IN DISTRIBUTION SYSTEMS

7

Amirsaman Arabali1, Mahmoud Ghofrani2, James B. Bassett2, My Pham2 and Moein Moeini-Aghtaei3 1

LCG Consulting, Los Altos, CA, United States 2University of Washington, Bothell, WA, United States 3 Sharif University of Technology, Tehran, Iran

7.1 INTRODUCTION The increasing demand for reliable power poses a challenge for traditional power generation models. As the demand increases, the consumption of finite resources and emission of climate altering CO2 escalate the urgency to integrate renewable power sources into the distribution system [1]. Distributed generators (DG) are relatively small power sources that can be powered by renewable and nonrenewable sources. Renewable-energy-based DGs have the potential to reduce the environmental impact of traditional power generation, but their integration into the distribution system must be carefully planned to optimize their benefit. The material in this chapter addresses the siting and sizing of DG units within distribution systems. The power output of DG units ranges from a few kW to a few hundred kW and are classified in Table 7.1 by their output. The technologies associated with DG are broadly classified by their use of renewable or nonrenewable sources. Renewable sources include wind, solar photovoltaic (PV), solar thermal, hydro, biomass, geothermal, and tidal. Nonrenewable sources include reciprocating engines, gas turbines, combustion turbines, and fuel cells [3]. The economic impacts of using DG include the deferment of investment in building new power generation plants, and upgrading aging facilities. Certain DG technologies such as PV and wind have lower operation and maintenance (O&M) costs than traditional sources. In addition, reducing expenditures on fuel can provide enormous economic benefits to power providers. The diversification of the energy source profile may help insulate an economy to disruptions and fuel scarcity. The environmental benefits of DG include reducing emissions of pollutants. This can have an indirect benefit in terms of reduced healthcare costs in high-pollution areas. When implemented properly, there are technical benefits to the adoption of renewable DG within the power grid. These benefits include the minimization of system losses, improvement of the voltage profile, and enhanced system reliability, stability, and loadability. However, when not properly implemented, the technical benefits may become a liability. Stability issues, bidirectional power flow, harmonic instability, and islanding difficulties are challenges to overcome when integrating DG. Optimization in Renewable Energy Systems. DOI: http://dx.doi.org/10.1016/B978-0-08-101041-9.00007-7 Copyright © 2017 Elsevier Ltd. All rights reserved.

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Table 7.1 Classification of DG Units by Power Output [2,3] Class

Size

Micro distributed generation Small distributed generation Medium distributed generation Large distributed generation

1 W # 5 kW 5 kW # 5 MW 5 MW # 50 MW 50 MW # 500 MW

In the sections that follow, we will examine power generation models for renewable DG. We will examine the effects of DGs on the distribution network by considering the impact on voltage profile, power loss, reliability indices, economic cost and benefits, and voltage stability. Fault current, harmonic distortion, and reactive power supply will also be considered. We will then develop an objective function that can be used with a variety of solution techniques to model the siting and sizing of DG units within the distribution network. Finally, we will provide and discuss a case study that utilizes the theory to simulate the ideal placement of DG units.

7.2 RENEWABLE-ENERGY-BASED DG MODELS For the purpose of predicting power output, power generation models of DG units can be categorized into deterministic and stochastic. Deterministic DG models are those in which the output can be highly controlled by increasing or decreasing the energy supply sources. Stochastic DG models have an inherent degree of variability due mostly to the unpredictability of regional weather conditions. For both types of DG models, it is desirable to model the net power output as a function of gross input. Gross input can be measured in terms of dollars per hour, cubic meters of gas per hour, incoming solar radiation per hour, or any other unit that accurately describes the operation of the facility. For stochastic models, gross input must include some form of probability distribution function (PDF) that models the unpredictable nature of the input. Net output is the electrical power that is available to the larger utility system.

7.2.1 DETERMINISTIC MODELS Nonrenewable power generation technologies include reciprocating engines, gas turbines, combustion turbines, and others. For these technologies, the energy sources are not dependent upon fluctuations in the weather and therefore can be represented by deterministic models. Renewable power generation technologies include geothermal and biomass. Because the energy supply in these technologies is stable and dispatchable to meet demand, they are considered to be deterministic, and their power models will not require PDFs. Geothermal and biomass power production will be analyzed in further detail.

7.2.1.1 Geothermal Models The internal heat of the earth can be used to produce electricity through the use of steam turbines. The temperature T at a depth D is given by T 5 Tsurface 1 DΓ , where Tsurface is the average

7.2 RENEWABLE-ENERGY-BASED DG MODELS

235

temperature at the surface, andΓ is the temperature gradient. The temperature gradient is related to the heat flow Q and rock conductivity K by Q 5 KΓ . The values for the temperature gradient and heat flow vary depending on the age of the geological formation, but are typically 20
C/km. Electricity production from geothermal sources are liquid based, where hot, compressed water fills the fractured and porous rocks of a reservoir. As the fluid moves upward toward the surface, the pressure decreases until the saturation pressure is reached corresponding to the geofluid temperature. The fluid begins to boil creating a two-phase liquidvapor mixture that will flow through the upper section of the well. A separator will divert steam to turbine generators for electricity production and brinewater for either direct heating uses or will be returned to the reservoir through injection wells. After the steam has been used for electricity production, it passes through a condenser, is cooled, and is returned to the geothermal reservoir as shown in Fig. 7.1 [4]. The potential power output of a geothermal plant depends primarily on two factors: The temperature of the geofluid and the fluid flow rate. Figs. 7.2 and 7.3 show these dependencies [5].

7.2.1.2 Biomass Models Organic plant matter, known as biomass, can be converted to energy by burning, gasification, fermentation, or other processes. Wood products are the most common form of biomass power, but agricultural waste, yard-clippings, and municipal solid waste can also be used. To generate power, biomass is typically burned in a boiler, which produces high-pressure steam that is used to drive a turbine that generates electricity. Turbine

Transmission lines

Generator

Steam

Air/water vapour

Condenser

Hot water

Cooling tower Cold water

Air

Pump

Seperator Steam Excess cold water

Pump

Direct heat use

Brine Waste brine Impermeable strata Production well

Geothermal reservoir

Injection well

FIGURE 7.1 Electricity production from geothermal sources. T.M. Letcher, Future energy: improved, sustainable and clean options for our planet, Second ed., Elsevier, 2014, Chapter 22.

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FIGURE 7.2 Specific power output in kW/(kg/s) for low to moderate temperature geofluids as a function of inlet (T1 and T2) shown in degrees Celsius. M. Kubik, The future of geothermal energy: impact of enhanced geothermal systems (EGS) on the United States in the 21st century. 2016, from http://geothermal.inel.gov.

FIGURE 7.3 Optimized power output for a 1-flash plant as a function of geofluid temperature in degrees Celsius for geofluid flow rates of 100 and 1000 kg/s. M. Kubik, The future of geothermal energy: impact of enhanced geothermal systems (EGS) on the United States in the 21st century., 2016, from http://geothermal.inel.gov.

7.2 RENEWABLE-ENERGY-BASED DG MODELS

237

The power output potential for biomass depends on the heat content of the biomass fuels which is highly dependent on the plant species. Some combustion technologies can accept a wide range of biomass feedstock, while others require a much more homogenous feedstock. The heat content can also vary depending on the moisture content of the biomass fuel and the conversion technologies.

7.2.2 STOCHASTIC MODELS Two of the primary DG models that rely on variable input are wind energy and PV solar. Other renewable energy sources that have stochastic elements are solar thermal, microhydro, wave, and tidal. The wind energy and PV will be the focus of this chapter as they have seen a significant increase in deployment in recent years.

7.2.2.1 Wind Unit Models The power generated by a wind turbine is calculated by a simple application of the kinetic energy equation KE 5 1=2mv2 through a cross sectional area A. The available power in Watts can be shown to be 1 1 πd 2 P 5 ρv3 A 5 ρv3 4 2 2

(7.1)

where v is the wind speed in meter per second (m/s), ρ is the density of air in kg/m3 (typically 1.225 kg/m3 at 15.55
C and 101,325 Pa), andd is the rotor diameter in m. Clearly, as the wind speed increases, the power output will increase with the cube of the wind speed. This model must be adapted to account for cut-in speed, rated speed, and cut-out speed. Low-wind speeds do not provide sufficient torque to the turbine blades to make them rotate. Power generation typically begins at wind speeds between 3 m/s and 4 m/s (6.71 and 8.95 mi/h). The wind speed at which power generation begins is called the cut-in speed. As the wind speed exceeds the cut-in speed, power generation will rise steeply as a function of the cube of the wind speed. However, when the speed reaches a threshold, the wind turbine will reach its rated power output and will no longer increase its power as the velocity increases. This typically occurs between 12 m/s and 17 m/s (26.84 and 38.03 m/h). With large turbines, the wind power is usually maintained at a constant level by adjusting the angle of the blades. As the speed increases above the rated speed, there is a danger that excessive forces will cause damage to the turbine. A braking system will engage and bring the rotor to a standstill. This occurs at the cut-out speed which is typically around 25 m/sÐ (Fig. 7.4). Energy production, measured in kW h, is E 5 PðtÞdt. Since the power output of a wind turbine depends on the wind speed, predicting energy production requires the use of PDF to account for the variances in wind speed in time. There are two commonly used PDFs that are used to describe wind velocity variation: The normal (Gaussian) distribution and the Weibull distribution. The Weibull distribution is often a better fit for modeling wind speeds due to the asymmetry of measured distributions. The Weibull PDF applied to wind speeds is defined as f ðv;c; kÞ 5

k  v k21 2ðv=cÞk e c c

where k is the shape factor, c is the scale factor, and v is a vector of the measured wind speeds.

(7.2)

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CHAPTER 7 OPTIMUM SITING AND SIZING OF RENEWBLE DGs

FIGURE 7.4 Power Generated as a function of wind speed. Wind Power Program UK. ,http://www.wind-power-program.com/wind_statistics.htm., 2016 (accessed 21-06-16).

The value of the shape factor k determines the width of the wind speed distribution around the average. Smaller values indicate a relatively wider wind speed distribution, while a larger indicates the opposite. The scale factor c defines where the majority of the distribution lies and how stretched out the distribution is. Fig. 7.5 shows a Weibull PDF for wind speed data collected during a 1-year period in Gilbert, IA and includes the shape and scale factors. MATLAB was used to analyze the data. Fig. 7.6 shows the fitted Weibull PDF with c 5 5.0298, k 5 1.9579 compared to the empirical PDF plotted using historical hourly wind speed for the Ames site. The Weibull curve fitting is used to obtain the scale and shape factors using historical wind speed data. As shown, the fitted Weibull PDF is a suitable fit and an appropriate probabilistic model for the wind speed data.

7.2.2.2 PV Unit Models Solar irradiance is the amount of power that reaches the surface of the earth per unit area. For a given location, this amount will vary on a daily and yearly basis due to the motion of the earth relative to the sun. The amount of solar radiation also depends on the geographical location (latitude and longitude) and the climatic conditions. Cloud cover is the main factor affecting the amount of solar radiation that reaches the surface of the earth. Fig. 7.7 shows the seasonal and hourly variability in solar irradiance measured at Ames, Iowa for the year 2015.

7.2 RENEWABLE-ENERGY-BASED DG MODELS

FIGURE 7.5 Weibull PDF for Gilbert, IA (2009).

FIGURE 7.6 Fitted Weibull PDF (c 5 5.0298, k 5 1.9579) versus Empirical PDF for Ames, IA (2009).

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CHAPTER 7 OPTIMUM SITING AND SIZING OF RENEWBLE DGs

FIGURE 7.7 Measured Solar Irradiance for Ames, IA in 2015.

There are a variety of models that are used to estimate the solar irradiance and PV power generation for example, the Meneil, Hottel, and Iqbal models for irradiance. Two models will be examined: The ASHRAE model [6,7] and a model developed by Tina et al. [8]. The ASHRAE model was developed by the American Society of Heating, Refrigerating, and Air Conditioning engineers to estimate the beam portion of the irradiation reaching the earth’s surface for a clear sky [6,7]. It is based on original empirical work and has the advantage that it does not depend on any atmospheric data; yet, it has been widely used in many solar applications. The direct beam radiation at sea level, Io , is determined by the following equation:  Io 5 A  exp 2

 B  cosðθZ Þ ½W=m2  cosðθZÞ

(7.3)

where A represents the apparent extraterrestrial flux, A 5 1140 1 75 sinðnÞ, B 5 0.132 1 0.023 cosðnÞ; θZ is the solar zenith angle (depends on declination, latitude, and hour for a given location), and n is the day number. Fig. 7.8 shows the maximum direct beam radiation as a function of the day of the year for latitudes ranging from 0
to 60
. To account for the differences between the maximum solar radiation available and the actual amount reaching earth’s surface, an hourly clearness index, kt , has been defined as the ratio of the irradiance on a horizontal plane, It [kW/m2] to the extraterrestrial maximum solar irradiance Io [kW/m2]: kt 5

It Io

(7.4)

7.2 RENEWABLE-ENERGY-BASED DG MODELS

241

FIGURE 7.8 Irradiance values for day of year and latitudes (0, 15, 30, 45, and 60
N).

Since most PV units are installed at an angle relative to the horizontal plane, the irradiance can be modeled by an equation that accounts for this angle and the reflectivity of the ground.      1 1 cosβ 1 2 cosβ Iβ 5 R b 1 2 Rb  k 1 ρ   It kW=m2 2 2

(7.5)

where Rb is the ratio of beam radiation on the tilted surface to that on a horizontal surface, k is the fraction of the hourly radiation on the horizontal plane which is diffused, and ρ is the reflectance of the ground. The power output of a PV can then be modeled using the following equation: Ppv 5 AC  η  Iβ

(7.6)

where AC is the surface area of the PV array (m ), η is the efficiency of the PV in realistic reporting conditions (RRC), and Iβ is irradiance on a surface with an angle to the horizontal plane (kW/m2). The efficiency of a PV system will depend on a number of factors such as the temperature of the solar cell, intensity, and spectrum of the incident sunlight. The open circuit voltage is highly 2

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CHAPTER 7 OPTIMUM SITING AND SIZING OF RENEWBLE DGs

sensitive to changes in temperature, which could be due to variation in ambient temperature, irradiance, or wind speed at the panel site [9]. Like all electronics, PV systems have lower efficiency in the heat than the cold. Because Iβ will vary throughout the year based on latitude, the day of the year, and the hour angle, sunset hour angle, and reflectance of the ground, the parameters T and T0 will be used to account for these variations. Thus, the expressions for the power output generated by a PV array is given by the following equation: 

0 Ppv 5 AC  η  T  kt 2 T  kt2

where T5

(7.7)

     1 2 cosβ 1 1 cosβ Ho 1 2 R b ρ  rd  Rb 1 ρ 2 2 3600   1 1 cosβ Ho 0 2 Rb  q  rd  T 5 2 3600

in which Rb is a geometric factor, the ratio of beam radiation on the tilted surface to that of a horizontal surface at any time, ρ is the reflectance of the ground, β is the tilt angle measured in degrees, rd is the ratio of the diffuse radiation in an hour to the diffuse radiation in a day, Ho is the extraterrestrial radiation on a horizontal surface (mJ=m2 ), and q is the parameter in the functional form for k (kt ). The hourly clearness index,kt , can be considered a random variable due to weather conditions such as cloud cover and visibility factors such as mist and haze. From sampled values of kt over a period of time, it is possible to obtain a probability density function fp pv ðPpvÞ for Ppv. There are four PDFs typically used for solar radiation. In all cases, the value of x represents the hourly clearness index.

Normal Distribution f ðx;μ; σÞ 5

2 1 2 pffiffiffiffiffiffi e2ðx2μÞ =2σ σ 2π

(7.8)

where μ is the mean of the distribution, and σ is the standard deviation of the distribution.

Weibull Distribution f ðx;a; bÞ 5 b  a2b xb21 e2ðx=aÞ

2

(7.9)

where a is the shape parameter of the distribution, and b is the scale parameter of the distribution.

Extreme Value Distribution f ðx;μ; σÞ 5 σ21  e½ðx2μÞ=σ  e2e½ðx2μÞ=σ

(7.10)

where μ is the location parameter of the distribution, and σ is the scale parameter of the distribution.

7.3 EFFECTS OF DGs ON DISTRIBUTION NETWORKS

243

Beta Distribution f ðx;α; β Þ 5

Γ ðα 1 β Þ ð12xÞβ21  xα21 Γ ðαÞΓ ðβ Þ

(7.11)

where α is the first shape parameter of the distribution, and β is the second shape parameter of the distribution. It has been observed that the choice of an adequate PDF model may depend on latitude [10].

7.3 EFFECTS OF DGs ON DISTRIBUTION NETWORKS Correct siting and sizing DG units is crucial when implemented within an existing distribution network. In this section, we will examine how integrating DG units may have an impact on the distribution network in terms of voltage profile, power losses, system reliability, economic and environmental cost and benefits, and voltage stability of the distribution network. We will suggest several objective functions used to model optimal siting and sizing of DG units in this section.

7.3.1 VOLTAGE PROFILE Random placement of DG units can produce many technical problems such as voltage rise, voltage fluctuations, and the introduction of harmonics and transients. When installed in a distribution network, there can be a mismatch between the reactive power requirements of the load and the reactive power supplied by the system. In weakly loaded systems, DG integration may result in high-voltage problems and interfere with standard voltage regulation practices [3]. The stochastic nature of renewable energy sources can further worsen the voltage conditions due to their intermittency and uncertainty. Intermittency can be mitigated with a combination of energy storage, spinning reserves, and demandresponse management. However, carefully siting and sizing DG units may actually lead to an improvement in the system voltage profile, which can lead to improvement in overall system performance. One of the key objectives in choosing the size and placement of a DG unit is to minimize the voltage deviation compared to the nominal voltage (1.0 p.u.). There are several strategies that can be used to develop an objective function for the voltage profile. One technique is to use the voltage index (VIi ) at each node in the distribution network. The location and sizing of the DG should provide the best voltage profile. The voltage deviation index is then calculated using the following equation: VIi 5

nbus X

ð12Vk Þ2

(7.12)

k

where at bus i, a DG unit is assumed to be installed. Then, k varies from 1 to the number of all or critical buses in the system. Using Eq. (7.12), Vk is the per-unit voltage at the kth bus and nbus is the number of buses. The optimal placement and sizing of the DG would result in the smallest value for VIi [11].

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CHAPTER 7 OPTIMUM SITING AND SIZING OF RENEWBLE DGs

Another technique is to calculate the voltage profile index (VPI) [12]. This index penalizes a sizelocation pair which gives higher voltage deviations from the nominal value (Vnom). The VPI is defined as follows: 

VPI 5 maxni51

jVnom j 2 jVi j jVnom j



(7.13)

where n is the number of buses. Using the VPI, values closer to zero indicate better network performance. Instead of using the voltage index or VPI as objective functions, it is common to use the voltage profile as a constraint so that Vlower bound , jVj , Vupper bound to ensure that it remains within acceptable operational limits. In a weakly loaded system, overvoltage problems can be mitigated by reducing the primary substation voltage, using auto transformers and voltage regulators, increasing conductor size, or shutting down DG units during light load conditions [3,13]. While day-ahead forecasts are useful for pricing within the energy market, hour-ahead forecasts are necessary for grid operators to manage and schedule spinning reserves to manage the voltage profile [4]. This poses a challenge for renewable, stochastic DG units such as wind and solar PV.

7.3.2 POWER LOSSES Power loss due to the resistances in overhead lines and underground cables should be minimized in the distribution network branches. The real power loss, commonly referred to as exact loss in a system is given by (7.14)(7.16) PL 5

N X N  X

 αij Pi Pj 1 Qi Qj 1 β ij ðQi Pj 1 Pi Qj Þ

(7.14)

i51 i5j

In this expression, losses are a function of the net active (P) and net reactive (Q) power injected on each bus of the network [14]. N is the number of buses. Here, the loss coefficients are defined by αij 5

Rij cosðδi 2 δj Þ Vi Vj

(7.15)

β ij 5

Rij sinðδi 2 δj Þ Vi Vj

(7.16)

where Pk is the net real power at bus k, Qk is the net reactive power at bus k, Rij is the real part of the element at row i and column j of the bus impedance matrix, Vk is the voltage at bus k, and δk is the voltage angle at bus k. Calculating the exact power loss is computationally expensive. When developing models that include power loss, approximation formulas will often suffice. One such approximation technique is the B Matrix Power Loss Formula given by (7.17). Using the B Matrix approximation eliminates, the need to calculate the loss for each transmission line provided that the system structure remains reasonably constant [15]. This approximation works well for wide changes in the loading pattern of the system. We refer our readers to Ref. [15] for a discussion on the techniques to compute the values for the B matrix.

7.3 EFFECTS OF DGs ON DISTRIBUTION NETWORKS

PL 5 PT ½BP 1 BTo P 1 Boo

245

(7.17)

where P is the vector of all generator buses’ net real power, ½B is the square matrix of the same dimension as P, Bo is the vector of the same length as P, and Boo is a constant. This can be written in terms of row and column terms given by the following equation: PL 5

XX i

j

Pi Bij Pj 1

X

Bio Pi 1 Boo

(7.18)

i

The penetration of DG units within a network can cause significant changes in voltage magnitudes and power flows. These changes will affect power system losses. If DG units are strategically allocated, they can cause a significant reduction in losses of vulnerable feeders [16]. If a DG unit is sited near a large load, the network losses will be reduced since the load will be fed by both real and reactive power, and conversely, the network losses will increase if the generator is placed far away from the load center. When optimally sized and placed, DGs can reduce system losses by 1020% [2]. DG technologies such as solar PV, micro turbines, and fuel cells are capable of reducing active power losses, but are incapable of delivering reactive power [2]. DG units with power electronic interfaces (depending on the inverter technology used) are capable of supplying and consuming reactive power when required. The management of the apparent power may require sophisticated control mechanisms during DG implementation [3].

7.3.3 POWER DISTRIBUTION SYSTEM RELIABILITY 7.3.3.1 An Introduction on Importance of Reliability Studies in Distribution Systems Over the past 100 years, the role of electricity has evolved. In today’s information age, reliable electricity is no longer a luxury; it is now a basic requirement of many customers. It is critical to all aspects of safely operating our cities, businesses, and homes. However, the electric grids have not kept pace with surging demand. Even with substantial improvements in energy-efficient buildings, electricity demand has increased from 1500 billion kW h in 1970 to over 3700 billion kW h in 2004 and is projected to reach 5600 billion kW h by 2030 [17]. With the growing demand and increasing dependence on electricity energy, the necessity to achieve an acceptable level of reliability becomes a crucial issue for decision makers of power systems [18]. Therefore, the utilities in electricity industry have to improve systems continuously to meet customers’ reliability requirements. Distribution systems, as the final level in delivery of electricity to the end users, have an undeniable role in providing reliable electricity to the costumers. However, the technical studies show that more than 90% of the customer service interruptions occur due to failures in different equipment of these systems [19]. This represents the importance of this sector in reliability level of customers and emphasizes on the reliability improvement of distribution systems. Reliability is a measure of the system’s ability to meet the electricity needs of customers, which can be used as a means of assessing the past performance of a system or predicting its future. For distribution systems, at the present time, reliability evaluations are more widely used in assessing the system performance. Assessment of system performance is a valuable procedure for three important reasons.

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It establishes the chronological changes in system performance and therefore helps to identify weak areas and the need for reinforcements. It establishes existing indices which serve as a guide for acceptable values in future reliability assessments. It enables previous predictions to be compared with actual operating experience.

Reliability assessment necessitates evaluation techniques and indices to quantify current system status and compare the merits of various reinforcement schemes.

7.3.3.2 Reliability Evaluation Methods Power system reliability evaluation methods can be divided into two categories: (1) Analytical methods, and (2) simulation methods. In addition, the reliability evaluation may be a qualitative study, in which the main factors that impact system reliability can be determined and prioritized, or a quantitative study, where the reliability is assessed through different parameters and indices defined and calculated for the system or load points [20]. Analytical techniques represent the system by a mathematical model and evaluate the reliability indices from this model using direct numerical solutions. They generally provide expectation indices in a relatively short computing time. There are several methods to analytically evaluate reliability, including fault tree analysis, failure mode, effect, and criticality analysis, Markov processes, minimum cut-set method, and network reduction method [21]. The most common evaluation techniques using a set of approximate equations are failure mode analysis and minimum cut set analysis. Analytical approaches are based on assumptions concerning the statistical distributions of failure rate and repair times. On the contrary, assumptions are frequently required in order to simplify the problem and produce an analytical model of the system. This is particularly the case when complex systems and complex operating procedures have to be modeled. The resulting analysis can therefore lose some or much of its significance. The use of simulation techniques is very important in the reliability evaluation of such situations. Simulation methods estimate the reliability indices by simulating the actual process and random behavior of the system. Monte Carlo technique is the widely used simulation method which has two different categories [21]: (1) Sequential, and (2) nonsequential. In a nonsequential Monte Carlo simulation (MCS), the samples are taken without considering the time dependency of the states or sequence of the events in the system. Therefore, using this method, a nonchronological state of the system is determined. On the other hand, a sequential MCS can address the sequential operating conditions of the system, and may be used to include time correlated events and states such as output generation of renewable-based generating units, demand profile, and customer decisions, which is more applicable for power system reliability studies. The MCS methods are classified based on the methods used for sampling. Three common sampling approaches in MCS are: (1) State sampling approach; (2) system state transition sampling approach; and (3) state duration sampling approach [21].

7.3.3.3 Reliability Indices Reliability indices are used by system planners and operators as a comparative tool to improve the quality level of service to customers. Planners use these parameters to determine the requirements

7.3 EFFECTS OF DGs ON DISTRIBUTION NETWORKS

247

for capacity additions, while operators use them to ensure that the system is robust enough to withstand possible failures without catastrophic consequences. The vast majority of reliability indices reflect faults and failures in the distribution system. These metrics are based upon the customers’ outage data and the unserved load. They describe how often electrical service are interrupted, how many customers are involved with each outage, how long the outages lasts, and how much load goes unsupplied. Distribution reliability indices falls into two major categories, costumer-oriented and load- and energy-oriented, which both provide detailed information about the system [18]. 1. Customer-oriented indices a. System average interruption frequency index (SAIFI) SAIFI indicates how often an average customer is subjected to sustained interruption over a predefined time interval: SAIFI 5

P Total Number of Customer Interruptions λi Ni 5 P Total Number of Customers Served Ni

(7.19)

where λi is the failure rate, and Ni is the number of customers of load point i. b. System average interruption duration index (SAIDI) It indicates the total duration of interruption an average customer is subjected for a predefined time interval: SAIDI 5

P Ui Ni Sum of Customer Interruption Durations 5 P Total Number of Customers Ni

(7.20)

where Ui is the annual outage time. c. Customer average interruption duration index (CAIDI) This index indicates the average time required to restore the service: CAIDI 5

P Ui Ni Sum of Customer Interruption Durations 5P Total Number of Customer Interruptions λi Ni

(7.21)

d. Average service availability/unavailability index (ASAI/ASUI) ASAI specifies the fraction of time that a customer has received the power during the predefined interval of time and is vice versa for ASUI: P Ui Ni Customer Hours of Available Service P 512 Customer Hours Demanded 8760 3 Ni P Customer Hours of Unavailable Service Ui Ni P 5 ASUI 5 1 2 ASAI 5 8760 3 Ni Customer Hours Demanded ASAI 5

(7.22.a) (7.22.b)

where 8760 is the number of hours in a calendar year. 2. Load- and energy-oriented indices a. Energy not supplied (ENS) ENS specifies the average energy the customer has not received in the predefined time: ENS 5 Total Energy Not Supplied by the System 5

where LαðiÞ is the average load connected to load point i.

X

LαðiÞ Ui

(7.23)

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b. Average energy not supplied (AENS) AENS 5

Total Enery Not Supplied 5 Total Number of Customers Served

P L U PαðiÞ i Ni

(7.24)

c. Average customer curtailment index (ACCI) ACCI 5

Enery Not Supplied Total Number of Customers Affected

(7.25)

This index differs from AENS in the same way that CAIFI differs from SAIFI. It is therefore a useful index for monitoring the changes of average energy not supplied between one calendar year and another.

7.3.3.4 DGs and Electric Distribution System Reliability Today, changing consumer needs, combined with advances in DG technologies making them more efficient and less costly, have opened new opportunities for consumers to use DGs to meet their own energy needs, as well as electric utilities to explore possibilities to meet electric system needs with distributed generation. Implementation of DGs offers potential benefits to distribution systems. They provide an enhanced power quality, reduce peak loads, and provide ancillary services such as reactive power and voltage support. Using DGs to meet these local system needs can add up to improvements in overall electric system reliability. Therefore, DGs can be used by electric system planners and operators to improve reliability in both direct and indirect ways. 1. Direct Effects DGs could be used directly to support local voltage levels and avoid an outage that would have otherwise occurred due to excessive voltage sag. In addition, they can add to supply diversity and thus lead to improvements in overall system adequacy. Multiple analyses have shown that a distributed network of smaller sources provides a greater level of adequacy than a centralized system with fewer large sources, reducing both the magnitude and duration of failures [22]. However, it should also be noted that a single stand-alone distributed unit without grid backup will provide a significantly lower level of adequacy. A study of actual operating experience determines how DG units perform in the field [23]. The study results include forced outage rates, scheduled outage factors, service factors, mean time between forced outages, and mean down times for a variety of DG technologies and duty cycles. The availability factors collected during this study are summarized in Fig. 7.9. Although the sample size for the DG equipment has been smaller than that for the central station equipment, the availability of the DG is generally comparable to that of central station equipment. Modeling DGs of varying failure and repair rates, using a Monte Carlo evaluation technique and the unserved load as a reliability measure, has showed that DG can enhance the overall capacity of the distribution system and can be used as an alternative to the substation expansion in case of expected demand growth [24]. Integration of DGs can also help the distribution system reliability by serving a portion of the load, and also enabling feeder tie operations that were previously blocked by high-load levels. A study on an actual utility system has also shown that the addition of DG on just one feeder has led to SAIDI improvements ranged from 5 to 22% in all feeders [25].

7.3 EFFECTS OF DGs ON DISTRIBUTION NETWORKS

249

100

Availability factor (%)

95

90

85

Reciprocating engines

Steam turbines

Gas turbines

Fuel cell

All DG

Hydro

Combined cycle

Gas turbine

Nuclear

Fossil boiler