Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Optimization of the operation of a two-resevoir hydropower system Nash, Garth Andrew 2003

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2003-860558.pdf [ 55.65MB ]
Metadata
JSON: 831-1.0063632.json
JSON-LD: 831-1.0063632-ld.json
RDF/XML (Pretty): 831-1.0063632-rdf.xml
RDF/JSON: 831-1.0063632-rdf.json
Turtle: 831-1.0063632-turtle.txt
N-Triples: 831-1.0063632-rdf-ntriples.txt
Original Record: 831-1.0063632-source.json
Full Text
831-1.0063632-fulltext.txt
Citation
831-1.0063632.ris

Full Text

OPTIMIZATION OF THE OPERATION OF A TWO-RESERVOIR HYDROPOWER SYSTEM by GARTH ANDREW NASH B.A.Sc, The University of Toronto, 1993 M.A.Sc., The University of Toronto, 1995 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF CIVIL ENGINEERING) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 2003 ® Garth Andrew Nash, 2 003 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C ^ / \ A / £i1/jfV\_-r-fb'i'*ia The University of British Columbia Vancouver, Canada Date /{lAj IAS}" 2041 DE-6 (2/88) A b s t r a c t A method for optimizing the operation of a system of two hydraulically separate reservoirs serving the same demand area for hydropower production is described. The reservoir system is assumed to be operated, and import and export decisions made, so as to maximize the value of energy produced while considering the value of water stored in the reservoirs at the end of the model time horizon. The optimization considers uncertain reservoir inflows, energy demands, and electricity prices, and is subject to physical and operational constraints. The proposed method consists of two cascaded models. A longer-term monthly model based upon dynamic programming and linear programming is used to estimate the value of water stored in each reservoir as a function of the storage in both reservoirs, as well as the marginal values of water storage in the two reservoirs. Linear programming is used to evaluate the recursive equation in the dynamic program by making tradeoffs between releasing water, making energy trades, and keeping water in storage for the next month. The monthly energy value functions are input to the shorter-term model, which is based upon stochastic linear programming with recourse. The shorter-term model allows for the planning of operations and the calculation of marginal water values over periods shorter than one month. The time horizon in the shorter-term model is divided into time steps that may be of variable duration. Uncertainty in the model is handled through a scenario tree. Scenarios describe the values assumed by the inflows, demands, and prices in each time step. Sub-periods allow for the consideration of on- and off-peak periods. Application of the proposed model is made to a system based roughly on the two main river systems in the B C Hydro system—the Peace and Columbia. It is found that the marginal value of storage in the Columbia Reservoir is generally dependent upon the storage in both the Columbia and Peace Reservoirs, and vice versa. Regions of storage existed in which the marginal energy value in one reservoir was independent of storage in the second, although no general rules for identifying these regions were found. 11 Table of Contents Abstract i i Table of Contents..: i i i List o f Tables iv List of Figures vi 1 Introduction 1 2 Literature Review 7 2.1 Introduction 7 2.2 Techniques 7 2.2.1 Dynamic Programming 8 2.2.2 Linear Programming 17 2.2.3 Other Methods 23 2.2.4 Simulation : . . 2 8 2.3 Summary and Conclusions 29 3 Stochastic Dynamic Programming and Linear Programming Based Model 31 3.1 Introduction 31 3.2 D P and L P Based Model Overview 32 3.2.1 Terminology 32 3.2.2 Model Overview 34 3.3 Model Details 37 3.3.1 Dynamic Programming Model 37 3.3.2 Linear Programming Model 46 3.4 Case Study 55 3.4.1 Description 55 3.4.2 Data 56 3.4.3 Results 61 3.5 Summary 144 4 Short-term Marginal Value Model 150 4.1 Introduction 150 4.2 S T M V M Overview 151 4.2.1 Stochastic Linear Programming with Recourse 153 4.2.2 Choice of Here-and-Now Variables 156 4.3 Model Details r 157 4.3.1 Mathematical Formulation 157 4.3.2 Solution Methodology 166 4.4 Case Study 167 4.4.1 Overview 167 4.4.2 Scenario Definitions 168 4.4.3 Objective Function Values 179 4.4.4 Peace Marginal Energy Values 197 4.4.5 Columbia Marginal Energy Values 216 4.4.6 Summary 235 5 Summary and Conclusions 243 6 Literature Cited 255 in List of Tables Table 3-1: Average Monthly Inflows 56 Table 3-2: Average Monthly Market Electricity Prices 57 Table 3-3: Average Monthly Electricity Demand 57 Table 3-4: Monthly Duration of H L H and L L H Periods 58 Table 3-5: Monthly Maximum Import and Export Transmission Limits 58 Table 3-6: Monthly Min imum Import and Export Transmission Limits 59 Table 3-7: Plant Related Limits 59 Table 3-8: Monthly Min imum Plant Discharge 60 Table 3-9: Slopes of HK-Storage Function for Peace Plant 60 Table 3-10: Slopes of HK-Storage Function for Columbia Plant 61 Table 3-11: Storage Intercepts for HK-Storage Functions 61 Table 3-12: Miscellaneous Data 61 Table 3-13: Minimum Plant Turbine Release Violation Penalties 61 Table 3-14: Min imum and Maximum Monthly Base Case Columbia Marginal Energy Values 69 Table 3-15: Minimum and Maximum Peace Marginal Energy Values 70 Table 3-16: Definition of Five-Scenario Cases 79 Table 3-17: Unit Normal Function Values for Five-Scenario Cases 80 Table 3-18: Min imum Peace Marginal Energy Values for Demands Only Scenarios 81 Table 3-19: Maximum Peace Marginal Energy Values for Demands Only Scenarios 82 Table 3-20: Minimum and Maximum Columbia Marginal Energy Values for Demands Only Scenarios 83 Table 3-21: Minimum and Maximum Peace Marginal Energy Values for Inflows Only Scenarios 85 Table 3-22: Minimum and Maximum Columbia Marginal Energy Values for Inflows Only Scenarios 86 Table 3-23: Minimum and Maximum Peace Marginal Energy Values for Perfect Positive Correlation Scenarios 88 Table 3-24: Min imum and Maximum Columbia Marginal Energy Values for Perfect Positive Correlation Scenarios 88 Table 3-25: Min imum and Maximum Peace Marginal Energy Values for Perfect Correlation—Positive for Demands and Inflows and Negative for Prices 90 Table 3-26: Min imum and Maximum Columbia Marginal Energy Values for Perfect Correlation—Positive for Demands and Inflows and Negative for Prices 90 Table 3-27: Min imum and Maximum Peace Marginal Energy Values for Perfect Correlation—Positive for Demands and Prices and Negative for Inflows 92 Table 3-28: Minimum and Maximum Columbia Marginal Energy Values for Perfect Correlation—Positive for Demands and Prices and Negative for Inflows 93 Table 3-29: Minimum and Maximum Peace Marginal Energy Values for Perfect Correlation—Positive for Inflows and Prices and Negative for Demands 94 Table 3-30: Minimum and Maximum Columbia Marginal Energy Values for Perfect Correlation—Positive for Inflows and Prices and Negative for Demands 94 Table 3-31: Modified Monthly Minimum Plant Discharge 136 iv Table 4-1: Length of Sub-Periods for January Studies 168 Table 4-2: Limits for January Studies 169 Table 4-3: January Base Case Scenario-Dependent Parameters 169 Table 4-4: Scenario-Dependent Demands for January Studies 170 Table 4-5: Scenario-Dependent Inflows for January Studies 170 Table 4-6: Scenario-Dependent Prices for January Studies 171 Table 4-7: Length of Sub-Periods for May Studies 172 Table 4-8: Limits for May Studies 172 Table 4-9: May Base Case Scenario-Dependent Parameters 173 Table 4-10: Scenario-Dependent Demands for May Studies 173 Table 4-11: Scenario-Dependent Inflows for May Studies 174 Table 4-12: Scenario-Dependent Prices for May Studies 175 Table 4-13: Length of Sub-Periods for September Studies 175 Table 4-14: Limits for September Studies 176 Table 4-15: September Base Case Scenario-Dependent Parameters 176 Table 4-16: Scenario-Dependent Demands for September Studies 177 Table 4-17: Scenario-Dependent Inflows for September Studies 177 Table 4-18: Scenario-Dependent Prices for September Studies 178 v List of Figures Figure 3-1: Base Case January Storage Value Function 63 Figure 3-2: Base Case Apr i l Storage Value Function 63 Figure 3-3: Base Case June Storage Value Function 64 Figure 3-4: Base Case October Storage Value Function 64 Figure 3-5: Storage Value Function Slices for Columbia Reservoir Empty 65 Figure 3-6: Storage Value Function Slices for Columbia Reservoir 50% Ful l 66 Figure 3-7: Storage Value Function Slices for Columbia Reservoir Full 66 Figure 3-8: Storage Value Function Slices for Peace Reservoir Empty 67 Figure 3-9: Storage Value Function Slices for Peace Reservoir 50% Ful l 67 Figure 3-10: Storage Value Function Slices for Peace Reservoir Ful l 68 Figure 3-11: Peace Marginal Energy Values for Columbia Reservoir Empty 71 Figure 3-12: Peace Marginal Energy Values for Columbia Reservoir 50% Ful l 72 Figure 3-13: Peace Marginal Energy Values for Columbia Reservoir Ful l 72 Figure 3-14: Columbia Marginal Energy Values for Peace Reservoir Empty 73 Figure 3-15: Columbia Marginal Energy Values for Peace Reservoir 50% Ful l 74 Figure 3-16: Columbia Marginal Energy Values for Peace Reservoir Ful l 74 Figure 3-17: M a y Marginal Peace Energy Value Functions 75 Figure 3-18: May Marginal Columbia Energy Value Functions 76 Figure 3-19: November Marginal Peace Energy Value Functions 77 Figure 3-20: November Marginal Peace Energy Value Functions—Detail 77 Figure 3-21: November Marginal Columbia Energy Value Functions 78 Figure 3-22: November Marginal Columbia Energy Value Functions—Detail 78 Figure 3-23: Min imum Peace Marginal Energy Values for Demands Only Scenarios.... 81 Figure 3-24: Maximum Peace Marginal Energy Values for Demands Only Scenarios ... 82 Figure 3-25: Min imum Columbia Marginal Energy Values for Demands Only Scenarios : 83 Figure 3-26: Maximum Columbia Marginal Energy Values for Demands Only Scenarios 84 Figure 3-27: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.40 96 Figure 3-28: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.25 97 Figure 3-29: Maximum Columbia Marginal Energy Values for Coefficient of Variation ofO.10 97 Figure 3-30: Min imum Columbia Marginal Energy Values for Coefficient of Variation of 0.40 98 Figure 3-31: Min imum Columbia Marginal Energy Values for Coefficient of Variation of 0.25 99 Figure 3-32: Minimum Columbia Marginal Energy Values for Coefficient of Variation of 0.10 99 Figure 3-33: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.40 100 Figure 3-34: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.25 100 vi Figure 3-35: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.10 101 Figure 3-36: Min imum Peace Marginal Energy Values for Coefficient of Variation of 0.40 102 Figure 3-37: Min imum Peace Marginal Energy Values for Coefficient of Variation of 0.25 102 Figure 3-38: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.10 103 Figure 3-39: Maximum Columbia Marginal Energy Values for Demands Only Scenarios 105 Figure 3-40: Maximum Peace Marginal Energy Values for Demands Only Scenarios . 106 Figure 3-41: Min imum Columbia Marginal Energy Values for Demands Only Scenarios 107 Figure 3-42: Min imum Peace Marginal Energy Values for Demands Only Scenarios.. 107 Figure 3-43: Maximum Columbia Marginal Energy Values for Inflows Only Scenarios 109 Figure 3-44: Maximum Peace Marginal Energy Values for Inflows Only Scenarios.... 109 Figure 3-45: Min imum Columbia Marginal Energy Values for Inflows Only Scenarios 110 Figure 3-46: Minimum Peace Marginal Energy Values for Inflows Only Scenarios 110 Figure 3-47: Maximum Columbia Marginal Energy Values for Prices Only Scenarios 111 Figure 3-48: Maximum Peace Marginal Energy for Prices Only Scenarios 112 Figure 3-49: Minimum Columbia Marginal Energy Values for Prices Only Scenarios. 113 Figure 3-50: Min imum Peace Marginal Energy Value for Prices Only Scenarios 114 Figure 3-51: Maximum Columbia Marginal Energy Values for Perfectly Positively Correlated Demand, Inflow, and Price Scenarios 115 Figure 3-52: Maximum Peace Marginal Energy Values for Perfectly Positively Correlated Demand, Inflow, and Price Scenarios 115 Figure 3-53: Min imum Columbia Marginal Energy Values for Perfectly Positively Correlated Demand, Inflow, and Price Scenarios 116 Figure 3-54: Min imum Peace Marginal Energy Values for Perfectly Positively Correlated Demand, Inflow, and Price Scenarios 117 Figure 3-55: Maximum Columbia Marginal Energy Values for Demand and Inflow Perfectly Correlated and Price Perfectly Negatively Correlated Scenarios 118 Figure 3-56: Maximum Peace Marginal Energy Values for Demand and Inflow Perfectly Correlated and Price Perfectly Negatively Correlated Scenarios 119 Figure 3-57: Minimum Columbia Marginal Energy Values for Demand and Inflow Perfectly Correlated and Price Perfectly Negatively Correlated Scenarios 120 Figure 3-58: Minimum Peace Marginal Energy Values for Demand and Inflow Perfectly Correlated and Price Perfectly Negatively Correlated Scenarios 121 Figure 3-59: Maximum Columbia Marginal Energy Values for Demand and Price Perfectly Correlated and Inflow Perfectly Negatively Correlated Scenarios 122 Figure 3-60: Maximum Peace Marginal Energy Values for Demand and Price Perfectly Correlated and Inflow Perfectly Negatively Correlated Scenarios 123 Figure 3-61: Min imum Columbia Marginal Energy Values for Demand and Price Perfectly Correlated and Inflow Perfectly Negatively Correlated Scenarios 123 vn Figure 3-62: Min imum Peace Marginal Energy Values for Demand and Price Perfectly Correlated and Inflow Perfectly Negatively Correlated Scenarios 124 Figure 3-63: Maximum Columbia Marginal Energy Values for Inflow and Price Perfectly Correlated and Demand Perfectly Negatively Correlated Scenarios 125 Figure 3-64: Maximum Peace Marginal Energy Values for Inflow and Price Perfectly Correlated and Demand Perfectly Negatively Correlated Scenarios 126 Figure 3-65: Min imum Columbia Marginal Energy Values for Inflow and Price Perfectly Correlated and Demand Perfectly Negatively Correlated Scenarios 126 Figure 3-66: Minimum Peace Marginal Energy Values for Inflow and Price Perfectly Correlated and Demand Perfectly Negatively Correlated Scenarios 127 Figure 3-67: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.25 129 Figure 3-68: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.10 130 Figure 3-69: Min imum Columbia Marginal Energy Values for Coefficient of Variation of 0.25 131 Figure 3-70: Minimum Columbia Marginal Energy Values for Coefficient of Variation of 0.10 131 Figure 3-71: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.25 133 Figure 3-72: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.10 133 Figure 3-73: Min imum Peace Marginal Energy Values for Coefficient of Variation of 0.25 '..„. 134 Figure 3-74: Minimum Peace Marginal Energy Values for Coefficient of Variation 0.10 . 135 Figure 3-75: Effect of Peace Ice Restrictions on Minimum Peace Marginal Energy Values 137 Figure 3-76: Effect of Peace Ice Restrictions on Maximum Peace Marginal Energy Values 138 Figure 3-77: Effect of Peace Plant Ice Restrictions on Minimum Columbia Marginal Energy Values 139 Figure 3-78: Effect of Peace Ice Restrictions on Maximum Columbia Marginal Energy Values 139 Figure 3-79: Effect of Peace Ice Restrictions on M a y Peace Marginal Energy Values.. 140 Figure 3-80: Effect of Peace Ice Restrictions on November Peace Marginal Energy Values 141 Figure 3-81: Effect of Peace Plant Restrictions on November Peace Marginal Energy Values—Detail 142 Figure 3-82: Effect of Peace Ice Restrictions on May Columbia Marginal Energy Values 142 Figure 3-83: Effect of Peace Ice Restrictions on November Columbia Marginal Energy Values 143 Figure 4-1: Two-Stage Recourse Problem 154 Figure 4-2: Multiple-Stage Recourse Problem 155 Figure 4-3: Objective Function Values for One-Scenario Base Cases 179 vni Figure 4-4: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for N o Non-Anticipative Constraints Assumption in January 181 Figure 4-5: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for Non-Anticipative Turbine Release Constraints Assumption in January 182 Figure 4-6: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for Non-Anticipative Ending Storage Volume Constraints Assumption in January 183 Figure 4-7: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for N o Non-Anticipative Constraints Assumption in May 184 Figure 4-8: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for Non-Anticipative Turbine Release Constraints Assumption in May 185 Figure 4-9: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for Non-Anticipative Ending Storage Volume Constraints Assumption in May 186 Figure 4-10: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for No Non-Anticipative Constraints Assumption in September 187 Figure 4-11: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for Non-Anticipative Turbine Release Constraints Assumption in September 188 Figure 4-12: Difference Between Base Case and Scenario-Dependent Case Objective Function Values for Non-Anticipative Ending Storage Volume Constraints Assumption in September 189 Figure 4-13: E V P I Under Non-Anticipative Turbine Release Assumption for January. 191 Figure 4-14: E V P I Under Non-Anticipative Ending Storage Volume Assumption for January 192 Figure 4-15: E V P I Under Non-Anticipative Turbine Release Assumption for May 193 Figure 4-16: E V P I Under Non-Anticipative Ending Storage Volume Assumption for M a y 194 Figure 4-17: E V P I Under Non-Anticipative Turbine Release Assumption for September 195 Figure 4-18: E V P I Under Non-Anticipative Ending Storage Volume Assumption for September 196 Figure 4-19: Peace Marginal Energy Value for One-Scenario Base Cases 198 Figure 4-20: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for N o Non-Anticipative Constraints Assumption in January 199 Figure 4-21: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Turbine Release Constraints Assumption in January 201 Figure 4-22: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Ending Storage Volume Constraints Assumption in January 202 ix Figure 4-23: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for N o Non-Anticipative Constraints Assumption in May 203 Figure 4-24: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Turbine Release Constraints Assumption in May 204 Figure 4-25: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Ending Storage Volume Constraints Assumption in M a y 205 Figure 4-26: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for N o Non-Anticipative Constraints Assumption in September 206 Figure 4-27: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Turbine Release Constraints Assumption in September 207 Figure 4-28: Difference in Peace Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Ending Storage Volume Constraints Assumption in September 208 Figure 4-29: Difference in Peace Marginal Energy Value Under Non-Anticipative Turbine Release Assumption for January 210 Figure 4-30: Difference in Peace Marginal Energy Value Under Non-Anticipative Ending Storage Volume Assumption for January : 211 Figure 4-31: Difference in Peace Marginal Energy Value Under Non-Anticipative Turbine Release Assumption for M a y 212 Figure 4-32: Difference in Peace Marginal Energy Value Under Non-Anticipative Ending Storage Volume Assumption for M a y '. 213 Figure 4-33: Difference in Peace Marginal Energy Value Under Non-Anticipative Turbine Release Assumption for September 214 Figure 4-34: Difference in Peace Marginal Energy Value Under Non-Anticipative Ending Storage Volume Assumption for September 215 Figure 4-35: Columbia Marginal Energy Value for One-Scenario Base Cases 217 Figure 4-36: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for No Non-Anticipative Constraints Assumption in January 219 Figure 4-37: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Turbine Release Constraints Assumption in January 220 Figure 4-38: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Ending Storage Volume Constraints Assumption in January 221 Figure 4-39: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for No Non-Anticipative Constraints Assumption in May 222 Figure 4-40: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Turbine Release Constraints Assumption in M a y 223 Figure 4-41: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Ending Storage Volume Constraints Assumption in May 224 Figure 4-42: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for N o Non-Anticipative Constraints Assumption in September 225 Figure 4-43: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Turbine Release Constraints Assumption in September 226 Figure 4-44: Difference in Columbia Marginal Energy Values Between Base Case and Scenario-Dependent Case for Non-Anticipative Ending Storage Volume Constraints Assumption in September 227 Figure 4-45: Difference in Columbia Marginal Energy Value Under Non-Anticipative Turbine Release Assumption for January 229 Figure 4-46: Difference in Columbia Marginal Energy Value Under Non-Anticipative Ending Storage Volume Assumption for January 230 Figure 4-47: Difference in Columbia Marginal Energy Value Under Non-Anticipative Turbine Release Assumption for May 231 Figure 4-48: Difference in Columbia Marginal Energy Value Under Non-Anticipative Ending Storage Volume Assumption for May 232 Figure 4-49: Difference in Columbia Marginal Energy Value Under Non-Anticipative Turbine Release Assumption for September 233 Figure 4-50: Difference in Columbia Marginal Energy Value Under Non-Anticipative Ending Storage Volume Assumption for September 234 xi 1 Introduction The problem of operating a reservoir, or system of reservoirs, in order to achieve some set of specified objectives is a fascinating and difficult problem. In order to appreciate some of the complexities, first consider the case of a single-purpose reservoir with natural inflows that is operated independently of any other reservoirs. In determining how to operate such a reservoir, decisions need to be taken as to how water should be stored and released in order to satisfy the stated objective. The main complication is that inflows to the reservoir are inherently uncertain. Thus, the decisions as to how much water to release must be made without knowing how much water w i l l be flowing into the reservoir. The difficulty of the situation led one author to refer to the operation problem as "a game against nature" (Lindqvist, 1962). If the reservoir in question is operated for hydropower production, additional complications arise. When the power plant is connected to an electricity grid with interconnections, energy transactions become possible, and thus part of the operation problem. The expanded problem consists of making decisions as to when, and how much, water to draft or store, and when, where, and what quantity of energy to buy or sell. The operating strategy, which provides answers to these questions, must meet the demand and respect all of the physical and operational constraints on the system. Uncertainty in the load and market factors, such as energy prices, becomes important factors to consider (Nash et al., 1999). Uncertainty in the inflow is important, as inflows affect the amount of power that can be generated. Uncertainty in the demand is important, as the firm demand must be met, regardless of cost. Uncertainties in the market prices are important as they determine the amount of energy that can be economically bought or sold on external markets. The reservoir operation problem increases in difficulty again i f the reservoir in question is not operated independently, but, rather, is one component of a system of reservoirs. A l l of the decisions just described for one reservoir must be made for each reservoir in the system. Determining the optimal operation of the system as a whole is different from determining the optimal operation of the individual components of the system. Furthermore, in most cases, the reservoir operation problem wi l l be complicated by multiple, and generally, conflicting, objectives. The difficulties that must be considered in scheduling the operation of a reservoir, or reservoir system, "well"—which is to say towards satisfying some goal or set of goals— have been part of the impetus for the development of computer models that can provide decision support in making the operating decisions. The literature reveals that such models can be descriptive simulation models, prescriptive optimization models, or some blend of the two. In addition, the literature (e.g., Perera and Codner, 1998; Turgeon and Charbonneau, 1998; Philbrick and Kitanidis, 1999) notes the relative lack of recent development of major new reservoir projects and the growing influence of a market-based approach in the hydropower industry, which together, exacerbate the pressure to operate reservoir systems as "wel l " as possible. 1 The British Columbia Hydro and Power Authority ( B C Hydro), the electrical utility serving the majority of the province of British Columbia, Canada provides a good example of an electrical system dominated by hydropower. In the day to day operations of this system, the difficulties and uncertainties in inflows, demands, prices, etc. outlined above must be addressed. B C Hydro has developed two main computer models in order to provide decision support for the system operators. The first model is a very detailed optimization model of the system on an hourly basis for the next week (Shawwash et al., 1999). Within this short "look ahead" period, loads, demands, prices, and inflows to the reservoirs are considered to be deterministic. The second model employed by B C Hydro is a long term stochastic dynamic programming (SDP) model for the main reservoir/hydro project on the Peace River, the largest and most flexible project in the B C Hydro system (Druce, 1989, 1990). The S D P model provides discounted present values for water stored in the reservoir, the slopes of which give marginal values for the value of storage. These values are taken as representative of the marginal value of energy throughout the system and are used in the short term optimization model just described, for purposes of trading off the value of energy generated in the next period against the value of the water i f it is retained and stored in a reservoir for use in another period, and as decision support for marketing decisions. The SDP model can take into account uncertainty in inflows and future prices; this is done at a monthly time step. To complement the existing modelling of a system such as that of B C Hydro, it is desirable to be able to consider more than the one large reservoir at a time (the B C Hydro system has one other major storage project in addition to that on the Peace River), as is done at present, in order to better understand the interactions between multiple reservoirs and generating plants. Further, it is desirable from an operating viewpoint to be able to include more detail, while maintaining the ability to consider uncertainty in inflows, demands, and prices in the intermediate period between the immediate short-term covered by the deterministic model and the longer term future addressed by the SDP. This thesis addresses these two needs, with the aim of filling an important gap in the modelling capability needed for decision support for the operation of a complex hydro system in today's competitive, yet uncertain environment. In this thesis, a method for optimizing the operation of a reservoir system for hydropower production, for both the long-term and intermediate-term, is developed. It is assumed that the reservoir system is to be operated to maximize the value of energy produced while taking into consideration the value of water remaining in the reservoirs at the end of the model time horizon. The model is developed for two non-hydraulically connected reservoirs that serve the same demand area. While this may seem limiting, aggregation techniques such as those developed by Turgeon and Charbonneau (1998) and Valdes et al. (1992) expand the potential range of application. Aggregation techniques can be used to form a single composite reservoir that simulates the behaviour of a set of reservoirs. 2 The method developed in this thesis is able to address uncertainty in market-related parameters such as energy prices, as well as in demands, in addition to the typically considered inflow uncertainty. The model allows the energy prices to be subdivided into several periods within the day, matching the current nature of energy markets in which energy prices vary within the day as well as across months. This ability to simultaneously consider uncertainty in energy market parameters and inflows for the operation of a multiple-reservoir hydropower system is vital, as the importance of uncertainty in the former can be of equal or greater importance than the latter ( K i m and Palmer, 1997; Russell and Campbell, 1996). Models reported in the literature typically either ignore uncertainty in energy prices, fail to consider energy price differentiation within a day, or only consider a single reservoir. These simplified models fail to address the main problem actually faced by the operator of a two-reservoir hydropower system. The proposed method involves the use of two cascaded models. A longer-term model is used to estimate the value of month end storage in the two reservoirs over a time horizon of several years, and to generate the marginal values of water stored in the reservoirs. The results produced by this model feed into a shorter-term model that is able to better deal with uncertainty in the first time step used in the longer-term model through the use of variable time-steps and scenario trees. The length of the time steps in each of the two models is variable. It is proposed that the time step used in the longer-term model be on the order of a month, and the time steps in the shorter-term model range from the order of a day for the first time step to the order of several weeks for the last time step. Through the use of the two models, marginal energy values for two reservoirs that cover a time horizon stretching out several years, but which cover the near-term in sufficient detail are developed. These marginal energy values are crucial to price-based dispatching of a hydropower system, and can be input to an hourly dispatch model. The longer-term model employs the optimization techniques dynamic programming (DP) and linear programming (LP). D P is employed to link the decisions made during one time interval with decisions during other time periods. D P deals with "stages" and "states"; here, the stages are on the order of months, and the states are vectors of the volume of water stored in the two reservoirs. A recursive equation that gives the optimal decision for a given starting state in a stage, in the form of the release from each reservoir, that maximizes the expected value of the sum of the value of the operation during the next time period and the value of being at the ending state for the next stage. In order to begin calculations, which move backwards in time, an assumption about being in any given state must be made. If this point is far enough in the future, its value is inconsequential in the results obtained by the DP . Uncertainty in the D P model is handled through the use of scenarios, which describe possible values for the scenario-dependent inflows, demands, and energy prices. The number of possible scenarios can vary with the stage in the model. Associated with each scenario is an occurrence probability. For any given state in a given stage, the future can unfold in any one of the manners described by the set of scenarios for that stage. The model finds a turbine release policy that maximizes the expected value of system 3 operation over all scenarios—that is, for a given reservoir and stage, the single best turbine release is found. Other variables, such as the ending storage for each reservoir, spill from each reservoir, imports to the system, and exports from the system can vary with the scenario. The values of these other variables are determined by the L P such that the value of system operation under the scenario subject to the policy-specified turbine release is maximized. The L P model performs the tradeoffs between releasing water and making energy trades during a stage and the value of keeping that water in storage until the next stage that are contained within the recursive equation. The value of water stored beyond the current time interval is represented in the L P by storage value curves, the slopes of which are the marginal water values. Piecewise-linearization is used to model these terminal value functions, which in general are dependent upon the month, as well as the storage in both reservoirs. As the true hydropower production problem is not strictly linear, the use of linear and piecewise linear relationships in the model requires that a series of closely related LPs be solved successively until convergence is obtained. The use of L P to evaluate the recursive D P equation departs from the traditional method. Typically, in D P the optimal transition is from a discrete state in one stage to a discrete state in the next stage. For the current problem, the discrete states would be vectors of the volume of water stored in the two reservoirs. In the formulation outlined herein, the ending storage volumes in the second stage are decision variables in the L P algorithm, and are thus not restricted to discretized state values. The information generally contained in the ending storage state vectors is contained in the piecewise-linear storage value curves. In the storage value curves, the discretized storage states are the breakpoints between linear segments. Between discretized storage volume points, the marginal value of stored water is constant. The use of L P to evaluate the D P recursive equation alleviates the problem of artificial spilling of water that can result from the transition between discretized storage volume vectors. This problem can be particularly acute i f a coarse state space discretization is used to increase the speed of a typical D P algorithm. The monthly storage value functions, and the associated marginal values, produced by the monthly D P and L P based model are input to the shorter-term model, which is based upon a technique known as stochastic linear programming with recourse (SLPR). S L P R is reported in the water resources literature as being very promising (Yeh, 1985) but has not been widely used. The application of S L P R in the shorter-term model described herein demonstrates its suitability for complicated hydropower reservoir operation problems. The shorter-term, S L P R based, model allows operations to be planned over time steps shorter than those used in the longer-term model (typically on the order of one month) and to generate short-term marginal values over each of these time steps. While the time step employed in the D P and L P based model is appropriate for some types of studies, the longer time step of necessity neglects variations in the scenario-dependent parameters within the time step. The shorter-term variable time step model provides a way in which 4 these important departures from the mean values can be taken into consideration for both the calculation of marginal values and operation of the system. The shorter-term marginal values are of particular importance when operation of the reservoir system is constrained. In S L P R uncertainty is handled through the use of a scenario tree. It is easiest to understand the idea of a scenario tree using the implied analogy. A scenario tree branches from a single root at the start of the modelled time horizon to leaves at the end of the time horizon being considered. The tree branches at locations where decisions must be made. A scenario in the tree is a direct path from the root to one of the leaves. Consider the simple case of a model with one reservoir that is to be operated for two time periods, in which the inflow in each of the time periods can assume one of two discrete values. A t the start of each time step a decision must be made as to the release from the reservoir. The scenario tree representing this simple problem has two branches growing out of the root at the start of the first period: one for the high inflow and one for the low inflow. The tree then branches again at the start of the second time period for both of the first period branches, giving four leaves at the end of time period two. There are thus four scenarios—direct paths from the root to a leaf—in the tree: high inflow in both time periods; high inflow in the first time period and low inflow in the second; low inflow in the first time period and high inflow in the second; and low inflow in both time periods. The scenario tree just described outlines all of the possible ways in which the future can unfold in the simple model. The tree structure is also instructive in illustrating how decisions are made using SLPR. As described above, each branching point, or node, in the tree represents a place at which a decision must be made. A t each node a probability is assigned to each branch, and the sum of the probabilities of all branches from a node must equal one. A t each of these nodes a decision, in this case the quantity of water to release, must be made. This decision must be made prior to knowing in which one of the possible manners the future wi l l unfold; therefore, the decision must be feasible over all possible evolutions of the future, and should be the optimal decision considering all possible outcomes. Such a decision, being made in the face of uncertainty is known as a "here-and-now" decision. Once the future has been revealed at the end of the time period, "recourse" decisions can be made based upon which of the possible futures occurred. Decisions of this latter type are referred to as "wait-and-see." The decision making approach just described results in decisions that do not anticipate the future, and can therefore be implemented. In the shorter-term, variable time step, SLPR-based model, scenarios in which reservoir . inflows, demands, and energy market prices vary are defined by a scenario tree. The here-and-now decisions at the start of each time period are either the quantities of turbine release from the two reservoirs or the ending storage volume. The link between the shorter- and longer-term models is through the storage value curves generated by the longer-term model, which are used as terminal value functions in the shorter-term model. 5 Both the longer- and shorter-term models take into consideration that the energy produced is differentiated by when it is produced, with energy being more valuable during some portions of the day. This aspect of the models allows the division of generation into on- and off-peak times. Having the model recognize different energy products is particularly important with the inclusion of external market opportunities in the model. For instance, in some cases it may be optimal during one period of time to import energy during the lower priced period in the day and to export during the higher priced period. The models presented in this thesis allow for the optimization of hydropower reservoir systems over a time horizon stretching from several hours or days in the future to years into the future. The selected approach, in which results from a monthly model feed in to a shorter-term variable time step model, allows sufficient detail to be considered such that the uncertain reservoir operating problem actually faced by the operator of a hydropower system, in which energy varies in price both within the day, between days, and between months, in which inflows and demands are also uncertain, and when generation can come from different sources, to be solved successfully. Typically, parts of the true operating problem have been ignored, either by dealing with only a single reservoir, or by not addressing uncertainty in all of the important parameters. The final outputs from the models are marginal energy values for each period modelled, with the short-term considered at a finer time resolution, as well as operating decisions for each period in the shorter-term model. It is proposed that these short-term marginal values be used as input to a very-short term, very detailed, but deterministic reservoir operation model, with a time-step of an hour or less, and a total time horizon of a week or less, such as the one actually in use by B C Hydro (Shawwash et al., 1999). The remainder of this thesis is organized in the following manner. Chapter 2 presents a review of the pertinent literature. The focus of the review of the application of systems analysis techniques to reservoir operations is on D P and LP , with particular attention given to reservoirs operated predominantly for hydropower production. Chapter 3 presents the D P and L P based model used for estimating the value of water stored behind hydropower dams over the longer-term at a monthly time step. In Chapter 4 the shorter-term, variable time-step, SLPR-based model is presented. Both chapters 3 and 4 contain case studies demonstrating the applicability and efficacy of the models developed to a hydropower system based upon two reservoirs with significant multiple-year storage from the B C Hydro system. Conclusions, recommendations for future research, and a summary of the work are presented in Chapter 5. 6 2 L i terature R e v i e w 2.1 Introduction In this chapter, the literature is reviewed for cases of the application of optimization and simulation techniques to the operation of reservoirs. The development of optimization techniques and their application to reservoir operations have been much intertwined (Yakowitz, 1982). As such, any review of the literature in this area must be limited in scope. The present review is focussed in several respects. Firstly, of the many available optimization techniques only the two that have been most widely used, dynamic programming (DP) and linear programming (LP), are considered in detail. A second restriction imposed upon the scope of the literature review is to place attention primarily on reservoir systems operated for hydropower production. The operation of even one single-purpose reservoir is not a straightforward task. In operating such a reservoir, decisions must be made as to how water should be stored, or released, in order to satisfy some objective. The major complicating factor in the operation problem arises owing to the uncertainty of natural inflows into the reservoir. That is, decisions as to how much water should be released must be made without knowing how much water wi l l flow into the reservoir as the result of naturally stochastic hydrological processes. The difficulty of the situation led one author to refer to the operation problem as "a game against nature" (Lindqvist, 1962). In addition to handling uncertainty in inflows and other important parameters, the reservoir operation problem can be complicated by multiple, often conflicting, objectives. When, as is often the case, a reservoir is part of a larger system, and so cannot be considered in isolation, the difficulty of the problem is further amplified. The desire to operate a reservoir system "well"—where the meaning of well can be taken to be satisfying some set of objectives—has resulted in the expenditure of a great deal of effort in the development of tools and techniques to aid the decision maker in making operating decisions. Further, as noted in the literature (e.g., Perera and Codner, 1998; Turgeon and Charbonneau, 1998; Philbrick and Kitanidis, 1999) the relative lack of recent development of reservoir projects and the growing influence of a market-based approach by some reservoir operators are increasing the pressure to operate reservoir systems as well as possible. For this reason, it is useful to examine the literature to shed some light upon the techniques available to aid the reservoir system operator. 2.2 Techniques As noted above, the body of previous work in which simulation and optimization techniques have been applied to reservoir operations is vast. Previous reviews of the literature have been conducted by Yeh (1985), Wurbs (1991, 1993), Reznicek and Cheng (1991), and Simonovic (1992). A literature review of D P techniques has been published 7 by Yakowitz (1982). The interested reader is referred to these earlier reviews for a more comprehensive examination of the topic. The remainder of this chapter addresses particular systems analysis techniques applied to the optimization of reservoir operations. Dynamic programming is discussed first, followed by linear programming, other optimization techniques, and, lastly, simulation. 2.2.1 Dynamic Programming Dynamic programming, introduced by Bellman (1957), is an approach to optimization that has been widely employed in the area of water resources. Wurbs (1993) notes that D P is one of the two optimization methods most widely applied to reservoir operation problems. D P is an attractive method for solving sequential decision processes in an efficient manner. The main benefits of D P are its ability to incorporate non-linear constraints and objectives, and the fact that the method can be used to handle stochastic inputs. The main limitation regarding D P is that the computational effort increases exponentially with the number of state variables in the problem, thus restricting the number of dimensions that can be considered practically; this limitation is commonly known as "the curse of dimensionality." Another drawback of D P cited by Wurbs (1993) is that as D P is not a precise problem solving algorithm, but rather an approach, commercial D P packages are not available, and each application must be custom built. Despite the above limitations, Yeh (1985) states that D P is well suited to both short-term and long-term reservoir operation optimization problems; for the latter stochastic D P (SDP) is recommended, as compared to deterministic D P for the former. Literature reviews describing the application of D P to water resources problem in addition to Yeh (1985) include Yakowitz (1982), Reznicek and Cheng (1991), Simonovic (1992), and Wurbs (1993). Applications of D P to the problem of optimizing reservoir operations are discussed below. Deterministic applications are discussed first, followed by stochastic applications. 2.2.1.1 Deterministic DP Many applications of D P to reservoir operations problems assume that all model inputs are completely known, or deterministic. Yakowitz (1982) notes that the literature contains two justifications for applying deterministic methods: (i) in particular cases, flows are well-enough known that the assumption is justifiable; and, (ii) through applying deterministic methods to known inflows, knowledge of reservoir operations can be gained that wi l l be useful in the future. Applications of deterministic D P are discussed next. Becker and Yeh (1974) employ a combination of L P and D P (LP/DP) in order to optimize the operation of a multiple-reservoir system so as to minimize the loss of stored 8 potential energy. D P is used to select the optimal policy path, and L P is used for optimization between stages. Satisfactory ranges of ending reservoir levels must be specified. In the application of the model to the California Central Valley Project, the difference between on- and off-peak pricing is included in the formulation. In comparing L P / D P with successive LP and an optimal control algorithm, Grygier and Stedinger (1985) note that Bellman's principle of optimality does not apply, meaning that a global optimum cannot be guaranteed; and, furthermore, concluded that L P / D P is dominated by the other two methods which are able to find more optimal solutions in less time. Karamouz and Houck (1982, 1987) describe a method of combining deterministic D P , regression analysis, and simulation (DPR) to obtain reservoir operating policies, noting that there is no guarantee that D P R w i l l converge to the optimal solution. Efthymoglou (1985) uses deterministic D P to determine the optimal operation of a reservoir to produce hydroelectric energy in a hydro-thermal system. The objective is to minimize the thermal fuel cost over an annual period, with a monthly time step. Water values and short-run marginal costs are derived. The state variable is stored energy, summed over all reservoirs. The composite reservoir start-of-year and end-of-year energy must be the same. With regard to the deterministic nature of the model, Efthymoglou states: "Because of the stochastic nature of basic variables of the problem (demand, capacity availabilities, water inflows), the optimal solution could be considered in practice as providing operating decisions only for the first stage." Al len and Bridgeman (1986) apply deterministic D P to three separate hydropower scheduling problems at the instantaneous, hourly, and seasonal time scales. The latter two cases are of the most interest. For the hourly load dispatch problem the objective is to meet the forecast demand using water as efficiently as possible. Storage is the state variable. The seasonal load dispatch problem considers two reservoirs, the storage in each being a state variable, and the two releases as the decision variables. D P is used to minimize the cost of imported power and energy on an annual basis. Market prices are deterministic. With regard to the deterministic framework, Al len and Bridgeman (1986), state: "These model results are optimistic because the methodology of analyses assumes complete foreknowledge of future hydrology, while a real-time system operation must at best rely on forecasted hydrology for implementation of a flexible operational strategy." Braga et al. (1991) apply both deterministic D P and S D P to optimize the operation of a multiple-reservoir system for hydropower production. The deterministic D P model is used to calculate the value of water stored in each reservoir as a function of the reservoir storage quantities and the stage. The deterministic D P model is referred to as "an off-line, one time-only" problem. 2.2.1.2 Stochastic DP In their review of the literature, Reznicek and Cheng (1991) found SDP to be one of the two stochastic techniques most often applied to reservoir operation optimization 9 problems, and further that SDP has been applied to the greatest extent in longer-term studies in which variation in the inflows must be considered. Similarly, Yeh (1985) notes that SDP is particularly suitable for long-term reservoir operation optimization problems. However, in his review, Yeh (1985) also notes that the multiple-purpose and multiple-reservoir nature of many reservoir systems has presented problems for the application of SDP, due to the "curse of dimensionality"—the well-used phrase, coined by Bellman (1957), describing the exponential increase in computation with an increasing number of states. Reznicek and Cheng (1991) assert that SDP is unable to handle more than two state variables. Yakowitz (1982) cite the upper limit as two to three dimensions. Philbrick and Kitanidis (1999) place the limit at up to seven states. This latter estimate exceeds the number of states typically reported in the literature. It is for these reasons that Yeh (1985) notes that decomposition is "essential" for the application of SDP to multiple reservoir systems, and that the application of stochastic methods has been to either single reservoir systems, or to multiple reservoir systems operated for a single purpose. Additionally, Yeh (1985) and Reznicek and Cheng (1991) report that in using SDP for multiple reservoir systems, the assumption of no cross-correlation between natural inflows into the system is typically made. Reznicek and Cheng (1991) state that, as a result, the results of such analyses must be considered "rough estimates of the real situation." Despite these limitations, SDP remains a very popular tool for including inflow uncertainty in the optimization of reservoir operations. Yakowitz (1982) traces the application of stochastic D P to reservoir operations back to Little (1955), and notes, interestingly enough, that this stochastic application preceded the first deterministic application. Applications of SDP, beginning with Little (1955), are discussed next. Little (1955) applies SDP to a single-reservoir operation problem. In addition to the reservoir, the system is comprised of a known demand and a source of thermal generation. State variables are the reservoir level and the flow during the preceding stage. The objective is to minimize the expected cost of operation; towards this end, water is considered to be free, but a known cost function is used for thermal generation. Load curtailment is also handled as a cost. Inflows are assumed to follow a lag-one Markov process, and a time step of two weeks is used. Given the time step, Little (1955) states that "the assumption of complete independence is untenable for river flow. The flow this week is an indicator of next week's flow, since a drainage basin charged with water wi l l maintain river flow for some time." The cost function is a power series of supplemental energy. Average head is used in computing hydroelectric generation. Application is made to the Grand Coulee reservoir on the Columbia River. Summer flows are high enough that the assumption is made that at some point the objective function is equal to zero. Quadratic and linear cost functions are used. Policies are obtained, and the performance of the system is simulated. A comparison of the policies thus obtained with the operation specified by a rule curve yielded a 1 % reduction in cost. In some years the rule curve performed better. Linear cost operation was found to be less smooth than quadratic cost operation. Emphasis is placed on the use of non-linear relationships. In conclusion, Little (1955) states, presciently, that "It would seem that 10 expected value methods for the storage water problem are worthy of future development and of application to other models." Stage and Larsson (1961) investigate the problem of determining the incremental cost of hydropower. Their technique, the "incremental water value method" uses SDP. Lindqvist (1962) also uses the incremental water value method to determine the operating decisions for a reservoir system so as to minimize the operating cost of a hydro-thermal system. The stochastic element in the formulation is the inflow; all other elements are deterministic. The inflows in consecutive months are assumed independent of one another. The application is to an aggregated reservoir. The efficiency is assumed to be constant. Lindqvist (1962), in discussing the extension of the model to the multiple-reservoir case, notes that theoretically the incremental water value in one reservoir would depend upon the storage level in all reservoirs. He suggests, as an approximation, that the incremental water value of a reservoir could be expressed as a function of the storage in that reservoir and the sum of the storage in the rest of the system. Gjelsvik et al. (1992) also describe the use of the incremental water value method to minimize average operating costs with consideration given to the value of water remaining in storage at the end of the model time horizon. Butcher (1971) describes the use of a stationary S D P model for the optimization of the operation of a single multi-purpose reservoir. The monthly inflows are modelled by a lag-one Markov process that is stationary from year to year. Flood control objectives are incorporated through the use of constraints that specify the maximum reservoir storage in each month such that a portion of the reservoir is available for flood reservation storage. Recreation objectives are taken into account by penalizing deviations from target levels. Water released for irrigation purposes is assumed to have a value in July through August, and all hydroelectric generation throughout the year is valued at a single price. Convergence is achieved in 30 months. Takeuchi and Moreau (1974) employ a combination of L P and SDP to minimize the expected losses from a reservoir system operated for low-flow augmentation and water supply. The losses in one reservoir are a function of the storage in all reservoirs, and are specified by convex piecewise-linear functions. The L P component is used to make operating decisions within a stage—i.e., making the tradeoffs between immediate losses and the expected value of future losses. Simulation of the policies is used to find the required conditional probabilities in an iterative process. Klemes (1977) investigates the problem of determining the necessary discretization of reservoir storage, and compares two types of storage representation schemes. In determining the required discretization, Klemes considers the probability of filling or completely drafting the reservoir. Stedinger et al. (1984) introduce the use of the current inflow (as opposed to the previous period's inflow) as a hydrologic state variable in SDP. The problem that they consider is one in which targets for release, generation, and irrigation are established, and deviations from these targets are penalized. It is found that the resulting policies, when used in 11 simulation, produce better outcomes, as measured by the objective function. Stedinger et al. (1984) also discuss the relative merits of stationary and non-stationary approaches. Pros and cons are cited for both approaches, but the main point made is that improved results can be obtained by using "better hydrologic state variables" in the analysis. Goulter and Tai (1985) consider a stationary two reservoir problem, and investigate the effect of changing the discretization of the storage state variables on the gain of the system, and the amount of time to convergence to steady state. It is found that when too few storage states are used, states at the low end of the storage range can act as "trapping states", causing the probability distributions to be unrealistically skewed. A s the discretization of the storage states is increased, the gain increases. Wang and Adams (1986) describe a two-step approach to the application of D P to the optimization of the operation of a reservoir. The method is centered on the idea of describing the time horizon by two phases. A stochastic D P model is used to describe the phase closest to the start of the model. A deterministic D P is used to find future steady state results that feed into the SDP. Karamouz and Houck (1987) compare the results obtained using S D P with a lag-one Markov process describing the inflows to a combination of D P and regression. The application is to a single reservoir that is to be operated so as to minimize the losses from a piecewise exponential loss function, which depends only on the release. The time step is monthly. SDP is only found to have better performance when the reservoir, in comparison to annual inflows, is small. In addition, the S D P policy is found to be more sensitive to the discretization of the storage state. This result would seem to agree with that of Goulter and Tai (1985). Tai and Goulter (1987) apply SDP and heuristics to solve a three-reservoir, "Y-shaped", problem. The reservoirs are considered separately, with downstream reservoirs providing targets for upstream reservoirs. In each sub-problem, the inflow and storage are the state variables. The objective is to maximize revenue, where energy prices vary with the month. Inflows are described by a lag-one Markov process. In the application, the SDP results slightly under-perform the historical operation. Druce (1989,1990) applies SDP, and an associated simulation model, to operate a storage reservoir and downstream run-of-river reservoir so as to maximize expected net export revenue. Inflows and loads are stochastic through weather sequences. The time step is monthly. There are firm and interruptible energy markets, specified by price and quantity. In the formulation, perfect foresight is assumed for the current month, but not beyond. Reservoir storage is the state variable. Marginal water values are discussed and calculated. The model was developed as electric utilities were in the transition between energy conservation policies and profit maximization policies. There is no assumption of stationarity. Inflows are independent, not lag-one Markov, and the previous inflow is not a state variable. The state variables are the reservoir level and historical weather year. The model is run for longer than necessary to dampen end effects. 12 Johannesen and Flatabo (1989) describe a set of cascaded models for planning hydropower operation. For the long-term, the S D P water value method is used. The unit commitment problem is solved using forward dynamic programming. Kelman et al. (1990) present the method known as sampling stochastic dynamic programming (SSDP). In SSDP inflow stochasticity is handled via streamflow sequences. The idea behind the method is to base the optimal decisions on joint consideration of all scenarios. The objective is to maximize benefits including a terminal value function for water remaining in storage. Application of the method is made to a one-reservoir system. Kelman et al. (1990) find that there is greater value in including hydrologic forecasts in the model when the value, as opposed to the quantity, of energy is maximized. Braga et al. (1991) apply both deterministic D P and SDP to optimize the operation of a multiple-reservoir system for hydropower production. The S D P model makes use of the water storage values calculated by the D P model in determining the optimal releases to maximize system benefits. States in the SDP model are the previous monthly inflows and starting storage. The SDP model is applied to one reservoir at a time, with iterations until convergence. Application was made to a three-reservoir system. Inflows were discretized into ten states. Convergence is reported to have generally been achieved in three or four iterations. The time step in the application is one month. Valdes et al. (1992) investigate the hydro-thermal coordination problem using an SDP model for an aggregated reservoir. State variables in the model are the energy storage and the previous inflow. The decision variable is the release; the stages are months; and the objective is to minimize the total cost of energy production. The results from this model are disaggregated at a daily time step. Several aggregation/disaggregation methods using L P are explored. In order to reduce the dimensionality of the problem, disaggregation is carried out in separate space and time steps, instead of in a single step, at the cost of sub-optimal results. It is found that it is better to disaggregate first in time, and then in space. Karamouz and Vasiliadis (1992) propose Bayesian stochastic dynamic programming (BSDP) as a means to incorporate Bayesian decision theory into SDP. The method is applied to a single-reservoir problem in which the state variables are the current inflow, forecast inflow for the next period, and the current storage. Inflows are described by a lag-one Markov process. The objective is to minimize the expected loss. The stages are months, and the decision variables are the releases. The results of the method are compared to D P and SSDP. The effect of using Bayesian decision theory is to update prior probabilities to posterior probabilities as new information becomes available; in particular, the conditional occurrence probabilities of forecasts are updated given actual flows. The results, as measured by simulating the reservoir system operation based on the developed policies, indicate that B S D P outperformed both SSDP and DP. K i m and Palmer (1997) also apply B S D P in a formulation which has one reservoir and the current storage, current inflow, and snowmelt forecast as state variables. The reservoir is to be operated so as to maximize the revenue produced by generation. The energy prices and 13 loads are deterministic. The B S D P results are compared to SDP without hydrologic state variables. It is found that there is value in incorporating hydrologic state information into the problem formulation, and that this value increases with the storage capacity of the reservoir. With regard to the deterministic demand and price, the authors conclude: "The energy-related variables, such as energy demand and price, may be as significant as the hydrologic state variables." Similarly, K i m and Palmer (1997) found that the effect of energy prices on operation increases with demand. Tejada-Guibert et al. (1993) consider a stationary, infinite horizon, problem of optimizing the operation of a two reservoir system so as to maximize the expected benefits from hydropower production. The state variables are the two storage volumes, and one variable for the current period inflow, where it is assumed that the two inflows are perfectly correlated. One question addressed by Tejada-Guibert et al. (1993) is whether the use of multi-dimensional cubic spline interpolation is superior to multi-dimensional linear interpolation in evaluating the terminal value function; the conclusion is that cubic spline interpolation is better. The other major facet of this research involves the method used for determining policies in simulating the policies. The authors found that it was better to determine the policy release decisions by reoptimizing, using the terminal value function, within the simulation, than it was to interpolate within the policy tables. The two reasons cited for reoptimizing within each period in the simulation are: (i) the optimal policy for similar system states can be quite different; and, (ii) employing the actual current state of the system ensures that all constraints associated with these starting conditions wi l l be respected. Tejada-Guibert et al. (1995) investigate the value of incorporating hydrologic information into an S D P model of a two reservoir system. The decision variables are the releases from the two reservoirs. The results of the study show that when the benefit function is to maximize energy, the choice of hydrologic state variable is of little importance. Hydrologic state variables are found to be of greater importance for objective functions that penalize deviations from targets. Russell and Campbell (1996) incorporate fuzzy logic into implicit S D P to investigate the problem of operating a reservoir so as to maximize the value of energy generated plus the value of water remaining in storage at the time horizon of the model. In the model, prices and inflows are both stochastic. The state variables are the storage, price, and inflow, and the release is the decision variable. Russell and Campbell (1996) conclude that the use of fuzzy logic does nothing to reduce the effect of the curse of dimensionality. With regard to the importance of uncertainty, Russell and Campbell (1996) conclude that foreknowledge of the prices is of greater importance than it is for the inflows. Turgeon and Charbonneau (1998) employ S D P to produce an optimal reservoir operation, and associated marginal water values, for a reservoir system that is run to maximize the expected profits. Prices are given by deterministic piecewise-linear curves. The method employs aggregation/disaggregation to solve the problem. The entire reservoir system is first aggregated into a single reservoir, and SDP is used to determine the optimal operation. Another SDP model is then solved for each river system; in these problems 14 there is one state variable for the total storage in the river system under consideration and a second state variable for the storage in the remaining river systems. Finally, the operation of each river in all o f the river systems is determined. The calculated marginal values are stated to be functions of the storage in all reservoirs. The loads used in the model are deterministic, in this regard Turgeon and Charbonneau (1998) state: "ideally the randomness of electricity demand should be taken into account." The results produced by the model cannot be guaranteed to be the global optimum. Perera and Codner (1998) investigate computational methods to increase the speed with which SDP can find solutions. The problem considered by Perera and Codner is one of water supply, where the demand is considered deterministic. The objective is to maximize the value of release. Storage and inflow are the state variables. The stages are months. Two separate computational methods are suggested. The first is to limit streamflow cross-correlations to a narrow band. The second is to use a corridor approach to eliminate consideration of "infeasible and/or inferior storage volume combinations" in solving the recursive equation based on discrete differential DP . M o et al. (1998) describe a stochastic dynamic programming/stochastic dual dynamic programming (SDP/SDDP) model used to combine the operation of a reservoir system with the management o f a portfolio of energy futures. The model includes a Markov process for spot prices. Convergence problems are experienced in the example. The model is able to calculate marginal water values. The objective of the model is to maximize expected benefits, with penalties included for revenue target shortfalls. The states in the model are reservoir storages, profit periods, trading periods, inflows, and prices. Gjelsvik et al. (1999) describe a method employing both SDP and S D D P for incorporating spot price uncertainty into the operation, over the medium-term, of a hydro-thermal system. Independent stochastic processes are used for price and inflow. The demand is deterministic. Incremental water values are obtained for each reservoir. The terminal value function is evaluated using results from a longer-term aggregated model, making the terminal value function dependent on the total storage. The authors note that this assumption means that the calculated value of water may underestimate the value of water in well-regulated reservoirs. Philbrick and Kitanidis (1999) discuss the limitations of applying deterministic techniques to the optimization of reservoir operations with stochastic inflows. Deterministic methods are stated to produce sub-optimal results unless the system under consideration is "certainty equivalent". Certainty equivalence occurs when the objective function is quadratic; system dynamics are linear; there are no inequality constraints; and uncertain inputs are independent parameters with normal distributions. Reservoir systems operated primarily for hydropower production are stated to be the reservoir operating problems that are closest to certainty equivalent, but still depart significantly from the requirements. When reservoirs are operated according to deterministically determined policies the average cost is greater (assuming minimization) than the average cost resulting from stochastically determined policies. When the actual inflow sequence 15 is very close to the assumed inflow sequence in the deterministic analysis, deterministic policies can outperform stochastic policies. Philbrick and Kitanidis (1999) assert that SDP is limited to models with up to seven state variables. Lund and Guzman (1999) in a discussion of operating rules for reservoir systems, including those in series and parallel, note that SDP can be successfully used for making operating decisions. The problems with which they are concerned involve short-term operations for water supply, water quality, and energy production. The biggest drawbacks about the application of S D P stated by Lund and Guzman (1999) are "extreme computational demands" for problems with many state variables, and the description of streamflows by explicit probabilistic methods. 2.2.1.3 DP Approximation Methods As noted above, the primary difficulty in applying D P to problems of reservoir operations is the "curse of dimensionality" that limits the number of state variables that can be included in a model. The result has been that the number of reservoirs and hydrological state variables that can be successfully modelled is quite limited. It comes as no surprise then, that a number of methods for lessening the effects of the curse of dimensionality have been developed for both deterministic and stochastic problems. These methods are discussed next. Askew (1974) describes a method known as reliability-constrained DP . This method allows the operation of a reservoir to be optimized while taking into consideration the maximization of the discounted net benefits as well as a limit on the number of system failures over the model horizon. The ability to place restrictions on the number of system failures arises from the desire to be able to handle "noneconomic aversion to failure." Chance-constraints are incorporated through the use of a penalty term in the objective function that takes system failures into account; the value of this penalty term is found through an iterative process. Inflows are described by probability density functions, and are considered independent between periods. Heidari et al. (1971) describe a method known as discrete differential D P (DDDP) , which is a generalization of state increment D P , introduced by Larson (1968). D D D P is stated to have been developed to ease the computational burden (computer memory and computer time) associated with traditional DP. D D D P is an iterative technique requiring an initially feasible reservoir trajectory. D P is then applied to a corridor surrounding the trial solution. The improved solution, i f found, becomes the trial solution for the next iteration. The method continues until convergence. D D D P is found to be most effective for cases where the order of the state vector is equal to the order of the decision vector. Young (1967), in what Yakowitz (1982) notes as the first application of deterministic D P to reservoir operations, describes the use of a combination of D D D P and regression analysis used to produce reservoir operation rules. In the study, the reservoir is to be operated so as to minimize the losses given by a loss function. The operating rules are found by applying regression to the results produced by the DP . 16 Gal (1979) describes the "parameters iteration method," which is a technique for approximating the optimal policy in the stochastic case. The idea behind the method is to assume that the recursive functions depend on the vector of state variables as well as a vector of parameters. Solving for the parameters allows the approximation of the optimal policy. The method is tested for a three-state problem, and is found to produce results comparable to those found by DP. Application of the method is then made to a water supply system, having several reservoir volumes and previous inflows as state variables. Turgeon (1980) describes two possible approaches for dividing a computationally infeasible stochastic multi-reservoir weekly operation problem into smaller computationally feasible pieces. The first of these methods, termed the "one-at-a-time" method, breaks the original multi-state problem into a number of single state problems that can be solved by SDP. The second method, termed the "aggregation/decomposition" method, breaks the original multi-state problem into a number of problems with two state variables: one for the storage in the reservoir under consideration, and a second for the storage in the remainder of the system. Neither of the two models is able to guarantee a global optimum. The two techniques are applied to a system comprised of six reservoirs in parallel. The aggregation/decomposition method is found to obtain a more optimal operation policy. Turgeon (1981) presents a method for decomposing the reservoir operation problem of n reservoirs in series into n-1 S D P problems of two state variables that can be solved by SDP. The two state variables are the storage in the reservoir under consideration and the amount of stored potential energy in the remainder of the system. The method is demonstrated on a system of four reservoirs in series. Trezos and Yeh (1987) describe an analytical method that can be applied to D P problems under a limiting set of assumptions. For appropriate problems, the method provides an analytical solution, thus eliminating the computational problem associated with the curse of dimensionality. In order to apply the method the problem must have a quadratic objective function, linear system dynamics, and inflows that can be fully described using the first and second moments of the probability distributions. The process is iterative, requiring a feasible initial solution. The solutions at each iteration are obtained by quadratic programming. The method can be thought of as an extension of differential D P (DDP) (Jacobson and Mayne, 1970) to the case of stochastic inflows. Constraints in the model apply to the expected values of variables. 2.2.2 Linear Programming Next to D P , linear programming (LP) is the optimization technique that has been applied most often to reservoir operations problems (Yeh, 1985; Wurbs, 1993; Simonovic, 1992). The application of L P to the water resources field is traced back to the early 1960s (Simonovic, 1992). Yeh (1985) reviews the "state-of-the-art" in L P models; his examination includes stochastic L P models, stochastic programming with recourse, 17 chance-constrained L P , and linear decision rules. Based on the review, it is Yeh's conclusion that L P is a useful tool for the optimization of reservoir operations. In particular he cites the fact that, through the use of linearization techniques (e.g., piecewise-linearization and Taylor series expansion), L P can be used to successfully model non-linear constraints and objectives. Additional reasons for the popularity of L P cited by Yeh (1985) and Wurbs (1993) are: (i) it is a well-defined and easily-understood technique; (ii) the ability with which problems of relatively large dimension, as compared to other methods, can be solved; (iii) global optima are obtained; (iv) an initially feasible trial policy is not required; and, (v) commercial programs are widely available, so that the method does not need to be developed from scratch for each application The main disadvantage of LP , as stated by Wurbs (1993) is that the modelled problem must be strictly linear, thus, generally making it necessary to approximate the physical problem through linearization methods. A n additional limitation of the method, cited by Yeh (1985) is that in some cases, for very large reservoir systems, decomposition techniques may still be required. One of Yeh's (1985) three recommended areas of future research with respect to the use of L P in the field of the optimization of reservoir operations is the development and implementation of decomposition techniques. Linear programming is a broad field, within which many distinct branches have developed in addition to "traditional" L P . Typical applications of L P techniques to the optimization of reservoir operations problems selected from the literature are presented next. The previous work described is only presented as being representative of the field, not as an exhaustive review. The interested reader is referred to Yeh (1985), Wurbs (1993), Reznicek and Cheng (1991), and Simonovic (1992) for further examples of the use of L P in the water resources field. Linear programming methods have been developed for addressing problems in which model parameters are uncertain, as well as models where all parameters are deterministic. A s noted above, applications of L P in water resources date back to the 1960s. The focus in this section is on some of the more recently reported studies. Deterministic applications of L P are discussed first. Becker and Yeh (1974) describe the use of a combination of L P and D P to optimize a reservoir system for energy production. The system is aggregated into an equivalent reservoir, with storage described in terms of energy. The objective is to minimize the loss of potential energy, L P is used to generate possible feasible stage transitions, and D P is used to choose the optimal transition. A combination of LP and D P is also used by Takeuchi and Moreau (1974) to optimize the operation of a reservoir system for irrigation low flow augmentation and water supply. L P is used to compute immediate economic losses, and the stages are linked by DP . Grygier and Stedinger (1985) revisited the combination of L P and D P used by Becker and Yeh (1974) and concluded that the method was dominated—that is a solution with a higher objective function could be found in less time—by successive L P . Grygier and Stedinger (1985) also include an optimal control algorithm ( O C A ) in their comparison of 18 methods. The conclusion that they reach is that both successive L P and the O C A reach the global optimum of the problem considered. The advantage of L P is that it is fast to implement, owing to the existence of commercial L P codes; on the other hand, the O C A method is slower to implement, but finds results in less time. Martin (1986) applies successive L P , in a deterministic context, to maximize the benefits of operating a multipurpose reservoir system on a daily time step. The decision variables in the model are the reservoir releases. The objective function is to minimize the penalties associated with deviations from targets. Non-linear terms are approximated by first-order Taylor series approximations and bounds are placed on the decision variables to ensure that the approximation remains valid. The algorithm is applied successively until convergence. In order to improve execution time, separate models are solved for each reservoir, and the results coupled; the expense of this approach is not having a guarantee of achieving a global optimum. Tao and Lennox (1991) apply deterministic successive L P to optimize the operation of the High Aswan Dam. Barritt-Flatt and Cormie (1989) describe the use of deterministic L P to solve the problem of determining the optimal operation of reservoirs in a hydro-thermal electrical system so as to maximize net revenues, including the value of water in storage at the end of the model time horizon. Non-linear functions, such as those describing the relationship between head, generation, and discharge, are replaced with piecewise-linear functions. Piekutowski et al. (1994) employ deterministic L P to schedule a reservoir system over the short-term so as to minimize the value of energy used, both as turbine discharge and spill, in meeting demand over the course of the study period. Incremental costs for each unit, as well as for the entire system, are produced. The problem is solved using a commercial L P package. Christoforidis et al. (1995) use mixed integer linear programming to optimize the operation of a reservoir system, including energy trades, over the short-term so as to minimize the sum of energy costs and penalties for the violation of soft constraints. The energy generation functions of the power stations are described by piecewise-linear functions. The integer variables are used to describe the start-up and shut-down of units. Lund and Guzman (1999), in a discussion of operating rules for reservoir systems, note that L P can be used for making operating decisions. The problems with which they are concerned involve short-term operations for water supply, water quality, and energy production. Lund and Guzman (1999) state that one advantage that L P has over simply applying "rules of thumb" is that it can be used to incorporate operational constraints into the analysis, with the obvious limitation that the constraints must be linear, or piecewise-linear. The optimization of the operation of reservoir systems is dependent on a number of parameters that are inherently uncertain. The most readily apparent sources of uncertainty are reservoir inflows. In addition, when a reservoir system is to be operated for the generation of hydropower, energy demands and market prices for electricity are 19 also uncertain. A number of techniques have been developed that allow the analyst to account for the incorporation of these sources of uncertainty with an L P formulation. The use of mean values of uncertain parameters such as inflows generally results in overly optimistic policies—overestimated system benefits or underestimated system costs—that result from the failure to properly consider large-impact, low-probability, events (Reznicek and Cheng, 1991; Philbrick and Kitanidis, 1999). The exceptions to the above rule are those systems that are "certainty equivalent"; that is, for certainty equivalent systems, the results obtained using a deterministic model with the values of uncertain parameters taken at their mean values are equal to the results for a full stochastic examination. Certainty equivalence occurs for systems exhibiting the following properties: (i) the objective function is quadratic; (ii) the system dynamics are linear; (iii) there are no inequality constraints; and, (iv) the uncertain parameters are independent and normally distributed. It is because no reservoir system operation problems satisfy these requirements that stochastic techniques have been developed and employed (Philbrick and Kitanidis, 1999). Linear programming approaches that take uncertainty into account include chance-constrained LP ( C C L P ) , stochastic L P (SLP) , stochastic LP with recourse (SLPR), and the combined use of L P and S D P (LP/SDP) , all of which have been applied to reservoir operation optimization problems (Yeh, 1985; Reznicek and Cheng, 1991; Simonovic, 1992). O f the above methods, Reznicek and Cheng (1991) report that C C L P is the method that has been used most often. The main benefit and main drawback of C C L P , reported by Reznicek and Cheng (1991), respectively, are the lack of dimensionality problems as compared with SDP, and the inability to assess the impact of failure of the probabilistic constraints. O f the three areas of recommended future research in applying L P to the optimization of reservoir operations reported by Yeh (1985), two deal with uncertainty: addressing the dimensionality problem of stochastic L P ; and, continuing the development and application of stochastic programming with recourse. One manner of addressing uncertainty in model parameters, such as inflows, is to include constraints that are considered to be satisfied i f they are only violated a certain percentage of the time. In chance-constrained programming (CCP) , model constraints are specified along with the probability with which they must hold—or, conversely, the probability with which failure of a constraint is acceptable. Assuming that the cumulative density functions of the uncertain parameters are known, the probabilistic constraints can be written in deterministic form. If the remainder of the model only contains linear constraints and objectives, then the model can be solved using L P ; in such cases the technique is known as chance-constrained linear programming ( C C L P ) . The main drawback of C C L P outlined in Yeh (1985) is that violations of constraints are not explicitly penalized, nor can corrective actions to address the violations be taken. Finding the necessary data to specify the acceptable level with which constraints can be exceeded is another issue in employing C C P (Reznicek and Cheng, 1991). Despite these limitations, C C L P has been successfully applied to the optimization of reservoir operations. Reznicek and Cheng (1991) note that C C L P is one of the two most widely used techniques for including inflow uncertainty into the optimization of reservoir operations. 20 The use of C C L P in reservoir system optimization is traced back to ReVelle et al. (1969) by Reznicek and Cheng (1991). Datta and Houck (1984) employ chance constrained programming with linear decision variables to optimize the operation of a reservoir so as to minimize the expected deviations from storage and release targets. Inflows are described by conditional distribution functions relating the actual inflow to the forecast inflow. Time horizons that can be considered by the model range from several days to a month. Bhaskar and Whitlatch (1987) report the use of C C L P to generate monthly release policies for the operating of a multipurpose reservoir. In the same study, a combination of D P and regression is also used to determine operating rules. The two sets of resulting policies are compared using a simulation model. For the case studied, Bhaskar and Whitlatch (1987) find that the combination of D P and regression produced better policies, in that the average annual loss was lower while reliability levels were higher, than did the C C L P model. Another approach that has been taken for dealing with input parameter uncertainty in L P is the use of scenarios. A scenario describes the values assumed by each of the uncertain parameters in each time step. L P methods have been developed that describe uncertainty using a finite number of scenarios, each having a discrete probability of occurrence. A stochastic L P (SLP) model for optimizing the operation of a single reservoir is described by Loucks (1968). Inflows in the model are described by a lag-one Markov process. The objective function is to minimize a loss function that penalizes the deviation of the storage and release from targets. The model is formulated such that the decision variables are the joint probabilities of given starting reservoir volumes, inflows, and releases. Loucks (1968) notes that the formulation employed results in very large problems, as the problem size grows with the product of the discretized volumes, inflows, and times. One technique that utilizes scenarios to describe uncertainty in model parameters is stochastic linear programming with recourse (SLPR). S L P R is a method that appears suitable for adding consideration of uncertainty to the L P framework in a manner that retains the positive features of LP . In fact, one of the conclusions reached by Yeh (1985) based on his review of the state-of-the-art of the application of L P techniques to water resources is that one of the areas in which future research should be directed is the continued development and application of S L P R . Reznicek and Cheng (1991) also discuss the potential of S L P R for incorporating uncertainty into the L P framework. Grygier and Stedinger (1985) recommend the "multiple futures" method, as described by Pereira and Pinto in an unpublished manuscript from 1984, as a promising means for incorporating inflow uncertainty into L P by describing future inflows using a decision tree. Based on the description, the multiple futures method appears to be equivalent to S L P R . A simplified version of S L P R , referred to as "two-stage linear programming" is briefly described by Loucks et al. (1981) for application to allocating flows from a river to different uses. Gjelsvik et al. (1992) apply two-stage stochastic programming to the problem of operating a reservoir system so as to minimize operating costs when the value of water at the end of the model time horizon is included in the analysis. The application 21 of Gjelsvik et al. (1992) employs Benders' method (Benders, 1962) to piece together sub-problems. The authors state: "The two-stage stochastic programming approach has the drawback that each scenario is considered too optimistically since the future inflows are not really known." With S L P R , decisions are divided into those that must be made at a given point in time, in the face of uncertainty—termed "here-and-now" decisions—and those that can be made in the future, after the values of the unknown parameters are known—termed "wait-and-see" decisions. The here-and-now decisions must take into account the fact that the uncertain parameters can assume a number of different possible values, and that for each one of these possible outcomes an optimal recourse wait-and-see decision wi l l be made; this is done by basing the decision on the expected value over all of the possible outcomes. Constraints must be included in the model that enforce the fact that at each node, nothing is known about how the future wi l l unfold. Failure to include these "non-anticipative" constraints is equivalent to assuming perfect foresight of the future. When uncertainty is described by a scenario tree, a here-and-now decision must be made at each node in the tree. In a scenario tree, a scenario describes the outcome of the uncertain parameters at each branch in the tree from the root to a leaf; where a branch is a point at which decisions must be made, the root is the point in time when the first decision must be made, and a leaf is the point in time when all uncertainty has been revealed. Assuming that the problem for which the scenario tree represents uncertainty is completely linear, there is one linear programming problem associated with each scenario. The non-anticipative constraints link the individual linear programs for the individual scenarios into one large L P problem that can be solved as a typical L P problem. Alternatively, some type of aggregation/disaggregation method can be used. Yeh (1985) and Reznicek and Cheng (1991) note that applying S L P R can be difficult, as the size of the equivalent deterministic problem including all o f the scenarios can be quite large. Dupacova (1980) discusses the application of stochastic programming with recourse to water resources systems, noting in particular that the method can be used as a means of handling uncertain inflows. Dupacova states that while, from the viewpoint of the user, chance-constrained models may appear to be more attractive, their failure to consider the dependencies between the chance constraints makes it necessary to consider the use of alternative methods such as SLPR. In Dupacova (1980) the problem of operating a reservoir so as to minimize the penalties, described by piecewise-linear penalty functions, arising from deviating from storage targets is formulated, but not solved. Fleten and Wallace (1998) employ S L P R to jointly address the problems of optimal reservoir operations for an aggregated reservoir operated for hydropower generation and risk management, through the use of financial instruments, of the generated revenue. Risk aversion is included in the model through the use of revenue targets and piecewise-linear penalties for failure to achieve these targets. The objective is to maximize the sum of the revenue less the penalty costs. The scenarios employed in the model are used to describe the uncertain inflows as well as the uncertain prices for electricity and the 22 financial instruments. The deterministic equivalent of the problem described by the scenario tree is solved. A subset of linear programming problems can be termed "network flow" problems. Network flow problems can be described in terms of "nodes" and "arcs", where the arcs join nodes. In order to have a pure network programming problem, the only decision variables in the problem must be the flows along the arcs, and the only allowable constraints specify continuity of flow at the nodes. For problems that can be described by such a formulation, commercial L P solvers have algorithms that are able to find solutions in much less time than for more general L P formulations. Yeh (1985) notes that some water resources optimization problems have structures that allow network algorithms to be successfully applied. Wurbs' (1993) review of simulation and optimization models includes a section on network flow programming. It is noted by Wurbs (1993) that in some ways, a network flow model can be thought of as straddling the line between simulation and optimization. Given the speed of the technique, single-period or multiple-period, multiple-reservoir problems can be quickly solved. As with other L P methods, the network formulation allows for piecewise-linear penalty functions and objectives. In order to handle non-linearities, such as those associated with hydropower generation, Wurbs (1993) suggests the use of successive iterative algorithms. A recent example of the application of a network formulation is described in Christoforidis et al. (1996), who address the problem of optimally scheduling a system of reservoirs for hydropower production over the medium- to long-term. The objective of the optimization is to minimize the sum of energy transaction costs and penalties for the violation of soft constraints. The time horizon in the model is divided into a deterministic stage and a stochastic stage. In the stochastic period the inflows are described by scenarios that have discrete occurrence probabilities. In the example problem, four scenarios are used to describe possible evolutions of the future. The model is solved using the interior point method, which is an alternative to the well-known simplex algorithm. Christoforidis et al. (1996) conclude that very large problems with similar structure can be solved using the interior point method. 2.2.3 Other Methods Although dynamic programming and linear programming are the two optimization techniques employed most often in the optimization of reservoir operations, the use of other methods, including analytical techniques, genetic algorithms, goal programming, non-linear programming, optimal control algorithms, reliability programming, and stochastic dual dynamic programming have been reported in the literature. A selection of applications of the alternative methods is reviewed in this section. The techniques discussed, as well as the individual papers cited, are intended to be representative of those found in the literature. The review is not intended to be exhaustive. 23 2.2.3.1 Analytical Techniques Included in the reservoir operation literature are examples of applying analytical solutions to optimization problems. Examples of the use of such techniques are presented in this section. Gessford and Karlin (1958) use induction to obtain an optimal policy for the operation of a reservoir in a hydro-thermal system. The reservoir is to be operated so as to minimize the expected cost of supplying energy, where hydropower is considered to be free and the marginal cost of thermal power is non-decreasing, and inflows are described by a known probability function. In order to obtain a solution, it must be assumed that the reservoir has an infinite capacity, the turbine discharge has no upper bound, and head effects are not important. Gl imn and Kirchmayer (1958) employ numerical integration of non-linear differential equations, obtained using calculus, to optimize the operation of hydroelectric plants with variable head. In the demonstration problem it is necessary to assume that the reservoir is a vertical-sided tank, and that the relationship between discharge and generation can be described as the product of a constant, a quadratic function of net head alone, and a quadratic function of generation alone. Morel-Seytoux (1999) uses calculus to calculate marginal water values, as continuous functions, for the case in which a multiple-reservoir system is operated so as to minimize costs, or maximize benefits. The problem is formulated in continuous time, implying that input data, such as inflows, can be described as continuous functions. The model is limited to a deterministic description of inflows. The advantage in obtaining an analytical solution is that do so is fast. However, in order for analytical techniques to be applicable, the problem under consideration must meet stringent limiting properties. Typically, the required properties severely inhibit the applicability of the models. It is for this reason that analytical techniques have seen limited use for the optimization of reservoir operations. 2.2.3.2 Decomposition Methods In recent years, several techniques have been proposed in the literature for addressing the very large problems that can result when stochastic parameters, such as inflow, demand, and energy price, are added to the solution of reservoir operation optimization problems. Typically, these techniques use decomposition to break the larger problem into smaller, more manageable pieces, the solutions of which are then reassembled to provide an approximation of the optimal solution to the underlying stochastic problem. Note that such methods cannot guarantee optimal solutions. Two recently discussed methods, stochastic dual dynamic programming (Pereira, 1989; Pereira and Pinto, 1991) and Dantzig and Infanger's (1997) combination of importance sampling and Benders' 24 decomposition (Benders, 1962) are discussed briefly in order to provide an overview of such methods. Pereira (1989) and Pereira and Pinto (1991) introduce the method known as stochastic dual dynamic programming (SDDP). S D D P was developed as a method for solving stochastic multiple-reservoir, multiple-stage, operation optimization problems, and represents a means by which stochasticity can be introduced into a problem without discretizing the state space. Since the state space is not discretized, S D D P can be used to solve problems with a large number of decision variables. A scenario tree representing all of the possible combinations of the random variables through time is used to model stochasticity. The idea behind the method is to approximate the "cost-to-go" function with a piecewise-linear function obtained from dual solutions of the single-stage sub-problems. The approximation "may be interpreted as Benders' cuts in a stochastic, multistage decomposition algorithm" (Pereira, 1989). Monte Carlo methods are used to select a subset of the scenarios on which to carry out forward simulations. The results of these simulations feed back into the cost-to-go functions, and optimization and simulation are run in sequence until convergence is achieved. Pereira (1989) and Pereira and Pinto (1991) apply S D D P to the problem of hydro-thermal coordination of the Brazilian power system. Gorenstin et al. (1992) extend the work of Pereira (1989) and Pereira and Pinto (1991) by including the transmission network and an optimal power flow (OPF) model in the hydro-thermal coordination problem. Gorenstin et al. (1992) note that the use of S D D P allows the calculation of the expected marginal costs of various components of the model. Rotting and Gjelsvik (1992) apply S D D P to the problem of the seasonal scheduling of a reservoir system operated for hydropower production. The system is to be operated so as to minimize the thermal operating costs while taking the terminal value of water in storage into account. The inflows form the stochastic part of the model. The paper extends S D D P , as described by Pereira (1989) and Pereira and Pinto (1991), to some extent by using relaxation to solve the sub-problems. Gjelsvik et al. (1992) also apply S D D P to minimize the average reservoir operating costs, with consideration given to the value of water remaining in storage at the end of the model time horizon. Gjelsvik et al. (1999) describe a method employing both SDP and S D D P for incorporating spot price uncertainty into the operation, over the medium-term, of a hydro-thermal system. Some implementations of S D D P have exhibited numerical problems, such as failure to converge to a reasonable solution (e.g., M o e t a l . 1998). Dantzig and Infanger (1997) apply importance sampling and Benders' decomposition (Benders, 1962) for the optimization, and intelligent control, of a reservoir system that is to be operated so as to minimize expected costs. In the method, as in S D D P , a scenario tree is used to describe the uncertain inflows; errors in the control instrument readings are also described by the scenario tree. The time horizon of several hours is discretized into ten-minute increments, leading to a very large multi-stage decision problem. Importance sampling is used to evaluate a subset of all of the scenarios in the scenario tree, with the subset being chosen in the areas that are predicted to have the greatest effect on the objective function. 25 2.2.3.3 Genetic Algorithms Genetic algorithms (GA) are an optimization technique based on the theory of natural selection. The idea behind the method is that a potential solution to a problem can be described by numerical "chromosomes", and that these chromosomes determine the "fitness", or objective function value, of the possible solution. Functions simulating the chance of survival, reproduction, and mutation of "individuals" are then applied to a "population", or possible solutions, over a number of "generations", or iterations. At the end of the iterations, the chromosomes of the individual with the highest fitness describe the optimal values of the decision variables. Wardlaw and Sharif (1999) apply G A s to determine the optimal operation of a multiple-reservoir system for a deterministic problem of finite length. The objective of operating the reservoir system is to maximize system benefits less penalties for constraint violations. Penalties must be included in the objective function because in the G A implementation employed, the values of the decision variables generated are not checked for feasibility elsewhere; in alternative G A formulations only feasible sets of decision variables are generated. The results generated by the G A model for a problem with four release decision variables at twelve points in time, for a total of 48 decision variables, are compared with D D D P , and with LP . The G A model was found to be faster than D D D P , but slower than L P . The largest problem that Wardlaw and Sharif (1999) report solving has a total of roughly 400 decision variables. A drawback of the method is the need to calibrate the parameters that describe the operators on the chromosomes. A combination of G A s and Monte Carlo analysis is applied by Otero et al. (1995) to a multiple-reservoir system. 2.2.3.4 Goal Programming Goal programming (GP) is a method that can be applied to multi-objective problems in which it is possible to establish a firm order in which the objectives are to be accomplished. Goals for higher-ranking objectives are fulfilled completely before lower ranked objectives are addressed. Reznicek and Simonovic (1991) employ G P to optimize the operation of a reservoir so as to minimize a loss function based on storage and release targets. The authors state that the advantages of G P over an L P model in which conflicting objectives are weighed against one another are the lesser data requirements and simpler model formulation. Mohan and Keskar (1991) use goal programming to optimize the monthly operation of an irrigation supply reservoir so as to meet storage and release targets. The conclusion reached was that, at least for the system studied, release targets are superior to storage targets for determining the policies for the operation of a multi-purpose reservoir system. A n obvious limitation on GP is that most reservoir operation problems are complex enough that it is not possible to clearly articulate a set of ranked objectives with corresponding target values. 26 2.2.3.5 Non-linear Programming Yeh (1985) discusses the use of non-linear programming (NLP) in his review, noting in particular the use of conjugate gradient and Lagrangian gradient methods. Simonovic (1992) includes quadratic programming, geometric programming, and separable programming as N L P techniques that have been applied. Impediments cited by Yeh (1985) and Simonovic (1992) to more widespread use of N L P methods are convergence problems, computer requirements, non-viability for non-separable problems, and the fact that they are difficult to apply to problems involving stochastic inflows. Reasons to use N L P include the ability to handle non-linear constraints and objectives, and the ability with which they can be linked to complex simulation models (Wurbs, 1993; Yeh, 1985). Recent applications of N L P for the optimization of reservoir operations are by Tejada-Guibert et al. (1990) and Syalla (1994). Syalla (1994) uses a sub-gradient N L P technique for the optimization of a multiple-reservoir system that is to be operated so as to minimize the power deficit relative to a target, and to have this deficit be uniform through time. The model is deterministic, and employs a node-arc formulation. The benefit of directly including non-linear terms in the model is stressed. Syalla (1994) notes that while the model employed is reliable, the computational burden for larger problems is "a major drawback." Tejada-Guibert et al. (1990) employ M I N O S (Murtagh and Saunders, 1983) to operate a reservoir system so as to maximize the value, as opposed to the amount, of energy generated. The value of energy depends upon the time of year and time of day. A s with other N L P methods, the optimal solution cannot be guaranteed to be the global optimum, and depends upon the initial estimates of the decision variables. 2.2.3.6 Optimal Control Algorithms The use of optimal control algorithms ( O C A ) to optimize the operation of reservoir operations has been reported in the literature. Grygier and Stedinger (1985) compare the performance of an O C A to a combination of L P and D P and to successive L P for optimizing the operation of a multiple-reservoir system, with a deterministic description of the inflows, over the medium-term. The objective is to maximize the value of power generated, taking into account the value of water in storage at the end of the model time horizon. The O C A cannot guarantee an optimal solution. Grygier and Stedinger (1985) find that, at least for simple systems, the O C A is significantly faster than successive L P , but requires much more time to implement. Georgakakos et al. (1997) apply an O C A to optimize the operation of a reservoir system that is part of a combined hydro-thermal system. The objective function includes terms for thermal cost savings, dependable capacity, as well as storage bounds and targets. The time step of the model is hourly, and the time horizon is up to several years; Georgakakos 27 et al. (1997) report that the method is computationally efficient for such problems. The framework of the model is deterministic. The particular O C A employed, the Extended Linear Quadratic Gaussian control method, cannot guarantee a global optimum, and is dependent on an initial feasible solution. Hayes et al. (1998) apply O C A to the optimal operation of a sub-system of the Tennessee Valley Authority hydroelectric system under water quality constraints. 2.2.3.7 Reliability Programming Reliability programming can be described as an extension of chance-constrained programming (Reznicek and Cheng, 1991). The extension is that in reliability programming the probability with which constraints must be respected are decision variables, whereas in chance-constrained programming (CCP) they are input parameters. A s a result of the extension, reliability programming contains non-linear relationships. Just as with C C P , reliability programming requires knowledge of the risk-loss function, which can be challenging to obtain (Reznicek and Cheng, 1991). Srinivasan and Simonovic (1994) describe the application of reliability programming to a composite reservoir operated for multiple purposes, including hydropower production, placing emphasis on the incorporation of hydropower production into the reliability programming formulation. The reservoir is to be operated so as to maximize the value of hydropower produced less the losses described by a risk-loss function, arising from failing to meet required reliabilities for energy production and flood control. In the piecewise-linear model of the hydropower production function it is assumed that production is independent of the tailwater. In finding a solution, the algorithm cycles through non-linear search to find the probability limits and linear programming to solve the C C P . Rangarajan et al. (1999) also apply reliability programming to the problem of operating a reservoir for hydropower production while taking into account the losses arising from failure to meet the required load and provide the required flood control. The emphasis in Rangarajan et al. (1999) is placed on the determination of the risk-loss functions, and the sensitivity of the optimal model results to these functions. 2.2.4 Simulation Although optimization models have received a great deal of attention in the research literature, in particular in the academic community, the institutions responsible for actual reservoir operations have been slow to use optimization, relying instead on simulation models (Yeh, 1985; Wurbs, 1993). Simulation models include widely available, general purpose models such as H E C - 3 (Hydrologic Engineering Center, 1971) and HEC-5 (Hydrologic Engineering Center, 1979); custom-built project specific models developed in programming languages such as F O R T R A N and C; and models constructed using general purpose software such as Excel, or modelling packages such as S T E L L A (Wurbs, 1993). Other simulation models employed by operating agencies such as the U.S. Army 28 Corps of Engineers include S S A R R (U.S. Army Corps of Engineers, 1975), H Y S S R (U.S. Army Corps of Engineers, 1985), and the Acres model (Sigvaldason, 1976; Bridgeman et al., 1989). Reasons cited for the preference of decision makers for simulation models over optimization models include: (i) the non-involvement of operators in optimization model development; (ii) the simplifications and abstractions required to apply optimization techniques to an actual system; (iii) the generally poor documentation of optimization models reported in the literature; and, (iv) the institutional constraints that impede reservoir operator-research interactions (Yeh, 1985; Simonovic, 1992; Wurbs, 1993; Russell and Campbell, 1996). The line between simulation and optimization models can be a blurry one, in that some simulation models incorporate optimization techniques, and optimization model results are often interpreted through simulation (Yeh, 1985; Wurbs, 1993). The definition adopted here is that a simulation model can be thought of as a model having a primarily descriptive, as opposed to prescriptive, purpose—that is, the model is not to make recommendations on how a system should be operated, but is to model how the system would respond to specific operating rules. It is this difference in focus that allows simulation models to incoiporate details that are of necessity ignored in optimization models, making them important for verifying the results of optimization (Yeh, 1985; Simonovic, 1992). A s this thesis is focused upon the application of optimization methods, the literature on simulation models, both project-specific and general, is not reviewed here. The interested reader is referred to the reviews of Wurbs (1993), Yeh (1985), and Simonovic (1992) for an in-depth treatment of this topic. 2.3 Summary and Conc lus ions In this chapter the literature has been reviewed for applications of optimization techniques to the problem of reservoir operation optimization problems. Simulation has also been discussed briefly. The review of the literature reveals that the two most widely used optimization techniques are dynamic programming and linear programming. A number of other methods have been employed to a lesser extent. The major drawback regarding D P is the exponential increase in computational effort associated with an increase in the number of state variables. Despite this limitation, S D P is one of the two most often applied techniques for the consideration of uncertainty in reservoir operation problems. One of the main advantages of D P is its ability to easily handle non-linearities. Linear programming has also been widely applied, at least partly due to the widespread availability of L P solvers. The major drawback in the application of L P is the need for the problem to be completely linear; the use of linearization techniques can make this drawback less severe than it may initially appear. A number of techniques have been developed for incorporating uncertainty into L P formulations. One technique that shows 29 much promise, and has been recommended as a future research area by Yeh (1985), is stochastic linear programming with recourse. From the review of the literature it can be concluded that D P and L P , particularly S L P R , remain very useful techniques for the optimization of reservoir operations. In the remainder of this thesis the use of D P and S L P R models for the optimization of a two-reservoir system operated to maximize the value of hydropower generated, while considering the value of water remaining in storage, is described. 30 3 Stochastic Dynamic Programming and Linear Programming Based Model 3.1 Introduction The operator of a hydroelectric system with significant storage is faced with the problem of trading off the release of water to generate power at any given point in time with storing that water for potential release at a later date. In today's environment of active markets for electricity trade, a comparison of the marginal value of the water in storage and the value of releasing water can be used in making the tradeoffs. In general, the marginal value of stored water in a reservoir is dependent upon the time of year, the volume stored in the reservoir, and the volumes stored in all other reservoirs in the system. For example, the market electricity price for the current hour may be $50/MWh. A t this price, the operator may want to consider generating electricity for export. Taking into account transmission losses and wheeling charges, the.net price for exports that the operator would realize would be less, say $45/M Wh. This energy price can be converted into a water price using the conversion rate between power production and turbine discharge. Supposing that the appropriate conversion rate is 1.5 MW/cms , the water price would be $30/cmsh, or $720/cmsd. The current export price for water could then be compared against the marginal value of water in storage. If the marginal value of water in storage exceeds the export water price, then a sale should not be made. On the other hand, i f the export water price exceeds the marginal value of water in storage, then a sale could be profitably made. Similarly, a comparison could be made between an import water price and the marginal value of water in storage. If the import water price is less than the marginal value of water in storage, then the import could be profitably made. Conversely, i f the import water price is greater than the marginal value of water in storage, then the purchase should not be made. Another aspect involved in making the tradeoffs is weighing a certain price today against an uncertain marginal value in the future. The selected discount rate should reflect this fact. Uncertainty in the future marginal value arises because the estimate relies upon assumptions about future demands, inflows, and prices—all of which are inherently uncertain. For example, i f inflows significantly exceed the forecast values, it may be impossible to operate a reservoir such that it does not spill. When a reservoir is spilling, the marginal value of water w i l l be zero (and could actually be negative i f spill causes damage) which wi l l , in general, be less than the marginal value for non-spill conditions. A model has been developed to estimate the marginal values of stored water over the medium- to long-term in order to aid the decision-maker in making the operating decisions discussed above. The purpose of the model is to generate storage water value curves (the slopes of which are the marginal water values) that can be used to value the water remaining in storage for models with a shorter time horizon. The model combines the use of backward-moving dynamic programming (DP) to link time periods together, 31 and the use of linear programming (LP) to determine the operation of the hydropower system and electricity trades within each time period. The LP also determines the end of period storage volumes. Uncertainty is introduced into the model through scenarios. The scenario-dependent parameters are demands, inflows, and prices. The model is limited to the consideration of two reservoirs, which must be hydraulically separate from one another. The limited number of reservoirs that can be considered is a function of the well known "curse of dimensionality" that afflicts dynamic programming (DP). The apparently restricted range of applicability of the model can be extended using aggregate reservoirs, a technique that has been used in previous research (e.g., Turgeon and Charbonneau, 1998; Valdes et al., 1992). For example, the two reservoirs could be used to represent two separate river systems, each reduced into a single aggregate reservoir. Alternatively, one reservoir could represent either a single reservoir or the aggregate reservoir for a river system, and the second reservoir could represent an aggregate reservoir that replaces the remainder of the plants in the hydroelectric system. The remainder of this chapter outlines the model and its application to two multiple-year storage facilities that are part of the B C Hydro reservoir system. The case study demonstrates the value of the model through its ability to calculate the marginal values for both reservoirs, as well as identifying regions of storage over which the reservoirs could be modelled individually while addressing the uncertainty in demands, electricity prices, and inflows. 3.2 DP and L P Based Model Overview A model overview that w i l l set a framework under which the model details can be discussed is presented in this section. In order to help elucidate the description of the model, terminology is presented first. 3.2.1 Terminology The D P model considers a time interval that is divided into an ordered set of time periods. The length of each time period in the set can differ. For example, one natural division of a time interval into a set of time periods is that of dividing a year into months. In this case, the time interval would be one year, there would be 12 time periods in the set, and the length of each time period would be the number of days in the month. The fact that the set is ordered means, for example, that January can be specified to chronologically precede February. In every DP iteration each time period in the time interval is considered. Continuing the example, one D P iteration would consider each of the 12 months in the year. The use of a recurring time interval divided into a set of fixed time periods means that the length of the time interval and time periods cannot vary with the D P iteration. Thus, in the example, each year must have the same number of days, as must each February. The D P algorithm is backward moving, meaning that the ordered set of time periods is moved through in anti-chronological order. In the example, i f 32 calculations start for the month of February, then the next time period considered is January, and the last time period considered in an iteration is March. In standard D P terminology, the points at which decisions are made are known as stages. For the case here, the stages correspond to the start of each time period. In the example, a stage would correspond to the start of a month. Uncertainty in the D P model is handled through scenarios. The number of scenarios in each time period can vary. A scenario is defined by the values specified for the scenario-dependent parameters. The scenario-dependent parameters are demands, inflows, and import and export prices. Scenario probabilities specify the probability with which each scenario is expected to occur. Note that since the scenarios are related to the time periods, the scenarios and scenario probabilities also recur in each time period in each time interval. In the DP , the storage volume in each reservoir is described by discretized storage volumes. The discretized storage volumes for a reservoir are contained in the set of possible storage volumes at the beginning of each time period. The storage volume for a reservoir at the end of a time period is found by the L P algorithm, and is not restricted to the set of discretized storage volumes. In standard D P terminology, the values that can be assumed by a decision variable are known as states. In the current case, the discretized storage volumes are the states. The L P algorithm is run for a time horizon, equal in length to one of the D P time periods, in order to evaluate the D P objective function. For example, the L P algorithm may be run for a time horizon equal to the number of days in the month of February. The time horizon in the LP can be divided into sub-periods, thus allowing the L P to consider time in greater detail than does the D P . For example, the time horizon in an L P may be divided into two sub-periods: one for the heavy load hours ( H L H ) and a second for the light load hours ( L L H ) . Note that in this example, all of the monthly H L H would be grouped into one sub-period and all of the monthly L L H would be grouped into a second sub-period. The outputs from the L P algorithm are an objective function value, and a policy. The objective function value is equal to the maximized scenario probability weighted sum of the export revenues, import and penalty costs, and the discounted terminal value of water remaining in storage. The policy is the vector of turbine releases that is optimal over all scenarios. Within each scenario, subject to the optimal turbine release, the values of ending storage and spill are found for each reservoir, and imports and exports for each sub-period, are calculated assuming perfect foresight within the scenario. The storage values are derived from the objective function values. The storage value for a particular discretized state (in the two-reservoir case, the state is two-dimensional) is equal to the objective function value for that discretized state minus the minimum objective function value for any discretized state. 33 3.2.2 Model Overview Data that must be input by the user include the time interval, the number of time periods, the length of each time period, the number of sub-periods, the length of each sub-period, and the period from which the run should commence. The user must provide the number of scenarios for each time period, and the scenario probability and scenario-dependent parameters for each scenario. In addition, all of the remaining sets and parameters outlined in sections 3.3.1.1 and 3.3.2.1 below must be provided. The D P algorithm starts at some point that is far enough in the future such that the water remaining in storage at the end of the first time period can be considered valueless—i.e., the system operator is indifferent to the terminal storage values at the end of this time period. A storage value is found for each discretized state. In order to calculate the storage values, objective function values must be calculated for each discretized storage volume combination. Each objective function value is found by iteratively solving an L P until convergence. Iteration is required in the L P solution as the problem being modelled is non-linear. The rate at which turbine discharge can be converted into power is a non-linear function of reservoir storage, and is approximated in the model by a piecewise-linear function. In the L P model, a single conversion rate is used for each reservoir for the entire time horizon. The rate used is an average of that for the known starting reservoir volume and that for the reservoir volume at the time horizon, which is one of the decision variables in the LP model. Convergence is recognized when the ending storage volume used in calculating the conversion rate and the optimal ending storage volume found by the L P fall within a specified tolerance of one another. The estimate of the ending storage volume used in the first iteration is equal to the starting storage volume, which is one of the discretized storage volumes. In subsequent iterations, the estimate of the ending storage volume is the ending storage volume found by the L P in the previous iteration. The value of water in storage at the time horizon also depends upon the ending storage volumes. The value of water stored in each reservoir is a function of the ending storage volumes of both reservoirs. The storage value curves used in the L P model are functions of the estimated storage volumes at the end of the time horizon. When convergence is recognized, the forecast ending reservoir volumes (equal to the ending reservoir volumes found by the previous iteration of the LP) are equal to the ending reservoir volumes found by the L P . Once the objective function value has been calculated for each discretized storage volume state, the storage values can be determined. The storage values are calculated by finding the minimum objective function value for any of the states, and subtracting this value from all of the objective function values. The reason for performing the subtraction is to remove a constant value from the optimization. The storage value can be thought of as the amount by which the value of a particular storage volume state exceeds the value of the worst state. The worst (in terms of objective function values) case w i l l occur for the lowest discretized storage volumes. 34 After the storage values have been calculated, they are checked to ensure that convexity requirements are satisfied. Convexity problems may arise when the L P iterations end because a maximum number have been performed. If the convexity check fails, either the storage values, or the slopes of the line segments connecting storage values for the discretized storage volumes—the marginal storage values—are modified in order to restore convexity. A t this point, the storage value table can be used to generate piecewise-linear curves that can be used to value the end of period storage when the D P steps backward in time by one period. The storage value table consists of storage values for each possible combination of the discretized storage volume for the two reservoirs. The information in the table is a storage value that can be achieved for a given combination of discretized storage volumes. Thus, the storage value is associated with a pair of storage volumes. The table specifies a single storage value that is obtained when each of the two reservoirs is at a discretized storage volume; the storage value table does not indicate the respective contributions of the two reservoirs to the total storage value. However, the piecewise-linear curves that are used in the LP to provide the value of water remaining in storage at the end of the time horizon must be functions of a single storage volume. Thus, two piecewise-linear storage value curves must be produced—one for each reservoir. Each curve gives the contribution of one reservoir towards the total storage value as a function of that same reservoir's storage volume. A s the total storage value is a function of the storage volume in both reservoirs, each curve is valid for a particular storage volume in the other reservoir. The requirement for these curves is that i f a pair of end of time horizon storage volumes corresponds to a pair of discretized storage volumes, then the sum of the two piecewise-linear functions should equal the value in the storage value table corresponding to the two storage volumes. In other words, the results from the two piecewise-linear curves should correspond to the results of the storage value table. The two piecewise-linear storage value functions are derived from the storage value table by leaving the marginal storage values in the storage value table unchanged, and altering the storage value intercepts. (A storage value intercept is the ordinate axis intercept on a plot of storage value versus storage in one reservoir for a specified storage in the second reservoir.) The storage value intercepts must be changed such that for the forecast ending storage volumes, the sum of the piecewise-linear functions is equal to the appropriate value in the storage value table. In this method, it is assumed that the two storage value intercepts for the two reservoirs are equal. The sum of the two revised storage value intercepts is set equal to the sum of the two unaltered storage value intercepts less the appropriate storage value table entry. The above D P process is repeated for the next time period as the D P steps backwards. The newly calculated storage value curves become input to this next time period through the LP . The process is repeated for the set of time periods in the D P time interval. D P iterations continue until a convergence criterion is met. 35 The convergence criterion is based upon changes in the storage values between D P iterations. For each time period, the percentage change in the storage value from the previous iteration is calculated for each discretized storage volume state. The maximum percentage change for each time period is stored. In order for convergence to be recognized, all of the maximum percentage changes must be less than a specified tolerance. Once convergence has been achieved (or a limiting number of D P iterations have been performed) one final D P iteration is performed. In this final step, during each time period, the optimal policy is stored for each discretized state. The policy specifies the turbine discharges, and the remainder of the optimal operation of the system over all scenarios, including the storage volumes at the time horizon, and spills, generation, imports and exports for each sub-period. The manner in which scenarios are used has implications for the types of correlations that can be handled in the model. Correlations between scenario-dependent parameters—e.g., inflows to the two reservoirs—within a time period are handled through scenarios and scenario probabilities. The model does not allow for the correlation of a scenario-dependent parameter between time periods. That is, the inflow to a reservoir in one period cannot depend upon the inflow in the previous period; the demand in one period cannot depend upon the inflow in the previous period; nor can the price in one period depend upon the price in the previous period. This restriction is a limitation of the model, and a departure from the typical use (e.g., Little, 1955; Butcher, 1971; Karamouz and Houck, 1987; Tai and Goulter, 1987; and Karamouz and Vasiliadis, 1992) of a lag-one Markov process to represent reservoir inflows. Some recent models have also used a lag-one Markov process to represent energy prices on the spot market (Mo et al., 1998). The simplification has been made in order to reduce the number of state variables, and thus reduce the effects of the curse of dimensionality; similar assumptions have been made in other work (Druce, 1989, Druce 1990). One additional difference between a typical D P model and the formulation employed here must be noted. Normally, D P is used to find the optimal transition from a state in one time step to a state in an adjacent time step. In the case of reservoir operations, the states would typically include discretized reservoir storage volumes. In the formulation employed here, the ending storage volumes are decision variables in the L P algorithm, and are not restricted to discretized values. The information generally contained in the ending storage states is contained in the piecewise-linear storage value curves. In the storage value curves, the discretized storage volumes are the breakpoints between linear segments. Thus, at each discretized storage volume the marginal value of stored water can potentially change. Between discretized storage volume points, the marginal value of stored water is a constant. The stepwise constant marginal values are used in the L P model in determining the optimal use of water over the time horizon. When the end of period storage is restricted to discretized state values, the potential arises for forcing the "artificial" spilling of water. This problem would be particularly acute if, 36 in trying to increase the speed of the program, a coarse state space discretization was used. Such a spill could occur i f the inflow to a reservoir exceeded the maximum discharge capacity of the turbines. Suppose that the inflow is higher than the turbine capacity, the turbines are generating at maximum capacity with any excess generation above domestic demand being exported, and that there is no spill. In this case, the continuity equation would specify that the amount of water stored in the reservoir is increasing with time. The additional storage volume is equal to the product of the time step and the amount by which the inflow rate exceeds the turbine discharge rate. If this additional storage volume is less than the increment in the discretized storage volumes, then the next discretized state point cannot be reached. A s a result, all of the additional storage volume would need to be spilled. Such a spill would not be representative of actual reservoir operations, and thus artificial. In the formulation employed here, the additional storage volume could be stored, and the objective function would be greater by the product of the marginal value of stored water and the additional storage volume. This section has described the D P and L P based model in general terms. In section 3.3, the details of the model are provided. 3.3 Model Details In section 3.2, the D P and L P based model was outlined in general terms. In this section, the details of the model, including a mathematical formulation, are presented. As implied by the name, the D P and L P based model contains both D P and L P components. The D P component operates at a higher level than the L P component, in that the LP component is used to evaluate the D P objective function. The uncertain natures of reservoir inflows, energy prices, and energy demands are handled through the use of scenarios at the D P level. A scenario is defined by the values specified for the scenario-dependent parameters—the energy demands, energy prices, and reservoir inflows. The probability of occurrence of each scenario is defined by a scenario probability. The scenario probabilities must sum to one for each stage. Correlations between scenario-dependent parameters—e.g., inflows to the two reservoirs, or energy demand and reservoir inflow—within a time period are handled through scenarios and scenario probabilities. The model does not allow for the correlation of a scenario-dependent parameter between time periods. That is, the inflow to a reservoir in one period cannot depend upon the inflow in the previous period; the demand in one period cannot depend upon the inflow in the previous period; nor can the price in one period depend upon the price in the previous period. The simplification has been made in order to reduce the number of state variables, and thus lessen the effects of the curse of dimensionality. 3.3.1 Dynamic Programming Model 37 The use of dynamic programming in water resources problems is well established. For examples of the use of dynamic programming in the field, see Yeh (1985), Yakowitz (1982), Wurbs (1993), and Esogbue (1986). Dynamic programming is used here to calculate the value of water and the marginal value of water for each reservoir, at the end of each period. In general, the value of stored water in each reservoir depends upon the amount of water stored in all reservoirs, as well as the time of year. The mathematical formulation of the D P model is presented next. 3.3.1.1 Mathematical Formulation Prior to presenting the mathematical formulation of the D P , some terminology is presented. Terminology Dynamic programming is best thought of as an approach to optimization, not as a particular technique. A s a much-used optimization method, D P has developed its own terminology; this jargon is presented here. D P is a method that can be used to make optimal decisions at a number of decision-making points. Often these points are times, but this is not a requirement. The points at which decisions must be made are referred to as stages. In DP , the decisions that are to be made are functions of some important model components. The decisions are of the form: "If, for a given stage, the values of the key model components have these particular values, then take that particular action." The important model components are referred to as state variables. In D P , each state variable is typically discretized into a finite number of values. A vector composed of one discretized value for each state variable in the model is known as a state. In D P , it is common to refer to being in a particular state at a particular stage. It is important to note that an optimal decision is found for each state. Associated with each optimal decision is an ending state; that is, an optimal decision defines the best transition from a particular state for a given stage to a particular state at an adjacent stage, as measured by an objective function. The optimal transition from one state to another is typically referred to as a policy. A policy simply specifies the action to take i f a particular state occurs during a particular stage. Stages In section 3.2.1, it was outlined that in the D P model a time interval is divided into a set of time periods. The duration of the time periods can differ; for example, a time interval of one year may be divided into twelve monthly time periods. The stages in the model correspond to the beginning of the time periods. States In D P , optimal policies are found as functions of the state variables. The state variables used in the D P model are the storage volumes in the reservoirs. There is one state variable for each reservoir in the modelled system. Notation The sets, parameters, and state variables used in the D P model are defined next. Loucks et al. (1981) has been used as a guide for establishing some of the following notation. Sets Sets are used to index components of the model. The sets used are presented here. Let the set of reservoirs in the system be represented by {R}. In the model, {Rj is limited to two hydraulically separate reservoirs. Let the set of scenarios for stage / be represented by {Ot}. A scenario specifies values for the scenario-dependent demands, inflows, and prices. Let {D,} represent the set of discretized storage volumes for reservoir re {R}. Let {G,} represent the set of discretized inflows for reservoir r e {Rj. Let {J,} represent the set of releases for reservoir re {R}. Parameters Let NR represent the number of reservoirs included in the model. Let gr represent the inflow volume for reservoir r e {1, NR} in stage /; gr e {G,}. Let jr represent the reservoir release volume for reservoir r e {1, NR} in stage t; jr e (Jrl Let wlr represent the reservoir storage volume for reservoir r e (1, NR} under scenario (oe{Q} in stage /+/. 39 Let n represent the number of stages to go until T+J, including the current period, t. Let NP represent the number of stages (time periods) in the time interval considered in one iteration. Let NSr represent the number of discretized storage volumes for reservoir re {R}. Let aP' represent the probability of occurrence of scenario CD for stage /. Let SV" represent the storage value for stage t. Let T represent the number of stages at which the objective function has been evaluated when convergence is recognized. The D P model is not concerned with operation of the reservoir system beyond stage T+l. Let £ represent the D P convergence recognition tolerance. Let /? represent the discount factor for stage t. Let O) e (Qtj represent a scenario for stage /. State Variables Let kr represent the reservoir storage volume for reservoir r e {1, NRj in stage t; kr e fDr}. Recursive Equation In dynamic programming, the aim is to find the policy maximizing the sum of the return from the current period and the return from the end of the current period until the model horizon. Let mB a a represent the return during the current period given that the storage kl,...,km, /, lm,i volume in reservoir r for stage t is kr; the storage volume in reservoir r for stage t+\ is mlr; the stage is t; and the scenario is co. The return during the period is given by the sum of the net revenue generated through energy trades and penalties for constraint violations. Let (&,, . . . , km) represent the total value of system performance with n periods to go until the horizon T+l, including the current period /, given that in period t the initial storage volume in reservoir re (Rjis kr and that the scenario is CO. 40 The objective function, which combines the return from the current period and the return from the end of the current period until the model horizon, is given as f!"(kl,...,kNR) = max j\ >—Jm a {a,\ (3-1) V (ki, km);(jj, ...JNR) feasible for (gi, gm), (kh km), and lNR), (o, and subject to: ki+gi-ji=li. (3-2) In equation (3-1), note that it is assumed that returns in the current period occur at the end of the period. Further, note that the total return for stage t is dependent upon the total return for stage t+1. This dependence between stages has led to equations of the form of (3-1) being referred to as "recursive equations". The approach taken to evaluating (3-1) is presented below in section 3.3.1.2. 3.3.1.2 Solution Methodology In section 3.3.1.1, a mathematical formulation of the D P model is presented. In this section, an approach to solving the model is presented. Algorithm Dynamic programming describes a general approach to decision making, not a specific technique. D P algorithms can be classified as either forward- or backward-moving. This categorization specifies the order in which stages are considered by the model. In forward-moving D P , the recursive equation is evaluated for stage t prior to stage t+1, whereas the converse is true for backward-moving DP . For stochastic dynamic programming, only backward-moving D P is applicable (Yeh, 1985). In the backward-moving D P algorithm employed here, the total returns for stage t+1 are calculated before the total returns for stage t. Consequently, an assumption needs to be made regarding the values of a f?+x (kx,..., k m ) when calculations begin at stage T. The assumption used is that fr+i (h\' • • • > ^ NR ) = 0' ^ co,kx ,...,km. (3-3) Equation (3-3) implies that at stage T+1, the system operator is indifferent to the volume stored in the system reservoirs. A t this stage, the total value of system performance for all remaining periods is equal to zero under all scenarios. Obviously, stage T+1 occurs at some point far in the future. When the D P calculations begin, it is unknown how far into 41 the future stage T+l lies. The recursive equation (3-1) is evaluated a sufficient number of times such that the assumption expressed in (3-3) can be held to be valid. Details of the criteria required for convergence are specified later in this section. Evaluation of the iterative equation begins at stage T. Under the assumption of equation (3-3), the iterative equation for stage T becomes f}(kl,...,km)= max h •~Jm max h <—Jm B, * i bNR,'"l\,...,"lNi!,T (3-4) Following the evaluation of equation (3-4) for all discretized storage volume states (ki, kNR), calculations proceed for stage / = T-l using equation (3-1). Equation (3-1) is then used to successively evaluate stages until convergence recognition. Convergence Convergence should be recognized when equation (3-1) has been evaluated for a sufficient number of stages such that the assumption of equation (3-4) is valid. The assumption contained in equation (3-4) is that the system operator is indifferent to all ending states of the system. In order to determine i f a sufficient number of stages have been evaluated, the concept of "storage values" is used. The storage value is the amount by which the recursive equation for a particular state in a given stage exceeds the minimum value, over all states, of the recursive equation for that stage. Thus, storage value, SV", is defined as: 5 ^ ^ , . . . , ^ ) = / " ^ , . . . , ^ ) - min (ftn(kx,...,km)\ (3-5) Convergence is recognized when max SVt (kx,...,kNR) SVI+NP (kt,...,kNR) sv;(kx,...,km) ' ^ NR ' ^  ' {l,...,NP} (3-6) where £ is a specified tolerance. Equation (3-6) states that convergence is recognized when the maximum percentage change in storage value for each state in each stage within the iteration is less than a specified tolerance. A n upper limit is set on the number of iterations that are to be performed. Thus, in some cases, the assumption of equation (3-4) may not strictly hold. Discussion 42 The DP recursive equation defined by (3-1) differs in several respects from what one may term a "typical" DP problem (e.g., Loucks et al., 1981). These differences are described next. Ending State Discretization One important difference between a typical formulation and the one employed here has to do with the discretization of the ending states. Generally, the ending states are limited to the set of discretized starting states. Thus, a policy for a particular starting state would typically indicate which member of the finite set of feasible ending states is optimal. In the formulation used here, the ending state is not limited to a finite set of states; rather the values of the ending state variables are continuous. To this point, little attention has been given to the two components of the recursive equation, or to the means by which the maximization in equation (3-1) is performed. The immediate return function, mB a a , takes into account the revenue obtained * 1 NW> <l'---> 'NR>1 through the export of energy; the cost incurred through the import of energy; and the cost of any soft constraint violations. These costs and revenues should be those associated with the optimal operation of the reservoir system, subject to all appropriate constraints. The value of the system being in a particular state at the end of stage t, mfil\l Ch>-- -?1HR ) > takes into account the value of water in storage. Stored water has value as it can be used in future periods. The maximization in equation (3-1) is performed using linear programming. The objective function in the LP contains terms for the three components of the immediate return function. The LP objective function contains additional terms for the value of water remaining in storage at the end of the stage. Details of the LP model are presented in section 3.3.2. Linear programs employ continuous variables; hence the ending state variables cannot be limited to discretized states. The use of continuous ending state variables is discussed next. For each starting state, (kj, kpjR), a linear program is solved in order to obtain a value for the total return function. The LP calculates the optimal operation of the reservoir system for the stage, starting from the given state, for each scenario. The optimal operation includes the reservoir operations and energy trades to make during the period, as well as the optimal ending values for the state variables at the end of the stage. In order to determine the optimal ending values of the state variables, it is necessary to specify functions giving the value of ending the stage with a particular state variable value. That is, functions providing the value of ending a stage with particular volumes of water stored in the reservoirs must be input to the LP. The types of functions that can be used in a linear program are limited to linear and piecewise-linear functions of the decision variables. Piecewise-linear functions are used to specify the value of ending a stage with particular volumes of water stored in the reservoirs. The functions are generated by linearly 43 interpolating between the objective function values calculated for the states in the previous stage. That is, the objective function, as specified by equation (3-1) is first evaluated for all states, (ki, km), in stage t+1. The result is an objective function value for each discrete starting state. Piecewise-linear functions are then obtained by linearly interpolating between the discrete points. The intercepts of the objective function axes of the piecewise-linear functions for the two reservoirs are adjusted so as to avoid double counting in the objective function. These functions are used by the L P to assign a value to the ending states when equation (3-1) is solved for each discrete starting state in stage t. The procedure is repeated as the D P proceeds backward through the stages in anti-chronological order. For the calculations at stage T (the stage at which recursion begins), the assumption is made that the value of system operation at stage T+1 is equal to zero for all possible values of the state variables. This assumption is a re-stating of equation (3-3). The maximization in equation (3-1) that is performed using L P contains a term for the immediate benefit and a term that accounts for the value of the state variables at the start of the next stage. O f these two terms, consider the latter, which is mf,l\~l (alx ,6>l2)- The term states that the value of system operation from stage t+\ until the horizon is a function of all of the state variables, which are the two reservoir storage volumes. In the L P , it is necessary to represent this term in the objective function. A difficulty arises, as in L P it is not possible to specify, using a piecewise-linear function, that the value of future system operation is a function of two reservoir volumes, each of which is a decision variable in the LP . A s a result, an approximation is made. The approximation used is: CO r n—\ /CO f co-i \ V 1 CO n-\ /COr \ f% m\ fl+l ( A>"-> lm)= Zu e'+l.r( lr)> (3"?) where < ye"~,1 r('u/ r) is a piecewise-linear function giving the future value of system operation solely dependent on the storage in reservoir r. Equation (3-7) states that the value of future system operation commencing from a particular state is approximated by a sum of piecewise-linear functions. The sum contains one piecewise-linear function for each reservoir. Each piecewise-linear curve gives the value of future system operation as a function of one reservoir—i.e., it is assumed that the value of future system operation in any reservoir can be assumed to be independent of the volume stored in all other reservoirs. In reality, the piecewise-linear function describing the storage value of one reservoir should be a function of the storage in the second reservoir. That is, for different ending levels of the second reservoir, there would be different piecewise-linear curves for the first reservoir. The piecewise-linear curves used depend upon forecasts of the ending volumes in the two reservoirs. Thus, for each starting state, the recursive equation must be repeatedly evaluated by solving instances of the linear program described in section 3.3.2 until the forecast and optimal ending volumes fall within a specified tolerance of one another. 44 Scenarios As outlined in section 3.3.1.1, the DP described herein makes use of scenarios. A scenario is described by the values assumed by the scenario-dependent parameters—the demands, inflows, and energy prices. The occurrence probability of each scenario must be specified. The sum of the occurrence probabilities in each stage must sum to one: In equation (3-1), optimal policies, in the form of a vector of turbine releases, are found for each state {k\, ..., ICNR). As explained previously, apart from the turbine release vector, which is the same over all scenarios, the reservoir system operation as given by spills, ending reservoir storage volumes, and energy trades, can vary with the scenario, and are found assuming perfect foresight within the scenario. Typically (e.g., Loucks et al., 1981), one optimal ending state is found for each starting state. This optimal ending state is the one that maximizes the sum of the current period return, defined by start and end storage, and the probability weighted terminal value function. An additional dimension (or dimensions) would typically be added to the state in order to account for the scenario-dependent parameters. For example, in the case of a one-reservoir system, the system state may be described by a discretized reservoir volume and a discretized inflow value. In this instance, the policy would be a function of both the starting volume and an inflow. The "curse of dimensionality" restricts the number of state variables that can be feasibly included. Here, the decision has been made to only use the reservoir volumes as state variables. Including additional state variables would severely affect the practicality of the algorithm for the two-reservoir case considered here. Having the ability to handle two reservoirs is vital to the present work, so other state variables have been excluded from the model. A consequence of this exclusion is that the future value of system performance is solely dependent upon the volume stored in the two reservoirs in the system. Or, to restate matters, the future value of system performance is not a function of the current demand, inflow, or price. This restriction is a limitation of the model. The purpose of the model is to estimate the value of water stored in each reservoir at some future point. Once calculated, these future water values are to provide end conditions to models with a shorter time horizon. Given that storage values are to be calculated for some point in the future, the restriction is justifiable. In an ideal case, the size of the state space could be expanded to include the reservoir storage volumes as well as the scenario-dependent parameters. However, the curse of dimensionality appears to preclude this possibility at present. The preceding discussion raises a point that warrants elaboration. Because reservoir inflow is not a state variable, inflows cannot be assumed to follow a lag-one Markov process as is typical (e.g., Little, 1955; Butcher, 1971; Karamouz and Houck, 1987; Tai and Goulter, 1987; and Karamouz and Vasiliadis, 1992). Similarly, Markov processes cannot be used for either the demands or prices. Thus, the assumption is made that the (3-8) 45 demands, inflows, and prices in one stage are independent of the values in the preceding stage. This assumption is necessitated by the "curse of dimensionality". The scenarios can be defined in such a way as to capture cross-correlations between scenario-dependent parameters within a stage. For example, correlations between the inflows to two reservoirs can be captured, as can a correlation between inflow and price. Yeh (1985) notes that cross-correlations are generally ignored in SDP in order to reduce the dimension of the problem. Implementation The D P model described above has been implemented in a computer program written in A M P L (Fourer et al., 1993). The evaluation of the objective function is performed using a sub-routine that solves a series of linear programs. C P L E X ( C P L E X Optimization, Inc., 1995) has been used as the L P solver. The linear programming sub-routine is described in section 3.3.2. 3.3.2 Linear Programming Model A s described in section 3.3.1, a linear programming (LP) routine is used to evaluate the dynamic programming (DP) recursive equation. This section describes the L P model. 3.3.2.1 Mathematical Formulation In order to define the L P problem mathematically, the parameters, sets, and variables must first be defined. Following these definitions, the model constraints and objective are presented. Sets Sets are used to index parameters, variables, and constraints. Sets must be input to the L P . The sets in the L P model are specified below. Let the set of reservoirs in the system be represented by {R}. The membership of {R} is limited to two hydraulically separate reservoirs. In order to differentiate between energy prices during different times within a period— e.g., heavy load hours and light load hours—the time horizon must be divided into sub-periods. Let the set of sub-periods be represented by {U}. Let the set of scenarios be represented by (Qj. Parameters 46 Model parameters are values that must be input to the LP , either through data files, or through pre-processing. The values of the decision variables found by the L P depend upon the parameters. The parameters in the LP model are specified below. Let Atu represent the length in days of sub-period u e {Uj. Let wpu represent the market price, prior to discounting, of energy in $ /MWh, under scenario CO, during sub-period u e {U}. Let wpr represent the occurrence probability of scenario (O. Let wdu represent the firm power demand, including losses, in M W , under scenario CO, during sub-period u e {U}. Let b represent the annual discount rate as a decimal fraction. Y > , „ / 3 6 5 ) Let Prepresent the discount factor: j3 - (1 + b)"ei'ri Let qr represent the maximum turbine discharge in cms from reservoir r e (Rj and qr the minimum turbine discharge in cms from reservoir re {R}. Let sr represent the maximum spill in cms from reservoir r e {Rj and sr the minimum spill in cms from reservoir re {Rj. Let or represent the maximum total plant discharge in cms from reservoir r e {Rj and or the minimum total plant discharge in cms from reservoir r e {Rj. Let vr represent the maximum storage volume in cmsd in reservoir re {Rj, and vr the minimum storage volume in cmsd in reservoir re {R}. Let v° represent the initial storage volume in reservoir re {R}. Let gr represent the maximum generation limit in M W for reservoir r e {R} and gr the minimum generation limit in M W for reservoir re {Rj. Let mu represent the maximum import limit in M W during sub-period u e {Uj, and mu the minimum import limit. Let xu represent the maximum export limit in M W during sub-period u e {U}, and xu the minimum export limit. 47 Let <aHKr represent the conversion factor between turbine discharge and power generation in MW/cms , under scenario CO, for reservoir re {R}. Let y represent the decimal fraction by which the purchase price of energy should be increased above the market price to account for wheeling and losses. Let z represent the decimal fraction by which the sale price of energy should be decreased below the market price to account for wheeling and losses. Let miUir represent the inflow to reservoir re {R}, under scenario CO, during sub-period u e {U}. Variables The L P finds values of the decision variables that maximize the objective function, subject to the specified constraints. In other words, the decision variables are the results being sought from the model. The decision variables in the L P model are specified below. Let V, represent the storage volume in cmsd in reservoir re {R} at the start of the time period. Let coQu,r represent the turbine discharge in cms from reservoir re {Rj, under scenario CO, during sub-period u e {U}. Let mSu,r represent the spill in cms from reservoir r e {Rj, under scenario CO, during sub-period u e {U}. Let mOu,r represent the total plant discharge in cms from reservoir re {Rj, under scenario co, during sub-period u e (Uj. Let ^G^r represent the generation in M W from reservoir re {R}, under scenario co, during sub-period u e {U}. Let 0)Vr represent the storage volume in reservoir re {R}, under scenario co, at the end of the time period. Let WMU represent the power imported in M W , under scenario co, during sub-period u e {U}. Let WXU represent the power exported in M W , under scenario co, during sub-period u e {U}. 48 Constraints The non-linear problem of scheduling reservoirs for hydropower production is modelled using an iterative linear model. The model is subject to linear constraints that limit the values that can be assumed by the decision variables in the search for an optimal objective function value. The constraints in the L P model are specified below. Initial Volume Constraint The initial volume constraint specifies the value of the start of period volume for each reservoir, r e {RJ: V?=v°r. (3-9) Continuity Constraint The continuity constraint specifies the conservation of mass equation for each reservoir, r e {Rj, under each scenario co e {Q}\ Vt+Yj&AXs-O^V,. (3-10) Load Balance Constraint The load balance constraint specifies that the demand must balance the net generation, where the net generation is equal to the generation plus the imports less the exports. The load balance constraint, under each scenario co e {Q}, for each sub-period, u e {Uj, can be written as: " ^ ( X > " K = X + X - (3-11) re{R] Storage Limit Constraints The storage limit constraints specify the allowable operational range for each reservoir, r e {R}, under each scenario co e {Q}\ v<MVr<Vr. (3-12) Turbine Limit Constraints The turbine limit constraints specify the allowable turbine discharge range for each reservoir, re {R}, under each scenario CO e {£2}, and sub-period, u e {U}: q<aQlKr<q~r. (3-13) 49 Spill Limit Constraints The spill limit constraints specify the allowable spill range for each reservoir, re {R}, under each scenario co e {Q}, and sub-period, u e {U}\ s<«Su,r<7r. (3-14) In practice, the upper bound, sr, is not used. Import Limit Constraints The import limit constraints specify the allowable range of power imports, under each scenario co e {Q}, for each sub-period, u e {U}: mu<»Mu<m~u. (3-15) In the objective function imports are priced in a step-wise manner—that is, a block of energy has a price associated with it. The most expensive block of energy corresponds to load curtailment, which can be viewed as an expensive, special energy import case. As such, in practice the upper bound, mu , is not used for the most expensive import block. Export Limit Constraints The export limit constraints specify the allowable range of power exports, under each scenario co e (Qj, for each sub-period, u e {U}\ x<mXu<T„. (3-16) In the objective function exports are priced in a step-wise manner—that is, a block of energy has a price associated with it. Total Plant Discharge Constraints The total plant discharge constraints specify the allowable total plant discharge range for each reservoir, re {R}, under each scenario CO e {Q}, and sub-period, u e {U}. The total plant discharge constraints can be handled as either "hard" or "soft." The hard case is: * ^ - (3-17) In the soft constraint case, an additional variable is introduced. The variable Ou r , which represents the violation in cms of the minimum total plant discharge, or , is added. The soft constraint formulation is: 50 o^O^O^ <or. (3-18) In practice, the upper bound, or , is not used. Generation Limit Constraints The generation limit constraints specify the allowable range of power generation, from a power plant at a given reservoir, r e {Rj, under each scenario CO e {12}: Total Plant Flow Constraints The total plant flow constraints define total plant flow for each reservoir, r e {R}, under each scenario CO e {Qj, and sub-period, u e {U}, as the sum of the respective turbine discharge and spill flow: • 0 „ , ( 3 " 2 ° ) Power Generation Constraints The power generation constraints define the power generated from the power plant at reservoir re {R}, under each scenario 0) e {Qj, during sub-period u e {Uj as a function of the turbine discharge from plant r during sub-period u: coGur-0}Qur-(0HKr. (3-21) Note that the conversion factor is a function of the average storage volume—i.e., aHKr ^mHKr ( v r ° V r ) . The value of mHKr used in a solution of the linear program is calculated based upon a forecast reservoir ending volume. Scenario Probability Constraints The occurrence probabilities of the scenarios must sum to one: £ V = l-0. (3-22) Policy Decision Constraints The policy decision constraints specify that the policy decision variable must be the same over all scenarios. With the turbine release as the policy decision variable, the constraint for reservoir re {R}, under scenario C0i, where Oil, Ohe {Q} and CO] ^ Oh, during sub-period u e {U} are: 51 u,r' (3 Non-negativity Constraints A l l variables are subject to the constraint that they cannot be negative: Vr°, mQ«s, aSur, a O u r , a O ' u r , C0Gur, » V r , X . m X u ^ 0 ; re {R};ue {U};6>e {Q}. (3 Objective Function The objective function seeks to maximize the value of operating the hydroelectric system, where the operation of the system is understood to include making energy trades. The objective function consists of four terms: the income generated through exporting energy; the expense incurred through importing energy; the value of water remaining in storage at the end of the time horizon; and the penalty costs associated with violation of soft constraints. The only soft constraint is the minimum total plant discharge, outlined in equation (3-18). The objective function could be modified to include penalties associated with spill, or spill above a certain level. Penalties for spill could be attributed to physical damage that may occur downstream, or to perceived damages that may accompany a spill. In order to understand the objective function, it is useful to discuss each of the four terms separately. The entire function is.then presented. The income generated through the export of energy is given by: In equation (3-25), ^lfXu) is a piecewise-linear function describing the revenue earned from the sale of energy. The slopes of the segments in a-wi(°Xu) are equal to the net export prices ($/MWh). The net export price is given by apu-(l-z); where mpu is the market energy price, and z is a factor used to account for wheeling and losses. Describing the export revenue as a piecewise-linear function of the power generated allows for a block price structure to be used. That is, an export price can be associated with a quantity of energy. For example, using two blocks in the structure would mean that all of the energy exported up to a certain limit would earn revenue at one rate, and all of the energy exported above this threshold would earn revenue at a lower rate. Note that in order to preserve the necessary convexity properties, the blocks of export prices must be in the shape of a descending staircase. Also note that the above formulation, in which Atu multiplies ^ m wx {01Xu), implies that the power generation is constant over each (3 COe{£l] U<E{U} sub-period. 52 The cost incurred through the import of energy is given by: P-2A- X J K ( X ) ^ J (3-26) In equation(3-26), a>W2(0>MH) is a piecewise-linear function describing the cost of energy purchased. The slopes of the segments in a>W2(0)Mll) are the net import prices of energy ($/MWh), where the net import price is given by apu •(! + y). The parameter y is a factor used to account for the wheeling and losses associated with importing energy. Convexity requirements specify that the blocks of import prices must be in the shape of an ascending staircase. (The difference between this requirement and that for the export price blocks arises because the cost of purchasing energy is multiplied by negative one in the objective function.) The block pricing structure for import prices allows load curtailment to be modelled as an expensive import. A benefit o f including load curtailment in the model—essentially making the need to meet the fixed load a soft constraint—is that the D P algorithm does not need to be concerned with the possibility of "power-infeasible" cases. (The term "power-infeasible" is used to differentiate those cases in which infeasibility is caused by the load balance equation from those where infeasibility is caused by the continuity equation.) Without load curtailment in the model, i f a reservoir starts a period empty, or nearly so, the sum of the generation from all of the available water plus the maximum imports may be less than the firm demand. In such a case, there would be no feasible ending states, as defined for equation (3-1), for the starting state. The D P algorithm would have to recognize, and handle this situation, which would prove difficult with the use of piecewise-linear curves representing the future value of system operation. Al lowing load curtailment in the L P model alleviates this potential difficulty in the D P algorithm. The value of water remaining in storage in a reservoir is given by: P- £ (3-27) <De{Q.} r<B{R} In equation (3-27), the piecewise-linear function describing the value of water remaining in storage is a function of reservoir storage. In order for convexity requirements to be satisfied, the value of storage must decrease with increasing storage—that is the storage must have a decreasing marginal value. Note that while the above equation describes the value of water in storage as a function of a single reservoir storage, the actual dependency is on all reservoir storage volumes. The slopes of the segments used are only valid for forecast ending volumes. These estimates upon which the slopes are based, are updated in an iterative procedure, as described in section 3.3.1.2. The piecewise-linear functions W3,r(Vr) in equation (3-27) are equivalent to the piecewise-linear functions ffle^'r(/f) in equation (3-7). The final term included in the objective function accounts for the penalty associated with violating the minimum total plant discharge constraints. The cost of the penalty terms is: P- X I i k r O - A / J (3-28) 53 Treating the total plant discharge constraint as a soft constraint introduces flexibility into the continuity equation; this flexibility is useful in the D P algorithm as it eliminates the need to consider "water-infeasible" cases. (The term "water-infeasible" is used to differentiate cases in which infeasibility is caused by the continuity equation from cases where the infeasibility is caused by the load balance equation.) Without flexibility in the continuity equation, the situation may arise where there is not enough water in the reservoir to meet a minimum plant discharge requirement. In such an instance, there would be no feasible ending states for a starting state. As discussed above, requiring the D P algorithm to recognize infeasible states would create difficulties with the use of L P to evaluate the value of future system operation. The slopes of the segments in the piecewise-linear curves are the costs ($/cms) associated with violation of the minimum plant discharge constraint. A block cost model is used. A s the costs of penalty violations are multiplied by negative one in the objective function, the shape of the block costs must be that, of an ascending staircase. The entire objective function comprised of the parts described by equations (3-25) to (3-28), summed over all scenarios, is: max-^  p f f I z I V reW J J <ae{0} rs(R) (3-29) Note that it is assumed that revenues, costs, and penalty costs all occur at month end. The means by which the objective function described by (3-29) is optimized, subject to the constraints described by equations (3-9) to (3-24), is described in section 3.3.2.2. 3.3.2.2 Solution Methodology The linear programming problem described by equations (3-9) to (3-29) is modelled using the language A M P L (Fourer et al., 1993). The main benefit of using A M P L is the ease with which sets can be defined and used. For example, sets of sub-periods and reservoirs can be defined, and parameters, variables, and constraints can then be indexed using these sets. The result is a much more compact notation than would result i f it were necessary to individually specify each parameter, variable, and constraint. Obviously, having a more compact model facilitates making changes, and thereby maintenance of the model. In A M P L , the model and data are specified individually. The model files specify a general L P model in terms of parameters, variables, constraints, and objective functions. The data files specify the values for the parameters and the objective function to use in solving a particular instance of the general model. Once a model has been developed, the input data can be modified without changes being made to the model itself. 54 A M P L is a modelling language, not a solver. When A M P L is run, provided that the model and data are valid, the data is reformatted to the specifications of the solver. A M P L can also be used to pass specific instructions to the solver. After the data have been properly reformatted, the solver is executed to find the optimal solution. The solver used to solve the linear program defined by (3-9) to (3-29) is C P L E X ( C P L E X Optimization, Inc., 1995). C P L E X is primarily a linear programming solver, with the ability to handle integer and quadratic problems. Once the run of the solver is complete, control is returned to A M P L . Another feature of A M P L that proved useful in model development is the ability to run in batch mode. In batch mode, a text file containing a number of A M P L commands can be specified as input. Batch mode allows another application to call A M P L and solve a particular instance of a model. This feature allows the D P computer program described in section 3.3.1 to call A M P L in order to evaluate the D P objective function values through use of the L P model described in section 3.3.2. Batch mode also makes it possible to iteratively solve the L P model until the forecast and optimal ending reservoir storage volumes agree within a specified tolerance. 3.4 Case Study In this section, in order to demonstrate the ability of the model to address the uncertainty faced by the operator of a large hydropower system with considerable flexibility, the D P and L P based model described in sections 3.2 and 3.3 is applied to a test system. The test system is based upon a sub-system of the British Columbia Hydro and Power Authority ( B C Hydro) system. 3.4.1 Description The system used as a case study to demonstrate the D P and L P based model is derived from part of the B C Hydro system. The B C Hydro electric generation system, at the time of this study, included 29 hydroelectric plants, one conventional thermal station, and two combustion turbine stations. Approximately 75% of the total installed generation capacity of the B C Hydro system is at stations on two river systems: the Columbia and the Peace ( B C Hydro, 1995). B C Hydro has two generating plants on the Peace River: G . M . Shrum and Peace Canyon. The Williston reservoir associated with the G . M . Shrum plant has significant over-year storage, whereas the Dinosaur reservoir associated with the Peace Canyon plant has some storage, but is largely run in hydraulic balance with G . M . Shrum. Similarly, B C Hydro generation facilities on the main-stem of the Columbia River include an upstream plant (Mica) with a reservoir having significant storage, and a downstream plant (Revelstoke) associated with a reservoir with less storage. Plants on other rivers in the Columbia basin contribute to the 75% figure cited above, but their contribution is significantly less than that of the M i c a and Revelstoke stations. Together, the G . M . Shrum, Peace Canyon, Mica , and Revelstoke stations account for roughly 70% of the installed hydroelectric generation capacity in the B C Hydro system ( B C Hydro, 1995). 55 In the test system used in the case study, generation on each of the two river systems has been modelled with a single plant. On the Columbia River, the "Columbia" aggregated plant replaces the M i c a and Revelstoke plants. The "Peace" aggregated plant replaces the G. M . Shrum and Peace Canyon plants. B y combining reservoirs, the effect of the curse of dimensionality, to which dynamic programming is subject, is diminished. The practice of aggregating reservoirs for this purpose has been widely reported in the literature (e.g., Turgeon and Charbonneau, 1998; Valdes et al., 1992). The data for the aggregated Peace and Columbia plants are presented in the following section. 3.4.2 Data The data for the case study are presented in this section. Data relating to the Columbia and Peace plants are based, respectively, upon data for the M i c a and Revelstoke, and the G. M . Shrum and Peace Canyon facilities. As the intent of this work is to examine the relationship between the operation of two hydroelectric generating river systems at a gross scale, many of the details of the operation of the physical plants are neglected. The aim was to work with data that reasonably approximated the major characteristics of the underlying facilities. The above is to state that the Columbia and Peace plants, and the associated data, should be considered as conceptual only; there is not a direct correspondence between the data for them and the Mica , Revelstoke, G . M . Shrum, and Peace Canyon facilities. 3.4.2.1 Inflows The average monthly inflows to the Columbia and Peace reservoirs are presented in Table 3-1. Peace Reservoir Inflow (cms) Columbia Reservoir Inflow (cms) January 300 150 February 220 140 March 220 140 Apr i l 430 280 May 2230 1100 June 3490 2190 July 2200 2170 August 1010 1510 September 780 820 October 810 450 November 480 270 December 330 180 Table 3-1: Average Monthly Inflows 56 3.4.2.2 Prices The average monthly prices for heavy load hour (HLH) and light load hour ( L L H ) electricity assumed for the case study are presented in Table 3-2. The market is assumed to be deep enough such that transmission constraints w i l l be reached prior to market constraints. H L H Price ($/MWh) L L H Price (S/MWh) January 29.0 25.0 February 29.0 27.0 March 27.0 26.0 Apr i l 30.0 21.0 May 29.0 17.0 June 27.0 16.0 July 30.0 22.0 August 32.0 26.0 September 42.0 31.0 October 35.0 31.0 November 34.0 31.0 December 36.0 29.0 Table 3-2: Average Monthly Market Electricity Prices 3.4.2.3 Demands The average H L H and L L H domestic demands that are assumed to be served by the Columbia and Peace generating stations is presented in Table 3-3. H L H Demand (aMW) L L H Demand (aMW) January 6100 4200 February 6000 4200 March 5600 3900 Apr i l 5000 3500 May 4800 3300 June 4700 3300 July 4700 3300 August 4800 3300 September 4800 3400 October 5100 3500 November 5700 4000 December 6000 4200 Table 3-3: Average Monthly Electricity Demand 57 3.4.2.4 Miscel laneous In the model, each time period—month in this case study—is divided into H L H and L L H portions. It is assumed that the H L H period consists of 16 hours a day, Monday through Saturday; thus the L L H period includes 8 hours a day Monday through Saturday and all day Sunday. The number of H L H and L L H days in each month is shown in Table 3-4. H L H Period L L H Period (days) (days) January 17.7 13.3 February 16.0 12.0 March 17.7 13.3 Apr i l 17.1 12.9 May 17.7 13.3 June 17.1 12.9 July 17.7 13.3 August 17.7 13.3 September 17.1 12.9 October 17.7 13.3 November 17.1 12.9 December 17.7 13.3 Table 3-4: Monthly Duration of H L H and L L H Periods Given the division of time into H L H and L L H periods, parameters related to energy trade must also be differentiated on this basis. Table 3-5 presents the assumed upper transmission limits on imports and exports for the H L H and L L H periods of each month; the lower transmission limits are shown in Table 3-6. Max H L H Max L L H Max H L H Max L L H Imports Imports Exports Exports (aMW) (aMW) (aMW) (aMW) January 1620 1230 1900 1420 February 1500 1140 1720 1300 March 1740 1320 1720 1300 Apr i l 1950 1470 1600 1200 May 2010 1500 1660 1240 June 1950 1530 1600 1200 July 2100 1590 1840 1380 August 2100 1590 1840 1380 September 2040 1530 1780 1340 October 2100 1590 1900 1420 November 1860 1380 1840 1380 December 1620 1200 1900 1420 Table 3-5: Monthly Maximum Import and Export Transmission Limits 58 Min H L H M i n L L H M i n H L H M i n L L H Imports Imports Exports Exports (aMW) (aMW) (aMW) (aMW) January 0 0 0 0 February 0 0 0 0 March 0 0 0 0 Apr i l 0 o . 0 0 May 0 0 0 0 June 0 0 0 0 July 0 0 0 0 August 0 0 0 0 September 0 0 0 0 October 0 0 0 0 November 0 0 0 0 December 0 0 0 0 Table 3-6: Monthly Minimum Import and Export Transmission Limits Limits related to turbine releases, spills, total plant releases, generation, and reservoir storage volume must be specified for each plant. Plant related limits are shown in Table 3-7 and Table 3-8. Peace Plant Columbia Plant Max Turbine Discharge (cms) 2000 1500 Min Turbine Discharge (cms) 0 0 Min Spill (cms) 0 0 Max Reservoir Volume (cmsd) 500000 200000 M i n Reservoir Volume (cmsd) 0 0 Max Generation (M W) 3400 3500 M i n Generation (MW) 0 0 Table 3-7: Plant Related Limits Peace Plant Minimum Discharge (cms) Columbia Plant Minimum Discharge (cms) January 1600 85 February 200 85 March 200 85 Apr i l 200 85 May 200 85 June 200 85 July 200 85 August 200 85 September 200 85 59 October 200 85 November 200 85 December 1600 85 Table 3-8: Monthly Minimum Plant Discharge In Table 3-8 note that an operational constraint on the Peace plant discharge is in effect for December and January. The effects of this constraint are explored below in 3.4.3.4. The relationship between turbine release and generation is described by the " H K " factor, which takes into consideration efficiency, the specific weight of water, and the gross head—see (3-21). H K , which has units of MW/cms , is modelled as a piecewise-linear function of reservoir storage. The slopes of the segments for the Peace and Columbia plants are given, respectively, in Table 3-9 and Table 3-10. The storage intercepts for the two plants are given in Table 3-11. Range Number Minimum Storage Maximum Storage Slope of H K vs. (cmsd) (cmsd) Storage (MW/(cms) 2d) 1 0.0 50000.0 8.8395805668E-07 2 50000.0 100000.0 7.8826138426E-07 3 100000.0 150000.0 7.1926311364E-07 4 150000.0 200000.0 6.5153642709E-07 5 200000.0 250000.0 5.9569165768E-07 6 250000.0 300000.0 5.4044050367E-07 7 300000.0 350000.0 5.0455005931E-07 8 350000.0 400000.0 4.7493755667E-07 9 400000.0 450000.0 4.5268005924E-07 10 450000.0 500000.0 4.4196592180E-07 Table 3-9: Slopes of HK-Storage Function for Peace Plant Range Number Minimum Storage Maximum Storage Slope of H K vs. (cmsd) (cmsd) Storage (MW/(cms) 2d) 1 0.0 20000.0 3.9717525790E-06 2 20000.0 40000.0 3.5876743972E-06 3 40000.0 60000.0 2.9298473017E-06 4 60000.0 80000.0 2.5590057745E-06 5 80000.0 100000.0 2.4924027772E-06 6 100000.0 120000.0 2.2931980996E-06 7 120000.0 140000.0 2.0492320665E-06 8 140000.0 160000.0 1.8812578849E-06 60 9 160000.0 180000.0 1.6475791195E-06 10 180000.0 200000.0 1.5000000000E-06 Table 3-10: Slopes of HK-Storage Function for Columbia Plant Columbia Plant Peace Plant Storage Intercept -553096.134848 -1773335.270998 Table 3-11: Storage Intercepts for HK-Storage Functions Miscellaneous data for the case study are shown in Table 3-12. The curtailment cost is the cost of failing to supply demand. The loss factor takes into account transmission losses; for an import the loss factor is added to the cost, whereas it is subtracted for an export, as discussed in the paragraph following equation (3-25). Parameter Value Discount rate (%) Wheeling/loss factor (%) Curtailment Cost ($/MWh) 8 7 1000 Table 3-12: Miscellaneous Data Violation of the minimum plant release is penalized in the model, as shown in equations (3-28) and (3-29). The penalties are given in Table 3-13. The penalty rates in Table 3-13 equal the $1000/MWh curtailment cost for average H K values. Peace Plant Columbia Plant Penalty for Violation of Minimum Plant Turbine Release ($/cmsd) 42000 60000 Table 3-13: Minimum Plant Turbine Release Violation Penalties 3.4.3 Results The results of applying the D P and L P based model described in sections 3.2 and 3.3 to the case study data specified in section 3.4.2 are presented in this section. Results are first presented in 3.4.3.1 for a base case in which there is only one scenario per month, with the scenario-dependent parameters assuming their mean values. This one-scenario case establishes a basis for comparing the marginal energy values for the Peace and Columbia reservoirs. In section 3.4.3.2 eight cases covering different combinations of the scenario-dependent parameters are solved under the non-anticipative turbine release constraints specified by equation (3-23). In each of these eight cases there are five scenarios in the months of 61 May through October, and five scenarios in the months of November through Apr i l . In each of the cases the scenario-dependent parameters, and their correlations, during the months with five scenarios vary. The eight scenarios are: demands only; inflows only; prices only; demands, inflows, and prices all perfectly positively correlated; demands and inflows perfectly correlated and prices perfectly negatively correlated; demands and prices perfectly correlated and inflows perfectly negatively correlated; and inflows and prices perfectly correlated and demands perfectly negatively correlated. The above cases demonstrate the effect of uncertainty in the scenario-dependent parameters on the marginal energy values for the two reservoirs. Section 3.4.3.3 presents results for the eight cases outlined in the previous paragraph under the assumption of perfect foresight. Perfect foresight is obtained by removing equation (3-23) from the model. A comparison of the marginal energy values for the two reservoirs is made between the results under the assumptions of non-anticipative turbine releases and perfect foresight in section 3.4.3.3. In section 3.4.3.4 the effect of an operational constraint on the marginal energy values for the two reservoirs is examined. In all instances of the model solved in sections 3.4.3.1 through 3.4.3.3 the Peace plant releases are specified to have a minimum value during December and January that is significantly higher than the minimum in the other months. In section 3.4.3.4 a one-scenario case is solved for the case in which the December and January minimum Peace plant releases are the same as they are for the other months; no other model parameters change. A comparison is made between the results thus obtained and those in section 3.4.3.1. 3.4.3.1 One-Scenario Base Case A s described earlier in this chapter, the model can accommodate different scenarios for each time period—month in this case. Between scenarios, the scenario-dependent demands, inflows, and prices for each time period can vary. In later sections, scenarios employing different combinations of, and correlations between, the scenario-dependent parameters are explored. The scenario-dependent parameters in these later sections are symmetrically distributed about the mean value. In this section, a base case in which there is only one scenario in each month, and where the scenario-dependent parameters assume their mean values is studied. This base case establishes a comparison point for the multiple-scenario cases The storage value—that is the amount by which the objective function exceeds the objective function with both reservoirs empty, as defined in (3-5)—is a function of storage in both the Columbia and Peace reservoirs, as well as the month. The storage value functions for January, Apr i l , June, and October are shown, respectively, in Figure 3-1 through Figure 3-4. 62 Storage Value Function for January Figure 3-1: Base Case January Storage Value Function Figure 3-2: Base Case April Storage Value Function Storage Value Function for June Figure 3-3: Base Case June Storage Value Function There are several features to note in Figure 3-1 through Figure 3-4. First, it is apparent that the storage values, and hence the objective function values from which they are derived, are dependent upon the quantity of water stored in both reservoirs. Second, note that the maximum storage values for January and October are an order of magnitude greater than those for Apr i l and June, indicating that the value of having water in storage is very much dependent upon the time of year. Finally, note that in Apr i l the value of having both reservoirs empty is dramatically different than having either of the reservoirs empty and the other as much as 90% empty. The three dimensional plots in the previous figures is useful for gaining an appreciation of the nature of the storage value surface. Further information is available by displaying 64 "slices" through the storage value surfaces. Figure 3-5 displays slices taken through the storage value functions for all months for an empty Columbia reservoir. 14000000000 > 8000000000 6000000000 C=0 NonAnt QT 1 SCENARIO - J A N - F E B MAR - A P R - M A Y - J U N - J U L - A U G - S E P OCT NOV DEC 50000 100000 1 SO000 200000 250000 300000 350000 400000 450000 600000 Peace Reservoir Storage (cmsd) Figure 3-5: Storage Value Function Slices for Columbia Reservoir Empty In Figure 3-5 observe that the storage value when the Peace reservoir is full is greatest in the early winter and late fall, is lowest in the spring and early summer, and is between these extremes during the late winter and late summer. These results agree with intuition, as they show that it is more valuable having the Peace reservoir full when in the low inflow/high demand/high price portion of the year, than it is during the high inflow/low demand/low price portion of the year. The slopes of the line segments comprising the storage value vs. Peace reservoir storage in Figure 3-5 are the marginal values of water stored in the Peace reservoir; the decreasing marginal value of this water is apparent. The same features are evident in Figure 3-6 and Figure 3-7, which, respectively, present the storage value functions for the Columbia reservoir half-full and full. C=100000 NONANT QT 1 SCENARIO 16000000000 14000000000 — AFR -JUN O C T N O V 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 Peace Reservoir Storage (cmsd) 65 Figure 3-6: Storage Value Function Slices for Columbia Reservoir 50% Full In Figure 3-6 notice the increased storage value intercepts for a given month as compared with those in Figure 3-5; these increased intercept values demonstrate the value of having more storage in the Columbia reservoir. Also note that the curves start to flatten out at lower storage volumes for the Peace reservoir, illustrating that as the volume stored in the Columbia reservoir changes, the marginal value of water stored in the Peace reservoir is affected. These trends are further evident in Figure 3-7. C=200000 N O N A N T QT 1 S C E N A R I O 16000000000 14000000000 loouoo'jo'iurj 1 > 8000000000 MAR — APR - M A Y - J U N —JUL —AUG —SEP OCT NOV DEC 200000 250000 300000 Peace Reservoir Storage (cmsd) Figure 3-7: Storage Value Function Slices for Columbia Reservoir Full Just as the marginal value of water in the Peace reservoir can be affected by storage in the Columbia reservoir, the marginal water value in the Columbia reservoir can depend on storage in the Peace reservoir. Figure 3-8, Figure 3-9, and Figure 3-10, which present storage value functions, respectively, for the Peace reservoir empty, half full, and full, illustrate this fact. 66 P = 0 N O N A N T Q T 1 S C E N A R I O 12000000000 0 20000 -10000 60000 80000 100000 120000 140000 160000 1B0000 200000 Columbia Reservoir Storage (cmsd) Figure 3-8: Storage Value Function Slices for Peace Reservoir Empty Similar to the Peace reservoir, the worth of having the Columbia reservoir full is greatest in the early winter and late fall, is lowest in the spring and early summer, and is in between these extremes during the late winter and late summer. Again, these results agree with intuition, as they show that it is more valuable to have the Peace reservoir full when in the low inflow/high demand/high price portion of the year, than it is during the high inflow/low demand/low price portion of the year. I.JMJI-OOO'JOO , P = 2 5 G 0 0 0 N O N A N T 1 S C E N A R I O 12000000000 -I _ 10000000000 A tSIXKlOIHXKX! J - J A N - F E B MAR —APR -MAY - J U N - J U L —AUG — SEP OCT NOV DEC 40000 60000 80000 100000 120000 140000 160000 180000 200000 Columbia Reservoir Storage (cmsd) Figure 3-9: Storage Value Function Slices for Peace Reservoir 50% Full The dependence of the storage value on the storage in both reservoirs is evident in a comparison of Figure 3-8 and Figure 3-9. In Figure 3-9 the storage value intercepts for a 67 given month are greater than those in Figure 3-8. The increased intercept values demonstrate the value of having more storage in the Peace reservoir. The flattening of the curves at lower storage volumes for the Columbia reservoir, illustrate that as the volume stored in the Peace reservoir changes, the marginal value of water stored in the Columbia reservoir is affected. These trends are further evident in a comparison of Figure 3-9 and Figure 3-10. P=500000 NONANT QT 1 SCENARIO —JAN — FEB MAR —APR — M A Y - J U N - J U L —AUG —SEP OCT NOV DEC 40000 60000 60000 100000 120000 140000 160000 180000 200000 Columbia Reservoir Storage (cmsd) Figure 3-10: Storage Value Function Slices for Peace Reservoir Full A s noted above, the slopes of the line segments in Figure 3-5 through Figure 3-10 give the marginal value of stored water, with units of $/cmsd. While these values can be of use in and of themselves, in the case of a hydroelectric system, the marginal values of stored energy, with units of $ /MWh, are of greater use. The conversion of marginal water to marginal energy values is straightforward. From equation (3-21) recall that the generation of power is modelled as the product of the constant HK with units of MW/cms and the turbine release in cms, and that HK is assumed dependent on reservoir storage, but not on turbine release. Thus, the marginal energy value ($/MWh) for a reservoir is equal to the marginal water value ($/cmsd) in that reservoir divided by H K (MW/cms) divided by the number of hours in a day (h/d). The HK value used in the conversion should be representative of the range of reservoir storage over which the conversion is being performed. In the current case of converting the slope of a line segment from one of the storage value function slices, the representative HK is equal to the average value of HK associated with the reservoir storage at each end of the line segment. Table 3-14 presents the range, over all of the storage states, of Columbia marginal energy values for each month for the one-scenario base case. It can be observed that, with the exceptions of May and June, there is a very considerable range between the minimum and maximum marginal energy values in each month. From Figure 3-5 through Figure 3-7 it is apparent that the high marginal Columbia energy storage values are associated 68 with low storage in both the Columbia and Peace reservoirs, while the low marginal Columbia energy storage values are associated with high storage in both reservoirs. Min Columbia Marginal Value ($/MWh) Max Columbia Marginal Value ($/MWh) January 23.87 1084.66 February 18.49 1078.02 March 4.90 1067.13 Apr i l 0.00 948.87 May 0.00 48.39 June 0.00 75.02 July 0.00 1067.16 August 18.66 1143.96 September 31.72 1125.67 October 31.02 1117.20 November 28.30 1110.95 December 26.04 1098.45 Table 3-14: Minimum and Maximum Monthly Base Case Columbia Marginal Energy Values In Table 3-14 note that for some spring and summer months the minimum marginal value of energy stored in the Columbia reservoir is equal to zero, indicating that there is water in excess of the turbine capacity. In reality, spills could potentially have negative value associated with them; these damages have not been considered in the model, possibly overestimating the minimum marginal energy values. Furthermore, note that with the exception of the months of May and June, the maximum marginal Columbia energy value is on the order of the curtailment cost of $1000/MWh. Recall from Table 3-1 that the freshet begins in May. This result suggests that i f both reservoirs are at their respective minimums in any month other than the period between the onset of the freshet and one month following its onset, curtailment w i l l be necessary. Table 3-15 presents the range of Peace plant marginal energy values for each month for the one-scenario base case over all of the storage states. Similar to the Columbia, with the exceptions of M a y and June, there is a very large range between the minimum and maximum Peace marginal energy values. Min Peace Marginal Value ($/MWh) Max Peace Marginal Value ($/MWh) January 24.97 1947.31 69 February 20.25 1065.03 March 9.11 1051.11 Apri l 1.17 554.03 May 0.00 45.56 June 0.00 64.56 July 8.53 1102.03 August 30.84 1483.03 September 31.42 1694.98 October 30.93 1868.19 November 29.09 2034.20 December 27.52 2111.38 Table 3-15: Minimum and Maximum Peace Marginal Energy Values The maximum marginal Peace energy values for August through January are considerably higher than the maximum marginal Columbia energy value for the corresponding month. These differences are attributable to the penalty that accrues for failing to meet the minimum Peace plant release, as detailed in Table 3-8. For low Peace reservoir levels there is insufficient water to maintain the higher December and January minimum releases. Note that the effect of the penalty for violation of the minimum Peace plant flow restraint is felt in months other than December and January, as it is necessary to have stored the water required to maintain this minimum release ahead of time. For February, March, and July, the maximum marginal Peace energy value is attributable to load curtailment, and is comparable to the maximum marginal Columbia energy value for the respective month. There is no consequential difference between the minimum marginal Peace and Columbia energy values. The minimum marginal Columbia energy values are, in general, slightly lower. This small difference is likely attributable to the fact that the ratio of average annual inflow to maximum reservoir storage is greater for the Columbia than it is for the Peace. The marginal water values for Figure 3-5, Figure 3-6, and Figure 3-7 have been converted to marginal energy values using the appropriate H K factors. The results are presented, respectively, as Figure 3-11, Figure 3-12, and Figure 3-13. The decreasing marginal value of stored energy is apparent. 70 C =0; Ice Restrictions; 1 Scenario :>i!00 50000 100000 150000 200000 250000 300000 350000 400000 450000 Peace Reservoir Storage (cmsd) Figure 3-11: Peace Marginal Energy Values for Columbia Reservoir Empty A comparison of Figure 3-11 and Figure 3-12 yields interesting results. In Figure 3-11, the effects of both the minimum Peace plant release penalty and curtailment cost are evident (i.e., the marginal Peace energy value is much greater than $ 1000/MWh), in at least some months, for Peace reservoir storage less than 150000 cmsd, and the effects of load curtailment are felt (i.e., marginal Peace energy value is much greater than market prices), at least in some months, for Peace reservoir storage less than 400000 cmsd. B y way of comparison, in Figure 3-12, the effects of both the minimum Peace plant release penalty and curtailment cost are also felt, at least in some months, for Peace reservoir storage less than 150000 cmsd; the magnitude of the effects are lesser in Figure 3-12. The difference in magnitude is attributable to the additional 100000 cmsd of Columbia reservoir storage, which serves to reduce the curtailment cost component of the marginal Peace energy value. The effect of the additional Columbia reservoir storage is seen more dramatically in the 250000 cmsd Peace reservoir storage level above which curtailment costs are not felt in any month. 71 C = 100000; Ice Restrictions; 1 Scenario 150000 200000 250000 300000 Peace Reservoir Storage (cmsd) Figure 3-12: Peace Marginal Energy Values for Columbia Reservoir 50% Full In comparing Figure 3-12 and Figure 3-13, neither the cost of load curtailment nor the minimum Peace plant release penalty costs are components of the marginal Peace energy value above Peace reservoir storage of 200000 cmsd. In fact, there is little or no curtailment cost component at any level of Peace reservoir storage with the additional Columbia storage. C = 200000; Ice Restrictions; 1 Scenario moo 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 Peace Reservoir Storage (cmsd) Figure 3-13: Peace Marginal Energy Values for Columbia Reservoir Full Marginal Columbia energy value functions for the Peace reservoir empty, half full, and full, respectively are shown in Figure 3-14, Figure 3-15, and Figure 3-16. 72 P = 0; Non Ant QT ice Restrictions; 1 Scenario 1400 20000 40000 60000 80000 100000 120000 140000 160000 1B0O00 200000 Peace Reservoir Storage (cmsd) Figure 3-14: Columbia Marginal Energy Values for Peace Reservoir Empty Figure 3-14 contains interesting information. The marginal Columbia energy values are very similar in nature for the late fall and early winter. In these months, the effect of the Columbia reservoir storage on the marginal Columbia energy value becomes much stronger beyond 120000 cmsd; below this volume the marginal Columbia energy value curves are relatively flat. The nature of these curves reflects the fact that in these months, flows are on the receding limb of the inflow hydrograph, and the high demand season is approaching. Beyond the 120000 cmsd point, the curtailment cost component is reduced somewhat, but cannot be eliminated—that is, i f the late fall and early summer months are started with the Peace reservoir empty, at some point load curtailment wi l l be necessary. For the late winter and early spring months, with sufficient Columbia storage it is possible to avoid curtailment even with an empty Peace reservoir. During this period, the later into the year, the lower the Columbia reservoir storage at which the curtailment effect is lessened. For May and June only reduced curtailment effects are felt at very low Columbia storage. August, and in particular, July, demonstrate different characteristics as well , with the effect of Columbia storage on the marginal Columbia energy value decreasing with increasing storage as the reservoir heads towards full. In these months the peak of the freshet is past, yet the high demand season is still some time away. 73 P = 250000; Ice Restrictions; 1 Scenario 0 20000 40000 60000 80000 100000 120000 140000 160000 180000 Peace Reservoir Storage (cmsd) Figure 3-15: Columbia Marginal Energy Values for Peace Reservoir 50% Full When the Peace reservoir is half full as shown in Figure 3-15, the effect of curtailment costs on the marginal Columbia energy value is greatly diminished as compared to when the Peace reservoir is empty, as shown in Figure 3-14; the diminishment is further evident in Figure 3-16 for which the Peace reservoir is full. When the Peace reservoir is full, and the Columbia reservoir is approaching full, for some months the marginal Columbia energy value is equal to zero, indicating that water wi l l be spilled. P = 500000; Ice Restrictions; 1 Scenario 7 0 0 T . Figure 3-16: Columbia Marginal Energy Values for Peace Reservoir Full In Figure 3-11 through Figure 3-16 there appear to be ranges o f storage in both reservoirs over which the marginal energy values are not too dependent upon the reservoir storage; however, the vertical scales used in there figures obscure the details. A closer examination reveals that there are regions over which the marginal energy storage value in one reservoir is independent of the storage in the other reservoir, and may in fact not 74 be all that dependent on the storage in the reservoir. Examples are shown below in Figure 3-17 through Figure 3-22. May Marginal Peace Energy Value ($/MWh) 0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 Peace Reservoir Storage (cmsd) Figure 3-17: May Marginal Peace Energy Value Functions In Figure 3-17, the slopes of the curves indicate the sensitivity of the marginal Peace energy value to Peace reservoir storage and the vertical distance between the curves indicates the sensitivity of the marginal Peace energy value to Columbia reservoir storage. Over a Peace reservoir storage range of 150000 to 300000 cmsd, the Peace marginal energy value is largely independent of the Columbia reservoir storage, for Columbia reservoir storage of at least 100000 cmsd. For example, for Peace reservoir storage o f 300000 cmsd, the marginal Peace energy value is $31.75/MWh for the Columbia reservoir half full and $31.45/M W h for the Columbia reservoir full. Similarly, over this Peace reservoir storage range, the influence of Peace reservoir storage on the Peace marginal energy value is lower than that for other Peace reservoir storage. A s noted, the marginal Peace energy values for May are least sensitive to changes in the Columbia reservoir storage over a "mid-range" of Peace storage (150000 to 300000 cmsd) as well as when the Peace reservoir is full. In March through July a similar behaviour is observed. Figure 3-17 contains another feature worth noting. In the figure, there are some cases in which the marginal Peace energy value is non-decreasing with increasing Peace reservoir storage. This discrepancy results from the maximum number of iterations being performed in the search for convergence recognition. The model handles non-convex storage value functions by applying an algorithm to smooth them such that they are convex. 75 May Marginal Columbia Energy Value (S/MWh) Figure 3-18: May Marginal Columbia Energy Value Functions In Figure 3-18, the slopes of the curves indicate the sensitivity of the Columbia marginal energy value to Columbia reservoir storage and the vertical distance between the curves indicates the sensitivity of the Columbia marginal energy value to Peace reservoir storage. Observe that when the Columbia reservoir storage is at least 140000 cmsd, the marginal Columbia energy value is essentially independent of Peace reservoir storage below 60% full. For Columbia reservoir storage of 140000 cmsd, the Columbia marginal energy value is $18.22/MWh when the Peace reservoir is empty and $17.90/MWh when it is 60% full. The reduced dependence of the Columbia marginal energy value on Peace reservoir storage as the Columbia reservoir approaches full illustrated in Figure 3-18 is representative of the behaviour in all months. Figure 3-19 displays the marginal Peace energy values for the month of November, while Figure 3-20 "zooms i n " on the lower marginal Peace energy values shown in Figure 3-19. 76 November Marginal Peace Energy Value Figure 3-19: November Marginal Peace Energy Value Functions In Figure 3-19, observe that for the Peace reservoir up to half full, the Peace reservoir exhibits little dependence on Columbia reservoir storage less than 60000 cmsd, relative to the other values. Additionally, note that for Peace reservoir storage of at least 350000 cmsd, the marginal Peace energy value does not exhibit much dependence on either Peace or Columbia reservoir storage at the scale of the figure. 5 <>} , , , , , , , , — — I 0 50000 1OOO00 150000 200000 250000 300000 350000 400000 450000 500000 Peace Reservoir Storage (cmsd) Figure 3-20: November Marginal Peace Energy Value Functions—Detail From the detail in Figure 3-20 observe that there is still some influence of both Peace and Columbia reservoir storage on the marginal Peace energy value. The reduced dependence of the Peace marginal energy values on Columbia reservoir storage as the Peace reservoir approaches full is demonstrated in both Figure 3-19 and Figure 3-20; this behaviour also occurs for the months of August through February. 77 Figure 3-21 presents the Columbia marginal energy value for November as a function of both Columbia and Peace reservoir storage. Figure 3-22 presents details at lower marginal Columbia energy values that cannot be readily seen in Figure 3-21. November Columbia Marginal Energy Values 1200 -, , Figure 3-21: November Marginal Columbia Energy Value Functions In Figure 3-21 note that the marginal Columbia energy value is much less sensitive to Peace reservoir storage above half full than it is to Peace reservoir storage below half full over the entire range of possible Columbia reservoir storage. In addition, for lower Columbia reservoir storage, the marginal Columbia reservoir storage is less dependent on Peace reservoir storage levels equal to 60000 cmsd or less than it is for other values of Columbia reservoir storage. Figure 3-22 shows that there is still a slight dependence of the marginal Columbia energy value upon both Columbia and Peace reservoir storage. Figure 3-22: November Marginal Columbia Energy Value Functions—Detail 78 In this section a one-scenario base case has been studied, establishing the behaviour of the storage values and marginal storage values for the mean values of the scenario-dependent demands, inflows, and prices. In the following section, scenarios in which the monthly scenario-dependent parameters vary symmetrically around their mean values are presented. B y comparing the results in the following section with those in this section, the effect of varying the scenario-dependent parameters is determined. 3.4.3.2 Results for Five-Scenario Cases with Non-Anticipative Turbine Release In this section, a number of five-scenario cases are studied. In each case, some subset of the scenario-dependent parameters assume specified values. A s outlined above, the scenario-dependent parameters are the demands, inflows, and prices. The specification of the scenarios includes correlations between the scenario-dependent parameters. The seven five-scenario cases outlined in section 3.3 above were examined for coefficient of variation values of 0.10, 0.25, and 0.40. In the months of November through Apr i l there is only one scenario for all cases; in the months of May through October, there are either five scenarios or one scenario as indicated in Table 3-16. Case Description Demand Inflow Price A Demands Only 5 values 1 value 1 value B Inflows Only 1 value 5 values 1 value C Prices Only 1 value 1 value 5 values D Perfect Positive Correlation 5 values 5 values 5 values E Demands and Inflows Perfectly Correlated and Prices Perfectly Negatively Correlated 5 values 5 values 5 values F Demands and Prices Perfectly Correlated and Inflows Perfectly Negatively Correlated 5 values 5 values 5 values G Inflows and Prices Perfectly Correlated and Demands Perfectly Negatively Correlated 5 values 5 values 5 values Table 3-16: Definition of Five-Scenario Cases The values assumed by the scenario-dependent parameters under the five-scenario cases are dependent upon the mean value of the parameter and the coefficient of variation of the parameter. For the cases in which all of the scenario-dependent parameters assume five different values, the coefficient of variation is assumed to be the same for all of the parameters—that is, i f the coefficient of variation for the demand is 0.10, then the coefficients of variation for the inflows and prices are also 0.10. Under a scenario, the scenario-dependent parameter is equal to the parameter mean plus the product of the parameter mean, the coefficient of variation, and the characteristic values for a unit normal function for the scenario. The unit normal characteristic values and the discrete occurrence probabilities of the scenarios are defined in Table 3-17. The values in Table 3-17 are from K i m and Palmer (1997). 79 Unit Normal Value Probability Very Low -1.72 0.107 Low -0.76 0.245 Average 0.00 0.296 High 0.76 0.245 Very High 1.72 0.107 Table 3-17: Unit Normal Function Values for Five-Scenario Cases A l l of the cases studied in this section are for non-anticipative turbine releases—that is, equation (3-23) specifies that the turbine release must be the same over all of the scenarios. The effect of these non-anticipative constraints is to prevent the model from acting with perfect knowledge of the future. A Cases: Scenario-Dependent Demands Only In the scenario-dependent demands only cases, as shown in Table 3-16, the inflows and prices do not vary with the scenario, while the values assumed by the demand and the scenario probabilities, are defined by the characteristic values in Table 3-17. Perfect positive correlation is assumed between heavy load hour and light load hour demands. The cases studied are for coefficients of variation of 0.10, 0.25, and 0.40. The mean demands are shown in Table 3-3. In order to assess the effect of varying the coefficient of variation, a comparison of the minimum and maximum marginal energy values, for both the Columbia and Peace reservoirs, is made. For each coefficient of variation value, the monthly marginal values of energy stored in the Peace reservoir were calculated at an increment of 50000 cmsd from 0 to 500000 cmsd for the Peace reservoir and a 20000 cmsd increment in the Columbia reservoir between 0 and 200000 cmsd. For each month, and for each coefficient of variation value, the minimum Peace marginal energy values ($/MWh) are shown in Table 3-18, in which " C V " denotes the value of the coefficient of variation. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 J A N 24.97 24.34 22.10 8.25 F E B 20.25 16.68 4.11 0.24 M A R 9.11 3.23 0.41 0.21 A P R 1.17 0.07 0.26 0.00 M \ ^ 0 0 0 )()() DOO 0 0 0 .11 \ ( H m OIK) 0 0 0 0 0 0 J L I 3 So - _. o 0 0 0 0 0 80 A U G 30.84 27.79 21.78 9.50 S E P 31.42 31.47 34.65 19.23 O C T 30.93 31.11 34.87 25.55 N O V 29.09 30.21 30.93 25.32 D E C 27.52 27.75 27.70 20.98 Table 3-18: M i n i m u m Peace Marg ina l Energy Values for Demands Only Scenarios In Table 3-18, the "CV=0.00" scenario is the one-scenario base case from section 3.4.3.1, and the cells for which there are five scenarios are shaded. The data in Table 3-18 are presented graphically as Figure 3-23. Minimum PCE Marginal Energy Values for Demands Only Scenarios 40.00 -I JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-23: M i n i m u m Peace Marg ina l Energy Values for Demands On ly Scenarios The seasonal shape of the minimum Peace marginal energy value curves in Figure 3-23 is representative of those for cases B through G as well . Note that, for the most part, the higher the coefficient of variation—that is, the larger the range of demands covered by the scenarios—the lower the minimum Peace marginal energy value; the exceptions occur in the fall months. Thus, the one-scenario case is, in general, an upper bound on the minimum marginal Peace energy value. In months other than January, February, August, and September, the range in the minimum marginal Peace energy values over all of the C V values is less than $10/MWh. Table 3-19 presents the corresponding maximum Peace reservoir marginal energy values ($/MWh) for the three demands only scenarios, as well as for the one-scenario base case. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 J A N F E B M A R 1947.31 1065.03 1051.11 1948.38 1064.49 1051.56 1948.89 1064.52 1053.48 1947.30 1064.48 1051.21 81 APR 554.03 570.20 639.19 557.86 M A Y 45.56 76.06 210.63 53.72 JUN 64.56 193.08 362.84 160.20 J U L 1102.03 995.08 922.85 843.17 A U G 1483.03 1478.79 1466.76 1454.23 SEP 1694.98 1695.31 1707.04 1708.10 O C T 1868.19 1860.66 1866.72 1867.49 NOV 2034.20 2034.06 2033.31 2034.11 D E C 2111.38 2111.34 2111.22 2111.37 Table 3-19: Maximum Peace Marginal Energy Values for Demands Only Scenarios The data in Table 3-19 are presented graphically as Figure 3-24. Again, the seasonal shape of the maximum marginal Peace energy values in Figure 3-24 is representative of cases B through G as well . Max PCE Marginal Energy Values for Demands Only Scenarios Figure 3-24: Maximum Peace Marginal Energy Values for Demands Only Scenarios Note that the seasonal pattern of maximum marginal Peace energy values is independent of the coefficient of variation. However, the coefficient of variation value can affect the maximum marginal Peace energy value in a month by up to $300/MWh; in some other months, the range over all coefficient of variation values is less than $ l / M W h . The influence of the coefficient of variation value is greatest for the Apr i l through July period. Note that the one-scenario base case can either under- or over-estimate the marginal Peace energy value over all of the other coefficient of variation values. For each coefficient of variation value the monthly Columbia marginal energy value has been calculated at an increment of 20000 cmsd from 0 to 200000 cmsd for the Columbia reservoir and at a 50000 cmsd increment in Peace reservoir storage between 0 and 500000 cmsd. For each month, and for each scenario, the minimum and maximum marginal Columbia energy values ($/MWh) are shown in Table 3-20. 82 C V = 0.00 C V = 0.10 CV = 0.25 C V = 0.40 Min Max Min Max Min Max Min Max J A N 23.87 1084.66 24.33 1084.70 22.70 1084.87 5.99 1084.68 F E B 18.49 1078.02 15.10 1078.01 4.35 1077.95 0.26 1078.02 M A R 4.90 1067.13 1.48 1067.13 0.23 1067.13 0.13 1067.13 APR 0.00 948.87 0.00 954.41 0.11 973.19 0.00 950.48 M A Y 0.00 48.39 0.00 93.59 0.00 246.65 0.00 61.56 JUN 0.00 75.02 0.00 239.49 0.00 400.68 0.00 209.38 J U L 0.00 1067.16 0.00 964.38 0.00 847.24 0.00 746.11 A U G 18.66 1143.96 13.37 1131.56 0.00 1107.85 0.00 1090.67 SEP 31.72 1125.67 31.72 1124.69 32.73 1121.37 14.93 1119.35 O C T 31.02 1117.20 31.25 1117.41 31.95 1118.17 25.89 1116.79 N O V 28.30 1110.95 30.32 1111.19 30.71 1112.05 26.20 1111.01 D E C 26.04 1098.45 27.52 1098.58 27.13 1099.04 21.53 1098.49 Table 3-20: Minimum and Maximum Columbia Marginal Energy Values for Demands Only Scenarios The minimum Columbia marginal energy values shown in Table 3-20 are presented graphically as Figure 3-25. The shape of the minimum Columbia marginal energy values in Figure 3-25 are representative of cases B to G . Minimum COL Marginal Energy Values for Demands Only Scenarios Figure 3-25: Minimum Columbia Marginal Energy Values for Demands Only Scenarios The seasonal pattern displayed by the minimum Columbia marginal energy values is similar to that for the minimum Peace marginal energy values. Again, in general, the minimum marginal energy value decreases with an increasing coefficient of variation. With the exception of the autumn months, the one-scenario base case sets an upper bound on the minimum Columbia marginal energy value. The range of difference explained by the coefficient of variation values only exceeds $10/MWh in the months of January, 83 February, August, and September. In some summer months, the minimum Columbia marginal energy value does not differ amongst the coefficient of variation values considered. The maximum marginal Columbia energy values shown in Table 3-20 are presented graphically as Figure 3-26. The seasonal shape of the maximum Columbia marginal energy values in Figure 3-26 are representative of cases B to G . Maximum COL Marginal Energy Values for Demands Only Scenarios 1400.00 -, , 1200.00 0.00 -I • 1 , , , , , , , , r JAN FEB MAR APR MAY JUN JUL AUG S E P OCT NOV DEC Figure 3-26: Maximum Columbia Marginal Energy Values for Demands Only Scenarios As for the maximum Peace marginal energy values, the seasonal shape of the maximum Columbia energy value curve is independent of the coefficient of demand variations, and it is in the spring and summer that the maximum marginal energy values are most sensitive to the C V . At one extreme, over the range of coefficients of variation from 0.0 to 0.4, the monthly maximum value can vary by over $325/MWh; at the other extreme, the variation is less than $ l / M W h . The general observations to be taken from this section are that: the minimum marginal energy values are most sensitive to variation in demand over the autumn and winter; the maximum marginal energy values are most sensitive to variation in demand over the spring and early summer; there is the potential for the maximum, as compared to the minimum, marginal energy values to be more strongly affected by the coefficient o f demand variation; for all but the autumn, the base case sets an upper bound on the monthly minimum marginal energy values; and the base case can set either a upper or lower bound on the monthly maximum marginal energy values. B Cases: Scenario-Dependent Inflows Only In these cases the demands and prices do not vary between scenarios in any month. There are five scenarios for the inflows in the months of May through October, and there is only one scenario in the months of November through Apr i l . The cases studied are for 84 coefficient of variation values of 0.10, 0.25, and 0.40. Table 3-1 defines the mean inflows, and the inflow values in the five scenarios are given by Table 3-17. Perfect positive correlation is assumed between inflows into the Columbia and Peace reservoirs. The minimum and maximum marginal energy values for the Peace and Columbia are given, respectively in Table 3-21 and Table 3-22. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max Min Max Min Max M i n Max J A N F E B M A R A P R M A Y JW^ J U L ~ A U G si r ( K T * N O V D E C 24.97 20.25 9.11 1.17 11 00 0 00 s ? 3 >0S4 31 42 M)»V, 29.09 27.52 1947.31 1065.03 1051.11 554.03 45^ 56 64 5,6^  * n 02 03,1 1^*483*03 4694 98 v?868 19? 2034.20 2111.38 25.10 20.21 10.05 1.25 V-0 oo' 10 34 f ?3026j V3rn 30 61 29.63 27.52 1948.25 1064.48 1051.13 554.84 • 47 14 f\06\ 15 I4M) ^ 2 1690 48! % I860 981 2034.26 2111.37 24.47 19.61 11.17 2.46 V6*22 *0 25' 1 I 16 28 60 '30 48 ?29 94 28*91* 27.07 1948.39 1064.49 1051.60 571.48 M)61 180 56 1001 67 1478 34s Y69f 16 1860 33 2034.07 2111.34 23.71 18.73 12.08 4.72 1 *7 1 92 11 74 2 " " 2 . 29 2 ^ 4 0 28.19 26.58 1949.59 1064.73 1055.45 709.56 151 16 ^ : M .990 1473 67' 1690 ^ 2 1858 93 2032.39 2111.04 Table 3-21: Minimum and Maximum Peace Marginal Energy Values for Inflows Only Scenarios For the coefficient of variation values studied, the minimum marginal Peace energy value is not affected to a great extent in absolute terms. In each month, the four values for the minimum Peace marginal energy value are within $5 /MWh of one another. In comparison with case A , overall the coefficient of variation has less effect on the minimum Peace marginal energy values for case B ; Apr i l , May, and June are exceptions. Depending upon the month, the minimum marginal Peace energy value for the one-scenario base case can be either an upper or lower bound. The maximum Peace marginal energy value for the one-scenario base case can also be an upper or lower bound, depending upon the month. For the spring and early summer months, the maximum Peace marginal energy value increases with an increasing coefficient of variation—that is, when there is a wider range of possible inflows, the maximum marginal energy value is greater. Within a month, variation in the coefficient of variation can cause differences in the maximum Peace marginal energy value of more than $300/MWh or less than $ l / M W h ; for each of the months from November through March the spread in maximum marginal energy values is less than $5 /MWh. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max Min Max Min Max M i n Max J A N F E B M A R 23.87 18.49 , 4.90 1084.66 1078.02 1067.13 24.41 18.91 6.63 1084.66 1078.02 1067.13 23.87 18.26 8.96 1084.70 1078.01 1067.13 22.93 17.45 10.43 1085.16 1077.89 1067.13 85 A P R 0.00 948.87 0.09 949.19 1.31 954.27 3.66 1003.94 %Xv 01)11 18 T 3§| ooo 150% "W* fe* 9244 - 1 13 1 22 o00 ^ 02 ooo 51)1537^ 0 02 - 210 3b 3S1 3S II i 0 00 IOf.7 16 ooo 1064 04 1 61 1003 fo + 24 ONO 04 \i <. 18 66 1 143 06 19 43 1 141 50 IN r 1 1 30 3X 16 05 1 1 37 25 SI p :QCI 31 ^ 2 1 12* 6" 31 25 1 126 00 1 1 26 SS 20 00 *. 1/130 43 3|?02~ 30 1 $ 7 ^ 1 9 ' M l "M-l a!i9 3a , 1 120 30 NOV 28.30 " 1110.95*" 29.70 l f l 0*97 28.84 1111.19 28.06 Tim 6 D E C 26.04 1098.45 27.21 1098.45 26.82 1098.57 26.27 1099.75 Table 3-22: Minimum and Maximum Columbia Marginal Energy Values for Inflows Only Scenarios As with the minimum marginal Peace energy values, the minimum marginal Columbia values over all o f the coefficient of variation values studied lie within a narrow band. In all months except for March, the range is less than $5 /MWh. Also, as with the Peace values, there is less spread than for the case A values except for in the spring/early summer period. Depending on the month, the minimum Peace marginal energy value for the one-scenario base case can either be an upper or lower bound. For the maximum Columbia marginal energy values, the one-scenario base case can set either an upper or lower bound. Similarly to the Peace, over the spring and early summer months the maximum Columbia marginal energy value increases with an increasing coefficient of variation. The range of coefficients of variation considered can account for a difference in the maximum Columbia marginal energy value of as much as $450/MWh, or as little as $0 /MWh. The general observations to be taken from this section are that: the minimum marginal energy values, in general, exhibit less sensitivity to a range of coefficients of inflow variation than for an equal variation in the coefficient of demand variation; over the spring and early summer months the maximum marginal energy values for both reservoirs increase with an increasing coefficient of variation; and, the base case can set either an upper or lower bound on the monthly minimum or maximum marginal energy values for either the Peace or Columbia. C Cases: Scenario-Dependent Prices Only In the C cases the demands and inflows do not vary between scenarios. For each month from November through Apr i l there is only one scenario for the price, and in the remaining months, there are five scenarios. Perfect positive correlation between heavy load hour and light load hour prices is assumed. The mean prices are defined in Table 3-2, and the prices for the five scenarios are defined by these average prices and the unit normal values specified in Table 3-17. The cases studied are for coefficient of variation values of 0.10, 0.25, and 0.40. 86 In this section, the non-anticipative turbine release constraints ensure that the turbine release is the same over all of the scenarios. As the only parameters that are changing with the scenario are the prices, the non-anticipative constraints ensure that the turbine release is independent of the energy prices. As the releases are the same for each scenario, so is the generation, as are the demands and inflows. Recall from the formulation presented in (3-1) to (3-29) that the H L H and L L H prices are only present in the objective function (3-29), where they are used to determine the export revenues and import costs. Since the generation and demand are the same under all scenarios, so w i l l be the imports and exports, as well as the end of period storage volumes. Thus, the only difference between the objective function values under each scenario wi l l be a linear function of the H L H and L L H energy prices. So, the probability weighted objective function value over all of the price scenarios is equal to the objective function value for the mean price. In other words, given the symmetrical distribution, for these prices only cases, the results are independent of the coefficient of variation, and are equal to the results for the one-scenario base case, reported in section 3.4.3.1. D Cases: Scenario-Dependent Demands, Inflows, and Prices—All Perfectly Positively Correlated For the D cases, each of the five scenarios for May through October has unique demands, inflows, and prices. There is only one scenario for the months of November through Apr i l . The assumption is made that demand, inflow, and price are all perfectly positively correlated—that is, when the demands are "Very Low" , as defined in Table 3-17, the inflows and prices are also "Very Low" ; similarly, i f the demands are "Very High", so are the other two scenario-dependent parameters. Mean inflows, prices, and demands are respectively given by Table 3-1, Table 3-2, and Table 3-3. Perfect positive correlation between H L H and L L H prices is assumed, as is perfect positive correlation between H L H and L L H demands, and between Peace and Columbia inflows. The D cases are "good news/bad news cases"—if the demand is high, the inflows w i l l also be high allowing for more generation, but the price of any imports w i l l be high. If the demand is low, the inflows w i l l be low, but the prices w i l l also be low. Coefficient of variation values of 0.10, 0.25, and 0.40 are studied. The purpose of studying these perfect correlation cases is to present an extreme case of possible results. The minimum and maximum marginal energy values for the Peace and Columbia are given, respectively in Table 3-23 and Table 3-24. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max M i n Max Min Max M i n Max J A N 24.97 1947.31 24.43 1948.45 22.59 1949.20 5.33 1949.54 F E B 20.25 1065.03 17.32 1064.49 9.64 1064.69 0.73 1064.76 M A R 9.11 1051.11 3.96 1051.84 0.16 1054.75 0.05 1055.58 A P R 1.17 554.03 0.17 580.21 0.00 684.41 0.04 714.44 VI V'S 0.00 45.56 0.00 i;7.84 0 00 303.52 O.oo 362 S2 J l \ l).(K) 64 56 o.os 198.52 0 00 403.56 ().()() 87 J M ?SEPJ | | N O V D E C 8 53 30 84 31 42 M) 93 | 29.09 27.52 1 102 01 4 19 964 S2 1 ^6 86^ 96 o 14 814 ^S 1483 03 2 ' 7 6 l I4SSS6 19 2" 145^ ^0 N 6 - 1427 S"7 C ; J 6 9 / W 8 11 1 ^ p o i 1 ^08 06 16 16 ;^i7do55-^1868 19~ V 3 0 ^ i 186^ 61 22 ' ;1858J62 2034.20 29.81 2033.97 30.61 2032.86 * 25.15* 203I4V 2111.38 27.57 2111.33 26.94 2111.14 18.92 2111.05 Table 3-23: Minimum and Maximum Peace Marginal Energy Values for Perfect Positive Correlation Scenarios With the exception of the late fall and early winter, the minimum Peace marginal energy values decrease with an increasing coefficient of variation. In the D cases, a higher coefficient of variation translates into a wider range of demands, inflows, and prices covered over the scenarios. So, the wider the range of scenario-dependent parameters, the lower the minimum Peace marginal energy value. The maximum range in monthly minimum Peace marginal energy values over the coefficient of variation values tested was approximately $20/MWh, while the minimum range was $0 /MWh. Over the range of coefficient of variation values tested, the monthly Peace maximum marginal energy value varies by as much as $340/MWh, or as little as less than $ l / M W h . Over the spring and early summer, the maximum Peace marginal energy value increases with an increasing coefficient of variation. However, over the late summer and early fall, the maximum Peace marginal energy value decreases with an increasing coefficient of variation. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max Min Max Min Max M i n Max 2.57 1085.12 0.00 1077.90 0.00 1067.13 0.00 999.92 0 00 464 49 0 00 4 28 48 0 00 89 lip U i i K 8 0 ~ f 121 -93< 11 19 20 99 1 1 19 81 24.88 1113.12 17.34 1099.67 J A N F E B M A R A P R M \ \ .11 N JUL AU<, SEP OCT NOV D E C 23.87 18.49 4.90 0.00 0 00 0 00 0 00 ,18 66 II 72 11 02 28.30 26.04 1084.66 1078.02 1067.13 948.87 48 19 02 1067 16 :|W*43'?6 I 12^ 6^ I 1 r 20 1110.95 1098.45 24.18 15.71 2.15 0.00 0 00 0 00 0 00 •15 16 11 r 30 78 29.77 27.29 1084.73 1078.00 1067.13 956.74 I 12 56 21^ IS 955 06 1 132 24 1 124 85 I I 17 52 1111.29 1098.64 22.62 7.97 0.00 0.00 0 00 0 00 0 00 28'37 11 H) 30.32 26.97 1084.97 1077.92 1067.13 984.36 11"* •'I 432 M 85^ 86 I09S II 1122*:22 1119 :27 1112.59 1099.32 Table 3-24: Minimum and Maximum Columbia Marginal Energy Values for Perfect Positive Correlation Scenarios For the Columbia, as for the Peace, with the exception of the late fall and early winter, the minimum marginal energy value decreases with an increasing coefficient of variation. 88 The range of coefficient of variation values studied results in a monthly range in the minimum marginal energy value of between $0 /MWh and $22/MWh. Over the range of coefficient of variation values tested, the monthly maximum Columbia marginal energy value can vary by as much as $415/MWh, or as little as less than $ l / M W h . A s with the Peace, during the spring and early summer, the maximum Columbia marginal energy value increases with an increasing coefficient of variation, while during the late summer and early fall, the maximum Columbia marginal energy value decreases with an increasing coefficient of variation From this section, the general observations that can be taken are: except for the late fall/early winter period, the minimum marginal energy values for both the Columbia and Peace decrease as the coefficient of variation increases; during the spring and early summer the maximum marginal energy values for both reservoirs increases with the coefficient of variation; and, during the late summer and early fall, the maximum energy values for the Columbia and Peace decrease with an increasing C V . E Cases: Scenario-Dependent Demands, Inflows, and Prices—Demands and Inflows Perfectly Positively Correlated and Prices Perfectly Negatively Correlated In the E cases there are five scenarios for the months of M a y through October, with each scenario in a month having unique demands, inflows, and prices. There is only one scenario for each of the months from November through Apr i l . The assumptions are that the demands and inflows are perfectly positively correlated and the prices are perfectly negatively correlated with the demands and inflows—for example, i f the demands are " L o w " as defined by Table 3-17 then the inflows are also " L o w " , and the prices are "High". Mean inflows, prices, and demands are given, respectively, in Table 3-1, Table 3-2, and Table 3-3. Perfect positive correlation between H L H and L L H prices is assumed, as is perfect positive correlation between H L H and L L H demands, and between Peace and Columbia inflows. The E cases are "generally good news" cases. If the demand is high, the inflows wi l l also be high allowing for more generation, and the prices wi l l be low i f imports are required. On the other hand, i f the demand is low, the inflows wi l l also be low, but the prices w i l l be high i f there is any energy for export. Coefficient of variation values of 0.10, 0.25, and 0.40 are studied. The same value of the coefficient of variation is assumed for each scenario-dependent parameter. The minimum and maximum marginal energy values for the Peace and Columbia are given, respectively, in Table 3-25 and Table 3-26. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max Min Max Min Max M i n Max J A N F E B 24.97 20.25 1947.31 1065.03 24.38 17.26 1948.45 1064.49 22.64 9.66 1949.21 1064.69 5.47 1.00 1949.56 1064.76 89 M A R 9.11 1051.11 4.02 1051.84 0.18 1054.79 0.04 1055.64 A P R 1.17 554.03 0.16 580.29 0.00 686.07 0.05 716.48 .11 \ .11 1 0 0 0 } © ¥ 5 6 1 I02T /3' 0 0 0 ^98 00 iC.306 8Ch l |Wo) 3(,6 84 n 00 « 8 5 ' o 08 4 1^  199-22 065 41 -•fB?or| 1 56 | S > 7 1 43? II 00 0 13 307.52 21 02 \ l (. 30 S4 1483 03 2n 52 U S S S 6 10 2^ I45S 45 *$&7pS 1420 01 SI l» >l 42 1094 OS 31 02 r 0 3 32 30 03 1 OS 0 0 I'. II 1 700 55 O f ' i l l 30 93 1S6S 19 3 0 " 0 1N6N SO 32 0 0 1 S65 5ft ' %2LV» 1JOX 60 N O V "^29.09 2034.20 29.76 2033.96 30.61 " 2032.84^ 25A5 2032.43 D E C 27.52 2111.38 27.57 2111.33 26.94 2111.14 18.87 2111.05 Table 3-25: Minimum and Maximum Peace Marginal Energy Values for Perfect Correlation—Positive for Demands and Inflows and Negative for Prices The minimum Peace marginal energy value decreases with an increasing coefficient of variation value except for in the late fall and early winter. For the range of coefficient of variation values considered, the maximum range of minimum Peace marginal energy values in any month is $22/MWh, while the minimum range is $0 /MWh. The range, over the coefficient of variation values studied, of maximum Peace marginal energy values in a month ranges from a high of $340/MWh to a low of less than $ l / M W h . Over the spring and early summer, the maximum Peace marginal energy value increases with an increasing coefficient of variation; whereas, during the late summer and early fall, the maximum Peace marginal energy value decreases with an increasing coefficient of variation. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max Min Max Min Max M i n Max J A N 23.87 1084.66 24.18 1084.73 22.70 1084.97 2.57 1085 13 F E B 18.49 1078.02 15.64 1078.00 8.05 1077.92 0.00 1077 90 M A R 4.90 1067.13 2.53 1067.13 0.00 1067.13 0.00 1067 13 A P R 0.00 948.87 0.00 956.77 0.00 984.80 0.00 1000 55 ;M^YX r ^%;o:oo- 4S 391 i>o6; 4 1 12 M ^ O ' o i . 31120 o 00 ^ 4 6 9 67-' M ? 0 p0 f <»2t . O I H . " o ooj 43' ' 24 mo oo; 4 53 63 JL Lf | | | f o o o n x s f 16: IKK) • ^ 9 5 6 03 ^ 7 s \ 3 (•00 ,765 s , A U ; 1 s 66 1 143 06 15 OS 1 1 32 20 6 5 3 1000 04 2 72 1073 3 3 SI l» 31 "72 1 125 ft- 3 1 02 I r 1 ^5 2S 3"1 1 I22 2> 1 1 1" 1 122 02 O C 1 3 1 02 t : ;.l;fF7 20 30 63 11 r >2 31 hn 1 110 3| £ 2 t § 9 ; 1 110 SS N O V 28.30 TllO.95 29.70 1111.30 3(132 ' 1112.6V '"24.88 1113 14 D E C 26.04 1098.45 27.29 1098.64 26.97 1099.33 17.18 1099 69 Table 3-26: Minimum and Maximum Columbia Marginal Energy Values for Perfect Correlation—Positive for Demands and Inflows and Negative for Prices Similarly to the Peace, the minimum Columbia marginal energy value decreases with an increasing coefficient of variation. In addition, the maximum marginal Columbia energy 90 value increases with an increasing coefficient of variation during the spring and early summer; whereas, during the late summer and early fall, the maximum marginal Columbia energy value decreases with an increasing coefficient of variation. The range of variation in the monthly minimum Columbia marginal value attributable to the range of coefficients of variation tested is between $21/MWh and $0 /MWh; this range is very similar to that for the Peace. The range, over the coefficient of variation values studied, of maximum marginal Columbia energy values in a month ranges from a high of $425/MWh to a low of less than $ 1/M Wh. From this section, the general observations are: except for the late fall/early winter period, the minimum marginal energy values for both the Columbia and Peace decrease as the coefficient of variation increases; during the spring and early summer the maximum marginal energy values for both reservoirs increases with the coefficient of variation; and, during the late summer and early fall, the maximum energy values for the Columbia and Peace decrease with an increasing C V . Note that these observations are the same as those for the D cases. The difference between cases D and E is that in case E , the prices are perfectly negatively correlated with the demands and inflows, while in case D all of the scenario-dependent parameters are perfectly positively correlated. F Cases: Scenario-Dependent Demands, Inflows, and Prices—Demands and Prices Perfectly Positively Correlated and Inflows Perfectly Negatively Correlated The cases considered here all have five scenarios for the months of M a y through October, with each scenario in these months having unique demands, inflows, and prices. The months of November through Apr i l each have a single scenario. It is assumed that the demands and prices are perfectly positively correlated and the inflows are perfectly negatively correlated with the demands and prices. For example, i f the demands are "Very High" as defined by Table 3-17 then the prices are also "Very High", and the inflows are "Very Low". Perfect correlation between H L H and L L H prices is assumed, as is perfect positive correlation between H L H and L L H demands, and between Peace and Columbia inflows. The F cases studied here can be thought of as "bad news" cases—if the demand is high, the inflow w i l l be low, and the prices w i l l be high i f imports are needed. Conversely, i f the demand is low, there wi l l be high inflow, but prices in the export market w i l l be lower. Coefficient of variation values of 0.10, 0.25, and 0.40 are considered. The same value of the coefficient of variation is assumed for each scenario-dependent parameter. The minimum and maximum marginal energy values for the Peace and Columbia reservoirs are given, respectively, in Table 3-27 and Table 3-28. 91 C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max M i n Max M i n Max M i n Max J A N 24.97 1947.31 24 .38 1948.46 23.22 1949.21 16.14 1949.50 F E B 20.25 1065.03 17.29 1064.49 11.83 1064.69 7.82 1064.74 M A R 9.11 1051.11 4.18 1051.88 2.88 1054.78 3.36 1055.46 A P R 1.17 5*4 01 0.10 581.51 0.88 685 S"7 1.68 709.89 M \ \ 0 00 4 * 56 0 00 100 40 0 41 106 42 0 6S 151 81 J l N 0 00 64 *6 0 00 201 9S 0 00 404 14 0 29 • 187 10 J l 1. 8 ^ 1 1 102 01 4 20 961 So 1 74 S o - 7 09 1 810 1 1 u<; 10 S4 I 4 M 01 27 61 I4SS ss 19 81 \45n IS Knr 142* -() SEP 3 1 42 1694 9,K 31 15 |7()1 29 30 48 17(F t>n 19 77 P O O 60 O C T 10 93 1868.19 30.79 1868 S - 7 31.87 1865 .16 25 24 I85N.74 N O V 29.09 2034.20 29.85 2033.95 30.75 2032.85 26 W 2032.52 D E C 27.52 2111 .38 27.57 2111.33 27.34 2111.14 23 .49 2111 .06 Table 3-27: Minimum and Maximum Peace Marginal Energy Values for Perfect Correlation—Positive for Demands and Prices and Negative for Inflows The maximum range of minimum Peace marginal energy values in any month is $20/MWh, while the minimum range is $0 /MWh. The range, considering all months, over the coefficient of variation values studied is tighter than all previously considered cases, except for Case B , for which only the inflows varied. The minimum marginal Peace energy value decreases with increasing coefficient of variation for the late winter and the late summer and early fall. The maximum Peace marginal energy value increases with an increasing coefficient of variation over the spring and early summer, while over the late summer and early fall the maximum Peace marginal energy value decreases with an increasing coefficient of variation. The range, over the coefficient of variation values studied, of maximum Peace marginal energy values in a month ranges from a high of $320/MWh to a low of less than $ l / M W h . C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max M i n Max Min Max Min Max J A N 23.87 1084.66 24.18 1084.73 22.54 1084.97 12.90 1085.10 F E B 18.49 1078.02 15.68 1078.00 9.79 1077.92 5.72 1077.90 M A R 4.90 1067.13 2.49 1067.13 2.08 1067.13 2.08 1067.13 A P R 0 00 948.87 0.02 9 5 ^ 0 6 0.72 9S4.67 0.73 997.94 M A Y 0 00 48 19 0.00 115 13 0.00 340 20 0 13 448.39 J I N 0 00 "\s 02 0 00 218 0.00 412 15 0 14 414 o-7 J l ' L 0 00 1067 16 0 00 9 5 1 4 1 0 01 8S2 17 0 91 754 04 u<; 18 66 1 141 96 P 08 1 112 08 " '46 \wn 88 4 12 K P I 08 SEP 11 "*2 112* <r 11 17 1 124 SS 28 68 1 122 23 14 is 1 121 91 OCT 11 02 IIP 20 10 7S 1 1 P M 11 ~2 1 1 19 29 21 12 1 1 19 2^ N O V 28.30 ' 1110.95 29.85 1111.31 30.16 1112.61 25.73 1113.01 92 D E C 26.04 1098.45 27.29 1098.65 26.90 1099.33 21.38 1099.63 Table 3-28: Minimum and Maximum Columbia Marginal Energy Values for Perfect Correlation—Positive for Demands and Prices and Negative for Inflows As was the case for the Peace, the minimum Columbia marginal energy values have less spread, considering all months, over the coefficient of variation values studied for all cases except for B , the case with inflows as the only scenario-dependent parameters. The range of variation in the monthly minimum Columbia marginal value attributable to the coefficients of variation tested is between $18/MWh and less than $ l / M W h . In addition, the minimum Columbia marginal energy value decreases with an increasing coefficient of variation for the late winter and the late summer. The maximum Columbia marginal energy value increases with an increasing coefficient of variation over the spring and early summer, while over the late summer and early fall the maximum Columbia marginal energy value decreases with an increasing coefficient of variation; this was also true for the maximum Peace marginal energy value. The range, over the coefficient of variation values studied, of maximum Columbia marginal energy values in a month ranges from a high of $400/MWh to a low of less than $3 /MWh The observations that can be taken for the F cases are: the minimum marginal energy value for both reservoirs decrease with an increasing coefficient of variation for the late winter and the late summer; the minimum marginal energy value curves are closer together for all cases studied so far except for the case with inflows as the only scenario-dependent parameters; during the spring and early summer the maximum marginal energy values for both reservoirs increases with the coefficient of variation; and, during the late summer, the maximum energy values for the Columbia and Peace decrease with an increasing coefficient of variation. G Cases: Scenario-Dependent Demands, Inflows, and Prices—Inflows and Prices Perfectly Positively Correlated and Demands Perfectly Negatively Correlated The G cases considered here have five scenarios in each of the months from May through October, with each scenario in these months having unique values for the demands, inflows, and prices. The months of November through Apr i l each have a single scenario. In this section it is assumed that the inflows and prices are perfectly positively correlated, and that both of these parameters are perfectly negatively correlated with the demands. B y way of example, i f the prices are "High" , as defined by Table 3-17, then the inflows are also "High" and the demands are "Low" . Perfect positive correlation between H L H and L L H prices is assumed, as is perfect positive correlation between H L H and L L H demands, and between Peace and Columbia inflows. The G cases are "good news/bad news cases". If the demand is high, the inflows wi l l be low, but the price of any necessary imports w i l l be low. On the other hand, i f the demand is low, the inflows wi l l be high, and any exports w i l l be sold at a high price. 93 Values for the coefficient of variation of 0.10, 0.25, and 0.40 are studied. The same coefficient of variation value is assumed for each scenario-dependent parameter. The minimum and maximum marginal energy values for the Peace and Columbia reservoirs are given, respectively in Table 3-29 and Table 3-30. C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max M i n Max M i n Max M i n Max J A N 24.97 1947.31 24.34 1948.46 23.40 1949.23 16.14 1949.51 F E B 20.25 1065.03 17.20 1064.49 12.02 1064.70 7.83 1064.75 M A R 9.11 1051.11 4.07 1051.88 2.90 1054.87 3.36 1055.51 A P R 1.17 554 01 0.09 581.57 0.86 688.89 1.68 711.75 M \ \ 0 00 45 *6 0 00 100*4 0 00 112 r 0 6* 357 *0 J l \ 0 00 64 *6 0 00 202 *3 0 00 409 7s 0 29 19| ir J l L S53 1 102 03 4 IS 964 54 I. "6 S 0^ 77 1 SO S15 SO U G 10.84 1483 03 2 7 4S I4SSS1 20.0S I45S 11 10 W 1427 r S I T 11 42 1694 98 11 06 T01 29 10 SS nr 91 19 77 1700 69 ()( 1 10 91 IS6S 19 10 70 IS6S S^  12 21 186* 4S 2* 24 IS*S 1^ N O V 29.09 2034.20 . 29.76 .. 2033.95 31.02 2032.82 26.67 2032.50 D E C 27.52 2111.38 27.52 2111.33 27.43 2111.13 23.49 2111.06 Table 3-29: Minimum and Maximum Peace Marginal Energy Values for Perfect Correlation—Positive for Inflows and Prices and Negative for Demands C V = 0.00 C V = 0.10 C V = 0.25 C V = 0.40 Min Max Min Max Min Max M i n Max J A N 23.87 1084.66 24.10 1084.73 22.78 1084.98 12.90 1085.11 F E B 18.49 1078.02 15.59 1078.00 9.96 1077.92 5.73 1077.90 M A R 4.90 1067.13 2.54 1067.13 2.09 1067.13 2.08 1067.13 A P R 0.00 94S S"7 0 00 U S ' 7 08 0 - 0 9S* 16 0 1^ 99S 54 M \ \ 0 00 48 19 0 00 11 * 16 0 00 145 "N 0 12 451 26 J l \ 0 00 75 02 0 00 219 56 0 00 418 00 0 14 41S 79 . III . ' 0.00 lorr 16 0 00 954 36 0 01 856 2^ 0 94 7*9 61 U ( . IS 66 1 143 96 15 OS 1112 IS -> 54 I09S 77 4 12 HP2 so S I P 11 n2 1 12* 67 11 09 1 I24S* 29 00 1122 K) 14 1* 1121 99 O C T 11 02 1 1 P 20 10 61 1 1 P 54 11 95 1119 16 21 12 1 1 19 "»N NOV 28.30 1110.95 29.77 1111.31 30.39 1112.64 25.73 1113.04 D E C 26.04 1098.45 27.29 1098.65 26.90 1099.35 21.38 1099.65 Table 3-30: Minimum and Maximum Columbia Marginal Energy Values for Perfect Correlation—Positive for Inflows and Prices and Negative for Demands The minimum and maximum Peace marginal energy values for case G under the coefficient of variation values considered are virtually identical to those under case F, as are those for the minimum and maximum Columbia marginal energy values. The 94 observations for case F apply to case G as well . In both cases F and G the inflows and demands are perfectly negatively correlated, the difference between the two is that the prices are perfectly positively correlated with the demands in case F and the inflows in case G . Comparing the results for cases A through G over the three coefficient of variation values yields the following observations. For the two cases in which the demands and inflows are perfectly positively correlated, as well as the demands only case, with the exception of the late fall and early winter, the minimum marginal energy values decrease with an increasing coefficient of variation. For the two cases in which the demands and inflows are perfectly negatively correlated, during the late winter and late spring, the minimum marginal energy values decrease with an increasing coefficient of variation. It was found that with the exception of the demands only case, during the spring and early summer the maximum marginal energy values increase with the coefficient of variation. For the two cases in which the demands and inflows are perfectly positively correlated, during the late summer and early fall the maximum marginal energy values decrease with an increasing coefficient of variation. In the two cases in which the demands and inflows are perfectly negatively correlated and the demands only case, during the late summer, the maximum marginal energy values decrease with an increasing coefficient of variation. A final point should be made with regard to the dependence of the marginal energy value in one reservoir on storage in the other reservoir as the storage in the first reservoir changes. Peace reservoir storage affects the dependence of the Peace marginal energy values on the Columbia reservoir storage to a greater extent than Columbia reservoir storage affects the dependence of the Columbia marginal energy values on Peace reservoir storage. For example, for a coefficient of variation value of 0.25, the ranges of Columbia storage over which the Columbia marginal energy values were least sensitive to changes in Peace storage did not change with the case for any month. Similarly, during August through February, the ranges of Peace reservoir storage over which the Peace marginal energy values were least sensitive to changes in Columbia storage did not change; however, changes with the case were observed for March through July. Comparison Over Coefficient of Variation Values Above, the manner in which the minimum and maximum marginal energy values for the Columbia and Peace vary with the coefficient of variation has been studied for each of the seven different scenario-dependent parameter cases. Below, a similar comparison over the seven cases is made for each coefficient of variation value. Figure 3-27 plots the maximum Columbia marginal energy value for the one-scenario base case as well as for cases A to B and D through G for a coefficient of variation value of 0.40. In Figure 3-27 it can be seen that all cases have similar maximum Columbia marginal energy values over the fall and winter. Cases D to G—the cases in which there 95 is variation in all three of the scenario-dependent parameters—have very similar marginal values over the entire year. Case B , in which only the inflows vary, is similar to cases D to G over the fall, winter, and spring. During the fall, winter, and spring, the one-scenario base case is quite similar to case A , in which only the demands vary; while during the summer case B is very similar to the one-scenario base case. Comparison of Max Columbia Plant Marginal Energy Values for CV = 0.40 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-27: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.40 Figure 3-28 presents the maximum Columbia marginal energy values for each case for a coefficient of variation value of 0.25. For the months of January to March and September through December there is little difference in the maximum marginal Columbia energy values between the cases. Cases D through G—the cases in which all three scenario-dependent parameters vary with the scenario—all have very similar maximum marginal Columbia energy values. The maximum marginal Columbia energy values for case A , in which only the demands vary, are closer to those for cases D through G than they are to case B , in which only the inflows vary, or the one-scenario base case. Case B is closest to the one-scenario case. There are distinct differences between the results shown in Figure 3-27, for which the coefficient of variation is 0.40, and Figure 3-28. 96 Comparison of Max Columbia Plant Marginal Energy Values for CV • 0.25 1200 o4 , , , , , , , , , , , JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-28: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.25 Figure 3-29 presents the maximum Columbia marginal energy values for each case for a coefficient of variation value of 0.10. In comparing the maximum Columbia energy values with those for the two higher coefficient of variation values shown in the previous two figures, note the extent to which variation amongst the cases is reduced in Figure 3-29. A s was the case for a coefficient of variation value of 0.25, during the months of January to March and September through December there is little difference in the maximum marginal Columbia energy values between the cases; cases D through G , for which all of the scenario-dependent parameters vary, all have very similar maximum Columbia marginal energy values; the maximum Columbia marginal energy values for case A , which has scenario-dependent demands, are closer to those for cases D through G than they are to case B , which has scenario-dependent inflows, or the one-scenario base case; and case B is closest to the one-scenario case. Comparison of Max Columbia Plant Marginal Energy Values for CV = 0.10 1400 -i 120  0-1 , , , , , , , , , , , JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-29: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.10 97 Figure 3-30 shows the minimum Columbia marginal energy values for a coefficient of variation value of 0.40. From Figure 3-30 it can be observed that: the one-scenario base case is closest to case B , which has scenario-dependent inflows; cases F and G are virtually identical, as are cases D and E ; case A , which has scenario-dependent demands is more similar to cases D through G than it is to case B or the one-scenario base case; and, over most of the year, the one-scenario base case is least similar to cases D and E. Comparison of Minimum Columbia Plant Marginal Energy for CV = 0.40 35 . _ JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-30: Minimum Columbia Marginal Energy Values for Coefficient of Variation of 0.40 Figure 3-31 plots the minimum marginal Columbia energy values for a coefficient of variation value of 0.25. In comparison to a coefficient of variation value of 0.40, some minimum marginal values are decreased, while others increase. However, the increase in the range, over the seven cases, of minimum marginal energy values that occurs for an increase in the coefficient of variation is obvious in a comparison of Figure 3-30 and Figure 3-31. As for the previous figure, the one-scenario base case and the scenario-dependent inflows case B are quite similar to one another; and cases D and E are virtually identical, as are cases F and G ; case A , with scenario-dependent demands, is most similar to cases D through G. The one-scenario base case is least similar to case A . 98 Comparison of Minimum Columbia Plant Marginal Energy for CV = 0.25 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-31: Minimum Columbia Marginal Energy Values for Coefficient of Variation of 0.25 Figure 3-32 presents the minimum Columbia marginal energy values for a coefficient of variation of 0.10. Again, the effect of a reduced coefficient of variation on the similarity of the minimum Columbia marginal energy values for the cases is very noticeable through comparison with Figure 3-30 and Figure 3-31. Once again, case B , with scenario-dependent inflows, is most similar to the one-scenario base case, while case A , with scenario-dependent demands, is least similar to the one-scenario base case. Cases D through G , each with all three scenario-dependent parameters varying, are all virtually identical to one another. Comparison of Minimum Columbia Plant Marginal Energy for CV = 0.10 Figure 3-32: Minimum Columbia Marginal Energy Values for Coefficient of Variation of 0.10 Figure 3-33 presents the maximum Peace marginal energy value for all of the cases studied for a coefficient of variation value of 0.40. In Figure 3-33 it can be observed that 99 over the months of January through March, and September through December all of the scenarios are very similar. Cases D through G are virtually identical to one another, and are also quite similar to case B , with its scenario-dependent inflows, over the spring. Over the spring and early summer, the one-scenario base case is closest to case A , which has scenario-dependent demands, while over the late summer, the one-scenario base case is closest to case B . Comparison of Max PCE Marginal Energy Values for CV=0.40 0-1 . , , , " , , , , , , , 1 JAN FEE MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-33: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.40 In Figure 3-34 the maximum Peace marginal energy values for a coefficient of variation of 0.25 are presented for each case. A s with the coefficient of variation value of 0.40, over the autumn and winter all of the cases have similar maximum Peace marginal values in any given month. During the spring and summer the one-scenario base case is most similar to case B , which has scenario-dependent inflows. Again cases D through G are all very similar. Comparison of Max PCE Marginal Energy Values for C V-() :.>.<:•, 0-1 , , , , , , , , , , JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-34: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.25 100 Figure 3-35 presents the maximum Peace marginal energy values for a coefficient of variation value of 0.10. A comparison of Figure 3-35 with Figure 3-33 and Figure 3-34 illustrates the effect of the coefficient of variation on the maximum Peace marginal energy value. A s with a coefficient of variation value of 0.25, all scenarios are very similar over the autumn and winter, while over the spring and summer scenario B , which has scenario-dependent inflows, and the one-scenario base case are very close to one another, and cases D through G are also very similar to one another. Comparison of Max PCE Marginal Energy Values for CV=0.10 2500 -j • • • . , . • • . JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-35: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.10 Figure 3-36 presents the minimum Peace marginal energy values for a coefficient of variation value of 0.40. Observe that the one-scenario base case is most consistent with case B , which has scenario-dependent inflows. Cases D and E are virtually identical, as are cases F and G . For the coefficient of variation value of 0.40, for all but the spring and early summer, the minimum Peace marginal energy values for the one-scenario base case and case B are greater than those for the three scenario-dependent parameters cases or case A , which has scenario-dependent demands. 101 Comparison of Min PCE Plant Marginal Energy Values for CV=0.40 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-36: Minimum Peace Marginal Energy Values for Coefficient of Variation of 0.40 In Figure 3-37 the minimum Peace marginal energy values for a coefficient of variation of 0.25 are presented. The effect of the reduced coefficient of variation on the minimum Peace marginal energy values is noticeable in a comparison with Figure 3-36. As was true for the case in which the coefficient of variation value is 0.40, the one-scenario base case and case B , with scenario-dependent inflows, are most similar, cases D and E are virtually identical, as are cases F and G . For the late winter, spring, and summer, the minimum marginal energy values for case B and the one-scenario base case are greater than those for cases D to G or case A , which has scenario-dependent demands. Comparison of Min PCE Plant Marginal Energy Values for CV=0.25 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-37: Minimum Peace Marginal Energy Values for Coefficient of Variation of 0.25 102 Figure 3-38 further illustrates the effect of a reduced coefficient of variation and the similarity of the minimum Peace marginal energy values over the cases studied. Comparison of Min PCE Plant Marginal Energy Values for CV=0.10 35. . Figure 3-38: Maximum Peace Marginal Energy Values for Coefficient of Variation ofO.10 The comparison of the maximum and minimum marginal energy values over the seven cases for coefficient of variation values of 0.10, 0.25, and 0.40 yields some interesting insights. Over the fall and winter period, there is little variation in the maximum Columbia marginal energy value over the seven different cases; the same is true for the maximum Peace marginal energy value. Throughout the remainder of the year, there is comparatively little variation in the Columbia marginal energy values over the four cases in which the three scenario-dependent cases all vary; again, this is also true of the maximum Peace marginal energy values. For both the Columbia and Peace reservoirs it was found that during the spring and summer as the coefficient of variation decreases, the case in which only the demands vary approached those for the cases in which all three of the scenario-dependent parameters vary. It was also found that during the spring and summer as the coefficient of variation decreases the maximum marginal energy value for the case in which only the inflows vary approaches that for the one-scenario base case. For all of the coefficient of variation values studied, it was found that, for both the Columbia and Peace, the one-scenario base case underestimates the maximum marginal energy value, with respect to the other cases, during the spring and early summer, and overestimates the maximum marginal energy value during the late summer. For both reservoirs, and all coefficient of variation values studied, it was found that: the minimum marginal energy values under the one-scenario base case are closest to those under the inflows only case; the minimum marginal energy values under the two cases in which the demands and inflows are perfectly positively correlated are virtually identical, regardless o f which of these two parameters is perfectly negatively correlated with the prices; the minimum marginal energy values under the two cases in which the demands 103 and inflows are perfectly negatively correlated are virtually identical, regardless of which of these two parameters is perfectly positively correlated with the prices; and, that as the coefficient of variation value decreases, the minimum marginal energy values under the one-scenario base case are furthest from those under the demands only case. For most of the year, when the coefficient of variation is 0.40, the one-scenario base case overestimates the minimum marginal energy values in both reservoirs as compared to the other cases. When the coefficient of variation is equal to either 0.10 or 0.25, the one-scenario base case overestimates the minimum marginal energy values for both reservoirs—with the exception of the inflows only case—in the late winter and late summer, and underestimates these values in the late fall and early winter. The above results suggest that the minimum energy values are more sensitive to a variation in demand than they are to equal variation in the inflows. Recalling that the minimum energy values occur when both reservoirs are full, note that under healthy storage conditions, the marginal energy values are affected to a greater extent by a variation in the demand than by an equal variation in the inflow. Similarly, past the peak of the freshet, for the maximum marginal energy values—which occur under adverse storage conditions—variation in the demand was of greater influence on the marginal energy values than an equal variation in the inflows over all o f the coefficient of variation values studied. In spring, leading up to, and including, the peak freshet month, it was found that for the two lower coefficient of variation values studied, the marginal energy values are more sensitive to demand variation than inflow variation. However, when the coefficient of variation value was 0.40, inflow variation was of greater importance than demand variation; this was confirmed for a C V value of 0.55. 3.4.3.3 Results for Five-Scenario Cases with Perfect Foresight In this section, the results under the assumption of non-anticipative turbine releases from the previous section are compared against the case of operating without any non-anticipative constraints—that is, operating with perfect foresight. Perfect foresight is attained by removing constraint (3-23) from the model formulation. The same cases of scenario-dependent parameters, outlined in Table 3-16, are studied for coefficient of variation values of 0.10 and 0.25. A Cases: Scenario-Dependent Demands Only In the scenario-dependent demands only cases, as shown in Table 3-16, the inflows and prices do not vary with the scenario, while the values assumed by the demand, and the scenario probabilities, are defined by the characteristic values in Table 3-17. Perfect positive correlation is assumed between H L H and L L H demands. The cases studied are for coefficients of variation of 0.10 and 0.25; the mean demands are shown in Table 3-3. For each coefficient of variation value studied, the monthly marginal values of energy stored in the Peace reservoir were calculated at an increment of 50000 cmsd from 0 to 500000 cmsd for the Peace reservoir and a 20000 cmsd increment in the Columbia reservoir between 0 and 200000 cmsd. The maximum monthly marginal energy values 104 for the Columbia and Peace are presented, respectively, as Figure 3-39 and Figure 3-40 In the figures, the symbol " N A Q T " is used to represent the maximum marginal energy values under the non-anticipative turbine release constraints studied in section 3.4.3.2, and " N N A " represents the case of no non-anticipative constraints. Maximum C O L Marginal Energy Values for Demands Only Scenarios 1400.00 T 1200.00 0.00 -I 1 1 , , , , , , , , , JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-39: Maximum Columbia Marginal Energy Values for Demands Only Scenarios In Figure 3-39 observe that for all coefficient of variation values, during the late summer the maximum Columbia marginal energy values under the no non-anticipative constraints are greater than those for the non-anticipative turbine release constraints. During the spring and early summer the maximum Columbia marginal energy values under the perfect foresight case are exceeded by those for the non-anticipative turbine release constraints case. A s Figure 3-39 illustrates, the change of constraints regarding future knowledge on the problem can have a dramatic influence on the maximum marginal Columbia energy values in the spring and summer months. For a coefficient of variation of 0.10, the maximum marginal Columbia energy values under the no non-anticipative constraints in a month can be as much as $ 154/MWh lower, or as much as $95/MWh higher, than the corresponding value under the non-anticipative turbine releases; for a coefficient of variation value of 0.25, the corresponding values are $264/MWh and $180/MWh. 105 Max P C E Marginal Energy Values for Demands Only Scenarios 2500.00 0.00 -I 1 1 1 , ~ , , , , , , JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-40: Maximum Peace Marginal Energy Values for Demands Only Scenarios From Figure 3-40, observe that during the spring and early summer the maximum Peace marginal energy values are greater under the non-anticipative turbine release constraints case, while during the late summer, the values are greater under the no non-anticipative constraints case. For a coefficient of variation value of 0.25, the maximum Peace marginal energy values under the non-anticipative turbine release constraints in a month can exceed those under the no non-anticipative constraints by as much as $252/MWh, or be exceeded by as much as $124/MWh. 106 Minimum C O L Marginal Energy Values for Demands Only Scenarios S 20.00 -CV=0.10NNA -CV=0.25 NNA CV=0.10 NAQT CV=0.25 NAQT - 1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-41: Minimum Columbia Marginal Energy Values for Demands Only Scenarios Figure 3-41 demonstrates that during the winter the minimum Columbia marginal energy values for the case of no non-anticipative constraints exceed those for the non-anticipative turbine release constraints case. When the coefficient of variation is 0.25, the minimum monthly Columbia energy values can change by as much as $15/MWh between the perfect foresight and non-anticipative turbine release cases. Minimum P C E Marginal Energy Values for Demands-Only Scenarios -CV=0.10 NNA -CV=0.25NNA CV=0.10 NAQT •CV=0.25 NAQT -1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-42: Minimum Peace Marginal Energy Values for Demands Only Scenarios 107 Figure 3-42 demonstrates that during the winter and late summer, the minimum Peace marginal energy values for no non-anticipative constraints exceed those for non-anticipative turbine releases. During the fall the minimum Peace marginal energy values for non-anticipative turbine releases exceed those for perfect foresight. For a coefficient of variation value of 0.25, the difference in the monthly maximum Peace marginal energy value for the two assumptions can be as much as $15/MWh. The general observations to be taken for the perfect foresight demands only cases are: during the spring and early summer the maximum marginal energy values for both reservoirs increase with an increasing coefficient of variation value; in the late summer, the maximum marginal energy values decrease with an increasing coefficient of variation; in the late summer the maximum marginal energy values are greater under the perfect foresight case, while in the spring and early summer the maximum marginal energy values are greater for the non-anticipative turbine releases; and, during the winter and late summer the minimum marginal energy values are greater in the perfect foresight case. B Cases: Scenario-Dependent Inflows Only In these cases the demands and prices do not vary between scenarios in any month. There are five scenarios for the inflows in the months of May through October and there is only one scenario in the months of November through Apr i l . The cases studied are for coefficient of variation values of 0.10 and 0.25. Table 3-1 defines the mean inflows, and the inflow values in the five scenarios are given by Table 3-17. Perfect positive correlation is assumed between inflows to the Columbia and Peace reservoirs. The maximum marginal energy values for the Columbia and Peace reservoirs are given, respectively, in Figure 3-43 and Figure 3-44, the corresponding minimums are given in Figure 3-45 and Figure 3-46. 108 Maximum C O L Marginal Energy Values for Inflows Only Scenarios -CV=0.10NNA -CV=0.25 NNA CV=0.10 NAQT •CV=0.25 NAQT -1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3 - 4 3 : M a x i m u m C o l u m b i a M a r g i n a l Energy Values for Inflows O n l y Scenarios In Figure 3-43 observe that for the inflows only case, with minor exceptions, the maximum marginal Columbia energy values under the non-anticipative turbine release constraints are greater than those under the no non-anticipative constraints case. For the maximum Peace marginal energy values shown in Figure 3-44 there is no real pattern as to the effect of the differing constraints. Max P C E Marginal Energy Values for Inflows Only -CV=0.10NNA -CV=0.25 NNA CV=0.10 NAQT CV=0-25 NAQT -1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-44: M a x i m u m Peace M a r g i n a l Energy Values for Inflows O n l y Scenarios 109 Minimum C O L Marginal Energy Values for Inflows Only Scenarios I CV=0.10 NNA I—CV=0.25 NNA CV=0.10 NAQT >-CV=0.25 NAQT - - I SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-45: Minimum Columbia Marginal Energy Values for Inflows Only Scenarios During the fall and early winter for both the Peace and Columbia, the minimum marginal energy values for the no non-anticipative constraints case exceed those for the non-anticipative turbine release case, with the converse being true for spring and early summer. The magnitude of these differences is not large, being on the order of $ l / M W h . Minimum Marginal P C E Energy Values for Inflows Only Scenarios S> 15.00 -CV=0.10NNA -CV=0.25 NNA CVO.10NAQT •CV=0.25 NAQT -1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-46: Minimum Peace Marginal Energy Values for Inflows Only Scenarios 110 The observations for the inflows-only scenarios under perfect foresight are: during the spring and early summer the maximum marginal energy values for both reservoirs increase with an increasing coefficient of variation value; with minor exceptions, the maximum Columbia marginal energy values are greater under the non-anticipative turbine releases; and, the minimum marginal energy values during the fall and early winter are greater for the perfect foresight case, while during the spring and early summer the minimum marginal energy values are greater for the non-anticipative turbine releases. C Cases: Scenario-Dependent Prices Only In the C cases the demands and inflows do not vary between scenarios in any month. For each month from November through Apr i l there is only one scenario for the price; in the remaining months, there are five scenarios. Perfect positive correlation between H L H and L L H is assumed. The mean prices are defined in Table 3-2, and the prices for the five scenarios are defined by these average prices, and the unit normal values specified in Table 3-17. The cases studied are for coefficient of variation values of 0.10 and 0.25. In this section, the assumption of perfect foresight, in contrast to the previous section, allows the turbine releases to vary between scenarios. As a result, there are differences between case C and the one-scenario base case. Maximum C O L Marginal Energy Values for Prices Only Scenarios Figure 3-47: Maximum Columbia Marginal Energy Values for Prices Only Scenarios In Figure 3-47 observe that for the prices only case the differences in the monthly maximum Columbia marginal energy values between the non-anticipative turbine 111 releases and perfect foresight are minor. For a coefficient of variation value of 0.25, the maximum difference in the maximum monthly marginal value between the two cases is $2.18/MWh. During the fall and winter there are essentially no differences between the two cases, while in the spring and summer the maximum marginal Columbia energy values for the perfect foresight case exceed those for the assumption of non-anticipative turbine release. Max P C E Marginal Energy Values for Prices Only Scenarios 2500 -I — — — • —• JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-48: Maximum Peace Marginal Energy for Prices Only Scenarios For the maximum Peace marginal energy values, shown in Figure 3-48, the largest difference between maximum marginal energy values for the two cases is $2.27/MWh. During the spring, early summer, and early winter the maximum Peace marginal energy values are greater for the perfect foresight case for both coefficient of variation values, and the converse is true for February. For a coefficient of variation value of 0.25 there is little difference in the summer and fall values, while during this period for a coefficient of variation value of 0.10, the non-anticipative turbine release values are greater. The effects of the assumptions regarding foresight on the minimum Columbia marginal energy values are shown in Figure 3-49. This assumption can affect the minimum monthly Columbia marginal energy value by up to $7.78/MWh for a coefficient of variation value of 0.25. During the spring and early summer there is no difference between the two cases, while in the late summer, the maximum Columbia marginal energy values under the assumption of non-anticipative turbine releases exceed those for the perfect foresight case. During the late fall and winter for a coefficient of variation value of 0.10 the maximum Columbia marginal energy values for the perfect foresight 112 case exceed those for non-anticipative turbine releases, while the opposite is true for a coefficient of variation value of 0.25. Minimum COL Marginal Energy Values for Price Only Scenarios -CV=0.10NNA -CV=0.25 NNA CV=0.10 NAQT •CV=0.25 NAQT -1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-49: Minimum Columbia Marginal Energy Values for Prices Only Scenarios Figure 3-50 illustrates the effect of assumptions regarding foresight on the minimum Peace marginal energy values. For a coefficient of variation value of 0.25 the assumption made can affect the minimum monthly Peace marginal energy value by up to $1.84/MWh. During the late spring and early summer the assumption made has no effect. Furthermore, for a coefficient of variation value of 0.25 the minimum Peace marginal energy values for all other periods, with the exception of July and the early spring, under the assumption of non-anticipative turbine releases exceed those under perfect foresight. For a coefficient of variation value of 0.10, the minimum marginal Peace energy values under the assumption of non-anticipative turbine releases exceed those under perfect foresight, with the exceptions of July and the late fall and early winter. 113 Minimum P C E Marginal Energy Values for Prices Only Scenarios JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-50: Minimum Peace Marginal Energy Value for Prices Only Scenarios The observations to be taken for the prices only scenarios under the assumption of perfect foresight is that in comparison to the demands only and inflows only cases the changes in the maximum marginal energy values are minor for the prices only cases. D Cases: Scenario-Dependent Demands, Inflows, and Prices—All Perfectly Positively Correlated For these cases, each of the five scenarios for May through October has unique demands, inflows, and prices. There is only one scenario for the months of November through Apr i l . The assumption made is that demand, inflow, and price are all perfectly positively correlated—that is, when the demands are "Very Low", as defined in Table 3-17, the inflows and prices are also "Very Low"; similarly, i f the demands are "Very High", so are the other two scenario-dependent parameters. Mean inflows, prices, and demands are respectively given by Table 3-1, Table 3-2, and Table 3-3. Perfect correlation between H L H and L L H prices is assumed, as is perfect positive correlation between H L H and L L H demands, and between Peace reservoir and Columbia reservoir inflows. The maximum marginal energy values for the Columbia and Peace reservoirs are given, respectively, in Figure 3-51 and Figure 3-52, and the corresponding minimums are given in Figure 3-53 and Figure 3-54. 114 Maximum C O L Marginal Energy Values for Inflows, Prices, and Demands Scenarios «-**- • -• " — V . . . -CV=0.10 NNA -CV=0.25 NNA CV=0.10 NAQT CV=0.25 NAQT -1 SCEN Figure 3-51: Maximum Columbia Marginal Energy Values for Perfectly Positively Correlated Demand, Inflow, and Price Scenarios In Figure 3-51 observe that during the spring and early summer the maximum Columbia marginal energy values are greater under the case of non-anticipative turbine releases, while the converse is true during the late summer. For a coefficient of variation value of 0.25, the assumption regarding constraints can affect the maximum monthly marginal energy value by as much as $294/MWh. Maximum P C E Marginal Energy Values for Inflows, Prices, and Demands Scenarios -CV=0.10 NNA -CV=0.25 NNA CV=0.10 NAQT •CV=0.25NAQT -1 SCEN Figure 3-52: Maximum Peace Marginal Energy Values for Perfectly Positively Correlated Demand, Inflow, and Price Scenarios 115 The observations for the maximum Peace marginal energy values are very similar to those for the Columbia. Again, for all coefficient of variation values studied, the maximum marginal energy values under the assumption of non-anticipative turbine releases are greater than those for perfect foresight during the spring and the early summer, with the opposite being true during the late summer. For a coefficient of variation of 0.25, the different constraints can affect the maximum monthly Peace marginal energy value by as much as $281/MWh. Minimum C O L Marginal Energy Values for Inflows, Prices, and Demands Scenarios 35-00 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3 - 5 3 : Minimum Columbia Marginal Energy Values for Perfectly Positively Correlated Demand, Inflow, and Price Scenarios A s can be observed in Figure 3-53 during the winter and late summer the minimum Columbia marginal energy values are greater under the assumption of perfect foresight for all coefficient of variation values studied. In a similar manner, as shown in Figure 3-54, during the winter, spring, and summer the minimum Peace marginal energy values are greater under the assumption of perfect foresight for all coefficient of variation values studied. 116 Minimum PCE Marginal Energy Values for Inflow, Prices, and Demands Scenarios Figure 3-54: Minimum Peace Marginal Energy Values for Perfectly Positively Correlated Demand, Inflow, and Price Scenarios The observations to be taken for the perfect positive correlation case under the assumption of perfect foresight are: the minimum marginal energy values decrease with an increasing coefficient of variation; during the spring and early summer the maximum marginal energy values increase with an increasing coefficient of variation, while during the later summer and early fall, the maximum marginal energy values decrease with an increasing coefficient of variation; during the spring and early summer the maximum marginal energy values are greater for the non-anticipative turbine releases, while during the late summer the maximum marginal energy values are greater under the assumption of perfect foresight; during the winter and late summer the minimum Columbia marginal energy values are greater for perfect foresight; and during the spring, winter, and summer the minimum Peace marginal energy values are greater for the assumption of perfect foresight. E Cases: Scenario-Dependent Demands, Inflows, and Prices—Demands and Inflows Perfectly Positively Correlated and Prices Perfectly Negatively Correlated In the E cases there are five scenarios for the months of May through October, with each scenario in a month having unique demands, inflows, and prices. There is only one scenario for each of November through Apr i l . The assumptions for the cases studied here are that the demands and inflows are perfectly positively correlated and the prices are perfectly negatively correlated with the demands and inflows—for example, i f the demands are " L o w " as defined by Table 3-17 then the inflows are also " L o w " , and the prices are "High". Mean inflows, prices, and demands are respectively given by Table 117 3-1, Table 3-2, and Table 3-3. Perfect correlation between H L H and L L H prices is assumed, as is perfect positive correlation between H L H and L L H demands, and between Peace and Columbia inflows. Coefficient of variation values of 0.10 and 0.25 are studied. The same value of the coefficient of variation is assumed for each scenario-dependent parameter The maximum marginal energy values for the Columbia and Peace are given, respectively, in Figure 3-55 and Figure 3-56, the corresponding minimums are given in Figure 3-57 and Figure 3-58. Maximum C O L Marginal Energy Values for Inflows and Demands; Prices Scenarios ' CV=0.10 NNA I CV=0.25 NNA CV=0.10 NAQT l - -CV=0.25NAQT 1—1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-55: Maximum Columbia Marginal Energy Values for Demand and Inflow Perfectly Correlated and Price Perfectly Negatively Correlated Scenarios In Figure 3-55 observe that during the spring and early summer the maximum Columbia marginal energy values are greater under the case of non-anticipative turbine releases, while the converse is true during the late summer. These observations are the same as for case D. For a C V value of 0.25, the assumption regarding constraints can affect the maximum monthly Columbia marginal energy value by as much as $296/MWh. 118 Maximum PCE Marginal Energy Values for Inflows and Demands; Prices 2500.00 0.00 -I i 1 . 1 — — i , , , , , 1 1 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3 - 5 6 : Maximum Peace Marginal Energy Values for Demand and Inflow Perfectly Correlated and Price Perfectly Negatively Correlated Scenarios The observations for the maximum Peace marginal energy values are very similar to those for the Columbia. Again, for all coefficient of variation values studied, the maximum marginal energy values under the assumption of non-anticipative turbine releases are greater than those for perfect foresight during the spring and the early summer, with the opposite being true during the late summer. For a coefficient of variation of 0.25, the different constraints regarding future knowledge can affect the maximum monthly Peace marginal energy value by as much as $284/MWh. 119 Minimum COL Marginal Energy Values for Inflows and Demands; Prices Scenarios S 20.00 -CV=0.10 NNA -CV=0.25NNA CV=0.10NAQT CV=0.25NAQT -1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-57: Minimum Columbia Marginal Energy Values for Demand and Inflow Perfectly Correlated and Price Perfectly Negatively Correlated Scenarios A s can be observed in Figure 3-57 during the winter and late summer the minimum Columbia marginal energy values are greater under the assumption of perfect foresight for all coefficient of variation values studied. In a similar manner, as shown in Figure 3-58, during the winter, spring, and summer the minimum Peace marginal energy values are greater under the assumption of perfect foresight for all coefficient of variation values studied. These observations are again the same as those for case D , in which there is perfect positive correlation between all three scenario-dependent parameters. 120 Minimum P C E Marginal Energy Values for Inflows and Demands; Prices Scenarios 35.00 25.00 10.00 5.00 K / / • CV=0.10 NNA — • CV=0.25 NNA CV=0.10 NAQT <*— 1 SCEN \ \ //" , \ // V \ _ /J JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-58: Minimum Peace Marginal Energy Values for Demand and Inflow Perfectly Correlated and Price Perfectly Negatively Correlated Scenarios The observations to be taken for the case of perfect positive correlation for the demands and inflows and perfect negative correlation for the prices under the assumption of perfect foresight are: the minimum marginal energy values decrease with an increasing coefficient of variation; during the spring and early summer the maximum marginal energy values increase with an increasing coefficient of variation, while during the later summer and early fall, the maximum marginal energy values decrease with an increasing coefficient of variation; during the spring and early summer the maximum marginal energy values are greater for the non-anticipative turbine releases, while during the late summer the maximum marginal energy values are greater under the assumption of perfect foresight; during the winter and late summer the minimum Columbia marginal energy values are greater for perfect foresight; and during the spring, winter, and summer the minimum marginal energy values are greater for the assumption of perfect foresight. F Cases: Scenario-Dependent Demands, Inflows, and Prices—Demands and Prices Perfectly Positively Correlated and Inflows Perfectly Negatively Correlated The cases considered here all have five scenarios for the months of May through October, with each scenario in these months having unique demands, inflows, and prices. The months of November through Apr i l each have a single scenario. It is assumed that the demands and prices are perfectly positively correlated and the inflows are perfectly negatively correlated with the demands and prices. For example, i f the demands are "Very High" as defined by Table 3-17 then the prices are also "Very High", and the inflows are "Very Low". Perfect correlation between H L H and L L H prices is assumed, 121 as is perfect positive correlation between H L H and L L H demands, and between Peace and Columbia inflows. Coefficient of variation values of 0.10 and 0.25 are considered. The same value of the coefficient of variation is assumed for each scenario-dependent parameter. The maximum marginal energy values for the Columbia and Peace are given, respectively, in Figure 3-59 and Figure 3-60; the corresponding minimums are given in Figure 3-61 and Figure 3-62. As was true for the E cases, the observations for the F cases on the effects of the assumptions regarding foresight on the minimum and maximum marginal energy values for the two reservoirs are the same as those for case D , in which all three scenario-dependent parameters are perfectly positively correlated. Maximum COL Marginal Energy Values for Prices, Demands; Inflows Scenarios ' C V = 0 . 1 0 N N A I — C V = 0 . 2 5 N N A C V = 0 . 1 0 N A Q T | > - - C V = 0 . 2 5 N A Q T | — 1 S C E N J A N F E B M A R A P R M A Y J U N J U L A U G S E P O C T N O V D E C Figure 3-59: Maximum Columbia Marginal Energy Values for Demand and Price Perfectly Correlated and Inflow Perfectly Negatively Correlated Scenarios In Figure 3-59 and Figure 3-60 observe that during the spring and early summer the maximum marginal energy values are greater under the case of non-anticipative turbine releases, while the converse is true during the late summer. For a coefficient of variation value of 0.25, the assumption regarding constraints on utilizing knowledge about the future can affect the maximum monthly Columbia marginal energy value by as much as $117/MWh and the maximum monthly Peace marginal energy value by up to $121/MWh 122 Max P C E Marginal Energy Values for Prices, Demands; Inflows Scenarios CV=0.10NNA CV=0.25 NNA CV=0.10 NAQT CV=0.25 NAQT 1 SCEN Figure 3-60: Maximum Peace Marginal Energy Values for Demand and Price Perfectly Correlated and Inflow Perfectly Negatively Correlated Scenarios A s shown in Figure 3-61 during the winter and late summer the maximum Columbia marginal energy values are greater under the assumption of perfect foresight, and, as shown in Figure 3-62, during the winter, spring, and summer the maximum Peace marginal energy values are greater under the assumption of perfect foresight. Minimum C O L Marginal Energy Values for Prices, Demands; Inflows Scenarios — CV=0.10 NNA I—CV=0.25 NNA CV=0.10NAQT >--CV=0.25NAQT i—1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-61: Minimum Columbia Marginal Energy Values for Demand and Price Perfectly Correlated and Inflow Perfectly Negatively Correlated Scenarios 123 Minimum P C E Marginal Energy Values for Prices and Demands; Inflows 35.00 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-62: Minimum Peace Marginal Energy Values for Demand and Price Perfectly Correlated and Inflow Perfectly Negatively Correlated Scenarios The observations to be taken for the case of perfect positive correlation for the demands and prices and perfect negative correlation for the inflows under the assumption of perfect foresight are: the minimum marginal energy values decrease with an increasing coefficient of variation during the late summer, fall, and early winter, whereas they increase with an increasing coefficient of variation during the spring and early summer; during the spring and early summer the maximum marginal energy values increase with an increasing coefficient of variation, while during the later summer and early fall, the maximum marginal energy values decrease with an increasing coefficient of variation; during the spring and early summer the maximum marginal energy values are greater for the non-anticipative turbine releases, while during the late summer the maximum marginal energy values are greater under the assumption of perfect foresight; during the winter and late summer the minimum Columbia marginal energy values are greater for perfect foresight; and during the spring, winter, and summer the minimum Peace marginal energy values are greater for the assumption of perfect foresight. G Cases: Scenario-Dependent Demands, Inflows, and Prices—Inflows and Prices Perfectly Positively Correlated and Demands Perfectly Negatively Correlated The G cases considered here have five scenarios in each of the months from May through October, with each scenario in a month having unique values for the demands, inflows, and prices. The months of November through Apr i l each have a single scenario. For this section it is assumed that the inflows and prices are perfectly positively correlated, and that both of these parameters are perfectly negatively correlated with the demands. B y 124 way of example, i f the prices are "High", as defined by Table 3-17, then the inflows are also high and the demands are "Low" . Perfect correlation between H L H and L L H prices is assumed, as is perfect positive correlation between H L H and L L H demands, and between Peace and Columbia inflows. Values for the coefficient of variation of 0.10 and 0.25 are studied. The same value of the coefficient of variation is assumed for each scenario-dependent parameter. The maximum marginal energy values for the Columbia and Peace are given, respectively, in Figure 3-63 and Figure 3-64; the corresponding minimums are given in Figure 3-65 and Figure 3-66. Maximum C O L Marginal Energy Values for Inflows and Prices; Demands Scenarios 1400.00 T — •—-i 1200.00 0.00 -I 1 1 1 1 , , , , , , , JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-63: Maximum Columbia Marginal Energy Values for Inflow and Price Perfectly Correlated and Demand Perfectly Negatively Correlated Scenarios Once more, the observations of the effects of the assumptions regarding foresight on the minimum and maximum marginal energy values for the two reservoirs for the G case are the same as those for case D , in which all three scenario-dependent parameters are perfectly correlated. In Figure 3-63 and Figure 3-64 observe that during the spring and early summer the maximum marginal energy values are greater under the case of non-anticipative turbine releases, while the converse is true during the late summer. For a coefficient of variation value of 0.25, the assumption regarding constraints can affect the maximum monthly Columbia marginal energy value by as much as $118/MWh and the maximum monthly Peace marginal energy value by up to $125/MWh 125 Maximum P C E Marginal Energy Values for Inflows and Prices; Demands Scenarios -CVO.10 NNA - CV=0.25 NNA CV=0.10 NAQT CV=0.25 NAQT -1 SCEN Figure 3-64: Maximum Peace Marginal Energy Values for Inflow and Price Perfectly Correlated and Demand Perfectly Negatively Correlated Scenarios Minimum C O L Marginal Energy Values for Inflows and Prices; Demands Scenarios -CV=0.10 NNA -CV=0.25 NNA CV=0.10 NAQT •CV=0.25 NAQT -1 SCEN JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-65: Minimum Columbia Marginal Energy Values for Inflow and Price Perfectly Correlated and Demand Perfectly Negatively Correlated Scenarios A s shown in Figure 3-65 during the winter and late summer the minimum Columbia marginal energy values are greater under the assumption of perfect foresight, while, as shown in Figure 3-66, during the winter, spring, and summer the minimum Peace marginal energy values are greater under the assumption of perfect foresight. 126 Minimum P C E Marginal Energy Values for Inflows, Prices; Demands 35.00 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-66: Minimum Peace Marginal Energy Values for Inflow and Price Perfectly Correlated and Demand Perfectly Negatively Correlated Scenarios The observations to be taken for the case of perfect positive correlation for the inflows and prices and perfect negative correlation for the demands under the assumption of perfect foresight are: the minimum marginal energy values decrease with an increasing coefficient of variation during the late summer, fall, and early winter, whereas they increase with an increasing coefficient of variation during the spring and early summer; during the spring and early summer the maximum marginal energy values increase with an increasing coefficient of variation, while during the later summer and early fall, the maximum marginal energy values decrease with an increasing coefficient of variation; during the spring and early summer the maximum marginal energy values are greater for the non-anticipative turbine releases, while during the late summer the maximum marginal energy values are greater under the assumption of perfect foresight; during the winter and late summer the minimum Columbia marginal energy values are greater for perfect foresight; and during the spring, winter, and summer the minimum Peace marginal energy values are greater for the assumption of perfect foresight. Comparing the results for cases A through G over the two coefficient of variation values yields the following results. The variation in the maximum marginal energy values for the two reservoirs with the coefficient of variation was much less for the prices only case. For all but the prices only case, during the spring and early summer the maximum marginal energy values increase with the coefficient of variation. For the demands only case, during the late summer the maximum marginal energy values decrease with the coefficient of variation. For the four cases in which all three of the scenario-dependent parameters vary with the scenario, the maximum marginal energy values decrease with increasing coefficient of variation for the late summer and early spring. For the two cases in which the demands and inflows are perfectly positively correlated, the minimum 127 marginal energy values decrease with an increasing coefficient of variation. Similarly, for the two cases in which the demands and inflows are perfectly negatively correlated, the minimum marginal energy values decrease with an increasing coefficient of variation for the late summer through early winter, while they increase with an increasing coefficient of variation during the spring and early summer. In terms of a comparison with the marginal values under the non-anticipative turbine releases and perfect foresight assumptions, the following results were found. For the demands only case, as well as for the four cases in which all three scenario-dependent parameters vary, the maximum marginal energy values during the late summer are greater under the assumption of perfect foresight; for these same cases, the maximum marginal energy values are greater under the assumption of non-anticipative turbine releases for the spring and early summer. With minor exceptions, the maximum marginal energy values for the inflows only case are greater in all months under the assumption of perfect foresight. For the demands only case, the minimum marginal energy values during the winter and late summer are greater under the assumption of perfect foresight. For the four cases in which all three scenario-dependent parameters vary, the minimum Columbia marginal energy values are greater under the assumption of perfect foresight during the winter and late summer. For these same four cases, the minimum Peace marginal energy values are greater under the assumption of perfect foresight during the spring, summer, and winter. For the inflows only case, the minimum marginal energy values during the fall and early winter are greater under the assumption of perfect foresight, while the opposite is true during the spring and early summer. A final note should be made as to how the assumption regarding foresight affects the dependence of the marginal energy value in one of the reservoirs on the storage in the second reservoir with storage in the first reservoir. The region of Columbia reservoir storage over which the marginal Columbia energy values are least dependent on Peace reservoir storage do not change from the case of non-anticipative turbine releases in any month. However, in the months of March through June, the region of Peace reservoir storage over which the marginal Peace energy values are least dependent on Columbia reservoir storage change in the perfect foresight case. Comparison Over Coefficient of Variation Values Above, the manner in which the minimum and maximum marginal energy values for the Columbia and Peace reservoirs change with the coefficient of variation have been studied for each of the seven different scenario-dependent parameter cases under the assumption of perfect foresight. Below, a similar comparison over the seven cases is made for each coefficient of variation value for the perfect foresight case. The monthly maximum Columbia marginal energy values for coefficient of variation values of 0.25 and 0.10 are presented as Figure 3-67 and Figure 3-68. 128 In Figure 3-67 it can be observed that during the fall and winter there is little difference in the maximum Columbia marginal energy values between scenarios. The maximum Columbia marginal energy values for cases D and E—the two cases for which the demands and inflows are perfectly positively correlated—are virtually identical throughout the year. Similarly, cases F and G—the two cases for which the demands and inflows are perfectly negatively correlated—are very similar throughout the year. During the spring and summer the maximum Columbia marginal energy values for case A—the case in which only the demands vary—are similar to those for the two cases for which the demands and inflows are perfectly positively correlated. During the late summer, maximum Columbia marginal energy values for case B—the inflows only case—are similar to those for the two cases in which the demands and inflows are perfectly negatively correlated. In the spring and summer, the maximum marginal energy values for the one-scenario base case are closest to those for the prices only case. During the spring and summer the maximum Columbia marginal energy values for the two cases in which the demands and inflows are perfectly negatively correlated are furthest from those for the one-scenario base case. During the spring and early summer, the one-scenario base case provides the lowest maximum Columbia marginal energy values, and the cases in which the demands and inflows are negatively correlated provides the highest maximum Columbia marginal energy values. In the late summer, the one-scenario base case yields the highest maximum Columbia marginal energy values, and the cases in which the demand and inflows are perfectly negatively correlated yield the lowest maximum Columbia marginal energy values. Compar ison of Max Columbia Plant Marginal Energy Values for C V = 0.25 1400 i 1 12011 0-1 , , , , , , , , , , JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-67: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.25 129 The observations regarding the relationships between the maximum Columbia marginal energy values under the seven cases for Figure 3-68 are essentially the same as those for Figure 3-67, indicating that the relationships are not dependent on the coefficient of variation. Compar ison of Max Columbia Plant Marginal Energy Values for C V = 0.10 "ST - l ONLY D ONLY -PONLY -I.P.D -|.D;P -|,P;D -P.D; I 1 SCEN Figure 3-68: Maximum Columbia Marginal Energy Values for Coefficient of Variation of 0.10 Comparing Figure 3-67 and Figure 3-68 the manner in which the differences between the maximum Columbia marginal energy values over the seven cases increase with the coefficient of variation is evident. A comparison of Figure 3-67 and Figure 3-68 and the corresponding figures for the non-anticipative turbine release case—Figure 3-28 and Figure 3-29—indicates that there is less variation between the maximum Columbia marginal energy values over the scenarios when there is perfect foresight. B y way of example, the maximum and average variation in the maximum Columbia marginal energy values for a coefficient of variation of 0.25 for the non-anticipative turbine releases are $363/MWh and $81.1/M Wh, while the corresponding values for the perfect foresight case are $284/Mwh and S54.2/MWh. The monthly minimum Columbia marginal energy values for coefficient of variation values of 0.25 and 0.10 are presented as Figure 3-69 and Figure 3-70. 130 Comparison of Minimum Columbia Plant Marginal Energy for C V = 0.25 Figure 3-69: Minimum Columbia Marginal Energy Values for Coefficient of Variation of 0.25 Compar ison of Minimum Columbia Plant Marginal Energy for C V = 0.10 Figure 3-70: Minimum Columbia Marginal Energy Values for Coefficient of Variation of 0.10 Comparing Figure 3-69 and Figure 3-70 the manner in which the differences between the minimum Columbia marginal energy values over the seven cases increase with the 131 coefficient of variation is evident. Note that the minimum Columbia marginal energy values for the prices only case do not closely track those for the one-scenario base case. A comparison of Figure 3-69 and Figure 3-70 and the corresponding figures for the non-anticipative turbine release case—Figure 3-31 and Figure 3-32— indicates that there is less variation between the minimum Columbia marginal energy values over the scenarios when there is perfect foresight. B y way of example, for a coefficient of variation value of 0.25, the average monthly variation in the minimum marginal Columbia energy values was $4.7/MWh for the non-anticipative turbine release case, and $3.3/IVlWh for the perfect foresight case. The monthly maximum Peace marginal energy values for coefficient of variation values of 0.25 and 0.10 are presented as Figure 3-71 and Figure 3-72. In Figure 3-71 it can be observed that during the fall and winter there is little difference in the maximum Peace marginal energy values between scenarios. The maximum marginal energy values for cases D and E—the two cases for which the demands and inflows are perfectly positively correlated—are virtually identical throughout the year. Similarly, cases F and G—the two cases for which the demands and inflows are perfectly negatively correlated—are very similar throughout the year. During the spring and summer the maximum Peace marginal energy values for case A—the case in which only the demands vary—are similar to those for the two cases for which the demands and inflows are perfectly positively correlated. During the late summer, maximum Peace marginal energy values for case B—the inflows only case—are similar to those for the two cases in which the demands and inflows are perfectly negatively correlated. In the spring and summer, the maximum Peace marginal energy values for the one-scenario base case are closest to those for the prices only case. During the spring and summer the Peace marginal energy values for the two cases in which the demands and inflows are perfectly negatively correlated are furthest from those for the one-scenario case. During the spring and early summer, the one-scenario base case provides the lowest maximum Peace marginal energy values, and the cases in which the demands and inflows are negatively correlated provide the highest maximum Peace marginal energy values. In the late summer, the one-scenario base case yields the highest maximum Peace marginal energy values, and the cases in which the demand and inflows are perfectly negatively correlated yield the lowest maximum Peace marginal energy values. The above observations for the relationships between the maximum Peace marginal energy values for the different cases are the same as those that existed amongst the cases for the maximum Columbia marginal energy values. 132 Compar ison of Max P C E Marginal Energy Values for CV=0.25 2500 0 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-71: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.25 The observations regarding the relationships between the maximum Peace marginal energy values under the seven cases for Figure 3-72 are the same as those for Figure 3-71, which in turn are the same as those for the maximum Columbia marginal energy values, indicating that the relationships are not dependent on the coefficient of variation or the reservoir. Compar ison of Max P C E Marginal Energy Values for CV=0.10 o> 1500 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-72: Maximum Peace Marginal Energy Values for Coefficient of Variation of 0.10 133 Comparing Figure 3-71 and Figure 3-72, the manner in which the differences between the maximum Peace marginal energy values over the seven cases increase with the coefficient of variation is evident. A comparison of Figure 3-71 and Figure 3-72 and the corresponding figures for the non-anticipative turbine release case—Figure 3-34 to Figure 3-35—indicates that there is less variation between the maximum Peace marginal energy values over the scenarios when there is perfect foresight. B y way of example, the maximum and average variation in the maximum Peace marginal energy values for a coefficient of variation of 0.25 for the non-anticipative turbine releases are $345/MWh and $86.7/MWh, while the corresponding values for the perfect foresight case are $246/MWh and $55.3/MWh. The monthly minimum Peace marginal energy values for coefficient of variation values of 0.25 and 0.10 are presented as Figure 3-73 and Figure 3-74. Compar ison of Min P C E Plant Marginal Energy Values for CV-0 .25 APR MAY SEP OCT NOV DEC Figure 3-73: Minimum Peace Marginal Energy Values for Coefficient of Variation of 0.25 134 Comparison of Min P C E Plant Marginal Energy Values for CV=0.10 1 15 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-74: Minimum Peace Marginal Energy Values for Coefficient of Variation 0.10 Comparing Figure 3-73 and Figure 3-74, the manner in which the differences between the minimum Peace marginal energy values over the seven cases increase with the coefficient of variation is evident. A comparison of Figure 3-73 and Figure 3-74 and the corresponding figures for the non-anticipative turbine release case—Figure 3-36 through Figure 3-38—indicates that there is less variation between the minimum Peace marginal energy values over the scenarios when there is perfect foresight. B y way of example, for a coefficient of variation value of 0.25, the average monthly variation in the minimum Peace marginal energy values was $5 .7 /MWh for the non-anticipative turbine release case, and S2.4/MWh for the perfect foresight case. The corresponding figures for the minimum Columbia marginal energy values provide similar results. The comparison of minimum and maximum marginal energy values for the two reservoirs over all of the cases for the three coefficient of variation values studied yield the following observations. Having perfect foresight—that is not enforcing any non-anticipative constraints—diminishes the variation between cases of the different assumptions regarding the scenario-dependent parameters for the marginal energy values. Again, as for the non-anticipative turbine releases, there is little variation in the maximum marginal energy values during the fall and winter, and the maximum marginal energy values under the two cases in which the demands and inflows are perfectly positively correlated are similar, as are the values for the two cases in which the demands and inflows are perfectly negatively correlated. In the spring and summer, the maximum marginal energy values for the demands only case approach those for the two cases in which the demands and inflows are perfectly positively correlated. In the spring and early summer the maximum marginal energy values under the one-scenario base case are the lowest, while during the late summer they are the highest. During the spring and 135 summer the maximum marginal energy values for the one-scenario base case are closest to those for the prices only case. Finally, in the spring and summer the maximum marginal energy values for the one-scenario base case are furthest from those for the two cases in which the demands and inflows are perfectly negatively correlated. 3.4.3.4 Winter Peace Operation Constraint vs. Relaxed Peace Winter Operation Constraint for Non-Anticipative Turbine Releases The base case studied in section 3.4.3.1 used the minimum monthly Peace discharges specified in Table 3-8, which specifies minimum discharges in January and December that significantly exceed the allowable minimum in the other months. These higher minimum discharges are set for operational purposes. Setting a high minimum discharge during the freeze up of the river establishes an ice bridge at a stage established at the high flow; flows can be dropped below the level associated with this stage, but cannot be increased above it. In this section, the effects of having to keep releases from the Peace high during December and January on the marginal energy values are briefly examined through a relaxation of the high minimum flow constraints. The effects are studied through a one-scenario case with identical parameters to the base case studied in section 3.4.3.1 except for the relaxed Peace plant minimum discharges specified in Table 3-31. Monthly Peace Plant Monthly Columbia Plant Minimum Minimum Discharge (cms) Discharge (cms) January 200 85 February 200 85 March 200 85 Apr i l 200 85 May 200 85 June 200 85 July 200 85 August 200 85 September 200 85 October 200 85 November 200 85 December 200 85 Table 3-31: Modified Monthly Minimum Plant Discharge The effect of relaxing the Peace ice constraints on the minimum monthly Peace marginal energy values is shown in Figure 3-75, in which the "Base Case" is for the minimum Peace plant releases given in Table 3-8 and the "Relaxed Case" is for the minimum releases in Table 3-31. 136 Effect of Peace Plant Ice Restrictions on Minimum Peace Marginal Energy Values JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3-75: Effect of Peace Ice Restrictions on Minimum Peace Marginal Energy Values It is immediately evident that by relaxing the Peace ice constraints, the minimum Peace marginal energy values are reduced in all months except for May and June where they were already equal to zero under the base case. With relaxation of the ice constraints, the minimum Peace marginal energy value is equal to zero in all months except for August through November. As the minimum Peace marginal energy values occur when both the Peace and Columbia reservoirs are full, it is apparent that the effect of having to maintain a high release in two winter months affects the Peace marginal value energy in all months except for May and June, even under the best possible storage situation. The maximum Peace marginal energy values occur under the worst possible storage conditions—when both reservoirs are empty. The effect of relaxing the ice constraints on the maximum Peace marginal energy values is presented in Figure 3-76. 137 Effect of Peace Plant Ice Restrictions on Maximum Peace Marginal Energy Values J A N F E B MAR APR MAY J U N JUL A U G S E P OCT NOV D E C Figure 3-76: Effect of Peace Ice Restrictions on Maximum Peace Marginal Energy Values From Figure 3-76 it is apparent that relaxation of the ice constraints is very significant on the maximum Peace marginal energy values. The effect of relaxing the constraints is particularly marked in the months when the higher minimum Peace releases held— December and January—and in the preceding late summer and autumn months when the Peace marginal energy values were high, sending the price signal that it is valuable to store water in order to meet the minimum releases. The two previous figures indicate that the higher minimum Peace plant releases that must be maintained during the winter raise the marginal Peace energy values above what they would be otherwise under all possible reservoir conditions in both the Peace and Columbia. Figure 3-77 and Figure 3-78, which present the effect of relaxing the ice constraints, respectively, on the minimum and maximum Columbia marginal energy values, show that the ice constraints also affect the Columbia marginal energy values. 138 Effect of Peace Ice Restriction on Minimum Columbia Marginal Energy Values 35 00 J A N F E B MAR A P R MAY J U N J U L A U G S E P O C T NOV DEC Figure 3-77: Effect of Peace Plant Ice Restrictions on Minimum Columbia Marginal Energy Values Figure 3-77 shows that relaxing the Peace ice constraints reduces the Columbia marginal energy values in all months in which they were non-zero in the base case. A comparison of Figure 3-77 and Figure 3-75 reveals that the effect of the ice constraints on the minimum Columbia marginal energy values is less than that for the minimum Peace marginal energy values. Effect of Relaxed Peace Ice Constraints on Maximum Columbia Energy Values 1200.00 0 . 0 0 -I T — , 1 , , 1 , , , , | J A N F E B MAR APR MAY J U N JUL A U G S E P OCT NOV DEC Figure 3-78: Effect of Peace Ice Restrictions on Maximum Columbia Marginal Energy Values As compared to the maximum Peace marginal energy values, the effect of the ice constraints on the maximum Columbia marginal energy values shown in Figure 3-78 is 139 slight. However, in all months except for February, March, and August, the ice constraints do raise the Columbia marginal energy value above what it would be otherwise; in February and March the maximum marginal energy values are unaffected, and in August there is a decrease. The effects of the ice constraints on the Peace marginal energy values in May and November are shown, respectively, in Figure 3-79 and Figure 3-80 for the Columbia reservoir empty, half full, and full. The letter " R " is used in the legends of the Figures to indicate the relaxed ice constraints case. Effect of Relaxed Peace Ice Constraints on May Marginal Peace Energy Value 5 0 . . 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 Peace Reservoir Storage (cmsd) Figure 3-79: Effect of Peace Ice Restrictions on May Peace Marginal Energy Values As both minimum and maximum M a y Peace marginal energy values are reduced by relaxing the ice constraints, it is not surprising to see that relaxation of the constraints reduces the Peace marginal energy values in May regardless of the storage in the two reservoirs. O f particular note is that relaxation of the minimum Peace discharge constraints reduces the dependence of the marginal Peace energy values on the amount of water stored in the Columbia reservoir, as evidenced by the vertical distance between the curves for the relaxed case. This decoupling of the Peace marginal energy value and Columbia storage also occurs in June. In January through Apr i l and July there is a similar reduced dependence on the Columbia storage, provided that the storage in the Peace is above a certain level, which is between 100000 and 350000 cmsd depending upon the month. 140 Effect of Peace Ice Constraints on November Marginal Peace Energy Values 2500 T 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 Peace Reservoir Storage (cmsd) Figure 3-80: Effect of Peace Ice Restrictions on November Peace Marginal Energy Values Figure 3-80 indicates that during November the Peace marginal energy values are also reduced (with some minor exceptions) for all levels of storage in both reservoirs. The most significant effects of the relaxation of the ice constraints occur when storage in the Peace reservoir is less than 150000 cmsd; for greater storage amounts in the Peace reservoir, the effects of the relaxation are more muted. As for May, relaxing the ice constraints changes how both the Columbia storage and Peace storage affect the Peace marginal energy value. Figure 3-81 presents some of the obscured details in the lower right corner of Figure 3-80. Together, Figure 3-80 and Figure 3-81 show that the relaxation of the ice constraints can both increase and decrease the dependence of the marginal Peace energy values on both the Peace and Columbia storage. In contrast, in May, the relaxation of the ice constraints reduces the dependence of the Peace marginal energy values on the Peace reservoir storage as evidenced by the slope of the curves for the relaxed case as compared to those for the base case. 141 Effect of Peace Ice Constraints on November Marginal Peace Energy Values Figure 3-81: Effect of Peace Plant Restrictions on November Peace Marginal Energy Values—Detail Figure 3-82 and Figure 3-83 show, respectively, the effect of the ice constraints on the Columbia marginal energy values in May and November for the Peace reservoir empty, half full, and full. Effect of Relaxed Peace Ice Constraints on May Marginal Columbia Energy Value 60 , , 50 Columbia Reservoir Storage (cmsd) Figure 3-82: Effect of Peace Ice Restrictions on May Columbia Marginal Energy Values Regardless of the storage in the Columbia reservoir, the three May Columbia marginal energy curves for the relaxed case are closer to one another than they are for the base case, indicating that relaxation of the ice constraints reduces the dependence of the May Columbia marginal energy value on the Peace reservoir storage. No general conclusion can be drawn regarding how relaxing the ice constraints affects the dependence of the May Columbia marginal energy value on the Columbia reservoir storage—in some cases 142 the dependence increases, while in others it decreases. The results for the other Peace freshet months of June and July are similar to those for May. Effect of Peace Ice Constraints on November Marginal Columbia Energy Values o J , , , , = , , • " • 1 20000 40000 60000 B00OO 100000 120000 140000 160000 180000 200000 Columbia Reservoir Storage (cmsd) Figure 3-83: Effect of Peace Ice Restrictions on November Columbia Marginal Energy Values The effect of the relaxation of the ice constraints on the marginal Columbia energy values in November is very different than for May. In November, relaxation of the constraints does not, to any great extent, change the dependence of the Columbia marginal energy values on Peace storage. The behaviour of the other Peace non-freshet months is similar to that for November. To summarize, it can be seen that the higher minimum Peace plant discharges in January and December have implications for both the Columbia and Peace marginal energy values. Relaxation of these minimums reduces both the minimum and maximum Peace marginal energy values in all months, with the greatest reduction in the maximum occurring for the period of July through January. In addition, the minimum Columbia marginal energy value is reduced in all months by a relaxation of the ice constraints. The less constrained operation has a reduced impact on the maximum Columbia marginal energy values, with the largest impacts occurring during May through July. The most interesting observation is that when the ice constraints are relaxed, the Peace marginal energy value is essentially decoupled from the Columbia reservoir storage for May and June. In other months provided that the storage in the Peace reservoir exceeds a certain threshold, the Peace marginal energy value is largely independent of the Columbia reservoir storage. Similarly, during the period of May through July, the Columbia marginal energy value is much less dependent on Peace reservoir storage, and becomes independent when there is sufficient storage in the Columbia reservoir. 1 4 3 3.5 Summary In this chapter, a model has been proposed to estimate the storage values and marginal storage values of water over the medium- to long-term for a two-reservoir system. These marginal water values can be converted into marginal energy values which can be used by the operator of a hydroelectric system in making dispatch decisions while operating the system, and in making transactions on the energy markets, as well as for input to shorter-term models. The model combines the use of backward-moving dynamic programming (DP) to link time periods together and the use of linear programming (LP) to determine the operation of the hydropower system and electricity trades within each time period, as well as the end of period storage for the two reservoirs. Uncertainty is introduced into the model through scenarios. The scenario-dependent parameters are demands, inflows, and prices. The model is limited to the consideration of two reservoirs, which must be hydraulically separate from one another. The limited number of reservoirs that can be considered is a result of the "curse of dimensionality" afflicting dynamic programming (DP). The apparently restricted range of applicability of the model can be extended using aggregate reservoirs, a technique that has been used in previous research (e.g., Turgeon and Charbonneau, 1998; Valdes et al., 1992), For example, the two reservoirs could be used to represent two separate river systems; each reduced into a single aggregate reservoir. Alternatively, one reservoir could represent either a single reservoir or the aggregate reservoir for a river system, and the second reservoir could represent an aggregate reservoir that replaces the remainder of the plants in the hydroelectric system. In a case study, the model is applied to the two main river systems in the B C Hydro system: the Peace and the Columbia. The case studied approximates the actual hydro installations. B C Hydro has two generating plants on the Peace River: G . M . Shrum and Peace Canyon. The reservoir for the G . M . Shrum plant has significant over-year storage, whereas Peace Canyon is largely run in hydraulic balance with the upstream G . M . Shrum facility. Similarly, B C Hydro generation facilities on the main-stem of the Columbia River include an upstream plant with significant storage (Mica), and a downstream plant that is generally run in hydraulic balance with the upstream project (Revelstoke). In the case study, generation on each of the two river systems is represented by a single plant. On the Columbia River, the "Columbia" plant replaces the M i c a and Revelstoke plants. The "Peace" plant replaces the G . M . Shrum and Peace Canyon plants. The case study demonstrates the ability of the model to calculate the marginal values for two multiple-year storage reservoirs, while taking into account the uncertainty in hydrology, demand, and market prices faced by the operator of a large hydropower system with significant storage flexibility. The insights gained on the reservoirs examined in the case study are discussed below to illustrate both the flexibility of the model and its ability to provide critical decision support information for the operator of a complex hydropower operation. A key insight is that there are ranges of reservoir storage over which there is not much interdependence of the marginal value of storage in one reservoir with the storage in the second reservoir. The ability to identify such areas 144 allows more complex modelling of the two individual reservoirs within such storage ranges. The insights gained are discussed below. In the case study the marginal energy value functions for the Peace and Columbia, which in general depend upon the storage in each of the two reservoirs, are determined under a number of different assumptions. First, a "base case" in which there is only one scenario in each month, and thus the scenario-dependent parameters—the demand to be met by the sum of Peace and Columbia generation, the inflows to the two reservoirs, and the energy prices—assume their mean values, was considered. The model was also applied to a number of cases in each of which there were different assumptions regarding the correlations between the scenario-dependent parameters. In these cases, there was one scenario during the months of November through Apr i l , and five scenarios during the remaining months. During the months with five scenarios, the turbine release was specified to be independent of the scenario—that is, it was "non-anticipative". The cases studied, described in terms of the scenario-dependent parameters, were: demands only, with inflows and prices assuming their mean values in all scenarios; inflows only; prices only; demands, inflows, and prices all perfectly positively correlated; demands and inflows perfectly positively correlated and demands perfectly negatively correlated; inflows and prices perfectly positively correlated and demands perfectly negatively correlated; and demands and prices perfectly positively correlated and inflows perfectly negatively correlated. The same coefficient of variation was assumed to apply to each of the scenario-dependent parameters, perfect correlation was assumed between the inflows to the two reservoirs, and perfect correlation was assumed for H L H and L L H energy prices and demands. The model was also applied to each of the seven cases just described under the assumption of perfect foresight—that is, allowing the turbine release to depend upon the scenario. Finally, a one-scenario case in which the operational ice constraints which restrict the minimum Peace release during December and January was examined. For the base case it was observed that, in general, the marginal energy value for each of the two reservoirs is dependent on the storage in both reservoirs. However, in some months for particular storage ranges in a reservoir, the dependence of the marginal energy value for that reservoir is essentially independent of the storage in the other reservoir, providing that the storage in the second reservoir is in a particular range. For example, during May when the Peace reservoir is 60% full, the marginal Peace energy value varies by $0.30/MWh for a range of Columbia storage between half full and full. A second example, also for May, is that when the Columbia reservoir is 70% full, the marginal Columbia energy value only varies by $0.32/MWh over the lower 60% of storage in the Peace reservoir. These differences are small enough to be ignored for operational and trade decision purposes. The Columbia marginal energy values were found to display the least sensitivity to the entire range of Peace storage when the Columbia reservoir was approaching full in all months. The Peace marginal energy values for the months of August through February were found to be the least sensitive to changes over the entire range of Columbia storage when the Peace reservoir was approaching full. In the months of March through July, the Peace marginal energy values were found to display the least 145 sensitivity to changes over the entire range of Columbia storage when the Peace reservoir was in some mid-range, and in some of these months, when the Peace reservoir was full. In the base case, for both the Peace and Columbia, in all months except for May and June there were very large differences between the minimum marginal energy values— occurring when both reservoirs are full—and the maximum energy values, which occur with both reservoirs empty. For the Columbia reservoir, the maximum marginal energy values were on the order of the curtailment costs—$1000/MWh in the case study. The maximum marginal energy values reflect both the curtailment costs and the cost of the ice constraint flows, making the maximum marginal Peace energy values greater than those for the Columbia for July through January. There was no great difference between the minimum Peace and Columbia marginal energy values. The minimum Columbia marginal energy values were, in general, slightly lower. In the seven cases with non-anticipative turbine releases it was found that for the two cases in which the demands and inflows are perfectly positively correlated, as well as the demands only case, the minimum marginal energy values decrease with an increasing coefficient of variation except for in the late fall and early winter. Similarly for, the two cases in which the demands and inflows are perfectly negatively correlated, it was observed that during the late winter and late spring, the minimum marginal energy values decrease with an increasing coefficient of variation. With the exception of the demands only case, it was found that during the spring and early summer the maximum marginal energy values increase with the coefficient of variation. For the two cases in which the demands and inflows are perfectly positively correlated, it was observed that during the late summer and early fall the maximum marginal energy values decrease with an increasing coefficient of variation value. In the two cases in which the demands and inflows are perfectly negatively correlated as well as the demands only case, during the late summer, the maximum marginal energy values were found to decrease with an increasing coefficient of variation. For the seven cases with non-anticipative turbine releases it was observed that the dependence of the Peace marginal energy values on the Columbia reservoir storage with Peace reservoir storage was affected by the case more than was the dependence of the Columbia marginal energy values on the Peace reservoir storage with Columbia reservoir storage. It was observed that over the fall and winter period, there is little variation in the maximum Columbia marginal energy value over the seven different cases with non-anticipative turbine releases; the same is true for the maximum Peace marginal energy value. Throughout the remainder of the year, there is comparatively little variation in the Columbia marginal energy values over the four cases in which the three scenario-dependent cases all vary; again, this is also true of the maximum Peace marginal energy values. For both the Columbia and Peace, it was found that during the spring and summer as the coefficient of variation is reduced, the case in which only the demands vary approaches the cases in which all three of the scenario-dependent parameters vary. It was also found that during the spring and summer as the coefficient of variation is 146 reduced the case in which only the inflows vary approaches the one-scenario base case. For all o f the coefficient of variation values studied, it was found that, for both the Columbia and Peace, the one-scenario base case underestimates the maximum marginal energy value, with respect to the other cases, during the spring and early summer, and overestimates the maximum marginal energy value during the late summer. For both reservoirs, and all coefficient of variation values studied, it was found that the minimum marginal energy values under the one-scenario base case are closest to those under the inflows only case. It was also observed that the minimum marginal energy values under the two cases in which the demands and inflows are perfectly positively correlated are virtually identical, regardless of which of these two parameters is perfectly negatively correlated with the prices, and that the minimum marginal energy values under the two cases in which the demands and inflows are perfectly negatively correlated are virtually identical, regardless of which of these two parameters is perfectly positively correlated with the prices. It was also noted that as the coefficient of variation value is reduced, the minimum marginal energy values under the one-scenario base case are furthest from those under the demands only case. For most of the year, when the coefficient of variation is 0.40, the one-scenario base case overestimates the minimum marginal energy values in both reservoirs as compared to the other cases. When the coefficient of variation is equal to 0.10 or 0.25, the one-scenario base case overestimates the minimum marginal energy values for both reservoirs—with the exception of the inflows only case—in the late winter and late summer, and underestimates these values in the late fall and early winter. The results for the seven cases with non-anticipative turbine releases suggest that the minimum marginal energy values are more sensitive to a variation in demand than they are to an equal variation in the inflows. Recalling that the minimum marginal energy values occur when both reservoirs are full, note that under good storage conditions, the marginal energy values are affected to a greater extent by a variation in demand than by an equal variation in the inflow. Similarly, past the peak of the freshet, for the maximum marginal energy values—which occur under adverse storage conditions—variation in the demand is of greater influence on the marginal energy values than is an equal variation in the inflows over all of the coefficient of variation values studied. In spring, leading up to, and including, the peak freshet month, it was found that for the two lower coefficient of variation values studied, the marginal energy values are more sensitive to demand variation than inflow variation. However, when the coefficient of variation value is 0.40, inflow variation is of greater importance than demand variation; this was confirmed for a coefficient of variation value of 0.55. For the seven cases with perfect foresight it was found that the variation in the maximum marginal energy values for the two reservoirs with the coefficient of variation was much less for the prices only case than it was for the others. For all but the prices only case, during the spring and early summer the maximum marginal energy values are found to increase with the coefficient of variation. For the demands only case, during the late summer the maximum marginal energy values are found to decrease with the coefficient of variation. For the four cases in which all three of the scenario-dependent parameters change with the scenario, the maximum marginal energy values are observed to decrease 147 with an increasing coefficient of variation for the late summer and early spring. For the two cases in which the demands and inflows are perfectly positively correlated, the minimum marginal energy values are found to decrease with an increasing coefficient of variation. Similarly, for the two cases in which the demands and inflows are perfectly negatively correlated, the minimum marginal energy values decrease with an increasing coefficient of variation for the late summer through early winter, while they increase with an increasing coefficient of variation during the spring and early summer. In a comparison of the marginal values under assumptions of non-anticipative turbine releases and perfect foresight the following results were found. For the demands only case, as well as the four cases in which all three scenario-dependent parameters vary, the maximum marginal energy values during the late summer are found to be greater under the assumption of perfect foresight; for these same cases, the maximum marginal energy values are greater under the assumption of non-anticipative turbine releases for the spring and early summer. With minor exceptions, the maximum marginal energy values for the inflows only case are greater in all months under the assumption of perfect foresight. For the demands only case, the minimum marginal energy values during the winter and late summer are greater under the assumption of perfect foresight. For the four cases in which all three scenario-dependent parameters vary, the minimum Columbia marginal energy values are greater under the assumption of perfect foresight during the winter and late summer. For these same four cases, the minimum Peace marginal energy values are greater under the assumption of perfect foresight during the spring, summer, and winter. For the inflows only case, the minimum marginal energy values during the fall and early winter are greater under the assumption of perfect foresight, while the opposite is true during the spring and early summer. For the seven cases with perfect foresight, the region of Columbia reservoir storage over which the marginal Columbia energy values are least dependent on Peace reservoir storage do not change from the case of non-anticipative turbine releases in any month. However, in March through June, the region of Peace reservoir storage over which the Peace marginal energy values are least dependent on Columbia reservoir storage do change in the perfect foresight case. The comparison of minimum and maximum marginal energy values for the two reservoirs over all of the cases for the coefficient of variation values studied, under the assumption of perfect foresight, yield the following observations. Having perfect foresight—that is not enforcing any non-anticipative constraints—diminishes the variation between cases for the marginal energy values. Again, as for the non-anticipative turbine releases, there is little variation in the maximum marginal energy values during the fall and winter, and the maximum marginal energy values under the two cases in which the demands and inflows are perfectly positively correlated are similar, as are the values for the two cases in which the demands and inflows are perfectly negatively correlated. In the spring and summer, the maximum marginal energy values for the demands only case approach those for the two cases in which the demands and inflows are perfectly positively correlated. In the spring and early summer, the maximum marginal energy values under the one-scenario base case are the lowest, while 148 during the late summer, they are the highest. During the spring and summer the maximum marginal energy values for the one-scenario base case are closest to those for the prices only case. Finally, in the spring and summer the maximum marginal energy values for the one-scenario base case are furthest from those for the two cases in which the demands and inflows are perfectly negatively correlated. It was found that relaxing the higher minimum Peace plant discharges in January and December reduces both the minimum and maximum Peace marginal energy values in all months, with the greatest reduction in the maximum occurring for the period of July through January. In addition, the minimum Columbia marginal energy value is reduced in all months by relaxation of the Peace ice constraints. The less constrained operation has a smaller impact on the maximum Columbia marginal energy values, with the largest impacts occurring during May through July. When the ice constraints are relaxed, the Peace marginal energy value is essentially decoupled from Columbia reservoir storage during May and June. In other months, provided that storage in the Peace reservoir exceeds a certain threshold, the marginal Peace energy value is largely independent of Columbia reservoir storage. Similarly, during the period of May through July, the Columbia marginal energy value is much less dependent on Peace reservoir storage, and becomes independent when there is sufficient storage in the Columbia reservoir. The storage value curves calculated by the model described in this chapter serve as input for the shorter-term marginal value model described in the following chapter. 149 4 Short-term Marginal Value Model 4.1 Introduction The previous chapter presented a method for estimating the values and marginal values of energy stored in a two-basin hydroelectric system. The appropriate time step for that model is on the order of one month. The motivation for developing the medium-term D P and L P based model was to provide energy values for a two-reservoir system that could be used to provide the value of storage at the end of a shorter-term model. Such a model is described in this chapter. The shorter-term model has been developed to allow reservoir operations, and energy trades, for a two-basin system to be planned over a period shorter than one month and to generate short-term marginal values over this period. These marginal values can be used in supporting both reservoir operation and energy marketing decisions. While the monthly time step employed in the longer-term D P and L P based model is appropriate for longer-term planning and decision making, the time step neglects within-month variations in the scenario-dependent parameters. The shorter-term multiple-time step model provides a way in which these shorter-term departures from the monthly means can be taken into consideration for both the calculation of marginal values and operation of the system. The time steps in the model can be of variable length. For example, the model could be used with a total time horizon of one month. This month could be divided into a daily time step for three days (three time steps), a 3-day weekend time step, and a fifth time step could be used to model the balance of the days in the month. To summarize, the short-term marginal value model ( S T M V M ) serves as a bridge between the very-short term (on the order of hours) and the medium-term (greater than one month into the future). As was true for the D P and L P based model described in the previous chapter, uncertainty in the demands, inflows, and prices is taken into account by the shorter-term model. Uncertainty in all of these parameters is handled through the use of a scenario tree, which describes how the future can unfold. It is easiest to understand the idea of a scenario tree using the implied analogy. A scenario tree branches from a single root at the start of the first time step considered by the model to leaves at the end of the last time step considered by the model. The tree branches at locations where a decision must be made—that is, at the start of each time step. A scenario in the tree is a direct path from the root to one of the leaves. Consider the simple case of a model with one reservoir that is to be operated for two time steps, in which the inflow in each of the time steps can assume one of two values. A t the start of each time step a decision must be made as to the release from the reservoir. The scenario tree representing this simple problem has two branches growing out of the root at the start of the first step: one for high inflow in the first time step and one for low inflow in the first time step. The tree then branches again at the start of the second time step for both of the first period branches, giving four leaves at the end of time step two. There are thus 150 four scenarios—direct paths from the root to a leaf—in the tree: high inflow in both time steps; high inflow in the first time step and low inflow in the second time step; low inflow in the first time step and high inflow in the second time step; and low inflow in both time steps. The scenario tree just described outlines all of the possible ways in which the future can unfold in the simple model. The tree structure is also instructive in illustrating how decisions are made using the technique of stochastic linear programming with recourse which is used in the S T M V M described in this chapter. Each branching point, or node, in the tree represents a place at which a decision must be made. A t each node a probability is assigned to each branch; the sum of the probabilities of all branches from a node must equal one. At each of these nodes a decision, such as the quantity of water to release, must be made. Note that this decision must be made prior to knowing in which one of the possible manners the future wi l l unfold; therefore, the decision must be feasible over all possible evolutions of the future, and should in fact be the optimal decision considering all possible outcomes. Such a decision, being made in the face of uncertainty is known as a "here-and-now" decision. Once the future has been revealed at the end of the time period, corrective, or "recourse", decisions can be made based upon which of the possible futures held. These latter decisions are referred to as "wait-and-see". The decision making approach just described results in decisions that are "implementable"—that is they do not anticipate the future, and can therefore be implemented. It is proposed that the short-term marginal values produced by the model described here be used as input to a very-short term reservoir operation model, with a time-step of an hour or less, and a total time horizon of roughly one day (e.g., Shawwash et al., 1999; Piekutowski et al., 1993). The S T M V M and its application are described in greater detail in the remainder of the chapter. Application of the model to a two-reservoir system based upon the two major storage projects in the B C Hydro system in order to demonstrate its ability to produce the price signals required by an operator with significant storage and market opportunities while taking into account uncertainty in the demands and prices as well as the typically considered inflow uncertainty. 4.2 STMVM Overview The S T M V M has been designed for application to the hydroelectric system described in the previous chapter. The system consists of two, non-hydraulically linked, hydroelectric projects which serve the same load area. The system is assumed to be connected through tie lines to other regions making energy trades possible. The system is to be operated so as to maximize the value of its operation over the model time horizon—that is the sum of the net energy trade revenue earned over the modelled period, any penalties for constraint violation, and the value of water remaining in storage at the end of the model is to be as large as possible. The value of water remaining in storage is given by the storage value curves developed by the D P and L P based model described in the previous chapter. The maximization of the value of system operation is subject to physical and operational constraints. The maximization yields the marginal values of water for each reservoir for 151 each time step. The marginal energy values can then be directly calculated from the marginal water values. The maximization is performed by selecting optimal values for the decision variables in the problem. The decision variables are the turbine releases, spills, total facility releases, reservoir storage volumes, generations, imports, and exports. The search for the optimal values of these decision variables must take into account the fact that some of the model parameters are uncertain. The parameters in the S T M V M that are subject to uncertainty are the demands, inflows, and prices. The values that can be assumed by the uncertain parameters are described by a scenario tree, leading to the parameters being referred to as being "scenario-dependent". A scenario describes the values assumed by all of the scenario-dependent parameters for each time step in the model. The time steps can be of variable length, and each time step can be divided into sub-periods, for each of which the scenario-dependent parameters can assume different values. A t the start of a time step all of the here-and-now decisions must be made, while recourse wait-and-see decisions can be made at the end of each time step. The here-and-now decisions consider all of the possible outcomes for the scenario-dependent parameters as described by the scenario tree. The model is solved—that is, the values of the here-and-now and wait-and-see variables that maximize the value of system operation are found—by optimizing the stochastic linear programming problem with recourse (SLPR) described by the scenario tree. A s the underlying problem is not strictly linear, the L P must be solved iteratively. The sources of non-linearity that must be addressed concern the functions which convert turbine discharge into power and the functions describing the value of water remaining in storage. The relationship between turbine discharge and power production is approximated by a piecewise-linear function of storage. For each time step and each scenario, the reservoir storage used to calculate the relationship is assumed to be an average of that for the start and end of the period. As demonstrated in Chapter 3, the value of water remaining in storage in either reservoir at the end of the model time horizon in general depends upon the volume of water stored in both reservoirs. In order to provide the model with a piecewise-linear function for each reservoir that describes the value of water in storage as a function of the storage in that reservoir, estimates of the reservoir storage in the two reservoirs at the end of the model time horizon must be made. (These storage value functions and their derivation are described in Chapter 3.) However, the reservoir storage volumes upon which these two types of functions depend are decision variables. For the initial iteration, values of the storage in each reservoir under each scenario at the end of each time step are assumed. These assumed values are then used to calculate the conversion factors for each time step, reservoir, and scenario. The assumed values for the final time step are used to estimate the storage value functions. The model is then solved, and the marginal water values are compared with those from the previous iteration. If the difference in marginal value for either reservoir exceeds an allowable tolerance, another iteration is performed using the optimized reservoir storage levels to 152 estimate the conversion factors. The procedure is repeated until convergence or until a maximum number of iterations has been performed. At the completion of a model run, the values of all the decision variables— turbine releases, spills, total facility releases, reservoir storage volumes, generations, imports, and exports—for each modelled time step are available, as are the marginal water and energy values. O f particular interest for decision support are the marginal energy values for each time step, and the values of the here-and-now variables for the first time step. The here-and-now decision variable values for the first time are of interest as they represent the optimal values, over the scenarios modelled, for the most pressing decisions. For example, i f the here-and-now decisions are the turbine releases for the two reservoirs, the most pressing question is how much to release through the turbines for the first time step, which may be the coming day. The model provides results that can be used to help guide this decision for the two reservoirs. The optimal decisions for later time periods are of less immediate importance, as the model can be re-run with updated data before such decisions wi l l need to be taken. The marginal energy values are of importance over all of the time steps as they can be used to support energy trade decisions and for input to models dealing with a time step on the order of one hour. Trade decisions are not limited to the first time step in the model. For example, i f the first period represents the coming day, trades can be, and are, made beyond this period. The marginal energy values produced by the S T M V M for subsequent time periods represent estimates incorporating the currently available information. A s the model is run for subsequent periods, the marginal energy values wi l l be recalculated and new energy trade decisions can be made. Prior to presenting the details of the S T M V M , the stochastic linear programming with recourse technique is introduced, and the division of decision variables into the here-and-now and wait-and-see categories is discussed. 4.2.1 Stochastic Linear Programming with Recourse The S T M V M makes use of the technique known as stochastic linear programming with recourse (SLPR). Yeh 's 1985 review of the optimization of water resources problems literature noted the application of S L P R to water resources problems as a promising area for future research. This section provides a brief introduction to S L P R . Stochastic linear programming with recourse, for a two-stage problem, can be described mathematically as (Kai l and Wallace, 1994): mm{cx + Q(x)} (4-1) such that Ax = b, x > 0; (4-2) where 153 Q(x)= Y.'pr-QWd' ae{Q.} (4-3) Q(x,{) = mm{q&y\ Wi£)y = h(g) - T(£)x;y > o}, (4-4) and h(Q = h0 + H^=ho + T(Q = T0 + 2% and q(Q = q0 + Zq&. hi (4-1) to (4-4) x are the "here-and-now" decision variables—i.e., those that must be made in the face of uncertainty—and y are "wait-and-see" decision variables—i.e., those that can be made once the uncertainty has been revealed. The function Q(x) is called the expected recourse function, and the function Q(x, £,) is known as the recourse function. A scenario, £ which is indexed by CO over the set of scenarios {Q}, occurs with probability Figure 4-1 illustrates a simple one-time step, two-stage, recourse problem for a case in which there are two scenarios—H and L . The here-and-now variables are determined at t-1, prior to knowing which of the two scenarios w i l l occur, while the wait-and-see variables are determined at t, with knowledge of which scenario has occurred. / Here-and-now (x) \ ( variables determined at }-\ t - l . J >< • < / Wait-and-see (y) \ —( variables determined at J t-1 t Figure 4-1: Two-Stage Recourse Problem In essence, (4-1) says that the function to be minimized by the L P is the product of the objective function coefficients and the decision variables plus some function of the decision variables. Given the L P framework, the function of the decision variables—the expected recourse function—must be linear. Equation (4-2) presents the usual L P formulation. Equation (4-3) states that the expected recourse function is equal to the probability-weighted average of the recourse function evaluated for each scenario. Equation (4-4), which presents the recourse function, contains both the wait-and-see and here-and-now decisions; however, at the point when the wait-and-see decisions are to be found, the here-and-now variables have already been determined, and thus have fixed values. In equation (4-4), note that the matrices and vectors are functions of the scenario. 154 The extensions required to write a multiple-stage S L P R problem are straightforward. For instance, the objective function can be written as (Dantzig and Infanger, 1997): minjc^x, + ... + E(c2x2 + ... + E(cT_xxT_x + E(cTxT)))}. (4-5) For the two-stage problem, (4-5) reduces to min {c/Xi + E(c?x?)} which is equivalent to (4-1), with cjxj replacing cx and E(ciX2) replacing Q(x). t = T-l t=T Figure 4-2: Multiple-Stage Recourse Problem Figure 4-2 illustrates the multiple-stage problem for a case in which there are two possible outcomes in each s tage—H and L. A scenario consists of a path from the root of the tree to a leaf—i.e., S to either H T or L T . For example, a scenario in which H always occurs would be (S, H i , H 2 , H-r-i, HT) , while a scenario in which L always occurs would be (S, L i , L 2 , L r - i , LT). The boxes indicate scenario groups for which the here-155 and-now decisions at the previous stage must be the same for all scenarios in the group. For example, all scenarios (S, H i , ...) and (S, L i , ...)—that is all scenarios—must have the same here-and-now decision at t=0. Similarly, all scenarios (S, H i , H 2 , ...) and (S, H i , L 2 , ...) must have the same here-and-now decision at /=/ . The choice of here-and-now and wait-and-see variables is an important one. The following section examines the choice for the reservoir system operation problem under consideration. 4.2.2 Choice of Here-and-Now Variables In the previous section, the concept of dividing decision variables into here-and-now and wait-and-see variables was presented. This section discusses the division of the S T M V M decision variables into these two categories. The decision variables in the problem are the turbine flow, the spill, the total plant release, the end of time step storage, and the generation for each of the two reservoirs, as well as system imports and exports. The decision variables can be divided into water variables and power variables. The water variables are the turbine, spill, total plant flows, and the ending storage; the power variables are the generation, imports, and exports. It is not immediately obvious which variables should be here-and-now variables and which should be wait-and-see variables. Considering the operation, including energy trades, of the two river basin hydroelectric system over one time step is of assistance in making the categorization. To examine what the choice of here-and-now variable implies about the assumed operation of the system, consider an uncertain inflow which can assume either a high or a low value. Suppose that at the start of the time period a decision must be made as to what the ending storages for the time step wi l l be—i.e., operation to achieve ending reservoir target levels. The continuity equations link the starting and ending storage, the inflow, and the turbine and spill flows. The starting and ending storage for the time step would then be fixed, while the inflows would vary between scenarios. Thus, any difference between inflows would need to be accommodated by the turbine and spill flows, meaning that these two are candidates for recourse variables. It is unlikely that spills would be decided upon at the start of the time step, so they should be recourse variables. If the turbine flows were also to be treated as here-and-now variables, then any difference in inflows between scenarios would need to be spilled, resulting in wasted water. So, i f the end of time step storage is selected as a here-and-now variable, then it wi l l be the only here-and-now variable from the set of water decision variables. If the end of period storage is the only water decision to be made before the uncertainty is revealed, what does this mean with regard to the power variables? In the S T M V M , as in the D P and LP based model described in Chapter 3, H K (which relates generation to turbine discharge) is assumed to depend upon storage only, not on turbine discharge, so it is determined based solely on the start and end of time step reservoir levels. These levels are found iteratively. So, i f the end of time step storage is determined at the start of the time step, H K wi l l be fixed, and generation during the period wi l l only depend upon the turbine discharge. Therefore, generation must also be a recourse variable. Decisions 156 regarding imports and exports should be made in conjunction with generation decisions, making them recourse decisions as well. Now, consider what would happen should turbine discharges be here-and-now variables. Again, to avoid spilling water, spills wi l l not be here-and-now variables. If the discharges are specified, the ability to end at different storage levels must be maintained, so the end of time step storages wi l l also be wait-and-see variables. The generation depends upon turbine discharge and end of period storage. Since the end of time step storages are recourse decisions, the generation should also be recourse variables. And , as energy transactions are decided upon in concert with generation, these wi l l also be recourse decisions. Thus, a second possibility is to have turbine discharge as the here-and-now variables, and the remaining decision variables as wait-and-see variables. Two potential here-and-now decision variables have been identified: end of time step storage levels and turbine flows. For the choice of either of these as the wait-and-see variables, the other, along with imports, exports, and generation wi l l be recourse decisions. While it is believed that selecting the turbine flows as the here-and-now decision variables most closely represents the actual decision making used in operation of a hydroelectric system, both possible choices are explored in the case study in section 4.4. With the concepts of S L P R and the choice of here-and-now and wait-and-see variables having been explored, the model details are presented in the following section. 4.3 Model Details The preceding sections provide an overview of the S T M V M . The hydroelectric system under consideration was described, and the concept of stochastic linear programming with recourse (SLPR) and choice of here-and-now and wait-and-see decisions were introduced. In this section a mathematical formulation of applying S L P R to the S T M V M problem is presented, and the means of solving the model are discussed. 4.3.1 Mathematical Formulation This section provides a mathematical formulation of the S T M V M problem. A s noted at the end of Chapter 3, the S T M V M is essentially a multiple-period version of the L P component of the D P and L P based model. A s such, much of the mathematical description contained in this section is the same as that in section 3.3.2.1. In order to define the S T M V M problem mathematically, the parameters, sets, and variables must be specified. After these have been defined, the model constraints and objective are presented. 157 Sets Sets are used to index parameters, variables, and constraints. The sets in the model are specified below. Let the set of reservoirs in the system be represented by {R}. {R} is limited to two reservoirs, which must be hydraulically separate from one another. In order to differentiate between energy prices during different times within a time