HEDGING RULE FOR RESERVOIR OPERATION

HEDGING RULE FOR RESERVOIR OPERATION: HOW MUCH, WHEN AND HOW LONG TO HEDGE

BY JIING-YUN YOU B.S., National Taiwan University, 1997 M.S., National Taiwan University, 1999

DISSERTATION

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2008 Urbana, Illinois Doctoral Committee: Assistant Professor Ximing Cai, Chair Associate Professor Richard J. Brazee Professor Barbara S. Minsker Professor Hayri Önal

ABSTRACT Reservoirs have a significant role in the development of human civilization by regulating natural inflow for various human uses. After the peak period of dam and reservoir construction during the 20th century, effort has focused on reservoir operation, which is to regulate natural streamflow for storage and release utilization using existing reservoir systems.

Traditional reservoir operation policies for water supply seek to

minimize the water supply deficit in the current time period. However, with increased water demand in many regions, uncertain inflow in the future period can cause large water shortages.

It is then not always rational to satisfy the full current demand.

Hedging rule policies are designed for rationing water supply when it is below water demand. These polices make some delivery deficit at present to reduce the probability of greater water shortage in the future.

Hedging policies have increasingly attracted

attention because of growing water demand, increasing uncertainty in water sources, and more frequent drought events than before. This dissertation starts from the hedging policy in reservoir operation to illustrate the dynamicity of water systems, which addresses the allocation of resources under a dynamic framework. Previous studies mostly focused on applying numerical approaches to test hedging rules. Few studies have addressed the theoretical issues that need to take into account economic principles and hydrologic uncertainty systematically.

This

research attempts to establish systematic approaches to address fundamental questions of hedging in terms of when, how much and how long to hedge. These questions are related to the timing, magnitude and temporal scale of applying hedging rules, respectively. The dissertation explores how the decision making of hedging is influenced by physical conditions, hydrological characteristics and economic incentives.

The framework

especially addresses the importance of explicit stochastic approach and the impact of uncertainty on the decision making of hedging. The achievements of this dissertation include both theoretical and numerical developments, as well as a case study with a real reservoir operation problem. We developed a conceptual two-period model for reservoir operation with hedging that ii

includes explicitly future reservoir inflow. This theoretical analysis of the two-period model provides an updated basis for further theoretical studies and is used to enhance the numerical modeling. Extended analysis of the model properties and influencing factors is presented with a general utility function, addressing 1) the starting and ending water availability for hedging, 2) the range of hedging that is related to water demand levels, 3) inflow uncertainty, and 4) evaporation loss. Some intuitive knowledge on reservoir operation is proved or re-confirmed analytically; and new knowledge is derived. A numerical model which integrates derived hedging rules verifies the findings from theoretical analyses and also explores the knowledge on some influencing factors on reservoir operations, which are difficult to address quantitatively by theoretical analyses. Results show utility improvement with the hedging policy compared to the standard operation policy (SOP), considering factors such as reservoir capacity, inflow level and uncertainty, price elasticity and discount rate.

Following the theoretical analysis

presented in the companion paper, the condition for hedging application - the starting water availability (SWA) and ending water availability (EWA) for hedging are reexamined with the numerical example; the probabilistic performance of hedging and SOP regarding water supply reliability is compared; and some findings from the theoretical analysis are verified numerically. Following the development of a simple two-period model, the time scale (horizon) for hedging decision making is discussed by extending the horizon theory from operational research study to reservoir operation problems with continuous states. A stopping rule is purposed for detecting forecast horizon under a given decision horizon by using the marginal utility concept. This study conducted order of magnitude analysis and numerical modeling to identify the impact of various factors such as water stress level (the deficit between water availability and demand), reservoir size, inflow uncertainty, evaporation rate and discount rate. Three types of inflow time series are used: stationary, non-stationary with seasonality and random walk.

Results show that inflow

characteristics and reservoir capacity have major impacts on FH when water stress is modest; larger reservoir capacity and the deterministic component of inflow such as seasonality require a longer FH. Economic factors have strong impacts when water stress levels are high. iii

Considering the complexity of real world reservoir operation problems, this dissertation also developed a dual approach to determine the error between the myopic decision with a limited forecast horizon and the optimal decision with the ideal forecast horizon as long as the dynamic optimality requires. Applying this approach to numerical examples, it is found that the error generally shows power-law convergence. When the study horizon is extended, the error decreases quickly and approaches to a negligible level before the forecast horizon is achieved. The results also show that the error with a myopic decision increases with reservoir capacity and uncertainty and slightly decreases with the evaporation and discount rate for stationary inflow. But the trend becomes ambiguous due to the complexity of the different driving forces for non-stationary cases. In summary, the error bound method can be used to estimate the potential error which exists in actual operations with limited forecasts. The last part of this dissertation applied the theoretical and numerical developments to a real reservoir operation case, Lake Okeechobee in Southern Florida. Both inter-year and intra-year hedging model were developed for Lake Okeechobee to verify the generic findings of hedging rules. The results show that hedging rules could significantly reduce the utility loss compared to the standard operation policy (SOP). Hedging could be applied in an earlier time than that triggering the current hedging policy. The early hedging may mitigate the risk of severe water shortage and potential loss. It is also found that the operation of Lake Okeechobee should consider both short term and long term hedging which should be nested with each other. The study also shows that the rolling decision making, which uses a forecast horizon that is updated with period, could provide a practical way to approach to the optimal policy. However, the length of forecast and the accuracy of forecast are still important limitation to the application of hedging rules This dissertation adopts an interdisciplinary approach for water resources systems analysis by integrating knowledge, principles and methods form hydrology, engineering, economics and operational research. The research addresses fundamental questions of hedging such as when, how much and how long to hedge through both theoretical analysis and numerical modeling. These questions are related to the timing, magnitude and temporal scale of applying hedging rules, respectively. Detailed results show how iv

the decision making of hedging is influenced by physical conditions, hydrological characteristics and economic incentives.

The framework especially addresses the

importance of explicit stochastic dynamic approach and the impact of uncertainty on the decision making of hedging. The outcomes of this dissertation include 1) an analytical framework and a numerical model for hedging rule analysis, 2) an extended horizon theory and numerical procedures for estimating forecast horizon for optimal reservoir operation under hedging, 3) an examination of the error bound between the decision with the actual available forecast and the decision with idea perfect forecast, and 4) an application on a real reservoir operation problem. These are expected to contribute to the understanding of benefits, limits and impact factors on hedging rule and provide guideline to the application of hedging rules for rational reservoir operations.

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To My Parents, Ming-Yu Yu and Chao-Hua Hsu

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ACKNOWLEDGMENTS I still remember that, five years ago, I walked back from school in another snow day. I asked myself, is the PhD worth it? In these days, I was told I am not eligible to this program. I had no office, I had no funding, and I did not know what the future is going to be. At that moment, I was struggling and suffering. This question continuously pops up in my mind till now. Five years goes by, however, I am still not the one who deserved everything as others. I am paying more for this degree. And I still have no answer for whether this PhD worth it. No matter am I aware of the value of the PhD, or not. I finally finish this dissertation. This dissertation is the result of five years of work whereby I have been accompanied and supported by many people. It is my pleasure to have the opportunity to express my gratitude for all of them. First and foremost I sincerely thank my parents, Ming-Yu Yu and Chao-Hua Hsu. I really appreciate your patience and tolerance to allow this willful son spent so many years without any real achievement. Thank for your understanding all the time. Without you, I will have never been who I am. Thank for Hui-Hsia, one of the most important people in my life. Thank for so many years to be with me. Sorry for those dreams I have never realized. And sorry for those promises I have never make them come true. I feel a deep sense of gratitude for Dr. Yanqing Lian and Illinois State Water Survey. Your support financially and emotionally helped me passing the coldest winter in my life. Without your help, I could have already quit from University of Illinois. These days I worked in State Survey is a fun, educational and memorable experience. I am so glad I have the chance to work with you. I am very grateful for my Master’s advisor, Dr. Hong-Yong Lee, and other teachers at National Taiwan University, Dr. Tim-Hau Lee and Dr. Liang-Hsiung Huang, who encouraged me to pursuit the doctoral degree, and also be my mentors when I met difficulty during this period. I also want to thank group members, Mohamad, Dingbao, Yi-Chen, and Jihua. This group has been a source of friendships as well as good advice and collaboration. vii

We share the joy and frustration with each other. I appreciate you guys have contributed immensely to my personal and professional time here. Besides the group, I want to thank I-Chi. During the most difficult time, we walked through the streets together to overcome the depression. I also want to thank Hsi-Heng for these lunchtimes we shared our dream of the future. And I am so happy that you are already moving ahead in your way. I would like to thank Dr. Richard J. Brazee who plays a very significant role in my research. Your course opened a new door for me and gave me different perspectives. I also very appreciate your encouragement in all stages of my PhD study. It is very important for a student who always wonders the value of himself. I am also grateful to the other members in the committee, Dr. Barbara S. Minsker, Dr. Hayri Önal, and Dr. Paven Kumar. Thanks for you to monitor my work and take effort in reading and providing me with valuable comments on earlier versions of this thesis. I sincerely thank for my advisor, Dr. Ximing Cai. Thanks for your kindness to let me stay and prove myself. You taught me the inestimable value of knowledge. Thanks for your generosity from the beginning to the end. Thanks for your patience, understanding, time and effort although I am not a good enough student at all. As I always said, if I am dying tomorrow, I will never regret I do not get my doctoral degree. I will regret I did not spend enough time with people whom I love and who love me. However, I am so fortunate and blessed to keep breathing till I receive my PhD. Thanks for God’s mercy. And I believe, anything we face and experience today is for a better tomorrow. I beg and I pray.

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TABLE OF CONTENTS LIST OF TABLES .......................................................................................................................... xi LIST OF TABLES ......................................................................................................................... xii CHAPTER 1. 

INTRODUCTION ............................................................................................... 1 

1.1. 

Motivation ...................................................................................................................... 1 

1.2. 

Summary of the research objectives ........................................................................... 2 

CHAPTER 2.  ANALYSIS

HEDGING RULE FOR RESERVOIR OPERATIONS: A THEORETICAL ............................................................................................................................. 8 

2.1. 

Introduction ................................................................................................................... 9 

2.2. 

Conceptual Model and Analysis ................................................................................ 11 

2.3. 

Extended Analysis ....................................................................................................... 16 

2.4. 

Conclusions .................................................................................................................. 23 

Appendix I ................................................................................................................................. 26  Appendix II ................................................................................................................................ 27  References .................................................................................................................................. 28  CHAPTER 3.  HEDGING RULE FOR RESERVOIR OPERATION OPERATIONS: A NUMERICAL MODEL................................................................................................................. 38  3.1. 

Introduction ................................................................................................................. 39 

3.2. 

Model Specification and Extended Analysis ........................................................... 40 

3.3. 

Numerical Model - The Multi-State Markov Model with Hedging Rule ............. 42 

3.4. 

Results of the Analysis ................................................................................................ 47 

3.5. 

Conclusions .................................................................................................................... 51 

Appendix I: Derivation of price elasticity based on the utility function .................................... 54  Appendix II: Derivation of delivery ratio based on the utility function .................................... 55  Appendix III: Derivation of operational condition by replacing the economic discount with evaporation................................................................................................................................. 56  References .................................................................................................................................. 58 

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CHAPTER 4.  DETERMINING FORECAST AND DECISION HORIZONS FOR RESERVOIR OPERATIONS UNDER HEDGING POLICIES ................................................... 72  4.1. 

Introduction ................................................................................................................. 73 

4.2. 

Problem Formulation and Justification ................................................................... 75 

4.3. 

Order of Magnitude Analysis ..................................................................................... 80 

4.4. 

Numerical Examples ................................................................................................... 82 

4.5. 

Discussions and Conclusions ..................................................................................... 89 

Appendix I ................................................................................................................................. 93  References .................................................................................................................................. 97  CHAPTER 5.  ERROR BOUND OF THE OPTIMAL RESERVOIR OPERATION DECISION UNDER A LIMITED FORECAST HORIZON .......................................................................... 115  5.1. 

Introduction ............................................................................................................... 116 

5.2. 

Methodology .............................................................................................................. 118 

5.3. 

Numerical Examples ................................................................................................. 123 

5.4. 

Results ........................................................................................................................ 124 

5.5. 

Discussion and Conclusion ...................................................................................... 129 

References ................................................................................................................................ 131  CHAPTER 6.  IMPROVE HEDGING RULES FOR THE OPERATION OF LAKE OKEECHOBEE IN SOUTHERN FLORIDA ............................................................................. 147  6.1. 

Introduction ............................................................................................................... 148 

6.2. 

Methodology .............................................................................................................. 154 

6.3. 

Results ........................................................................................................................ 164 

6.4. 

Conclusions ................................................................................................................ 172 

References ................................................................................................................................ 175  CHAPTER 7. 

CONCLUSION ................................................................................................ 210 

References ................................................................................................................................ 219  CURRICULUM VITAE .............................................................................................................. 220 

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LIST OF TABLES Table 3.1. Numerical equations of hedging under various conditions ........................................... 61  Table 4.1. Inventory example for Bes and Sethi’s method. ......................................................... 100  Table 4.2. The relationships between FH and the impact factors ................................................ 101  Table 4.3. Parameters used in the numerical example, basic scenario and other scenarios ......... 102  Table 5.1. Parameters used in the numerical example, basic scenario and other scenarios ......... 133  Table 6.1. LOSA irrigated acreage (SEIS, 2007) ........................................................................ 178  Table 6.2. Agricultural land uses in EAA (SEIS, 2007) .............................................................. 178  Table 6.3. Agricultural land uses in the Caloosahatchee basin(SEIS, 2007) ............................... 178  Table 6.4. Agricultural land uses in the Caloosahatchee basin (SEIS, 2007) .............................. 178  Table 6.5. Profit functions for inter-year hedging model............................................................. 179  Table 6.6. Crop Price for each month in Southern Florida (SFWMM) ....................................... 179  Table 6.7. Water stress response coefficents of major crops in Southern Florida ....................... 179  Table 6.8. Crop evapotranspiration coefficient (kc) of major crops in Southern Florida (AFRIS manual) ........................................................................................................................................ 179  Table 6.9. Pearson correlation coefficient from the linear regression model .............................. 180  Table 6.10 Pearson correlation coefficient from the artificial neural network (ANN) model for each month ................................................................................................................................... 181  Table 6.11. Pearson correlation coefficient from the artificial neural network (ANN) model with month indicator for whole period ................................................................................................ 182  Table 6.12. Ex post hedging analysis during Oct-71 to Sep-74 (36 months) .............................. 183  Table 6.13. Ex post hedging analysis during Oct-81-Sep-82 (24 months) .................................. 183  Table 6.14. Ex post hedging analysis during Oct-89 – Apr 92 (32 month) ................................. 183 

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LIST OF FIGURES Figure 1.1. A research framework on hedging rule for reservoir operation..................................... 7  Figure 2.1. Trajectory of Pareto efficiency .................................................................................... 31  Figure 2.2. Optimal release policy ................................................................................................. 32  Figure 2.3. Standard operating policy ............................................................................................ 33  Figure 2.4. Hedging Rule Operation Policy................................................................................... 34  Figure 2.5. The situation of not applying hedging rule .................................................................. 35  Figure 2.6. Equilibrium under uncertainty. .................................................................................... 36  Figure 2.7. The influence of uncertainty to hedging rule ............................................................... 37  Figure 3.1. The relationship of hedging ratio and price elasticity under different discount rates .. 62  Figure 3.2. Diagram of the multi-state Markov model embedded with the hedging rule .............. 63  Figure 3.3. Starting water availability (SWA) and ending water availability (EWA) under three water stress levels (   0 , -0.5, and -1). ....................................................................................... 64  Figure 3.4. Cumulative distribution function (CDF) of release under hedging and SOP under two water stress levels. Other major parameters are K=2, Cv=1, n=2................................................. 65  Figure 3.5. Probability distribution function (PDF) of storage under hedging and SOP under two water stress levels. Other major parameters are K=2, Cv=1, n=2................................................. 66  Figure 3.6. Utility improvement in percentage from SOP to hedging under various uncertainty levels when  =-1 .......................................................................................................................... 67  Figure 3.7. Utility improvement in percentage from SOP to hedging under various uncertainty levels when  =-2 .......................................................................................................................... 68  Figure 3.8. Utility improvement from SOP to hedging vs. water stress levels under various uncertainty levels when K=5 ......................................................................................................... 69  Figure 3.9.The influence of price elasticity on hedging and SOP ................................................. 70  Figure 3.10. The influence of discount rates on hedging and SOP ................................................ 71  Figure 4.1. The concept of DH/FH .............................................................................................. 103  Figure 4.2. The pattern of a non-stationary time series with consideration of seasonality .......... 104  Figure 4.3. Flow chart of procedures to determine FH (T) with a given DH (t) .......................... 105 

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Figure 4.4. Relationship between FH and water stress index under different reservoir capacities (CAP), with inflow uncertainty variance coefficient (CV) = 0.2................................................. 106  Figure 4.5. Relationship between FH and water stress index under different uncertainty levels (variance coefficient, CV), CAP = 10 .......................................................................................... 107  Figure 4.6. Relationship between FH and water stress index under different discount rates (r), CAP=10, CV=0.2 ......................................................................................................................... 108  Figure 4.7. Relationship between FH and water stress index under different evaporation rates (EVP), CAP=10, CV=0.2 ............................................................................................................ 109  Figure 4.8. Relationship between FH and water stress index under different utility concavity values (n), CAP=10, CV=0.2 ....................................................................................................... 110  Figure 4.9. Relationship between FH and water stress index under different amplitudes (AMP), CAP=10 ....................................................................................................................................... 111  Figure 4.10. Relationship between FH and water stress index under different seasonality lengths (P), CAP=10................................................................................................................................. 112  Figure 4.11. Relationship between FH and water stress index under different randomness levels (R), CAP =10 ............................................................................................................................... 113  Figure 4.12. Relationship between FH and inflow for speed of prorogation (SP), CAP ............. 114  Figure 5.1. The concept of rolling horizon .................................................................................. 134  Figure 5.2. The two boundary conditions (free-end and a targeted storage) with a limited study horizon. ........................................................................................................................................ 135  Figure 5.3. The two boundary conditions (free-end and the maximum storage) with a limited study horizon................................................................................................................................ 136  Figure 5.4. Computation procedures to evaluate for the error bound with limited forecast horizons ..................................................................................................................................................... 137  Figure 5.5. Relationship between study horizon T and water stress index under different reservoir capacities (CAP), inflow uncertainty variance coefficient (CV) = 0.2 ........................................ 138  Figure 5.6. Relationship between study horizon T and water stress index under different uncertainty levels (variance coefficient, CV), CAP = 10 ............................................................ 139  Figure 5.7. Relationship between study horizon T and water stress index under different discount rates (r), CAP=10, CV=0.2 .......................................................................................................... 140  Figure 5.8. Relationship between study horizon T and water stress index under different evaporation rates (EVP), CAP=10, CV=0.2 ................................................................................ 141 

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Figure 5.9. Relationship between study horizon T and water stress index under different utility concavity values (n), CAP=10, CV=0.2 ...................................................................................... 142  Figure 5.10. Relationship between study horizon T and water stress index under different amplitudes (AMP), CAP=10 ........................................................................................................ 143  Figure 5.11. Relationship between study horizon T and water stress index under different seasonality lengths (P), CAP=10 ................................................................................................. 144  Figure 5.12. Relationship between study horizon T and water stress index under different randomness levels (R), CAP =10 ................................................................................................. 145  Figure 5.13. Utilities from different terminal boundary conditions vs. water stress levels ......... 146  Figure 6.1. Conceptual framework of EAA hedging ................................................................... 184  Figure 6.2. Water balance for EAA ............................................................................................. 185  Figure 6.3. Frequency analysis of I LOK ....................................................................................... 186  Figure 6.4. Frequency analysis of AL .......................................................................................... 187  Figure 6.5. Conceptual diagram of intra-year hedging model ..................................................... 188  Figure 6.6. Schematic diagram of the network for intra-year model ........................................... 189  Figure 6.7. Statistical parameters of monthly evaporation data ................................................... 190  Figure 6.8. Comparison of inflow between SFWMM and linear regression model ................... 191  Figure 6.9. Comparison of water demand between SFWMM and linear regression model ....... 192  Figure 6.10. Comparison of Lake Okeechobee stage between SFWMM and intra-year hedging model. .......................................................................................................................................... 193  Figure 6.11. Loss reduction from SOP to hedging ...................................................................... 194  Figure 6.12. The probability of different operational bands between hedging and SOP ............. 195  Figure 6.13. Loss reduction for different water stress (Inflow to LOK) ...................................... 196  Figure 6.14. Loss reduction for different uncertainty (Inflow to LOK)....................................... 197  Figure 6.15. Loss reduction for different water stress (EAA water availability) ......................... 198  Figure 6.16. Comparison between SOP and ex post analysis of hedging rules: Oct-71 to Sep-74 (36 months) .................................................................................................................................. 199  Figure 6.17. Comparison between SOP and ex post analysis of hedging rules: Oct-81-Sep-82 (24 months) ........................................................................................................................................ 200 

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Figure 6.18. Comparison between SOP and ex post analysis of hedging rules: 16 Oct-89 – Apr 92 (32 month)............................................................................................................................... 201  Figure 6.19. Marginal loss of water shortage: Oct-71 to Sep-74 (36 months) ............................. 202  Figure 6.20. Marginal loss of water shortage: Oct-81-Sep-82 (24 months) ................................ 203  Figure 6.21. Marginal loss of water shortage: Oct-89 – Apr 92 (32 month) ............................... 204  Figure 6.22. Comparison between analysis of hedging and SOP in four different time points ... 205  Figure 6.23. Comparison between marginal losses of hedging and SOP in four different time points ............................................................................................................................................ 206  Figure 6.24. Release of ex post and rolling policies and demand for 2000-2001 drought .......... 207  Figure 6.25. Water shortage of SOP, ex post analysis and rolling policy for 2000-2001 drought ..................................................................................................................................................... 208  Figure 6.26. Error bound examination for rolling policy in 2000 water year .............................. 209 

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CHAPTER 1. 1.1.

INTRODUCTION

Motivation Reservoirs have a significant role in the development of human civilization by

regulating natural inflow for various human uses. In the early and middle 20th century, human beings made a great effort in developing techniques for structural design and construction of reliable dams.

However, because of significant social impacts and

environmental concerns, decisions on new reservoirs have become very controversial and dam construction in many regions has been almost impossible. Nowadays, attention focuses on how to operate existing reservoir systems for effective storage use. Reservoir operations use certain operation rules.

Most operation rules are

designed to minimize current water supply deficit or maximize the profit in the current time period. During normal periods when inflow is plentiful these polices do not pose problems. However, during a drought period, a reservoir may need to reduce release in the current period to ensure an appropriate amount of water to avoid the possible shortage in the future. Hedging rule policies are designed to rationally allocate water resources over time. Bower et al. (1962) first provided a systematic economic description of hedging rules for water resources systems operations. Their classic work addressed the inter-temporal resource allocation and profit optimization within a dynamic framework. Hedging rules normally used for rationing operation to smooth the deficit fluctuation of water shortage. Hedging operating policies reduce the intensity of a more severe shortage which likely occurs in future by only providing portion of the target yield in advance. However, the inherent uncertainty of reservoir operation complicates the decision making of hedging. The random nature of reservoir inflow and other related hydrologic variable make the implementation of hedging rules difficult. Practically, the reservoir manager would be interested in answering questions, including when, how much, and how long to hedge, which are about the timing, magnitude and temporal characteristics of applying hedging rules, respectively. This dissertation studies hedging rule policies for reservoir operations by addressing fundamental questions including 1) when to hedge; 2) how much to hedge and 1

3) how long to hedge? (Figure 1.1) Given growing water stress and more frequent droughts than before in many areas of the world, insights on hedging rules regarding the benefits, limits and impact factors are still in need in order to provide guidelines for rational reservoir operations with consideration of future water demand and risk of drought loss, as well as those of present. Previous studies mostly focused on applying numerical approaches to test hedging rules. Few studies have addressed the theoretical issues that need to take into account economic principles and hydrologic uncertainty systematically. Moreover, hedging analysis needs to use the concept of forecast horizon, i.e. how long we should look into future. This concept has been developed in operations research and applied mainly in business management, but not in water and environmental systems.. 1.2.

Summary of the research objectives This thesis research establishes a systematic approach for addressing hedging

rules and provides a more comprehensive understanding of the fundamental questions and related issues, including when, how much, and how long to hedge, in terms of the timing, magnitude and temporal scale of applying hedging rules, respectively. The analysis involves the discussion on how the decision making of hedging is influenced by physical conditions, hydrological characteristics and economic incentives. In particular, the lack of perfect foresight and/or the reliability of forecast will make hedging operation complex and difficult. This thesis especially addresses the importance of an explicit stochastic approach and the impact of uncertainty on the decision making of hedging. The outputs include both theoretical and numerical developments, a case study with a real reservoir, result interpretations and policy implications.

Theoretical and numerical

findings are translated into practical operating rules for real world reservoir operations. The major research topics included in this thesis research are described as below. To establish a conceptual model for hedging analysis This dissertation firstly expands a theoretical analysis and develops a conceptual two-period model for reservoir operation with hedging that includes uncertain future reservoir inflow explicitly. Extended analysis of the model properties and influencing factors is presented with a general utility function, addressing 1) the starting and ending 2

water availability for hedging, 2) the range of hedging that is related to water demand levels, 3) inflow uncertainty, and 4) evaporation loss. Some intuitive knowledge on reservoir operation is proved or re-confirmed analytically and new knowledge is derived. This theoretical analysis provides an updated basis for further theoretical study and the theoretical findings can be used to improve numerical modeling for reservoir operation. To develop a numerical model for hedging rule analysis Following, this dissertation presents a method that derives a hedging rule from theoretical analysis with an explicit two-period Markov hydrology model, a particular form of nonlinear utility function, and a given inflow probability distribution. The unique procedure is to embed hedging rule derivation based on the marginal utility principle into reservoir operation simulation. The simulation method embedded with the optimization principle for hedging rule derivation will avoid both the inaccuracy problem caused by trial-and-error with traditional simulation models and the computational difficulty (“curse of dimensionality”) with optimization models. Results show utility improvement with the hedging policy compared to the standard operation policy (SOP), considering factors such as reservoir capacity, inflow level and uncertainty, price elasticity and discount rate. Following the theoretical analysis presented in the previous object, the condition for hedging application - the starting water availability (SWA) and ending water availability (EWA) for hedging are re-examined with the numerical example; the probabilistic performance of hedging and SOP regarding water supply reliability is compared; and some findings from the theoretical analysis are verified numerically. Extend the theory of horizon for reservoir operations under hedging rules One of the critical questions for hedging research is how long the forecast period should be taken so that reliable inflow forecast in a given period can be used for decision making under hedging. Decision makers always hope to look further into the future; however, the longer the forecast period, the more uncertain and less reliable the involved information, which will have a diminishing influence on decision making. For dynamic reservoir operation optimization models, when the decisions in the initial few periods are not affected by forecast data beyond a certain period, the period is known as forecast 3

horizon (FH), and the number of the initial periods is known as the decision horizon (DH). This study detects FH with given DH for dynamic reservoir operation problems through both theoretical and numerical analysis. Through an order of magnitude analysis and numerical modeling, forecast horizon is examined under various impact factors such as water stress level, reservoir size, inflow uncertainty, evaporation rate and discount rate. Two stochastic cases are tested with respect to the impact of inflow uncertainty: 1) stationary inflow time series; and 2) non-stationary inflow time series considering seasonality. According to the numerical example, inflow characteristics and reservoir capacity have major impacts on FH when water stress is modest; larger reservoir capacity and the deterministic component of inflow such as seasonality always requires a longer FH. Economic factors have strong impacts with high water stress levels. Examine the error bound of reservoir operation with limited forecast horizon: The error bound of reservoir operation is examined with rolling horizon decision making, which is a practical procedure in dynamic system problems, including in reservoir operation. The rolling-horizon procedure is practical for reservoir operation due to the nature of the problem. In real-world operations, reservoir managers approve the operation plan at the beginning of each period according to the best available forecast data; this process is undertaken at a weekly or monthly basis using rolling climate and hydrologic forecasts. It is supposed, mostly being true, that the choice of study horizon in each period is shorter than the forecast horizon required by optimal decision making, as discussed in previous part of this dissertation. Myopic horizons chosen in the rollinghorizon procedure may lead to suboptimal decisions.. However, for practicality, how to choose a myopic study horizon with reasonable error bound to the decision can be useful exercise. To conduct an application study of hedging rules with Lake Okeechobee Hedging rule policies are designed to rationally allocate water resources over time. Previous studies have addressed fundamental questions on hedging rules for reservoir operations including when, how much and how long to hedge. This study applies the theoretical findings of hedging policies to Lake Okeechobee in south-central Florida to explore the potential of hedging rule policies for the operation of Lake Okeechobee. 4

Lake Okeechobee is located in south-central Florida. The lake is the major water supply source to agricultural and urban users and the Everglades National Park in the southern Florida.

The Lake also contributes to functions such as flood control, navigation,

recreation, and fish and wildlife protection. Extensive studies, including those on hedging rules, have been conducted for the Lake. Since July 2000, Water Supply and Environment (WSE) regulation, approved by the U.S. Army Corps of Engineers (USACE) and the Sothern Florida Water Management District (SFWMD), was used for Lake Okeechobee to balance the different, always competing objectives. Most recently the Tentatively Selected Plan (TSP), that is supposed to improve the reservoir operations, replaced WSE in August 2008. Compared to WSE, TSP, a daily-based regulation schedule with new bands, new release magnitudes, and new forecasting indices, is believed to be more effective in decreasing the risk to public health and safety, providing a reduction of damaging events and salinity violations to the estuaries, and providing critical flexibility to perform water management operations. This study applies hedging policies to Lake Okeechobee in to explore the potential of hedging rule policies for the operation of the Lake Okeechobee. Given the complexity in real problems, the key to apply the theoretical and numerical developments are to introduce reasonable assumptions for specific problems and appropriately simulate the reservoir systems.

Additional procedures are needed to translate the theoretical

developments into practical problem solving with judgments on the rationality of the assumptions involved in specific applications. By real practice, this study is expected to discover the knowledge gap between theoretical studies and real-world implementations. Accordingly, we hope the findings of new knowledge of hedging rules could help to improve reservoir operation in the real world. This Chapter has laid out the motivation and goals of hedging rules and also introduces the conceptual framework of this dissertation. Chapter 2 describes the development a conceptual model for hedging analysis. Chapter 3 describes the development a numerical model for hedging rule analysis. Chapter 4 develops the theory of horizon for reservoir operations under hedging rules and demonstrates its application in the numerical examples. Chapter 5 examines the error bound of reservoir operation with limited forecast horizon. In Chapter 6 presents an application study of hedging 5

rules with Lake Okeechobee.

Finally, Chapter 7 provides some conclusion and

discusses potential future works. As a summary, the content of this thesis research includes: 1) an analytical framework and a numerical model for hedging rule analysis, 2) an extended horizon theory and numerical procedures for estimating forecast horizon for optimal reservoir operation under hedging, 3) an examination of the error bound between the decision with the actual available forecast and the decision with idea perfect forecast, and 4) an application on a real reservoir operation problem. Through studying these issues, this thesis research is expected to contribute to the understanding of benefits, limits and impact factors on hedging rule and provides guideline on the application of hedging rules for rational reservoir operations.

Moreover, this thesis illustrated the necessity and

effectiveness of inter-disciplinary studies (economics – hydrology and water resources engineering – system operations) for water resources system analysis.

6

  Figure 1.1. A research framework on hedging rule for reservoir operation.

7

CHAPTER 2.

HEDGING RULE FOR RESERVOIR

OPERATIONS: A THEORETICAL ANALYSIS Summary Hedging rule policies for reservoir operations accept small deficits in current supply to reduce the probability of a severe water shortage later. This chapter presents a theoretical analysis and develops a conceptual two-period model for reservoir operation with hedging that includes uncertain future reservoir inflow explicitly. Extended analysis of the model properties and influencing factors is presented with a general utility function, addressing 1) the starting and ending water availability for hedging, 2) the range of hedging that is related to water demand levels, 3) inflow uncertainty, and 4) evaporation loss.

Some intuitive knowledge on reservoir operation is proved or re-confirmed

analytically and new knowledge is derived. This theoretical analysis provides an updated basis for further theoretical study and the theoretical findings can be used to improve numerical modeling for reservoir operation.

8

2.1.

Introduction Reservoir operation for water supply is not always rational to satisfy the full

current demand, due to the possibility of larger water shortages in the future. Hedging rule policies are designed for rationing water supply in appropriate preparation for potential low inflows in the near future. These policies accept some present delivery deficit to reduce the probability of greater water or energy shortage in the future (Bower et al, 1962). The application of hedging in finance offers insight for hedging in reservoir operations. The objective of hedging in finance is to avoid risk through purchasing an additional portfolio; the perfect hedge ends with a portfolio that eliminates the risk completely. A comparable application of hedging in reservoir operations is to keep water for later use to reduce the risk of water shortage in the current and future time periods. Following a procedure similar to Rippl’s method, the standard operating policy (SOP) (Mass et al. 1962; Loucks et al. 1981; Stedinger 1984) was developed for reservoir operation under a fixed water delivery target. SOP releases water as close to the delivery target as possible, saving only surplus water for future delivery. When the objective of reservoir operation is to minimize the expected value of deficit over the decision horizon or any other linear function of deficit (Klemes 1977, 1979), SOP is the optimal operating policy (Burness and Quirk 1978; Hashimoto et al 1982). SOP is practical during periods of operation when inflow is plentiful.

However, it neglects potential shortage

vulnerability during later periods. Hedging was first introduced for rationing operation according to time preference of water storage in the field of natural resources economics. Masse (1946) analyzed reservoir operation problems using the economic concept of marginal value.

He

introduced the use of ‘reserves’ or ‘stocks’ of water to avoid or reduce shortages in the future. Following Masse, Gessford and Karlin (1958) presented a general mathematical analysis for optimal reservoir release policy and discussed the conditions of the existence of an optimal policy. Bower et al. (1962), during the Harvard Water Program in the 1960s, were the first to provide a systematic economic description of hedging rule in water resources systems operations, which addressed the allocation of resources over time and sought the inter-temporal maximum benefit under a dynamic framework. Bower’s study laid the economic basis for hedging. 9

Hedging for reservoir operations did not draw much attention during the decades following Bower’s study during the Harvard Water Program. It was not until the late 1970’s that hedging received attention from the hydrologic engineering community (Klemes 1977, Stedinger 1978, Loucks et al. 1981, and Hashimoto et al. 1982). These studies used various forms of planning and management objectives to optimize reservoir operations and demonstrated the concept explained by Bower et al. (1962) with explicit consideration of hydrologic and engineering constraints. Hedging was interpreted with stronger implications for operations than for economic efficiency - “providing only portion of the target release, when in fact all or at least more of the target volume could be provided” (Hashimoto et al. 1982).

Following these studies, hedging has been

explored to address reservoir operation problems focusing on minimizing utility loss or water supply deficit over drought periods (e.g., Shih and Revelle, 1994 and 1995). Most recently, hedging was again addressed in the context of economic water operation due to increasing water stress over the world.

Draper and Lund (2003)

expanded the objective of hedging from reducing gross loss to increasing net benefit by replacing water supply deficit with water use benefit. They argued that hedging rules curtail deliveries to retain water in storage for use in later periods. This procedure obtains a larger overall benefit so that “hedging provides insurance for higher-valued water uses where reservoirs have low refill potentials or uncertain inflows.”

Under such an

economic analytical framework, Draper and Lund illustrated dynamic equilibrium under hedging and demonstrated the efficacy of hedging in balancing current and future utility with a given utility function (also see details in Draper [2001]). Draper and Lund’s findings have stimulated further research on how the balance is influenced by various factors such as hydrologic and economic uncertainties. Following Draper and Lund (2003), this paper expands a conceptual model with a focus on analyzing optimal decision conditions by the marginal utility principle. Based on the model properties, several questions about the concept and realization of hedging are addressed, including 1) at what level of current water availability should hedging be considered, and how is the level affected by water demand and reservoir inflow? 2) Under what conditions is hedging trivial? 3) What is the effect of inflow uncertainty on

10

hedging? 4) What is the effect of reservoir evaporation on hedging?

Through

overviewing these questions, we can better understand hedging for reservoir operations. 2.2.

Conceptual Model and Analysis The uncertainty of future reservoir inflow complicates the analysis of hedging.

This study explicitly considers the uncertainty in a conceptual model. The model is based on the marginal utility principle.

The “ideal” hedging rule is first derived;

following that by an integrated economic-engineering approach, the realistic solution is discussed subject to physical constraints such as reservoir storage capacity and water delivery capacity. 2.2.1. Expansion of a conceptual model to incorporate uncertain future inflow A classic two-period model in natural resources economics is constructed, considering a single consumer and inter-temporal allocation of water as first introduced by Fisher (1930). This model defines economic value by a utility function of a composite user. According to Draper and Lund (2003), the condition for the optimal hedging is stated as “at optimality the marginal benefit of storage must equal the marginal benefits of release”:  B ( D ) C ( S )  D S

(2.1)

Where B (.) is the current water delivery benefit, D is the water delivery, C (.) is the carryover storage value function representing the expected value of water stored now (S) for all future time periods. Both B (.) and C (.) are non-linear utility functions. Note that when the utility or loss function is linear, SOP is the optimal operating policy (Hashimoto et al., 1982). The sum of D and S represents the current water availability (A):

AS D

(2.2)

Eq. (2.2) does not explicitly account for future inflow to the reservoir but implicitly considers this information in C ( S ) . The benefit from carryover storage value (C) can be a nonlinear function of the second and future periods of inflow too. According to the principle of decreasing marginal utility, the marginal value of carryover 11

storage will decline with additional water availability. Therefore, neglecting future water availability may not lead to an optimal total benefit over two periods. To account for the effect of the second-period inflow, the model proposed by Draper and Lund (2003) is modified by introducing the second period inflow ( I f ). The total water availability in two periods is: Atotal  A  I f

(2.3)

in which A represents water availability in the current period, which is a known item in the two-period model. For simplicity, evaporation is temporarily ignored in Eq (2.3), but it will be considered later in this chapter. We introduce another utility function, E (.) to represent the expected utility generated for future delivery. The two-period model is to maximize the profit in two periods, subject to the mass balance relation:. Max B ( D )  E ( S  I f ) D ,S

(2.4)

s.t. Atotal  S  D  I f

Both B (.) and E (.) are concave functions expressed in present value.

The

concavity of utility for water stored for the future was originally proved by Gal (1979) and illustrated further recently by Draper and Lund (2003). It should be noted that the definition of the storage variable (S) differs from that in Draper and Lund (2003), who defined S as the water available for the future time period including future inflow. In our study, S is the reservoir storage saved from current period into the future. 2.2.2. Economic solution – Ideal hedging rule Eq. (2.4) represents a pure economic model without considering engineering constraints. Eq (2.4) can be written in a Lagrange form: L= B ( D )  E ( S  I f )   ( Atotal  S  D  I f )

(2.5)

where L is Lagrange function and  is Lagrange multiplier. We further assume that I f has an independent identical distribution (IID). IID means that a collection of random variables ( I f ), for example, annual reservoir inflows in different years, have the same probability distribution, but the random variables (inflows) 12

in different years are

mutually independent.

The IID assumption is not suitable for some cases such as

multiple-year nested droughts, which imposes a restriction for the following analysis. However, a classical two-period model usually assumes the non-existence of correlations between the two periods. Following the assumption of IID, a partial derivative of I f over any variables is zero due to its independence, i.e., decisions cannot influence the occurrence of I f . In Eq. (2.5), it is also known that   0 for an equality constraint. Applying the first-order condition to the Lagrange function: L  B ( D ) =   0 D D L E ( S  I f ) =   0 S S

(2.6)

from which a new optimal condition can be represented as:

B( D) E ( S  I f ) = D S

(2.7)

Compared to Eq. (2.1), Eq. (2.7) explicitly accounts for uncertain future inflow in the next period. For a two-period model, the water availability to the second period ( S  I f ) is the water delivery in the period. This implies that no water should be left at

the end of the second period if the total water availability in the two periods is less than the total demand. This reflects the decision mechanism but also the limitation (i.e., shortsighted decision) of the classic 2-period model commonly used in natural resources economics. Thus, the right hand side of Eq. (2.7) represents the marginal value of water delivery in the second period. Eq. (2.7) represents the principle of marginal value, which can be expressed in general for two consecutive periods:

B( D) B( D)  D t1 D t 2

1

Since I f  0, we can write S   ( S  I f )

13

(2.8)1

This is consistent with a general marginal analysis conducted by Masse (1946). It also represents a well-known “Pareto efficiency” which states that no period can be improved without making sacrifices from another period. In this model, the two periods compete for water. When they have the same marginal utility, the total utility from two periods reaches the optimal value. When the periods (t1, t2 …) are long (e.g., annual) and discount rates are significant, the marginal utility in future periods is expressed in present value. Considering the duality of Eq.(2.4), the original problem can be converted into a dual problem that minimizes water supply under a given total utility in these periods. Figure 2.1. shows the isoquant curves associated with different utility levels (dash lines labeled as U 1 , U 2 ..). If the isoquant curve is smooth and convex, the minimum water use can be located at the tangent point of line Atotal  S  D  I f (dotted lines labeled as Atotal1 , Atotal 2 .. ) to the isoquant. By linking these points, the path of optimal solutions under various levels of water availability is presented in Figure 2.1.. The optimal release policy in Figure 2.1. follows the trajectory of Eq. (2.7). 2.2.3. Combined economic-engineering solution – realistic hedging rule The trajectory in Figure 2.1. represents the expansion path for different water availability levels, which represent the optimal solutions for various water availability levels, i.e., the interior solution of a dynamic reservoir operation problem. The expansion path exists only when the solution is not constrained by a bound or constraint. This occurs only when the reservoir is partially filled and unbounded by storage capacity and minimum storage constraint. In the real world, the levels of inflow and delivery and reservoir capacity could lead to an empty or full reservoir. If such constraints are imposed, part of the curve will be located in the infeasible region. To identify the infeasible region and determine a practical optimal release policy, the economic solution is compared with SOP, which is a feasible policy under hydrologic and engineering conditions such as reservoir capacity and storage-release relationships. Figure 2.1. is redrawn by the shifting horizontal axis from the second period inflow ( S  I f ) to current water available ( A ), i.e., moving the horizontal coordinates by ( I f  D ).

14

The

relationship between current water available and current optimal delivery is demonstrated in Figure 2.2. In Figure 2.2, the shaded region represents the infeasible solution space because the current delivery exceeds current water available (transferring water from a future period to the current is not possible). Physical constraints added to the economic model characterize a feasible solution space for the economic optimization problem, which is bounded by the solid line before the critical point ( D0 , A0 ) . The decision represented by the solid line would deliver all available water, which is the same policy as SOP in the lower range of water availability. The delivery equals the current water availability, D = A, before the point, while the delivery is less than the current water availability, D < A beyond the critical point. After the critical point, water is stored in the reservoir. The critical point is discussed further later in this chapter. Next we compare hedging to SOP from a conceptual perspective. The SOP (1) delivers the target demand if water is available; (2) retains extra water in storage before reservoir is full; and (3) spills when capacity is exceeded, as plotted in Figure 2.3. As Klemes (1979) and Hashimoto et al (1982) pointed out, SOP provides the optimal solution to a dynamic reservoir operation problem if the objective is to minimize the total release shortfall. However, SOP does not have a mechanism for either rationing supplies for future demand when water is insufficient or releasing more water when a surplus exists (Stedinger, 1984). Hedging, a combination of the economic principle (Eq. (2.7)) and the physical conditions embedded with SOP, is supposed to improve SOP where there are convex costs for shortage (concave value for delivery) and sufficient hydrologic persistence of dry periods. To plot a complete diagram for the hedging policy, Figure 2.2 needs to be modified by adding the constraints for delivery. The delivery should be non-negative ( D  0 ) and less or equal to current water available (D ≤ A). Moreover, the upper bound of delivery should not exceed demand, denoted as Dm . Combining these conditions, we have:

D  [0, min( A, Dm )] 15

(2.9)

With the constraints on delivery, Figure 2.2 is redrawn to have Figure 2.4, which shows a modified SOP as shown in Figure 2.3. The hedging rule exactly follows SOP in curve sections 1, 2, and 3. But in section 4, the release under the hedging rule is less than the SOP (the level of current demand). Hedging starts from the “critical point” ( D0 , A0 ) , which is also marked as the point of “starting water availability” (SWA) and ends with the point of “ending water availability (EWA).” In this section, water is “shared” between release for current demand and storage for future demand following the hedging rule. When current water availability is too small (less than the critical availability or A0 or SWA) or too large (greater than EWA), there is no need for hedging and reservoir operation follows SOP. Draper and Lund (2003) presented a similar analysis. The effect of reservoir capacity on the hedging rule is shown in Figure 2.4, making a larger range of section 4 applicable to a larger reservoir capacity. Smaller reservoir capacity limits the role of hedging. In particular, Figure 2.5 shows a situation where hedging is not applicable for reservoir operation. The upper curve, hedging, from Figure 2.2, is the ideal solution from the economic model, which is above the bold line representing SOP under physical constraints. Then section 4 as shown in Figure 2.4 does not exist in Figure 2.5. Such a situation could occur when future inflow (by average) is large enough and hedging is not needed. The release is constrained by the maximum demand and delivery capacity and the operation follows SOP. 2.3.

Extended Analysis Following the conceptual model presented above, some properties of the model

will be derived and an extended analysis will characterize the starting and ending conditions (EWA and SWA) of hedging and examine the major factors of hedging, including water demand, reservoir inflow uncertainty and evaporation loss for this twoperiod model. First, a ratio of water delivery to total water availability is defined as below:

H 

D Atotal

Atotal , D  { Atotal , D : S  Atotal

B ( D ) E ( S  I f ) }  I f  D,  D S

16

(2.10)

in which here  H is the “delivery ratio” under the hedging rule. The current delivery (D) will need to satisfy the conditions given above, which are related to current available water, future inflow and the utility function. To maintain the generality of the analysis, a general utility function is assumed.  H is characterized by the following two properties and the proofs of both are given as appendices of this chapter. Property 1:  H exceeds 0.5 for a stationary utility function (i.e., the form of the function does not change with period). Property 2:  H is a constant if B "( D ) E "( S  I f ) is constant and the third and higher order of differential is negligible (e.g., if the utility function of both periods is quadratic). Property 1 interprets the time preference for inter-temporal water allocation. That is, premium will be given to water demand in the current or near future periods, compared to next or further future periods.  H  0.5 means zero or negative time preference, which is usually unreasonable. Property 2 provides the condition for the socalled linear hedging rules. Among different hedging rules, linear hedging has been frequently examined (e.g., Shih and ReVelle, 1994; Bayazit and Unal, 1990). Draper and Lund (2004) showed that linear hedging is applicable when a quadratic utility function is assumed. Property 2 provides a more general condition for linear hedging. We derived

 H with a particular utility form in the next chapter. In the following, these two properties are used for some extended analyses of the conceptual model. 2.2.4. SWA and EWA- the range of current water availability suitable for hedging The concept of SWA and EWA was discussed by Bayazit and Unal (1990), who extended the reservoir operation example presented by Loucks et al. (1981). The study related SWA to the emptying period and EWA to the filling period and tested several scenarios of SWA and EWA by trial and error. Although the study demonstrated the existence of SWA and EWA, it did not provide a rigorous approach for identifying the optimal values of SWA and EWA with hedging. We derive SWA and EWA using the conceptual model and its properties presented above. Recalling Figure 2.4, SWA and EWA are associated with two points (starting point P1 and ending point P 2 ), respectively: 17

at P1 ,  H ( Atotal ) 

A0 A0  I f

(2.11)

at P 2 ,  H ( Atotal ) 

Dm A I f

(2.12)

And

Further, at P1 , D0  A0 and Atotal  A0  I f (referring to Figure 2.2). P1 implies a threshold condition when the marginal value of water use in the second period (responding to one unit of water is stored) is equal to the marginal value of water use in the first period. Before P1 , the former is higher than the latter, which leads to the decision of releasing an additional unit of water rather than holding it in storage and current water availability (A) should be all delivered.

Mathematically P1 has by the following

relationship:

E ( I f ) B( D)  D D  A S S 0

(2.13)

When water availability goes beyond A0 , for the additional availability A  A0 , hedging determines how much is allocated to “delivery” and “storage”, respectively, before reaching to another critical point P2, D   H  A0 , S  (1   H )  A0 .

At P 2 ,

Dm D   D  Dm , it is the upper bound for hedging due to the delivery capacity Atotal A  I f or maximum demand.

After P 2 , it may be preferable to deliver more but it is

constrained by delivery capacity. The hedging rule is justified by the condition of the current water availability and future inflow. With knowledge of A and I f , we may evaluate the range for applying the hedging rule for the two-period model, which is:

 H ( Atotal ) 

Dm A and  H ( Atotal )  A I f A I f

or 18

(2.14)

H I f D  A m If (1   H ) H

(2.15)

The left hand side is the SWA and the right hand side is the EWA. By Eq. (2.15), higher inflow and higher  H will increase the value of SWA; higher inflow, lower demand and higher  H will decrease the value of EWA. Both sets of conditions will reduce the range of A for hedging. The role of hedging is reduced when water demand is low relative to water availability. In Eq. (2.15),  H is a constant for particular forms of the utility function (Property 2 and further discussion in Chapter 2). However, If is unknown, which is assumed to be characterized by IID. Thus, Eq. (2.15) presents a conceptual relationship for EWA and SWA but it can not explicitly determine EWA and SWA. In Chapter 3, we present the procedures to determine the value of EWA and SWA using a numerical model based on this relationship and a specific utility function. 2.3.1. Water demand effects

From Eq. (2.15), we have: Dm If Dm  H ( Atotal )   I f  Dm 1  Dm If

(2.16)

If I f is replaced by a statistical mean value of inflow I , then an index   Dm I is introduced to represent the level of development, a common index for water supply systems (Vogel and Stedinger, 1987). Combining Property 1 and Eq. (2.16), we have the range of  H as: 0.5   H 

 1 

(2.17)

which provides a feasible range of water delivery at the current period under different water demand levels, and forms a constraint to the optimal solution determined from Eq. (2.10) (an economic solution). The lower bound represents time preference and the upper 19

bound is based on delivery capacity. To be feasible, Eq. (2.17) requires  >1, i.e., Dm>I, which is reasonable since if maximum demand is generally less than the inflow, there is no need to reduce current delivery and carryover water to the future periods, except for storing spills in wet periods to regulate seasonal variability. Thus, for areas in which water requirement is not high and water resource shortage is unlikely (small  ), Eq. (2.17) is less likely to be true, and the hedging rule is trivial. Moreover, the RHS increases with  , i.e., the range of  H increases with  . When  goes to infinite, the upper bound of  H approaches to 1.0. This shows that when the maximum demand is high (relative to long-term average inflow), delivery can be a large fraction of Atotal, depending on the marginal utility principle (Eq. (2.7)). 2.3.2. The impact of uncertainty

Uncertainty is the major factor on hedging.

If the forecast of future water

availability and demand is perfect, then hedging becomes easy. Here we analyze the impact of reservoir inflow uncertainty on the hedging rule. Considering the random variable of reservoir inflow in the second period, I f , we express it as I f  I   , where I is the mean value of inflow  is an unbiased noise variable whose mean is zero. Eq.

(2.7) is then rewritten as:

B( D)  EV [ E(S  I   )]

(2.18)

EV [ ]  0

Here EV [] represents the expected value. Expand the RHS of Eq. (2.18) in terms of  :

EV [ E(S  I   )]  EV [ E( S  I )  E( S  I )   

E(S  I ) 2 E(S  I ) 3      ] 2! 3! (2.18)

By definition, EV [ ]  0 and EV [ 2 ]   2 (  is standard derivation of  ). Neglecting the high order terms, Eq. (2.18) becomes

EV [ E(S  I   )]  E(S  I )  E(S  I )  0 

20

E(S  I ) 2  2

(2.19)

For a general utility function of water, the 1st derivative represents the marginal profit; the 2nd derivative is negative for diminishing marginal utility. To further explain the third derivative of benefit, we introduce the prudence defined by Kimball (1990), which represents the strength of the precautionary saving motive. Following Leland (1968) and Sandmo (1970), economists have paid much attention to the precautionary saving motive, which states that people take action of saving not only to smooth consumption, but also because they are uncertain about the future. This is also the purpose of hedging for reservoir operations or broad water stress management - people save water for both reasons. Kimball (1990) showed the prudence of the value function as Eq. (2.20).

 (.)  

E (.) 0 E (.)

(2.20)

which measures the convexity of the marginal value function, i.e., the intensity of the precautionary saving motive. Substitute Eq. (2.20) in to Eq. (2.19), we have

1 EV [ E ( S  I   )]  E ( S  I )   2 E ( S  I )   ( S  I ) 2

(2.21)

Since E (.)  0 (diminishing marginal utility) and  (.)  0 (positive precautionary saving motive), the second item in the RHS is greater than zero, and

B' ( D)  EV [ E ' ( S  I   )]  E ' ( S  I f ) , which shows the marginal utility in the future period will be reduced if uncertainty is considered, compared to a perfect forecast. To convert the stochastic form into a deterministic one, we may introduce a positive, deterministic term () to Eq. (2.21):

B' ( D)  EV [ E(S  I   )]  E(S  I   )

(2.22)

Because hydrological uncertainty increases the future marginal utility, as shown in Figure 2.6 graphically, the future utility function is given a level of consumption moving left and upward, and the equilibrium shifts from point A to B. That is to say, delivery ratio will decrease due to uncertainty with the assumption of diminishing marginal benefit and positive precautionary saving motive rules. Under this mechanism, hedging will reduce the current delivery to face the future risk. Figure 2.7 shows the shift 21

of delivery policy curve, where Point A represents the equilibrium under a deterministic resource level, and point B represents the equilibrium under uncertainty. The shift depends on the concavity and the precautionary saving motive of utility function. Assuming the marginal benefit will be the price of water supply, this can be related to price elasticity, which controls the concavity of a utility function.

A higher price

elasticity means a less concave of a utility function; whereas precautionary saving in response to risk requires the convexity of the marginal utility function. Therefore, with high price elasticity (i.e., lower concavity of the utility function), low precautionary saving motivation (i.e., weak convexity, of the marginal utility function), and low natural variability (i.e., small variance [  ] of inflow), the impact of uncertainty on hedging rule will be less significant, according to Eq. (2.21). 2.3.3. Evaporation loss effects

Hedging is motivated by economic incentives such as risk aversion, but is also influenced by physical conditions. Evaporative loss from the reservoir surface could affect reservoir operation policies including hedging (Burness and Quick, 1978; Booker and O’Neill, 2006). We use the conceptual model presented above to analyze the effect of evaporation on hedging decisions. Defining evaporation loss as l ( S ) , a function of storage, Eq. (2.7) is rewritten as:

B( D) E ( S  I f  l ( S ))  D S

(2.23)

Use the chain rule in derivation, B( D) E ( S  I f  l ( S ))  ( S  I f  l ( S ))   D  ( S  l ( S )) (S ) l ( S ) B( D)  E ( S  I f  l ( S ))  (1  ) S

On the RHS of Eq. (2.24),

(2.24)

l ( S ) l ( S ) represents the evaporation loss and (1  ) S S

represents the fraction of the water that can be delivered for use in the second stage; each unit of water to be delivered is worth E ' ( S  I f  l ( S )) . Eq. (2.24) has the same form as the Euler Equation in natural resources economics (Hotelling, 1931). 22

Suppose the

reservoir manager contemplates whether to save one more unit of water for next time period, which will reduce current consumption by one unit, at utility cost of B ( D ) . That unit of water is subject to evaporation loss at a rate of

l ( S ) l ( S ) . Thus, only (1  ) unit S S

of water can be used in next period at a marginal value of E ' ( S  I f  l ( s)) (in present value).

Eq. (2.24) shows the involvement of the variation item (1 

l ( S ) ) in the S

condition of optimal decision (essentially the first-order condition). Further, we relate evaporation loss to the surface area , A , of a reservoir, B ( D )  E ( S  I f  l ( S ))  (1 

Let Ep 

l ( A )  A ) A S

(2.25)

l (A ) A , which is the evaporation rate. And represents the topologic S A

relationship of a reservoir, and a larger value of this term reflects a larger reservoir surface area for a given reservoir capacity. Eq. (2.25) shows that higher evaporation rates and larger increase of reservoir surface area with storage (reservoir topology) will aggravate the effect of evaporation loss on the marginal values of water in the current and future periods. Evaporation discounts the future marginal benefit and reduces optimal hedging. An extreme, but unrealistic case occurs when

l ( S ) approaches to 1. This S

means the total water storage will decrease due to 100% of evaporation. This leads to B( D )  0 in Eq. (2.22). This is the 1st order condition for a static decision in the present

period. This implies that the dynamic framework reduces to a static decision mechanism because water storage for the future is inefficient. 2.4.

Conclusions

This chapter expands a theoretical analysis presented by Draper and Lund (2003) and develops a conceptual two-period model for reservoir operation with hedging that includes future reservoir inflow explicitly. While this theoretical analysis provides an updated basis for further theoretical studies, the theoretical findings can be used to enhance the numerical modeling for reservoir operation with a specific utility function. 23

Extended analysis of the model properties and influencing factors is presented with a general utility function, through which, some intuitive ideas of reservoir operation have been re-confirmed. For example, hedging is trivial when water demand is small relative to water availability and/or reservoir capacity is small and evaporation loss reduces the role of hedging rule in reservoir operation. Some insights derived from the analysis include 1) due to time preference, priority should be always given to the current demand even under the hedging rule (the delivery ratio is greater than 0.5 in the twoperiod operation, Property 1); 2) hedging can be implemented by a simple way with a constant delivery ratio (linear hedging rule, Property 2) when the second order of the utility function in the current period is linearly proportional to that in the future period; 3) given water demand and a predicted future inflow, hedging is only applicable within a range of the current water availability and outside of the range, i.e., the water availability is either too low or too high, the role of hedging is trivial; 4) optimal hedging is related to the level of water resources development in a region and a wider range of hedging corresponds to higher water demand levels relative to water availability in the region; 5) risk due to future uncertainty motivates the adoption of hedging to reduce the probability of more severe water shortage in future periods but high price elasticity and low precautionary saving motivation diminish the effect of uncertainty on hedging rule. These findings, some of which are extended from Draper and Lund (2003), provide general guidelines to develop reservoir operation rules and policies under hedging, as demonstrated further in next chapter with a specific form of utility function. The two-period model presented in this chapter is a simplified case of the real world. However, the simple model provides implications for the real case of three or more periods. When the third period is wet enough to satisfy all third period water uses, fill the reservoir, and then spill, then the two-period model represents the true and optimal case. When inflow is insufficient in the third period, then the second period storage will be needed for another period, i.e., the future inflow after the second period will only increase the tendency of store water (Lund and Draper 2003). In practice, the two-period model can be rolled over any number of future periods if it is assumed that the decision horizon is limited to the current and next period (Fisher 1930; Samuelson 1958; Sethi and Sorger 1991). By this way, the effect of coming periods can be considered and a near24

optimal operation can be obtained. Moreover, a two-period model is suitable for the cases with long periods (e.g., annual), for which, the marginal utility of the current period is equal to the present value of the future period. Moreover, due to the limitation of theoretical derivation, some analysis is left without an explicit result, for example the bounds of water availability for hedging. Even with a specific utility function, the complexity of the problem will not allow us to derive a closed-form analytical solution for the reservoir operation program. In next chapter, we develops a numerical model based on the theoretical analysis presented in this chpater to further examine these limitations, as well as to verify the theoretical derivations presented in this chapter.

25

Appendix I

Property 1:  H is greater than 0.5 for a homogenous utility function Proof Assume the utility function is identical for two periods, then according to the definition of  H in Eq (2.7),

B( D) B( S  I f )  D S Then D  S  I f and  H 

D  0.5 Atotal

However, due to the time value, the current value is always higher than future value for the same input, that is to say B (.)  E (.) . Moreover, according to marginal decreasing principle, B '(.)  E '(.) if B (.)  E (.) . Therefore, B ' ( D )  E ' ( S  I f ) (Eq. (2.9)), only if D  S  I f which leads  H 

D  0.5 . DS  If

26

Appendix II

Property 2:  H is a constant if B "( D ) E "( S  I f ) is constant and the third and higher order of variance are negligible Proof: Given B ' ( D)  E ' ( S  I f ) (Eq. (2.7)), assuming very small change of D and S, D , and S , Atotal  D  S , then, B '( D  D )  E '( S  S  I f )

Applying the Taylor expansion and assuming the third and higher order of variance are negligible,

B '( D  D)  E '(S  S  I f ) B '( D)  B "( D)D  E '(S  I f )  E "(S )S Since B ' ( D )  E ' ( S  I f ) , B" ( D ) D  E" ( S  I f )S Given B " ( D) E" ( S  I f ) =constant, D S  E" ( S  I f ) B" ( D ) =constant

Then

D (S  D)  D Atotal =constant, i.e, Considering a boundary condition: if Atotal  0 then D  0 , we have

D  c  Atotal and c is a constant Therefore,

 H  D Atotal  c =constant

27

References

Bayazit, M. and E. Unal (1990), Effects of hedging on reservoir performance, Water Resour. Res., 26(4), 713–719. Booker, J. F. and J. C. O’Neill (2006), Can reservoir storage be uneconomically large?, J. Water Resour. Plann. Manage., 132(6), 520-523 Bower, B. T., Hufschmidt, M. M., and W. W. Reedy (1962), Operating procedures: Their role in the design of water-resources systems by simulation analyses, Design of Water Resources Systems, A. Maass and Maynard M. Hufschmidt et al., Harvard University Press, Cambridge, M. A. Burness, H. S. and J. P. Quirk (1980), The theory of the dam: an application to the Colorado River, in Essays in Honor of E. T. Weiler, edited by Horwich, G. and J. P. Quirk(eds.), pp.107-130, Purdue University Press, Lafayette, IN. Draper, A. J. (2001), Implicit Stochastic Optimization with Limited Foresight for Reservoir Systems, PhD dissertation, Univ. of California Davis, CA. Draper, A. J. and J. R. Lund (2003), Optimal hedging and carryover storage value. J. Water Resour. Plann. Manage., 130(1), 83-87 Fisher, Irving (1930). The Theory of Interest: As Determined by Impatience to Spend Income and Opportunity to Invest It, Macmillan, New York; reprinted. Gal, S. (1979), Optimal management of a multireservoir water supply system, Water Resour. Res., 15(4), 737–748. Gessford, J. and S. Karlin (1958), Optimal policy for hydroelectric operations, in Studies on the Mathematical Theory of Inventory and Production, pp. 179-200, Stanford University Press, Stanford, CA, Hashimoto, T., J. R. Stedinger and D. P. Loucks (1982), Reliability, resilience, and vulnerability criteria for water resource system performance evaluation, Water Resour. Res., 18(1), 14-20. Hotelling, H. (1931), The economics of resources or the resources of economics? American Economic Review Paper and Proceedings, 64, 1-14 28

Kimball, M. S. (1990), Precautionary Saving in the Small and in the Large, Econometrica, 58, 53-73 Klemes, V. (1977), Value of information in reservoir optimization, Water Resour. Res., 13(5), 857–850. Leland, H. E. (1968), Saving and uncertainty: the precautionary demand for saving, Quarterly Journal of Economics, 82, 465-473. Loucks, D. P., J. R. Stedinger and D. A. Haith (1981), Water resources systems planning and analysis, Prentice-Hall, Englewood Cliffs, NJ. Lund, J.R. and R.U. Reed (1995), Drought water rationing and transferable rations. J. Water Resour. Plann. Manage. 121:429-437. Maass, Arthur, and Maynard M. Hufschmidt et al. (1962), Design of Water-Resources Systems, Harvard University Press, Cambridge, MA. Masse, P. (1946), Les reserves et la regulation de l’avenir dans la vie economique. I: Avenir determine, Hermann & Cie, Publishers, Paris (in French),. Rippl, W. (1883), The capacity of storage-reservoirs for water supply. Minutes Proc., Inst. Civ. Engrg., 71, 270–278. Samuelson, P.A. (1958), An exact consumption-loan model of interest with or without the social contrivance of money, Journal of Political Economy, 66 (5), 46782.Sandmo, A. (1970), The Effect of Uncertainty on Saving Decisions, Rev. Econ. Stud., 37, 353-60. Shih, J. S. and C. ReVelle (1995), Water supply operations during drought: A discrete hedging rule, Eur. J. Oper. Res.,, 82, 163–175. Shih, J.-S. and C. ReVelle (1994), Water-supply during drought: continuous hedging rule, J. Water Resour. Plann. Manage., 120(5), 613-629. Sethi, S.P. and G. Sorger (1991), A Theory of Rolling Horizon Decision Making, Ann. Oper. Res.,, 29, 387-416. Stedinger, J. R. (1978), Comment on value of information in reservoir optimization, Water Resource. Research, 14(5), 984–986. 29

Stedinger, J. R. (1984), The performance of LDR models for preliminary design and reservoir operation, Water Resour. Res., 20(2), 215-224. Stein, J. L. (1961), The simultaneous determination of spot and futures prices, American Economic Review, 51, 1012-1025. Vogel, R. M. and J. R. Stedinger (1987), Generalized storage-reliability-yield relationships, J. Hydrol., 89, 303–327. You, J-Y. and X. Cai (2008a), Hedging rule for reservoir operations: (1) A theoretical analysis. Water Resour. Res., 44, W01415, doi:10.1029/2006WR005481. You, J-Y. and X. Cai (2008b), Hedging rule for reservoir operations: (2) A numerical model. Water Resour. Res., 44, W01415, doi:10.1029/2006WR005482.

30

U4

U3

U2

U1 Atotal 4

Atotal 3

Atotal 2 Atotal1

S  If Figure 2.1. Trajectory of Pareto efficiency

31

( D0 , A0 )

Figure 2.2. Optimal release policy

32

Figure 2.3. Standard operating policy

33

( D0 , A0 )

Figure 2.4. Hedging Rule Operation Policy.

34

Figure 2.5. The situation of not applying hedging rule

35

Figure 2.6. Equilibrium under uncertainty.

36

Figure 2.7. The influence of uncertainty to hedging rule

37

CHAPTER 3.

HEDGING RULE FOR RESERVOIR OPERATION OPERATIONS: A NUMERICAL MODEL

Summary

Optimization models for reservoir operation analysis usually use a heuristic algorithm to search for the hedging rule. This chapter presents a method that derives a hedging rule from theoretical analysis in the previous chapter with an explicit two-period Markov hydrology model, a particular form of nonlinear utility function, and a given inflow probability distribution. The unique procedure is to embed hedging rule derivation based on the marginal utility principle into reservoir operation simulation. The simulation method embedded with the optimization principle for hedging rule derivation will avoid both the inaccuracy problem caused by trial-and-error with traditional simulation models and the computational difficulty (“curse of dimensionality”) with optimization models. Results show utility improvement with the hedging policy compared to the standard operation policy (SOP), considering factors such as reservoir capacity, inflow level and uncertainty, price elasticity and discount rate.

Following the theoretical analysis

presented in the previous chapter, the condition for hedging application - the starting water availability (SWA) and ending water availability (EWA) for hedging are reexamined with the numerical example; the probabilistic performance of hedging and SOP regarding water supply reliability is compared; and some findings from the theoretical analysis are verified numerically.

38

3.1.

Introduction

We develops a new numerical method that explicitly incorporates hedging rules derived from the theoretical analysis, taking advantage of the theoretical analysis on hedging rule policy presented in Chapter 2. While Chapter 2 focuses on the derivation of optimal conditions and influencing factors, this chapter will mainly examine utility improvement under hedging, compared to standard operation policy (SOP). Furthermore this part extends the theoretical analysis of hedging with a particular form of utility function. The findings from the theoretical analysis will be verified through a numerical example with detailed results. Numerical models for reservoir operation analysis include optimization and simulation models (Yeh, 1985; Wurbs, 1993; Labadie, 2004). Optimization models such as linear, nonlinear and dynamic programming have been used to identify the hedging rules with respect to the gross economic return or other system outcomes such as water supply reliability (Hashimoto et al., 1982; Shih and ReVelle, 1994, 1996; Neelakantan and Pundarikanthan, 1999; Shiau and Lee, 2005). Stochastic optimization techniques are required for hedging analysis since a perfect foresight for the future is usually unavailable. Challenges with the stochastic optimization models lie in the solution difficulties such as the “curse of dimensionality” for dynamic programming models and nonlinearity for mathematical programming models (Yeh, 1985; Labadie, 2004). This limits the use of optimization models to solve large-scale reservoir operation problems, although from a research perspective, optimization models have been used for identifying rational hedging rules. Simulation models are widely used as decision support tools to answer what-if questions regarding reservoir operation policies. Simulation models are driven by predetermined operating rules, including the standard operation policy (SOP) (Klemes, 1977; Moran, 1959; Pegram 1980; McMahon 1993) or hedging rules (Bayazit and Unal 1990; Srinivasan and Philipose 1996, 1998). These studies tested various hedging rules using trial-and-error numerical experiments. It might be assumed that the combination of optimal hedging rule and simulation procedures will make numerical modeling more acceptable.

This motivates the study presented in this chapter –to develop a new 39

numerical method that embeds hedging rules derived from the optimization principle – the marginal utility principle, which is developed in Chapter 2. In the rest of this part, a specific form of the utility function is described for a two-period reservoir operation model.

Then some derivations extended from the

theoretical analysis in Chapter 2 under a given utility function are presented. Following that, the formulation of a numerical model and the results are provided. 3.2.

Model Specification and Extended Analysis

Before the numerical modeling method is presented, some extended analysis is needed to describe the hedging rule for a particular form of utility function and of reservoir inflow uncertainty, following the theoretical analysis in Chapter 2. 3.2.1. Utility function, inflow probability distribution, and delivery ratio

The model adopts a specific form of the utility function to represent the benefit of reservoir water releases:

 D 1n ( ) U ( D)   Dm  1 

D  Dm

(3.1)

otherwise

in which U ( D) stands for the economic utility of water delivery D . Dm represents the maximum water delivery (usually limited by the delivery capacity). Extra reservoir release beyond Dm , i.e., the spill, is assumed to have zero utility. The value of the utility by Eq. (3.1) is normalized to the range of 0 to1. When n>1, the utility function is concave, which assures the principle of diminishing marginal benefit with increasing water supply, a general performance of water economics. The price elasticity of demand that measures the rate of demand change due to price change can be represented as the function of n as below and the derivation is given in Appendix I:



n D D D D   Price Price U ( D) U ( D) 1  n

(3.2)

Hashimoto et al. (1982) presented a similar function by defining a loss function to represent an economic loss from long term reservoir operation. 40

The function is

formulated as l ( R)  [(T  R) / T ] when R  T , R denotes the release and T denotes the supply target (demand); l ( R ) =0 when R  T . U ( D) defined in this study and l ( R ) used by Hashimoto et al.(1982) are both normalized to 0 to 1.  characterizes the shape of the functions for convexity of loss and n for the concavity of utility. When   1 , the loss function behaves linearly with respect to R and SOP will be the optimal policy; and for a convex loss function if   1 , the optimal policy will exhibit hedging. Similarly, in Eq. (3.1), n=1 results in a linear function and n >1 specifies a nonlinear utility function that will make hedging be an issue. Given the utility function (Eq. (3.1)), the delivery ratio (  H ), the ratio of delivery over total available water in two consecutive periods, defined in Chapter 2 is derived in Appendix II as: 1

n 1    H  1  (1  r )1n   1  (1  r )   

(3.3)

in which r is the discount rate representing time preference, a key concept underlying the theory of inter-temporal choice, i.e., current value is always larger than future value. Figure 3.1 shows the relationship between  H and price elasticity () under different levels of r. Higher r leads to higher  H (larger amount of delivery) because of the larger reduction of value over time. Lower n (i.e., higher price elasticity) also results in higher  H . This implies that for high price elasticity (water demand more sensitive to price change), leads to less potential for hedging and more water should be delivered for the current time. If n approaches 1 from a number greater than 1 (n>1 for a concave utility function),  H =1, which theoretically implies that the release at the current period will be the total water availability in the two periods (Atotal), which is not realistic since the maximum delivery at the current period cannot take water from the future period. This is a limitation of the pure economic solution as explained in theatrical analysis (Chapter 2). When n approaches 1, the delivery approaches the current water availability if the maximum delivery does not exceed the current water availability. This is the SOP

41

(curve section 4 of Figure 2.5). Again, we second the conclusion of Hashimoto et al. (1982), who claimed SOP as the optimal policy for a linear utility function (n=1). 3.2.2. Effects of time preference and evaporation

As discussed above, greater time preference of utility (represented by discount rate r ) reduces the merit of hedging. Here the effect of time preference is compared to that of evaporation loss from reservoir storage, which demonstrates the similar effect of these factors, one economic and the other natural. Intuitively evaporation loss reduces the useful storage for the future use and then poses a discount on the value of water saved for the future. For simplicity in mathematical derivation but without loss of generality, it is assumed that evaporation is a linear function of water storage, the function are defined by the evaporation rate,  . The increase of  or r increases the delivery in the first time

period (D), reducing the hedging storage. Assuming a very ~ small  , A  S  I f /(1   )  S  I f  Atotal  D , Appendix III shows that under the optimality condition (Eq. 2.7), we have:

r

1 (1   )1 n

1

(3.4)

by which a small  results in a small r . The discount rate determined by Eq. (3.4) can substitute the effect on hedging from a small evaporation rate (  ). Thus, for some cases in which it is difficult to account for evaporation, discount rate can approximately substitute for the effect of evaporation. 3.3.

Numerical Model - The Multi-State Markov Model with Hedging

Rule

To accommodate the hedging rule derived from Chapter 2 and the extended analysis presented above, a modeling framework is needed to simulate the two-period decision process accounting the uncertainty of inflow explicitly. This recalls a classic two-period model in natural resources economics, first described by Fisher (1930), which incorporates inflow uncertainty represented by the Markov chain. The model considers a single composite consumer and inter-temporal allocation of water under a dynamic mechanism. The dynamics of decision making is limited to two consecutive periods, but 42

the decision process is “rolled over” every two periods and extended into the infinite future. Water delivery is the decision variable subject to current water availability and predicted future inflow. The objective of the model is to maximize the overall utility from all periods. The two-period Markov model has been widely applied to study reservoir operations considering uncertainty (Klemes, 1977; Moran, 1959; Pegram, 1980; McMahon 1993). Markov models are among the so-called explicit stochastic models that incorporate explicit probability transition between periods. As indicated by Yeh (1985), explicit stochastic models may be more germane for reservoir operation, because they are designed to model probabilistic streamflow processes directly, not necessarily assuming perfect foresight. Therefore, the two-period Markov model is chosen for further hedging analysis in numerical analysis, which requires handling inflow uncertainty explicitly. 3.3.1. Two-period Markov models and applications for reservoir operation

Markov models applied for reservoir operation analysis include two-state and multiple-state models.

The two-state models assume the performance of reservoir

operation is either failure or failure-free; and the reliability of reservoir operation is defined as the probability of the failure-free state. The multiple-state models depict the probability distribution of multiple levels of reservoir storage and releases. Three typical storage states have been studied: empty, full and between empty and full; and the third one can be further divided into multiple states. Klemes (1977) used 10-state Markov chain in his numerical model to analyze the reliability indices. Through a rigorous theoretical analysis, Pegram (1980) demonstrated that multi-state Markov chain could estimate a more accurate reservoir storage–yield-reliability relationship. The basic procedure involved in Markov models is the transition of states, which are simulated as discrete state Markov processes. Given a state transition probability, a Markov process is characterized by:

Pt 1 ( St 1 )  P( St 1 St )  Pt ( St )

(3.5)

in which reservoir storage ( S ) ranges between 0 and Sc , Pt ( St ) denotes the probability density function (pdf) over S t and Pt 1 ( St 1 ) over St 1 .

43

As the time step approaches infinity, Pt ( St ) will reach the steady state Psteady ( S ) : lim Pt 1 ( St 1 )  lim Pt ( St )  Psteady ( S ) t 

t 

(3.6)

The Markov system eventually approaches a steady state after running the system for some number of time periods. Under the steady state, the system’s behavior is independent of the initial state.

This is also defined as the equilibrium condition,

implying that the long term behavior of a reservoir responding to inflow uncertainty will be steady. For an irreducible, positive recurrent Markov chain, the steady-state is unique. The steady distribution of a Markov system provides meaningful information for stochastic reservoir operation problems in terms of identifying a long-term stable policy for operations. Previous studies of reservoir operations (e.g., Jackson, 1975; Hashimoto et al., 1982; Pegram 1980; Vogel and Bolognese 1995) demonstrated that the model simulation could approach a steady state solution after running over some number of time periods. Pegram (1980) used linear differential equations to solve for Psteady ( S ) . The study showed that reasonable approximations could be computed by well documented numerical methods, although the analytical solution was available for a few special cases. A few other studies attempted to derive a closed form analytical solution for the equilibrium condition but only with highly simplified conditions (e.g., Moran, 1959; Burness and Quick, 1987; Buchberger and Maidment, 1989). Regarding the hedging policy, the steady state implies a policy of how much storage to carry to the next time period under various estimates of inflow occurrences. This policy may change from one time period to next, but reservoir managers need a general guide to decision making. In this chapter we extends multi-state Markov model of Pegram (1980) will be expanded to study hedging rules for reservoir operation. Pegram (1980) used a linear utility function and thus the SOP as an operation policy. This study replaces the linear utility function with a nonlinear utility function and thus replaces the SOP with the hedging rule. The key procedure is to incorporate explicit hedging rules derived from theoretical analysis (Chapter 2) to solve for the equilibrium state of reservoir storage transitions defined as Markov processes, as further described in the following. 44

3.3.2. Transition probability and stationary probability distribution

The stationary probability is computed by a numerical approach searching

Pt ( St ) that is identical to Pt 1 ( St 1 ) under a transition probability Pt ( St 1 St ) (Eq. (3.6)), which is to be determined by the explicit hedging rule derived in Chapter 2. Therefore, the approach integrates the fundamental relationship of the Markov model (Eq. (3.5)) and the hedging decision making, as further described below. First, hydrologic uncertainty – the probability distribution of reservoir inflowneeds to be specified.

Pegram (1980) tested various inflow distributions including

normal, lognormal and discrete inputs, both independent and serially correlated, for a multiple-state Markov model for reservoir operation study. Vogel and Wilson (1996) showed that annual streamflow could be approximated by either a Gamma or a lognormal distribution. Following this, the log-normal distribution [ LN ( I ,  ) , with I as the mean value and  as the standard deviation of annual inflow] is selected to characterize the distribution of inflows within a period. For the distribution of inflows between time periods, following the assumption given in Chapter 2, the numerical study will assume an independent identical distribution (IID). The numerical model follows the four components of hedging policy as shown in Figure 2.4. Table 3.1 presents the equations used to compute the decision variable (water delivery, D) and state variable (reservoir storage, S) under each component. Under components 1, 2, and 3, hedging policy is identical to SOP, and water delivery can be calculated explicitly by the equations provided in Table 3.1. Component 4 is special for hedging, by the definition of  H , D   H  ( St  It  It 1 ) , where St represents the reservoir storage at the beginning of period t, I t and I t 1 represent inflows in periods t and t  1 , respectively. Difficulties arise with incoming inflow, I t 1 , which needs to be predicted with imperfect information. Therefore, for a given value of  H , D can not be determined explicitly. The uncertainty with It+1s handled by the Monte Carlo method In this study.

45

For a discrete multi-state Markov model, the continuous storage variable (S) is divided into multiple discrete states, including empty storage, full storage. Pegram (1980) suggested that 30 states would be enough to obtain reasonable accuracy for a multi-state Markov chain. Since Pegram (1980) studied SOP only and a different decision policy may affect the shape of the stationary probability distribution, 41 states are chosen for the numerical model presented in this chapter. The essential component of a Markov model is the transition matrix. Different from most Markov models in hydrology, the transition probabilities in the reservoir operation model are controlled by both stochastic inflow process and the operation policy. In this study, the transition probabilities are quantified following a reservoir operation policy, either SOP or hedging. Applying the Monte Carlo simulations, 5000 realizations of two period inflows ( I t , I t 1 ) are generated under the given lognormal distribution of the inflow, LN ( I ,  ) . For each of the paired realizations, I t and I t 1 are applied to calculate the delivery ( D ) and storage (S+1) using the formula provided in Table 3.1. 5000 samples of St+1, with given I t and I t 1 , can be calculated according to different current state variables, St. Using frequency analysis, the 41*41 transition matrix for the 41-state Markov model is generated. Once the transition probability matrix is determined, the stationary probability can be identified according to the Markov Model (Eq. (3.5)) and the definition of the stationary probability (Eq. (3.6)). Figure 3.2 shows a diagram of the multi-state Markov model embedded with the hedging rule expressions shown in Table 3.1. The numerical model was implemented in Microsoft Excel, and a macro program was developed within Excel to obtain Psteady ( S i ) i  0...40 by trial-and-error. This is implemented with an Excel program which can directly solve a goal-seeking problem. For the Markov model, water delivery depends on the state variable (S) and uncertain inflow in the future period. Since the inflow is assumed to be I.I.D. with the specified distribution, LN ( I ,  ) , the probability distribution of delivery at equilibrium

Psteady ( D) is a function of Psteady ( S ) , the stationary probability distribution of storage.

46

Psteady ( D) describes the long-term characteristics of decision making on delivery under the equilibrium status. As illustrated above, the determination of Psteady (D) depends on water demand, reservoir capacity, and hydrologic parameters such as the mean and variance of inflow. Psteady (D) can also be interpreted as the probability of water stress given a certain level of water demand, i.e., the probability of a certain deficit between a delivery and demand (Hashimoto et al., 1982). At last Psteady (D) is used to calculate the expected utility:

EU   U ( D)  P( D)dD

(3.7)

  U ( Di )  P( Di ) i

Some parameters to be used in the procedures outlined in Table 3.1 are defined as below: 

Coefficient of variation of inflow (uncertainty level): Cv 

 I

Sc



Standardized reservoir capacity:  



Demand  , which is assumed constant for all time periods although a change



can be easily incorporated. A standardized form is used:  

I 



, which has

been used by Hurst 1951; McMahon 1993; Pegram 1980; Buchberger and Maidment, 1989, etc. 3.4.

Results of the Analysis

The modeling results will be presented regarding how much utility improvement the hedging rule can generate, considering the influences of various factors, including reservoir capacity, inflow level and uncertainty, price elasticity and discount rate. Before presenting the results on utility improvement, we re-examine a condition for hedging application - the starting water availability (SWA) and ending water availability (EWA) 47

for hedging, and compare the probabilistic performance of hedging and SOP regarding water supply reliability under the two operation policies.

3.4.1. Quantification of SWA and EWA

Chapter 2 provides the analytic form of SWA and EWA for hedging (Eq. (2.15)). The value of SWA and EWA can provide a direct guide for decision making under hedging. In this chapter, with the particular form of utility function and probability distribution function (PDF) of inflow, these two items are quantified. Using Eq. (3.3) derived above, the delivery ratio  H is determined; and the realization of predicted inflow If associated with a certain probability can be identified, given the log-normal distribution LN ( I ,  ) .

With the value of  H and If, SWA and EWA are calculated

according to Eq. (2.15).

In Figure 3.3, SWA and EWA are represented as water

availability (A) relative to the average future inflow ( I ); and they are plotted vs. a range of inflows associated with a series of percentiles under three standardized inflows,  =0, -0.5 and -1.0, respectively. For all three cases, SWA decreases, EWA increases and the range of water availability suitable for hedging (difference between SWA and EWA) declines with the increase of the inflow realization (If) characterized by LN ( I ,  ) . As shown in Figure 3.3, the two values converge when If reaches to a high percentile value, 70%, 80% and 85% under three water stress levels (), respectively. For higher levels of water stress (corresponding to lower levels of average inflow), the difference between SWA and EWA is larger for a given inflow realization, which shows that hedging can be applied to a wider range of current water availability when predicted inflows are at lower levels. 3.4.2. Probability distribution of water delivery and reservoir storage

Figure 3.4 presents the cumulative probability distributions of water delivery under a positive (  =0.125) and a negative drift of standardized inflow (  =-0.5). The probability represents the chance of water deliveries higher than particular fractions of maximum demand. For the positive drift, where the average inflow is larger than the demand, the probability of water delivery that satisfies the maximum demand is 74% and 48

60% under SOP and hedging, respectively; for the negative drift, the probability is 30% and 10% under SOP and hedging, respectively. For both cases, with high delivery levels, hedging does not produce a reliability value as high as SOP; however, at medium and low delivery levels, hedging policy performs better in reliability than SOP. This shows that, compared to SOP, hedging policy trades off reliability at high delivery levels with that at medium and low levels so as to reduce the risk of severe water supply deficit. In addition, comparing the positive and negative drift, a more noticeable difference between SOP and hedging can be seen under a water stress status (a negative  ) than a water surplus status (a positive  ). This verifies the analysis in Chapter 2 that hedging rule is a more efficient policy under water stress situations; whereas its role is trivial when water is abundant. Correspondingly, Figure 3.5 plots the probability distributions of reservoir storage states under the water stress and surplus status, respectively. These two figures show the same shape as the storage distribution curve presented by Buchberger and Maidment (1989). The reservoir tends to accumulate storage under positive standardized inflow and deplete storage under negative standardized inflow. For both inflow cases, hedging and SOP result in similar distributions for higher storage levels; whereas the performance of the two policies become complex for low storage: under hedging low storage occurs with higher possibilities the probability of “empty reservoir” (zero active storage) is lower significantly than that under SOP. Thus, compared to SOP, hedging tends to manipulate storage to prevent zero active storage under which no water is carried to the future. 3.4.3. Utility improvement from hedging

Utility improvement from hedging compared to SOP is demonstrated under various levels of inflow uncertainty and magnitude Figure 3.6 plots the utility improvements represented in percent vs. reservoir sizes under four levels of uncertainty (represented by Cv), respectively, with  =-1 (a water stress status).

The positive

percentages under all the uncertainty levels show higher utilities from hedging than SOP. The improvements of expected utility from SOP to hedging are larger under higher uncertainty levels. Moreover, the improvements increase with reservoir size but at a declining rate. When the reservoir capacity reaches to a certain level, further larger capacity will not lead to any larger utility improvement from SOP to hedging. With large 49

reservoir sizes, the marginal increase of the utility improvement is close to zero. This size may be taken as a reference for determining optimal reservoir capacity. Greater uncertainty (Cv from 0.5 to 4.0) increases this reference reservoir capacity, which implies that higher hydrologic uncertainty requires larger capacity in order to maximize the utility under hedging. Figure 3.7 plots the same curves as Figure 3.6 for a higher water shortage level (  =-2 compared to  =-1 in Figure 3.6). Very similar results can be found from Figure 3.6 and Figure 3.7 but with larger utility increase from hedging compared to SOP under a higher level of water shortage (  =-2). This again shows a more significant role of hedging under a more serious water stress status. Moreover, regarding the reference reservoir capacity beyond which the improvement does not increase with storage, it is larger for  =-2 than  =-1, under all the uncertainty levels. For example, when Cv=4.0, the standardized storage k=5.0 for  =-2 and k=4.0 for  =-1. This implies that, as higher hydrologic uncertainty, higher water shortage also requires larger capacity in order to maximize the utility under hedging. As illustrated before, when water is plentiful, hedging becomes unnecessary and the impact of hedging becomes prominent in the effective use of limited water resources under water stress situations. However, under extreme water scarcity conditions, policies become useless.

The most extreme condition is that the reservoir is dry during

consecutive drought periods, in which no reservoir operation policies are applicable. The influence of hedging under a series of inflow levels (which are indicators of water stress) is shown in Figure 3.8. The utility improvements from SOP to hedging vs. inflow levels are plotted for five levels of inflow uncertainty, respectively.

For each curve

corresponding to an uncertainty level, the improvement first increases with water stress levels; however, after reaching a peak, the improvement decreases. For the most severe water stress level (  =-9) shown in the figure, the improvements approach to zero, which indicates slight or no difference between hedging and SOP. This also demonstrates the appropriate range of inflow levels for applying hedging. Hedging is only meaningful for a particular range of resources stress and it is trivial for either too much or too little water availability. 50

The range of inflow levels for applying hedging is not only affected by quantity (average inflow) but also by uncertainty. As can be seen from Figure 3.8, larger variance of utility improvement exists with higher levels of uncertainty. The role of inter-temporal storage regulation is more significant when the inflow uncertainty is higher. Thus under the same level of water stress, larger utility improvement from SOP to hedging occurs with higher uncertainty. Additionally, the utility improvement is affected by economic parameters such as price elasticity and discount rate. Figure 3.9 shows that utility resulting from both SOP and hedging declines with price elasticity; furthermore, the utility improvement from SOP to hedging becomes smaller under higher levels of price elasticity. When the price elasticity is higher, the demand quantity change is higher for one unit change of price. In other words, the price change is smaller in response to one unit change of demand. Hedging carries water storage to next time period, which causes the change of the demand in both current and next time periods. Such demand change will cause lower price change when price elasticity is high. Relatively small price change implies the small change of utility. Therefore, under higher price elasticity, hedging policy is less significant in changing utility. This recalls the relation between the delivery ratio and price elasticity represented by Eq. (3.3), which shows higher price elasticity results in higher delivery ratio and lower hedging storage. Regarding the impact of discount rate, Figure 3.10 plots the expected utility vs. discount rate for SOP and hedging, respectively. As expected, the utility of hedging declines with discount rate. High discount rate diminishes the future value (of water use in this study) and motivates users to delivery more water for current use. This recalls the relation between the delivery ratio and discount rate as shown in Eq. (3.3) and Figure 3.1. 3.5.

Conclusions

Following the theoretical analysis presented in Chapter 2, this part describes a method for integrating hedging policy with stochastic simulation processes for reservoir operations, with a particular form of utility function and of inflow probability distribution. A two-period multi-state Markov model is extended to explicitly incorporate the optimal hedging rules derived from theoretical analysis and solved for the equilibrium condition 51

under the steady state. The method derives the transition probability and steady state probability based on both the property of Markov chain and the hedging theory. The simulation method embedded with the optimization principle to derive hedging rules will avoid both the inaccuracy problem caused by trial-and-error with traditional simulation models and the computational difficulty (“curse of dimensionality”) with optimization models. The results show that hedging policy generally has greater efficacy than SOP when water stress exists and future reservoir inflow is more uncertain. Hedging policy improves economic utility but creates lower reliability especially at high water delivery levels. Specific findings and policy implications include: 

Compared to the SOP, the hedging trades off the reliability of high deliveries levels for greater reliability at low levels of deliveries and significantly lower possibility of “empty reservoir”.



The merit of hedging compared to SOP regarding utility improvement depends on the level of water stress, inflow uncertainty, reservoir capacity, price elasticity, discount rate and evaporation loss.

Both higher hydrologic

uncertainty and higher water shortage requires larger reservoir capacity in order to maximize the utility under hedging. The range of inflow for applying hedging excludes extremely high or low inflow and is not only affected by quantity (average inflow) but also by uncertainty.

A hedging policy is particularly

beneficial in utility improvement from SOP to hedging under higher hydrologic uncertainty. 

The scope of hedging characterized by the difference between starting water availability (SWA) and ending water availability (EWA) depends on future inflow characteristics. A larger difference between SWA and EWA occurs with lower inflows with both lower-percentile inflows under a given flow prediction series and lower average inflow with various flow predictions. Thus the scope of hedging depends on the status of inflow prediction. A wider scope of hedging can be applicable for more serious water stress levels.



The performance of hedging varies with different levels of reservoir capacity 52

and water stress, and an upper bound of storage is found, beyond which the utility improvement from SOP to hedging is trivial. Further study is worthwhile to relate the storage bound to reservoir storage choice, which can address some practical problems. For example, for those areas with water stress, is the current storage too little or too much from a utility improvement perspective? This study is limited to a two-period model; the predicted inflows are I.I.D. with foresight limited to the next period.

Although the model and the assumption are

commonly used in dynamic optimization problems, they make the numerical model presented in this chapter mainly suitable for analyzing over-year hedging, which concerns only larger reservoirs with over-year flow regulation capacity.

Even for over-year

hedging, the two-period model with an assumption of I.I.D. for inflow prediction is not suitable for droughts lasting more than two years. Also the model and policy discussion presented here is not for intra-year reservoir operation, which may take account hedging season to season, month to month, or with even shorter time intervals. For intra-year operations, reservoir inflows over time periods are highly correlated and the I.I.D. assumption is not valid. The two-period foresight will not be long enough for problems with shorter time intervals, which suggests further research to overcome its limitation for solving intra-year reservoir operation problems.

53

Appendix I: Derivation of price elasticity based on the utility function According to the definition of price elasticity () and assume the price equals to marginal

utility



D Price

D D D  Price U ( D ) U ( D )

(A1)

Substituting U ( D) from Eq. (3.1)

 (

D 1 D ) / (1 n ) / n (1 n ) / n 1 D D 1  1/ n Dm n n Dm1/ n

(A2)

After simplification



n 1 n

54

(A3)

Appendix II: Derivation of delivery ratio based on the utility function According to the theoretical framework discussed in Chapter 2, Atotal , D  { Atotal , D : S  Atotal  I f  D,

B ( D) E ( S  I f ) }  D S

1 U ( S  I f ) U ( D)  1  r S  I f D

(A4)

Substituting U ( D) from Eq. (3.1) Atotal  D 1n   D 1n 1 )  ( ( ) 1  r  ( Atotal  D ) D Dm Dm

(A5)

Simplifying Eq. (A5), n

Atotal  (1  (1  r ) 1 n ) D

By definition of the delivery ratio,  H 

(A6)

D , thus Atotal

n D 1 n 1 H   (1  (1  r ) ) Atotal

55

(A7)

Appendix III: Derivation of operational condition by replacing the economic discount with evaporation. According to the theoretical framework discussed in Chapter 2,

Atotal , D  { Atotal , D : (1   ) S  Atotal  I f  D, U ( D) U ((1   ) S  I f )  D (S )

B( D) E ((1   ) S  I f )  } D S

(A8)

Because I f is an independent variable,  ( S )   ( S  I f /(1   )) , Eq. (3.4) can be rewritten as:

U ( D) U ((1   ) S  I f )  D  ( S  I f /(1   ))

(A9)

Substituting U (.) from Eq. (1)

S  I f /(1   ) 1n  D 1n 1/ n  ( )  (1   ) ( ) Dm D Dm S

(A10)

Let S  I f /(1   )  A , and S  A  D 1n  A 1n ( )  (1   )1/ n ( ) D Dm A Dm

(A11)

Recalling Eq. (A5) on the derivation of delivery ratio: A  D 1n   D 1n 1 )  ( total ( ) 1  r  ( Atotal  D ) D Dm Dm

(A12)

Comparing Eq. (A11) and Eq. (A12), the increase of  or r will decrease the values resulting from the right hand side (RHS) of both equations, and so as to the left hand side (LHS). Since the LHS of Eq. (A11) and Eq. (A12) are identical, and for a very small  , ~ A  S  I f /(1   )  S  I f  Atotal  D , to make the RHS of Eq. (A11) and Eq. (A12) identical, we need to have: (1   )1/ n 

56

1 1 r

(A13)

which can be rewritten as

r

1 (1   )1 n

57

1

(A14)

References

Bayazit, M. and E. Unal (1990), Effects of hedging on reservoir performance, Water Resour. Res., 26(4), 713–719. Buchberger, SG, and DR Maidment (1989), Diffusion approximation for equilibrium distribution of reservoir storage, Water Resour. Res., 25(7), 1643-1652. Burness, H. S. and J. P. Quirk (1980), The theory of the dam: an application to the Colorado River, in Essays in Honor of E. T. Weiler, edited by Horwich, G. and J. P. Quirk(eds.), pp.107-130, Purdue University Press, Lafayette, IN. Fisher, Irving (1930). The Theory of Interest: As Determined by Impatience to Spend Income and Opportunity to Invest It, Macmillan, New York; reprinted. Hashimoto, T., J. R. Stedinger and D. P. Loucks (1982), Reliability, resilience, and vulnerability criteria for water resource system performance evaluation, Water Resour. Res., 18(1), 14-20. Hirsch, R.M. and J.R. Stedinger (1987), Plotting positions for historical floods and their precision, Water Resour. Res., 23(44), 715–727. Hurst, H. E.(1951) Long term storage capacities of reservoirs, Trans. ASCE, 116, 776808. Jackson, B.B. (1975), Markov Mixture Models for Drought Lengths, Water Resour. Res., 11(1), 64-67. Klemes, V. (1977), Discrete Representation of Storage for Stochastic Reservoir Optimization, Water Resour. Res., 13(1), 149-158. Labadie, J (2004) Optimal Operation of Multi-Reservoir Systems: State of the Art Review. J. Water Resour. Plann. Manage., 130(2), 93-111. McMahon, T. A(1993), Hydrologic design for water use, Ch. 27 in Handbook of Hydrology, edited by D. Maidment, McGraw-Hill, New York. Moran, PAP (1959), The theory of storage. Methuen & Co., London. Neelakantan, T.R and N.V. Pundarikanthan (1999), Hedging rule optimization for water supply reservoirs system, J. Water Resour. Plann. Manage., 13(2), 409-426. 58

Pegram, GGS (1980), On reservoir reliability, J. Hydrol., 47, 269-296. Shiau, J. T. and H. C. Lee (2005), Derivaiton of Optimal Rules for a water supply reservoir through compromise programming, Water Resources Management, 19, 111-132. Shih, J. S. and C. ReVelle (1995), Water supply operations during drought: A discrete hedging rule, European Journal of Operational Research, 82, 163–175. Shih, J.-S. and C. ReVelle (1994), Water-supply during drought: continuous hedging rule, J. Water Resour. Plann. Manage., 120(5), 613-629. Srinivasan, K. and M. C. Philipose (1996), Evaluation and selection of hedging policies using stochastic reservoir simulation, Water Resources Management, 10(3), 163– 188. Srinivasan, K. and M. C. Philipose (1998), Effect of hedging on over-year reservoir performance, Water Resources Management, 12(2), 95–120. Vogel, R. M. and J. R. Stedinger (1987), Generalized storage-reliability-yield relationships, J. Hydrol., 89, 303–327. Vogel, R. M.(1987), Reliability indices for water supply systems. J. Water Resour. Plann. Manage., 113(4), 563–579. Vogel, R. M., and I. Wilson (1996), Probability distribution of annual maximum, mean, and minimum streamflows in the United States, Journal of

Hydrologic

Engineering, 1(2), 69–76. Vogel, R. M., and R. A. Bolognese (1995), Storage-reliability-resilience-yield relations for over-year water supply systems, Water Resour. Res., 31(3), 645–54. Wurbs, R. A (1993), Reservoir system simulation and optimization models, J. Water Resour. Plann. Manage., 119(4), 455–472. Yeh, WW-G (1985), Reservoir management and operations models: a state-of-the-art review, Water Resour. Res., 21(12), 1797-1818 Water Resour. Res.You, J-Y. and X. Cai (2008a), Hedging rule for reservoir operations: (1) A theoretical analysis. Water Resour. Res., 44, W01415, doi:10.1029/2006WR005481.

59

You, J-Y. and X. Cai (2008b), Hedging rule for reservoir operations: (2) A numerical model. Water Resour. Res., 44, W01415, doi:10.1029/2006WR005482.

60

Table 3.1. Numerical equations of hedging under various conditions

61

0.55

r= 0.01

r= 0.02

r= 0.05

r= 0.1

Delivery ratio (κh)

0.54 0.53 0.52 0.51 0.5 0.49 0

-1

-2

-3

-4

-5

-6

-7

-8

-9

Price elasticity

Figure 3.1. The relationship of hedging ratio and price elasticity under different discount rates

62

LN (  ,  )

LN (  ,  )

10%

10%

30%

30%

P ( X ti10~ k X t j  0~ k )

70% 60% 45%

X

70% 60% 45%

X 20% 10%

30% 20% 10%

30%

Figure 3.2. Diagram of the multi-state Markov model embedded with the hedging rule

63

6

5

4

EWA (e=‐1)

3

EWA (e=‐0.5)

2

SWA (e=‐1)

EWA (e=0) 1

SWA (e=‐0.5) SWA (e=0)

0 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Percentile of future inflow

Figure 3.3. Starting water availability (SWA) and ending water availability (EWA) under three water stress levels (   0 , -0.5, and -1).

64

100 90

Probability (%)

80 70 60 50 40 30 20 10 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Dm

Release (as a fraction of Dm)

Figure 3.4. Cumulative distribution function (CDF) of release under hedging and SOP under two water stress levels. Other major parameters are K=2, Cv=1, n=2.

65

70

60

Probability(%)

50

40

30

20

10

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Full

Reservoir Storage (relative to full storage)

Figure 3.5. Probability distribution function (PDF) of storage under hedging and SOP under two water stress levels. Other major parameters are K=2, Cv=1, n=2.

66

Utility improvement from SOP to hedging (%)

12 10

CV=4

8

CV=3

6 CV=2

4 2

CV=1 CV=0.5

0 0

1

2

3

4

5

6

7

Standardized reservoir capacity (K)

Figure 3.6. Utility improvement in percentage from SOP to hedging under various uncertainty levels when  =-1

67

8

Utility improvement from SOP to hedging (%)

12 CV=4

10 8

CV=3 6

CV=2

4 CV=1 2 CV=0.5 0 0

1

2

3

4

5

6

7

Standardized reservoir capacity (K)

Figure 3.7. Utility improvement in percentage from SOP to hedging under various uncertainty levels when  =-2

68

8

Utility improvement from SOP to hedging (%)

12

10

CV=4

8 CV=3 6 CV=2 4

CV=1

2

CV=0.5 0 0

-1

-2

-3

-4

-5 -6 -7 -8 Standard Inflow (declining from 0 to -9)

-9

-10

Figure 3.8. Utility improvement from SOP to hedging vs. water stress levels under various uncertainty levels when K=5

69

Expected Utility

0.8 0.6 0.4

hedging

0.2

SOP

0 0

-1

-2

-3

-4

-5

Price elasticity

Figure 3.9.The influence of price elasticity on hedging and SOP

70

-6

Expected Utility

0.40

hedging

0.35

0.30

SOP

0.25

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Discount Rate

Figure 3.10. The influence of discount rates on hedging and SOP

71

0.16

CHAPTER 4.

DETERMINING FORECAST AND DECISION

HORIZONS FOR RESERVOIR OPERATIONS UNDER HEDGING POLICIES Summary

Hedging policies for reservoir operations make a small deficit in current supply to reduce the probability of a severe water shortage in the future. One of the critical questions for hedging research is how long the forecast period should be so that reliable inflow forecast in the period can be used for decision making under hedging. Decision makers always hope to look further into the future; however, the longer the forecast period, the more uncertain and less reliable the involved information, which will have a diminishing influence on decision making. For dynamic reservoir operation optimization models, the decision horizon (DH) may be defined as the initial periods in which decisions are not affected by forecast data beyond a certain period, defined as the forecast horizon (FH). This chapter determines FH with given DH for dynamic reservoir operation problems through both theoretical and numerical analysis. We use order of magnitude analysis and numerical modeling to identify the impact of various factors such as water stress level (the deficit between water availability and demand), reservoir size, inflow uncertainty, evaporation rate and discount rate. Three types of inflow time series are used: stationary, non-stationary with seasonality and random walk.

Results show that inflow

characteristics and reservoir capacity have major impacts on FH when water stress is modest; larger reservoir capacity and the deterministic component of inflow such as seasonality require a longer FH. Economic factors have strong impacts when water stress levels are high. Keywords:

reservoir

operation,

decision

programming, hedging rule

72

horizon,

forecast

horizon,

dynamic

4.1.

Introduction

Hedging policies for reservoir operations decide how much water should be released for the current or near future period and how much should be carried over to the future. Such policies make a small deficit in current supply to reduce the probability of a severe water shortage in the future. Bower et al. (1962) first provided a systematic economic description of hedging rules for water resources systems operations. Following that, hedging has been explored through various numerical models and theoretical analyses (e.g., Klemes, 1977; Stedinger, 1978; Loucks et al., 1981; Hashimoto et al., 1982; Shih and ReVelle, 1994, 1995; Draper and Lund, 2004). In previous chapters, we applied the microeconomic theory and derived some properties in terms of when and how much water to reserve to mitigate water shortage in the future (Also see You and Cai 2008a). Although the concept of hedging is straightforward, the lack of foresight of reservoir inflow makes it difficult to analyze. One of the critical questions for hedging research is how long the forecast period should be so that reliable inflow forecast in the period can be used for decision making under hedging. Decision makers always hope to look further to the future, but the longer the forecast period, the more uncertainty and less reliable information to be involved, which will have a diminishing influence on decision making. For a multi-period reservoir operation optimization problem, it is usually the release (the decision) in the first or first few periods that is of immediate importance to the reservoir manager. These imminent release decisions depend, in general, upon inflow forecasts for subsequent periods. Forecasts of inflow further into the future are usually less reliable and even unavailable due to the limitation of current technology. It is therefore of utmost interest to quantify the diminishing effect, if any, of future data on the initial decisions in order to know if distant forecasts have negligible impact on initial decisions. For dynamic reservoir operation optimization models, when it happens that the decisions in the initial few periods are not affected by forecast data beyond a certain period, the period is known as forecast horizon (FH), and the number of the initial periods is known as the decision horizon (DH) (Bes and Sethi, 1988). For example, if we want to decide the release of a reservoir for the coming month, we find the inflow forecast in the next three months, at most, is useful for the decision, and the impact of any 73

forecast beyond the next three months is trivial underlying a stochastic dynamic optimization framework, then DH=1 and FH=3. The theory of time horizon originated from inventory studies. Modigliani and Hohn (1955), Charnes et al. (1955), Johnson (1957), and Wagner and Whitin (1958) initialized studies on planning horizon for production-inventory problems. Since then, numerous theoretical and application studies have been conducted.

For theoretical

establishment, Lundin and Morton (1975) systematically analyzed the planning horizon methods for deterministic problems. Morton (1981) later extended deterministic studies to stochastic problems, which provided a detailed explanation of FH and an algorithm to test the horizons.

However, a rigorous theoretical framework was still missing for

horizon studies until Bes and Sethi (1988) developed a more solid mathematical basis for determining the existence of DH and FH. Bes and Sethi refined the concept of DH and FH for dynamic decision problems with discrete modeling time periods and states, under both deterministic and stochastic forecasts. Bes and Sethi’s framework was applied to various fields involving dynamic decision-making such as inventory management, production planning/scheduling, plant location, machine replacement, cash management, capacity expansion, and food trading, as can be seen in a comprehensive review provided by Chand et al. (2002). However, as pointed out by Chand et al., most studies dealt with deterministic, single dimensional, multi-period planning problems in operations management. The stochastic natures of DH/FH in dynamic decision problems have not been fully explored yet. For the operation of reservoirs and other water systems, managers often need to deal with some questions related to both DH and FH, for example, should the weekly weather forecast be used for the release decision of next day, or the forecast of next month for the decision of next week? Few studies have rigorously addressed the horizon issues for reservoir operation problems. It is helpful to distinguish DH & FH from critical period (CP) (Hall et al., 1969), which is commonly used in the planning, design and operation of water storage reservoirs. CP is interpreted as a historical hydrologic period that includes the lowest flow or the most severe drought. CP is based on the fullempty cycle of reservoir storage, i.e., for a given water supply firm yield and a specified streamflow record, the reservoir will be full at the beginning and empty at the end of the 74

CP. Therefore, “the events outside this period cannot affect the events within it” (Oguz and Bayazit, 1991). However, conceptually CP is not FH. CP is defined based on a single historical time series, assuming deterministic hydrologic information, and supporting static decision makings. FH, which is defined as a future time period, adopts stochastic hydrologic information and supports dynamic decision making (You and Cai, 2008c). A challenge for applying the horizon theory to dynamic reservoir operation problems is that none of those studies from other fields can be directly applied to water systems problems.

Although Bes and Sethi’s concept has been widely used,

unfortunately, the theoretical derivation assumes both discrete time and discrete state of decision variables, e.g., logical decisions or decision tree structures in different time periods. For reservoir operation, it is common to use discrete time periods, i.e., dividing the continuous time period into discrete time intervals; however, we need to maintain a continuous state, i.e., the storage of a reservoir should be continuous conceptually because of hydrologic continuity. The state transition function is the core relationship of any reservoir operation models. In other words, the storages at different time points cannot be defined as finite, independent states since they are continuous in the real world. In this chapter, we will expand the theoretical framework provided by Bes and Sethi (1988) from problems with discrete states to dynamic reservoir operation optimization problems that are characterized by continuous states. In this chapter, we first introduce Bes and Sethi’s framework. Then we propose and prove a new condition to determine FH with given DH for dynamic reservoir operation problems. Following that is an order of magnitude analysis and a numerical example to show the relations among different parameters involved in the definition of the condition for determining FH, such as water stress level, reservoir size, inflow uncertainty, evaporation rate, and discount rate. Finally some discussions about the results and conclusions from this study are provided. 4.2.

Problem Formulation and Justification

The concept of DH and FH proposed by Bes and Sethi (1988) is adopted for the study of dynamic reservoir operation problems in this study. The essential idea of Bes

75

and Sethi’s work is described as follows. Define N (an integer) as the terminal period or the length of the problem horizon and (t, T) as a pair of time points, with 1  t  T  N  1 . If the optimal decisions in the periods in [1, t] are not affected by the model parameters (e.g., inflow, water demands) during periods [T + 1, N], then t is called a DH and T is the corresponding FH (Chand et al., 2002), as shown in Figure 4.1. The key for this problem is to determine T under a given t. Bes and Sethi proposed a general stopping rule. In decision theory, a stopping rule is characterized as a mechanism for deciding whether to continue or stop a procedure according to the present and past states. Underlying the stopping rule proposed by Beth and Sethi, the minimum forecast horizon for a given decision horizon is determined in the context of dynamic optimality.

This can be

explained as below: Assuming the optimal and next optimal solution of the N-horizon [1, N] optimization problem are found, let C(n) and C’(n), n=1, 2, …, N, be the objective value (e.g., cost) of the best and next best solution in period n, respectively; if existing T which satisfies T

T

n 1

n 1

 C ' ( n)   C ( n) 

N

 C ( n)

(4.1)

n T 1

Then the smallest T is the forecast horizon. This condition means if the best solution is changed to next best one, the additional cost is higher than the total discounted cost over periods [T  1, N ] . If the loss of changing the best solution to next best is larger than the possible gain in the future beyond a certain time period ( T ), then the current decision should stay with the best with time horizon of T. In other words, if T is the FH, then the optimal solution within periods [1, T ], which results from the information obtained only within periods [1, T ], is sufficiently close to the optimal solution within periods [1, T ] determined from the N-horizon optimization problem. The concept can be illustrated by a simplified stock-inventory example as given below. A manager runs a company and needs to determine the initial stock for a 10month period. A variable demand and constant production for each year are given in Table 4.1. In each year, if the production is higher than the demand, the excess is stored with a cost of $1,000 per unit. If the supply (production and storage) cannot meet the 76

demand, the deficit is ordered from other sources with a cost of $5,000 per unit. The manager needs to determine the initial stock in order to minimize the cost. By trial-anderror, the best solution is to put 9 units of initial stock and the next best is 8 units. Table 4.1 shows the cost difference between the best and next best solution by period (n) (the left hand side [LHS] of (4.1)), the sum of the cost over all periods after a particular period (the right hand side [RHS] of equation (4.1)), and the difference between LHS and RHS. As can be seen, when T=6, the additional cost to move from the best to next best is higher than the cost during the rest of the periods (7 –10). In other words, the best solution within time horizon [1, 6] is approximately close to the solution within a longer horizon (e.g., [1, 10]). Thus, by definition of (4.1), T=6 is the FH for this problem. See Sethi and Thompson (2000) for other typical examples. However, for a continuous domain, there is a technical difficulty to implement the concept of next best solution defined in Bes and Sethi’s framework. It is difficult, if not impossible, to find next best solution since for a continuous state problem, between the best and next best solution, there always exists another next best solution. Instead of the next best concept, this study introduces the concept of marginal value to modify Bes and Sethi’s method. In the following sections, we first introduce the notation and formulation for stochastic dynamic reservoir problems and then prove a condition that can be used to determine FH with a given DH for dynamic reservoir operation problems. 4.2.1. Problem formulation

We adopt a discrete multi-period, continuous state, dynamic optimization model for a simplified operation problem of a water supply reservoir. The economic value of water supply is defined by a utility function of a composite user. For the n-th time period, the utility is written as

bn  bn (rn , sn )

(4.2)

where bn is a concave, monotonically increasing utility function with respect to sn and

rn ; sn is the reservoir storage at the beginning of the n-th period, and is the state variable; rn  rn (sn , I n ) is the water release, which is the decision variable depending on storage, sn , and inflow, I n . For simplification, we combine release and spill together and assume 77

zero-utility for the spill.

This study only considers the uncertainty with reservoir

inflow, I n , which is defined as a random variable. The state transition equation is

sn1  f n (sn , rn , I n )

(4.3)

which follows the mass conservation principle. The reservoir storage ( sn ) is constrained by reservoir capacity. Other factors such as evaporation can be added to the transition function; however, they are not shown in (4.3) for simplicity but without affecting the generality of the following derivation. Assuming the objective of the reservoir operation problem is to maximize the utility over all time periods, the expected objective value can be written as  N n  Max B ( r ,  , N ) EV  bn ( rn , sn )       N r  n 1  

(4.4)

where B (r ,  , N ) is the N -period benefit, EV [ ] represents the expected value,  is the discount factor, r N is the decision variable vector, r N  r1 , r2 , r3 , , rN  , and r N  R N is the N-dimension Euclidean space, or N-space. There exists an optimal release vector, r N * , which satisfies

N  B(r N * ,  , N )  sup B(r ,  , N )  EV    n bn (r N * , sn )   s1   n 1

(4.5)

with a given initial storage s1 . Furthermore, for t  N , we define r T * as the optimal release during period [1, T] and define B (r T * ,  , T d )  sup B(r ,  , T d )

(4.6)

r d

where T d represents theT –horizon with a given forecast d . Equation (4.6) gives the optimal solution for the T -horizon problem, with optimal release decision ( r T * ) over the feasible domain of a policy (  d ) underlying a forecast (d). According

to

Beth

and

Sethi’s

theorem,

if

T

is

the

FH, B(r T * , , T d ) should be sufficiently close to B(r N * , ,T ) over periods [1, T], i.e., the

78

projection of B(r N * ,  , N ) in periods [1, T]. In the following, we derive and justify a condition for determining FH under a given DH. 4.2.2. Derivation of the condition of the stopping rule

Theorem: Assume B (r ,  , N ) is continuous and piecewise differentiable (i.e., each piece of the function is differentiable, e.g., a piecewise linear utility function) to r and s , and for n  Z , an upper bound ( Ln ) of the utility function in any period ( n ) exists, i.e.,

Ln  EV [bn (.| d )] . If there exists a time horizon ( T ) which satisfies 

B(r T * ,  , t d ) st 1

sT 1  sup( n Ln ), T  n  N st 1

(4.7)

then T is the forecast horizon for decision horizon t . The proof of this theorem is given in Appendix I. Note that the utility function is not necessary to be continuously differentiable (e.g., piecewise linear function is acceptable), although it is supposed to be concave and monotonically increasing with reservoir release. In (4.7),

B(r T * ,  , t d ) st

represents the utility change per one unit of

storage change under the current decision policy;

sT (1, the utility function is concave, which assures the principle of diminishing marginal benefit with increasing water supply, a general performance of water economics. Some properties of the utility function used for hedging analysis were discussed in previous chapters; also see You and Cai ( 2008b). This study considers a stochastic reservoir inflow forecast. We assume that a reliable probabilistic forecast is available for representing the statistics of reservoir inflows, i.e., the probability distribution of the inflow is perfect. The numerical model tests some typical types of inflow series [ I n ] as described below. An inflow forecast can be represented in general as

It  Tt  Pt  t

(4.15)

in which Tt is the long term trend, Pt is the seasonal variability also known as periodicity, and  t is the random noise. In this study, without losing any significance Tt is set as a constant.

83

Two stochastic cases are tested following (4.15), stationary and non-stationary time series. Stationary inflow time series are free of trends, shifts, or periodicity and the statistical parameters of the series, such as the mean and variance, remain constant through time; generally hydrologic time series defined on annual time scale are stationary when ignoring large-scale climatic variability (Salas, 1992). For stationary inflow time series, seasonal variability ( Pt ) drops from (4.15). In this study, for the stationary case, a yearly time step is set for an inter-year reservoir operation model. Non-stationary time series are usually defined on time scales smaller than a year such as monthly series, and they include seasonality, Pt in (4.15). Seasonality is defined with parameters such as period, amplitude, and randomness of noise, as shown in Figure 4.2. The period is the time interval between the two highest values of periodic function; amplitude is a measure of the magnitude of oscillation; randomness is represented by the standard deviation of noise. In this study, for the non-stationary case, a monthly or weekly time step is set for an intra-year reservoir operation model.

More detailed

description of trend, seasonality and randomness can be found in Salas (1992). Furthermore, we consider a case that uncertainties propagate with the time due to forecast limitation, which is even closer to the real world situation than the two cases given above. We apply the random walk time series to study the effect of uncertainty propagation. A time series I t is a random walk if it satisfies

It 1  It  t

(4.16)

where the uncertainty item,  t , is a white noise which is normally distributed N (0,  ) . The standard deviation item (δ) reflects the propagation speed of the uncertainty, i.e., a larger value of δ leads to a higher prorogation speed. As expressed in (4.16), the uncertainty is accumulative and diffusive, which adopts the stochastic nature. Assigning values of δ that increase with time, we have a time series that involves uncertainty growing with time, which can mimic the diminishing forecast capability. Figure 4.3 shows a flowchart of the numerical modeling procedures. A dynamic programming algorithm is implemented and applied to a continuous state Markov decision process (MDP) (Bellman, 1961; Howard, 1960; Bertsekas, 1995). A typical 84

backward methodology with SDP is used to solve the dynamic programming problem from period T to 1 according to the Bellman equation

Vn* (s)  Max rn

EV

{bn (rn )   (Vn*1 (sn1 )}

I n { I n ,1 ,.. I n ,100 }

where n is period index and V is the residual value of storage.

(4.17) Piecewise linear

approximation is used for the residual value and nonlinear objective function. Monte Carlo simulations are applied to generate the inflow time series, which may also be produced by an external hydrologic simulation model. For each period, given a normal distribution of  t , 100 samples of  t are generated and (4.15) or (4.16) is then used to calculate 100 inflow inputs for the SDP model. The finite difference method is used to solve the differential equation involved in (4.7). Following the procedure shown in Figure 4.3, the condition by (4.7) is used as the stopping rule to determine T, the forecast horizon. The procedure is to test T starting from T  t . If the condition (4.7) is not satisfied, set T  T  1 and continue the test; otherwise, the procedure stops and T is the FH. The parameters of the baseline scenario and other scenarios tested in the numerical model are listed in Table 4.3. 4.4.1. Numerical model results from stationary inflow time series

We examine the difference between FH and DH (i.e., FH-DH) under factors assuming the DH only includes one period ( t  1 ). Water stress index (WSI) is defined as a normalized form:

WSI 

I  Dm I

(4.18)

where I is the average inflow and Dm is the maximum water delivery. The negative value of the WSI means water shortage while the positive value means water excess. Figure 4.4 shows the value of (FH-DH) under different water stresses and reservoir capacities. When there is consistent water excess, there is no difference between FH and DH (FH - DH = 0), i.e., decision making is static. This is because in most periods (or on average), there is sufficient water for the demand with very little or no incentive to reserve water for the future. This situation explains why standard operation policy (SOP), 85

recognized as a static optimization policy, is suitable for a condition with both relatively high average inflow and low inflow uncertainty. However, when inflow cannot fully satisfy the demand, either now or potentially in the future, reservoir operation needs to take a longer foresight as shown in Figure 4.4. Larger reservoir capacity will always lead to a longer FH than a smaller capacity, which is consistent with the order analysis based on (4.11). FH increases with the level of water stress and then stabilizes when water stress reaches to a level (-0.3 ~ -0.6, depending on reservoir capacity). When the level of water stress is extremely high and water availability is extremely low, the marginal value of water in the current period is large and there is no incentive for the purpose of hedging. A reservoir cannot improve the operation much by regulating a very limited amount of water. Thus, it will not be beneficial to have a long foresight under extreme water stress situations. This is not shown in Figure 4.4 and can also be justified by (4.7) . When water stress approaches negative infinity, both sides of (4.7) will approach identical values (negative infinity) for any FH under a given DH, the minimum FH can then be the same as DH, i.e., FH – DH =0. Inflow uncertainty plays a significant role in decision making and the results in the following show how inflow uncertainty affects the forecast horizons. As shown in Figure 4.5, the impact of uncertainty (represented by inflow variance coefficient) on FH changes according to different water stress levels (4.18). Even under the condition of water excess, decision making under high inflow uncertainty needs to take a longer FH so that future potential water shortage can be mitigated. When water demand is close to the average inflow (the value of WSI is about zero), the probability of reservoir fullness is high and fullness reduces the capability to regulate inflow in the future.

Higher

uncertainty reduces the probability and therefore leads to a longer FH. Buchberger and Maidment (1989) also discussed how the probability of reservoir fullness was influenced by demand, average inflow and reservoir capacity. Under the status of water stress, the length of FH increases with the level of water stress and decreases with uncertainty. Uncertainty affects the length of FH by two factors, the probability of reservoir fullness, as discussed above, and risk premium.

86

Risk

premium is the difference between the expected utility and the risk-free utility, which can be understood as the loss due to uncertainty. Higher uncertainty leads to higher risk premium, reduces total utility value, and increases the marginal utility for both current and future. With high water stress, the impact of reservoir fullness declines and risk premium becomes a dominating factor. In this case, lower uncertainties can mitigate risk premium and result in longer FH. The compelling impacts of deterministic variability (existing with the mean level of water stress) and randomness on the length of FH are further discussed with the case of non-stationary inflow with seasonality, presented later in this chapter. Higher discount rate diminishes the future value and therefore shortens the FH, as shown in Figure 4.6. Evaporation loss affects the FH in a similar way (Figure 4.7). High evaporation rates decrease reservoir storage and the value of water for the future, and shorten FH. The impact of utility concavity (n) is shown in Figure 4.8. If the concavity equals to unity, which results in a linear utility function, the reservoir operation will become a static problem, and the solution will be SOP (Hashimoto et al., 1982). When the utility function is close to a linear form (n=1.1), FH is reduced significantly compared to a higher level of utility concavity. High concavity of the utility function, which allows more space for inter-temporal resource allocation, will enhance the dynamic feature of reservoir operation (You and Cai, 2008b) and then increases the FH. 4.4.2. Non-stationary inflow time series considering seasonality

The Monte Carlo procedure is used to generate a sequence of inflows to test the impact of seasonality. As can be seen from Figure 4.9, the highest FH for the nonstationary case occurs when inflow is close to demand, while the longest FH occurs when the water stress ranges between -0.3 and -0.6 with the stationary inflow series (Figure 4.9). This difference can be explained as follows. When inflow is close to demand, i.e., the water stress level is around zero, the likelihood of reservoir fullness under the nonstationary case will be lower than the stationary case, because the knowledge on seasonality can guide reservoir storage and release across high-flow and low-flow periods. Thus the non-stationary case results in the longest FH around zero water stress. For

87

stationary inflow case, the impact of reservoir fullness is strong when inflow is around the demand level or there is low level of water stress. This is because the random, unpredicted high inflow can make a reservoir full and incapable to store water for future low-flow periods. However, with the increase of water stress level, the impact of risk premium becomes stronger, which leads to longer FH at a higher level of water stress (but extreme high levels of water stress will decline FH, as previously discussed). FH increases with the level of the amplitude of seasonality. This implies that high amplitude of seasonal inflow will require longer FH, particularly when inflow and release demand are nearly the same level. The amplitude of inflow represents the intensity of extreme events; large amplitude means a large fluctuation of water availability so that high inflows occurring in some periods could be stored for low flow period. The length of FH is less sensitive to the period of seasonality, as can be seen from Figure 4.10. This might be related to the fact that the period of seasonality is relatively deterministic and depends less on future forecast information, although it determines the temporal operation cycle of a reservoir. The period length of seasonality directly affects neither the probability of reservoir fullness nor the risk premium. The only impact is related to reservoir storage and evaporation loss. Longer seasonality period may keep more water in storage and increase evaporation. Thus, a longer period of seasonality can result in shorter FH. On the other hand, under severe water stress conditions, a long period of seasonality may go with a long period of water scarcity, which may extend FH to prevent future loss. The balance of the two factors results in a slightly higher FH with a longer period of seasonality. The impact of randomness of seasonal inflow time series on FH depends on the level of randomness, as shown in Figure 4.11. When the randomness level is high, the seasonality effect will be reduced as illustrated in Figure 4.4. Thus, the case with nonstationary inflow will be similar to the case with stationary inflow, under which the probability of reservoir fullness will dominate the impact on FH (e.g., R=2 in Figure 4.11). When the randomness is small (R=0.5 or 1), the seasonality effect will be strong and the change of FH with water stress will be similar to what is shown under Case 2 (Figure 4.9). When comparing the impact of the three levels of randomness shown in

88

Figure 4.11, it can be seen that higher randomness results in longer FH. This is because randomness enhances the severity of the potential low flow and increases the risk premium in future periods. 4.4.3. Inflow with random walk inflow time series with uncertainty propagation

The random walk inflow time series (4.16) is applied to test how FH is influenced by uncertainty propagation speed. The speed of propagation (SP) represents the rate of the standard derivation (  ) increase with time. As can be seen from Figure 4.12, a higher SP results in a longer FH. As discussed before, larger uncertainty causes larger risk premium in the future, which motivates the decision maker to take a longer plan. Also, compared to other factors, the speed of uncertainty propagation has less impact on the current decision, because the information in certain early periods will not much affected by the uncertainty propagation. 4.5.

Discussions and Conclusions

This study extends Bes and Sethi’s framework on decision and forecast horizons to reservoir operation problems under hedging policies.

The stopping rule for

determining FH is modified using the marginal utility concept for problems with continuous states. This new condition is proved with an assumption of a concave, utility function monotonically increasing with reservoir flow release.

Order analysis and

numerical modeling are conducted to analyze the impact of various reservoir parameters and inputs on the length of FH. In general, the results show that larger marginal utility gfor the current period leads to a shorter FH; while larger marginal utility for the future (corresponding to smaller future utility, following the property of diminishing marginal utility) needs longer FH.

Reservoir capacity, inflow uncertainty (including stationary uncertainty and

seasonality), and economic parameters (e.g., discount rate, utility function concavity) all affect the marginal utility value of current and future periods under water stress conditions and impose various requirements of FH. When water stress is at a modest level (0 to -0.2), reservoir capacity and inflow forecast uncertainty, both of which influence the probability of reservoir fullness, have a more important role in determining FH. Large reservoir capacity requires a longer FH although the impact is complicated by 89

other factors such as uncertainty and high level of uncertainty shortens the FH since larger priority is required for the current water use utility. However, when water stress level increases, reservoir capacity and inflow uncertainty become less important since when the average inflow is low, the effect of storage regulation and inflow uncertainty is less significant. Under severe water stress, economic factors have a stronger influence on FH through dealing with risk premium, whereas under the condition of water excess, higher level of uncertainty increases the FH. With both water excess and low inflow uncertainty, a static decision mechanism (under which FH equals DH), such as SOP, is reasonable for optimal reservoir operations (You and Cai, 2008a, b). Thus, at modest water stress levels, sufficient reservoir capacity and reliable inflow forecasts are important for preventing large utility loss under possible severe droughts; whereas under severe water stress levels, economic incentives are worth more attention. It should be noted that today, in more areas, water stress is a growing concern and water management based on economic incentives is important. The comparison of the FH with three types of inflow time series shows the impact of hydrologic uncertainty and variability on the determination of FH. The three inflow series represent stationary inflow, non-stationary inflow with seasonality and inflow characterized by random walk without considering seasonality. It is found that the impact of stationary and non-stationary inflow time series on FH depends on the water stress levels. The highest FH for the non-stationary case occurs when inflow is close to demand, while the longest FH occurs when higher water stress levels are relatively high with the stationary inflow series. The difference is caused by the compelling impact of reservoir fullness probability and risk premium. However, in general the non-stationary inflow with the seasonal pattern leads to a longer FH than the stationary inflow. This helps understand the limitation of implicit stochastic optimization, which uses scenarios of inflow within a certain future time period and each scenario represents a deterministic case with an occurrence probability.

These scenarios, which include the “perfect”

forecast of deterministic information such as seasonality, may represent a long, overestimated and unrealistic forecast horizon for stochastic dynamic reservoir operation optimization. As pointed out by Draper (2001), the attribute of perfect foresight causes

90

unrealistic storage operations and it is more realistic to use a limited forecast horizon for implicit stochastic reservoir operation model. The result from the random walk time series shows that the diminishing forecast capability requires a longer FH for the case with a higher speed of uncertainty propagation. This follows the common knowledge that one may need a longer plan for the unknown future. However this is also kind of counterintuitive since the longer the FH, the less reliable of the forecast information. Thus in the real world, we have to deal with such a situation that involves conflicts between the request for reliable information in the future and the diminishing quality of information beyond a certain horizon due to the existing, limited forecast capability. As shown in this study the various factors can affect the FH. We explore the impact of each factor on FH in the way of assuming a constant level of other factors. However from a practical perspective it is more useful to identify the dominating factors in a complex context in which all factors are variables. For example, a small reservoir serving a demand of nearly the reservoir capacity takes only a few periods to refill, which will usually require a short forecast horizon. The large reservoir capacity generally requires long FH. However, as shown by the results, the various factors such as inflow and demand can make a complex situation. For a large reservoir serving a demand much smaller than the reservoir capacity, if the demand is even below the average inflow (i.e., no water stress), FH can be as short as DH; while if the demand is still higher than the average inflow (i.e., water stress existing), then the large reservoir capacity will be beneficial in dealing with inflow uncertainty to reduce risk premium, which will end with a long FH. Systematic analysis of such complex situations represents a meaningful future work. The theoretical derivation and numerical modeling presented in this chapter can be applied to not only to reservoir operation problems but also to various other civil and environmental engineering systems, which are characterized by the feature of storage and dynamic state transition. For example, the operation of detention tanks or ponds in drainage systems, the schedule of additional capacity for sewage or drinking water treatment under extreme conditions, the design and control of some highway lanes for

91

traffic problems during peak hours, etc. All these problems have common need for proper hedging rules to decide how much resource or facility should be carried over to the future and how long the future period needs to be in order to avoid serious economic and environmental risk in the future, while maintaining the efficient system operation at present.

92

Appendix I

Theorem: Assume B (r ,  , N ) is continuous and piecewise differentiable to r and s ; also assume for n  Z , an upper bound ( Ln ) of the utility function in any period n exists, i.e., Ln  EV [bn (.| d )] . If there exists a time horizon ( T ) which satisfies



B(r T * ,  , t d ) st 1

sT 1  sup( n Ln ), T  n  N st 1

then T is the forecast horizon for decision horizon t . Proof:

We use proof by contradiction. Define P as a condition and Q as a conclusion; we want to prove P => Q. To apply the method of proof of contradiction, we suppose that P and not-Q are true and look for a contradiction with known facts, i.e.,“P is true” => “not-Q is false”. In our proof, we use a variation of the proof by contradiction, which is called the contra-positive method, by which if we can show not-Q => not-P, then we can conclude that P => Q (Lay, 2004). For our problem, we assume T is not FH and

B(r T * ,  , T d ) is not the projection of B(r N * ,  , N ) (the optimal solution of the N-period problem) in period [1, T] ; with this assumption, if (4.7) is false, then we prove the theorem. If T is not the FH and B(r T * ,  , T d ) is not the projection of B(r N * ,  , N ) in period [1, T], then there exists a decision rˆ T , rˆ T  r T * , satisfying:

 N  B(rˆT ,  , T d )  EV    nbn (rn , sn )   n T 1  sT 1 ( rˆT )  N  B(r T * ,  , T d )  EV    nbn (rn , sn )  n T 1

 sT 1 ( r T * )  

(A1)

As defined before, r T is the decision variable vector, r T  r1 , r2 , r3 , , rT  .

B (rˆT ,  , T d ) and B(r T * ,  , T d ) are the total utility in T periods under the two release

93

decisions, rˆ T and r T * , respectively; and we assume B (rˆT ,  , T d ) is close to B(r T * ,  , T d ) . (A1) can be rearranged to

B(r T * ,  , T d )  B(rˆT ,  , T d )





 N   EV    n (bn (rn , sn ) ˆT  bn (rn , sn ) s ( rT * ) )  sT 1 ( r ) T 1  n T 1 

(A2)

The LHS represents the difference of outcome over periods [1, T] between rˆ T and r T * , and the RHS represents the difference of the utility over periods [T+1,N] resulting

from rˆ T and r T * , respectively. It is generally reasonable to assume that the reservoir storage at the end of period T resulting from the two decisions is different, i.e., sT 1 ( rˆT )  sT 1 ( r T * ) ≠ 0. Divide the two sides of (A2) by sT 1 ( rˆT )  sT 1 ( r T * )

B ( r T * ,  , T d )  B ( rˆT ,  , T d )  sT 1 ( rˆtT )  sT 1 ( r T * ) EV [

N

 [

n T 1

n

(bn ( rn , sn )

sT 1 ( rˆtT )

(A3)

 bn ( rn , sn ) s

T 1 ( r

T*

)

)]] [ sT 1 ( rˆtT )  sT 1 ( r T * )]

Note sT 1 ( rˆT )  sT 1 ( r T * )  0 . Because rtT * is the optimal decision for T-period problem (by definition), B(r T * ,  , T d )  B(rˆT ,  , T d ) . Additionally, the utility function in each period ( bn ) is concave, monotonically increasing with release.

B(r T * ,  , T d )  B(rˆT ,  , T d ) leads to

r

T*

Thus

  rˆT . By the continuity law, storage

monotonically decreases with release and we have sT 1 ( rˆT )  sT 1 ( r T * )  0 . Given that B(r T * ,  , T d ) and B (r T * , , t d ) are the optimal output in periods [1, T] and [1, t], respectively. The difference of these two items represents the optimal output during periods [t, T]. This follows the principle of dynamic optimality, which states that the decision at one period constitutes an optimal policy in that period if the decision is part of the overall optimal solution. Therefore, the utility in periods [t, T] from the overall optimal solution in period [1, T] should not be inferior to any other solution, i.e.,

B(r T * ,  , T d )  B(r T * ,  , t d )  B(rˆT ,  , T d )  B(rˆT ,  , t d )

94

(A4)

which is rearranged to:

B(r T * ,  , T d )  B(rˆT ,  , T d )  B(r T * ,  , t d )  B(rˆT ,  , t d )

(A5)

Following the LHS of (A3), we have

LHS 

B(r T * ,  , T d )  B(rˆT ,  , T d ) B(r T * ,  , t d )  B(rˆT ,  , t d )  sT 1 (rˆT )  sT 1 (r T * ) sT 1 (rˆT )  sT 1 (r T * )

(A6)

Following the principle of differentials, if sT ( r T * ) is close to sT ( rˆT ) and B (rˆT ,  , T d ) is close to B(r T * ,  , T d ) , then the LHS of (A3) can be expressed by a differential form:

LHS 

B (r T * ,  , t d )  B(rˆT ,  , t d ) sT 1 (rˆT )  sT 1 (r T * )



B(r T * ,  , t d )

(A7)

sT 1

B(r T * ,  , t d )  st 1

sT 1 st 1

Some derivation is conducted for the RHS of (A3). According to the continuity law, sT 1 ( rˆT )  sT 1 ( r T * ) 

N

 [r

n T 1

n sT ( rˆT )

 rn

sT ( r T * )

]

(A8)

which means that the additional storage save by period T must be greater or equal to the additional release during periods [T+1, N] between the two release decisions under the same inflow forecast. Thus following the RHS of (A3), we have:  N  n   [ (bn (rn , sn ) sT 1 ( rˆT )  bn (rn , sn ) sT 1 ( rT * ) )]   RHS  EV  n T 1 N   (rn s ( rˆT )  rn s ( rT * ) )  T 1 T 1   n T 1

95

(A9)

k k For all a series of ratios, n  0 , sup n  ln n ln

k l

n

n

n

 inf n

kn . n=1, 2… Thus (A9) will ln

n

lead to:   bn (rn , sn ) s ( rˆT )  bn (rn , sn ) s ( rT * )   T 1 T 1  RHS  sup   n EV    r r T T*    n n ˆ sT 1 ( r ) sT 1 ( r )    sup EV [ n bn '(rn , sn )]

(A10)

 sup( n Ln )

In which all items have been defined before. Finally (A7) and (A10) lead to:



B(r T * ,  , t d ) st

sT  sup( n Ln ) st

(A11)

This is to say, if T is not the FH, (A11) is true and (4.7) is false. This proves that if (4.7) is true, then T is the derived FH for a given DH ( t ).

96

References

Bes, C. and S. P. Sethi (1988), Concepts of forecast and decision horizons: Applications to dynamic stochastic optimization problems. Mathematics of Operations Research, 13, 295–310. Bower, B. T., Hufschmidt, M. M., and W. W. Reedy (1962), Operating procedures: Their role in the design of water-resources systems by simulation analyses, Design of Water Resources Systems, A. Maass and Maynard M. Hufschmidt et al., pp443– 458, Harvard University Press, Cambridge, M. A. Buchberger, S., and D. Maidment (1989), Diffusion Approximation for Equilibrium Distribution of Reservoir Storage, Water Resour. Res., 25(7), 1643-1652. Chand, H., V. N. Hsu and S.P. Sethi (2002), Forecast, solution and rolling horizons in operations management problems: a classified bibliography, Manufacturing & Service Operations Management, 4(1), 25-43. Charnes, A., W. W. Cooper and B. Mellon (1955), A model for optimizing production by reference to cost surrogates, Econometrica, 23, 307–323. D. P. Bertsekas (1995), Dynamic Programming and Optimal Control, Athena Scientific, Belmont, MA. Draper, A. J. (2001), Implicit Stochastic Optimization with Limited Foresight for Reservoir Systems, PhD dissertation, Univ. of California Davis, CA. Draper, A. J. and J. R. Lund (2004), Optimal hedging and carryover storage value. J. Water Resour. Plann. Manage., 130(1), 83-87 Hall, W. A., A. J. Askew and W. W.-G. Yeh (1969), Use of the critical period in reservoir analysis, Water Resour. Res., 5(6), 1205-1215. Hashimoto, T., J. R. Stedinger and D. P. Loucks (1982), Reliability, resilience, and vulnerability criteria for water resource system performance evaluation, Water Resour. Res., 18(1), 14-20. Johnson, S. M. (1957), Sequential production planning over time at minimum cost, Management Science, 3, 435–437.

97

Klemes, V. (1977), Value of information in reservoir optimization, Water Resour. Res., 13(5), 857–850. Lay , S.R. (2004). Analysis with an Introduction to Proof, Fourth Edition. Prentice-Hall, Inc., New Jersey. Loucks, D. P., J. R. Stedinger and D. A. Haith (1981), Water resources systems planning and analysis, Prentice-Hall, Englewood Cliffs, NJ. Lund, J.R. (2006), Drought Storage Allocation Rules for Surface Reservoir Systems. J. Water Resour. Plng. and Mgmt. 132, 395. Lundin, R. A. and T. E. Morton. (1975), Planning horizons for the dynamic lot size model: Zabel vs protective procedures and computational results, Operations Research, 23, 711–734. Modigliani, F. and Hohn, F. E. (1955), Production planning over time and the nature of the expectation and planning horizon, Econometrica, 23, 46-66 Morton, T. E. (1981), Forward algorithms for forward thinking managers. Applications of Management Science, Vol. I, JAI Press, Greenwich, CT, 1–55. Oguz, B. and Bayazit, M., (1991), Statistical properties of the critical period. J. Hydrol., 126, s. 183-194. R. E. Bellman (1957), Dynamic Programming. Princeton University Press, Princeton, NJ. R. Howard (1960), Dynamic Programming and Markov processes, The M.I.T. Press, Wiley, NY. Salas J.D. (1992). Analysis and modeling of hydrologic time series, in Handbook of Hydrology, Maidment, D. (ed.), chap. 19, pp. 19.1– 19.72, McGraw-Hill, Inc. Sethi, S. P. and G. L. Thompson, (2000) Optimal Control Theory: Applications to Management Science and Economics, Kluwer, 2nd Edition. Shih, J. S. and C. ReVelle (1995), Water supply operations during drought: A discrete hedging rule, Eur. J. Oper. Res.,, 82, 163–175.

98

Shih, J.-S. and C. ReVelle (1994), Water-supply during drought: continuous hedging rule, J. Water Resour. Plann. Manage., 120(5), 613-629. Stedinger, J. R. (1978), Comment on value of information in reservoir optimization, Water Resour. Res., 14(5), 984–986 Wagner, H.M., T.M. Whitin. (1958), Dynamic version of the economic lot size model, Management Science, 5, 89–96. You, J-Y. and X. Cai (2008a), Hedging rule for reservoir operations: (1) A theoretical analysis. Water Resour. Res., 44, W01415, doi:10.1029/2006WR005481. You, J-Y. and X. Cai (2008b), Hedging rule for reservoir operations: (2) A numerical model. Water Resour. Res., 44, W01415, doi:10.1029/2006WR005482. You, J-Y. and X. Cai (2008c), A Reexamination of Critical Period for Reservoir Design and Operation, J. Water Resour. Plann. Manage. (in press).

99

Table 4.1. Inventory example for Bes and Sethi’s method.

* Monthly discount factor is 0.85 ** T=6 (noted by bold text) is the shortest forecast horizon

100

Table 4.2. The relationships between FH and the impact factors Forecast horizon ( T )* Decision horizon ( t )



Discount factor ( 0    1 )**



Evaporation rate (e)



Probability of fullness ( prob[ X c ] )



Marginal utility of future inflow ( b( I min ) I min )



Marginal utility of current water use( b( Dn ) Dn



Future inflow ( I min )



Normal demand ( Dn )



  *  represents the monotonic increasing relationship and  represents the monotonic decreasing relationship.

** a higher discount factor (closer to 1.0) means that future utilities are discounted less.

101

Table 4.3. Parameters used in the numerical example, basic scenario and other scenarios

102

Figure 4.1. The concept of DH/FH

103

Inflow data

14

Amplitude

Stationary seasonality

12

Inflow

10 8

Randomness

6

Seasonal period

4 2 0 0

2

4

6 Time periods

8

10

12

Figure 4.2. The pattern of a non-stationary time series with consideration of seasonality

104

Figure 4.3. Flow chart of procedures to determine FH (T) with a given DH (t)

105

Figure 4.4. Relationship between FH and water stress index under different reservoir capacities (CAP), with inflow uncertainty variance coefficient (CV) = 0.2

106

Figure 4.5. Relationship between FH and water stress index under different uncertainty levels (variance coefficient, CV), CAP = 10

107

Figure 4.6. Relationship between FH and water stress index under different discount rates (r), CAP=10, CV=0.2

108

Figure 4.7. Relationship between FH and water stress index under different evaporation rates (EVP), CAP=10, CV=0.2

109

Figure 4.8. Relationship between FH and water stress index under different utility concavity values (n), CAP=10, CV=0.2  

110

Figure 4.9. Relationship between FH and water stress index under different amplitudes (AMP), CAP=10

111

Figure 4.10. Relationship between FH and water stress index under different seasonality lengths (P), CAP=10

112

Figure 4.11. Relationship between FH and water stress index under different randomness levels (R), CAP =10

113

Figure 4.12. Relationship between FH and inflow for speed of prorogation (SP), CAP

114

CHAPTER 5. ERROR BOUND OF THE OPTIMAL RESERVOIR OPERATION DECISION UNDER A LIMITED FORECAST HORIZON Summary

This chapter provides an approach to analyze the error bound with a limited forecast horizon for dynamic reservoir operation problems. It is a practical approach for real world applications since current weather predictability may not extend as long as the “perfect forecast” determined in Chapter 4. A method is proposed to measure the error and error bound according to a terminal stage boundary condition.

Additionally,

numerical examples are applied to testing two stochastic time series, stationary and nonstationary inflows. It is found that the error generally shows power-law convergence. This confirms the promise of rolling horizon decision making for water resources systems. When the perfect foresight is not available, the rolling decision procedure with regularly updated forecast information could avoid serious loss due to shortsighted policies. The results show that the error with a myopic decision increases with reservoir capacity and uncertainty and slightly decreases with the evaporation and discount rate for stationary inflow. But the trend becomes ambiguous due to the complexity of different driving forces for non-stationary cases. The error bound method can be used to estimate the potential error which exists in actual operations with limited forecasts.

115

5.1.

Introduction

One question for hedging operations is how long to hedge for particular operational frameworks. This is related to the question of how far to forecast to the future. For dynamic reservoir operation models, when the decisions in the initial few periods are not affected by forecast data beyond a certain period, that period is known as forecast horizon (FH), and the number of the initial periods is known as the decision horizon (DH). Chapter 4 presented a theoretical analysis for determining the length of the ideal FH for a given DH under the optimality condition of dynamic programming (Also see You and Cai, 2008a).

However, from a practical viewpoint, two major

concerns should be addressed. The first is the assumption of a reliable probabilistic forecast of reservoir inflows. Hence, the reliability and availability of the hydrologic forecast is limited by current forecast technology. With the current forecast technology, reliable hydrologic forecasts are even not available for next few months and reliable long-range hydrological forecasts are difficult, if not impossible, to obtain (Somerville, 2004). The second concern is the sensitivity or necessity of the information to decision making. The forecast horizon includes future information that can only have a very small influence to current policy; in other words, the current policy may not be very sensitive to distant future information (You and Cai, 2008a).

A hypothesis is that the distant

information may be neglected when reasonable error is allowed. Given these concerns, the forecast horizons derived from the horizon theory might be too long to be of practical use (Chand et al., 2002). Rather than trying to determine the perfect forecast horizon, this study considers a dual approach to determine how far the actual decision is away from the optimality with a limited, but more realistic forecast horizon. The approach could be more practical for real world applications since current predictability of weather and climate may not extend as long as the “perfect forecast”. In real world practices, limited foresight models have been applied to deriving reservoir operations (Draper, 2001). For example, the rollinghorizon model with limited foresight is a common tool in dynamic operation analyses with consideration of both operation efficiency and practicality (Chand et al, 2002).

116

Under a rolling horizon procedure, at the beginning of the first period, as shown in Figure 5.1(a) , a T1 -horizon problem is setup according to current conditions such as initial storage and forecasted inflow, and release decision is made for the current period. Following that, at the beginning of the second period, the result from the first period decision applies (i.e., the ending storage of period one is the starting storage of period two).

The system observation and forecast are updated.

The problem can be re-

formulated as the recourse problem of the problem in last period with an updated horizon T2 , as shown in Figure 5.1(b), which may or may not be equal to T1 . This procedure is repeated with updated horizons from period to period. The forecast horizons rolling over periods are called rolling horizon or study horizon (Bean et al., 1987), which is the forecast horizon that is actually available and used in practical dynamic system analysis. The rolling-horizon procedure is practical for reservoir operations since weather forecasts and streamflow predictions are updated from period to period. For example, since December 1994, Climate Prediction Center (CPC) of the U.S National Oceanic and Atmospheric Administration (NOAA) has been using the rolling forecast of temperature and precipitation. CPC issues forecast near the middle of each month for the following 3 calendar month periods. The forecast is updated each month and the 3-month period overlaps to the next. Although each reservoir may use a different forecast model, most of them use forecast framework similar to the CPC’s. In real-world operations, reservoir managers approve the operation plan at the beginning of each period according to the best available forecast data; this process is undertaken at a weekly or monthly basis using rolling climate and hydrologic forecasts. A study horizon shorter than the forecast horizon required by optimal decision making may lead to suboptimal decisions due to limited foresight. For example, in Figure 5.1, T1 , a study horizon which is shorter the FH. However, to be realistic with the current forecast technology, a myopic study horizon with acceptable error bound to the optimal decision can be a useful exercise. A framework that analyzes rolling-horizon optimality and describes the suboptimal condition with an error bound can be used to choose a realistic forecast horizon by considering the tradeoff among cost, data reliability, and decision accuracy. 117

Lundin and Morton (1975) introduced the concept of error/error bound by defining  optimality of a dynamic decision problem. This concept has been well accepted but a common definition with illustrative details is still needed. Morton (1981) identified the significance of the issue of error bound and its applications. The same argument was also addressed by Kleindorfer and Kunreuther (1978) and Sethi and Sorger (1991). However such studies have not been well followed, and so far, error bound is still not systematically studied from a theoretical perspective due to the complexity of developing a general analytical framework to evaluate rolling horizon procedures (Sethi and Sorger, 1991). Recently, Chand et al. (2002) re-emphasized the importance of error bound problem, which was versioned as a promising research topic that deserved much more attention. Following the concept proposed by Lundin and Morton (1975), this study develops a methodology to analyze error bound of rolling-horizon optimality for reservoir operation problems. We modify the original concept of horizon theory in Chapter 4 to explore the properties of error bound for dynamic reservoir operation problems. In the rest of this chapter, we first introduce the method to measure the error and error bound according to a terminal stage boundary condition. Then numerical examples are applied to testing the error bound method, followed by discussions and conclusions. 5.2.

Methodology

Lundin and Morton (1975) defined the  -optimality to present the upper error bound of suboptimal condition with limited foresight horizon. The methodology yields a solution with the desired accuracy/tolerance characterized by  , a prescribed threshold. Various studies focus on deriving an error bound based on more or less different definitions that seem to be convenient for different purposes (Lee and Denardo 1986, Denardo and Lee 1991, and Chen et al. 1995). To address the error bound problem, we use the same approach described in Chapter 4, which formulates the problem as a dynamic optimization model with discrete multi-periods and a continuous state for a water supply reservoir. The economic value of water supply is defined by a utility function of a composite user. For the n-th time period, the utility is written as

118

bn  bn (rn , sn )

(5.1)

where bn is a concave, monotonically increasing utility function; rn is the water release,.

sn is the reservoir storage at the beginning of the n-th period, which is the state variable. The expected objective value which maximizes the utility over all time periods is written as

 N n  Max B ( r ,  , N ) EV  bn (rn , sn )       N r  n 1  

(5.2)

where n=1, 2, ... N , the index of time periods. N represents the length of the time horizon for the problem analysis, called problem horizon. B (r ,  , N ) is the utility over all time periods. EV [ ] represents the expected value,  is the discount factor, and r N is the decision variable vector. There exists an optimal release vector, r N * , which satisfies N  B (r N * ,  , N )  sup B (r ,  , N )  EV    nbn (r N * , sn )   s1   n 1

(5.3)

with a given initial storage s1 . Furthermore, for T  N , we define r T * as the optimal release during period [1, T ] and define B (r T * ,  , T d )  sup B (r ,  , T d )

(5.4)

r d

where T d represents the T –horizon with a given forecast d . Equation (4.6) gives the optimal solution for the T -horizon problem, with optimal release decision ( r T * ) over the feasible domain of a policy (  d ) underlying a forecast (d). More detailed description of the formulation could be found in Chapter 4 (also see You and Cai, 2008a). Following the Lundin and Morton’s definition, we extend the  -optimality condition as: A policy rtT ( , T , t d ) under a given error threshold (  ), a study horizon ( T , i.e., a limited forecast horizon) and decision horizon ( t ), and a given forecast ( d ), is

119

t

conditionally optimal, if for all n  T , B ( r T * ,  , t d )  B ( r TF * ,  , t d )    n   is n 1

When   0 , T

irrespective to periods beyond period n (n+1, n+2, ..).

approaches to the “full” forecast horizon under optimality (defined as TF to be different from T ), which was discussed in Chapter 4. The concept can be visualized from Figure 2. The upper part shows the path of decision variable, r , and the lower part is the path of state variable, s , for sub-optimality with study horizon T and optimality with forecast horizon TF.

t

 n 1

n

, which represents

the sum of error, is the difference between the outcome from the suboptimal policy r T * and the optimal policy r TF * , as shown in the hatched area in the figure.  is the upper

bound of the error. In the following, the error bound  is further discussed considering the properties of reservoir operation problems. First, by replacing the problem horizon (N) with the perfect forecast horizon (TF), Eq. (5.3) can be re-written as TF    Max B ( r ,  , T ) E [  n bn ( rn , sn )]   F TF r n 1  

(5.5)

In order to determine the error bound  , here we update the definition of optimal policy provided in Eq. (5.4). We convert a free-end dynamic problem to a fix-end dynamic problem by introducing the ending state variable sN , which is expressed by Eq. (5.6) (similar with Eq. (5.4)). N

sup B(r ,  , N )  E  nbn (r N * ) n 1

(5.6) ( s0 , sN )

where r N * represents the global optimal solution with given initial storage so and ending state sN , which is further described as follows. For a problem with study horizon T , the suboptimal decision is written as r T * . Giving a decision horizon, t , the sum of utility over the periods [1, d ] is written as

120

B(r T * ,  , t d ) . If there is a T which falls in the range of t  T  TF , optimal decisions within the forecast horizon (TF) with a given forecast d is represented as r TF * , then we can represent the total optimal utility with TF by two items, the utility within T and the expected optimal utility between T and TF: Max {B ( r ,  , TF d )  B ( r T * ,  , T d ) T r

F

( sT

{ r  r TF * })

 Max ( E

TF



n T 1

n

bn ( r TF * , sn 1 ,  n ))

(5.7) ( sT { r  r TF * })

where sT {r  r T * } means that the state (storage) at T can be specified according to the F

optimal release policy derived under the perfect forecast horizon, which is set as the ending condition of the first part and the initial condition of the second part on the RHS of Eq. (5.7). Eq. (5.7) divides the original free-end dynamic problem in Eq. (5.3), within the forecast horizon) to two individual optimization problems, one with a fixed terminal state within the study horizon and the other is free during the period between T and TF (note that sT {r  r T * } is set as the initial condition of the problem). The state variable at F

T , sT {r  r T * } , conveys the information needs for both problems. As shown in Figure F

5.2, without solving the problem for the whole time horizon, the identical optimal policy r TF * can be calculated with given correct terminal state boundary condition. In this way,

we can simplify the TF -period problem to T -period problem with a given terminal state condition, sT {r  r T * } . F

B(r TF * ,  , T d )  B(r T * ,  , T d )

( sT {r  rTF * })

(5.8)

Compared to Eq. (5.7) which describes the optimal solution with the forecast horizon ( TF ), (5.4) describes the optimal solution with the study horizon (T) if an appropriate ending condition ( sT {r  r T * } ) is provided. F

By the continuity law, storage monotonically decreases with release, we could have the following property: Both B(r T * ,  , t d ) and B(r T * ,  , T d ) are a monotonically decreasing function of sT .

121

This property describes the monotonically decreasing relationship between the decision vector and the terminal state condition. It can be intuitively explained as follows. If we store more water for the future beyond the study horizon, the utility within the study

horizon

will

decrease.

Following

this

property,

the

error,

B(r T * ,  , t d )  B(r TF * ,  , t d ) , can be expressed as: B (r T * ,  , t d )  B (r T * ,  , t d )

t

( sT { r  r TF * })

 n

(5.9)

n 1

where  n represents the error between the optimal solutions within the decision horizon (t) underlying the study horizon (T) and the forecast horizon ( TF ). Further, if we assume that the perfect forecast horizon is known, we can obtain the value of sT {r  r T * } . F

However, the assumption is not realistic and this paper deals with the case that TF is not available. Under this case, there are two extreme situations 1) the decision makers make the reservoir full at the end of the study horizon to prepare for a potential water shortage; and 2) the decision makers just make a myopic decision fully neglecting the need beyond the study horizon. Figure 3 demonstrates these two extreme situations. The solid and the dash lines represent the conservative policy under the first and second situation, respectively.

Under the first situation, the utility in the decision horizon (t) will

be B ( r T * ,  , t d ) (s

T

 smax )

, where Smax is the maximum storage. Under the second situation,

the utility is B(r T * ,  , t d ) , as defined above. The difference between the utility under these two situations represents the largest error that can be involved in the decision process. By definition, B ( r T * ,  , t d ) (s

T {r  r

TF * })

 B (r T * ,  , t d )

B(r T * ,  , t d )  B(r T * ,  , t d ) B(r T * ,  , t d )  B(r T * ,  , t d )

( sT  smax )

( sT  smax )

, we have:

  t

( sT {r  rTF * })

 n

(5.10)

n 1

As shown in Figure 5.3,   t is the outcome of the hatched area in Figure 5.3. By considering the difference between the most conservative and myopic polices,   t is the

122

maximum potential value of

t

 n 1

n

. Therefore, it could be defined as the upper bound of

error when the optimality is unknown. By this definition, the approach described above could be applied to determining the error bound of rolling-horizon decision making in reservoir operations. However, using the largest error to estimate the error bound may not be reasonable when it is unnecessary or impossible to maintain a full storage, for example, when the reservoir capacity is very large. In this case, an alternative terminal state rather than the largest value would be more realistic. Further discussion of the choice of an appropriate terminal state is presented in the later part of this chapter. The main concept of this approach is to estimate the error bound by choosing the terminal state variable ( sT ). When we have a concern for the periods after the study horizon, we consider the terminal state variable (storage) and its residual utility value. A limited foresight can make the decision suboptimal and myopic. However, if the periods after the study horizon need to be incorporated to achieve optimality, the decision maker is motivated to conserve resources for the future rather than to consume all of them. The measure is to adjust the terminal state variable. The value of the terminal state will affect the distance between the suboptimal and optimal solutions. Thus the error bound can be estimated according to the choice of the terminal state value. 5.3.

Numerical Examples

This study applies the same numerical examples which was used in forecast horizon to illustrate this approach.

A dynamic programming model developed for

forecast horizon is used to estimate the error/error bound for identical scenarios which were analyzed in the forecast horizon study (Table 5.1). Following the same framework, two stochastic time series, stationary and non-stationary, are tested. The numerical model is applied to two cases as follows: 1) Information after study horizon T can be obtained to determine the error  using heuristic approaches (as shown in Figure 5.2) 2) Information after study horizon T is not available and the error bound  is estimated according to the difference between the optimal and the worst case (as shown is Figure 5.3) 123

In general, the procedure for these two cases is to solve the DP problem by assigning different terminal state variables, then measure the error bound. Figure 5.4 shows the diagram of the procedure. Intuitively, the error bound depends on the length of information, which is related to the difference between the study horizon and the perfect forecast horizon. However, the relationship of error bound and length of information may not be homogenous due to the system complexity. In this study, a numerical analysis with consideration of the various factors is conducted to illustrate the relationship of error, error bound and different study horizons. 5.4.

Results

We examine the error bound with given DH and study horizon ( T ) using the same numerical example and the same dynamic programming framework that were used in Chapter 4 for the forecast horizon study. The error bound under the given study horizon is assessed using different methods, while also considering the impact of inflow characteristics, reservoir capacity and economic factors. Assuming the DH only includes one period (t=1) and the study horizon ( T ) respectively equals to 3, 6, 9 and 12 periods, this study examine the error bound (  ) and error (  ) under different factors. Water stress index (WSI) is defined as a normalized form: WSI 

I  Dm I

(5.11)

where I is the average inflow and Dm is the demand. The negative value of the WSI means water shortage while the positive value means excess water. t

 n 1

n

represents the difference between the suboptimal solutions underlying the

study horizon and the optimality due to limited foresight. In this case study, we set the decision horizon as DH =1, so

1

 n 1

n

. It can also be understood as the error when

decision making fails to hedge the amount of water required beyond study horizon. From

124

the mathematical derivation, the error bound (  ) is the maximum potential error. However, the condition is determined by assigning the maximum terminal storage. Therefore,  also represents the error due to hedging too much water for the future. 5.4.1. Numerical model results under stationary inflow time series

Figure 5.5 demonstrates a series of plots for the error and error bound under different water stress, reservoir capacities and study horizons. The result shows that  and  decay with the increasing study horizon T . Both  and  approach to zero when

T is close to the forecast horizon, TF . When there is water excess, decision making of reservoir operation tends to be static. Hedging decision making with limited foresight is not different from that with perfect foresight. When there is no incentive to hedge, the forecast is not useful information because that does not contribute to the current decision making. However, when the water stress increases, failing to consider enough future periods will lead to myopic operation and associated loss. Figure 5.5 shows that when WSI is negative,  simultaneously increases with the water shortage level. From the result, it also can be observed that a larger reservoir capacity is associated with a higher

 , for example, when T  3 ,  is around 6% for CAP=20 and 2% for CAP=10. This is because larger reservoir capacity provides more opportunity for hedging. Therefore, the operation more closely achieves the dynamic optimality without physical constraints. On the other hand, the error could be also larger when the decision fails to take advantage of capacity for hedging. Error bound,  , generally has the same trend as 

under various reservoir

capacity and water shortage situations. As mentioned above,  means the error due to over hedging. When the water shortage is significant, over-hedging in order to maintain high final stage sacrifices the current water demand and cause unnecessary loss. This loss monotonically increases with reservoir capacity because more water is hedged. Both  and  converge to zero when the study horizon ( T ) approaches to the forecast horizon. For most cases,  converges to zero when T is larger than 9 periods. The error bound  , which is higher than  , converges to zero more slowly comparing to

 . However, it can be observed that  is negligible for most cases when the study

125

horizon equals to 12. Roughly estimating from the result, the speed of coverage is governed by power law. Error and error bound decay to around 1/2 or 1/3 for each additional three periods of study horizon. Figure 5.6 shows the impact of inflow uncertainty (represented by inflow variance coefficient).

Similarly, the result shows both  and  decrease with T for different

variance coefficients. Additionally, higher uncertainty leads to higher  , except that the CV is very high (CV=1). In Chapter 2, we indicate that the delivery ratio will decrease due to uncertainty because of the diminishing marginal benefit and positive precautionary saving motive (see You and Cai, 2008b). Limited foresight operation does not hedge enough water for future. Therefore, for higher uncertainty which needs to hedge more,

 is also higher. The uncertainty also increases the probability of reservoir fullness. The effect reduces the ability for hedging. For this reason, the value of  when CV=1 is a little smaller than that when CV=0.5. For  , uncertainty also has a similar impact. Over-hedging reduces current water supply and causes unnecessary high state. It also increases the probability of reservoir fullness to spill additional water. Both increase the error bound,  . From Figure 5.6, when CV=1, the value of  is much higher than that under a small CV even if there is excess water. For CV=0.5, the difference is only significant when the water excess is high or water shortage is low. This is because when CV=1, the probability of reservoir fullness plays a significant role for reservoir operation and therefore makes the plot of the decision error different from other cases. A higher discount rate diminishes the future value and reduces the effect of hedging. Correspondingly, high discount rate also leads to smaller value of  as shown in Figure 5.7. The short foresight could be more tolerant when the future value is less important. The impact of discount rates to over hedging is not very significant. This is because, when compared to other factors, discount rate is less important for decision making for reservoir operation (You and Cai, 2008ab). Similarly, a higher evaporation rate also reduces the motivation for hedging and allows a shorter foresight decision making. Higher evaporation rate leads to lower  In the same way, evaporation rate also has reverse impact to  . Because the evaporation

126

loss monotonically increases with the storage, over hedging causes additional evaporation loss. Therefore,  increases with evaporation rate. The impact of utility concavity, which is associated with economic incentives, is shown in Figure 5.9. The result shows, for n=1.1, which is very close to a linear utility function, the error is very small because the operation is close to a static case. Future information for decision making is not necessary. Over hedging also causes higher error bound. With the increase of n, higher concavity of the utility function will enhance the dynamic feature of reservoir operation (You and Cai, 2008c) and therefore decrease  . However, the result does not show a clear trend for the impact of elasticity. This is because when the utility concavity changes, the utility function also changes. Therefore the comparison cannot be performed on the same basis. When the utility function is not consistent, each scenario is based on a different standing point and difficult to compare to each other. 5.4.2. Non-stationary inflow time series considering seasonality

The impact of seasonality which has been discussed in the forecast horizon study is also tested with different study horizons.

As shown in Figure 5.10, due to the

fluctuation of inflow time series, limited foresighted error with non-stationary inflow does not always have a smooth trend with respect to water stress level. A higher amplitude of seasonal inflow requires reserving water for the future to avoid loss. So a myopic decision making will lead to larger errors. However, the outcome is different when inflow and release demand are nearly the same. In this case, the probability of reservoir fullness plays an important role in the performance of operation.

Higher

amplitude of inflow results in additional water beyond the demand, which refills the reservoir till it exceeds the capacity of reservoir and extra water will be spilled. The fullness of reservoir reduces the capability to regulate inflow in the future. Under this situation, some degree of myopic decision is acceptable. Both AMP=4 and AMP=5 have less deviation from the optimal decision with 3 period limited foresight. The error decays very fast when the T extends but still keeps the same sharpness. For the non-stationary case, error bound  shows a reverse trend from  . The error bound decreases with higher amplitudes for the short foresight period (T=3). Over-

127

hedging of water keeps the reservoir storage high and leads to the same probability of fullness no matter whether the amplitude is large or not. In this situation, hedging more water first reduces the loss when an extreme drought event occurs, therefore, overhedging benefits the higher amplitude case with a smaller error bound. However, when the study horizon extends, convergence speeds are not homogeneous and the trend becomes ambiguous. Figure 5.11 shows the impact of seasonal period length. When the foresight is relatively short (e.g., T=3), the long seasonal period (P=12) has less error due to myopic decision making. This phenomenon probably is due to the situation that fluctuation is not obvious when the decision horizon is relatively small, compared to the seasonal period. Additionally, the result shows the error could be significantly eliminated if the forecast could include a cycle of seasonal period. For P=4 and 8,  is very close to zero when T=9; while when T=12,  under P=12 approaches to zero, too. The result of error bound under different seasonal period lengths also provides other useful information. 

does not significantly or monotonically change

corresponding to the seasonal length. The convergence speeds are also different. As discussed above, this might be related to the fact that the period of seasonality is relatively deterministic and depends less on future forecast information. Additionally, from Figure 5.12, the randomness only has a very slight impact to both error and error bound for the non-stationary time series case. 5.4.3. Sensitive analysis of the terminal condition

As discussed above, it is difficult to select an appropriate error bound when no knowledge about the actual error is available. In particular using the reservoir capacity as the bound (assuming full storage for the future) could overestimate the error especially when the reservoir capacity is large and/or the uncertainty level of inflow is high. Sensitivity analysis is conducted with alternative error bounds. Figure 5.13 shows the utility outcome of the numerical example under different terminal conditions (assuming the study horizon is 3-period long). The result shows that the optimal final stage is around 30% of the reservoir capacity for this particular case. The sensitivity analysis can

128

identify a reasonable range of the terminal state to estimate the error bound for specific cases. 5.5.

Discussion and Conclusion

This study discusses the potential of decision making error due to limited foresight.

Assuming the optimal solution could be obtained, this study conducts a

numerical analysis to explore the characteristics of myopic decision error. Additionally, we also propose a method to measure the error bound according to the difference of conservative and over-hedge policies, when the optimal solution is unknown. The numerical example shows that the error decreases when the study horizon extends toward the perfect forecast horizon determined by the horizon theory presented in the previous chapter (also see You and Cai, 2008a). In general, the error shows powerlaw convergence to zero. It quickly approaches to zero or a negligible range before the forecast horizon is achieved for most cases. This chapter also discusses the value of information regarding the significance of the information to decision making at present. Compared to the result from the forecast horizon theory, the length of forecast horizon may not be proportional to the error with a limited study horizon. This is because the forecast horizon includes any information even it just slightly or potentially changes current policy. Based on the results of numerical examples, this study finds that a myopic decision error mainly increases with the reservoir capacity and uncertainty for stationary case and slightly decreases with the evaporation and discount rate. For nonstationary case, the trend becomes more ambiguous due to the complexity of the different driving forces. Generally speaking, failing to consider the future hydrological condition is the main reason for the short-sighted decision failure for reservoir decision making. The result of this study also confirms the promise of rolling decision making for water resources systems. The error decays very fast when the study horizon has an appropriate length leading to the future in numerical examples.

When the perfect

foresight is not available, the rolling decision procedure with regularly updated forecast information could avoid serious loss due to shortsighted policies. The error bound method can be used to estimate the potential error. When the reservoir capacity is large and/or the uncertainty is high, using maximum capacity as terminal condition may

129

overestimate the potential error and an alternative terminal condition can be determined by sensitivity analysis. The knowledge of error and error bound is more useful for practical reservoir operation problems because the perfect foresight is always not available in the real world. Error bound can help us evaluate the actual operations with limited forecasts.

By

understanding the potential error of operation, reservoir managers could evaluate the value of information and further improve their decision making.

130

References

Bean, J., R. Smith and C. Yano (1987), Forecast horizons for the discounted dynamic lot size model allowing speculative motive, Naval Research Logistics, 6, 761-774 Chand, H., V. N. Hsu and S.P. Sethi (2002), Forecast, solution and rolling horizons in operations management problems: a classified bibliography, Manufacturing & Service Operations Management, 4(1), 25-43 Chen, C.D., C.Y. Lee. 1995. Error bound for the dynamic lot size model allowing speculative motive. IIE Trans. 27, 683–688. Denardo, E.V., C.Y. Lee. 1991. Error bounds for the dynamic lot size model with backlogging. Ann. Oper. Res. 28, 213–229. Draper, A. J. and J. R. Lund (2003), Optimal hedging and carryover storage value, J. Water Resour. Plann. Manage., 130(1), 83-87 Kleindorfer, P., H. Kunreuther. 1978, Stochastic horizons for the aggregate planning problem. Management Science. 24 485–497. Lee, C.Y., E.V. Denardo. 1986. Rolling planning horizon: Error bounds for the dynamic lot size model. Math. Oper. Res. 11 423–432. Lundin, R. A. and T. E. Morton. (1975), Planning horizons for the dynamic lot size model: Zabel vs. protective procedures and computational results, Operation. Research, 23, 711–734. Morton, T.E. 1981, Forward algorithms for forward thinking managers. Applications of Management Science, Vol. I, JAI Press, Greenwich, CT, 1–55. Sethi, S.P. and Sorger, (1991), A theory of rolling horizon decision making. Ann. Oper. Res. 29 387–416. Somerville, R, (2004), The predictability of weather and climate. Climatic Change, 11, 239-246. You, J-Y. and X. Cai (2008a), Forecast and decision horizons for under hedging policies. Wat. Resour. Res. (in press).

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You, J-Y. and X. Cai (2008b), Hedging rule for reservoir operations: (1) A theoretical analysis. Wat. Resour. Res., 44, W01415, doi:10.1029/2006WR005481. You, J-Y. and X. Cai (2008c), Hedging rule for reservoir operations: (2) A numerical model. Wat. Resour. Res., 44, W01415, doi:10.1029/2006WR005482.

132

Table 5.1. Parameters used in the numerical example, basic scenario and other scenarios

133

(a) First period

(b) Next period

Figure 5.1. The concept of rolling horizon

134

rT*

r TF *

 t

TF*

sT {r  r

}

Figure 5.2. The two boundary conditions (free-end and a targeted storage) with a limited study horizon.

135

r TF *

 t r T * |{sT  smax }

Figure 5.3. The two boundary conditions (free-end and the maximum storage) with a limited study horizon.

136

Figure 5.4. Computation procedures to evaluate for the error bound with limited forecast horizons

137

Limited foresight error bound(  )

Limited foresight error (  )

T

20%

18%

CAP=5

16%

CAP=8

14%

CAP=10

12%

CAP=20

Limited f oresight Error bound (ε)

3

Limited f oresight Error (δ)

20%

10% 8% 6% 4% 2% 0%

18%

CAP=5

16%

CAP=8

14%

CAP=10

12%

CAP=20

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

CAP=5

16%

CAP=8

14%

CAP=10

12%

CAP=20

10% 8% 6% 4% 2% 0%

18%

CAP=5

16%

CAP=8

14%

CAP=10

12%

CAP=20

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

16%

CAP=8

14%

CAP=10

12%

CAP=20

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

CAP=5

10% 8% 6% 4% 0%

18%

CAP=5

16%

CAP=8

14%

CAP=10

12%

CAP=20

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

0

0.5

Water stress 20%

18%

CAP=5

16%

CAP=8

14%

CAP=10

12%

CAP=20

Limited f oresight Error bound (ε)

20%

Limited f oresight Error (δ)

0.5

20%

18%

2%

12

0 Water stress

20%

9

0.5

20%

18%

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

20%

6

0 Water stress

10% 8% 6% 4% 2% 0%

18%

CAP=5

16%

CAP=8

14%

CAP=10

12%

CAP=20

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

Water stress

-1

-0.5

0

0.5

Water stress

Figure 5.5. Relationship between study horizon T and water stress index under different reservoir capacities (CAP), inflow uncertainty variance coefficient (CV) = 0.2

138

Limited foresight error bound(  )

Limited foresight error (  )

T

20%

18%

CV=0.1

16%

CV=0.2

14%

CV=0.5

12%

CV=1

Limited f oresight Error bound (ε)

3

Limited f oresight Error (δ)

20%

10% 8% 6% 4% 2% 0%

18%

CV=0.1

16%

CV=0.2

14%

CV=0.5

12%

CV=1

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

CV=0.1

16%

CV=0.2

14%

CV=0.5

12%

CV=1

10% 8% 6% 4% 2% 0%

18%

CV=0.1

16%

CV=0.2

14%

CV=0.5

12%

CV=1

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

16%

CV=0.2

14%

CV=0.5

12%

CV=1

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

CV=0.1

10% 8% 6% 4% 0%

18%

CV=0.1

16%

CV=0.2

14%

CV=0.5

12%

CV=1

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

0

0.5

Water stress 20%

18%

CV=0.1

16%

CV=0.2

14%

CV=0.5

12%

CV=1

Limited f oresight Error bound (ε)

20%

Limited f oresight Error (δ)

0.5

20%

18%

2%

12

0 Water stress

20%

9

0.5

20%

18%

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

20%

6

0 Water stress

10% 8% 6% 4% 2% 0%

18%

CV=0.1

16%

CV=0.2

14%

CV=0.5

12%

CV=1

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

Water stress

-1

-0.5

0

0.5

Water stress

Figure 5.6. Relationship between study horizon T and water stress index under different uncertainty levels (variance coefficient, CV), CAP = 10

139

Limited foresight error bound (  )

Limited foresight error (  )

T

20%

18%

r=0.01

16%

r=0.02

14%

r=0.03

12%

r=0.05

Limited f oresight Error bound (ε)

3

Limited f oresight Error (δ)

20%

10% 8% 6% 4% 2% 0%

18%

r=0.01

16%

r=0.02

14%

r=0.03

12%

r=0.05

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

r=0.01

16%

r=0.02

14%

r=0.03

12%

r=0.05

10% 8% 6% 4% 2% 0%

18%

r=0.01

16%

r=0.02

14%

r=0.03

12%

r=0.05

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

16%

r=0.02

14%

r=0.03

12%

r=0.05

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

r=0.01

10% 8% 6% 4% 0%

18%

r=0.01

16%

r=0.02

14%

r=0.03

12%

r=0.05

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

0

0.5

Water stress 20%

18%

r=0.01

16%

r=0.02

14%

r=0.03

12%

r=0.05

Limited f oresight Error bound (ε)

20% Limited f oresight Error (δ)

0.5

20%

18%

2%

12

0 Water stress

20%

9

0.5

20%

18%

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

20%

6

0 Water stress

10% 8% 6% 4% 2% 0%

18%

r=0.01

16%

r=0.02

14%

r=0.03

12%

r=0.05

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

Water stress

-1

-0.5

0

0.5

Water stress

Figure 5.7. Relationship between study horizon T and water stress index under different discount rates (r), CAP=10, CV=0.2

140

Limited foresight error bound(  )

Limited foresight error (  )

T

20%

18%

EVP=0.2

16%

EVP=0.5

14%

EVP=0.5

12%

EVP=1

Limited f oresight Error bound (ε)

3

Limited f oresight Error (δ)

20%

10% 8% 6% 4% 2% 0%

18%

EVP=0.2

16%

EVP=0.5

14%

EVP=0.5

12%

EVP=1

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

EVP=0.2

16%

EVP=0.5

14%

EVP=0.5

12%

EVP=1

10% 8% 6% 4% 2% 0%

18%

EVP=0.2

16%

EVP=0.5

14%

EVP=0.5

12%

EVP=1

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

16%

EVP=0.5

14%

EVP=0.5

12%

EVP=1

Limited f oresight Error bound (ε)

Limited foresight Error (δ)

EVP=0.2

10% 8% 6% 4% 0%

18%

EVP=0.2

16%

EVP=0.5

14%

EVP=0.5

12%

EVP=1

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

0

0.5

Water stress 20%

18%

EVP=0.2

16%

EVP=0.5

14%

EVP=0.5

12%

EVP=1

Limited f oresight Error bound (ε)

20% Limited f oresight Error (δ)

0.5

20%

18%

2%

12

0 Water stress

20%

9

0.5

20%

18%

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

20%

6

0 Water stress

10% 8% 6% 4% 2% 0%

EVP=0. 2 EVP=0. 5 EVP=0. 5

18% 16% 14% 12% 10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

Water stress

-1

-0.5

0

0.5

Water stress

Figure 5.8. Relationship between study horizon T and water stress index under different evaporation rates (EVP), CAP=10, CV=0.2

141

Limited foresight error (  )

Limited foresight error (  )

T

20%

18%

n=1.1

16%

n=2

14%

n=3

12%

n=5

Limited f oresight Error bound (ε)

3

Limited f oresight Error (δ)

20%

10% 8% 6% 4% 2% 0%

18%

n=1.1

16%

n=2

14%

n=3

12%

n=5

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

n=1.1

16%

n=2

14%

n=3

12%

n=5

10% 8% 6% 4% 2% 0%

18%

n=1.1

16%

n=2

14%

n=3

12%

n=5

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

16%

n=2

14%

n=3

12%

n=5

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

n=1.1

10% 8% 6% 4%

18%

n=1.1

16%

n=2

14%

n=3

12%

n=5

10% 8% 6% 4% 2% 0%

0% -1

-0.5

0

-1

0.5

-0.5

0

0.5

Water stress

Water stress

18%

n=1.1

16%

n=2

14%

n=3

12%

n=5

Limited f oresight Error bound (ε)

20%

Limited f oresight Error (δ)

0.5

20%

18%

2%

12

0 Water stress

20%

9

0.5

20%

18%

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

20%

6

0 Water stress

10% 8% 6% 4% 2% 0%

14%

n=1.1

12%

n=2 n=3

10%

n=5

8% 6% 4% 2% 0%

-1

-0.5

0

0.5

Water stress

-1

-0.5

0

0.5

Water stress

Figure 5.9. Relationship between study horizon T and water stress index under different utility concavity values (n), CAP=10, CV=0.2

142

Limited foresight error (  )

Limited foresight error (  )

T

20%

18%

AMP=1

16%

AMP=2

14%

AMP=4

12%

AMP=5

10% 8% 6% 4% 2%

Limited f oresight Error bound (ε)

3

Limited f oresight Error (δ)

20%

0%

18%

AMP=1

16%

AMP=2

14%

AMP=4

12%

AMP=5

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

AMP=1

16%

AMP=2

14%

AMP=4

12%

AMP=5

10% 8% 6% 4% 2% 0%

18%

AMP=1

16%

AMP=2

14%

AMP=4

12%

AMP=5

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

16%

AMP=2

14%

AMP=4

12%

AMP=5

10% 8% 6% 4%

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

AMP=1

18%

AMP=1

16%

AMP=2

14%

AMP=4

12%

AMP=5

10% 8% 6% 4% 2% 0%

0% -1

-0.5

0

-1

0.5

-0.5

0

0.5

Water stress

Water stress

20%

18%

AMP=1

16%

AMP=2

14%

AMP=4

12%

AMP=5

10% 8% 6% 4% 2% 0%

Limited f oresight Error bound (ε)

20% Limited f oresight Error (δ)

0.5

20%

18%

2%

12

0 Water stress

20%

9

0.5

20%

18%

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

20%

6

0 Water stress

18%

AMP=1

16%

AMP=2

14%

AMP=4

12%

AMP=5

10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

Water stress

-1

-0.5

0

0.5

Water stress

Figure 5.10. Relationship between study horizon T and water stress index under different amplitudes (AMP), CAP=10

143

Limited foresight error (  )

Limited foresight error (  )

T 20%

3

Limited f oresight Error (δ)

18%

Limited f oresight Error bound (ε)

20% P=4

16%

P=8

14%

P=12

12% 10% 8% 6% 4% 2%

18%

P=4

16%

P=8

14%

P=12

12% 10% 8% 6% 4% 2% 0%

0% -1

-0.5

0

-1

0.5

-0.5

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

18%

P=4

16%

P=8

14%

P=12

12% 10% 8% 6% 4% 2% 0%

18%

P=4

16%

P=8

14%

P=12

12% 10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

P=4

16%

P=8

14%

P=12

12% 10% 8% 6% 4% 2% 0%

18%

P=4

16%

P=8

14%

P=12

12% 10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

0

0.5

Water stress 20%

18%

Limited f oresight Error bound (ε)

20%

Limited f oresight Error (δ)

0.5

20%

18%

12

0 Water stress

20%

9

0.5

20%

20%

6

0 Water stress

Water stress

P=4

16%

P=8

14%

P=12

12% 10% 8% 6% 4% 2% 0%

18%

P=4

16%

P=8

14%

P=12

12% 10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

Water stress

-1

-0.5

0

0.5

Water stress

Figure 5.11. Relationship between study horizon T and water stress index under different seasonality lengths (P), CAP=10

144

Limited foresight error (  )

Limited foresight error (  )

T

20%

3

Limited f oresight Error (δ)

18%

Limited f oresight Error bound (ε)

20% R=0.5

16%

R=1

14%

R=2

12% 10% 8% 6% 4% 2% 0%

18%

R=0.5

16%

R=1

14%

R=2

12% 10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

Limited f oresight Error (δ)

R=0.5

16%

R=1

14%

R=2

12% 10% 8% 6% 4% 2% 0%

18%

R=0.5

16%

R=1

14%

R=2

12% 10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

Limited f oresight Error bound (ε)

Limited f oresight Error (δ)

R=0.5

16%

R=1

14%

R=2

12% 10% 8% 6% 4% 2% 0%

18%

R=0.5

16%

R=1

14%

R=2

12% 10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

-1

-0.5

Water stress

0

0.5

Water stress

20%

18%

Limited f oresight Error bound (ε)

20% Limited f oresight Error (δ)

0.5

20%

18%

12

0 Water stress

20%

9

0.5

20%

18%

Limited f oresight Error bound (ε)

20%

6

0 Water stress

R=0.5

16%

R=1

14%

R=2

12% 10% 8% 6% 4% 2% 0%

18%

R=0.5

16%

R=1

14%

R=2

12% 10% 8% 6% 4% 2% 0%

-1

-0.5

0

0.5

Water stress

-1

-0.5

0

0.5

Water stress

Figure 5.12. Relationship between study horizon T and water stress index under different randomness levels (R), CAP =10

145

0.98

Utility

0.93

0.88

Free-end 25% 50% 75%

0.83

Smax Optimal

0.78 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Water stress

Figure 5.13. Utilities from different terminal boundary conditions vs. water stress levels

146

CHAPTER 6.

IMPROVE HEDGING RULES FOR THE

OPERATION OF LAKE OKEECHOBEE IN SOUTHERN FLORIDA

Summary

Lake Okeechobee is the major water source in Southern Florida. As a multi-functional reservoir, Lake Okeechobee serves multi-purposes including water supply, flood control, navigation, recreation, and fish and wildlife protection.

Environmental and economic

objectives have been in conflicts among water uses in this area. The management of the Lake faces many challenges today due to variable inflow and growing demand. Hedging policy has been studied and implemented for the operation of Lake Okeechobee, which makes acceptable deficit for current and reserve water to reduce the risk of water shortage in the future. Following those, this chapter applies the methods presented in the former chapters of this dissertation to Lake Okeechobee to explore optional hedging policies. Both inter-year and intra-year models are developed, as a planning tool for long-term operations and management tool dealing with seasonal variability. The results show that hedging rules could significantly reduce the profit loss compared to the standard operation policy (SOP). Hedging could be applied in an earlier time than that triggering the current hedging policy, the so-called Supply-Side Management, to mitigate the risk of severe water shortage and profit loss.

It is also found that the operation of Lake

Okeechobee should consider both short term and long term hedging which should be nested with each other. In this chapter, through a real world example, we also show that the rolling decision making, which uses a forecast horizon that is updated with period (rolling horizon), could provide a practical way to approach to the optimal policy. However, the length of forecast and the accuracy of forecast are still important limitations to the application of hedging rules.

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6.1.

Introduction

Lake Okeechobee is the major water source in Southern Florida. As a multifunctional reservoir, Lake Okeechobee serves water supply, flood control, navigation, recreation, and fish and wildlife protection.

Several environmental and economic

objectives have been in conflicts among water uses in this area.

Recently, a new

operation policy, revised Tentative Selected Policy (revised TSP), was approved in April 2008. This policy mainly addresses the potential safety and environmental problems during a wet condition which is caused by heavy rainfall or hurricanes. It is believed to be more effective in public safety and environmental consideration but it may make agricultural and municipal water supply face higher possibility of water stress in drought periods. With the increasing possibility of water stress, how to manage water during drought periods becomes essential for the operation of Lake Okeechobee. Hedging rule policies are designed to rationally allocate water resources over time. These policies accept some present delivery deficit to reduce the probability of greater water or energy shortage in the future (Bower et al., 1962). By this way, hedging resolves reservoir operation problems focusing on minimizing profit loss or water supply deficit over drought periods. Bower et al. (1962) first provided a systematic economic description of hedging rules for water resources systems operations. Their classic work addressed the inter-temporal resource allocation and profit optimization within a dynamic framework. The implementation of hedging rules needs to address the issues of magnitude (how much to hedge) and temporal scale (how long to hedge) in water resources allocation. The theoretical properties of hedging rules have been examined in Chapter 2 (also see You and Cai, 2008a).

According to the theoretical derivations, a two-period

numerical model is developed, which can be used to analyze hydrologic uncertainty explicitly in Chapter 3 (also see You and Cai, 2008b). To analyze time scale issue, the theory of horizon in reservoir operation, which addresses the question of how far to forecast to the future, has been proposed (Chapter 4; also seeYou and Cai, 2008c). Furthermore, considering the actually available forecast horizons due to forecast limitation, a dual approach has been developed to determine how far the actual decision

148

under a limited forsight is away from the optimality with an ideal foresight (Chapter 5; also see You and Cai, 2008d). While these studies focused on theoretical and methodological developments of hedging rules for reservoir operations, including when, how much and how long to hedge, they were based on hypothetical studying examples, without paying sufficient attention on the complexity of real world problems. This chapter extends the previous work to the operation problem of Lake Okeechobee, a real world case, and addresses the challenges facing the implementation of hedging policy. It should be noted that many previous studies explored hedging rules for real reservoir operation problems, but those studies focused on developing numerical models which determined the hedging rule by heuristic algorithm or tested optional hedging rules by performing trial-and-error experiments (Bayazit and Unal, 1990; Shih and ReVelle, 1994, 1995; Srinivasan and Philipose 1996, 1998; Neelakantan and Pundarikanthan, 1999; Shiau and Lee, 2005). Most studies used similar procedures as those provided by Shih and Revelle (1994, 1995) with more or less different numerical schemes. Moreover, to deal with uncertainty, those studies applied the implicit stochastic approach, which is based on synthetically generated, unregulated inflow time series, and did not consider the effect of forecast limitation. In this case study, we will use an explicit stochastic framework developed in Chapters 2 and 3 (also see You and Cai 2008a, b), which is based on probabilistic descriptions of random streamflow time series. Given the complexity in real problems, the key to apply the theoretical developments to those problems is to introduce reasonable assumptions for specific problems. Additional procedures are developed in this chapter to apply the theoretical developments to practical problems, with judgments on the rationality of the assumptions involved in the specific application. By this case study, we address the knowledge gap between theoretical studies and real-world implementations. Meanwhile, we suppose the findings of the new knowledge could help improve the operation of Lake Okeechobee. In the following sections, we first introduce the current conditions and potential issues of Lake Okeechobee. Then we develop an inter-year hedging model and an intrayear hedging model to analyze the operation of the Lake with long and short time 149

intervals, respectively. Based on the model results, we demonstrate the merits of optional hedging rules for the Lake. Finally some discussions and conclusions are provided at the end of this chapter. 6.1.1. Background

Lake Okeechobee has an area 730 square miles with an average depth of 9 feet. The lake is enclosed by a 20-foot (6 m) high dike built by the U.S. Army Corps of Engineers after a severe hurricane in 1928. The drainage basin of the Lake covers more than 4,600 square miles. The preliminary uses of Lake Okeechobee water includes: (1) agricultural water supply to the Lake Okeechobee Service Area (LOSA); (2) backup of water supply and prevention of saltwater intrusion to the Lower East Coast Service Areas (LEC-SA); (3) non-regulatory environmental deliveries to Water Conservation Area (WCA) and Everglades National Park (ENP); and (4) regulatory environmental supply to downstream ecosystems including the Caloosahatchee and St. Lucie Estuaries and the remnant Everglades. Following, these water uses are introduced in sequence (SFWMD, 2003). LOSA is the collection of areas whose primary supplemental water supply needs are met by Lake Okeechobee. This area is composed of several major basins including the Everglades Agricultural Area (EAA), the Caloosahatchee River (C-43) Basin, the St. Lucie River (C-44) Basin, and many other, smaller basins located around the Lake. In this region, the majority of the supplemental demands is for agricultural irrigation. Table 6.1 shows the area of irrigated agriculture in the major subareas of LOSA. Everglades Agricultural Area (EAA) encompasses an area south and southeast of Lake Okeechobee, covering approximately 593,000 acres of land of which 542,000 acres for agricultural production (SFWMM, 2003).

The agricultural land use of EAA is

presented in Table 6.2. Table 6.3 and Table 6.4 shows agricultural land uses for other two major areas, Caloosahatchee and St. Lucie basins. The water consumption of Lower East Coast Service Areas (LEC-SA) includes public water supply needs, agricultural supplemental requirements, and deliveries which maintain LEC canals at desired levels. Lake Okeechobee functions as back-up source for LEC service area. 150

The Water Conservation Areas (WCAs) are located south of Lake Okeechobee and the EAA. WCAs function as small downstream reservoirs in the Southern Florida water supply system and have the operating rules governing the water management. The Everglades National Park (ENP), located to the south of the Florida peninsula, is operated by the National Park Service. Water consumption of ENP depends on the environmental flow from WCA.

The WCAs and the ENP, which were set for environmental

conservation, are commonly known as the Everglades Protection Area (EPA) (SFWMD, 1992). Another major environmental water demand is the estuarine systems which comprise South Florida’s complex and vital ecosystems. The estuary area is important to the environmental and economic well-being of the Southern Florida. 6.1.2. Current hedging polices for the operation of Lake Okeechobee

Due to the critical water use of Lake Okeechobee, extensive studies have been conducted for Lake Okeechobee. Water Supply and Environment (WSE) regulation, approved by the U.S. Army Corps of Engineers (USACE) and the Sothern Florida Water Management District (SFWMD) since July 2000, was used in past several years to balance the different, always competing objectives. During the period of implementing WSE, Lake Okeechobee experienced unexpected wet years caused by heavy rainfall and several hurricanes leading to continuous high water level in the Lake. Because of the high stages, the issues about structural safety of water control facilities and the adverse environmental effect have been raised.

The WSE tends to hold water in Lake

Okeechobee. The storage is beneficial to water supply but may end with extra amount of freshwater release to Caloosahatchee and St. Lucie Estuary, which might be harmful for the ecosystems. Moreover, the high lake stage also increases the risk of Herbert Hoover Dike and other potential dangers during the hurricane season. As a result, a series of operational deviations from the WSE has been approved and implemented between 2003 and 2006 to ease the stress of the current schedule. Considering the public safety and environmental effects associated with high lake stage, Lake Okeechobee Regulation Schedule Study (LORSS) has been conducted to develop alternative schedules to address issues of high lake levels, high estuarine discharges, estuary ecosystem conditions, and lake ecology conditions. In November 2007, LORSS announced the revised Tentatively Selected Plan (revised TSP) and the 151

Supplemental Environmental Impact Statement (SEIS, 2007) for a 30-day public comment period. The revised TSP has been implemented since May 2008. The revised TSP is supposed to be more effective in decreasing the risk of public safety and reducing the damaging events and salinity violations to the estuaries by lower the average reservoir stage. However, the decrease of reservoir storage may make agricultural and municipal water supply face water stress in drought periods. Among the water supply areas, the revised TSP affects the supplementary supply to EAA most significantly (SEIS, 2007). Current guiding policy on the agricultural water shortage management in LOSA and LEC-SA is the Supply-Side Management (SSM) plan. In contrast to the WSE and revised TSP schedule, SSM is used to manage lower stages in Lake Okeechobee. Temporal allocation of water under SSM is designed to avoid Lake levels lower than the reference elevation at the end of the dry season, although this may not be met due to the severity of a drought event. SSM could be considered as a hedge policy but for a relatively short period. It is triggered only when the Lake level is possibly to be lower than the operating band at the end of the dry season. The planning horizon of SSM is within-year and therefore it is not able to deal with inter-year hedging operation. From the viewpoint of annual water budget planning, current operation of Lake Okeechobee actually tends to follow SOP. However, longer period hedging such as inter-year hedging could benefit the operation of Lake Okeechobee. According to the historical data, over-year drought happened not only once, such as those during 1981-83, 1989-91 and 2000-2001. Considering the severity, hedging the water supply for later use before the drought could provide an opportunity to reduce the potential overall loss. Also the current SSM focuses on maintaining the reference level without considering the utility of water use. The difference of water value in different seasons is not considered either. For some seasons, the value of water supply may be more than anything else. When applying hedging rules, it is also important to know which period to apply hedging. Without considering these issues, SSM may not be effective. This chapter examines the potential of optional hedging rules for the operation of Lake Okeechobee. First we use an inter-year model to analyze the reservoir operation 152

based on annual water budget. As discussed in Chapter 4, the inter-year model treats reservoir inflow as independent identical distribution (IID) (also see You and Cai, 2008a). The inter-year model is used as a planning tool which determines the water shortage band to apply hedging rules. Following the inter-year model, a monthly intra-year hedging model is developed to explore the hedging operation with seasonal variability.

The intra-year model

discusses more details of hedging operations for different drought events. The impact of hydrological variability, water demands, and forecast capability on hedging rules are discussed. Useful hydrologic forecast is only accessible for next few months in the real world (You and Cai, 2008d). When the long range forecast is not possible to obtain, the concern becomes how far the actual decision with limited but available foresight is away from the optimality. For Lake Okeechobee, this horizon issue will be explored through a rolling-over decision making procedure. In this chapter, we discuss the implementation of hedging for the agricultural water use in Southern Florida, which is supposed to be affected most significantly by water stress. The inter-year model addresses the hedging policy to EAA while intra-year model extends the objective to the whole LOSA water supply including both EAA and non-EAA areas. Both models assume a single, composite user, while the regulation on flood control and environmental protection defined in revised TSP is formulated as “hard constraints” to the optimization problem. Water supply to municipal and industrial sectors follows the standard operation policy (SOP), i.e., releasing water close to the delivery target as much as possible. Because of this setting, hedging rules are applied to portion of water supply and the effect of hedging policy on the whole water system could be questionable. In this study, from the analysis of water budget of LORSS, the annual LOSA (EAA and non-EAA) agricultural water demand is about 1/2~2/3 of total annual release form Lake Okeechobee. Considering only EAA supplementary supply, it is around 1/3~1/4 of the total annual release. Although the agricultural water supply does not dominate the release the Lake, the hedging policy applied to EAA and LOSA will still have a significant impact of the operation of Lake Okeechobee, given the limited purpose of this study, in which the hedging is only explored for agricultural water use.

153

6.2.

Methodology

6.2.1. Inter-year hedging model

This study applies the two-period hedging model developed in Chapter 3 (also see, You and Cai, 2008b). The dynamics of decision making is limited to two consecutive periods, but the decision process is “rolled over” every two periods and extended into infinite future. The hedging policy is solved under the steady state of the model, which implies a policy of how much storage to carry to next time period under various estimates of inflow occurrences. The conceptual framework of EAA inter-year hedging model is built on a simplified schematic diagram of water storage and release in the Southern Florida as shown as Figure 6.1. The consumption of EAA water supply is based on the crop evapotransportation (ET) requirement. Lake Okeechobee supplements rainfall and local storage to meet total ET requirements (SFWMM, 2002). Total annual ET for EAA only slight fluctuates. According the LORSS result, the mean value of annual ET for EAA is 1948 thousand acre-feet (taf), the standard derivation is only 37.9 taf and coefficient of variance is only 0.02.

Therefore, the inter-year variation and uncertainty of water

demand is minor. Thus it is proper that the inter-year hedging model treats annual water demand as a constant value in its profit function.

The model determines the

supplementary supply besides the local water source. With given profit function and probability distribution of inflow, the inter-year hedging model derives the hedging rules to the equilibrium condition, which represents the long term behavior of reservoir operation responding to inflow uncertainty will be consistent (Chapter 3, also see You and Cai, 2008b). The over-year hedging model analyzes the supplementary supply from Lake Okeechobee to EAA as shown in Figure 6.1. According to the simulation result from the SFWMM, 75% of EAA irrigation requirement comes from local inflow and storage and the rest 25% is provided by Lake Okeechobee. Current supply policy on the Lake Okeechobee supplementary supply follows the SOP but rune to a hedging policy when Supply-Side management (SSM) is triggered. This study explores the hedging rules to

154

determine the supplementary supply delivery to EAA to maximize the overall profit of agricultural water supply in EAA. This model considers the decision making within the operational band specified in the beginning of each year by the revised TSP, which I sin the range of 11.4 feet to 17.25 feet. The active reservoir capacity use is around 2430 taf. The formulation for EAA supplementary supply problem is given as below:

Max U (CU )  E ( St 1  I f ) D,S

s.t. St 1  St  I LOK  WLOK

(6.1)

CU  Pe    IW IW  WLOK  WLS where U (.) , current profit, and E (.) , expected future profit, are concave functions expressed in present value. S is the storage of Lake Okeechobee, I LOK is the available inflow to Lake Okeechobee for EAA and WLOK is the supplementary supply delivered by Lake Okeechobee to EAA crop water use ( CU ). The EAA crop water demand comes from two parts, effective rainfall Pe and applied irrigation, IW . IW is firstly provided by the local source, WLS , the rest of IW is provided by Lake Okeechobee supplement supply. Here  is the irrigation efficiency coefficient. The implementation of hedging rule is based on the expectation of water availability for EAA both from Lake Okeechobee and local area. The water requirement is estimated by water balance equations. According to Equation (6.1), CU , EAA crop water needs can be rewritten as,

CU  Pe   2WLS   1WLOK

(6.2)

 1 is the irrigation efficiency for Lake Okeechobee supplementary supply and  2 is the local irrigation efficiency.

Both  1 and 

2

have been implicitly considered in the

LORSS data. The water budget result of LORSS has been adjusted and does not show the loss of irrigation and delivery. Pe   2WLS is the crop water requirement satisfied by 155

local water availability, defined as AL . AL is estimated according to the water balance relationship shown as Figure 6.2. Equation (6.3) shows the water balance equation for Lake Okeechobee. I LOK  I total   D j  P  ET

(6.3)

j

where I total is the total inflow to Lake Okeechobee, D j is the demand of different users ( j ), P is the rainfall, and ET is the evaporation. This over-year hedging model considers the explicit stochastic methodology. The probability distributions of I LOK and AL need to be determined for the hedging decision making. The study uses the simulation data of SFWMM from 1965-2000 to calculate I LOK and AL . For the decision process under explicit uncertainty, both I LOK and AL are stochastic variables. Figure 6.3 and Figure 6.4 show the result of frequency analysis on these two variables. The result demonstrates that I LOK and AL both fit normal distribution, with N(372, 791) and N(1523,160), respectively. The correlation coefficient between I LOK and AL is 0.45. Therefore, this study assumes that the two items are independent. The hedging model is based on the marginal utility principle (You and Cai, 2008a), which requires appropriate definition of profit function to decide the amount of supply. We test three forms of profit functions 1) linear function equation, 2) quadratic function, and 3) power function as listed in Table 6.5. The economic data of water shortage is very limited for Lake Okeechobee. This study relies on SFWMM result of historical drought events to estimate the profit function. The parameters of linear profit function equation are directly from annual benefit and the water supply. For the quadratic and the power function, the parameters are estimated based on several assumptions: 1) the profit is zero when the water supply is zero; 2) the profit equals to annual maximum revenue when the water demand is totally satisfied and 3) the marginal profit is zero when the water supply equals the demand.

156

6.2.2. Intra-year hedging model

Reservoir operations exhibit a mixture of both within-year and over-year behaviors, albeit to varying degree depending on the hydrologic and physical condition (Montaseri and Adeloye, 1999). Intra-year reservoir operation is important to avoid the seasonal shortfall due to the variability of water availability and demand. When the water availability is not consistent with the demand, intra-year hedging could regulate the temporal water stress by time within a year. To analyze the intra-year hedging, decision makers need information which can capture the seasonal patterns of hydrological condition and/or economic factors. This information can be obtained by simulating/reproducing the hydrological process with given conditions. Usually we rely on well-developed hydrologic model to access the information. The South Florida Water Management Model (SFWMM) is a regional-scale computer model that simulates the hydrology and the management of the Southern Florida water resources system. The SFWMM is accepted by SFWMD as the best available tool for analyzing regional-scale structural and/or operational changes to the complex water management system in Southern Florida.

Other than SFWMM, the

Natural System Model (NSM) and Florida Regional Routing Model (SFRRM) (Trimble and Marban, 1988) are also used for planning and management purposes. SFWMM provides a comprehensive analysis for the revised TSP.

An ideal

framework is to integrate the intra-year hedging model with SFWMM. However, this framework is difficult to implement. SFWMM is a deterministic simulation model which cannot consider the impact of uncertainty. Additionally, the SFWMM is a very complex model, the effort and the integration will cause a great difficulty, which will be left for future research.

For these reasons, this study uses a response function of current

hydrologic process to approximate the role of SFWMM within our proposed framework. This study develops a monthly intra-year hedging model. The general concept of the intra-year hedging` model is demonstrated by Figure 6.5. With given hydrological forecast, the inflow and demand could be estimated. According to the estimation, the intra-year hedging model determines how much water is released at present use and how much water is stored for future use. 157

As the inter-year model, the intra-year hedging model focuses on the agricultural water supply but extends to whole LOSA. Other water supply demands are considered as hard constraints to avoid the complexity of water resource competition. Hedging policy functions to minimize the potential loss of LOSA. Equation (6.4) shows the formulation of the model to determine the hedging policy T

min L(r )   ln (rn ) n 1

Smin  S  Smax

(6.4)

Sn 1  Sn  rn  I n  Pn  ETn   D j ,n where l represents the monthly loss function of production due to water shortage and

r is the decision variable, the release to LOSA (EAA and non-EAA) . Storage of Lake Okeechobee, S , is bounded by the operation band of schedule. I n is the inflow to LOK while P and ET are monthly rainfall and evaporation respectively. D j , n represents the monthly demand for different user j, other than the LOSA. Figure 6.6 presents a schematic diagram of the network for different users considered in this study. Non-EAA LOSA can withdraw water from Lake Okeechobee as supplementalary supply; meanwhile the region also can recharge the Lake when additional runoff is available. For EAA, the surplus runoff only can flow to downstream WCAs, which might be shared among different WCAs. The water demand of WCAs depends on the evapotranspiration, precipitation, demand of downstream users, and the regulation among the WCAs. The loss function SFWMM uses Economic Post Processor (EPP) to estimate the economic loss of agriculture under water stress. EPP calculates the reduction of agricultural production under water stress by the FAO equation (Doorenbos and Kassam, 1979). 1  (Yact / Ymax )  k y (1  ETa / ETm )

158

(6.5)

where Yact is the annual crop yield per acre (simulated), Ymax is the maximum crop yield per acre. k y is the crop response coefficient. ETa is actual evapotranspiration and

ETm is the potential evapotranspiration. However, the FAO equation is linear to water shortage. For a linear loss/revenue function, SOP is the optimal operating policy and no hedging is needed (Hashimoto et al., 1982). For this reason, we modified the FAO equation to a quadratic form as follows:

P  Y max ETm  ETa 2 [k y ( ) ] N ETm ETm  ETa    ( D j  rj )

l

(6.6)

where P is price,  is the irrigation efficiency, N is number of crop growth months. D j  r j , Demand minus supply, could be considered as the water shortage which is not

met. This study considers different crops as listed in Table 6.2, Table 6.3 and Table 6.4. Monthly crop price and k y are obtained from SFWMM as shown in Table 6.6 and Table 6.7, respectively.

The ETm is estimated by ETm  Kc  ETo .

coefficient (Table 6.8).

Kc is the water use

ETo , the reference ET , is obtained from is obtained from

SFWMM. It should be noticed that Equation (6.6) could only roughly estimate the magnitude of the economic loss. Even for the EPP that has been tested and calibrated for use in SFWMM by SFWDM, the estimation of economic loss is still coarse to accurately assess the economic impacts (SEIS 2007). Additionally, both SFWMM and this model calculate crop yield effect on a monthly basis. For shortage of a duration of several months, the reduction may repeatedly be counted and it could overestimate the effects on crop yield and revenue because each month is treated independently (SEIS, 2007). Besides the modified FAO equation, this study also applies SI, shortage index (USACE, 1981), as another loss function to test the hedging policy.

D r ) l  SI  ( D j

j

j

j

j

159

2

(6.7)

SI reflects the observation that economic effects of shortages are roughly proportional to the square of the degree of shortage. SI has a merit over shortage frequency alone as a measure of severity because shortage frequency considers neither magnitude nor duration. SI can be used to multiplied by a constant to obtain a rough estimation of crop loss. Stochastic optimization model Rainfall and ET are the main driving forces of the hydrology in Southern Florida (SFWMM, 2002). The intra-year hedging model estimates the monthly runoff and water demand in each area according to the amount of monthly rainfall and ET. To simplify, we assumed rainfall and ET have the same temporal distribution over space occurred historically as SFWMM does. In order to consider the explicit uncertainty to capture the seasonal variability for ex ante analysis, this model applies the “Climate Outlook” produced by the Climate, Prediction Center (CPC), National Oceanic and Atmospheric Administration (NOAA). The predictions of precipitation updated every month by the CPC include 13 monthly windows: an outlook for the next 13 months into the future with one-month time step overlapping (http://www.cpc.ncep.noaa.gov). Because rainfall applied to the model is assumed to be temporal and spatial uniform distributed, CPC forecast data cannot be directly used. For the model use, CPC is statistically biased, which means the expected value of rainfall is not consistent between forecast and historical data. This is because both temporal and spatial average monthly rainfall at the local scale is much lower than the forecasted value. In this study, we conduct the bias-correction procedure to CPC forecast data. The bias-correction is performed according to historical data for each individual month. The monthly mean value of CPC forecast is adjusted to be the same as the historical average rainfall for that month. We multiply the regional precipitation forecast by the ratio of the historical average month rainfall to the expected value of forecasted value. Thus the forecast will be adjusted according to historical data, and meanwhile the variance coefficient of forecast can be maintained to reserve the uncertainty information from CPC. By this approach, we could keep the statistic property consistent.

160

We conduct a stochastic optimization analysis using the monthly rainfall of CPC forecast. Equation (6.4) can be rewritten to formulate a stochastic dynamic optimization problem as follows. T

min L(r )  EX ( ln (rn )) P

n 1

Smin  S  S max

(6.8)

Sn 1  S n  rn  I n ( P )  P  ET   D j ( P )

where EX represents expected value, P is the random variable of rainfall which we obtain from CPC forecast (  indicates the uncertainty). This model applies the Monte Carlo method to decompose the stochastic program approach. In each period, the CPC issues a resolution of rainfall outlook that contains 13 exceedence threshold values for given probability percentile levels. Based on the CPC outlook, 14 intervals in the probability density function, 0–2, 2–5, 5–10, 10–20, 20–30, 30–40, 40–50, 50–60, 60–70, 70–80, 80–90, 90–95, 95–98, 98–100, are calculated as the samples for Monte Carlo method. The original random variable is replaced by a set of these samples. The expected loss function can also be calculated by the probability and the rainfall value in each interval. By this way, the model solves the stochastic program to determine the optimal policy. SFWMM accounts the evaporation for three distinct zones of the Lake, an open water zone, a marsh zone, and a no-water zone. The evaporation is dominated by the open-surface area. From the SFWMM output, evaporation shows consistent trend for each month with small fluctuation. Figure 6.7 shows the mean value and the range of the variability of evaporation.

The upper and lower bounds represent the mean value

plus/minus the standard derivation. It exhibits a small range and the coefficient of variance is only around 0.04 for each month. This study uses the monthly mean value of historical evaporation data from SFWMM to estimate the evaporation loss for Lake Okeechobee and WCAs.

161

Estimation of inflow and demands As mentioned above, this study uses a response function to mimic the role of SFWMM. As shown by Miralles-Wilhelm et al. (2005), the monthly precipitation and runoff in Southern Florida present a linear relationship. This is because with a monthly interval, the main mechanism of hydrology is mass conservation and flow routing is not a major issue. Therefore, we apply linear regression to reproduce the inflow I n and LOSA demand from SFWMM depending on precipitation. Linear regression precipitationrunoff model is a statistical model, which is considered as a black-box model. The blackbox model is often criticized because it lacks the consideration of physical mechanism. However, such simple models still make sense in relating hydrologic inputs and outputs (Diskin 1970 and Troutman 1985). When the hydrological process is not significantly non-linear, the application of monthly linear regression models is approved to be able to estimate the precipitation-runoff relationship by previous studies such as Raman and Mohan (1995) and Xu and Singh (1998). Equation (6.9) shows the general form of linear regression model

E ( y | x )  E ( y )   ( x, y )

 ( y) ( x  E ( x))  ( x)

(6.9)

where x is input and y is the output. E ( y | x) is the unbiased estimation with given x .

 is variance and  is the covariance. The response functions are developed using Equation (6.9). With given rainfall P as input x and inflow, LOSA’s runoff and demand as output y , respectively, the

response functions are calculated individually for each sub-areas of LOSA and each month. In general, the linear regression model is found be able to catch the significant seasonal pattern of hydrological conditions. Table 6.9 shows that the Pearson correlation coefficients for these response functions are different by month, and different by basin ranging between 0.6 and 0.95 with an average of 0.73. Figure 6.8 shows the comparison of the inflows and Figure 6.9 shows the comparison of the LOSA demand between SFWMM and linear regression estimation.

162

To verify the linear precipitation-runoff relation, artificial neural network (ANN) model is used to simulate runoff as a function of rainfall. The model uses two ANN designs suggested by Garbrecht (1996) and uses the Levenberg-Marquardt algorithm for training (Levenberg 1944, Marquardt 1963; Nocedal 1999).

For the first design, a

separate and independent ANN was developed for each calendar month, thereby accounting explicitly for seasonal variations. Table 6.10 shows the Pearson correlation coefficients for this ANN design in different months and sub-basins. Compared to the results from the linear regression as shown in Table 6.9, the two asspraoches exhibit similar performance in terms of the correlation between the estimated values and the values from SFWMM. The average of Pearson correlation coefficient is around 0.71 which is slightly smaller than the linear regression model. This comparison confirms the feasibility of using the linear regression model for estimating the relationship between monthly monthly rainfall and runoff in Southern Florida. The shortcoming of first design was the comparatively smaller training dataset for each network. For this reason, we conduct the other design which using dataset for the whole 36 years period without separating them to different calendar months. In addition, 12 binary identifiers of calendar months were used as the second input layer in the ANN model. The results from ANN design are showed in Table 6.11. Compared to the first design, the second design does not lead to better simulation results. Actually the first design explicitly reflects the seasonal rainfall and runoff variations, while the second design corresponding to a single ANN involving all calendar months implicitly accounts for the seasonal variations, ANN is supposed to be better than the simple linear regression model in catching non-linear hydrological relationship.

However, two possible reasons reduce the

capability of using the ANN model for this particular case. The first is that the dataset is relatively too small for the neural network training when considering the month interval. ANN requires a large amount of data for the machine-learning process. The second reason is related to our purpose which is limited to determine the rainfall-runoff relationship without involving other parameters such as soil moisture, evaporation.etc., which contribute more to the nonlinearity of the rainfall-runoff hydrological process. A multi-layer ANN is promising to improve the estimation. 163

However, because linear

regression model already provides a reasonable result and the rainfall-runoff relationship is not the major concern of this study, we just use the linear regression model without further discussing this issue. Water releases from Lake Okeechobee through WCAs also depend on water demand of other regions such as LEC-SA and ENP. This model just follows the monthly average value of the demands obtained from the SFSAM report (1998) for LEC-SA, ENP and also Caloosahatchee and St. Lucie Estuaries water demand. Note that the water demand of LEC-SA and ENP is first supplied by WCAs, and Lake Okeechobee works as a backup source when WCAs cannot fully meet the demand. Figure 6.10 shows the simulation of Lake Okeechobee’s monthly stage for each water year using the linear regression model under standard operating policy. Although the result from our model does not perfectly reproduce that of SFWMM, they are reasonable close. 6.3.

Results

6.3.1. Results of the Inter-year Hedging Model

The model results show that with the linear profit function, the difference between SOP and hedging is not obvious. The loss of applying SOP is 16.22 million dollar while the loss of hedging is 15.89 million dollars. However, with the quadratic profit function, the loss of hedging is 11.99 million dollars compared to 16.86 million dollars from SOP. Hedging policy reduces the loss around 28.8% compared to SOP. For the power function, the reduction percentage of loss by hedging is 13.2%, which reduces the loss from 8.98 under SOP to 7.79 million dollars under hedging. This result confirms that when the utility or loss function is linear, SOP is the optimal operating policy (Hashimoto et al., 1982). Figure 6.11 shows the loss reduction from SOP to hedging under different flow conditions. The hedging policy accepts some delivery deficit when the inflow in the middle range (decile 0.4~0.7) to reduce the probability of greater shortage (decile 0.1~0.4). Compared to SOP, hedging policy trades off water supply reliability at high delivery levels with that at medium and low levels so as to reduce the risk of severe water supply deficit.

164

The purpose of applying the revised TSP is to operate Lake Okeechobee in a lower stage to avoid the structural safety problem and the adverse environmental effects. Hedging rule conflicts to this purpose because it tends to store and carryover water for the future shortage. The environmental demand has been considered as a hard constraint in this model; while the high stage could still provoke the issue of structural safety. The average stage of Lake Okeechobee by applying SOP is around14.1 feet. The average stages for quadratic, power and linear profit functions are 14.6, 14.5 and 14.1 feet, respectively.

The result shows hedging rules only slightly raise the water level.

Therefore, it does not impact the structural safety in a significant way. Figure 6.12 shows the difference of the probability of the five operational bands with the quadratic function. Hedging increases the probability of reaching high lake management band (A) from 16% to 19%; meanwhile, the chance of water shortage band (E) drops from 20% to 9.6%. Thus hedging rules slightly lift the risk of structural safety but significantly reduces the risk of water shortage. As mentioned above, the revised TSP puts higher priority on environmental demands and structural safety and thus decreases the water availability for agricultural and municipal demand. However, comparing the annual water budget between revised TSP and WSE, the data does not show significant difference of annual water supply between the two polices. To examine the impact of increasing water stress in this region, this model assumes the different levels of I LOK . Figure 6.13 shows the loss reduction from SOP to the hedging under the different levels of I LOK . This result shows a more significant improvement of hedging under a more serious water stress level. Figure 6.14 shows the loss reduction under different uncertainty levels. Larger profit loss reduction from SOP to hedging occurs with higher uncertainty of the I LOK . Figure 6.15 shows the effects of EAA local water availability ( AL ) to hedging. The influence of water shortage stress of AL is similar to I LOK . However, the fluctuation of AL is less than I LOK . Consequently, the uncertainty of AL does not show a significant role for implementing hedging rules.

165

This study also explores the impact of reservoir capacity and discount rate. Both parameters are not significant for the implementation of hedging rules for Lake Okeechobee. The reservoir capacity of Lake Okeechobee is not a constraint for the implementation of hedging rules thus could also be understood from previous discussion of water stage. Hedging rule policy does not tend to hold much water in this case. Therefore, it is not necessary to have a larger reservoir storage from a limited view of water supply only.

However, re-examination of current operational bands may be

necessary to incorporate the inter-year hedging. Storing more water in the mid-level operational bands and starting to hedging water earlier comparing to current water shortage band for SSM could provide opportunity to avoid severe economic loss. 6.3.2. Ex Post analysis of intra-year hedging

The hedging model is first applied to conduct the ex post analysis using historical drought events.

Three significant drought periods, October-71 to Septmeber-74,

October-81 to Septmebr-82, and October-89 to April-92 in the Southern Florida region are modeled. For this ex post analysis, the model obtains the values of water availability and demand from LORSS SSM result, and takes these as deterministic input to the model. The model tests different objective functions (Equation [6] and [7]). Table 6.12, Table 6.13 and Table 6.14 show the comparison of the outcomes from the current SSM and different hedging models. The results of hedging rules are affected much by objective chosen for the analysis. Specifically, hedging policies according to the modified-FAO production equation can minimize the total and also the maximum loss, while the SI index equation leads to a policy which achieves lower maximum and average water cutback ratio. The current SSM is designed to avoid water level in the Lake lower than the reference elevation at the end of the dry season. Comparing these hedging polices to the SSM, SSM does not consider economic incentives or severity and then lead to a higher loss; while hedging results in the longer water stress periods but less severity. The overall loss is mitigated by hedging. Figure 6.16 shows the levels of water shortage by SSM and hedging (with modified – FAO equation), plot of SSM and the potential hedging rule policy driven by modified166

FAO in the period between October 1971 and September 1974, covering three water year, respectively. During this period, droughts occurred in each of the summers of each year. The drought in 1974 is the most serious one. As discussed in Chapter 2, the hedging rule accepts some deficit of present delivery to reduce the risk of greater water shortage in the future. The Ex post analysis assumes the perfect knowledge of coming droughts, although this is not possible in the real world, the hedging rule avoids the severity of water shortage in 1974 by applying minor cutback of water supply in previous periods. Note that hedging is applied to both inter- and intra-year operation. Water is hedged in the first two years and carried over to reduce the high vulnerability with drought event in the third year. Additionally, the impact of seasonality also can be observed from Figure 6.16. Hedging rule tends to reduce the potential water shortage in the beginning of each year (January, February, and March). This period is the flowering stage of the major crops in the study area and the crops are most sensitive to water stress. As can be seen in Table 6.7, that higher response coefficients, ky, exist in the first several months. Therefore, hedging water in other stages reduces water stress in the flowering stage to avoid large crop yield loss. A more typical inter-year hedging is demonstrated in Figure 6.17. The drought starts from April, 1980 and continued till the summer of 1981. The potential policy from model analysis is to hedge water use in the beginning of 1981 water year. This amount of water hedged is reserved for 1982 water year to mitigate the severity of water shortage. Over-year hedging is the main component for the operational policy. However, the case showed in Figure 6.18 tends to be intra-year hedging. In 1990 water year, hedging is to decrease the degree of the water shortage occurring in the summer. After the summer drought, the hedging policy contributes to smooth the fluctuation of water shortage with a shorter time scale. Figure 6.16, Figure 6.17 and Figure 6.18 demonstrate the advantage of hedging policies during the various historical periods. Figure 6.19, Figure 6.20 and Figure 6.21 compare the marginal values of water uses under SSM and hedging. As can be seen, hedging regulates storage to smooth the marginal values by time. The marginal value in August and September is relatively smaller than those in other months because in these 167

two months, LOSA demand is small so the economic loss due to water stress will not be significant. This fact confirms that hedging rule presented here achieves the Pareto efficiency by time based on the marginal utility principle as discussed in Chapter 2. Although the same principle works for all time periods, the hedging rule varies different periods due to the variability of water demand and the length of drought periods. In particular, when the period of droughts is not long and the recurrence frequency is short, intra-year hedging is the major rule like drought in 1990. When the drought period is long, inter-year hedging is needed for reservoir operation like the drought in 1980~1981. However, for most cases with Lake Okeechobee, the inter-year and intra-year hedging coexists while the long-term and short-term droughts also simultaneously occur through the operation of Lake Okeechobee. Thus, the long term hedging can be nested with short term hedging to reduce the vulnerability of water shortage. 6.3.3. Hedging with CPC forecast

Perfect information is not realistic for reservoir operation in the real world. The operation usually relies on the uncertain forecast of weather or climate condition. The CPC prediction is applied to studying the intra-year hedging under imperfect information. The CPC 13-month outlook is used to determine the potential hedging policy. Figure 6.21 shows the hedging rule using to CPC prediction made for year 2000. The ex ante simulation demonstrates the hedging policies at four different time points in the year, starting in October, January, April, and July, respectively. Figure 6.22(a) is the hedging policy according the CPC forecast provided in October, 1999. The dotted line represents the expected value of release under hedging while solid line is the expected value of the estimated demand. The dashed line is the expected value of water availability. In this case, the water delivery will be equal to the water availability when applying SOP because under the water stress, not additional water can be stored. So the dashed line could be understood as the release of the SOP, standard operation policy. Because that CPC forecast told a drought around the end of 2000, the hedging rule suggests some early cutback except for January, February and March. After these three months, as showed in (b), CPC updated their forecast for drought at the end of 2000, which was not as serious as reported earlier. The amount of 168

hedge is reduced while the hedging during May to September is still suggested to reserve water for September to December when a drought might occur. From Figure 6.22 (c), April 2000 indicated that the drought could be very severe after June 2000 and it may continue to next year. Correspondingly, in the beginning of 2001, our hedging rule suggests the water cutback in April to protect more valuable water use later on. In July 2000, the drought actually occurred, and CPC predicted that the drought could continue up to April 2001. To reduce the potential loss in the beginning of 2001, hedging is suggested at the end of 2000 to reserve water for the spring of 2001, which is more important for crop production. Figure 6.23 shows the marginal value of water under SOP and hedging,

respectively, at the four time points. In general, the decision making under hedging with consideration of uncertainty still follows the marginal utility principle. Especially, these figures demonstrate the expectation of decision making following economic incentives. In October 1999 and January 2000, the marginal utility of water uses is low under both the hedging and SOP. Without being aware of the severity of drought, the decision making seems to be optimistic without storing much water. Later, the decision turns to be conservative when the severity of drought is realized. A larger amount of water is hedged and the marginal loss is increased. The expected loss for the planning horizon at this time point now is higher comparing to that at the first two time points. The hedging with CPC forecast shows an ex ante analysis following the expectation and awareness of coming droughts. When the forecast tells a coming drought, hedging rule tries to mitigate the severity of water shortage during the drought, particularly during the stage, in which crop production is most sensitive to water stress such as flowing stage. 6.3.4. Rolling decision making

Rolling decision making (RDM) with limited foresight is a common practice (You and Cai, 2008d). Under a rolling decision making procedure, the optimization problem is setup and solved according to current conditions such as initial storage and forecasted inflow, and release decision is made for the current period. The initial periods which the decision will be realized according to the solution is called decision horizon. 169

While the system observation and forecast are updated, the problem can be re-formulated as the recourse problem of the previous problem with an extended horizon.

This

procedure is repeated with updated forecast from period to period. The forecast horizons rolling over periods are called rolling horizon, also called study horizon (Bean et al., 1987). This part applies the RDM to Lake Okeechobee to examine the effectiveness of hedging rules under a given forecast horizon usually with a limited length. The hedging decision is made for the current month using the CPC forecast of the coming 13 months. This procedure is repeated with updated forecast from month to month. For this case study, the decision horizon for the rolling procedure is one month and the study horizon is 13 months. The 2000-2001 drought is a major one in the Southern Florida area. We apply the RDM with hedging as an ex ante analysis. An ex post analysis which assumes perfect information of the drought is also conducted and the result is compared to the RDM. As the ex post analysis presented earlier, the observed rainfall data is also obtained from the LORSS study. However, LORSS only provides data till 2000. The rainfall data after 2000 is obtained from the SFWMD website. It should be noticed that the data does not go through quality assurance/quality control (QA/QC) (SFWMM, 2002). Figure 6.24 compares the estimated demand, and the releases of ex post policy and RDM. Figure 6.25 compares the water shortage under SOP and policy from ex post analysis and RDM. With the perfect knowledge of the coming drought, ex post policy hedges water starting from October 1999 till the flowering season of year 2000 to avoid severe water shortage in the future. However, RDM, which is not aware of the coming drought with a limited study horizon, just hedges a small amount of water, which does not help avoid the coming water shortage around April 2000 and peaking in the summer of 2001. Even for the flowering stage, a serious water shortage occurs with the RDM. The shortage in this period is supposed to be avoided if the CPC forecast was longer and more reliable. The forecast horizon required for the optimal policy is usually unavailable for the practice of water reason system operation. 170

Chapter 5 proposed a methodology to

estimate the error bound with a limited study horizon (e.g. CPC 13 month) by assigning appropriate terminal state boundary condition. The terminal boundary condition of the state variable (reservoir storage) represents the water needed to carry over to the future but being realized with the limited study horizon. For Lake Okeechobee, the capacity (around 3000 taf) is much larger than the average LOSA demand (558 taf). So it is not reasonable to use the storage capacity as the potential terminal boundary condition. This study tests the error bound of the rolling policy by assigning the amount of the water need beyond the study horizon as 50, 100, 150 and 200 taf as the terminal boundary, respectively. The analysis answers the following question: if a certain amount of water is carried over to the period after the study horizon, how the current policy will be changed? Figure 6.26 shows the additional water cutback under these boundary conditions compared to water hedge under original RDM varies by time. In January and February of 2000, the crop flowering stage, the additional cutback is minor; while in April, July, and September of 2000, the original RDM already suggests a large amount of hedging (Figure 6.24). Therefore, there is not much additional cutback in these months. For the month in which more cutback is possible with less loss, the impact of additional hedge is significant. Thus the RDM based on the current CPC forecast shows the result from the RDM is still sensitive to the boundary conditions, which means that the RDM based on the current forecast is different from the theoretical optimum if we consider longer study horizons. The gap could be large especially for those months in which a large amount of water can be hedged. Because of its huge storage capacity, Lake Okeechobee actually can deal with a long term hedging operation. Due to the multiple year or consecutive droughts in the region as occurring in the history and possibly revisiting the region, long-term hedging is necessary as showed by the ex post analysis. However, the long-term hedging needs reliable forecast with a long forecast horizon. An additional concern is the quality of long term forecasts. For a stochastic forecast such as that of CPC. The evaluation of its quality should be made continuously for a long period.

According to a comment from NOAA, CPC provides climate

prediction instead of weather forecast; the lead time has much less influence on the 171

predictions than it has on forecasts. Most variability at the seasonal scales is due to the influence of climate regimes that are unpredictable beyond about 14 days. The forecast probability distribution function mostly reflects long term means and variances. The current CPC forecast has a limited capacity to predict the seasonal variability which is usually most relevant to droughts. 6.4.

Conclusions

The operation of Lake Okeechobee faces many challenges today due to variable inflow and growing demand. Climate change and variability in the future may worsen the situation. The coming revised TSP which operates Lake Okeechobee under lower stage will cut down the margin for agricultural and municipal water supply especially when drought occurs. The users possibly face more frequent cutback of water supply and more severe shortage. This study first develops an inter-year model to explore the potential for over-year hedging. Assuming a purpose of risk aversion, the results show that hedging rules could significantly reduce the profit loss compared to SOP. According to the model result, the hedging of water supply could be applied in an earlier time than that triggering the current Supply-Side Management.

That is to say, the reservoir

operation could apply hedging in advance, for example, in the base flow sub-band or the beneficial use sub-band. The early hedging may help users mitigate the risk of severe water shortage and potential loss. The intra-year hedging model considers the impact of seasonal variability. It has been found that the operation of Lake Okeechobee should consider both short term and long term hedging which are nested with each other. Hedging rules extend the water stress period but reduce the potential vulnerability. Using different objective functions, the hedging rule policy will achieve different operational goals. When the modified-FAO crop yield water relation is used, hedging rule policy tends to avoid the water shortage in the beginning of each year and cut back water supply during other seasons. While the SI index equation leads to a policy focus on the severity of water stress rather than economic loss. It achieves lower maximum and average water cutback ratio. The study also shows that the rolling decision making could provide a practical way to approach to the optimal policy. However, hedging rule policy still relies on the quality of the predication of 172

future condition. The length of forecast and the accuracy of forecast are still important limits. Current operational schedule adopts climate outlook information for decision making of water supply but is still based on the concept of SOP. The cutback and hedging are only applied when Supply-Side Management is triggered.

SSM is a

relatively short hedging policy and its objective is to maintain the reference elevation at the end of the dry season. However, the economic vulnerability is not considered by SSM. This study shows the merits for a longer-term hedging policy. SFWMD could consider incorporating an economic objective and taking a longer planning horizon for shortage management. Besides improving the short-term management, a long-term and high quality forecast system is also worthwhile to invest. The modeling framework used in the study is subject to some limitations. First, the model does not consider the reduction of other water demands under drought conditions. In the real operation of Lake Okeechobee, other demands are also adjusted by SSM criteria. SSM tries to maintain the prescribed storage level at the end of the dry season, but it still allows the stage of Lake lower than the reference depending upon the severity of a drought event. The flexibility of water supply can reduce the severity of water shortage when lake stage is lower than the operation bands. However, in our model, we just cut down all LOSA water supplies but maintaining other non-agricultural demands when the water level is lower than the operation band. LOSA is only allowed to use the water hedged for the region. These constraints may lead an overestimate of the water shortage situation. Second, to avoid the computation burden of integrating a current hydrologic model with the hedging model, linear response function is used to reproduce the hydrologic inputs.

Additionally, the current shortage management policy is very

complex and it needs a lot of information to represent the full policy in our model. Because of these simplifications, the results of our model may have limited value for real world implementation. However, this study demonstrates the possible effectiveness of optional hedging rule for Lake Okeechobee operation and shows an economic gain for shortage management with the Lake. For more practical study of hedging rule for the 173

Lake, the integration of current hydrologic model with hedging policy analysis will be beneficial.

174

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177

Table 6.1. LOSA irrigated acreage (SEIS, 2007) LOSA Sub Area 1. EAA 2. North Shore 3. Caloosahatchee Basin 4. St. Lucie Basin Total LOSA

Irrigated Acreage 541,878 13,380 138,337 49,073 742,558

Table 6.2. Agricultural land uses in EAA (SEIS, 2007) Crop Sugarcane Miscellaneous Row Crops Sod Total EAA

Acreage 436,856 18,514 21,107 26,912 493,389

Percent of Total 86.8% 37% 4.2% 5.3% 100%

Table 6.3. Agricultural land uses in the Caloosahatchee basin(SEIS, 2007) Crop Citrus Sugarcane Vegetables SOD Ornamentals Total

Acreage 78,113 50,359 8,091 1,296 478 138,517

Percent of Total 56% 36% 6% 1%