c Copyright by Hung Khanh Nguyen 2017 All Rights Reserved

Transcription

1 c Copyright by Hung Khanh Nguyen 2017 All Rights Reserved

2 BIG DATA OPTIMIZATION FOR DISTRIBUTED RESOURCE MANAGEMENT IN SMART GRID A Dissertation Presented to the Faculty of the Electrical and Computer Engineering University of Houston in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Electrical and Computer Engineering by Hung Khanh Nguyen May 2017

3 BIG DATA OPTIMIZATION FOR DISTRIBUTED RESOURCE MANAGEMENT IN SMART GRID Hung Khanh Nguyen Approved: Chair of the Committee Dr. Zhu Han, Professor, Electrical and Computer Engineering Committee Members: Dr. Kaushik Rajashekara, Professor, Electrical and Computer Engineering Dr. Miao Pan, Assistant Professor, Electrical and Computer Engineering Dr. Amin Khodaei, Associate Professor, Electrical and Computer Engineering, University of Denver Dr. Hamed Mohsenian-Rad, Associate Professor, Electrical Engineering, University of California at Riverside Dr. Suresh K. Khator, Associate Dean, Cullen College of Engineering Dr. Badrinath Roysam, Professor and Chair, Electrical and Computer Engineering

4 Acknowledgements First, I would like to express my great appreciation to my advisor, Professor Zhu Han, for his esteemed guidance, constant encouragement, and continuous support during my graduate studies at University of Houston. His deep academic background, keen insights and cheerful personality have helped me achieve significant improvement in my Ph.D. research and to be well prepared for future professional development, which will be invaluable assets for my future career. It has been extremely lucky for me to work and learn many things from him. Furthermore, I would like to express my sincere gratitude to Professor Amin Khodaei and Professor Hamed Mohsenian-Rad, who provided me many precious technical discussions, comments, and advices for my research. I would also like to thank the rest of my dissertation committee, Professor Kaushik Rajashekara and Professor Miao Pan for their valuable time and support on this dissertation. My appreciation also goes to my dear colleagues at Wireless Networking, Signal Processing and Security Lab, Huaqing Zhang, Yunan Gu, Yanru Zhang, Yong Xiao, Fahira Sangare, Radwa Aly, Xunsheng Du, Kevin Tsai, Qiuyang Sheng, Ali Arab, Erte Pan, Lanchao Liu, Yingyu Li, Yun Hu, Jingyi Wang, Xinyue Zhang, Debing Wei, Mounika Sai, Hui Chen, Ye Yu, Jingjing Zhao, Liangliang Zhang, Dr. Haitao Xu, Dr. Li Xin, Dr. YiFei Wei, Dr. Shaohua Qin, and many others for creating such a wonderful working environment. I also would like to thank Dr. Duy Nguyen at San Diego State University and Dr. Hien Van Nguyen at University of Houston for spending time with me to share experience about many different perspectives in daily life. I am grateful to all my friends who gave me strong support during my Ph.D. life in Houston. Last but by no means least, I would like to thank my lovely mom, dad, brothers, sisters, aunts and uncles for their permanent love and encouragement throughout my life. Their moral support is the motivation to assist me in overcoming the difficulty of studying as well as living far away from my home country. v

5 BIG DATA OPTIMIZATION FOR DISTRIBUTED RESOURCE MANAGEMENT IN SMART GRID An Abstract of a Dissertation Presented to the Faculty of the Electrical and Computer Engineering University of Houston In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Electrical and Computer Engineering by Hung Khanh Nguyen May 2017

6 Abstract Electric power grids are experiencing the increasing adoption of distributed energy resources, which can bring huge economical and environmental benefit. However, the large-scale penetration of distributed energy resources will make both operations and long-term planning to be more and more complex due to the higher degree of output variability than traditional centralized sources. This variability creates irresistible challenges for grid operators to ensure system security and reliability. In addition, traditional optimization algorithms are no longer applicable for such integrated and complex systems in which economic efficiency, grid reliability, and privacy need to be simultaneously satisfied. Therefore, an innovative optimization framework is critical to tackle the emerging challenges due to the large-scale and independent decision-making nature of distributed resource management problem in the future power system. In this dissertation, we focus on the application of big data optimization methods for distributed resource management problem in smart grid to improve the reliability and security of the distribution system. First, we propose an incentive mechanism design to motivate microgrids to participate in the peak ramp minimization problem for the system to mitigate the ramping effect due to the high penetration of distributed renewable generations. Distributed algorithms to achieve the optimal operation point are proposed, which allow microgrids to execute their computation in either synchronous fashion or asynchronous fashion. Second, a large-scale optimization problem for microgrid optimal scheduling and the load curtailment problem is formulated. We propose a decomposition algorithm and implement parallel computation for the proposed algorithm to run on a computer cluster using the Hadoop MapReduce software framework. Third, a decentralized reactive power compensation model is studied to reduce the power losses and improve the voltage profile for distribution networks. Finally, we consider big data optimization methods for resource allocation problem in wireless network virtualization to prevent traffic disruption against physical network failures. vii

7 Table of Contents Acknowledgements v Abstract vii Table of Contents viii List of Figures xii List of Tables xv 1 Introduction Smart Grid Big Data Optimization for Distributed Resource Management Dissertation Contributions and Organization Incentive Mechanism Design for Integrated Microgrids in Peak Ramp Minimization System Model Microgrid s Payoff DSO s Payoff Social Welfare Maximization Nash Bargaining Solution Distributed Algorithms for NBS An Introduction to ADMM Synchronous Distributed Algorithm Asynchronous Distributed Algorithm viii

8 2.4 Simulation Results Conclusions A Big Data Scale Algorithm for Optimal Scheduling of Integrated Microgrids System Model System Description Islanded Operation Optimal Scheduling Problem of Integrated Microgrids Decomposition Algorithm based on ADMM Simulation Results Hadoop MapReduce Implementation MapReduce Programming Model ADMM Implementation using Hadoop MapReduce Performance Results Conclusions Decentralized Reactive Power Compensation System Model System Description User s Payoff Modeling Electric Utility Company s Payoff Modeling Network Social Welfare Maximization Sequential Bargaining One-To-One Nash Bargaining Solution ix

9 4.2.2 Generalized Sequential Bargaining for Multiple Users Concurrent Bargaining Simulation Results Conclusions Distributed Resource Allocation with Minimum Traffic Disruption for Wireless Network Virtualization Network Model and Assumptions Wireless Network Virtualization Routing Model for Virtual Network Joint Resource and routing Optimization for Virtual Wireless Networks Preventive Traffic Disruption with Link failures Decomposition Algorithms using ADMM Parallel Algorithm using ADMM Distributed Algorithm using ADMM Simulation Results Convergence and Computational Performance System Performance Conclusions Conclusions and Future Works Conclusions Future Works Transactive Energy x

10 6.2.2 Local Energy Trading for Prosumers References 109 xi

11 List of Figures 1.1 A conceptual model of smart grid framework with electricity and information flow A conceptual structure of a microgrid including loads and DER units Overview of big data application in energy services Ramping effect of the renewable generation on the system net load in California The model of power system with microgrids and a DSO The information exchange between the DSO and microgrids Illustration of synchronous parallel computing for Algorithm 2.1. New iteration starts after the slowest microgrid finishes its computation Illustration of asynchronous parallel computing for Algorithm 2.2. New iteration starts when any microgrid finishes its computation The convergence performance of the proposed algorithms The net load of the system with and without incentive mechanism for microgrids Extra cost incurred and reimbursement for microgrids The model of microgrids with islanding operation The illustration of information exchange between the normal operation subproblem and islanded operation subproblems The convergence performance of the proposed algorithm The effect of τ on the percentage of load curtailment when microgrids switch into the islanded operation mode The effect of τ on the generation cost of the system in the normal operation xii

12 3.6 The fraction of load shedding of each individual microgird in the islanded operation mode for the 9-microgrid system The flow chart of MapReduce programming model Data sharing for iterative ADMM using Hadoop MapReduce and the detailed illustration of Map tasks and Reduce task in each MapReduce job in each iteration A distribution network with local reactive power compensation The active power demand and generation profiles The reactive power demand and generation profiles The reimbursement of users in sequential bargaining The reimbursement of users in concurrent bargaining The effect of DG unit penetration level on voltage deviation The effect of DG unit penetration level on power factor The model of multiple virtual networks operate on top of a single substrate network The model of substrate link failure The illustration of information exchange between the normal state sub-problem and link failure state sub-problems The structure of auxiliary variables define for each substrate link and corresponding virtual links operate on top of the substrate link The convergence performance of the proposed algorithms Computation time of two proposed algorithms versus percentage of link failure Average computation time of the normal state sub-problem and link failure state sub-problem in Algorithm xiii

13 5.8 Average computation time of each substrate link and each service provider in Algorithm The comparison of percentage of traffic reduction at individual service provider when substrate links fail with and without incorporating preventive traffic disruption model The effect of τ on the operation cost of the substrate network and the percentage of traffic disruption of SPs The effect of percentage of link failure incorporated into the model on the total bandwidth allocated to virtual networks and the operation cost of the substrate network Transactive agents and interactions xiv

14 List of Tables 2.1 Hourly Market Price Microgrid daily demand Generation of renewable resources Demand of microgrids Total generation cost and load curtailment comparison Running time on the cluster Performance comparison with centralized control Operation cost and bandwidth utilization comparison xv

15 Chapter 1 Introduction 1.1 Smart Grid The energy industry across the world is trying to address numerous challenges, including generation diversification, optimal deployment of expensive assets, demand response, energy conservation, and reduction of the overall carbon footprint. It is evident that such critical issues cannot be addressed within the conventional operation nature of the existing electricity grid [1]. The traditional electrical power grid is unidirectional in nature, where the electricity flows from central power generation facilities to end users. However, it has been subjected to government deregulation and has suffered from several technical, economic, and environmental issues. The next-generation electric power system, known as the Smart Grid, is expected to revolutionize electricity generation, transmission, and distribution by allowing two-way flows of electricity and information [2]. In general, a smart grid is the combination of a traditional distribution network and a two-way communication network for sensing, monitoring, and dispersion of information on energy consumptions. By employing two-way communications, the smart grid will not only allow dynamic monitoring of the use of electricity but also open up possibilities of autonomous scheduling of electricity consumption, provides utility companies with full visibility and pervasive control over their assets and services as illustrated in Fig. 1.1 [3]. The electric power system is undergoing a profound change driven by a number of requirements such as environmental compliance, grid reliability and operational efficiencies while dealing with an aging infrastructure. These changes are important for the electricity distribution grid, where manual operations will need to be transformed into an automatic and digitalized intelligent grid [4]. This transformation will be necessary to meet environmental targets, to accommodate a greater application for demand response, and to support plug-in hybrid electric vehicles (PHEVs) as well as distributed generation and storage capabilities. Moreover, it also presents the power industry with 1

16 Figure 1.1: A conceptual model of smart grid framework with electricity and information flow. the biggest challenge it has ever faced. On one hand, the transition to a smart grid has to be evolutionary to keep the lights on; on the other hand, the issues surrounding the smart grid must be significant enough to drive major changes in power systems operating methodologies. Distributed generation has become extremely important in recent years due to the growing global interest in reliable and sustainable electric power supply, to incorporate more renewable and alternative energy sources and to reduce the stress and loss in existing transmission system [5]. Motivated by the rapid falling cost of renewable technology and governmental financial incentives, the total worldwide capacity for distributed generation from renewable resources is expected to grow even more in the next several years [6]. By using local renewable energy resources which are close to loads, distributed generation has been considered as a promising solution to improve energy efficiency as well as to enhance security by diversifying energy resources [7 9]. Although providing considerable environmental and sustainability benefits, the variable and uncertain nature of the renewable generation would significantly challenge the supply-demand balance and can potentially jeopardize grid stability and reliability. This challenge cannot be efficiently addressed 2

17 Figure 1.2: A conceptual structure of a microgrid including loads and DER units. unless adequate flexibility is available in the grid. Tacking the above challenges requires an innovative optimization approach to handle large-scale nature of the problem size as well as independent decision-making property. Recent technological advancement on distributed energy resources management helped creating a new grid paradigm, the smart microgrid distribution network. A microgrid encompasses a portion of an electric power distribution system that is located downstream of the distribution substation, and it includes a variety of distributed energy resource (DER) units and different types of end users of electricity and/or heat, as illustrated in Fig DER units include both distributed generation and distributed storage units with different capacities and characteristics [10]. The electrical connection point of the microgrid to the utility system, at the low-voltage bus of the substation transformer, constitutes the microgrid point of common coupling (PCC). The microgrid serves a variety of customers, e.g., residential buildings, commercial entities, and industrial parks. A microgrid can dynamically respond to the changes in energy supply by self-adjusting the demand and generation to ensure an uninterrupted power supply in the most efficient and economic configuration. Moreover, 3

18 by deploying power generation close to end users, microgrids are becoming promising solution to improve reliability and power quality [11 13]. In addition to the aforementioned characteristics, one of the salient features of microgrids is islanded operation, which is defined as the capability to disconnect from the main distribution network and locally supply their loads [14]. By rapidly disconnecting from the main grid, microgrids can protect their components from upstream disturbances or voltage fluctuations. More importantly, the islanded operation mode allows microgrids to ensure energy supply for critical loads by increasing the generation output of local generators when the main distribution network is faulty. This capability has been advocated as a viable solution to achieve high system resiliency during major outages [15]. For microgrids to work properly, an upstream switch must be disconnected (typically during an unacceptable power quality condition), and the DER must be able to carry the critical load on the islanded state. This includes maintaining suitable voltage and frequency levels for all islanded loads. Power flow analysis of islanded scenarios should be performed to insure that proper voltage regulation is maintained. The DER must be able to supply the real and reactive power requirements during islanded operation and to sense if a fault current has occurred downstream of the switch location [15]. Microgrids largest impact will be in providing higher reliability electric service and better power quality to the end customers. Microgrids can also provide additional benefits to the local utility by providing dispatchable power for use during peak power conditions and alleviating or postponing distribution system upgrades. In summary, the aforementioned requirements and capabilities of smart grid call for significant enhancements to the tools and management approaches of the power delivery and distribution grid operations. Traditionally, power system operations have been governed by economic dispatch of wholesale generation to meet the forecasted demand while ensuring the reliability of the transmission grid. It has been assumed that the load can be statistically forecasted based on the past consumption patterns and the distribution grid have adequate capacity to deliver the power to consumers. However, this operation methodology may not be adequate to meet the emerging smart grid operational challenges. The emerging environmental requirements and the need for greater pene- 4

19 tration of renewable resource, is creating challenges for generation dispatch due the variable nature of the primary fuel for renewable generation, e.g., wind and solar power. To maintain reliability of the power grid, the increased levels of variable generation calls for greater levels of regulation, ramping, load following capabilities that may not be available through the existing generation fleet. Furthermore, with the expansion of behind-the-meter smart appliances, roof-top solar generation, PHEVs, storage and demand response technologies, the load will no longer follow the traditional statistical patterns, and new tools and capabilities are needed to forecast and manage demand. 1.2 Big Data Optimization for Distributed Resource Management The recent blackout experiences have demonstrated the vulnerability of the interconnected electric power system to grid failure caused by natural disasters and unexpected phenomena. Customer service expansion, additional stress due to liberalized electricity markets, and a high degree of dependency of today s society on sophisticated technological services also intensify the burden on traditional electric systems, which derive for a more reliable and resilient power delivery infrastructure. A restructured electric distribution network that employs a large number of small DER units can improve the level of system reliability and provide service differentiations [16]. The conventional planning methods are designed based on electricity delivery from centralized power generation stations to customers through passive transmission and distribution networks. Under this operation structure, all customers supplied by the same distribution substation will obtain the same level of power quality [16]. Although the current distribution system allows small-scale integration of DER, the overall penetration level is constrained to a certain level to prevent adverse impact of bidirectional power flow on system operation coordination. Hence, DER has limited capability in providing any type of grid support including voltage regulation, reactive power control, and power frequency stabilization. The microgrid approach promotes many advantages such as highly efficient energy delivery and supply system based on co-locating DER and loads, secure and reliable power supply config- 5

20 uration with service differentiations based on customer technology preference and power quality desires, and an energy delivery structure that has sufficient power generation and balancing sources to operate independent from the main grid in an autonomous manner during power outages or an energy crisis. Restructuring the electricity network based on microgrid architectures can facilitate large-scale DER interconnection into medium/low voltage distribution systems and provides a mechanism to fully utilize benefits of DER [17]. The microgrid design methodology also offers a systematic approach for planning, large-scale deployment, and autonomous control of DER rather than dealing with individual generation sources with diverse technologies. Furthermore, microgrids utilize three sets of resources for power balancing and energy management including dispatchable DER controls, demand response management, and control of power exchange with the main grid. It is now apparent that DERs can significantly affect the design, operation, organization, and regulation of the distribution network. It is also widely admitted that distribution regulation has to be modified substantially to prevent these DERs from driving serious economic distortions. The extent to which DERs will affect all the activities of the power sector from the future generation mix to how voltage is controlled at the distribution level to the expansion of the capacity of the transmission network is a matter of debate today. Therefore, the impacts of DERs on the design, operation, and regulation of the transmission network need to be systematically investigated to provide better understanding of power systems, and electricity networks in particular. The future impacts of DERs on transmission networks may be large, difficult to predict, and diverse in nature. Therefore, the critical question is not how to estimate the scale of the impact of DERs but how to ensure that the combined presence of distributed and centralized generation is managed efficiently and reliably, while creating a level playing field that enables all capable agents to have efficient, active roles in the power sector. There are several key topics to study including: the need for a comprehensive and internally consistent system of economic signals with an adequate level of granularity in space and time, the influence of networks in determining the competitiveness among centralized and decentralized resources, the need and the requirements of coordination between the distribution system operators and transmission system operators, and the 6

21 impact of DERs on the capacity expansion and the operation of the transmission network. The changing operation nature of the current power network results in an unstructured multitude of agents interacting with one another with a diversity of roles, producing and consuming different electricity services, as well as ubiquitous connectivity in the electric grid, either at transmission or distribution level. The only way of achieving an efficient outcome is to make use of a comprehensive system of economic signals, spanning the entire grid, accounting for the effects of network losses and technical constraints, and adequately valuing each service at the location and time where it is delivered. Concerning another driving factor of smart grid implementation aims at encouraging endusers to actively take part in grid operation by promoting incentive methods such as real-time consumption information or real-time electricity pricing transported over communication networks, to make them behave as real actors of the entire electricity production and distribution system. The investments in different types of DERs, the volume of deployment, the form of their operation, and their impact on the electric grid will critically depend on the prices and charges that apply to them. The economic signals will allow the large diversity of agents to compete and collaborate in the efficient provision of the various services, while maintaining the reliability of the power system. The critical issue to be discussed here is how precise the economic signals that arrive at the countless DERs connected in the distribution network should be to guarantee their efficient participation. The focus is now moving toward optimizing the system, including distributed and centralized resources, to operate more efficiently and at a higher reliability and resiliency in an existing and aging grid infrastructure. These technological and economic developments are currently developing an integrated grid vision that integrates these resources into a distributed way that uses distributed intelligence in the integration process. The vision includes an integrated power system that is highly flexible and resilient against events that optimizes the different energy resources and assets. Grid operations are moving in such an integrated grid from hundreds of control nodes to millions or billions of them. 7

22 The need for distributed intelligent automation and control becomes evident in such an integrated and complex grid. System operators are not able to manage these high numbers of nodes, which further require the need for automatic distributed intelligence. Also clear from such an integrated grid vision is that new players are already entering the power industry that provide deregulated services to the traditional utility customers, sometimes eroding revenue from the traditional utility and in competition with the power utilities. These services are associated with DER services including distributed generation on customers premises, demand response, and energy-efficiency-related services. More services are currently being delivered and planned by third parties that go back to the core operations of power utilities like ancillary services including load smoothing, frequency regulation, voltage regulation, real-time power balancing, blackstart, and spinning reserves. As the aforementioned discussions, the energy industry is in the middle of a transformation driven by the adoption of distributed generation, and more localized energy through DERs. When utilized effectively, DERs can create benefits for both utilities and customers. However, the complexity of handling different types of DERs with varying energy capacity, in diverse, networkconstrained areas presents challenges for many utilities in planning, integration, operation and maintenance. In particular, the forthcoming presence of distributed generation in electrical distribution systems has strongly modified the nature of current power grids, which is predicted to reach an active role by the implementation of the typical functions of load management, demand side management, demand response and generation curtailment. However, the distributed generation penetration also creates many technical problems in electrical systems that must be faced and solved rapidly to fully utilize the potential benefits of distributed energy resources. The inevitable coupling between reliability and economic efficiency poses unique challenges as well as opportunities for smart grid. On one hand, the smart electrical distribution grids represent the needed evolution of the actual networks by means of a deeper implementation of automation functions, with high a level of information and communication technology applications in order to increase the power quality, ensuring adequate grid capacity as well as guaranteeing the security in electric energy supplying. On the other hand, concerning the third driving factor of smart grid 8

23 Figure 1.3: Overview of big data application in energy services. implementation aims at encouraging end-users to actively take part in grid operation by promoting incentive methods to make them behave as real actors of the entire electricity production and distribution system, as illustrated in Fig Smart grid deployment also aims at changing end-user behavior by providing reliable services during high-demand periods. The anticipated smart grid data deluge, generated by users, generation and distribution system, provides huge opportunities to design robust big data optimization algorithms in implementing a scalable management model with distributed decision-making processes to facilitate gird operation optimization in terms of economic efficiency, reliability, sustainability and privacy. Therefore, new computational and mathematical models must be explored to effectively operate the emerging integrated and complex power networks. 1.3 Dissertation Contributions and Organization In this dissertation, we mainly focus on the big data optimization for distributed resource management problem in smart grid to improve the reliability and efficiency of the distribution system. By tacking large-scale and distributed nature of the resource management problem in smart grid, the 9

24 main contributions of this dissertation can be summarized as follows: Efficient and scalable models to manage distributed resource with security and resilience consideration: First, we propose an incentive mechanism design to mitigate the ramping effect due to the penetration of DERs as well as to reduce the power losses and voltage fluctuation for decentralized reactive power compensation. Second, the optimal scheduling model to minimize the operation cost of microgrids with security constraint consideration has been also proposed. New computational frameworks to handle the emerging integrated and complex management systems: We propose scalable algorithms which can be performed in parallel and distributed computing fashion based on the large-scale decomposition technique. The algorithms ensures the optimal operation point for the overall system as well as preserve the privacy for individual users. Implementation of the parallel and distributed computing framework by using big data optimization technique: We facilitate practical application of our distributed resource management models by providing detailed implementation of the algorithms using big data optimization technique. Particularly, in Chapter 2, we propose an incentive mechanism design to motivate microgrids to participate in the peak ramp minimization problem for the system. By offering reimbursement for each microgrid to deviate from the original optimal operation point, the ramping capability requirement to match supply-demand can be significantly reduced. We model and analyze the economic interaction between the distribution system operator (DSO) and microgrids using the Nash bargaining theory. The Nash bargaining solution (NBS) can be obtained by solving the centralized social welfare maximization problem. However, due to the distributed topology of the power network as well as independent decision-making nature of microgrids, the centralized design is not suitable for practical implementation. Therefore, we propose two distributed algorithms to achieve the NBS 10

25 using the alternating direction method of multipliers (ADMM) decomposition technique, which can execute in either synchronous fashion or asynchronous fashion. In Chapter 3, we propose a new model for the microgrids optimal scheduling and load curtailment problem. The proposed problem determines the optimal schedule for local generators of microgrids to minimize the generation cost of the associated distribution system in the normal operation. Moreover, when microgrids have to switch into the islanded operation mode due to reliability considerations, the optimal generation solution still guarantees for the minimal amount of load curtailment. Due to the large number of constraints in both normal and islanded operations, the formulated problem becomes a large-scale optimization problem and is very challenging to solve using the centralized computational method. Therefore, we propose a decomposition algorithm using the ADMM that provides a parallel computational framework. The simulation results demonstrate the efficiency of our proposed model in reducing generation cost as well as guaranteeing the reliable operation of microgrids in the islanded mode. We finally describe the detailed implementation of parallel computation for our proposed algorithm to run on a computer cluster using the Hadoop MapReduce software framework. In Chapter 4, we consider a distributed reactive power compensation problem in a distribution network, in which users locally generate reactive power using distributed generation units to contribute to the local voltage control. We model and analyze the interaction between one electric utility company and multiple users by using the Nash bargaining theory. On one hand, users determine the amount of active and reactive power generation for their distributed generation units. On the other hand, the electric utility company offers reimbursement for each user based on the amount of reactive power dispatched by that user. We first quantify the benefit for the electric utility company and users in the reactive power compensation problem. Then we derive the optimal solution for the active and reactive power generation as well as reimbursement for each user under two different bargaining protocols, namely sequential bargaining and concurrent bargaining. Numerical results show that both electric utility company and users benefit from the proposed decentralized reactive power compensation mechanism, and the overall system efficiency is improved. 11

26 Chapter 5 proposes a new formulation for bandwidth allocation and routing problem for multiple virtual wireless networks that operate on top of a single substrate network to minimize the operation cost of the substrate network. We also propose a preventive traffic disruption model for virtual wireless networks to minimize the amount of traffic that service providers have to reduce when substrate links fail by incorporating l 1 -norm into the objective function. Due to the large number of constraints in both normal state and link failure states, the formulated problem becomes a large-scale optimization problem and is very challenging to solve using the centralized computational method. Therefore, we propose the decomposition algorithms using ADMM that can be implemented in a parallel and distributed fashion. The simulation results demonstrate the computational efficiency of our proposed algorithms as well as the advantage of the formulated model in ensuring the minimal amount of traffic disruption against substrate link failure. Finally, conclusions and some possible future works about local energy trading under Transactive Energy paradigm are mentioned in Chapter 6. 12

27 Chapter 2 Incentive Mechanism Design for Integrated Microgrids in Peak Ramp Minimization Electric power grids are experiencing an unprecedented and growing proliferation of customeradopted variable renewable generation, motivated primarily by the rapidly falling cost of the renewable technologies combined with a wide range of state and governmental incentives. Although the integration of renewable generation offers environmental and economical benefits, it also poses significant challenges for grid operations. A fundamental challenge is to handle the intermittent nature of renewable generation. In order to balance demand and supply, the grid operator must be able to either drive down the generation output when renewable generation units start producing power, or ramp up generation when renewable generation drops. For example, Fig. 2.1 shows the current and future estimated ramping effect of renewable penetration on the system net load in California [18], which is calculated by taking the forecasted load and subtracting the forecasted electricity production from variable generation resources, mainly from solar generation. Due to the large amount of solar power available during midday when distributed generation is at the highest capacity, the net load of the system is pulled down to extremely low levels. Then, later in the day when solar generation is declining, the net load of the system ramps up dramatically. Studies conducted by the California ISO show that the grid operators must be able to ramp up system generation MW in three hours to satisfy customers demand [19]. Recently, microgrids have also been advocated as a viable solution to mitigate ramping effects in distribution networks by rescheduling their energy generation and storage resources to compensate for the system net load. However, due to the variability and uncertainty of the large-scale renewable integration, the grid operators need to deploy advanced scheduling algorithms that can coordinate dispersed and individual resources of microgrids to guarantee the steady-state stability for the power system during the period of sudden increase in demand or loss of generation. Moreover, since deviating from the original optimal operation point may incur extra costs for microgrids, 13

28 Megawatts (actual) 2013 (actual) Hour 2020 ramp need ~ MW in three hours Figure 2.1: Ramping effect of the renewable generation on the system net load in California. they will not be willing to participate in reducing ramping effects unless they receive proper financial incentives from DSOs. In this chapter, we propose an incentive mechanism that the DSO needs to deploy for motivating microgrids to participate in the peak ramp minimization problem. Specifically, microgrids reschedule their energy generation and storage resources to reduce grid ramping capability requirement. By deviating from the original optimal operation point, the DSO offers reimbursement to microgrids as economic incentives. The main contribution of this work lies in the fact that we model and analyze the coordinated peak ramp minimization problem between the DSO and microgrids using the Nash bargaining theory [20]. We quantify the benefits for both microgrids and the DSO as well as investigate the connection between the Nash bargaining solution (NBS) and the social welfare of the network. The NBS can be obtained by solving the centralized problem with complete information from all microgrids. However, due to the privacy concerns as well as the distributed topology of the power network, the centralized method is not suitable for practical implementation. Therefore, we propose two distributed algorithms to achieve the NBS using the ADMM decomposition technique. The remainder of this chapter is organized as follows. A model of the peak ramp minimiza- 14

29 power link communication link DSO Residential load. Microgrid 1 Microgrid 2 Microgrid N Figure 2.2: The model of power system with microgrids and a DSO. tion problem is formulated in Section 2.1. We investigate the incentive mechanism using the Nash bargaining theory in Section 2.2. Section 2.3 provides distributed algorithms to achieve the NBS. Simulation results are presented in Section 2.4, and Section 2.5 concludes the chapter. 2.1 System Model We consider a distribution network consisting of N microgrids, denoted by the set N {1, 2,..., N}, and a DSO as illustrated in Fig Each microgrid can acquire power from the main grid and/or generate using distributed energy resources (DERs) to supply its local load demand. The energy scheduling problem is considered in a one-day period which is divided into a set of T equal time slots, denoted by the set T {1, 2,..., T }. Let D n = {D n,1, D n,2,..., D n,t } be the predetermined demand vector over T time slots of microgrid n. To supply load demand at each time slot t T, microgrid n generates g n,t amount of energy, which is bounded as 0 g n,t g max n, (2.1) where g max n is the energy generation capacity of microgrid n. Moreover, microgrid n can decide to charge e + n,t, or discharge e n,t amount of energy for its storage system, which must satisfy the 15

30 maximum charging and discharging rate e max n in each time slot 0 e + n,t emax n and (2.2) 0 e n,t emax n. (2.3) Based on the amount of energy charging and discharging, we can calculate the energy level in the storage system of microgrid n at each time slot as s n,t+1 = s n,t + β + n e + n,t β n e n,t, (2.4) where 0 < β + n 1 and 1 β n are the charging efficiency and discharging efficiency, respectively. The energy level in the storage system must be greater than a certain threshold, and cannot exceed the maximum storage capacity, which can be expressed as the following constraint where B min n B min n is the desired minimum storage level, and B cap n s n,t B cap n, (2.5) is the storage capacity. Depending on the amount of energy generation and charging/discharging for the storage system, microgrid n needs to purchase from the main grid d n,t amount of energy to satisfy its load at each time slot, which can be calculated as d n,t = D n,t g n,t + e + n,t e n,t W n,t, (2.6) where W n,t is the available amount of energy that microgrid n obtains from the renewable resource. We assume that microgrids central controller can predict available renewable generation for the scheduling horizon using historical data or machine learning methods [21]. Let f n (g n,t ) be the convex cost function of microgrid n for generating g n,t at time slot t, and p = {p 1, p 2,..., p T } is the day-ahead price for purchasing energy from the main grid. Microgrid n can calculate its daily energy cost as T T C n (d n, g n ) = p t d n,t + f n (g n,t ). (2.7) t=1 t=1 Note that, the daily energy cost of each microgrid will depend on variables (d n, g n, e n ). However, since e n is just the internal variable to determine d n and g n, we do not explicitly express e n in the daily energy cost C n (d n, g n ) in (2.7) for the sake of notational convenience. 16

31 To minimize the total cost, each microgrid individually schedules its generation and storage resources by solving the following optimization problem min T T p t d n,t + f n (g n,t ) (2.8) t=1 t=1 s.t. d n,t = D n,t g n,t + e + n,t e n,t W n,t, t, (2.9) s n,t+1 = s n,t + β n + e + n,t β n e n,t, t, (2.10) B min n variables: {d n, g n, e n }, s n,t B max n, t, (2.11) 0 e + n,t emax n, t, (2.12) 0 e n,t emax n, t, (2.13) g n,t g n,t 1 R max n, t, and (2.14) 0 g n,t g max n, t, (2.15) where (2.14) is the ramping constraint for local generator of microgrid n, which cannot exceed ramping limit Rn max. Let {d o n, g o n} be the optimal generation and storage profiles obtained by solving the the optimization problem in (2.8)-(2.15). Then the daily energy cost C o n when microgrid n does not participate in the peak ramp minimization problem can be calculated as C o n = T p t d o n,t + t=1 T f n (gn,t). o (2.16) Note that, we use superscript o to signify the original optimal point, i.e., when microgrids do not participate in the peak ramp minimization problem with the DSO. Based on the demand request from all microgrids as well as the fixed demand from residential users, the total net load of the system at t=1 each time slot can be determined as l t = P r,t + d n,t, (2.17) n=1 where P r,t is the total amount of energy demand of residential users at time slot t, which is assumed to be known in advance. This information can be obtained from forecasting or historical data due to 17

32 repeated daily energy usage patterns. Then the ramp between two consecutive time slots, t and t 1 is determined as r t = l t l t 1 = P r,t P r,t 1 + (d n,t d n,t 1 ), (2.18) where for the case t = 1, the ramp r 1 can be calculated as the difference between the net load at time slot t = 1 and the net load at the last time slot of previous day, which is assumed to be known at the beginning of the optimization process. We further define the ramp vector of the system over the scheduling horizon as r = {r 1, r 2,..., r T }. (2.19) n=1 Then, we can determine the maximum ramp that the system has to suffer over the scheduling horizon as the infinity-norm of the ramp vector r = max{ r t : t = 1, 2,..., T }. (2.20) By offering incentives for microgrids to reschedule their generation and storage resources over the scheduling horizon, the DSO can reduce the peak ramp of the system Microgrid s Payoff Each microgrid can reschedule its generation and storage resources to reduce the peak ramp in the distribution network. However, this causes a higher total energy cost to satisfy demand for its users due to the deviation from the original optimal operation point. By deviating from the original optimal generation and storage profiles {d o n, g o n}, the extra energy cost of microgrid can be calculated as C n = C n (d n, g n ) Cn, o (2.21) where C n (d n, g n ) is determined in (2.7), and C o n is as in (2.16). To incentivize peak ramp minimization, the DSO offers a reimbursement z n to microgrid n for its deviation from the original optimal operation point {d o n, g o n}. Therefore, we define the microgrid 18

33 n s payoff as the energy cost reduction as follow U n (d n, g n, z n ) = z n C n = z n (C n (d n, g n ) C o n). (2.22) From (2.22), we realize that when microgrid n does not participate in peak ramp minimization, its payoff is U o n = DSO s Payoff By offering financial incentives to encourage microgrids to reschedule their generation and storage resources, the peak ramp of the system over the scheduling horizon can be reduced, which also benefits for the DSO. We assume that the DSO suffers a convex cost function of the peak ramp, h( ), to guarantee sufficient ramping flexibility during the scheduling horizon. Then, we can calculate the amount of cost reduction, h, for the DSO when microgrids participate in the peak ramp minimization h = h( r o ) h( r ). (2.23) Note that, in this work, we model the benefit of the DSO as a function of the peak ramp. However, our work is by no means limited to a particular choice of h( ) in (2.23). In general, our incentive mechanism can be applied to more general functions as long as the DSO can quantify its benefit by reducing the peak ramp. Then the DSO s payoff can be defined as the saving cost in the peak ramp minimization problem as U dso (r, z) = h z n = h( r o ) h( r ) z n. (2.24) n=1 n=1 When the DSO does not participate in the peak ramp minimization problem, its payoff is U o dso = Social Welfare Maximization We define the social welfare as the aggregate payoff of the DSO and microgrids in the network Ψ(r, d, g, z) = U dso (r, z) + U n (d n, g n, z n ) n=1 19

34 = h( r o ) h( r ) z n + (z n (C n (d n, g n ) Cn)) o n=1 n=1 = h( r o ) h( r ) (C n (d n, g n ) Cn) o n=1 Ψ(r, d, g). (2.25) Note that, total amount of reimbursement of microgrids cancels out the amount of reimbursement that the DSO has to make, the social welfare of the system is independent with variable z in (2.25). as Based on the definition in (2.25), the social welfare maximization problem can be formulated max Ψ(r, d, g) (2.26) s.t. r t = P r,t P r,t 1 + (d n,t d n,t 1 ), t, (2.27) n=1 d n,t = D n,t g n,t + e + n,t e n,t W n,t, t, n, (2.28) s n,t+1 = s n,t + β n + e + n,t β n e n,t, t, n, (2.29) B min n variables: r, {d n, g n, e n } n. s n,t B max n, t, n, (2.30) 0 e + n,t emax n, t, n, (2.31) 0 e n,t emax n, t, n, (2.32) g n,t g n,t 1 R max n, t, n, and (2.33) 0 g n,t g max n, t, n, (2.34) By solving the problem in (2.26)-(2.34), we can obtain the maximum social welfare for the system. However, solving (2.26)-(2.34) cannot provide the information of reimbursement for microgrids. Therefore, in the next section, we use the Nash bargaining theory to investigate the incentive mechanism for the peak ramp minimization problem. 20

35 2.2 Nash Bargaining Solution In this work, we try to find a bargaining solution that satisfies four axioms proposed by John Nash [22]: Pareto efficiency, symmetry, invariance to affine transformations, and independence of irrelevant alternatives. The solution satisfying four axioms above, called Nash bargaining solution, is the optimal solution of the following optimization problem [22] max N (U dso Udso o ) (U n Un) o (2.35) n=1 s.t. (2.27) (2.34), U dso Udso o 0, and U n Un o 0, n, where U o dso is the disagreement point of the DSO, and U o n is the disagreement point of microgrid n. By solving the optimization problem in (2.35), we obtain the NBS as the following theorem. Theorem 2.1. The bargaining problem in (2.35) is feasible only if the social welfare of the system Ψ(r, d, g) is positive, and the NBS {r, (d n, g n, e n, z n) n } is as follow: The reimbursement is ( zn = 1 h( r o ) h( r ) N + 1 ) (C n Cn) o + (C n Cn), o n N, (2.36) and r, (d n, g n, e n) n maximizes the social welfare problem in (2.26)-(2.34). n=1 Proof. Since the disagreement points Udso o = 0, U n o = 0, and by taking ln of the objective function in (2.35), we obtain an equivalent optimization problem ( ) max ln h( r o ) h( r ) z n + (ln(z n (C n (d n, g n ) Cn)) o (2.37) s.t. (2.27) (2.34), U dso 0, and U n 0, n. n=1 n=1 21

36 The optimization problem in (2.37) can be solved by decomposing into the following two steps. First, for fixed r, {d n, g n, e n } n, we can obtain the optimal solution for z n by setting the first derivative of the objective function (2.37) with respect to z n to zero Or equivalent to 1 h( r o ) h( r ) 1 N n=1 z + n z n (C n Cn) o = 0, n. (2.38) h( r o ) h( r ) z n = z n (C n Cn), o n. (2.39) n=1 Solving the set of N equations in (2.39), we can obtain the expression of z n as ( ) z n = 1 h( r o ) h( r ) (C n C o N + 1 n) + (C n Cn), o n. (2.40) n=1 In order to guarantee for the feasibility of the bargaining problem in (2.37), the optimal reimbursement has to satisfy the conditions U dso 0, and U n 0, n. By substituing z n into (2.22) and (2.24), we can calculate the payoff of microgrids and the DSO as ( ) U n = 1 h( r o ) h( r ) (C n C o N + 1 n) and n=1 ( U dso = 1 h( r o ) h( r ) N + 1 ) (C n Cn) o. Therefore, the conditions U dso 0 and U n 0 can be satisfied when we have the condition ( h( r o ) h( r ) ) N n=1 (C n Cn) o 0, which is equivalent to the social welfare of the system to be positive, Ψ(r, d, g) 0. n=1 By substituting (2.40) into the objective function in (2.37), we obtain max [ ( 1 (N + 1) ln h( r o ) h( r ) N + 1 )] (C n Cn) o n=1 (2.41) s.t. (2.27) (2.34). 22

37 Since ln is a strictly increasing function, we can rewrite (2.41) as ( ) max h( r o ) h( r ) (C n Cn) o n=1 (2.42) s.t. (2.27) (2.34). variables: r, {d n, g n, e n } n. By solving the optimization problem in (2.42), we obtain optimal solution for r, {d n, g n, e n } n. Moreover, we realize that the objective function in (2.42) is the social welfare of the system. Therefore, the NBS maximizes the social welfare of the system, which concludes the proof. From the result in Theorem 2.1, we observe that the bargaining problem in (2.35) is feasible when the social welfare of the system is positive, i.e., the saving cost from the peak ramp reduction of the DSO must be greater than the total extra cost of microgrids due to the deviation from the original optimal point. Then, the reimbursement for each microgrid covers the extra cost incurred due to the deviation from the original optimal operation point, C n Cn, o and the social welfare of the system is equally divided among all microgrids and the DSO. Therefore, the net payoff of each microgrid can be determined as in the first term of (2.36). From Theorem 2.1, the NBS can be obtained by solving the social welfare maximization problem in (2.26)-(2.34), which requires the DSO to collect all information from microgrids such as detailed cost function of local generation units, daily energy demand, renewable generation, energy storage system information, etc. However, this centralized approach is impractical due to the enormous amount of signaling as well the privacy concern of microgrids. Therefore, in the next section, we propose distributed algorithms to achieve the NBS with minimal information exchange between microgrids and the DSO. 2.3 Distributed Algorithms for NBS In this section, we first provide an overview of the ADMM method to solve a convex optimization problem. Then we propose the synchronous and asynchronous distributed algorithms to 23

38 achieve the NBS An Introduction to ADMM Consider an optimization problem with the general form as min f(x) + g(z) (2.43) s.t. Ax + Bz = c. The augmented Lagrangian function of the problem in (2.43) is L(x, z, λ) = f(x) + g(z) + λ T (Ax + Bz c) + ρ 2 Ax + Bz c 2, (2.44) where λ is the Lagrangian multiplier, and ρ is a penalty parameter. The iterative procedure of ADMM to solve the problem in (2.43) can be expressed as follows [23]: x[t + 1] := arg minl(x, z[t], λ[t]), (2.45) x z[t + 1] := arg minl(x[t + 1], z, λ[t]), and (2.46) z λ[t + 1] := λ[t] + ρ (Ax[t + 1] + Bz[t + 1] c), (2.47) where in each iteration, the augmented Lagrangian function is minimized over x and z in an alternating fashion Synchronous Distributed Algorithm In this subsection, we propose a synchronous distributed algorithms to achieve the NBS for the system. Since the NBS is the optimal solution of the social welfare maximization problem as shown in Theorem 2.1, we propose a synchronous distributed algorithm to solve the optimization problem in (2.26)-(2.34), in which microgrids individually determine their energy generation and storage strategies with minimal information exchange with the DSO. The problem in (2.26)-(2.34) contains a large number of constraints. However, we realize that constraints (2.28)-(2.34) are separated into each microgrid. The constraints in (2.27) couples 24

39 microgrids and the DSO together. In order to decouple microgrid s demand profile variables from the ramping calculation in (2.27), we define the auxiliary variable as ˆd n,t = d n,t, t, n, (2.48) where d n,t is the amount of energy purchased by microgrid n from energy providers to satisfy its demand, while each auxiliary variable ˆd n,t can be interpreted as the amount of energy that the DSO recommends for microgrid n to achieve the minimal peak ramp. Then, by enforcing consensus constraints as in (2.48), the requested energy and allocated energy reach an agreement. The problem in (2.26)-(2.34) can be rewritten as max h( r o ) h( r ) (C n Cn) o (2.49) n=1 s.t. r t = P r,t P r,t 1 + ˆd n,t = d n,t, t, n, and (2.28) (2.34), variables: r, { ˆdˆdˆdn, d n, g n, e n } n. n=1 ( ˆdn,t ˆd n,t 1 ), t, In order to facilitate presentation, we further define the feasible set for each microgrid as F n = {(d n, g n, e n ) (2.28) (2.34)}. (2.50) Since r o and C o n are constants, the problem in (2.49) can be equivalently rewritten as min h( r ) + C n (d n, g n ) (2.51) n=1 s.t. r t = P r,t P r,t 1 + ˆd n,t = d n,t, t, n, and n=1 (d n, g n, e n ) F n, n N, variables: r, { ˆdˆdˆdn, d n, g n, e n } n. ( ˆdn,t ˆd n,t 1 ), t, 25

40 Due to the l-infinity norm in the objective function, solving the problem in (2.51) is difficult. We transform it into an equivalent problem by introducing an auxiliary variable as follow min h(γ) + C n (d n, g n ) (2.52) n=1 s.t. Γ r t Γ, t, r t = P r,t P r,t 1 + ˆd n,t = d n,t, t, n, and n=1 (d n, g n, e n ) F n, n N, variables: Γ, r, { ˆdˆdˆdn, d n, g n, e n } n. ( ˆdn,t ˆd n,t 1 ), t, The Lagrangian function of the problem in (2.52) with respect to consensus constraints is given by [23] L = h(γ) + C n (d n, g n ) + n=1 n=1 t=1 T λ n,t ( ˆd n,t d n,t ) + ρ 2 ( T = h(γ) + λ n,t ˆdn,t + C n (d n, g n ) n=1 t=1 n=1 t=1 ˆdˆdˆdn d n 2 n=1 ) T λ n,t d n,t + ρ 2 ˆdˆdˆdn d n 2, (2.53) where λ n,t is the Lagrangian multiplier, and ρ > 0 is a penalty parameter. By using the ADMM decomposition technique, the problem in (2.52) can be decomposed into subproblem for the DSO as min h(γ) + n=1 t=1 T λ n,t ˆdn,t + ρ 2 n=1 ˆdˆdˆdn d n 2 (2.54) n=1 s.t. Γ r t Γ, t, r t = P r,t P r,t 1 + variables: Γ, r, { ˆdˆdˆdn } n, n=1 ( ˆdn,t ˆd n,t 1 ), t, and the subproblem for microgrid n as min T C n (d n, g n ) λ n,t d n,t + ρ 2 ˆdˆdˆdn d n 2 (2.55) t=1 26

41 Algorithm 2.1 Synchronous ADMM 1: initialize: k = 0, λ n = 0, n 2: repeat 3: At the DSO: 4: repeat 5: wait 6: until receive updates λ n, d n from all microgrids 7: 1) solve local problem in (2.54) for optimal solution Γ, { ˆdˆdˆdn } n 8: 2) send ˆdˆdˆdn to corresponding microgrid 9: 10: At each individual microgrid: 11: repeat 12: wait 13: until receive the update ˆdˆdˆdn from the DSO 14: 1) solve problem (2.55) for the optimal solution d n 15: 2) update dual variables: 16: ( λ [k+1] n = λ [k] n + ρ ˆdˆdˆd[k+1] n 17: 3) send λ n, d n to the DSO 18: 19: k k : until a stopping criterion is met ) d [k+1] n s.t. (d n, g n, e n ) F n, variables: {d n, g n, e n }. Algorithm Implementation: The whole procedure for synchronous algorithm to achieve NBS can be summarized in Algorithm 2.1. First, the DSO solves the optimization problem in (2.54) to obtain the optimal solution (Γ, r, { ˆdˆdˆdn } n ). Then it sends the value of ˆd n to the corresponding microgrid. Each microgrid, after receiving the update value of ˆd n from the DSO, solves the local problem in (2.55) to obtain the optimal solution for (d n, g n, e n ). Finally, based on the current values of ˆdˆdˆdn and d n, the microgrid updates the dual variables as in line 15 in Algorithm 2.1 and sends the values of d n and λ n to the DSO. After receiving updates from all microgrids, the DSO will start a new iteration. The amount of information exchange between the DSO and microgrids is illustrated in Fig Note that, in the synchronous Algorithm 2.1, the DSO must wait to receive updates from all microgrids before it can start a new iteration, which is shown in Fig The following theorem demonstrates the convergence of the synchronous ADMM Algorithm 2.1. The synchronous ADMM Algorithm 2.1 converges to the optimal solution. Since the objective function in (2.52) is closed, proper, and convex, tt suffices to prove the convergence of Algorithm 27

42 DSO Solve local problem for, ^, Send ^ to microgrids ^ ^ ^, Microgrid 1 Microgrid 2 Microgrid N Solve local problem for Update Send, to the DSO, Solve local problem for Update Send, to the DSO, Solve local problem for Update Send, to the DSO Figure 2.3: The information exchange between the DSO and microgrids. DSO Microgrid 1 Microgrid 2 idle idle idle idle Microgrid N Iteration k = 0 Iteration k = 1 : Computation time of the DSO and microgrids Figure 2.4: Illustration of synchronous parallel computing for Algorithm 2.1. New iteration starts after the slowest microgrid finishes its computation. 2.1 by showing the strong duality holds for the problem in (2.52) [23, Section 3.2.1]. In fact, the feasible region in (2.52) defined by linear constraints is convex and bounded. Therefore, there exists a strictly feasible point, which guarantees Slater s condition and that the strong duality property holds [24] Asynchronous Distributed Algorithm In this subsection, we propose a distributed algorithm that can solve the optimization problem in (2.52) in an asynchronous fashion by using the asynchronous parallel coordinate updates method [25]. Particularly, the DSO and microgrids do not need to wait for the slowest agent to finish computation to start a new iteration. 28

43 We first provide an overview of the asynchronous parallel coordinate update framework by considering an optimization problem with the general form as min f(x) + g(y) (2.56) s.t. Ax + By = b. The asynchronous parallel ADMM framework to solve the problem in (2.56) can be performed as [25] y k+1 := arg min g(y) z k, By b + γ y 2 By b 2, (2.57) w k+1 g = z k γ(by k+1 b), (2.58) x k+1 := arg min f(x) 2wg k+1 z k, Ax + γ x 2 Ax 2, (2.59) w k+1 f = 2w k+1 g z k γax k+1, and (2.60) zn k+1 = zn k + η(w k+1 f,n wk+1 g,n ), (2.61) where (2.58)-(2.61) normally can be decomposed into each agent, and (2.57) can be performed in the asynchronous fashion. By applying the procedure in (2.57)-(2.61), we can solve the problem in (2.52) in the asynchronous distributed fashion. Specifically, the DSO solves the following local optimization problem min h(γ) + z n, ˆdˆdˆdn + γ 2 ˆdˆdˆdn 2 (2.62) n=1 n=1 s.t. Γ r t Γ, t, variables: Γ, r, { ˆdˆdˆdn } n. r t = P r,t P r,t 1 + ˆd n,t ˆd n,t 1, t, n=1 n=1 Each microgrid performs its own local computation in the asynchronous fashion including w [k+1] g,n = z [k] n + γ ˆdˆdˆd[k+1] n, (2.63) min C n (d n, g n ) 2w [k+1] g,n z [k] n, d n + γ 2 d n 2 (2.64) 29

44 Algorithm 2.2 Asynchronous ADMM 1: initialize: k = 0, z n = 0, ˆdˆdˆdn = 0, n 2: repeat 3: At the DSO: 4: repeat 5: wait 6: until receive any update z n from microgrid n 7: 1) update z n into global memory 8: 2) solve (2.62) for the optimal solution Γ, { ˆdˆdˆdn } n, r 9: 3) send ˆdˆdˆdn to microgrid n 10: update global counter: k k : 12: At any microgrid n: 13: if receive ˆdˆdˆdn from the DSO 14: 1) calculate w g,n as in (2.63) 15: 2) solve (2.64) for {d n, g n, e n } 16: 3) calculate w f,n as in (2.65) 17: 4) calculate z n as in (2.66) and send to the DSO 18: 19: until a stopping criterion is met s.t. {d n, g n, e n } F n, w [k+1] f,n = 2w [k+1] g,n z [k] n γd [k+1] n. (2.65) Then microgrid n updates its dual variable z n and sends to the DSO to start new iteration z [k+1] n = z [k] n + η(w [k+1] f,n w [k+1] g,n ). (2.66) Algorithm Implementation: The whole procedure for the asynchronous distributed algorithm can be described in Algorithm 2.2. The implementation of the asynchronous algorithm can be done similarly to Algorithm 2.1. However, the difference is the DSO will start a new iteration when it receives update from any microgrid instead of waiting for the update of all microgrids as illustrated in Fig Note that, this asynchronous property of Algorithm 2.2 is more suitable for the practical deployment than Algorithm 2.1 since it will achieve faster convergence speed. Moreover, Algorithm 2.2 will outperform Algorithm 2.1 in large distributed power networks, in which the effect of communication delay on the convergence performance due to information exchange between microgrids and the DSO will be minimal. The asynchronous ADMM framework in (2.57)-(2.61) guarantees the convergence if both function f(x) and g(y) in (2.56) are closed proper convex [25, Section 2.5], which is satisfied for 30

45 DSO Microgrid 1 Microgrid 2 Microgrid N Iteration k = : Computation time of the DSO and microgrids Figure 2.5: Illustration of asynchronous parallel computing for Algorithm 2.2. New iteration starts when any microgrid finishes its computation. the optimization problem in (2.52). Therefore, the asynchronous ADMM Algorithm 2.2 converges to the optimal solution. 2.4 Simulation Results In this section, we provide the numerical simulations to demonstrate the performance of the proposed model and algorithms. We test a system with N = 10 microgrids, and the period of scheduling is divided into T = 24 time slots. The daily energy demand and the available renewable generation of each microgrid are uniformly generated from the typical demand and renewable generation in Table 2.2, and Table 2.3, respectively. The capacity of energy storage system of each microgrid is 4 MWh, where the maximum charging/discharing rate is generated randomly from [0.5 1] MW. The hourly day-ahead market price is provided in Table 2.1 [26]. We assume the convex cost function for local generation units of microgrids as f n (x) = ax 2 + bx, where a and b are generated from uniform distributions a [0.1, 0.5] ($/MWh 2 ), b [20, 50] ($/MWh) [27, 28]. We also assume the convex cost function of the peak ramp for the DSO as h(γ) = α 1 Γ 2 + α 2 Γ, where α 1 = 5 ($/MW 2 ), and α 2 = 30 ($/MW). All tests are conducted on a Windows 7 64-bit personal computer with Intell i GHz CPU and 16GB of RAM using Matlab. Each subproblem in our proposed algorithms is solved using CVX package [29]. We show the convergence behavior of our proposed algorithms in Fig Since the both al- 31

46 Table 2.1: Hourly Market Price Time (h) Price ($/MWh) Time (h) Price ($/MWh) Time (h) Price ($/MWh) Time (h) Price ($/MWh) Table 2.2: Microgrid daily demand Time (h) Load (MW) Time (h) Load (MW) Time (h) Load (MW) Time (h) Load (MW) gorithms are implemented and performed on a single computer, the delay time due to local information exchange can be ignored. We plot the trend of objective value versus the number of iterations. From the result, we can see that synchronous Algorithm 2.1 converges after about 70 iterations, while asynchronous Algorithm 2.2 needs about 250 iterations to achieve the optimal solution. The faster convergence behavior of Algorithm 2.1 is due to the synchronous updating framework among all microgrids. However, the computation time for each iteration in Algorithm 2.2 is much faster than Algorithm 2.1. Particularly, the average computation time for one iteration in Algorithm 2.2 is 1.3 seconds, while it is 7.1 seconds for Algorithm 2.1. We evaluate the capability of our proposed model in reducing the peak ramp by plotting the net load of the system in Fig. 2.7 with and without the deployment of the incentive mechanism. Due to the large amount of distributed renewable generation during the period from 8:00 to 18:00, residential users and microgrids use the available renewable energy to serve their load. Therefore, the total net load of the system is significantly reduced during this period, which leads to the increased need for ramping capability requirement when renewable generation shuts down in the late 32

47 Objective value (k$) Objective value (k$) Table 2.3: Generation of renewable resources Time (h) Load (MW) Time (h) Load (MW) Time (h) Load (MW) Time (h) Load (MW) Iteration Asynchronous Algorithm Iteration Synchronous Algorithm 2.1 Figure 2.6: The convergence performance of the proposed algorithms. afternoon in the case of without economic incentives offering to microgrids. In contrast, by using our proposed model, microgrids have incentives to reschedule their energy generation and storage resources to reduce the peak ramp of the system. Particularly, without incentives, the peak ramp is 20.9 MW, while in our model, the peak ramp is 9.76MW, which reduce about 53.3%. This can significantly release stress of the ramping flexibility requirements for the grid operators. Although the proposed model can reduce the peak ramp of the system, microgrids will not be willing to reschedule their energy generation and storage resources if they do not have benefit. To investigate this perspective, we show the reimbursement that microgrids receive from the DSO and the extra cost incurred over the entire scheduling horizon due to the deviation from the original 33

48 without incentive mechanism with incentive mechanism Energy demand (MWh) Hour Figure 2.7: The net load of the system with and without incentive mechanism for microgrids. optimal operation point. From the results in Fig. 2.8, we realize that the reimbursement from the DSO for each microgrid covers the extra cost, which guarantees that microgrids achieve the positive payoff (the difference between the reimbursement received and extra cost incurred). This demonstrates that microgrids will have financial encouragement to participate in the peak ramp minimization problem. 2.5 Conclusions In this chapter, an incentive mechanism design for integrated microgrids in the peak ramp minimization problem has been proposed. We investigate the economic interaction between the DSO and microgrids using the Nash bargaining theory, which achieves the maximum social welfare of the overall system. Due to the privacy concerns of microgrids, we propose two distributed algorithms to achieve the NBS, in which each microgrid individually schedules its energy generation and storage resources. The first algorithm requires a synchronous update from all microgrids, i.e., the new iteration starts only after all microgrids finish their computations. On the other hand, the 34

49 Payoff ($) Extra cost Reimbursement Microgrid Figure 2.8: Extra cost incurred and reimbursement for microgrids. second algorithm can perform in the asynchronous framework, i.e., the DSO starts new iteration when any microgrid in the system finishes its computation. The simulation results demonstrate the convergence performance of the proposed distributed algorithms as well as the efficacy of our model in reducing the peak ramp of the power system. 35

50 Chapter 3 A Big Data Scale Algorithm for Optimal Scheduling of Integrated Microgrids Microgrids have been proposed as one of the key components for grid modernization, which operate as single controllable entities to supply a group of interconnected loads [30]. With the capability of integrating renewable energy resources and energy storage devices, microgrids are expected to reduce large capital investment by meeting increased energy demand using locally generated power. In addition, by deploying power generation close to end users, microgrids are becoming promising solution to improve reliability and power quality [11 13]. In addition to the aforementioned characteristics, one of the salient features of microgrids is islanded operation, which is defined as the capability to disconnect from the main distribution network and locally supply their loads [14]. By rapidly disconnecting from the main grid, microgrids can protect their components from upstream disturbances or voltage fluctuations. More importantly, the islanded operation mode allows microgrids to ensure energy supply for critical loads by increasing the generation output of local generators when the main distribution network is faulty. This capability has been advocated as a viable solution to achieve high system resiliency during major outages [15]. However, due to the limitation of ramping ability of local generators in increasing generation output from the operating point in the normal operation before switching into the islanded operation, microgrids may not be able to satisfy all load demand for local customers. In order to overcome supply deficiency, the work in [31] investigates the real-time pricing for a power grid operator to incentivize aggregators to reschedule energy consumption when experiencing contingency. Another approach is to propose a power management algorithm for islanded microgrids using energy storage and demand response program [32]. However, since these works either focus on the operation of microgrids in the islanded mode solely or tackle the problem when contingencies have already been occurred, the proposed models present limitations in maintaining proper level of reliability that can be improved by incorporating certain requirements of the islanded operation into the 36

51 optimal scheduling problem for microgrids. The work presented in this chapter is to fill the gap in considering the operation for integrated microgrids in both grid-connected and islanded modes. More specifically, the optimal scheduling of local generators in the normal operation needs to take into account the requirement for satisfying critical loads when switching into the islanded operation. The objective is to minimize the generation cost of the associated distribution system in the normal operation while ensure the minimal amount of load curtailment when microgrids switch into the islanded operation. Due to the large number of constraints incorporated into the model, the formulated problem becomes a large-scale optimization problem, and may not be scalable to solve by the centralized method. Therefore, we apply the ADMM decomposition technique [23] to efficiently solve the problem. By jointly tackling the above discussed challenges, our main technical contributions can be summarized as follows: Minimal Load Curtailment Modeling in Islanded Operations: We formulate a microgrid optimal scheduling problem in the normal operation mode. Moreover, we also incorporate l 1 - norm into the objective function to obtain the minimal amount of load curtailment when microgrids are disconnected from the main grid. The model demonstrates that only a sparse number of microgrids have to curtail load when switching into the islanded operation. Parallel Algorithm: The formulated problem consists of a large number of constraints and becomes a large-scale optimization problem. Therefore, we propose a parallel algorithm using the ADMM decomposition technique to efficiently solve the optimization problem. Big Data Framework Implementation: We provide a detailed implementation of the parallel computing for our proposed algorithm using the Hadoop MapReduce software framework to run on a computer cluster to reduce the computational complexity. The remainder of this chapter is organized as follows. The optimal scheduling problem with the islanded operation constraints is formulated in Section 3.1. Section 3.2 provides the decomposition algorithm using ADMM. Simulation results are presented in Section 3.3. Detailed imple- 37

52 Main grid Microgrid 1 Microgrid 2 Microgrid N Figure 3.1: The model of microgrids with islanding operation. mentation of parallel computing for the proposed algorithm on a computer cluster using the Hadoop MapReduce is described in Section 3.4, and Section 3.5 concludes the chapter. 3.1 System Model In this section, we describe the model of the microgrid system, and formulate the optimal scheduling problem of integrated microgrids System Description Consider a distribution network consisting a set N {1, 2,..., N} of microgrids, which are connected to the main power grid as in Fig Each microgrid i N is required to serve a group of customers having demand D i. To fully satisfy demand requested from users, each microgrid can locally generate power using its generator and/or acquires power from the main grid. Moreover, each microgrid has direct connections with a group of other microgrids and can exchange power locally. Let N i be the set of neighboring microgrids connected to microgrid i through transmission lines (including the microgrid i itself), with cardinality N i = N i. Define x o i = {xo ij } j N i be the 38

53 vector of power generation that microgrid i generates to exchange with its neighbors, where x o ii is the amount of power that microgrid i generates to supply its own customers. We use superscript o for all variables in the normal operation to differentiate with variables in the islanded operation, which will be defined later. Then the total amount of power that microgrid i has to generate during the normal operation using its local generation is 1 T x o i = j N i x o ij, where 1 = [1, 1,..., 1]T is a column vector of ones. In order to satisfy demand for its users, microgrid i acquires y o i amount of power from the main grid. Any power transfer between a microgrid and the main grid is accompanied with the loss of power over the distribution lines. The power loss due to the power exchange between the main power grid and microgrid i, Ploss,i o o, can be calculated as [33] Ploss,i o o = R oi(yi o)2 Vo 2 + αyi o, (3.1) where R oi is resistance of the transmission line connecting microgrid i and the main grid, α is the power loss constant due to transformer, and V o is operating voltage between microgrid i and the main grid. Similarly, the power loss due to power exchange between two microgrids i and j, P o loss,i j, can be calculated as P o loss,i j = R ij(x o ij )2, (3.2) V1 2 where R ij is the resistance of the transmission line connecting microgrid i and j, and V 1 is the operating voltage between microgrids. The difference between (3.1) and (3.2) is that there is no power loss due to transformer in (3.2) since all microgrids operate at the same voltage level. Then the total power that a microgrid obtains to supply its customers includes: its local generation, power acquired from the main grid and/or from its neighbors. We have the following power balance constraint in the normal operation x o ii + (y o i P o loss,i o ) + j N i,j i (x o ji Ploss,j i o ) = D i, i. (3.3) 39

54 3.1.2 Islanded Operation When microgrid k is disconnected from the main grid, the power that it obtains to supply its customers can only be generated locally from its generator and/or from the power exchange with its neighbors. However, it may be possible that the total amount of available power when a microgrid is islanded cannot fully satisfy the demand for its customers. Therefore, the load demand that microgrid k is able to serve will be reduced. Let ɛ k denote the fraction amount of load demand that microgrid k has to curtail if it is islanded from the main grid (we use superscript k to denote all variables in islanded operation case k, which is corresponding to situation that microgrid k is islanded from the main grid). Since microgrids serve different types of demand, they typically have different requirements for load satisfaction when switching into the islanded operation. Therefore, we have the following constraint for load curtailment of microgrid k 0 ɛ k ɛ k max, (3.4) where ɛ k max is the predefined maximum allowable fraction of load curtailment. The amount of load demand that microgrid k has to satisfy in the islanded operation mode is (1 ɛ k )D k. Then we have the following constraint for microgrid k x k kk + (x k jk P loss,j k k ) (1 ɛk )D k. (3.5) j N k,j k Note that, in (3.5), there is no variable y k k since microgrid k is islanded and cannot obtain the power from the main grid. Moreover, we assume that each microgrid is located at a different geographical area, and therefore, when the upstream disturbance or voltage fluctuation happens, only one microgrid is disconnected from the main grid at any given time instance. The more general case of more than one microgrid are islanded from the main grid can be directly applied without changing the structure of this work by constructing and adding constraint (3.5) into the problem for any microgrids switch into the islanded operation. All remaining microgrids i k that are still connected to the main grid must fully satisfy their users demand as the following power balance constraint x k ii + (y k i P k loss,i o ) + j N i,j i (x k ji Ploss,j i k ) = D i. (3.6) 40

55 When microgrid k is islanded from the main grid, all microgrids in the network have to adjust the power generation output compared to the normal operation mode to help satisfy power demand for users. However, the adjustment of power generation output must be constrained by the ramping limit 1 T x k i 1 T x o i max i, i N, (3.7) where max i is the maximum ramping rate of local generator at microgrid i. Similarly, the main grid also needs to reschedule its generation output when a microgrid is disconnected from the main distribution network. Therefore, we have the following ramping constraint 1 T y k i 1 T y o i max o, (3.8) where y k = [y1 k, yk 2,..., yk N ] denotes the power flow vector from the main grid to microgrids when microgrid k is islanded, and max o is the maximum ramping rate of generators at the main grid. Note that, all constraints (3.5), (3.6), (3.7), and (3.8) must be constructed for all possible islanded operation cases by considering each microgrid is disconnected from the main grid one at a time Optimal Scheduling Problem of Integrated Microgrids In this subsection, we formulate an optimization problem to obtain the optimal scheduling for all microgrids in the normal operation. In addition, we want the optimal solution to guarantee the reliable operation of microgrids as well. Specifically, when a microgrid is disconnected from the main grid, the local generator at each microgrid is able to adjust its generation output to a new operation point compared to that in the normal operation so that to minimize the amount of load curtailment. Let C i (x) be the convex cost function for generating x amount units of power at microgrid i, or at the main grid if i = 0. We also define ɛ D = [ɛ 1 D 1, ɛ 2 D 2,..., ɛ k D k,..., ɛ N D N ] T as the load curtailment vector of microgrids. Then the optimal scheduling problem for microgrids and the main 41

56 grid can be formulated as follows: min C o (y o ) + C i (x o i ) + τ ɛ ɛ D 1 (3.9) s.t. i=1 x o ii + (y o i P o loss,i o ) + F L max i j F L max i x k kk + j N k,j k j N i,j i (x o ji Ploss,j i o ) = D i, i, (3.10) x o ij + x o ji F L max i j, i, j, (3.11) y o i F L max i, i, (3.12) x k ii + (y k i P k loss,i o ) + F L max i j F L max i (x k jk P loss,j k k ) (1 ɛk )D k, k, (3.13) j N i,j i (x k ji Ploss,j i k ) = D i, i k, k, (3.14) x k ij + x k ji F L max i j, i, j, k, (3.15) y k i F L max i, i, k, (3.16) 0 ɛ k ɛ k max, k, (3.17) 1 T x k i 1 T x o i max i, i, k, and (3.18) 1 T y k 1 T y o max o, k, (3.19) where τ is a positive weighted parameter to capture the trade-off between generation cost minimization and minimal amount of load curtailment. The third term in (3.9), ɛ ɛ D 1, is l 1 -norm of vector ɛ D, which determines the amount of load curtailment in the islanded operation mode def ɛ ɛ D 1 = k ɛ k D k. Incorporating l 1 -norm, ɛ ɛ D 1, into the objective function (3.9) allows us to obtain the optimal solution that minimizes the amount of load curtailment when microgrids switch into islanded operation mode. The constraints (3.11), (3.12), (3.15), and (3.16) are the line flow limit where F L max is the flow limit. The optimization problem in (3.9)-(3.19) is convex and can be solved in a centralized fashion to obtain the global optimal solution. However, since a large number of constraints are coupled over the normal operation and the islanded operation, the centralized computation scheme is not scalable. 42

57 3.2 Decomposition Algorithm based on ADMM In this section, we propose a parallel algorithm for the optimal scheduling of microgrids using the ADMM decomposition method. The problem in (3.9)-(3.19) contains a large number of constraints and becomes a large-scale optimization problem. However, we realize that constraints are separable into the different islanded operation cases for different microgrids. In order to make the problem in (3.9)-(3.19) to be more compact and ready for using ADMM, we define the feasible set power generation vectors in both the normal operation case and in the islanded operation case k as follows: F o = {(x o, y o ) (3.10), (3.11), (3.12)}, F k = {(x k, y k, ɛ k ) (3.13), (3.14), (3.15), (3.16), (3.17)}. We further introduce auxiliary variables y o,k and x o,k i as the local copies of y o and x o i in the normal operation at each islanded operation case y o,k = y o, k, x o,k i = x o i, i, k. Then, the problem in (3.9)-(3.19) can be reformulated as follows: min C o (y o ) + C i (x o i ) + τ i=1 s.t. (x o, y o ) F o, ɛ k D k (3.20) k=1 (x k, y k, ɛ k ) F k, k = 1,..., N, max o max i y o,k = y o, k, and x o,k i = x o i, i, k. 1 T y k 1 T y o,k max o, k, 1 T x k i 1 T x o,k i max i, i, k, 43

58 The augmented Lagrangian function of the problem in (3.20) is given by [23] L =C o (y o ) + + γ 2 C i (x o i ) + τ i=1 ɛ k D k + k=1 y o,k y o 2 + γ 2 k=1 =C o (y o ) + + C i (x o i ) i=1 k=1 i=1 [ τɛ k D k + (λ k ) T (y o,k y o ) + k=1 + γ 2 yo,k y o 2 + γ 2 i=1 (λ k ) T (y o,k y o ) + k=1 x o,k i x o i 2 i=1 x o,k i x o i 2 (µ k i ) T (x o,k i x o i ) ] k=1 i=1 (µ k i ) T (x o,k i x o i ), (3.21) where λ, µ are the Lagrangian multipliers, and γ is a penalty parameter. Define the primal variables z = (y o, {x o i } i), which is the decision variable vector in normal operation, and w = ({w k, ɛ k } k ), where w k = (y k, {x k i } i) is the decision variable vector in islanded operation case k. Then the ADMM decomposition technique can be used to solve the problem in (3.20) in an iterative procedure. Specifically, at the t th iteration, the primal variables and dual variables are updated sequentially as z[t + 1] = arg min L(z, w[t], λ[t], µ[t]), (3.22) w[t + 1] = arg min L(z[t + 1], w, λ[t], µ[t]), (3.23) ( ) λ k [t + 1] = λ k [t] + γ y o,k [t + 1] y o [t + 1], k, and (3.24) ( ) µ k i [t + 1] = µ k i [t] + γ x o,k i [t + 1] x o i [t + 1], i, k. (3.25) Based on the Lagrangian function in (3.21), we decompose the problem in (3.20) into the following N + 1 optimization problems. The first problem is associated with variables in the normal operation mode only and corresponding to the primal variables update in (3.22) min C o (y o ) + C i (x o i ) (y o ) T N λ k i=1 k=1 44

59 k=1 i=1 (µ k i ) T x o i + γ 2 s.t. (y o, {x o i } i ) F o. y o,k y o 2 + γ 2 k=1 k=1 i=1 x o,k i x o i 2 (3.26) By fixing the values of {x o,k i } i, y o,k, λ, µ, and then solving the problem in (3.26), we obtain the optimal solution for (y o, {x o i } i). The remaining N problems are associated with variables in the islanded operation cases and corresponding to the primal variables update in (3.23). For each islanded operation case, we decompose into the following problem min τɛ k D k + (λ k ) T y o,k + s.t. (x k, y k, ɛ k ) F k, max o max i i=1 1 T y k 1 T y o,k max o, and 1 T x k i 1 T x o,k i (µ k i ) T x o,k i + γ 2 yo,k y o 2 + γ 2 max i, i=1 x o,k i x o i 2 (3.27) variables: {x k i } i, y k, ɛ k, {x o,k i } i, y o,k. By fixing (y o, {x o i } i, λ, µ) and then solving the problem in (3.27), we obtain the optimal solution ( for {x k i } i, y k, {x o,k i } i, y o,k, ɛ ). k Algorithm Implementation: The whole procedure for solving the problem in (3.20) using ADMM is described in Algorithm 3.1. First, a master computing node solves the optimization problem in (3.26) to obtain the optimal solution (y o, {x o i } i). Then it broadcasts the optimal solution in the normal operation mode to all distributed computing nodes. Each distributed computing node ( ) solves the optimization problem in (3.27) to obtain the optimal solution {x k i } i, y k, {x o,k i } i, y o,k, ɛ k in the islanded operation. Finally, based on the local values of ( y o,k, {x o,k i } i ), and (y o, {x o i } i), the dual variables can be updated as in line 15 in Algorithm 3.1. Note that, N optimization problems associated with the islanded operation cases are decoupled and can be solved in a parallel fashion at different computing nodes without affecting the others. This parallel implementation reduces the computation time for the proposed Algorithm

60 Algorithm 3.1 ADMM Decomposition 1: Initialization: t = 0, λ = 0, {µ i = 0} i 2: repeat 3: At master computer: 4: repeat 5: wait 6: until receive updates λ, µ i, y o,k, x o,k i from all distributed computers 7: solve (3.26) for optimal solution (y o [t + 1], {x o i [t + 1]} i) 8: broadcast (y o [t + 1], {x o i [t + 1]} i) to all distributed computers 9: 10: At each distributed computer k 11: repeat 12: wait 13: until receive updates (y o [t + 1], {x o i [t + 1]} i) from master computer 14: solve (3.27) for the optimal solution {x k i } i, y k, {x o,k i } i, y o,k, ɛ k 15: update dual variables: ( ) λ k [t + 1] = λ k [t] + γ y o,k [t + 1] y o [t + 1] ( ) µ k i [t + 1] = µ k i [t] + γ x o,k i [t + 1] x o i [t + 1], i ( ) 16: send λ, µ i, y o,k, x o,k i to the master computer 17: 18: t t : until a stopping criterion is met The amount of information exchange between the master node and distributed computing nodes is depicted in Fig The master node broadcasts the same solution in the normal operation ( ) to all distributed nodes. Each distributed node needs to send the local information λ, µ i, y o,k, x o,k i to the master. Note that in the proposed algorithm, all microgrids are required to exchange information with the master computer to solve subproblems. This can be performed via certain entities that are designed to operate distribution networks such as distribution system operators. Besides the responsibility of controlling and operating the distribution grids, distribution systems operators will play a role as information hubs to facilitate for data exchange as well as data aggregation [34 37]. 3.3 Simulation Results In this section, we use computational experiments to evaluate the performance of our proposed algorithm. We use two modified IEEE 9-bus and 14-bus power systems [38] to obtain the physical connections as indicators for communication lines between microgrids, in which each bus 46

61 Master computer Normal operation Find, Broadcast,,,,,,,,,,,,,,,,,,, Computer 1 Islanded operation (k=1) Find,,, Update, Computer 2 Islanded operation (k=2) Find,,, Update, Computer N Islanded operation (k=n) Find,,, Update, Figure 3.2: The illustration of information exchange between the normal operation subproblem and islanded operation subproblems. Table 3.1: Demand of microgrids Microgrid Demand (MW) Microgrid Demand (MW) Microgrid Demand (MW) is considered as a microgrid. The operation voltage between the main grid and microgrids, V 0 = 50 kv, while the operation voltage between microgrids is V 1 = 22 kv [39]. The power loss constant α = 0.02 [39]. The generation capacity of power generator for each microgird is generated as 10% greater than its total demand, and is able to adjust up to 10% of its maximum output capacity when switching to the islanded operation mode. To avoid the infeasibility of the problem, we set the maximum allowable fraction amount of load curtailment, ɛ k max = 1 for all microgrids. We select τ = 10 in all simulations, unless otherwise stated. The power demand of microgrids is given in Table 3.1. We assume the convex cost function for main grid and microgrids as C(x) = ax 2 + bx, where a and b are generated randomly from a uniform distribution a [0.01, 0.5] ( $/MW 2), b [10, 40] ($/MW ). All tests are conducted on a Windows 7 64-bit personal computer with Intel i GHz CPU and 16GB of RAM using Matlab. Each sub-problem in our proposed algorithm is solved using CVX [29]. 47

62 0.1 9 microgirds 14 microgrids 0.01 Relative error 1E-3 1E-4 1E Iteration index Figure 3.3: The convergence performance of the proposed algorithm. To demonstrate the advantage of the ADMM decomposition, we show the number of iterations required for the proposed algorithm to converge in Fig For the system with 9 microgrids, the relative error approaches to 10 4 in about 40 iterations, while 14 microgrids system needs 60 iterations to yield the same relative error. This is due to the fact that a larger system leads to a greater number of constraints in the islanded mode, and consequently produces more sub-problems when using ADMM decomposition. Further, notice that the computational time required for each iteration is varied for different systems. The average computational time for each iteration in the system with 9 microgrids is 8.2 seconds, while it is 41.3 seconds for the system with 14 microgrids. It is an important task to find an approximate value for the parameter τ in problem (3.9)- (3.19). In general, higher values of τ increase the weight for l 1 -norm term in the objective function, and the resultant optimal solution achieves smaller amount of the load curtailment in the islanded operation. Particularly, Fig. 3.4 plots the average percentage of load curtailment as a function of τ, and the result shows that we can significantly reduce the amount of load shedding by setting a higher value for τ, which can achieve about 1% of the total load of the system. Even a higher value of τ leads to a smaller amount of load shedding, it may incur in general 48

63 Percentage of Load Curtailment (%) microgrids 14 microgrids Figure 3.4: The effect of τ on the percentage of load curtailment when microgrids switch into the islanded operation mode. Table 3.2: Total generation cost and load curtailment comparison No l 1 -norm With l 1 -norm Cost($) Cut(%) Cost($) Cut(%) 9 microgrids microgrids a higher generation cost in normal operation. To investigate the impact of τ on the operational cost of the system, we plot the generation cost versus τ in Fig It shows that the system total generation cost does not increase too much when we increase τ. Based on the results from Fig. 3.4 and Fig. 3.5, we can select an appropriate value of τ to satisfy the design criteria for power systems. This selection will obtain the trade-off between the generation cost in the normal operation and the amount of load shedding in the islanded mode. To further study the effectiveness of our model in improving the reliability of power systems, we report the generation cost and the amount of load shedding as in Table 3.2, with and without l 1 - norm in the objective function. The column with l 1 -norm denotes the results when we incorporate l 1 -norm into the objective function. It shows that with l 1 -norm in the model does not increase too much generation cost in the normal operation while still obtains large reduction on the percentage of 49

64 microgrids 14 microgrids Total cost (k$) Figure 3.5: The effect of τ on the generation cost of the system in the normal operation. load shedding compared to the model without l 1 -norm. This indicates that incorporating l 1 -norm in our model can improve the power system reliability without significantly increasing the generation cost in the normal operation. Moreover, our model not only reduces the percentage of load shedding, but also produces a sparse solution, which means that only several microgrids have to reduce their loads. To demonstrate this, Fig. 3.6 plots the fraction of load shedding for each individual microgrid in the system with 9 microgrids. It can be noticed that only microgrids 2, 3, 5, 6, 8, 9 will reduce their loads when switching into the islanded operation. 3.4 Hadoop MapReduce Implementation In this section, we introduce an overview of MapReduce programming model and describe the detailed implementation of ADMM Algorithm 3.1 using Hadoop MapReduce framework. 50

65 Fraction of Load Curtailment Without l 1 -norm With l 1 -norm Microgrid Figure 3.6: The fraction of load shedding of each individual microgird in the islanded operation mode for the 9-microgrid system MapReduce Programming Model MapReduce is a programming model for distributed processing of very large datasets using a large cluster of commodity machines [40]. It has been widely used to perform special-purpose computations both in industry and academia [41]. A MapReduce computation consists of a set of Map tasks and Reduce tasks. The input data will be split into independent blocks and processed by the Map tasks in a completely parallel manner to produce a set of intermediate key-value pairs. Then, all outputs of the mapping operation that share the same intermediate key will be grouped together and passed to the same Reducer. The MapReduce work flow is shown in Fig Generally, the Map and Reduce steps can be conceptually expressed as [40] map (k 1, v 1 ) list(k 2, v 2 ) reduce (k 2, list(v 2 )) list(v 3 ). Apache Hadoop is an open-source software framework written in Java for easily writing application to process massive amount of data on computer clusters in reliable, fault-tolerant man- 51

66 Input data Shuffle HDFS MAP Output data Split 1 Split 2 MAP REDUCE HDFS Output file Split N MAP REDUCE Output file Figure 3.7: The flow chart of MapReduce programming model. ner [42]. The core of Hadoop consists of a storage part, which provides the Hadoop Distributed File System (HDFS) architecture, and a processing part which implements the MapReduce computation paradigm. The HDFS manages the storage of data across an entire cluster of machines by splitting files into blocks and distributing them amongst the nodes in the cluster. Then, the data at each node is divided into fixed-size piece called splits. Each split of data is processed in the Map tasks based on the user-defined Map function to produce a list of key-value pairs. The process of sorting key-value pairs of map tasks and sending them to reducers is handling internally by Hadoop. This allows Hadoop to reduce many complexities such as data partitioning, scheduling tasks across many machines, handling machine failures and performing inter-machine communication [43] ADMM Implementation using Hadoop MapReduce Each iteration of ADMM Algorithm 3.1 can be represented as a MapReduce job as illustrated in Fig The parallel computations for islanded operation sub-problems in (3.27) are performed by Map tasks, and the normal operation sub-problem computation in (3.26) is performed by a Reduce task. We have total N Mappers, one for each islanded operation sub-problem. Each Mapper solves the optimization problem in (3.27) to obtain y o,k, {x o,k i } i. However, solving the problem in (3.27) for y o,k [t + 1], x o,k i [t + 1] on iteration t + 1 needs to use y o [t], x o i [t] and λk [t], µ k [t] from the previous iteration. Since MapReduce is not designed to support iterative applications, we facilitate 52

67 Input data HDFS read Iteration 1 MapReduce Job HDFS write HDFS read Iteration 2 MapReduce Job HDFS write HDFS read Iteration T MapReduce Job HDFS write Final result HDFS,,,,,, MAP Islanded sub-problem 1 Solve (32) with k=1 Islanded sub-problem 2 Solve (32) with k=2 Emit,,,,,, REDUCE,,, Emit,,, Normal operation sub-problem Solve (31) HDFS Output data:,,,,,, Islanded sub-problem N Solve (32) with k=n Emit,,,,,, Figure 3.8: Data sharing for iterative ADMM using Hadoop MapReduce and the detailed illustration of Map tasks and Reduce task in each MapReduce job in each iteration. iterative computation for Algorithm 3.1 by writing the output data at each iteration to the HDFS, which will be used as the input data for Mappers in the next iteration. Particularly, each Mapper uses splitid provided by Hadoop to identify which islanded problem is and loads the corresponding λ k [t], µ k [t] from HDFS in the previous iteration, the y o [t], x o i [t] are the same for all Mappers. After solving the optimization problem, each Mapper updates values for λ k and µ k using (3.24) and (3.25), respectively. Then each Mapper emits an intermediate key-value pair, which is 1, {y o,k, x o,k i, λ k, µ k } to the Reducer. Since in our problem, there is a single reducer, which plays a role of performing the normal operation computation, all the keys in all Map tasks are selected as 1 to force all information from the Mappers is sent to a unique Reducer. Based on all information received from the Mappers, the Reducer solves (3.26) to obtain y o, {x o i } i. The values of y o, {x o i } i, λ, µ are written out to HDFS directly by the Reducer, which will be used as the input data for MapReduce job in the next iteration. The detailed Map tasks and Reduce task in each iteration is illustrated in Fig The pseudo code for implementing Algorithm 3.1 using Hadoop MapReduce is described in Algorithm

68 Algorithm 3.2 ADMM using Hadoop MapReduce 1: function MAP(islanded ID, inputdata) 2: 3: Load data of previous iteration from HDFS corresponding to islanded ID Solve islanded sub-problem (3.27) 4: Update λ k, µ k using (3.24) and (3.25) 5: EMIT 1, {y o,k, x o,k i, λ k, µ k } 6: end function 7: 8: function REDUCE(key, Data from Mappers) 9: 10: Concatenate {y o,k, x o,k i, λ k, µ k } Solve normal operation sub-problem (3.26) for y o, x o 11: EMIT {y o, x o, λ, µ} 12: end function 13: 14: function MAIN(inputPath, outputpath) 15: Initialization 16: while ( notconverged and k < maxiterations ) do 17: run MapReduceJob (inputpath, outputpath) 18: t t : end while 20: end function Performance Results We build a Hadoop cluster with 8 computers in which each computer has a 2.33GHz Intel processor, 4GB of RAM. Algorithm 3.2 is written in Java. Each Mapper and Reducer solve the optimization problem using Gurobi, which provides an interface to construct and solve optimization problem in Java programming language [44]. We run Algorithm 3.2 on the cluster with different total number of microgrids in the system and report the running time as in Table 3.3. We can see that the running time for one iteration, which is determined as the time for reading input data, processing Map and Reduce tasks, and writing out data, on the cluster does not increase significantly when the number of microgrids increase. Specifically, the time for Map and Reduce tasks, which solve subproblems, is relatively small for the three different systems. However, the total time for convergence increases since systems with large number of microgrids need more iterations to converge to the optimal solution. 3.5 Conclusions In this chapter, we propose a new model for the microgrid generation schedule problem with the islanded operation constraints. The proposed problem produces an optimal generation sched- 54

69 Table 3.3: Running time on the cluster 9 Microgrids 14 Microgrids 30 Microgrids One iteration (sec.) MAP (sec.) REDUCE (sec.) Total time (min.) ule with a minimal amount of load curtailment when microgrids have to switch into the islanded operation. To achieve this, we incorporate the l 1 -norm into the objective function of the problem. We apply the ADMM-based decomposition technique to decompose the large-scale centralized optimization problem into multiple sub-problems in which each sub-problem corresponds to the optimization problem in each islanded case and can be solved simultaneously at different computing nodes. Numerical results are conducted to demonstrate the convergence performance of our proposed algorithm. Moreover, the results also show that our model reduces the generation cost in the normal operation and achieves the minimal load curtailment when microgrids switch to the islanded mode. Finally, we describe the detailed implementation of parallel computing for the proposed algorithm using the Hadoop MapReduce software framework to run on a computer cluster. 55

70 Chapter 4 Decentralized Reactive Power Compensation Reactive power compensation is necessary in power systems in order to assure power quality and voltage support [45]. In traditional power systems, reactive power is provided by synchronous generators or shunt capacitor banks installed at specific locations of the distribution networks [46 48]. However, this centralized reactive power compensation can be costly and it can also increase the power loss on transmission and distribution lines [49,50]. In addition, due to the increasing number of inductive residential appliances, such as microwaves, washing machines, air conditioners, and refrigerators, there is a need to explore new reactive power compensation options at distribution level [51 55]. With the introduction of distributed generation (DG), an alternative approach to compensate reactive power is to utilize the power electronics interfaces at DG units, c.f., [56,57]. In [58], the optimal control schemes for reactive power dispatch to achieve the trade-off between distribution loss reduction and voltage variation minimization using distributed photovoltaic generators are proposed. A stochastic optimization model for real and reactive power management under the uncertainty of solar power generation has been proposed in [59] to maximize the sum utility of users while maintaining the voltage at every node at safe levels. In [60], a novel reactive power management strategy under stochastic parameters of the system has been proposed to allow system operators to detect the reactive power vulnerable part of the power grid. The work in [61] develops a convex optimization framework for reactive power compensation. The authors in [62] propose an online reactive power control scheme considering the stochastic nature of reactive demand and renewable generation. Although end-user reactive power compensation via DG units has been advocated as a viable solution to achieve high system efficiency [52, 53, 63, 64], users will not actively participate in generating reactive power unless they have proper financial incentives from electric utility companies or distribution network operators. Recent research has applied game theory to propose incentive 56

71 mechanisms to encourage users to participate in power management systems in the smart grid. For instance, the works in [65] and [66] investigate the demand side management problem as a noncooperative game and propose a smart pricing model to encourage users to participate in energy consumption scheduling program. In [67], the authors formulate the reactive power compensation as a Stackelberg game and derive a pricing scheme to encourage plug-in electric vehicles in generating and consuming reactive power. In this chapter, we consider the problem of controlling reactive power generation from DG units in a radial network, and focus on economic incentives that a utility company needs to provide for users to achieve high system efficiency. Specifically, each user individually controls its DG unit to determine the amount of active and reactive power generation to partially satisfy its own demand. Based on the amount of reactive power compensation, the electric utility company offers reimbursement to users as financial encouragement. The main contribution of this work lies in the fact that we model and analyze the interaction between the electric utility company and users using the Nash bargaining theory [20]. We quantify the benefit for users and electric utility company in collaborative reactive power compensation. The closed-form optimal solutions for reactive and active power generation as well as the amount of reimbursement offered to users are derived under both sequential bargaining and concurrent bargaining. We also investigate the connections of the optimal solutions with the social welfare of the network. The remainder of this chapter is organized as follows. A model of decentralized reactive power compensation is formulated in Section 4.1. We solve the reactive power compensation problem using the Nash bargaining theory under sequential and concurrent bargaining protocols in Section 4.2 and Section 4.3, respectively. The simulation results are provided in Section 4.4. Section 4.5 presents our conclusions of this chapter. 57

72 4.1 System Model In this section, we describe the topology of the power distribution network considered for reactive power compensation and define the payoff functions for users and utility company System Description Without loss of generality, consider a linear distribution network as in Fig The set of nodes is denoted by N = {1,..., N}. The reference node is connected to the substation, denoted by node 0. Let P n and Q n represent active and reactive power flowing down the network from node n to node n + 1. At each node n, the complex power demand is denoted by p d n + jqn, d where p d n and qn d are active and reactive power demand, respectively. The exact values of p d n and qn d depend on the demand condition of each node over time. However, since our focus in this work is on pertime-slot analysis, we assume that the demand of each node remains unchanged during the period of study. The length of each time slot is a design parameter. In general, a shorter time slot may improve the accuracy in predicting demand; however, such improvement may also come at the expense of increasing computational complexity due to the need for solving the optimal reactive power compensation problem more frequently. In this work, the length of each time slot is assumed to be equal to the length of each time slot of power price so that power price does not change during the period of study. However, our analysis can apply to any particular choice for the length of time slots. We further assume that each node n N has a DG unit, e.g., a solar panel or wind turbine, which is capable of generating p g n active and qn g reactive power respectively. The amount of power that a DG unit can generate must satisfy the following constraint: (p g n) 2 + (qn) g 2 s 2 n, (4.1) where s n is the maximum apparent power that the DG unit can support. Note that, our model can be applied to systems that have only a subset of nodes have DG units by setting s n = 0 for any node which is not equipped with a DG unit. For the distribution network illustrated in Fig. 4.1, the power 58

73 0 n-1 n n+1 N P n + jq n P n+1 + jq n+1 demand generation Figure 4.1: A distribution network with local reactive power compensation. flow and voltage for each link between nodes n and n + 1 satisfies the following equations [49] [50] P n+1 = P n r n P 2 n + Q 2 n V 2 n p d n+1 + p g n+1, (4.2) Q n+1 = Q n x n P 2 n + Q 2 n V 2 n qn+1 d + q g n+1, and (4.3) Vn+1 2 = Vn 2 2(r n P n + x n Q n ) + (rn 2 + x 2 n) P n 2 + Q 2 n Vn 2, (4.4) where r n + jx n is the complex impedance of the link between node n and node n + 1, V n is the voltage at node n. Since the quadratic terms in (4.2), (4.3), and (4.4) are relatively small [49] [50], we can approximate (4.2), (4.3), and (4.4) as linear equations as P n+1 = P n p d n+1 + p g n+1, (4.5) Q n+1 = Q n qn+1 d + q g n+1, and (4.6) V n+1 = V n (r n P n + x n Q n )/V 0. (4.7) Since V 0 is constant, we can absorb it into the voltage at each node and define the voltage variation V n between node n and node n + 1 as V n = V n+1 V n = (r n P n + x n Q n ). (4.8) 59

74 Then, the total voltage deviation of the system with respect to the reference bus can be calculated as N 1 n=0 N 1 n = (r n P n + x n Q n ) = = n=0 n=1 n=1 [ ] (p d n p g n)(r r n 1 ) + (qn d qn)(x g x n 1 ) [(p d n p g n)ˆr n + (q d n q g n)ˆx n ], (4.9) where ˆr n = n 1 k=0 r k and ˆx n = n 1 k=0 x k, which are the cumulative resistance and reactance from the substation to node n User s Payoff Modeling Each user has a DG unit that can generate both active and reactive power to satisfy its demand. However, it can decide not to generate reactive power if there is no incentive from electric utility company for reactive power compensation. Next, we quantify the benefit that each user can receive if it decides to participate in reactive power dispatch. We focus on the benefit for each user from reactive power dispatch due to the reduction of payment to the electric utility company. We first calculate the payment of each user n N if it decides not to participate in reactive power compensation. In this case, user n only generates active power from its DG unit to meet its own load demand p d n. Let p g n,max be the maximum available active power of the DG unit n, which depends on solar irradiance and temperature for solar panels or wind speed for wind turbines. Based on the amount of available capacity p g n,max, user n will generate p g0 n its own active power demand. The amount of p g0 n is determined by user n as amount of active power to serve p g0 n min{p d n, p g n,max}. (4.10) Note that, in (4.10), user n can predict its power demand p d n and maximum available active power generation p g n,max at the current period of study. Therefore, p g0 n is a fixed parameter and not a control variable in this problem. Moreover, equation (4.10) means that user n only generates active power to satisfy its demand and does not inject surplus active power back to grid even if there 60

75 is active power available. The cost for user n to buy the remaining active power from the electric utility company is obtained as C 0 n = λ(p d n p g0 n ), (4.11) where λ is the unit price of active power at the current period of study. The unit price of active power may vary during the day. However, we assume that λ will be unchanged during each decision making time slot. Given the price information from the electric utility company, each user n can calculate its payment to the electric utility company since p d n and p g0 n of study. do not change during the period To incentivize reactive power compensation, the electric utility company offers a reimbursement z n to user n for its amount of q g n reactive power dispatch. Therefore, the user n s payment to the electric utility company if participating in reactive power compensation can be calculated as the cost of purchasing remaining active power minus reimbursement: C n = λ(p d n p g n) z n. (4.12) In (4.12), the remaining amount of active power that user n purchases from the electric utility company is (p d n p g n). Also, by generating qn g reactive power, user n may potentially reduce the amount of active power p g n, constrained by (4.1). Moreover, the amount of active power generation p g n cannot be greater than the amount of active power generation in case the user does not participate in reactive power compensation: p g n p g0 n. (4.13) User n controls the amount of active and reactive power generation {p g n, q g n} so that it can reduce payment to the electric utility company. Then we define the user n s payoff as the payment reduction when compensating reactive power for the electric utility company, denoted by V n V n (p g n, q g n, z n ) = C 0 n C n = z n λ(p g0 n p g n). (4.14) From (4.14), we realize that when user n does not participate in reactive power compensation, its payoff is V 0 n = 0. 61

76 4.1.3 Electric Utility Company s Payoff Modeling By offering financial incentive for users to locally generate reactive power, the utility company can reduce the amount of remaining reactive power it has to provide, and thus reduce the cost for reactive power compensation. Let Q nocomp inj = N n=1 qd n and Q inj = N n=1 (qd n q g n) be the total amount of reactive power that node 0 has to inject into the distribution feeder to satisfy all reactive power demand of the overall system, without and with reactive power compensation from users, respectively. We further define f(q) = πq as the cost for the electric utility company to compensate Q units of reactive power at node 0, where π is a constant parameter [62]. Then the saving cost for reactive power compensation can be calculated as f cost f(q nocomp inj ) f(q inj ) = π qn. g (4.15) n=1 Moreover, by locally compensating for reactive power, the total voltage deviation along the network can be reduced. Based on (4.9), we first determine the total voltage deviation of the network in case users do not participate in local reactive power compensation as o n = [ˆr n (p d n p g0 n ) + ˆx n qn]. d (4.16) n=1 n=1 From (4.9) and (4.16), the reduction of voltage deviation by locally compensating for reactive power can be computed as f vol o n n = [ˆr n (p g n p g0 n ) + ˆx n qn]. g (4.17) n=1 n=1 n=1 Then the electric utility company s payoff can be defined as the saving cost for reactive power compensation and the reduction of voltage deviation along the distribution network ( ) U(q g, p g, z) = f cost z n + α f vol n=1 = π qn g z n + α [ˆr n (p g n p g0 n ) + ˆx n qn], g (4.18) n=1 n=1 n=1 where α is a positive weighted parameter to capture the trade-off between saving cost and voltage deviation. 62

77 4.1.4 Network Social Welfare Maximization We define the social welfare as the aggregate payoff of electric utility company and users in the network Ψ(p g, q g, z) = U(q g, z) + V n (p g n, qn, g z n ) n=1 = π qn g λ (p g0 n p g n) + α n=1 n=1 n=1 [ˆr n (p g n p g0 n ) + ˆx n qn] g Ψ(p g, q g ). (4.19) Then, the social welfare maximization problem can be formulated as max Ψ(p g, q g ) (4.20) s.t. {q g n, p g n} X n, n N, where X n is the set of feasible {q g n, p g n} of user n, which is defined as X n {q g n, p g n q g n [0, q d n], p g n [0, p g0 n ], constraint (4.1)}. Given the complete knowledge and centralized control of the network, we can solve the network social welfare maximization problem in (4.20) to obtain the optimal active and reactive power generation. However, this requirement is difficult to fulfill in practice due to the distributed nature of the network topology. Moreover, solving (4.20) is not able to determine the amount of reimbursement for users. Therefore, in the next section, we use the Nash bargaining theory to determine the optimal solutions for power generation and reimbursement in a distributed fashion. 4.2 Sequential Bargaining In this section, we first analyze the NBS of the decentralized reactive power compensation under sequential bargaining protocol for a simple network consisting of one electric utility company and one user. Then we use this result to generalize the solution for multi-user network. 63

78 4.2.1 One-To-One Nash Bargaining Solution In this subsection, we determine the NBS for a simple two-person bargaining, one electric utility company and one user. Let Z n be the sets of feasible z n Z n {z n z n [0, + )}. (4.21) Then the NBS is the solution of the following optimization problem [ max U(q g n, p g n, z n ) U 0] [V n (qn, g p g n, z n ) Vn] 0 (4.22) s.t. {q g n, p g n} X n, z n Z n, where U 0 and V 0 n are the disagreement points of the electric utility company and user n, respectively. From (4.18), we can calculate the disagreement point of the electric utility company, which is the electric utility company s payoff without reactive power compensation U 0 = U(0, 0, 0) = 0. Then we can explicitly express the optimization problem (4.22) as [ max πq g n z n + αˆr n (p g n p g0 n ) + αˆx n qn g ] [ zn λ(p g0 n p g n) ] (4.23) s.t. 0 q g n q d n, 0 p g n p g0 n, (p g n) 2 + (q g n) 2 s 2 n, and z n 0. By solving the optimization problem (4.23), we obtain the NBS for the two-person bargaining as the following theorem. Theorem 4.1. The NBS (q g n, p g n, z n) for the one-to-one bargaining is If (p g0 n ) 2 + (q d n) 2 s 2 n q g n = q d n, (4.24) p g n = p g0 n, and (4.25) z n = 1 2 πqd n + α 2 ˆx nq d n. (4.26) 64

79 If (p g0 n ) 2 + (q d n) 2 > s 2 n q g n (π + αˆx n )s n = min{ (λ + αˆrn ) 2 + (π + αˆx n ), 2 qd n}, (4.27) p g n (λ + αˆr n )s n = min{ (λ + αˆrn ) 2 + (π + αˆx n ), 2 pg0 n }, and (4.28) z n = λ(p g0 n p g n ) [πqg n λ(p g0 n p g n ) + α[ˆr n (p g n p g0 n ) + ˆx n qn g ]]. (4.29) Proof. We can rewrite the optimization problem (4.23) as an equivalent optimization problem by taking ln of the objective function max ln[πq g n z n + αˆr n (p g n p g0 n ) + αˆx n q g n)] + ln [ z n λ(p g0 n p g n) ] (4.30) s.t. 0 q g n q d n, 0 p g n p g0 n, (p g n) 2 + (q g n) 2 s 2 n, and z n 0. The optimization problem (4.30) can be solved by decomposing into the following two steps. First, for fixed q g n, p g n, solve for optimal z n by setting the first derivative of the objective function (4.30) to zero, we obtain z n = 1 2 πqg n + λ 2 (pg0 n p g n) + α 2 [ˆr n(p g n p g0 n ) + ˆx n q g n]. (4.31) By substituting (4.31) into the problem (4.30), we obtain the following subproblem for decision variables {q g n, p g n} max q g n,p g n s.t. 2 ln πqg n λ(p g0 n 0 p g n p g0 n, 0 q g n q d n, and (p g n) 2 + (q g n) 2 s 2 n. p g n) + α[ˆr n (p g n p g0 n ) + ˆx n q g n] 2 (4.32) We now solve (4.32) to find the optimal {q g n, p g n }. Let consider two cases: 65

80 If (p g0 n ) 2 +(q d n) 2 s 2 n (total demand of active power and reactive power is less than generation capacity), the user n can generate both active and reactive power to fully satisfy its own demand. Therefore, the optimal reactive power generation is q g n = q d n and (4.33) p g n = p g0 n. (4.34) Then we can easily obtain the optimal value for reimbursement z n = 1 2 πqd n + α 2 ˆx nq d n. (4.35) If (p g0 n ) 2 + (q d n) 2 > s 2 n, we realize that the objective function (4.32) is an increasing function of p g n and q g n. Therefore, the constraint (p g n) 2 + (q d n) 2 = s 2 n must be hold at the optimality. Then we can express p g n as a function of decision variable qn g as p g n = s 2 n (qn) g 2. (4.36) By substituting (4.36) to the objective function (4.32) and taking the first derivative of the objective function to zero, we can find the optimal solution for q g n and p g n as q g n = (π + αˆx n )s n and (4.37) (λ + αˆrn ) 2 + (π + αˆx n ) 2 p g n = (λ + αˆr n )s n (λ + αˆrn ) 2 + (π + αˆx n ) 2. (4.38) Since the reactive power and active power that user n generated cannot exceed its own demand, therefore we have (π + αˆx n )s n = min{ (λ + αˆrn ) 2 + (π + αˆx n ), 2 qd n} and (4.39) q g n p g n (λ + αˆr n )s n = min{ (λ + αˆrn ) 2 + (π + αˆx n ), 2 pg0 n }. (4.40) Moreover, we rewrite the (4.31) for purpose of analysis as z n = λ(p g0 n p g n ) [πqg n + α[ˆr n (p g n p g0 n ) + ˆx n qn g ] λ(p g0 n p g n )]. (4.41) From the result in Theorem 4.1, we realize that the reimbursement covers the cost incurred by reducing the active power generation λ(p g0 n contributed to the system, i.e., 1 2 [πqg n λ(p g0 n p g n ) and a half of its portion of social welfare p g n ) + α[ˆr n (p g n p g0 n ) + ˆx n qn g ]]. 66

81 4.2.2 Generalized Sequential Bargaining for Multiple Users In this subsection, we find the NBS for a general model of reactive power compensation with multiple users under sequential bargaining protocol. The electric utility company will bargain with each user n N sequentially to determine (q g n, p g n, z n ). Without loss of generality, we assume that the electric utility company will bargain with users in the order of 1, 2,..., N to obtain the NBS. We first assume that at the current bargaining stage, the electric utility company already finished bargaining with prior users 1, 2,..., n 1, and starts bargaining with user n. Then the NBS (q g n, p g n, z n) between the electric utility company and user n is obtained via solving the following optimization problem max [ ] U [n] U[n] 0 [V n (qn, g p g n, z n ) Vn 0 ] (4.42) s.t. {q g n, p g n} X n, z n Z n. Note that in (4.42), we use the subscript [n] to denote the bargaining stage index. Moreover, the disagreement point of the electric utility company at the current bargaining stage is U[n] 0 rather than U 0, which is calculated as the payoff that electric utility company achieved after bargaining with prior users 1, 2,..., n 1. From (4.18), we can determine U[n] 0 as the following equation n 1 U[n] 0 = π i=1 q g i n 1 i=1 n 1 zi + α i=1 [ˆr i (p g i p g0 i ) + ˆx i q g i ]. (4.43) We further calculate the payoff of the electric utility company at the current bargaining stage [n] as from (4.18) n 1 U [n] = π i=1 q g i n 1 i=1 n 1 zi + α i=1 [ˆr i (p g i p g0 i ) + ˆx i q g i ] + πqn g z n + α[ˆr n (p g n p g0 n ) + ˆx n qn]. g (4.44) Therefore, from (4.43) and (4.44) the payoff gain U [n] U[n] 0 that the electric utility company receives if bargaining with user n is U [n] U 0 [n] = πqg n z n + α[ˆr n (p g n p g0 n ) + ˆx n q g n]. (4.45) By substituting (4.45) into (4.42), we obtain [ max πq g n z n + α[ˆr n (p g n p g0 n ) + ˆx n qn] g ] [z n λ(p g0 n p g n) ] (4.46) 67

82 s.t. {q g n, p g n} X n, z n Z n. It is readily to realize that the optimization problem (4.46) at the bargaining stage [n] is identical in case of one-to-one bargaining in (4.23). Therefore, the optimal solution for {q g n, p g n, z n} is similar to Theorem 4.1. Furthermore, we analyze the connection between the bargaining result and social welfare problem as the following theorem. Theorem 4.2. The NBS {q g n, p g n } n=1,2,...,n under the sequential bargaining maximizes the social welfare problem (4.20). Proof. From (4.32), we realize that the NBS in sequential bargaining is the optimal solution of the following optimization problem {q g n, p g n } = arg max (q g n,p g n) X n [πq g n + α[ˆr n (p g n p g0 n ) + ˆx n q g n] λ(p g0 n p g n)], n N. (4.47) We need to show that {q g n, p g n } n=1,2,...,n also maximize the social welfare optimization problem (4.20). First, we decouple the objective function of the social welfare maximization problem (4.20) into Ψ(p g, q g ) = π qn g + α [ˆr n (p g n p g0 n ) + ˆx n qn] g λ = = n=1 n=1 n=1 (p g0 [πqn g + αˆr n (p g n p g0 n ) + αˆx n qn g λ(p g0 n p g n)] n=1 Ψ n (p g n, qn), g n=1 n p g n) (4.47) as where Ψ n (p g n, qn) g = [πqn g + αˆr n (p g n p g0 n ) + αˆx n qn g λ(p g0 n p g n)]. Then we can rewrite {q g n, p g n } = arg max (q g n,p g n) X n Ψ n (p g n, q g n), n N. (4.48) From (4.48), for any {q g n, p g n } {q g n, p g n}, we have 68

83 Ψ n (p g n, qn g ) Ψ(p g, q g ) Ψ n (p g n, qn) g Ψ(p g, q g ). (4.49) n=1 Therefore, the NBS in sequential bargaining maximizes the social welfare. n=1 4.3 Concurrent Bargaining In this section, we find the NBS for the decentralize reactive power compensation under the concurrent bargaining protocol, where the electric utility company bargains with users concurrently. We also analyze the connection between the NBS and the network social welfare problem. The generalized NBS under concurrent bargaining is the solution of the following optimization problem max [ U(q g n, p g n, z n ) U 0] N [ Vn (qn, g p g n, z n ) Vn] 0 n=1 (4.50) s.t. {q g n, p g n} X n, z n Z n. By solving the optimization problem (4.50), we obtain the following result. Theorem 4.3. The NBS under concurrent bargaining {q g n, p g n } n=1,2,...,n also maximizes the social welfare problem (4.20) and is identical to NBS under sequential bargaining. However, the reimbursement for each user is given by [ z n = λ(p g0 n p g n) + 1 π qn g λ N + 1 n=1 (p g0 n n=1 p g n) + α ] [ˆr n (p g n p g0 n ) + ˆx n qn] g. n=1 (4.51) Proof. Since the disagreement point U 0 = 0, V 0 n = 0, n, and by taking ln of the objective function in (4.50), we obtain max s.t. [ ] ln π qn g z n + α [ˆr n (p g n p g0 n ) + ˆx n qn] g + ln [ z n λ(p g0 n p g n) ] (4.52) n=1 n=1 n=1 {q g n, p g n} X n, z n Z n, n. n=1 69

84 We can solve the optimization problem (4.52) using similar method of the proof in Theorem 4.1. Given the fixed q g n, p g n, the optimal solution z n can be obtained by setting the first derivative of the objective function (4.52) with respect to z n to zero 1 π N n=1 qg n N n=1 z n + α N n=1 [ˆr n(p g n p g0 n ) + ˆx n qn] g + 1 z n λ(p g0 n p g n) = 0, n. (4.53) Solving the set of N equations (4.53), we can obtain the expression of z n as [ z n = λ(p g0 n p g n) + 1 π N + 1 qn g λ n=1 (p g0 n n=1 p g n) + α ] [ˆr n (p g n p g0 n ) + ˆx n qn] g. n=1 (4.54) By substituting (4.54) into the objective function of (4.52), we obtain [ ( )] 1 max(n + 1) ln π qn g λ (p g0 n p g N + 1 n) + α[ˆr n (p g n p g0 n ) + ˆx n qn] g. (4.55) n=1 n=1 n=1 From (4.55), the optimal solution {p g n, q g n } n=1,2,...,n maximizes the inner term π N n=1 qg n λ N n=1 (pg0 n p g n) + N n=1 α[ˆr n(p g n p g0 n ) + ˆx n qn], g which is identical to the objective function of the social welfare problem (4.20). Therefore, the optimal solutions are as in Theorem 4.1. And the reimbursement is given in (4.54). Remark: From the above results, we conclude that the NBS {q g n, p g n } n=1,2,...,n in both sequential bargaining and concurrent bargaining will maximize the social welfare problem (4.20). The reimbursement for user n covers the cost incurred by reducing the active power generation λ(p g0 n p g n) and a half of its portion of social benefit contributed to the system, under sequential bargaining. While in concurrent bargaining, the social welfare of the system is equally divided among all users and the electric utility company. Based on Theorems 4.2 and 4.3, the NBS for the decentralized reactive power compensation can be implemented by two steps as follows. First, each user individually determines the amount of active and reactive power generation, which maximizes the social welfare problem. Then, in the second step, depending on the amount of power generation from users and which bargaining 70

85 protocol is selected, the electric utility company determines the amount of reimbursement offered to each user. Due to the distributed topology of power distribution networks as well as lacking coordination among users, the sequential bargaining is more practical to be deployed in realistic applications. For the concurrent bargaining, it can be applied in the scenario in which a group of users located in the same geographical area acts as a single entity and negotiates with the electric utility company in the reactive power compensation problem. 4.4 Simulation Results In this section, we use numerical simulations to demonstrate the effectiveness of decentralized reactive power compensation. We test a distribution network with N = 250 users/nodes. The voltage at the node 0 is V 0 = 7.2kV. The line impedance is ( j0.38)ω/km, and the distances between neighboring nodes are drawn from a uniform distribution from 0.2km to 0.3km. Each node has the active power demand uniformly generated in the range [1kW, 3kW ], and the corresponding reactive power demand is generated in the range of [0kV AR, 1.8kV AR]. The number of users equipped with DG units is selected randomly and accounts for 50% of the total users in all simulations, unless otherwise stated, while the other users do not have DG units to participate in the reactive power compensation. The maximum apparent capacity for all DG units is s n = 2.2kV A. The amount of available active power generation using renewable resources is generated randomly from a uniform distribution with lower and upper limits [0.75s n, s n ]. The active power price is λ = 6.6/kW h [62] and the constant parameter π = 0.25 λ/kv ARh. We set α = 1. In Figs. 4.2 and 4.3, we plot the demand and generation profiles for active power and reactive power, respectively, of 20 randomly selected users from the set of users equipped with DG units. We compare active power generation when users do participate and when users do not participate in reactive power compensation. For users who have the generation capacity greater than demand, they generate as much reactive power and active power as possible to satisfy their power demand. Moreover, some users reduce the amount of active power generation to increase the amount of 71

86 Active power demand Active power generation without compensation Active power generation with compensation kw User ID Figure 4.2: The active power demand and generation profiles. reactive power generation. For instance, users 11 and 17 even decrease the active power generation when they participate in reactive power compensation since at the current period, compensating for reactive power brings higher reimbursement than generating active power. We further compare the reimbursement of 20 randomly selected users under sequential bargaining and concurrent bargaining protocols in Figs. 4.4 and 4.5. Specifically, the reimbursement that each user received covers the cost incurred due to the reduction of active power generation and its net payoff. For instance, users 11, 17 reduce active power generation to reserve capacity for reactive compensation. Therefore, the reimbursements that they received cover these reduction costs. Specifically, under the sequential bargaining protocol, the net payoff that each user received will be determined from a half portion of social welfare that the user contributed to the system, as shown in (4.29). Thus, each user has a different payoff. However, under the concurrent bargaining protocol, the net payoff each user received is equally divided from the total social welfare of the system for all users and the electric utility company, and hence the same for all users. We study the effect of DG unit penetration level on the system reliability and efficiency. Note 72

87 kvar Reactive power demand Reactive power generation with compensation User ID Figure 4.3: The reactive power demand and generation profiles. that, for all simulations so far, we assume that the number of users equipped with DG units accounted for 50% of the total number of users. In Fig. 4.6, we plot the percentage of voltage deviation of the system when the DG unit penetration level varies from 10% to 80%. Furthermore, three types of weather conditions are considered, namely, sunny, partly cloudy, and cloudy. For each of these weather types, p g n,max is generated from a uniform distribution within the ranges [0.75s n, s n ] [0.5s n, 0.75s n ] [0, 0.25s n ], corresponding to sunny, partly cloudy, and cloudy. From the figure, we realize that when the percentage of DG unit penetration level increases, the voltage variation of the system will decrease consequentially. This improvement happens due to local compensate for reactive demand, which renders the voltage variation to decrease. In addition, in Fig. 4.7, we plot the power factor of the system, which is calculated by P F = P 0 / P0 2 + Q2 0, when the DG unit penetration level varies from 10% to 80% for three different types of weather as well. The figure reveals that the power factor increases correspondingly to the increase of the penetration level. Finally, we compare the performance of our proposed method with the centralized reactive power compensation control. The results are shown in Table 4.1. To obtain the centralized control solution, we assume that the electric utility company has the ability to fully control the amount of re- 73

88 Reimbursement( /h) Net payoff each user received Cost incurred by reducing active generation User ID Figure 4.4: The reimbursement of users in sequential bargaining Net payoff each user received Cost incurred by reducing active generation 12 Reimbursement ( /h) User ID Figure 4.5: The reimbursement of users in concurrent bargaining. active and active power generation of all DG units of all users and to solve the optimization problem in (4.20). Since the DG units are controlled by the electric utility company, the amount of surplus active power and reactive power can be injected back to the power grid in centralized control solu- 74

89 Percentage of voltage deviation(%) Sunny Partly Cloudy Cloudy Percentage of DG unit penetration level (%) Figure 4.6: The effect of DG unit penetration level on voltage deviation. Power Factor Sunny Partly Cloudy Cloudy Percentage of DG unit penetration level (%) Figure 4.7: The effect of DG unit penetration level on power factor. tion. We compare the amount of reactive power reduction and the percentage of voltage deviation along the distribution network between the centralized control and our proposed model for the case with 50% users equipped with DG units. From the results in Table 4.1, the centralized solution can help the electric utility company reduce 71.5% amount of reactive power generation. Similarly, a 75

90 Table 4.1: Performance comparison with centralized control Reactive power reduction(%) Voltage deviation(%) Centralized Our approach higher power quality in terms of voltage deviation is shown for the centralized solution, where the total voltage deviation is only 6.8% while it is 8.5% in our proposed decentralized solution. Such performance improvement for centralized control is obtained due to the fact that the surplus reactive power and active power from users equipped with DG units can be utilized to supply demand to neighbor users who do not have DG units. 4.5 Conclusions In this chapter, the decentralized reactive power compensation problem in a distribution network has been studied. Each user independently determines the amount of active and reactive power generation for its DG unit to locally compensate for reactive power. Based on the amount of power dispatch, users will receive reimbursement from the electric utility company. We investigate the economic interaction between users and electric utility company using the Nash bargaining theory. Optimal solutions of power generation and reimbursement are derived under both sequential bargaining and concurrent bargaining protocols. Numerical results in a distribution network with 250 nodes/users are conducted to illustrate the effectiveness of the decentralized reactive power compensation in enhancing system reliability. 76

91 Chapter 5 Distributed Resource Allocation with Minimum Traffic Disruption for Wireless Network Virtualization The rapid growth of traffic demand and application proliferation creates irresistible challenges for traditional wireless networks to ensure the qualify of service (QoS) and quality of experience of subscribers [68,69]. However, due to the inefficient resource utilization and the tightly coupling between hardware and wireless protocols caused by the inherent design, the current wireless networks and Internet can hardly meet such great expectations without fundamentally changing network architectures [70, 71]. Recently, wireless network virtualization has been proposed as one of the key enablers to overcome the ossification of the current Internet by allowing diverse services and applications coexist on the same infrastructure [72 75]. In wireless network virtualization, the traditional Internet service providers are decoupled into infrastructure providers (InPs) who own and manage only infrastructure resources, and service providers (SPs) who lease resources from InPs and concentrate on providing services to subscribers [76]. The physical resources that belong to different InPs are virtualized into a single physical substrate network. Consequently, multiple virtual wireless networks are deployed and operated on top of the single substrate network [77]. As a result, multiple experiments can be performed and tested simultaneously on isolated virtual networks without affecting the operation of the others. Therefore, wireless network virtualization offers great opportunities to shorten the process of evaluating and deploying innovative technologies. Moreover, by sharing the same infrastructure resources, expenses of wireless network expansion and operation can be significantly reduced [78]. Unpredictable wireless network events such as link failures may happen anytime. Any substrate link failure will affect the services of SPs, who have virtual wireless networks that operate on top of that substrate link. In order to guarantee nonstop services even after substrate link failures, resource allocation and routing problems for virtual wireless networks need to take into account the requirement for maintaining quality of services of SPs. This motivates us to consider a resource 77

92 allocation and routing problem for multiple virtual wireless networks to achieve efficient resource utilization for InPs as well as ensure QoS for SPs. Particularly, the objective is to minimize the operation cost of the substrate network. Moreover, we incorporate the preventive traffic disruption model into the resource allocation and routing problem to guarantee the minimal amount of traffic reduction of SPs when substrate links fail. Due to the large number of constraints incorporated into the model, the formulated problem becomes a large-scale optimization problem, and can be intractably solved by the centralized computational framework. Therefore, we apply the ADMM-based decomposition technique to efficiently solve the problem. By jointly tackling the above discussed challenges, our main technical contributions can be summarized as follows: Preventive Traffic Disruption Modeling: We propose a preventive traffic disruption model for virtual networks when a substrate link failure event happens. We also incorporate l 1 -norm into the objective function to ensure the minimal amount of traffic reduction of SPs. Parallel and Distributed Implementation: We propose two algorithms based on the ADMM decomposition technique. The first algorithm provides a parallel computational framework that can be solved concurrently at different computing nodes, and the second algorithm allows SPs and substrate links distributively solve local problems to achieve the global optimal solution. Performance Evaluation: We evaluate the performance of our proposed algorithms using various system parameters. We also demonstrate the efficacy of our preventive model in reducing the amount of traffic reduction. The remainder of this chapter is organized as follows. We explain the network model and assumptions in Section 5.1. The resource allocation and routing problem for virtual wireless networks is formulated in Section 5.2. Section 5.3 describes the preventive traffic disruption model with link failures for virtual networks. We propose two decomposition algorithms using ADMM in Section 5.4. Simulation results are presented in Section 5.5, and Section 5.6 concludes the chapter. 78

93 Virtual Network 1 Virtual Network 2 Virtual Network K... Service providers Substrate Network : Substrate node : Substrate link : Virtual node : Virtual link Figure 5.1: The model of multiple virtual networks operate on top of a single substrate network. 5.1 Network Model and Assumptions Wireless Network Virtualization We consider a wireless network with a set of InPs. Each InP possesses and operates a physical network, also call substrate network. The physical network is composed of physical nodes connected by physical links that form the physical topology. Based on the virtualization frameworks, the physical networks of all InPs are virtualized into a unique physical topology, denoted by a directed graph G s = (N s, L s ), where N s is the set of physical nodes and L s is the set of physical links. Suppose that there is a set K {1, 2,..., K} of SPs request K different virtual networks, which is composed of a set of virtual nodes and virtual links, each established over the same physical network, denoted by a directed graph G k = (N k, L k ). Note that each virtual network is operated by each SP, we use virtual network index and SP index interchangeably. K virtual networks coexist and operate over the same physical network as illustrated in Fig We assume the virtual network mapping result from each G k to G s is already known and focus on resource allocation for virtual 79

94 networks. Depending on the resource request from SPs, InPs will allocate bandwidth capacity of each substrate link l L s to virtual links of SPs. For each substrate link l L s, let w l,k be the bandwidth that substrate link l allocates to virtual link of virtual network k. Then, we have bandwidth allocation vector w l {w l,1, w l,2,..., w l,k }. (5.1) For any virtual network k that does not have virtual link operates on top of the substrate link l, the bandwidth allocation must be equal to zero w l,k = 0, l / L k. (5.2) The total bandwidth allocated to all virtual links must be less than the bandwidth capacity of the physical link, which can be expressed as the following constraint K k=1 w l,k W max l, (5.3) where W l is the maximum capacity of physical link l. Note that the capacity of each physical link will change over time, which depends on the power control and adaptive modulation scheme deployed at the physical layer. However, we consider capacity of physical links as the achievable average capacity, which is assumed to achieve through optimizing the physical layer. Therefore, the maximum capacity of each physical link changes slowly and is treated as a constant during the period of study Routing Model for Virtual Network Each virtual network k, denoted by a directed graph G k = (N k, L k ), has a collection of N k virtual nodes that can send, receive, and relay data across virtual communication links. The network topology with respect to the interactions between virtual nodes and virtual links of virtual network k can be compactly represented by a node-link incidence matrix A k R N k L k. An entry A k [n k ][l k ] 80

95 of the matrix A k associated with node n k N k and link l k L k, is given by [79] 1 if node n k is the start node of link l k, A k [n k ][l k ] = 1 if node n k is the end node of link l k, 0 otherwise. (5.4) We consider a network flow model for routing data to a single destination in each virtual network. The data flows are assumed to be lossless in each virtual link and flow conservation law is assumed to be satisfied at each virtual node in each virtual network. In addition, a virtual source node may need a number of relay nodes to route the data stream to its destination node. We assume each SP uses multi-path routing protocol where the traffic from each virtual source node is split into several flows which follow different multi-hop paths to reach the desired destination. We assume that d k is the destination node of virtual network k. Each source node n k N k (n k d k ) generates data with an average rate of r nk to destination d k. Then the total data rate at the destination node d k is r dk = n k d k r nk. (5.5) For each virtual network k K, we also define a source-sink vector r k R N k as r k {r 1, r 2,..., r nk,..., r Nk }, (5.6) whose the n k -th (n k d k ) entry r nk denotes the amount of data that virtual source node n k injects into the network and destined for virtual destination nodes d k, and the d k -th entry is the total data rate at the destination node d k, determined as in (5.5). On each virtual link l k L k of virtual network k, we let f k,lk 0 be the aggregate flow for destination node d k. The aggregate on each link may come from different virtual source nodes under the multi-path routing model. At each virtual node n k N k, the total flow going into a virtual node is the same as the total flow going out of that virtual node f k,lk f k,lk = r nk, (5.7) l k O(n k ) l k I(n k ) where O(n k ) be the set of outgoing links of node n k, and I(n k ) be the set of incomming links to virtual node n k. The compact expression for the flow conservation law across the whole virtual 81

96 network k can be expressed as A k f k = r k, (5.8) where f k be the flow vector in virtual network k, which can be defined as f k {f k,lk } lk L k. (5.9) The total amount of traffic on each virtual link must be no more than the bandwidth that substrate link allocates to virtual links in virtual network k f k,lk w l,k, l L k. (5.10) 5.2 Joint Resource and routing Optimization for Virtual Wireless Networks Based on the above definitions, a resource and routing optimization problem for multiple virtual networks operate on top of a single substrate network can be formulated. In the considered system, each InP has a cost function for operating substrate link l L s, which is assumed to be a strictly convex function on the total bandwidth, motivated by energy consumption cost [80]. Then the total cost for operating the substrate network is L s l=1 C l(w l ), where C l ( ) denotes the cost function for operating physical link l. Given K virtual networks operate on top of the substrate network and fixed traffic demand from source nodes of each virtual network, the objective is to find an optimal bandwidth allocation for virtual links such that all traffic demand injected from source nodes is delivered to the desired destination in each virtual network with a minimum operation cost of the substrate network. The operation cost minimization problem can be formulated as min L C l (w l ) (5.11) l=1 s.t. A k f k = r k, k K, (5.12) f k,lk w l,k, l k L k, k K, and (5.13) k w l,k W max l, l L s, (5.14) 82

97 variables: {f k } k, {w l } l. The constraints in (5.12) represent the flow conservation low for each virtual network. The constraints in (5.13) ensure the amount of traffic on each virtual link to be less than the bandwidth that substrate link allocates for that virtual link, and the constraints in (5.14) represent the maximum amount of bandwidth of each substrate link. Remark 5.1. The optimization problem in (5.11)-(5.14) involves resource allocation and routing, which may be executed at different time scales. Traditionally, the resource allocation process needs to run at a fast time scale due to the rapid time-variation of wireless channels. The routing process, by contrast, may have a slower time scale due to the low dynamic of data traffic flows. However, the underlying channel capacity in this work is treated as the achievable average capacity, which changes slowly and is assumed to be constant during the period of study. Therefore, the joint resource allocation and routing can be performed on the same time scale as in [81, 82]. The problem in (5.11)-(5.14) is convex and can be solved using the convex optimization techniques to obtain the optimal solution. The optimal solution will fully satisfy for all traffic demands to be delivered from sources to destinations in all virtual networks. However, when there is a substrate link fails, all traffic on virtual links that operate on top of that substrate link will be disrupted. This influences multiple virtual networks that have virtual links mapped on the failure link, and leads to discontinuation on service to end-users. In the next section, we will address how SPs can avoid this service discontinuation when substrate links fail. 5.3 Preventive Traffic Disruption with Link failures The bandwidth allocation model in previous section can satisfy traffic demands for all virtual networks only when all substrate links are fully available, which we will refer as the normal state. However, unpredictable wireless network events such as link failures may occur anytime. Although when a link failure event happens, the network controller can reformulate the problem in (5.11)- (5.14) with new system parameters to reallocate bandwidth for all virtual networks, it will take a 83

98 New routing flow Substrate link failure Figure 5.2: The model of substrate link failure. certain amount of time to wait for network re-convergence. Since different SPs target different types of services and may have stringent reliability and QoS requirements, this performance degradation and severe discontinuation will be intolerable to end-users. Therefore, in this section, we propose a preventive traffic disruption model for virtual networks to provide nonstop reliable services to end-users. Particularly, the system allows virtual networks to continue operation, possibly at an allowable reduced performance level, rather than failing completely, when some part of substrate links fail. Let J be a set of substrate links may possibly be failed. When a substrate link in the substrate network fails, all virtual links that operate on top of that substrate link are no longer available. Consequentially, the physical topology of each virtual network also changes. SPs have to find another routing path to carry data traffic across virtual networks as illustrated in Fig However, SPs may not be able to fully satisfy all data demand as in the normal state. Therefore, SPs may reduce a fraction amount of traffic demand. Let A j k be the node-link incidence matrix of virtual network k when substrate link j fails. Note that we use superscript j to denote all variables and system parameters associated with link j failure event, which we also refer as link failure state j. In this work, we consider a set of J J substrate links can be possibly failed. However, we assume that only one substrate link fails at a time. Multiple-link failure scenarios can be applied without changing the structure of this work 84

99 by constructing all corresponding constraints associated with multiple-link failure scenarios and incorporating into the original problem. The only difference is the node-link incidence matrix will be more sparse, which makes the routing solution has limited routing paths to carry data traffic from the source to the destination, and consequently may lead to the larger amount of traffic disruption. We define r j k be the new traffic demand vector that SP k can support when substrate link j fails. Similar to the normal state, the new traffic demand must satisfy flow conservation law in all virtual networks as A j k f j k = rj k, k K. (5.15) Moreover, the new traffic flow across each virtual network must satisfy virtual link capacity in the normal state, i.e., f j k,l k w l,k, l k L k, k K. (5.16) Note that, in constraints (5.15) and (5.16), only traffic demand and flow across virtual links change. The bandwidth that substrate links allocate to virtual links does not change since when a substrate link fails, SPs still use the existing available resource that has been already allocated to virtual networks in the normal state to continue operation in failure states. Since SPs have to decrease demand to satisfy with the current available resource, we can calculate the amount of traffic reduction at each virtual network k K when substrate link j fails as j k = n k d k (r nk r j n k ). (5.17) We further define the traffic reduction vector of all virtual networks when substrate link j fails as j { j 1,..., j k,..., j K }. (5.18) Since SPs target different services and typically have different requirements for traffic satisfaction when a substrate link fails, each SP has a predefined maximum threshold for data reduction. Let max k be the maximum allowable traffic reduction of SP k. Then we have a constraint for SP k when substrate link j fails j k max k. (5.19) 85

100 We can express the above constraint for all SPs compactly in vector-form as j max. (5.20) The objective is to find the optimal resource allocation that can minimize the cost of operating substrate network in the normal state. Moreover, when any substrate link fails, we also want to guarantee for minimal amount of traffic disruption in virtual networks. The optimization problem for the preventive traffic disruption model can be formulated as min L s l=1 C l (w l ) + τ j J j 0 (5.21) s.t. A k f k = r k, k, (5.22) f k,lk w l,k, l k L k, k K, (5.23) k w l,k W max l, l L s, (5.24) A j k f j k = rj k, k K, j J, (5.25) f j k,l k w l,k, l k L k, k K, j J, (5.26) j k = n k d k (r nk r j n k ), k K, j J, and (5.27) j k max k, k K, j J, (5.28) where τ is a positive parameter to capture the trade-off between cost minimization and minimal traffic disruption. The second term in (5.21), j 0, is l 0 -norm of vector j, which determines the number of nonzero entries in j j def 0 = #{k : j k 0}. (5.29) Incorporating l 0 -norm j 0 into the objective function allows us to obtain the optimal solution that leads to a minimal number of SPs have to reduce traffic in link failure states. Constraints (5.22), (5.23), and (5.24) are for the normal state, while constraints (5.25), (5.26), (5.27), and (5.28) are for all link failure states. Due to the non-convexity of j 0, exactly solving the problem in (5.21) is computationally difficult. To avoid computational burdens, we use l 1 -norm, which is the convex approximation to 86

101 l 0 -norm j 1 def = k j k. (5.30) The problem in (5.21)-(5.27) can be reformulated as min L s l=1 C l (w l ) + τ j J j 1 (5.31) s.t. (5.22), (5.23), (5.24), (5.25), (5.26), (5.27), (5.28). The problem in (5.31) is convex and can be solved using several convex optimization techniques. However, directly solving (5.31) can be intractable due to the large-scale nature of the original problem. Remark 5.2. By incorporating the preventive traffic disruption constraint into the resource allocation and routing problem in (5.31), the optimal solution provides a proactive mechanism to cope with physical network failures, i.e., reserves the backup resource before any failure happens. Specifically, InPs will allocate a certain amount of redundant bandwidth for virtual links so that when physical links fail, SPs still can use the redundant bandwidth from the other virtual links to flow the data traffic. 5.4 Decomposition Algorithms using ADMM In this section, we propose two algorithms to solve the centralized problem in (5.31) using the ADMM-based decomposition technique. The first algorithm provides a parallel computational framework, and the second one can be implemented in a distributed fashion at each SP and each substrate link Parallel Algorithm using ADMM In this subsection, we use the ADMM decomposition method to propose a parallel algorithm. This algorithm decomposes the original problem in (5.31) into a master problem corresponding 87

102 to the normal state, and J problems corresponding to J link failure states, which can be solved concurrently at different computing facilities. The problem in (5.31) contains a large number of constraints. However, we realize that almost all the constraints are separable into the normal state and different link failure states. Particularly, constraints (5.22), (5.23), and (5.24) are associated with the normal state only, while constraints (5.25), (5.26), (5.27), and (5.28) can be separated into each link failure state, except for variables {w l,k } l, k belong to the normal state. In order to make constraints (5.26) to be separable from the normal state, we define auxiliary variables w l = w j l, l L s, j J, (5.32) where each auxiliary variable w j l can be interpreted as the local copies of w l in the normal state at each link j failure state. We now can rewrite constraint (5.26) with respect to only local variables at each link failure state as f j k,l k w j l,k, l k L k, k. (5.33) The link failure state constraints (5.25), (5.27), (5.28), and (5.33) are now decoupled from the normal state. To facilitate for presentation, we further define the feasible set for the normal state, F 0, and each link failure state, F j as F 0 = {({f k } k, {w l } l ) (5.22), (5.23), (5.24)}, F j = {({f j k, rj k } k, {w j l } l) (5.25), (5.27), (5.28), (5.33)}, j J. Then the problem in (5.31) can be rewritten as min L s l=1 C l (w l ) + τ j J s.t. ({f k } k, {w l } l ) F 0, j 1 (5.34) ({f j k, rj k } k, {w j l } l) F j, j J, and w l = w j l, l L s, j J. (5.35) The set of equality constraints in (5.35) represents the consensus constraints, i.e., it enforces the local copies of bandwidth allocation variables to be agreement with the corresponding variables in 88

103 the normal state. The augmented Lagrangian function of problem in (5.34) with respect to consensus constraints in (5.35) is given by L s J K J L s L 1 = C l (w l ) + τ j k + (λ j l )T (w j l w l ) + ρ 2 l=1 j=1 k=1 j=1 l=1 [ L s J L s J = C l (w l ) (λ j l )T w l + τ l=1 j=1 l=1 j=1 k=1 l=1 J L s w j l w l 2 j=1 l=1 ] K L s j k + (λ j l )T w j l + ρ L s w j 2 l w l 2, l=1 (5.36) where {λ j l } l, j is the Lagrangian multiplier, and ρ > 0 is a penalty parameter. Defining the primal variables x = ({f k } k, {w l } l ), which is the decision variable vector in the normal state, and z j = ({f j k, rj k } k, {w j l } l) is the decision variable vector in link j failure state. Then the ADMM decomposition technique can be used to solve the problem in (5.34) in an iterative procedure. Specifically, at the t-th iteration, the primal variables and dual variables are updated sequentially as x[t + 1] = arg min L 1 (x, z[t], λ[t]), (5.37) z j [t + 1] = arg min L 1 (x[t + 1], z j, λ j [t]), j, and (5.38) ) λ j l [t + 1] = λj l (w [t] + ρ j l [t + 1] w l [t + 1], l, j. (5.39) Based on the Lagrangian function (5.36), we decompose the problem in (5.34) into the following J + 1 optimization problems. The first sub-problem is associated with variables in the normal state only and corresponding to the primal variables update in (5.37) min L s C l (w l ) l=1 j=1 l=1 J L s (λ j l )T w l + ρ 2 J L s w j l w l 2 (5.40) j=1 l=1 s.t. ({f k } k, {w l } l ) F 0. By fixing the values of {w j l, λj l }, and then solving the problem in (5.40), we obtain the optimal solution for ({f k } k, {w l } l ). The remaining J sub-problems are associated with variables in link failure states and corresponding to primal variables update in (5.38). For each link j J failure state, we decompose into 89

104 the following problem min τ K L s j k + (λ j l )T w j l + ρ L s w j 2 l w l 2 (5.41) k=1 l=1 l=1 s.t. ({f j k, rj k } k, {w j l } l) F j. By fixing (w l, λ j l ) l and then solving the problem in (5.41), we obtain the optimal solution for ({f j k, rj k } k, {w j l } l). Algorithm Implementation: The whole procedure for parallel algorithm using ADMM is described in Algorithm 5.1. First, a master computing node solves the optimization problem in (5.40) to obtain the optimal solution ({f k } k, {w l } l ). Then it will broadcast the bandwidth allocation solution of the whole network {w l } l to all J distributed computing nodes. Each distributed computing node solves the optimization problem in (5.41) to obtain the optimal solution ({f j k, rj k } k, {w j l } l). Finally, based on the local value of {w j l } l and {w l } l received from the master node, the dual variables can be updated as in line 16 in Algorithm 5.1. Note that, J optimization problems associated with link failure states are decoupled and can be solved in a parallel fashion at different computing nodes without affecting the others. This parallel implementation reduces the computation time for the proposed Algorithm 5.1. The amount of information exchange between the master node and distributed computing nodes is depicted in Fig The master node broadcasts the bandwidth allocation solution in the normal state, which is same to all distributed nodes. Each distributed node needs to send the local information including dual variable {λ j l } l and {w j l } l to the master node. This can be done by using Message Passing Interface (MPI), which is widely used for high-performance computing paradigm [83]. Note that in Algorithm 5.1, all SPs and InPs need to exchange information with the master computing node to perform computation in each sub-problem. This can be done via certain entities that are proposed to operate the wireless network virtualization such as mobile virtual network operators (MVNOs), who lease the network resources from InPs and create virtual resources based on the requests from SPs [72]. 90

105 Algorithm 5.1 Parallel Algorithm based on ADMM Decomposition 1: Initialize: t = 1, {λ j l } l, j = 0 2: repeat 3: At master computing node: 4: repeat 5: wait 6: until receive updates w j l, λj l from all J distributed computing nodes 7: step 1: solve (5.40) for optimal solution ({f k } k, {w l } l ) 8: step 2: broadcast {w l } l to all J distributed computing nodes 9: step 3: t t : 11: At each distributed computing node: 12: repeat 13: wait 14: until receive the update {w l } l from the master 15: step 1: solve (5.41) for optimal solution ({f j k, rj k } k, {w j l } l) 16: step 2: update dual variables: λ j l [t + 1] = λj l [t] + ρ (w j l [t + 1] w l [t + 1] 17: step 3: send (λ j l, wj l ) l to the master 18: until a stopping criterion is met ), l Distributed Algorithm using ADMM Even Algorithm 5.1 decomposes the original problem in (5.31) into J + 1 sub-problems and provides a parallel computational framework, it requires to have a central controller to collect all information of the whole network to solve each sub-problem. However, in practice, it is hard to be fulfilled due to enormous amount of signaling. Therefore, in this subsection, we propose a fully decentralized algorithm for solving the problem in (5.31), in which each SP and each substrate link independently solve their own problems to obtain the optimal solution, while requires a limited amount of information exchange between SPs and substrate links. The objective function in (5.31) involves operation cost of substrate links and amount of traffic reduction of SPs. However, we can separate into the individual cost for each substrate link as well as the amount of traffic reduction of each SP. Besides, constraints (5.22), (5.23), (5.25), (5.26), (5.27), and (5.28) are separable into each SP. The only constraints in (5.24) are coupled between substrate links and SPs. In order to make the constraints in (5.24) are decoupled between individual substrate link and SPs, we consider w l,k as the local variables at SP k and define auxiliary variables as w l = w s l, l L s. (5.42) 91

106 Master computer Link normal state Find (, ) Broadcast,,, Computer 1 Computer 2 Computer J Link failure state (j=1) Find (,, ) Update Send, to master Link failure state (j=2) Find (,, ) Update Send, to master Link failure state (j=j) Find (,, ) Update Send, to master Figure 5.3: The illustration of information exchange between the normal state sub-problem and link failure state sub-problems., {,,,,,, },..,,,,, =,, =,, =, : Substrate link : Virtual link Figure 5.4: The structure of auxiliary variables define for each substrate link and corresponding virtual links operate on top of the substrate link. Each w l,k has a local copy wl,k s at the substrate link, as illustrated in Fig We can interpreter w l,k as the bandwidth that virtual link requests to satisfy its service, while wl,k s as the true bandwidth that the substrate link can allocate to virtual link. Then, by adding consensus constraints, the requested bandwidth and allocated bandwidth reach an agreement. The problem in (5.31) can be rewritten as min L s l=1 C l (w s l ) + τ k K j k (5.43) j J s.t. A k f k = r k, k, (5.44) 92

107 f k,l w l,k, l L k, k, (5.45) A j k f j k = rj k, j, k (5.46) f j k,l w l,k, l L k, j, k, (5.47) j k = n k d k (r nk r j n k ), j, k, (5.48) j k max k, k, j, (5.49) w s l = w l, l L s, and (5.50) K k=1 w s l,k W max l, l L s. (5.51) The cost function in (5.43) is substituted by local variables of each substrate link. The equality constraints in (5.50) represent the consensus constraints. Constraints (5.51) are maximum bandwidth capacity constraints with local variables at each substrate link. We further define feasible set for each SP as F k = {(f k, {f j k, rj k } j, {w l,k } l ) (5.44) (5.49)}, k K. Then, the problem in (5.43)-(5.51) can be compactly expressed as min L s l=1 C l (w s l ) + τ k K j k (5.52) j J s.t. (f k, {f j k, rj k } j, {w l,k } l ) F k, k, w s l = w l, l L s, and K k=1 w s l,k W max l, l L s. The Lagrangian function of the problem in (5.52) with respect to consensus constraints is given by L s K L 2 = C l (w s l ) + τ l=1 k=1 j=1 l=1 L s { = Cl (w s l ) + (µ l ) T w s } K l + τ J L s j k + (µ l ) T (w s l w l ) + γ L s w s l 2 w l 2 l=1 k=1 j=1 l=1 l=1 J L s j k (µ l ) T w l + γ L s w s l 2 w l 2, (5.53) l=1 93

108 where {µ l } l is the Lagrangian multiplier, and γ > 0 is a penalty parameter. Defining primal variable u = (f k {f j k, rj k } j, {w l,k } l ) k, which is the decision variable vector for SPs, and v = {w l } l, which is the decision variable vector for substrate links. Then the ADMM decomposition technique can be applied to solve the problem in (5.52) in an iterative procedure. Specifically, at the t-th iteration, the primal variables and dual variables are updated sequentially as u[t + 1] = arg min L 2 (u, v[t], µ[t]), (5.54) v[t + 1] = arg min L 2 (u[t + 1], v, µ[t]), and (5.55) µ l [t + 1] = µ l [t] + γ (w s l [t + 1] w l [t + 1]), l. (5.56) From the Lagrangian function in (5.53), we decompose the problem in (5.52) into the service provider-level problem and substrate link-level problem. The service provider-level problem is associated with primal variable update in (5.54) and can be expressed as min s.t. K τ J L s j k (µ l ) T w l + γ L s w s l 2 w l 2 (5.57) k=1 j=1 l=1 l=1 (f k, {f j k, rj k } j, {w l,k } l ) F k, k. The substrate link-level problem is associated with primal variable update in (5.55) and can be expressed as min s.t. L s l=1 K k=1 { C l (w s l ) + (µ l ) T w s l + γ 2 ws l w l 2} (5.58) wl,k s W l max, l. After solving (5.57) and (5.58), the dual variable is update as in (5.56). Moreover, we realize that the problem in (5.57) is completely separable into each SP and can be solved by each individual SP. Each SP k K solves it own problem as follow min τ J L s j k µ l,k w l,k + γ L s (wl,k s 2 w l,k) 2 (5.59) j=1 l=1 l=1 94

109 s.t. (f k, {f j k, rj k } j, {w l,k } l ) F k. Similarly, the problem in (5.58) can be decomposed to solve at each substrate link as min C l (w s l ) + (µ l ) T w s l + γ 2 ws l w l 2 (5.60) s.t. K k=1 w s l,k W max l. Algorithm Implementation: We propose a distributed algorithm to solve the problem in (5.31) as in Algorithm 5.2. The decomposition structure of the solution process is clearly visible: SPs and substrate links perform optimization independently. Particularly, SPs solve optimization problem in (5.59) simultaneously and send the updated value of w l,k to the corresponding substrate link. Each substrate link will solve its local optimization problem in (5.60) after receiving all information from SPs, and then updates dual variable µ l. The new values of {w s l, µ l } will be sent to SPs. This process is repeated until convergence. Note that, each substrate link sends exchange information to only SPs who have virtual links operate on top of it. The distributed nature of Algorithm 5.2 in which substrate links and SPs are completely decoupled facilitates for the capability of adapting the dynamic behavior of wireless links. When physical link capacity varies, each substrate link updates the constraint in sub-problem (5.60) and performs the calculation independently. SPs continue executing the computation in (5.59) without any modification. 5.5 Simulation Results In this section, we use the computational experiment to evaluate the performance of our proposed algorithms. We generate a random substrate network topology comprising 20 physical nodes and 70 physical links. The bandwidth capacity of substrate links are generated randomly with a uniform distribution from [50, 100] Mb/s. We also deploy 10 virtual networks on top of the substrate network, each virtual network has the random number of nodes from [5, 10], and the random number of links from [10, 15]. We select one node as a destination for each virtual network, and source nodes inject data with an average rate randomly generated from [10, 15] Mb/s. We assume 95

110 Algorithm 5.2 Distributed Algorithm based on ADMM Decomposition 1: initialize: t = 1, µ l = 0, l 2: repeat 3: At each virtual network: 4: repeat 5: wait 6: until receive updates w s l,k, µ l,k from all substrate links 7: 1) solve (5.59) 8: 2) broadcast w l,k to substrate link 9: 10: At each substrate link: 11: repeat 12: wait 13: until receive the update w l,k from all SPs 14: 1) solve (5.60) for optimal solution w s l 15: 2) update dual variables: µ l [t + 1] = µ l [t] + γ (w s l [t + 1] w l [t + 1]) 16: 3) send µ l, w s l to SPs 17: 18: t t : until a stopping criterion is met the convex operating cost function of each substrate link is C l (w l ) = a l ( k w l,k) 2, where a l is generated randomly for each substrate link from a uniform distribution a l [0.002, 0.004] $/Mb 2. The link failure set J is selected randomly from 10% of the total substrate links and τ = 0.3 in all simulations, unless otherwise stated. Virtual networks are allowed to reduce half of the traffic data when substrate links fail. All tests are conducted on a Windows 7 64-bit personal computer with Intel i GHz CPU and 16GB of RAM. Each sub-problem in our proposed algorithms is solved using CVX [29] Convergence and Computational Performance We show the convergence behavior of our proposed algorithms in Fig Since both algorithms are implemented and performed on a single computer, the delay time due to local information exchange can be ignored. We plot the relative error of the objective function versus the number of iterations without applying any specific termination condition for the proposed algorithms. For parallel Algorithm 5.1, the relative error approaches to 10 3 in about 15 iterations, while distributed Algorithm 5.2 needs about 20 iterations to yield the same relative error. The faster convergence behavior of Algorithm 5.1 is due to the smaller number of sub-problems compared to those of dis- 96

111 1 0.1 Algorithm 5.1 Algorithm 5.2 Relative error E-3 1E-4 1E Iteration index Figure 5.5: The convergence performance of the proposed algorithms. tributed Algorithm 5.2, and consequently reach to an agreement at a faster speed. Specifically, in this simulation, Algorithm 5.1 produces a master problem and 7 parallel problems corresponding to 7 link failure states, while for Algorithm 5.2, the number of decomposed problems is 80 (10 problems for SPs and 70 problems for substrate links). A larger set of link failure J leads to a greater number of constraints in the optimization problem, and generally increases the computation time of two algorithms. Therefore, we investigate the computation time of the proposed algorithms by varying the percentage of link failure. For each percentage of link failure, we select the set of substrate links can possibly failed randomly and run 100 realizations to average out the result. In Fig. 5.6, we plot the computation time of two algorithms to reach the relative error < Since Algorithm 5.1 is proposed to be implemented in a parallel fashion, the total computation time of each iteration is determined as the total computation time in the normal state sub-problem and the maximum amount of computation time among J different link failure sub-problems. For Algorithm 5.2, the computation time of each iteration is the maximum amount of computational time among SPs and the maximum computation time among substrate links. We ignore the delay time due to information exchange. From the result in Fig. 5.6, 97

112 Algorithm 5.1 Algorithm 5.2 Computational Time (min.) Percentage of Link Failure (%) Figure 5.6: Computation time of two proposed algorithms versus percentage of link failure. a higher percentage of link failure increases the computational time of both algorithms, especially in Algorithm 5.1. This due to the fact that a larger percentage of link failure does not increase the number of consensus constraints in Algorithm 5.2; while in Algorithm 5.1, the number of consensus variables increases significantly. Therefore, much more iterations are required to reach the global solution. We further plot the computation time for different layers of decomposition structure in two algorithms versus the percentage of link failure. In Fig. 5.7, we show the average computation time of the master node, who solves the normal state problem, and average computation time of distributed computing nodes, who solve the link failure state problems in parallel fashion. As it can be seen in Fig. 5.7, the computation time of the master node increases as the percentage of link failure increases, while the computation time at each distributed computing node is not affected. This can be explained by the decomposition structure in Algorithm 5.1, when the number of link failure states increase, the number of consensus constraints in (5.35) increase accordingly. This leads to a higher complexity of the normal state problem in (5.40), and a larger number of sub-problems in (5.41). However, J sub-problems in (5.41) are performed in parallel fashion by distributed comput- 98

113 Average Computational Time (min.) Normal state subproblem Failure state subproblem Percentage of Link Failure (%) Figure 5.7: Average computation time of the normal state sub-problem and link failure state subproblem in Algorithm 5.1. ing nodes, the complexity does not grow. A similar behavior can be observed for Algorithm 5.2 in Fig. 5.8, we can see that the computation time at each substrate link is not affected by the percentage of link failure, while the computation time for each SP increases accordingly. This can be explained by the independence of the substrate link-level problem in (5.60) to link failure states. Particularly, the decomposition structure in Algorithm 5.2 produces the optimization problem for each substrate link in (5.60) whose complexity does not depend on the percentage of link failure. In contrast, the optimization problem that each SP performs in (5.59) grows the complexity as the percentage of link failure increases. Note that, this property can facilitate for the practical implementation of Algorithm 5.2 since each substrate link normally has less computation capability, while SPs with better computation capability can perform higher complexity computing tasks. 99

114 Average Computational Time (min.) Substrate Link Subproblem Service Provider Subproblem Percentage of Link Failure (%) Figure 5.8: Average computation time of each substrate link and each service provider in Algorithm 5.2. Percentage of Traffic Reduction (%) Without preventive traffic disruption With preventive traffic disruption Service Provider Figure 5.9: The comparison of percentage of traffic reduction at individual service provider when substrate links fail with and without incorporating preventive traffic disruption model. 100

115 Table 5.1: Operation cost and bandwidth utilization comparison Problem in (5.11) Problem in (5.31) Bandwidth (Mb/s) Cost ($/s) System Performance We evaluate the efficacy of our model in guaranteeing the minimal amount of traffic reduction when substrate links fail by plotting the percentage of traffic reduction of individual SP in Fig We compare the result with and without incorporating preventive traffic disruption model into the resource allocation problem. We realize that incorporating preventive traffic disruption model can significantly reduce the amount of traffic reduction. Moreover, the result also indicates that l 1 -norm in the objective function ensures a sparse number of SPs have to reduce traffic in link failure states. Particularly, only SPs 3, 4, 5, 7, and 8 have to reduce traffic demand when substrate links fail. Furthermore, we examine the effect of preventive traffic disruption model to the operation cost of the substrate network. Note that the problem in (5.11)-(5.14) and the problem in (5.31) can both fully satisfy for traffic demand of all virtual networks in the normal state, i.e., all substrate links are fully operational. However, the problem in (5.31) takes into account the constraints that ensure the minimal amount of traffic disruption of SPs when substrate links fail. This enforces the substrate network to allocate a certain amount of surplus bandwidth to some virtual links in each virtual network so that when substrate links fail, SPs can find another routing path to deliver data from source to destination. Table 5.1 lists the operation cost of the substrate network, the total bandwidth capacity allocated to virtual networks for the optimization problem in (5.11)-(5.14) and the problem in (5.31). It can be easily inferred from Table 5.1 that the optimal solution of the problem in (5.11)- (5.14) utilizes less bandwidth capacity, and consequently the lower operation cost of the substrate network is achieved. We next evaluate the effect of parameter τ on the operation cost of the substrate network as well as the amount of traffic disruption of SPs. In general, higher values of τ increase the weight for l 1 -norm term in the objective function, and the resultant optimal solution will ensure smaller 101

116 Traffic Reduction (%) Cost ($/s) Figure 5.10: The effect of τ on the operation cost of the substrate network and the percentage of traffic disruption of SPs. amount of traffic disruption of SPs when substrate links fail as illustrating in Fig However, it also leads to higher operation costs for the substrate network. Based on the result from the figure, the network operator can select appropriate values of τ to satisfy the design criteria for the whole system. The selection will obtain the trade-off between the operation cost for InPs and quality of service for SPs. Finally, we study the effect of percentage of link failure on the operation cost of the substrate network. Although incorporating a higher percentage of link failure will provide a better preventive level for traffic reduction, it may incur in general a higher operation cost in the normal state. In Fig. 5.11, we show the amount of surplus bandwidth that the substrate network allocates to virtual networks when we increase the percentage of link failure. We can see from Fig that the larger amount of surplus bandwidth the substrate network has to allocate to virtual networks to ensure the minimal amount of traffic disruption when we incorporate higher percentage of link failure. However, the operation cost of the substrate network increases accordingly. 102

117 Surplus Bandwidth(%) Cost increase (%) Percentage of Link Failure (%) Figure 5.11: The effect of percentage of link failure incorporated into the model on the total bandwidth allocated to virtual networks and the operation cost of the substrate network. 5.6 Conclusions In this chapter, a resource allocation and routing for wireless network virtualization has been studied. We propose a new formulation for preventive traffic disruption for SPs in wireless network virtualization. The proposed model minimizes the operation cost of the substrate network in the normal state while still guarantees for the minimal amount of traffic reduction when substrate links fail. Due to the large-scale nature of the formulated model, directly solving the optimization problem can be intractable. We then apply the ADMM decomposition technique to propose two algorithms, namely parallel algorithm and distributed algorithm. The parallel algorithm decomposes the centralized problem into multiple sub-problems that can be solved concurrently at different computing nodes, while the distributed algorithm allows each SP and substrate link distributively solve the local problem to converge to the global optimal solution. Numerical results are conducted to demonstrate the convergence behavior as well as computational efficiency of the proposed algorithms. 103

118 Chapter 6 Conclusions and Future Works 6.1 Conclusions In this dissertation, we have investigated the application of big data optimization methods for distributed resource management problem to improve the reliability and security of the overall system. First, an incentive mechanism design for integrated microgrids in the peak ramp minimization problem has been proposed. We investigate the economic interaction between the DSO and microgrids using the Nash bargaining theory, which achieves the maximum social welfare of the overall system. Due to the privacy concerns of microgrids, we propose two distributed algorithms to achieve the NBS, in which each microgrid individually schedules its energy generation and storage resources. The first algorithm requires a synchronous update from all microgrids, i.e., the new iteration starts only after all microgrids finish their computations. On the other hand, the second algorithm can perform in the asynchronous framework, i.e., the DSO starts new iteration when any microgrid in the system finishes its computation. Second, we propose a new model for the microgrid generation schedule problem with the islanded operation constraints. The proposed problem produces an optimal generation schedule with a minimal amount of load curtailment when microgrids have to switch into the islanded operation. To achieve this, we incorporate the l 1 -norm into the objective function of the problem. We apply the ADMM-based decomposition technique to decompose the large-scale centralized optimization problem into multiple sub-problems in which each sub-problem corresponds to the optimization problem in each islanded case and can be solved simultaneously at different computing nodes. Then, we describe the detailed implementation of parallel computing for the proposed algorithm using the Hadoop MapReduce software framework to run on a computer cluster. Third, the decentralized reactive power compensation problem in a distribution network has been studied. Each user independently determines the amount of active and reactive power gen- 104

119 eration for its DG unit to locally compensate for reactive power. Based on the amount of power dispatch, users will receive reimbursement from the electric utility company. We investigate the economic interaction between users and electric utility company using the Nash bargaining theory. Optimal solutions of power generation and reimbursement are derived under both sequential bargaining and concurrent bargaining protocols. Finally, we propose a new formulation for preventive traffic disruption for SPs in wireless network virtualization. The proposed model minimizes the operation cost of the substrate network in the normal state while still guarantees for the minimal amount of traffic reduction when substrate links fail. Due to the large-scale nature of the formulated model, directly solving the optimization problem can be intractable. We then apply the ADMM decomposition technique to propose two algorithms, namely parallel algorithm and distributed algorithm. The parallel algorithm decomposes the centralized problem into multiple sub-problems that can be solved concurrently at different computing nodes, while the distributed algorithm allows each SP and substrate link distributively solve the local problem to converge to the global optimal solution. Numerical results are conducted to demonstrate the convergence behavior as well as computational efficiency of the proposed algorithms. 6.2 Future Works Transactive Energy Traditional power system is not designed to incorporate power generation and storage at the distribution level. It is also not designed to allow the distributed energy sources to supply the power to the customers directly. Interconnecting and integrating distributed energy sources to power grid, therefore, is a challenging task due to the involvement of significant and critical technical issues associated. Recent reports and studies have discussed the significant transformations occurring in the electric power system [84,85]. These transformations include growth in the use of renewable energy resources in the bulk power system, proliferation of DERs of various capacities in both transmission 105

120 and distribution systems, an increasing number of installations of local renewable resources at enduse points, and load growth through electrification of transportation and other end-uses. The future scenario of power system requires coordinated management of large numbers of distributed and demand response resources, intermittent resources, while maintaining high degrees of grid reliability and improving operational economics. This will involve information exchange between many entities, systems and devices for scheduling and monitoring. New methods are needed for real-time and end-to-end management of such complex system such as incentives, retail markets, and similar mechanisms. The growing utilization of variable resources will require an integrated and coordinated operational paradigm for dynamic balancing of supply and demand across the entire electrical infrastructure to provide safe and reliable electric service [86]. Recently, Transactive Energy has been proposed as a potential solution for the aforementioned challenges the power grid is facing today, which is defined by the GridWise Architecture Council s Framework as follows: A set of economic and control mechanisms that allows the dynamic balance of supply and demand across the entire electrical infrastructure using value as a key operational parameter [87]. The first key part in the definition is using value as a key operational parameter, which defines what makes the approach transactive : operational decisions are made through an exchange of value-based information captured in transactions between participants. The other, across the entire electrical infrastructure, declares this approach feasible across the entire electricity system, from the transmission level with its bulk generation and transport of electricity down to the distribution system and the variety of connected customers, as illustrated in Fig. 6.1 [88]. The transactive control approach offers distinct advantages in integrating flexible devices in the electricity operations. Many participants such as smart homes, buildings, and industrial sites engage in automated market trade with others at the distribution system level and with representation of the bulk system. Communications are based on prices and energy quantities in a two-way negotiation. Analogous to the price reaction approach, the operation of the flexible devices is optimized economically by a local intelligent controller under the control of the end user [89]. This controller 106

121 Figure 6.1: Transactive agents and interactions. receives price information and takes the device state and user preferences into account to operate local demand and supply resources. Under transactive conntrol approach, the local controller communicates the available flexibility combined with their preferences and conditions to an electronic market place through a market transaction. Consuming devices communicate their willingness to pay, while producing devices communicate the price for which they are willing to produce. Since all resources participating in the market communicate their intended reaction to a range of price levels, the pool reaction to a range of price signals is known up front and the market mechanism can determine the price for an appropriate balance of supply and demand. From the end user s or energy consumer s point of view, the local energy management system agent acts on behalf of the user or consumer to bid into the market and reacts to the resulting market price signals. Unlike the centralized optimization approach, no direct outside control is involved here. However, from a system perspective, the participants engage in coordinated control actions. With this approach, demand response moves from influencing, with an uncertain overall response, into market-based control with a collaboratively derived dynamic price as a control signal to trigger a predictable system reaction. When properly implemented, the market bids sent by the end users energy management systems can be aggregated together. Hence, the resulting bid represents the preferences of the two participants together. The message size of the aggregated bid curve is a simple combination of the 107