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The value of one month ahead inflow forecasting in the operation of a hydroelectric reservoir Zhou, Dequan 1991

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T H E V A L U E OF O N E M O N T H A H E A D I N F L O W F O R E C A S T I N G I N T H E O P E R A T I O N OF A H Y D R O E L E C T R I C RESERVOIR by Dequan Zhou B. Eng. Tsinghua University,1982 M. Eng. Tsinghua University, 1985  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR T H E DEGREE OF M A S T E R 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  April 1991  © Dequan Zhou, 1991  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.  CIVIL ENGINEERING The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date:  ABSTRACT  The research assesses the value of forecast information in operating a hydro-electric project with a storage reservoir. The benefits are the increased hydro power production, when forecasts are available. The value of short term forecasts is determined by comparing results obtained with the use of one month ahead perfect predictions to those obtained without forecasts but a knowledge of the statistics of the possible flows. The benefits with perfect forecasts provide an upper limit to the benefits which could be obtained with actual less than perfect forecasts. The effects of generating capacity and flow patterns are also discussed. The operation of a hypothetical but typical project is modelled using stochastic dynamic programming. A simple model of streamflow is formulated based on the historical statistics ( means and deviations). The conclusions are: The inflow forecasts can improve the operational efficiency of the reservoir considerably because of the reduction in forecasting uncertainty. The maximum release constraints affect the additional expected values. The benefits from the forecasts increase as the discharge limits reduce. Flow predictions in the high flow season are most valuable when the runoff in that time period dominates the annual flow pattern. However flow predictions at other times of the year also have value.  u  Table of Contents  ABSTRACT  ii  List of Tables  vi  List of Figures  viii  Acknoledgement 1  INTRODUCTION  1  1.1  STATEMENT OF T H E PROBLEM  1  1.2  METHODOLOGY  3  1.3  SOME OTHER PROBLEMS  4  1.3.1  SELECTION OF OBJECTIVE FUNCTION  4  1.3.2  T H E WORK ENVIRONMENT OF T H E HYDRO POWER PLANT  5  1.3.3  E F F E C T OF T H E MAXIMUM GENERATOR DISCHARGE ON  1.3.4  2  ix  T H E OPERATION OF RESERVOIR  6  OPERATIONAL POLICIES WITH VARIOUS FLOW PATTERNS  6  1.4  LITERATURE REVIEW  1.5  SUMMARY OF T H E THESIS  7 10  T H E STOCHASTIC DYNAMIC PROGRAMMING MODEL  11  2.1  STOCHASTIC DYNAMIC PROGRAMMING MODEL  11  2.1.1  T H E STATE TRANSITION EQUATION  13  2.1.2  STAGE RETURN EQUATION  14  in  2.1.3  THE RECURSIVE EQUATION  2.1.4  THE DYNAMIC PROGRAMMIMG MODEL FOR ONE MONTH  16  AHEAD PERFECT INFLOW FORECAST 2.1.5 2.2  17  MATHEMATICAL STATEMENT FOR LONG TERM ANALYSIS  OPTIMAL OPERATION OVER ONE YEAR PERIOD  24  2.2.1  SOLUTION TECHNIQUE  .  2.2.2  CALCULATION OF STATE TRANSITION PREOBABILITY MATRICES  2.3  ANALYSIS OF LONG TERM RESERVIOR OPERATION  2.4  MEASURING ADDITIONAL VALUES OF ONE MONTH AHEAD PER-  28  30  INFLOW F O R E C A S T  31  3.1  PROBLEM DESCRIPTION  31  3.2  THE INFLOW FORECASTING MODEL  34  3.2.1  T H E INFLOW INTERVAL  34  3.2.2  LOWER AND UPPER VALUES  34  3.2.3  STREAM FLOWS AND THEIR PROBABILITIES  35  3.3 3.4 4  24  24  FECT FORECAST 3  18  THE INFLOW STATISTICS  38  THE OUTCOME OF INFLOW FORECAST  43  RESULTS  51  4.1  RESERVOIR DESCRIPTION  51  4.1.1  T H E OBJECTIVE FUNCTION  51  4.1.2  RESERVOIR VOLUME AND DISCHARGE  52  4.1.3  T H E ADDITIONAL EXPECTED VALUES OF ONE MONTH AHEAD PERFECT FORECAST iv  53  4.2  4.3 5  OPERATION RESULTS FOR FLOW PATTERN I  56  4.2.1  GENERAL MAXIMUM DISCHARGE (D =180 Mm )  . . . .  56  4.2.2  RESULTS WITH MAXIMUM RELEASE 7J =210 Mm . . . .  62  4.2.3  RESULTS WITH FLOW PATTERN I, D  70  3  max  3  max  max  = 150Mm  3  THE OPERATION RESULTS WITH FLOW PATTERN II  70  CONCLUSIONS  79  5.1  ADDITIONAL BENEFITS OF PERFECT INFORMATIOM  79  5.2  THE EFFECT OF MAXIMUM RELEASE ON THE POTENTIAL BENEFIT  5.3  81  THE ROLE OF THE FLOW PATTERN IN THE RESERVOIR OPERATION  83  5.4  THE CONSEQUENCES OF PART TIME PERFECT FORECASTING  86  5.5  RECOMMENDATIONS FOR FURTHER RESEARCH  87  Bibliography  89  v  List of Tables  3.1  Flow Statistics of Flow Pattern I  39  3.2  Inflow Statistics of Flow Pattern II  39  3.3  Comparing Flow Characteristics  42  3.4  Inflows and Probabilities ( Flow Pattern I)  44  3.5  Results of Inflow Forecast (Flow Pattern II)  48  3.6  Comparison of Mean Monthly Flow ( Flow Pattern I )  50  3.7  Comparison of Mean Monthly Flow ( Flow Pattern II )  50  4.1  Reservoir Constraints and Characteristics  52  4.2  Average Reservoir Operation Processes with £> .=180 Mm  56  4.3  The Year End Values of States with D =180 Mm  58  4.4  The Additional Values of States with D =180 Mm  4.5  Reservoir Operation Processes for D =210 Mm  4.6  Expected Values of States with D  4.7  Additional Values of States D  4.8  Operation Processes of Flow Pattern I, D  4.9  The Year End Values of States D  3  maa  3  max  60  3  max  62  3  max  =210 Mm  63  3  max  =210 Mm  65  3  max  max  = 150Mm  = 150Mm  max  4.10 The Additional Expected Values D  max  67  3  68  3  = 150Mm  69  3  4.11 Average Operation Pattern with General Flow Pattern II and D 180Mm  max  = 75  3  4.12 The Year End Values of States D  max  = 180Mm  76  3  4.13 The Additional Values of Forecasts with Flow Pattern II, D  max  vi  = 180Mm 77 3  5.1  Summary of Additional Benefits with Forecasts  80  5.2  Additional Expected Values of Different D  82  5.3  The Additional Values with Flow Pattern I  84  5.4  The Additional Values of Flow Pattern II  85  vn  max  List of Figures  2.1  Sequential Reservoir Operation Process  12  2.2  The Flow Chart of the SDP Problem  25  2.3  The Flow Chart of the Perfect Forecast Dynamic Program  26  2.4  Schematic of Iterative Cycle Used in the Study  29  3.1  The Distribution of Inflow  33  3.2  The Flow Chart of Inflow Forecasts  37  3.3  The Mean Annual Hydrograph  40  3.4  Mean Flow Hydrograph of Flow Pattern II  41  3.5  Probability Distribution of Generated Flow  45  4.1  Reservoir Elevation-volume Curve  54  4.2  The Reservoir Operation Processes with D =180Mm  57  4.3  The Expected Values of States, D  4.4  The Expected Values of States, D  4.5  The Reservoir Operation Processes, D  4.6  The Operation Processes, D  4.7  The Expected Values of States, D  4.8  The Reservoir Operation Processes, Flow Pattern II, D  4.9  The Expected Values of States, Flow Pattern II, D  3  max  max  max  max  = 180Mm  3  59  = 210Mm  3  64  max  = 210Mm  — 150Mm Max  66  3  71  3  = 150Mm  72  3  max  vm  — 180Mm . . 3  = 180Mm . . . . 3  max  73 74  Acknoledgement  I would like to take this opportunity to thank my advisor, Dr. S. O. Russell of the Civil Engineering Department of UBC for helpful suggesting the topic and for providing advice throughout the course of my research.  ix  Chapter 1  INTRODUCTION  1.1  STATEMENT OF THE PROBLEM  Achieving the most efficient operation of hydroelectric projects is a complex task. Firstly, there are uncontrollable and uncertain elements that affect the projects' operation, such as stream flow which is variable and difficult to forecast.  The value of optimization  techniques is dependent on the accuracy and the availability of information regarding the magnitude of future basin inflows. Secondly, there are multiple components ( reservoirs, canals, river diversions, power plants . . . etc.)  which must operate jointly. Thirdly,  conditions such as electricity demand and regulations are continuously changing due to the inherent dynamic nature of society and technology. Fourthly, there exist many conflicting, interests and constraints that influence the management of the system. Research efforts to improve the efficiency of hydro-electric power production have been mainly concentrated on mathematical models and solution techniques during the past few decades. Many different models and solution approches, such as Linear Programming (LP), Deterministic Dynamic Programming (DDP), Linear Programmig-Dynamic Programming (LP-DP), Stochastic Dynamic Programming (SDP), State Increament Dynamic Programming (SIDP), and Markov Decision Process (MDP) have been studied by researchers. Hydrologists are interested in building and studying stream flow forecasting models.  1  Chapter 1.  INTRODUCTION  2  Their purpose is to provide more accurate predictions about inflows. Many models relating to this issue have been built. Less efforts have been made to study the role of stream flow forecasting models in the optimization of hydro-electric power plant operation, specifically the extent to which more accurate inflow forecasts result in an improvement in operating efficiency. But, now, more and more researchers have noted that to justfy accurate information about inflows, the benefits and costs from using accurate runoff forecasts compared with using relatively inaccurate inflow information should be considered. In another words, the use of a complicated forecasting model must be based not only on the technical possibilities but also on the expected additional benefits it may provide. The expected additional returns give a clue as to how much money can be spent on research to find better stream flow forecasting models. It is well known that forecasting systems can be very costly, and the question that frequently arises is whether or not the benefits outweigh the cost. If we know the value of inflow information, it will help to make decisions about research proposals which aim at finding accurate inflow forecasting models and how much money one can spend on the project. When the budget for research and data gathering is higher than the additional benefits from the information, the work is difficult to justfy. Otherwise, the research work is worth doing. Unfortunately, the process of reservoir operation is complex, and the value of improved forecasting is very difficult to estimate. This is mainly because perfect predictions of stream flow are never available, especially for long term forecasting. There are inherent inaccuracies and these are difficult to quantify in a meaningful way. Stream flow, a stochastic process, is determined by many random factors which can not be foreseen. Short term forecasting can be more accurate than long term prediction. In some cases, such as when winterflowsresult entirely from groundwater, short term forecasts of runoff can be almost completely accurate, so they may be termed as 'perfect forecasts'. Of course, 'short term', here, is a fuzzy word. It may refer to hourly, daily, weekly or  Chapter 1. INTRODUCTION  3  even monthly prediction. In this study a hypothetical hydro project with realistic characteristics is used. Calculations are done on a monthly basis as it usual in power studies. The study first finds a long term discounted expected value for the output from the project when it is operated in an optimal way with no forecasts. Then various forecasts are assumed and their values assessed, with most effort concentrated on the one month ahead perfect forecast. It is much simpler to determine the value of perfect information or a perfect forecast than of information with some inaccuracies. The value of perfect information gives a useful upper bound on the value of information or of a forecasting system. The aim of this thesis is to develop a methodology for operating a typical hydro electric plant with a storage reservoir in an optimal way and to assign a value to perfect short term forecasts.  1.2  METHODOLOGY  The approach adopted to achieve the stated objectives was to use a stochastic dynamic programming model to optimize the operation of a hypothetical hydro-electric power station and then to obtain the long term discounted expected benefit of its output. Next, the process was repeated assuming perfect forecasts one month ahead. For the purpose of calculating the additional benefits due to the use of one month ahead perfect inflow information, some modifications had to be made to the pure stochastic dynamic programming model. The benefit increment shown by the model reflected the reduction of uncertainty in the inflow prediction. Thus the difference between the energy production with the stochastic dynamic programming model and the modified one assuming perfect information measure the value of the perfect stream flow forecasts. The major steps in the research are listed below: • Build the dynamic programming model for stochastic and one month ahead perfect  Chapter 1. INTRODUCTION  4  stream flow prediction. • Run the stochastic dynamic programming to get the one year optimal operating policy for the reservoir. • Repeat the optimizing process using the value of various water levels obtained from the last run until the results stablized in order to obtain the expected long term value of the output from the project— and the long term optimization policy for the hydro-electric projcet. • Run the modified dynamic programming model and assume that perfect one month ahead forecasts were available. Again repeat until the results stablized to find the value of operating with the accurate forecasts. • Analyze the results and draw conclusions. Details of the procedures are explained in the following chapters.  1.3  SOME OTHER PROBLEMS  Several aspects of the study are discussed here to define the problems encountered and the scope of the research.  1.3.1  SELECTION OF OBJECTIVE  FUNCTION  As mentioned above, water resource projects usually serve multiple purposes or multiple objectives.  Single objective projects rarely exist in the real world. Fortunately, our  purpose is to find the value of perfect stream flow information. Also the hydro-power plant which is studied is a hypothetical one, so there is no need to consider multiple objectives.  This helps concentrate our interests on the main purpose. On the other  Chapter 1. INTRODUCTION  5  hand, multiple objective problems are more complicated than single objective. If two or more objectives are involved in the study, it is very difficult to judge the value of perfect information, since the trade-off among objectives affects the optimal solution. Although there are some techniques which deal with multiple objective optimization problems successfully, these techniques are usually very complicated and are difficult to use in large scale optimization problems. Therefore, using only one objective to study the value of perfect inflow forecasting simplifies the study. The objective considered in this study is the energy production from the hypothetical hydro-power project. In order to measure the value of the perfect inflow and to make comparisons more meaningful, the energy production is transformed into monetary terms. However, the value of perfect information in this research refers only to the value in terms of increased energy production due to the reduction of uncertainty in predictions and the resulting increased operational efficiency of the reservoir. 1.3.2  THE WORK ENVIRONMENT OF T H E HYDRO POWER PLANT  As described in Section 1.1, a hydro-electric project must work as a part of a whole system which contains multiple reservoirs, hydro power plants, canals . . . etc., and it should be optimized under a given energy demand curve. Furthermore, in the real time operation of a hydro power station, the energy price varies with time which can cause complications in optimizing the operation of the reservoir. Such optimization problems require large scale systems analysis models and decomposition techniques will usually be needed. However, as indicated above, this study is only for a hypothetical project.  It is  assumed that the project will operate independently to simplify the model and its calculation. That is, the hydro-electric power plant can be optimized to generate maximum enery at all times without considering the shape of the energy demand curve and without  Chapter 1. INTRODUCTION  6  dealing with the combined optimization of multiple reservoirs operation. This is quite realistic for a research project. 1.3.3  E F F E C T OF T H E M A X I M U M GENERATOR DISCHARGE O N T H E OPERATION OF RESERVOIR  The constraint of the maximum amount of water which can be released from a reservoir through the power plant can affect the operational efficiency of a hydro-electric plant. This is because the limitation of the maximum discharge influences the total amount of water spilled from the reservoir during the high flow period. The amount of spilled water from the reservoir is an indicator of energy loss. The smaller the amount of water spilled from a reservoir, the better the operational efficiency for the hydro power project is. If the upper bound of the release is too low, it causes a larger amount of water to be spilled than does the case which has a higer upper bound of release. This will, of course, reduce the operating benefits from the hydro electric plant. On the other extreme, if there is no upper bound to the release of water through the turbines, there will be no water being spilled over the spillway. It will produce maximum energy for a given reservoir size and no energy loss any time during the high flow period. It is the ideal case, from an energy production point of view. However the maximum discharge constraint is determined by the capacity of the turbines installed in the power station.  This study examines the effects of changing  the upper bound of discharges by setting several release limits and then comparing the results. 1.3.4  O P E R A T I O N A L POLICIES W I T H V A R I O U S F L O W P A T T E R N S  Another interesting problem is the effect of different stream flow patterns on the operation policies of the hypothetical hydro power project. Here, a different flow pattern means that  Chapter 1.  INTRODUCTION  7  for a given amount of annual inflow, the distribution of the mean monthly stream flows and their deviations are different. Finding out how the operation policies and the output changes with the flow patterns is of the research significance. In this study, two different flow distributions have been examined, one typical of the British Columbia interior and one typical of the B.C. Coast. The results will be shown in the related section.  1.4  LITERATURE  REVIEW  Stream flow forecasts and predictions play a fundamental role in the operation of hydropower projects. The degree to which forecasts aid the operator in improving the operational efficiency is dependent on the quality of the forecasts.  Many researchers have  noted that it is worth deriving quantitative measures of the value of flow forecasts. Joanna Mary Barnard(1989)[l] investigated the value of inflow forecasting models in the operation of a hydro-electric reservoir. She considered how conceptual hydrologic forecasts could be used in combination with optimizing techniques to improve the operational efficiency of a hydro-electric project.  Her thesis used a stochastic dynamic  programming model and a simulation model to study the problem. Barnard compared the role of three different inflow forecasting models: a naive forecast, a conceptual forecast and a perfect forecast (actually recorded inflows), in the operation of a hydro-power station. Some interesting results were found. She concluded that the accuracy of the forecast is more important in influencing the value of the conceptual forecast than the magnitude of the flows. However, since there is no way of determining when a forecast will be accurate, a policy must be developed to decide when forecasting could be of use in long term power production planning. The approach used in this study is somewhat similar but with a different emphasis. Aris P. Georgakakos (1989) [2] evaluated the benefits of stream flow forecasting in  Chapter 1. INTRODUCTION  8  three specific systems: the Savannah River System in the state of Georgia, The High Aswan Dam, and the Equatorial Lake System. The approach taken was to simulate the performance of the systems under Extended Linear Quadratic Gaussian (ELQG) control with several stream flow models of varying forecasting power. The research concluded that probabilistic stream flow forecasting can considerably improve reservoir operation but the benefits are system specific. Nabeel R.Mishalani and Richard N.Palmer (1988) [3] studied the benefits of forecasting to a water supply system. Questions relating operational losses to forecast period and accuracy were addressed. Some simple available forecasting techniques were assessed for their accuracy and applicability. The issues were addressed through the use of a simulation model, where the system was modelled as a single purpose reservoir supplying municipal and industrial water. Their conclusions were: (1) reservoir operation deteriorates markedly with the loss of forecast accuracy; (2) the optimal length of forecasting period is five months; (3) reservoir operation efficiency may be improved by as much as 88 percent if perfect predictive abilities are available; (4) the mean of the historic data is not recommended for predicting future flows because Markov methods are always superior; and (5) lag one autoregressive Markov schemes exhibit about a 9 percent improvement in operation over no forecasting. Roman Krzysztofowicz (1983) [4] considered several fundamental questions about forecasting: How to optimally use categorical and probabilistic forecasts? What opportunity losses are expected to be incurred when forecast uncertainty is ignored? Why the classical contingency analysis is suboptimal? and what economic gains are to be expected from probabilistic forecasts.  Analytic solutions were derived for the optimal  and a nonoptimal (one that ignores forecast uncertainty) formulation of a single period quadratic decision problem with a categorical and probabilistic forecast of the state. He  Chapter 1. INTRODUCTION  9  concluded that: (1) Probabilistic forecasts can be at least as valuable as categorical forecasts, and categorical forecasts always have a nonnegative value if the decision maker accounts for the forecast uncertainty and employs the optimal (Bayesian) decision procedure. (2) No matter which forecasting method is used, an opportunity loss is always incurred by a decision maker who does not account for the forecast uncertainty. As a result, the actual value of a forecast may be negative.  (3) A classical procedure of  accounting for uncertainty of categorical forecasts by means of contingency analysis is suboptimal. It usually results in an opportunity loss and may also result in a negative actual value of the forecast. (4) Probabilistic forecasts are likely to be more valuable than categorical forecasts, even if used in suboptimal decision procedures. (5) The relative gains from probabilistic forecasts (over categorical forecasts) are likely to be greater for decision makers who employ suboptimal decision procedures ( which ignores the categorical forecast uncertainty) than for these who already employ optimal decision procedures (which accounts for the categorical forecast uncertainty). William W-G.Yeh et.al. (1982) [5] discussed the worth of inflow forecasts for reservoir operation. They used a simulation model to examine the benefits over a range of forecast accuracies and forecast periods of one month and longer.  The conclusions obtained  were that using the historical monthly means for estimates of streamflow rather than attempting any prediction, and releasing water on the basis of those estimates to generate hydro-power and minimize spill, produced amazingly good results which were only slightly worse than with reasonably good predictions. Thus, hydro-power benefit gains of several percent can be made either with good predictions on a month-to- month basis, or by using the historical means for estimates of stream flow and taking the uncertainty into account. Thus they suggested that until high confidence predictions can be established, use of the historical means is preferable. Labadie, et.al. (1981) [6] investigated the worth of short term rainfall forecasting for  Chapter 1.  10  INTRODUCTION  combined sewer overflow control. They addressed the question "what levels of forecast error can be tolerated before it is better to abandon adaptive ( anticipatory ) control policies utilizing forecast information in favor of simple reactive ( myopic ) control methods ?" Their study demonstrated that the expected forecast model errors are generally lower than the error threshold above which reactive policies become more attractive.  1.5  SUMMARY OF THE  THESIS  The thesis begins with a short introduction to the problem at hand, including some related research issues, as well as describing some of the work which has been done by other researchers. In addition to this introductary chapter, the thesis consists of four other chapters as summarized below: Chapter 2 establishes the Stochastic Dynamic Programming model used in this study. Next, it presents the modification used for analyzing the effects of the one month ahead perfect inflow forecast on the optimal operation of reservoir. Then, the solution techniques developed to solve the problem are discussed. The way to obtain the state transition probability matrices, both monthly and yearly, is also examined. Chapter 3 describes the inflow estimating method used in this research and some related problems. The results of forecasting inflow are also shown in the chapter. Chapter 4 presents the results of the optimization model and compares the results of the dynamic program for both stochastic and perfect inflow information. The analysis is then discussed. Chapter 5 presents the conclusions obtained from the research and some suggestions for further study.  Chapter 2  T H E STOCHASTIC DYNAMIC P R O G R A M M I N G MODEL  The hydro-electric reservoir under consideration is modelled with stochastic dynamic programming (SDP) to optimize the operation of the reservoir along similar lines to Barnard (1989). The solution technique is presented in the following sections. Since there are differences in reservoir operation between the pure stochastic stream flow situation and with the forecast inflows, the operation policies are decided in different ways. Therefore, they are modelled separately.  2.1  STOCHASTIC DYNAMIC PROGRAMMING M O D E L  Fig. 2.1 is a pictorial representation of the single resrvoir system having inflow Qt and making release decision D in each time period t, t=l,2, t  T.  The model uses periods of time, t, as the stage variables. In this case months are used ( a time period of a month can be used without loss of generality ), so that t=l,2, ..-,12. The stream flow input, Q , to the reservoir is a stochastic variable. Its probability t  distribution can be any type which is suitable to the inflow historial records. Assume that, for a particular period of time t, a set of inflows {Q\} and their corresponding probabilities, {-P[Qf]} , i=l,2,  I , are available. t  The problem is that in each month, t, with the stochastic stream flow input Q , the t  amount of water released from the reservoir, D , has to be determined so as to maximize t  11  Chapter 2. THE STOCHASTIC  DYNAMIC  PROGRAMMING  MODEL  12  Figure 2.1: Sequential Reservoir Operation Process the energy generation.  Therefore, the decision variable in the model is the quantity  of water to release during stage t. There are usually upper and lower bounds on the quantity of water which can be released from the reservoir. A n upper bound might be due to physical size limitations of the stream below the reservoir in question or, more likely, the maximum capacity of the turbines; while a lower bound might be required to maintain navigation or ecological water balance. In this case, the quantity of water that can be discharged through the turbines provides the upper bound and zero flow the lower bound. For calculation convenience, the decision variable is discretized with K-1 intervals between - D  m m  and  D . max  D = {Df} , k=l,2, k  Where Di = D  m i n  DK — Dmax  A release D?, which is one of the set of all possible releases, and which satisfies the  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  13  system govening equation ( which will be described below ) is called a feasible release. Let V be the state variable, which represents the volume of water in storage. The t  reservoir considered in this research has a predetermined maximum and minimum allowable volume, and they are represented by V  max  and V i m  n  .  For calculating convenience,  the state variables are discretized with the same unit of discretization as the decision variable D^. There are N discrete units or states. That is  n= 1,2, . . , N . where the minimum and maximum volumes are equivalent to Vi and VN respectively. That is v, = v  min  VN = V  max  With above variables definition, the stochastic dynamic programming model may be formulated as follows. 2.1.1  T H E STATE TRANSITION EQUATION  As indicated in Fig.2.1, according to the principle of continuity (or mass balance), the governing equation of the reservoir state transition relationship is Vt  +1  (2.1)  = V + Qt - D t  t  subject to V v  • < V< V  771171  t  _  *t  'max  Dmin < D < D t  m a x  which states that the storage volume at the begining of month t+1 is equal to the sum of the initial storage volume and stream flow input of month t minus the release made at the end of month t. If the final storage Vt+i is greater than V  max  ,  in order to  Chapter 2. THE STOCHASTIC  DYNAMIC  PROGRAMMING  14  MODEL  keep it within the state constraints, the volume of spill, St, will be required. S is given t  by:  V S = { t  t+1  - Vmax  0  when V  > V  t+1  maa  (2.2)  otherwise  Thus the state transition equation is  V  t+1  = V + Q - D - S t  t  t  (2.3)  t  where: V = total volume of water in storage at time period t; t  Qt = total volume of inflow at time interval t; D = volume of water released through the turbines at stage t; t  S = volume of spill required to stay within constraints at stage t. t  The decision process is to decide upon the release D to be made after observing the t  state V and V t  t+1  of the system, and maximize the objective function. The complexity of  the stochastic process optimization is that each of the possible inflows must be analyzed for each state and discharge. Therefore several potential end states V i exist correspondt+  ing to a initial state V and discharge D\. Once each possible inflow has been examined, t  expected values of discharges, spills, and hence energy generation are calculated.  2.1.2  STAGE RETURN EQUATION  The stage return function represents the benefit or cost, R(d), of the decision being made. The objective function of this study is maximum energy generation. Therefore, maximization of the objective function is needed during the decision process. The production of hydro-electric energy during any time period t is dependent on the installed plant capacity; the flow through the turbines; the average productive storage head; the number  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  15  of hours in the period; the plant factor; and a constant for converting the product of flow, head, and plant efficiency to kilowatt-hours of electrical energy, KWH. In order to calculate the energy generation, the mean volume should be computed first. Then it will be used to determine the average head. That is  H  averagett  = f{(V  t  + V )/2) t+1  (2.4)  Then energy can be determined as a function of average head and discharge as following  E = ^average* Q Tl = PT1 t  t  (2.5)  where P= power; E = energy generated in time period t; t  n — efficiency factor; 7 = density of water; H  aveTa  ge,t  =  average head of water above power house at t;  Qt = flow of water through turbines during time period t; T l = Number of hours in time period t. The value of 77, the efficiency factor, takes into account turbine efficiency, generation efficiency and other losses. Efficiency will actually vary with head, but can be considered constant if the range of head is relatively small. The value used in this study is 87% and would apply to ±20% of the design head.  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  2.1.3  MODEL  16  THE RECURSIVE EQUATION  In stochastic dynamic programming, the past is knomn but the future is uncertain. Thus the analysis has to proceed backwords in time. The stochastic dynamic programming recursive equation determines the maximum or minimum of the objective function which is dependent upon the value of the stage return function and the discounted value of the state at the previous stage. It starts with known or assumed values of the possible states at the end of the year. It is assumed that future returns are less valuable than present returns and hence, they are discounted by a discount factor 8 , 0 < 8 < 1 ; where 8 is the monthly discount factor. The discount factor is applied to the second part of the expression. Max  Ft{Vk) = dk  *=£«  E(W*  + ^-i(V«)))]  i=i subject to: V  min  < V ,V k  ki  < V  max  dmin < d < dmax k  where Pi= Probability of inflow qi in period t. Ft{Vk) =  Expected value of state 14 with optimal operation of the  reservoir with t remaining time periods; Rik = Return from generating power from discharge d with the reservoir in state k ( at time t ) and inflow Ft-i(Vki)  = Expected value of being in state V/y ( the state  reacted from state k with inflow g,- and discharge d ) with t-1 remaining time periods; dk — Discharge. Various values are tried to find the maximum value  (2.6)  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  17  oiF (V ). t  2.1.4  k  THE DYNAMIC PROGRAMMIMG MODEL FOR ONE MONTH AHEAD PERFECT INFLOW FORECAST  Assume that for a given time interval t, one month ahead perfect stream flow forecasts are available, or in other words, we know the stream flow input to the reservoir that will actually occur one month in advance. Problems are what is the value of this perfect inflow prediction, how to operate the reservoir based on the perfect flow information to improve the operational efficiency, and what are the extra benefits provided by the perfect forecast. Assume that all of the variable definitions are the same as above, and the one month ahead perfect forecasts are given in the form of a set of inflows [Qi}t and a set of probabilities [Pi] corresponding to the stream flow input, i=l,2, t  I . The t  recursive equation of this problem is: V= ' J  FtiYk) = £[  M a x d  k (Rik+f3F -i(Vki)]Pi t  (2.7)  i=l  where definitions as before. The difference between eq. (2.6) and eq. (2.7) is, for a given reservoir state, equation (2.7) first finds the optimal release policy for every forecast flow and the correspanding expected value for state 14 given inflow qi] and then takes the expected value of the state with all the possible inflows. Equation (2.6) finds the expected value first, then takes the maximized expect value as the optimal state value. The expected values calculated from eq.  (2.7) should be higher than the results  computed from eq. (2.6) since there is less uncertainty with the forecasts.  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  18  M A T H E M A T I C A L S T A T E M E N T F O R L O N G T E R M ANALYSIS  2.1.5  Long term reservoir operation is complicated unless the reservoir empties each year. This section formulates the problem when the reservoir does not empty each year. Assume that n years operation will be analyzed, n=l,2, • • •. and that it is possible to obtain the long term monthly state transition probability matrices in the form P£ (d) T+t  = Prob ( transition to state j in time t-fl  from state i in month t of the n'th year of the process where the release was d ) and it is possible to obtain monthly transition rewards of the form r? (d) = expected immediate return when making the monthly T+t  transition from state i in month t to month t+1, in the n'th year of the process, when the release is d. Assumption A: The expected immediate return is independent of the year of the process. That is  rf {d)=r\{d)in +t  (2.8)  Assumption B: The stream flow input to the reservoir is a stochastic process and is periodic in nature with a period of 12 months, T=12.  Pf (d) +t  = P%  (2.9)  When the releases D becomes stationary, that is the set of releases remain the same for all years of the process n=l,2,  the steady long term operation condition is reached.  D is the yearly release policy such that  D - (d ,d , • • • ,di+r-i) t  t+1  (2.10)  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  19  where d is the vector of discharges in month t ( one discharge each state). t  The monthly state transition probability matrices at time t will be  V  P M )  P^di)  PM)  PkW  P M )  PINM  PNIWN)  PUdN)  •  \ (2.11)  PNNVN)  j  where t=l, 2, • • - ,12, and Pij(d) is the probability of going from state i in time period t to state j in time period t-fl given discharge d^. It will be shown that the yearly transition returns and probabilities are functions of the monthly transition returns and probabilities. It is convenient to use matrix notation to show the functional relationship between the yearly and monthly transition returns and probabilities. The functional relationship between the yearly and monthly transition probabilities is  IP\D)  = P\d )P (d )-.-P (d ) 2  1  (2.12)  12  2  12  where IP (D) is the yearly transition probability of year one. 1  The proof is quite simple: a yearly transition is comprised of 12 individual monthly transitions, hence, the yearly transition probability matrix is the product of the twelve monthly transition probability matrices. Generally, for month t, the yearly transition probability matrix is  IP\D)  = P (d )P (d )--l  t+1  t  t+1  P'+r-^dt+T-MT  and hence the long term transition probability is  + 1= 1  (2.13)  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  20  P\d )P (d )---P (d ) 2  T  1  2  t  IP(D) = P (d )P\d )---P -\d _ ) T  IP\D)  j  T  T  l  T  l  0  0  (2.14)  IP (D) 2  IP {D) T  0 where IP(D) represents long term state transition probability matrix. The functional relationship between the yearly and monthly transition return, IR , l  of month t is  IR^D)  = R (d ) + 8P (d )R +i(d ) t  t  t  t  t  t+1  + --- +  f-'P'id^^d^)-.P - {d ^)R -\d + - it, t+T  2  T + 1= 1  t+T  t+T  t  T  ir  (2.15)  and hence IR(D) = (  R\d^)+  6P\d )R (d )- ---  +  2  1  2  r  /3 - P (d )P (d ) r  1  1  2  1  2  P - (d _ )i2 (d ) r  1  r  r  1  r  (2.16) R (d )+ T  T  6P (d )R (d ) T  1  B -'P {d )P\d ) T  V  + --- +  1  T  T  t  l  •••P -\d _ )R -\d _ ) T  T  T  2  T  l  )  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING MODEL  21  where IR(D) is the yearly transition return vector given release policy D. The proof goes as follows: the yearly transition return is comprised of twelve monthly transition returns, each of which must be discounted and multiplied by the probability of obtaining these monthly returns. Thus, we have a stochastic decision process where the expected immediate return vector IR(D) is defined by equation (2.16) and the transition probability matrix IP(D) is defined by equation (2.14). The long-run expected discounted return vector F ( where F = ( Fi,F ,--2  Ft =  ,F  T  ),  (Fl,~-,F!,~.,F ), t  N  and Ff = long term expected discounted return when in state i at month t ) satisfies the following, for every feasible release policy D ( a feasible policy D is one which assigns a feasible release to each state).  F = IR(D) + /3 IP(D)F T  (2.17)  and furthermore that the optimal long-run expected discounted return vector satisfies the functional equation  F" = D [IR(D) + /3 IP{D)F~] M  T  (2.18)  Equation (2.17) can be simplified as shown by the following derivations. Equation (2.11) F -rIR(D) + B IP(D)F T  is equivalent to the following set of equations:  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  F  1  F  F  2  T  =  R\d )^BP\d )F  =  R (d ) + BP (d )F  =  R (d ) + BP {d )F\{T  1  2  2  (2)  z  2  2  T  22  (1)  2  l  MODEL  + 1-1)  T  T  T  (T)  (2.19)  Proof: for some t  F = IR (D) + B IP (D)F l  t  t  t  (2.20)  t  remembering that  IR\D)  = R (d )+^ (*)^ (^+i) + --t  t  J  t+1  4  /9 - P'(*)P' (d ).--' r  1  +1  t+1  +P " ( i f+r  2  <  _ )^ - (d _ )(2.15) +:r  t+T  1  2  t+r  1  and JP'(Z>) =  P'(d )P (*+i) • • • P'^-H^+r-i)  (2-13)  f+1  t  so that equation (2.20) becomes  F  l  =  R (d ) + BP (d R (d ) t  t  t  B -'P\d ) T  t  + --- +  t+l  t+l  t+1  •••  P -\d _ )R^ -\d ^) t+T  +(3 P (d )-..P - (d _ )F T  t  t+T  t  T  t+T  1  t  t+T  l  Now assuming that equation (2.19) is correct  2  t+T  (2.21)  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  23  F* = R'idt) + 8P\d )F  (2.19-t)  t+1  t  pt+i  t -i  F  +T  Rt+ifa^) + 0pt+i(d )F  =  (2.19-t+l)  t+2  t+1  =  jR  *+i'-i(  dt+T  _) 1  +  i  gpt r-i +  ( d t + T i ) F  t  (2.19-t+T-l)  substitute the last equation (2.19 t+T-1) in the prior equation (2.19 t+T-2) and continue repecting this process until the final substitution for F  t+1  in equation (2.13 t)  has been made, in which case the following is obtained  F* =  R (d ) + 3P (d )R (d )- --t  t  t  B -'P\d ) T  t  +  t+1  t  t+l  •••  r  P - (d _ )R -i(d _,) t+T  2  t+T  t+T  2  t+T  +3 P\d ) • • • P'+r-^dt+T^F*  (2.22)  T  t  Since equation (2.22) is identical to equation (2.21) the proof is complete.  It is  obvious that equations (2.19) are in a simplier form than those of equation (2.17), the most important simplification being that each equation in (2.19) is a function only of monthly release policy while each equation in (2.17) is a function of a yearly release policy. Thus, the results can be extended to the simpler case, that is  F* = R {d ) + /3P'(rf )F l  t  F  r  = dt  (2.23)  t+1  t  [R (dt)-rf3P\d )F yt t  t+i  t  (2.24)  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING MODEL  2.2  24  O P T I M A L OPERATION O V E R ONE Y E A R PERIOD  2.2.1  SOLUTION TECHNIQUE  The operation of the reservoir over one year period was optimized on a monthly inflow in conventional fashion with dynamic programming to find the optimal release policies {D }, t=l,2,• • -,12, and to obtain the year end value of each state. t  Fig.2.2 is the flow chart of the SDP problem andfig.2.3presents the flow chart of the SDP with the one month ahead perfect forecast. Solution of the dynamic programming model yields optimal one year operating policies in the form of release sequences, the monthly state transition probability matrices, the monthly reservoir state sequences, and the year end values of states. The one year operation analysis was carried out starting with initial state and end with another state. The terminal state values should be estimated before beginning. After one year's operation a new group of terminal values will be produced. Obviously, this analysis will not lead to the steady optimal operating policies. So that the long term analysis is needed. This will be discused in the next section.  2.2.2  C A L C U L A T I O N OF STATE TRANSITION  PREOBABILITY M A -  TRICES An important problem in the dynamic programming is to obtain the state transition probability matrices for each stage of the calculation. This is because the long term operation analysis is based on information from the state transition probability matrices and the associated operating policies. The state transition probability P^ is defined as: the probability that state i, at the beginning of stage t, changes to state j , at the end of stage t, in the course of operating the reservoir.  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING MODEL  stsrt  • read reservoir information stages.statesy y^v^ dmlnd max  read inflow and its prob. X T=1,stages do for each time period  nextt  do for each potential volume v I do for each potential discharges d  next v  pass daily flow and calculate spill < yes  calculate new final volume  1  compute the energy 1save Information on optimal discharge  output results stop  Figure 2.2: The Flow Chart of the SDP Problem  25  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  start T  read reservoir Information: stages, states discharges, v,, vu ,vw , d - , d _ X  read inform and It's probability q, p  I  do for each time period, t=1,stage  I  do for each potential volume  nextt  I  next v  1  next q  do for each potential inflow do for each potential discharge^  pass dailly Inflow and calculate spill  1 1  yes  compute new final volum  T  compute the energy T  save Information on optimal discharge  output results stop Figure 2.3: The Flow Chart of the Perfect Forecast Dynamic Program  26  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  1  Pii  MODEL  27  • J  •• j  1  Pn  •  i  Pu  • •• Pii • • Pu  I  Pn  • ••  •  Pn  • •  Pu  Pu  According to the state transition equation V  t+l  =V + Qt-Dt  (2.1)  t  it is known that, for a given initial state V , when release variable, D , has been t  t  decided, the probability of the state transitioning to V i is equal to the probability of t+  the input variable Pg . When V \ > V 4  t+  , in order to keep the reservoir volume within  max  the maximum allowable volume, the spill variable, S , is needed. t  S = V t  t+l  - V  (2.2)  max  when V > t+ 1  therefore (2.3) V + Qt-D -St t+i In this case, a problem is arises from the spill water S . That is, there may be several t  t  t  Qt corresponding to V  . In this situation, the transition probability P - from V to  max  tJ  t  V x,t+i is defined as: ma  Pu  where Q E high flow , when S > 0 t  t  -  XI PQT  (2.25)  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  2.3  MODEL  28  ANALYSIS O F L O N G T E R M RESERVIOR O P E R A T I O N  Assume that information from one year's operation is available. The next step is to get the long term reservoir operation policies. W.F.Caselton and S.O.Russell developed an iteration techenique to solve this problem (1974). However, a simpler iteration process is to use the information obtained from the previous year's calculation as the new input data, and then perform another round of calculation with the stochastic dynamic program model. This process is repeated until the operation policies stabilize. This means that if the reservoir's operation starting from water level A, after one year's operation the terminal values assigned to each state at the beginning of operation should be equal to the year end value computed from the iteration. Thus the terminal values calculated for one year's operation became the initial values for the next year's iteration. After several iterations, the terminal values reach a steady state, suggesting that the initial estimated values are no longer influencing the optimal operation policy decision and the optimal release policy (i.e. action) will be stationary; that is, the set of releases D will remain the same for all years (iterations) n = 0,1,2, • • • of the process. Fig.(2.4) shows the iteration process to obtain the long term operating policies. After one year's operation, 12 state transition probability matrices, {PijY t=l,2,- - -, 12 were obtained. Based on these matrices, the yearly state transition probability matrix may be calculated by multiplying:  {^•}-{^} {^} ---{^} 1  2  12  (2-26)  Multiplying of the yearly state transition probability matrices yields the long term state transition probability matrix when the results of multiplying remain stationary.  {Pij}opt =  {Pij}{Pij}'--{Pij}  (2.27)  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  29  Input data Optimization over one year: optimizing operation for given annual Inflow using the revised Information from II as year end state values  Revised long term reservoir state values  Optimized year end values reservoir state changes and associated probabilities  Analysis of long term operation. Computes revised long term reservoir state values on basis of one year value and state change Information from I  output  Figure 2.4: Schematic of Iterative Cycle Used in the Study  Chapter 2. THE STOCHASTIC DYNAMIC PROGRAMMING  MODEL  30  where {Pij}opt denotes the long term steady state transition probability matrix. Thus the long term optimal year end values of states will be:  F" = {PiiUtF  (2.28)  where F" is the long term steady state terminal values while F is the one year's state terminal values. 2.4  MEASURING PERFECT  ADDITIONAL VALUES OF ONE M O N T H  AHEAD  FORECAST  Based on the information given from both the stochastic dynamic program and the optimization with the one month ahead perfect forecast, two different expected values for the reservoir states are obtained. Let F denotes the expected value obtained from e  the stochastic dynamic program and F represents the expected value of the one month p  ahead perfect forecast, then, the additional benefit due to using perfect information can be calculated by:  V Value = F — F, p  (2.29)  for each state. Alternatively, the additional benefit may by expressed by the form of percentage:  V Value =  F  p  F  F  s  " x %  (2.30)  Chapter 3  INFLOW FORECAST  3.1  P R O B L E M DESCRIPTION  The streamfiow forecast is a very important component of the dynamic programming model. The modelling of catchment behavior is quantitative whether reconstructing past precipitation-to-runoff behaviour or forecasting future runoff behaviour. However, all forecasts have some degree of uncertainty due to factors, such as rainfall, snowmelt, temperture, wind, solar radiation which are all random variables, which cannot be forecast exactly. Also catchment physical characteristics, such as size and shape, geology and soil type can not be accurately determined. There are two basic types of forecasting used to analyze or to model catchment behaviour: stochastic and deterministic. Deterministic models can be further divided into two main categories. One uses mathematical descriptions of the relevant catchment processes to estimate discharges, and is termed a 'conceptual' model. The other way considers the whole catchment as a system of subsystems from which output occurs as responses to input. The latter concept features a system 'box', that is characterized by a unique response function, h(t). The system approach concentrates on the operation performed by h(t) on an input, x(t), to produce an output, y(t), rather than considering the underlying physical reasoning.  Many deterministic catchment models have been  estiblished, such as the UBC watershed model, the MIT model etc. The statistical approach follows another line of thinking. It views streamfiow as  31  Chapter 3. INFLOW FORECAST  32  a stochastic process. The purpose is to generate an inflow series which has the same statistical characteristics as the historical observations. Two basic techniques are used for streamfiow generation. If the streamfiow population can be described by a stationary stochastic process, whose parameters do not change over time, then a statistical model may be fitted to the streamfiow. However the assumption that the process is stationary is not always plausible due to changes of runoff characteristics. An alternative scheme is to assume that precipitation is a stationary process and to predict the runoff sequence through an appropriate rainfall-runoff model of the river basin. The model used in this research is a simple statistical one. Because a typical watershed is studied, it is impossible to generate inflow sequences by setting up a relationship between rainfall and runoff. The method provides probability distributions of monthly inflows using statistical data about the inflow pattern in the form of monthly means and standard deviations. The forecast inflows take the form of a probability distribution for the monthly streamflows Q for each particular time interval t, (here a month). Q is assumed as a normally t  t  distributed random variable with mean pQ and standard deviation o~Q . In some cases t  t  the streamflows cannot be completely normally distributed because of the impossibility of negative flows. Thus an alternative treatment to avoid negative values is needed. Q  t  is a continuous variables, as shown in Fig.3.1. However in order to keep the calculations within manageable limits, the variables need to be discreted. In another words, the inflows are generated as a set of discrete values, each with an associated probability. The details of the approach are discussed in the next section.  Chapter 3. INFLOW FORECAST  33  Chapter 3. INFLOW FORECAST  3.2  34  T H E INFLOW FORECASTING M O D E L  3.2.1  T H E INFLOW INTERVAL  For convenience of analysis, the forecast streamfiow interval, AQ, has the same unit as the discretized decision variable, AD, and the state variable, AV. This simplification makes the calculation of the state transition equation in the dynamic programming much easier and more accurate. In fact, the real world reservoir volume will change continuously all the time. However, if the interval value be chosen carefully, the operation optimization can be quite accurate.  3.2.2  LOWER A N D UPPER VALUES  According to the characteristics of the normal distribution, the probability of variables occuring within the range of (pq —  3<TQ,PQ  +  3CTQ)  is equal to 99.73%. Thus it includes  almost all possible inflows which may occur during the time period and the lower value can be set as  QL  =  PQ  -  3CTQ  (3.1)  subject to: QL > 0 and the upper value will be  Qu = PQ + 3<r<5  (3.2)  But the values of QL and Qu usually do not meet the needs of inflow discretization. Thus the real lower and upper value used in this study are obtained by the following modification. For the lower value, firstly let  Chapter 3. INFLOW FORECAST  35  PQ - SCTQ N  ~  l  AQ  ( 3  -  3 )  then, if Nr, is not in integer, set Njj equal to the next lower integer. When NL < 0, then set NL = 0. For the upper value, let (3.4)  N = ^ V  if iVj/ is not an integer, set Nu equal to the next higher integer. Thus the real lower value will be  Q = Nx L  AQ  (3.5)  x AQ  (3.6)  L  the upper value will be  Q'u = N 3.2.3  v  S T R E A M FLOWS A N D THEIR PROBABILITIES  For each integer I, N  L  < I < N , let V  E  I  =  PQ  —  I x AQ Q  A  e  ;  = " « -  Po ei =  /  / x  X  A  ° - ^  AQ +  (3.8)  ^ —  (3.9)  Chapter 3. INFLOW FORECAST  36  (3.10)  Pi =  (3.11)  P = e-'-t  (3.12)  Qi = I x A g  (3.13)  3  Then the inflow,Qj is  and its probability, P(Qi),  is  Pprob(Ql)  2P +P X  2  + P  3  4  (3.14)  in order to ensure that the sum of the probabilities of all possible stream flows is equal to 1.0, the following modification for each probability is needed  Pprob{Ql) Yl Pprob{Ql)  (3.15)  The calculation procedures are indicated by the flow chart. For each month t, repeating the same calculation gives the set of streamflows and their probabilities of occuring. The model described above is based on the assumption that the monthly inflows are seasonally independent. There is no serial correlation among streamflows.  This is  generally not the real situation in a particular watershed, since the flow in one period might give some hints of the likely flows in the following periods, especially in short term flow forecasting. In real world problems, any assumption regarding to time series  Chapter 3. INFLOW FORECAST  37  begin  I  Fori = N to N 1 u epsilonl = (mu - l*deltaq )/sigma  epsilon2 = (mu - l*deltaq - 0.5delta)/ sigma epsilob3 = (mu - l*deltaq -0.5deltaq)/ sigma  P1 = e  (epsilon? 12)  I  p2 = e  -(epsilon? 12)  I  P3 = e  - (epsilonl 12)  Q(l) =l*deltaq  P(l) = (2*P1 + P2 +p3)/4 P'(I) = P(I)/^P(I)  stop Figure 3.2: The Flow Chart of Inflow Forecasts  o c  Chapter 3. INFLOW FORECAST  38  should be proved through statistical test. But in our test problem, the assumption was reasonable as the main interest lay in assessing the effects of stochastic forecasting on improving the operating efficiency of the reservoir, not on making accurate flow forecasts.  3.3  T H E I N F L O W STATISTICS  The research for this problem was carried out with a hypothetical hydroelectric project. For the purpose of convenience and comparison with other research, it was decided to use the data on reservoir size and inflows which had been used by Joanna Mary Barnard in her study (Barnard,1988). She described the watershed, the reservoir and its hydropower project and the inflow characteristics in detail. Here, the descripations are not repeated except for the main characteristics of the flow which have significant influence on the research. As already indicated in chapter I, two quite different flow patterns have been analyszed in this study. Table 3.1 presents the basic statistics describing the inflow sequence of flow pattern I. The mean annual runoff volume is 1245.5 M m (million cubic meters). 3  Figure 3.2 shows the mean annual hydrograph being used. The second flow pattern is presented below in table 3.2 and figure 3.3. Table 3.2 shows the distribution of mean monthlyflowsand their standard deviations during a one year period. Figure 3.3 is the corresponding mean annual flow hydrograph. It is named flow pattern II for comparing with flow pattern I. Note that: % means both the mean monthly flows and the standard deviations are in % of the total annual flow in table 3.2. Comparing flow pattern II with pattern I, it may be seen that, although the total annual flow for the two flow patterns is the same, the distribution of mean flows and their standard deviations are quite different. Table 3.3 shows the comparison of the two  ei 3. INFLOW FORECAST  Month  Table 3.1: Min inflow Mm 13.4 12.4 11.4 20.5 99.3 238.8 182.5 108.2 55.2 34.4 21.1 14.8 3  January February March Aprile May June July August September October November December  Flow Statistics of Flow Pattern I Max inflow Mean inflow Standard deviation Mm Mm Mm 34.1 17.7 4.3 20.8 16.2 2.6 29.7 4.4 17.5 79.1 44.0 16.6 262.1 182.0 45.0 491.2 334.2 67.1 373.0 273.0 57.6 294.2 155.2 38.1 178.1 91.0 31.5 95.9 54.7 14.0 64.9 37.1 11.9 41.5 22.9 5.5 3  3  3  Table 3.2: Inflow Statistics of Flow Pattern II mean Q standard deviation <r Month M m Mm % % January 76.72 6.16 4.31 53.68 February 5.97 74.36 2.89 35.99 March 5.91 73.61 2.59 32.26 April 99.64 8.00 2.03 25.28 May 12.68 157.93 2.89 35.99 June 12.62 157.18 3.51 43.72 8.62 July 107.36 3.51 43.72 August 4.56 2.22 27.65 56.79 September 5.54 69.00 3.20 39.86 October 10.59 131.90 5.17 64.39 November 9.85 122.68 4.62 57.54 December 9.84 122.56 3.82 47.58 3  3  Chapter 3. INFLOW FORECAST  Figure 3.3: The Mean Annual Hydrograph  40  Chapter 3. INFLOW FORECAST  Figure 3.4: Mean Flow Hydrograph of Flow Pattern II  41  Chapter 3. INFLOW FORECAST  42  Table 3.3: Comparing Flow Characteristics Flow Pattern I Flow Pattern II month January Feburary March April May June July August September October November December  Q 17.7 16.2 17.5 44.0 182.0 334.2 273.0 155.2 91.0 54.7 37.1 22.9  4.3 2.6 4.4 16.6 45.0 67.1 57.6 38.1 31.5 14.0 11.9 5.5  *IQ{%) Q 24.3 16.0 25.1 37.7 24.7 20.1 21.1 24.5 34.6 25.6 32.1 24.0  76.72 74.36 73.61 99.64 157.93 157.18 107.36 55.79 69.0 131.9 122.68 122.56  53.68 35.99 32.26 25.28 35.99 43.72 43.72 27.68 39.86 64.4 57.54 47.58  70.0 48.4 43.8 25.37 22.8 27.8 40.7 48.7 57.8 48.8 46.9 38.8  inflows. The peak flow of pattern I, which is equal to 334.2 Mm is much higher than the 3  peak flow of flow pattern II, which is 157.93 M m . There is only one peak flow period 3  that happens from May to September for flow pattern I and its peak flow period is quite obvious and different from the low flow. The total flow during that period is 83.1% of the total annual flow. The differences between Q  (334.2Mm ) and Q i 3  max  m  (16.2Mm ) 3  n  is 318 Mm . But forflowpattern II, there are two peak flow periods. Its first peak flow 3  period starts from May and ends with July. The second one is from October to December. The total peak flow is only 64% of the total annual flow, although the whole peak flow length is six months which is one month longer than the flow pattern I. The difference between Q  (157.93 Mm ) and Q 3  max  (56.79 Mm ) is only 101.14 Mm . That means 3  min  3  that its variance of mean monthly inflow is smaller than pattern I. On the other hand, the distribution of monthly deviations of flow pattern II shows that the monthly deviations in the low flow period are much greater compared with flow  Chapter 3. INFLOW FORECAST  43  pattern I. For example, in January, the deviation of flow pattern II is 70% of its monthly mean flow, while it is only 24.3% for pattern I at the same month. This means there is more dispersion of values in the set of low flows for pattern II. The next chapter will show that these flow patterns have a significant influence on the optimization results of the reservoir operation. 3.4  T H E O U T C O M E OF INFLOW FORECAST  Using the method described in section 3.2, the inflow set was derived based on the information given in the section 3.3. A computer program has been developed to calculate the stream flows and their probabilities. Table 3.4 represents the results for flow pattern I. It can be seen that there are only a few of possible flows which can occur during the low flow period due to the small values of the monthly mean and deviation. For example, there are four possible flows in December of the forecasting year, and they are (0,0.005), (15,0.471), (30,0.516), (45,0.008). Their mean value is 22.9 M m which is exactly equal 3  to the input mean value. On the other hand, the range of possible flows in the high flow period is larger than during the low flow period. In June of the forecasting sequence, for example, the flow ranges from 120 Mm to 540 Mm , resulting in 29 discrete values. 3  3  The mean value of the flows is equal to 334.0 Mm , slightly less than the input mean 3  flow which is equal to 334.2 Mm . 3  Table 3.5 shows the flows and probabilities for stream flow pattern II. The generated inflow set shows some differences between input mean monthlyflowsand mean values of the generated flows. Because of the larger values of deviations and the relatively high values of mean inflow in the low flow period, the range of possible flows is wider than flow pattern I. In December of the forecasting year, for example, the flow ranges from 0 to 270 Mm , which is much wider than the same month's possible flow with pattern I. 3  Chapter 3. INFLOW FORECAST  Table 3.4: Inflows and Probabilities ( Jan. Feb. March Apr. Q 0 0.021 0.002 0.026 0.014 15 0.086 0.780 0.970 0.783 30 0.199 0.028 0.191 0.249 45 0.000 0.000 0.343 60 0.226 75 0.071 90 0.011 .105 0.001 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330  44  Flow Pattern I) May June  0.001 0.004 0.008 0.017 0.031 0.052 0.077 0.103 0.123 0.132 0.127 0.109 0.084 0.058 0.036 0.020 0.010 0.004 0.002 0.001  0.001 0.001 0.002 0.004 0.006 0.010 0.016 0.024 0.033 0.045 0.056 0.068 0.078 0.085 0.089  Fig.3.5 shows the probability distribution of the generatedflowsfor pattern I in July. Tables 3.6 and 3.7 show the means of the input data and the generated values.  Chapter 3. INFLOW FORECAST  P(Q)  0.12  r  90  150 120  210 270 330 390 450 180 240 300 360 420 Q  Figure 3.5: Probability Distribution of Generated Flow  Chapter 3. INFLOW FORECAST  Q  Jan.  Feb.  46  March  Apr.  May  June  345  0.088  360  0.083  375  0.074  390  0.063  405  0.051  420  0.040  435  0.029  450  0.020  465  0.013  480  0.009  495  0.005  510  0.003  525  0.002  540  0.001 Aug.  Sept.  Oct.  Nov.  Dec.  0  0.002  0.003  0.000  0.008  0.005  15  0.005  0.011  0.012  0.110  0.471  30  0.013  0.030  0.103  0.395  0.516  45  0.029  0.067  0.324  0.382  0.008  60  0.056  0.117  0.375  0.099  75  0.090  0.165  0.160  0.006  Q  July  90  0.001  0.126  0.187  0.025  105  0.002  0.150  0.170  0.001  Table 3.4 Results of Inflow Forecast ( Flow Pattern I), continued  Chapter 3. INFLOW FORECAST  Q  July  Aug.  Sept.  120  0.003  0.154  0.124  135  0.006  0.136  0.073  150  0.011  0.103  0.034  165  0.018  0.067  0.013  180  0.028  0.038  0.004  195  0.042  0.018  0.001  210  0.057  0.007  225  0.073  0.003  240  0.088  0.001  255  0.099  270  0.103  285  0.101  300  0.093  315  0.080  330  0.064  345  0.048  360  0.033  375  0.022  390  0.013  405  0.008  420  0.004  435  0.002  450  0.001  Oct.  Nov.  Dec.  Table 3.4 Results of Inflow Forecast ( Flow Pattern I), continued  Chapter 3. INFLOW FORECAST  Table 3.5: Results of Inflow Forecast (Flow Pattern II) Jan. Feb. Mar. Apr. May June Q 0 0.043 0.021 0.015 15 0.061 0.044 0.037 0.001 0.001 30 0.081 0.079 0.076 0.006 0.002 45 0.099 0.120 0.126 0.025 0.001 0.005 0.112 0.154 0.169 0.071 0.004 0.012 60 75 0.118 0.166 0.184 0.147 0.012 0.024 0.114 0.152 0.162 0.216 0.029 0.042 90 105 0.103 0.117 0.116 0.227 0.057 0.067 120 0.085 0.076 0.067 0.170 0.096 0.095 135 0.066 0.042 0.032 0.091 0.135 0.120 150 0.047 0.019 0.012 0.035 0.161 0.134 165 0.031 0.007 0.004 0.009 0.162 0.134 180 0.019 0.002 0.001 0.002 0.137 0.119 195 0.011 0.001 0.098 0.094 210 0.006 0.059 0.066 225 0.003 0.030 0.042 240 0.001 0.013 0.023 255 0.005 0.012 270 0.001 0.005 285 0.002 300 0.001  48  Chapter 3. INFLOW FORECAST  Q  July  Aug.  Sept.  Oct.  Nov.  Dec.  0  0.007  0.028  0.035  0.012  0.011  0.006  15  0.015  0.072  0.062  0.018  0.018  0.012  30  0.029  0.137  0.096  0.027  0.029  0.023  45  0.050  0.196  0.128  0.039  0.042  0.039  60  0.077  0.213  0.149  0.051  0.058  0.060  75  0.104  0.174  0.151  0.064  0.075  0.084  90  0.126  0.107  0.134  0.076  0.089  0.106  105  0.136  0.050  0.103  0.086  0.100  0.121  120  0.131  0.017  0.069  0.093  0.105  0.125  135  0.112  0.005  0.040  0.094  0.103  0.118  150  0.086  0.001  0.020  0.090  0.094  0.101  165  0.058  0.009  0.083  0.080  0.078  180  0.035  0.003  0.071  0.064  0.054  195  0.019  0.001  0.058  0.048  0.035  210  0.009  0.045  0.033  0.020  225  0.004  0.033  0.022  0.010  240  0.001  0.023  0.013  0.005  255  0.015  0.008  0.002  270  0.010  0.004  0.001  285  0.006  0.002  300  0.003  0.001  315  0.002  320  0.001  Table 3.5 Results of Inflow Forecast (Flow Pattern II), continued  Chapter 3. INFLOW FORECAST  Table 3.6: Comparison Jan. 17.7 I flQ 17.7 0 flQ AQ{%) 0.0 July 273.0 I p 272.9 0 po Q  A<2(%)  -0.04  50  of Mean Monthly Flow ( Flow Pattern I ) Feb. Mar. Apr. May June 16.2 16.2 0.0  17.5 17.5 0.0  44.0 44.2 0.5  182.0 181.9 -0.05  334.2 334.0 -0.06  Aug.  Sept.  Oct.  Nov.  Dec.  155.2  91.0 91.0 0.0  54.7 54.7  57.1 57.1 0.0  22.9 22.9 0.0  155.1 -0.06  0.0  Table 3.7: Comparison of Mean Monthly Flow ( Flow Pattern II ) Jan. Feb. Mar. Apr. May June o p  Q  AQ(%) I flQ O flQ AQ(%)  76.7 80.5 4.9  74.4 75.4 1.4  73.6 74.2 0.8  99.6 99.7 0.06  157.9 158.0 0.04  157.2 157.3 0.08  July  Aug.  Sept.  Oct.  Nov.  Dec.  107.4 107.7 0.3  55.8 57.6 3.2  69.0 71.6 3.8  131.9 134.1 1.7  122.7 124.3 1.3  122.6 118.7 -3.1  Chapter 4  RESULTS  4.1  RESERVOIR  DESCRIPTION  The hypothetical reservoir and hydroelectric power plant used in this study is the same as that used by and described by Joanna Mary Barnard (1989). The project's characteristics and the data used in the study are presented in this section. 4.1.1  THE OBJECTIVE FUNCTION  As mentioned before, for the purpose of simplification, the only objective of the hydro power project considered in this study is energy production. Other goals, such as flood control or water supply are not considered directly. However, goals of water supply and environmental protection can be included in the minimum discharge requirement, (that is the discharge must be greater than or equal to a predetermined minimum release in any time period), while flood control may be incorporated into the constraints of maximum reservoir volume and maximum release. The values chosen for use in comparing policies was the total year end values of each state over the long term optimal operation. These values are appropriate only because of the assumption that the reservoir is operated independently and therefore the energy is of the equal value at all times of the operating year. Energy generation is given in Gigawatt Hours (Gwh) of energy generated over the whole year. The percentage of potential increase in energy production from using the one month ahead perfect stream flow forecast compared to the stochastic stream flow  51  Chapter 4. RESULTS  52  Table 4.1: Reservoir Constraints and Characteristics Item Value Reservoir Design Volume, Mm 495 Minimum Volume, Mm 270 Maximum Volume, Mm 765 Minimum discharge, Mm 15 Maximum discharge, Mm 150, 180, 210 State Incremental Value, Mm 15 Discharge Incremental Value, Mm 15 3  3  3  3  3  3  3  can be transformed into dollars. The value of energy is assumed at 20 mills or 2 cents per KWH which is equivalent to $20,000/Gwh. 4.1.2  RESERVOIR V O L U M E A N D DISCHARGE  The minimum reservoir volume used in this study is 270 Mm  3  (Million cubic meters)  while the maximum volume is 765 Mm and the reservoir's live storage volume is 495 3  Mm . 3  The minimum discharge is assumed to be 15 Mm per month. In order to examine the 3  effect of maximum discharge on the value of stream flow information, three alternative maximum discharges were used: 150 Mm , 180 Mm , and 210 Mm . 3  3  3  Table 4.1 is the  summary of the reservoir constraints and its characteristics. Both the state and decision variables, that is the reservoir volume and the release, were discretized with the same incremental value of 15 Mm . The number of the reservoir 3  states is 33. The state incremental value was chosen to obtain reasonable accuracy yet keep the number of states manageable. The reservoir elevation-volume curve is as shown in figure 4.1. In the computer program for calculating energy production, it is convenient to have a mathematical equation to express the relationship between the elevation and the reservoir  Chapter 4. RESULTS  53  volume. The best-fit mathematical expression is:  (4.1)  Elev = C + Ci * V + C * V  2  0  2  where Co = 32.7308 C = 0.078263 l  C =-0.00001 2  4.1.3  T H E ADDITIONAL E X P E C T E D VALUES OF ONE M O N T H A H E A D PERFECT  FORECAST  To obtain the additional value of the one month ahead perfect stream flow forecast, the pure stochastic dynamic programming model was first used to optimize the reservoir's operation. The expected value of the power output with this type of operation provided the basis of comparison. The same optimization procedure was then used with the one month ahead perfect runoff forecasts and the expected value again computed. The results of the two sets of computations showed the differences between stochastic (i.e. with no forecasts) and operation with the forecast. Equation (2.29) and (2.30) given the definition of the additional benefit of the perfect stream flow forecast. AValue = F - F p  (2.29)  t  or AValue =  (2.30)  The values of the different optimized situations are shown in the following sections. The following runs of the study models were performed 1. Stochastic forecast: for each of the maximum releases of 150 Mm , 180 Mm , and 3  3  210 Mm with the two inflow patterns, the stochastic dynamic programming model was 3  Chapter 4. RESULTS  54  Elevation, meters 140 i  120  100  80  60  40  0  200  400  600  800  1000  1200  1400  1600  Water volume, million cubic meters Figure 4.1: Reservoir Elevation-volume Curve  1800  Chapter 4. RESULTS  55  run to find the optimal reservoir operation policies and their values. Thus six operation policies were analyzed. 2. Perfect forecast: each of the cases corresponding to the stochastic dynamic programming were analyzed with the one month ahead perfect forecast dynamic programming model to make the comparison. In real world problems, making accurate predictions for the peak flow period is of great interest because the total amount of flow in the high flow period has a significant influence on the reservoir operation. The dynamic programming model using a combination of stochastic and perfect short term forecasts was developed for the purpose of investigating the additional expected values of longer term perfect stream flow forecasts.  That is,  assuming that one ( or two, three, • • • ) months of one month ahead perfect forecasts were available, and using an appropriate combination of the stochastic and perfect forecast models to find the additional expected values. For flow pattern I, the analyses performed were: one month perfect inflow prediction for June; two months of perfect inflow predictions for June and July; three months of perfect inflow predictions for June, July, and August; four months of perfect inflow predictions for June, July, August and September. For stream flow pattern II, optimization was performed for: one month perfect inflow prediction for May; two months of perfect inflow predictions for May and June; three months of perfect inflow predictions for May, June and October; four months of perfect inflow predictions for May, June, October and November. For these cases the general maximum release constraint was Z) =180 Mm /month. 3  max  Chapter 4. RESULTS  56  Table 4.2: Average Reservoir Operation Processes with D =lS0 Mm 2 4 12 1 6 7 9 11 3 5 8 10 Mm Sto. 405 405 405 405 451 645 735 705 635 569 492 405 I 405 405 405 405 451 645 765 705 635 569 492 405 II 405 405 410 410 465 645 765 765 635 585 500 405 III 420 420 420 420 480 645 765 765 635 585 500 420 IV 420 420 435 435 480 645 765 765 635 600 500 420 Perf. 435 435 435 435 495 645 765 765 765 699 607 435 3  max  3  4.2  OPERATION RESULTS F O R F L O W P A T T E R N I  4.2.1  GENERAL M A X I M U M DISCHARGE (£>  m o x  =180  Mm ) 3  The long-term optimal operation policy for stochastic dynamic programming shows that on average the reservoir starts with a water volume of 405 M m at the beginning of 3  January and ends at the same volume at the end of the operating year. The maximum volume, which occurs at July, reached 735 Mm , that is two states less than full. Figure 3  4.2 and Table 4.2 show the average operating regime of the reservoir. The table also contains the operating policies for the corresponding one month ahead perfect forecast and all the other runs. The average starting point for the one month ahead perfect forecast was 435 M m , two states higher than with no forecasts. 3  The  water level reaches the allowable highest point, that is the volume of 765 M m , in July 3  and keeps level until September.  At the end of the operating year, it returns to the  starting water level. Fig.4.2 shows the average operation with all runs. The figure shows that the less uncertainty in stream flow prediction, the higher the average water level is and the more stages where the reservoir reaches the full state. These results suggest that accurate inflow predictions do improve the operating efficiency of the hydro-electric plant. Table 4.3 shows the year end expected values of the states. Table 4.4 shows the  Chapter 4.  RESULTS  57  water volume, milliom cubic meters 800  i  700  600  500  300 U  I  I  I  I  I  I  I  I  I  I  I  L  time, month D=180 Sto.  One month perfect two months perfect  three months perfectfour months perfect one month ahead per.  Figure 4.2: The Reservoir Operation Processes with  D =180Mm  3  max  Chapter 4. RESULTS  Table 4.3: The Year state Sto. Mm 270 195.6 285 197.5 300 200.0 202.4 315 330 204.9 345 207.4 360 209.9 375 212.4 390 215.0 405 217.7 420 220.3 223.0 435 225.7 450 228.4 465 480 231.0 495 233.6 510 235.6 525 238.9 540 241.5 555 244.3 570 247.0 585 249.8 252.6 600 615 255.5 630 258.4 645 261.3 264.2 660 675 267.1 690 270.1 705 273.1 276.2 720 279.2 735 750 282.3 765 285.5 3  58  End Values of States with D =lS0 Mm I II III rv Perf. Gwh 199.0 203.9 206.7 210.0 212.9 200.9 205.8 208.5 211.9 214.9 203.4 208.3 211.1 214.4 217.4 205.8 210.8 213.5 216.8 219.9 208.3 213.3 216.1 219.3 222.4 210.8 215.8 218.6 221.9 224.9 213.4 218.3 220.1 224.4 227.4 215.9 220.8 223.6 226.9 230.0 218.5 223.4 226.3 229.5 232.6 221.2 226.2 229.0 232.2 235.3 223.8 228.8 231.6 234.8 237.9 226.5 231.5 234.3 237.6 240.7 229.2 234.2 237.1 240.3 243.5 232.0 236.9 239.8 243.0 246.3 234.6 239.6 242.4 245.6 249.0 237.2 242.2 245.0 248.2 251.7 239.3 244.2 247.0 250.2 253.7 242.6 247.5 250.4 253.6 257.1 245.2 250.1 253.0 256.2 259.7 248.0 252.9 255.5 259.0 262.5 250.7 255.7 258.5 261.8 265.3 253.5 258.5 261.3 264.6 268.1 256.3 261.3 264.2 267.4 271.0 259.2 264.2 267.1 270.3 274.0 262.1 267.1 270.0 273.3 277.0 265.1 270.1 272.9 276.2 280.0 268.0 273.0 275.9 279.1 283.0 270.9 275.9 278.8 282.1 286.0 273.9 278.9 281.8 285.1 289.1 276.9 282.0 284.8 288.2 292.2 280.0 285.1 288.0 291.3 295.4 283.0 288.1 291.0 294.3 298.5 286.1 291.2 294.1 297.4 301.7 289.3 294.4 297.4 300.6 305.4 max  3  59  Chapter 4. RESULTS  The expected values of state, Gwh  200  300  400  500  600  700  800  Water volume, million cubic meters, D=180 sto. forecast  one month perfect two months perfect  three months perfectfour months perfect one month ahead per.  Figure 4.3: The Expected Values of States, D  max  = 180Mm  3  Chapter 4. RESULTS  60  4.4: The Add itional Values of States with D =180 Mm perfect Stoch. state, Mm Sto., Gwh AV,Gwh values, $ % 270 195.6 17.3 334,000 8.8 17.4 285 197.5 348,000 8.8 300 17.4 348,000 8.7 200.0 315 202.4 17.5 350,000 8.6 330 204.9 17.5 350,000 8.5 207.4 8.4 345 17.5 350,000 360 209.9 17.5 350,000 8.3 375 212.4 17.6 352,000 8.3 8.2 390 215.0 17.6 352,000 405 8.1 217.7 17.6 352,000 420 220.3 352,000 17.6 8.0 435 17.7 354,000 7.9 223.0 450 225.7 17.8 356,000 7.9 228.4 465 17.9 358,000 7.8 480 360,000 231.0 18.0 7.8 495 233.6 18.1 362,000 7.8 510 235.6 18.1 362,000 7.8 18.2 525 238.9 . 364,000 7.7 540 241.5 18.2 364,000 7.7 18.2 555 244.3 364,000 7.6 570 247.0 18.3 366,000 7.6 585 249.8 18.3 366,000 7.5 18.4 600 252.6 368,000 7.5 615 255.5 370,000 7.4 18.5 258.4 630 18.6 372,000 7.3 645 261.3 18.7 374,000 7.3 660 264.2 7.2 18.8 376,000 7.2 675 267.1 378,000 18.9 7.1 690 270.1 19.0 380,000 705 273.1 19.1 382,000 7.1 276.2 720 19.2 384,000 7.0 735 279.2 19.3 386,000 7.0 750 19.4 388,000 282.3 6.9 765 285.5 19.5 390,000 6.8 max  3  3  Chapter 4. RESULTS  61  expected values with the one month ahead perfect stream flow forecast compared to the stochastic forecast. The additional values lie between 16.8 Gwh for the minimum volume of 270 Mm to 19.6 Gwh for the maximum volume of 765 M m ( see table 4.4 column 3 3  3  for details ). And the higher the water level is, the greater the additional expected value gains. When arranged as percentages, the additional values decrease with increasing reservoir state, from 8.6 percent of volume 270 Mm to 6.9 percent of volume 765 Mm . 3  3  This is because the base value in calculating percentage incremental value of volume 270 Mm  3  is equal to 195.6 Gwh, which is much lower than the base value of volume 765  Mm , which is equal to 285.5 Gwh. 3  Table 4.2 and 4.3 also show the results with one, two, three and four months of perfect forecasts by the dynamic program model. It can be concluded that along with the reduction in uncertainty, on average the reservoir water level will be higher and the values of the states will be greater, that is, the expected state values increase with the number of months which have perfect stream flow predictions in high flow season. Fig 4.3 shows the state's year end value increment curve. All of the expected values for part time perfect flow forecasting operations He between the pure stochastic flow operation and the complete year with one month ahead flow forecasting optimization. The outcomes with the peak stream flow period (i.e. four months ) perfect forecast indicated that the additional values of the state are very near to the state values with the whole year of one month ahead perfect forecasts. This suggests that perfect forecasts for the high flow period are more valuable than perfect forecasts for the low flow season. That is a reasonable result. First, from table 3.1, it is known that the amount of water input into the reservoir during that period is very high. The water volume and water level vary most in that time, and the energy production is obviously higher than in the low flow season. Thus accurate prediction of inflow during the high flow period has a significant influence in the operational efficiency of the reservoir. Second, from table 3.1,  Chapter 4. RESULTS  62  Table 4.5: Reservoir Operation Processes for D =210 Mm 1 2 4 3 5 7 11 6 8 9 10 Mm Sto. 480 480 480 495 510 660 735 690 670 660 622 I 480 480 480 495 510 665 735 740 740 699 650 II 480 480 480 495 510 665 765 745 740 699 650 III 480 480 480 495 510 665 765 765 745 699 650 IV 480 480 480 495 510 665 765 765 765 699 650 Per. 495 495 495 495 510 743 765 765 765 765 650 3  maj  12  3  480 480 480 480 480 495  it is seen that the flow variations during the high flow period is much greater than in the low flow season, which indicates the wide range of flow variance. Therefore perfect flow forecasts, which reduce uncertainty about the flows, should cause more improvement with high flows. 4.2.2  R E S U L T S W I T H M A X I M U M R E L E A S E D =210 max  Mm  3  For the case where the maximum release equals 210 Mm , the reservoir volume starts 3  on average at 480 Mm  3  and ends with the same state except that with the one month  ahead perfect forecast, which on average starts from water volume of 495 Mm and 3  returns to the same volume at the end of operating year. For the case of stochastic operation and the one month perfect forecast during the high flow period, the highest average water levels during the operation are both 735 Mm . In the other months during 3  the high flow period, the water levels with the one month perfect forecast are slightly higher than with the stochastic forecast. But in the other cases, the highest water level reaches the reservoir full state, that is V  = 765 Mm . Table 4.5 shows the differences 3  max  among the different forecasting cases. Comparing the corresponding water levels between having the maximum release limited to 210 Mm  3  and equal to 180 Mm , it was found that the water level in the low 3  Chapter 4. RESULTS  Table 4.6: Expected Values of States with state Sto. I II III Mm Gwh 270 202.3 205.1 208.3 210.2 285 205.6 208.4 211.6 213.5 300 208.4 211.2 214.4 216.3 315 212.5 215.4 218.5 220.4 330 215.0 217.9 221.0 223.0 345 217.9 220.8 224.0 225.9 360 221.0 223.9 227.1 229.0 375 223.1 226.0 229.2 231.1 390 226.3 229.2 232.4 234.3 405 228.9 231.9 235.0 236.9 232.2 235.2 238.3 240.3 420 435 233.6 236.6 239.8 241.7 450 236.7 239.7 242.9 244.8 465 239.8 242.8 246.0 247.9 480 242.2 245.3 248.4 250.3 495 244.6 247.7 250.8 252.7 510 247.3 250.4 253.5 255.4 525 250.4 253.5 256.6 258.7 540 252.6 255.7 258.8 260.9 555 255.8 259.0 262.0 263.9 570 258.5 261.7 264.8 266.7 585 261.1 264.3 267.4 269.3 600 263.4 266.6 269.7 271.7 266.2 269.4 272.5 274.3 615 630 268.2 271.5 274.5 276.5 645 270.5 273.8 276.7 278.9 660 273.3 276.6 279.5 281.6 675 276.4 279.7 282.7 284.8 690 279.4 282.7 285.8 287.6 705 282.4 285.8 288.8 290.6 720 285.6 289.0 292.0 293.9 735 287.7 290.1 294.2 296.0 750 291.9 294.1 298.4 301.0 765 295.1 298.5 301.6 303.6  63  D IV  max  =210 Mm Per.  3  214.7 218.1 218.6 221.4 221.4 224.3 225.5 228.5 227.9 231.0 229.6 234.0 232.7 237.1 235.2 239.3 237.0 242.5 240.5 245.2 244.3 248.2 248.0 250.1 250.1 253.2 253.3 256.4 255.7 258.8 257.3 261.3 259.6 264.0 262.6 267.2 264.3 269.5 268.8 272.7 271.7 275.4 275.4 278.1 277.6 280.4 280.5 283.2 283.2 285.3 285.6 287.7 287.5 290.6 289.3 293.8 291.4 296.8 293.4 299.8 297.5 303.1 300.1 305.3 303.2 309.6 307.3 312.9  3  Chapter 4. RESULTS  64  The expected values of state, Gwh 320  300  280  260  240  220  200  200  300  400  500  600  700  800  Water volume,million cubic meters d=210 sto. forecast  one month perfect two months perfect  three months perfectfour months perfect one month ahead per.  Figure 4.4: The Expected Values of States, D  max  - 210Mm  3  Chapter 4. RESULTS  65  Table 4.7: Additional Values of States D ,=210 Mm state Sto. Perfect Mm Gwh AV Gwh Value $ % 270 202.3 15.8 316,000 7.8 285 205.6 15.8 316,000 7.7 208.4 15.9 300 318,000 7.6 315 212.5 16.0 320,000 7.5 330 215.0 16.1 322,000 7.4 345 217.9 16.1 322,000 7.4 360 221.0 16.1 322,000 7.3 375 223.1 16.2 324,000 7.3 390 226.3 16.2 324,000 7.2 405 228.9 16.3 326,000 7.1 232.2 16.4 420 328,000 7.1 435 233.6 16.5 330,000 7.1 236.7 16.5 450 330,000 7.0 465 239.8 16.6 332,000 6.9 480 242.2 16.6 332,000 6.9 16.7 495 244.6 332,000 6.8 510 247.3 16.7 334,000 6.8 250.4 16.8 525 336,000 6.7 252.6 16.9 540 338,000 6.7 555 253.8 16.9 338,000 6.6 570 258.6 16.9 338,000 6.5 585 261.1 17.0 340,000 6.5 263.4 17.0 600 340,000 6.5 615 266.2 17.0 340,000 6.4 268.2 17.1 630 342,000 6.4 645 270.5 17.2 344,000 6.4 660 273.3 17.3 346,000 6.3 276.4 17.4 675 348,000 6.3 690 279.4 17.4 348,000 6.2 282.4 17.4 705 348,000 6.1 720 285.6 17.5 350,000 6.1 287.7 17.6 735 352,000 6.1 750 291.9 17.7 354,000 6.0 765 295.1 17.8 356,000 6.0  3  ma  3  er 4.  RESULTS  water volume, million cubic meters 800  750  700  650  600  550  500  450  1  2  3  sto. forecast  4  5  6  7  8  9  time, month d=210 one month perfect  10  11  12  two months perfect  three months perfectfour months perfect one month ahead per. Figure 4.5: The Reservoir Operation Processes, D  max  = 210MTO  3  Chapter 4. RESULTS  67  Table 4.8: Operation Processes of Flow Pattern I, D 1 2 4 3 6 7 9 5 8 Mm Sto. 360 360 360 389 488 660 715 660 564 I 360 360 360 389 488 665 720 665 564 II 360 360 360 389 488 665 735 665 564 III 360 360 360 389 488 665 750 665 580 IV 375 375 375 389 488 665 765 670 580 Pec. 390 390 390 420 541 665 765 670 645  max  " 150Mm 12 10 11 3  3  516 516 516 535 535 550  360 360 360 360 375 390  495 495 495 495 510 540  flow period for the former case was higher than with the latter operation. For example, the average starting point for the former is 480 M m , but for the latter is 405 Mm . 3  3  The reason is that the ability to adjust high flows for the former is stronger than the latter. Because it allows the maximum release of 210 M m , the reservoir water level can 3  be kept high to provide more energy production without the risk of reservoir spill. However, when the maximum allowable release is equal to 180 M m , 30 M m lower 3  3  than 210 M m , its adjustable ability for flood flows is weaker. ' Because of the lower 3  discharge capacity, the water level should keep lower before the high flow period. As a result, the energy production will be smaller than when the maximum allowable release is 210 M m . In the next section, it can be seen that the average starting reservoir volume 3  for release limit of 150 M m case is even lower. Therefore, for the purpose of providing 3  more energy production, a high release limit is recommanded if possible. Tables 4.6 and 4.7 show the state values. For the stochastic forecast operation (see column 2 of table 4.6), the state values range from 202.3 Gwh for the minimum reservoir volume of 270 M m to 295.1 Gwh for the maximum water volume of 765 M m . The state 3  3  long term year end values with the one month ahead perfect forecast (see column 7,table 4.6) vary from 218.1 Gwh to 312.9 Gwh corresponding to the minimum and maximum  Chapter 4. RESULTS  Table 4.9: The Year End Values of States state Sto. I II III Mm Gwh 270 194.2 197.0 203.3 207.1 285 196.1 200.0 205.8 209.5 198.5 202.5 208.4 211.2 300 315 200.9 203.6 210.2 214.7 330 203.3 205.9 213.7 216.4 345 205.7 207.5 215.5 218.9 360 208.0 211.8 217.8 220.1 210.5 213.2 219.3 223.5 375 390 213.3 216.4 221.5 227.3 405 215.7 218.3 224.2 229.9 218.2 220.6 226.6 232.7 420 435 220.5 223.3 230.0 233.4 450 223.1 226.0 233.5 235.9 465 225.6 229.2 235.5 238.2 480 228.3 232.1 237.7 241.5 495 230.6 234.7 240.0 243.9 510 233.5 237.4 243.1 247.0 525 235.9 239.6 247.4 249.6 238.5 242.1 250.1 252.4 540 241.2 245.6 253.3 256.5 555 570 243.9 248.8 255.4 259.2 246.6 250.3 257.8 262.7 585 249.4 252.6 259.3 264.4 600 252.2 256.0 262.0 265.9 615 630 255.0 259.2 265.3 267.8 645 257.9 262.1 268.6 269.7 660 260.8 264.4 271.1 273.2 263.7 267.6 274.0 276.1 675 690 266.7 269.9 276.8 279.3 705 269.6 273.3 279.3 281.5 272.6 275.9 281.6 284.1 720 735 275.7 277.8 284.5 287.3 750 278.8 280.0 286.3 289.8 765 281.9 284.0 288.6 292.5  68  D IV  max  = 150Mm Per.  3  210.5 212.9 215.4 217.6 219.5 221.2 223.3 226.4 228.6 231.2 234.3 237.7 240.0 242.4 244.8 247.7 249.8 252.6 256.8 259.0 262.1 264.4 266.2 269.1 271.5 274.2 276.8 279.2 282.7 285.2 287.9 291.8 294.7 297.9  214.8 217.1 219.5 221.9 224.3 223.3 229.1 231.6 234.4 236.9 239.4 241.7 244.4 247.0 248.7 252.1 255.0 257.5 260.2 263.0 265.8 268.5 271.4 274.3 277.1 280.1 283.1 286.2 289.3 292.3 295.4 298.5 301.7 304.9  3  Chapter 4. RESULTS  69  Table 4.10: The Additional Expected Values D = 150Mm state Sto. Perfect Mm Gwh A V Gwh Value $ % 194.2 20.6 270 412,000 10.6 285 196.1 21.0 420,000 10.7 300 198.5 21.0 420,000 10.6 315 200.9 21.0 420,000 10.5 330 203.3 21.0 420,000 10.3 205.7 345 21.0 420,000 10.2 360 208.0 21.1 422,000 10.1 375 210.8 21.1 422,000 10.0 213.3 21.1 390 422,000 9.9 405 215.7 21.2 424,000 9.8 218.2 21.2 420 424,000 9.7 435 220.5 21.2 424,000 9.6 223.1 21.3 450 426,000 9.5 225.6 21.4 465 428,000 9.5 228.3 21.4 480 428,000 9.4 495 230.6 21.5 430,000 9.3 510 233.5 21.5 430,000 9.2 525 235.9 21.6 432,000 9.2 540 238.5 21.7 434,000 9.1 241.2 21.8 555 436,000 9.0 570 243.9 21.9 438,000 9.0 585 246.6 21.9 438,000 8.9 249.4 22.0 600 440,000 8.8 252.2 615 22.1 442,000 8.8 630 255.0 22.1 442,000 8.7 257.9 22.2 645 444,000 8.6 660 260.8 22.3 446,000 8.6 675 263.7 22.5 450,000 8.5 266.7 22.6 690 452,000 8.5 269.6 22.7 705 454,000 8.4 720 272.6 22.8 456,000 8.4 275.7 22.8 735 456,000 8.3 750 278.8 22.9 458,000 8.2 765 281.9 23.0 460,000 8.2 max  3  3  Chapter*  RESULTS  70  reservoir volume. Comparing the two results it was found that the additional expected values with the perfect forecasts range from 7.8 % to 6.0 %.  4.2.3  R E S U L T S W I T H F L O W P A T T E R N I, D  MAX  = 150Mm  3  Tables 4.8, 4.9 and 4.10 tabulate the results of the production runs for maximum release limited to 150 Mm . In the case with the stochastic flow forecast, the reservoir volume 3  starts at an average of 360 Mm at the begining of January and ends at the same point 3  at the end of December.  The maximum volume of the year's operation is 715 Mm  3  (not reaching the reservoir full state). But with the perfect inflow forecast, the average starting point for the reservoir operation is 390 Mm and the highest water level reaches 3  the reservoir full state. The long term year end expected state values vary from 194.2 Gwh for water volume 270 Mm to 281.9 Gwh for water level 765 Mm in the case of 3  3  the stochastic inflow forecast, and from 214.8 Gwh to 304.9 Gwh in the case of perfect flow prediction. The additional expected values due to the perfect forecast go from 20.6 Gwh to 23.0 Gwh, which are 10.6% to 8.2% improvements compared to the stochastic flow forecast.  4.3  T H E OPERATION RESULTS W I T H F L O W P A T T E R N n  This section describes the results with the second general flow pattern. This pattern is typical of the B.C. Coast. As shown in section 3.3, although the mean annualflowis the same as flow pattern I, its distribution of mean monthly flows and standard deviations are quite different. The operation results demonstrated that the mean stream flow Qi, i=l,2,  12 and their standard deviations DQ have significant influence on the energy {  production. Table 4.11 shows the operating pattern of the reservoir. With the stochastic dynamic  Chapter 4. RESULTS  71  water volume, million cubic meters 800  i  700  600  500  400  1  2  3  4  5  6  7  8  9  10  11  12  time, month d=150 sto. forecast  one month perfect  two months perfect  three months perfectfour months perfect one month ahead per.  Figure 4.6: The Operation Processes, D  max  = 150Mm  3  oter 4. RESULTS  The expected values of state, Gwh 320  300 280 260 240 220 200 180 200  300  400  500  600  700  800  Water volume, million cubic meters D=150 sto. forecast  one month perfect  two months perfect  three months perfectfour months perfectone month ahead per.  Figure 4.7: The Expected Values of States, D  Max  = 150M771  3  Chapter 4.  RESULTS  water volume, million cubic meters 800  750  700  650  600  550  500  450  1  2  3  sto. forecast  4  5  6  7  8  9  10  11  12  time.month d=180 one month perfect  two months perfect  three months perfectfour months perfect one month ahead per.  Figure 4.8: The Reservoir Operation Processes, Flow Pattern II, D  max  — 180Mm  3  Chapter 4. RESULTS  Flow Pattern II The expected values of state, Gwh 340  I  320 -  200  300  400  500  600  700  800  Water volume, million cubic meters D=180 sto. forecast  one month perfect  two months perfect  three months perfectfour months perfect one month ahead per.  Figure 4.9: The Expected Values of States, Flow Pattern II, D  max  = 180Mm  3  Chapter 4. RESULTS  75  Table 4.11: Average Operation Pattern with D = 180Mm 1 2 4 3 5 6 7 Mm Sto. 750 650 585 510 594 650 705 I 750 650 590 510 630 675 720 II 750 660 605 510 630 675 735 III 750 660 605 525 630 675 735 IV 750 660 605 525 630 690 750 Per. 750 660 645 570 660 705 765  General Flow Pattern II and  3  max  8  9  10  11  12  632 632 632 632 645 645  600 615 615 615 615 615  675 675 690 705 705 705  750 750 750 750 750 750  750 750 750 750 750 750  3  program, the average reservoir operation begins from volume 750 M m , which is near 3  the reservoir full state. This is because inflows are high and range from 131.7 Mm to 3  122.56 Mm from October to December in the previous operating year. There are two 3  peak flow periods during one year. The other peak flow starts from May and ends in July. Therefore there are two high water level periods in the reservoir operation process. In the first high flow period, that is from May to July, the highest water storage reaches 705 Mm at July. And then it becomes lower until September, when the water volume 3  reduces to 600 M m . At the end of December it returns to 750 M m again. 3  3  In the case with the one month ahead perfect forecast, the average starting water volume at the begining of January is also 750 M m , and it ends with the same volume 3  at the end of the operating year. The highest water volume is 765 M m , which is the 3  reservoir full state, and that happens in July. It can be seen that the end states with the perfect forecast exceeded or were equal to the monthly end states of corresponding months in the stochastic forecast case due to reducing the uncertainty. Note that there are similar outcomes in the case of the operation with flow pattern I. This is a clear indication of the improvement in reservoir operational efficiency. In the other operation cases, that is with part time perfect stream flow forecasts, the month end states become  Chapter 4. RESULTS  Tab: e 4.12: state Mm 270 285 300 315 330 345 360 275 390 405 420 435 460 475 480 495 510 525 540 555 570 585 600 615 630 645 660 675 690 705 720 735 750 765 3  76  The Year End Values Sto. I II Gwh Gwh Gwh 197.2 198.6 201.3 199.2 200.6 203.3 203.0 204.4 207.1 205.4 206.9 209.5 208.4 209.9 212.5 211.5 213.0 215.7 214.6 216.1 218.8 217.2 218.8 221.4 220.6 222.2 224.9 223.9 225.5 228.1 227.0 228.6 231.3 230.1 231.7 234.4 233.3 234.9 237.6 236.3 237.9 240.6 239.4 241.1 243.7 242.5 244.2 246.9 245.7 247.4 250.1 249.4 251.1 253.8 253.0 254.7 257.4 256.7 258.4 261.2 259.5 261.2 264.0 263.5 265.2 268.0 266.1 267.9 270.6 269.4 271.2 273.9 272.3 274.3 276.9 275.2 277.0 279.8 278.4 280.2 283.0 281.6 283.4 286.3 284.2 286.0 288.9 287.1 289.0 291.8 290.2 292.1 294.9 293.4 295.3 298.3 295.7 297.6 300.5 298.8 300.7 303.6  of States D III IV Gwh Gwh 203.5 206.1 205.5 208.1 209.3 211.9 211.7 214.3 214.7 217.3 217.9 220.5 221.0 223.6 223.6 226.2 227.1 229.6 230.4 232.9 233.5 236.0 236.6 239.1 239.8 242.4 242.9 245.4 246.0 248.5 249.1 251.6 252.3 254.9 256.0 258.6 258.6 262.2 263.4 265.9 266.2 268.8 270.2 273.8 272.8 275.4 276.1 278.7 279.1 281.6 282.0 284.5 285.2 287.7 288.4 291.0 291.0 293.6 294.0 296.5 297.1 299.6 300.3 302.9 302.6 305.2 305.7 308.3 max  = 180Mm Per. Gwh 214.4 216.5 220.7 222.8 225.9 229.0 232.1 234.8 238.3 241.7 244.9 248.1 251.4 254.5 257.7 260.8 264.1 267.9 271.6 275.4 278.2 282.3 285.0 288.4 291.4 294.4 297.7 301.0 303.6 306.6 309.8 313.1 315.5 318.7  3  Chapter 4. RESULTS  Table 4.13: The Additional Values of Forecasts with Flow Pattern II, D state Sto. Perfect Mm Gwh AV (Gwh) Value ($) % 270 197.2 17.2 344,000 8.7 285 199.2 17.3 8.7 346,000 300 202.3 17.4 348,000 8.6 315 205.4 17.4 348,000 8.5 208.4 17.5 330 8.4 350,000 345 211.5 17.5 350,000 8.3 360 214.6 17.5 350,000 8.2 375 217.7 17.6 352,000 8.1 390 220.6 17.7 354,000 8.0 405 223.9 17.8 356,000 7.9 420 227.0 17.9 358,000 7.9 435 230.1 18.0 360,000 7.8 450 233.3 18.1 362,000 7.8 465 236.3 18.2 364,000 7.7 480 239.4 18.3 366,000 7.6 495 242.5 18.3 366,000 7.6 245.7 18.4 510 368,000 7.5 525 249.4 18.5 7.4 370,000 540 253.0 18.6 372,000 7.4 555 256.7 18.7 374,000 7.3 570 259.5 18.7 374,000 7.2 585 263.5 18.8 376,000 7.1 600 266.1 18.9 378,000 7.1 269.4 19.0 615 380,000 7.1 630 272.3 19.1 382,000 7.0 645 275.5 19.2 384,000 7.0 660 278.9 19.3 386,000 6.9 675 281.6 19.4 388,000 6.9 284.2 19.4 690 388,000 6.8 705 287.5 19.5 390,000 6.8 720 296.2 19.6 6.7 392,000 293.4 19.7 735 394,000 6.7 750 295.7 19.8 396,000 6.6 765 298.8 19.9 398,000 6.6  max  3  = 180M-  Chapter 4. RESULTS  78  higher as the number of months which have perfect inflow predictions increase (see table 4.11 for details). Table 4.12 represents the year end expected values of the states for all operating runs. Table 4.13 shows the additional expected values with one month ahead perfect inflow forecasts compared to the results with pure stochastic stream flow prediction. From the second column of table 4.12, it can be seen that the state year end expected values, for stochastic flow forecast, range from 197.2 Gwh for 270 Mm to 298.8 Gwh for 2  765 Mm . Column 7 shows the state's year end expected values with perfect stream flow 3  forecasting. The minimum and maximum expected values are 216.4 Gwh and 315.9 Gwh, which correspond to the lower and upper bounds of water volume respectively. Fig (4.9) plots the year end expected values against the reservoir states. This shows that there is a similar relationship between states and their year end values as with flow pattern I.  Chapter 5  CONCLUSIONS  This chapter presents the general conclusions obtained from the research. In Chapter I, several questions were posed: • What is the value of one month ahead perfect inflow forecasts to reservoir operation? The answer could be used as a guide to whether or not it is worth making stream flow predictions. • What benefits can be gained from varying term perfect streamflowforecasts in the highflowseason? • What is the effect of the maximum power plant capacity on the opportunities for improving the reservoir operating efficiency? • How do different stream flow patterns affect the expected benefits which could be obtained by reducing the uncertainty about future flows with forecasting? The results presented in chapter IV have already partially answered these questions. They are summarized in this chapter and the results of some further analyses and conclusions are presented.  5.1  ADDITIONAL BENEFITS OF P E R F E C T INFORMATIOM  As stated in chapter I, the main purpose of this study is to find an upper limit to the value of flow forecasts by computing the expected value of perfect short term inflow 79  Chapters.  CONCLUSIONS  80  Table 5.1: Summaty of Additional Benefits with Forecasts Flow Pattern I, $ Flow Pattern II, $ D ax m  I II III IV Per.  forecasts.  = 150 Mm  3  Dmax = 180Mm  70,000 170,000 226,000 290,000 354,000  76,000 196,000 242,000 318,000 422,000  3  D  max  = 210Mm  3  62,000 124,000 162,000 270,000 334,000  Dmax = 180Mm  3  38,000 96,000 138,000 190,000 396,000  The expected values of the perfect forecast is the value of the extra energy  production possible with the perfect inflow predictions compared to the operation with pure stochastic flow estimates. The value of energy is assumed to be $20,000/Gwh, so that the benefits can be evaluated approximately in dollar terms. Table 5.1 summarizes the expected additional benefits with perfect inflow information, which correspond to the optimal long term operational policies of the hyphothetical hydroelectric power project for each production run. It may be seen from the table that the perfect flow forecasts, both for one month ahead perfect forecasting and for part time perfect flow forecasting in the high flow season, improve the operational efficiency considerably by reducing the forecasting uncertainty. For example, when the allowable maximum release D x ma  —  180Mm , in the case offlowpattern I, the perfect forecasting 3  could generate $354,000 more than the stochastic forecast. The potential improvement due to perfect forecasting with flow pattern II could achieve $396,000. The potential additional expected value of perfect inflow information forflowpattern I is slightly lower than with flow pattern II. It is obvious that the effort of making pefect stream flow forecasting is worth doing when the cost of the research work is not greater than the extra expected benefits, that is $354,000 forflowpattern I and $396,000 forflowpattern II when the maximum allowable  Chapter 5. CONCLUSIONS  81  discharge is equal to 180 Mm . 3  5.2  T H E E F F E C T OF M A X I M U M RELEASE ON T H E POTENTIAL BENEFIT  This section discusses the role of the maximum allowable release (which depends on the generating capacity) in the improvement of reservoir operational efficiency. Table 5.2 compares the percentage and the net potential additional expected state values with different maximum release capacities for flow pattern I. It is interesting to note, from both Table 5.1 and 5.2, that the largest additional expected value among the three situations is when the maximum allowable release is limited to 150 Mm . The additional expected state values 3  vary from 10.6 % of reservoir volume 270 Mm to 8.2 % of volume 765 Mm . When 3  3  the maximum discharge is 180 Mm , The extra values He between 8.6 % and 6.9 % . In 3  the case of release limitation being 210 Mm , the potential expected benefits go from 3  7.8 % to 6.0 % respectively.  That is, the additional expected values reduce when the  discharge limitation increases. The results indicated that perfect inflow information is more valuable when the maximum allowable release is smaller for the same reservoir size. The explanation of this phenomenon is as follows: The 'adjusting ability' of the reservoir is limited by the maximum release capacity. When D  max  is small, the energy  production is affected greatly by the release constraint and a lot of water may be spilled through spillway. Thus the perfect inflow prediction provides more improvement in reservoir operational efficiency. If D  max  is large, the effect of improving operational efficiency  by perfect inflow forecasting will be smaller relatively which suggests less additional expected values may be obtained. It can be concluded, from the above results, that if a hydroelectric power project has a strict limitation on discharges, it is more worthwhile  Chapter 5. CONCLUSIONS  82  Table 5.2: Additional Expected Values of Different D. = 180Mm D = 210Mm state D = 150Mm D Mm $ $ $ % % % 10.6 412,000 270 8.8 334,000 7.8 316,000 10.7 420,000 285 8.8 348,000 7.7 316,000 10.6 420,000 8.7 348,000 7.6 318,000 300 315 10.5 420,000 8.6 350,000 7.5 320,000 10.3 420,000 8.5 350,000 7.4 322,000 330 10.2 420,000 8.4 350,000 345 7.4 322,000 10.1 422,000 360 8.3 350,000 7.3 322,000 375 10.0 422,000 8.3 352,000 7.3 324,000 8.2 352,000 390 9.9 422,000 7.2 324,000 405 9.8 424,000 7.1 326,000 8.1 352,000 9.7 420 424,000 8.0 352,000 7.1 328,000 435 9.6 424,000 7.9 354,000 7.1 330,000 9.5 450 426,000 7.9 356,000 7.0 330,000 9.5 428,000 465 7.8 358,000 6.9 332,000 9.4 480 428,000 7.8 360,000 6.9 332,000 9.3 495 430,000 7.8 362,000 6.8 334,000 9.2 510 430,000 7.8 362,000 6.8 334,000 9.2 525 432,000 7.7 364,000 6.7 336,000 540 9.1 434,000 7.7 364,000 6.7 338,000 9.0 555 436,000 7.6 364,000 6.6 338,000 9.0 570 438,000 7.6 366,000 6.5 338,000 8.9 585 438,000 7.5 366,000 6.5 340,000 8.8 600 440,000 7.5 378,000 6.5 340,000 8.8 7.4 370,000 6.4 340,000 615 442,000 8.7 630 442,000 7.3 372,000 6.4 342,000 645 8.6 444,000 7.3 374,000 6.4 344,000 7.2 376,000 660 8.6 446,000 6.3 346,000 675 8.5 450,000 7.2 378,000 6.3 348,000 8.5 452,000 7.1 380,000 6.2 348,000 690 8.4 705 454,000 7.1 382,000 6.1 348,000 8.4 720 456,000 7.0 384,000 6.1 350,000 735 8.3 456,000 7.0 386,000 6.1 352,000 8.2 750 458,000 6.9 388,000 6.0 354,000 8.2 765 460,000 6.8 390,000 6.0 356,000 3  3  max  3  max  max  3  Chapter 5. CONCLUSIONS  83  to make inflow forecasts. 5.3  T H E R O L E O F T H E F L O W P A T T E R N IN T H E R E S E R V O I R O P E R ATION  Comparing the results from the different flow patterns leads to an interesting conclusion. That is , the year end expected additional values with part time perfect inflow information depend on the pattern of mean flows and their deviations. Table 5.3 and 5.4 show the degree of improvement with part time perfect flow forecasts in the form of incremental percentages of the additional values over the one month ahead perfect forecast. A P represents the additional values with one month ahead perfect forecasting and thus may be viewed as the possible maximum improvement with perfect inflow information. A i , i=l,2, 3 and 4, is the additional values with one, two, three and four months of perfect forecasting. Thus  Ai/ A P is the operational improvement of part time perfect inflow forecasts in terms of percentage. A comparison of the two tables shows that the additional values are about 20%, 47%, 63% and 80% of the possible maximum improvements respectively for part time perfect forecasting withflowpattern I, whereas, the corresponding improvement are only about 9%, 23%, 35%, and 50% forflowpattern II. The reasons for these results are as follows. On the one hand, the total amount of inflow in the highflowperiod dominates the stream flow of the forecasting year with flow pattern I. It may be seen from chapter three that the stream flow in that period is about 83% of the total annual flow, so the energy production during that time is very high. Consequently, the improvement in reservoir operation efficiency due to the availability of part time perfect inflow prediction should also be great. But for the flow pattern II, the sum of the inflow in the two highflowperiods is only about 64% of the total annual flow,  Chapter 5. CONCLUSIONS  Table 5.3: The Additional Values with Flow Pattern I State ( A l / A P ) (A2/AP) (A3/AP) (A4/AP) A P Mm Gwh % % % % 19.7 270 48.0 64.2 83.2 17.3 285 19.5 47.7 17.4 63.8 82.8 300 19.5 47.7 63.8 17.4 82.8 19.4 315 48.0 63.4 82.3 17.5 19.4 330 48.0 64.0 82.3 17.5 19.4 345 48.0 64.0 82.9 17.5 360 20.0 48.0 82.9 64.0 17.5 375 19.9 47.7 82.4 63.6 17.6 390 19.9 47.7 64.2 82.4 17.6 19.9 405 48.3 64.2 82.4 17.6 420 19.9 48.3 64.2 82.9 17.6 19.8 435 48.0 63.8 82.5 17.7 450 19.7 47.8 64.0 82.0 17.8 465 20.1 47.5 63.7 81.6 17.9 480 20.0 47.8 63.3 81.1 18.0 19.9 495 47.5 63.0 80.6 18.1 510 20.4 47.5 63.0 81.2 18.1 525 20.3 47.3 63.2 18.2 80.8 540 20.3 47.3 63.2 80.8 18.2 555 20.3 47.3 63.2 81.3 18.2 20.2 570 47.5 62.8 80.9 18.3 20.2 585 47.5 62.8 80.9 18.3 600 20.1 47.3 63.0 80.4 18.4 615 20.0 47.0 62.7 80.5 18.5 630 19.9 46.8 62.4 80.1 18.6 645 20.3 47.1 62.0 79.7 18.7 660 20.2 46.8 62.2 79.3 18.8 675 20.1 46.6 61.9 79.4 18.9 690 20.0 46.3 61.6 78.9 19.0 705 19.9 46.6 61.3 79.1 19.1 46.4 720 19.8 19.2 61.5 78.6 735 19.7 46.1 61.1 78.2 19.3 750 19.6 45.9 60.8 77.8 19.4 765 19.5 45.6 77.4 61.0 19.5 3  84  Chapters.  CONCLUSIONS  Table 5.4: The Additional Values of F: ow Pattern II State ( A l / A P ) (A2/AP) (A3/AP) (A4/AP) AP Mm Gwh % % % % 17.2 270 8.1 23.8 36.6 51.7 285 8.1 23.7 36.4 51.4 17.3 300 8.0 36.2 23.6 51.1 17.4 315 8.6 23.6 36.2 17.4 51.1 23.4 330 8.6 36.0 50.9 17.5 8.6 345 24.0 36.6 51.4 17.5 360 8.6 24.0 36.6 51.4 17.5 375 9.1 36.4 23.9 51.1 17.6 9.0 390 24.3 36.7 50.8 17.7 405 8.9 24.2 36.5 50.6 17.8 8.9 420 24.0 36.3 50.3 17.9 435 8.9 23.9 36.1 50.0 18.0 450 8.8 23.8 35.9 50.3 18.1 465 8.8 23.6 18.2 36.3 50.0 480 9.3 23.5 36.1 49.7 18.3 495 9.3 24.0 36.1 49.7 18.3 9.2 510 23.9 35.9 18.4 50.0 525 9.2 23.8 35.7 49.7 18.5 540 9.1 23.7 35.5 49.5 18.6 555 9.1 24.1 49.2 35.8 18.7 24.1 570 9.1 18.7 35.8 49.7 585 9.0 23.9 35.6 49.5 18.8 600 9.5 35.4 49.2 23.8 18.9 23.7 615 9.5 35.3 48.9 19.0 630 9.4 24.1 35.6 48.7 19.1 9.4 645 24.0 35.4 19.2 48.9 35.2 660 9.3 23.8 48.2 19.3 24.2 675 9.3 35.1 19.4 48.5 690 9.3 24.2 35.1 19.4 48.5 9.7 705 24.1 35.4 48.2 19.5 720 9.7 35.2 24.0 48.0 19.6 735 9.6 23.9 35.0 48.2 19.7 750 9.6 24.2 34.8 48.0 19.8 765 9.5 24.1 34.7 47.7 19.9 3  85  Chapter 5. CONCLUSIONS  86  and thus has a relatively smaller influence on the improvement of reservoir operation than the former. The monthly standard deviation of the flow in the low flow season with flow pattern II, on the other hand, is much greater than flow pattern I. Therefore, its possible flows vary more widely during low flow periods, which means more uncertainty about the flow occurence. For example, there are only 4 to 8 possible flow values with flow pattern I in the low flow season, whereas there are at least 12 possible flows in the low flow months with flow pattern II. As a result, there is only small additional value in reservoir operational efficiency improvement in the low flow periods with flow pattern II if perfect runoff information is available. It can be concluded, from the above results, that more additional value may be obtained when the greater amount of stream flow occurs in the wet season. In other words, it is more valuable to make perfect inflow forecasts in the high flow period if the stream flow in that time dominates the total flow of the year as it does with flow pattern I. Where there is not much difference in the amount of water between the dry season and the high flow period as in flow pattern II for instance, it is better to get forecasts throughout the whole year. Although there are some influences on the additional values of perfect forecasting owing to the different inflow patterns, that isflowpattern I and pattern II, the differences in the potential additional expected values between the twoflowpatterns is not very much as has been already pointed out in section 5.1.  5.4  T H E CONSEQUENCES OF PART TIME P E R F E C T FORECASTING  Making perfect stream flow forecasts for a whole year is a costly and difficult task usually. An alternative is to predict the inflows as accurately as possible during the high flow  Chapter 5. CONCLUSIONS  87  period. This section examines the effects of part time perfect forecasting to the improvement of reservoir operational efficiency. Table 5.1, 5.3 and 5.4 tabulate the results of all operation runs. From these tables, it is clear that the perfect forecasts in the wet season have significant influence on the improvement in potential additional expected values. The more months which have perfect flow information in high flow season, the more improvement is possible. The potential for additional expected values with four months of perfect predictions may reach about 80% or 50% of the maximum possible improvement with flow pattern I and pattern II respectively, whereas the one, two or three month perfect forecasts have relatively smaller additional benefits, about 20%, 47% or 63% for flow pattern I and 9%, 23% or 35% for flow pattern II. Therefore, it is recommended to try to make as accurate flow predictions as possible for the whole wet season instead just part of the high flow period to get the best results.  5.5  RECOMMENDATIONS  FOR FURTHER RESEARCH  From the experience gained in this study, the following further research is recommended. • Find the additional expected values with two month ahead perfect forecasting. This is a more difficult issue to formulate into the optimization model than the model in the present study but it would be an interesting task both in theory and in practice. • Examine the effects of different reservoir sizes on the improvement in expected values with one month ahead perfect flow forecasts. Reservoir size is usually one of the key elements in the control of its operation. Different reservoir sizes might cause different reactions to the perfect inflow forecasts. • Study the role of one month ahead perfect forecasts in the operation of a system of reservoirs. Generally speaking, a hydroelectric power project is operated in a large  Chapter 5. CONCLUSIONS  88  scale electric network, rather than as an individual project. • Examine the effects of less than perfect forecasts in reservoir operation using a methodology similar to the perfect study.  Bibliography  Barnard, Joanna Mary 1989. The Value of Inflow Forecasting in the Operation of a Hydroelectric Reservoir. M.A.Sc. Thesis Department of Civil Engineering, The University of British Columbia. Bellman, R.E. 1957. Dynamic Programming.Princeton N.J. Princeton University press. Caselton, William F. and Samuel 0. Russell 1976. Long-term Operation of Storage Hydro Projects. Journal of the Water Resources Planning and Managemaet. Division Proceedings of American Society of Civil Engineers. Vol.l20(WRl) 163-176 Georgakakos, Aris P. 1989. The Value of Streamfiow Forecasting in Reservoir Operation. Water Resources Bulletin. Vol.25 No.4:789-800 Krzysztofowicz, Roman 1983. Why Should a Forecaster and a Decision Maker Use Bayes Theorem. Water Resources Research vol.19 No.2 327-336 Labadie, J. W., R. C. Lazaro and D. M. Morrow, 1981. Worth of Short-term Rainfall Forecasting for Combined Sewer Overflow Control. Water Resources Research Vol.17 No.5. 1489-1497 Nabeel R. Mishalani and Richard N. Palmer 1988. Forecast Uncertaint in Water Supply Reservoir Operation. Water Resources Research Vol.24 No.6 1237-1245 Stedinger Jery R., Bola F. Sule and David P. Loucks 1984. Stochastic Dynamic Programming Models for Reservoir operation Optimization. Water Resources Research Vol.20 No.ll 1499-1505 Trezos, Thanos and William W-G Yeh 1987. Use of Stochastic Dynamic Programming for Reservoir Management. Water Resources Research Vol.23 No.6, 983-996 Yakowitz, Sidney 1982. Dynamic Programming Application in Water resources. Water Resources Research Vol.18 N0.4 673-696 Yeh, William W-G, Leonard Becker and Robert Zettlemoyer 1982. Worth of Inflow Forecast for Reservoir Operation. Journal of Water Resources Planning and Management. Division Proceedings of American Society of Civil Engineers Voll08(WR3) 257-269  89  Bibliography  90  [12] Yeh, William W-G 1985. Reservoir Management and Operation Models. A Stateof-the-Art Review. Water Resources Research Vol.21 No.12 1797-1818  

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