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Stochastic dynamic programming optimization model for operations planning of a multireservoir hydroelectric… Ayad, Amr 2018

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  STOCHASTIC DYNAMIC PROGRAMMING OPTIMIZATION MODEL FOR OPERATIONS PLANNING OF A MULTIRESERVOIR HYDROELECTRIC SYSTEM   by Amr Ayad M.Sc., Alexandria University, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) January 2018      © Amr Ayad, 2018ii  Abstract This thesis presents a Stochastic Dynamic Programming (SDP) modeling algorithm to model six hydropower plants in British Columbia (BC), Canada. The main output of the algorithm is the water value function for the two biggest reservoirs in BC, Williston and Kinbasket reservoirs. The AMPL programming language was used to implement the algorithm. Extensive testing has shown that the program is able to solve the problem producing acceptable water value and marginal value functions up to a problem size of ~ 164 million states per time step using the computing resources available on one of the BC Hydro’s servers. The objective of the work presented here was to assess the sensitivity of solution efficiency and precision for several storage state and decision space discretizations. The impact of introducing a storage state-space corridor, as an alternative of the traditional fixed storage state-space, was investigated. In addition, the sensitivity of the modeling results to different spill penalty values was analyzed. It was found that finer state-space increments give more precise results but the granularity was limited to the computing resources available. Introducing the storage state-space corridor provided several advantages; nevertheless, care should be taken in the design of such corridors so that the solution efficiency and accuracy are not jeopardized. Also, recommendations on the use of suitable spill penalty value are provided. Flexibility is one important feature of the modeling algorithm. This flexibility is a result of optimizing the algorithm and the organization of the code, which provided control over the increment of the state-spaces and the storage corridor, the ability to run the problem for one storage reservoir while fixing the state of the other storage reservoir and the ability of the user to iii  run the model either directly on a personal computer/server using the command prompt or by using a scheduling program to optimize the use and sharing of computing assets. Further enhancements of the algorithm will enable the model developed in this thesis to handle much larger problems but will likely still suffer from the limitations due to the inherent curse of dimensionality in modeling using the SDP algorithm.  iv  Lay Summary The researcher has developed a computer program that uses advanced mathematical modeling technique, called Stochastic Dynamic Programming, to produce price signals representing the value of water stored in British Columbia’s biggest reservoirs such as Williston Reservoir in the Peace region in British Columbia (BC) and Kinbasket Reservoir in the Columbia region in BC. These price signals are intended to inform the operators of the generating stations downstream of these reservoirs of the optimal way to dispatch the generation, within a certain time window, through comparing the value of the energy produced to the value of energy in the wholesale electricity markets connected to BC. To ensure that this computer program is working properly, the researcher has tested the program using several case studies with different input parameters. The results of the tests have shown that the program gives acceptable results within certain limits.   v  Preface The work presented in this research is part of BC Hydro’s Water Value Project in collaboration with a research team from the University of British Columbia and with financial support from BC Hydro and the Natural Sciences and Engineering Research Council (NSERC) in Canada for the research supervisor, Dr. Ziad Shawwash of the University of British Columbia.  This thesis is based on the development and testing of a model applied to the BC Hydro system. The author, Amr Ayad, is the lead investigator responsible for model design, development, and analysis of the results and composition of the manuscript in Chapter 1 through Chapter 6.  A version of Chapter 3 is a fully detailed documentation of the modeling developed as part of this research. This documentation is used internally at BC Hydro (Ayad, et al., 2012).  Part of this documentation could be also found in (Abdalla, et al., 2013). A version of Chapter 5 was published in the conference proceedings of HydroVision International 2013 (Ayad, et al., 2013).  Dr. Shawwash provided guidance throughout the building of the model, gathering the data and analyzing the results. Dr. Shawwash also has contributed to editing and review of all the chapters included in this manuscript. Many others at BC Hydro and University of British Columbia have provided guidance, and the author hereby acknowledges their support and contributions. Much of the computations and analysis was performed using the hardware and software resources at BC Hydro.   vi  Table of Contents Abstract ......................................................................................................................................................... ii Lay Summary ............................................................................................................................................... iv Preface .......................................................................................................................................................... v Table of Contents ......................................................................................................................................... vi List of Tables ................................................................................................................................................ x List of Figures .............................................................................................................................................. xi List of Symbols .......................................................................................................................................... xiv List of Abbreviations ................................................................................................................................ xvii Glossary ..................................................................................................................................................... xix Acknowledgments ....................................................................................................................................... xx Dedication .................................................................................................................................................. xxi Chapter 1: Introduction ............................................................................................................................. 1 1.1. Research Goals .......................................................................................................................... 1 1.2. Challenges and Gaps ................................................................................................................. 1 1.3. Contributions of the Research ................................................................................................... 2 1.4. Implementation ......................................................................................................................... 3 1.5. Organization of the Thesis ........................................................................................................ 3 Chapter 2: Survey of Literature ................................................................................................................ 5 2.1. Introduction ............................................................................................................................... 5 2.2. Modeling of Reservoir Operation and Management ................................................................. 5 2.3. Bellman’s Principle of Optimality ............................................................................................ 7 2.4. Principle of Progressive Optimality .......................................................................................... 7 2.5. Advantages and Challenges of Dynamic Programming (DP) Technique ................................. 8 2.6. Deterministic Dynamic Programming Techniques ................................................................. 10 2.6.1. Incremental Dynamic Programming Models (IDP) ........................................................ 10 2.6.2. Differential Dynamic Programming Models .................................................................. 11 2.6.3. Dual Dynamic Programming Models (DDP) .................................................................. 12 2.7. Stochastic Dynamic Programming Techniques ...................................................................... 13 2.7.1. Conventional Stochastic Dynamic Programming (SDP) ................................................ 13 vii  2.7.2. Stochastic Dynamic Programming with Function Approximation ................................. 16 2.7.3. Dynamic Programming with Successive Approximation (DPSA) ................................. 17 2.7.4. Aggregation and Decomposition SDP ............................................................................ 17 2.7.5. Chance-constrained Programming Model and the Linear Decision Rules ..................... 18 2.7.6. Stochastic Dual Dynamic Programming (SDDP) ........................................................... 19 2.7.7. Sampling Stochastic Dynamic Programming (SSDP) .................................................... 21 Chapter 3: Development of a Stochastic Dynamic Programming Optimization Model for Operations Planning of a Multireservoir Hydroelectric System................................................................................ 24 3.1. Background on the BC Hydro System .................................................................................... 24 3.1.1. BC Hydro’s System ........................................................................................................ 24 3.1.2. Columbia River Treaty .................................................................................................... 25 3.1.3. Representation of the BC Hydro System in the Modeling Algorithm ............................ 26 3.2. Approach and Context ............................................................................................................ 27 3.3. Objectives of the Model Development ................................................................................... 28 3.4. Modeling of the Problem ........................................................................................................ 28 3.4.1. State Variables ................................................................................................................ 28 3.4.2. Decision Variables .......................................................................................................... 29 3.4.3. Stochastic Variables ........................................................................................................ 30 3.4.4. Space Discretization and Transitions .............................................................................. 31 3.4.4.1. Discretization of State-space and Decision-space ...................................................... 31 3.4.4.2. Transition Matrix and State Transitions...................................................................... 32 3.4.5. Storage-Generation Curves (HK Curves) ....................................................................... 33 3.4.6. Storage-Forebay Curves .................................................................................................. 33 3.4.7. Representation of Unit Outage ........................................................................................ 34 3.4.8. Load-Resource Balance .................................................................................................. 35 3.4.9. Representation of Prices .................................................................................................. 37 3.4.10. State-space Discretization and Generation of Discretized Values .................................. 38 viii  3.4.11. Approximations ............................................................................................................... 38 3.5. Problem Formulation and Solution Algorithm ....................................................................... 39 3.5.1. Objective Function and Calculation of the Value Function ............................................ 39 3.5.2. Main Constraints ............................................................................................................. 42 3.5.3. Solution Algorithm ......................................................................................................... 44 3.6. Capabilities ............................................................................................................................. 47 3.7. Limitations .............................................................................................................................. 48 3.8. Issues Experienced in Model Development and Code Run .................................................... 48 Chapter 4: Results ................................................................................................................................... 50 4.1. Introduction ............................................................................................................................. 50 4.2. Sample Results for Two Cases of State-space Discretization ................................................. 50 4.3. Results of Intoducing the Storage State-space Corridor ......................................................... 62 Chapter 5: Assessing the Impact of Storage State and Decision Space Discretization on Solution Efficiency and Precision of a Stochastic Dynamic Programming Algorithm in a Multireservoir Operations Planning Model .................................................................................................................... 67 5.1. Introduction ............................................................................................................................. 67 5.2. Assessing the Impact of Introducing a Storage State-space Corridor ..................................... 67 5.3. Assessing the Impact of State-space Discretization ................................................................ 71 5.4. Sensitivity of Results to Spill Penalty Values ......................................................................... 73 Chapter 6: Conclusions, Recommendations and Future Work ............................................................... 75 6.1. Conclusions ............................................................................................................................. 75 6.2. Proposed Future Enhancements to the Model Developed ...................................................... 76 6.3. Future Work and Recommendations ....................................................................................... 77 Bibliography ............................................................................................................................................... 79 Appendices .................................................................................................................................................. 85 Appendix A: Running the SDPOM6RM Model ..................................................................................... 85 A.1. Code and Computation Details ....................................................................................................... 85 A.1.1. Discretizer Model ..................................................................................................................... 85 A.1.1.1. Code .................................................................................................................................. 85 A.1.1.2. Computation Details .......................................................................................................... 90 A.1.2. SDP Model ............................................................................................................................... 90 ix  A.1.2.1. Code .................................................................................................................................. 90 A.1.2.2. Computation Details ........................................................................................................ 103 A.1.3. Value Iteration Model ............................................................................................................ 107 A.1.3.1. Code ................................................................................................................................ 107 A.1.4. Other Modules ........................................................................................................................ 108 A.1.4.1. Code ................................................................................................................................ 108 1. HK Model ......................................................................................................................... 108 2. HK ROTR Model .............................................................................................................. 108 3. HK Rough Model .............................................................................................................. 109 4. Inflow ROTR Model ......................................................................................................... 110 5. Maximum Generation Limits Model ................................................................................ 111 6. Maximum Turbine Limits Model...................................................................................... 112 7. FB Model .......................................................................................................................... 112 8. Prices Model ..................................................................................................................... 113 A.1.4.2. Computation Details ........................................................................................................ 114 A.2. Main Sets and Parameters of the Model ....................................................................................... 116 A.3. How to Run the Model .................................................................................................................. 130    x  List of Tables Table 1: Conversion of Binary Outages to a Generation Factor ................................................................. 35 Table 2: Storage State Discretization Cases ............................................................................................... 71 Table 3: Different Cases for the Penalty Function ...................................................................................... 73 Table 4: Calculations Details in the Discretizer Model .............................................................................. 90 Table 5: Calculations Details in the SDP Model ...................................................................................... 103 Table 6: List of the Other Modules and Their Functions .......................................................................... 114 Table 7: Main Sets and Parameters of the Model in an Alphabetical Order ............................................. 116 Table 8: The Main Parameters Used in the SDP Model in an Alphabetical Order ................................... 117    xi  List of Figures Figure 1: SDP Procedure (Nadalal & Bogardi, 2007)................................................................................. 16 Figure 2:  A Map of British Columbia Illustrating the Main Power Generation Plants and Local and Interconnected Transmission Lines, (BC Transmission Corporation, 2010) .............................................. 24 Figure 3: Probability Distribution for the Inflows to the Williston Reservoir for the Month of June ........ 31 Figure 4: GMS and MCA Storage-Forebay Curves .................................................................................... 34 Figure 5: Elements in the Load-Resource Balance Equation ...................................................................... 36 Figure 6: Prices for a Forecast Water Year for Different Total System Inflow Scenarios ......................... 37 Figure 7: Backward Recursion Value iteration Procedure .......................................................................... 41 Figure 8: Flowchart of the SDPOM6R Model ............................................................................................ 46 Figure 9: Three Dimensional Water Value for the Williston Reservoir and the Kinbasket Reservoir for the Month of December- CASE A .................................................................................................................... 51 Figure 10: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of October- CASE A ....................................................................................................................................... 52 Figure 11: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of January- CASE A ........................................................................................................................................ 52 Figure 12: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of May- CASE A ...................................................................................................................................................... 53 Figure 13: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of August- CASE A......................................................................................................................................... 53 Figure 14: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of October- CASE A ....................................................................................................................................... 54 Figure 15: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of January- CASE A ........................................................................................................................................ 54 Figure 16: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of May- CASE A ...................................................................................................................................................... 55 Figure 17: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of August- CASE A......................................................................................................................................... 55 Figure 18: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of October- CASE B ....................................................................................................................................... 56 Figure 19: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of January- CASE B ........................................................................................................................................ 56 xii  Figure 20: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of May- CASE B ....................................................................................................................................................... 57 Figure 21: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of August- CASE B ......................................................................................................................................... 57 Figure 22: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of October- CASE B ....................................................................................................................................... 58 Figure 23: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of January- CASE B ........................................................................................................................................ 58 Figure 24: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of May- CASE B ....................................................................................................................................................... 59 Figure 25: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of August- CASE B ......................................................................................................................................... 59 Figure 26: Value of Water in Storage of the Williston Reservoir and the Kinbasket Reservoir along the Water Year (October to September)- CASE A ........................................................................................... 60 Figure 27: Value of Water in Storage of the Williston Reservoir and the Kinbasket Reservoir along the Water Year (October to September)- CASE B ........................................................................................... 61 Figure 28: Percentage of Difference between the Marginal Value of Water in CASE A and CASE B for the Williston Reservoir and the Kinbasket Reservoir ................................................................................. 61 Figure 29: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of October- FIXED Case ................................................................................................................................. 63 Figure 30: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of January- FIXED Case ................................................................................................................................. 63 Figure 31: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of May- FIXED Case ................................................................................................................................................ 64 Figure 32: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of August- FIXED Case .................................................................................................................................. 64 Figure 33: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of October- FIXED Case ................................................................................................................................. 65 Figure 34: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of January- FIXED Case ................................................................................................................................. 65 Figure 35: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of May- FIXED Case ................................................................................................................................................ 66 xiii  Figure 36: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of August- FIXED Case .................................................................................................................................. 66 Figure 37: Simulated historical storage bands for the Williston Reservoir ................................................ 68 Figure 38: Simulated historical storage bands for the Kinbasket Reservoir ............................................... 69 Figure 39: Run Times for FIXED Case vs. CORRIDOR Case .................................................................. 70 Figure 40: Problem Size vs. Run Time for Different Cases ....................................................................... 72 Figure 41: Schematic of the Main Steps Needed to Run the SDPOM6RM Model .................................. 130    xiv  List of Symbols 𝑇𝑆: Starting time step in months, 𝑇𝐸: Ending time step in months, t: Time step, where t ∈ 𝑇𝑆, … , 𝑇𝐸 in months, j: Plant number; since there are two plants in the model, j of GM Shrum =1 while j for Mica=2, QP_Max j, t: Maximum physical plant j release limit for each time step t in m3/sec, QP_Min j, t: Minimum physical plant j release limit for each time step t in m3/sec, Qturb j, t: Turbine release in m3/sec, Qturb_Max j, t: Maximum physical limit on turbine release in m3/sec, Qmin_Max j, t: Maximum physical limit on turbine release in  m3/sec, Qspill j,t: Spill outflow in m3/sec, 𝑎𝑗,𝑡: Action (Decision) which represents the releases for each time step t from each plant j in m3/sec,  W_upper j,t: Absolute maximum storage limit for plant j for time step t in m3/sec-day which is deduced from the dynamic storage trajectory,  W_lower j,t: Absolute minimum storage limit for plant j for time step t in m3/sec-day which is deduced from the dynamic storage trajectory,  𝑆𝑗,𝑡: Discretized starting storage value for the plant j in the current time step t in m3/sec-day, 𝑊𝑗,𝑡+1 , 𝑆𝑗,𝑡: Discretized terminal storage value for the plant j in the next time step t+1 for the starting storage 𝑆𝑗,𝑡 in m3/sec-day,  𝑖𝑗,𝑡: Natural inflow for each plant in each time step t in m3/sec, 𝐿𝑡: Domestic Load for each time step t in MWh, xv  𝐻𝐾𝑗,𝑡,𝑆𝑗,𝑡,𝑊𝑗,𝑡+1 , 𝑆𝑗,𝑡: Coefficient governing the generation storage relationship,  𝐺𝑗,𝑡,𝑆𝑗,𝑡,𝑖𝑗,𝑡,𝑊𝑗,𝑡+1 , 𝑆𝑗,𝑡,𝑎𝑗,𝑡 : Energy generation from plant j in time t for the pair of storage states 𝑆𝑗,𝑡 𝑎𝑛𝑑 𝑊𝑗,𝑡+1 , 𝑆𝑗,𝑡 and inflow value i𝑗,𝑡 and decision 𝑎𝑗,𝑡 in MWh, 𝐺_𝑚𝑖𝑛𝑗,𝑡 : 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑙𝑖𝑚𝑖𝑡 𝑖𝑛 𝑀𝑊, 𝐺_𝑚𝑎𝑥𝑗,𝑡 : 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑙𝑖𝑚𝑖𝑡 𝑖𝑛 𝑀𝑊, G_PCNt: Energy generation from Peace Canyon plant in MWh, G_REVt: Energy generation from Revelstoke plant in MWh, G_ARDt: Energy generation from Arrow Lakes plant in MWh, G_IPP_Thermt: Energy generation from the independent power producers , thermal plants and other sources in MWh, Contr_Exp t: Forward energy sales for each time step t in MWh, Contr_Imp t: Forward energy purchases for each time step t in MWh, 𝑆𝑝𝑜𝑡_𝐵𝑢𝑦𝑡,𝑆1,𝑡,𝑆2,𝑡,𝑖1,𝑡,𝑖2,𝑡,𝑊1,𝑡+1 ,𝑆1,𝑡 ,𝑊2,𝑡+1 ,𝑆2,𝑡 ,𝑎1,𝑡,𝑎2,𝑡 : Amount of energy that ought to be bought from the spot market in MWh,  𝑆𝑝𝑜𝑡_𝑆𝑒𝑙𝑙𝑡,𝑆1,𝑡,𝑆2,𝑡,𝑖1,𝑡,𝑖2,𝑡,𝑊1,𝑡+1 ,𝑆1,𝑡 ,𝑊2,𝑡+1 ,𝑆2,𝑡 ,𝑎1,𝑡,𝑎2,𝑡 : Amount of energy that could be sold to the spot market in MWh,  Trans_Exp_Limit t: Transmission lines limits on exporting in MW, Trans_Imp_Limit t: Transmission lines limits on importing in MW, Inr: Interest rate, : Discount rate, 𝐸𝑥𝑝_𝑃𝑟𝑖𝑐𝑒t,i1,𝑡,i2,𝑡: Export price according to the incoming inflows to the plants in $/MWh, 𝐼𝑚𝑝_𝑃𝑟𝑖𝑐𝑒t,i1,𝑡,i2,𝑡: Import price according to the incoming inflows to the plants in $/MWh, xvi  𝐶𝑜𝑛𝑡_𝑃𝑟𝑖𝑐𝑒𝑡,𝑖1,𝑡,𝑖2,𝑡: Contract price according to the incoming inflows to the plants in $/MWh, 𝐸𝑋_𝐼𝑚𝑝_𝐶𝑜𝑠𝑡𝑡,𝑆1,𝑡,𝑆2,𝑡,𝑎1,𝑡,𝑎2,𝑡 : The expected cost for importing from the spot market in million $, 𝐸𝑋_𝐸𝑥𝑝_𝑅𝑒𝑣:𝑡,𝑆1,𝑡,𝑆2,𝑡,𝑎1,𝑡,𝑎2,𝑡 : The expected revenue from exporting to the spot market in million $, 𝐶𝑜𝑛𝑡_𝑅𝑒𝑣_𝑅𝑒𝑣𝑡,𝑆1,𝑡,𝑆2,𝑡,𝑎1,𝑡,𝑎2,𝑡 : The revenue/ cost of the forward long-term contracts in million $, 𝑃𝑜𝑙𝑖𝑐𝑦_𝐼𝑛𝑐𝑜𝑚𝑒𝑡,𝑆1,𝑡,𝑆2,𝑡,𝑎1,𝑡,𝑎2,𝑡 : The expected revenue/ cost of operation in million $, 𝑇_𝑃𝑡,𝑆1,𝑡,𝑆2,𝑡,𝑖1,𝑡,𝑖2,𝑡,𝑊1,𝑡+1 ,𝑆1,𝑡 ,𝑊2,𝑡+1 ,𝑆2,𝑡 ,𝑎1,𝑡,𝑎2,𝑡 : The state transition probability, 𝑃𝑉𝑡,𝑆1,𝑡,𝑆2,𝑡 : Value of water in storage in million $,  𝑀𝑉𝑊𝑡,𝑆1,𝑡,𝑆2,𝑡 : Marginal value of energy in $/ MWh, 𝐹𝐵_𝑖𝑗,𝑡,𝑆𝑗,𝑡: The forebay corresponding to certain starting storage state in m, 𝐹𝐵_𝑓𝑗,𝑡,𝑆𝑗,𝑡,𝑖𝑗,𝑡,𝑊𝑗,𝑡+1 , 𝑆𝑗,𝑡: The forebay corresponding to certain terminal storage state in m,   xvii  List of Abbreviations AMPL A Mathematical Programming Language ARW Arrow Lakes Reservoir BC British Columbia BC Hydro The British Columbia Hydro and Power Authority CCP Chance-Constrained Programming cms-d/cms-day Cubic meters per second days CPU Central Processing Unit CRO/ Flocal Commercial Resource Optimization/ Flow calculation Software CRT The Columbia River Treaty CSUDP Colorado State University Dynamic Programming Software DDP Dual Dynamic Programming DP Dynamic Programming DPSA Dynamic Programing with Successive Approximation HK Energy Conversion Factor GWh Giga Watt Hour IDP Incremental Dynamic Programming IPP Independent Power Producers KBT Kinbasket Reservoir LDR Linear Decision Rules MCM Marginal Cost Model LP Linear Programming xviii  MW Mega Watt PCN Dinosaur Reservoir REV Revelstoke Reservoir RLROM Reinforcement Learning Reservoir Optimization Model ROTR Run-of-the-river plant SDPOM6R Stochastic Dynamic Programming Optimization Model for 6 Reservoirs SDP Stochastic Dynamic Programming SDDP Stochastic Dual Dynamic Programming SLP Stochastic Linear Programming SSDP Sampling Stochastic Dynamic Programming STC Site-C UBC University of British Columbia WSR Williston Reservoir      xix  Glossary Capacity Maximum sustainable power that could be produced at any instant, usually measured by MW Energy Electricity produced/consumed over a period of time, usually measured by GWh Forebay The water level in a given water reservoir as measured at the generating station Freshet The period during which the snowpack melts and the resulting inflows feed into watersheds and reservoirs. In British Columbia and Northwest United States, this period is typically from late April to July Hydroelectric  A system that generates electricity from free-falling water through turbines and generators Multipurpose Usually associated with water reservoirs that serves more than one purpose such as generating hydroelectricity and agriculture…etc. Multireservoir A group of water reservoirs Planning Horizon The length of time in the future the model is run for Plant/ Project A group of units (generators and turbines) connected to a man-made reservoir for the purpose of generating electricity Power Electricity produced/consumed at any instant in time, usually measured by MW Run-of-the-river plant A plant that has a reservoir with  little to no storage capability Shoulder Months Months between seasons where neither domestic load nor prices are high; such as September and October Stage  A unit of time over which an optimization process is undertaken in a DP or SDP model State A unit in a given space, for example storage space within a reservoir   xx  Acknowledgments I sincerely thank my supervisor Dr. Ziad Shawwash for his support, guidance and invaluable advice throughout my studies. He shared with me his experience and knowledge in both academia and in life. I am also grateful to Dr. Alaa Abdalla for his support during the time he was the manager of Reliability and Planning Group at BC Hydro. I wish to express my thanks to all the graduate students who were involved in the BC Hydro research program at the Department of Civil Engineering, UBC. The casual and formal discussions with them have provided me with invaluable insights in research and in life. I would like to express my deep gratitude to Dr. Dave Bonser, my team lead at BC Hydro, for his advice, continuous support and most importantly, for listening to me and giving me the time to express my thoughts in both good times and bad times. I also thank my colleagues at work, Tim Blair and Andrew Keats, for their encouragement and support. Special thanks go to my previous manager, Kelvin Ketchum, and current manager at BC Hydro, Darren Sherbot, for supporting me and accommodating my needs during the ups and downs in my research and in my life. Last but not least, I would like to thank my wife, Hend, from the bottom of my heart for her continuous support, encouragement and dedication. She has sacrificed her health and her career for me to succeed. Her sacrifices are unquantifiable and invaluable. Thank you my lovely wife.   xxi  Dedication      To my late mother, who supported me and prayed for me all the time up until the moment she left this world while I was thousands of kilometres away from her…    1  Chapter 1: Introduction 1.1. Research Goals The main goal of this research is to develop an algorithm to solve the medium-term/long-term stochastic optimization problem using a practically acceptable representation of the inherent stochasticity and uncertainty in the modeled reservoir system. The algorithm is meant to be used as a potential decision support tool for operations planning of large-scale multireservoir systems such as the BC Hydro system. The second goal for the research is to provide a benchmarking tool for other more sophisticated models developed by the UBC/ BC Hydro research team. A further goal is to test the limits of the Stochastic Dynamic Programming technique that is used to develop the algorithm using the computer resources and programing capabilities available at the time this research was conducted. The driver is to provide guidance for future implementation of algorithms based on the same technique. The development of the modeling in this research was done in consideration of some of the challenges and the gaps outlined in the next section. 1.2. Challenges and Gaps  There are several shortcomings to the current models that were surveyed in the literature and the ones developed in-house and currently in use at BC Hydro. Some of the gaps identified are: 1. It is hard to reasonably represent the inherent stochasticity and uncertainty in reservoir systems without extensive computation cost; 2  2. Some of the best models that are currently used still need some manual guidance and/or several trial and error simulations in order to achieve the best possible outputs, which might jeopardize the final product by introducing human and other inherent errors; 3. Due to the curse of dimensionality and/or other modeling shortcomings, many of the models currently in-use cannot cover the desirable planning horizon or the actual state-space especially for long/medium-term planning purposes without jeopardizing the accuracy or the proper representation of  the system modeled; 4. Several models and techniques seem very promising and have good potential, such as heuristic techniques, but unfortunately they have not been tried on large systems which typically entail more challenges; and 5. There is a need to develop more accurate estimates of the value of water in storage reservoirs for use in long-term capacity expansion planning studies and to improve the system operation in operations planning. It is not claimed that the model developed in this research is able to cover all of the gaps mentioned above, but rather it is thought to add to the pool of knowledge of the UBC/BC Hydro team and reasonably represent the complexity of the systems modeled within the expected limiting factors of availability of computing resource and shortcomings of the technique and programming language used. The next section lists the contributions of this research. 1.3. Contributions of the Research The following contributions are thought to be achieved by the current research: 1. Representing the stochasticity and uncertainty in the system in an acceptable form; 3  2. Concurrent modeling of six of the main generating facilities in the BC Hydro system on the Peace and Columbia Rivers which results in good representation of the system; 3. Providing practically acceptable representation of the water value functions that reflect the value of water in storage; 4. Preforming proper and extensive testing of the limits of the Stochastic Dynamic Programming technique and the AMPL programming language; 5. Extending the planning horizon up to 36 months with a monthly time step which is not possible for some of the models used currently that have comparable problem size; and 6. Developing generic and flexible code that could be easily enhanced, extended and used for different purposes including benchmarking and sensitivity analysis. 1.4. Implementation An implementation of the Stochastic Dynamic Programming technique is used to develop the core SDPOM6R model. AMPL programming language is used to develop the model. The details of the approach and its implementation can be found in Chapter 3 of this manuscript. 1.5. Organization of the Thesis The majority of Chapter 2 is dedicated to the survey of the dynamic programming optimization technique, which is the technique used in developing the model in this research. The development of the modeling approach is detailed in Chapter 3. The source of most of the materials included in this chapter is a report written by the author and co-authored by his supervisor and the author’s manager at BC Hydro (Ayad, et al., 2012). A briefing of the same materials is also included in (Abdalla, et al., 2013). Samples of the output and the results of the 4  model are laid out and briefly discussed in Chapter4. Chapter 5 includes the results and the discussion of the extensive testing of the model developed in this research. The material of this chapter is adopted from a paper that was included in the proceedings of the HydroVision International Conference (Ayad, et al., 2013). Finally, the conclusion, and recommendations for future work are discussed in Chapter 6.    5  Chapter 2: Survey of Literature 2.1. Introduction In this chapter, a survey is conducted on the different optimization techniques used in the fields of reservoir operation and operations planning. A number of the techniques are briefly introduced while others are thoroughly investigated due to their relative importance and relevance to the technique applied in this research.  Following this introduction, a brief and general review of the reservoir operation and management models is conducted in section 2.2.  Since the Dynamic Programming technique is used to develop the model in this research, the rest of the sections in this chapter are dedicated to this technique. The first few sections discuss the theories and principles the technique is based on. The last two sections of this chapter discuss the different variations and applications of the technique. They are sorted into two main categories: Deterministic Dynamic Programming techniques and Stochastic Dynamic Programming techniques. 2.2. Modeling of Reservoir Operation and Management  Scientists and engineers were, and still, interested in optimizing the operation of storage reservoirs. This interest ramped up in the early 70’s with the increased access to computers. Computers enabled the development of variety of new approaches aiming at deriving the optimal operating policy (least cost/ highest profit). There are other reasons that drove the development of various reservoir optimization techniques for the most efficient use of water (Wyatt, 1996) 6  such as: the increases complexity of the reservoir systems, the rise of energy prices in the 70’s and the emerging public awareness of the ecological issues and their relation to water resources.  (Yeh, 1985) stated that the adoption of optimization techniques to be used in planning, management and design studies of water systems is one of the most important advances in the field of water resources during the 60’s through 80’s. Many of the studies conducted were successful in practice, especially for planning purposes while the same level of success was not attained in operations optimization (Yeh, 1985). Before that time, the most used approach to handle the operation of simple reservoirs systems, such as a single-purpose single reservoir, was the Critical Period Analysis (Hall, et al., 1969; Duranyildiz & Bayazit, 1988; Christensen & Soliman, 1989; Wyatt, 1996). Despite being simple, the single-purpose single reservoir system faces several challenges in operation either technical challenges, seasonal variations in parameters of the system, or running against license constraints. For the more complicated systems, such as multi-purpose multireservoir systems, several optimization techniques were developed. These include simulation, linear, nonlinear, and dynamic programming, applied separately or in combinations. (Wyatt, 1996) stated that “The advantage of these methods over critical period analysis can be attributed to their considering operating costs over the entire whole of the flow series simulated, rather than just minimising costs during the most critical periods of reservoir draw-down”.  Nowadays, optimization techniques are adopted for the operations planning of all the reservoir systems regardless of their complexity. The selection of the most appropriate method is yet challenging. The main issue is that the deficiencies of these techniques are hard to quantify and it depends to large extent on the characteristics of each system (Wyatt, 1996). When the complexity of the system increases the number of possible operating policies and variable 7  combinations increase exponentially, which increases the computational effort associated (Wyatt, 1996).  In order to make sure of the feasibility of the optimal solutions deduced by the optimization models or on other words, simulation modeling studies should also be performed (Labadie, 2004). For that, using a combination of both simulation and optimization models would be of a great benefit and in some cases a necessity in order to obtain the optimal policy. 2.3. Bellman’s Principle of Optimality From (Bellman, 1957),  Bellman’s principle of optimality is such that “ An optimal policy has the property that whatever the initial state and the initial decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision”. Bellman also defined Dynamic Programming as” the theory of multistage decision processes”. The word “Dynamic” here means that this approach can handle the sequential or multi-stage decision problem and that is why it is efficient in making sequences of interrelated decisions (Nadalal & Bogardi, 2007). 2.4. Principle of Progressive Optimality  (Howson & Sancho, 1975) were the first to suggest this principle to use it to solve multistate dynamic programming problems.  It is a successive approximation using a general two overlapped stages solution. One of the advantages of the algorithm is that it requires little storage resources. As might be inferred, it depends on or could be considered as an extension of the Bellman’s Principle of Optimality. The Principle of Progressive Optimality states that "The 8  optimal path has  the property  that  each pair of decision sets  is  optimal  in  relation  to  its  initial  and  terminal  values" (Howson & Sancho, 1975). Using the principle of progressive optimality makes it unnecessary to discretize the state space (Yeh, 1985).  (Turgeon, 1981a) applied this principle to solve the short-term multireservoir operations scheduling problem.  A case study was performed using the application of the principle on four hydroelectric plants in series and it proved effective. Head variations, spills and time delays between upstream and downstream reservoirs were all taken into consideration (Turgeon, 2007). The author summarized the characteristics of the problem he was tackling as follows: nonlinear objective function, non-separable production functions, state and decision variables are bounded and the problem is stochastic due to the inflows into the system and the electricity demand. The advantages of this method compared to the traditional Principle of Optimality according to the same researcher are: 1. No discretization is required for the state variables; 2. Dimensionality problem is non-existing; 3. Non-convexity (such as in in production function) and discontinuity (like in cost function); are solvable using this technique unlike other techniques; 4. Convergence is monotonic and global optimum is reached; and 5. Relatively easy programming and fast execution. 2.5. Advantages and Challenges of Dynamic Programming (DP) Technique Dynamic Programming (DP) has the capabilities to decompose the problem into sub-problems that can be solved sequentially over the planning periods (Abdalla, 2007). The DP 9  approach is based on the Bellman’s Principle of Optimality. The number of discrete DP variables equals the number of state values times the number of decision variables which is guaranteed to be less than in LP (Yakowitz, 1982). Applications of the DP are very broad; however, the technique suffers from two curses that limit its applications to solve problems. The first curse is the curse of dimensionality which means that the problem size increases exponentially with increasing of the state-space which makes solving the problem in reasonable time very computationally expensive and time consuming (Bellman, 1957).  (Yakowitz, 1982) stated that “the exponential growth in memory and CPU time requirements with increase in dimension of the state vector (i.e., the 'curse of dimensionality') is the greatest single hindrance to dynamic programming solution of large-scale optimal control problems”. Some attempts were made to overcome the curse of dimensionality such as: making coarse grid, use of dynamic programming successive approximation, incremental dynamic programming, differential dynamic programming (Labadie, 2004). These variations will be discussed in later sections of this chapter. The advances in the computational capabilities of modern computers are one of the best solutions to the dimensionality problem. With those advances, the impact of this curse is alleviated but not eliminated. The second curse is the curse of modeling which means that when the real system that is being modeled gets complex, it is hard to model it using the DP technique (Bertsekas, 1995; Wyatt, 1996). The solution is to limit the number of storage states employed in the model when dealing with two or three reservoir systems (Wyatt, 1996). In other words, the solution to this problem is to under-represent the system or to approximate it. According to (Wyatt, 1996; Nadalal & Bogardi, 2007; Abdalla, 2007; Pereira & Pinto, 1991), advantages of DP include: 10  1. It  can be extended to multistage problems as well as stochastic case; 2. It handles discrete values and the nonlinearity; 3. The computational effort increases linearly when increasing the number of stages in the model; 4. It is suitable for problems where the decision variable takes a discrete or an integer form; and 5. It can handle nonlinearity, non-convexity, and even the discontinuity of the relations between the objective functions and constraints. DP is widely applied and well-suited to the reservoir operation and operations planning problems. Its popularity comes from the possibility of translating of the water resources features such as nonlinearity and stochasticity into a DP formulation (Yeh, 1985).  2.6. Deterministic Dynamic Programming Techniques 2.6.1. Incremental Dynamic Programming Models (IDP) In conventional DP, the state variables (usually set as reservoir storage or forebay in the reservoir operation problem) are discretized. Simultaneous derivation of operation policies for all the reservoirs and having a dense discretization is required in order to have close-to-global optimum operation policy in these systems (Nadalal & Bogardi, 2007). The disadvantage of this is that it makes it hard to use the conventional DP because of the curse of dimensionality (Nadalal & Bogardi, 2007) as previously mentioned. The Incremental Dynamic Programming (IDP) technique was introduced by (Larson, 1968). Instead of using the entire state-space to search for the optimal solution as the DP does, IDP uses a pre-specified number of state variables to visit. In other words, the IDP algorithm restricts the 11  state space to a corridor around the current given solution (Labadie, 2004). This idea has inspired the author of the research at hand in developing some solutions to increase the capacity of his model, which is discussed in Chapters 3, 4 and 5. IDP uses the recursive equation of DP to search for a better trajectory starting with some arbitrary feasible solution (initial trial trajectory) which serves as the first approximation of the optimal trajectory. The IDP creates what is called “corridor” around this initial trajectory. The corridor specifies the state variables to be visited in each time step in which the width of the corridor is the difference between the upper and lower bounds created around the state variable based on the initial trajectory. The trajectory obtained from this iteration is used as the new trial trajectory for the new iteration. The computation cycle continues until a convergence to the global optimal solution occurs. The convergence criterion is pre-specified for the system to prevent infinite calculations as the IDP solution might exhibit monotonic behavior (Nadalal & Bogardi, 2007). IDP has some shortcomings such as: hardship of interpolation over the corridor and selection of discretization intervals and sensitivity of the IDP to the initially assumed storage trajectories (Labadie, 2004).  2.6.2. Differential Dynamic Programming Models (Jacobson & Mayne, 1970) developed the Differential Dynamic Programming technique for the purpose of overcoming the dimensionality problems in DP. This technique uses an analytical solution, such as Taylor’s series expansion, instead of discretization of the state space (Labadie, 2004; Abdalla, 2007), which makes it more suitable for application on the multireservoir systems. (Yeh, 1985) stated that when the system dynamics are not linear and the objective function is not quadratic, then the Differential Dynamic Programming is one of the best 12  options. The differentiability of both the objective functions and the constraints is required to apply this technique (Labadie, 2004) which limits the application of this approach. 2.6.3. Dual Dynamic Programming Models (DDP) DDP is inspired by the Benders’ Decomposition Algorithm. (Pereira & Pinto, 1991) summarized the steps of the two-stage DDP algorithm as follows: 1. Set the initial value of approximate future value (cost) function, upper bound and lower bound; 2. Solve the approximate first stage problem; 3. Calculate the lower bound, if the convergence criterion is satisfied: stop otherwise, go to the next step; 4. Solve the second stage problem (calculate the approximate future value function and update the value of the upper bound); 5. Increment the number of vertices through which the approximate future value function is constructed; and 6. Go to the Step 2 again The advantages of the DDP compared to other techniques such as the conventional DP are:  1. Discretization is not necessary; 2. It provides upper and lower bounds for each iteration; 3.  It could be extended to solve multistage problems; and 4. It could be also extended to the stochastic case (SDDP), which will be discussed later in this chapter. 13  2.7. Stochastic Dynamic Programming Techniques 2.7.1. Conventional Stochastic Dynamic Programming (SDP)  Stochastic Dynamic Programming (SDP) is one of the most powerful and commonly used techniques to aid decision making in reservoir operation. SDP is well-established in long-term planning of multireservoir systems (Yeh, 1985). The inflows, electricity demands, and market prices are examples of stochastic variables that may be considered in the reservoir operations planning problem. The optimal operating policy in SDP is derived using the Bellman’s backward recursive relationship (Bellman, 1957). The convergence is determined by two criteria (Nadalal & Bogardi, 2007): stabilization of the incremental change in the optimal value according to the Bellman recursive formula and stabilization of the operating policy. The objective is usually to maximize the total benefit, which consists of current benefits coming from operations at present plus the discounted value coming from future use of stored water within the given planning/operating horizon. As mentioned before, there are two major problems with using the SDP technique to solve large-scale problems: the curse of dimensionality and the curse of modeling. (Arvanitidis & Rosing, 1970) developed one of the earliest applications of SDP in reservoir operation which had a primary goal of determining the optimal monthly hydropower generation of a hydroelectric  system. The authors focused on the most important variables to alleviate the curse of dimensionality. The model output was compared to a well-established rule-curve operation. 14   (Stedinger, et al., 1984) introduced a medium-term monthly SDP model that forecasted the current period inflows using available information at that period. The Aswan High Dam on the Nile river basin in Egypt was used as a case study. (Tejada-Guibert, et al., 1993) applied the SDP technique for three reservoirs and five thermal plants using a Markov Chain1 and a discrete distribution that approximated a normal distribution. Penalty functions were used for power and water shortages. (Druce, 1989; Druce, 1990) developed the Marginal Cost Model (MCM) for operations planning of the BC Hydro system using the SDP technique. The model uses weather sequences with equal probabilities to develop the monthly marginal value of water in the Williston Reservoir for a medium-term planning horizon. The uncertainty in inflows and market prices is accounted for in the model. At the time the model was created the Williston reservoir marginal value derived from the model results was used as a proxy for the system marginal price2. Later, and after several years of development by the System Optimization Group at BC Hydro, this model is now part of a bigger suite of models where it is coordinated with other models representing the other components of the system in order to derive the system marginal price.                                                  1 It entails the assumption that: the probability of an occurrence happening at a given stage in time depends only on the previous stage 2 The word” Price” is usually used to refer to the marginal value of energy as opposed to value of water used to produce this energy 15  (Wyatt, 1996) developed two models, one for power-generation reservoir systems and another for water supply reservoir systems. SDP was used in the two models along with a simulation model. (Turgeon, 2005; Turgeon, 2005) investigated the effect of incorporating multi-lag autocorrelation of inflows and the potential use of multi-lag autocorrelation for a single hydrologic variable for  the solution of the SDP problem. One of the findings is that the flood and shortage risks decreased as power generation increased. (Nadalal & Bogardi, 2007) applied the SDP technique to maximize the expected power generation from the Rantembe Reservoir in Sri Lanka. Operating policies were derived from an SDP model and then reservoir operation was simulated using historical inflow data. An improvement to the objective function was noted when storage discretization was refined but with the shortcoming of experiencing a polynomial increase in computational time. Unlike other mathematical programming techniques, such as linear and non-linear programming, very few general purpose dynamic programming (DP) solvers are available. An example of software available for solving DP and SDP problems is the CSUDP model, which is generalized dynamic programming software developed at the Colorado State University (USA). This software can handle “multidimensional problems, stochastic problems, and certain classes of Markov decision processes” (Labadie, 2003). The general SDP procedure, considering the inflow as the only stochastic variable, is illustrated in Figure 1. 16   Figure 1: SDP Procedure (Nadalal & Bogardi, 2007) 2.7.2. Stochastic Dynamic Programming with Function Approximation One of the most successful approaches used to alleviate the complexity of the reservoir system modeling problem is function approximation. Also, it is considered one of the most effective solutions to the dimensionality problem. In this method, state-space discretization is not needed any more as the near optimal value is expressed at each state/stage point in functional form. The value function can be approximated in many ways using linear function, polynomial function, piecewise-linear piecewise-polynomial or splines. 17  (Lamond, 2003) used the piecewise-polynomial functions to approximate the future value function. He applied this algorithm on a single hydroelectric reservoir with finite and discrete time horizon assuming a piecewise -inear concave reward for the production function.  2.7.3. Dynamic Programming with Successive Approximation (DPSA) Stochastic Dynamic Programming using Successive Approximation (DPSA in short) is used to handle the problem of reservoirs in parallel (Turgeon, 1980; Christensen & Soliman, 1989). It optimizes one reservoir at a time. Unfortunately, the major drawback of this approach is that it does not take the dependence of operation of reservoirs on each other’s energy content (storage) (Christensen & Soliman, 1989); in addition, with DPSA, the computation time and resources needed for the problem to converge are relatively large. 2.7.4. Aggregation and Decomposition SDP (Arvanitidis & Rosing, 1970) and later  (Turgeon, 1981b) adopted the method of aggregation/decomposition of a group of reservoirs in series into one equivalent reservoir. Each reservoir contributions were weighted according to its energy conversion factor (HK). The aggregation procedure was performed on storage, inflows and outflows. This approach was proposed as a solution to the computational infeasibility problem the authors faced when applying a conventional SDP algorithm on more than three reservoirs. The criticism to this method is that it does not account for parameters such as local constraints of reservoirs, which limits the application of this approach to the most systematic reservoir systems. In spite of that, the Aggregation and Decomposition SDP technique proved 18  effective in long-term planning studies in cases where decomposed reservoir systems are sufficiently similar (Christensen & Soliman, 1989). (Archibald, et al., 1997) added a conditional probability that allows switching between inflow scenarios at the beginning of each week. The modeling was done using four-state variables instead of two in the work of (Turgeon, 1981b). 2.7.5. Chance-constrained Programming Model and the Linear Decision Rules Chance-constrained stochastic Programming (CCP) is a technique that applies the probability conditions on constraints. It is mostly suited for application on multipurpose reservoirs. The main advantage of this technique is alleviation of the problem of estimating the cost function, (Yeh, 1985). The CCP implicitly converts the stochastic problem to an equivalent deterministic problem that could be solved more easily (Abdalla, 2007).  Linear Decision Rules (LDR) can be considered as an add-in to the CCP. They relate the releases to storage and remove the dependency on random storage levels which allows the releases to be specified at the beginning of each time period (Yeh, 1985; Labadie, 2004). In other words, the optimization is no longer dependent on storage variables; alternatively, it depends on a decision parameter. In addition, LDR eliminates the mathematical complexity in CCP formulation. On the other hands LDR is considered as an additional constraint in itself and it does not take the complete stochasticity of the streamflow into consideration. Applying the LDR, the number of constraints gets smaller which reduces the probability of converging to an optimal policy, (Yeh, 1985). 19  2.7.6. Stochastic Dual Dynamic Programming (SDDP) The SDDP is a combination of Stochastic Dynamic Programming (SDP) and stochastic linear or nonlinear programming using the Duality Theory3 (with conservation of the convexity condition). The algorithm is based on the approximation of the cost-to-go functions (value functions) of SDP using a piecewise-linear function.  The approximation mechanism can be done in two ways. The first is derived from Benders’ decomposition method as in (Pereira & Pinto, 1991). The second method is performing the approximation on a grid as in (Read & George, 1990; Tilmant, et al., 2008). Using SDDP with the latter type requires it to be performed on a relatively coarse grid to avoid increasing the number of inflow alternatives exponentially (Lamond & Boukhtouta, 1996)4.  The approximated cost-to-go function is obtained from the dual solutions of the problem at each stage. One of the most remarkable features of the algorithm is that it does not require the state- space to be discretized. By this, dimensionality problem is alleviated.  SDDP can be described as a two-stage problem. In the first stage, a decision is taken given a trial decision, and then a number m of second stage problem will exist (Pereira & Pinto, 1991; Lamond & Boukhtouta, 1996).  For the multistage problems of the second stage, each sub-problem represents one stage corresponding to one period. At each stage, sub-problems of one period are being solved.                                                   3 It states that an optimization problem is viewed as a primal problem or a dual problem. Solving the dual problem provides a lower bound to the solution of the primal one. 4 Assuming that inflows are the only stochastic variable in the problem 20  (Pereira & Pinto, 1991) applied the SDDP model on 39 hydroelectric plants in Brazil, 22 of which have reservoirs while the rest are run-of-the-river plants. The planning horizon used was 10 periods. Inflows were represented as independent random variables. The total number of variables and constraints were close to 150,000 for each stage problem. (Tilmant, et al., 2008) applied a SDDP model on the Euphrates River. An assumption was made that the system is interconnected and fully integrated between Turkey and Syria which is not the case in reality. The reason, as the authors stated, is that they wanted to show how much benefits can be achieved from an integrated system planning approach. The modeled water usage included power generation and irrigation.  (Guan, et al., 2017) implemented the SDDP technique for the BC Hydro System. The model uses stochastic inflows and the Columbia River Treaty (CRT)5 and other agreements to generate the water value function. For this purpose, two independent models were developed: the inflow model to generate forecasts of inflow volumes in the freshet period and monthly inflows and the CRT model to model operations for storage for flood control and other accounts. The SDDP implementation was benchmarked against several operations planning studies. An extensive testing and sensitivity analysis was performed to ensure the robustness and practicality of the model. (Dias, et al., 2010) stated that “Nowadays, the SDDP methodology is used in many countries, as in the case of the Brazilian power system, where the SDDP with aggregated reservoirs is still the official methodology used for determining the long-term hydrothermal system operation, the short run marginal cost, among other applications.”                                                  5 Details on the CRT can be found in Chapter 3 21  (Lamond & Boukhtouta, 1996)  stated that it is not recommended to use SDDP in cases of nonlinearity or non-convexity. Also, (Dias, et al., 2010) explained that although the SDDP is one of the fastest techniques when it comes to computer time, it might give solutions that are far from the optimal solution, obtained by other techniques such as SDP, in case of not estimating the cost-to-go function properly for all the important parts of the problem’s state-space. 2.7.7. Sampling Stochastic Dynamic Programming (SSDP) (Kelman, et al., 1990) were the first to propose the Sampling Stochastic Dynamic Programming technique (SSDP). They defined it as “a technique that captures the complex temporal and spatial structure of the streamflow process by using large number of sample streamflow sequences”. The authors presented this technique as a solution to the problems of poor representation of the system stochasticity and computation limitations that are inherently existent in traditional techniques .SSDP was originally designed for online operation using forecasted stream flows but later was extended to operations planning using historical stream flows (Lee & Labadie, 2007). The technique uses streamflow scenarios to represent the stochastic inflow processes. Like the deterministic optimization techniques, this approach still assumes the perfect foreknowledge in updating the optimal value function (Lee & Labadie, 2007). In other words, the current scenario continues with certainty into the future and the optimal value function is developed for the specific streamflow scenario.  Moving from one inflow scenario to another requires the knowledge of the transition probability from each flow to another. One of the challenges in using the SSDP is initializing 22  the terminal optimal value function as a boundary condition otherwise the model will empty all the reservoirs by the end of the time horizon (Lee & Labadie, 2007).  (Kelman, et al., 1990) developed a SSDP model that handles the complexity of the streamflow process by using a large number of sample streamflow sequences. The authors included what they called the “best inflow forecast” in the model as a hydrologic state variable to improve the reservoir operating policy. The model was applied on a case study to check its effectiveness on a hydroelectric system at the Feather River in California. (Lee & Labadie, 2007), in their comparative state-of-the-world study, used the SSDP as one of the benchmarking techniques. The SSDP performance was good in some aspects while performed poor in others. To enhance its performance, the authors suggested using more reliable inflow forecasting models to be fed to the SSDP model which shows how sensitive the technique is to the inflow scenarios used. (Schaffer, 2015) developed a SSDP model to maximize the value of water in storage in the BC Hydro system. The author investigated the use of different hydrologic inputs on the SSDP model performance such as: historical record data, inflows and forecasts generated from an autoregressive lag-1 model, and BC Hydro’s ensemble streamflow prediction forecasts. Results revealed the significance of using forecasts earlier in the freshet period compared to the rest of the water year. (Blair, 2017) developed a SSDP model for the Columbia River System, BC, Canada. Building on (Faber & Stedinger, 2001), the model (MUREO) incorporates probabilistic persistent reservoir constraints. The constraints are non-optional, persist over multiple stages, and are either a function of historic inflows, or a function of seasonal volume forecasts from 23  a future stage. The model has two state variables: the non-treaty storage account6 and the Kinbasket Reservoir storage. The model could be run for horizon for up to 6 years on a monthly time step. The implementation took advantage of the recent cloud computing and storage capabilities such that the user is able to run it either on a local computer or on the Amazon Cloud. To optimize the operations planning of the BC Hydro system, MUREO is run in an iterative fashion to coordinate with the aforementioned MCM (Druce, 1989; Druce, 1990) model and other models representing the rest of the BC Hydro system.                                                    6 Treaty here refers to the CRT 24  Chapter 3: Development of a Stochastic Dynamic Programming Optimization Model for Operations Planning of a Multireservoir Hydroelectric System 3.1. Background on the BC Hydro System 3.1.1. BC Hydro’s System The Province of British Columbia is one of the leading producers of hydroelectric power in Canada. The total installed generating capacity of the BC Hydro system is 12.05 GW (BC Hydro, 2017) of which more than 90% is hydropower. BC Hydro serves 95% of the population in British Columbia and produces about 80% of the total power generated in the province (BC Hydro, 2013; BC Hydro, 2017). There are 61 dams and more than 30 hydro plants in the BC Hydro system. Figure 2:  A Map of British Columbia Illustrating the Main Power Generation Plants and Local and Interconnected Transmission Lines, (BC Transmission Corporation, 2010) 25  The major river systems in BC are: the Peace system meeting 34% of electrical demand, the Columbia system meeting 31% of electrical demand, the Kootenay Canal and Seven Mile plants meeting 13% of electrical demand, and 23 small hydropower plants meeting 16% of electrical demand (BC Hydro, 2000)7. As of 2013, the remaining 6% of demand is served by independent power producers (IPPs) and thermal generating facilities (gas-fired and combustion turbines). The majority of the energy produced by the power system is from renewable sources with close to 2,000 MW coming from run-of-river projects, biomass projects and other renewable resources. BC Hydro meets the domestic electrical load of its service area and also trades energy in regional markets in Alberta, the Northwest USA and California through its subsidiary Powerex (BC Hydro, 2000; BC Hydro, 2013). Optimizing the operation of the main storage reservoirs in the BC Hydro system is quite challenging due to the uncertainties that must be dealt with given the significant multi-year reservoir storage capabilities. The existence of a transmission network connecting the system with regional markets adds one more dimension to the complexity of the system.  It is not an easy task to optimize the planning of operations of the system under the various constraints that the BC Hydro system encounters such as: the physical generation constraints, environmental and non-power requirements, water licenses and international treaties, to name a few.  3.1.2. Columbia River Treaty                                                  7 These percentages are averages and vary from year to year 26  The Columbia River Treaty (CRT) between Canada and the United States was ratified in 1964. The implementation of the treaty is the responsibility of the Canadian entity (BC Hydro) and the two American entities (Bonneville Power Administration and the US Army Corps of Engineers). The main features of the CRT include: building large storage reservoirs (completed in the 60’s and 70’s), streamflow regulation, sharing flood control benefits, sharing power generation benefits, determining the authorities on evacuation of flood control space, water diversion, mechanisms to resolve emerging disputes and the options to terminate or extend the treaty. Of concern to this research, the CRT imposes a number of constraints on Canadian reservoir operations and these constraints are included in operations planning models developed/used by BC Hydro. 3.1.3. Representation of the BC Hydro System in the Modeling Algorithm Six plants and their associated reservoirs on two river systems are explicitly included in the optimization model. Three on the Peace River: G.M. Shrum (GMS) and Williston Reservoir, Peace Canyon (PCN) and Dinosaur Reservoir downstream of GMS, and Site-C8 (STC) and Site-C Reservoir downstream of PCN. The other three are on the Columbia River: Mica (MCA) and Kinbasket Reservoir, Revelstoke (REV) and Revelstoke Reservoir downstream of MCA, and Arrow Lakes Hydro (ALH) and Arrow Lakes Reservoir downstream of REV. All these plants are optimized except ALH due to the complexity inherent in modeling the CRT. ALH generation is fixed along with power generation from other sources in BC.                                                  8 Site-C is currently under construction and is included in some long-term resource forecasts. 27  A storage plant9  is defined as a plant that has multi-year storage capacity and thus its storage is modeled as a state variable in the optimization model. GMS and MCA are modeled as storage plants because they are immediately downstream of the two largest reservoirs in the BC Hydro system. A run-of-the-river plant (ROTR) is a plant that does not have much storage capability and it simply passes all the water it receives within a period that is less than the time step modeled. PCN, STC, REV and ALH 10 are modeled as run-of-the-river (ROTR) plants. 3.2. Approach and Context The model developed as part of the current research is particularly concerned with the two largest reservoirs in the BC Hydro system, Williston and Kinbasket reservoirs. In addition to these two reservoirs there are four other reservoirs that are represented as ROTR. Several constraints and characteristics that are related to the seix reservoirs in particular or to the BC Hydro system in general are included in the model. The name of the model is Stochastic Dynamic Programming Optimization Model for Six Reservoirs or SDPOM6R for short.  The development of this model was part of a capital project at BC Hydro, The Water Value Project (Abdalla, et al., 2013), to develop multi-reservoir stochastic optimization models to generate water value and marginal value functions that best represent the expected                                                  9 The words “plant” and “project” are used synonymously to refer to the same facility. 10 ALH is an exception here since although as it has quite large storage capabilities it was modeled as a ROTR plant to simplify the algorithm. 28  value of water in storage and the marginal value of the multi-year storage reservoirs in the system. The objective is to obtain the optimum operating policies to maximize the revenue of BC Hydro from reservoir operation. This model was developed primarily to be used as a benchmarking tool for multiagent reinforcement learning model (MARL), which is under development by the same team as part of the Water Value Project, as well as against other already-developed or under-development models such as Reinforcement Learning Reservoir Optimization (RLROM) Model, Stochastic Dual Dynamic Programming (SDDP) model, Sampling Stochastic Dynamic Programming (SSDP) model and Stochastic Linear Programming (SLP) model.   3.3. Objectives of the Model Development The SDPOM6R model is developed to: 1. Improve on currently used models of multireservoir long/medium term operations planning; 2. Increase the number of reservoirs taken into consideration in the model (better representation of the real system); 3. Evaluate the marginal value of water for multi-reservoirs (better operations planning); and 4. Capture part of the complexity and the inherent uncertainties of the system. 3.4. Modeling of the Problem  3.4.1. State Variables 29  State variables in the model are the storage states, which are structured in two groups in the model: initial storage states and terminal storage states.  The initial storage state is fixed along all the time steps to enable the backward recursion value iteration procedure. The range of the initial storage states for each reservoir is controllable and can be changed to obtain a measure of the sensitivity of reservoir operation within a certain range of storage. For normal operations planning, the range is usually chosen to cover the storage state trajectory that will be described later in this manuscript.  The terminal storage states for each reservoir vary with initial states, time steps, release decision and inflow values. For each time step and initial state, each single release decision and each single inflow value, there is a unique range of terminal states covering all the possible states that can be visited given the historic operation and the physical limits. The bounds of the range of the terminal storage states are also controllable and usually lie within the storage state trajectory. 3.4.2. Decision Variables There is one decision variable per reservoir used in the SDPOM6R, which is the total release from each plant. The discretized values of the total release are calculated from the plant spill limits which are provided to the model in a data file. For each set of total release data, another set of values is deduced from the turbine release limits. This set is then used to calculate the generation corresponding to that release given the condition that it cannot exceed the following two values: the discretized plant spill value (the decision variable), and the difference in storage between the starting storage state and the terminal storage state given a specific value of the inflow. 30  3.4.3. Stochastic Variables  The expected inflow to each reservoir per time step is represented by a discrete probability distribution functions. This distribution is developed from 60 years of historical monthly inflows. These historical values are obtained from the BC Hydro records, CRO/Flocal and other sources internal to BC Hydro. The number of bins used in each discrete distribution is variable depending on the range of inflow values at hand and the inflow step (increment) adopted. A frequency analysis was carried out for different discretized inflow increments that were determined by the range of historical monthly inflows. In the current implementation of the algorithm, these increments determine the discretization of other state spaces such as the storage state-space and plant release decision-space. Several references, such as (Nadalal & Bogardi, 2007), recommend that if the actual distribution does not have a zero bin then a zero bin with very low probability should be added to the distribution to achieve a representative state probability transition matrix. In winter months  (December, January and February) the range of historical inflows is normally narrow and 4 to 5 bins were found to be sufficient to cover the distribution, whereas in the freshet months (May and June), when the inflow is snowmelt-dominated, the distribution range is large and the number of bins range from 8 to 11. At all times the minimum number of bins was set to 4. Figure 3 illustrates an example of the discretized June inflow probabilities for the Williston Reservoir. 31  Discretization of the inflow-space is performed only for the storage reservoirs, i.e. Williston and Kinbasket reservoirs. The inflow to ROTR reservoirs was calculated using monthly regression equations relating their inflow to the inflow and probability of their associated upstream storage projects. 3.4.4. Space Discretization and Transitions 3.4.4.1. Discretization of State-space and Decision-space  One of the important considerations affecting the accuracy and shape of the value function in the SDP algorithm is how to properly discretize the state and decision space variables in the optimization problem (Nadalal & Bogardi, 2007). For the case discussed here those include: the storage state-space, the inflow-space, and the decision-space of plant releases. It is well known that finer discretization yields better results, but unfortunately this is limited by the available computational resources and the time needed to solve the problem. The sensitivity of the algorithm and the quality of the solution, to the discretization increment are discussed in Chapter 5. Figure 3: Probability Distribution for the Inflows to the Williston Reservoir for the Month of June 32  3.4.4.2. Transition Matrix and State Transitions As discussed earlier, inflow is the only stochastic variable that is explicitly represented in the SDP solution algorithm of the SDPOM6R model. The inflow probability can be used to calculate each reservoir’s state transition probabilities for a given initial storage state. To calculate these state transition probabilities, the procedure outlined in the following steps was followed: 1. For each reservoir and for a given hydraulically feasible transition from one storage state to another, the probability of the state transition is set equal to the inflow probability and is used to calculate the state transition probability. If the transition is not hydraulically feasible then its probability is set equal to zero; 2. The global11 state transition probability (joint probability) for the system from a global state to another global state is equal to the product of the transition probabilities; and 3. The global transition probabilities are then used to calculate the expected values of the different terms in the value function equation12. Calculation and storage of the transition matrix is one of the major challenges that arise when applying the SDP technique to this type of problems. This challenge was primarily addressed by using a dynamic storage range corridor, which is discussed later in the manuscript.                                                  11 It involves all storage reservoirs modeled (e.g., the Williston and Kinbasket reservoirs in this context) 12 Discussed in Problem Formulation later in this chapter 33  3.4.5. Storage-Generation Curves (HK Curves) One of the most important aspects of the hydropower problem is how the HK values are accurately calculated as that level of accuracy affects the generation calculations and accordingly the optimum policy. For each reservoir and from historic data, a 3rd degree polynomial regression equation is generated and used to deduce the proper HK values for each transition storage state. The equation used in the model calculates the HK value as a function of both starting storage state and terminal storage state (linearly interpolated between the two states). 3.4.6. Storage-Forebay Curves Forebay elevation is not used in any of the core calculations or constraints in the model; instead, all the calculations are done using the storage volume of the reservoirs. However, the forebay elevation is calculated as a by-product of the model. A regression equation between the storage and the forebay elevation for each reservoir is developed as can be seen in Figure 4. The sources of the data used to develop the regression equations is the CRO/Flocal. These equations are then used in the model to calculate the forebay elevation as a function of the storage for both the initial and terminal states. 34    Figure 4: GMS and MCA Storage-Forebay Curves 3.4.7. Representation of Unit Outage It is important to note that the outages that are tackled here are the scheduled/planned outages of the plants and not forced outages. The latter are covered through a coefficient for contingency reserves/availability on generation.  The main source of the outage schedule data used for the outage representation in the SDPOM6R is the data files of the GOM model (Fane, 2003). First, the numbers from that file are mapped to the binary system to represent the outages of units and the duration of each outage. After that, the outages are aggregated to monthly time steps. Table 1 shows the outage schedule for one of the plants (GMS). The outage percentage is multiplied by the generation to calculate the maximum possible generation for each time step per plant.   35  Table 1: Conversion of Binary Outages to a Generation Factor Decimal 0 383 495 511 831 999 1007 1023 Total Generation Factor Binary 0 101111111 111101111 111111111 1100111111 1111100111 1111101111 1111111111 Month 1 0% 0% 0% 0% 0% 0% 68% 32% 100% 0.932 2 0% 0% 0% 0% 0% 0% 0% 100% 100% 1.000 3 0% 0% 0% 0% 0% 0% 0% 100% 100% 1.000 4 0% 0% 0% 0% 0% 0% 0% 100% 100% 1.000 5 0% 0% 0% 0% 0% 0% 0% 100% 100% 1.000 6 0% 0% 0% 0% 0% 0% 23% 77% 100% 0.977 7 0% 0% 0% 0% 0% 100% 0% 0% 100% 0.800 8 0% 48% 23% 0% 10% 19% 0% 0% 100% 0.807 9 0% 23% 0% 13% 63% 0% 0% 0% 100% 0.816 10 0% 0% 0% 35% 0% 0% 0% 65% 100% 0.965 11 0% 0% 0% 0% 0% 0% 0% 100% 100% 1.000 12 0% 0% 0% 0% 0% 0% 0% 100% 100% 1.000 3.4.8. Load-Resource Balance Calculation of generation in the SDPOM6R model is governed by the load resource balance equation. This equation can be simplified as: For each time step,  ∑ Generation (modeled hydroplants + un-modeled hydroplants + IPPs + thermal) – Load ± Forward contracts = Net Export (surplus or deficit)   Equation 1 36  ∑ Generation: represents the sum of all energy feeding into the system including the modeled hydroelectric plants (GMS, PCN, MCA, REV, ALH, and STC), the Independent Power Producers (IPPs), thermal plants and other sources of energy to the system including the un-modeled hydroplants.  Load: represents the domestic demand that has to be satisfied as a first priority by the BC Hydro system. Import/ Export: The sum of left hand side (LHS) of the equation is considered an import if it is a negative number and an export if it is a positive number, and is subject to the transmission line limits which are inputs to the model as well. Forward contracts: are the forward sales that BC Hydro is committed to fulfill during the planning horizon, which could be either imports or exports for each time step. Figure 5: Elements in the Load-Resource Balance Equation Figure 5 illustrates the load, summation of IPPs, thermal, and other sources of generation and the import and export limits. Any deficit will be covered by the generation 37  from the hydroelectric plants and imports if needed; any surplus will be exported or spilled if needed. For instance, the area under the light blue line with red markers represents the deficit that should be covered by the hydroelectric plants and the imports (Abdalla, 2007). 3.4.9. Representation of Prices The approach followed in this work is to adjust the average forecasted monthly energy prices at the Mid-C trading hub by applying a price multiplier that captures the effect of the variability of inflow conditions from the average water conditions in the Pacific Northwest region. In wet water years, using the forecast total seasonal flow at The Dalles near the mouth of the Columbia River, the price multiplier is less than 1 and the corresponding regional electricity market prices are less than average. Under dry conditions the multiplier is greater than one and the prices are above the average. The total inflow to the system is calculated and correlated with the Columbia River inflows at The Dalles to calculate the price multipliers which are used to scale the Mid-C market prices for different scenarios using monthly regression equations as shown in Figure 6.    Figure 6: Prices for a Forecast Water Year for Different Total System Inflow Scenarios 38  The coefficients of those equations are extracted for each month in each future water year and are then used to calculate a new set of regional price scaling factors. Wheeling charges are then added or subtracted to create import and export prices, respectively, at the BC-US border. 3.4.10. State-space Discretization and Generation of Discretized Values One of the governing factors in the current model and in the SDP technique in general is how to properly discretize the state space, as the value function and its shape are directly impacted by how the sta-e space is discretized. Furthermore, from what had been discovered through the SDPOM6R modeling process, not only the state-space but also both of the inflows and the releases need to be optimally discretized. Also, they all need to be of the same discretization step (increment) at least in each time step for each reservoir. The size of the discretization step is limited by the computation capacity of the computer/server that the model execution is performed on as well as the capacity of the coding language (AMPL in this case).  The code was written and indexed in a way that the discretization could be hybrid, which means that each reservoir can have its own discretization step and also for each reservoir the discretization step can vary by each time step. 3.4.11. Approximations  The approximations that are used in developing the code itself, in creating the data or in developing any regression equations are:  39  1. The maximum and minimum bounds of the state space are inputted as rough values; and these limits are first rounded to nearest discretization step; and 2. Several regression equations are used in the model such as price-inflow regression equation, HK regression equations and Forebay-storage regression equations. 3.5. Problem Formulation and Solution Algorithm 3.5.1.  Objective Function and Calculation of the Value Function The objective is to maximize the value of the hydropower resources. This is accomplished by optimizing the system dispatch to capture electricity market opportunities in the planning horizon while satisfying the domestic load. The objective function is expressed in the following equation13. PV𝑡(𝐒𝑗,𝑗∗,𝑡)= 𝑚𝑎𝑥𝐚𝑗,𝑗∗,𝑡{B𝑡(𝐒𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡) + γ ∗ ∑ [Pr𝑡(𝐒𝑗,𝑗∗,𝑡, 𝐒𝑗,𝑗∗,𝑡+1, 𝐚𝑗,𝑗∗,𝑡) ∗ PV𝑡+1 (𝐒𝑗,𝑗∗,𝑡+1)] 𝐒𝑗,𝑗∗,𝑡+1 } Equation 2   𝑤ℎ𝑒𝑟𝑒,  B𝑡(𝐒𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡)=  CR𝑡 (𝐒𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡)+ IC𝑡 (𝐒𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡) + ER𝑡 (𝐒𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡) + DR𝑡 − SP𝑡(𝐒𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡)   Equation                                                  13 Characters used in equations are defined in the List of Symbols in the beginning of this manuscript. 40   3 𝑤ℎ𝑒𝑟𝑒, 𝑗, 𝑗∗ ∊ 𝐽 𝑓𝑜𝑟 𝑗 ≠ 𝑗∗  As can be seen from Equation 2 that the present value of water in storage, PV (∙) is calculated as the sum of two terms: the expected income B (∙) for the set of decisions in the current period, and the discounted expected future value of water in storage in the next stage. Equation 3 shows that the expected income, or policy income, is the expected value of contract sales revenue (CR), cost of imports (IC), revenue from exports (ER), and the revenue/cost of satisfying the domestic load (DR). Also, a spill penalty function (SP) is added to discourage solutions requiring spill. There are three classical methods that can be used to solve Equation 2: policy iteration, linear programming, and value iteration. Value iteration is the most commonly used method and is adopted in this work. Figure 7 shows the application of the value iteration method in the model through a process called Backward Recursion which updates the value function starting with the last time step (stage) in the planning horizon and moving backwards. After the value function converges, the marginal value of energy can be computed by differentiating the value function with respect to storage for a given state, as shown in the following equation, Equation 4. MVW𝑡(𝐒𝑗,𝑗∗,𝑡) =𝜕PV𝑡(𝐒𝑗,𝑗∗,𝑡)𝜕𝐒𝑗,𝑡 ∗ HK𝑗,𝑡(𝐒𝑗,𝑡)    Equation 4   41   Figure 7: Backward Recursion Value iteration Procedure  42  3.5.2. Main Constraints The objective function above is subjected to the following constraints: 1. Load-resources balance: 𝑠. 𝑡. ∑ 𝐺𝑗,𝑡(𝐒𝑗,𝑡, 𝐒𝑗,𝑡+1, 𝐚𝑗,𝑡)𝐉𝑗=1+ ∑ g𝑘,𝑡𝐊𝑘=1(𝐈𝑗,𝑡, 𝐚𝑗,𝑡) + 𝐺𝑅𝑡 + 𝐶𝐼𝑡 − 𝐶𝐸𝑡− ST𝑡(𝐒𝑗,𝑗∗,𝑡, 𝐈𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡, 𝐒𝑗,𝑗∗,𝑡+1)=  L𝑡                                                                            Equation 5    2. Mass (hydraulic) balance: For the storage projects (WSR and KBT), s. t.  𝐒𝑗,𝑡+ 𝐈𝑗,𝑡 − 𝑇𝑄𝑗,𝑡(𝐒𝑗,𝑡, 𝐒𝑗,𝑡+1, 𝐚𝑗,𝑡) − 𝑆𝑄𝑗,𝑡(𝐒𝑗,𝑡, 𝐒𝑗,𝑡+1, 𝐚𝑗,𝑡) = 𝐒𝑗,𝑡+1                                                  Equation 6 For ROTR projects, 𝑠. 𝑡.   𝐢𝑘,𝑗,𝑡(𝐈𝑗,𝑡) +  𝐚𝑗,𝑡 − 𝑡𝑞𝑘,𝑗,𝑡(𝐈𝑗,𝑡, 𝐚𝑗,𝑡) − 𝑠𝑞𝑘,𝑗,𝑡(𝐈𝑗,𝑡, 𝐚𝑗,𝑡) = 0    Equation 7 where  𝑖𝑘,𝑗,𝑡, 𝑡𝑞𝑘,𝑗,𝑡 , and 𝑠𝑞𝑘,𝑗,𝑡 are computed only when storage project (j ∊ 𝐉) and ROTR project (k ∊ 𝐊) are hydraulically connected. 3. Storage limits: 43  𝑠. 𝑡.  𝐿𝑆𝑗,𝑡 ≤ 𝐒𝑗,𝑡 ≤ 𝑈𝑆𝑗,𝑡   Equation 8  3. Turbine flow limits: For the storage projects, 𝑠. 𝑡.  𝑇𝑄_𝑚𝑖𝑛𝑗,𝑡(𝐒𝑗,𝑡, 𝐒𝑗,𝑡+1) ≤ 𝑇𝑄𝑗,𝑡(𝐒𝑗,𝑡, 𝐒𝑗,𝑡+1, 𝐚𝑗,𝑡) ≤ 𝑇𝑄_𝑚𝑎𝑥𝑗,𝑡(𝐒𝑗,𝑡, 𝐒𝑗,𝑡+1)   Equation 9 For ROTR projects, 𝑠. 𝑡.  𝑡𝑞_𝑚𝑖𝑛𝑘,𝑡 ≤ 𝑡𝑞𝑘,𝑗,𝑡(𝐈𝑗,𝑡, 𝐚𝑗,𝑡) ≤ 𝑡𝑞_𝑚𝑎𝑥𝑘,𝑡  Equation 10 4. Total plant discharge limits: 𝑠. 𝑡.  𝑃𝑄_𝑚𝑖𝑛ℎ,𝑡 ≤ 𝑃𝑄ℎ,𝑡(𝐚𝑗,𝑡) ≤ 𝑃𝑄_𝑚𝑎𝑥ℎ,𝑡  Equation 11 5. Transmission limits: 𝑤ℎ𝑒𝑛 𝑡ℎ𝑒 𝑠𝑝𝑜𝑡 𝑡𝑟𝑎𝑑𝑒 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑖𝑠 𝑖𝑚𝑝𝑜𝑟𝑡,   s. t.  ST𝑡(𝐒𝑗,𝑗∗,𝑡, 𝐈𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡, 𝐒𝑗,𝑗∗,𝑡+1) +  CI𝑡 ≤ 𝐿𝑇𝐼𝑡   Equation 12  44  𝑤ℎ𝑒𝑛 𝑡ℎ𝑒 𝑠𝑝𝑜𝑡 𝑡𝑟𝑎𝑑𝑒 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑖𝑠 𝑒𝑥𝑝𝑜𝑟𝑡,s. t.  ST𝑡(𝐒𝑗,𝑗∗,𝑡, 𝐈𝑗,𝑗∗,𝑡, 𝐚𝑗,𝑗∗,𝑡, 𝐒𝑗,𝑗∗,𝑡+1) +  CE𝑡 ≤ 𝐿𝑇𝐸𝑡    Equation 13  6. Generation limits: For the storage projects, 𝑠. 𝑡.   𝐺_𝑚𝑖𝑛𝑗,𝑡 ≤ 𝐺𝑗,𝑡(𝐒𝑗,𝑡, 𝐒𝑗,𝑡+1, 𝐚𝑗,𝑡) ≤ 𝐺_𝑚𝑎𝑥𝑗,𝑡  Equation 14  For ROTR projects,  𝑠. 𝑡.   𝐺_𝑚𝑖𝑛𝑘,𝑡 ≤ 𝑔𝑘,𝑡(𝐈𝑗,𝑡, 𝐚𝑗,𝑡) ≤ 𝐺_𝑚𝑎𝑥𝑘,𝑡                                                                                                               Equation 15 3.5.3.  Solution Algorithm The solution algorithm is illustrated in Figure 8 and it consists of three main modules. The first is the Discretizer that discretizes the storage state-space and the release decision-space. The main data sets used by this module are the storage corridor for the storage projects for the planning horizon, which will be detailed in later sections, the discrete inflow probability distribution for the planning horizon, and the limits on plant discharge for the storage projects for the planning horizon. This module prepares the discretized starting 45  and terminal storage states for each time step as well as the discretized releases. These outputs are used in the second module. The second module is the SDP model which implements the SDP algorithm and applies the constraints in each time step. These outputs include: turbine flow, spill, total system inflow, generation, transition probability, forebay, spot electricity market trade, policy income, and the marginal values of water and energy. The third module is the Value Iteration module which develops the water value functions for the time horizon considered. In addition to the main modules discussed above, there are eight smaller modules that perform other calculations including the inflow regression analyses for ROTR, HK, price multiplier calculations, and a module to calculate the capacity limits for plant generation and turbine discharge limits. The details and the code of the modules mentioned above can be found in Appendix A.1 46   Figure 8: Flowchart of the SDPOM6R Model 47  3.6. Capabilities  Extensive testing of the model has shown that it is able to solve the problem for up to 36 monthly time steps (3 years) producing practicaly acceptable water value and marginal value functions up to a problem size of ~ 164 million states per time step. It is expected that, with further enhancements of the algorithm, the model could handle a much larger problem and could also be extended to include more state variables. The development of a dynamic storage-state corridor using the simulated historical data significantly accelerated the convergence of the algorithm and allowed the solution of larger problems, but care must be taken to ensure that the derived solutions are robust and globally optimal. Flexibility is one of the most important features of the current model, and it is well known that SDP solution algorithms are not typically very flexible and are custom built for specific systems. This flexibility is a result of enhancements made to the algorithm and the formulation of the model coding in AMPL, which provided the following advantages: 1. The increment of all the state-spaces and the storage corridor can be controlled and changed easily for each reservoir for each time step; 2. The problem can be solved for one storage reservoir while fixing the states of the other storage reservoir14 simply by changing few control parameters; and                                                  14 This option allows the model to solve for one storage reservoir given a fixed state of the other reservoir. 48  3. The model can be run either directly on a personal computer (or a server) using the command prompt or by using a scheduling program that uses simple scripts to optimize the use and sharing of computing assets. 3.7. Limitations Because of the nature of the SDP technique, as well as the complexity of the system investigated herein, there are several limitations of the algorithm. Some of these limitations could be overcome with the advances in computing resources and the programming languages capabilities. Other limitations are likely to persist in future versions of the model or extension of it. Some of these limitations are: the stochasticity of domestic electricity load and prices is not currently represented, the accuracy of the existing regression equations representing the prices and other variables could be improved, and the model only contains variables representing six major reservoirs in the BC Hydro system while fixing the output of other resources and therefore it simplifies the real system; in addition, several other environmental or operational constraints are not modeled. 3.8. Issues Experienced in Model Development and Code Run Several problems were experienced in developing the model as well as in its outputs: 1. The shapes of the value function and the marginal value of water function heavily depend on the discretization used and the inflow values and their probabilities. It was a challenge to find the right combination of this data that produce a practically acceptable shape of each function; 49  2. As the problems of dimensionality and modeling are inherent in the SDP modeling, it was expected to experience problems related to these two curses; 3. Sometimes the resulting output files are too big to store; 4. Indexing is cumbersome which is partially due to the nature of coding in AMPL and partially due to the size of the problem at hand; and 5. Both the transition probability and value iteration calculations are sensitive to changes in indexing or other changes and tracing these sensitivities is challenging.   50  Chapter 4: Results 4.1. Introduction In this chapter few samples from the model results are illustrated in graphical form. All the results illustrated are normalized for readability and data confidentiality reasons. Discussion of the results is kept to minimum as more details are presented in Chapter 5. 4.2. Sample Results for Two Cases of State-space Discretization There are several cases that have been thoroughly investigated and tested (please see Table 6) in Chapter 5. For brevity, only samples of two cases of them are shown in the following graphs. The cases are: CASE A: Storage state increment is 500 cms-day for the Williston Reservoir and 1000 cms-day for the Kinbasket Reservoir (CASE A); and  CASE B: Storage state increment is 1000 cms-day for both of the Williston Reservoir and the Kinbasket Reservoir. Figure 9 below shows a three dimensional graphical representation of the water value functions of both of the Williston Reservoir and the Kinbasket Reservoir. As expected, the value of water in storage is lower when there is less water in both reservoirs. This value increases with any incremental increase in the amount of water in storage in one or both of the reservoirs. This increase continues until a certain point where the surface almost levels. This means that the incremental increase of water in storage has little to no effect on the total value of water in storage. It could be noted that the farthest tip of the surface (top right) 51  slightly drops after the surface has leveled and that could be attributed to the boundary conditions.  Figure 9: Three Dimensional Water Value for the Williston Reservoir and the Kinbasket Reservoir for the Month of December- CASE A  Figure 10 to Figure 13 show the normalized value of water (left hand side) and the normalized marginal value of water (right hand side) in the Kinbasket Reservoir for different storage-states at the Williston Reservoir for CASE A. Each figure represents one of the selected months. Those months are selected to represent different stages of the water year and energy demand; Namely, October (a shoulder month), January (a winter month), May (a freshet month) and August (a summer month). 52    Figure 10: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of October- CASE A     Figure 11: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of January- CASE A  53    Figure 12: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of May- CASE A    Figure 13: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of August- CASE A  Figure 14 to Figure 17 show the normalized value of water (left hand side) and the normalized marginal value of water (right hand side) in the Williston Reservoir for different 54  storage-states at the Kinbasket Reservoir for CASE A. The months selected for illustration are the same as discussed before.   Figure 14: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of October- CASE A     Figure 15: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of January- CASE A   55    Figure 16: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of May- CASE A     Figure 17: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of August- CASE A  Figure 18 to Figure 21 show the normalized value of water (left hand side) and the normalized marginal value of water (right hand side) in the Kinbasket Reservoir for different 56  storage-states at the Williston Reservoir for CASE B. The months selected for illustration are the same as discussed before.   Figure 18: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of October- CASE B      Figure 19: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of January- CASE B   57    Figure 20: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of May- CASE B     Figure 21: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of August- CASE B  Figure 22 to Figure 25 show the normalized value of water (left hand side) and the normalized marginal value of water (right hand side) in the Williston Reservoir for different 58  storage-states at the Kinbasket Reservoir for CASE B. The months selected for illustration are the same as discussed before.   Figure 22: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of October- CASE B     Figure 23: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of January- CASE B   59     Figure 24: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of May- CASE B     Figure 25: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of August- CASE B   60  Figure 26 shows the variation in the monthly value of water in storage for selected storage combinations of both storage reservoirs over the water year for CASE A. Similarly, Figure 27 shows these variations for CASE B. Comparing these two cases, it could be concluded that CASE A which has the finer discretization yields higher value of water on average as well as smoother change in the value of water from month to month.  Figure 26: Value of Water in Storage of the Williston Reservoir and the Kinbasket Reservoir along the Water Year (October to September)- CASE A On the other hand, comparing the marginal value of water in CASE A and CASE B, setting CASE A as the base case as illustrated in Figure 28, shows that the marginal value of water in generally higher in CASE B which might be due to the coarser grid and hence the higher value of the derivatives of the value function. It could be also noticed that there is a trend to that change as it gets smaller in the freshet period and grow bigger towards the shoulder months. 61   Figure 27: Value of Water in Storage of the Williston Reservoir and the Kinbasket Reservoir along the Water Year (October to September)- CASE B   Figure 28: Percentage of Difference between the Marginal Value of Water in CASE A and CASE B for the Williston Reservoir and the Kinbasket Reservoir   62  4.3. Results of Intoducing the Storage State-space Corridor One of the contributions of the algorithm used in the modeling is the use of the state-space corridor as opposed to using a fixed state-space for all the stages. This is a way to limit the state-space points visited and hence alleviating the computation effort. The details of this corridor are described in Chapter 5.  At an earlier stage of the model development, the fixed state-space was used until it was realized that a different approach is needed to alleviate the dimensionality problem to be able to solve a bigger and more complex problem. In this section, the results of using the corridor are compared to the results of using a fixed state-space. For brevity, only one case of state-space discretization is discussed which is CASE B in the previous section- Case of Storage State Increment of 1000 cms-d for both of the Williston Reservoir the Kinbasket Reservoir .These results are laid out in a graphical form. Since the graphs of the value function and the marginal value of water for the CORRIDOR case have been illustrated in the previous section 4.2  (Figure 18 to Figure 25), the graphs that are shown in the following figures (Figure 29 to Figure 36) are only for the FIXED case. These graphs show that both of the value of water and the marginal value curves are sparser and more flat compared to the COORIDOR case where the same curves are more clustered. This suggests that using the CORRIDOR enabled the model to deduce a better operating policy and value of water for the different storages state-space combinations. Other advantages of the CORRIDOR are discussed in details in Chapter 5. 63    Figure 29: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of October- FIXED Case     Figure 30: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of January- FIXED Case   64    Figure 31: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of May- FIXED Case     Figure 32: Water Value and Marginal Value of Water for the Kinbasket Reservoir for the Month of August- FIXED Case  65    Figure 33: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of October- FIXED Case    Figure 34: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of January- FIXED Case  66    Figure 35: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of May- FIXED Case    Figure 36: Water Value and Marginal Value of Water for the Williston Reservoir for the Month of August- FIXED Case    67  Chapter 5: Assessing the Impact of Storage State and Decision Space Discretization on Solution Efficiency and Precision of a Stochastic Dynamic Programming Algorithm in a Multireservoir Operations Planning Model 5.1.Introduction The SDPOM6R model has gone through extensive testing. In this chapter, the results of sensitivity analyses, which were done as part of this extensive testing of the model, are discussed. The sensitivity analysis results discussed are for impact of introducing the state-space corridor, the impact of the state-space discretization and the sensitivity of results to the spill penalty function. The core material included in this chapter has been published in (Ayad, et al., 2013). 5.2.  Assessing the Impact of Introducing a Storage State-space Corridor Seventy three years of historical data were simulated to generate potential monthly storage states for the Williston and Kinbasket reservoirs. From this data, the upper and lower bounds of monthly storage levels were determined. Figure 37 shows the storage corridor for the Williston Reservoir while Figure 38 shows the storage corridor for Kinbasket Reservoir. Storage buffers15 are added to the storage bands shown in the figure above. The resulting data is then used to generate discretized storage states for both reservoirs for each time step using the Discretizer module as discussed earlier.                                                  15 Usually one state up and one state down at each stage; the exact value depends on the chosen increment. 68  It can be noted that both reservoirs are drafted during the winter period and then refilled during the freshet period. This drafting/refilling operation takes into consideration the prevailing hydrologic regime in these basins, the domestic electrical demand, and the seasonal trends in market prices.  Figure 37: Simulated historical storage bands for the Williston Reservoir Using this state-space corridor provides several advantages: the first is to have a realistic state-space for each time step which allows better representation of the real system and the second is to have a smaller problem size therefore reducing the required computational resources needed to solve the problem. By using the state-space corridor, it is also possible to extend the planning horizon, to run the model with finer space discretization, and to add more state variables in future implementations. However, care must be taken to 69  ensure that the selected state-space corridor does not cause the algorithm to choose sub-optimal solutions or result in infeasibilities for some state transitions, particularly near the upper and lower storage limits.  Figure 38: Simulated historical storage bands for the Kinbasket Reservoir A comparison between running the model for a fixed state-space16 (FIXED case) versus using the storage corridor (CORRIDOR case) was performed to assess the impact of using the corridor on the problem solution efficiency and precision of the output. The storage states-space was discretized at 1000 cms-d for both reservoirs. The reason for using this increment was that it was not possible to run the FIXED case for smaller increments because                                                  16 By using only the physical maximum and minimum storage values at all stages in the entire horizon. 70  of the problem’s dimensions. The following results compare runs for 12, 24 and 36 time steps17.  Comparing the two cases, it was found that it took on average about double the time to run FIXED cases as compared to CORRIDOR cases and that the trend of the increase in time was linear for both cases as illustrated in Figure 39. Setting the FIXED CASE As the base case, and for a 12 time-step run, the average difference in the value of water in storage was -1.12% between the two cases, while the average difference in the marginal value of energy was 8.96% and -5.81% for Williston and Kinbasket respectively. Repeating the same analysis for the 24 time steps run, the corresponding differences were -1.04%, 10.36%, and -3.42%. For the 36 time-step trial, the corresponding differences were -1.04%, 10.87% and -0.12%. It should be noted that the differences in the marginal values of energy for Williston Reservoir were                                                  17 All the runs in this manuscript are done on a server with 48 GB of RAM. Figure 39: Run Times for FIXED Case vs. CORRIDOR Case 71  higher than those for the Kinbasket Reservoir. This is due to the larger operation range of the Williston as compared to Kinbasket (approximately 1.65 larger). As a result of this work, it is apparent that using the storage state-space corridor can significantly impact the accuracy of the results, but care must be taken in defining the corridor and specifying the state discretization increments for such problems. 5.3.Assessing the Impact of State-space Discretization The solution efficiency of the problem as well as the precision of the output is impacted by the state-space discretization in the SDP technique. The smaller the space increments, the more precise the output, but that comes at the cost of increasing the dimensions of the problem and hence may jeopardize the solution efficiency of the problem. A trade-off between the accuracy of the output and the solution efficiency of the problem is tested in this section. A group of six cases were investigated, all for a planning horizon of 24 months and all using the selected storage state-space corridor discussed above. The discretization Table 2: Storage State Discretization Cases Case Storage state increment for the Williston Reservoir, cms-d  Storage state increment for the Kinbasket Reservoir, cms-d  Problem size per time step, million 1 500 1000 23.33 2 750 750 13.12 3 750 1000 6.62 4 1000 500 14.02 5 1000 750 5.04 6 1000 1000 2.54 72  parameter tested is the increment of the storage states, which is the same as the increment of the inflow discrete distribution and the increment of the discretized decision space. The different cases and the corresponding problem size are shown in Table 2. Comparing the run time for each case showed an exponentially increasing trend - i.e. as the problem size increases the run time increases exponentially as shown in Figure 40. Case 1, with the largest problem size, was taken as the base case to compare the other cases to. It should be noted that cases with larger problem sizes were tested (up to ~164 million states per time step) but they are not analyzed here. It was noted that refining the discretization of the problem space remarkably enhances the smoothness, curvature, and shapes of both of the value function and the marginal value of energy function. Given the above results, it can be concluded that finer discretization of the state-space will yield more accurate estimates of the value of water in storage and marginal value of water functions. Figure 40: Problem Size vs. Run Time for Different Cases 73  5.4.Sensitivity of Results to Spill Penalty Values As discussed earlier, a penalty in the form of an import price multiplier is used to estimate the cost of spill. Eight cases were tested for the same storage state-space corridor for 24 stages at an increment of 1000 cms.d for both storage reservoirs to investigate the sensitivity of the model to the penalty used. Table 3 shows the cases tested. Case 1 was taken as the base case to compare the other cases to. In general, the higher the penalty the lower the value of water in storage is. At the higher storage states the effect became more pronounced because, at those states, the penalty had more effect as the reservoir was more likely to spill in those states. It was also noticed that the value of water in storage was 85% lower than the base case, on average, when the penalty value was set to very high values (cases 7 and 8). On the other hand the marginal value of energy for both reservoirs increased for cases 2 through 6 with the highest increase for case 5, which corresponds to a penalty multiplier of 1.5. Because the penalty is very high for cases 7 and 8, the marginal value of energy decreased for both reservoirs for these cases. One important note is that, if high and low storage states are excluded, the ratio of the marginal value of energy in Kinbasket to the marginal value of energy of Williston lies Table 3: Different Cases for the Penalty Function Case Penalty Multiplier 1 0 2 0.20 3 0.5 4 1 5 1.50 6 2.00 7 10.00 8 50.00 74  within reasonable limits for Cases 4, 5, and 6 18 while the ratio is significantly higher for other cases, particularly Cases 1, 2, and 3.  It could be concluded, based on the tested cases, that using a penalty multiplier of 1 to 2 is the best option to obtain reasonable19 marginal values of water of storage as compared to historically observed results.                                                    18 Reasonable limits are assumed to be the ratio of total HK of the Columbia River system to the total HK of the Peace River system, which is equal to ~1.23 on average. 19 Comparing them to the actual market prices and looking into actual trade schedules for the spot market and other historically observed results. 75  Chapter 6: Conclusions, Recommendations and Future Work 6.1. Conclusions Extensive testing of the model developed as part of the current research has shown that it is able to solve the problem for up to 36 monthly time steps (3 years) producing practical water value and marginal value functions up to a problem size of ~ 164 million states per time step. It is expected that, with further enhancements of the algorithm, the model could handle a much larger problem and could easily be extended to include more state variables. The development of a dynamic storage-state corridor using the simulated historical data significantly accelerated the convergence of the algorithm and allowed the solution of larger problems but care must be taken to ensure that the derived solutions are robust and globally optimal. Flexibility is one of the most important features of the current model. It is well known that SDP solution algorithms are not typically very flexible as they are custom-built for specific reservoirs and systems. The flexibility of the SDPOM6R model is a result of enhancements made to the algorithm and the formulation of the model coding in AMPL, which provided the following advantages: 1. The increment of all the state spaces and the storage corridor can be controlled and easily changed for each reservoir for each time step; 76  2. The problem can be run for one storage reservoir while fixing the states of the other storage reservoir20; 3. The modeling algorithm is generalized so that the user can easily adapt it and change model mode simply by changing few control parameters; and 4. The model can be run either directly on a personal computer (or a server) using the command prompt or by using a scheduling program that uses simple scripts to optimize the use and sharing of computing assets. Because of the nature of the SDP technique, as well as the complexity of the system investigated herein, there are several limitations of the algorithm. Some of these limitations might be overcome while others are likely to persist in future versions of the model or adaptation of the same technique. Some of these limitations are: the stochasticity of load and prices is not currently represented, the accuracy of the existing regression equations representing the prices and other variables could be improved, and the model only contains variables representing six major plants in the BC Hydro system while fixing the output of other resources and therefore it simplifies the real system. 6.2. Proposed Future Enhancements to the Model Developed As discussed in Chapter 3 and Chapter 5, there are several features of the model that need to be enhanced along with some features that could be added in order to get a more robust and representative model and obtain more practical functions of the water value and                                                  20 This option allows the model to solve for one storage reservoir given a fixed state of the other reservoir. 77  the marginal value of water. Several functions and features need to be added to the SDPOM6R such as: 1. To split the state variable for Kinbasket into two state variables: Kinbasket Reservoir storage and non-treaty storage as well as to represent the Kinbasket and Arrow Lakes flex storage accounts to better represent the Columbia River Treaty operation of the Columbia River system; 2. To include new storage limits derived from flood control curves from the Columbia River Treaty operating plans, and 3. To introduce sub-time step functionality such as: peak load hours (PLH), heavy load hours (HLH), and light load hours (LLH) in order to better capture the electricity market depth and price variability. The following list presents the features that are included in the current modeling but need to be better represented, enhanced or modified: 1. Representation of prices to replace the currently used regression equations; 2. Representation of load stochasticity; 3. Representation of inflows; 4. Representation of the transmission limits need to be revised and enhanced; and 5. HK calculation procedure could be enhanced. 6.3. Future Work and Recommendations An extensive literature survey has been conducted, but not included in the current manuscript, on both reinforcement learning and the multiagent reinforcement learning 78  techniques. These two techniques are closely related and are based on the SDP technique that was used to develop the SDPOM6R Model.  Below is a list of the work which has been conducted by the author of this manuscript, but not included in it: 1. Enhancement and testing work that was performed on the models developed by (Abdalla, 2007; Shabani, 2009); 2. A framework for the use of Multi-agent Reinforcement Learning (MARL) technique. Based on both of the literature survey and the modeling work mentioned above, the following is recommended: 1. Explore the possibility of developing a full working version of MARL model that better represent the system and its inherent stochasticity and uncertainty and ensure the proper coordination between components of the system through the artificial intelligence and the MARL approach adopted in this model; 2. Benchmark the model above against the SDPOM6R model as well as the other models developed as part of the Water Value Project.   79  Bibliography Abdalla, A. E., 2007. A Reinforcement Learning Algorithm for Operations Planning of a Hydroelectric Power Multireservoir System. s.l.:Ph.D. Thesis, Department of Civil Engineering, University of British Columbia. Abdalla, A. et al., 2013. Water Value Project Definiton Phase, Burnaby, BC: BC Hydro. Archibald, T. W., Mckinnon, K. I. M. & Thomas, L. C., 1997. An Aggregate Stochastic Dynamic Programming Model of Multiple Reservoir Systems. Water Resources Research, 33(2), pp. 333-340. Arvanitidis, N. & Rosing, J., 1970. 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Code and Computation Details The model is composed of three main modules, the Discretizer , the SDP Model and the Value Iteration Model. In addition to these main modules, there are several small modules that calculate specific variables. The code and the computation detauils in each one of these modules are listed below. All code is listed in the AMPL syntax. A.1.1. Discretizer Model A.1.1.1. Code 86   87    88    89     90  A.1.1.2. Computation Details Table 4: Calculations Details in the Discretizer Model Step Number Parameter Calculated Notes 1 Delta_States, Delta_Staters. Delta_Releases The increment of the starting, terminal states and plant release respectively. They are calculated as multiples of the inflows increment. When the multiple equals 1, they are exactly the same value as the inflow increment. 2 Max_States_Act, Min_States_Act A rounded up/down  number for the maximum/minimum starting state to the nearest “Delta_States” 3 N_States The number of starting states between the” Max_States_Act” and the “Min_States_Act” 4 Max_Staters_Act A rounded up number for the maximum terminal state to the nearest “Delta_Staters” capped by the maximum inflow in a given stage. 5 Min_Staters_Act A rounded down number for the minimum terminal state to the nearest “Delta_Staters” capped by the minimum inflow and the maximum plant release in a given stage. 6 N_Staters The number of terminal states between the” Max_Staters_Act” and the “Min_Staters_Act” in a given stage. 7 QP_Max_Act ,QP_Min_Act A rounded up/down number for the maximum/minimum plant release in a given stage to the nearest “Delta_States”. 8 N_Releases The number of plant releases in a given stage.  A.1.2. SDP Model  A.1.2.1. Code 91   92   93   94   95   96   97   98   99   100   101   102     103  A.1.2.2. Computation Details Table 5: Calculations Details in the SDP Model Step Number Parameter Calculated Function 1 Total_Sys_Inflow Sums up the inflows to the plants included in the model including the storage and non-storage plants. 2 Exp_Price, Imp_Price, Cont_Price Calculate the prices for import, export and contract prices using regression equations that relate the energy prices to the total inflow of the system. For a given inflows combination, the difference between import and export prices is fixed while the contract prices are the average of them. 3 Term_W For each point of starting storage state, release decision and inflow, this parameter calculates the actual terminal storage that corresponding to those values. 4 Turbine_Release Calculates the turbine flow for the storage plants (i.e. GM Shrum and Mica) which is capped by the release decision and the maximum turbine flow (QT_Max). 5 Spill The difference between the release decision and the turbine release. 6 Hydraulic_Balance The difference between the actual terminal storage and the discretized terminal storage. It is mainly used to assign the state probability and the transition probability. 7 Tot_Inflow_ROTR Sums up the natural inflow to a given run-of-the-river (ROTR) plant to the flows coming from all the plants at the upstream of this plant. 104  Step Number Parameter Calculated Function 8 Turbine_Release_ROTR Calculates the turbine flow for the ROTR plants (i.e. PCN, REV and STC) from the total flow to each plant capped by absolute maximum turbine flow (Abs_QT_Max). 9 Spill_ROTR The difference between the ROTR total flow and the ROTR turbine release. 10 Generation The storage plants generation calculated as the turbine release times the HK capped by the Max_Gen_Limits and the Abs_Max_Gen_Capacity considering both of the outage and the availability factors. 11 Gen_ROTR The ROTR plants generation calculated as the turbine release times the HK capped by the Abs_Max_Gen_Capacity considering both of the outage and the availability factors. 12 Tot_Gen Total energy generated from the system in MWhr including the storage plants, ROTR plants, IPPS, thermal plants and fixed generation from the non-optimized plants such as ARD. 13 Trade Is the amount of energy surplus or deficit after satisfying the long-term contracts and meeting the domestic load. 14 State_Prob The state transition probability from a given starting state to a given terminal states for each storage reservoir using the logic discussed earlier in the document. The size of the resulting matrix  equals the number of stages times number of starting states times number of inflows times the number of ending states times the number of release decisions. 105  Step Number Parameter Calculated Function 15 Spot_Buy, Spot_Sell Is the trade but capped by the export/import transmission limits  16 Trans_Prob Is the State_Prob of a storage reservoir times the State_Prob of the other reservoir. The size of the matrix equals the size of State_Prob matrix of GM Shrum times the size of the State_Prob matrix of Mica. 17 Trans_Prob_S Is the summation of the Trans_Prob matrix over different combination of inflows to the storage plants 18 Load_Reseource_Balance Is the energy balance of the system calculated as the total generation from the system minus the domestic load and the trade. 19 EX_Imp_Cost Expected import cost calculated as Spot_Buy times the Trans_Prob times the Imp_Price and summing the outcome over different combinations of inflows and terminal states for the storage reservoirs. 20 EX_Exp_Rev Expected export revenue calculated as Spot_Sell times the Trans_Prob times the Exp_Price and summing the outcome over different combinations of inflows and terminal states for the storage reservoirs. 21 Cont_Rev  Expected revenue/cost related to the long-term contracts calculated as contract trade times the Trans_Prob times the Cont_Price and summing the outcome over different combinations of inflows and terminal states for the storage reservoirs. 22 Policy_Income Is the current total expected revenue/cost for a given time step for different combinations of release decisions and starting states of the storage reservoirs, 106  Step Number Parameter Calculated Function calculated as the summation of EX_Imp_Cost, EX_Exp_Rev and Cont_Rev. 23 Value iteration calculations Calculates the value of water in storage along a pre-determined time horizon considering a pre-defined tolerance for convergence. The details of the procedure are discussed earlier in the document. 24 Optimum_Policy_GMS, Optimum_Policy_MCA Parameters that pick the release decisions for the storage plants that gives the maximum value of water. 25 MVW_GMS, MVW_MCA Is the marginal value of water which is the slope of the water value function for the storage reservoir. 26 MVE_GMS, MVE_MCA Is the marginal value of energy in $/MWhr calculated as the marginal value of water for GMS/ MCA divided by the total HK of the plants on the Peace River system (GMS, PCN, STC)/ the Columbia River system (MCA, REV). 107  A.1.3. Value Iteration Model  A.1.3.1. Code   108  A.1.4. Other Modules A.1.4.1. Code 1. HK Model   2. HK ROTR Model  109  3. HK Rough Model  110  4. Inflow ROTR Model  111  5. Maximum Generation Limits Model  112  6. Maximum Turbine Limits Model  7. FB Model  113  8. Prices Model    114  A.1.4.2. Computation Details Table 6: List of the Other Modules and Their Functions Module Parameter Calculated Function HK_"&v&".mod" HK Calculates the HK values for the storage plants as a function of each pair of starting and terminal storage states through regression equations capped by the maximum absolute value of the HK for the storage plants HK_ROTR_"&v&".mod HK_ROTR Calculated the HK values of the run-of-the-river plants as a function of the total flow that passes by those plants. HK_rough_"&v&".mod" HK_rough Calculates the total HK values on a river system for a given starting storage state of the storage plants. For instance, HK_rough for peace river would be the summation of the GMS’s HK, PCN’s HK and STC’s HK for a given GMS starting state. Inflow_ROTR_"&v&".mod Inflow_ROTR Calculated the natural inflows to the run-of-the-river plants as a function of the upstream storage plants through regression equations deduced from historical inflows. Max_Gen_Limits_"&v&".mod" Max_Gen_Limits Calculates the maximum generation for the storage plants as a function of the starting and ending storage capped by the absolute maximum generation capacity that dictates the generation curve after a certain point. The calculation is made through regression equations that relate the storage to generation. 115  Module Parameter Calculated Function QT_Max__"&v&".mod" QT_Max Calculates the maximum turbine flow for the storage plants as a function of the starting and terminal storage capped by the absolute turbine flow capacity that is when reached the flow must be reduced to account for the generator capacity. The calculation is made through regression equations that relate the turbine flow to generation. "FB_"&v&".mod" FBi, FBf Calculates the forebay corresponding to staring states “FBi” and the forebay corresponding to terminal states “FBf”. The calculation is done through regression equations that relate the storage to the forebay. “Prices__"&v&".mod" Imp_Price, Exp_Price, Cont_Price Calculates the prices as a function of total system  inflows    116  A.2. Main Sets and Parameters of the Model In the tables below, Table 7 and Table 8, the details of the main sets used in the SDPOM6R are presented.  Table 7: Main Sets and Parameters of the Model in an Alphabetical Order Set Definition Indexed Over Files21 Notes counter A counter for the number of starting storage states at each time step Reservoirs  The range is 1 to number of starting storage states  counterel A counter for the number of water release decisions Reservoirs & Months  The range is 1 to number of release decisions Counter A counter for the number of terminal storage states at each time step Reservoirs, Months, State & Inflows  The range is 1 to number of terminal storage states Inflows                                                                                                                                                                                                           Set of Inflow values foreach time step  Reservoirs & Months Inflows.dat Units in cms. Therange is The starting month to the end month ( controllable)                                                  21 Other than the main modules (Discretizer, Value Iteration and SDP) 117  Set Definition Indexed Over Files21 Notes Load  Electricity local demand for each time step Months Load.dat Units in MWh Months Time step (monthly)    Rel_Decision Set of total water release decisions for each time step Reservoirs & Months Rel_Decision.dat Total here means the sum of spills and turbine releases, units in cms Reservoirs  Set of reservoirs involved in the optimization process  Horizon.dat  State Set of starting storage  Reservoirs States.dat Units in cms.day Stater  Set of terminal storage states for each reservoir for each time step Reservoirs, Months, State & Inflows Staters.dat Units in cms.day Study_Years Set of the future years used in the study  YMD.dat   Table 8: The Main Parameters Used in the SDP Model in an Alphabetical Order Parameter Definition Indexed Over Files Notes a Coefficient provided in a flat file to calculate Months Price_Coef.dat  118  Parameter Definition Indexed Over Files Notes Import/Export/ contracted prices  B Coefficient provided in a flat file to calculate import/export/ contracted prices Months Price_Coef.dat   Cont_Price  Prices of contracted energy Months& Inflows   Cont_Rev Revenue from contracted energy  Months, Load, StateRel_Decision   Contr_Trans Contracted amount of energy (imports/exports) Months Contr_Trans.dat Units in MWh Days_Months  Number of days in each month for the entire planning horizon Study_Years & Months YMD.dat  Delta_Releases  Increment of total releases Reservoirs & Months  Units in cms Delta_Staters  Increment of terminal storage states Reservoirs & Months  Units in cms.day Delta_States  Increment of starting Reservoirs  Units in cms.day 119  Parameter Definition Indexed Over Files Notes storage states Desctz_Releases  Generated values of releases according to the pre-calculated release increment and maximum/minimum plant releases provided Reservoirs, Months &  counterel  Units in cms.day Desctz_Staters  Generated values of terminal storage states according to the pre-calculated state increment, maximum/ minimum storage values, starting states and inflow values Reservoirs, Months, State, Inflows & counter  Units in cms.day Desctz_States  Generated values of starting storage states according to the pre-calculated state increment and maximum/ minimum storage values provided Reservoirs & counter  Units in cms.day End_Months The last monthly time step   Horizon.d  120  Parameter Definition Indexed Over Files Notes in the planning horizon at EX_Exp_Rev Expected revenue from the spot market exports Months, Load, State & Rel_Decision  Units are in million $ EX_Imp_Cost Expected cost from the spot market imports Months, Load, State & Rel_Decision  Units are in million $ EX_Income Expected total income from export, import and contracts Months, Load, State & Rel_Decision  Units are in million $ Exp_Imp_Margin Difference between export and import prices   default 9.11, units are in $ Exp_Price  Prices of exported energy Months & Inflows  Units are in $ FBf Final forebay at each time step corresponding to a certain storage Reservoirs, Months, State, Inflows & Stater  Units are in m FBi Starting forebay at each time step corresponding to a certain storage Reservoirs, Months & State  Units are in m Generation Calculated generation of each plant  Reservoirs, Months, State,  Units in MWh 121  Parameter Definition Indexed Over Files Notes Inflows, Stater& Rel_Decision HK_rough a rough estimate for HK to use in MVW Reservoirs, Months & State   HK   Reservoirs, Months, State, Inflows & Stater   Imp_Price Prices of imported energy   Units in $ Inflow_Step   Reservoirs & Months  Units in cms IPP_Therm Generation of the independent power producers (IPP) and thermal plants. Added to the plants generation to get the total generation of the system involved  Months  Units in  MWh Iter Counter for number of iterations used to convergence in the value iteration procedure    Max_Gen_Limits  Limits on maximum Reservoirs & Gen_Limit Units in MWh 122  Parameter Definition Indexed Over Files Notes generation of each plant Months s.dat Max_Staters_Act Rounded number for maximum terminal storage states Reservoirs, Months, State &Inflows  Units in cms.day Max_Staters Provided value for maximum terminal storage states for each reservoir in each time step Reservoirs & Months State_Space.dat Units in cms.day Max_States_Act Rounded number for maximum starting storage states Reservoirs  Units in cms.day Max_States Provided value for maximum starting storage states for each reservoir  Reservoirs State_Space.dat Units in cms.day Min_Gen_Limits  Limits on minimum generation of each plant Reservoirs & Months Gen_Limits.dat Units in MWh Min_Staters_Act Rounded number for minimum terminal storage states Reservoirs, Months & State  Units in cms.day Min_Staters Provided value for minimum terminal storage Reservoirs & Months State_Space.dat Units in cms.day 123  Parameter Definition Indexed Over Files Notes states for each reservoir in each time step Min_States_Act  Reservoirs  Units in cms.day Min_States Provided value for minimum starting storage states for each reservoir Reservoirs State_Space.dat Units in cms.day MVW_GMS Marginal value of water at GMS  Months, State & State  Units are in $/MWh MVW_MCA Marginal value of water at GMS  Months, State & State  Units are in $/MWh N_Releases Number of releases generated according to the release increment and maximum/minimum releases for each plant  Reservoirs & Months   N_Staters Number of terminal storage states generated according to the storage  increment, maximum/minimum storage states , starting Reservoirs, Months, State & Inflows   124  Parameter Definition Indexed Over Files Notes storage states and inflows for each plant in each time step N_States Number of starting storage states generated according to the storage  increment and maximum/minimum storage states ,for each plant  Reservoirs   Outage Factors <=1 representing the maximum generation capacity ratio taken the outage schedule into consideration Reservoirs & Months Outage.dat  Plant_Release Plant release for each time step depending on the hydraulic balance and plant release limits Reservoirs, Months, State, Inflows, Stater & Rel_Decision  Units in cms Policy_Income  The total income of the policy which is the sum of the revenue from import, Months, Load,  State & Rel_Decision  Units in million $ 125  Parameter Definition Indexed Over Files Notes export and contract transactions.  Prob_Inflow  The probability of each inflow value provided for each plant in a certain time step Reservoirs, Months & Inflows Inflows.dat  PV_diff_total, PV_diff The total and sequential difference of the present value of water; used in the value iteration procedure Months & State  Units in million $ PV_Fin , PV_Max, PV_Temp Maximum present value of water at a certain time step over the terminal storage states and inflow scenarios Months & State  Units in million $ PV  Present water value Months, Load, State &Rel_Decision  Units in million $ QP_Max Maximum plant release  Reservoirs & Months QP_Limits.dat Units in cms QP_Max_Act Rounded value for maximum plant release Reservoirs & Months  Units in cms 126  Parameter Definition Indexed Over Files Notes QP_Min Minimum plant release Reservoirs & Months QP_Limits.dat Units in cmc QP_Min_Act Rounded value for minimum plant release Reservoirs & Months  Units in cms QT_Max Maximum turbine release Reservoirs & Months QT_Limits.dat Units in cmc QT_Min Minimum turbine release Reservoirs & Months QT_Limits.dat Units in cms Rate Interest rate   Default 0.05858 (monthly rate) SiW The surplus/deficit  of water in each reservoir  due to a transition from a storage state to another including the inflows and excluding the releases (Starting storage+Inflow-terminal storage) Reservoirs, Months, State,  Inflows, Stater & Rel_Decision  Units in cms.day Spill Forced spills ( Non-power spills) Reservoirs,  Months, State, Inflows,  Stater  Units in cmc 127  Parameter Definition Indexed Over Files Notes & Rel_Decision Spot_Buy   Months, Load, State,  Inflows, Stater& Rel_Decision  Units in million $ Spot_Sell  Months, Load, State,  Inflows, Stater& Rel_Decision  Units in million $ Start_Months The first monthly time step  in the planning horizon  Horizon.dat  State_Prob The probability attached to each storage transition state according to the values of the inflow and releases used Reservoirs,  Months,  State,  Inflows, Stater & Rel_Decision   Total_Sys_Inflow Summation of the system inflows for each time step Months & Inflows  Units in cms Tot_Gen Total system generation including thermal plants and the IPPs Months,  State,  Inflows, Stater &  Rel_Decision  Units in MWh 128  Parameter Definition Indexed Over Files Notes Trade  Export or Import spot energy transactions Months, Load, State, Inflows, Stater & Rel_Decision  Units in MWh Trade_Exp_Limit Export limits according to the capacity of transmission lines and other considerations  Months TradeLimits.dat Units in MWh Trade_Imp_Limit Import limits according to the capacity of transmission lines and other considerations Months TradeLimits.dat Units in MWh Trans_Prob  The joint probability of the transition from a combination storage state of the reservoir considered to another combination of terminal storage states Months, Load , State, Inflows, Stater & Rel_Decision   Trans_Prob_S  Sum of transition probability over the terminal storage states and Months, Load, State & Rel_Decision   129  Parameter Definition Indexed Over Files Notes inflows Turbine_Release Outflows coming through turbines to generate electricity (power spills) Reservoirs, Months, State, Inflows,  Stater & Rel_Decision  Units are in cms   130  A.3. How to Run the Model The main steps that the user has to follow are illustrated in the following workflow diagram.  Figure 41: Schematic of the Main Steps Needed to Run the SDPOM6RM Model 9 10 Check these files if interested in developing statistics about the run you have just finished User can pick the version desired to run according to the desired level of details. To make things easier, there is a brief documentation on each version in the same file. Where user can determine the length of the time horizon he/she likes to run the model for. Also, user can select the run-of-the-river plants considered in the run. Some of the plants have expansion plans in the near future, so if the run involves future years beyond 2018 user is advised to go and check the number of units used for each plant in this file. Having a look at these files representing the main data files used will give the user an idea about the different data sets involved in the run. When the user looks into the file Data_1_(# of version).dat, he/she will find out which inflow data file is currently activated and then he/she can go and open this files to check the inflow data used and the increment the model space is discretized by. Optional: the user can go and check all the other model files in the folder (i.e. filename.mod) to check how the regression equations of several parameters in the main model are formulated. The output files in this sub-folder are named in a way that enables the user to understand what data is included ineach file. The Inflows file Other Model Files if desired  The output folder will be created as a sub-folder inside the same folder that has the data files. The name of this sub-folder includes the number of stages included and the version used and the time of the run. Data_1_ (# of version).dat, Data_3_ (# of version).dat, and Data_3_ (# of version).dat   Run the file “SDP.run” in AMPL on the server or using the AMPL Job Manager Aftermath files (screen.out and Run_Times_Stats.amr) Version.dat Horizon_ (# of version).dat Units.dat State_Space.dat This file identifies the limits of the storage state-space of the storage plants. The user can control these limits according the scope of the study he/she is performing. In the folder, user can find different versions of the Inflows files; each has a unique name that identifies its content. The increment of the inflows in a given inflows file sets the increment of the decision-space and the state-space for each storage reservoir. 2 3 4 5 6 7 8 1 

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