Uncertainties in Modern Power Systems

By Ahmed F. Zobaa and Shady Abdel Aleem

Uncertainties in Modern Power Systems combines several aspects of uncertainty management in power systems at the planning and operation stages within an integrated framework. This book provides the state-of-the-art in electric network planning, including time-scales, reliability, quality, optimal allocation of compensators and distributed generators, mathematical formulation, and search algorithms. The book introduces innovative research outcomes, programs, algorithms, and approaches that consolidate the present status and future opportunities and challenges of power systems. The book also offers a comprehensive description of the overall process in terms of understanding, creating, data gathering, and managing complex electrical engineering applications with uncertainties.

This reference is useful for researchers, engineers, and operators in power distribution systems.

• Includes innovative research outcomes, programs, algorithms, and approaches that consolidate current status and future of modern power systems
• Discusses how uncertainties will impact on the performance of power systems
• Offers solutions to significant challenges in power systems planning to achieve the best operational performance of the different electric power sectors

• Language: English
• Publisher: Academic Press
• Release date: Oct 26, 2020
• ISBN: 9780128208939

Ahmed F. Zobaa

Ahmed Faheem Zobaa received his B.Sc.(Hons), M.Sc. and Ph.D. degrees in electrical power and machines from Cairo University, Egypt. After being a senior lecturer in renewable energy at University of Exeter, U.K, he became associate professor and then Professor at Cairo University. Currently, he is a Senior Lecturer in power systems, an MSc course director and a full member of the Institute of Energy Futures at Brunel University London, U.K. His main areas of expertise are power quality, (marine) renewable energy, smart grids, energy efficiency, and lighting applications. Dr. Zobaa is involved in the editorial activities of many international journals, including as editor-in-chief, editor and associate editor. He is a registered Chartered Engineer, Chartered Energy Engineer, European Engineer, and International Professional Engineer. He is also a registered member of the Engineering Council U.K., Egypt Syndicate of Engineers, and the Egyptian Society of Engineers. He is a senior fellow of the Higher Education Academy of U.K. He is a fellow of the Institution of Engineering and Technology, the Energy Institute of U.K., the Chartered Institution of Building Services Engineers, the Royal Society of Arts, the African Academy of Science, and the Chartered Institute of Educational Assessors. He is a senior member of the Institute of Electrical and Electronics Engineers. Also, He is a member of the International Solar Energy Society, the European Power Electronics and Drives Association, and the IEEE Standards Association.

Book preview

Uncertainties in Modern Power Systems

Editors

Ahmed F. Zobaa, BSc(Hons), MSc, PhD, DSc

College of Engineering, Design and Physical Sciences, Brunel University London, Uxbridge, United Kingdom

Shady H.E. Abdel Aleem, BSc, MSc, PhD

Mathematical, Physical and Engineering Sciences Department, 15th of May Higher Institute of Engineering, Cairo, Egypt

Table of Contents

Cover image

Title page

Copyright

Contributors

Preface

Chapter 1. Overview of uncertainties in modern power systems: uncertainty models and methods

1. Introduction

2. Uncertainty models of parameters in power systems

3. Uncertainty modeling methods

4. Future trends

Chapter 2. Expansion planning of transmission networks

1. Introduction

2. Definition of uncertainties in the structure of modern power systems

3. Deterministic programing–based expansion planning of transmission networks

4. Stochastic programing–based expansion planning of transmission networks

5. Robust programming–based expansion planning of transmission networks

6. Hybrid robust/stochastic programming–based expansion planning of transmission networks

7. Conclusion

Chapter 3. A stochastic robust approach to deal with the generation and transmission expansion planning problem embedding renewable sources

1. Introduction

2. Mathematical uncertainty

3. Problem formulation

4. Solution approach

5. Study case

6. Conclusions

Chapter 4. Data-driven robust stochastic optimization for power systems operations

1. Uncertain optimization in the operation of modern power systems

2. RO-based distribution network reconfiguration

3. Data-driven ACOPF with wind power uncertainty

4. Wasserstein metric-based DRO for unit commitment problem

5. RSO model for dynamic economic dispatch problem

6. Conclusion

Chapter 5. Optimal power flow for distribution systems with uncertainty

1. Introduction

2. Distributed generator's benefits and uncertainty

3. Problem formulation

4. The uncertainties of load demand, solar, and wind power resources

5. Equilibrium optimizer

6. Simulation results

7. Conclusion

Chapter 6. Distributed demand-side management for microgrids in modern power system

1. Introduction

2. Distributed DR literature review

3. Distributed demand response in campus microgrids

4. Distributed demand response for microgrid-aided restoration

5. Case study

6. Summary

A. Section in appendix

Chapter 7. Evaluation of DG impacts on distribution networks

1. Introduction

2. Network losses

3. Voltage profile of the system

4. Reliability

Chapter 8. A comprehensive review of islanding detection methods

1. Introduction

2. International standards

3. Criteria to evaluate islanding detection methods

4. Islanding detection methods

5. Conclusions and future scope

Chapter 9. Protection system failure and power system blackout

1. Introduction

2. Performance evaluation of distance relay under power system disturbances

3. Performance of conventional techniques during system stressed conditions

4. Available solutions

5. Conclusion

Chapter 10. Testing of influence of SVC and energy storage device’s location on power system using GAMS

1. GAMS

2. SVC devices

3. Energy storage systems

4. Optimal location of SVC and ESS devices in a power network

Chapter 11. DG investment and allocation in active distribution networks

1. Introduction

2. Allocation of DGs in distribution networks

3. Modeling and simulation

4. Modifications in Newton–Raphson load flow

5. Average load demand and renewable generation pattern with uncertainty modeling

6. Objective functions with technical constraints

7. Mixed discrete particle swarm optimization

8. Uncertainty analysis with renewable generation and load

9. Annual cost–benefit analysis for DG placement

10. The methodology of DGs placement using the proposed method

11. Essential assumptions and parameters

12. Case study (33 bus RDNR)

13. Conclusions

Appendix

Chapter 12. Technical assessment of the impacts of distributed energy resources on distribution feeders

1. Introduction

2. Framework for impact assessment studies

3. A stochastic–probabilistic approach

4. Case studies

5. Conclusion

Chapter 13. Large-scale integration of distributed generation in reconfigured distribution networks considering load uncertainty

1. Introduction

2. Problem statement

3. Problem formulation

4. Simulation results and discussions

5. Conclusions

Chapter 14. Steady-state and time-varying harmonics in distribution system

1. Introduction

2. Harmonic data acquisition systems

3. Steady-state harmonic analysis

4. Design of harmonic analyzer

5. Time-varying harmonics analysis

6. Results and discussions

7. Estimation of time constant, time varying, and interharmonics

8. Mitigation of harmonics

9. Conclusions

Chapter 15. A case study with power quality analysis on building integrated PV (BIPV) system

1. Introduction to power quality issues

2. Solar photovoltaic as a renewable source of energy

3. Power quality standards

4. Voltage disturbances on electrical equipment

5. Guilty party for the power quality issues

6. Solar photovoltaic energy system incorporation

7. A case study on power quality analysis

8. Discussion of building integrated photovoltaic system issues of chosen building—Chennai automobile showroom

9. System and component description

10. Connected load for automobile showroom

11. Battery design consideration and sizing method

12. Procedure to estimate energy delivered by a photovoltaic panel

13. Example case

14. Conclusion

Chapter 16. Reliability evaluation of Li-ion batteries for electric vehicles applications from the thermal perspectives

1. Introduction

2. Electric vehicles trend

3. Effects of the temperature and control strategies on the reliability of the Li-ion batteries

4. Passive and active battery thermal management systems and their limitations

5. Case study: thermal management system and reliability assessment in Li-ion batteries

6. Conclusion

Chapter 17. Role of compensators' hybrid participation for isolated wind-driven electrical system in presence of input and load uncertainties

1. Introduction

2. Operational issues for wind-driven renewable energy system

3. Policy framework for wind-driven renewable energy–based system

4. Introduction to wind-diesel based isolated hybrid electrical system

5. Reactive power compensation issues in isolated hybrid electrical system

6. Reactive power compensation as techno-economic issues

7. Mathematical study of isolated hybrid electrical system

8. Challenges in load modeling

9. Modeling for composite loads

10. Effect of load and input uncertainties on system dynamics

11. Load and input uncertainties’ related findings for system

12. Modeling of load and input uncertainties for system

13. Role of compensators to mitigate the load and input disturbances

14. Importance of tuning methods for probabilistic load model

15. Investigations of hybrid compensation for economic benefits

16. Conclusion

Chapter 18. Bidding strategies of a power producer in power market: measurement indices and evaluation

1. Introduction

2. Problem formulation

3. Sinusoidal function–enabled chaotic grasshopper optimization algorithm

4. Sinusoidal function–enabled chaotic grasshopper optimization algorithm

5. Results analysis

6. Conclusion

Author Index

Subject Index

Copyright

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Contributors

Naeem Abas,     Department of Electrical Engineering, University of Gujrat, Hafiz Hayat Campus, Gujrat, Pakistan

Almoataz Y. Abdelaziz,     Faculty of Engineering and Technology, Future University in Egypt, Cairo, Egypt

Shady H.E. Abdel Aleem,     Department of Mathematical, Physical and Engineering Science, 15th of May Higher Institute of Engineering, Helwan, Cairo, Egypt

Ziad M. Ali

Electrical Engineering Department, College of Engineering at Wadi Addawaser, Prince Sattam Bin Abdulaziz University, Al-Khar, Saudi Arabia

Electrical Engineering Department, Aswan Faculty of Engineering, Aswan University, Aswan, Egypt

Dhafer Almakhles,     Renewable Lab, College of Engineering, Prince Sultan University, Riyadh, Saudi Arabia

Xiaoqing Bai,     Key Laboratory of Guangxi Electric Power System Optimization and Energy-saving Technology School of Electrical Engineering, Guangxi University, Nanning, China

Ramesh C. Bansal,     Department of Electrical Engineering, University of Sharjah, Sharjah, United Arab Emirates

Soumyabrata Barik,     Department of Electrical and Electronics Engineering, Birla Institute of Technology and Science Pilani, K. K. Birla Goa Campus, Zuarinagar, Goa, India

Hamidreza Behi

ETEC Department & MOBI Research Group, Vrije Universiteit Brussel, Brussel, Belgium

Flanders Make, Heverlee, Belgium

Mohammadreza Behi

The University of Sydney, School of Chemical and Biomolecular Engineering, Sydney, NSW, Australia

Department of Energy Technology, KTH Royal Institute of Technology Stockholm, Sweden

Bernard Bekker,     Department of Electrical and Electronic Engineering, Stellenbosch University, Stellenbosch, South Africa

Maitane Berecibar

ETEC Department & MOBI Research Group, Vrije Universiteit Brussel, Brussel, Belgium

Flanders Make, Heverlee, Belgium

Monalisa Biswal,     Department of Electrical Engineering, NIT Raipur, Chhattisgarh, India

Martin Ćalasan,     Faculty of Electrical Engineering, University of Montenegro, Podgorica, Montenegro

B.K. Chaitanya,     Department of Electrical Engineering, National Institute of Technology, Raipur, Chhattisgarh, India

M. Justice Chihota,     Department of Electrical and Electronic Engineering, Stellenbosch University, Stellenbosch, South Africa

Debapriya Das,     Department of Electrical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India

Ibrahim Mohamed Diaaeldin,     Engineering Physics and Mathematics Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt

Mohamed Ebeed,     Department of Electrical Engineering, Faculty of Engineering, Sohag University, Sohag, Egypt

Ahmed El-Rafei,     Engineering Physics and Mathematics Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt

Foad H. Gandoman

ETEC Department & MOBI Research Group, Vrije Universiteit Brussel, Brussel, Belgium

Flanders Make, Heverlee, Belgium

Wenzhong Gao,     Department of Electrical and Computer Engineering, University of Denver, Denver, United States

Agustina Hernandez-Tolentino,     Electrical Engineering Department, CINVESTAV del IPN – Unidad Guadalajara, Zapopan Jal., Mexico

M. Jagabar Sathik

SRM Institute of Science and Technology, Kattankulathur Campus, Kattankulathur, Tamilnadu, India

Renewable Lab, College of Engineering, Prince Sultan University, Riyadh, Saudi Arabia

Joris Jaguemont

ETEC Department & MOBI Research Group, Vrije Universiteit Brussel, Brussel, Belgium

Flanders Make, Heverlee, Belgium

Ali Raza Kalair,     Department of Telecommunications, Electrical, Robotics and Biomedical Engineering, Swinburne University, Australia

Anam Kalair,     Department of Electrical Engineering, Iowa State University, Ames, IA, United States

Salah Kamel,     Department of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan, Egypt

Katarina Kecojević,     Faculty of Electrical Engineering, University of Montenegro, Podgorica, Montenegro

Nasrullah Khan,     Department of Electrical Engineering, COMSATS University Islamabad, Pakistan

Wei Kou,     Department of Electrical and Computer Engineering, University of Connecticut, Storrs, CT, United States

Ashwani Kumar,     National Institute of Technology, Haryana, Kurukshetra, India

Rajesh Kumar,     Department of Electrical Engineering MNIT, Jaipur, Rajasthan, India

Ognjen Lukačević,     Faculty of Electrical Engineering, University of Montenegro, Podgorica, Montenegro

M.A. Mahmud,     School of Engineering, Deakin University, Geelong, VIC, Australia

Jose Antonio Marmolejo-Saucedo,     Universidad Panamericana, Facultad de Ingeniería, Augusto Rondin 498, Ciudad de Mexico, México

Hesam Mazaheri,     Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

Moein Moeini-Aghtaie,     Faculty of Energy Engineering, Sharif University of Technology, Tehran, Iran

Loai Nasrat,     Department of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan, Egypt

Sung-Yeul Park,     Department of Electrical and Computer Engineering, University of Connecticut, Storrs, CT, United States

Mohammad Pazoki,     School of Engineering, Damghan University, Damghan, Iran

P. Prem,     Department of Electrical and Electronics Engineering, Bannari Amman Institute of Technology, Sathyamangalam, Tamilnadu, India

Ashraf Ramadan,     Department of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan, Egypt

Juan M. Ramirez,     Electrical Engineering Department, CINVESTAV del IPN – Unidad Guadalajara, Zapopan Jal., Mexico

Hossein Ranjbar,     Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

N. Rishikesh,     Department of Electrical and Electronics Engineering, Bannari Amman Institute of Technology, Sathyamangalam, Tamilnadu, India

Hossein Saber,     Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

Muhammad Shoaib Saleem,     Department of Electrical Engineering, University of Management and Technology Lahore, Sialkot Campus, Sialkot, Pakistan

Akash Saxena,     Department of Electrical Engineering SKIT, M&G, Jaipur, Rajasthan, India

Nitin Kumar Saxena,     KIET Group of Institutions, Delhi-NCR, Ghaziabad, India

J. Senthil Kumar,     Department of Electrical and Electronics Engineering, Bannari Amman Institute of Technology, Sathyamangalam, Tamilnadu, India

Mehdi Seyedmahmoudian,     Department of Telecommunications, Electrical, Robotics and Biomedical Engineering, Swinburne University, Australia

Alex Stojcevski,     Department of Telecommunications, Electrical, Robotics and Biomedical Engineering, Swinburne University, Australia

Xiangyang Su,     Key Laboratory of Guangxi Electric Power System Optimization and Energy-saving Technology School of Electrical Engineering, Guangxi University, Nanning, China

C. Trevor Gaunt,     Department of Electrical Engineering, University of Cape Town, Cape Town, South Africa

Joeri Van Mierlo

ETEC Department & MOBI Research Group, Vrije Universiteit Brussel, Brussel, Belgium

Flanders Make, Heverlee, Belgium

Kasimala Venkatanagaraju,     Department of Electrical Engineering, NIT Raipur, Chhattisgarh, India

Anamika Yadav,     Department of Electrical Engineering, National Institute of Technology, Raipur, Chhattisgarh, India

Rujie Zhu,     Key Laboratory of Guangxi Electric Power System Optimization and Energy-saving Technology School of Electrical Engineering, Guangxi University, Nanning, China

Ahmed F. Zobaa,     College of Engineering, Design and Physical Sciences, Brunel University London, Uxbridge, United Kingdom

Preface

Currently, uncertainty in energy systems is an ongoing issue, particularly in modern renewable energy systems, where it is difficult to accurately describe current or future states of parameters in power systems because of the many uncertainty issues that could expose the systems to potential issues. This is why identifying uncertainties is not a simple task or a straightforward issue. Nowadays, there are different strategies to model these uncertainties. Each has its strengths and limitations. Hence, much attention should be paid to managing uncertainty in energy systems and mitigating their potential risks both at the planning and operational stage. Although predicting uncertainty at the planning stage can deviate from the real situation, it can always be liberated at the operational stage. As well, new tools and techniques have to be investigated to examine all possible (and sometimes impossible) risks and consequences of all scenarios to accurately quantify the importance of uncertainty sources, in a practical sense, to help understand how uncertainties will impact power systems performance. This will enable solving the significant challenges in power systems planning and help to achieve the best operational performance in power systems operation. In this regard, this book introduces good practice to researchers who are interested in modern power systems with numerous research outcomes, and ideas that address several uncertainty issues, opportunities, challenges, and solutions to many problems in today's power systems. The book is sorted out and organized in 18 chapters. Each chapter begins with the fundamental structure of the problem required for a simple understanding of the methods described.

Chapter 1: Models of the uncertainty of parameters in modern power systems such as uncertainty models of load demand, wind energy, solar energy, plug-in electric vehicles, electricity price, and load growth are summarized to bring together several aspects of uncertainty management in power systems in planning and operation stages within an integrated framework. Then, uncertainty modeling methods such as probabilistic (numerical and analytical methods), possibilistic (α-cut and defuzzification), hybrid probabilistic-possibilistic (fuzzy-scenario and fuzzy-Monte Carlo), information gap decision theory, robust optimization, and interval analysis are presented in detail. Finally, the related future trends are explored.

Chapter 2: Transmission networks play an inevitable role in the secure and effective operation of power systems. This role calls for well-organized and comprehensive procedures designed to be used in the operational and expansion of these complicated networks. In this regard, many different frameworks and optimization algorithms have been introduced to deal with the complex nature of the transmission expansion planning (TEP) problem. Moreover, recently some events including the presence of renewable energy resources in the generation sector and deregulation of power systems have intensified the complexity and randomness of the TEP problem. Therefore, some strategies have been employed to handle the uncertainties in these studies. Furthermore, stochastic, robust models, and hybrid (robust/stochastic) models are introduced as effective tools to simplify the complexities experienced in TEP studies of renewable-based power systems. This chapter focuses on the application of these techniques in the TEP problem.

Chapter 3: The objective of this chapter is to propose and deal with an optimization model that helps the system planner to identify the optimal investments in generation and transmission projects. To this end, a dynamic stochastic adaptive robust approach is proposed to solve the problem of generation and transmission expansion planning. The advantage of the dynamic stochastic adaptive robust approach is that it considers long- and short-term uncertainties. Long-term uncertainty refers to year-to-year changes, including maximum demand and available generation capacity in the system during each year of the planning horizon. In contrast, short-term uncertainty represents the production of renewable capacity dependent on climate and load during the planning horizon base year. The dynamic stochastic adaptive robust approach is formulated through a three-level problem. The three-level outline is embedded into a two-level problem. Finally, this last one is figured out by the Benders decomposition method. The proposed model is applied to an equivalent network of the Mexican electricity system. The results show that the expansion plans achieved allow solving the real expansion needs of the system, considering long- and short-term uncertainties.

Chapter 4: The uncertainty of renewable energy brings significant challenges to the operations of the power system with the increasing penetration of renewable energy, such as wind power generation and photovoltaic power generation. The traditional optimization methods are not suitable to solve the problems of the power system operations anymore. Robust optimization (RO) method and stochastic optimization (SO) method are usually used to solve uncertain optimization problems. However, RO often suffers conservativeness because of using an uncertainty set to cover all possible values for the uncertainty variable, while the computation cost of SO is prohibitive since SO considers random variables follow a predefined probability distribution based on the historical data. To overcome these weaknesses, distributionally robust optimization (DRO) model, as well as a robust stochastic optimization (RSO) method, is proposed by combining RO with SO. In this chapter, the applications of using RO, DRO, and RSO to solve the distribution network reconfiguration problems considering the uncertain connected renewable energy resources, ACOPF (alternating current optimal power flow) problems with uncertain wind power, the unit commitment problems considering the uncertainties causing by wind power connected, and the dynamic economic dispatch model considering the uncertainty of wind power are presented.

Chapter 5: Renewable energy-based distributed generators (DG) are widely integrated into the distribution network for several technical, environmental, and economical tasks. In this chapter, the optimal allocation of solar-wind based DG in the distribution system using a novel physical-based algorithm named equilibrium optimizer. The probabilities of the solar irradiance, wind speed, and load level are considered where several scenarios are generated by using a probabilistic generation of combining all possible operating conditions of PV and wind-based DG with load levels. The renewable DG units are optimally allocated for minimizing the expected power loss in IEEE 85 bus network. The simulation results confirm that the optimal inclusion of solar and wind units minimizes the power loss considerably. Moreover, the proposed optimization algorithm is effective to solve the allocation problem of the hybrid solar-wind DG in terms of the expected power loss considering the uncertainties.

Chapter 6: This chapter introduces the distributed demand-side management strategies for the modern power system with microgrids (MGs). The first distributed demand response (DR) application is direct load control. It has been an open issue to address the peak load shedding in a network of multiple types of buildings, like a campus MG, where each building is concerned about maximizing its own end user's satisfaction level. A decentralized DR model is developed to realize peak load shaving under the incentive of minimizing the affected population caused by the load interruption. It is solved by an ADMM (alternating direction method of multipliers) based DR algorithm considering complementary building consumption patterns. A simulated system based on the consumption data of the University of Connecticut co-generation plant is built to validate the effectiveness of the proposed DR control. The second distributed DR application is emergency DR, which is critical for increasing power system resilience in disaster recovery. This chapter presents a game-theoretic DR approach for MG-aided restoration service. The transmission line sensitivity analysis is applied to quantify the physical impacts of energy trading on the system stability level. Taking these quantified impacts as the stability incentive rewards earned by MGs aside from the economic incentive rewards, the interactions between the utility and MGs become highly combinatorial. A two-level game structure is built to model energy trading between the utility and MGs for the restoration service. The first level is a Nash game to formulate the noncooperative relationship among MGs. The second level is a Stackelberg game led by the utility for determining the electricity price which could maximize restoration service cost efficiency. Case studies on IEEE 6-bus and 57-bus test systems demonstrated the effectiveness of the proposed distributed DR approach on energy trading decision support for MG-aided restoration service.

Chapter 7: DG units are specifically designed to support the distribution network operation. These DG units cannot be included with transmission grid operation as the size of these units is smaller but they can be connected at any point near the load. The siting and sizing of DG systems impact the distribution system parameters such as network losses, voltage profile, and reliability. The optimal location and size of DG are identified through the analytical method. In paving way for advanced research in protection studies under uncertain conditions, the effects of DG penetration on voltage profile of the system, network losses, and reliability are studied. The reliability of the network is studied through energy not supplied of the system. The impact of DG on distribution system performance provides higher significance toward the uncertain conditions.

Chapter 8: With the integration of renewable energy resources into a distribution network, the power system is witnessing a drastic transformation. In comparison to conventional resources, renewable energy resources have a strong preference for the consumer end due to their availability and their positive influence on the environment. Despite their advantages, renewable resources bring some challenges to the distribution networks, where islanding is one of the critical issues in which the distributed generator continues to feed the network even after the utility grid is isolated from the distribution system. Planned islanding does not pose any risk to the maintenance personnel, but unplanned islanding does. Many research articles have been proposed to deal with the problem to help mitigate the issues related to islanding detection. This chapter presents a comprehensive review of different islanding detection methods available in the literature. Islanding detection methods can be broadly categorized as remote methods and local methods. The remote methods deal with the detection of islanding on the utility grid side that they rely on communication links/channels to disconnect the distributed generator, in contrast, the local methods deal with the islanding detection on the distributed generator side. The local methods are further classified as passive, active, hybrid, signal processing, and computational intelligence-based methods. In contrast to the passive methods used the electrical parameters at the DG, the active methods intentionally inject small disturbance at the DG side to sense the disturbance and the hybrid methods are the combinations of both passive and active methods. The passive methods are easy and more advantageous to implement due to their low cost and low power quality problems, but are associated with larger nondetection zone and are vulnerable toward threshold setting; whereas, the active methods have a very smaller nondetection zone but are associated with power quality problems. To overcome these problems involved in islanding detection, signal processing and computational intelligence-based methods have been used, which have the capability of dealing with complex nonlinear systems. An insight into different methods based on various criteria such as detection time, nondetection zone, and detection accuracy are tabulated and summarized to assist the field engineers and researchers in establishing the suitability of the method for the system. Lastly, the present status with a suggestion toward the future direction of research for islanding detection has also been presented.

Chapter 9: Distance relay is designed with different overlapping zones to provide complete protection to the associated lines and other lines. From the three different operating zones, the third zone is the delayed operative zone as its operation is followed by the nonresponding operation of the other two zones. Although the operation of zone-3 is not so important, the false operation of the same is a leading factor behind the power system blackout. Maloperation of the zone-3 unit of distance relays during system stressed conditions, such as voltage instability, power swing, and load encroachment, results in such a disaster. Zone-3 unit of distance relay is heavily affected by such events due to the large reach area and longtime delay. During symmetrical fault in zone-3 section and system stressed events, the impedance trajectory enters into the setting characteristics. So, relay unable to make proper discrimination between the different balanced phenomena and operate. As system stressed events are nonfault events, the relay should remain silent. These unwanted consequences lead to cascaded events and further large-area power outage. So, to maintain security, selectivity, proper detection, and discrimination of symmetrical fault from the system stressed events are essential. In this book chapter, a complete survey report behind zone-3 maloperation and power system blackout is provided. The main objectives of this book chapter are to discuss the impact of uncertainty measures, root causes behind power system blackout, and what are the available solutions to mitigate this issue. The simulation work is conducted in EMTDC/PSCAD platform to generate the voltage, current, active power, reactive power signal during system stressed, and fault conditions. The IEEE 39-bus test system is considered for the study and to verify the performances of conventional and available techniques.

Chapter 10: In this chapter, the testing of the influence of Static Var compensator (SVC) and energy storage device's location on the power system using the general algebraic modeling system (GAMS) will be presented. First of all, the program GAMS will be described. The structural organization of the program model, the speed of convergence, the comparison with other optimization techniques, and some of GAMS's solvers will be discussed. On the other side, the basic information about SVC and energy storage devices will be presented. Concretely, different types of energy storage as well as their connection with the power network, and the trend of use will be described shortly. Also, the structure of SVC devices and their impact on the network will be discussed. The main part of this chapter will be oriented on the location of the storage and SVC device's influence on power system losses and voltage deviation, respectively. For these tests, the IEEE 9-test bus system will be used. By using GAMS, for all network nodes, the optimal value of storage active power, as well as SVC's reactive power, will be found for two cases: case without, and case with the wind farm. All results will be graphically presented, while certain parts of the realized GAMS program will be inserted.

Chapter 11: The existing modern power grid is an intricate human-made system that delivers power to the consumers unidirectionally from the central power station via transmission and distribution networks. However, due to the technical issues as well as environmental hazards caused by the burning of fossil fuels, researchers are trying to develop the new strategies and technologies to deliver powers to the consumers with minimizing the technical and environmental issues. The concept of the centralized power supply is changing to the distributed power supply with the emerging concept of distributed generation/distributed energy resources (DG/DER), for example, solar, wind, biomass, fuel cell, etc. The integration of the DGs makes the distribution networks active from passive, where the power flows are bidirectional. The DGs are purposely installed in the distribution networks to gain the advantages such as power loss reduction, voltage profile improvement, less power exchange with the main grid, profit maximization, etc.; however, if the DGs are not correctly placed, then instead of gaining the advantages it may cause some adverse effects. A lot of techniques are present in the literature; however, in this chapter, two novel DG allocation and sizing techniques with the uncertainty analysis are presented from different aspects.

Chapter 12: This chapter provides a framework for evaluating the probable technical performance of electrical distribution feeders with distributed energy resources (DERs). It covers the key aspects of the underlying problems associated with assessing the technical impacts of DERs: the scope of the technical variables, data requirements and modeling, the management of uncertainty, and methodologies for simulating impacts. The analysis of practical feeders with DER penetration is demonstrated through case studies utilizing a state-of-the-art tool combining a probabilistic load flow approach and a stochastic simulator. Analyzing the analysis of the simulation results using risk factors leads to conclusions regarding the susceptibility of networks to technical issues. These outcomes are useful to network operators in determining the capacity of existing networks to host DERs, which informs the regulations, and feeder planning and design.

Chapter 13: Large-scale integration of distributed generation (DG) has been promoted as a cleaner resource of energy, and also its ability to cover up uncertainties in the demanded powers. Besides, the broad integration of power electronic devices has been demanded extensively due to their ability to interface renewable resources with the central AC grids. In this work, large-scale integration of DGs is based on smart inverters that can provide both active and reactive power simultaneously. The main objective of this work is to maximize the hosting capacity of a large distribution system without violating its thermal and voltage limits. The proposed optimization approach is based on both DGs allocation along with network reconfiguration. A recent optimization algorithm called Harris hawks optimization is employed to solve the proposed nonlinear optimization problem. Two case studies are carried out on the 415-node distribution system to maximize its HC (hosting capacity) for DGs with/without load uncertainty consideration. Results showed up the effectiveness of the proposed optimization approach to maximize both the deterministic and probabilistic HCs for the 415-node large distribution system by 98.345% and 58.68%, respectively.

Chapter 14: Widespread use of power electronics devices causes harmonics in distribution networks due to their nonlinear nature. Mitigation strategies using active or passive filters require exact information of current and voltage harmonics. Loads are time-varying in nature due to switching of load equipment, and the pattern of harmonics changes accordingly. Intermittent harmonics are also present in some cases. This study overviews the progress in data acquisition devices and describes the computation of voltages and currents in the presence of harmonics to monitor waveforms and estimate total harmonic distortion (THD) using Arduino/Raspberry Pi microcomputer interfaced voltage and current transducers. The results of the study provide an economic solution for the power quality issues including harmonics, distortions, transients, and flickers. A hardware model has been developed to sense current and voltage from a line that energizes a nonlinear load. The harmonic number and the relative magnitudes of steady-state and time-varying harmonics for appropriate mitigation strategies are presented.

Chapter 15: Building applied photovoltaic system (BAPV) and building integrated photovoltaic system (BIPV) are two separate ways of integrating solar photovoltaics into a commercial or domestic building. BIPV methodology is considered as a case study in this chapter. A grid-tied BIPV system has been designed for an automobile showroom, which is governed to act as an active filter to provide reactive and harmonic compensation for the loads at the system's coupling point with a proper filtering algorithm. The voltage fluctuation issue due to the indeterminant nature of the solar PV has been addressed by a proper battery management system and the results in terms of active power-sharing and THD comparison were analyzed using MATLAB/Simulink.

Chapter 16: Electric vehicles (EVs) are the most capable technologies replacing the internal combustion engine in the transport system over the last few decades. The battery is a source of energy that combines two or more cells where electrochemical reactions occur to provide power to the EVs. Lithium-ion (Li-ion) batteries are one of the most significant power sources owing to their advantages, including high-energy storage density, long-cyclic lifespan, and low self-discharge. However, Li-ion batteries due to chemical reactions and ohmic resistance in the process of charge/discharge generate a considerable amount of heat that may cause problems such as overheating, swelling, and even explosion. Therefore, designing an efficient battery thermal management system has become a significant challenge in EVs to control and remove the generated heat by the cells from the reliability evaluation of the Li-ion batteries' points of view. Many cooling systems are applied in EVs using different active and passive cooling technologies. Active cooling systems, including forced air cooling and liquid cooling, need an external source of energy. On the other hand, the passive cooling systems comprise natural air cooling, phase change material, and heat pipe that do not consume any external energy [8]. As many researchers mention, the safe temperature range for Li-ion batteries is between 25°C and 45°C that results in a balance for performance and lifetime. In this chapter, the EVs trend is described shortly. Aging mechanisms in Li-ion batteries from temperatures effects and control strategies are explained afterword. Active and passive battery thermal management system and their limitations are illustrated. The thermal management system and reliability assessment in cylindrical Li-ion batteries are then presented in detail.

Chapter 17: Combustion-based generators are the first choice for isolated electrical systems because of its highly proven technology, low installation and capital cost, easy in maintenance, and simply approachable to the remote area located end-users’ market where no utility supply exists. However, availability, cost, and environment-related issues are forcing users toward the use of renewable energy sources. Therefore, an isolated wind-driven electrical system becomes more reliable for continuous power from it if it is being coupled with reserve diesel generators. Such systems are called hybrid electrical systems. The load and input both have intermittent nature, and such fluctuating behaviors affect the voltage of the system. For any electromechanical conversion device, the adjustment in real and reactive power can help in controlling the frequency and voltage response, however, the frequency is more influenced with real power, and voltage is more influenced by reactive power. So, an additional device is required that can adjust reactive power for the system. This device, which is mainly used to generate or absorb reactive power for the system, is known as reactive power compensator. The system voltage can be controlled by adjusting the reactive power output from reactive power compensator. Compensating device selection depends on the cost and their characteristics toward voltage performance. The compensator performance becomes more important with input and load penetrations in the system because, to suppress the effect of these penetrations, fast-acting compensator (commonly referred to as dynamic compensator) is required. However, such a dynamic compensator is very costly. Hybrid participation of reactive power compensators can optimize the rating of compensators in terms of cost and performance. Recapitulating the facts elaborated above, this chapter focuses on the necessity and modeling of the wind-driven electrical system in which wind power is synchronized with a diesel generator for developing hybrid electrical system especially for nongrid connected isolated areas. The techniques to control the voltage for such an isolated hybrid wind-driven electrical system are explained based on two parameters: (1) system performance and (2) cost of compensation. To optimize the system between these two parameters, hybrid participation of compensator is proposed and elaborated in detail with the help of MATLAB codes and Simulink model interfacing. The study is further expanded to identify and cope-up with the presence of input and load penetrations in the system.

Chapter 18: The formulation of bidding strategies in a competitive energy market can be a profit-making way for power producers. With the evolution of smart grids and focus on the con-summer centric policies, companies are coming with competitive strategies so that they can provide better services to consumers and make a profit from market conditions. In this chapter, an anticipator of market sentiment is proposed through the evaluation of a forecasting algorithm derived from the radial basis function neural network (RBFNN) and Chaotic Variant of Grasshopper Algorithm. In the introductory part of the chapter, a critical review of the literature is presented, which dealt with different market clearing mechanisms, different strategies pertaining to make profit, and different market structures, and their evolution. After a crisp review of these, an optimization routine and supervised architecture (RBFNN-SFECGOA)-based framework for predicting the market sentiments is proposed. The anticipator receives inputs from rival bidders and models them with the help of mathematical formulation. A recently published chaotic variant of Grasshopper Optimization Algorithm, namely Sinusoidal Function Enabled Chaotic Grasshopper Optimization Algorithm (SFECGOA), is employed to serve optimization task. The results obtained from this anticipator and other approaches are evaluated and validated through the calculation of various error indices, and the decisive evaluation of the profit-making strategies is carried out. Different scenarios of uncertainty are simulated to validate the proposed approach. The results reveal that the proposed market anticipator can keep a close watch on market sentiments and provide profitable results for a power producer.

The editors

Shady H. E. Abdel Aleem, and Ahmed F. Zobaa

Chapter 1: Overview of uncertainties in modern power systems: uncertainty models and methods

Mohamed Ebeed ¹ , and Shady H. E. Abdel Aleem ²       ¹ Department of Electrical Engineering, Faculty of Engineering, Sohag University, Sohag, Egypt      ² Department of Mathematical, Physical and Engineering Science, 15th of May Higher Institute of Engineering, Helwan, Cairo, Egypt

Abstract

Currently, uncertainty in energy systems is an ongoing issue, particularly in modern renewable energy systems, where it is difficult to accurately describe current or future states of parameters in power systems because of the many uncertainty issues that could expose the systems to potential issues. This is why identifying uncertainties is not a simple task or a straightforward issue. Nowadays, there are different strategies to model these uncertainties. Each has its strengths and limitations. Hence, much attention should be paid to managing uncertainty in energy systems and mitigating their potential risks both at the planning and operational stage. Although predicting uncertainty at the planning stage can deviate from the real situation, it can always be liberated at the operational stage. In this chapter, models of the uncertainty of parameters in modern power systems such as uncertainty models of load demand, wind energy, photovoltaic energy, plug-in electric vehicles, electricity price, and load growth are summarized to bring together several aspects of uncertainty management in power systems in planning and operation stages within an integrated framework. Then, uncertainty modeling methods are presented in detail. Finally, the related future trends are explored.

Keywords

Electricity price; Information gap decision theory; Interval analysis; Linearization methods; Load demand; Load growth; Monte Carlo simulation; PDF approximation Methods; Photovoltaic energy; Plug-in electric vehicles; Possibilistic optimization; Probabilistic optimization; Robust optimization; Scenario-based method; Uncertainty; Wind energy

1. Introduction

2. Uncertainty models of parameters in power systems

2.1 Load demand uncertainty model

2.2 Wind energy uncertainty model

2.3 PV energy uncertainty model

2.4 PEVs uncertainty model

2.5 Electricity price uncertainty model

2.6 Load growth uncertainty model

3. Uncertainty modeling methods

3.1 Probabilistic methods

3.1.1 Numerical methods

3.1.2 Analytical methods

3.2 Possibilistic methods

3.2.1 α-Cut method

3.2.2 Defuzzification

3.3 Hybrid methods

3.3.1 Fuzzy scenario

3.3.2 Fuzzy MCS

3.4 Information gap decision theory

3.5 Robust optimization

3.6 Interval analysis

4. Future trends

References

1. Introduction

Nowadays, the development of power systems and the appearance of advanced and new energy concepts such as smart grids, renewable energy resources with energy storage systems, microgrids and nanogrids have introduced many operation and control opportunities but also caused many challenges in planning, investment, scheduling, and operation of modern power system networks. In this regard, uncertainty in energy systems is an ongoing issue, particularly in modern renewable energy systems, where it is difficult to accurately describe deterministic states of parameters in power networks because of the increase in the uncertainty associated with high penetration of renewables, or others, that could expose the systems to potential risks. Hence, from operation and management perspectives, the need for accurate decisions is necessary to decide between alternatives, calculate costs, expect revenue, and mitigate possible risks safely and reliably. The simple way is to have enough data available to help take the right decision in a time-effective framework; unfortunately, uncertainty means the absence of the confirmed data. Therefore, to manage the uncertainties, decision-making techniques have been investigated under uncertainties in many works to examine possible (and sometimes impossible) risks and consequences of all scenarios to accurately quantify the importance of uncertainty sources in practical terms to help understand how uncertainties will impact the performance of power systems [1,2].

Generally speaking, the uncertainty of parameters can be categorized into two main categories, namely uncertainty of technical and economic parameters. The technical parameters set describes parameters from two perspectives: topology and operation. The topology group indicates failure or outage of any element (generator, line or others) in power networks, while the operational group indicates uncertainty models of load demand alteration, load growth, renewables output (wind, PV, etc.), fluctuation, and uncertain penetration of plug-in electric vehicles (PEVs). The economic parameters set describes variations in electricity market price, gross domestic product, employment and unemployment rates, and economic growth. Recently, a third category of uncertainty has arisen in 2020, which is epidemics, pandemics, and disasters that affect all the energy sector from both techno-socio-economic perspectives, as over the past few months in 2020, the Covid-19 pandemic has caused an unprecedented global economic and social crisis that has significantly affected all aspects of life across the globe, including the energy sector [3]. Although electrical network operators were well prepared to cope with the immediate effects of the Covid-19 pandemic, it remains difficult to deal with and predict the wide range of consequences for the energy sector.

In this chapter, models of the uncertainty of parameters in modern power systems such as uncertainty models of load demand, wind energy, PV energy, PEVs, electricity price, and load growth are summarized to bring together several aspects of uncertainty management in power systems in planning and operation stages within an integrated framework. Then, uncertainty modeling methods such as probabilistic (numerical and analytical methods), possibilistic (α-cut and defuzzification), hybrid probabilistic-possibilistic (fuzzy scenario and fuzzy Monte Carlo), information gap decision theory, robust optimization, and interval analysis are presented in detail. Finally, the related future trends are explored.

2. Uncertainty models of parameters in power systems

2.1. Load demand uncertainty model

Generally, the uncertainty of load demand can be modeled using the normal or the Gaussian probability density functions (PDFs). Thus, the PDF of load demand can be expressed as follows [4,5]:

(1.1)

are the standard deviation and the mean values of the load demand, respectively.

2.2. Wind energy uncertainty model

The Weibull PDF is usually applied to define wind speed distribution [6]. It can be described as follows:

(1.2)

). α denotes the scale parameter and β denotes the shape parameter for the PDF of the Weibull function.

. Hence, the Rayleigh PDF is modeled as follows [5,7–12]:

(1.3)

Further, the output power from a wind power unit is expressed using the following models.

• Model #1

(1.4)

, respectively [11,12].

• Model #2

(1.5)

where ρ is the air density, A is the swept area of the rotor, V is the wind velocity (m/s), and C p is the coefficient of performance [13].

• Model #3 [13,14]

(1.6)

• Model #4 [15]

(1.7)

2.3. PV energy uncertainty model

The lognormal probability density is used to express the solar irradiance [16–18]. It can be described as follows:

(1.8)

denotes the standard deviation.

The probability of solar irradiance can be also represented using the Beta distribution function as follows [4,19]:

(1.9)

are parameters of the beta probability function. The parameter of the Beta PDF can be evaluated using the mean and standard deviation of the random variable as follows:

(1.10)

(1.11)

Further, the output power of PV units can be calculated as a function of solar irradiance as expressed in the following models.

• Model #1

(1.12)

denotes the rated power of the PV denotes the solar irradiance under standard environment conditions (about 1000   W/m²). X c represents a specific irradiance point [20,21].

• Model #2 [22]

(1.13)

• Model #3

(1.14)

denotes the average output power from the PV module, P(G s denotes the PDF [23].

• Model #4

(1.15)

is the area of the PV module (m²), respectively [24].

• Model #5

(1.16)

(1.17)

(1.18)

(1.19)

is the charge of the electron [25].

• Model #6

(1.20)

(1.21)

(1.22)

(1.23)

is the voltage temperature coefficient, S a is the voltage at MPP [26,27].

2.4. PEVs uncertainty model

The PEVs also have a random nature to be considered. Three random variables are associated with the PEVs as the daily arrival time (initial parking time), the initial state of charge (SOC) of the EV battery, and vehicle travel concerning the distance, which can be represented using Eqs. (are the standard deviation and mean value of the traveling distance, respectively.

(1.24)

(1.25)

(1.26)

2.5. Electricity price uncertainty model

The electricity price that is purchased from the grid is also one of the vital random parameters in power systems. The PDF of the electric price is given using are the standard deviation and mean value of the electricity price.

(1.27)

2.6. Load growth uncertainty model

represents the initial power at the base year at an iis the incremental load growth at year t. Hence, the load at year t of bus i . Its PDF can be represented by is the mean value of the load growth.

(1.28)

3. Uncertainty modeling methods

Various uncertainty modeling methods are presented in the literature, and they are categorized into six main categories:

• Probabilistic approaches: The input parameters of a problem are random with a PDF identified for them. The commonly known probabilistic approaches or uncertainty modeling methods are categorized into numerical and analytical methods, such as Monte Carlo simulation (MCS), sequential MCS, Markov Chain MCS, pseudosequential MCS, and nonsequential MCS methods for the numerical-based methods and linearization, scenario-based, and PDF approximation methods for the analytical-based methods.

• Possibilistic approach: It rests on the fuzzy sets where the input parameters are represented using a membership function. The common methods in this group are the α-cut and defuzzification methods.

• Hybrid possibilistic-probabilistic approaches: The input parameters are mixed between both the possibilistic and probabilistic approaches. The common methods in this group are the Fuzzy-scenario and fuzzy-MCS methods.

• Information gap decision theory: Information gap decision theory (IGDT) approach specifies as to what extent the uncertain parameter can change while assuring the minimum income for the decision maker, in which robustness and opportuneness are the two basic features of IGDT.

• Robust optimization: The achieved decisions are taken based on solving a problem considering the worst-case scenario of a given uncertain dataset.

• Interval analysis: The input parameters are assumed to be taken from a known interval.

The listed uncertainty modeling methods help assist the decision maker in evaluating the consequences of different aspects of the problem under attention in the presence of uncertain parameters. Fig. 1.1 shows the well-known methods that have been applied for modeling uncertainty in electric power systems.

3.1. Probabilistic methods

3.1.1. Numerical methods

3.1.1.1. Monte Carlo simulation methods

The MCS method is a numerical method that uses random numbers associated with their PDFs to solve problems [represents the power system decision variables including the sizing or locations of DGs, energy storage system, FACTS, and charging stations of EVs, etc.

Figure 1.1 Methods used in the literature for modeling uncertainty in electric power systems.

MCS is an iterative process, in which the model is iteratively evaluated over a maximum iteration number (Max.Iteration, then the output function is calculated for the produced vector.

(1.29)

After performing the simulation up to the maximum iteration number, the approximate distribution of the output can be formulated in which it includes the mean and standard deviation of the output vector.

(1.30)

(1.31)

The steps procedure of the MCS method is illustrated in Fig. 1.2. The merits and demerits of MCS are summarized as follows [38,39]:

The main merits of MCS are as follows:

• It is a simple and flexible method to apply.

• It can be easily applied for nonconvex or nondifferentiable and complex problems.

• It can be applied to systems that have many uncertain variables.

• It can deal with all PDF types.

Figure 1.2 Flowchart of MCS.

On the other side, the demerits of MCS are as follows:

• It is an approximation method.

• This method is applied only when the PDF is defined.

• This method is iterative and repetitive, which indicates that the main feature of MCS is its high computation burden. Thus, several methods have been proposed in the literature for lessening the computation burden.

3.1.1.2. Sequential MCS method

Sequential MCS (SMCS) method or particle method is a numerical set of Monte Carlo methods that can provide an attractive and convenient method for computing posterior distributions [40,41]. Besides, they have considered sampling and resampling techniques designed to be derived from a sequence of PDFs and are applied for simulating the chronological behavior of a system. The main feature of this method is its flexibility, easiness, excellent computational power, and generality that help support its application in a wide range of applications with general settings.

In [42], the reliability indices with the presence of DG have been assessed considering the random interruption duration at the load point that has been modeled using the SMCS. Arabali et al. [43] presented a stochastic framework to assess the optimal integration of hybrid systems including wind, PV, storage system, and HVAC loads while SMCS was employed to obtain the sequential samples of system states for the chronological progress. Li et al. [44] presented probabilistic wind storm models for risk analysis of the distribution systems and SMCS was applied to simulate the impacts of wind storms. In Refs. [45], Lopes et al. assessed the integration of wind and hydropower units in the transmission system, and the SMCS was employed to obtain the time series of the wind and river inflows. In Ref. [46], SMCS was applied to assess the system reliability considering the effects of faults on load in distribution systems. In Ref. [47], SMCS has been utilized for evaluating the reliability of a smart grid considering real-time thermal rating electrical conductors under different weather conditions. In Ref. [48], SMCS was utilized for the reliability assessment of an electrical transmission system combining with a probabilistic prediction interval with the integration of wind power system. In Ref. [49], a well-designed method for the maximum power point tracking was proposed for a PV system considering the partial shading. SMCS has been applied to undertake the nonlinearity of time-varying at the adjustment of the step size voltage among the incremental conductance approach.

3.1.1.3. Pseudosequential MCS

The pseudosequential MCS (PSMCS) has been presented by Leite da Silva in 1994 [50]. This method is based on nonsequential sampling and chronological simulation of the subsequences associated with failed states only. The step procedure of this method is depicted in Fig. 1.3. In Ref. [51], PSMCS has been applied to evaluate customers' nodal reliability under PV power fluctuations. Celli et al. [52] presented an approach for assessing the smart distribution network reliability based on the adoption of the PSMCS. In addition, the authors in Ref. [53] presented an approach based on PSMCS-chronological simulation to evaluate the loss of load indices, with particular prominence with time-varying loads.

Figure 1.3 Flowchart of the PSMCS method.

3.1.1.4. Markov chain MCS

Markov chain MCS (MCMCS) is a dynamic MCS method and is usually utilized to handle the uncertainty of parameters of the system. In MCMCS, a Markov chain with MCS is used to create the samples depending on the probability distribution, in which the probability of producing a novel state in the chain is based on the current state only [54,55].

. The step procedure of MCMCS based on the Metropolis–Hastings approach is depicted in Fig. 1.4 [54,56,57].

In [58], MCMCS was applied to generate a synthetic time series of wind power output. In Refs. [59], MCMCS was applied for forecasting the generated power of the PV system using statistics of the historical data. In Ref. [60], the reliability indices were captured by MCMCS to obtain the wind power time series for the evaluation of the generating capacity adequacy.

3.1.1.5. Nonsequential MCS

In the nonsequential MCS, the states are obtained randomly sampling regardless of the chronology of the events, unlike the sequential MCS that is based on the time-dependent states. The steps of this method are illustrated in Fig. 1.5.

In [61], the nonsequential MCS method was applied to minimize the harmful emissions of thermal generation units with wind power generation unit's behavior of hourly loads and wind generation. Amaral et al. [62], presented an efficient method based on the nonsequential MCS for the evaluation of a composite system. In addition, the authors in Ref. [63] have applied the nonsequential MCS method for evaluating well-being indices for composite generation and transmission systems. In Ref. [64], the reliability evaluation of power systems has assigned based on the sequential nonsequential MCS method and particle swarm optimization.

3.1.2. Analytical methods

The analytical methods are based on performing arithmetic with PDF of the uncertain input parameters. These methods are categorized as follows:

3.1.2.1. Scenario-based method

The scenario-based method is an efficient and simple method to model probabilistic uncertainties where the continuous space of uncertain function