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Linear Power Discretization and Nonlinear Formulations for Optimizing Hydropower in a Pumped-Storage System

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Linear Power Discretization and Nonlinear Formulations for Optimizing Hydropower in a Pumped-Storage System Linear Power Discretization and Nonlinear Formulations For Optimizing Hydropower in a Pumped-Storage System Craig S. Moore Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science in Civil and Environmental Engineering G.V. Loganathan, Chair David Kibler Tamim Younos November 2, 2000 Blacksburg, Virginia Keywords: electric system, hydropower, optimal scheduling, optimization, pumped storage, nonlinear programming, linear programming Copyright 2000, Craig S. Moore Linear Power Discretization and Nonlinear Formulations For Optimizing Hydropower in a Pumped Storage System Craig S. Moore (Abstract) Operation of a pumped storage system is dictated by the time dependent price of electricity and capacity limitations of the generating plants. This thesis considers the optimization of the Smith Mountain Lake-Leesville Pumped Storage-Hydroelectric facility. The constraints include the upper and lower reservoir capacities, downstream channel capacity and flood stage, in-stream flow needs, efficiency and capacity of the generating and pumping units, storage-release relationships, and permissible fluctuation of the upper reservoir water surface elevation to provide a recreational environment for the lake shore property owners. Two formulations are presented: (1) a nonlinear mixed integer program and (2) a discretized linear mixed integer program. These formulations optimize the operating procedure to generate maximum revenue from the facility. Both formulations are general and are applicable to any pumped storage system. The nonlinear program retains the physical aspects of the system as they are but suffers from non-convexity related issues. The linear formulation uses a discretization scheme to approximate the nonlinear efficiency, pump, turbine, spillway discharge, tailrace elevation-discharge, and storage- elevation relationships. Also, there are binary unit dispatch and either/or constraints accommodating spill and gated release. Both formulations are applied to a simplified scheme of the Smith Mountain Lake and Leesville pumped storage system. The simplified scheme uses a reduced number of generating and pumping units at the upper reservoir to accommodate the software limitations. Various sensitivity analyses were performed to test the formulations. The linear formulation consistently performs better than the nonlinear. The nonlinear solution requires a good starting point for optimization. It is most useful as a verification tool for the solution from the linear program on all occasions. The formulations yield the best schedules for generating and pumping. A coarse time interval limits the use of all pumps in the presence of the spill constraint. A sufficiently large difference in the diurnal unit price encourages short-term pump back as opposed to a weekly cycle. The Leesville (downstream) reservoir affects the power production schedule with its large (approx. 9 ft) forebay rise for every foot drop at the Smith Mountain Lake. The linear formulation provides a valuable tool for studying the system under a wide range of conditions without having to worry about the computational difficulties associated with the nonlinear formulation. ii Acknowledgement I would like to thank Dr. G.V. Loganathan, for his time, guidance, advice, and patience. He has provided many hours of support. Not only has he provided a foundation for my current research, but he his given me great insights for a long life of learning. I would like to thank and Dr. David Kibler and Dr. Tamim Younos for serving as members of my committee. Also, I would like to thank Dr. Kibler and Dr. Younos for the comments and direction that they provided. I would like to thank the employees of AEP at the Smith Mountain Dam, the Leesville Dam, the Claytor Dam, and the Roanoke office for sharing their time and knowledge. I would like to specifically thank the late Mr. Randy Agnew (AEP - Power Dispatching Supervisor), Mr. Jim Thrasher (AEP - Hydro Generation), and Ms. Teresa Rogers (AEP - Hydro Generation) for sharing their knowledge and providing the needed data. I would like to thank Jake and Catherine Moore, my parents, for their support and encouragement to excel in all that I do. They have provided a great environment to learn and grow. I can never express my complete appreciate of all their efforts in my life. The faculty and students of the Hydrosystems and GIS/CAD divisions have encouraged and supported me throughout my time at Virginia Tech. I would like to thank them for their assistance. I would like to thank Linette Dudley, James Moore, Dale Moore, Carl Moore and Donna Brown, my siblings, for encouraging me with all aspects of my life. I greatly appreciate their love and support. I would like to thank Paula Moore, my wife, for her support through this time of work and research. She has been very patient with me. She has helped to keep me going in difficult times. Without her support, I would not have been able to finish this thesis. iii Table of Contents (Abstract)...................................................................................................................................... ii Acknowledgement ..................................................................................................................... iii Table of Contents....................................................................................................................... iv List of Figures ............................................................................................................................ vi List of Tables............................................................................................................................. vii Chapter 1 Introduction......................................................................................................... 1 1.1 Problem Overview...................................................................................................................................... 1 1.2 Project History............................................................................................................................................ 3 1.3 Smith Mountain Lake................................................................................................................................. 4 1.4 Leesville Lake ............................................................................................................................................ 5 1.5 Problem Description................................................................................................................................... 6 1.6 Objectives................................................................................................................................................... 7 Chapter 2 Electrical System ................................................................................................. 8 2.1 Introduction ................................................................................................................................................ 8 2.2 Power Generation..................................................................................................................................... 10 2.3 Transmission and Distribution ................................................................................................................. 13 2.4 System Control......................................................................................................................................... 16 Chapter 3 Formulation of Pumped Storage Optimization Problem .............................. 21 3.1 Introduction .............................................................................................................................................. 21 3.2 List of Notations....................................................................................................................................... 24 3.3 Nonlinear Mathematical Formulation ...................................................................................................... 28 3.4 Solution Methodology.............................................................................................................................. 38 Chapter 4 Mixed-integer Linear Formulation ................................................................. 39 4.1 Introduction .............................................................................................................................................. 39 4.2 List of Notation ........................................................................................................................................ 40 4.3 Linear Mathematical Formulation............................................................................................................ 43 4.4 Implementation of the Linear Formulation............................................................................................... 57 Chapter 5 Results and Discussion...................................................................................... 62 5.1 Overview .................................................................................................................................................. 62 5.2 Nonlinear Formulation Implementation ................................................................................................... 62 5.3 Linear Formulation Implementation........................................................................................................
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