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The value of inflow forecasting in the operation of a hydroelectric reservoir Barnard, Joanna Mary 1989

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T H E VALUE OF INFLOW FORECASTING IN T H E OPERATION OF A HYDROELECTRIC RESERVOIR By Joanna Mary Barnard B. Eng. (Civil Engineering) Memorial University of Newfoundland A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1989 (c) Joanna Mary Barnard, 1989 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\M\ 1 ^ j/w <L r> ^ The University of British Columbia Vancouver, Canada Date A^"SI yJJ DE-6 (2/88) A b s t r a c t The present study examines the value of conceptual hydrologic forecasting in the op-eration of a hydroelectric generating project. The conceptual forecasting method used is the UBC Watershed Model. The value of the conceptual forecast is determined by comparing results obtained by use of the forecast to those obtained by use of a forecast based purely on the historic record. The effect of the size of the reservoir on the value of the forecast is also considered. The operation of a hypothetical project is modelled using dynamic programming. The operation of the project is optimized using the conceptual and historic forecasts to generate a variety of operating policies. The operation of the project is then simulated using the derived operating policies and several test years of real data, to determine the potential energy generation for each scenario. The analysis is performed for several reservoir sizes and for deterministic and two stochastic representations of the data. The analysis concludes that conceptual forecasting is most useful when the annual flow is significantly different from the average annual flow of the basin. If an historic forecast is used, a deterministic representation of the data is most valuable. If a conceptual forecast is used stochastic analysis gives the most efficient operation. Forecasting of either kind is valuable for reservoir sizes greater than 25% of the mean annual flow, but the value decreases as the volume approaches 100% of the mean annual flow. ii Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgement x 1 INTRODUCTION 1 1.1 OBJECTIVE 1 1.2 BACKGROUND ' 1 1.3 METHODOLOGY 2 1.4 LITERATURE REVIEW 4 1.5 SCOPE . 7 1.5.1 Value of the Forecast 7 1.5.2 Hydroelectric Project as IPP . 7 1.5.3 Effect of Reservoir Size 8 1.5.4 Forecast Uncertainty 9 2 RESERVOIR MODELS 11 2.1 BASIN DESCRIPTION 11 2.2 HYDROELECTRIC DEVELOPMENT DESCRIPTION 15 2.3 OPTIMIZATION MODELS 19 2.3.1 Overview of Dynamic Programming . 20 in 2.3.2 Dynamic Programming Models 22 2.3.3 Terminal State Values 29 2.4 SIMULATION MODELS 30 2.4.1 Model Operation 30 3 INFLOW FORECASTS 34 3.1 DEFINITIONS 34 3.1.1 Conceptual Hydrologic Models 35 3.1.2 UBC Watershed Model . 37 3.2 GENERATING FORECASTS 39 3.2.1 Naive Sequence 40 3.2.2 Conceptually Forecast Flow Sequences 43 3.3 COMPARISON OF INFLOW FORECASTS . 44 3.3.1 Forecast Year: 1966 . 47 3.3.2 Forecast Year: 1968 48 3.3.3 Forecast Year: 1969 49 3.3.4 Forecast Year: 1970 . . 50 3.3.5 Uncertainty 51 4 RESERVOIR OPERATION RESULTS 53 4.1 RESERVOIR OPERATION 53 4.2 IDEAL OPERATION 55 4.2.1 Reservoir Size 56 4.2.2 Magnitude of Inflow 58 4.2.3 Uncertainty 60 4.3 CASE STUDIES 64 4.3.1 1970 64 iv 4.3.2 1968 68 4.1 CONCLUSIONS 72 5 RESULTS 74 5.1 PRODUCTION RUNS 74 5.2 TABULATED RESULTS 75 5.3 OBSERVATIONS 79 5.3.1 Forecast Year: 1966 79 5.3.2 Forecast Year: 1968 80 5.3.3 Forecast Year: 1969 82 5.3.4 Forecast Year: 1970 83 5.4 COMMENTS 84 5.4.1 Relative Value of Forecasts 84 5.4.2 Value of Stochastic Analysis . 85 5.4.3 Use of Second State Variable 87 5.4.4 Influence of Reservoir Size . 88 5.4.5 Conclusions 89 6 CONCLUSIONS 91 6.1 WHEN TO FORECAST 93 6.2 FORMAT OF OPERATING POLICY 96 6.3 SIZE OF RESERVOIR 97 6.4 RECOMMENDATIONS FOR FUTURE RESEARCH . 97 Bibliography 99 APPENDICES 102 v A Dynamic Programming Models 102 A.1 Deterministic Model 102 A.2 One State Stochastic Model 106 A. 3 Two State Stochastic Model I l l B Simulation Models 118 B. l Naive Forecast Simulation Models 118 B.2 Conceptual Forecast Simulation Models 124 C Historic Forecast 133 D Programs To Calculate Probabilities 134 D.l Probabilites 134 D. 2 Conditional Probabilities 135 E Conceptual Hydrologic Forecasts 137 E. l Forecast Year: 1966 137 E.2 Forecast Year: 1968 139 E.3 Forecast Year: 1969 142 E.4 Forecast Year: 1970 144 F Simulation Results 148 vi L i s t of Tables 2.1 Inflow Volume Information, Goldstream River (Based on 24 years of record) 12 2.2 Sizes of Hypothetical Goldstream Reservoirs 17 2.3 Reservoir Constraints 18 2.4 Reservoir Starting Volumes 32 3.5 Naive, Observed and Conceptual Inflow Volumes (in Mm3), Goldstream River 44 3.6 Statistics for Naive and Conceptual Forecast Inflow Volumes 45 3.7 Standard Deviations of Naive and Conceptual Forecasts 52 5.8 Raw Data: Total Yearly Energy Generation, Gwh 77 5.9 Loss in Generation Due to Forecast Uncertainty, % 78 5.10 Serial Correlation Coefficients 88 6.11 Benefit of Use of Different Operating Policies, % 92 6.12 Potential Profits from Different Operating Policies 94 vii List of Figures 2.1 Map of British Columbia, Showing Goldstream River 13 2.2 Mean Annual Hydrograph, Goldstream River 14 2.3 Hydrograph for Goldstream River, 1964 to 1987 14 2.4 Average Annual Precipitation, Revelstoke Met. Station 15 2.5 Elevation Volume Curve for Goldstream Basin 16 2.6 Schematic of Dynamic Programming Structure 21 2.7 Flowchart of Deterministic Optimization Model 24 2.8 Flowchart of One State Stochastic Optimization Model 25 2.9 Flowchart of Two State Stochastic Optimization Model 26 2.10 Flowchart of Simulation Models 31 3.11 Simplified Flow Chart of the UBC Watershed Model 38 3.12 Annual Inflow Volumes, Goldstream River, 1964 to 1987 41 3.13 Hydrographs for Test Years: 1966, 1968, 1969, 1970 . 42 3.14 Comparison of Perfect, Naive and Conceptual Forecast Hydrographs: 1966 47 3.15 Comparison of Perfect, Naive and Conceptual Forecast Hydrographs: 1968 48 3.16 Comparison of Perfect, Naive and Conceptual Forecast Hydrographs: 1969 49 3.17 Comparison of Perfect, Naive and Conceptual Forecast Hydrographs: 1970 50 4.18 Comparison of Reservoir Operation for each Reservoir Size . . . . . . . . 57 4.19 Comparison of 750 Mm3 Reservoir Operation for each Forecast Year . . . 59 4.20 750 Mm3 Reservoir Operation: High Inflow Case (196S) 61 4.21 750 Mm3 Reservoir Operation: Medium Inflow Case (1969) 62 vm 4.22 750 Mm3 Reservoir Operation: Low Inflow Case (1970) 63 4.23 Reservoir Operation for 1970, 375 Mm3 Reservoir: Volumes 66 4.24 Reservoir Operation for 1970, 375 Mm3 Reservoir: Flows 67 4.25 Reservoir Operation for 1970, 375 Mm3 Reservoir: Energy Generation . . 68 4.26 Reservoir Operation for 1968, 375 Mm3 Reservoir: Volumes . 70 4.27 Reservoir Operation for 1968, 375 Mm3 Reservoir: Flows 71 4.28 Reservoir Operation for 1968, 375 Mm3 Reservoir: Energy Generation . . 72 5.29 Efficiency Losses Due to Uncertainty, 1966 ( g ^ = 1.06) 79 5.30 Efficiency Losses Due to Uncertainty, 1968 ( g ^ = 1.18) 81 5.31 Efficiency Losses Due to Uncertainty, 1969 ( g ^ = 0.96) 82 5.32 Efficiency Losses Due to Uncertainty, 1970 ( ^ = 0.82) 83 i x Acknowledgement I would like to take this opportunity to thank my two advisors. Firstly I would like to thank Dr. Michael Quick of the Civil Engineering Department of UBC for suggesting the topic and for providing advice throughout the course of my research. Secondly, I would like to thank Don Druce of the Operation Control Department of BC Hydro for assisting in focussing the research topic, for generating the inflow forecasts and for help and encouragement throughout the reseach period. I would also like to acknowledge the financial assistance provided for this research by the BC Disaster Relief Fund. x Chapter 1 INTRODUCTION 1.1 OBJECTIVE The objective of this study is to answer the following questions: ' • Would it be worthwhile for an Independent Power Producer developing the Gold-stream River for hydroelectric generation, to obtain conceptual hydrologic forecasts to aid in the project operation? • How does the capacity of the reservoir affect this decision? 1.2 BACKGROUND This study considers how conceptual hydrologic forecasts may be used in combination with optimizing techniques to improve operational efficiency of hydroelectric projects. This research compliments aspects of the Twenty-Year Resource Plan released re-cently by the British Columbia Hydro and Power Authority (BC Hydro, 1989a) which outlines BC Hydro's polices designed to meet the load growth in British Columbia until the year 2010. The plan includes several new generating projects, but emphasizes other programs aimed at minimizing capital outlay. These programs include: 'Power Smart' programs which are designed to reduce load growth by energy conservation, 'Resource Smart' programs which aim to produce more electricity from existing plants by increas-ing efficiency, coordination and purchase agreements with neighbouring utilities and the purchase of power from Independent Power Producers (IPP's). 1 Chapter 1. INTRODUCTION 2 The value of optimization techniques is dependent on the accuracy and availability of knowledge regarding the magnitude of future basin inflows. Though perfect fore-knowledge of inflows is never available, for inland basins in which there is a large inflow contribution made by snowmelt, forecasting may be effective in reducing uncertainty and thus be useful in increasing the efficiency of operation. Forecasts range between the sim-ple and the very complex. Naive forecasts are based only on historic records from the basin under consideration. Empirical forecasts are generated by models using regression techniques to correlate historic data to basin characteristics and generate only a seasonal volume forecast. Conceptual hydrologic models generate seasonal volume forecasts and also provide indication of the time distribution of the flows. These models can also give updated forecasts as the season progresses and more information becomes available. The cost to calibrate and run forecasting models increases as their complexity increases and therefore it is necessary to ensure that the model will give increases in energy generation sufficient to justify the forecasting expense. The research described here endeavours to estimate the value of a conceptual hydro-logic forecast to an Independent Power Producer. The UBC Watershed Model will be used to derive the conceptual forecasts. 1.3 METHODOLOGY The approach used to achieve the stated objectives was to model the operation of a hypo-thetical hydroelectric generating scheme using several computer programs. Optimization was used to generate operating policies for the hypothetical projects. The operating policies indicate the optimal operation of the project based on information regarding the expected inflows. Optimal policies were derived using several inflow scenarios: a conceptual forecast, a naive forecast and a perfect forecast, i.e. actual recorded inflows. Chapter 1. INTRODUCTION 3 The perfect forecast, of course is never attainable, however its use gives a measure of the absolute maximum potential generation. The operation of the system was then simu-lated to determine the energy that would be generated by the system using the various operating policies, under actual inflow conditions. The quantity of the energy produced using the forecast flows relative to the theoretical maximum energy produced by perfect foreknowledge, measures the value of the forecast. The major steps in the research are listed below. • Select and model a hypothetical project and hydroelectric development. Several different project and plant sizes were modelled to measure the effect of size on the value of the forecast. • Generate inflow sequences for the conceptual forecast and naive forecast cases. • Optimize to derive operating policies for the project based on the various forecast scenarios: naive, conceptual and perfect. • Simulate the operation of the project using observed inflow for the test years and the various operating policies. • Analyze results and draw conclusions. Details of the procedures will be explained in the following chapters. Chapter 2 describes the basin used in this study and the optimization and simulation models created to analyse its operation. Chapter 3 addresses several aspects of inflow forecasting. The various forecasting techniques used in the research are described and the forecasts for the test years are presented and compared. Chapter 4 and 5 contain the results of the research. Chapter 4 describes the resulting operation of the projects by the models and Chapter 5 presents the results in terms of the energy generated. Chapter 6 presents the conclusions of the research, and recommendations for further work. Chapter 1. INTRODUCTION 4 1.4 LITERATURE REVIEW As new forecasting techniques and models are developed, numerous studies are presented which make assessments of model accuracy. Fewer studies, however, actually apply the forecasting method to a practical application to derive any quantitative measure of the value of the model or more importantly the marginal value of the model over existing practice. Several authors have noted the need to assess the value of conceptual hydrologic fore-casting. Yeh et al. (1978) in a paper assessing the value of remote sensing in reservoir operation restate the problem considered in this study by saying that "an accurate deter-mination of hydropower benefits that can be gained through better sensing methods and inflow information requires both an operational watershed runoff model and a short-term hydropower optimization model to effectively use the better information". They go on to say that few studies utilize both these models. In the introduction to a text discussing river flow forecasting and modelling, Schultz (1986) emphasizes that the decision to use a complex forecasting model rather than a simpler model, must be based not only on the technical possibilities of the model, but also on the expected benefit gained. In the same text, P.E. O'Connell et al. (1986) state that "The case for using a more complex model should be established in terms of the additional benefits which can accrue from its use in preference to a simpler model, rather than accepting a priori that the more complex model is best". Unfortunately, though there seems to be agreement that this research is necessary, few papers directly address the issue of the practical value of conceptual forecasting. Some studies evaluate what aspects of the problem affect the value of the forecast, but few will quantify the actual forecast value. A brief review of the related literature will be presented here. Chapter 1. INTRODUCTION 5 In a review of operational water supply forecasting techniques, Zuzel and Cox (1978) say that relationships between forecast accuracy and monetary benefits have not been determined due to the complexity and interactions of various water uses. They go on to quote several studies that have at least attempted to make some quantitative assessment. One study estimated values of water supply forecasts as ranging from a few cents to US$6/acre, depending on the economic value of the water. A second study estimated the value of forecasts for irrigation purposes in ten Western states and found it to be greater than US$22 million/year. In 1980, Kitanidis and Bras studied conceptual hydrologic models and found that projects can be operated more efficiently if the uncertainty of the forecast is acknowledged and if frequent updates are made in the optimization process, using observed data as it becomes available. In 1982, Yeh et al. performed a study similar to the one described in this report but with a different emphasis. They studied a hypothetical multipurpose reservoir with forecasts that were merely statistically perturbed historic inflow sequences. The reservoir studied was operated without any forecasting. For hydropower benefits, they concluded that the use of historical monthly means for estimates of streamfiow produced benefits only slightly less than were achieved using good prediction from a forecasting model. They therefore recommended that until high confidence prediction was established, that operation using historical means was preferable. Similar conclusions were drawn regard-ing other benefits such as water conservation or flood control. Benefits could be gained by using historical estimates; similar benefits were found using prediction techniques, but only if the predictions were accurate. The reservoir used in the 1982 study had a ratio of reservoir storage to mean annual flow of 0.8. Yeh et al. predict that: Chapter 1. INTRODUCTION 6 To a great extent, larger ratios would result in greatly diminished bene-fits in percentage terms from prediction (or estimation of streamflow by the mean) simply because there would be considerably less spill and excess wa-ter. Smaller ratios also would tend to yield fewer benefits due to inadequate opportunities prior to the estimated high inflow to release water beneficially before minimum storage constraints are met. In this case, it would be ex-pected that accurate inflow prediction would result in benefits appreciably greater that those obtainable from estimation by the mean. They also suggest that system constraints such as flood control reservations tend to depreciate potential benefits. In 1982, Braga et al. use a perfect forecast to assess the upper limit to forecasting value and therefore to provide a method to rate various forecasting systems. They used a range of multipurpose reservoirs, with volumes corresponding to 10, 20, 50 and 100% of mean annual streamflow. They found that the value of the perfect forecast decreases with increasing reservoir size. They conclude that a forecasting system will be most useful in reservoirs having volumes of up to 50% of the mean annual volume. Datta and Burges (1984) studied how the shape of the loss function and the accuracy of the forecast influenced the value of the forecast. One interesting conclusion was that a single bad forecast in a series of generally good forecasts could offset all the benefits derived from using a fairly accurate forecast model. They also state that the accuracy of the forecast can influence the system performance in several ways and therefore it is necessary to always account for potential forecast errors when evaluating the optimal pol-icy. Their study concluded that the performance of an operating policy using forecasted inflows could be correlated with the shape of the assumed loss function. Chapter 1. INTRODUCTION 7 1.5 SCOPE Several other aspects of the study must be discussed here to furthur define the scope of the research. Items discussed are: how the value of the forecast is defined, how the hydroelectric project operates wi th relation to other power producers, the expected effect of the size of the reservoir on the results of the study and the consideration of forecast uncertainty. 1.5.1 Value of the Forecast In this study the value of the forecast has a fairly narrow definition. It refers only to the value in terms of increased energy generation due to increased operational efficiency. Aside from this narrow definition, flow forecasts could st i l l be useful for overall product ion planning, as well as for flood prevention methods, design considerations etc. 1.5.2 Hydroelectric Project as IPP The hypothetical hydroelectric project studied here is assumed to be operated by an Independent Power Producer. B C Hydro's policy statement on IPP ' s states that: In it 's effort to achieve the most economic supply of electricity, B C H is turn-ing to Independent Power Producers for a port ion of its electricity supply requirements. Cost effective independent power product ion should allow de-ferral of larger, potential ly more expensive projects on the integrated system ( BC Hydro, 1989b). B C Hydro intends to invite proposals for the supply of electricity each spring as new generation is required. Thus, the assumption of a single reservoir acting independently of any others in a system is quite realistic. The policy also states that a standard purchase Chapter 1. INTRODUCTION 8 price will apply. This means that projects can be optimised to generate maximum energy at all times. Operation of large BC Hydro systems such as the Mica Project or the G.M. Shrum and Peace Canyon Power Projects are made complicated because the value of energy varies with time, giving added incentive for storage. IPP's will not have to deal with this added complication. In addition, IPP's do not have to match a demand curve, rather the energy will be purchased by BC Hydro whenever it is generated. Operation according to a demand curve would change the priorities of the operation which might also affect the value of the forecast. 1.5.3 Effect of Reservoir Size A forecast can only be valuable if it is used effectively. Accurate foreknowledge of flows can be used to increase the efficiency of hydroelectric project operation through reduc-tion in spill or maintenace of higher reservoir levels, but only if reservoir optimization procedures are used. The potential to increase efficiency through reducing spill or altering average reservoir levels is dependent on the original design of the reservoir. Large reservoirs often have the capacity to store several times the mean annual inflow to the basin so the operation is not dependent on the day to day inflow. Water levels can be kept fairly high through most of the year while still maintaining sufficient storage for flood control if high inflows are experienced. Reservoirs of this magnitude have sufficient storage that spill is very rare. These factors suggest that short term forecasting would not be useful in extremely large reservoirs. Large reservoirs benefit more from knowledge of long term flows. On the other extreme, reservoir storage may be minimal or non-existent. In these systems, called 'run of the river' plants, energy is generated with whatever inflow occurs, since no storage is possible. If inflows are high, the excess must be spilled. Thus forecasting would not be operationally useful in extremely small reservoirs, either. Chapter 1. INTRODUCTION 9 Presumably however, there is some range of reservoir size between the extremes of magnitude described here, in which forecasting would be useful in increasing the efficiency of operation. It is one of the aims of this study to try and quantify this size range. 1.5.4 Forecast Uncertainty Many inflow forecasts are deterministic, i.e. they neglect all uncertainty in the estimate and present one number as the forecast. After examining analytic solutions to decision problems with various characteristics, Krzysztofowicz (1983) concluded that "No matter which forecast is used, an opportunity loss is always incurred by a decision maker who does not account for the forecast uncertainty." The first step toward accounting for the uncertain nature of the reservoir inflows is to treat them as a stochastic series in which the input is represented by several potential values and their associated probabilities, rather than by one expected value. The effect of each possible inflow can be examined and then an expected outcome can be determined, again with reference to the individual probabilities. The single state stochastic representation assumes that each period's flow is indepen-dent of every other periods flow. This clearly is not the case, since it is obvious that flows in one period will give some indication of the likely flows in the next period. The degree of this serial correlation is of course dependent on the length of the period in question. Serial correlation between daily flows is much higher than between monthly or seasonal flows. The validity of the assumption of independence has not been clearly established. Little (1955) (considering a bi-weekly analysis) states that "the simpler assumption of complete independence is untenable for river flow" while in a later study Su and Deninger (1974) conclude "The assumption that the monthly inflows are seasonally independent, instead of serially correlated, does not seem to change the minimal value significantly". Druce (1986) suggests that when using a simple historic sequence, the serial correlation Chapter 1. INTRODUCTION 10 should be considered and therefore the assumption of independence would not be valid. When using a conceptual hydrologic model however, serial correlation is considered in the generation of the inflow sequence and thus the assumption of independence of the flows during the optimization phase is reasonable. In the present study three optimization models are analysed to examine this question further. Chapter 2 RESERVOIR MODELS 2.1 BASIN DESCRIPTION The research for this study was carried out through a case study of a hypothetical hy-droelectric power development. In order for the study to be as realistic as possible it was decided to model the hypothetical system on a real watershed. Geographic and hydro-logic data describing the real watershed were used, but no attempt was made to conform rigidly to any additional constraints that would exist in that basin. The catchment chosen to use as the model for the hypothetical system was the basin of the Goldstream River, in the interior of British Columbia. See Figure 2.1 for a map showing the location of the river. The use of this basin was purely for convenience; it in no way indicates any intention to develop this basin for power generation. The factors contributing to the selection of this basin are as follows: • It is located in the interior of British Columbia, and therefore has a large proportion of it's runoff contributed by snowmelt. This means that the runoff volume for the annual freshet is forecastable from snowpack information. In addition, the calibration required for this basin had previously been completed by B C Hydro. • Comparatively long hydrologic records were available from a Water Survey of Canada (WSC) gauge, located near the mouth of the river. • Meteorologic data were available from three stations near the river, for the periods required. 11 Chapter 2. RESERVOIR MODELS 12 Min Max Mean Serial Corr Month Inflow Inflow Inflow S.D. Coefficient Mm3 Mm3 Mm3 January 13.4 34.1 17.7 4.3 .721 February 12.4 20.8 .16.2 2.6 .441 March 11.4 29.7 17.5 4.4 .075 Apri l 20.5 79.1 44.0 16.6 .111 May 99.3 262.1 182.0 45.0 .253 June 238.8 491.2 334.2 67.1 .062 July 182.5 373.0 273.0 57.6 .161 August 108.2 294.2 155.2 38.1 .516 September 55.2 178.1 91.0 31.5 .305 October 34.4 95.9 54.7 14.0 .215 November 21.1 64.9 37.1 11.9 .317 December 14.8 41.5 22.9 5.5 .342 Table 2.1: Inflow Volume Information, Goldstream River (Based on 24 years of record) The Goldstream River is approximately 60 K m long. The catchment basin has an area of 938 K m 2 . The river is in the Selkirk Mountains and is tributary to the Columbia River. Streamflow information for this river was taken from WSC gauge #08ND012. The mean annual runoff volume of the river, based on the full twenty-four years of record is 1245 M m 3 (million cubic meters). Some basic statistics describing the flow are given in Table 2.1. Figure 2.2 shows the average annual hydrograph for the river, Figure 2.3 shows the hydrograph for the entire period of record. Figure 2.4 shows the average annual precipitation in the form of both rain and snow, recorded at Revelstoke, which is the nearest first order meteorological station, about 75 km south of the Goldstream River. Comparison of the annual hydrograph, which peaks in June, and the precipitation record, which peaks from November to January, emphasizes the influence of snowmelt on this catchment basin. Figure 2.1: Map of British Columbia, Showing Goldstream River Chapter 2. RESERVOIR MODELS 14 340 0 - j 1 1 1 1 1 1 1 1 1 1 1 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC MONTH Figure 2.2: Mean Annual Hydrograph, Goldstream River 240 -r 220 - \ 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 19M 1982 198J 198+ 1985 1986 1987 Figure 2.3: Hydrograph for Goldstream River, 1964 to 1987 Chapter 2. RESERVOIR MODELS 15 190 I i ~ r — l : i i i i i i t i JAN FEB MAR APR MAY JUN JUL AUG S E P OCT NOV DEC Figure 2.4: Average Annual Precipitation, Revelstoke Met. Station 2.2 HYDROELECTRIC DEVELOPMENT DESCRIPTION The hypothetical reservoir was formed by placing a dam on the Goldstream River, at the sight of the Water Survey of Canada stream flow gauge, just downstream of Old Camp Creek. The gauge is located at latitude 51° 40' 09" North and longitude 118° 35' 58" West, approximately 220 m upstream of the point where the river flows into the Columbia River. This location for the dam was chosen for convenience; it means that the flows given in the WSC records are the total flows into the reservoir, but is also a fairly reasonable location for a dam, based on examination of the topography. At the point where Old Camp Creek exits Goldstream River there is a low elevation channel leading from the river. A second dam was placed at a narrow section of the channel to prevent the loss of water from the reservoir, through this channel. Details of the dams are unimportant; they were assumed sufficiently large to create whatever head was required. In a similar way it was assumed that a spillway existed that has sufficient capacity to pass any spill Chapter 2. RESERVOIR MODELS 16 140 -1 • ACTUAL DATA 130 -APPROXIMATING EQUATION RANGE OF SIZES USED IN STUDY 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 RESERVOIR VOLUME. MILLION CUBIC METRES Figure 2.5: Elevation Volume Curve for Goldstream Basin rates encountered. Since this basin was used for a model only, no consideration of problems created by flooding roads, towns etc., was necessary for this case study. various contoured elevations. This data was used to derive an equation relating elevation to volume which is used in all the mathematical models of the basin in this study. Figure 2.5 is a graph of the elevation volume curve for the reservoir. The actual elevation volume values, and the curve defined by the approximating equation are shown. Several different live storage capacities were studied to determine how the size of the reservoir affects the value of the inflow forecast. For comparison purposes, the reservoir volumes can be expressed by the ratio of storage capacity to mean annual flow (MAF). The volumes studied were decided after performing a standard Rippl analysis (Chow, 1964). The firm flow, i.e. mean monthly flow in the river was calculated to be 40 m3/s. From 1:50,000 topographic maps an estimate was made of areas corresponding to Chapter 2. RESERVOIR MODELS 17 Volume Percent Firm Flow Percent Ratio Mm3 Ideal m3/s Ideal to MAF 1000 >100% 40 100% .80 750 85% 38 96% .60 500 56% 34 85% .40 375 42% 30 75% .30 250 28% 23 57% .20 Table 2.2: Sizes of Hypothetical Goldstream Reservoirs To maintain this firm flow year round, a Rippl analysis suggests a live storage of 886 Mm3. Reservoirs are rarely designed to provide 100% of the firm flow year round so the max-imum practical reservoir volume was selected to be 750 Mm3. However in order to see the value of a forecast on a larger reservoir, an additional reservoir of 1000 Mm3 was also considered. Smaller reservoirs, representing 20%, 30% and 40% of the mean annual flow were also studied. The reservoir sizes studied, as well as the firm flows they can provide and their size with respect to the mean annual flow is given in Table 2.2. The minimum and maximum reservoir volumes needed to allow for the live storages required are listed in Table 2.3. These volumes were determined by constraining the reservoir head to vary no more than ±20% during the operation. The energy generation from a hydroelectric plant is dependent not only on the size of the reservoir but also on the constraints placed on the discharge from the reservoir, through the turbines. A minimum flow is often enforced for secondary downstream uses of the water such as water supply, recreation or environmental uses. Here the minimum is assumed to be 15 Mm3/month. The maximum discharge is determined by the capacity of the generating equipment, and is assumed to be twice the average firm flow. Table 2.3 lists these constraints, for each reservoir studied. To create a truly representative system it would be necessary to model the power Chapter 2. RESERVOIR MODELS 18 Reservoir Design Volume, Mm3 250 375 500 750 1000 Minimum Volume, Mm3 90 210 270 495 780 Maximum Volume, Mm3 345 585 765 1245 1785 Minimum Discharge, Mm3/month Maximum Discharge, Mm3/month 15 15 15 15 15 120 165 180 195 210 Table 2.3: Reservoir Constraints generation from a knowledge of turbine and generator performance curves to estimate efficiency at each head encountered. This level of accuracy was not felt to be necessary for this study so a simplifying assumption was made, that whatever decision is made regarding releases, the equipment will be locally operated to obtain maximum efficiency. This allows the use of the standard power equation: P = njHQ (2.1) and then: E = PT (2.2) where: P = power E = energy n = efficiency 7 = density of water H = head of water above power house Q = flow of water through turbine T = time The value of n, the efficiency, takes into account turbine efficiency, generation effi-ciency and other losses. Efficiency will actually vary with head, but can be considered constant if the range of heads is relatively small. The value used in this study was 87% Chapter 2. RESERVOIR MODELS 19 and would apply to heads ± 20% of the design head. Several other relatively minor assumptions were made. Preliminary calculations were made to estimate expected evaporation losses. Data from evaporation pans in the area were used to calculate expected evaporation for the range of surface areas likely to be encountered in the study. Losses were found to be very small relative to the volumes being considered and so were neglected. Calculations of the expected range in tailwater levels were completed and showed that the range will be fairly small compared to the change in head so constant tailwater levels were assumed. If the programs used for this study were to be used for a study of the potential for generation at this site or adapted for use at any other site these assumptions would have to be re-examined. 2.3 OPTIMIZATION MODELS To generate the most power possible for every unit of water discharged from the reservoir, it is necessary to determine optimal operating policies. Any method which is utilized to determine the optimal operating policy is called an optimization technique. Many different optimization techniques are available and the choice depends on the formulation of the problem and on the characteristics of the objective function. The technique to be used in this study is dynamic programming. Dynamic pro-gramming is a well established technique and several descriptive references are available. Bellman (1967) describes the general formulation of a dynamic programming approach, Yarkowitz (1982) gives an excellent review of water resources applications of dynamic programming, Butcher (1971) describes the use of stochastic dynamic programming in reservoir optimization, and Allen and Bridgeman (1986) describe three recent commercial applications of dynamic programming for reservoir operation. Chapter 2. RESERVOIR MODELS 20 2.3.1 Overview of Dynamic Programming Dynamic programming was popularized by Bellman and is based on what he calls the 'theory of optimality' which states that: ... an optimal policy had the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision (Bellman, 57). In simpler terms this implies that at any one stage in the process, the optimal decision is independent of the decisions in the previous stages. The use of dynamic programming for the optimization of reservoir operation was first presented in English by Little in 1955 and most present applications still follow the same format. Dynamic programming is especially useful in project operation analysis since there is no requirement of linearity on the objective function. All dynamic programming problems are set up in the same basic format. The steps in the problem, at which a decision must be made are called stages, the characteristics of the problem at that stage are described as states. The value of the decision variable determined to be optimal at each stage determines the progression from state to state in subsequent stages. The decision in each stage is made by maximizing or minimizing the value of a recursion equation, which is a function of the value of the decision at this stage, and of the accumulated value of decisions previously made. Figure 2.6 is a schematic of the standard dynamic programming set up. The vertical lines represent the boundaries of each stage, where several values of the state variables are possible. At one state and stage on the diagram, three possible decisions are shown, leading to three possible states at the next stage. At another state and stage the infor-mation derived and saved by the analysis is indicated. This is the value of the recursion function at that state and stage (Fn(sn)), the optimal decision if the system reaches that Chapter 2. RESERVOIR MODELS 21 stage 1 stage 2 stage 3 stage 4 state 1 state 2 state 3 state 4-state s stage n time Figure 2.6: Schematic of Dynamic Programming Structure condition (<f), and the resulting state in the following stage, if that decision is made (*n+l). The standard form of the recursive equation is a follows: Fn(sn) = max[R(d) + Fn^(sn^)] d (2.3) where: Fn(sn) = maximum value of the recursion function from state s in stage n, to the end of the analysis n= index of stage s= index of state in stage n R(d) = value of stage return function as a function of the decision variable, d Fn-\{sn-\) — maximum value of the recursion function from state 5n_x in stage n — 1, Chapter 2. RESERVOIR MODELS 22 to the end of the analysis. The above formulation is for the simplest form of dynamic programming and is re-ferred to as a one dimensional deterministic model. More complex dynamic programming adds other states and functions but the equations have the same overall form. The main disadvantage of the dynamic programming approach is the limitation on the size of the problem that can be realistically handled. Bellman termed this the 'curse for dimensionality'. Because the analysis proceeds in a step by step fashion evaluating the effect of many states and values of the decision variable, it consumes a lot of computer time and space. Thus systems with more than one or two reservoirs are computationally difficult. Many refinements and adaptations are constantly being made to dynamic program-ming analysis to cope with more detailed and complex problems. See papers by Turgeon (1980), Trezos and Yeh (1987), and Yakowitz (1982) for recent examples of work in this area. 2.3.2 Dynamic Programming Models The following section describes the dynamic programming models used. The three dif-ferent programs reflect the differences in the assumptions made about the inflow data. The simplest program is a one state deterministic program which considers expected values. The second program is a one state stochastic program which assumes the inflow data to be represented by a series of inflows and their probabilities. The most complex program used in this study is a two state stochastic dynamic program which considers the previous time period's inflow as an indicator of the present inflow and thus the input data also contains conditional probabilities. All models use discrete dynamic program-ming, meaning that while an infinite number of possible states, stages and decisions are possible, only a discrete number of steps, at previously determined volume increments Chapter 2. RESERVOIR MODELS 23 were considered. The following paragraphs describe the set up of the models, in the traditional dynamic programming format. Figures 2.7 through 2.9 show flow charts of the three programs. Appendix A contains a listing of each of the programs. The programs were written to be as flexible as possible, leaving most descriptive parameters to be input through data files. Define Stages. The stages in this problem are periods in time, in this case months. Each month a release decision has to be made. For reasons inherent in the dynamic programming technique, the analysis progresses from the future, backwards in time, to the present. This is only necessary for the stochastic model, but will be used here in the deterministic model also to avoid confusion. Define States. The states describe the possible conditions of the system at the start of a specific stage. All programs have the volume of water in storage as a state variable, the two state stochastic model also has a state which represents the previous stage's inflow. Each of the various sized reservoirs to be considered in this study has a predetermined maximum and minimum allowable volume. The allowable reservoir states are at intervals of 15 Mm3 between these constraints. Thus the different reservoir sizes have different numbers of states, between eighteen for the 250 Mm3 reservoir and sixty-eight for the 1000 Mm3 reservoir. The incremental value of 15 Mm3 was chosen to get the best precision possible without allowing the number of states to get unmanageable. The second state in the two state stochastic case is an index of the inflow in the previous stage. The value of the second state then, is dependent on the inflow series being considered, not on the operation of the reservoir. It is used to allow consideration of different probability distributions of inflow in one time period, depending on the previous months inflow. Chapter 2. RESERVOIR MODELS 24 start r Input reservoir | Information Input Inflow volumes £ do for each time step do for each \ potential volume/ u. do for each potential ^ discharge calculate final volume N pass dally > • Inflow calculate energy save Info on optimum discharge I calculate new final volume Figure 2.7: Flowchart of Deterministic Optimization Model Chapter 2. RESERVOIR MODELS 25 ( • U r t ) I Input nNMrvolr InfbnTudlon I InpH Inflow volumaa < do far uoh tbnaatap > ILL do for aach potantlal volumt. <do for Mch \ pottntlal r Jlachargt / i f do for aacft potential Inflow oalnlat* final volunt Y ^•^la volumi\ within ^ \eona!ralnta^-'' N pan dally Inflow and oalcUata spill 1 oalcUata 1 naw final voluma oaletiatt anargy oalnaatt axpactad valua aavalnfo on optimum dlacharga Figure 2.8: Flowchart of One State Stochastic Optimization Model Input reservoir Herniation £ H k M W t R H dolor each time step do tor each potential vokma > < Tt do lor each previous Mow > dolor each ont Lai discharge < i f do tor each potential Inflow > calculals ftiai volume Y ^-^TaypJumi>\ ^ ^ c o n a r a i n U ^ ^ H ^ pass daay Mow and calculata sp* T k w calculate T new anal volume calculata energy calculata expected save Moon optimum discharge to Chapter 2. RESERVOIR MODELS 27 Define Decision Variables. The decision variable describes the action taken at the beginning of a stage. In this study the decision variable is the quantity of water to release during the time period of the stage. This release can be discharge through the turbines and/or spill over the spillway. Minimum and maximum discharges are specified and the program decides what discharge, within these constraints is optimal. Again the allowable discharge decisions are in multiples of 15 Mm3. Formulate State Transition Equation. The state transition equation defines which state in the following stage will result from a certain decision being made at the present state and stage. Though the two state stochastic program has two states, the second state is not a part of the transformation process. Thus the state trans-formation equation is essentially the same in all three programs; it is based on a simple continuity equation. The final volume (i.e. the state in the following time period) is the present volume (state), plus the inflow, minus the discharge (decision variable). where: V= total volume of water in storage Q= total volume of inflow D= volume of water released through the turbines S= volume of spill required to stay within constraints The stochastic process is slightly more complex in that each of the possible inflows must be analyzed for each state and discharge. Therefore several potential end states exist. Once each possible inflow has been considered, expected values of discharge, spill and hence energy generation are calculated. Formulate Stage Return Function. The stage return function calculates the value Vfinal = Vinitial + Q — D — S (2.4) Chapter 2. RESERVOIR MODELS 28 or cost of the decision being made, R(d). This study uses the maximization of energy generation as the objective function, thus the stage return function must calculate the energy produced by the discharge decision. In order to do this, the program must first calculate the mean volume over the period which is then used to determine the head. Energy can then be calculated as a function of head and discharge as described in Section 2.2. Formulate the Recursion Equation. The recursion equation calculates the maximum or minimum of the objective function which is dependent on the values of the stage return function and the recursion equation at the previous stage. In this analysis each program has a unique recursion equation, however they are all similar to the form previously described. In each equation, a discount factor is applied to the second part of the expression. This factor a represents the time value of money and discounts the value of energy generated in the future with respect to the energy that can be generated in the present stage. In this study the discount factor was assumed to be 5% per annum. One state deterministic Fn(sn) = max[R(d) + aFn_1(5n_1)] (2.5) a One state stochastic Q—Qjnax Fn(aB) = max[ £ (P(qn)*{R{d) + aFn.1(sn-1))]. (2.6) d 9=<lmin Two state stochastic <t=Qmax Fn{sn,qn+1) = max[ £ (P(<7„k„+i) * (R(d) + a / ^ - i K - i , ? » ) ) ) ] (2.7) a ?=7min where: Fn(sn) o r Fn(sn, <7n+i) = expected return from the optimal operation of the system Chapter 2. RESERVOIR MODELS 29 which has n remaining time periods (dependent on either one or two state vari-ables) sn = index of volume state qn — index of inflow R(d) = return from the system obtained by releasing a quantity of water d in the n'th time period P(qn) = probability of the system receiving an inflow of q during the nth period P(qn\qn+i) = probability of system receiving an inflow of q in the n'th time period, if it received a flow of qn+i during the n + l'th period a = discount factor = 1/1 -f r where r = the interest rate. 2.3.3 Terminal State Values In order to initialize the optimization models it was necessary to provide terminal values to represent the value of the water remaining in storage at the end of the analysis period. Selection of these values is important because poor values can dominate all the results of a dynamic programming analysis by biasing the decision process. Studies by Caselton and Russell (1976) and Caselton et al (1981) developed a complex method of selecting terminal values which recognizes the unbounded time horizon nature of storage reservoir operation decision processes. Analysis of this complexity was not felt to be required for this study, because of the simplicity of the system studied. Instead, an initial estimate of the terminal values was made by assuming that at the end of the analysis period, all inflows to the system ceased but power could be generated at a constant rate until the reservoir emptied. These values were input to the optimization models. During each, optimization process, the system was modelled for several years, working backwards in time. As each year was analyzed, the values calculated for that year's operation became Chapter 2. RESERVOIR MODELS 30 the terminal values for the next year. Within three years, the terminal values reached a steady state, suggesting that the initial estimates were no longer influencing the optimal policy decisions. The final year policy was then used for the simulation runs and the new steady state, terminal values were recorded for use later in the analysis for the comparison of runs in which the reservoirs ended the year at different final volumes. 2.4 SIMULATION MODELS Several simulation programs were written, one for the deterministic and one state stochas-tic models and one for the two state stochastic model, for both the naive and forecast scenarios. The simulation models perform what is known in dynamic programming lan-guage as 'optimal state trajectory recovery'. The program reads the optimal policy specified as well as the actual inflows to the system. The programs then model the oper-ation of the reservoir, by stepping through the time periods, and calculating the release of the reservoir by consideration of the actual inflow and the optimal policy. All the models are essentially the same but the two state models include consideration of the previous months inflow, and the models for the forecast scenario consult several operating policies, one generated for each months updated forecast. The algorithms of the simulation programs and the input required are very similar to the dynamic programs already described. Figure 2.10 shows a simplified flow chart of the analysis. Full listings of the programs are included as Appendix B. 2.4.1 Model Operation The simulation models operate the reservoir in real time. In order to commence operation the model needs an initial condition representing the volume of water in the reservoir on January 1, the beginning of the study period. For this study the reservoirs were assumed Chapter 2. RESERVOIR MODELS 31 •tart Input reservoir Information Input 1 Dor 20 operating policy do for each \ time period / oalouleta final volume pase dally Inflow and oaloulate •pill energy oaloulata new final volume final volume -Initial volume of next time step output results Figure 2.10: Flowchart of Simulation Models Chapter 2. RESERVOIR MODELS 32 Reservoir Design Volume, Mm3 Reservoir Starting Volume, Mm3 250 165 375 255 500 330 750 495 1000 670 Table 2.4: Reservoir Starting Volumes to start two thirds full, a realistic value. Table 2.4 lists these values. The simulation phase is slightly different for the two forecast cases. The simulation using the naive forecast scenario uses the same operating policy for the entire year, however, the simulation for the forecast cases uses a different forecast for each month from January through August (the August forecast is used for the remainder of the year), resulting in an updated operating policy. This reflects the fact that in real time operation of a reservoir, the forecasts would be updated for each planning period to make use of more accurate information that becomes available as the year progresses. An additional routine had to be added to the simulation model for the two state analysis. The operating policies are derived considering a specific set of inflows and therefore of inflow and antecedent inflow combinations. During simulation however it is possible that an inflow and antecedent inflow combination may occur during the real time period, which never occurred during the forecast inflow series used to generate the operating policies. The fewer years of historic data that are available to generate the operating policies, in either of the forecast scenarios, the worse this problem will be. An unfamiliar inflow will not cause problems during the time step in which it occurs, however when the analysis moves to the next inflow and looks for the optimal decision at that time step depending on the unfamiliar previous stage's inflow, it will not find a recommended Chapter 2. RESERVOIR MODELS 33 discharge. In this case the model uses as the optimal discharge, an interpolation of the optimal discharges at the next lowest and next highest previous inflow states. If the unfamiliar inflow is lower than the extreme values encountered in the optimization the recommended discharge of the lowest encountered previous inflow is used. Similarly, if the unfamiliar inflow is higher than any previous inflow encountered during optimization the optimal discharge for the next highest experienced flow will be used. Chapter 3 INFLOW FORECASTS 3.1 DEFINITIONS The term hydrologic forecasting can encompass techniques from simple statistical meth-ods to complex models that endeavors to exactly model the response of the basin to the external input. In this study forecasting will be restricted to real-time flow forecasting which is: ... the hour by hour or day by day assessment of flow sequences based on (1) measurement of rainfall, snow accumulation, inflow and other meteoro-logic variables, up to the time of forecast, and (2) predictions of meteorologic variables expected to occur in the immediate future (Fleming, 1975). An ideal forecast would completely and accurately predict the future state of the system, however all forecasts have some degree of uncertainty. The inflow to a reservoir is equal to the basin runoff which is a function of the input to the system, such as rainfall, snowmelt, temperature, wind, solar radiation etc., and also the physical characteristics of the basin such as size and shape, geology and soil type etc., none of which can be forecast or determined with complete accuracy. There are two main categories of forecasts: empirical and conceptual. Empirical models use regression analysis or other mathematical techniques to correlate volumes of seasonal runoff to any number of hydrometeorologic or physiographic parameters. These models are primarily black-box models, the emphasis is on the input and the output 34 Chapter 3. INFLOW FORECASTS 35 of the model, the actual basin processes are not considered. These statistically based models only produce an estimate of total seasonal runoff and do not give estimates of seasonal flow distribution. The second major class of forecast models, conceptual models, attempt to model the actual response of the basin. Conceptual models use a variety of mathematical algorithms to model the different potential pathways of the water through the basin. Superposition is used to combine the various flow components into an outflow. Utilities are still primarily relying on empirical regression type forecasts, but conceptual forecasts are becoming more common. If no forecast model has been created for a basin, a reservoir operator can use any existing record of the historical inflows to the system, to gain some knowledge of prob-able flow. Given no other information about the present season, an arithmetic mean of previous inflows, during a specified time period would give an estimate of expected inflows during.the equivalent time period in the coming season. This can be called a naive forecast. 3.1.1 Conceptual Hydrologic Models A conceptual hydrologic forecast is derived from a model that attempts to model the structure and reaction of the catchment to the precipitation input. T. O'Donnell (1986) describes conceptual models (which he calls non-linear deterministic catchment models) as follows: The general pattern of such modelling techniques is to postulate a general model of catchment processes and behavior, the structure of the model and its functioning being based partly intuitively and subjectively, partly empir-ically, partly theoretically, on what is known or assumed about catchment Chapter 3. INFLOW FORECASTS 36 processes and behaviour. The basis of operation of such models is the prin-ciple of continuity, i.e. maintaining at all times a complete water balance between all inputs, outputs and changes in storages. Such a general model is fitted or matched to any specific catchment in some systematic way using recorded input and output data for that catchment. The fitting is done by making adjustments to the parameters used in algebraic equations purporting to describe the processes and interconnections of the catchment model. Such parameter adjustment is continued until the output computed by the model when supplied with the recorded catchment input agrees with the recorded catchment output (to within some specified tolerance). The most difficult aspects of conceptual modelling are the selection of the appropriate model for the catchment and the fitting (or calibration) of the model to the specific basin. Not all models are able to forecast flow in all basins due to the different parameters affecting the flow. Even once the overall processes in the model are established to be appropriate, the selection of the correct parameter values can still be quite complex. Often several sets of varying parameter values will produce a good fit using historic data and it is the task of the model designer or user to determine which values will most accurately represents the system and hence will give the most accurate prediction of runoff when the model is used in real time. There are several advantages of a good conceptual forecast over an empirical fore-cast. A forecast based on purely historical data such as a series of historical means or a regression analysis is liable to be accurate only within the recorded range of values. In contrast, a conceptual model represents more closely the actual characteristics of the basin and should therefore be able to model the response to previously unrecorded sys-tem input. This is only true if the model is accurate and has been well calibrated. In Chapter 3. INFLOW FORECASTS 37 addition conceptual models give not only an estimate of the seasonal runoff volume, as empirical formulas do, but also give an estimate of the time distribution of that runoff. There are however, disadvantages to conceptual models. They often require large quantities of data to describe the physiography and hydrology of the basin being modelled and also expertise is needed to calibrate and operate the model for a specific basin. It can therefore not be assumed that because a model is more complex, that it is automatically 'better1, or that it is worthwhile. These models are costly to develop, calibrate and operate and the marginal value of conceptual models over straightforward regression models, or over naive forecast operation must be examined. In hydroelectric power generation scenarios, this marginal value can be measured in terms of the increase in energy that can be generated by improved foreknowledge of the inflows. 3.1.2 UBC Watershed Model The conceptual hydrologic forecasting model used in this study is the UBC Watershed Model. This model was developed by Quick and Pipes of the University of British Columbia in 1964 and has been extensively modified and updated since that time. The main structure of the model is illustrated in Figure 3.11 (from Quick and Pipes, 1976). In contrast to other conceptual models, the UBC model was designed for use on a sparse data base and in mountainous basins where a significant portion of the basin runoff is due to snowmelt. A key aspect of the UBC Watershed Model is that it is based on the area elevation characteristics of the catchment. This approach is effective because both precipitation and temperature are elevation dependent. These effects are modelled using relationships describing lapse rates and orographic precipitation. In addition, many physical charac-teristics of a catchment basin are also elevation dependant. For calibration, the model requires daily temperature and precipitation values as well Chapter 3. INFLOW FORECASTS METEOROLOGICAL DATA INPUT DISTRIBUTION BY ELEVATION ZONE TEMPERATURE E V A P O -TRANSPIRATION LOSSES WATERSHED MOISTURE BALANCE COMPUTATIONS RUNOFF COMPONENT ALLOCATION TIME DISTRIBUTION OF RUNOFF MODIFICATION BY WATERSHED STORAGES SLOW RUNOFF I GROUNDWATER SNOWFALL SNOWMELT MEDIUM RUNOFF I INTERFLOW FLASH BEHAVOIR FROM HIGH INTENSITY RUNOFF FAST RUNOFF SURFACE RUNOFF EVALUATION WITH RECORDED STREAMFLOW DATA GENERATED STREAMFLOW (INPUT TO CHANNEL SYSTEM) Figure 3.11: Simplified Flow Chart of the UBC Watershed Model Chapter 3. INFLOW FORECASTS 39 as flow data for the appropriate rivers. Once the model is calibrated, it can be run for several years without outside influence. Given appropriate starting conditions and input of precipitation and temperature data, it will model the response of the basin, determining rainfall runoff as well as snow accumulation or melt. Streamflow forecasts can be obtained for periods of several days to several months. The major control parameter of the model is the soil moisture deficit. It is used to control the amount of runoff released to the various component of runoff; direct runoff, interflow and groundwater flow. Routing of the flows is accomplished using unit hydro-graph convolution. Another important aspect of the UBC Watershed Model is its ability to be updated regularly as more information becomes available. Each time the model is run a forecast is generated for the duration of the study period, however, well before the end of the time period, a new forecast will be generated which uses the latest streamflow measurements, observed temperature and precipitation values and snowpack information. The UBC Watershed Model has been used since its development for research at UBC and for reservoir operation by BC Hydro (Druce, 1984). In 1979-1983 it was analyzed as part of the World Meteorological Organization's Intercomparison of Models of Snowmelt Runoff (WMO, 1986). 3.2 GENERATING FORECASTS In order to derive the inflow sequence based on the historic flows and also the forecasts from the conceptual hydrologic model it is necessary to use some historic information from the catchment basin. The naive sequence uses the actual means of observed inflows as the inflow forecast, assuming that the past experience of the catchment gives an estimate of the future conditions. The conceptual forecasting model uses historic weather years Chapter 3. INFLOW FORECASTS 40 to predict the range of conditions that might occur during the years to be forecast. A series of historic weather conditions are imposed on the 'present' snowpack conditions to obtain an estimate of the flow, given those weather conditions. It was decided to use seventeen years, from 1971 to 1987, as the historic record to use in deriving,the forecasts. The years chosen to forecast were 1966, 1968, 1969, and 1970. These represent a high flow year, a medium high year, a medium low year and a low flow year, as can be seen in Figure 3.12 which plots the total annual volume inflow, in comparison to the average annual inflow volume for the full hydrologic record. Hydro-graphs for the four selected years are plotted, along with the mean annual hydrograph for comparison, in Figure 3.13. All inflow values are considered in volume units of million cubic meters per month and are rounded to the nearest 15 Mm3 to correspond to the volume states considered in the analysis. 3.2.1 Naive Sequence If no complex forecast of inflow is available for a catchment the operators will usually operate the reservoir with reference to historic data. The runoff values for the previous years of record will give some idea of the magnitude of flow likely to occur in the upcoming season. For the river used in this study, there are twenty-four years of data available, seventeen of which were used for the flow forecasting. A table showing the historic inflow data used to derive the naive forecasts is provided in Appendix C. For the deterministic programs, a simple arithmetic mean of each period's data gives a 'mean year' of flow. For the one state stochastic model a flow series was determined using a simple Fortran program which calculates the frequency of occurrence of each observed flow for each period and calculates corresponding probabilities. For the two state stochastic model the program was modified to perform the same analysis but for each observed flow in the previous time period. Listings for both programs are provided in Appendix D. Chapter 3. INFLOW FORECASTS 41 "1 i i i i i i i i i i i i i i i — i — i — i — i — i — i — i — I 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 Figure 3.12: Annual Inflow Volumes, Goldstream River, 1964 to 1987 Chapter 3. INFLOW FORECASTS 42 Figure 3.13: Hydro-graphs for Test Years: 1966, 1968, 1969, 1970 Chapter 3. INFLOW FORECASTS 43 3.2.2 C o n c e p t u a l l y Forecast Flow Sequences The conceptually forecast flow sequences were generated using the UBC Watershed Model, calibrated and run by BC Hydro. The calibration statistics calculated by the model suggested that the model accurately predicts the response of the basin. The av-erage of the explained daily variances (R2) was 93.9%. The annual computed volumes, analyzed to indicate bias, showed overall minimal bias, with an average of 99.7% though the values for each year varied between 89.5% and 111.8%. The monthly computed volumes showed similar lack of bias: Jan = 107% Feb = 104% Mar = 103% Apr = 101% May = 102% Jun = 101% Jul = 98% Aug = 100% Sep = 100% Oct = 95% Nov = 100% Dec = 98% The spring shows higher bias than the remainder of the year, but the average is near perfect at 100.8%. The model was run in the forecast mode, as if it was operating in real time. For each of the test years, the January 1 conditions of the system were entered and the meteorologic data of the seventeen years of historical data are applied in turn to the basin. The resulting seventeen years of runoff predicted form a stochastic representation of the predicted inflow for the basin for that forecast year. The initial run is the best guess for the inflows for the entire year that can be made on January 1. Subsequently an updated forecast was generated representing February 1. This would be a new forecast using observed information for January. In the same way the forecast was updated six more times representing March 1 to August 1 forecasts, each run using observed information about the system that would be available at that time. The resulting output then was four years of eight updated monthly forecasts each consisting of seventeen Chapter 3. INFLOW FORECASTS 44 Month Naive 1966 1968 1969 1970 Cone. Obs. Cone. Obs. Cone. Obs. Cone. Obs. January 15 15 30 15 15 15 15 15 15 February 15 15 15 15 15 15 15 15 15 March 15 15 15 15 15 15 15 15 15 April 45 60 45 60 30 60 60 45 15 May 195 210 195 210 180 195 225 165 120 June 330 375 315 315 375 285 345 300 345 July 270 360 300 330 375 165 180 180 210 August 165 165 165 150 165 120 120 120 120 September 90 90 90 90 120 90 75 90 60 October 45 45 60 45 75 45 60 45 45 November 30 30 45 30 45 30 45 30 30 December 15 15 15 30 30 15 30 15 15 Total 1230 1395 1290 1305 1440 1050 1185 1035 1005 Table 3.5: Naive, Observed and Conceptual Inflow Volumes (in Mm3), Goldstream River possible inflow sequences based on seventeen year of historic weather patterns. A full listing of each forecast is provided in Appendix E. Stochastic inflow sequences for input to the optimization models were then derived using the same programs as were written for the naive input. Table 3.5 gives the generated deterministic conceptual forecasts for each of the test years. The forecast given consists of the best forecast available for each month, i.e. the January value is from the January forecast, the February value is from the February fore-cast and so on. This composite forecast is used throughout the study when comparisons are made between forecasts. 3.3 COMPARISON OF INFLOW FORECASTS Table 3.5 listed the observed inflows as well as the naive and conceptual inflow series for the Goldstream River for the test years. Figures 3.14 through 3.17 are plots of the Chapter 3. INFLOW FORECASTS 45 1966 1968 1969 1970 Naive Cone. Naive Cone. Naive Cone. Naive Cone. Ratio .94 1.07 .85 .90 1.04 .89 1.22 1.03 MPE (%) 14.4 19.9 20.6 25.2 21.5 13.9 31.9 26.3 MAE {Mm3/month) 8.8 17.5 22.5 22.5 21.3 13.8 21.3 15.0 RMSE (Mm3/month) 13.0 26.7 36.2 28.7 32.1 21.7 33.3 23.7 C p or E .98 .93 .92 .95 .89 .95 .88 .94 Table 3.6: Statistics for Naive and Conceptual Forecast Inflow Volumes three forecast scenarios for each study year. Table 3.6 gives some standard comparative statistics that measure the goodness of fit of the forecasts. The definitions of these parameters are given below. Ratio is the ratio of the total forecast volume to the observed inflow volume. Ratio = E j i f Qo(i) (3.8) MPE is the mean percent error. M P E = [ g 0 / ( 0 - f - ( 0 1 ] , M (3.9) Qo(i) J N MAE is the mean absolute error, which is in the units of the numbers being compared, in this case Mm3/month. MAE = Egf I Q / ( 0 - Q Q ( 0 I -N (3.10) RMSE is the root mean squared error, also in Mm3/month. RMSE = TZ?(Qf(i)-Q.(i)y N (3.11) In addition, Lettenmaier (1984) suggests the use of the coefficient of prediction, Cr Chapter 3. INFLOW FORECASTS 46 C p - l - [ — (3.12) The coefficient of prediction is a measure of the same variance as the coefficient of efficiency described by Nash and Sutcliff (1970) and recommended by Kitinadis and Bras (1980), where E = ^ - (3.13) in which s = fW) - Qod))2 i=l and So = E(g.(0 - Q~o)2 where: Qf = forecast (conceptual or naive) inflow Q0 = observed inflow Q0 = mean of observed inflow iV = number of observations cr2 = variance of the observed inflow Examination of the table shows that the forecast statistics do not always unanimously conclude that one forecast is superior to the other. This is because each statistic measures a different aspect of the results and their variations. Some are dependent on the total magnitude of the results only, others also depend on the time distribution. Several of these statistics have dimensions, i.e. are a function of the magnitude of the numbers they describe and while these statistics can be used to compare the different forecasts of individual years, they are not appropriate for comparison between years as they are a function of the actual inflow. The simplest statistics for comparison of the forecasts are: Chapter 3. INFLOW FORECASTS 47 V) Ml <Z t-LU 5 U 00 o z o o 400 350 300 -250 -200 -150 -100 -50 -• OBSERVED - r NAIVE FORECAST o CONCEPTUAL FORECAST 1 1 1 1 1 i 1 1 1 1 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3.14: Comparison of Perfect, Naive and Conceptual Forecast Hydrographs: 1966 the ratio for an overall estimate of the volume accuracy and the coefficient of prediction or efficiency and the marginal percent error, for estimates of how the forecast performs over time. The following sections assess the accuracy of each years forecast. 3.3.1 Forecast Year: 1966 The inflow volume during 1966 was only fractionally higher than the average annual inflow. The peak in June was lower than average but the flow remained higher in July than usual. The historic mean in this year is a good estimate of the inflow, giving a coefficient of prediction of 98% and a marginal percent error of only 14%. The conceptual hydrologic forecast for 1966 overestimates the flow for most of the year. The shape of the forecast hydrograph is very similar to the observed hydrograph Chapter 3. INFLOW FORECASTS 48 UJ 5 o CO 5 o 400 350 -300 250 200 150 100 -50 • OBSERVED + NAIVE FORECAST o CONCEPTUAL FORECAST I I I I I 1 1 1 1 1 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3.15: Comparison of Perfect, Naive and Conceptual Forecast Hydrographs: 1968 shape but during the peak in June and July the magnitude is overestimated by almost 25%. The coefficient of prediction for the conceptual forecast is 93% and the mean percent error is 20%. 3.3.2 Forecast Year: 1968 The inflow volume during 1968 was significantly higher than average and is the second highest annual inflow on record for this basin; the year 1976 had a higher inflow. The peak occurred in June as is usual but the same magnitude of inflow was maintained through July. The naive forecast is a fair estimate of the inflows resulting in a coefficient of prediction of 92% and a mean percent error of 21%. The conceptual forecast for 1968 also underestimates the inflow for most of the year, but not as severely. The shape of the forecast hydrograph is fairly accurate having high Chapter 3. INFLOW FORECASTS 49 UJ CC. r-UJ S o CO 3 t> z o 5 o _! Z 400 350 300 250 -200 150 100 50 -• OBSERVED + NAIVE FORECAST o CONCEPTUAL FORECAST 1 1 1 1 1 1 1 1 1 1 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3.16: Comparison of Perfect, Naive and Conceptual Forecast Hydrographs: 1969 flow in both June and July. The forecast statistics indicate however that the conceptual forecast is similar in accuracy to the historic mean, giving a coefficient of prediction of 95% and a mean percent error of 25%. It can be seen that the other forecast statistics are somewhat mixed as to which forecast is superior. 3.3.3 Forecast Year: 1969 The inflow volume during 1969 was slightly below the average annual inflow. The rising limb and peak of the observed hydrograph is very similar to the mean, however the lag portion of the hydrograph shows significantly lower inflows than average through July to September. The historic mean is a fair estimate of the actual inflow, giving forecast statistics of 89% for the coefficient of prediction and 22% for the mean percent error. Chapter 3. INFLOW FORECASTS 50 v> UJ cc r -UJ 2 o CQ "3 O z o _1 5 o 400 350 -300 -250 -200 -150 100 -50 -• OBSERVED + NAIVE FORECAST o CONCEPTUAL FORECAST 1 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 3.17: Comparison of Perfect, Naive and Conceptual Forecast Hydrographs: 1970 The conceptual forecast for 1969 was quite good. The shape of the forecast hydro-graph and the magnitude of the rising and lagging sections are very accurate, however the peak was underestimated by about 17%. The conceptual forecast is significantly better than the naive forecast, having a coefficient of prediction of 95% and a mean percent error of only 14%. It is interesting to note that as an estimate of volume inflow only the mean is a far better estimate than is the conceptual forecast. The ratio of forecast inflow to actual inflow volume is 1.04 for the naive forecast and 0.S8 for the conceptual forecast. 3.3.4 Forecast Year: 1970 The inflow volume for 1970 is significantly lower than the annual mean and is in fact the lowest year on record, which means that it has a lower inflow than anyof the years used Chapter 3. INFLOW FORECASTS 51 to derive the naive forecast. The peak of the observed hydrograph is almost coincident with the peak on the mean hydrograph, however the rising and lagging section of the hydrograph show much lower inflows during the majority of the year. The naive forecast then is a poor predictor of the actual inflow. The coefficient of prediction is only 88% and the mean percent error is 32%. The conceptual forecast for 1970 gives a good estimate of the overall volume inflow for the year but the shape of the hydrograph is not as good. The flow is overestimated during the spring but the peak is underestimated. Overall though, the conceptual forecast is fairly good and has a coefficient of prediction of 94% and a mean percent error of 26%. 3.3.5 U n c e r t a i n t y The previous discussion has considered only the accuracy of the deterministic form of the forecasts. Much of this study however, is concerned with the uncertainty of the inflow forecasts. A determinisic representation considers only one value and assumes it to be accurate. Stochastic representations recognizes that unceratinty exists and account for it by considering a range of numbers of varying probabilities, rather than a single number. A superior forecast then, should have not only a more accurate mean in the determinisitc sence but should have a narrower range of values in the stochastic representation. The hypothesis used in this study is that the uncertainty of the conceptual forecast will be lower than that of the naive forecast. The naive forecast was based on seventeen different years of flow data, wheras the conceptual forecast was developed by considering the same seventeen years of weather data, but on a known snowpack which eliminates a large source of variability in the runoff estimates. In addition, the conceptual forecast is updated as the season progresses and more information becomes available so that the uncertainty should also reduce as the season progresses. The uncertainty in the forecasts can be measured by the standard deviation. The Chapter 3. INFLOW FORECASTS 52 Month Naive Forecast Conceptual Forecasts 1966 1968 1969 1970 January 3.5 0 0 3.5 3.5 Febraury 0 0 0 0 0 March 5.7 6.4 7.2 7.2 7.2 April 16.5 16.9 16.9 17.9 14.5 May 45.2 54.1 52.5 49.9 47.7 June 66.4 47.1 86.3 33.3 33.8 July 53.7 37.1 33.0 22.1 30.9 August 44.6 16.2 16.3 14.5 17.0 September 35.4 30.2 29.5 30.1 30.1 October 12.5 15.5 15.5 13.6 14.3 Novermber 35.3 6.8 8.2 8.2 7.7 December 20.3 7.5 7.5 7.4 6.8 Mean 28.3 19.8 22.7 18.0 17.8 Table 3.7: Standard Deviations of Naive and Conceptual Forecasts monthly standard deviations for the seventeen years of the naive and four conceptual forecasts are presented in Table 3.7. The standard deviation of all forecasts increase to a peak in May or June. The standard deviation of the naive forecast remains quite high for the remainder of the year, however the standard deviation of the forcasts generally decreases as the year progresses, showing the value of updating the forecasts. Chapter 4 RESERVOIR OPERATION RESULTS The purpose of this study was to determine the value of conceptual forecasting as a function of reservoir size. The optimization and simulation models of the hypothetical hydroelectric project were used only as tools for comparing the naive and conceptual forecasts. Examination of the operation of the projects is valuable for several reasons: (1) to ensure that the optimization and simulation models produce reasonable results and (2) to see how the actual operation differs between the various operating policies and thus determine how the value of the forecast may be reflected in the operation of the project. This chapter is divided into three sections. The first section describes the way in which the reservoir models operate the study reservoir. The operation is dependent on the capacity of the project and on the inflow sequence and format used, however general observations pertaining to most runs can be made. In the second section, the project operation is studied under certain ideal conditions. Patterns will be observed in these idealized conditions which should be useful in interpreting, in later chapters, the results of the production runs. The last section studies in detail two production runs, to demonstrate the operation under real conditions. 4.1 RESERVOIR OPERATION The month to month operation of the reservoirs is a direct result of the assumptions made regarding the operating objectives and on the constraints imposed on the system. 53 Chapter 4. RESERVOIR OPERATION RESULTS 54 Specifically, the operation of this hypothetical reservoir is affected by the assumption that the operator is acting as an Independent Power Producer, meaning that energy generated can be sold to the grid at any time and that the value is constant, regardless of supply or demand. The operation is also a function of the minimum and maximum reservoir volumes and discharge capabilities. These limits were described in Section 2.2. In the production runs, the reservoirs were assumed to be two thirds full at the beginning of January (the start of the simulation period) which was thought to be a realistic value. The volume at the end of the simulation in December was not constrained by the model but in most cases simulation runs ended with the reservoir full, or one state below full. The exceptions were all in 1970, which was a year of extremely low flows. From January to March or April, the reservoirs maintain a fairly constant volume, passing whatever inflow they receive directly through the turbines. The volumes remain constant and the discharges remain low until provision has to be made for the freshet flows. The operating policies of the smaller reservoirs require that the reservoir volumes must be decreased by increasing releases, during April, May and sometimes June to create storage for the high freshet flow. The reservoirs rarely completely empty, though it can happen. The larger reservoirs have sufficient storage and discharge capacity to continue passing inflow until summer. In June or July the reservoir levels start to rise due to heavy summer inflows. Discharges are therefore also high during this period. By September or October the reservoirs are full and then stay full for the remainder of the year, by again releasing inflow. During simulation the operation will pass the optimal discharge if possible. If the reservoir cannot provide the required flow without breaking volume constraints, the actual flow is increased or decreased until the resulting final volume is within the constraints. This adjustment is usually only necessary when the reservoir is full, and results in a higher than recommended discharge, or if the discharge is already at maximum limits, a Chapter 4. RESERVOIR OPERATION RESULTS 55 higher spill. Many of the operating policies contained some spill in the high flow summer months, because of reservoir storage limitations. In many of the simulations however, the observed inflows are lower than anticipated by the forecasts and therefore the spill can be reduced. Only in 1968, a very high flow year, is spill significant. As would be expected, the amount of spill required decreases as the reservoir size increases, regardless of the inflow. In nearly all simulation cases there were magnitudes of flow in the observed inflows that were not present during those time periods in the inflow series used to generate the operating policies. This resulted in some months of the two state stochastic analysis having an undefined operating policy. In these cases the model approximated an optimal policy by interpolating from decisions linked to other previous inflows. Energy generation is a function of reservoir level and discharge and so follows the same trends as discussed for these two parameters. Generation is low during the winter when flows and reservoir volumes are low, increases rapidly during early spring as inflows and volumes increase and then more slowly until a peak in August or September when the reservoir fills. As inflows and thus discharges decrease, energy production again gradually decreases, but remains higher than in the spring, due to higher reservoir volumes. 4.2 IDEAL OPERATION The runs for the following section were made assuming perfect knowledge of inflows, i.e. the optimization and simulation phases of the analysis used the same inflow se-ries, whether it was the forecast or the observed flows. For these runs, the reservoirs were assumed to start full. These simplifications were made in order to produce results that could be used to observe patterns to assist in interpreting the more complex runs, discussed later in the report. Chapter 4. RESERVOIR OPERATION RESULTS 56 4.2.1 Reservoir Size Figure 4.18 shows the effect of reservoir size on the operation of the reservoir. Each reservoir was optimized and simulated using the deterministic models and assuming per-fect foreknowledge of an inflow series comprised of the mean flow for each month. The volume of the drawdown for each size reservoir is plotted for each month. This volume is defined as: where: Volume = Volume at the beginning of the month Maximum Volume = Maximum volume as defined by the constraints discussed in Section In general the drawdown increases as the size of the reservoir decreases. The only exception to this pattern is the smallest reservoir size, 250 Mm3, which is prevented from drawing down further by volume constraints. The trend occurs because of the relationship between the reservoir size and the maximum allowable discharge, as discussed in Section 2.2, namely that the maximum discharge is twice the firm flow provided by the reservoir and thus increases as the size of the reservoir increases. Therefore in this study, the larger reservoirs can usually discharge the inflow through the turbines during the period in which it occurs and thus little storage is necessary. In the smaller reservoirs, inflows during the freshet are much greater than the turbine capacity of the project and thus the reservoir must be drawn down before the freshet in order to store the expected flows for later release. The 250 Mm3 reservoir creates an exception to this pattern because it cannot create enough storage for the freshet flow, even if it empties completely, which it Drawdown = Volume — M aximumV olume (4.14) 2.2 Chapter 4. RESERVOIR OPERATION RESULTS 57 50 > UJ W -350 -UJ OC -400 -j 1 1 1 ; 1 1 1 1 1 1 1 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 4.18: Comparison of Reservoir Operation for each Reservoir Size Chapter 4. RESERVOIR OPERATION RESULTS 58 does in May. This results in the reservoir filling very quickly as the inflows increase and also in significant quantities of spill. It can also be observed that the larger reservoirs refill more quickly after the freshet. Again this is because the additional discharge capacity means that it is not necessary to maintain storage for potential high flows. Consequently the operation of the project appears to be more a function of the discharge capacity than of the size of the reservoir. 4.2.2 Magnitude of Inflow Figure 4.19 shows the same parameter as the previous figure but this time the size of the reservoir is kept constant, at 750 Mm3, and the annual inflow to the reservoir is varied. The reservoir is optimized and simulated using the deterministic models and assuming perfect foreknowledge of the observed flows. This is the perfect forecast case as described in previous sections and represents the ideal operation of the system under the experienced conditions. The results are as expected; the amount of drawdown increases as the magnitude of the annual inflow increases. 1968 was a high flow year and the reservoir reacts to this by drawing the reservoir well down in May and June ready for the high flows in July and August. The years 1966 and 1969 are both fairly close to the average in magnitude, and therefore would be expected to operate fairly similarly, however the time distribution of the flows is quite different. The flows in 1969 peak in June and recede quickly so the reservoir can refill quickly. The high flows in 1966 start earlier and remain high longer so the overall drawdown needs to be greater. In 1970, the peak flow is quite high, but occurs later than usual and only lasts for one month and therefore the reservoir needs to be drawn as low as in 1969, but not until slightly later in the year and the reservoir can refill quickly afterwards. Chapter 4. RESERVOIR OPERATION RESULTS 59 50 Figure 4.19: Comparison of 750 Mm3 Reservoir Operation for each Forecast Year Chapter 4. RESERVOIR OPERATION RESULTS 60 4.2.3 Uncertainty Figures 4.20 through 4.22, plot the drawdown of the 750 Mm3 reservoir, as recommended by the deterministic, and the one and two state stochastic models. The reservoir was optimized and simulated using the updated forecasts generated by the UBC Watershed Model assumed to be correct (i.e. the forecast flows rather than the observed flows were used in the simulation). A high, medium and low year were chosen, namely 1968, 1969, and 1970. Because the forecasts, rather than the actual inflows, were used in this analysis, the spread between the inflows was not as high. 1968 was underestimated by the forecast and 1969 and 1970 were overestimated. Each year was operated by the deterministic model, and the one and two state models to examine how uncertainty affects the operation of the reservoir. The shapes of the drawdown curves follow fairly closely the observations made in the previous section, the differences are due to the inaccuracies of the forecasts. In each case, the deterministic model caused the reservoir to be drawn down more than the stochastic models. Of the two stochastic models, the two state model generally suggested more drawdown than the one state model during the freshet, but less during the recession part of the year. The stochastic representations operate more conservatively than the deterministic, by drawing down the reservoir further in expectation of the freshet and then operating the reservoir at a lower level for the remainder of the year. This characteristic is a result of hedging. The deterministic model considers the inflows to be known definitely and therefore takes more aggressive action. The stochastic models consider several different potential inflows for each period and must decide on discharges that minimize risk and maximize profit, despite the uncertainty. How this affects the operation of the reservoir depends on the loss function and the constraints applied to the reservoir. In this scenario, the water remaining in storage at the end of the year is Chapter 4. RESERVOIR OPERATION RESULTS 61 Figure 4.20: 750 Mm3 Reservoir Operation: High Inflow Case (1968) Chapter 4. RESERVOIR OPERATION RESULTS 62 50 D OC O -300 -> cc Ul CO -350 -UJ OC -400 -\ 1 1 1 1 1 1 1 1 1 1 ' 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 4.21: 750 Mm3 Reservoir Operation: Medium Inflow Case (1969) Chapter 4. RESERVOIR OPERATION RESULTS 63 Figure 4.22: 750 Mm3 Reservoir Operation: Low Inflow Case (1970) Chapter 4. RESERVOIR OPERATION RESULTS 64 evaluated in terms of its potential for future generation (see Section 2.3). Beyond this there is no penalty for the reservoir ending the year less than full and so there is no incentive to withhold discharge in order to fill the reservoir. Additionally spill is strongly discouraged as it represents a complete loss of potential energy generation. Thirdly, the value of energy generated is constant over time, except for the discounting due to the time value of money, so the tendency would be to release higher discharges to generate energy now rather than later. These factors all cause the hedging of the stochastic models to tend toward operating the reservoir at a slightly lower level than that indicated by the deterministic model. It must be remembered however that other reservoirs with different constraints and loss functions may tend to operate differently. 4.3 CASE STUDIES Appendix F tabulates the results of all the production runs of this study. This section studies the operation of two reservoir scenarios in detail. The operation of the 375 Mm3 reservoir for the forecast years 1970 and 1968 was chosen. The flow in 1970 was extremely low and the flow in 1968 was extremely high, making the naive forecast poor in both years. The conceptual forecast was excellent for 1970 but was less good in 1968. The operation of the reservoir in real time under these flow conditions will be discussed in the next sections. 4.3.1 1970 The naive forecast for 1970 was very poor, and the conceptual forecast was very accurate. In both forecast cases the deterministic operating policy gave better results than either of the stochastic operating policies so this discussion will consider only the deterministic results. Figures 4.23 through 4.25 show various aspects of the reservoir operation. The Chapter 4. RESERVOIR OPERATION RESULTS 65 values are plotted for the perfect forecast, i.e. the maximum potential efficiency, and for the deterministic representations of the two forecast cases. Figure 4.23 plots the initial monthly volumes of the reservoir. The reservoir maintains a constant volume during early spring, and then starts to empty in time to receive the freshet flows. In June, the reservoir volume begins to increase as a result of high inflows. By August or September, the reservoir is full and remains so for the remainder of the year. Operation of the reservoir according to the conceptual forecast is more efficient than operation resulting from the naive forecast. Because the inflows during 1970 are so much lower than the mean, operation by the naive forecast causes the reservoir to be maintained at a lower volume than is ideal. The reservoir nearly empties in May and June, and subsequently fills several months later than is ideal. Operation of the reservoir according to the conceptual forecast keeps the volumes very close to the ideal levels obtained using the perfect forecast. The discharges plotted in Figure 4.24 are the actual discharges calculated by the sim-ulation model. The discharges are identical to those required by the operating policy in all months except for July when discharges for both forecast scenarios must be increased to prevent overtopping. A discharge value above the maximum discharge line indicates that spill is occurring. It can be seen that the conceptual operating policy gives results closer to the ideal operation than does the naive. The naive operating policy starts to increase discharges too early, because it anticipates higher freshet inflows than actually occur. Discharges are then lower in June because the reservoir volume is too low. The observed reservoir inflows are also plotted on Figure 4.19. When the inflow line is above a particular discharge line, water is being added to storage. When the inflow line is below a particular discharge line, water is being released from storage. The energy generation in each month, plotted in Figure 4.25, reflects the values of the volume and discharges already discussed. Energy generation using the naive operating Figure 4.23: Reservoir Operation for 1970, 375 Mm3 Reservoir: Volumes Figure 4.24: Reservoir Operation for 1970, 375 Mm3 Reservoir: Flows Chapter 4. RESERVOIR OPERATION RESULTS 68 Figure 4.25: Reservoir Operation for 1970, 375 Mm3 Reservoir: Energy Generation policy is higher in the spring because of the higher discharges, but is very low in June because of low discharges and head in the reservoir. Energy generation using the concep-tual policy is near ideal generation based on perfect foreknowledge throughout the year. Ideal generation gives a total of 145.3 Gwh (gigawatt hour) over the full year. Gener-ation according to the naive forecast generates 134.6 Gwh and generation according to the conceptual forecast generates 142.1 Gwh, a 5% improvement. 4.3.2 1968 The naive forecast for 1968 was very poor. The conceptual forecast was an improvement, but was still not very accurate. This discussion will focus on the one state stochastic Chapter 4. RESERVOIR OPERATION RESULTS 69 results, as they gave overall the best operation in both forecast cases. Figures 4.26 through 4.28 show various aspects of the reservoir operation. The values are plotted for the perfect forecast, i.e. the maximum potential efficiency, and for the single state stochastic representations of the two forecast cases. Figure 4.26 plots the initial monthly volumes of the reservoir. The reservoir maintains a constant volume during early spring, and then starts to empty in time to receive the freshet flows. In June, the reservoir volume begins to increase as a result of high inflows. By August the reservoir is full and remains so for the remainder of the year. Operation of the reservoir according to the conceptual forecast is more efficient than operation resulting from the naive forecast. Because the inflows during 1968 are so much higher than average, operation by the naive forecast causes the reservoir to be maintained at a higher volume than is ideal. Ideal operation, according to the perfect forecast empties the reservoir in May to provide maximum possible storage for the high freshet flows. Both forecasts keep a higher volume throughout the spring and summer, but the conceptual forecast is closer to the ideal. The discharges plotted in Figure 4.27 are the actual discharges calculated by the simulation model. Because of the high inflows, several of the actual releases had to be increased over the optimum for both forecast scenarios to prevent overtopping. In all cases, including the perfect forecast, the discharge curve is above the maximum discharge line for July, indicating that spill is occurring. Spill is slightly lower in the conceptual case than in the naive forecast case, indicating more efficient use of storage. The energy generation in each month, plotted in Figure 4.28, reflects the values of the volume and discharges already discussed. Energy generation is similar for each scenario throughout the year. In most months the naive and conceptual forecasts follow the same trends, which are slightly different than the ideal. In March the perfect forecast generates more energy than the other cases because a higher discharge was released. Chapter 4. RESERVOIR OPERATION RESULTS 70 600 200 -\ 1 1 1 1 1 1 1 1 1 1 J a n Feb Mar Apr M a y J u n J u l A u g S e p Oct N o v D e c Figure 4.26: Reservoir Operation for 1968, 375 M m 3 Reservoir: Volumes Figure 4.27: Reservoir Operation for 1968, 375 Mm3 Reservoir: Flows Chapter 4. RESERVOIR OPERATION RESULTS 72 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 4.28: Reservoir Operation for 1968, 375 Mm3 Reservoir: Energy Generation Generation using the naive and conceptual operating policies is higher in the spring and summer because of the higher overall water level, but is overall lower in the fall when the discharges and water levels are lower. Ideal generation gives a total of 192.5 Gwh over the full year. Generation according to the naive forecast generates 187.9 Gwh and generation according to the conceptual forecast generates 188.9 Gwh, a 0.5% improvement. 4.4 CONCLUSIONS This brief study of the actual operation of selected reservoirs under ideal and then re-alistic conditions has shown that the optimization and simulation models operate the reservoirs in a realistic manner. The system's response to the varying input and reservoir Chapter 4. RESERVOIR OPERATION RESULTS 73 characteristics is as expected, which suggests that the results of the analysis are reliable. Observed patterns in the operation of the reservoirs under the conditions studied here will be useful in interpreting the energy generation results. The next chapter compares the energy generated in real time analysis, by each reser-voir during each study year, and draws conclusions regarding the value of the forecasts. Chapter 5 RESULTS This chapter presents the results of the simulation runs in terms of the value of the energy generated over a years operation. 5.1 PRODUCTION RUNS The following runs of the study models were performed. Perfect Forecast. Each reservoir was optimized and then simulated with the observed inflow for each test year. This represents operation with complete foreknowledge of inflow and therefore gives the maximum energy generation possible given the applied constraints. 4 years * 5 reservoirs = 20 operating policies 20 operating policies * 1 inflow format = 20 simulation runs Naive Forecast. Each reservoir was optimized using the mean naive inflow sequence in deterministic, one state stochastic and two state stochastic format thus creating three operating policies per reservoir. The system was then simulated using each test year's observed flow with each operating policy to determine the potential energy production for each year under each operating policy. 5 reservoirs * 3 inflow formats = ', 15 operating policies * 4 years = 74 15 operating policies : 60 simulation runs Chapter 5. RESULTS 75 Conceptual Forecast. Each reservoir size and each test year was optimized individu-ally using the eight different forecasts, one for each month from January to August, again for the three inflow formats. The simulation then involved operating each year, using the appropriate operating policy for each month from January to Au-gust, the remainder of the year being operated using the August forecast. 5 reservoirs * 3 inflow formats * 4 years * 8 forecasts = 480 operating policies 480 operating policies -f- 8 operating policies/simulation = 60 simulation runs 5.2 TABULATED RESULTS The value chosen for use in comparison of the operating policies was the total energy generated over the test year, given the specified operating policy and the observed flow. This value is appropriate only because of the assumption that the operation is by an Independent Power Producer and therefore the energy is of equal value at all times of the year. In most of the runs, the reservoir volumes at the end of December were the same for each forecast scenario, independent of the operating policy used. In some cases the reservoir ended at a lower head using the stochastically derived operating policies, than when using the deterministically derived operating policies, which resulted in a higher overall power generation (because more water had passed through the turbines). In these cases, the total energy was adjusted by considering the value of the water remaining in the reservoir. If the reservoir volume at the end of the year was lower than in the base case, the potential future generation is lower and that difference in energy was subtracted from the total obtained in the simulation runs. In each scenario the base case used for comparison was the perfect forecast case. This case represents the maximum potential energy generation for the given inflows. The raw results of the runs are presented in Table 5.8. The energy generation is given in gigawatt Chapter 5. RESULTS 76 hours (Gwh) of energy generated over the whole year. The following notation will be used to distinguish between the operating policies: D = Deterministic Model 51 = One State Stochastic Model 52 = Two State Stochastic Model N = Naive Forecast C = Conceptual Forecast Table 5.9 lists the percentage loss in energy generation resulting from the lack of perfect foreknowledge in each of the forecast scenarios. This loss is calculated by: where: Epot = potential energy generation with complete foreknowledge Eact = energy generation, using specified forecast The following sections discuss each forecast year in turn and then make general com-ments regarding the results. The three key areas of consideration are the marginal value of the conceptual forecast over the naive forecast, the value of using a one or two state stochastic representation of the inflows rather than a deterministic representation, and the effect of reservoir size on the value of the forecasts. Loss = \EPot — Eact\ Epot * 100 (5.15) Chapter 5. RESULTS Test Reservoir Perfect Naive Forecast Conce )tual Forecast Year Size, Mm3 Forecast D-N Sl-N S2-N D-C Sl-C S2-C 1966 250 121.3 121.3 118.9 118.9 121.3 119.8 119.8 375 185.4 184.7 181.8 182.5 176.0 179.0 178.7 500 213.9 212.5 210.7 212.0 205.0 203.1 198.7 750 271.8 269.2 265.8 268.4 261.2 255.3 252.4 1000 •311.3 307.3 302.3 307.5 303.4 297.3 295.1 1968 250 123.9 123.8 122.8 122.6 123.8 123.6 123.8 375 192.5 186.8 187.9 183.9 186.9 18S.9 189.5 500 227.8 214.6 215.8 208.8 213.5 220.0 221.4 750 297.3 270.9 283.7 259.9 271.7 285.9 281.2 1000 348.9 309.1 336.1 303.8 311.4 334.7 335.5 1969 250 119.3 118.4 116.0 116.3 118.4 117.0 117.5 375 173.6 168.4 171.0 172.1 171.2 171.6 171.1 500 198.2 193.5 193.4 193.7 196.1 194.4 195.1 750 245.1 241.4 235.3 237.9 244.9 240.4 240.3 1000 274.9 272.8 263.3 268.7 270.3 271.4 272.2 1970 250 103.8 103.5 102.8 103.2 103.5 102.9 102.9 375 145.3 134.6 127.8 136.3 142.1 140.4 141.2 500 163.6 154.0 144.3 154.4 160.4 157.6 157.9 750 198.6 191.5 180.6 188.1 194.4 192.0 194.0 1000 215.9 212.9 201.8 210.5 215.7 211.9 213.9 Table 5.8: Raw Data: Total Yearly Energy Generation, Gwh Chapter 5. RESULTS 78 Test Reservoir Naive Forecast Conce )tual Forecast Year Size, Mm3 D-N Sl-N S2-N D-C Sl-C S2-C 1966 250 0.00 1.98 1.98 0.00 1.24 1.24 375 0.38 1.94 1.56 5.07 3.45 3.61 500 0.65 1.50 0.89 4.16 5.05 7.11 750 0.96 2.21 1.25 3.90 6.07 7.14 1000 1.28 2.89 1.22 2.54 4.50 5.20 1968 250 0.08 0.89 1.05 0.08 0.24 0.08 375 2.96 2.39 4.47 2.91 1.87 1.56 500 5.79 5.27 8.34 6.28 3.42 2.81 750 8.88 4,57 12.58 8.61 3.83 5.42 1000 11.41 3.67 12.93 10.75 4.07 3.84 1969 250 0.75 2.77 2.51 0.75 1.93 1.51 375 3.00 1.50 0.86 1.38 1.15 1.44 500 2.37 2.42 2.27 1.06 1.92 1.56 750 1.51 4.00 2.94 0.08 1.92 1.96 1000 0.76 4.22 2.26 1.67 1.27 0.98 1970 250 0.29 0.96 0.58 0.29 0.87 0.87 375 7.36 12.04 6.19 2.20 3.37 2.82 500 5.87 11.80 5.62 1.96 3.67 3.48 750 3.58 9.06 5.29 2.11 3.32 2.32 1000 1.39 6.53 2.50 0.09 1.85 0.93 Table 5.9: Loss in Generation Due to Forecast Uncertainty, % Chapters. RESULTS 79 250 375 500 750 1000 RESERVOIR SIZE Figure 5.29: Efficiency Losses Due to Uncertainty, 1966 ( g ^ = 1.06) 5.3 O B S E R V A T I O N S Figures 5.29 to 5.32 plot the results tabulated in Table 5.9, showing the value of each forecast scenario for each reservoir size and year. From the table and the corresponding plots, the following observations can be made. 5.3.1 Fo recas t Y e a r : 1966 • Use of the naive forecast gives generally less error than use of the conceptual hy-drologic forecast. The naive forecast gives losses of between 0 and 3%. Use of the conceptual forecast gives losses of between 0 and 7%, depending on the reservoir Chapter 5. RESULTS 80 size. • In either the naive or conceptual forecast scenarios, use of the deterministic model generally gives more efficient operation than the use of either the one or two state stochastic models. • For the smallest reservoir size of 250 Mm 3 , there is little loss in energy generation regardless of which forecast scenario is used. For larger sizes, the two forecasts show different trends. The naive forecast suggests an overall increase in the potential energy loss with increasing reservoir size, but with an optimum reservoir size in the mid range. The conceptual forecast however, provides better results for small and large reservoirs, but gives most errors in medium sized reservoirs. 5.3.2 Forecast Year: 1968 • The range of potential efficiency loss is quite high for both forecast scenarios, rang-ing form 0 to 11% for the conceptual forecast and from 0 to 13% for the naive forecast. • Distinction between the relative advantages of the two forecasts are not as clear as in 1966. The scenarios giving the best results are the one and two state stochastic representations of the conceptual forecast and the one state stochastic naive fore-cast. The two deterministic scenarios give almost identically poor results and the two state naive representation gives very high errors. • As in 1966, the operation of the smallest reservoir is close to optimal regardless of the policy used. The three best scenarios (Sl-N, Sl-C and S2-C) show an increase in losses from the smallest to the mid sized reservoir and then appear to stabilize Chapter 5. RESULTS Figure 5.30: Efficiency Losses Due to Uncertainty, 196S = 1.18) Chapters. RESULTS 82 o.o 250 375 500 750 1000 RESERVOIR SIZE Figure 5.31: Efficiency Losses Due to Uncertainty, 1969 (g**. = 0.96) to an error of approximately 4% for the larger reservoirs. The deterministic and S2-N models give errors which increase with reservoir size. 5.3.3 Forecast Year: 1969 • The results for 1969 are very irregular, the relative value of the various scenarios changing with the reservoir size. The range in efficiency loss is however quite low throughout, varying between 1 and 3% for the smaller to medium sized reservoirs and between 0 and 4.5% for the larger reservoirs. • In general the results of the conceptual forecast are an improvement on the naive forecast. Chapter 5. RESULTS 83 00 GO o > o z LU O -2 --3 --4 --5 -6 H -7 -8 -9 --10 --11 -- 1 2 -13 + o A X v 2 5 0 D-N 5 1 - N 5 2 - N D-C 5 1 - C 5 2 - C 3 7 5 1 5 0 0 RESERVOIR SIZE 7 5 0 1 0 0 0 Figure 5.32: Efficiency Losses Due to Uncertainty, 1970 = 0.82) • There is little evidence to suggest that the use of a stochastic representation is of any value in either the naive or the conceptual forecast cases. • Though the relative advantages of various forecasts and inflow scenarios vary with reservoir size, no trend is readily visible. 5.3.4 Forecast Year: 1970 • This year shows a clear advantage in the use of the conceptual hydrologic forecast in all reservoir sizes. The losses produced using the conceptual forecast are between 0 and 4% compared to between 0 and 12% for use of the naive forecast. Chapter 5. RESULTS 84 • In both forecast cases the use of the stochastic models gives lower operational efficiency than the deterministic model. • As with the other years, the operation for the smallest reservoir is near optimal, regardless of the policy used. In 1970 however, the errors are fairly constant for the three mid sizes, and decrease again for the larger reservoir sizes. 5.4 COMMENTS The following sections comment on the observations listed for each forecast year and consider whether the results are in agreement with the expectations and hypotheses. 5.4.1 Relative Value of Forecasts The primary objective of this study was to determine the value of an inflow forecast in the operation of a hydroelectric development. Inherent in this objective is the consideration of the accuracy of the forecast. The more accurately a forecast predicts the actual inflow to the basin, the more useful it will be in deriving an optimal operating policy. The discussion in Section 3.3 showed that for 1966, the naive forecast was a more accurate estimate of the actual flows than was the conceptual hydrologic forecast. Con-sequently it is not surprising that the naive forecast leads to a more efficient operating policy than does the conceptual forecast. Because the actual inflow series is so close to the historic mean, the loss in energy due to the forecast uncertainty is not great. The observed inflow during 1968 was considerably higher than average and it was shown that the conceptual forecast for 1968 was generally a better predictor than the historic mean, though the forecast statistics were not entirely consistent. Again, the value of the conceptual forecast exceeds the value of the naive forecast. The energy losses for 1968 are as great as 13% for the poor forecasts because the flows are so high above the Chapter 5. RESULTS 85 average annual inflow that there is more potential for loss due to uncertainty. The annual flow in 1969 was, as in 1966, very close to the mean but in this case a little less than average. Despite this, the conceptual forecast is still generally superior to the naive forecast, because it gives a better estimate of the time distribution of the flow. In most cases the loss caused by use of the naive forecast is greater than the loss caused by the use of the conceptual forecast. In either case the operation is near optimal; the worst case gives an efficiency loss of less than 4.5%. The forecast statistics for the 1969 forecasts sometimes suggest that the conceptual forecast is a better predictor of actual flows than the naive forecast, and sometimes the reverse. These mixed results explain why in some cases, the naive forecast may prove more valuable than the conceptual forecast. Inflow during 1970 was far below average and thus the naive forecast is not very accurate. The conceptual forecast however was fairly good and thus the conceptual forecast is more valuable. The naive forecast was sufficiently poor to give operation losses of up to 12%. The conceptual forecast was much better, giving operation losses of less than 4%. 5.4.2 Value of Stochastic Analysis One hypothesis of this research was that the use of a stochastic representation of the data during optimization would increase the operational efficiency of the forecast. This was partially based on the conclusion of Krzysztowicz, which was that not accounting for uncertainty always caused an opportunity loss (see Section 1.5.3). It would appear from the observations listed in Section 5.3 that Krzysztowicz's conclusion does not always apply in this case. This result will be discussed in the following paragraphs. In the following section, the value of a second state variable representing the predictive value of the previous months inflow is examined. In 1966 the naive forecast was found to be the more valuable, because the inflow was Chapters. RESULTS 86 so close to the annual mean. Within that forecast, the deterministic representation gave the least operational error followed by the two state stochastic and lastly the one state stochastic representations, however the marginal error between the scenarios was never more than 2%. Thus the use of the stochastic analysis does not decrease the errors as was expected. Since it will be seen that this result occurs in three of the four forecast years it is important to investigate why it happens. A possible explanation is discussed in the next paragraph. It was observed in Chapter 4 that when a stochastic policy is used, the attempt to consider the uncertainty of the estimate causes the reservoirs to be drawn down further in the spring and kept at a lower overall operating level for the remainder of the year. In 1966 the flow was very close to the mean and the deterministic operation appears to be more valuable. This suggests that if actual flows are similar or lower to those forecast, a deterministic forecast will appear to be more valuable. In contrast, if a year of higher than expected flows occurred, the more conservative stochastic policy, which allowed for more storage of potential high flows will appear to be more valueable. The relative values of the deterministic and stochastic policies as discussed here, are dependent on the loss function and the constraints applied to this reservoir model. Operation of actual reservoirs may favour a more conservative operation at all times. The year 1968 was a year of extremely high flows. The conceptual forecast was more accurate than the naive forecast but.still underestimated the flows. As anticiptated by the above discussion, the stochastic operating policies were more valuable. The hedging in the stochastic operation to allow for potential high inflows, proved valuable when the inflows were in fact higher than anticipated. The results for 1969 are somewhat mixed, probably because neither forecast was uni-formly superior according to the statistical parameters. In most cases the deterministic operating policies were more valuable than the stochastic versions. This is because the Chapter 5. RESULTS 87 flow was lower than average and thus the deterministic pol icy was most efficient. S im-ilary in 1970, the forecast overestimated the flows and thus operation according to the deterministic pol icy was more efficient. 5.4.3 Use of Second State Variable The previous section considered whether a stochastic analysis is valuable. This section wi l l focus on the second part of the problem; whether it is valuable to consider a second state variable to use the previous month's inflow as an indicator of the present month's inflow. If the flows are highly correlated, an optimizat ion policy which considers the value of the previous months inflow in making the present discharge decision, should be able to give a better decision. Though the results of this study show that to use the second state variable is at times valuable, other evidence suggests that the effect is not because of the desired improvement in decision making. Table 5.10 lists the lag-one serial correlation coefficients for each forecast case (January cannot be calculated because data for the corresponding Decembers is not available, serial correlations for February and March are undefined because al l the data points for February are identical, once they have been rounded to the interval used in this study). Statistically, a serial correlation greater than 0.23 indicates that the monthly inflow volume is not independent of the previous months, at the 95% level. Examinat ion of the serial correlation coefficients for the naive forecast and for each forecast year shows that the highest serial correlations are in May and then November and December for the conceptual forecasts and in August to October for the naive forecast. These months however tend to have l itt le variation from year to year so use of the second state variable would not be especially valuable in predicting the inflow. The mixed results obtained using the second state variable that were observed in Section 5.3 were probably due much to coincidence. Use of the second state variable did Chapter 5. RESULTS 88 Month Naive 1966 1968 1969 1970 January - - - - -February - - - - -March - - - - -Apr i l .121 .153 .170 .213 .135 May .132 .451 .485 .355 .420 June .076 .001 .115 .024 .000 July .138 .017 .021 .089 .112 August .624 .023 .103 .069 .001 September .320 .177 .304 .154 .226 October .391 .056 .071 .106 .133 November .128 .382 .440 .445 .244 December .143 .469 .454 .540 .616 Table 5.10: Serial Correlation Coefficients reduce the uncertainty of the estimate by reducing the number of potential inflows, but did not neccessarily make the estimate more accurate. In many cases, especially w i th the naive forecast, many undefined previous inflows were encountered during the simulation phase. The operating policies are dependent on the value of the previous month's inflow. If an observed inflow was not part of the forecast series, the simulation program had to interpolate to find an ideal discharge. This appeared to work reasonably well in some forecast years, but not in others. Again this suggests that this model does not give a valid indication of the value of a two state representation. 5.4.4 Influence of Reservoir Size The th ird objective of this study was to examine the influence of reservoir size on the value of a forecast. It was expected that forecasting would not be useful in very small or very large reservoirs. In small reservoirs, storage is not sufficient to allow for changes in operation dependent on inflow. The idealized analyis in Chapter 4 showed that even wi th perfect foreknowledge of average flows, the 250 M m 3 reservoir could not be operated to Chapter-5. RESULTS 89 efficiently use all of the available inflows. In large reservoirs, storage is great enough and allowable discharges are high enough that the operation is not dependent on inflows. In 1966 the naive forecast showed slightly greater value in forecasting for a reservoir size of 500 Mm3, and slightly higher losses for both smaller and larger reservoirs. This agrees with the expectations discussed. In 1968 the operation of the smallest reservoir is near optimal whatever policy is used. Errors increase as reservior size increases. The three best policies then show a uniform error for the largest reservoir sizes, whereas the poor policies show increasing error for all the remaining reservoir sizes. This may suggest that if flows are sufficiently high, even the largest reservoirs still require foreknowledge of inflows to be operated totally optimally. In 1969 the results are mixed but overall the errors are fairly uniform, whatever the reservoir size. Since this results is similar to that found in 1966, and both these years have near average inflow conditions it could be assumed that when flows are near average, effects are independent of reservoir size. In 1970, again the operation of the smallest reservoir is near optimal whatever policy is used. Losses are slightly higher and uniform for the three mid sized reservoir and then decrease again for the largest reservoir, when operation is again near optimal regardless of the policy. This agrees with the original hypothesis stated. 5.4.5 Conclusions With only four forecast years to study and those years producing varying and often conflicting results, it is hard to draw definitive conclusions. Some general conclusion are possible and are presented here. These conclusions, however are possible only because of the hindsight available only at the completion of the study. Some are correlated to the magnitude of the inflow and to the accuracy of the forecast; parameters not known Chapter 5. RESULTS 90 when the forecasting decision is required. The next chapter discusses the conclusions of this study in more detail and applies them to real-world situations. The conclusions are summarized as follows: • If an accurate forecast is available, it will be valuable in generating operating poli-cies for reservoir operation. • In extreme flow years, a conceptually forecast inflow sequence is a better predictor of observed inflows and is therefore more useful in reservoir operation than is the naive forecast. • For the reservoir in this study, if the observed flow is equal to or lower than expected a deterministic representation of the data is superior to a stochastic analysis. If the observed flow is much higher than expected, a stochastically derived policy is more efficient. • Forecasting does not increase operational efficiency of very small reservoirs, (less than 25% of the mean annual flow). In average years, forecasting is equally useful in all other reservoir sizes. In extreme years efficiency losses increase for medium sized reservoirs (30% to 80% of mean annual flow) but remain constant or decrease for extremely large reservoirs (greater than 80%). It is worthwhile obtaining a forecast for operating a medium sized reservoir but for operating a large reservoir, the value is variable. Chapter 6 CONCLUSIONS In order for the results of this analysis to be useful in a real-world situation it is nec-essary to formulate some more general conclusions than those listed in Chapter 5. The conclusions need to answer the questions posed at the beginning of this report : • Would it be worthwhile for an Independent Power Producer developing the Gold-stream River for hydroelectric generation, to obtain conceptual hydrologic forecasts to aid in the reservoir operation? • How does the capacity of the reservoir affect this decision? The results were discussed in Chapter 5 only in terms of losses in energy compared to potential generation attainable only with perfect foreknowledge. From the viewpoint of a power producer it would be more valuable to compute directly the marginal benefit of using the conceptual rather than the naive forecast. The deterministic naive forecast is therefore chosen as the basis for comparison since it would always be available, presuming that the site has been gauged, and would be the easiest and cheapest forecast to use. The discussion in Chapter 5 suggested that the results of the analysis concerning the two state stochastic model are not an accurate measure of the value of the second state variable. Consequently, the conclusions of the present study will be limited to the deterministic and single state stochastic models. Table 6.11 is similar to Table 5.9 but the value tabulated is redefined to represent the relative improvement of any forecast scenario over the basic deterministic naive forecast. 91 Chapter 6. CONCLUSIONS 92 Test Reservoir Ratio to Perfect Naive For. Cone. For. Year Size, Mm3 MAF For. Sl-N D-C Sl-C 1966 250 .2 0.00 -1.98 0.00 -1.24 375 •3 0.38 -1.57 -4.71 -3.09 500 .4 0.66 -0.85 -3.53 -4.42 750 .6 0.97 -1.26 -2.97 -5.16 1000 .8 1.30 -1.63 -1.27 -3.25 1968 250 .2 0.08 -0.81 0.00 -0.16 375 .3 3.05 0.59 0.05 1.12 500 .4 6.15 0.56 -0.51 2.52 750 .6 9.75 4.72 0.30 5.54 1000 .8 12.88 8.74 0.74 8.28 1969 250 .2 0.76 -2.03 0.00 -1.18 375 .3 3.09 1.54 1.66 1.90 500 .4 2.43 -0.05 1.34 0.47 750 .6 1.53 -2.53 1.45 -0.41 1000 .8 0.77 -3.48 -0.92 -0.51 1970 250 .2 0.29 -0.68 0.00 -0.58 375 .3 7.95 -5.05 5.57 4.31 500 .4 6.23 -6.30 4.16 2.34 750 .6 3.71 -5.69 1.51 0.26 1000 .8 1.41 -5.21 1.32 -0.47 Table 6.11: Benefit of Use of Different Operating Policies, % Chapter 6. CONCLUSIONS 93 A positive result indicates an improvement over the deterministic naive case, a negative result indicates that the deterministic naive forecast gave a superior operating policy. where: ED-N = energy generation, using the deterministic representation of the naive forecast Eact = energy generation, using specified forecast Table 6.12 is similar to Table 6.11 but instead of recording the percentage of potential increase or decrease in energy generation, it records the loss or gain in thousands of dol-lars. The value of energy is assumed to be 20 mills which is equivalent to $20,000/Gwh. The results indicate that the choice of operating policy greatly affects the value of the hydroelectric scheme. For example, looking at the 750 Mm3 reservoir, in 1968 a perfect forecast could generate $528,000 more than the naive forecast. The best policy available, the stochastic conceptual forecast, achieves $300,000 of this potential improvement. In 1966, when the perfect forecast can only improve slightly on the naive to gain $52,000, use of the stochastic conceptual forecast would result in a loss of revenue equal to $278,000. This brief examination shows that while significant profit can be obtained from an im-provement in operating policies, there is a corresponding risk of revenue loss if the wrong policy is chosen. Bene fit = Eact — EQ-N ED-N * 100 (6.16) 6.1 WHEN TO FORECAST A power producer who chooses to develop the Goldstream or indeed any other basin in the interior region of British Columbia has an advantage over the majority of reservoir users. Because a large proportion of the runoff is due to snowmelt, some idea of the erfJ. CONCLUSIONS Test Res'r Rat io to D-N Perfect Naive For. Conceptual For. Year Size M A F Value For. S l - N D-C S l - C Mm3 Thousands of Dollars 1966 250 .2 2,426 0 -48 0 -30 375 .3 3,694 14 -58 -174 -114 500 .4 4,250 28 -36 -150 -188 750 .6 5,384 52 -68 -160 -278 1000 .8 6,146 80 -100 -78 -200 1968 250 .2 2,476 2 -20 0 -4 375 .3 3,736 114 22 2 42 500 .4 4,292 264 24 -22 108 750 .6 5,418 528 256 16 300 1000 .8 6,182 796 540 46 512 1969 250 .2 2,368 18 -48 0 -28 375 .3 3,368 104 52 56 64 500 .4 3,870 94 -2 52 18 750 .6 4,828 74 -122 70 -20 1000 .8 5,456 42 -190 -50 -28 1970 250 .2 2,070 6 -14 0 -12 375 .3 2,692 214 -136 150 116 500 .4 3,080 192 -194 128 72 750 .6 3,830 142 -218 58 10 1000 .8 4,258 60 -222 56 -20 Table 6.12: Potential Profits from Different Operating Policies Chapter 6. CONCLUSIONS 95 relative magnitude of the coming seasons inflow is possible by examination of the relative magnitude of the snowpack. The results of this study suggest that if the inflow is significantly larger or smaller than the mean, then conceptual forecasting is valuable. In years close to the mean, conceptual forecasting may still be of value but not to the same extent. More important in influencing the value of the conceptual forecast than the magnitude of the flows, is the accuracy of the forecast. In the four years studied in this analysis, two of the conceptual forecasts were excellent and proved to be valuable, one was fair and proved occasionally valuable and the fourth was poor and was of negative value. In this study the UBC Watershed Model was more accurate in forecasting the low flow year than the high flow year. Clearly a more extensive series of years should be studied to test the various conclusions put forward in this study. Also since there is no way of determining when a forecast will be accurate, a policy must be developed to decide when it is worth risking forecasting. Taking an average of all reservoir sizes and assuming a conceptual forecast was used for all years the results are as follows: Maximum potential profit is: $141,000 Sl-N loses revenue of: $29,000 D-C generates no profit or loss Sl-C generates profit of: $16,000 (all values as compared to D-N case). The same analysis but considering forecasting only in the extreme years (1968 and 1970) yields: Maximum potential profit is: $232,000 Sl-N generates profit of: $4,000 Chapter 6. CONCLUSIONS 96 D-C generates profit of: $43,000 S l - C generates profit of: $112,000 (all values as compared to D-N case). This clearly recommends the use of conceptual forecasting only if the operators have reason to expect that the flow in the coming season wi l l be significantly higher or lower than average. This conclusion is based only on the consideration of the value of conceptual forecasting in monthly operations decision support. Apar t from the benefits considered here, forecasting wi l l be of use in longer term power production planning. 6.2 FORMAT OF OPERATING POLICY Once the decision has been made regarding which forecasting methodology to use, a further decision must be made as to whether a deterministic or a stochastic operating policy is liable to give the more efficient operation. The results given in the previous section suggest that if operation is to be based on a conceptual forecast, that the one state stochastic model gives on average the most profit. If the naive forecast is used for optimization, the results show that on average the deterministic format is preferred. The analysis of the results in Chapter 5 showed that the decision is more complex than appears here. The deterministic operating policies are generally superior if the inflow is average or below average, whereas the stochastic policies are superior in years of high flow. This result however is true only because of the assumptions made regarding this reservoir model, and would not necessarily hold true for a more complex model. Chapter 6. CONCLUSIONS 97 6.3 SIZE OF RESERVOIR The results clearly show that complex forecasting is redundant in attempting to increase reservoir operational efficiency for the smallest reservoir (250 Mm 3 ) , simply because no improvement can be made. For the remaining reservoir sizes, the value of the forecast was more dependent on the individual year or forecast case than on the size of the reservoir, but in general it would appear that forecasting is equally of value for the medium sized reservoirs and then slightly less valuable for the largest reservoir size (1000 Mm 3 ) . To generalize this result for other basins, this analysis would suggest that it is valuable to forecast for reservoirs with storages greater than 25% of the mean annual inflow. As the storage approaches 100% of the mean annual inflow, the value diminishes. This result is in keeping with the results of the studies reviewed in Section 1.4, namely Yeh et al. (1982) and Braga et al. (1982). 6.4 RECOMMENDATIONS FOR F U T U R E RESEARCH On the basis of the experience gained in this study, the following further research is recommended. • Examine ways to improve the use of the naive forecast. The naive forecast does produce valuable results in average flow years. If the historic data was used in a more complex manner it may be able to be made to yield better results in the remaining years. For example generating several forecasts for use in probable high, probable mean or probable low years or correlating expected flow to other hydrom-eteorologic parameters, would make the naive forecast more accurate in all years. The value of such forecasting methods could be analyzed by the same techniques used in this study. Chapter 6. CONCLUSIONS 98 • Further examine the use of a second state variable in the stochastic models. Mod-elling of a basin which has a longer flow record would reduce the approximations necessary. A basin showing more serial correlation of the monthly data, should also show more value of the second state. • Continue the present analysis using more forecast years so that more conclusive generalizations can be made regarding the relative values of the forecasts with respect to the magnitude of the yearly inflow. • Continue the present analysis using different loss functions and constraints to de-termine how much the constraints affect the value of the forecast. 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A Case Study presented at the Workshop on the WMO Project on Simulated Real-Time Intercomparison of Hydrologic Models. Fleming, George 1975. Computer Simulation Techniques In Hydrology. Environmental Science Series. New York, N.Y.: American Elsevier Publishing Company, Inc. Kitanidis, Peter K. and Raphael L. Bras. 1980. Real Time Forecasting with a Con-ceptual Hydrologic Model. 2. Application and Results, Water Resources Research 16(6): 1034-1044. Krzysztofowicz, Roman. 1983. Why Should a Forecaster and a Decision Maker Use Bays Theorem. Water Resources Research 16(2): 275-283. Lettenmaier, Dennis P. 1984. Synthetic Streamflow Forecast Generation. Journal of Hydraulic Engineering 110(3): 277-2S9. Little, John D.C. 1955. The Use of Storage in a Hydroelectric System, Operations Research 3: 187-197. Nash, J.E. and J.V. Sutcliffe. 1970. River Flow Forecasting Through Conceptual Mod-els. Part 1 — A Discussion of Principles. Journal of Hydrology 10: 282-290. O'Donnell, T. 1986. Deterministic Catchment Modelling, chap, in River Flow Modelling and Forecasting ed. D.A. Kraijenhoff and J.R. Moll, Holland: D. Reidel Publishing Company. O'Connell, P.E., G.P. Brunsdon, D.W. Reed and P.G. Whitehead. 1986. Case Studies in Real Time Hydrological Forecasting from the UK, chap, in River Flow Modelling and Forecasting ed. D.A. Kraijenhoff and J.R. Moll, Holland: D. Reidel Publishing Company. Quick, Michael C, and Anthony Pipes. 1976. A Combined Snowmelt and Rainfall Runoff Model. Canadian Journal of Civil Engineering 3(3): 449-460. Schultz, G.A. 1986. Introduction, chap, in River Flow Modelling and Forecasting ed. D.A. Kraijenhoff and J.R. Moll, Holland: D. Reidel Publishing Company. Su, Shiaw Yuan, and Rolf A. Deininger. 1974. Modeling the Regulation of Lake Superior Under Uncertainty of Future Water Supplies. Water Resources Research 10(1): 11-25. Turgeon, Andre. 1980. Optimal Operation of Multireservoir Power Systems with Stochastic Inflows. Water Resources Research 16(2): 275-283. Trezos, Thanos and William W-G. Yeh. 1987. Use of Stochastic Dynamic Programming for Reservoir Management. Water Resources Research 23(6): 983-996. World Meteorological Organization. 1986. Intercomparison of Models of Snowmelt Runoff WMO No. 646, Operational Report No. 23, Secretariat of the World Meteorological Organization, Geneva, Switzerland. Bibliography 101 Yakowitz, Sidney. 1982. Dynamic Programming Applications in Water Resources, Water Resources Research 18(4): 673-696. Yeh, William W-G., Leonard Becker and Robert L. Sohn. 1978. Information Re-quirements for Improving Hydropower. Journal of the Water Resources Planning and Management Division, Proceedings of American Society of Civil Engineers 104(WR1): 139-156. Yeh, William W-G., Leonard Becker and Robert Zettlemoyer. 1982. Worth of In-flow Forecast for Reservoir Operation. Journal of the Water Resources Planning and Management Division, Proceedings of American Society of Civil Engineers 108(WR3): 257-269. Zuzel, John F. and Lloyd M. Cox. 1978. A Review of Operations Water Supply Fore-casting Techniques in Areas of Seasonal Snowcover. Presented at the 46'th Annual Meeting of the Western Snow Conference. Colorado: Colorado State University. Appendix A Dynamic Programming Models The following listings are for the dynamic programming models used for the analysis in this study. The input describes the reservoir being modelled and the forecasted inflows to the system, either in deterministic or stochastic form. The input files are slightly different in some instances for the different dynamic programs so will be described separately. In each program there are two output files; one contains the bare operating policy and is for input to the simulation programs. The other file is for user examination and contains a summary of the input data, as well as the operating policy. The programs are fairly straightforward and the code is frequently commented for reference. A.1 Deterministic Model The input files are as follows: File 5: a general title to identify the run , the number of years for which the analysis is to be run, the number of days in the time interval, the inflow data File 7: a second title identifying the reservoir being modelled, information regarding the minimum and maximum volumes and discharges of the reservoir, the terminal values £*********************************************************** C* THIS IS THE FIRST LEVEL OF A SERIES OR DYNAMIC PROGRAMS * C* WRITTEN TO ANALYZE THE VALUE OF INFLOW FORECASTS FOR * C* OPERATION OF A HYDROELECTRIC DEVELOPMENT. * C* THIS PROGRAM IS DETERMINISTIC AND MAKES USE OF ONLY ONE * C* STATE VARIABLE. * c* * C* PROGRAMMED BY: JOANNA BARNARD * C* * c*********************************************************** c C FORMAT AND DIMENSION STATEMENTS, % , x , v DIMENSION Q(20),QV(20),V0L(100),DISCHV(40),DMAX(20) DIMENSION P(20,100),D0PT(20,100),SP0PT(20,100) DIMENSION S0PT(20,100),VMIN(20),VMAX(20),DMIN(20) REAL MVOL CHARACTER*70 TITLE,TITLE2 INTEGER S,SF,STAGES,DAYS(20) 100 FORMAT(A) 101 F0RMAT(I3) 102 F0RMAT(F3.0) 103 F0RMAT(10F6.0) 104 F0RMAT(2F8.0) 105 F0RMAT(5F8.0) 200 FORMAT(IX,/// * ' N INITIAL INFLOW DICHARGE SPILL POWER TOTAL FINAL *'/' VOLUME POWER V0LU *ME') 106 F0RMAT(1X,I2,7F9.2) 102 Appendix A. Dynamic Programming Models 107 F0RMAT(IX,312,2F8.0,2F6.0,2F8.2) 108 F0RMAT(7F8.1) 119 F0RMAT(1X,///'YEAR OF ANALYSIS IS YEAR',14) 501 FORMAT(13F6.0) C INPUT BASIC INFORMATION AND PERFORM BASIC CALCULATIONS READ (5,100) TITLE READ (7,100) TITLE2 WRITE (9,100) TITLE WRITE (9,100) TITLE2 WRITE (9,*) '************************************************' WRITE (12,100) TITLE WRITE (12,100) TITLE2 WRITE(12,200) C READ ALL ALLOWABLE STATES READ (7,102) SINT READ (7,101) NSTATE READ (7,105)(V0L(S),S=1,NSTATE) C READ ALL ALLOWABLE DISCHARGES READ(7,101) NDISCH READ(7,105)(DISCHV(K),K=1,NDISCH) C DETERMINE TIME SPANS AND INTERVALS READ (5,101)NYRS READ (5,101)NDAYS NSTAGE=365/NDAYS C NOTE THAT IN ORDER TO INCLUDE THE TERMINAL VALUE IT IS NECC. C TO ITERATE FROM 2 TO (NSTAGE+1), CALLED STAGES. IN ADDITION TO C ALLOW FOR THE BACKWARD ITERATION, THE VALUES MUST BE READ IN C REVERSE IE, FROM STAGES TO 2 C READ IN INFORMATION PERTAINING TO EACH STAGE STAGES=NSTAGE+1 DATA (DAYS(N),N=13,2,-l)/31,28,31,30,31,30,31,31,30,3 * 1,30,31/ READ (7,104) (VMIN(N),VMAX(N),N=STAGES,2,-1) VMIN(1)=VMIN(STAGES) VMAX(1)=VMAX(STAGES) READ (7,104) (DMIN(N),DMAX(N),N=STAGES,2,-1) DMIN(1)=DMIN(STAGES) DMAX(l)=DMAX(STAGES) C INITIALIZE VARIABLE AND INPUT TERMINAL VALUE DO 5 N=2,STAGES DO 5 S=l,NSTATE P(N,S)=0.0 DOPT(N,S)=0.0 SPOPT(N,S)=0.0 5 CONTINUE \/ f \ READQ7 105)(P(l S) S=l NSTATE) C THIS SECTION OF CODE'PRINTS AN ECHO OF THE INPUT VALUES WRITE (9,401) 401 FORMAT(/'THE FOLLOWING INFORMATION IS BEING USED',/,' * IN THIS ANALYSIS') WRITE (9,402) 402 FORMAT(//'NOTE: THE ANALYSIS GOES FROM FUTURE TO PAST SO',/,' * JANUARY IS N=13 AND DECEMBER IS N=2') WRITE (9,403) WRITE (9,404) 403 FORMAT(//,' N MINIMUM MAXIMUM MINIMUM MAXIMUM' ) 404 FORMATC VOLUME VOLUME DISCHARGE DISCHARGE') Appendix A. Dynamic Programming Models DO 6 N=STAGES,2,-1 WRITE (9,405)N,VMIN(N),VMAX(N),DMIN(N),DMAX(N) 6 CONTINUE 405 F0RMAT(1X,I2,4F10.0) WRITE (9,406) 406 FORMAT(///'THE STARTING VOLUMES TO BE CONSIDERED ARE:') WRITE (9,105) (VOL(S),S=1,NSTATE) WRITE(9,408) 408 FORMAT(/'THE TERMINAL POWER VALUES ASSIGNED IN THIS RUN ARE:') WRITE(9,105) (P(1,S),S=1,NSTATE) WRITE(9,407) 407 FORMAT(/'THE DISCHARGES TO BE CONSIDERED IN THIS ANALYIS ARE:') WRITE(9,105) (DISCHV(K),K=1,NDISCH) C PERFORM THE ANALYSIS C I REPRESENTS THE LOOP ON THE NUMBER OF YEARS IN THE ANALYSIS READ (5,103)CQ(N),N=STAGES,2,-1) DO 10 1= l.NYRS WRITE(9,119) I C N REPRESENTS THE LOOP ON THE TIME VARIABLE, IE THE STAGE. C VALUES AT N REPRESENT THE VALUES AT THE BEGINNING OF THE REAL C TIME PERIOD, BUT THE END OF THE PERIOD IN ANALYSIS DIRECTION. C J IS THE STAGE IN THE FOLLOWING REAL TIME PERIOD IE N-1 DO 20 N = 2,STAGES IF(I.EQ.NYRS) WRITE(9,200) J=N-1 C S INDICATES THE LOOP IN THE POTENTIAL VOLUMES, IE THE STATES DO 30 S=1,NSTATE IF (VOL(S) .LT. VMIN(N)) GO TO 30 IF (VOL(S) .GT. VMAX(N)) GO TO 30 PMAX=-1000.0 C K IS THE LOOP ON THE POSSIBLE DISCHARGES DO 40 K=1,NDISCH DISV=DISCHV(K) IF(DISV.LT.DMIN(N)) GO TO 40 IFCDISV.GT.DMAX(N)) GO TO 40 SPILLV=0.0 POWER=0.0 , x , • ENDVOL= VOL(S) + Q(N) - DISV C CHECK TO ENSURE THAT THE CALCULATED END VOLUME IS WITHIN THE LIMITS C OF THE STAGE FOLLOWING , x , x '% CALL VCHK(N,VMIN(J),VMAX(J),Q(N),VOL(S),DMAX(N),DMIN(N), * SINT,ENDVOL,SPILLV,DISV,IFAIL,DAYS(N)) MVOL = (ENDVOL +V0L(S))/2 DISCH=DISV*1.E06/(60.*60.*24.*DAYS(N)) C CALCULATE ELEVATION AS A FUNCTION OF VOLUME ELEV=ELVOL(MVOL) C CALCULATE POWER AS A FUNCTION OF ELEVATION AND DISCHARGE CALL TURB(DISCH,DAYS(N),ELEV,POWER) C CONVERT FINAL VOLUME TO A STATE INDEX v SF=INT((ENDV0L-V0L(1)T/SINT+.5)+l IF (SF.LE.l) SF=1 C IF IFAIL=2, CONTRAINTS ARE STILL BEING BROKEN SO PREVENT THIS C BEING OPTIMAL BY IF (IFAIL.EQ.2) POWER=0.0 C NEXT LINE IS ACTUAL RECURSIVE EQUATION,ALPHA IS DISCOUNT VALUE ALPHA=l./(l.+(.05/12.)) TOTPOW= POWER + ALPHA*P(J,SF) C SAVE INFO REGARDING OPTIMAL PATHWAY ONLY IF (TOTPOW .LE. PMAX) GO TO 40 PMAX= TOTPOW Appendix A. Dynamic Programming Models PPOWER=POWER D = DISV E=ENDVOL SPILL=SPILLV 40 CONTINUE P(N,S) «. PMAX PPP0W=PP0WER D0PT(N,S)=D S0PT(N,S)=E SPOPT(N,S)=SPILL IF(I.NE.NYRS) GO TO 30 IF(N.EQ.STAGES) THEN WRITE(12,106) N,V0L(S),Q(N),D0PT(N,S),SP0PT(N,S),PPP0W,P(N,S) * ,S0PT(N,S) ENDIF HRITE(9,106) N,V0L(S),Q(N),D0PT(N,S),SP0PT(N,S),PPP0W,P(N,S) * ,S0PT(N,S) 30 CONTINUE 20 CONTINUE DO 15 S=1,NSTATE P(1,S)=P(STAGES,S) 15 CONTINUE 10 CONTINUE C CREATE A LOOKUP TABLE OF THE DETERMINED OPERATING POLICY FOR C USE IN THE SIMULATION MODEL DO 70 S=1,NSTATE WRITE(11,501)VOL(S),(DOPT(N,S),N=STAGES,2,-1) 70 CONTINUE DO 80 S=1,NSTATE WRITE(ll,50l)V0L(S),(SPOPT(N,S),N=STAGES,2,-1) 80 CONTINUE STOP END C C SUBROUTINE TO ENSURE VOLUME AT END OF STATE IS ALLOWED SUBROUTINE VCHK(N,VMIN,VMAX,Q V,VOL,DMAX,DMIN,SINT,ENDVOL, * SPILLV,DISCHV,IFAIL,DAYS) INTEGER T,DAYS,N,FULL,EMPTY IFAIL=0 FULL=0 T=0 EMPTY=0 IF (ENDVOL .LE. VMAX) GO TO 5 FULL=1 GO TO 8 5 IF (ENDVOL .GE. VMIN) GO TO 7 IF (QV.GT.DMIN) GO TO 9 EMPTY=1 GO TO 8 C END VOLUME IS BEYOND CONSTRAINTS,IE IS FULL OR EMPTY SO GO TO C DAILY FLOW 8 IFOUND=0 DVOL=VOL OUT = 0.0 DO 12 L=1,DAYS DVOL=DVOL + (QV/DAYS) -(DISCHV/DAYS) IF(((DVOL.GE.VMAX).OR.(DVOL.LE.VMIN)).AND.(IFOUND.EQ.0)) THEN T=L IF0UND=1 0UT=0UT+QV/DAYS ELSE IF(IFOUND.EQ.l) OUT=OUT+QV/DAYS Appendix A. Dynamic Programming Models END IF 12 CONTINUE IF(T.Eq.O) THEN T=DAYS DISCH=(DISCHV*(T-1)/DAYS)+0UT DISCHV=DISCH DISCHV=SINT*(INT((DISCHV/SINT)+.5)) ENDV0L= VOL + QV - DISCHV IF (ENDVOL.GT.VMAX) THEN DISCHV=DISCHV+(ENDVOL-VMAX) ENDVOL=VMAX ENDIF IF(DISCHV.GT.DMAX) THEN SPILLV=DISCHV-DMAX DISCHV=DMAX ENDIF IF(DISCHV.LT.DMIN) GOTO 9 IF ((EMPTY.EQ.l).AND.(ENDVOL.LT.VMIN)) GO TO 9 7 RETURN 9 IFAIL=2 RETURN END C C FUNCTION TO CONVERT VOLUMES INTO RESERVOIR ELEVATIONS C NOTE: THIS EQUATION VALID ONLY FOR H>35 AND H<110 M FUNCTION ELVOL(MVOL) REAL MVOL ELEV=0.0 X=MVOL X2=MVOL*X C0=32.7308 Cl=.078263 C2=-.00001 ELV0L=C0+C1*X+C2*X2 RETURN END C C SUBROUTINE TO CALCULATE POWER FROM HEAD AND DISCHARGE SUBROUTINE TURB(DISCH,DAYS,ELEV,POW) INTEGER DAYS POW = ((.87)*(9.81)*(DISCH*DAYS*24.*ELEV))/1.E06 RETURN END A.2 One State Stochastic Model The input files are as follows: File 5: as for the deterministic program but does not include the inflow data File 7: as for the determinisic program File 8: probabilistic inflow data 3(c ?|c sfc sfc ?|c 3fc 3fc 3^ C 3(c 3|C 3|C 3|c 3^ C 3^ C 3f( sfc 3|C 3^ C 3(C 3fC 3|C 3fC 3^ C 5|C 5|C 3f( 3^ C 3fC 3|C 3^ C 3|C 3f( 3|C 3|C 3|C 3(C 3^ C 3(C 3|C 3fC 3fC 3fC 3fC 3f( 3fC 3fC 3fC 3fC C* THIS IS THE SECOND LEVEL OF A SERIES OF DYNAMIC PROGRAMS * C* WRITTEN TO ANALYSE THE VALUE OF INFLOW FORECASTS FOR * C* OPERATION OF A HYDORELECTRIC DEVELOPEMENT. * C* THIS PROGRAM IS STOCHASTIC AND MAKES USE OF ONE STATE * C* VARIABLE, THE VOLUME. * * C* PROGRAMMED BY: JOANNA BARNARD * C* * c C FORMAT AND DIMENSION STATEMENTS Appendix A. Dynamic Programming Models 107 DIMENSION PQ(20,25),qV(20,25),V0L(100),NU(54) DIMENSION P(20,100),PT(1,100) DIMENSION S0PT(20,100),VMIN(20),VMAX(20),DMIN(20),DMAX(20) DIMENSION DISCHV(50),D0PT(20,100),SP0PT(20,100) REAL MVOL, MPQ(20) INTEGER S.SF,STAGES,U,DAYS(20) CHARACTER*70 TITLE,TITLE2 100 F0RMAT(20I3) 101 F0RMAT(I3) 102 F0RMATCF3.0) 103 FORMAT(10F6.1) 104 FORMAT(2F8.0) 105 F0RMAT(5F8.0) 200 F0RMAT(1X,/// * 'N INITIAL EXPECTED DISCH SPILL POWER TOTAL EXPECTED FINAL'/ * ' VOLUME INFLOW POWER VOLUME') 106 F0RMAT(I2,7F8.1) 107 FORMAT(IX,412,F5.0,2F8.0,2F6.0,F8.2) 108 F0RMAT(7(F5.2,F5.0)) 109 F0RMAT(14I3) 110 F0RMATC7F8.1) 111 FORMATQA) 112 FORMAT(//'YEAR OF ANALYSIS IS YEAR',14) 301 F0RMAT(I2,7F8.2) 501 F0RMAT(13F6.0) 115 F0RMAT(I2,F7.0,I4,F7.3) C INPUT BASIC INFORMATION AND PERFORM BASIC CALCULATIONS READ(5,111)TITLE WRITE(9,111) TITLE READ(7,111)TITLE2 WRITE(9,111) TITLE2 WRITE(9,*)'**********************************^ WRITE(12,111) TITLE WRITE(12,111) TITLE2 WRITE(12,200) C READ ALL ALLOWABLE STATES READ (7,102) SINT READ (7,101) N1STAT READ (7,105)(VOL(S),S=1,N1STAT) C READ ALL ALLOWABLE DISCHARGES READ(7,101) NDISCH READ(7,105)(DISCHV(K),K=1.NDISCH) C DETERMINE TIME SPANS AND INTERVALS READ (5,101)NYRS READ (5,101)NDAYS NSTAGE=365/NDAYS C NOTE THAT IN ORDER TO INCLUDE THE TERMINAL VALUE IT IS NECC. C TO ITERATE FROM 2 TO (NSTAGE+l), CALLED STAGES. IN ADDITION TO C ALLOW FOR THE BACKWARD ITERATION, THE VALUES MUST BE READ IN C REVERSE IE, FROM STAGES TO 2 C READ IN INFORMATION PERTAINING TO EACH STAGE STAGES=NSTAGE+1 DATA (DAYS(N),N=13,2,-1)/31,28,31,30,31,30,31,31,30,3 * 1,30,31/ READ (7,104) (VMIN(N),VMAX(N),N=STAGES,2,-1) VMIN(1)=VMIN(STAGES) Appendix A. Dynamic Programming Models VMAX(1)=VMAX(STAGES) READ (7,104) (DMIN(N),DMAX(N),N=STAGES,2,-1) DMIN(1)=DMIN(STAGES) DMAX(1)=DMAX(STAGES) DO 13 N=STAGES,2,-1 NU(N)=0.0 13 CONTINUE U=l N=STAGES C READ INFLOW FROM PROBABILITY OUTPUT FILE, SOME DUMMY VARIABLES 8 IF(N.EQ.l) GO TO 7 READ(8,115) IX,QV(N,U),IY,Pq(N,U) MPQ(N)=MPQ(N)+QV(N,U)*PQ(N, U) NU(N)=NU(N)+1 TQ=TQ+PQ(N,U) IF(TQ.GT.0.995) THEN N=N-1 U=l TQ=0.0 GO TO 8 GO TO 8 END IF 7 CONTINUE C INITIALIZE POWER VARIABLE AND INPUT TERMINAL VALUE DO 5 N=l,STAGES DO 5 S=1,N1STAT P(N,S)=0.0 DOPT(N,S)=0.0 SPOPT(N,S)=0.0 5 CONTINUE w , x . READ(7,105)(P(l,S),S=1,N1STAT) C THIS SECTION OF CODE PRINTS AN ECHO OF THE INPUT VALUES WRITE (9,401) 401 FORMAT('THE FOLLOWING INFORMATION IS BEING USED',/,' * IN THIS ANALYSIS') WRITE (9 402) 402 FORMAT(//'NOTE: THE ANALYSIS GOES FROM FUTURE TO PAST SO',/,' •JANUARY IS N=13 AND DECEMBER IS N=2') WRITE (9,403) WRITE (9,404) 403 FORMAT(///' N MINIMUM MAXIMUM MINIMUM MAXIMUM' ) 404 FORMAT(' VOLUME VOLUME DISCHARGE DISCHARGE') DO 6 N=STAGES,2,-1 WRITE (9,405)N,VMIN(N),VMAX(N),DMIN(N),DMAX(N) 6 CONTINUE 405 FORMAT(IX,12,4F10.0) WRITE (9,408) 408 FORMAT(//'THE FOLLOWING STOCHASTIC INFLOWS ARE BEING CONSIDERED' *,/,'FOR EACH STAGE ONE OR MORE PAIRS OF PROBABILITIES',/, *'AND FLOWS ARE GIVEN') WRITE(9,410) 410 FORMAT(//'STAGE (PROBABILITY, INFLOW)') DO 1 N=STAGES,2,-1 WRITE(9,409)N,(Pq(N,U),qV(N,U),U=1,NU(N)) 1 CONTINUE 409 F0RMAT(I4,7(F5.2,F5.0)) Appendix A. Dynamic Programming Models WRITE ( 9 , 4 0 6 ) 406 F0RMAT(//'THE STARTING VOLUMES TO BE CONSIDERED ARE:') WRITE ( 9 , 1 0 5 ) (VOL(S),S=1,N1STAT) WRITE(9,411) 4 1 1 FORMAT(/'THE TERMINAL POWER VALUES ASSIGNED I N THIS RUN ARE:') WRITE(9,105) (P(1,S),S=1,N1STAT) WRITE(9,407) 407 FORMAT(//'THE DISCHARGES TO BE CONSIDERED I N THIS ANALYIS ARE:') WRITE(9,105) (DISCHV(K),K=1,NDISCH) C PERFORM THE ANALYSIS C I REPRESENTS THE LOOP ON THE NUMER OF YEARS I N THE ANALYSIS DO 10 1= l.NYRS WRITE(9,112)1 C N REPRESENTS THE LOOP ON THE TIME VARIABLE, I E THE STAGE. C VALUES AT N REPRESENT THE VALUES AT THE BEGINNING OF THE REAL C TIME PRIOD, BUT THE END OF THE PERIOD I N ANALYSIS DIRECTION. C J I S THE STAGE I N THE FOLLOWING REAL TIME PERIOD I E N-1 DO 20 N = 2,STAGES I F ( I.EQ.NYRS) WRITE(9,200) J=N-1 C S INDICATES THE LOOP I N THE POTENTIAL VOLUMES, I E THE '1ST' STATE DO 30 S=1,N1STAT I F ( VOL(S) .LT. VMIN(N)) GO TO 30 I F (VOL(S) .GT. VMAX(N)) GO TO 30 C T INDICATES THE LOOP I N THE PREVIOUS MONTHS FLOWS, I E THE 2'ND STATE PMAX=-1000.0 C K I S THE LOOP ON THE POSSIBLE DISCHARGES DO 40 K=1,NDISCH PPOW=0.0 PE=0.0 PPOWER=0.0 PSPILL=0.0 , V DISV=DISCHV(K) I F ( D I S V . L T . D M I N ( N ) ) GO TO 40 IF(DISV.GT.DMAX(N)) GO TO 40 DO 60 U=1,NU(N) SPILLV=0.0 POWER=0.0 '' ENDVOL= VOL(S) + QV(N,U) - DISV C CHECK TO ENSURE THAT THE CALCULATED END VOLUME I S WITHIN THE L I M I T S C OF THE STAGE FOLLOWING , % , , CALL VCHK(N,VMIN(J),VMAX(J),QV(N,U),VOL(S),DMAX(N), * SINT,DMIN(N).ENDVOL,SPILLV,DISV,IFAIL,DAYS(N)) MVOL = (ENDVOL + V 0 L ( S ) ) / 2 . D ISCH=DISV*l.E06/(60.*60.*24.* DAYS(N)) C CALCULATE ELEVATION AS A FUNCTION OF VOLUME ELEV=ELVOLTMVOL) C CALCULATE POWER AS A FUNCTION OF ELEVATION AND DISCHARGE CALL TURB(DISCH,DAYS(N),ELEV,POWER) C CONVERT FI N A L VOLUME TO A STATE INDEX SF=INT((ENDVOL-VOL(l))/SINT+.5)+l I F ( S F . L T . l ) SF=1 C PREVENT A BROKEN CONSTRAINT BEING PART OF THE OPTIMAL SOLN I F ( I F A I L . E Q . 2 ) POWER=0.0 C CALCULATE ACUMULATED POWER AND PROBABLE VALUES C FOLLOWING I S RECURSIVE EQN ALPHA I S DISCOUNT FACTOR PPOWER=PPOWER +PQ(N,U)*POWER ALPHA=1./(1.+.05/12.) TOTPOW= POWER + ALP H A * P ( J , S F ) PPOW=PPOW+PQ(N,U)+TOTPOW Appendix A. Dynamic Programming Models PE=PE+Pq(N,U)+ENDVOL PSPILL=PSPILL+PQ(N,U)+SPILLV 60 CONTINUE IF (PPOW .LE. PMAX) GO TO 40 PMAX= PPOW P0WER2=PP0WER D = DISV E=PE SPILL=PSPILL 40 CONTINUE P0WER3=P0WER2 P(N,S) = PMAX DOPT(N,S)=D SOPT(N,S)=E SPOPT(N,S)=SPILL IF(I.NE.NYRS) GO TO 30 IF(N.Eq.l3) THEN WRITE(12,106)N,VOL(S),MPq(N),DuPT(N,S),SPOPT(N,S),P0WER3, * P(N,S),SOPT(N,S) END IF WRITE(9,106)N,VOL(S),MPq(N),DOPT(N,S),SPOPT(N,S),P0WER3, * P(N,S),SOPT(N,S) 30 CONTINUE 20 CONTINUE DO 15 S=1,N1STAT P(1,S)=P(STAGES,S) 15 CONTINUE 10 CONTINUE C CREATE A LOOKUP TABLE OF THE DETERMINED OPERATING POLICY FOR C USE IN THE SIMULATION MODEL DO 70 S=1,N1STAT WRITE(11,501)V0L(S),(DOPT(N.S),N=STAGES,2,-1) 70 CONTINUE DO 80 S=1,N1STAT WRITE(11,501)V0L(S),(SPOPT(N,S),N=STAGES,2,-1) 80 CONTINUE STOP END C C SUBROUTINE TO ENSURE VOLUME AT END OF STATE IS ALLOWED SUBROUTINE VCHK(N.VMIN,VMAX,qV,VOL,DMAX,SINT,DMIN,ENDVOL, * SPILLV,DISCHV,IFAIL,DAYS) INTEGER TIME,DAYS,N,FULL,EMPTY IFAIL=0 FULL=0 TIME=0 EMPTY=0 IF (ENDVOL .LE. VMAX) GO TO 5 FULL=1 GO TO 8 5 IF (ENDVOL .GE. VMIN) GO TO 7 IF (qV.GT.DMIN) GO TO 9 EMPTY=1 GO TO 8 C END VOLUME IS BEYOND CONSTRAITS, IE IS FULL OR EMPTY SO GO TO C DAILY FLOW 8 IFOUND=0 DVOL=VOL OUT = 0.0 DO 12 L=1,DAYS DVOL=DVOL +(qV/DAYS) - (DISCHV/DAYS) IF(((DVOL.GE.VMAX).OR.(DVOL.LE.VMIN)).AND.(IFOUND.Eq.O)) THEN Appendix A. Dynamic Programming Models TIME=L IF0UND=1 OUT=OUT+QV/DAYS ELSE IF(IFOUND.EQ.l) 0UT=0UT+QV/DAYS END IF 12 CONTINUE IF(TIME.EQ.O) TIME=DAYS DISCH=(DISCHV*(TIME-1)/DAYS)+0UT DISCHV=DISCH DISCHV=SINT*(lNT((DISCHV/SINT)+.5)) ENDV0L= VOL + QV - DISCHV IF(ENDVOL.GT.VMAX) THEN DISCHV=(VOL+QV)-VMAX ENDVOL=VMAX ENDIF IF (DISCHV.GT.DMAX) THEN SPILLV=DISCHV-DMAX DISCHV=DMAX ENDIF IF(DISCHV.LT.DMIN) GOTO 9 IF ((EMPTY.Eq.1).AND.(ENDVOL.LT.VMIN)) GO TO 9 7 RETURN 9 IFAIL=2 RETURN END C C C SUBROUTINE TO CONVERT VOLUMES INTO RESERVOIR ELEVATIONS C NOTE EqUATION IS ONLY VALID FOR ELEV>35 AND ELEV<110 METERS FUNCTION ELVOL(MVOL) REAL MVOL X=MVOL X2=MV0L*X C0=32.7308 Cl=.078263 C2=-.00001 ELV0L=C0+C1*X+C2*X2 RETURN END C C SUBROUTINE TO CALCULATE POWER FROM HEAD AND DISCHARGE SUBROUTINE TURB(DISCH,DAYS,ELEV,POW) INTEGER DAYS POW = (.87)*(1000.)*(9.81)*DISCH*DAYS*24.*ELEV/1.E09 RETURN END A.3 Two State Stochastic Model The input files are as follows: File 5 : as for the deterministic program but does not include the inflow data File 7: as for the determinisic program File 8 : probabilistic inflow data C* ***************************************************************** C* THIS IS THE THIRD LEVEL OF A SERIES OF DYNAMIC PROGRAMS * C* WRITTEN TO ANALYSE THE VALUE OF INFLOW FORECASTS FOR * C* OPERATION OF A HYDORELECTRIC DEVELOPEMENT. * C* THIS PROGRAM IS STOCHASTIC AND MAKES USE OF TWO STATE * C* VARIABLES, THE VOLUME AND THE PREVIOUS INFLOW. * C* * Appendix A. Dynamic Programming Models C* PROGRAMMED BY: JOANNA BARNARD * C* * c C FORMAT AND DIMENSION STATEMENTS s , v DIMENSION Pq(20,35,35),QV(20,35,35),V0L(70),NU(20,35) DIMENSION P(20,70,35),PT(1,70) DIMENSION S0PT(20,70,35),VMIN(20),VMAX(20),DMIN(20),DMAX(20) DIMENSION DISCHV(35),D0PT(20,70,35) ,SP0PT(20,70,35) REAL MVOL, MPQ(20,35) INTEGER S,SF,STAGES,T,TF,U,DAYS(20),PI CHARACTER*70 TITLE,TITLE2 100 F0RMAT(20I3) 101 F0RMAT(I3) 102 F0RMAT(F3.0) 103 FORMAT(10F6.1) 104 F0RMAT(2F8.0) 105 FORMAT(5F8.0) 200 F0RMAT(1X,/// * ' N PREV INIT EXPECTED DISCH SPILL POWER TOTAL EXPD FINAL'/ * ' INF VOL INFLOW POWER VOLUME') 106 F0RMAT(I2,I5,8F7.2) 107 F0RMAT(1X,4I2,F5.0,2F8.0,2F6.0,F8.2) 108 F0RMAT(7(F5.2,F5.0)) 109 F0RMAT(25I2) 110 F0RMAT(7F8.1) 111 FORMAT(A) 112 FORMAT(//,'THE YEAR OF ANALYSIS IS YEAR',14) 501 F0RMATC4F6.0) 502 F0RMAT(10I4) C INPUT BASIC INFORMATION AND PERFORM BASIC CALCULATIONS READ (5,111) TITLE WRITE(9,111) TITLE READ (7,111) TITLE2 WRITE(9,111) TITLE2 WRITE(9,*)'************************************************** ' WRITE(12,111) TITLE WRITE(12,111) TITLE2 WRITE(12,200) C READ ALL ALLOWABLE STATES READ (7,102) SINT READ (7,101) N1STAT READ (7,105)(V0L(S),S=1,N1STAT) C READ ALL ALLOWABLE DISCHARGES READ(7,101) NDISCH READ(7,105)(DISCHV(K),K=1,NDISCH) C DETERMINE TIME SPANS AND INTERVALS READ (5,101)NYRS READ (5,101)NDAYS NSTAGE=365/NDAYS C NOTE THAT IN ORDER TO INCLUDE THE TERMINAL VALUE IT IS NECC. C TO ITERATE FROM 2 TO (NSTAGE+l), CALLED STAGES. IN ADDITION TO C ALLOW FOR THE BACKWARD ITERATION, THE VALUES MUST BE READ IN C REVERSE IE, FROM STAGES TO 2 C READ IN INFORMATION PERTAINING TO EACH STAGE STAGES=NSTAGE+1 DATA (DAYS(N),N=13,2,-l)/31,28,31,30,31,30,31,31,30,3 * 1,30,31/ Appendix A. Dynamic Programming Models READ (7,104) (VMIN(N),VMAX(N),N=STAGES,2,-l) VMIN(1)=VMIN(STAGES) VMAX(1)=VMAX(STAGES) READ (7,104) (DMIN(N),DMAX(N),N=STAGES,2,-1) DMIN(1)=DMIN(STAGES) DMAX(1)=DMAX(STAGES) C THIS BLOCK READS FLOWS AND PROBABILIES C INITIALIZE VARIABLES DO 13 N=STAGES,2,-1 DO 13 T=l,35 NU(N,T)=0 13 CONTINUE U=l N2STAT=0 NY=0 N=STAGES C READ INFLOWS FROM PROBABILITY OUTPUT FILE, SOME DUMMY VARIABLES READ READ(8,101)NP0INT 8 IF(N.EQ.l) GO TO 12 TOLD=T READ(8,997)IY,T,QVTEMP,IZ,IW.PQTEMP TNEW=T IF(T0LD.NE.TNEW)U=1 QV(N,T,U)=QVTEMP Pq(N,T,U)=PQTEMP NU(N,T)=NU(N,T)+1 997 F0RMAT(I2,I4,F7.0,2I4,F7.3) MPq(N,T)=MPq(N,T)+PQ(N,T,U)*QV(N,T,U) IF(T.GT.N2STAT) N2STAT=T NY=NY+IW IF(NY.Eq.NPOINT) THEN N=N-1 U=l NY=0 GO TO 8 ELSE U=U+1 GO TO 8 END IF 12 CONTINUE C INITIALIZE POWER VARIABLE AND INPUT TERMINAL VALUE DO 5 N=l,STAGES DO 5 S=1,N1STAT DO 5 T=1,N2STAT P(N,S,T)=0.0 DOPT(N,S,T)=0.0 SPOPT(N,S,T)=0.0 5 CONTINUE w , READ(7,105)(PT(1,S),S=1,N1STAT) DO 7 S=1,N1STAT DO 7 T=1,N2STAT P(1,S,T)=PT(1,S) 7 CONTINUE C THIS SECTION OF CODE PRINTS AN ECHO OF THE INPUT VALUES WRITE (9,401) 401 FORMAT('THE FOLLOWING INFORMATION IS BEING USED',/,' * IN THIS ANALYSIS') WRITE (9 402) 402 FORMAT(//'NOTE: THE ANALYSIS GOES FROM FUTURE TO PAST SO',/,' •JANUARY IS N=13 AND DECEMBER IS N=2') Appendix A. Dynamic Programming Models 114 WRITE (9,403) WRITE (9,404) 403 FORMAT(///' N MINIMUM MAXIMUM MINIMUM MAXIMUM' ) 404 FORMAT(' VOLUME VOLUME DISCHARGE DISCHARGE') DO 6 N=STAGES,2,-1 WRITE (9,405)N,VMIN(N),VMAX(N),DMIN(N),DMAX(N) 6 CONTINUE 405 F0RMAT(1X,I2,4F10.0) WRITE (9,408) 408 FORMAT(//'THE FOLLOWING STOCHASTIC INFLOWS ARE BEING CONSIDERED' *,/,'FOR EACH STAGE AND POTENTIAL PREVIOUS INFLOW ONE OR MORE',/, •'PAIRS OF PROBABILITIES AND FLOWS ARE GIVEN') WRITE(9,410) 410 FORMAT(//'STAGE PREV INF (PROBABILITY, INFLOW)') DO 1 N=STAGES,2,-1 DO 1 T=1,N2STAT IF (NU(N,T).EQ.O) GO TO 1 PI=15+T WRITE(9,409)N,PI,(PQ(N,T,U),qV(N,T,U),U=1,NU(N,T)) . 1 CONTINUE 409 F0RMAT(2I5,7(F5.2,F5.0)) WRITE (9,406) 406 FORMAT(//'THE STARTING VOLUMES TO BE CONSIDERED ARE:') WRITE (9,105) (VOL(S),S=1,N1STAT) WRITE(9,411) 411 FORMAT(//'THE TERMINAL POWER VALUES ASSIGNED ARE:') WRITE(9,105)(PT(1,S),S=l,N1STAT) WRITE(9,407) 407 FORMAT(//'THE DISCHARGES TO BE CONSIDERED IN THIS ANALYIS ARE:') WRITE(9,105) (DISCHV(K),K=1,NDISCH) C PERFORM THE ANALYSIS C I REPRESENTS THE LOOP ON THE NUMER OF YEARS IN THE ANALYSIS DO 10 1= l.NYRS WRITE(9,112)1 C N REPRESENTS THE LOOP ON THE TIME VARIABLE, IE THE STAGE. C VALUES AT N REPRESENT THE VALUES AT THE BEGINNING OF THE REAL C TIME PERIOD, BUT THE END OF THE PERIOD IN ANALYSIS DIRECTION. C J IS THE STAGE IN THE FOLLOWING REAL TIME PERIOD IE N-l DO 20 N = 2,STAGES WRITE(9,200) J=N-1 C S INDICATES THE LOOP IN THE POTENTIAL VOLUMES, IE THE '1ST' STATE DO 30 S=1,N1STAT IF (VOL(S) .LT. VMIN(N)) GO TO 30 IF (VOL(S) .GT. VMAX(N)) GO TO 30 C T INDICATES THE LOOP IN THE PREVIOUS MONTHS FLOWS, IE THE 2'ND STATE DO 50 T=1,N2STAT IF (NU(N.T).Eq.O) GO TO 50 PMAX=-1000.0 C K IS THE LOOP ON THE POSSIBLE DISCHARGES DO 40 K=l,NDISCH PPOW=0.0 PPOWER=0.0 PE=0.0 PSPILL=0.0 , vDISV=DISCHV(K) IF(DISV.LT.DMIN(N)) GO TO 40 IF(DISV.GT.DMAX(N)) GO TO 40 DO 60 U=1,NU(N,T) Appendix A. Dynamic Programming Models SPILLV=0.0 POWER=0.0 ENDVOL= VOL(S) + QV(N,T,U) - DISV C CHECK TO ENSURE THAT THE CALCULATED END VOLUME IS WITHIN THE LIMITS C OF THE STAGE FOLLOWING CALL VCHK(N,VMIN(J),VMAX(J),QV(N,T,U),V0L(S),DMAX(N), * SINT.DMIN(N),ENDVOL,SPILLV,DISV,IFAIL,DAYS(N)) MVOL = (ENDVOL +V0L(S))/2 DISCH=DISV*1,E06/(60*60*24* DAYS(N)) C CALCULATE ELEVATION AS A FUNCTION OF VOLUME ELEV=ELVOL(MVOL) C CALCULATE POWER AS A,FUNCTION OF ELEVATION AND DISCHARGE CALL TURB(DISCH,DAYS(N),ELEV,POWER) C CONVERT FINAL VOLUME TO A STATE INDEX SF=INT((ENDVOL-VOL(1))/SINT+.5)+1 IF (SF.LE.l) SF=1 TF=INT(QV(N,T,U)/SINT+.5) C CALCULATE ACUMULATED POWER AND PROBABLE VALUES C ALPHA IS DISCOUNT FACTOR ALPHA=1./(1.+.05/12.) IF (IFAIL.EQ.2) POWER=0.0 PPOWER=PPOWER+POWER*PQ(N,T,U) TOTPOW= POWER + ALPHA*P(J,SF,TF) PPOW=PPOW+PQ(N,T,U)*TOTPOW PE=PE+PQ(N,T,U)*ENDVOL PSPILL=PSPILL+PQ(N,T,U)*SPILLV 60 CONTINUE IF (PPOW .LE. PMAX) GO TO 40 PMAX= PPOW P0WER2=PP0WER D = DISV E=PE SPILL=PSPILL 40 CONTINUE P(N,S,T) = PMAX P0WER3=P0WER2 DOPT(N,S,T)=D SOPT(N,S,T)=E SPOPT(N,S,T)=SPILL PI=15.0*T IF(I.NE.NYRS) GOTO 50 IF(N.EQ.STAGES) THEN WRITE(12,106)N,PI,V0L(S),MPQ(N,T),DOPT(N,S,T),SPOPT(N,S,T), * P0WER3,P(N,S,T),S0PT(N,S,T) ENDIF WRITE(9,106)N,PI,V0L(S),MPQ(N,T),DOPT(N,S,T),SP0PT(N,S,T), * P0WER3,P(N,S,T),S0PT(N,S,T) 50 CONTINUE 30 CONTINUE 20 CONTINUE DO 15 S=1,N1STAT DO 15 T=1,N2STAT P(1,S,T)=P(STAGES,S,T) 15 CONTINUE 10 CONTINUE C CREATE A LOOKUP TABLE OF THE DETERMINED OPERATING POLICY FOR C USE IN THE SIMULATION MODEL WRITE(11,101)N2STAT WRITE(11,502)((NU(N,T),T=1,N2STAT),N=STAGES,2,-1) DO 80 N=STAGES,2,-1 DO 70 S=1,N1STAT Appendix A. Dynamic Programming Models DO 75 T=1,N2STAT IF(NU(N,T).EQ.O) GO TO 75 WRITE(11,501)V0L(S),D0PT(N,S,T),SP0PT(N,S,T) 75 CONTINUE 70 CONTINUE 80 CONTINUE STOP END C C SUBROUTINE TO ENSURE VOLUME AT END OF STATE IS ALLOWED SUBROUTINE VCHK(N,VMIN,VMAX,QV,VOL,DMAX,SINT,DMIN,ENDVOL, * SPILLV,DISCHV,IFAIL,DAYS) INTEGER TIME,DAYS,N,FULL,EMPTY IFAIL=0 FULL=0 TIME=0 EMPTY=0 IF (ENDVOL .LE. VMAX) GO TO 5 FULL=1 GO TO 8 5 IF (ENDVOL .GE. VMIN) GO TO 7 IF (QV.GT.DMIN) GO TO 9 EMPTY=1 GO TO 8 C END VOLUME IS BEYOND CONSTRAITS, IE IS FULL OR EMPTY SO GO TO C DAILY FLOW 8 IFOUND=0 DVOL=VOL OUT = 0.0 DO 12 L=1,DAYS DVOL=DVOL+qV/DAYS-DISCHV/DAYS IF(((DVOL.GE.VMAX).OR.(DVOL.LE.VMIN)).AND.(IFOUND.Eq.O)) THEN TIME=L IF0UND=1 , OUT=OUT+qV/DAYS ELSE IF(IFOUND.Eq.l) OUT=OUT+qV/DAYS END IF 12 CONTINUE IF(TIME.Eq.O) TIME=DAYS DISCH=(DISCHV*(TIME-1)/DAYS)+OUT DISCHV=DISCH DISCHV=SINT*(INT((DISCHV/SINT)+.5)) ENDVOL= VOL + qV - DISCHV IF (ENDVOL.GT.VMAX) THEN DISCHV=(VOL+qV)-VMAX ENDVOL=VMAX ENDIF IF (DISCHV.GT.DMAX) THEN SPILLV=DISCHV-DMAX DISCHV=DMAX END IF IF (DISCHV.LT.DMIN) GO TO 9 IF ((EMPTY.Eq.l).AND.(ENDVOL.LT.VMIN)) GO TO 9 7 RETURN 9 IFAIL=2 RETURN END C C FUNCTION TO CONVERT VOLUMES INTO RESERVOIR ELEVATIONS C NOTE EqUATION VALID FOR ELEV BTW 35 AND 110 METRES FUNCTION ELVOL(MVOL) REAL MVOL Appendix A. Dynamic Programming Models ELEV=0.0 X=MVOL X2=MV0L*X C0=32.7308 Cl=.078263 C2=-.00001 ELV0L=C0+C1*X+C2*X2 RETURN END C C SUBROUTINE TO CALCULATE POWER FROM HEAD AND DISCHARGE SUBROUTINE TURB(DISCH,DAYS,ELEV,POW) INTEGER DAYS, x , , POW = (.87)*(1000.)*(9.81)*DISCH*DAYS*24.*ELEV/1.E09 RETURN END Appendix B Simulation Models The following listings are for the simulation programs used in this analysis. They use primarily the same input files describing the reservoirs as the dynamic programs. In addition they require information regarding the observed inflow to the system as well as the optimal operating policy as determined by the optimization programs. The programs are fairly straightforward and the code is frequently commented for reference. B. l Naive Forecast Simulation Models The input files are as follows: File 8: general title of the run, observed inflows File 11: operating policy from dynamic programs File 12: title indicating reservoir size, number of years for which the analysis is to be run, minimum and maximum reservoir volumes and discharges C* THIS PROGRAM USES THE OPERATING POLICIES CREATED USING THE * C* THREE DYNAMIC PROGRAMS, TO OPERATE THE HYPOTHETICAL SYSTEM * C* USING REAL INFLOWS. THIS PROGRAM CAN BE USED FOR THE * C* DETERMINISTIC CASE AND THE ONE DIMENSIONAL STOCHASTIC CASE, * C* SINCE THE DISCHARGE DECISION IS BASED ONLY ON THE TIME OF * C* THE YEAR AND THE VOLUME OF THE RESERVOIR. SLIGHT MODIFICATIONS* C* HAVE TO BE MADE FOR THE TWO DIMENSIONAL CASE SINCE THE PREVIOUS* C* MONTHS INFLOW IS ALSO USED AS A GUIDE. * C* * c C FORMAT AND DIMENSION VARIABLES DIMENSION Q(20),VOL(IOO),D0PT(20,100),T0TP0W(100) DIMENSION VMIN(20),VMAX(20),DMIN(20),DMAX(20) DIMENSION SP0PT(20,100) INTEGER S,SF,STAGES,DAYS(20) CHARACTER*70 TITLE, TITLE2 100 F0RMAT(3I3) 110 F0RMAT(2F8.0) 111 F0RMAT(F5.0) 112 F0RMAT(F3.0) 120 F0RMAT(13F6.0) 130 F0RMAT(10F6.0) 140 F0RMAT(I3,7F9.0,2F9.1) 150 FORMAT(A) REAL MVOL C READ IN OPERATION PARAMETERS, AND VOLUME AND DISCHARGE CONSTRAINTS READ(8,150) TITLE WRITE(13,150)TITLE READ(12,150) TITLE2 WRITE(13,150)TITLE2 118 Appendix B. Simulation Models READ(12,112) SINT READ(12,100) NYRS,NSTATE,NSTAGE READ(12,111) TVOL DATA (DAYS(N),N=13,2,-1)/31,28,31,30,31,30,31,31,30,3 * 1,30,31/ STAGES=NSTAGE+1 READ(12,110)(VMIN(N),VMAX(N),N=STAGES,2,-1) VMIN(1)=VMIN(STAGES) VMAX(1)=VMAX(STAGES) READ(12,110)(DMIN(N),DMAX(N),N=STAGES,2,-1) DMIN(1)=DMIN(STAGES) DMAX(1)=DMAX(STAGES) C READ OPERATING POLICY FROM DP PROGRAMS DO 10 S=l,NSTATE READ(11,120)V0L(S),(DOPT(N,S),N=STAGES,2,-1) 10 CONTINUE DO 20 S=l,NSTATE READ(11,120)V0L(S),(SPOPT(N,S),N=STAGES,2,-1) 20 CONTINUE C START OPERATION READ(8,130) (Q(N),N=STAGES,2,-1) C TVOL IS TRIAL STARTING AND FINISHING VOLUME S=INT((TV0L-V0L(1))/15.)+1 HRITE(13,900)V0L(S) 900 FORMAT(///,'For I n i t i a l January Volume of,F6.0,':') WRITE(13,910) WRITE(13 911) 910 FORMAT(IX,/,'TIME INITIAL OPTIMAL OPTIMAL INFLOW ACTUAL •ACTUAL FINAL ENERGY TOTAL') 911 F0RMAT(1X, ' VOL DISCH SPILL DISCH •SPILL VOLUME ENERGY'/) TOTPOW(S)=0.0 C DO 30 1=1,NYRS VOLUME=VOL(S) SF=S DO 50 N=STAGES,2,-1 J=N-1 IFAIL=0 ' DISV=DOPT(N,SF) SPILL=SPOPT(N,SF) ENDVOL=VOLUME+Q(N)-DISV-SPILL C CHECK TO ENSURE THAT THE CALCULATED END VOLUME IS WITHIN THE LIMITS C OF THE STAGE FOLLOWING N , x , NCALL VCHK(N,VMIN(J),VMAX(J),Q(N),VOLUME,DMAX(N),DMIN(N), * SINT, ENDVOL,SPILL,DISV,DAYS(N),IFAIL) MVOL = (ENDVOL +V0LUME)/2 IF(DISV.LT.DMIN(N)) IFAIL=2 DISCH=DISV*1.E06/(60*60*24*DAYS(N)) C CALCULATE ELEVATION AS A FUNCTION OF VOLUME ELEV=ELVOL(MVOL) C CALCULATE POWER AS A FUNCTION OF ELEVATION AND DISCHARGE CALL TURB(DISCH,DAYS(N),ELEV,POWER) C IFAIL=2 INDICATES A DISCHARGE CONSTRAINT IS BROKEN, PENALIZE C BY GENERATIONS C IF(IFAIL.Eq.2) POWER=0. TOTPOW(S)= TOTPOW(S) + POWER WRITE (13,140) N,VOLUME,DOPT(N,SF),SPOPT(N,SF),Q(N),DISV, * SPILL,ENDVOL,POWER,TOTPOW(S) C CONVERT FINAL VOLUME TO A STATE INDEX Appendix B. Simulation Models SF=INT((ENDV0L-V0L(l))/SINT+.5)+l IF (SF.LE.l) SF=1 VOLUME=ENDVOL 50 CONTINUE 30 CONTINUE 40 CONTINUE END C C SUBROUTINE TO CHECK END VOLUMES SUBROUTINE VCHK(N,VMIN,VMAX,Q,VOLUME,DMAX,DMIN,SINT,ENDVOL, * SPILL,DISV,DAYS,IFAIL) INTEGER DAYS,T,N,FULL,EMPTY,IFAIL IF((SPILL.GT.0.0).AND.(ENDVOL.LE.VMAX))THEN SPILL=SPILL-(VMAX-ENDVOL) IF (SPILL.LT.0.0) SPILL=0.0 ENDVOL=VOLUME+q-DISV-SPILL ENDIF REL=SPILL+DISV IF (ENDVOL .LE. VMAX) GO TO 5 FULL=1 GO TO 8 5 IF (ENDVOL .GE. VMIN) GO TO 7 EMPTY=1 GO TO 8 C END VOLUME IS BEYOND CONSTRAINTS,IE IS FULL OR EMPTY SO GO TO C DAILY FLOW 8 IFOUND=0 DVOL=VOLUME OUT = 0.0 IFAIL=0 T=0.0 DO 12 L=1,DAYS DVOL=DVOL + (Q/DAYS) -(REL/DAYS) IF(((DVOL.GE.VMAX).OR.(DVOL.LE.VMIN)).AND.(IFOUND.EQ.O)) THEN T=L IF0UND=1 OUT=OUT+Q/DAYS ELSE IF(IFOUND.EQ.l) OUT=OUT+Q/DAYS END IF 12 CONTINUE IF(T.EQ.O) T=DAYS REL=(REL*(T-l)/DAYS)+OUT REL=SINT*(INT((REL/SINT)+.5)) ENDVOL= VOLUME+ Q - REL IF(ENDVOL.GT.VMAX) THEN REL=(VOLUME+q)-VMAX ENDVOL=VMAX ENDIF IF(REL.GT.DMAX) THEN SPILL=(REL-DMAX) DISV=DMAX IF(ENDVOL.LT.VMAX) THEN SPILL=SPILL-(VMAX-ENDVOL) ENDVOL=VMAX ENDIF ELSE DISV=REL IF(DISV.LT.DMIN) IFAIL=2 ENDIF 7 RETURN END Appendix B. Simulation Models C SUBROUTINE TO CONVERT VOLUMES INTO RESERVOIR ELEVATIONS C NOTE EQUATION VALID ONLY FOR ELEV>35 AND <110 FUNCTION ELVOL(MVOL) REAL MVOL ELEV=0.0 X=MVOL X2=MV0L*X C0=32.73080 Cl=.078263 C2=-.00001 ELV0L=C0+C1*X+C2*X2 RETURN END C C SUBROUTINE TO CALCULATE POWER FROM HEAD AND DISCHARGE SUBROUTINE TURB(DISCH,DAYS,ELEV,POW) INTEGER DAYS POW = ((.87)*(9.81)*DISCH*DAYS*24.*ELEV)/1.E06 RETURN END . C****************************************************************** C* THIS PROGRAM USES THE OPERATING POLICIES CREATED USING THE * C* THREE DYNAMIC PROGRAMS, TO OPERATE THE HYPOTHETICAL SYSTEM * C* USING REAL INFLOWS. THIS VERSION OF THE PROGRAM HAS BEEN * C* ADAPTED FROM THE ONE DIMENSIONAL DETERMINISTIC AND STOCHASTIC * C* PROGRAM, FOR USE WITH THE TWO DIMENSIONAL STOCHASTIC OPERATING * C* POLICY. THE STARTING VOLUME OF THE RESERVOIR AND THE INFLOW * C* DURING THE PREVIOUS STAGE ARE USED TO DETERMINE THE OPTIMAL * C* RELEASE FROM THE RESERVOIR. * C* * C****************************************************************** c C FORMAT AND DIMENSION VARIABLES DIMENSION Q(20),V0L(60),D0PT(20,100,40).TOTPOW(IOO),NU(20,40) DIMENSION VMIN(20),VMAX(20),DMIN(20),DMAX(20) DIMENSION SP0PT(20,100,40) INTEGER S,SF,STAGES,T,TS,TB,DAYS(20) CHARACTER*70 TITLE ,TITLE2 100 F0RMAT(4I3) 110 F0RMAT(2F8.0) 111 F0RMAT(F5.0) 112 F0RMAT(F3.0) 120 FORMAT(13F6.0) 130 FORMAT(10F6.0) 140 F0RMAT(I3,8F9.0,2F9.1) 150 FORMAT(A) REAL MVOL C READ IN OPERATION PARAMETERS, AND VOLUME AND DISCHARGE CONSTRAINTS READ(8,150) TITLE WRITE(13,150)TITLE READ(12,150) TITLE2 WRITE(13,150)TITLE2 READ(12,112) SINT READ(12,100) NYRS,NISTAT,NSTAGE READ(12,111) TVOL READ(11,100)N2STAT DATA (DAYS(N),N=13,2,-1)/31,28,31,30,31,30,31,31,30,3 * 1,30,31/ STAGES=NSTAGE+1 READ(12,110)(VMIN(N),VMAX(N),N=STAGES,2,-1) Appendix B. Simulation Models VMIN(1)=VMIN(STAGES) VMAX(1)=VMAX(STAGES) READ(12,110)(DMIN(N),DMAX(N),N=STAGES,2,-1) DMIN(1)=DMIN(STAGES) DMAX(1)=DMAX(STAGES) C READ OPERATING POLICY FROM DP PROGRAMS READ(11,503)((NU(N,T),T=1,N2STAT),N=STAGES,2,-1) 503 F0RMAT(10I4) DO 80 N=STAGES,2,-1 DO 70 S=1,N1STAT DO 75 T=1,N2STAT DOPT(N,S,T)=0.0 SPOPT(N,S,T)=0.0 IF(NU(N,T).EQ.O) GOTO 75 READ(11,501)VOL(S),DOPT(N,S,T),SPOPT(N,S,T) 75 CONTINUE 70 CONTINUE 80 CONTINUE 501 F0RMAT(4F6.0) C START OPERATION , , vREAD(8,130) (q(N),N=STAGES,2,-l) Q(STAGES+1)=Q(2) S=INT((TV0L-V0L(1))/15.0)+1 WRITE(13,900)V0L(S) 900 FORMAT(///,'For I n i t i a l January Volume o f ,F6.0,':') WRITE(13,910) WRITE(13 911) 910 F0RMAT(1X,/,'TIME INITIAL PREVIOUS OPTIMAL OPTIMAL INFLOW •ACTUAL ACTUAL FINAL ENERGY TOTAL') 911 F0RMAT(1X, ' VOL INFLOW DISCH SPILL •DISCH SPILL VOLUME ENERGY'/) TOTPOW(S)=0.0 VOLUME=VOL(S) SF=S DO 50 N=STAGES,2,-1 IFAIL=0, S l T=INT(q(N+l)/SINT+.5) DISV=DOPT(N,SF,T) SPILL=SPOPT(N,SF,T) C MAKE SURE THE PREVIOUS INFLOW NOW ENCOUNTERED HAS BEEN CONSIDERED C IN THE OPERATING POLICY. IF NOT INTERPOLATE OR USE MAX OR MIN 65 IF (NU(N,T).Eq.O) THEN TB=T+1 66 IF(NU(N,TB).Eq.O) THEN TB=TB+1 IF((TB-T).GT.5) THEN TB=TB-1 76 IF(NU(N,TB).Eq.O)THEN TB=TB-1 GOTO 76 ENDIF END IF GO TO 66 END IF TS=T-1 67 IF ((NU(N,TS).Eq.O).OR.(TS.LE.O)) THEN TS=TS-1 IF(((T-TS).GT.5).OR.(TS.LE.O)) THEN TS=TS+1 77 IF(NU(N,TS).Eq.O) THEN Appendix B. Simulation Models 123 TS=TS+1 GOTO 77 ENDIF ENDIF GO TO 67 END I F I F ( T B . E Q . T S ) THEN TI=TB GOTO 72 ENDIF T I = . 1 * ( R E A L ( ( 1 0 * T B - 1 0 * T ) / ( T B - T S ) ) ) 72 D I S V = D O P T ( N , S F , T S ) + ( T I * ( D O P T ( N , S F , T B ) - D O P T ( N , S F , T S ) ) ) S P I L L = S P O P T ( N , S F , T S ) + ( T I * ( S P O P T ( N , S F , T B ) - S P O P T ( N , S F , T S ) ) ) C ROUND RESULTING VALUES TO NEAREST INTERVAL D I S V = S I N T * ( I N T ( ( D I S V / S I N T ) + . 5 ) ) S P I L L = S I N T * ( I N T ( ( S P I L L / S I N T ) + . 5 ) ) END I F 69 CONTINUE J=N-1 ENDVOL=VOLUME+Q(N) -DISV-SPILL C CHECK TO ENSURE THAT THE CALCULATED END VOLUME I S WITHIN THE L I M I T S C OF THE STAGE FOLLOWING v , v , s C A L L V C H K ( N , V M I N ( J ) , V M A X ( J ) , Q ( N ) . V O L U M E , D M A X ( N ) , D M I N ( N ) , * S I N T , E N D V O L , S P I L L , D I S V , D A Y S ( N ) , I F A I L ) MVOL = (ENDVOL + V 0 L U M E ) / 2 I F ( D I S V . L T . D M I N ( N ) ) I F A I L = 2 D I S C H = D I S V * 1 . E 0 6 / ( 6 0 * 6 0 * 2 4 * D A Y S ( N ) ) C CALCULATE E L E V A T I O N AS A FUNCTION OF VOLUME ELEV= ELVOL(MVOL) C CALCULATE POWER AS A FUNCTION OF E L E V A T I O N AND DISCHARGE C A L L T U R B ( D I S C H , D A Y S ( N ) , E L E V , P O W E R ) C I F ( I F A I L . E Q . 2 ) POWER=0.0 TOTPOW(S)= TOTPOW(S) + POWER WRITE ( 1 3 , 1 4 0 ) N , V O L U M E , Q ( N + 1 ) , D O P T ( N , S F , T ) , S P O P T ( N , S F , T ) , Q ( N ) , * D I S V , S P I L L , E N D V O L , P O W E R , T O T P O W ( S ) C CONVERT F I N A L VOLUME TO A STATE INDEX % S F = I N T ( E N D V O L / S I N T - ( V O L ( 1 ) / S I N T - 1 . ) + . 5 ) I F ( S F . L E . l ) SF=1 VOLUME=ENDVOL 50 CONTINUE 40 CONTINUE END C C SUBROUTINE TO CHECK END VOLUMES SUBROUTINE V C H K ( N , V M I N , V M A X , Q , V O L U M E , D M A X , D M I N , S I N T , E N D V O L , * S P I L L , D I S V , D A Y S , I F A I L ) INTEGER T I M E , D A Y S , F U L L , E M P T Y , N I F ( ( S P I L L . G T . 0 . 0 ) . A N D . ( E N D V O L . L E . V M A X ) ) T H E N S P I L L = S P I L L - ( V M A X - E N D V O L ) I F ( S P I L L . L T . 0 . 0 ) S P I L L = 0 . 0 ENDVOL=VOLUME+Q-DISV-SPILL ENDIF R E L = S P I L L + D I S V I F (ENDVOL . L E . VMAX) GO TO 5 FULL=1 GO TO 8 5 I F (ENDVOL . G E . VMIN) GO TO 7 EMPTY=1 GO TO 8 C END VOLUME I S BEYOND C O N S T R A I N T S , I E I S F U L L OR EMPTY SO GO TO C D A I L Y FLOW 8 IFOUND=0 DVOL=VOLUME Appendix B. Simulation Models OUT = 0.0 IFAIL=0 TIME=0 DO 12 L=1,DAYS DV0L=DV0L + (Q/DAYS) -(REL/DAYS) IF(((DVOL.GE.VMAX).OR.(DVOL.LE.VMIN)).AND.(IFOUND.EQ.0)) THEN TIME=L IF0UND=1 OUT=OUT+Q/DAYS ELSE IF(IFOUND.EQ.l) OUT=OUT+q/DAYS END IF 12 CONTINUE IF(TIME.Eq.O) TIME=DAYS REL=(REL*(TIME-1)/DAYS)+OUT REL=SINT*(INT((REL/SINT)+.5)) ENDVOL= VOLUME+ Q - REL IF(ENDVOL.GT.VMAX) THEN REL=(VOLUME+q)-VMAX ENDVOL=VMAX ENDIF IF(REL.GT.DMAX) THEN SPILL=(REL-DMAX) DISV=DMAX IF(ENDVOL.LT.VMAX) THEN SPILL=SPILL-(VMAX-ENDVOL) ENDVOL=VMAX ENDIF ELSE DISV=REL IF(DISV.LT.DMIN) IFAIL=2 ENDIF 7 RETURN END C C SUBROUTINE TO CONVERT VOLUMES INTO RESERVOIR ELEVATIONS C EQN VALID FOR ELEV<110 AND >35 FUNCTION ELVOL(MVOL) REAL MVOL ELEV=0.0 X=MVOL X2=MV0L*X C0=32.7308 Cl=.078263 C2=-.00001 ELV0L=C0+C1*X+C2*X2 RETURN END C C SUBROUTINE TO CALCULATE POWER FROM HEAD AND DISCHARGE SUBROUTINE TURB(DISCH,DAYS,ELEV,POW) INTEGER DAYS POW = ((.87)*(9.81)*DISCH*24.*DAYS*ELEV)/1.E06 RETURN END B.2 Conceptual Forecast Simulation Models The input files are as follows: File 8: general title of the run, observed inflows File 12: title indicating reservoir size, number of years for which the analysis is to run, minimum and maximum reservoir volumes and discharges Appendix B. Simulation Models 125 Files 20 to 27: operating policies based on each monthly forecast, from the dynamic programs Q************************** **************************************** C* THIS PROGRAM USES THE OPERATING POLICIES CREATED USING THE * C* THREE DYNAMIC PROGRAMS, TO OPERATE THE HYPOTHETICAL SYSTEM * C* USING REAL INFLOWS. THIS PROGRAM CAN BE USED FOR THE * C* DETERMINISTIC CASE AND THE ONE DIMENSIONAL STOCHASTIC CASE, * C* SINCE THE DISCHARGE DECISION IS BASED ONLY ON THE TIME OF * C* THE YEAR AND THE VOLUME OF THE RESERVOIR. SLIGHT MODIFICATIONS* C* HAVE TO BE MADE FOR THE TWO DIMENSIONAL CASE SINCE THE PREVIOUS* C* MONTHS INFLOW IS ALSO USED AS A GUIDE. * C* * C* THIS VERSION OF THE PROGRAM IS FOR USE WITH THE FORECASTED * C* FLOWS. IT READS A DIFFERENT OPERATING POLICY FOR EACH MONTH * C* FROM JANUARY TO AUGUST. * Q****************************************************** ************ C C FORMAT AND DIMENSION VARIABLES DIMENSION Q(20),TOTPOW(100) DIMENSION VMIN(20),VMAX(20),DMIN(20),DMAX(20) COMMON VOL(IOO),D0PT(20,100),SP0PT(20,100) INTEGER S,SF,STAGES,DAYS(20) CHARACTER*70 TITLE, TITLE2 100 F0RMAT(3I3) 110 F0RMAT(2F8.0) 111 F0RMAT(F5.0) 112 FORMAT(F3.0) 120 F0RMAT(13F6.0) 130 FORMAT(10F6.0) 140 F0RMAT(I3,7F9.0,2F9.1) 150 FORMAT(A) REAL MVOL C READ IN OPERATION PARAMETERS, AND VOLUME AND DISCHARGE CONSTRAINTS READ(8,150) TITLE WRITE(13,150)TITLE READ(12,150) TITLE2 WRITE(13,150)TITLE2 READ(12,112) SINT READ(12,100) NYRS,NSTATE,NSTAGE READ(12,111) TVOL DATA (DAYS(N),N=13,2,-1)/31,28,31,30,31,30,31,31,30,3 * 1,30,31/ STAGES=NSTAGE+1 READ(12,110)(VMIN(N),VMAX(N),N=STAGES,2,-1) VMIN(1)=VMIN(STAGES) VMAX(1)=VMAX(STAGES) READ(12,110)(DMIN(N),DMAX(N),N=STAGES,2,-1) DMIN(1)=DMIN(STAGES) DMAX(1)=DMAX(STAGES) C START OPERATION READ(8,130) (Q(N),N=STAGES,2,-1) C WRITE(13,900)V0L(S) 900 FORMAT(///,'For I n i t i a l January Volume of,F6.0,':') WRITE(13,910) WRITE(13 911) 910 F0RMAT(1X,/,'TIME INITIAL OPTIMAL OPTIMAL INFLOW ACTUAL •ACTUAL FINAL ENERGY TOTAL') Appendix B. Simulation Models 911 FORMAT(IX, ' VOL DISCH SPILL DISCH *SPILL VOLUME ENERGY'/) C N IS THE LOOP ON TIME IE MONTHS DO 50 N=STAGES,2,-1 C READ OPERATING POLICY FROM DP PROGRAMS. IF STATEMENTS DETERMINE C WHICH MONTH AND THEREFORE WHICH OP IS REQUIRED. AUGUST'S OP IS USED C FOR AUGUST THROUGH DECEMBER IF(N.EQ.13) M=20 IF(N.EQ.12) M=21 IF(N.EQ.ll) M=22 IF(N.EQ.IO) M=23 IF(N.EQ. 9) M=24 IF(N.EQ. 8) M=25 IF(N.EQ. 7) M=26 IF(N.EQ. 6) M=27 IF(N.LT. 6) GOTO 25 CALL READER(M.NSTATE,STAGES) C ONLY LOOK AT REQUIRED RESERVOIR END VOLUME S=INT((TV0L-V0L(1))/15.)+1 IF(N.EQ.13) THEN VOLUME=VOL(S) SF=S ENDIF 25 J=N-1 IFAIL=0 DISV=DOPT(N,SF) SPILL=SPOPT(N,SF) ENDVOL=VOLUME+Q(N)-DISV-SPILL C CHECK TO ENSURE THAT THE CALCULATED END VOLUME IS WITHIN THE LIMITS C OF THE STAGE FOLLOWING CALL VCHK(N,VMIN(J),VMAX(J),Q(N),VOLUME,DMAX(N),DMIN(N), * SINT, ENDVOL,SPILL,DISV,DAYS(N),IFAIL) MVOL = (ENDVOL +V0LUME)/2 IF(DISV.LT.DMIN(N)) IFAIL=2 DISCH=DISV*1.E06/(60*60*24*DAYS(N)) C CALCULATE ELEVATION AS A FUNCTION OF VOLUME ELEV=ELVOL(MVOL) C CALCULATE POWER AS A FUNCTION OF ELEVATION AND DISCHARGE CALL TURB(DISCH,DAYS(N),ELEV,POWER) C IF IFAIL=2 INDICATES A DISCHARGE CONSTRAINT HAS BEEN BROKEN, C PENALIZE BY LETTING POWER = 0 C IF(IFAIL.EQ.2) POWER=0.0 TOTPOW(S)=TOTPOW(S) + POWER WRITE (13,140) N,VOLUME,DOPT(N,SF),SPOPT(N,SF),Q(N),DISV, * SPILL,ENDVOL,POWER,TOTPOW(S) C CONVERT FINAL VOLUME TO A STATE INDEX SF=INT((ENDV0L-V0L(l))/SINT+.5)+l IF (SF.LE.l) SF=1 VOLUME=ENDVOL 50 CONTINUE C 40 CONTINUE END C C SUBROUTINE TO CHECK END VOLUMES SUBROUTINE VCHK(N,VMIN,VMAX,Q,VOLUME,DMAX,DMIN,SINT,ENDVOL, * SPILL,DISV,DAYS,IFAIL) INTEGER DAYS,T,N,FULL,EMPTY IF((SPILL.GT.0.0).AND.(ENDVOL.LE.VMAX))THEN SPILL=SPILL-(VMAX-ENDVOL) IF (SPILL.LT.0.0) SPILL=0.0 Appendix B. Simulation Models ENDVOL=VOLUME+Q-DISV-SPILL ENDIF REL=SPILL+DISV IF (ENDVOL .LE. VMAX) GO TO 5 FULL=1 GO TO 8 .5 IF (ENDVOL .GE. VMIN) GO TO 7 EMPTY=1 GO TO 8 C END VOLUME IS BEYOND CONSTRAINTS,IE IS FULL OR EMPTY SO GO TO C DAILY FLOW 8 IF0UND=0 DV0L=V0LUME OUT = 0.0 IFAIL=0 T=0.0 DO 12 L=1,DAYS DVOL=DVOL + (Q/DAYS) -(REL/DAYS) IF(((DVOL.GE.VMAX).OR.(DVOL.LE.VMIN)).AND.(IFOUND.EQ.O)) THEN T=L IF0UND=1 , OUT=OUi+Q/DAYS ELSE IF(IFOUND.EQ.l) OUT=OUT+q/DAYS END IF 12 CONTINUE IF (T.Eq.O) T=DAYS REL=(REL*(T-1)/DAYS)+OUT REL=SINT*(INT((REL/SINT)+.5)) ENDVOL= VOLUME+ q - REL IF(ENDVOL.GT.VMAX)THEN REL=(VOLUME+q)-VMAX ENDVOL=VMAX ENDIF IF(REL.GT.DMAX) THEN SPILL=(REL-DMAX) DISV=DMAX IF(ENDVOL.LT.VMAX) THEN SPILL=SPILL-(VMAX-ENDVOL) ENDVOL=VMAX ENDIF ELSE DISV=REL IF(DISV.LT.DMIN) IFAIL=2 ENDIF 7 RETURN END C C SUBROUTINE TO CONVERT VOLUMES INTO RESERVOIR ELEVATIONS C NOTE EqUATION VALID ONLY FOR ELEV>35 AND <110 FUNCTION ELVOL(MVOL) REAL MVOL ELEV=0.0 X=MVOL X2=MV0L*X C0=32.73080 Cl=.078263 C2=-.00001 ELV0L=C0+C1*X+C2*X2 RETURN END C C SUBROUTINE TO CALCULATE POWER FROM HEAD AND DISCHARGE SUBROUTINE TURB(DISCH,DAYS,ELEV,POW) Appendix B. Simulation Models INTEGER DAYS , POW = ((.87)*(9.8l)*DISCH*DAYS*24.*ELEV)/l.E06 RETURN END C C SUBROUTINE TO READ THE APPROPRIATE OPERATING POLICY SUBROUTINE READER(M.NSTATE,STAGES) COMMON VOL(IOO),D0PT(20,100),SP0PT(20,100) INTEGER STAGES,S 120 F0RMAT(13F6.0) DO 10 S=1,NSTATE READ(M,120)V0L(S),(D0PT(N,S),N=STAGES,2,-1) 10 CONTINUE DO 20 S=1,NSTATE READ(M,120)V0L(S),(SP0PT(N,S),N=STAGES,2,-1) 20 CONTINUE RETURN END Q****************************************************************** C* THIS PROGRAM USES THE OPERATING POLICIES CREATED USING THE * C* THREE DYNAMIC PROGRAMS, TO OPERATE THE HYPOTHETICAL SYSTEM * C* USING REAL INFLOWS. THIS VERSION OF THE PROGRAM HAS BEEN * C* ADAPTED FROM THE ONE DIMENSIONAL DETERMINISTIC AND STOCHASTIC * C* PROGRAM, FOR USE WITH THE TWO DIMENSIONAL STOCHASTIC OPERATING * C* POLICY. THE STARTING VOLUME OF THE RESERVOIR AND THE INFLOW * C* DURING THE PREVIOUS STAGE ARE USED TO DETERMINE THE OPTIMAL * C* RELEASE FROM THE RESERVOIR. * C* * C* THIS VERSION IS TO BE USED WITH THE FORECAST SERIES. IT READS * C* A DIFFERENT OPTIMAL POLICY FOR EACH MONTH FROM JAN TO AUG. * C****************************************************************** c C FORMAT AND DIMENSION VARIABLES v DIMENSION Q(20),T0TP0W(100) DIMENSION VMIN(20),VMAX(20),DMIN(20),DMAX(20) COMMON NU(20,60),V0L(100),D0PT(20,100,60),SP0PT(20,100,60) INTEGER S,SF,STAGES,T,TS,TB,DAYS(20) CHARACTER*70 TITLE ,TITLE2 100 F0RMAT(4I3) 110 F0RMAT(2F8.0) 111 FORMAT(F5.0) 112 F0RMAT(F3.0) 120 F0RMAT(13F6.0) 130 F0RMAT(10F6.0) 140 F0RMAT(I3,8F9.0,2F9.1) 150 FORMAT(A) REAL MVOL C READ IN OPERATION PARAMETERS, AND VOLUME AND DISCHARGE CONSTRAINTS READ(8,150) TITLE WRITE(13,150)TITLE READ(12,150) TITLE2 WRITE(13,150)TITLE2 READ(12,112) SINT READ(12,100) NYRS,NlSTAT,NSTAGE READ(12,111) TVOL DATA (DAYS(N),N=13,2,-1)/31,28,31,30,31,30,31,31,30,3 * 1,30,31/ STAGES=NSTAGE+1 KVM READ(12,110)(VMIN(N),VMAX(N),N=STAGES,2,-1) Appendix B. Simulation Models VMIN(1)=VMIN(STAGES) VMAX(1)=VMAX(STAGES) READ(12,110)(DMIN(N),DMAX(N),N=STAGES,2,-1) DMIN(1)=DMIN(STAGES) DMAX(1)=DMAX(STAGES) C START OPERATION • READ(8,130) (Q(N),N=STAGES,2,-1) Q(STAGES+l)=q(2) C CONSIDER ONLY REQUIRED ENDING VOLUME, TVOL C WRITE(13,900)V0L(S) 900 FORMAT(///,'For I n i t i a l January Volume o f ,F6.0,':') WRITE(13,910) WRITE(13 911) 910 FORMAT(IX,/,'TIME INITIAL PREVIOUS OPTIMAL OPTIMAL INFLOW •ACTUAL ACTUAL FINAL ENERGY TOTAL') 911 FORMAT(IX, ' VOL INFLOW DISCH SPILL •DISCH SPILL VOLUME ENERGY'/) DO 50 N=STAGES,2,-1 C CLEAR THE OPTIMAL VALUES FOR READING THE NEW IF(N.GE.6) THEN DO 81 S=1,N1STAT DO 81 T=1,N2STAT DOPT(N,S,T)=0.0 SPOPT(N,S,T)=0.0 81 CONTINUE ENDIF C READ APPROPRIATE OPERATING POLICY IF(N.EQ.13) M=20 IF(N.EQ.12) M=21 IF(N.Eq.ll) M=22 IF(N.Eq.lO) M=23 IF(N.Eq. 9) M=24 IF(N.Eq. 8) M=25 IF(N.Eq. 7) M=26 IF(N.Eq. 6) M=27 IF(N.LT.6) GO TO 25 CALL READER(M,N1STAT,N2STAT,STAGES) C STARTING VOLUME IS AS INPUT AS TVOL S=INT((TVOL-VOL(1))/15.)+1 IF(N.Eq.l3) THEN VOLUME=VOL(S) SF=S ENDIF, , 25 T=INT(q(N+l)/SINT+.5) DISV=DOPT(N,SF,T) SPILL=SPOPT(N SF T) C MAKE SURE THE PREVIOUS INFLOW NOW ENCOUNTERED HAS BEEN CONSIDERED C IN THE OPERATING POLICY. IF NOT INTERPOLATE 65 IF (NU(N,T).Eq.O) THEN TB=T+1 66 IF(NU(N,TB).Eq.O) THEN TB=TB+1 s IF((TB-T).GT.5) THEN TB=T-1 76 IF(NU(N,TB).Eq.O) THEN TB=TB-1 GO TO 76 END IF END IF Appendix B. Simulation Models 130 GO TO 66 END IF TS=T-1 67 IF ((NU(N,TS).Eq.O).OR.(TS.LE.O)) THEN TS=TS-1 IF(((T-TS).GT.5).OR.(TS.LE.O)) THEN TS=T+1 77 IF(NU(N,TS).EQ.0) THEN TS=TS+1 GO TO 77 END IF ENDIF GO TO 67 END IF IF(TB.EQ.TS) THEN TI=TB GO TO 72 ENDIF ... TI=.1*(REAL((10*TB-10*T)/(TB-TS))) 72 DISV=DOPT(N,SF,TS)+(TI*(DOPT(N,SF,TB)-DOPT(N,SF,TS))) DISV=SINT*(INT(DISV/SINT+.5)) SPILL=SPOPT(N,SF,TS)+(TI*(SPOPT(N,SF,TB)-SPOPT(N,SF,TS))) SPILL=SINT*(INT(SPILL/SINT+.5)) END IF J=N-1 IFAIL=0 _ ENDVOL=VOLUME+Q(N)-DISV-SPILL C CHECK TO ENSURE THAT THE CALCULATED END VOLUME IS WITHIN THE LIMITS C OF THE STAGE FOLLOWING CALL VCHK(N,VMIN(J),VMAX(J),q(N),VOLUME,DMAX(N),DMIN(N), * SINT,ENDVOL,SPILL,DISV,DAYS(N),IFAIL) MVOL » (ENDVOL +V0LUME)/2 IF(DISV.LT.DMIN(N)) IFAIL=2 DISCH=DISV*1.E06/(60*60*24*DAYS(N)) C CALCULATE ELEVATION AS A FUNCTION OF VOLUME ELEV= ELVOL(MVOL) C CALCULATE POWER AS A FUNCTION OF ELEVATION AND DISCHARGE CALL TURB(DISCH,DAYS(N),ELEV,POWER) C IFAIL=2 INDICATES THAT A DISCHARGE CONSTRAINT HAS BEEN BROKEN, C SET POWER =0 AS A PENALTY C IF(IFAIL.EQ.2) POWER=0.0 TOTPOW(S)= TOTPOW(S) + POWER WRITE (13,140) N,VOLUME,Q(N+1),D0PT(N,SF,T),SP0PT(N,SF,T),Q(N), * DISV,SPILL,ENDVOL,POWER,TOTPOW(S) C CONVERT FINAL VOLUME TO A STATE INDEX v SF=INT(ENDVOL/SINT-(VOL(1)/SINT-1.)+.5) IF (SF.LE.l) SF=1 VOLUME=ENDVOL 50 CONTINUE 40 CONTINUE END C C SUBROUTINE TO CHECK END VOLUMES SUBROUTINE VCHK(N,VMIN,VMAX,Q,VOLUME,DMAX,DMIN,SINT,ENDVOL, * SPILL,DISV,DAYS,IFAIL) INTEGER TIME,DAYS,FULL,EMPTY,N IF((SPILL.GT.0.0).AND.(ENDVOL.LE.VMAX))THEN SPILL=SPILL-(VMAX-ENDVOL) IF(SPILL.LT.0.0) SPILL=0.0 ENDVOL=VOLUME+Q-DISV-SPILL ENDIF REL=SPILL+DISV Appendix B. Simulation Models IF (ENDVOL .LE. VMAX) GO TO 5 FULL=1 GO TO 8 5 IF (ENDVOL .GE. VMIN) GO TO 7 EMPTY=1 GO TO 8 C END VOLUME IS BEYOND CONSTRAINTS,IE IS FULL OR EMPTY SO GO TO C DAILY FLOW 8 IFOUND=0 DVOL=VOLUME IFAIL=0 TIME=0 OUT = 0.0 DO 12 L-l,DAYS DVOL=DVOL + (Q/DAYS) -(REL/DAYS) IF(((DVOL.GE.VMAX).OR.(DVOL.LE.VMIN)).AND.(IFOUND.EQ.O)) THEN TIME=L IF0UND=1 , OUT=OUT+Q/DAYS ELSE IF(IFOUND.EQ.l) OUT=OUT+Q/DAYS END IF 12 CONTINUE IF (TIME.EQ.O) TIME=DAYS REL=(REL*(TIME-1)/DAYS)+OUT REL=SINT*(INT((REL/SINT)+.5)) ENDVOL= VOLUME+ Q - REL IF (ENDVOL.GT.VMAX) THEN REL=(VOLUME+Q)-VMAX ENDVOL=VMAX ENDIF IF(REL.GT.DMAX) THEN SPILL=(REL-DMAX) DISV=DMAX IF(ENDVOL.LT.VMAX) THEN SPILL=SPILL-(VMAX-ENDVOL) ENDVOL=VMAX ENDIF ELSE DISV=REL IFCDISV.LT.DMIN) IFAIL=2 ENDIF 7 RETURN END C C SUBROUTINE TO CONVERT VOLUMES INTO RESERVOIR ELEVATIONS C EQN VALID FOR ELEV<110 AND >35 FUNCTION ELVOLCMVOL) REAL MVOL ELEV=0.0 X=MVOL X2=MV0L*X C0=32.7308 Cl».078263 C2=-.00001 ELV0L=C0+C1*X+C2*X2 RETURN END C C SUBROUTINE TO CALCULATE POWER FROM HEAD AND DISCHARGE SUBROUTINE TURB(DISCH,DAYS,ELEV,POW) INTEGER DAYS POW = C(.87)*(9.81)*DISCH*24.*DAYS*ELEV)/1.E06 RETURN Appendix B. Simulation Models END C C SUBROUTINE TO READ OPERATING POLICY FROM DP PROGRAMS SUBROUTINE READERCM,N1STAT,N2STAT,STAGES) COMMON NU(20,60),VOL(100),D0PT(20,100,60),SP0PTC20,100,60) INTEGER STAGES,S,T 502 FORMAT(13) 503 F0RMATC10I4) 501 F0RMAT(4F6.0) READ(M,502)N2STAT READ(M,503)((NU(N,T),T=1,N2STAT),N=STAGES,2,-1) C DO 85 N=STAGES,2,-1 C NU(N,0)=0 C85 CONTINUE DO 80 N=STAGES,2,-1 DO 70 S=1,N1STAT DO 75 T=1,N2STAT IF(NU(N,T).EQ.O) GOTO 75 READCM,501)VOL(S),DOPT(N,S,T),SPOPT(N,S,T) 75 CONTINUE 70 CONTINUE 80 CONTINUE RETURN END Appendix C Historic Forecast Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1971 15 15 15 30 225 330 240 165 75 45 30 15 1972 15 15 15 30 225 480 330 180 60 45 30 15 1973 15 15 15 30 165 285 225 120 60 45 30 15 1974 15 15 15 45 150 420 345 195 90 45 30 15 1975 15 15 15 45 150 315 300 135 75 60 60 30 1976 15 15 15 75 225 255 375 300 180 60 30 15 1977 15 15 15 60 150 300 195 150 75 30 30 15 1978 15 15 30 60 135 315 270 135 180 75 45 30 1979 15 15 15 30 150 300 255 135 75 45 15 30 1980 15 15 15 75 255 240 195 105 90 75 45 45 1981 30 15 30 60 255 270 300 180 90 45 45 30 1982 15 15 15 15 150 450 285 180 120 60 30 15 1983 15 15 15 60 195 300 315 150 90 45 60 15 1984 15 15 15 45 105 330 300 180 105 60 30 15 1985 15 15 15 45 225 255 210 105 60 45 30 15 1986 15 15 15 45 195 375 255 150 75 45 30 15 1987 15 15 30 60 240 315 195 120 75 30 30 15 133 Appendix D Programs To Calculate Probabilities D. l Probabilites The following program calculate probabilities for inflow data. The output is in a form directly readable by the dynamic programs. The program reads information from two data files. File 5 contains the number of years of data, the number of data points per year and the desired rounding interval (used only if the program is required to round the data, before performing the analysis). File 6 contains the inflow data, in the format specified. Output is written to file 7. C THIS PROGRAM READS A SERIES OF FLOW VALUES AND CALCULATES C PROBABILITIES C C PROGRAMMED BY: JOANNA BARNARD C PROGRAMME DATE: OCT. 1988 C DIMENSION Q(20,15) 100 FORMAT(213,F5.0) 101 F0RMAT(6F8.1) 102 F0RMAT(I2,F7.0,I4,F7.3) READ(5,100) NYRS,NSTS,STEP DO 10 1=1,NYRS READ(6,101)(Q(I,J),J=NSTS+1,2,-1) DO 20 J=NSTS+1 2-1 Q(I,J)=Q(I,J)*(60.0*60.0*24.0*(365.0/NSTS)/lE06) Q(I,J)=INT((Q(I,J)/STEP)+.5)*STEP IF(Q(I,J).GT.QMAX) QMAX=Q(I,J) 20 CONTINUE 10 CONTINUE DO 30 J=NSTS+1,2,-1 201 F0RMAT(6F6.2) Y=0.0 NY=0 50 DO 40 1=1,NYRS IF (ABS(Q(I,J)-Y).GE.1.0) GO TO 40 NY=NY+1 40 CONTINUE IF (NY.EQ.O) GO TO 70 PR0B=REAL(NY)/REAL(NYRS) WRITE(7,102) J,Y,NY,PR0B 70 IF (Y.LT.QMAX) THEN Y=Y+STEP NY=0 GO TO 50 ENDIF 30 CONTINUE STOP END 134 Appendix D. Programs To Calculate Probabilities 135 D.2 Conditional Probabilities The following program calculates conditional probabilities of a series of inflow data. The output file is in the format required for direct input into the dynamic programs. The input files are identical to those described for the previous program. File 7 is again the output file, but contains more data, since an indication of the previous months inflow is also necessary. C THIS PROGRAM READS A SERIES OF FLOW VALUES AND CALCULATES C CONDITIONAL PROBABILITIES C C PROGRAMMED BY: JOANNA BARNARD C PROGRAMME DATE: SEPT. 1988 C DIMENSION q(30,365) 100 FORMAT(213,F5.0) 101 F0RMAT(6F8.1) 102 F0RMAT(I2,I4,F7.0,2I4,F7.3) READ(5,100) NYRS,NSTS,STEP WRITE(7,103)NYRS 103 FORMAT(13) DO 10 1=1,NYRS READ(6,101)(q(I,J),J=NSTS+1,2,-1) DO 20 J=NSTS+1 2 -1 q(I,J)=q(I,J)*(60.0*60.0*24.0*(365.0/NSTS)/1E06) q(I,J)=INT((q(I,J)/STEP)+.5)*STEP iF(q(i,J).GT.qMAX) qMAX=q(i,J) 20 CONTINUE IF(I.GT.l). Q(I,NSTS+2)=q(I-l,2) i F ( i . E q . i ) q(i,NSTS+2)=q(i,2) 10 CONTINUE DO 30 J=NSTS+1,2,-1 201 F0RMAT(6F6.2) X=0.0 Y=0.0 NX=0 NY=0 50 DO 40 1=1,NYRS C SINCE ITERATING FORM DEC=2 TO JAN=13 PREVIOUS FLOW IS J+l IF (ABS(q(I,J+l)-X).GE.1.0) GO TO 40 NX=NX+1 IF (ABS(q(I,J)-Y).GE.1.0) GO TO 40 NY=NY+1 40 CONTINUE IF (NX.Eq.O) GO TO 60 IF (NY.Eq.O) GO TO 70 PR0B=REAL(NY)/REAL(NX) IX=INT(X/STEP+.5) WRITE(7,102) J,IX,Y,NX,NY,PROB 70 IF (Y.LT.qMAX) THEN Y=Y+STEP NY=0 NX=0 GO TO 50 ENDIF 60 IF (X.LT.qMAX) THEN X=X+STEP Y=0.0 NY=0 NX=0 Appendix D. Programs To Calculate Probabilities GO TO 50 E N D I F 30 CONTINUE STOP END Appendix E Conceptual Hydrologic Forecasts E . l Forecast Year: 1966 January Forecast Jan Feb Mar Apr May 15 15 15 30 225 15 15 30 30 180 15 15 15 45 180 15 15 15 45 165 15 15 15 30 135 15 15 15 45 195 15 15 15 60 165 15 15 15 60 150 15 15 15 45 180 15 15 15 75 300 15 15 15 60 225 15 15 15 30 150 15 15 30 75 240 15 15 30 60 135 15 15 15 45 225 15 15 30 75 195 15 15 30 75 240 February Forecast Jan Feb Mar Apr May 15 15 15 30 240 15 15 30 30 195 15 15 15 45 210 15 15 15 45 180 15 15 15 30 150 15 15 15 45 210 15 15 15 60 195 15 15 15 60 180 15 15 15 45 195 15 15 15 75 360 15 15 15 45 255 15 15 15 30 150 15 15 30 75 255 15 15 15 60 150 15 15 15 45 255 15 15 30 75 210 15 15 30 90 285 March Forecast Jan Feb Mar Apr May 15 15 15 30 225 15 15 30 30 180 15 15 15 45 210 15 15 15 45 165 15 15 15 30 150 15 15 15 45 210 15 15 15 60 195 15 15 15 60 180 15 15 15 45 195 15 15 15 75 345 Jun Jul Aug Sep Oct Nov Dec 330 270 195 75 60 30 15 465 345 240 75 45 30 15 270 225 135 60 45 30 15 390 300 195 105 30 30 15 330 360 135 75 60 45 30 240 315 225 180 45 30 30 285 210 135 75 30 30 15 270 225 120 150 75 30 30 315 285 135 75 45 30 15 270 195 105 90 90 45 30 210 240 165 75 60 45 30 480 315 180 135 60 30 15 315 300 135 75 45 45 30 345 330 210 120 45 30 15 285 225 120 75 60 30 15 390 240 135 60 45 30 15 315 210 150 75 30 30 30 Jun Jul Aug Sep Oct Nov Dec 330 255 180 75 60 30 15 465 345 240 75 45 30 15 300 270 150 60 45 30 15 390 285 195 90 30 30 15 345 375 150 75 60 45 30 240 330 255 180 45 30 30 345 240 165 75 30 30 15 315 285 135 150 75 45 30 360 360 165 90 45 30 15 315 240 120 90 90 45 30 240 315 195 75 60 45 30 480 315 180 135 60 30 15 330 315 150 75 45 45 30 345 330 195 120 45 30 15 345 300 120 75 60 30 15 420 270 150 60 45 30 15 345 240 150 75 30 30 30 Jun Jul Aug Sep Oct Nov Dec 345 270 195 75 60 30 15 450 330 210 75 45 30 15 300 285 150 60 45 30 15 390 285 195 90 30 30 15 345 390 150 75 60 45 30 240 330 255 180 45 30 30 345 255 180 90 30 30 15 345 300 150 150 75 45 30 345 330 165 90 45 30 15 330 255 120 90 90 45 30 137 Appendix E. Conceptual Hydrologic Forecasts 138 15 15 15 45 255 255 330 210 75 60 45 30 15 15 15 30 135 465 300 180 120 60 30 15 15 15 30 75 255 345 315 150 75 45 60 30 15 15 15 60 150 360 345 210 120 45 30 15 15 15 15 45 255 345 315 135 75 60 30 15 15 15 30 75 210 420 270 150 60 45 30 15 15 15 30 90 285 375 285 165 75 30 30 30 A p r i l Forecast an Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 45 225 330 270 195 75 60 30 15 15 15 15 30 180 420 315 195 75 45 30 15 15 15 15 45 210 300 300 165 60 45 30 15 15 15 15 60 165 375 285 180 90 30 30 15 15 15 15 30 150 345 390 150 75 60 45 30 15 15 15 45 210 240 330 255 180 45 30 30 15 15 15 60 195 360 270 180 90 30 30 15 15 15 15 60 180 345 315 150 150 75 45 30 15 15 15 45 195 345 345 165 90 45 30 15 15 15 15 75 345 330 255 120 90 90 45 30 15 15 15 60 270 270 345 240 90 60 45 30 15 15 15 30 135 465 300 180 120 60 30 15 15 15 15 75 255 360 330 150 75 45 60 30 15 15 15 60 150 375 360 240 120 45 30 15 15 15 15 60 255 360 345 135 90 60 30 15 15 15 15 75 225 435 270 150 60 45 30 15 15 15 15 90 285 375 285 165 75 30 30 30 May Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 45 225 345 300 210 90 60 30 15 15 15 15 45 180 420 315 195 75 45 30 15 15 15 15 45 210 315 315 180 60 45 30 15 15 15 15 45 150 405 300 195 105 30 30 15 15 15 15 45 150 360 420 165 90 60 45 30 15 15 15 45 210 255 345 270 180 45 30 30 15 15 15 45 195 390 300 195 90 30 30 15 15 15 15 45 180 375 345 165 150 75 45 30 15 15 15 45 180 360 375 180 90 45 30 15 15 15 15 45 345 375 285 135 90 90 45 30 15 15 15 45 255 270 345 225 75 60 45 30 15 15 15 45 135 465 315 180 135 60 30 15 15 15 15 45 255 390 360 165 75 45 60 30 15 15 15 45 150 390 375 240 120 45 30 15 15 15 15 45 255 360 330 135 75 60 30 15 15 15 15 45 225 450 285 150 60 45 30 15 15 15 15 45 285 420 300 165 75 30 30 30 June Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 45 180 360 315 225 90 60 30 15 15 15 15 45 180 405 315 210 75 45 30 15 15 15 15 45 180 330 330 180 60 45 30 15 15 15 15 45 180 405 285 180 90 30 30 15 15 15 15 45 180 360 420 165 90 60 45 30 15 15 15 45 180 285 345 270 180 45 30 30 15 15 15 45 180 390 300 195 90 30 30 15 15 15 15 45 180 390 330 150 150 75 45 30 15 15 15 45 180 360 360 165 90 45 30 15 15 15 15 45 180 420 345 150 90 90 45 30 15 15 15 45 180 285 375 255 90 60 45 30 15 15 15 45 180 450 300 180 135 60 30 30 15 15 15 45 180 390 405 210 90 45 60 30 15 15 15 45 180 375 345 225 120 45 30 15 15 15 15 45 180 375 390 150 90 60 30 15 15 15 15 45 180 435 315 180 60 45 30 30 15 15 15 45 180 450 345 180 90 30 30 30 Appendix E. Conceptual Hydrologic Forecasts 139 J u l y Forecast fan Feb Mar Apr May 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 August Forecast an Feb Mar Apr May 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 15 15 15 45 180 E.2 Forecast Year: 1968 January Forecast Jan Feb Mar Apr May 15 15 15 30 225 15 15 30 30 180 15 15 15 45 180 15 15 15 45 165 15 15 15 30 135 15 15 15 45 195 15 15 15 60 165 15 15 15 60 150 15 15 15 45 180 15 15 15 75 315 15 15 15 60 225 15 15 15 30 135 15 15 30 75 240 15 15 30 60 135 15 15 15 45 225 15 15 30 75 195 15 15 30 75 255 February Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 330 330 225 90 60 30 15 330 315 210 75 45 30 15 330 330 195 75 45 30 15 330 300 210 105 30 30 15 330 420 165 90 60 45 30 330 315 240 180 45 30 30 330 330 240 90 30 30 15 330 375 180 165 75 45 30 330 375 195 90 45 30 15 330 345 180 90 90 45 30 330 345 225 75 60 45 30 330 345 255 150 60 30 30 330 420 225 90 45 60 30 330 345 240 120 45 30 15 330 405 165 90 60 30 15 330 330 225 75 45 30 30 330 405 210 90 30 30 30 Jun J u l Aug Sep Oct Nov Dec 330 330 180 90 60 30 15 330 330 165 60 45 30 15 330 330 165 60 45 30 15 330 330 150 90 30 30 15 330 330 165 75 60 45 30 330 330 195 165 45 30 15 330 330 180 75 30 30 15 330 330 165 150 75 45 30 330 330 165 90 45 30 15 330 330 135 90 90 45 30 330 330 180 75 60 45 30 330 330 180 135 60 30 30 330 330 165 75 45 45 30 330 330 210 120 45 30 15 330 330 165 90 60 30 30 330 330 165 60 45 30 15 330 330 165 90 30 30 30 Jun J u l Aug Sep Oct Nov Dec 345 270 195 75 60 30 15 480 360 240 75 45 30 30 285 240 135 60 45 30 15 405 300 210 105 30 30 15 345 375 135 75 60 45 30 240 330 240 180 45 30 30 300 210 135 75 30 30 15 285 240 120 150 75 30 30 315 300 135 75 45 30 15 285 195 105 90 90 45 30 210 255 150 75 60 45 30 480 330 180 135 60 30 30 330 300 135 75 45 45 30 360 345 210 120 45 30 15 300 225 105 75 60 30 15 405 240 135 60 45 30 15 330 210 150 75 30 30 30 Jun J u l Aug Sep Oct Nov Dec Appendix E. Conceptual Hydrologic Forecasts 140 15 15 30 30 180 465 345 225 75 45 30 15 15 15 15 45 195 285 270 135 60 45 30 15 15 15 15 45 165 390 285 180 90 30 30 15 15 15 15 30 135 345 375 135 75 60 45 30 15 15 15 45 195 240 330 240 180 45 30 30 15 15 15 60 180 345 240 150 75 30 30 15 15 15 30 60 165 315 270 135 150 75 45 30 15 15 15 45 180 345 345 165 90 45 30 15 15 15 15 75 345 315 225 105 90 90 45 30 15 15 15 60 240 240 315 180 75 60 45 30 15 15 15 30 135 465 315 180 120 60 30 15 15 15 30 75 240 330 300 135 75 45 45 30 15 15 30 60 135 345 315 195 120 45 30 15 15 15 15 45 240 345 285 120 75 60 30 15 15 15 30 75 195 420 255 150 60 45 30 15 15 15 30 75 270 345 225 150 75 30 30 30 March Forecas t fan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 210 330 240 180 75 60 30 15 15 15 30 30 180 420 300 180 60 45 30 15 15 15 15 45 195 285 255 135 60 45 30 15 15 15 15 45 165 375 270 165 90 30 30 15 15 15 15 30 135 330 345 135 75 60 45 30 15 15 15 45 195 240 315 225 165 45 30 30 15 15 15 60 180 330 240 150 75 30 30 15 15 15 30 75 165 315 270 135 150 75 45 30 15 15 15 45 180 315 300 135 75 45 30 15 15 15 15 75 330 300 225 105 90 90 45 30 15 15 15 60 240 240 315 180 75 60 45 30 15 15 15 30 135 450 270 165 120 60 30 15 15 15 30 75 240 315 285 135 75 45 45 30 15 15 30 60 135 345 315 195 120 45 30 15 15 15 15 45 240 330 270 120 75 60 30 15 15 15 30 75 195 405 240 135 60 45 30 15 15 15 30 75 270 345 240 150 75 30 30 30 A p r i l Forecas t Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 30 45 210 315 240 180 75 60 30 15 15 15 30 30 165 405 285 165 60 45 30 15 15 15 30 45 195 285 270 150 60 45 30 15 15 15 30 60 150 375 270 165 90 30 30 15 15 15 30 30 120 330 360 135 75 60 45 30 15 15 30 45 195 225 315 225 165 45 30 30 15 15 30 60 180 345 240 165 75 30 30 15 15 15 30 60 165 330 285 135 150 75 45 30 15 15 30 45 180 330 315 150 75 45 30 15 15 15 30 75 330 315 225 120 90 90 45 30 15 15 30 60 255 255 345 195 75 60 45 30 15 15 30 30 120 450 270 165 120 60 30 15 15 15 30 75 240 330 315 135 75 45 45 30 15 15 30 60 135 360 345 210 120 45 30 15 15 15 30 60 240 345 315 120 75 60 30 15 15 15 30 75 195 405 240 135 60 45 30 15 15 15 30 90 270 360 255 150 75 30 30 30 May Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 30 45 210 330 270 180 75 60 30 15 15 15 30 45 165 390 285 165 60 45 30 15 15 15 30 45 210 300 285 150 60 45 30 15 15 15 30 45 150 390 285 165 90 30 30 15 15 15 30 45 135 345 390 150 75 60 45 30 15 15 30 45 195 240 330 240 180 45 30 30 Appendix E. Conceptual Hydrologic Forecasts 141 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 June Forecast Jan Feb Mar 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 July Forecast Jan Feb Mar 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 August Forecast Jan Feb Mar 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 15 15 30 45 180 375 270 165 90 30 30 15 45 165 360 315 135 150 75 45 30 45 180 345 345 150 75 45 30 15 45 330 345 255 120 90 90 45 30 45 240 255 330 180 75 60 45 30 45 135 435 285 165 120 60 30 15 45 255 375 330 150 75 45 45 30 45 150 375 360 210 120 45 30 15 45 255 345 285 120 75 60 30 15 45 225 435 255 135 60 45 30 15 45 270 390 270 150 75 30 30 30 Apr May Jun Jul Aug Sep Oct Nov Dec 45 195 315 285 195 75 60 30 15 45 195 360 285 165 60 45 30 15 45 195 300 285 150 60 45 30 15 45 195 360 240 150 90 30 30 15 45 195 105 360 135 75 60 45 30 45 195 255 315 225 165 45 30 30 45 195 360 255 165 75 30 30 15 45 195 120 285 135 150 75 45 30 45 195 330 315 150 75 45 30 15 45 195 375 300 135 90 90 45 30 45 195 240 345 210 75 60 45 30 45 195 405 270 165 120 60 30 30 45 195 360 375 165 75 45 60 30 45 195 330 300 180 120 45 30 15 45 195 345 330 135 75 60 30 15 45 195 390 270 150 60 45 30 30 45 195 405 300 150 75 30 30 30 Apr May Jun Jul Aug Sep Oct Nov Dec 45 195 360 300 195 75 60 30 15 45 195 360 300 180 60 45 30 15 45 195 360 300 150 60 45 30 15 45 195 360 270 165 90 30 30 15 45 195 360 375 135 75 60 45 30 45 195 360 300 195 165 45 30 30 45 195 360 300 180 90 30 30 15 45 195 360 330 150 150 75 45 30 45 195 360 330 165 90 45 30 15 45 195 360 315 135 90 90 45 30 45 195 360 315 180 75 60 45 30 45 195 360 330 210 135 60 30 30 45 195 360 405 180 75 45 60 30 45 195 360 300 195 120 45 30 15 45 195 360 360 135 90 60 30 15 45 195 360 315 180 60 45 30 30 45 195 360 360 180 90 30 30 30 Apr May Jun Jul Aug Sep Oct Nov Dec 45 195 360 345 165 90 60 30 15 45 195 360 345 150 60 45 30 15 45 195 360 345 135 60 45 30 15 45 195 360 345 135 90 30 15 15 45 195 360 345 135 75 60 45 30 45 195 360 345 180 165 45 30 15 45 195 360 345 150 75 30 30 15 45 195 360 345 150 150 75 45 30 45 195 360 345 150 90 45 30 15 45 195 360 345 120 90 90 45 30 45 195 360 345 165 75 60 45 30 Appendix E. Conceptual Hydrologic Forecasts 142 15 15 30 45 195 360 345 165 120 60 30 30 15 15 30 45 195 360 345 135 75 45 45 30 15 15 30 45 195 360 345 180 120 45 30 15 15 15 30 45 195 360 345 150 90 60 30 30 15 15 30 45 195 360 345 150 60 45 30 15 15 15 30 45 195 360 345 135 75 30 30 30 E.3 Forecast Year: 1969 January Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 240 345 300 195 90 60 30 30 15 15 30 30 195 480 360 255 75 45 30 30 15 15 15 45 180 285 255 135 60 45 30 15 30 15 15 60 165 405 315 225 105 45 30 15 15 15 15 30 150 345 405 150 75 60 45 30 15 15 15 45 210 255 330 255 180 45 30 30 15 15 15 60 165 330 225 150 75 30 30 15 15 15 30 75 165 300 255 120 150 75 45 30 15 15 15 45 180 330 315 150 75 45 30 15 15 15 15 75 330 300 210 105 90 90 45 30 15 15 30 60 240 225 270 165 75 60 45 30 15 15 15 30 150 495 345 195 135 60 30 30 15 15 30 75 255 345 315 150 75 45 60 30 15 15 30 60 150 360 360 225 120 45 30 30 15 15 15 45 240 315 255 120 75 60 30 15 15 15 30 75 210 420 255 150 60 45 30 30 15 15 30 90 255 345 240 150 75 30 30 30 Febraury Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 225 330 240 180 75 60 30 30 15 15 30 30 180 450 345 225 75 45 30 30 15 15 15 45 195 285 255 135 60 45 30 15 15 15 15 45 165 375 285 165 90 30 30 15 15 15 15 30 135 330 360 135 75 60 45 30 15 15 15 45 195 240 315 225 180 45 30 30 15 15 15 60 180 330 225 150 75 30 30 15 15 15 30 75 165 315 255 135 150 75 45 30 15 15 15 45 195 330 330 150 90 45 30 15 15 15 15 75 330 300 210 105 90 90 45 30 15 15 30 60 240 240 300 180 75 60 45 30 15 15 15 30 135 465 285 180 120 60 30 30 15 15 30 75 240 315 285 135 75 45 60 30 15 15 30 60 135 330 300 180 120 45 30 15 15 15 15 45 240 330 270 120 75 60 30 15 15 15 30 75 195 405 255 150 60 45 30 30 15 15 30 90 255 330 225 150 75 30 30 30 March Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 210 315 240 165 75 60 30 30 15 15 30 30 180 420 300 165 60 45 30 15 15 15 15 45 195 285 240 135 60 45 30 15 15 15 15 45 165 360 270 150 90 30 30 15 15 15 15 30 135 315 330 135 75 60 45 30 15 15 15 45 195 225 315 210 165 45 30 30 15 15 15 60 180 330 225 150 75 30 30 15 15 15 30 75 165 315 255 135 150 75 45 30 15 15 15 45 180 315 285 135 75 45 30 15 15 15 15 75 330 300 210 105 90 90 45 30 15 15 15 . 60 240 240 300 180 75 60 45 30 15 15 15 30 135 435 255 165 120 60 30 30 15 15 30 75 225 315 285 135 75 45 60 30 Appendix E. Conceptual Hydrologic Forecasts 143 15 15 30 60 135 330 315 180 120 45 30 15 15 15 15 45 240 330 255 120 75 60 30 15 15 15 30 75 195 390 240 135 60 45 30 30 15 15 30 90 270 345 240 150 75 30 30 30 A p r i l Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 210 315 225 165 75 60 30 15 15 15 15 30 165 390 270 150 60 45 30 15 15 15 15 45 195 285 240 135 60 45 30 15 15 15 15 60 150 345 255 150 90 30 30 15 15 15 15 30 135 315 330 135 75 60 45 30 15 15 15 45 180 225 300 210 165 45 30 30 15 15 15 60 180 315 225 150 75 30 30 15 15 15 15 60 165 315 255 135 150 75 45 30 15 15 15 45 180 315 285 135 75 45 30 15 15 15 15 75 315 300 210 105 90 90 45 30 15 15 15 45 240 240 315 180 75 60 45 30 15 15 15 30 135 420 255 165 120 60 30 30 15 15 15 75 225 315 285 135 75 45 60 30 15 15 15 60 135 345 315 195 120 45 30 15 15 15 15 60 240 330 285 120 75 60 30 15 15 15 15 75 195 390 225 135 60 45 30 15 15 15 15 90 270 330 225 150 75 30 30 30 May Forecast Jan Feb Mar Apr May. Jun J u l Aug Sep Oct Nov Dec 15 15 15 75 195 300 210 165 75 60 30 30 15 15 15 75 165 360 240 150 60 45 30 15 15 15 15 75 180 270 240 135 60 45 30 15 15 15 15 75 135 345 240 150 90 30 30 15 15 15 15 75 135 300 330 135 75 60 45 30 15 15 15 75 180 210 300 210 165 45 30 30 15 15 15 75 180 330 225 150 75 30 30 15 15 15 15 75 150 315 255 135 150 75 45 30 15 15 15 75 165 315 285 135 75 45 30 15 15 15 15 75 315 315 210 120 90 90 45 30 15 15 15 75 225 225 270 165 75 60 45 30 15 15 15 75 120 375 240 165 120 60 30 30 15 15 15 75 225 315 285 135 75 45 60 30 15 15 15 75 135 330 315 195 120 45 30 15 15 15 15 75 240 300 240 120 75 60 30 15 15 15 15 75 210 390 210 135 60 45 30 15 15 15 15 75 255 345 225 150 75 30 30 30 June Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 75 240 270 195 150 75 60 30 30 15 15 15 75 240 300 210 135 60 45 30 15 15 15 15 75 240 240 195 120 60 45 30 15 15 15 15 75 240 300 165 120 75 30 30 15 15 15 15 75 240 270 255 120 75 60 45 30 15 15 15 75 240 210 225 180 165 45 30 30 15 15 15 75 240 285 180 135 75 30 30 15 15 15 15 75 240 285 180 120 150 75 45 30 15 15 15 75 240 270 210 120 75 45 30 15 15 15 15 75 240 315 210 105 90 90 45 30 15 15 15 75 240 210 240 165 75 60 45 30 15 15 15 75 240 315 195 150 120 60 30 30 15 15 15 75 240 285 285 135 75 45 60 30 15 15 15 75 240 270 210 165 105 45 30 15 15 15 15 75 240 285 225 120 75 60 30 15 15 15 15 75 240 315 195 135 60 45 30 30 15 15 15 75 240 330 195 150 75 30 30 30 Appendix E. Conceptual Hydrologic Forecasts 144 J u l y Forecast Jan Feb Mar Apr May 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 August Forecast Jan Feb Mar Apr May 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 15 15 15 75 240 E.4 Forecast Year: 1970 January Forecast Jan Feb Mar Apr May 15 15 15 30 225 15 15 30 30 480 15 15 15 45 165 30 15 15 45 165 15 15 15 30 135 15 15 15 45 180 15 15 15 45 150 15 15 30 60 135 15 15 15 45 165 15 15 15 75 285 15 15 15 45 210 15 15 15 30 135 15 15 30 75 225 15 15 30 60 120 15 15 15 45 210 15 15 30 75 180 15 15 30 75 240 Febraury Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 360 150 135 75 60 30 15 360 150 120 60 45 30 15 360 150 105 60 45 30 15 360 150 105 75 30 15 15 360 180 105 75 60 45 30 360 150 150 165 45 30 15 360 150 120 75 30 30 15 360 150 105 150 75 30 30 360 165 105 75 45 30 15 360 165 90 90 75 45 30 360 150 120 75 60 45 30 360 180 150 120 60 30 30 360 240 120 75 45 45 30 360 150 135 105 45 30 15 360 165 105 75 60 30 15 360 165 120 60 45 30 15 360 180 135 75 30 30 30 Jun J u l Aug Sep Oct Nov Dec 360 150 120 75 60 30 15 360 150 120 60 45 30 15 360 150 105 60 45 30 15 360 150 105 75 30 15 15 360 150 105 75 60 45 30 360 150 135 165 45 30 15 360 150 120 75 30 30 15 360 150 105 150 75 45 30 360 150 105 75 45 30 15 360 150 90 90 75 45 30 360 150 120 60 60 45 30 360 150 135 120 60 30 15 360 150 105 75 45 45 30 360 150 150 105 45 30 15 360 150 120 90 60 30 30 360 150 105 60 45 30 15 360 150 105 75 30 30 30 Jun J u l Aug Sep Oct Nov Dec 330 240 180 75 60 30 15 450 330 225 75 45 30 30 255 210 120 60 45 30 15 375 285 180 90 30 30 15 330 330 135 75 60 45 30 225 300 210 180 45 30 30 270 195 135 75 30 30 15 270 210 120 150 75 30 30 300 270 135 75 45 30 15 270 165 105 90 90 45 30 210 225 150 75 60 45 30 465 300 180 120 60 30 30 300 285 135 75 45 60 30 345 300 195 120 45 30 15 270 195 105 75 60 30 15 375 225 135 60 45 30 15 300 195 150 75 30 30 30 Jun J u l Aug Sep Oct Nov Dec Appendix E. Conceptual Hydrologic Forecasts 145 15 15 15 30 210 300 210 165 75 60 30 15 15 15 30 30 180 435 315 195 75 45 30 15 15 15 15 45 165 270 210 120 60 45 30 15 15 15 15 45 150 360 255 150 90 30 30 15 15 15 15 30 135 315 300 135 75 60 45 30 15 15 15 45 180 225 300 210 165 45 30 30 15 15 15 60 165 285 210 135 75 30 30 15 15 15 30 60 150 270 225 120 150 75 45 30 15 15 15 45 180 315 285 135 75 45 30 15 15 15 15 75 300 270 180 105 90 90 45 30 15 15 15 60 225 225 255 165 75 60 45 30 15 15 15 30 135 420 255 165 120 60 30 15 15 15 30 75 210 285 255 135 75 45 45 30 15 15 30 60 120 315 270 165 105 45 30 15 15 15 15 45 225 300 225 120 75 60 30 15 15 15 30 75 180 375 225 135 60 45 30 15 15 15 30 75 240 300 180 150 75 30 30 30 March Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 195 285 195 150 75 60 30 15 15 15 30 30 165 390 255 150 60 45 30 15 15 15 15 45 165 255 210 120 60 45 30 15 15 15 15 45 135 345 225 135 90 30 15 15 15 15 15 30 120 300 270 120 75 60 45 30 15 15 15 45 165 210 270 195 165 45 30 30 15 15 15 60 150 285 195 135 75 30 30 15 15 15 30 60 150 270 210 120 150 75 45 30 15 15 15 45 150 285 240 120 75 45 30 15 15 15 15 75 285 270 180 105 90 90 45 30 15 15 15 60 210 210 240 150 75 60 45 30 15 15 15 30 120 375 225 150 120 60 30 15 15 15 30 60 195 270 240 120 75 45 45 30 15 15 30 60 120 315 255 165 105 45 30 15 15 15 15 45 210 285 210 105 75 60 30 15 15 15 30 75 180 360 195 120 60 45 30 15 15 15 30 75 240 300 195 150 75 30 30 30 A p r i l Forecast an Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 180 270 180 150 75 60 30 15 15 15 15 30 150 345 225 135 60 45 30 15 15 15 15 45 165 240 195 120 60 45 30 15 15 15 15 45 135 315 210 135 75 30 15 15 15 15 15 30 120 285 270 120 75 60 45 30 15 15 15 45 165 210 255 180 165 45 30 30 15 15 15 60 150 270 195 135 75 30 30 15 15 15 15 60 150 270 210 120 150 75 30 30 15 15 15 45 150 285 240 120 75 45 30 15 15 15 15 75 285 255 165 105 90 90 45 30 15 15 15 45 225 225 255 165 75 60 45 30 15 15 15 30 120 360 210 150 120 60 30 15 15 15 15 60 195 270 255 120 75 45 45 30 15 15 15 60 120 315 270 165 105 45 30 15 15 15 15 60 210 300 225 120 75 60 30 15 15 15 15 60 180 345 195 120 60 45 30 15 15 15 15 75 240 300 180 150 75 30 30 30 May Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 180 285 195 150 75 60 30 15 15 15 15 30 135 345 225 135 60 45 30 15 15 15 15 30 165 270 210 120 60 45 30 15 15 15 15 30 120 345 225 135 75 30 15 15 15 15 15 30 120 300 285 120 75 60 45 30 Appendix E. Conceptual Hydrologic Forecasts 146 15 15 15 30 165 210 285 195 165 45 30 30 15 15 15 30 150 315 210 135 75 30 30 15 15 15 15 30 135 300 225 120 150 75 30 30 15 15 15 30 150 300 255 120 75 45 30 15 15 15 15 30 285 285 195 105 90 90 45 30 15 15 15 30 210 225 240 150 75 60 45 30 15 15 15 30 105 360 210 150 120 60 30 15 15 15 15 30 210 300 255 120 75 45 45 30 15 15 15 30 120 330 285 180 105 45 30 15 15 15 15 30 210 285 210 105 75 60 30 15 15 15 15 30 195 375 195 120 60 45 30 15 15 15 15 30 240 330 195 150 75 30 30 30 June Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 135 285 210 165 75 60 30 15 15 15 15 30 135 315 225 150 60 45 30 15 15 15 15 30 135 270 210 120 60 45 30 15 15 15 15 30 135 315 195 120 75 30 15 15 15 15 15 30 135 285 270 120 75 60 45 30 15 15 15 30 135 225 255 180 165 45 30 15 15 15 15 30 135 300 195 135 75 30 30 15 15 15 15 30 135 300 210 120 150 75 30 30 15 15 15 30 135 285 240 120 75 45 30 15 15 15 15 30 135 330 225 105 90 90 45 30 15 15 15 30 135 225 270 165 75 60 45 30 15 15 15 30 135 345 210 150 120 60 30 15 15 15 15 30 135 315 315 135 75 45 45 30 15 15 15 30 135 285 240 165 105 45 30 15 15 15 15 30 135 300 255 120 75 60 30 15 15 15 15 30 135 330 210 135 60 45 30 15 15 15 15 30 135 345 225 150 75 30 30 30 Ju l y Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 135 345 165 135 75 60 30 15 15 15 15 30 135 345 165 120 60 30 30 15 15 15 15 30 135 345 165 105 60 45 30 15 15 15 15 30 135 345 150 105 75 30 15 15 15 15 15 30 135 345 195 105 75 60 45 30 15 15 15 30 135 345 165 150 165 45 30 15 15 15 15 30 135 345 165 120 75 30 30 15 15 15 15 30 135 345 180 105 150 75 30 30 15 15 15 30 135 345 180 105 75 45 30 15 15 15 15 30 135 345 180 90 90 75 45 30 15 15 15 30 135 345 165 120 75 60 45 30 15 15 15 30 135 345 195 150 120 60 30 15 15 15 15 30 135 345 255 120 75 45 45 30 15 15 15 30 135 345 165 150 105 45 30 15 15 15 15 30 135 345 195 105 75 60 30 15 15 15 15 30 135 345 180 120 60 45 30 15 15 15 15 30 135 345 195 135 75 30 30 15 August Forecast Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 15 15 15 30 135 345 195 135 75 60 30 15 15 15 15 30 135 345 195 120 60 30 30 15 15 15 15 30 135 345 195 105 60 45 30 15 15 15 15 30 135 345 195 105 75 30 15 15 15 15 15 30 135 345 195 120 75 60 45 30 15 15 15 30 135 345 195 150 165 45 30 15 15 15 15 30 135 345 195 120 75 30 30 15 15 15 15 30 135 345 195 120 150 75 30 30 15 15 15 30 135 345 195 120 75 45 30 15 15 15 15 30 135 345 195 90 90 75 45 30 Appendix E. Conceptual Hydrologic Forecasts 147 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 30 135 345 30 135 345 30 135 345 30 135 345 30 135 345 30 135 345 30 135 345 195 120 60 195 135 120 195 105 75 195 165 105 195 120 90 195 120 60 195 120 75 60 45 30 60 30 15 45 45 30 45 30 15 60 30 15 45 30 15 30 30 15 Appendix F Simulation Results Notes regarding the results table: • Abbreviations list: Init. Vol. = Volume at beginning of the month, Mm3 Inf. = Reservoir inflow, Mm3 Opt. Dis. = Discharge recommended by operating policy, Mm3 Opt. Spil. = Spill recommended by operating policy, Mm3 Act. Dis. = Actual discharge released during the month, Mm3 Act. Spil. = Actual spill released during the month, Mm3 Ener. Gen. = Energy generated during the month, Gwh • A 0 in the optimal discharge column indicates that during that period the operating policy is undefined, because of an undefined previous inflow. The spill during that period would also be undefined. 1966 250 P 1966 250 D-N 1966 Mon. I n i t V o l . .Inf.. Opt. Dis. Opt. Act. S p i l . D i s . Act.Ener Spil.Gen Jan 255 30 15 0 15 0 1. .9 Feb 270 15 15 0 15 0 1. .9 Mar 270 15 120 0 120 0 14. .0 Apr 165 45 120 0 120 0 12. .1 May 90 195 120 0 120 0 12. .1 Jun 165 315 120 15 120 15 14. .8 J u l 345 300 120 180 120 180 16. .7 Aug 345 165 120 45 120 45 16. .7 Sep 345 90 90 0 90 0 12 .5 Oct 345 60 60 0 60 0 8. .3 Nov 345 45 45 0 45 0 6. .2 Dec 345 30 30 0 30 0 4. .2 Jan 255 30 15 0 15 0 1. .9 Feb 270 15 15 0 15 0 1. .9 Mar 270 15 120 0 120 0 14. .0 Apr 165 45 120 0 120 0 12, .1 May 90 195 120 0 120 0 12, .1 Jun 165 315 120 30 120 15 14, .8 J u l 345 300 120 150 120 180 16, .7 Aug 345 165 120 45 120 45 16 .7 Sep 345 90 90 0 90 0 12 .5 Oct 345 60 45 0 60 0 8. .3 Nov 345 45 30 0 45 0 6, .2 Dec 345 30 15 0 30 0 4, .2 Jan 255 30 15 0 15 0 1. .9 Feb 270 15 15 0 15 0 1. .9 Mar 270 15 105 0 105 0 12 .4 Apr 180 45 105 0 105 0 11 .0 148 Appendix F. Simulation Results 1966 1966 250 D-C 1966 1966 May 120 195 120 0 120 0 12. .7 Jun 195 315 120 60 120 45 15. .1 J u l 345 300 120 150 120 180 16. .7 Aug 345 165 120 40 120 45 16. .7 Sep 345 90 120 .7 120 7 16. .3 Oct 315 60 45 0 45 0 6. .1 Nov 330 45 45 0 45 0 6. ,1 Dec 330 30 30 0 30 0 4. ,1 Jan 255 30 15 0 15 0 1. .9 Feb 270 15 45 0 45 0 5. .6 Mar 240 15 120 0 120 0 13. .4 Apr 135 45 60 0 60 0 6. .1 May 120 195 120 0 120 0 12. .7 Jun 195 315 120 68 120 45 15. .1 J u l 345 300 120 135 120 180 16, .7 Aug 345 165 120 45 120 45 16. .7 Sep 345 90 75 0 90 0 12, .5 Oct 345 60 75 0 75 0 10, .3 Nov 330 45 45 0 45 0 6. .1 Dec 330 30 30 0 30 0 4. .1 Jan 255 30 15 0 15 0 1. .9 Feb 270 15 15 0 15 0 1. .9 Mar 270 15 120 0 120 0 14, .0 Apr 165 45 120 0 120 0 12 .1 May 90 195 120 0 120 0 12 .1 Jun 165 315 120 75 120 15 14, .8 J u l 345 300 120 240 120 180 16, .7 Aug 345 165 120 45 120 45 16, .7 Sep 345 90 90 0 90 0 12 .5 Oct 345 60 45 0 60 0 8. .3 Nov 345 45 30 0 45 0 6. .2 Dec 345 30 30 0 30 0 4, .2 Jan 255 30 15 0 15 0 1. .9 Feb 270 15 15 0 15 0 1, .9 Mar 270 15 120 0 120 0 14 .0 Apr 165 45 105 0 105 0 10 .7 May 105 195 120 0 120 0 12 .4 Jun 180 315 120 98 120 30 15 .0 J u l 345 300 120 235 120 180 16 .7 Aug 345 165 120 50 120 45 16 .7 Sep 345 90 120 5 120 0 16 .3 Oct 315 60 60 0 60 0 8, .0 Nov 315 45 15 0 15 0 2 .0 Dec 345 30 30 0 30 0 4 .2 Jan 255 30 15 0 15 0 1, .9 Feb 270 15 0 0 15 0 1, .9 Mar 270 15 120 0 120 0 14 .0 Apr 165 45 105 0 105 0 10 .7 May 105 195 120 0 120 0 12 .4 Jun 180 315 0 0 120 30 15 .0 J u l 345 300 0 0 120 180 16 .7 Aug 345 165 0 0 120 45 16 .7 Sep 345 90 120 3 120 0 16 .3 Oct 315 60 60 0 60 0 8 .0 Nov 315 45 15 0 15 0 2 .0 Dec 345 30 30 0 30 0 4 .2 Mon. I n i t . Inf. .Opt. Opt. Act. Act. . Ene] Vol. Dis . S p i l D i s . S p i l Gen Jan 465 15 15 0 15 0 2 .4 1966 375 P Appendix F. Simulation Results Feb 465 15 15 0 15 0 2 .4 Mar 465 15 15 0 15 0 2, .4 Apr 465 15 75 0 75 0 11 .5 May 405 120 165 0 165 0 23 .9 Jun 360 345 165 0 165 0 25 .8 J u l 540 210 165 0 165 0 28 .8 Aug 585 120 120 0 120 0 21 .4 Sep 585 60 60 0 60 0 10 .7 Oct 585 45 45 0 45 0 8, .0 Nov 585 30 30 0 30 0 5. .3 Dec 585 15 15 0 15 0 2. .7 Jan 465 15 15 0 15 0 2. .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 75 0 75 0 11 .5 Apr 405 15 165 0 165 0 22 .5 May 255 120 135 0 135 0 16 .5 Jun 240 345 90 0 90 0 12 .8 J u l 495 . 210 165 15 165 0 27 .6 Aug 540 120 120 0 120 0 20 .5 Sep 540 60 45 0 45 0 7, .7 Oct 555 45 15 0 15 0 2, .6 Nov 585 30 30 0 30 0 5, .3 Dec 585 15 15 0 15 0 2 .7 Jan 465 15 15 0 15 0 2 .4 Feb 480 15 15 0 15 0 2 .4 Mar 480 15 60 0 60 0 9 .5 Apr 435 15 165 0 165 0 23 .7 May 315 120 165 0 165 0 22 .5 Jun 345 345 165 9 165 0 25 .0 J u l 495 210 165 31 165 45 28 .2 Aug 585 120 165 13 165 0 29 .4 Sep 585 60 165 2 165 0 28 .4 Oct 510 45 15 0 15 0 2, .5 Nov 555 30 30 0 30 0 5, .2 Dec 570 15 30 0 30 0 5. .3 Jan 465 15 15 0 15 0 2. .4 Feb 480 15 15 0 15 0 2 .4 Mar 480 15 45 0 45 0 7, .1 Apr 450 15 165 0 165 0 24 .1 May 330 120 165 0 165 0 22 .9 Jun 360 345 165 0 165 0 25 .4 J u l 510 210 165 30 165 60 28 .4 Aug 585 120 165 10 165 0 29 .4 Sep 585 60 75 0 90 0 16 .0 Oct 585 45 75 0 75 0 13 .3 Nov 570 30 45 0 45 0 7, .9 Dec 570 15 30 0 30 0 5, .3 Jan 465 15 15 0 15 0 2, .4 Feb 465 15 15 0 15 0 2, .4 Mar 465 15 15 0 15 0 2 .4 Apr 465 15 105 0 105 0 15 .9 May 375 120 150 0 150 0 21 .2 Jun 345 345 135 0 135 0 21 .1 J u l 555 210 150 0 165 15 29 .0 Aug 585 120 120 0 120 0 21 .4 Sep 585 60 90 0 90 0 15 .8 Oct 555 45 15 0 15 0 2 .6 Nov 585 30 30 0 30 0 5. .3 Dec 585 15 15 0 15 0 2 .7 Jan 465 15 15 0 15 0 2 .4 Feb 480 15 15 0 15 0 2. .4 Mar 480 15 135 0 135 0 20 .4 Appendix F. Simulation Results Apr 360 15 165 0 165 0 21.6 May 240 120 165 0 165 0 20.4 Jun 270 345 165 0 165 0 22.9 J u l 420 210 165 28 165 0 26.8 Aug 555 120 165 1 165 0 28.6 Sep 555 60 135 0 135 0 22.9 Oct 510 45 15 0 15 0 2.5 Nov 555 30 15 0 15 0 2.6 Dec 585 15 30 0 30 0 5.3 1966 375 S2-C Jan 465 15 15 0 15 0 2.4 Feb 480 15 0 0 15 0 2.4 Mar 480 15 165 0 165 0 24.6 Apr 330 15 150 0 150 0 19.1 May 225 120 150 0 150 0 18.6 Jun 270 345 0 0 165 0 22.9 J u l 420 210 0 0 165 0 26.8 Aug 555 120 0 0 165 0 28.6 Sep 555 60 120 0 120 0 20.5 Oct 525 45 30 0 30 0 5.1 Nov 555 30 15 0 15 0 2.6 Dec 585 15 30 0 30 0 5.3 1966 500 P 1966 Mon. I n i t . Inf. ,0pt. Opt. Act. Act. Enei Vo l . Di s . S p i l Dis . S p i l Gen. Jan 600 30 15 0 15 0 2.7 Feb 615 15 15 0 15 0 2.7 Mar 615 15 15 0 15 0 2.7 Apr 615 45 165 0 165 0 28.6 May 495 195 180 0 180 0 29.7 Jun 510 315 180 0 180 0 31.8 J u l 645 300 180 0 180 0 35.4 Aug 765 165 165 0 165 0 33.9 Sep 765 90 90 0 90 0 18.5 Oct 765 60 60 0 60 0 12.3 Nov 765 45 45 0 45 0 9.3 Dec 765 30 30 0 30 0 6.2 Jan 600 30 15 0 15 0 2.7 Feb 615 15 15 0 15 0 2.7 Mar 615 15 15 0 15 0 2.7 Apr 615 45 150 0 150 0 26.2 May 510 195 180 0 180 0 30.1 Jun 525 315 180 0 180 0 32.3 J u l 660 300 165 0 180 15 35.6 Aug 765 165 165 0 165 0 33.9 Sep 765 90 90 0 90 0 18.5 Oct 765 60 45 0 60 0 12.3 Nov 765 45 30 0 45 0 9.3 Dec 765 30 , 15 0 30 0 6.2 Jan 600 30 15 0 15 0 2.7 Feb 615 15 15 0 15 0 2.7 Mar 615 15 15 0 15 0 2.7 Apr 615 45 150 0 150 0 26.2 May 510 195 180 0 180 0 30.1 Jun 525 315 180 6 180 0 32.3 J u l 660 300 180 16 180 0 35.6 Aug 765 165 180 8 180 0 36.8 Sep 750 90 165 0 165 0 32.6 Oct 675 60 15 0 15 0 2.9 Nov 720 45 15 0 15 0 3.0 Dec 750 30 30 0 30 0 6.1 Jan 600 30 15 0 15 0 2.7 Appendix F. Simulation Results Feb 615 15 15 0 15 0 2. .7 Mar 615 15 15 0 15 0 2. .7 Apr 615 45 150 0 150 0 26-, .2 May 510 195 180 0 180 0 30. .1 Jun 525 315 180 0 180 0 32, .3 J u l 660 300 180 5 180 15 35, .6 Aug 765 165 180 0 180 0 36. .8 Sep 750 90 60 0 75 0 15, .3 Oct 765 60 75 0 75 0 15, .3 Nov 750 45 45 0 45 0 9. ,2 Dec 750 30 30 0 30 0 6. ,1 Jan 600 30 15 0 15 0 2. ,7 Feb 615 15 15 0 15 0 2. ,7 Mar 615 15 60 0 60 0 10, .8 Apr 570 45 180 0 180 0 29 .7 May 435 195 180 0 180 0 27, .9 Jun 450 315 180 0 180 0 30, .1 J u l 585 300 180 0 180 0 33, .7 Aug 705 165 105 0 105 0 21 .1 Sep 765 90 90 0 90 0 18, .5 Oct 765 60 45 0 60 0 12 .3 Nov 765 45 30 0 45 0 9. .3 Dec 765 30 30 0 30 0 6. .2 Jan 600 30 15 0 15 0 2. .7 Feb 615 15 15 0 15 0 2. .7 Mar 615 15 60 0 60 0 10, .8 Apr 570 45 180 0 180 0 29, .7 May 435 195 180 0 180 0 27 .9 Jun 450 315 180 0 180 0 30 .1 J u l 585 300 180 14 180 0 33 .7 Aug 705 165 150 0 150 0 29 .7 Sep 720 90 120 0 120 0 23 .6 Oct 690 60 15 0 15 0 3, .0 Nov 735 45 15 0 15 0 3, .1 Dec 765 30 30 0 30 0 6. .2 Jan 600 30 15 0 30 0 2 .7 Feb 615 15 0 0 15 0 2 .7 Mar 615 15 105 0 15 0 18 .4 Apr 525 45 165 0 45 0 26 .2 May 405 195 180 0 195 0 27 .0 Jun 420 315 0 0 315 0 29 .2 J u l 555 300 0 0 300 0 32 .9 Aug 675 165 0 0 165 0 23 .5 Sep 720 90 105 0 90 0 20 .8 Oct 705 60 30 0 60 0 6 .0 Nov 735 45 15 0 45 0 3 .1 Dec 765 30 30 0 30 0 6 .2 Mon. I n i t . Inf. .Opt. Opt. Act. Act . Ene] Vol. D i s . S p i l Dis . S p i l Gen, Jan 990 30 15 Feb 1005 15 15 Mar 1005 15 15 Apr 1005 45 30 May 1020 195 195 Jun 1020 315 195 J u l 1140 300 195 Aug 1245 165 165 Sep 1245 90 90 Oct 1245 60 60 Nov 1245 45 45 0 15 0 3. .6 0 15 0 3. ,6 0 15 0 3. .6 0 30 0 7, .2 0 195 0 47. .2 0 195 0 48, .8 0 195 0 51, .7 0 165 0 44, .9 0 90 0 24 .5 0 60 0 16, .3 0 45 0 12, .2 Appendix F. Simulation Results 1966 750 D-N 1966 1966 1966 750 D-C 1966 Dec 1245 30 30 0 30 0 8. 2 Jan 990 30 15 0 15 0 3. 6 Feb 1005 15 15 0 15 0 3. 6 Mar 1005 15 15 0 15 0 3. 6 Apr 1005 45 15 0 15 0 3. ,6 May 1035 195 195 0 195 0 47. .6 Jun 1035 315 195 0 195 0 49. .2 J u l 1155 300 180 0 195 15 51. .9 Aug 1245 165 165 0 165 0 44. .9 Sep 1245 90 90 0 90 0 24. .5 • c t 1245 60 45 0 60 0 16. .3 Nov 1245 45 30 0 45 0 12. .2 Dec 1245 30 15 0 30 0 8. ,2 Jan 990 30 15 0 15 0 3. ,6 Feb 1005 15 15 0 15 0 3. .6 Mar 1005 15 15 0 15 0 3. .6 Apr 1005 45 60 0 60 0 14. .3 May 990 195 195 0 195 0 46. .4 Jun 990 315 195 3 195 0 48. .0 J u l 1110 300 195 4 195 0 50. .9 Aug 1215 165 195 4 195 0 51. .9 Sep 1185 90 120 0 120 0 31. .5 Oct 1155 60 15 0 15 0 3. .9 Nov 1200 45 15 0 15 0 4. .0 Dec 1230 30 30 0 30 0 8. .1 Jan 990 30 15 0 15 0 3. .6 Feb 1005 15 15 0 15 0 3. .6 Mar 1005 15 15 0 15 0 3. .6 Apr 1005 45 75 0 75 0 17. .9 May 975 195 195 0 195 0 46. .0 Jun 975 315 165 0 165 0 40 .6 J u l 1125 300 180 0 180 0 47, .6 Aug 1245 165 180 0 180 0 48 .8 Sep 1230 90 60 0 75 0 20, .3 Oct 1245 60 75 0 75 0 20, .3 Nov 1230 45 45 0 45 0 12, .1 Dec 1230 30 30 0 30 0 8. .1 Jan 990 30 15 0 15 0 3. .6 Feb 1005 15 15 0 15 0 3. .6 Mar 1005 15 15 0 15 0 3, .6 Apr 1005 45 120 0 120 0 28 .2 May 930 195 195 0 195 0 44 .8 Jun 930 315 195 0 195 0 46 .4 J u l 1050 300 180 0 180 0 45 .8 Aug 1170 165 90 0 90 0 24 .0 Sep 1245 90 90 0 90 0 24 .5 Oct 1245 60 45 0 60 0 16 .3 Nov 1245 45 30 0 45 0 12 .2 Dec 1245 30 30 0 30 0 8 .2 Jan 990 30 15 0 15 0 3, .6 Feb 1005 15 15 0 15 0 3 .6 Mar 1005 15 15 0 15 0 3 .6 Apr 1005 45 150 0 150 0 34 .9 May 900 195 195 0 195 0 43 .9 Jun 900 315 195 0 195 0 45 .6 J u l 1020 300 195 0 195 0 48 .6 Aug 1125 165 90 0 90 0 23 .5 Sep 1200 90 120 0 120 0 31 .7 Oct 1170 60 15 0 15 0 4 .0 Nov 1215 45 15 0 15 0 4 .0 Dec 1245 30 30 0 30 0 8 .2 Jan 990 30 15 0 15 0 3 .6 Appendix F. Simulation Results Year S i z e Case Mon. Feb 1005 15 0 0 15 0 3 6 Mar 1005 15 45 0 45 0 10 7 Apr 975 45 150 0 150 0 34 3 May 870 195 195 0 195 0 43 1 Jun 870 315 0 0 195 0 44 8 J u l 990 300 0 0 165 0 40 8 Aug 1125 165 0 0 90 0 23 5 Sep 1200 90 105 0 105 0 27 8 Oct 1185 60 30 0 30 0 8 0 Nov 1215 45 15 0 15 0 4 0 Dec 1245 30 30 0 30 0 8 2 n. I n i t . Inf. Opt. Opt. Act. Act Ener Vol. D i s . S p i l . D i s . Spil.Gen. Jan 1440 30 15 0 15 0 4 4 Feb 1455 15 15 0 15 0 4 5 Mar 1455 15 15 0 15 0 4 5 Apr 1455 45 15 0 15 0 4 5 May 1485 195 90 0 90 0 27 6 Jun 1590 315 210 0 210 0 66 9 J u l 1695 300 210 0 210 0 69 0 Aug 1785 165 165 0 165 0 55 0 Sep 1785 90 90 0 90 0 30 0 Oct 1785 60 60 0 60 0 20 0 Nov 1785 45 45 0 45 0 15 0 Dec 1785 30 30 0 30 0 10 0 Jan 1440 30 15 0 15 0 4 4 Feb 1455 15 15 0 15 0 4 5 Mar 1455 15 15 0 15 0 4 5 Apr 1455 45 15 0 15 0 4 5 May 1485 195 75 0 75 0 23 1 Jun 1605 315 210 0 210 0 67 2 J u l 1710 300 195 0 210 15 69 2 Aug 1785 165 165 0 165 0 55 0 Sep 1785 90 90 0 90 0 30 0 Oct 1785 60 45 0 60 0 20 0 Nov 1785 45 30 0 45 0 15 0 Dec 1785 30 15 0 30 0 10 0 Jan 1440 30 15 0 15 0 4 4 Feb 1455 15 156 0 156 0 4 5 Mar 1455 15 15 0 15 0 4 5 Apr 1455 45 15 0 15 0 4 5 May 1485 195 165 0 165 0 49 9 Jun 1515 315 210 0 210 0 65 .1 J u l 1620 300 210 0 210 0 67 .4 Aug 1710 165 210 1 210 0 67 .9 Sep 1665 90 60 0 60 0 19 .3 Oct 1695 60 15 0 15 0 4 9 Nov 1740 45 15 0 15 0 5 0 Dec 1770 30 30 0 30 0 10 .0 Jan 1440 30 15 0 15 0 4 4 Feb 1455 15 15 0 15 0 4 5 Mar 1455 15 15 0 15 0 4 5 Apr 1455 45 15 0 15 0 4 5 May 1485 195 195 0 195 0 58 .7 Jun 1485 315 120 0 120 0 37 .4 J u l 1680 300 195 0 195 0 63 .9 Aug 1785 165 180 0 180 0 59 .8 Sep 1770 90 60 0 75 0 24 .9 Oct 1785 60 75 0 75 0 24 .9 Nov 1770 45 45 0 45 0 14 .9 1966 1000 P 1966 1000 D-N Appendix F. Simulation Results 1966 1000 D-C 1966 1000 Sl-C Dec 1770 30 30 0 30 0 10. .0 Jan 1440 30 15 0 15 0 4. .4 Feb 1455 15 15 0 15 0 4. .5 Mar 1455 15 15 0 15 0 4. .5 Apr 1455 45 15 0 15 0 4. .5 May 1485 195 195 0 195 0 58, .7 Jun 1485 315 210 0 210 0 64. .4 J u l 1590 300 165 0 165 0 52. .8 Aug 1725 165 105 0 105 0 34, .7 Sep 1785 90 90 0 90 0 30. .0 Oct 1785 60 45 0 60 0 20, .0 Nov 1785 45 30 0 45 0 15. .0 Dec 1785 30 30 0 30 0 10. .0 Jan 1440 30 15 0 15 0 4. .4 Feb 1455 15 15 0 15 0 4. .5 Mar 1455 15 15 0 15 0 4. .5 Apr 1455 45 45 0 45 0 13. .4 May 1455 195 210 0 210 0 62, .3 Jun 1440 315 210 0 210 0 63. .4 J u l 1545 300 180 0 180 0 56, .6 Aug 1665 165 90 0 90 0 29, .2 Sep 1740 90 120 0 120 0 39, .3 Oct 1710 60 15 0 .15 0 4. .9 Nov 1755 45 15 0 15 0 5. .0 Dec 1785 30 30 0 30 0 10, .0 Jan 1440 30 15 0 15 0 4. .4 Feb 1455 15 0 0 15 0 4. .5 Mar 1455 15 15 0 15 0 4. .5 Apr 1455 45 75 0 75 0 22, .2 May 1425 195 210 0 210 0 61, .5 Jun 1410 315 0 0 210 0 62 .6 J u l 1515 300 0 0 150 0 46, .9 Aug 1665 165 0 0 90 0 29 .2 Sep 1740 90 105 0 105 0 34 .4 Oct 1725 60 30 0 30 0 9. .9 Nov 1755 45 15 0 15 0 5. .0 Dec 1785 30 30 0 30 0 10, .0 Mon. , I n i t . Inf. .Opt. Opt. Act. Act, . Ener Vol. D i s . S p i l . D i s . Spil.Gen. Jan 255 15 15 0 15 0 1, .9 Feb 255 15 15 0 15 0 1, .9 Mar 255 15 90 0 90 0 10 .5 Apr 180 30 120 0 120 0 12 .3 May 90 180 120 0 120 0 11 .9 Jun 150 375 120 60 120 60 14 .6 J u l 345 375 120 255 120 255 16 .7 Aug 345 165 120 45 120 45 16 .7 Sep 345 120 120 0 120 0 16 .7 Oct 345 75 75 0 75 0 10 .4 Nov 345 45 45 0 45 0 6 .2 Dec 345 30 30 0 30 0 4 .2 Jan 255 15 15 0 15 0 1 .9 Feb 255 15 15 0 15 0 1 .9 Mar 255 15 105 0 105 0 12 .1 Apr 165 30 120 0 120 0 10 .6 May 90 180 120 0 120 0 11 .9 Jun 150 375 120 15 120 60 14 .6 J u l 345 375 120 150 120 255 16 .7 Aug 345 165 120 45 120 45 16 .7 Sep 345 120 90 0 120 0 16 .7 1968 250 P 1968 250 D-N Appendix F. Simulation Results 1968 1968 1968 250 D-C 1968 1968 Oct 345 75 45 0 75 0 10, .4 Nov 345 45 30 0 45 0 6, .2 Dec 345 30 15 0 30 0 4, .2 Jan 255 15 15 0 15 0 1, .9 Feb 255 15 15 0 15 0 1, .9 Mar 255 15 90 0 90 0 10, .5 Apr 180 30 105 0 105 0 10, .9 May 105 180 120 0 120 0 12, .3 Jun 165 375 120 39 120 75 14, .8 J u l 345 375 120 150 120 255 16, .7 Aug 345 165 120 40 120 45 16. .7 Sep 345 120 120 7 120 0 16, .7 Oct 345 75 75 0 75 0 10, .4 Nov 345 45 60 0 60 0 8. .3 Dec 330 30 30 0 30 0 4. .1 Jan 255 15 15 0 15 0 1. .9 Feb 255 15 30 0 30 0 3. .7 Mar 240 15 120 0 120 0 13. .4 Apr 135 30 60 0 60 0 6. .0 May 105 180 120 0 120 0 12. .3 Jun 165 375 0 0 120 75 14, .8 J u l 345 375 120 135 120 255 16, .7 Aug 345 165 120 180 120 45 16, T . 1 Sep 345 120 75 0 120 0 16, .7 Oct 345 75 60 0 75 0 10, A Nov 345 45 45 0 45 0 6. .2 Dec 345 30 45 0 30 0 6. .2 Jan 255 15 15 0 15 0 1. .9 Feb 255 15 15 0 15 0 1. .9 Mar 255 15 105 0 105 0 12 .1 Apr 165 30 120 0 105 0 10 .6 May 90 180 120 0 120 0 11, .9 Jun 150 375 120 0 120 60 14, .6 J u l 345 375 120 180 120 255 16 .7 Aug 345 165 120 30 120 45 16 .7 Sep 345 120 90 0 120 0 16, .7 Oct 345 75 45 0 75 0 10 .4 Nov 345 45 30 0 45 0 6. .2 Dec 345 30 30 0 30 0 4. .2 Jan 255 15 15 0 15 0 1. .9 Feb 255 15 15 0 15 0 1, .9 Mar 255 15 105 0 105 0 12 .1 Apr 165 30 105 0 105 0 10 .6 May 90 180 120 0 120 0 11 .9 Jun 150 375 120 0 120 0 14 .6 J u l 345 375 120 34 120 60 16 .7 Aug 345 165 120 204 120 255 16 .7 Sep 345 120 120 30 120 45 16 .7 Oct 345 75 90 4 90 0 12 .4 Nov 345 45 30 0 30 0 4, .1 Dec 345 30 30 0 30 0 4, .2 Jan 255 15 15 0 15 0 1, .9 Feb 255 15 15 0 15 0 1 .9 Mar 255 15 105 0 105 0 12 .1 Apr 165 30 0 0 105 0 10 .6 May 90 180 0 0 120 . 0 11 .9 Jun 150 375 0 0 120 60 14 .6 J u l 345 375 0 0 120 255 16 .7 Aug 345 165 0 0 120 45 16 .7 Sep 345 120 120 0 120 0 16 .7 Oct 345 75 60 0 75 0 10 .4 Nov 345 45 45 0 45 0 6 .2 Appendix F. Simulation Results 1968 375 P 1968 375 D-N 1968 1968 1968 375 D-C Dec 345 30 30 0 30 0 4. .2 Mon. I n i t . Inf. Opt. Opt. Act. Act. Enei Vol. D i s . S p i l . Dis. Spil.Gen Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2. .4 Mar 465 15 135 0 135 0 20 .1 Apr 345 30 165 0 165 0 21 .0 May 210 180 165 0 165 0 19 .3 Jun 225 375 165 0 165 0 22 .5 J u l 435 375 165 60 165 60 27 .4 Aug 585 165 165 0 165 0 29 .4 Sep 585 120 120 0 120 0 21 .4 Oct 585 75 75 0 75 0 13 .4 Nov 585 45 45 0 45 0 8, .0 Dec 585 30 30 0 30 0 5. .3 Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 75 0 75 0 11 .5 Apr 405 30 165 0 165 0 22 .7 May 270 180 150 0 150 0 19 .3 Jun 300 375 150 0 150 0 22 .5 J u l 525 375 165 45 165 150 28 .6 Aug 585 165 165 0 165 0 29 .4 Sep 585 120 90 0 120 0 21 .4 Oct 585 75 45 0 75 0 13 .4 Nov 585 45 30 0 45 0 8 .0 Dec 585 30 15 0 30 0 5 .3 Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 45 0 45 0 7. .0 Apr 435 30 165 0 165 0 23 .5 May 300 180 165 0 165 0 21 .8 Jun 315 375 165 4 165 0 25 .0 J u l 525 375 165 51 165 150 28 .6 Aug 585 165 165 13 165 0 29 .4 Sep 585 120 165 2 165 0 28 .8 Oct 540 75 30 0 30 0 5 .2 Nov 585 45 60 0 60 0 10 .6 Dec 570 30 30 0 30 0 5 .3 Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 30 0 30 0 4 .7 Apr 450 30 165 0 165 0 23 .9 May 315 180 165 0 165 0 22 .3 Jun 330 375 0 0 135 0 21 .1 J u l 570 375 165 75 165 195 29 .2 Aug 585 165 165 135 165 0 29 .4 Sep 585 120 75 0 120 0 21 .4 Oct 585 75 60 0 75 0 13 .4 Nov 585 45 45 0 45 0 8 .0 Dec 585 30 45 0 45 0 8 .0 Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 90 0 90 0 13 .7 Apr 390 30 165 0 165 0 22 .3 May 255 180 165 0 165 0 20 .6 Jun 270 375 135 0 135 0 19 .8 J u l 510 375 165 60 165 135 28 .4 Aug 585 165 150 0 165 0 29 .4 Sep 585 120 90 0 120 0 21 .4 Appendix F. Simulation Results Oct 585 75 45 0 75 0 13.4 Nov 585 45 30 0 45 0 8.0 Dec 585 30 30 0 30 0 5.3 1968 375 Sl-C Jan 465 15 15 0 15 0 2.4 Feb 465 15 15 0 15 0 2.4 Mar 465 15 75 0 75 0 11.5 Apr 405 30 165 0 165 0 22.7 May 270 180 165 0 165 0 21.0 Jun 285 375 165 0 165 0 24.1 J u l 495 375 165 69 165 120 28.2 Aug 585 165 165 2 165 0 29.4 Sep 585 120 165 0 165 0 28.8 Oct 540 75 45 0 45 0 7.8 Nov 570 45 30 0 30 0 5.3 Dec 585 30 30 0 30 0 5.3 1968 375 S2-C Jan 465 15 15 0 15 0 2.4 Feb 465 15 15 0 15 0 2.4 Mar 465 15 105 0 105 0 15.9 Apr 375 30 0 0 165 0 21.8 May 240 180 0 0 165 0 20.1 Jun 255 375 0 0 150 0 21.4 J u l 480 375 0 0 165 0 28.0 Aug 585 165 0 0 165 105 29.4 Sep 585 120 120 0 120 0 21.4 Oct 585 75 60 0 75 0 13.4 Nov 585 45 45 0 45 0 8.0 Dec 585 30 30 0 30 0 5.3 Year Size Case Mon. I n i t . Inf. .Opt. Opt. Act. Act. . Enei Vol. Di s . S p i l . D i s . Spil.Gen 1968 500 P Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 90 0 90 0 15.7 Apr 525 30 180 0 180 0 28.1 May 375 180 180 0 180 0 25.9 Jun 375 375 180 0 180 0 28.8 J u l 570 375 180 0 180 0 34.4 Aug 765 165 165 0 165 0 33.9 Sep 765 120 120 0 120 0 24.7 Oct 765 75 75 0 75 0 15.4 Nov 765 45 45 0 45 0 9.3 Dec 765 30 30 0 30 0 6.2 1968 500 D-N Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 15 0 15 0 2.7 Apr 600 30 135 0 135 0 23.2 May 495 180 165 0 165 0 27.2 Jun 510 375 165 0 165 0 30.2 J u l 720 375 180 45 180 150 36.4 Aug 765 165 165 0 165 0 33.9 Sep 765 120 90 0 120 0 24.7 Oct 765 75 45 0 75 0 15.4 Nov 765 45 30 0 45 0 9.3 Dec 765 30 15 0 30 0 6.2 1968 500 Sl-N Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 120 0 120 0 2.7 Apr 600 30 180 0 180 0 20.8 May 510 180 180 0 180 0 29.9 Jun 510 375 180 4 180 0 32.7 J u l 705 375 180 41 180 135 36.2 Appendix F. Simulation Results Aug 765 165 180 8 180 0 36.8 Sep 750 120 165 0 165 0 33.0 Oct 705 75 15 0 15 0 3.0 Nov 765 45 60 0 60 0 12.3 Dec 750 30 30 0 30 0 6.1 1968 500 S2-N Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 15 0 15 0 2.7 Apr 600 30 135 0 135 0 23.2 May 495 180 180 0 180 0 29.5 Jun 495 375 0 0 120 0 22.1 J u l 750 375 180 60 180 180 36.8 Aug 765 165 180 120 180 0 36.8 Sep 750 120 60 0 105 0 21.5 Oct 765 75 60 0 75 0 15.4 Nov 765 45 45 0 45 0 9.3 Dec 765 30 45 0 45 0 9.2 1968 500 D-C Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 15 0 15 0 2.7 Apr 600 30 180 0 180 0 30.3 May 450 180 180 0 180 0 28.1 Jun 450 375 120 0 120 0 21.2 J u l 705 375 180 60 180 135 36.2 Aug 765 165 150 0 165 0 33.9 Sep 765 120 90 0 120 0 24.7 Oct 765 75 45 0 75 0 15.4 Nov 765 45 30 0 45 0 9.3 Dec 765 30 30 0 30 0 6.2 1968 500 Sl-C Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 15 0 15 0 2.7 Apr 600 30 165 0 165 0 28.0 May 465 180 180 0 180 0 28.6 Jun 465 375 180 0 180 0 31.4 J u l 660 375 180 40 180 90 35.6 Aug 765 165 180 0 180 0 36.8 Sep 750 120 150 0 150 0 30.2 Oct 720 75 45 0 45 0 9.1 Nov 750 45 30 0 30 0 6.1 Dec 765 30 30 0 30 0 6.2 1968 500 S2-C Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.9 Mar 600 15 30 0 30 0 5.4 Apr 585 30 0 0 180 0 29.9 May 435 180 0 0 180 0 27.7 Jun 435 375 0 0 165 0 28.2 J u l 645 375 0 0 180 75 35.4 Aug 765 165 0 0 165 0 33.9 Sep 765 120 120 0 120 0 24.7 Oct 765 75 60 0 75 0 15.4 Nov 765 45 145 0 45 0 9.3 Dec 765 30 30 0 30 0 6.2 Year Si z e Case Mon. , I n i t . Inf. .Opt. Opt. Act. Act . Enei Vol. Di s . S p i l . Dis . Spil.Gen. 1968 750 P Jan 990 15 15 0 15 0 3.6 Feb 990 15 15 0 15 0 3.6 Mar 990 15 15 0 15 0 3.6 Apr 990 30 120 0 120 0 27.8 May 900 180 195 0 195 0 43.7 endix F. Simulation Results 1968 750 D-N 1968 750 1968 1968 750 D-C 1968 750 Jun 885 375 195 0 195 0 46. ,0 J u l 1065 375 195 0 195 0 50. ,8 Aug 1245 165 165 0 165 0 44. ,9 Sep 1245 120 120 0 120 0 32. .6 Oct 1245 75 75 0 75 0 20. .4 Nov 1245 45 45 0 45 0 12. .2 Dec 1245 30 30 0 30 0 8. 2 Jan 990 15 15 0 15 0 3. ,6 Feb 990 15 15 0 15 0 3. 6 Mar 990 15 15 0 15 0 3. 6 Apr 990 30 15 0 15 0 3. ,6 May 1005 180 165 0 165 0 39. ,8 Jun 1020 375 180 0 180 0 46. ,0 J u l 1215 375 195 45 195 150 52. .6 Aug 1245 165 165 0 165 0 44. ,9 Sep 1245 120 90 0 120 0 32. .6 Oct 1245 75 45 0 75 0 20. .4 Nov 1245 45 30 0 45 0 12. .2 Dec 1245 30 15 0 30 0 8. .2 Jan 990 15 15 0 15 0 3. ,6 Feb 990 15 15 0 15 0 3. .6 Mar 990 15 15 0 15 0 3. .6 Apr 990 30 45 0 45 0 10. .7 May 975 180 195 0 195 0 45. .8 Jun 960 375 195 0 195 0 48. .0 J u l 1140 375 195 10 195 75 51. .7 Aug 1245 165 195 6 195 0 52, .6 Sep 1215 120 150 0 150 0 39. .9 Oct 1185 75 15 0 15 0 4. .0 Nov 1245 45 60 0 60 0 16. .3 Dec 1230 30 30 0 30 0 8. .1 Jan 990 15 15 0 15 0 3. .6 Feb 990 15 15 0 15 0 3. .6 Mar 990 15 15 0 15 0 3. .6 Apr 990 30 60 0 60 0 14, .2 May 960 180 195 0 195 0 45, .4 Jun 945 375 0 0 75 0 18 .9 J u l 1245 375 195 60 195 180 53 .0 Aug 1245 165 195 105 195 0 52 .6 Sep 1245 120 45 0 90 0 24 .3 Oct 1245 75 60 0 75 0 20 .4 Nov 1245 45 45 0 45 0 12 .2 Dec 1245 30 45 0 45 0 12 .2 Jan 990 15 15 0 15 0 3 .6 Feb 990 15 15 0 15 0 3 .6 Mar 990 15 15 0 15 0 3 .6 Apr 990 30 45 0 45 0 10 .9 May 975 180 195 0 195 0 45 .8 Jun 960 375 135 0 135 0 33 .8 J u l 1200 375 195 60 195 135 52 .5 Aug 1245 165 150 0 165 0 44 .9 Sep 1245 120 90 0 120 0 32 .6 Oct 1245 75 45 0 75 0 20 .4 Nov 1245 45 30 0 45 0 12 .2 Dec 1245 30 30 0 30 0 8 .2 Jan 990 15 15 0 15 0 3 .6 Feb 990 15 15 0 15 0 3 .6 Mar 990 15 15 0 15 0 3 .6 Apr 990 30 75 0 75 0 17 .6 May 945 180 195 0 195 0 45 .0 Jun 930 375 180 0 180 0 43 .8 J u l 1125 375 195 17 195 60 51 .5 Appendix F. Simulation Results 1968 750 S2-C Aug 1245 165 180 0 180 0 48 .8 Sep 1230 120 150 0 150 0 40 .2 Oct 1200 75 45 0 45 0 12 .1 Nov 1230 45 30 0 30 0 8, .1 Dec 1245 30 30 0 30 0 8, .2 Jan 990 15 15 0 15 0 3, .6 Feb 990 15 15 0 15 0 3. .6 Mar 990 15 15 0 15 0 3. .6 Apr 990 30 0 0 120 0 27 .8 May 900 180 0 0 195 75 43 .7 Jun 885 375 0 0 120 0 28 .9 J u l 1140 375 0 0 195 0 51 .7 Aug 1245 165 0 0 165 0 44 .9 Sep 1245 120 120 0 120 0 32 .6 Oct 1245 75 60 0 75 0 20 .4 Nov 1245 45 45 0 45 0 12 .2 Dec 1245 30 30 0 30 0 8. .2 Mon. . I n i t . Inf. .Opt. Opt. Act. Act . Ener Vol. D i s . S p i l . Dis . Spil.Gen. Jan 1440 15 15 0 15 0 4, .4 Feb 1440 15 15 0 15 0 4, .4 Mar 1440 15 15 0 15 0 4, .4 Apr 1440 30 15 0 15 0 4, .4 May 1455 180 180 0 180 0 53 .5 Jun 1455 375 210 0 210 0 64 .4 J u l 1620 375 210 0 210 0 68 .2 Aug 1785 165 165 0 165 0 55 .0 Sep 1785 120 120 0 120 0 40 .0 Oct 1785 75 75 0 75 0 25 .0 Nov 1785 45 45 0 45 0 15 .0 Dec 1785 30 30 0 30 0 10 .0 Jan 1440 15 15 0 15 0 4 .4 Feb 1440 15 15 0 15 0 4 .4 Mar 1440 15 15 0 15 0 4. .4 Apr 1440 30 15 0 15 0 4, .4 May 1455 180 45 0 45 0 13 .7 Jun 1590 375 195 0 195 0 62 .9 J u l 1770 375 210 45 210 150 69 .8 Aug 1785 165 165 0 165 0 55 .0 Sep 1785 120 90 0 120 0 40 .0 Oct 1785 75 45 0 75 0 25 .0 Nov 1785 45 30 0 45 0 15 .0 Dec 1785 30 15 0 30 0 10 .0 Jan 1440 15 15 0 15 0 4 .4 Feb 1440 15 15 0 15 0 4 .4 Mar 1440 15 15 0 15 0 4. .4 Apr 1440 30 15 0 15 0 4, .4 May 1455 180 135 0 135 0 40 .5 Jun 1500 375 210 0 210 0 65 .5 J u l 1665 375 210 4 210 45 68 .7 Aug 1785 165 210 5 210 0 69 .5 Sep 1740 120 135 0 135 0 44 .3 Oct 1725 75 15 0 15 0 5 .0 Nov 1785 45 60 0 60 0 19 .9 Dec 1770 30 30 0 30 0 10 .0 Jan 1440 15 15 0 15 0 4. .4 Feb 1440 15 15 0 15 0 4 .4 Mar 1440 15 15 0 15 0 4 .4 Apr 1440 30 15 0 15 0 4 .4 May 1455 180 195 0 195 0 57 .8 1968 1000 P 1968 1000 D-N Appendix F. Simulation Results 1968 1000 D-C Jun 1440 375 0 0 45 0 14 1 J u l 1770 375 210 30 210 150 69 8 Aug 1785 165 210 90 210 0 69 5 Sep 1740 120 30 0 75 0 24 8 Oct 1785 75 60 0 75 0 25 0 Nov 1785 45 45 0 45 0 15 0 Dec 1785 30 45 0 45 0 15 0 Jan 1440 15 15 0 15 0 4 4 Feb 1440 15 15 0 15 0 4 4 Mar 1440 15 15 0 15 0 4 4 Apr 1440 30 15 0 15 0 4 4 May 1455 180 120 0 120 0 36 1 Jun 1515 375 135 0 135 0 42 9 J u l 1755 375 210 60 210 135 69 7 Aug 1785 165 150 0 150 0 55 0 Sep 1785 120 90 0 90 0 40 0 Oct 1785 75 45 0 45 0 25 0 Nov 1785 45 30 0 30 0 15 0 Dec 1785 30 30 0 30 0 10 0 Jan 1440 15 15 0 15 0 4 4 Feb 1440 15 15 0 15 0 4 4 Mar 1440 15 15 0 15 0 4 4 Apr 1440 30 15 0 15 0 4 4 May 1455 180 195 0 195 0 57 .8 Jun 1440 375 150 0 150 0 46 3 J u l 1665 375 210 11 210 45 68 7 Aug 1785 165 180 0 180 0 59 8 Sep 1770 120 150 0 150 0 49 5 Oct 1740 75 45 0 45 0 14 9 Nov 1770 45 30 0 30 0 10 0 Dec 1785 30 30 0 30 0 10 0 Jan 1440 15 15 0 15 0 4 4 Feb 1440 15 15 0 15 0 4 4 Mar 1440 15 15 0 15 0 4 4 Apr 1440 30 0 0 15 0 4 4 May 1455 180 0 0 195 0 57 .8 Jun 1440 375 0 0 150 0 46 3 J u l 1665 375 0 0 210 45 68 7 Aug 1785 165 0 0 165 0 55 0 Sep 1785 120 120 0 120 0 40 0 Oct 1785 75 60 0 75 0 25 0 Nov 1785 45 45 0 45 0 15 0 Dec 1785 30 30 0 30 0 10 0 Mon I n i t . Inf Opt. Opt. Act. Act Ener. Vol. Di s . S p i l . Dis. Spil.Gen. Jan 255 15 15 0 15 0 1 9 Feb 255 15 15 0 15 0 1 9 Mar 255 15 120 0 120 0 13 7 Apr 150 60 120 0 120 0 11 9 May 90 225 120 0 120 0 12 .4 Jun 195 345 120 75 120 75 15 1 J u l 345 180 120 0 120 0 16 7 Aug 345 120 120 0 120 0 16 7 Sep 345 75 75 0 75 0 10 4 Oct 345 60 60 0 60 0 8 3 Nov 345 45 45 0 45 0 6 2 Dec 345 30 30 0 30 0 4 2 Jan 255 15 15 0 15 0 1 9 Feb 255 15 15 0 15 0 1 9 Mar 255 15 105 0 105 0 12 1 1969 250 P endix F. Simulation Results Apr May Jun J u l Aug Sep Oct Nov Dec 1969 250 Sl-N Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 1969 250 S2-N Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 1969 250 D-C Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 1969 250 Sl-C Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 1969 250 S2-C Jan Feb Mar Apr May 165 60 120 0 120 0 12 .3 105 225 120 0 120 0 12 .7 210 345 120 75 120 90 15, .3 345 180 120 150 120 60 16 .7 345 120 120 45 120 0 16, .7 345 75 90 0 90 0 12 .4 345 60 30 0 45 0 6. .2 345 45 30 0 45 0 6, .2 345 30 15 0 30 0 4, .2 255 15 15 0 15 0 1, .9 255 15 5 0 5 0 1. .9 255 15 90 0 90 0 10 .5 180 60 105 0 105 0 11 .2 135 225 120 0 120 0 13 .4 240 345 120 102 120 120 15, .6 345 180 120 150 120 60 16 .7 345 120 120 40 120 0 16 .7 345 75 120 7 120 0 16 .2 300 60 30 0 30 0 4. .0 330 45 45 0 45 0 6. .1 330 30 30 0 30 0 4, .1 255 15 15 0 15 0 1, .9 255 15 30 0 30 0 3, .7 240 15 120 0 120 0 13 .4 135 60 60 0 60 0 6 .1 135 225 120 0 120 0 13 .4 240 345 120 109 120 120 15 .6 345 180 0 0 120 60 16 .7 345 120 0 0 120 0 16 .7 345 75 75 0 75 0 10 .4 345 60 60 0 60 0 8 .3 345 45 60 0 60 0 8 .3 330 30 30 0 30 0 4 .1 255 15 15 0 15 0 1 .9 255 15 15 0 15 0 1 .9 255 15 105 0 105 0 12 .1 165 60 120 0 120 0 12 .3 105 225 120 0 120 0 12 .7 210 345 120 135 120 90 15 .3 345 180 120 45 120 60 16 .7 345 120 120 0 120 0 16 .7 345 75 90 0 90 0 12 .4 330 60 30 0 45 0 6 .2 345 45 30 0 45 0 6 .2 345 30 15 0 30 0 4 .2 255 15 15 0 15 0 1 .9 255 15 15 0 15 0 1 .9 255 15 105 0 105 0 12 .1 165 60 105 0 105 0 10 .9 120 225 120 0 120 0 13 .1 225 345 120 43 120 105 15 .4 345 180 120 44 120 60 16 .7 345 120 120 4 120 0 16 .7 345 75 120 4 120 0 16 .2 300 60 30 0 30 0 4 .0 330 45 30 0 30 0 4 . 1 345 30 30 0 30 0 4 .2 255 15 15 0 15 0 1 .9 255 15 15 0 15 0 1 .9 255 15 105 0 105 0 12 .1 165 60 105 0 105 0 10 .9 120 225 0 0 120 0 13 .1 Appendix F. Simulation Results Jun 225 345 0 0 120 0 15.4 J u l 345 180 120 113 120 105 16.7 Aug 345 120 0 0 120 60 16.7 Sep 345 75 90 0 90 0 12.4 Oct 330 60 45 0 45 0 6.2 Nov 345 45 45 0 45 0 6.2 Dec 345 30 30 0 30 0 4.2 Mon. . I n i t . Inf. .Opt. Opt. Act. Act. Ener Vo l . Di s . S p i l . D i s . Spil.Gen. 1969 375 P Jan 465 15 15 0 15 0 2. .4 Feb 465 15 15 0 15 0 2. .4 Mar 465 15 45 0 45 0 7. .0 Apr 435 60 165 0 165 0 23, .9 May 330 225 165 0 165 0 23 .3 Jun 390 345 165 0 165 0 26 .6 J u l 570 180 165 0 165 0 29 .2 Aug 585 120 120 0 120 0 21, .4 Sep 585 75 75 0 75 0 13, .4 Oct 585 60 60 0 60 0 10, .7 Nov 585 45 45 0 45 0 8. .0 Dec 585 30 30 0 30 0 5. .3 1969 375 D-N Jan 465 15 15 0 15 0 2. .4 Feb 465 15 15 0 15 0 2. .4 Mar 465 15 75 0 75 0 11, .5 Apr 405 60 165 0 165 0 23 .1 May 300 225 165 0 165 0 22 .5 Jun 360 345 165 0 165 0 25 .8 J u l 540 180 165 60 165 0 28 .4 Aug 555 120 135 0 135 0 23 .2 Sep 540 75 45 0 45 0 7, .8 Oct 570 60 30 0 45 0 8, .0 Nov 585 45 30 0 45 0 8, .0 Dec 585 30 15 0 30 0 5, .3 1969 375 Sl-N Jan 465 15 15 0 15 0 2, .4 Feb 465 15 15 0 15 0 2, .4 Mar 465 15 45 0 45 0 7. .0 Apr 435 60 165 0 165 0 23 .9 May 330 225 165 0 165 0 23 .3 Jun 390 345 164 18 164 0 26 .6 J u l 570 180 165 90 165 0 29 .2 Aug 585 120 165 13 165 0 28 .8 Sep 540 75 135 0 135 0 22 .4 Oct 480 60 15 0 15 0 2 .5 Nov 525 45 15 0 15 0 2 .6 Dec 555 30 15 0 15 0 2 .6 1969 375 S2-N Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 30 0 30 0 4 .7 Apr 450 60 165 0 165 0 24 .4 May 345 225 165 0 165 0 23 .7 Jun 405 345 165 34 165 0 27 .0 J u l 585 180 0 0 165 15 29 .4 Aug 585 120 0 0 150 0 26 .3 Sep 555 75 45 0 45 0 7 .9 Oct 585 60 60 0 60 0 10 .7 Nov 585 45 60 0 60 0 10 .6 Dec 570 30 30 0 30 0 5 .3 1969 375 D-C Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 75 0 75 0 11 .5 Appendix F. Simulation Results Apr 405 60 165 0 165 0 23 .1 May 300 225 150 0 150 0 20 .6 Jun 375 345 165 15 165 0 26 .2 J u l 555 180 135 0 135 0 26 .3 Aug 585 120 120 0 120 0 21 .4 Sep 585 75 90- 0 90 0 15 .9 Oct 570 60 30 0 45 0 8, .0 Nov 585 45 30 0 45 0 8, .0 Dec 585 30 15 0 30 0 5, .3 Jan 465 15 15 0 15 0 2, .4 Feb 465 15 15 0 15 0 2, .4 Mar 465 15 60 0 60 0 9, .3 Apr 420 60 15 0 15 0 21 .6 May 330 225 165 0 165 0 23 .3 Jun 390 345 165 0 165 0 26 .6 J u l 570 180 165 4 165 0 29 .2 Aug 585 120 150 0 150 0 26 .3 Sep 555 75 135 0 135 0 22 .7 Oct 495 60 15 0 15 0 2, .5 Nov 540 45 15 0 15 0 2, .6 Dec 570 . 30 15 0 15 0 2, .7 Jan 465 15 15 0 15 0 2, .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 105 0 105 0 15 .9 Apr 375 60 135 0 135 0 18 .6 May 300 225 0 0 150 0 20 .6 Jun 375 345 0 0 150 0 24 .0 J u l 570 180 165 53 165 0 29 .2 Aug 585 120 0 0 150 0 26 .3 Sep 555 75 60 0 60 0 10 .5 Oct 570 60 45 0 45 0 8 .0 Nov 585 45 45 0 45 0 8, .0 Dec 585 30 30 0 30 0 5, .3 Mon. , I n i t . Inf. .Opt. Opt. Act. Act, . Ener Vo l . D i s . S p i l . Dis . Spil.Gen. 1969 500 P Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 15 0 15 0 2.7 Apr 600 60 105 0 105 0 18.6 May 555 225 180 0 180 0 31.8 Jun 600 345 180 0 180 0 24.8 J u l 765 180 180 0 180 0 37.0 Aug 765 120 120 0 120 0 24.7 Sep 765 75 75 0 75 0 15.4 Oct 765 60 60 0 60 0 12.3 Nov 765 45 45 0 45 0 9.3 Dec 765 30 30 0 30 0 6.2 1969 500 D-N Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 15 0 15 0 2.7 Apr 600 60 135 0 135 0 23.6 May 525 225 180 0 180 0 31.0 Jun 570 345 180 0 180 0 33.9 J u l 735 180 180 60 180 0 36.2 Aug 735 120 135 0 135 0 27.0 Sep 720 75 45 0 45 0 9.1 Oct 750 60 30 0 45 0 9.2 Nov 765 45 30 0 45 0 9.3 Dec 765 30 15 0 30 0 6.2 1969 500 Sl-N Jan 600 15 15 0 15 0 2.7 Appendix F. Simulation Results Feb 600 Mar 600 Apr 600 May 540 Jun 585 J u l 750 Aug 750 Sep 690 Oct 660 Nov 705 Dec 735 1969 500 S2-N Jan 600 Feb 600 Mar 600 Apr 600 May 525 Jun 570 J u l 735 Aug 735 Sep 735 Oct 765 Nov 765 Dec 750 1969 500 D-C Jan 600 Feb 600 Mar 600 Apr 600 May 525 Jun 585 J u l 750 Aug 765 Sep 765 Oct 750 Nov 765 Dec 765 1969 500 Sl-C Jan 600 Feb 600 Mar 600 Apr 600 May 525 Jun 570 J u l 765 Aug 765 Sep 735 Oct 675 Nov 720 Dec 750 1969 500 S2-C Jan 600 Feb 600 Mar 600 Apr 585 May 510 Jun 570 J u l 765 Aug 765 Sep 735 Oct 750 Nov 765 Dec 765 Year S i z e Case Mon.Init V o l . 15 15 0 15 15 0 60 120 0 225 180 0 345 180 19 180 180 75 120 180 6 75 105 0 60 15 0 45 15 0 30 15 0 15 15 0 15 15 0 15 15 0 60 135 0 225 180 0 345 180 26 180 0 0 120 0 0 75 45 0 60 60 0 45 60 0 30 30 0 15 15 0 15 15 0 15 15 0 60 135 0 225 165 0 345 180 30 180 150 0 120 120 0 75 90 0 60 30 0 45 30 0 30 15 0 15 15 0 15 15 0 15 15 0 60 135 0 225 180 0 345 150 0 180 180 4 120 150 0 75 135 0 60 15 0 45 15 0 30 15 0 15 15 0 15 15 0 15 30 0 60 135 0 225 0 0 345 0 0 180 180 53 120 0 0 75 60 0 60 45 0 45 45 0 30 30 0 Inf, .Opt. Opt. Dis. S p i l 15 0 2. ,7 15 0 2. .7 120 0 21. .1 180 0 31. .4 180 0 34. .4 180 0 36. .6 180 0 35. .8 105 0 20. .2 15 0 2. ,9 15 0 3. .0 15 0 3. .0 15 0 2. ,7 15 0 2. ,7 15 0 2. .7 135 0 23, .6 180 0 31, .0 180 0 33, .9 180 0 36, .2 120 0 24, .1 45 0 9. .2 60 0 12, .3 60 0 12, .3 30 0 6. .1 15 0 2. .7 15 0 2. .7 15 0 2. .7 135 0 23 .6 165 0 28 .6 180 0 34 .4 150 0 33 .7 120 0 24 .7 90 0 18 .4 45 0 9, .2 45 0 9. .3 30 0 6, .2 15 0 2 .7 15 0 2 .7 15 0 2 .7 135 0 23 .6 180 0 31 .0 150 0 28 .6 180 0 37 .0 150 0 30 .5 135 0 26 .5 15 0 2 .9 15 0 3 .0 15 0 3 .1 15 0 2 .7 15 0 2 .7 30 0 5 .4 135 0 23 .2 165 0 28 .2 150 0 28 .6 180 0 37 .0 150 0 30 .5 60 0 12 .1 45 0 9 .2 45 0 9 .3 30 0 6 .2 Act. Act. Ener. D i s . Spil.Gen. Appendix F. Simulation Results 167 1969 750 P Jan 990 15 15 0 15 0 3. ,6 Feb 990 15 15 0 15 0 3. .6 Mar 990 15 15 0 15 0 3. .6 Apr 990 60 15 0 15 0 3. .6 May 1035 225 165 0 165 0 41. .0 Jun 1095 345 195 0 195 0 51. .1 J u l 1245 180 180 0 180 0 48. .9 Aug 1245 120 120 0 120 0 32, .6 Sep 1245 75 75 0 75 0 20, .4 Oct 1245 60 60 0 60 0 16, .3 Nov 1245 45 45 0 45 0 12, .2 Dec 1245 30 30 0 30 0 8. .2 1969 750 D-N Jan 990 15 15 0 15 0 3. .6 Feb 990 15 15 0 15 0 3. .6 Mar 990 15 15 0 15 0 3. .6 Apr 990 60 15 0 15 0 3. .6 May 1035 225 195 0 195 0 48 .0 Jun 1065 345 195 0 195 0 50, .4 J u l 1215 180 195 45 195 0 52, .1 Aug 1200 120 120 0 120 0 31 .9 Sep 1200 75 45 0 45 0 12 .1 Oct 1230 60 30 0 45 0 12 .2 Nov 1245 45 30 0 45 0 12 .2 Dec 1245 30 15 0 30 0 8. .2 1969 750 Sl-N Jan 990 15 15 0 15 0 3, .6 Feb 990 15 15 0 15 0 3, .6 Mar 990 15 15 0 15 0 3 .6 Apr 990 60 45 0 45 0 10 .8 May 1005 225 195 0 195 0 47 .2 Jun 1035 345 195 9 195 0 49 .6 J u l 1185 180 195 31 195 0 51 .3 Aug 1170 120 195 2 195 0 50 .2 Sep 1095 75 30 0 30 0 7, .7 Oct 1140 60 15 0 15 0 3 .9 Nov 1185 45 15 0 15 0 4 .0 Dec 1215 30 15 0 15 0 4, .0 1969 750 S2-N Jan 990 15 15 0 15 0 3 .6 Feb 990 15 15 0 15 0 3 .6 Mar 990 15 15 0 15 0 3 .6 Apr 990 60 60 0 60 0 14 .3 May 990 225 180 0 180 0 43 .4 Jun 1035 345 195 19 195 0 49 .6 J u l 1185 180 0 0 195 0 51 .3 Aug 1170 120 0 0 75 0 19 .9 Sep 1215 75 45 0 45 0 12 .1 Oct 1245 60 60 0 60 0 16 .3 Nov 1245 45 60 0 60 0 16 .3 Dec 1230 30 30 0 30 0 8 .1 1969 750 D-C Jan 990 15 15 0 15 0 3 .6 Feb 990 15 15 0 15 0 3 .6 Mar 990 15 15 0 15 0 3 .6 Apr 990 60 15 0 15 0 3 .6 May 1035 225 165 0 165 0 41 .0 Jun 1095 345 195 45 195 0 51 .1 J u l 1245 180 165 0 180 0 48 .9 Aug 1245 120 120 0 120 0 32 .6 Sep 1245 75 90 0 90 0 24 .4 Oct 1230 60 30 0 45 0 12 .2 Nov 1245 45 30 0 45 0 12 .2 Dec 1245 30 15 0 30 0 8 .2 1969 750 Sl-C Jan 990 15 15 0 15 0 3 .6 endix F. Simulation Results 1969 1969 1000 P 1969 1000 D-N Feb 990 15 15 0 15 0 3 .6 Mar 990 15 15 0 15 0 3, .6 Apr 990 60 45 0 45 0 10 .8 May 1005 225 195 0 195 0 47 .2 Jun 1035 345 135 0 135 0 34 .9 J u l 1245 180 195 3 195 0 52 .8 Aug 1230 120 135 0 135 0 36 .3 Sep 1215 75 135 0 135 0 35 .7 Oct 1155 60 15 0 15 0 3. .9 Nov 1200 45 15 0 15 0 4. .0 Dec 1230 30 15 0 15 0 4. .1 Jan 990 15 15 0 15 0 3. .6 Feb 990 15 15 0 15 0 3. .6 Mar 990 15 15 0 15 0 3. .6 Apr 990 60 60 0 60 0 14 .3 May 990 225 0 0 180 0 43 .4 Jun 1035 345 0 0 150 0 38 .6 J u l 1230 180 195 23 195 0 52 .5 Aug 1215 120 0 0 120 0 32 .2 Sep 1215 75 60 0 60 0 16, .1 Oct 1230 60 45 0 45 0 12 .2 Nov 1245 45 45 0 45 0 12 .2 Dec 1245 30 30 0 30 0 8. .2 Mon. I n i t . Inf. .Opt. Opt. Act. Act, . Ener Vo l . Dis. S p i l . D i s . Spil.Gen. Jan 1440 15 15 0 15 0 4. .4 Feb 1440 15 15 0 15 0 4, .4 Mar 1440 15 15 0 15 0 4, .4 Apr 1440 60 15 0 15 0 4, .5 May 1485 225 60 0 60 0 18 .6 Jun 1650 345 210 0 210 0 68 .5 J u l 1785 180 180 0 180 0 60, .0 Aug 1785 120 120 0 120 0 40, .0 Sep 1785 75 75 0 75 0 25 .0 Oct 1785 60 60 0 60 0 20 .0 Nov 1785 45 45 0 45 0 15 .0 Dec 1785 30 30 0 30 0 10 .0 Jan 1440 15 15 0 15 0 4, .4 Feb 1440 15 15 0 15 0 4, .4 Mar 1440 15 15 0 15 0 4, .4 Apr 1440 60 15 0 15 0 4, .5 May 1485 225 75 0 75 0 23 .2 Jun 1635 345 210 0 210 0 68 .2 J u l 1770 180 210 45 210 0 69 .3 Aug 1740 120 120 0 120 0 39 .4 Sep 1740 75 45 0 45 0 14 .9 Oct 1770 60 30 0 45 0 15 .0 Nov 1785 45 30 0 45 0 15 .0 Dec 1785 30 15 0 30 0 10 .0 Jan 1440 15 15 0 15 0 4, .4 Feb 1440 15 15 0 15 0 4, .4 Mar 1440 15 15 0 15 0 4, .4 Apr 1440 60 15 0 15 0 4 .5 May 1485 225 165 0 165 0 50 .2 Jun 1545 345 210 2 210 0 66 .2 J u l 1680 180 210 6 210 0 67 .4 Aug 1650 120 165 0 165 0 52 .3 Sep 1605 75 15 0 15 0 4 .8 Oct 1665 60 15 0 15 0 4 .8 Nov 1710 45 15 0 15 0 4 .9 Appendix F. Simulation Results 1969 1000 D-C Dec 1740 30 15 0 15 0 4 9 Jan 1440 15 15 0 15 0 4 4 Feb 1440 15 15 0 15 0 4 4 Mar 1440 15 15 0 15 0 4 4 Apr 1440 60 15 0 15 0 4 5 May 1485 225 135 0 135 0 41 .3 Jun 1575 345 210 15 210 0 66 9 J u l 1710 180 0 0 195 0 63 .6 Aug 1695 120 0 0 60 0 19 .6 Sep 1755 75 45 0 45 0 14 .9 Oct 1785 60 60 0 60 0 20 .0 Nov 1785 45 60 0 60 0 19 .9 Dec 1770 30 30 0 30 0 10 0 Jan 1440 15 15 0 15 0 4 4 Feb 1440 15 15 0 15 0 4 4 Mar 1440 15 15 0 15 0 4 4 Apr 1440 60 15 0 15 0 4 5 May 1485 225 45 0 45 0 14 .0 Jun 1665 345 210 60 210 15 68 7 J u l 1785 180 165 0 180 0 60 0 Aug 1785 120 120 0 120 0 40 .0 Sep 1785 75 90 0 90 0 29 .9 Oct 1770 60 30 0 45 0 15 .0 Nov 1785 45 30 0 45 0 15 .0 Dec 1785 30 15 0 30 0 10 .0 Jan 1440 15 15 0 15 0 4 4 Feb 1440 15 15 0 15 0 4 4 Mar 1440 15 15 0 15 0 4 4 Apr 1440 60 15 0 15 0 4 5 May 1485 225 120 0 120 0 36 .8 Jun 1590 345 150 0 150 0 48 .5 J u l 1785 180 210 2 210 0 69 .7 Aug 1755 120 120 0 120 0 39 .6 Sep 1755 75 135 0 135 0 44 .2 Oct 1695 60 15 0 15 0 4 9 Nov 1740 45 15 0 15 0 5 0 Dec 1770 30 15 0 15 0 5 0 Jan 1440 15 15 0 15 0 4 4 Feb 1440 15 15 0 15 0 4 4 Mar 1440 15 15 0 15 0 4 4 Apr 1440 60 15 0 15 0 4 5 May 1485 225 0 0 120 0 36 8 Jun 1590 345 0 0 150 0 48 .5 J u l 1785 180 210 23 210 0 69 .7 Aug 1755 120 0 0 120 0 39 .6 Sep 1755 75 60 0 60 0 19 9 Oct 1770 60 45 0 45 0 15 .0 Nov 1785 45 45 0 45 0 15 .0 Dec 1785 30 30 0 30 0 10 .0 Mon I n i t . Inf Opt. Opt. Act. Act . Ener Vol. Dis. S p i l . D i s . Spil.Gen. 1970 250 P Jan 255 15 15 Feb 255 15 15 Mar 255 15 75 Apr 195 15 120 May 90 120 120 Jun 90 345 120 J u l 315 210 120 Aug 345 120 120 Sep 345 60 60 0 15 0 1 9 0 15 0 1 9 0 75 0 8 9 0 120 0 12 4 0 120 0 11 3 0 120 0 13 7 60 120 60 16 3 0 120 0 16 7 0 60 0 8 3 Appendix F. Simulation Results 1970 Oct 345 45 45 0 45 0 6. .2 Nov 345 30 30 0 30 0 4. .2 Dec 345 15 15 0 15 0 2. .1 Jan 255 15 15 0 15 0 1. .9 Feb 255 15 15 0 15 0 1. .9 Mar 255 15 105 0 105 0 12 .1 Apr 165 15 120 0 90 0 9. .1 May 90 120 120 0 120 0 11, .3 Jun 90 345 120 0 120 0 13 .7 J u l 315 210 120 120 120 60 16, .3 Aug 345 120 120 45 120 0 16, .7 Sep 345 60 90 0 90 0 12 .3 Oct 315 45 15 0 15 0 2, .0 Nov 345 30 30 0 30 0 4. .2 Dec 345 15 15 0 15 0 2, .1 Jan 255 15 15 0 15 0 1, .9 Feb 255 15 15 0 15 0 1, .9 Mar 255 15 90 0 90 0 10 .5 Apr 180 15 105 0 105 0 10 .7 May 90 120 120 0 120 0 11 .3 Jun 90 345 120 13 120 0 13 .7 J u l 315 210 120 120 120 60 16 .3 Aug 345 120 120 40 120 0 16 .7 Sep 345 60 120 7 120 0 16 .0 Oct 285 45 15 0 15 0 2 .0 Nov 315 30 30 0 30 0 4, .0 Dec 315 15 15 0 15 0 2 .0 Jan 255 15 15 0 15 0 1, .9 Feb 255 15 30 0 30 0 3 .7 Mar 240 15 120 0 120 0 13 .4 Apr 135 15 60 0 60 0 5 .9 May 90 120 120 0 120 0 11 .3 Jun 90 345 0 0 120 0 13 .7 J u l 315 210 0 0 120 60 16 .3 Aug 345 120 105 0 120 0 16 .7 Sep 345 60 75 0 75 0 10 .3 Oct 330 45 30 0 30 0 4 .1 Nov 345 30 60 0 60 0 8 .2 Dec 315 15 15 0 15 0 2 .0 Jan 255 15 15 0 15 0 1 .9 Feb 255 15 15 0 15 0 1 .9 Mar 255 15 105 0 105 0 12 .1 Apr 165 15 120 0 120 0 9 .1 May 90 120 120 0 120 0 11 .3 Jun 90 345 120 0 120 0 13 .7 J u l 315 210 120 30 120 60 16 .3 Aug 345 120 120 0 120 0 16 .7 Sep 345 60 90 0 90 0 12 .3 Oct 315 45 15 0 15 0 2 .0 Nov 345 30 30 0 30 0 4 .2 Dec 345 15 15 0 15 0 2 .1 Jan 255 15 15 0 15 0 1 .9 Feb 255 15 15 0 15 0 1 .9 Mar 255 15 105 0 105 0 12 .1 Apr 165 15 105 0 105 0 9 . 1 May 90 120 120 0 120 0 11 .3 Jun 90 345 120 0 120 0 13 .7 J u l 315 210 120 30 120 60 16 .3 Aug 345 120 120 6 120 0 16 .7 Sep 345 60 120 4 120 0 16 .0 Oct 285 45 15 4 15 0 2 .0 Nov 315 30 15 0 15 0 2 .0 Appendix F. Simulation Results 1970 Dec 330 15 15 0 15 0 2 .0 Jan 255 15 15 0 15 0 1. .9 Feb 255 15 15 0 15 0 1, .9 Mar 255 15 105 0 105 0 12 .1 Apr 165 15 105 0 90 0 9 .1 May 90 120 0 0 120 0 11 .3 Jun 90 345 0 0 120 0 13 .7 J u l 315 210 120 30 120 60 16 .3 Aug 345 120 0 0 120 0 16 .7 Sep 345 60 120 3 120 3 16 .0 Oct 285 45 15 0 15 0 2 .0 Nov 315 30 15 0 15 0 2 .0 Dec 330 15 15 0 15 0 2 .0 Mon. I n i t . Inf. Opt. Opt. Act. Act. . Ener Vol. D i s . S p i l . D i s . Spil.Gen Jan 465 15 15 0 15 0 2. ,4 Feb 465 15 15 0 15 0 2. ,4 Mar 465 15 15 0 15 0 2. .4 Apr 465 15 75 0 75 0 11, .5 May 405 120 165 0 165 0 23, .9 Jun 360 345 165 0 165 0 25, .8 J u l 540 210 165 0 165 0 28, .8 Aug 585 120 120 0 120 0 21, .4 Sep 585 60 60 0 60 0 10, .7 Oct 585 45 45 0 45 0 8. .0 Nov 585 30 30 0 30 0 5. .3 Dec 585 15 15 0 15 0 2, .7 Jan 465 15 15 0 15 0 2. .4 Feb 465 15 15 0 15 0 2. .4 Mar 465 15 75 0 75 0 11. .5 Apr 405 15 165 0 165 0 22. .5 May 255 120 135 0 135 0 16, .5 Jun 240 345 90 0 90 0 12 .8 J u l 495 210 165 15 165 0 27, .6 Aug 540 120 120 0 120 0 20 .5 Sep 540 60 45 0 45 0 7. .7 Oct 555 45 15 0 15 0 2. .6 Nov 585 30 30 0 30 0 5, .3 Dec 585 15 15 0 15 0 2. .7 Jan 465 15 15 0 15 0 2. .4 Feb 465 15 15 0 15 0 2. .4 Mar 465 15 45 0 45 0 7. .0 Apr 435 15 165 0 165 0 23 .3 May 285 120 150 0 150 0 18 .9 Jun 255 345 165 0 165 0 22 .9 J u l 435 210 165 6 165 0 26 .0 Aug 480 120 165 2 165 0 26 .0 Sep 435 60 30 0 30 0 4 .7 Oct 465 45 15 0 15 0 2 .4 Nov 495 30 15 0 15 0 2 .5 Dec 510 15 15 0 15 0 2 .5 Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2. .4 Mar 465 15 30 0 30 0 4 .7 Apr 450 15 165 .0 165 0 23 .7 May 300 120 150 0 150 0 19 .3 Jun 270 345 0 0 135 0 19 .4 J u l 480 210 0 0 150 0 24 .9 Aug 540 120 60 0 75 0 13 .1 Sep 585 60 75 0 75 0 13 .3 1970 375 P Appendix F. Simulation Results Oct 570 45 30 0 30 0 5 .3 Nov 585 30 60 0 60 0 10 .5 Dec 555 15 15 0 15 0 2 .6 Jan 465 15 15 0 15 0 2, .4 Feb 465 15 15 0 15 0 2, .4 Mar 465 15 15 0 15 0 2, .4 Apr 465 15 105 0 105 0 15 .9 May 375 120 150 0 150 0 21 .2 Jun 345 345 135 0 135 0 21 .1 J u l 555 210 150 0 165 15 29 .0 Aug 585 120 120 0 120 0 21 .4 Sep 585 60 90 0 90 0 15 .8 Oct 555 45 15 0 15 0 2, .6 Nov 585 30 30 0 30 0 5, .3 Dec 585 15 15 0 15 0 2 .7 Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 15 0 15 0 2. .4 Apr 465 15 120 0 120 0 18 .0 May 360 120 165 0 165 0 22 .7 Jun 315 345 120 0 120 0 18 .3 J u l 540 210 165 3 165 0 28 .8 Aug 585 120 165 0 165 0 28 .8 Sep 540 60 120 0 120 0 19 .9 Oct 480 45 15 0 15 0 2 .5 Nov 510 30 15 0 15 0 2 .5 Dec 525 15 15 0 15 0 2 .5 Jan 465 15 15 0 15 0 2 .4 Feb 465 15 15 0 15 0 2 .4 Mar 465 15 30 0 30 0 4 .7 Apr 450 15 120 0 120 0 17 .7 May 345 120 0 0 135 0 18 .6 Jun 330 345 0 0 135 0 20 .8 J u l 540 210 165 3 165 0 28 .8 Aug 585 120 0 0 165 0 28 .8 Sep 540 60 105 0 105 0 17 .6 Oct 495 45 15 0 15 0 2 .5 Nov 525 30 15 0 15 0 2 .5 Dec 540 15 15 0 15 0 2 .6 Mon. , I n i t . Inf. .Opt. Opt. Act. Act ..Ener Vol . D i s . S p i l . Dis . Spil.Gen. 1970 500 P Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 15 0 15 0 2.7 Apr 600 15 15 0 15 0 2.7 May 600 120 150 0 150 0 26.7 Jun 570 345 180 0 180 0 33.9 J u l 735 210 180 0 180 0 36.6 Aug 765 120 120 0 120 0 24.7 Sep 765 60 60 0 60 0 12.3 Oct 765 45 45 0 45 0 9.3 Nov 765 30 30 0 30 0 6.2 Dec 765 15 15 0 15 0 3.1 1970 500 D-N Jan 600 15 15 0 15 0 2.7 Feb 600 15 15 0 15 0 2.7 Mar 600 15 15 0 15 0 2.7 Apr 600 15 135 0 135 0 23.1 May 480 120 150 0 150 0 23.8 Jun 450 345 105 0 105 0 18.4 J u l 690 210 180 15 180 0 35.4 endix F. Simulation Results Aug 720 120 120 0 120 0 23 .9 Sep 720 60 45 0 45 0 9, .0 Oct 735 45 15 0 15 0 3. .1 Nov 765 30 30 0 30 0 6. .2 Dec 765 15 15 0 15 0 3, .1 Jan 600 15 15 0 15 0 2. .7 Feb 600 15 15 0 15 0 2, .7 Mar 600 15 15 0 15 0 2 .7 Apr 600 15 120 0 120 0 20 .6 May 495 120 180 0 180 0 28 .6 Jun 435 345 165 0 165 0 27 .8 J u l 615 210 180 4 180 4 33 .3 Aug 645 120 180 0 180 0 32, .9 Sep 585 60 15 0 15 0 2. .7 Oct 630 45 15 0 15 0 2, .8 Nov 660 30 15 0 15 0 2. .9 Dec 675 15 15 0 15 0 2. .9 Jan 600 15 15 0 15 0 2 .7 Feb 600 15 15 0 15 0 2. .7 Mar 600 15 15 0 15 0 2, .7 Apr 600 15 135 0 135 0 23 .1 May 480 120 135 0 135 0 21, .6 Jun 465 345 0 0 135 0 23 .7 J u l 675 210 0 0 165 0 32 .3 Aug 720 120 60 0 75 0 15 .2 Sep 765 60 75 0 75 0 15 .3 Oct 750 45 30 0 30 0 6 .1 Nov 765 30 60 0 60 0 12, .2 Dec 735 15 15 0 15 0 3. .0 Jan 600 15 15 0 15 0 2, .7 Feb 600 15 15 0 15 0 2. .7 Mar 600 15 15 0 15 0 2, .7 Apr 600 15 15 0 15 0 2. .7 May 600 120 165 0 165 0 29 .2 Jun 555 345 150 0 150 0 28 .3 J u l 750 210 165 0 180 15 36 .8 Aug 765 120 120 0 120 0 24 .7 Sep 765 60 90 0 90 0 18 .3 Oct 735 45 15 0 15 0 3, .1 Nov 765 30 30 0 30 0 6, .2 Dec 765 15 15 0 15 0 3, .1 Jan 600 15 15 0 15 0 2, .7 Feb 600 15 15 0 15 0 2. .7 Mar 600 15 15 0 15 0 2 .7 Apr 600 15 45 0 45 0 8, .0 May 570 120 180 0 180 0 30 .8 Jun 510 345 135 0 135 0 24, .7 J u l 720 210 180 2 180 0 36 .2 Aug 750 120 150 0 150 0 30 .2 Sep 720 60 120 0 120 0 23 .3 Oct 660 45 15 0 15 0 2, .9 Nov 690 30 15 0 15 0 2 .9 Dec 705 15 15 0 15 0 2 .9 Jan 600 15 15 0 15 0 2, .7 Feb 600 15 15 0 15 0 2. .7 Mar 600 15 15 0 15 0 2, .7 Apr 600 15 90 0 90 0 15 .7 May 525 120 0 0 150 0 24 .9 Jun 495 345 0 0 105 0 19 .2 J u l 735 210 180 3 180 0 36 .6 Aug 765 120 0 0 165 0 33 .4 Sep 720 60 105 0 105 0 20 .5 Appendix F. Simulation Results 1970 750 P 1970 750 D-N 1970 1970 1970 750 D-C Oct 675 45 15 0 15 0 2.9 Nov 705 30 15 0 15 0 3.0 Dec 720 15 15 0 15 0 3.0 Mon. I n i t . Inf. Opt. Opt. Act. Act, . Ener. Vol. Di s . S p i l . Dis . Spil.Gen. Jan 990 15 15 0 15 0 3.6 Feb 990 15 15 0 15 0 3.6 Mar 990 15 15 0 15 0 3.6 Apr 990 15 15 0 15 0 3.6 May 990 120 30 0 30 0 7.3 Jun 1080 345 195 0 195 0 50.8 J u l 1230 210 195 0 195 0 52.8 Aug 1245 120 120 0 120 0 32.6 Sep 1245 60. 60 0 60 0 16.3 Oct 1245 45 45 0 45 0 12.2 Nov 1245 30 30 0 30 0 8.2 Dec 1245 15 15 0 15 0 4.1 Jan 990 15 15 0 15 0 3.6 Feb 990 15 15 0 15 0 3.6 Mar 990 15 15 0 15 0 3.6 Apr 990 15 15 0 15 0 3.6 May 990 120 150 0 150 0 35.4 Jun 960 345 120 0 120 0 29.9 J u l 1185 210 195 15 195 0 51.7 Aug 1200 120 120 0 120 0 31.9 Sep 1200 60 45 0 45 0 12.0 Oct 1215 45 15 0 15 0 4.0 Nov 1245 30 30 0 30 0 8.2 Dec 1245 15 15 0 15 0 4.1 Jan 990 15 15 0 15 0 3.6 Feb 990 15 15 0 15 0 3.6 Mar 990 15 15 0 15 0 3.6 Apr 990 15 45 0 45 0 10.6 May 960 120 180 0 180 0 41.3 Jun 900 345 150 0 150 0 35.9 J u l 1095 210 195 2 195 0 49.4 Aug 1110 120 165 0 165 0 41.5 Sep 1065 60 15 0 15 0 3.8 Oct 1110 45 15 0 15 0 3.8 Nov 1140 30 15 0 15 0 3.9 Dec 1155 15 15 0 15 0 3.9 Jan 990 15 15 0 15 0 3.6 Feb 990 15 15 0 15 0 3.6 Mar 990 15 15 0 15 0 3.6 Apr 990 15 60 0 60 0 14.1 May 945 120 195 0 195 0 44.2 Jun 870 345 0 0 45 0 10.9 J u l 1170 210 0 0 180 0 47.6 Aug 1200 120 60 0 75 0 20.2 Sep 1245 60 75 0 75 0 20.3 Oct 1230 45 30 0 30 0 8.1 Nov 1245 30 60 0 60 0 16.2 Dec 1215 15 15 0 15 0 4.0 Jan 990 15 15 0 15 0 3.6 Feb 990 15 15 0 15 0 3.6 Mar 990 15 15 0 15 0 3.6 Apr 990 15 15 0 15 0 3.6 May 990 120 45 0 45 0 10.9 Jun 1065 345 165 0 165 0 42.9 J u l 1245 210 180 0 195 15 53.0 Appendix F. Simulation Results 1970 750 Sl-C 1970 750 S2-C Aug 1245 120 120 0 120 0 32 .6 Sep 1245 60 90 0 90 0 23 .4 Oct 1215 45 15 0 15 0 4, .0 Nov 1245 30 30 0 30 0 8, .2 Dec 1245 15 15 0 15 0 4, .1 Jan 990 15 15 0 15 0 3, .6 Feb 990 15 15 0 15 0 3, .6 Mar 990 15 15 0 15 0 3. .6 Apr 990 15 15 0 15 0 3 .6 May 990 120 120 0 120 0 28 .6 Jun 990 345 135 0 135 0 34 .1 J u l 1200 210 195 1 195 0 52 .1 Aug 1215 120 135 0 135 0 36 .1 Sep 1200 60 120 0 120 0 31 .5 Oct 1140 45 15 0 15 0 3, .9 Nov 1170 30 15 0 15 0 3. .9 Dec 1185 15 15 0 15 0 4. .0 Jan 990 15 15 0 15 0 3, .6 Feb 990 15 15 0 15 0 3 .6 Mar 990 15 15 0 15 0 3. .6 Apr 990 15 15 0 15 0 3 .6 May 990 120 0 0 120 0 28 .6 Jun 990 345 0 0 105 0 26 .7 J u l 1230 210 195 3 195 0 52 .8 Aug 1245 120 0 0 165 0 44 .4 Sep 1200 60 105 0 105 0 27 .6 Oct 1155 45 15 0 15 0 3, .9 Nov 1185 30 15 0 15 0 4, .0 Dec 1200 15 15 0 15 0 4, .0 Mon. I n i t . Inf. Opt. Opt. Act. Act . Ener Vol. Dis. S p i l . D i s . Spil.Gen. Jan 1440 15 15 0 15 0 4, .4 Feb 1440 15 15 0 15 0 4. .4 Mar 1440 15 15 0 15 0 4. .4 Apr 1440 15 15 0 15 0 4, .4 May 1440 120 15 0 15 0 4 .5 Jun 1545 345 105 0 105 0 33 .7 J u l 1785 210 210 0 210 0 70 .0 Aug 1785 120 120 0 120 0 40 .0 Sep 1785 60 60 0 60 0 20 .0 Oct 1785 45 45 0 45 0 15 .0 Nov 1785 30 30 0 30 0 10 .0 Dec 1785 15 15 0 15 0 5. .0 Jan 1440 15 15 0 15 0 4, .4 Feb 1440 15 15 0 . 15 0 4, .4 Mar 1440 15 15 0 15 0 4, .4 Apr 1440 15 15 0 15 0 4, .4 May 1440 120 30 0 30 0 9. .0 Jun 1530 345 135 0 135 0 42 .9 J u l 1740 210 210 15 210 0 69 .0 Aug 1740 120 120 0 120 0 39 .4 Sep 1740 60 45 0 45 0 14 .8 Oct 1755 45 15 0 15 0 5 .0 Nov 1785 30 30 0 30 0 10 .0 Dec 1785 15 15 0 15 0 5. .0 Jan 1440 15 15 0 15 0 4, .4 Feb 1440 15 15 0 15 0 4 .4 Mar 1440 15 15 0 15 0 4 .4 Apr 1440 15 15 0 15 0 4 .4 May 1440 120 120 0 120 0 35 .5 1970 1000 P 1970 1000 D-N Appendix F. Simulation Results Jun J u l Aug Sep Oct Nov Dec 1970 1000 S2-N Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 1970 1000 D-C Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 1970 1000 Sl-C Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 1970 1000 S2-C Jan Feb Mar Apr May Jun J u l Aug Sep Oct Nov Dec 440 345 165 0 165 0 50 .5 620 210 210 0 210 0 66 .4 620 120 150 0 150 0 47 1 590 60 15 0 15 0 4 7 635 45 15 0 15 0 4 8 665 30 15 0 15 0 4 8 680 15 15 0 15 0 4 8 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 120 120 0 120 0 35 .5 440 345 0 0 60 0 18 .7 725 210 0 0 195 0 63 .9 740 120 60 0 75 0 24 .8 785 60 75 0 75 0 24 .9 770 45 30 0 30 0 10 0 785 30 60 0 60 0 19 9 785 15 15 0 15 0 5 0 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 120 15 0 15 0 4 5 545 345 90 0 105 0 33 .7 785 210 180 0 210 0 70 .0 785 120 120 0 120 0 40 .0 785 60 90 0 90 0 29 .9 755 45 15 0 15 0 5 0 785 30 30 0 30 0 10 0 785 15 15 0 15 0 5 0 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 120 30 0 30 0 9 .0 530 345 135 0 135 0 42 .9 740 210 210 0 210 0 69 .0 740 120 120 0 120 0 39 .4 740 60 120 0 120 0 39 .1 680 45 15 0 15 0 4 9 710 30 15 0 15 0 4 9 725 15 15 0 15 0 4 .9 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 4 440 15 15 0 15 0 4 .4 440 120 0 0 15 0 4 5 545 345 0 0 120 0 38 .4 770 210 210 0 210 0 69 .7 770 120 0 0 150 0 49 .5 740 60 105 0 105 0 34 .3 695 45 15 0 15 0 4 .9 725 30 15 0 15 0 4 .9 740 15 15 0 15 0 4 .9 

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