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Marginal value controlled reservoir operation Smith, Gerald Keith 1979

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MARGINAL VALUE CONTROLLED RESERVOIR OPERATION by GERALD KEITH SMITH B.A.Sc, University of Br i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the - FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering WE ACCEPT THIS THESIS AS CONFORMING TO THE REQUIRED STANDARD The University of Briti s h Columbia A p r i l , 1979 © Gerald K. Smith, 1979 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be gr a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department n f Civil Engineering The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 )E-6 BP 75-51 1 E ABSTRACT Techniques have been developed for coordinating releases from a system of water reservoirs to maximize the expected returns from the system. The system i s decomposed into single reservoir subsystems linked by various common demands and interdependent releases and inflows. Demands for firm and secondary energy are considered, and thermal generation can be included i n the system. Two basic c r i t e r i a for reservoir operation have been defined according to principles of economic efficiency. In the operation of each individual reservoir, releases should be increased u n t i l the current marginal return equals the marginal loss i n future returns and, when several reservoirs supply a common demand, the marginal costs of supply from the reservoirs should be equal. Each single reservoir subsystem i s modelled and optimization proceeds on three levels. An iterative procedure calculates the expected value of long-term operation, and then employs dynamic program-ming to determine releases to maximize that value. Annual inflows are treated as serially independent probabilistic events, and a winter pre-diction based on snow surveys i s assumed to provide complete knowledge of inflows during the coming year. The model i s then extended to include reservoirs with interdependent releases and inflows. Releases from upstream reservoirs are influenced by monthly marginal values of water to downstream reservoirs i n a release adjustment cycle whi?h converges on sets of optimal releases for a l l reservoirs i n the series. Finally, monthly proportions of the common demands are satisfied by each subsystem, i i and t h e o p t i m a l p r o p o r t i o n s wh ich equa l i ze marg ina l cos ts o f p r o d u c t i o n are found. Computat ional f e a s i b i l i t y and o p t i m i z a t i o n c r i t e r i a e f f e c t i v e n e s s were t e s t e d w i t h s e v e r a l r e a l i s t i c a p p l i c a t i o n s . I^namlc programming d imens iona l problems a r e avoided and computa t iona l t ime inc reases l i n e a r l y as r e s e r v o i r s are added t o t h e system. The present model s imu la tes s t e a d y - s t a t e o p e r a t i o n w i t h annual I n f l o w as t h e on ly v a r i a b l e . T h e r e f o r e , a p p l i c a t i o n i s l i m i t e d t o problems i n v o l v i n g l o n g - t e r m r e s e r v o i r o p e r a t i o n , o r the des ign o f proposed p r o j e c t s . However, m o d i f i c a t i o n s are o u t l i n e d t o a l l o w a p p l i c a t i o n o f t h e system o p t i m i z a t i o n techn iques t o t r a n s i e n t o p e r a t i o n . I l l TABLE OF CONTENTS Page LIST OF TABLES v i LIST OF FIGURES v i i CHAPTER 1. INTRODUCTION 1.1 The Water Resources Development Objective 1 1.2 The Systems Approach 2 1.3 The Planning Problem 4 1.4 The Operations Problem 4 1.5 Scope of Study and Past Research 6 2. RELEVANT OPTIMIZATION THEORY 2.1 Reservoir Returns 12 2.2 Hierarchical Analysis 14 2.3 Reservoir System Costs and Benefits 15 2.4 Present Values of Future Benefits 20 2.5 Howard's Value Determination -Policy Improvement Method 22 2.6 Cptimization by Dynamic Programming 25 3. DESCRIPTION OF THE MODEL 3.1 Inflow Uncertainty 27 3.2 Optimization of Long-Term Operation 27 3-3 Optimization of Release Decisions 29 3.4 Series of Reservoirs 33 3-5 Computational Feasibility 37 i v TABLE OF CONTENTS (continued) Page CHAPTER 4. APPLICATIONS OF THE MODEL 4.1 Data 39 4.2 Application I: Monthly Load Dispatching 43 4.3 Application I I : Serially Connected Reservoir Operation 46 4.4 Application I I I : Hydroelectic Returns from Thermal Generation 49 4.5 Application IV: Project Timing 53 5. PROPOSED EXTENSIONS -5.1 Transient Conditions 56 5.2 Allocations of Demand 57 5.3 Risk of Failure 60 5.4 Reservoirs on Branching Rivers 6 l 5-5 Inflow Uncertainty 62 6. CONCLUSIONS 64 REFERENCES 68 APPENDICES A. COMPUTER PROGRAM LISTING • 71 B. SAMPLE OUTPUT SHOWING THE OPTIMAL ALLOCATION OF FIRM ENERGY DEMAND 85 v LIST OF TABLES Page TABLE 4.1.1 RESERVOIR CAPACITIES 40 4.1.2 ANNUAL RESERVOIR INFLOWS 40 4.1.3 MONTHLY GENERATING SYSTEM DATA 4l 4.2.1 IMPROVEMENTS IN SYSTEM PERFORMANCE WITH MONTHLY COORDINATION 46 v l LIST OF FIGURES FIGURE Page 2.3.1 FORMS OF HYDROELECTRIC ENERGY DEMANDS 16 2.3.2 FORMS OF RESERVOIR SYSTEM DEMANDS 19 2.5.1 THE VALUE DETERMINATION - POLICY IMPROVEMENT CYCLE 25 3.3.1 TYPICAL MARGINAL RETURNS AND MARGINAL COSTS OF RELEASES 31 3.4.1 RELEASE ADJUSTMENT CYCLE FOR A SERIES OF RESERVOIRS 36 4.4.1 RESERVOIR GENERATION RETURNS WITH VARIOUS AVAILABLE THERMAL GENERATION CAPACITIES 51 4.4.2 MAXIMUM RESERVOIR GENERATION RETURNS WITH VARIOUS AVAILABLE THERMAL GENERATION CAPACITIES 52 4.5.1 NET INCREASES IN EXPECTED ANNUAL RETURNS GAINED WITH THE REVELSTOKE PROJECT 54 5.2.1 IMPROVED REPRESENTATION OF MONTHLY HYDROELECTRIC GENERATION DEMAND 59 -v i i ACKNOWLEDGMENT The following research was supported by the Department of C i v i l Engineering at the University of B r i t i s h Columbia with financial assistance received from the National Research Council of Canada. The author i s especially grateful to Drs. W.F. Caselton and S.O. Russell for their advice and guidance during the last two years. v i i i 1. CHAPTER 1 : INTRODUCTION 1.1 THE WATER RESOURCES DEVELOPMENT OBJECTIVE The activities of people create water related demands which typically arise from needs for irri g a t i o n , energy, residential or industrial water supplies, recreation, flood control or low-flow augmentation. These demands grow as the regional economy expands, or as substitutes become scarce. In the case of hydroelectric energy, a large growth i n i t s demand w i l l eventually occur due to the inevitable shift away from o i l as a source of energy. Water supplied to each demand results i n benefits which depend on the quantity supplied, and water resources development i s an attempt to increase the net benefits that w i l l ultimately be enjoyed. Therefore, the overall development objective can be generally stated as maximization of the total value of benefits less costs. Of course, problems associated with the evaluation of costs and benefits are immediately apparent. Some effects may be aesthetic or intangible, while others may be uncertain future events. The values of future events must be discounted at some appropriate rate, and the equity of the distribution of costs and benefits must be evaluated. However, these are simply subproblems of the overall task of optimization. Projects are designed to optimize the above objective by altering the time, location or quality of water supplies. The purpose of storage reservoirs i s to alter the time of supply, and to develop head for hydroelectric generation. This thesis i s concerned with maximization of the total value of benefits, less costs, for a system of reservoirs. 2. 1.2 THE SYSTEMS APPROACH The performance of Individual water projects often depends on the operation of other projects. Releases from one project may affect Inflows to another, or the output from one may affect the demand for another's output. Projects with this interdependence form a system and the. overall optimization of this system can only be achieved with coordinated operation of a l l individual system components. In addition to their highly interrelated nature, reservoir systems are characterized by large numbers of interacting variables, and a high degree of uncertainty surrounding system inputs, demands and physical capacities. The evaluation of alternative policies can become a very d i f f i c u l t problem due to the complexity of the systems involved, and as the demands for our limited water resources continue to grow, incorrect management decisions become more and more costly i n terms of wasted potential benefits. Over the past two decades, systems analysis techniques combined with computers have become increasingly popular as an aid to the solution of water resources management problems. Since Maass, et a l . [ l ] adopted the systems approach in 1962, research i n this direction has produced many valuable results. The main elements of the approach are: i ) Identify a problem or need, i i ) Establish an objective to be optimized, i i i ) Define the boundaries of the system to be analyzed. Numbers In parentheses refer to literature cited i n the List of References. 3. iv) Describe the system interactions and identify pertinent decision variables. v) Gather necessary data, vi) Subject to physical and imposed policy constraints, determine the set of decision variable values that optimizes the objective. At this point, a problem is encountered i f a l l goals and responses cannot be measured in similar units. A l l system outputs that affect the objective must be directly comparable for optimization. Otherwise, the system analysis can only describe possible combinations of outputs, and some more qualitative decision process must select the best combination. Methods are being developed for the evaluation of project consequences such as changes in social, recreational or environmental conditions. However, some effects usually remain incommensurable and systems analysis can only be employed to provide other decision processes with Information about the quantifiable aspects of the overall problem. Mathematical models are used to simulate system response to decision variables, and to search for optimal solutions. With traditional methods, competent water project managers will eventually develop near-optimal operating plans, but systems analysis with mathematical models can be very helpful in the absence of experienced personnel, or in cases of rapidly changing systems and demands. Because of the complexity of reservoir systems and limits on compu-tational time, any existing model will include simplifications and inadequacies in its representation of reality. Consequently, improve-ments in the accuracy of solutions and increases in system returns will always be possible. 4. 1.3 THE PLANNING PROBLEM Reservoir system management problems can be generally classified as short-run or long-run in the terminology of economics. Long-run planning Is concerned with how and when capacity should be expanded, while short-run operation attempts to obtain rriaximum benefits from a fixed physical system. As demands grow, the potential for increased benefits grows, and the task of system planners is to evaluate this potential and the alternatives for development. The alternatives may be capital investment projects or changes in the long-term constraints and contract obligations under which the existing system operates. Capital costs, the level of demand and its rate of growth, future inflation in prices of system inputs or outputs, and the rate for discounting values of future benefits are factors of utmost importance to planners. A l l of these factors must be estimated for the operating l i f e of each proposed project. In addition, the value of a l l future costs and benefits must be estimated for each alternative in a realistic mode of operation, and this leads to the solution of the operations problem as a subproblem within the system planning process. 1.4 THE OPERATIONS PROBLEM In the short-run, managers seek to operate the existing system in a manner that maximizes its value. This reduces to the problem of determining the best allocation of available resources to the existing demands. 5. Because one primary function of a reservoir is the storage of water for future use, optimal operation must consider expected future, as well as current, demands. A solution of the reservoir system operations problem consists of a set of release quatities for each decision period. The decision period may be a month, week or hour depending on the fineness of the analysis, and the optimal solution at any time will depend on current and future demands, present storage levels, future inflows and individual reservoir production efficiencies. Because future events are always uncertain, the optimality of solutions will be influenced by the expected Value resulting from probable outcomes. For a system of reservoirs at a certain time with a certain set of future events about to occur, there exists a truly optimal set of releases that will yield the maximum total value. Any operating policy will attempt to specify releases that will ultimately yield this maxiirium total value and, occasionally, the policy may happen to be correct. However, i f a nonoptimal set of releases results from inaccuracies in either the prediction of future events or the evalu-ation of total system response, some of the potential niaxijTium value will be lost. Usually a particular policy will select a nonoptimal solution with some wasted potential value at each decision.. A better policy will select solutions which cause a smaller average loss of potential value. If the average saving achieved with each decision is signifi-cant, the total saving during continuous operation of the system 6. can easily justify the time and effort required to develop the Improved release policy. 1.5 SCOPE OP STUDY AND PAST RESEARCH The thesis presented here applies to the optimization of system operation or capacity expansion. An operations model is used to increase the value of an existing system, and to assist In the design and scheduling of successive additions to the system. The optimal sequencing of available alternatives for capacity expansion requires the evaluation of a l l promising sequences with changing and uncertain future events. This involves the comparison of many possible combinations of projects, usually including various non-commensurable considerations. The techniques described here are not as well suited to this task as methods which sacrifice simulation details for speed and versatility. The simulation and optimization of reservoir operation has been the subject of much recent research and significant advances have been made. However, some potential for improvements in computational speed, simulation accuracy, and general applicability s t i l l remains. The most difficult problems are usually encountered in the treatment of inflow uncertainty and the coordination of interdependent reservoir releases. Computational feasibility usually requires simulations to be based, as in this study, on a time interval of one month. A l l relevant research efforts have one feature in common; some form of simulation routine for relating inflows and releases to some response function. The simplest and most common simulation approach has been to calculate a value for the objective with a given release 7. policy and a deterministic sequence of inflows. Improved policies have then been obtained by entering the simulation model results as data in some external optimizing technique such as multivariate analysis [ 2 , 3 ] , conjugate gradient or gradient projection [ 4 ] , or linear prograirirnng [5]. Due to simplifying assumptions made necessary by the nonlinear and complex nature of system responses, simulation-optimization processes must be repeated iteratively until successive solutions converge. This type of analysis has been attempted for multipurpose-multireservoir systems [4,5] but computational time has limited the number of decision variables and constraints that were included. Consequently, the analyses were very coarse and covered a small number of time intervals. An alternative approach has been to define the optimizing criteria, and then to include in the simulation models some mechanism for deriving optimal operating policies according to the specified criteria. Dynamic prograraiiing has been by far the most popular internal optimization technique. Reservoir operation easily fits the dynamic prograrariing sequential decision framework with storage levels corresponding to states which depend on decisions in each time period, or stage. By 1955, while Bellman's source book on the method [6] was s t i l l in preparation, dynamic programming had been applied to a reservoir problem [ 7 ] . Use had become widespread by the mid-1960's [ 8 , 9 ] . However, problems arise with a number of interrelated reservoirs because each release decision depends on other simultaneous release decisions. In this situation, multidimensional decision vectors must be scanned for optimal solutions, and the computer memory requirements 8. and execution times become exponentially related to the number of interdependent decision variables. Practical simulation models have been limited to a maximum of three dimensions, although recent advances such as discrete differential dynamic programming [10,11] and state increment dynamic prograirming [12] allow the extension of the method to cases of more than three interacting decision variables. An interesting solution to the <±3Lmensionality problem was developed for a four reservoir system i n northern Califomia [13,14,15]. Using the principle of decomposition, each reservoir was defined as a subsystem with optimal operation determined by a dynamic prograimiLng algorithm. Then, l inear prograraning was employed to maximize the total firm production of the system. Shadow prices were obtained for firm outputs i n each decision period, and these prices became controls in the next dynamic prograrriTiing optimization of each subsystem. The process was iterative and tended to improve system performance although convergence d i f f i c u l t i e s were encountered [15]. A serious limitation of the deterministic approach i s that solutions are optimal only for a specific sequence of future inflows, not for the general operating conditions to be expected. Sometimes, as i n the northern California system studies described above, a recorded sequence of c r i t i c a l l y low inflows i s used with the intention of revealing firm output quantities that can be guaranteed under any circumstances. The result i s never an output that can be produced with absolute certainty, because there i s always a chance that a sequence of lower inflows w i l l occur and production w i l l f a i l to meet firm expectations. 9. Reservoir operation decisions must consider various future events and, in reality, complete knowledge of these future events is never available. To accurately represent reality, a simulation of reservoir operation must recognize the uncertain, or stochastic, nature of decisions. Stochastic analysis has been classified as either implicit or explicit [ 3 ] . In an implicit analysis [16], historical records are used to estimate the inflow probability distribution, and sequences of flows are generated according to this estimated dis t r i -bution. These sequences are then combined with a deterministic simulation model to represent operation with realistically varied hydrologic inputs. Alternatively, explicit stochastic analysis approximates the probability distribution with a number of discrete events and their associated probabilities of occurrence, and these are entered into the simulation to yield expected values of the objective. Compared to an implicit analysis, the number of sequences necessary to describe typical operating conditions is reduced, but computational effort at each stage is increased. Most explicit analysis work has involved single reservoir models with monthly inflows treated as a first-order Markov process [7 ,8,17,18]. Since water stored in a reservoir represents potential future benefits, release decisions must weigh immediate returns against the expected reduction in future returns. A current decision cannot be optimized until operation has been simulated far enough into the future so that the current decision no longer affects the future returns. If a reservoir has high storage capacity relative to the mean annual inflow volume, and releases are closely constrained, 10. current decisions may affect returns for years to come. Most past models have simply assumed target storage levels for the termination of simulation operation. A recent description of a model for planning monthly releases for one year [12, p.70] reports that: "The desired end of forecast period storage values, e.g., the end of a 12-month forecast, are estimated using the operator's experience and judgement." Another group of researchers [5 , p.108] states: "One of the major objectives in operating the system is to maximize electrical power benefits, while rrdnimizing spilling, adhering to a l l system constraints, and culminating in end-of-period reservoir storage states which are adequate for continuing operation." These attitudes do not seem to recognize the high variation in returns according to the time of release, or the extensive real-time experimentation required to gain the necessary 'experience and judgement'. The entire purpose of the systems approach is to aid such intuitive judgements in cases involving high degrees of complexity and uncertainty. Young [2] avoided this final storage state problem by discarding results from the last 100 of 1000 intervals simulated. This approach reduces the importance of final storage levels, but also wastes considerable computational effort. An alternative technique [19] has recently been developed for reservoirs with variable annual inflows providing hydroelectric energy to meet firm demands. Alternative thermal generation was included and optimal monthly releases were determined to nrinimize expected 11. thermal costs. Annual operation was described as a serially independent Markov process, and a method proposed by Howard [20] was adapted to rninimize the discounted present value of a l l thermal costs over an unbounded time horizon. Then, a higher level optimiza-tion scheme coordinated annual load dispatching in a multireservoir system to reduce total thermal generation. The study described here has extended this Markov process model to include monthly coordination between multipurpose reservoirs. Expected marginal values of reservoir releases have been combined with basic principles of economic efficiency to guide the operation of reservoirs linked by interdependent inflows and releases, or by common demands for firm and secondary energy. A portion of the British Columbia Hydro and Power Authority (BCHPA) system was simulated to test the proposed optimization techniques. 12. CHAPTER 2 : RELEVANT OPTIMIZATION THEORY This chapter Includes brief descriptions of some basic concepts of economics in a water resources context. A more complete description can be found in any text dealing with introductory microeconomics [21]. 2.1 RESERVOIR RETURNS The supply of each unit of water to a proposed use causes a user benefit and, in order to effectively compare the relative merits of supplying limited quantities to different uses, the values of the resulting benefits must be expressed in a consistent manner. The measure usually chosen is the monetary price users would willingly pay for each unit supplied, because this price reflects the value of alternative benefits that would be sacrificed for the proposed benefit, considering a l l other price levels at the time of proposed use. For any one source of demand, the value placed on successive units of supply typically declines as the most urgent demands are satisfied f i r s t . Of course, the price that would be paid for each unit supplied can only.be estimated because of the transient nature of demands and the inherent difficulty in accurately measuring quantities consumed at other than market prices. In addition, no market exists for some water resources development by-products such as recreational opportun-ities, environmental or ecological effects, and social impacts. Sometimes prices for these items can be estimated from close substit-utes, but often, particularly with ecological or social effects, 13. pricing is impossible or inappropriate. These are the noncommensurable effects that must be excluded from a decision analysis based on monetary values and considered later in some reconciliation of alternative values. The uncertainty about values of a l l quantities demanded can be partially avoided since most water resource allocation decisions involve changing supplies within some feasible limits. Therefore, prices need to be estimated only for quantities within the feasible supply range. For any total quantity supplied to a demand there exists a marginal return and a marginal cost. The marginal return consists of the value to users of the last unit, and this value will decline as demand is satisfied. The marginal cost corresponds to the value of benefits obtainable i f the resources required for the last unit of supply were diverted to their next best use. Marginal costs may also include negative returns such as flood or erosion damages that accompany releases. As water continues to be supplied to a particular demand, i t must be drawn from increasingly valuable uses and the marginal cost of supply will increase. For optimal economic efficiency, a demand should continue to be satisfied until the values of declining marginal returns and increasing marginal costs become equal. No further value can then be achieved by transferring more water from its next best use. This also means that, i f a demand is supplied from more than one source, the marginal costs of supply should be equal for a l l sources. The research presented here has employed these principles In the simulation of optimal monthly reservoir releases, 14. and In the coordination of releases supplying common demands. The annual return attributed to a water resources system consists of the total value of benefits resulting from the system's annual output. Annual variation in hydrology or demand will cause variation in annual returns, and these variable annual returns can be represented by an aggregate mean. Several discrete annual inflows and their probabilities were entered in the simulations performed here, and expected annual returns were obtained as estimates of true mean values. 2.2 HIERARCHICAL ANALYSIS The hierarchical analysis of water resources systems is well developed [ 2 2 ] and essentially consists of the decomposition of highly complex systems into more manageable units. The procedure, which is nonrigorous and often intuitive, attempts to isolate subsystems with interactions described by a few simple variables. System control variables are assigned values by a master program seeking to improve combined subsystem performance with respect to an overall system objective, and these values become constants in lower level subsystem optimizations. Two or more hierarchical levels can exist and overall optimization usually becomes an iterative process alternating between levels until no further significant improvement can be gained for the system. The system dealt with here includes several reservoirs combined with both thermal and hydroelectric generating facilities. The subsystems consist of individual reservoirs supplying various 15. powerplants and the optimal operation of these subsystems will contribute to the system monthly output in a way that maximizes the total system value. Interacting variables include downstream marginal returns from releases that ultimately enter other reservoir subsystems, proportions of energy demands assigned to subsystems, and marginal costs of production that result from these proportions. Manipulation of these variables according to the criteria given in the previous section ultimately provides the desired integration of the overall system operation. 2.3 RESERVOIR SYSTEM COSTS AND BENEFITS Two extreme varieties of demand are encountered in the following treatment of firm and secondary energy demands. Most hydroelectric energy is provided on a firm contract basis and these contract obli-gations form constraints on the monthly minimum amount of generation produced. The result is a perfectly Inelastic demand condition in which a particular quantity (FEMX) is demanded and produced in a month regardless of cost. This condition is illustrated in Figure 2.3.1a. If some thermal generating capacity is available, the hydro-electric system demand becomes discontinuous (Figure 2.3.1b). Assuming the marginal cost of thermal generation (PTH) to be constant up to the maximum available monthly capacity (THRMX), the total hydroelectric energy demanded is FEMX for prices below PTH, and FEMN for prices above PTH. The demand appears, perfectly elastic between FEMN and FEMX. In this range, the marginal return from hydroelectric generation is constant at PTH and the optimal level of output is 16. FEMX quantity a) Perfectly inelastic firm energy demand. PTH THRMX units of available thermal generation FEMN FEMX quantity b) Hydroelectric energy firm demand with available thermal capacity. PSG quantity c) Hydroelectr ic energy secondary demand. FIGURE 2.3. I FORMS OF HYDRAULIC ENERGY DEMANDS 17. reached when the marginal cost of supply equals PTH. The thermal plant produces any balance of firm energy not generated by the hydro-plants in the system. A perfectly elastic demand condition is also apparent in the case of secondary energy (Figure 2.3.1c). Any electricity available in excess of firm coirraitments can be sold at a price (PSG) established by the market for interruptable secondary generation. This price generally varies from month to month, but can be considered constant with respect to supply each month i f prevailing market prices will not be significantly affected by supplies from the system being modelled. For example, the BCHPA sells most of its surplus energy in the extensive market of the western United States and the selling price is due primarily to supply and demand conditions in the United States. This price will show l i t t l e response to supply quantities from British Columbia, considering the likely magnitudes of these supplies. The exclusion of secondary energy demand elasticity simplifies the analysis by eliminating the possibility of a reservoir release affecting the secondary returns of another reservoir. This is consistent with a systems approach in which a system boundary is defined and values of relevant, but external, variables are held constant. If the values chosen for external variables, such as secondary,energy prices, are finally found to be inappropriate, the analysis can be redone with adjusted values. This would constitute an extension of the system boundary and the inclusion of a higher level in the decomposition hierarchy. 18. A release from a multipurpose reservoir yields a total marginal return equal to the combined price that a l l users would pay for the last unit. If the water can be used for one purpose and then again for another (e.g. first to generate electricity and then to irrigate crops), the total marginal return depends on the return from each use of the entire last. unit. However, i f uses are mutually exclusive, as in the case of firm and secondary energy generation, the total marginal return results from proportioning the water between demands. In this study, the calculation of marginal returns from generation is not a problem because a l l energy is firm up to the monthly firm demand, and secondary returns arise only from generation exceeding this firm demand. The returns from water releases are related to the hydro-electric generation returns by a function which depends on generating plant characteristics and reservoir head at the time of release. The forms of the composite demand for hydroelectric energy and the result-ing demand for water are shown in Figure 2.3.2. Reservoir system costs can be classified as fixed or variable. The optimization of short-run reservoir operation is only concerned with variable costs which may include operating and maintenance costs, negative returns from releases, or lost future returns. Water reservoir and powerplant variable costs are usually small in comparison with available returns and can be safely neglected in the search for optimal operating policies. Water supplied for energy production does not cause direct negative returns, so the only cost to be considered is the loss of future benefits due to immediate water use. For any time period, an increased release will result in a decreased PTH PSG FEMN FEMX quantity a) Composite demand for hydroelectric energy. Decreasing returns due to decreasing reservoir head Maximum generating capacity quantity b) Related demand for generating water FIGURE 2 . 3 . 2 FORMS OF RESERVOIR SYSTEM DEMANDS. 2 0 . end-of-period storage level and a decreased value of future benefits as operation continues from this lower level. This additional water release will not be available for some later use, nor will i t contri-bute to future generating head. Therefore, the marginal cost of a release is the total value of future benefits lost with the last unit of water taken from future operation. To maximize reservoir value, water must always be released until the marginal immediate return less the marginal future cost is as close to zero as possible within feasible limits. .It should be noted that only the differences between feasible end-of-period storage state values, not their magnitudes, need to be known accurately for efficient operation. 2.4 PRESENT VALUES OF FUTURE BENEFITS At any point in time, the present value of a system is defined as the total of a l l future returns expressed in current units of value and discounted with respect to time at some appropriate rate. An estimation of the proper discount rate involves consideration of the prevalent market rate of interest on money, as this rate is an indication of society's preference for benefits enjoyed immediately rather than in the future. The discount rate can also include an allowance for any possible risk that expected future returns may not be realized. This possible risk is not considered in the following discussion, although a high penalty is introduced later for any failure to satisfy firm demand. If an expected return will occur • t years in the future with monetary value at that time of ER, and i f r is the rate of interest 21. required from Invested capital, then the present value of the return is ER/d+r)^. However, i f the expected return has a value of ER at present prices which are rising at an inflation rate I, then t years in the future the monetary value will be ER(1+I)t. Therefore, i f an expected return has a value of ER at present price levels and will be realized t years in the future, its present value (FV) is expressed as PV = ER ( 1 + I ) t (2.4.1) (l+r) t For typical values of I and r, the term (l+I)/(l+r) can be approximated by l/(l+r-I), or 1/(1+R) where R represents the real rate of interest and R = r - I (2.4.2) Equation 2.4.1 can now be approximated by the following equation: PV = m (2.4.3) a+Rr If a return is valued at ER, according to present prices, the present value of receiving this return annually forever is the sum of an infinite series. If the return is discounted annually after the current year, PV = ER + ^  + + + ... (2.4.4) 1 + R (1+R)2 (1+R)3 22. and this equation can be modified to read FV _ ER , ER ER . ER . (o h *\ 1+R 1+R ( 1 + R ) 2 ( 1 + R ) 3 ( 1 + R ) U The infinite series on the right in Equation 2.4.5 is identical to a l l but the first term of the series in the preceding equation. Therefore, by substitution, PV = ER + (2.4.6) and after rearranging, FV = ER + ^  (2.4.7) 2.5 HOWARD'S VALUE DETERMINATION-POLICY IMPROVEMENT METHOD In river basins where much of the runoff comes from melting snow, annual reservoir operation can be viewed as a Markov process with storage levels as states and state transitions occurring annually. With a specific release policy and annual inflow volume as the only nonfixed factor influencing transitions, reservoir operation results in various state transitions, and associated returns, occurring with the same probabilities as the concurrent inflows. In discrete form, a set of n feasible reservoir states is defined and annual transitions occur from beginning-state i to end-state j . Each transition occurs with a probability, p.., equal to the probability of the discrete annual Inflow causing the transition. Assurning that the annual inflow probability distribution is stationary with respect to time, and that no serial correlation 23-exists between these inflows, the set of state transition probabilities, p „ , for a l l values of i and j are also stationary. Howard [20] presented a method for maximizing the value of this sort of Markov process, with returns, for an unbounded time horizon, and this method has been successfully applied to minimize thermal generation costs with a mixed hydro and thermal generating system [19]. The application of Howard's method in this thesis is similar to that done previously, except that the objective is the maximization of reservoir returns, rather than the minimization of thermal generation. A two-phase iterative process is involved. To commence the cycle, a present value of returns from continuing future operation is estimated for each possible end-of-year state j . These end-state values, symbolized as A/^nd(j=l,n), are discounted and included as end-conditions in the operating policy objective of value maximization. For each discrete probable annual Inflow and each possible beglnning-of-year state i , a particular set of monthly releases will cause a transition to some end-state j , and an annual return, er „. If the value of each beginning-state is maximized for each probable Inflow, the total expected value of the system will be maximized. Eynamic prograimiing (Section 2.6) is used in the policy improvement routine to determine the monthly release policy that will maximize the present value of state i . This value is symbolized as v^ and defined by the following equation; v ? n d v. = er. . + =4R- (2.5-1) l I J 1+R 24. where R is the real rate of interest defined in Section 2 . 4 . An optimal policy, complete with annual return and end-state j , is identified for each possible combination of beginning-state i and discrete annual inflow. The following system of linear equations can then be written to express the expected present value of operation starting from each state; vend V ± = E[v i] = Z PyCer^ + - ^ - ) ; i=l,n. ( 2 . 5 . 2 ) This system of equations is a discrete probabilistic form of Equation 2 . 4 . 6 . If the reservoir operation will continue over an unbounded time horizon with stationary values of a l l p.. and er.., an end-of-year state value, V?n<^, will be equal to the beginning-of-year value, V . , for the same state. Howard's value determination routine 3 replaces v f n d in Equation 2 . 5 . 2 with V . to derive the following J J equations; v i " J P y ^ y + i^-> = 0 > 1 = 1> n- ( 2 ' 5 ' 3 ) With the values of p.. and er.. corresponding to each possible -LJ -LJ optimal policy, this system of n linear equations is solved for n values of V . . l The values of "V\ obtained in this way become new estimates of end-state values, V r n d , for use as end-conditions in another J policy improvement routine. The two-phase cycle (Figure 2 . 5 - 1 ) is repeated until the successive values of "V\ converge. 25. This method yields values of long-term operation from various end-of-period reservoir states, enabling the calculation of marginal costs of releases in terms of future returns (Section 2.3)- In addition, the final policy improvement routine demonstrates long-term optimal monthly releases for each possible combination of beginnlng-of-year state and discrete probable inflow. Enter Estimates o f V ? n d I W New Estimates o f V e n d j POLICY IMPROVEMENT ROUTINE VALUE DETERMINATION ROUTINE Values of Pi] and er 'J FIGURE 2.5.1 THE VALUE DETERMINATION-POLICY IMPROVEMENT CYCLE 2.6. OPTIMIZATION BY DYNAMIC PROGRAMMING To optimize release decisions in the above method, the policy improvement routine employs the mathematical technique of dynamic programming. This technique evolved during the early 1950's and, in 1957, the first extensive description [6] was published with the inclusion of this statement of the fundamental Principle of Optimality; 26. "An optimal policy has the property that whatever the Initial state and Initial decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision." Hall and Dracup [23] give a good Introduction to the methodology of dynamic programming in a water resources context. Monthly reservoir operation can easily be described as a sequential decision process that fits the general dynamic programniing framework. At monthly stages in time, decisions are made concerning releases during following time periods. Reservoir levels are designated as states, and any particular release decision will transform the current state into some resultant state at the next stage. In reality, system response to decisions is seldom totally linear, convex, or even continuous, but these properties are rarely a problem in dynamic programming. The only requirement is that optimal future releases depend only on the current state and stage and not on previous decisions causing that state. 27. CHAPTER 3 : DESCRIPTION OF THE MODEL 3.1 INFLOW UNCERTAINTY The model used in this research incorporates dynamic programming (Section 2.6) within Howard's value determination-policy improvement method (Section 2.5) to optimize long-term reservoir operation. Within this model, annual inflow volumes and their associated state transitions are the only uncertainties considered. The magnitude and monthly distribution of each annual inflow is assumed known one year in advance from a prediction based upon the observed snowpack and typical monthly hydrology. This assumption is quite realistic for regions, such as the Pacific Northwest, where most precipitation falls as snow which later melts at a predictable rate. With this knowledge of monthly inflows and estimates of end-of-year state values, the dynamic programming proceeds in a deterministic manner to identify an optimal state trajectory from each possible beginning-of-year state. The actual distribution of annual inflow volumes is represented in the model by a number of discrete probable events. With operation commencing in a certain state, each discrete probable inflow causes an annual state transition with an associated return that contributes to the expected value of occupying the original state. This treatment accounts for the stochastic nature of inflows on an annual basis. 3.2 OPTIMIZATION OF LONG-TERM OPERATION For each new set of end-of-year state values estimated with the value determination routine, an improved operating policy is 28. determined for each discrete annual inflow. With a given sequence of monthly inflows, the dynamic programming routine finds an optimal set of monthly releases for simulated operation beginning in each possible starting-state. The objective of this operating policy optimization is to maximize the expected value of a l l future returns from each annual starting-state. Computation begins at the last month of the operational year, and imximizes the expected value of releases from each possible state in that stage. The expected value consists of returns from the current monthly release plus the expected value of occupying the resulting end-of-month state. At the last stage of each annual policy improvement, end-of-month state values are calculated by discounting the available estimates of end-of-year state values. Once an optimal release and expected value are stored for each state of the last stage, computation continues at the next to last stage and expected values are calculated for each possible state. This recursive calculation and storage of optimal releases and state values continues backward in time to the beginning of the operational year. Then, for each beglnning-of-year state i , a state trajectory for optimal operation is traced, by means of the stored release decisions, to the end of the simulation year. Each of these optimal state trajectories defines an annual Markov process state transition from beginning-state i to end-state j with a probability of occurrence, p.., equal to the probability of the monthly inflow sequence specified for the current dynamic programming optimization. The value assigned to each beglnning-of-year 29. state consists of the annual return and the discounted end-of-year state value that results from optimal operation (Equation 2.5.1). Therefore, the annual state transition return, ret.., is calculated by subtracting the discounted end-of-year state value from the beginning-of-year state value. The dynamic programming routine is repeated for a l l discrete probable annual inflows, and values of ret.. and p.. are obtained for a l l probable state transitions within the discrete limits of the analysis. Then, the value determination routine uses this policy improvement output to construct equations similar to Equations 2.5-3 and this system of equations is solved to provide new estimates of state values for use as end-of-year targets in the next policy improvement routine. When successive estimates of end-of-year state values converge, the modelled release policy yields the maximum system, value with operation continuing over an unbounded time horizon. 3.3 OPTIMIZATION OF RELEASE DECISIONS During the dynamic programming described above, each monthly release decision seeks to maximize the sum of returns from current releases plus the expected value of the end-of-month state due to the release. The returns from releases will depend on the existing demand for water. Figure 2.3.2b shows the form of water demand due to the composite energy demand considered here, and this form is reproduced in Figure 3.3-1- In this figure, several significant release quantities are labelled as follows: 30. q l 3 the minimum feasible release. q 2, the release necessary to satisfy the firm energy demand after a l l available thermal generation is used. q 3, the release necessary to satisfy the firm energy demand without the use of any available thermal generation. q^, the release corresponding to the generating plant's maximum capacity. at q , the optimal (value maximizing) release. For a given inflow volume and beglnning-of-month state, the expected value of the subsequent end-of-month state can be expressed as a function of release quantity (Figure 3 . 3 .la). An increase in this expected value with an increased release would only occur i f negative returns such as flood damages could not be avoided from higher storage states. The negative slope of this end-state value function represents the marginal cost of current releases in terms of lost future returns. For optimality, the quantity released must be increased until the marginal cost equals the marginal return. Figure 3«3-l includes three examples of common marginal cost conditions. The marginal cost is low in Figure 3.3.1b, indicating an abundance of water of l i t t l e future value, so considerable energy is produced for sale at the secondary price. However, each unit of water has a higher marginal cost in Figure 3.3-lc and only the firm demand on the reservoir is satisfied (without using any available thermal energy). Finally, Figure 3-3-Id illustrates a situation in which more water should be saved but the rninimum release is constrained for some reason. 31. a) End - o f - month s ta te value as a f unc t i on of r e l e a s e quantity q release quanti ty b ) Opt imal secondary g e n e r a t i o n q q q q * q release quan t i t y marginal cost d) Minimum release is c o n s t r a i n e d and thermal genera t ion is n e c e s s a r y q 2 q * q 3 q 4 release quant i ty FIGURE 3.3. I TYPICAL MARGINAL RETURNS AND MARGINAL COSTS OF RELEASES. 32. Reservoir capacity or downstream miriimum flow requirements form typical minimum release constraints. The total return from a release is the sum of marginal returns from a l l unit releases up to the quantity released. Because the marginal returns from firm energy generation are undefined for quantities below the minimum requirement, returns from these quantities are based on the cost of alternative thermal generation. Violations of the minimum firm energy constraint were allowed in the model, but only with a high penalty cost. The simulated operation never accepts this penalty unless i t is absolutely unavoidable due to high firm requirements and a scarcity of water. This 'soft' treatment of constraints has appeared in previous research [15,24,25], and in realistic reservoir operation, failure to meet firm demand can actually occur under continuing conditions of high load or low runoff. Such failures would incur penalties due to the high costs of emergency energy purchases, or the disutilities associated with any necessary cuts in service. This realistic possibility of failure to meet firm demand is allowed here with a somewhat arbitrary penalty equal to twice the usual thermal generation cost of meeting the shortage. Optimal releases are found at each decision point in the dynamic programming by testing i f the minimum feasible release yields a marginal return greater than marginal cost. If i t does not, the release is increased incrementally and the test is repeated. This continues until the marginal cost surpasses the marginal return, or until a maximum release constraint becomes critical. If marginal cost minus marginal return becomes positive through an increment of 33. release, the value maximizing release is known to be within the last increment. This increment is then repeatedly split down to some prespecified size and the midpoint of the final increment containing the value maximizing release is chosen as the optimum. The computer calculations performed in this research split the final discrete state increment four times while searching for the optimal release. Linear Interpolation is used to approximate the expected value of end-ofHTionth states between discrete state values. More elaborate interpolation methods were tried, but the minor improvement in results did not seem to justify the additional computational time. At each decision, the optimal release quantity, amount of energy, generated, and marginal cost of generation are stored until the eventual recovery of interpolated state trajectories. Since the value determination routine requires purely discrete annual state transitions, each transition ending in a non-discrete state is represented by two transitions to the two nearest states, and the probability of the original state transition is divided proportionally between the two substitutes. After state trajectories are identified, expected marginal costs of generation and marginal returns of inflows are calculated and stored for later use in coordinating the operation of several reservoir subsystems. 3.4 SERIES OF RESERVOIRS In a series of reservoirs on one river, the releases of upstream reservoirs influence returns received downstream. To coordinate the operation of these reservoirs without resorting to multiple state 34 variable dynamic programming, the approach taken here is to give upstream releases an additional return equal to the downstream value of water at the time of release. If the probabilities of occupying each beginning-of-year storage state are known, the expected state at each stage can be determined from the state trajectories. In long-term reservoir operation with stationary values of state transision probabilities, p.., the end-of-year state occurrence probabilities will converge on J one set, PI i(i=l,n), regardless of the state at which operation commenced. This final set of probabilities can be found by solving the following system of linear equations; PI. = ? PI. • p , , ; j=l,n, (3-4.1) J i 1 with the additional requirement that I PI, = 1. (3-4.2) i 1 Upstream monthly releases are estimated and, once the long-term operating policy has been determined for the reservoir furthest downstream in the series, the model proceeds to find the set of PI i values for that reservoir according to the above equations. The expected operating state at each stage with each discrete annual inflow is then found using the state trajectories corresponding to the discrete inflow and the probabilities of operation cconencing from each beginning state. 35. At each stage of the dynamic programming routine for a given annual inflow, a value will have been assigned to each state. The expected marginal value of additional water at a particular stage and state is equal to the rate of change in state value with an additional unit of stored water. This rate is calculated at the expected operating state in each stage for the given annual inflow, and the resulting value is entered as an additional marginal return from releases in the policy improvement of the next upstream reservoir. Perfect correlation is assumed for the magnitudes of unregulated annual inflows occurring at the upstream and downstream reservoirs. The operation of each reservoir in the series is optimized with the Inclusion of expected marginal downstream values of water to influence release decisions. Releases expected from this operation are stored to provide new estimates of controlled inflows when the optimization cycle returns to the downstream reservoir in the series. This release adjustment cycle through the series is repeated until reservoir releases and expected operating states reach an equilibrium. The cycle is shown schematically in Figure 3-4.1 for two reservoirs in series. In effect, the downstream reservoirs purchase water from the upstream reservoir for a price that depends on the month of release and the annual runoff conditions. Each price is the marginal value of water to the downstream reservoir at the time of release. When the releases have stabilized, no upstream reservoir monthly return can be increased without a greater decrease in future returns at that reservoir and at downstream reservoirs. 36. probabilistic inflows new estimates of upstream releases enter with estimates of upstream releases expected downstream marginal values of water controlled releases probabi listic inflows ,d/s\ storageN 1 1 1 controlled releases FIGURE 3.4. I RELEASE ADJUSTMENT CYCLE FOR A SERIES OF RESERVOIRS. 37. 3.5 COMPUTATIONAL FEASIBILITY The research model was computer coded (Appendix A) and tested with data based on several units of the BCHPA system. Execution times quoted here resulted from using the Fortran H compiler and AMDAHL V/6-II computer at the University of British Columbia. Any necessary solutions to systems of linear equations were obtained by calling the routine FSLE provided by the U.B.C. Computing Centre. Value determination-policy Improvement cycles typically converged within four iterations to give state values within .01 percent of previous estimates. Occasionally, minor interpolation inaccuracies in the policy improvement routine caused expected state values to continue oscillating slightly with repeated iterations. While this inhibited convergence within a finer limit than .01 percent, the crucial differences between individual state values in each set of estimates were not affected and remained virtually unchanged. For a single reservoir subsystem including twenty-six storage states, five discrete annual Inflows, and four iterations for convergence of state values, the model's optimization of long-term operation required less than 1.5 seconds of computer execution time. For three similar subsystems in a series, five release adjustment cycles through the series involved fifteen separate subsystem optimlzations. This required less than 20 seconds of execution time. The adjusted releases calculated for three reservoirs in series were examined after each release adjustment cycle. By assuming a constant downstream marginal value for water from an upstream 38. reservoir in each month, the actual elasticity of the downstream demand for water was ignored. This was of no consequence once the set of releases reached an equilibrium, but adjustments tended to erratically overcompensate for the different expected downstream marginal values of water in different months. To improve the efficiency of the release adjustment process, a damping factor was applied to the downstream marginal values (Line 24Q, Appendix A). This damping factor acts to decrease the downstream marginal value of additional water as releases are increased past the previous estimate for the same month and runoff conditions, and then has the opposite effect for decreases in releases below previous estimates. Finally, as releases approach the previous estimates, the damping effect disappears completely. It was found that the most efficient damping technique for three units in series was to obtain an Initial quick but rough adjustment with very low damping through three cycles of release adjustments. A stable set of releases could then be found within two additional cycles with increased damping. 39. CHAPTER 4 : APPLICATIONS OF THE MODEL 4.1 DATA Once the model was programmed and debugged, four different aspects of reservoir system optimization were investigated. To add some realism, simulations were based on BCHPA projects which already existed, or were under construction. The modelled system included Williston Lake on the Peace River controlling flows to the G.M. Shrum and Site One generating plants, the Revelstoke reservoir and plant downstream of the Mica reservoir and plant on the Columbia River, and the 1000 Mw Burrard thermal plant. Tables 4 .1 .1 , 4 .1 .2 , and 4.1.3 are summaries of the system data. Personnel of BCHPA provided much of the necessary information including system monthly firm load distribution, reservoir head-storage data, maximum generating plant capacities, and inflow data for the three reservoirs. Each annual inflow distribution was approximated by a log-*iormal distribution and five discrete probable inflows were selected for each reservoir. Two volumes, each with discrete probab-i l i t i e s of .05, were chosen to represent high and low magnitude events and three intermediate values, each with a discrete probability of .30, represented the remaining range of inflows. The beginning of January was used as the beginning of annual operation In the model. For British Columbia, this seemed most compatible with the assumption that an annual prediction based on snow surveys would be available. Although the best prediction is 40. TABLE 4.1.1 RESERVOIR CAPACITIES Reservoir Maximum .Head f t (m) Maximum Storage b i l l i o n f t 3 ( b i l l i o n m3) Williston Site One * Mica Revelstoke 551 (167.9) 135 ( 41.1) 585 (178.3) 423 (128.9) 2622.2 (74.2) 874.5 (24.8) 173.4 ( 4.9) Run of river operation i s assumed for Site One TABLE 4.1.2 ANNUAL RESERVOIR INFLOWS Probability of Occurrence Discrete Annual b i l l i o n f t 3 Uncontrolled Inflow ( b i l l i o n m3) Williston Mica Revelstoke .05 910 (25.8) 528 (14.9) 210 (5.9) • 30 1040 (29.4) 582 (16.5) 238 (6.7) .30 1140 (32.3) 627 (17.8) 262 (7.4) .30 1270 (36.0) 680 (19.3) 287 (8.1) .05 1440 (40.8) 750 (21.2) 325 (9.2) 41. TABLE 4.1.3 MONTHLY GENERATING SYSTEM DATA MONTH Secondary Selling Price rrdlls/Kwh Distribution of Annual Firm Demand Distribution of Inflow Annual Williston Mica Revelstoke JAN 24 .0948 .0185 .0145 .0156 FEB 18 .0844 .0166 .0122 .0129 MAR 18 .0902 .0170 .0143 .0152 APR 12 .0818 .0367 .0295 .0328 MAY 6 .0803 .1821 .1204 .1316 JUN 6 .0761 .2743 .2328 .2377 JUL 6 .0757 .1640 .2353 .2256 AUG 12 .0786 .0898 .1641 .1506 SEP 12 .0775 .0627 .0852 .0808 OCT 15 .0824 .0646 .0467 .0482 NOV 18 .0864 .0446 .0266 .0288 DEC 24 .0918 .0281 .0184 .0202 42. actually available during April, observations of snowfalls during the early winter months will influence reservoir operation from January to March. The estimates of current secondary energy selling prices shown in Table 4.1.3 were derived by multiplying BCHPA estimates from 1974 [26, p.107] by an Inflation factor of 1.5. These estimates are very approximate and take no account of quantity supplied or annual runoff conditions. An estimate of 25 mills/kwh for the cost of alternative thermal generation, which defined the value of firm energy, was derived in the same way as were secondary energy prices. This estimate was based on the variable cost of energy from the Burrard Generating Station with fuel priced at parity with the cost of residual o i l . Although the BCHPA planning study in 1974 assumed natural gas to be the primary Burrard thermal fuel, alternative evaluations were done with fuel costs equal to the cost of residual o i l [25, p.100]. The following reason was given; "This recognizes the possibility of an increase in the price of natural gas, a restriction in the supply of natural gas so that residual o i l would be required to be substituted for i t , or of both an increase in price and a restriction in supply." In addition, i t is obvious that some external unpriced cost is related to generation from natural gas. Tne cost of energy generated at natural gas fuel prices is less than the price usually offered for secondary energy [26, p .106], but because the Burrard plant does not operate, even in winter, to meet secondary demands, i t is implied 43-that total operating costs are greater than even the potential winter returns. In a l l applications of the model, future returns were dis-counted using a real rate of interest (Equation 2.4.2) of 1 percent, based on annual rates of 10 percent for the cost of loans and 9 percent for price Inflation in recent years. This is quite a low discount rate and could easily range up to 10 percent i f annual inflation dropped to 5 percent, and i f investments required an annual return of 15 percent, including an allowance for risk. Of course, the discounted present value of future reservoir returns varies directly with the discount rate, but the actual returns show l i t t l e variation. An increase in the discount rate applied to future returns will cause a slight decrease in the long-term mean annual return as operation tends to sacrifice more future benefits for immediate returns. 4.2 APPLICATION I : MONTHLY LOAD DISPATCHING This application follows previous research [19] which demonstrated the optimal annual allocation of firm load within a mixed thermal and hydroelectric generating system. In that research, long-term operation of single reservoir subsystems was simulated with available thermal generating capacity and various firm energy loads. The expected annual cost of thermal generation could then be plotted for each subsystem as a function of firm energy demand. To minimize expected annual cost, total system firm demand was divided among the subsystems so as to equalize the marginal 44. increases in expected annual cost with the last unit of firm load assigned to each subsystem. In the present application, the value of allocating demand to equalize subsystem marginal costs on a monthly basis has been demonstrated. With an i n i t i a l t r i a l allocation of demand, long-term reservoir operation was modelled and an expected marginal cost of generation was calculated for each month in the final policy improvement routine of each subsystem. The marginal cost of hydro-electric generation was derived from the marginal cost of releases less marginal returns from any source other than subsystem generation, such as downstream reservoir returns. Only one allocation of demand was made in each month, regard-less of Inflow, and the expected marginal costs were mean values for a l l probable annual inflows and beginmng-of-year states. Each single allocation is an aggregation which represents the mean of a l l of the allocations that would be optimal for the various possible combinations of inflows and storage states in the system. Although some realism is lost in this aggregation, the needs for multiple state variable dynamic programming and multivariate regression analysis are eliminated. The aggregation is most valid with a high degree of correlation between the inflows to different reservoirs, because low inflows at low states causing high marginal costs of production at one reservoir are not likely to coincide with opposite conditions at another reservoir. To demonstrate the value of monthly allocations, firm demand was first divided between two subsystems so that the allocation did 45. not vary from month to month, and so that the marginal expected annual costs of generation from a l l subsystems were equal. Then, the monthly allocations were adjusted to equalize marginal costs of generation in each month, and the improvement in total system performance was noted. The modelled system consisted of Willlston Lake regulating flow to the G.M. Shrum and Site One plants, the Mica reservoir regulating flow to the Mica and Revelstoke plants, and 1000 Mw of generating capacity at the Burrard thermal plant. The Site One and Revelstoke plants were assumed to operate always at f u l l head. This system was decomposed into two subsystems, each with a reservoir and some thermal and hydroelectric generating capabilities. The total annual hydro-electric capacity of this system was eventually found to be slightly less than 32000 Gwh, and Table 4.2.1 shows a summary of the improved performance achieved in long-term system operation with three different levels of firm demand. Appendix B contains a sample copy of computer printout for the final allocation of 32000 Gwh. By equalizing the marginal costs of monthly rather than annual generation, the expected system return was increased by more than 1.0 million dollars annually for a l l three levels of firm demand. An annual Increase of 3.4 million dollars was possible with 28000 Gwh of firm demand because, with this firm load, nearly 4000 Gwh of excess hydroelectric capacity existed and additional returns could be obtained by shifting secondary sales to winter months, as well as by minimizing reservoir drawdown losses. 46. TABLE 4.2.1 IMPROVEMENTS IN SYSTEM PERFORMANCE WITH MONTHLY COORDINATION FIRM ENERGY DEMAND Gwh WITH ANNUAL COORDINATION $million/year WITH MONTHLY COORDINATION $million/year IMPROVEMENT $million/year 28000 775.8 779.2 3.4 32000 786.4 787.5 1.1 36000 790.4 792.0 1.6 4.3 APPLICATION II : SERIALLY CONNECTED RESERVOIR OPERATION Three different series combinations of reservoirs were investigated with the model according to the procedure outlined in Section 3-4. The first system Included the Revelstoke project downstream of Mica and 1000 Mw of available thermal generating capacity. Because the total expected annual hydro-generation was found to be over 15000 Gwh and annual firm demand was set at 14000 Gwh, some secondary generation was possible in most years. Initially, run of river operation was assumed for Revelstoke and an expected annual return of 374.9 million dollars was calculated for the combined output of Mica and Revelstoke. Then, drawdown was allowed at Revelstoke, as well as Mica, and release decisions for Mica considered the value of water downstream at Revelstoke. The long-term operation of tills system was repeatedly simulated as allocations of monthly firm load were adjusted in an effort to equalize the 47. expected monthly marginal costs of hydroelectric generation. Finally, when no further allocation improvement seemed possible, the total expected annual return was calculated to be 374.6 million dollars. A slight drawdown was predicted for the Revelstoke reservoir during the months of April to July in higher than average runoff years. The model involves considerable aggregation of possible storage levels and inflows, and these simplifications combined with slight inaccuracies in dynamic programming state value interpolations can account for the 0.1 percent decrease in expected annual returns with live storage operation at Revelstoke. The writer feels that drawdown of the Revelstoke reservoir during early summer of high inflow years would slightly increase the combined value of the Mica and Revelstoke projects. However, within the existing model's limits of accuracy, run of river operation is Indicated as optimal for the Revelstroke project. While the above information is of great practical interest, the model had not adequately demonstrated coordinated operation including downstream reservoir drawdown. Therefore, a hypothetical two reservoir system was created. The downstream reservoir was based on Mica head-storage data with Revelstoke inflows and maximum generating capacity, and for the upstream subsystem, Mica inflows and generating capacity were combined with the nmimum head of the Revelstoke dam. Maxirium live storage at the upstream reservoir was set at two-thirds the storage available downstream. The larger values of head and live storage were deliberately assigned to the downstream project to ensure that drawdown of both reservoirs would 4 8 . be optimal. System operation showed an expected annual return of 364.1 million dollars when downstream drawdown was Initially not allowed. Then, with a t r i a l allocation of demand, coordinated operation using the storage of both reservoirs was simulated and an expected annual return of 386.4 million dollars was indicated. An improved allocation of load increased this return to 388.6 million dollars, and several more adjustments increased the expected annual return to 392.4 million dollars. Drawdown was indicated as optimal for both reservoirs in this hypothetical system and almost a l l the live storage available downstream was required to contain larger than average summer flood flows. Finally, a third system was created by adding another reservoir to control three-quarters of the inflow received upstream of the previous hypothetical system. New generating capacity was not added because the additional storage was only Intended, to improve the efficiency of existing facilities, but a simple return function was included to account for benefits depending on monthly water levels in the new reservoir. Such benefits would realistically arise from demands for flood protection, recreation or scenic beauty, and without this sort of return, the reservoir operation would not show any tendency to seek higher stages once the quantity of water stored was adequate for efficient downstream operation. A small idealized return was defined to depend on the square of average monthly storage states, and the annual return that eventually came from this source was about -2.0 million dollars. This has not been included with the energy returns discussed below. 49. With coordinated releases from a l l three subsystems, a final adjustment of monthly loads resulted in an expected annual return of 400.2 million dollars after four release adjustment cycles. Then, with no change in load allocation, this return was found to vary from 398.4 million dollars after three cycles to 397.7 million dollars after five cycles. These figures, combined with observed calculations of expected monthly operating states and releases, indicate a slight oscillation of final releases. This oscillation is the result of overadjustments of upstream releases, based on expected downstream water values, and a more stable solution and improved model performance can only be achieved by increasing the number of states representing possible storage levels, or by Improving the accuracy of monthly water value estimates. 4.4 APPLICATION III : HYDROELECTRIC RETURNS FROM THERMAL GENERATION Because the monthly distribution of energy demand does not coincide with the monthly distribution of inflow, reservoirs must be drawn down to adequately control releases and, as a result, generating efficiency is decreased. If some additional generation is available from thermal sources, reservoir drawdown is decreased and hydro-electric returns are increased by meeting firm demand with thermal generation at appropriate times. In general, thermal generation is appropriate when natural flows of water are inadequate to generate required firm energy. By reducing reservoir output at these times, 50. more hydroelectric energy can be produced later when water is more plentiful. To demonstrate this potential for additional benefits, various levels of thermal energy capability were assumed and expected annual returns were calculated for Mica releases through the Mica and Revelstoke plants. The results show that, as firm demand grows, the addition of thermal generation capacity increases the value of existing hydro plants. The curves in Figure 4.4.1 illustrate these results. With growing firm demand and a limited thermal capacity, reservoir benefits reach a maximum and then decline as available thermal generation becomes insufficient during critical inflow periods. Greater reservoir drawdown becomes necessary, and eventually, penalties for firm output shortages become more likely. Figure 4.4.2 shows the maximum reservoir returns expected for various thermal capacities. As the annual load increases, an annual benefit is caused by system capacity expansion that allows thermal generation to improve hydroelectric output. For example, i f the output of the Mica reservoir is augmented by 1000 Mw from a proposed thermal project in order to increase annual firm energy capacity from 14000 Gwh to 21000 Gwh, the expected annual hydro returns will be increased by 12.5 million dollars. In any economic analysis of the proposed thermal project, this increase In returns should be included as an annual benefit. 4 0 0 THRMX = 6000MW 3 4 0 THRMX = 4000MW NOTE : GENERATION IS FROM MICA AND REVELSTOKE PLANTS WITH NO DRAWDOWN AT REVELSTOKE 1 10 15 2 0 25 3 0 35 4 0 45 50 55 F i rm e n e r g y demand ( x l 0 3 Gwh ) FIGURE 4.4.1 RESERVOIR GENERATION RETURNS WITH VARIOUS AVAILABLE THERMAL GENERATION CAPACITIES. 6 0 52. 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 Thermal complement-THRMX ( Mw) FIGURE 4 . 4 . 2 MAXIMUM RESERVOIR GENERATION RETURNS WITH VARIOUS AVAILABLE THERMAL GENERATION CAPACITIES. 53-4.5 APPLICATION IV : PROJECT TIMING With negligible "reservoir operating costs, the expansion of system capacity with a proposed hydroelectric project becomes beneficial when the potential Increase in annual system returns surpasses the annual fixed costs of the project. These fixed costs typically include the annual cost of required capital, depreciation and maintenance. For the Revelstoke project currently being constructed downstream of Mica, the total annual fixed cost was estimated in 1974 to be 88.1 million dollars [26, p.75]. With current price levels, this annual cost is estimated now to be 132.0 million dollars. The long-term expected annual return has been calculated for the Mica reservoir with 1000 Mw of available thermal capacity and 500 Gwh as an upper limit for secondary energy sales. The calculations were done initially with no generation available at Revelstoke, and then with additional energy production there due to the new project. Finally, annual returns which included Revelstoke production were decreased by 132.0 million dollars to account for the cost of the additional project. Figure 4.5.1 contains plotted curves resulting from sets of calculations for various firm loads. Based on long-term operation, this information provides an estimate of the optimal time at which the new project should be brought into the system. According to the plotted results, the Revelstoke project is beneficial once the Mica reservoir subsystem is required to meet firm 54. FIGURE 4.5.1 NET INCREASES IN EXPECTED ANNUAL RETURNS GAINED WITH THE REVELSTOKE PROJECT. demands exceeding 9500 Gwh. Often a project is delayed because of environmental or other nonpriced costs which are not included in the project's estimated annual fixed cost. In these cases, an analysis similar to this can provide an estimate of potential monetary benefits forgone each year in favour of the nonpriced savings. This information is of value in the qualitative decision concerning optimal project timing. Of course, a more accurate analysis would consider the effect of including the new project in the coordinated operation of the entire system according to the method discussed in Section 4.2. In addition, any new reservoir must be f i l l e d before expected annual returns are possible and the transient f i l l i n g period prior to steady-state operation is important in planning i n i t i a l reservoir operation. This transient operation is discussed in the next chapter. 56. CHAPTER 5 : PROPOSED EXTENSIONS 5.1 TRANSIENT CONDITIONS This research thesis has employed a reservoir model that is capable of simulating efficient reservoir system operation for steady-state load and storage conditions. However, because real systems always exist in transient conditions, the value of a steady-state model is limited. Once a reservoir is f i l l e d and operated for a few years within a large integrated system, the steady-state approximation is valid. If new units are added annually to accomodate a typical annual load growth of 8 percent, the load placed on any particular subsystem will only vary within 4 percent of the mean. Therefore, steady-state simulation can assist in the design of new units for maximum long-term annual returns, or i t can indicate the eventual operating policy that preliminary operation should evolve towards. Although the steady-state assumption prevents the present model's application to the optimization of short-term releases according to current storage levels and inflow predictions, end state values can be approximated for a short-term transient analysis extending over several years. Remembering that optimal current release decisions depend on only the difference between possible end-of-month state values and not the magnitudes of these values (end of Section 2.3), a transient analysis can be carried far enough ahead to eliminate any significant variation in current decisions 57. due to the ultimate steady-state assumption. The rate at which transient operation approaches steady-state will depend on system capacity in excess of firm demand, and some experimentation will be necessary to determine an adequate length of transient analysis. The same dynamic programming algorithm as presently used for policy "improvement will optimize releases, trace probable state trajectories for operation continuing from current states, and calculate probable monthly states and marginal costs for a l l reservoirs. These values can be used according to methods described in Sections 4.2 and 4.3 to coordinate the operation of a l l reservoirs, existing and new, up to assumed steady-state conditions. The accuracy of these coordination methods will improve in transient analysis because there is less uncertainty about future probable storage states when the current state is known. 5.2 ALLOCATIONS OF DEMAND Only firm and secondary energy demands have been divided between subsystems in this study, but any water demand expressible in monetary terms can be included. Simple mathematical approximations of benefits as functions of output, as in past research [ 4 , 8 ] , can be combined to yield composite demand functions. However, each monthly demand that must be separately allocated within the system adds to the number of decisions required in the subsystem coordination task. In Sections 4.2 and 4 .3 , each final allocation of demand required from four to six trials with load adjustments in a l l twelve months, and in an analysis with transient conditions, adjustments to subsystem loads 58. will be possible in each month of each year. The development of a computer routine for this adjusting process is highly desirable, especially i f transient conditions and several competing monthly demands are to be considered. At present, observations of calculated values of marginal costs indicate an inaccuracy of about 5 percent which causes load adjustment to proceed rather irregularly. The response of calculated marginal costs to changes In demand will be smoother i f interpolation accuracy is improved, possibly with quadratic approximations. This response will also be improved with the use of a more realistic continuous form of demand than that shown in Figure 2 .3 .2 . The 1974 BCHPA planning study [26, p.107] gives a constant price for secondary energy sales and a higher price for energy purchases in each month. However, as noted in that study, the prices will vary with different quantities supplied. As quantities sold or purchased in a month approach zero, the selling or purchase price will likely approach the same value except for a small transaction cost (Figure 5.2.1a). If some thermal energy capacity is available within the system at a constant marginal cost of PTH, the plot of marginal returns against quantity supplied by the hydroelectric plants will be similar to Figure 5-2.lb (FE represents the system firm load) .• Figure 5.2.1c illustrates a simplified continuous version of system demand that will adequately represent returns from hydroelectric generation. With improved interpolations and demand functions, a method can be developed to-make monthly load adjustments, within the computer 59. external equilibrium p purchase price selling price quantity exported or imported a) Energy prices external to the system . available thermal energy capaci ty PTH " FE quantity b) Real ist ic form of hydroelectric generation demand PTH -quantity c) Simplified representat ion of (b) FIGURE 5.2.1 IMPROVED REPRESENTATION OF MONTHLY HYDROELECTRIC GENERATION DEMAND. 60. program, between several repetitions of subsystem optimizations. The progress of adjustments can then be checked by the model user, and new instructions can be entered i f revised automatic adjustments are desired. In this way, more computer time but less man-hours will be required to find the optimal allocations of system demands. 5.3 RISK OF FAILURE One of the criteria faced by hydroelectric system planners is that firm quantities will be supplied. Past practice has been to estimate future demand, and then provide adequate capacity to meet this estimated demand with the recurrence of a critical inflow sequence selected from historical records. It must be realized that there will always be some probability that expected future demand will be surpassed or a more critical flow sequence will occur with the result that some firm demand cannot be satisfied. Indeed, there may occasionally be times when supply from outside the system is desirable because of high system operating costs. This situation would corres-pond to production in the range of marginal returns greater than PTH in Figure 5-2.1. An alternative to critical period analysis is the determination of system failure probabilities with specific firm loads and probable inflows. Reservoir state transition probabilities have previously been used for this purpose [27,28], and one study [25] has included penalties for failures to meet target releases. The probable state trajectories identified in the dynamic programming simulation previously described (Section 3-2) will enable calculation of the 61. probability that a l l live storage will be depleted and a supply shortage will develop. System capacity can then be provided at a rate that maintains the system failure probability below some, specified limit. If a monetary penalty or high cost of emergency purchases is defined, the problem simply involves a tradeoff between expected annual system failure losses and annual costs of additional capacity. 5.4 RESERVOIRS ON BRANCHING RIVERS As the development of a river basin continues, reservoirs are usually constructed on branching upstream tributaries. If the releases of these reservoirs combine and add to the inflow of a downstream reservoir, the optimization of upstream release decisions must recognize effects on downstream returns, just as in the case of reservoirs in series on one river. The method described in Section 3.4 will apply here with a slight reorganization of subsystem coordination. The expected marginal value of water to a downstream reservoir will be included in the returns from the reservoir immediately upstream on each tributary. Then, the releases from a l l upstream reservoirs contribute to estimates of downstream reservoir inflows for the next release adjust-ment cycle. After several of these cycles, the loads placed on the subsystems can be adjusted and the entire process repeated to ultimate-ly determine the optimal allocation of monthly loads, and the corresponding releases. 62. 5.5 INFLOW UNCERTAINTY Provided sufficient historical records of hydrology exist, streamflow magnitudes can be quite accurately described by probability distributions and computational requirements will Increase rapidly as simulation models become more detailed In their descriptions of these probability distributions. An additional problem exists in realistically sijriulatlng the decision-maker's knowledge of future events. In this study, only annual inflow uncertainty was modelled and complete knowledge of inflow during each operating year was assumed to be available at the start of the year. Monthly inflow uncertainty is most important in regions where most precipitation is experienced as rainfall and seasonal streamflow patterns are poorly defined, but the accuracy of simulations for any region will be affected by simplifications regarding future flows. If prediction outcome uncertainty is treated on a monthly basis, expected values of the objective are calculated for a l l possible outcomes of each possible release. Necessary computational effort is increased but state trajectories are s t i l l easily recoverable. However, i f the decisions are uncertain despite fixed operating distributions of possible decisions cause trajectories to divide into increasing numbers of possible trajectories at each stage. Consequently, computational effort is increased and trajectory recovery also becomes very difficult. It is proposed here that, because the value determination routine requires stationary annual state transition probabilities (Section 2.5), the distribution of annual inflow volumes should 63. continue to be represented by discrete serially independent events. If increased model realism is desired, monthly uncertainty can be considered within the dynamic programming of annual operation in the following way. Decision uncertainty is assumed to be nonexistent so that a particular monthly prediction of future events will be followed by one certain release decision outcome. In reality, any set of actual future events can be preceded by various likely predictions but, in order to hold computational time within feasible limits, these likely predictions will continue to be represented by a mean that is assumed accurate. However, In their consideration of a prediction, decision-makers will be aware of potential error and will estimate expected values according to the events they expect to follow the prediction. At any monthly decision stage, one perfectly predicted inflow will occur with certainty, but the release decision will be made according to the consequences of a l l the possible occurrences perceived by the decision-makers. These possible consequences will be weighted according to the estimated dependability of predictions. If information about prediction accuracy is available, this added degree of realism can easily be modelled. This will allow simulated annual operation in the Pacific Northwest to commence in autumn instead of January, thereby increasing the validity of the assumption that annual inflows are serially independent. The simulation will also be improved for operation during the mid-winter months in which high secondary energy prices coincide with high uncertainty about eventual inflows. 6H. CHAPTER 6 : CONCLUSIONS The proposed optimization techniques have been tested and shown to be effective in four applications involving the operation and planning of a system of reservoirs. Marginal costs of reservoir releases were defined in terms of lost future returns, and the total present value of the system consistently increased when reservoir releases were guided by these marginal costs and basic principles of economic efficiency. The divisions of system load between two reservoir subsystems were adjusted to equalize the marginal costs of monthly production in simulated steady-state operation and the resulting increases in expected annual system returns were found to be significant. Because of the single valued monthly demand allocation and the aggregate marginal cost values for a l l probable storage levels and inflows in each month, further increases in returns would be possible with the reduced uncertainty of storage and inflows in real-time operation. At present, the adjustment of monthly load allocations is a t r i a l and error process required between computer suboptimizations, but a computer assisted approach to the problem has been suggested. In the coordination of serially connected reservoirs, the release adjustment cycles converged quite quickly but the final solution continued to oscillate slightly due to the discontinuity of monthly demands and inaccuracies of necessary linear interpolations. Despite this oscillation, consistent improvements in system value 65. were achieved for two and three reservoir systems. For one system of two reservoirs in series, optimization indicated nearly run of river operation for the downstream reservoir. The optimality of upstream reservoir operation could then be verified with a single subsystem simulation in which the downstream reservoir was constrained to run of river operation. Virtually identical results were obtained. Long-term expected annual returns calculated for a hydroelectric plant increased as thermal capacity was added to the subsystem to accomodate growing demand. This increase was due to a reduction In the generating efficiency losses that accompany reservoir drawdown. For the reservoir studied,, a rapid rise in expected hydroelectric returns was achieved as available thermal generation increased the combined subsystem energy capacity by approximately 25 percent. Beyond this capacity, the rate of increase became more moderate as energy capacity was expanded with thermal plants. These additional hydroelectric benefits are a .direct result of thermal plants construct-ed to supply a growing demand, and therefore, the additional returns should be considered in any economic evaluation of thermal plants as alternatives for system expansion. The final application also concerned system planning by providing an estimate of the demand level at which a proposed project will become economically beneficial considering the annual fixed costs of the project. This solution to the project timing problem is a first approximation which would require an analysis with transient system conditions for improvement. 66. A l l mathematical models are merely representations of reality with many inaccuracies and simplifications in their construction and input data. Consequently, any solutions identified as optimal in a modelled system will only be approximations of the truly optimal solution for the real system. It is futile to search for ideal solutions with approximate models, and instead, the intention should be to provide as much guidance as possible for real system decisions. In this research, a reasonably realistic model of reservoir operation has been presented and methods for reservoir system optimization have been demonstrated. The model Is restricted to steady-state conditions with system coordination based on aggregated monthly operating conditions, but the simulation of long-term operation does provide information that is valuable In the planning and design of new projects. In addition, this simulation can quickly provide information about the ultimate steady-state operation that should evolve from i n i t i a l conditions. Without a model, this inform-ation can only be obtained from years of real system operation. A proposed extension of the existing model has been outlined with the intention of approximating optimal transient operation. By simulating several years of transient operation leading to steady-state conditions, values of current storage levels can be estimated for the approximation of optimal short-term releases. Without end-conditions described by a steady-state model, a transient analysis would need to include more years of future operation and any solution obtained would s t i l l be an approximation of the true 67. optimum. In short-term optimization, storage levels and Inflows are less uncertain for the immediate future and system coordination will be more precise than in steady-state analysis. Although the detail and flexibility of the present model is already adequate for practical purposes, the realism and usefulness of simulation results could be increased with the refinements suggested for the interpolation of state values and the treatment of inflow uncertainty. Very little computer time is currently required and more detail in the model could easily be accommodated. Maass, A., et al. Design of Water-Resource Systems, Harvard University Press, Cambridge, Mass., 1962. Young, G.K. "Finding Reservoir Operating Rules, " Journal of the Hydraulics Division, ASCE, Vol. 93, (Nov. 1967), p. 297-321. Roefs, T.G., and Bodin, L.D. "Multireservoir Operation Studies," Water Resources Bulletin, Vol. 6, (Apr. 1970), pp. 410-420. Lee, E.S., and Waziruddin, S. "Applying Gradient Projection and Conjugate Gradient to the Optimum Operation of Reservoirs," Water Resources Bulletin, Vol. 6, No. 5, 1970, pp. 713-724. Becker, L., et al. "Operations Models for Central Valley Project," Journal of the Water Resources Planning and Management Division, ASCE, Vol. 102, (Apr. 1976), pp. 101-115. Bellman, R.E. Dynamic Programming, Princeton University Press, Princeton, N.J., 1957-Little, J.D.C. "The Use of Storage Water in a Hydro-Eletric System," Journal of the Operations Research Society of America, Vol. 3(May 1955), pp. 187-197. Buras, N. "Conjuctive Operation of Dams and Aquifers," Journal of the Hydraulics Division, ASCE, Vol. 89, (Nov. 1963), pp. 111-131. Hall, W.A., and Roefs, T.G. "Hydropower Project Output Optimization," Journal of the Power Division, ASCE, Vol. 92, (Jan. 1966), pp. 67-79. Meredith, D.D. "Optimal Operation of Multiple Reservoir System," Journal of the Hydraulics Division, ASCE, Vol. 101, (Feb. 1975), pp. 299-312. Heidari, M. et al. "Discrete Differential Dynamic Programming Approach to Water Resources Systems Optimization," Water Resources Research, Vol. 7, (Apr. 1971), pp. 273-282. Fults, D.M., et al. "A Practical Monthly Optimum Operations Model," Journal of the Water Resources Planning and Management Division, ASCE, Vol. 102, (Apr. 1976), pp. 63-76. Parikh, S.C. "Linear Dynamic Decomposition Programming of Optimal Long-Range Operation of a Multiple Multipurpose Reservoir System," Proceedings, 4th International Conference on Operations Research, Boston, 1966. Hall, W.A., et al. "Optimization of the Operation of a Multiple Purpose Reservoir by Dynamic Programming, " Water Resources Research, Vol. 4, (June, 1968), pp. 471-477. Hall, W.A., et al. "An Alternate Procedure for the Optimization of Operations for Planning with Multiple River, Multiple Purpose Systems," Water Resources Research, Vol. 5 , (Dec. 1969), pp. 1367-1372. Hufschmidt, M.M., and Flering, M.B. Simulation Techniques for Design of Water Resource Systems, Harvard University Press, Cambridge, Mass., 1966. Loucks, D.P., and Falkson, L.M. "A Comparison of Some Dynamic, Linear and Policy Iteration Methods for Reservoir Operation," Water Resources Bulletin, Vol. 6, No. 3, 1970, pp. 384-400. Butcher, W.S. "Stochastic Dynamic Programming for Optimum Reservoir Operation," Water Resources Bulletin, Vol. 7, (Feb. 1971), pp. 115-123. Caselton, W.F., and Russell, S.O. "Long-Term Operation of Storage Hydro Projects," Journal of the Water Resources Planning and Management Division, ASCE, Vol. 102, (Apr. 1976), pp. 163-176. Howard, R.A. Dynamic Programming and Markov Processes, MIT Press, Cambridge, Mass. i960. James, D.L., and Lee, R.R. Economics of Water Resources Planning, McGraw-Hill, New York, 1971. Haimes, Y.Y. Hierarchical Analyses of Water Resources Systems, McGraw-Hill, New York, 1977. Hall, W.A., and Dracup, J.A. Water Resources System Engineering, McGraw-Hill, New York, 1970. Erickson, L.E., et al. "A Nonlinear Model of a Water Reservoir System with Multiple Uses and the Optimization by Combined Use of Dynamic Programming and Pattern Search Techniques," Water Resources Bulletin, Vol. 5> No. 3, 1969, pp. 18-36. Askew, A.J. "Optimum Reservoir Operating Policies and the Imposition of a Reliability Constraint," Water Resources Research, Vol. 10,. (Feb. 1974), pp. 51-56. British Columbia Hydro and Power Authority: Task Force on Future Generation and Transmission Requirements, Alternatives 1975 to 1990, British Columbia Hydro and Power Authority, Vancouver, B.C. 1975-Moran, P.A.P. "A Probability Theory of Dams and Storage Systems," Australian Journal of Applied Science, Vol. 5, (June, 1954), pp. 116-124. Mitchell, T.B. "Reservoir Yield Using TPM Method," Journal of the Hydraulics Division, ASCE, Vol. 103, (Feb. 1977), pp. 133-150. APPENDIX A COMPUTER PROGRAM LISTING 72. l 2 3 C 4 C O P T I M I Z A T I O N O F S T E A D Y - S T A T E O P E R A T I O N OF M U LT I R E S E R V O I R S U B S Y S T E M S 5 C 6 R E A L E F A C T / 8 . 7 6 6 / , C 2 / . 0 1 0 5 7 5 5 2 / , P V S Y S T / O . / , T O T A L / 0 . / , 7 * P 0 R T N ! 1 2 , 2 l , A N N V 0 L t 5 , 2 » , F E A L O C ( 1 2 , 3 I , T H A L O C ( I 2 , 3 1 « * , H ! 2 6 , 3 I , 0 ( 1 2 , 5 , 3 I , G E N I 1 2 , 5 , 3 ) , G E N M C C 1 2 , 5 , 3 1 , 9 * F E D M R ( 1 2 J , F E M N U 2 ) , P S G ( 1 2 ) , G N M C A V 1 1 2 I , E S T V I 2 7 , 3 I , E S T O ( 6 3 , 3 ) 1 0 D I M E N S I O N D E L T A I 3 ) , D O L M T ( U , N I ( 3 ) , S I ( 3 ) , C A P M A X ( J ) , D V D I ( 3 ( , HVJSi i > 1 1 C O M M O N N S , B F T A . N P D S , N S U B S , N V . P T H . M H L M T , P V O L ( 5 ) , P R B H 1 2 , 5 , 3 ) 1 2 C O M M O N / D E P / D V D S ( b O ) , N D S S , N U S S , K C H K O . D A M P 1 3 R E A D I 5 . 5 0 1 I R , N P D S . N V , N S U B S , F E D , T H R M X . P T H , M H L M T , [ T L M T , D A M P , 1 4 * F E D M R , P S G , P V O L 1 5 5 0 1 F 0 R M A T ( F 4 . 3 , 3 I 3 , F 8 . 0 , F 7 . 0 , F 5 . 1 , 2 I 3 , F 4 . 2 , / , 1 2 F 6 . 4 , / , 1 2 F 5 . 1 , / , 5 F 5 . 3 ) 1 6 DO 1 M R = 1 , N P D S 1 7 P S G ( M R ) = P S G ( M R ) / 1 0 0 0 . 1 8 1 F E D M R ( M R I = F E D M R ( M R ) * F E D 1 9 W R I T E ( 6 , 6 0 1 I 2 0 6 0 1 F O R M A T ! • I S Y S T E M D A T A : " ) 2 1 W R I T E I 6 . 6 0 2 ) R , N V . N S U B S , F E D , T H R M X , P T H , M H L M T , I T L M T , D A M P 2 2 6 0 2 F O R M A T ( • R = « , F 5 . 3 , ' N V = ' , I 2 , ' N S U B S = ' , I 2 / « F E D = ' , F 7 . 0 , • G W H T H R M 2 3 * X = • , F 6 . 0 , » M W • / • P T H = • , F 5 . 1 , ' M I L L S / K W H • / • M H L M T = • , I 2 , « I T L M T = ' , I 2 , 2 4 * • D A MP = ' , F 4 . 2 ) 2 5 3 E T A = l . / ( l . * R ) 2 6 P T H = P T H / 1 0 0 0 . 2 7 C M A X T H E R M A L G E N E R A T I O N D U R I N G O N E P E R I O D 2 8 T H R M X = T H R M X * E F A C T / F L O A T ( N ° D S ) 2 9 W R I T E ! 6 , 6 0 3 ) 3 0 6 0 3 F O R M A T 1 ' - S U B S Y S T E M D A T A : ' / 1 6 X , ' D E L T A D O L M T N I S I ( M K C F T ) CA>> 3 1 * M A X ( M W I N R U S ' ) 3 2 DO 2 N S = l , N S U f i S 3 3 R E A D ( 5 , 5 0 2 ) D E L T A ( N S ) , D Q L M T ! N S ) , N I ( N S ) , S l ( N S I , C A P M A X ( N S ) . O V O I ( N S ) , 3 4 * N R U S ( N S t , I P Q R T N I M R , N S I , M R = 1 . N P D S I 3 5 5 0 2 F 0 R M A T ( F 5 . 4 , F 6 . 4 , I 3 , F 7 . 1 . F 6 . 0 . F 7 . 2 , 1 3 , / , 1 2 F 6 . 4 ) 3 6 N I N S = N H N S ) 3 7 W R I T E ( 6 , 6 0 4 ) N S , O E L T A I N S I , D O L M T < N S ) , N I N S , S I ( N S t , C A P M A X < N S I , N R J S ( N S I 3 8 6 0 4 F O R M A T ! ' S U B S Y S T E M * , 1 2 , ' ! ' , 2 F 8 . 4 , I 5 , F 1 0 . l , F H . O , 1 8 1 3 9 C A P M A X ( N S ) = C A P M A X I N S ) * E F A C T / F L O A T ( N P O S ) 4 0 R E A D ( 5 , 5 0 3 ) { H ( I , N S ) , I = 1 , N I N S I 4 1 5 0 3 F O P M & T ! 1 1 F 6 . 1 , / , 1 1 F 6 . 1 , / , 1 1 F 6 . 1 ) 4 2 R E A D < 5 , 5 0 4 ) < A N N V O L < N , N S ) , N = I , N V ) 4 3 5 0 4 F O R M A T ! 5 F 7 . 1 ) 4 4 D O 3 N = 1 , N V 4 5 3 A N N V O L I N . N S t = A N N V O L i N , N S I / S I ! N S ) 4 6 R E A D ! 5 , 5 0 5 ) ( F E A L 0 C ! M R , N S I , M R = 1 , N P D S I , I T H A L O C ( M R , N S ) , M R = 1 , N P D S ) 4 7 5 0 5 F O R M A T ! 1 2 F 6 . 3 , / , 1 2 F 6 . 3 I 4 8 DO 4 M R = 1 , N P D S 4 9 F E A L O C ! M R , N S ) = F E A L O C ( M R , N S I * F E D M R I M R ) 5 0 4 T H A L 0 C I M R , N S ) = T H A L O C ! M R , N S I * T H R M X 5 1 2 0 E L T A 1 N S ) = D E L T A ( N S ) * R 5 2 5 3 C A P P L Y H O W A R D ' S M E T H O D T O E A C H S I N G L E R E S E R V O I R S U B S Y S T E M 5 4 N S = 0 5 5 1 3 N S = N S + 1 5 6 I F ( N S . G T . N S U B S ) G O T O 5 5 7 C I F R E S E R V O I R N S I S N O T AT T H E D / S E N D O F A S E R I E S , GO TO 6 5 3 I F ( N R U S I N S ) . E Q . 0 ) G O TO 6 5 9 N D S S = N S 6 0 N D S = N S - H 73. 6 1 N U S S = N R U S I N S ) - N S 6 2 N V N P D S = N V * N P D S 6 3 0 0 7 N S = N D S , N U S S 6 4 DO 7 M R N 0 = l . N V N P D S 6 5 7 E S T Q ( M R N 0 , N S ) = 0 . 6 6 DO 8 N S = N D S S . N U S S 6 7 N S R = N U S S + N D S S - N S 6 8 N I X = N I ( N S R l - 1 6 4 0 0 9 1 = 1 , N I X 7 0 9 E S T V ( I , N S R ) = F L O A T { I I * D V D I ( N S R I 71 I F ( N S R . E Q . N D S S ) G O TO 8 72 DO 10 N 0 = 1 , N V 7 3 V O L = A N N V O L ( N Q , N S R » * S I I N S R t / F L O A T ( N P DS t 74 DO 10 M R = l , N P D S 75 M R N O = M R - l N C - l l * N P D S 7 6 DO 10 N = N D S , N S R 77 10 E S T O l M R N Q . N ) = E S T Q ( M R N O , N l + V O L / S H N - l I 78 8 C O N T I N U E 7 9 K C H K 0 = 1 8 0 K C O U N T = 0 81 C R E P E A T H O W A R D ' S METHOD FOR D E P E N D E N T R E S E R V O I R S U N T I L E S T Q S T A B I L I Z E S 82 1 5 K C 0 U N T = K C C U N T - 1 8 3 DO 11 N S = N D S S , N U S S 84 11 C A L L D E P S U B I O E L T A I NS ) , N I , S I . C A P M A X I NS ) , H , C ? , P O R T N , A N N V O L , G E N , i) , 8 5 * G E N M C , F E A L O C . T H A L O C , F E D MR , P S G , E ST V , ES T a , P VS YS T ) 86 I F { K C H K O . E C . I I GO TO 12 8 7 W R I T E l 6 , 6 0 5 l KC OUN T 8 8 6 0 5 F O R M A T C O N O . OF R E L E A S E A D J U S T M E N T C Y C L E S ON D E P E N D E N T S U B S Y S T E M S 8 9 * = ' , I 2 / / t 9 0 GC T O 1 3 91 1 2 K C H K O = 0 9 2 0 0 14 N S = N D S , N U S S 9 3 S I P R O P = S I ( N S I / S K N S - l I 9 4 DC 14 N 0 = 1 , N V 9 5 N O N P D S = ( N O - 1 ) * N P D S 9 6 DO 14 M R = 1 , N P D S 9 7 M R N C = M R + N C N P D S 9 8 I F ( A S SI E S T C I M R N J . N S l / S I P R 3 P - Q ( M R , N O , N S I » . G T . D O L M T ( N S t I K C H K 3 = l 9 9 14 E S T O I M R N 3 , N S ) = 0 ( M R , N O , N S ) * S I P R O P 1 0 0 I F ( KCOUNT . E Q . I T L M T ) KCHK.Q=0 1 0 1 GO TO 1 5 1 0 2 C C l FOR C A L C OF G E N E R A T I O N FROM E A C H R E L E A S E 1 0 3 6 C 1 = S I ( N S I * C 2 1 0 4 C A P P L Y H O W A R D ' S METHOD ONLY ONCE FOR E A C H N O N S E R I E S R E S E R V O I R 1 0 5 C A L L I N D S U B t D E L T A ! N S ) . N I I N S I . C A P M A X ( N S ) , D V D I { N S ) . H , C l . P O R T N , 1 0 6 * A N N V O L , G E N , 0 , G E N M C , F E A L O C , T H A L O C , F E D M R , P S G . ° » 5 * N S ) 1 0 7 P V S Y S T = P V S Y S T ~ P W E R N S 1 0 8 GO TO 1 3 1 0 9 1 1 0 C 0 / P SUMMARY OF E X P E C T E D Q U A N T I T I E S AND M A R G I N A L C O S T S OF 1 1 1 C S U B S Y S T E M P R O D U C T I O N 1 1 2 5 DO 21 M R = 1 , N P D S 1 1 3 2 1 P S G I M R ) = P S G ! M R ) * 1 0 0 0 . 1 1 4 W R I T E ( 6 , 6 0 8 ) ( P S G ( M R ) , M R = 1 , N P D S ) 1 1 5 6 0 8 F O R M A T I ' I S U B S Y S T E M PROD IX T l O N : • / / , 1 3 X , • D E C NOV OCT S E P 1 1 6 * AUG J U L J U N MAY APR MAR F E B J A N ' / 1 1 7 * ' 0 P S G = ' , 1 2 F 8 . 1 , « M I L L S / K W H * I 1 1 8 DO 16 N S = l , N S U B S 1 1 9 DO 1 7 M R = 1 , N P D S 1 2 0 G N M C A V ( M R ) = 0 . 74. 1 2 1 1 7 F E M N ! M R > = F E A L O C ( M R , N S I - T H A L O C ( M R , N S t 1 2 2 W R I T E < 6 , 6 0 9 ) N S , ( F E A L O C ( M R , N S ) , M R = I , N P D S 1 , t F E M N ( MR ) , MR = l , N P O S ) 1 2 3 6 0 9 F O R M A T ( • 0 S U B S Y S T E M • , 1 2 , ' : ' / • F E A L O C = ' , 1 2 F 8 . 0 , ' G W H ' / ' F E M N = • , 1 2 4 * 1 2 F 8 . 0 , « G W H " / ) 1 2 5 G E N S U M = 0 . 1 2 6 0 0 1 8 N Q = 1 , N V 1 2 7 0 0 1 9 M R = 1 , N P D S 1 2 8 G E N S U M = G E N S U M - G E N l M R , N O , N S ) * P V O L ( N O ) 1 2 9 G E N M C 1 M R , N O , N S ) = G E N M C ( M R , N a , N S ) * 1 0 0 0 . 1 3 0 1 9 G N M C A V ( M R | = G N M C A V I MR I + G E N M C ! M R , NO , N S I * P V 01. I N 0 I 1 3 1 1 8 W R I T E I 6 , 6 1 0 I N 0 , ( P R 3 I ( M R , N O . N S ) , M R = 1 , N P D S ) . ( G E N ( M R , N O , N S ) , M R = l , N P D S 1 3 2 * > , < G E N M C ( M R , N O , N S I . M R = 1 , N P D S ) 1 3 3 6 1 0 F O R M A T ! • N Q = ' , I 1 / ' P R B 1 = « , 1 2 F 8 . 2 / ' G E N = ' , 1 2 F 8 . 0 , « G W H • / * G 1 3 4 * E N M C = ' , 1 2 F 3 . 1 , ' M I L L S/KWH' ) 1 3 5 G N M C N S = 0 . 1 3 6 0 0 2 0 M R = l , N P D S 1 3 7 T H A L O C ( M R , N S I = T H A L O C ( M R , N S l / T H R M X 1 3 8 F E A L O C ! M R , N S ) = F E A L O C t M R , N S ) / F E O M R ( M R ) 1 3 9 2 0 G N M C N S = G N M C N S * G N M C A V ( M R ) 1 4 0 G N M C N S = G N M C N S / F L 0 4 T ( N P 0 S t 1 4 1 T 0 T A L = T O T A L + G E N S L H 1 4 2 1 6 W R I T E ! 6 , 6 1 1 I ( G N M C A V ( M R » , M R = 1 , N P D S ( . G N M C N S , G E N S U M , I F E A L O C ( M R . N S I , 1 4 3 * M R = 1 , N P D S I , { T H A L O C ( M R , N S I , M R = 1 , N P D S J 1 4 4 6 1 1 F O R M A T ! * O M E A N ' / ' G E N M C = ' , 1 2 F 8 . 1 , ' M I L L S / K d H ' / ' O M E A N M A R G I N A L C D 1 4 5 * S . T O F S U 3 S Y S T E M G E N E R A T I O N = ' , F 5 . 1 , ' M I L L S / K W H ' / • M E A N A N N U A L S U B S 1 4 6 * Y S T E M G E N E R A T I O N = ' , F 7 . 0 , ' G W H • / ' A L L O C A T I O N S : ' / ' F I R M E = ' , 1 2 F 3 . 3 1 4 7 * / ' T H E R M L = « , 1 2 F 8 . 3 / ) 1 4 8 E X P A N R = P V S Y S T * 1 0 0 0 . * R / ( l . + R ) 1 4 9 W R I T E ( 6 , 6 1 2 l T O T A L . P V S Y S T , E X P A N R 1 5 0 6 1 2 F O R M A T ( • - M E A N A N N U A L S Y S T E M G E N E R A T I O N = « , F 3 . 0 . ' G W H ' / ' P R E S E N T V A 1 5 1 * L U E O F E X P E C T E D S Y S T E M R E T U R N = ' , F 7 . 3 , « B I L L I O N D O L L A R S ' / ' V A L U E 0 1 5 2 * F E X P E C T E D A N N U A L S Y S T E M R E T U R N = ' , F 7 . 1 , ' M I L L I O N D O L L A R S ' / ' 1 ' ) 1 5 3 S T O P 1 5 4 E N D E N D OF F I L E 75. 1 5 5 1 5 6 C 1 5 7 C H O W A R D ' S METHOD FOR R E S E R V O I R S IN S E R I E S ON THE SAME R I V E R 1 5 8 C 1 5 9 S U B R O U T I N E D E P S U B I / D E L T A / , N I N S , S I N S , / G E N M A X / , H H , / C 2 / , P O R T N , A N N V O L , 1 6 0 * G E N N S , Q N S , G N M C N S , F E A L O C . T H A L O C , F E D M R , P S G N , E S T V , E S T G , P V S Y S T ) 1 6 1 COMMON N S , B E T A , N P D S , N S U B S , N V . P T H , M H L M T , P V 0 L 1 5 ) . P R B I ( 1 2 . 5 , 3 I 1 6 2 COMMON / D E P / D V D S ( 6 0 ) , N D S S . N U S S . K C H K O , D A M P 1 6 3 R E A L MR V O L , F E D M R ( N P D S t , F E A L O C t N P D S , N S U B S t . T H A L O C ( N P D S , N S U B S I , 1 6 4 * S I N S ( N S U B S ) , A N N V O L ( N V , N S U B S ) , P Q R T N I N P D S , N S U B S I , H H < 2 6 , N S U 3 S I , H ( 2 7 » , I 6 5 * U L T J < 6 1 , 2 7 ) , G E N l < 6 1 , 2 6 ) , G E N M C 1 1 6 1 , 2 6 ) , O i l 6 1 , 2 6 ) , EL R I 2 6 ) . EL 0 ^ 1 2 6 , 1 6 6 * 2 ) , V J ( 2 7 , 2 I . D V J D O I 2 7 ) , V ( 2 7 I , P I ( 2 7 ) , E R ( 2 7 ) , P ( 2 7 , 2 7 I , A ( 2 7 , 2 7 ) , 1 6 7 * E S T V ( 2 7 , N S U B S ) , E S T Q ( 6 0 , N S U 8 S I , V D S ( 2 7 , 6 0 ) , P S G N < N P D S > , 1 6 8 * G N M C N S ( N P D S , N V . N S U B S ) . G E N N S I N P D S , N V , N S U B S ) , O N S ( N P O S , N V , N S U B S ) 1 6 9 I N T E G E R N I N S ( N S U B S I , I P E R M ( 5 4 ) , I A / I / , I B I I I 1 7 0 N I = N I N S ( N S ) 1 7 1 S I = S I N S ( N S ) 1 7 2 I F ( N S . E O . N D S S ) G O TO 1 0 1 1 7 3 S I P R 0 P = S I / S I N S ( N S - 1 > 1 7 4 1 0 1 R N I = F L O A T ( N I 1 - 1 . E - 6 1 7 5 C 1 = S I * C 2 1 7 6 N I X = N I + 1 1 7 7 DO 1 0 2 1 = 1 , N T 1 7 8 EL R ( I ) = 0 . 1 7 9 1 0 2 HI I ) = H H ( I , N S ) 1 8 0 H ( N I X ) = H ( N I ) 1 3 1 C O M I T QMN C O N S T R A I N T D U R I N G I N I T I A L MODEL D E V E L O P M E N T 1 8 2 0 " N = 0 . 0 1 8 3 C C A L C S T A T E V A L U E E S T I M A T E S 1 8 4 DO 1 0 3 1 = 1 , N I X 1 8 5 1 0 3 V I I ) = E S T V ( I . N S ) 1 8 6 D S C M R - D . 1 8 7 K V = 1 1 8 8 1 C 0 U N T = 0 1 8 9 2 1 2 I C 0 U N T = I C 0 U N T - 1 1 9 0 DO 1 0 4 1 = 1 , N I X 1 9 1 E P ( I ) = 0 . 1 9 2 0 0 1 0 5 J = 1 , N I X 1 9 3 1 0 5 P< I , J ) = 0 . 1 9 4 1 0 4 C O N T I N U E 1 9 5 C 1 , N V P R O B A B L E A N N U A L I N F L O W S 1 9 6 DO 2 0 1 N 0 = 1 , N V 1 9 7 C E N D - O F - Y E A R ST AT F V A L U E S 1 9 8 DO 1 0 6 1 = 1 , N I X 1 9 V 1 0 6 V J ( I , I A ) = VI I ) * 3 E T A 2 0 0 I F ( ( K V . E O . I ) . O R . I K C H K Q . E O . l U G O TO 1 0 7 2 0 1 DO 1 0 8 1 = 1 , N I 2 0 2 1 0 3 EL O C R ( I , I A ) = 0 . 2 0 3 C l . N P D S P E R I O D S 2 0 4 1 0 7 DO 2 0 2 M R = 1 , N P D S 2 0 5 P S G = P S G N ( M P ) 2 0 6 M P N O = M R + ( N O - 1 ) * N P D S 2 0 7 DO 1 0 9 K = 2 , N I 2 0 8 1 0 9 D V J D O < M = V J I K - 1 , 1 A I - V J I K , IA ) 2 0 9 D V J O O ! 1 1 = D V J D 0 ( 2 I 2 1 0 D V J D O I N l X ) = D V J D 0 1 N U 2 1 1 F E M X = F E A L O C ( M R , N S ) 2 1 2 F E M N = F E M X - T H A L O C ( M R , N S ) 2 1 3 I F ( F E M N . G T . G E N M A X ) F E M N = G E NMAX 2 1 4 C I N F L O W THAT W I L L OCCUR I N P E R I O D MR G I V E N A N N V O L ( N O ) 76. 2 15 MR VOL = ANN V O L ( N O . N S I * P O R T N ( M R , N S 1 2 1 6 C C A L C . T O T A L E X P E C T E D INFLOW F R O M U / S R E L E A S E AND 2 1 7 C I N T E P . R E S E R V O I R R U N O F F 2 1 8 1 F ( N S . N E . N U S S I MRVOL = M R V O L + E S T Q ( M R N O , N S + 1) 2 1 9 C C A L C . M A R G I N A L R E T U R N F R O * WATER R E L E A S E D TO D / S S U B S Y S T E M 2 2 0 I F ( N S . G T . N D S S >DSOMR = D V D S < M R N O ) * S I 2 2 1 S T O R E J = R N I 2 2 2 O P T J = R N I 2 2 3 C I , N I X R E S E R V O I R S T A T E S 2 2 4 0 0 2 0 3 I R = 1 , N I X 2 2 5 1= N I X - I R + 1 2 2 6 H I = H ( I I 2 2 7 R J M X = F L O A T I I l + M R V O L 2 2 8 R J Q F = R J M X - C M N 2 2 9 I F ( ( OPT J . E C . I . I . O R . ( R J O F . L T . l . 0 M G O TO 1 1 0 2 3 0 K T 0 P = N I X - I F I X ( 0 P T J + . 5 ) 2 3 1 DO 2 0 4 K R = K T O P , N I 2 3 2 K 1 = N I X-KR+ 1 2 3 3 O P T J = F L O A T ( K l ) - . 5 2 3 4 R L S = R J M X - O P T J 2 3 5 H J = ( H ( K l - l ) * H ( K l ) » / 2 . 2 3 6 D G E N D Q = C l * ( H I * H J t 2 3 7 G E N = R L S * D G E N D O 2 3 8 D V D C = O V J D C ( K l I + GEN MR I P T H , G E N » FEMX » P S G > G E N M A X , F E M N , D G E N D O l 2 3 9 I F ( N S . E O . N D S S » G 0 TO 1 1 1 2 4 0 D V D 0 = D V D 0 + D S 0 M R * ( l . - f D A M P * ( E S T 0 ( M R N 0 , N S I / S I P R D P - R L S ) * * 3 ) 2 4 1 1 1 1 I F ( D V D O . L E . O . » G 0 TO 1 1 2 2 4 2 2 0 4 C O N T I N U E 2 4 3 GO TO 1 1 3 2 4 4 1 1 0 0 P T J = 1 . 2 4 5 GO TO H 4 2 4 6 1 1 2 I F ( K R . E O . K T O P ) G O TO 1 1 3 2 4 7 R J 1 = 0 P T J 2 4 8 R J 2 = R J 1 » 1 . 2 4 9 R K 1 = R J 1 2 5 0 K 2 = K 1 - 1 2 5 1 1 1 6 O P T J = < R J 1 + R J 2 1 / 2 . 2 5 2 I F ( R J 2 - R J 1 . L T . 0 . 1 0 ) G O TO 1 1 3 2 5 3 J 1 = I F I X I 0 P T J ) 2 5 4 J 2 = J 1 + 1 2 5 5 O J = O P T J - F L C A T ( J l l 2 5 6 H J = H< J l l + ( H ( J 2 I - H U 1 ) l * D J 2 5 7 R L S = R J M X - O P T J 2 5 8 D G E N D Q = C 1 * ( H I + H J ) 2 5 9 G E N = R L S * D G E N D O 2 6 0 D V J D O J = D V J D O I K l » - ( D V J D O I K 2 I - D V J D O ( K I ) ) » ( O P T J - R K 1 ) 2 6 1 D V D O = D V J D O J * G E N M R ( P T H , G E N , F E M X , P S G , G E N M A X , F E M N , D G E N D O I 2 6 2 I F i N S . E O . N D S S I G O TO 1 1 5 2 6 3 D V D O = D V D O + D S O M R * ( l . + D A M P * ! E S T O I M R N O , N S ) / S I P R O P - R L S ) * * 3 ) 2 6 4 1 1 5 1 F < D V D 0 . L E . 0 . ) R J 1 = 0 P T J 2 6 5 I F ( D V D O . L E . O . > G 0 TO 1 1 6 2 6 6 R J 2 = Q P T J 2 6 7 GO TO 1 1 6 2 6 8 1 1 3 I F I O P T J . L T . l . i O P T J = l . 2 6 9 I F ( O P T J . G T . R J O F * 0 P T J = R J 0 F 2 7 0 I F ( O P T J . G T . S T O R E J > O P T J = S T O R E J 2 7 1 1 1 4 J l = I F I X ( 0 P T J 1 2 7 2 J 2 = J l * l 2 7 3 RL S = R J M X - O P T J 2 7 4 D J = OP T J - F L C A T ( J l l 77. 2 7 5 H J = H ( J l • • ( H ( J 2 ) - H [ J l ) » * D J 2 7 6 D G E N D Q = C 1 * ( H I - H J | 2 7 7 G E N = R L S * D G E N D O 2 7 8 I F ( G E N . G T . G E N M A X t G E N = 3 E N M A X 2 7 9 V J END = V J ( J l , ] A I + ( V J ( J 2 , I A I - V J ( J l , ! A ) ) * D J 2 8 0 I F I G E N . G T . F E M X I G E N B E N = F E M X * P T H - < G E N - F E * X I * P S G 2 8 1 I F I G E N . L E . F E M X I G E N B E N = G E N * P T H 2 8 2 C H I G H P R I C E TO BE P A I D I F E N E R G Y TO MEET F E M N MUST BE 2 8 3 C O B T A I N E D O U T S I D E S Y S T E M OR NOT AT A L L 2 84 I F I G E N . L T . F E M N I G E N B F N = G E N 6 E N - ( F E M N - G E N 1 * 2 . * PT H 2 8 5 V J I I , I B l = G E N B E N * V J E N D 2 8 6 I F ( N S . E Q . N D S S ) G 0 TO 1 1 7 2 8 7 V J ( I , I B ) = V J ( I , I B ) * D S O M R * R L S 2 8 9 1 1 7 IF I I . E Q . N I X ) G O TO 2 0 3 2 8 9 G E N I I M R N Q . I l = G E N 2 9 3 01 I M R M Q , I | = R L S 2 9 1 K 1 = I F I X ( 0 P T J - . 5 I 2 9 2 D V J D Q J = D V J D Q < K l ) + ( D V J D Q I K 1 + 1 l - D V J D Q I K 1 I ) * ( OPT J+ . 5 - F L O A T I K 1 I ) 2 93 GE KMC 1 1 M R N Q , I ) = - I D V J D O J * D S QM R ) / D G E N DO 2 9 4 I F ( ( K V . E O . 1 ) . O R . I K C H K Q . E O . I I I G O TO 2 0 3 2 9 5 C C A L C . E X P E C T E D L O C A L R E T U R N W I T H O U T D/S V A L U E OF rfATER 2 9 6 E L R E N D = E L O C R ( J l , I A ) • I z L 0 C R ( J 2 , I A I - E L O C R ( J l , I A) > * D J 2 9 7 E L O C R I I , I B ) = G E N B E N + E L R E N D 2 9 8 2 0 3 U L T J ( M R N O . I I = O P T J 2 9 9 S T O R E J = O P T J 3 0 0 I F | K V . E O . I » G 0 TO 1 1 8 3 0 1 C E S T A B L I S H S E T OF V D S TO USE I N O P T I M I Z A T I O N OF N E X T U / S R E S E R V O I R 3 0 2 DO 1 1 9 1 = 1 , N I X 3 0 3 1 1 9 V D S ( I , M R N U I = V J ( I , I B I 3 0 4 C R E V E R S E V A L U E S OF F I R S T S U B S C R I P T S OF V J S E T S 3 0 5 1 13 IC =I A 3 0 6 I A = I B 3 0 7 2 3 2 I B = I C 3 0 3 C E X P E C T E D A N N U A L R E T U R N A NO S T A T E T R A N S I T I O N P R O B A B I L I T I E S FROM E A C H 3 C ? C S T A T E I A F T E R ONE P R O B A B L E A N N V O L 3 1 0 N O N P = N O * N P D S 3 1 1 DO 1 2 0 1=1 . N I X 3 1 2 E R ( I I = E R ( I I - V J ( I , I A » * P V O H N Q l 3 1 3 J 1 = I 3 1 4 D J = 0 . 3 1 5 J 2 = J 1 3 1 6 DO 1 2 1 M = l , N P D S 3 1 7 M R N O = N O N P * 1 - M 3 1 8 E N D J = U L T J ( M R N O , J l I + D J * ( U L T J < M R N Q . J 2 ) - U L T J ( M R ^ Q , J I ) ) 3 1 9 J 1 = I F I X l E N D J t 3 2 0 D J = E N D J - F L G A T I J 1 ) 3 2 1 1 2 1 J 2 = J 1 - 1 3 2 2 P J 2 = P V 0 L < N 0 » * D J 3 2 3 P ( I , J 2 I = P ( I , J 2 > * P J 2 3 2 4 1 2 0 P ( I , J 1 ) = P ( I , J U + P V D L ( N 0 ) - P J 2 3 2 5 I F ( ( K V . E O . l » . O R . ( K C H K Q . E Q . 1 I » G 0 TO 2 0 1 3 2 6 C C A L C . E X P E C T E O L O C A L R E T U R N W I T H O U T D / S V A L U E OF W A T E R 3 2 7 DO 1 2 2 1 = 1 , N I 3 2 8 1 2 2 E L R ( l » = E L R 1 1 » * E L O C R ( I , I A » * P V O L ( N O » 3 2 9 2 0 1 C O N T I N U E 3 3 0 C NEW V E S T I M A T E S 3 3 1 DO 1 2 3 1 = 1 , N I X 3 3 2 E R ( I l = E R ( I > / B E T A 3 3 3 DO 1 2 4 J = 1 , N I X 3 3 4 P J = P < I , J ) 78. 3 3 5 E R I I I = E R ! I l - P J * V I J l 3 3 6 1 2 4 A ! I , J I = - P J 3 3 7 1 2 3 A ( I , 1 ( = A 1 I , I 1+1 . / B E T A 3 3 8 C S O L V E S Y S T E M O F L I N E A R E Q U A T I O N S 3 3 9 C A L L F S L E ( N I X , 2 7 , A , I , 2 7 , E R . P I , I P E R M . 2 7 , A , D E T , J E X P I 3 4 0 I F ( K V . L T . l l G O T O 2 1 1 3 4 1 K V = 0 3 4 2 D O 1 2 5 1 = 1 , N I X 3 4 3 C C H E C K V A L U E I M P R O V E M E N T 3 4 4 I F ( A B S ( P I ( I l - V ( I ) ) . G T . D E L T A * P I ( I ) 1 K V = 1 3 4 5 1 2 5 V ( I ) = P I ( I ) 3 4 6 I F ( I C O U N T . E O . M H L M T » K V = - I 3 4 7 G O T O 2 1 2 3 4 8 2 1 1 0 0 1 2 6 1 = 1 , N I 3 4 9 D O 1 2 7 J = l , N I 3 5 0 1 2 7 A l I , J ) = P ( J , I ) 3 5 1 A ( I , I ) = A ( I , 1 ) - l . 3 5 2 A t 1 , I l = A l 1 , 1 I - l . Q 3 5 3 E S T V ( I , N S ) = P I < I » 3 5 4 1 2 6 P I ( I t = 0 . 3 5 5 E S T V ( N I X , N S ) = P I ( N I X J 3 5 6 P I ( 1 ) = 1 . 3 5 7 C A L L F S L E I N I , 2 7 , A , 1 , 2 7 , P I , P I , 1 P E R M , 2 7 , A , D E T , J E X P | 3 5 8 I F ( K C H K O . E O . l J G O T O 2 1 3 3 5 9 W R I T E ( 6 , 6 3 2 » N S . I C O U N T 3 6 0 6 3 2 F O R M A T ( • - S U B S Y S T E M 1 2 , • V A L U E D E T E R M I N A T I O N : ' / ' N O . OF C Y C L E S =• , 3 6 1 * I 2 / ' S T A T E ( I ) P I I I I E R I S M I L L / Y R ) V A L U E I $ M I L L t ' » 3 6 2 DO 1 2 8 1 = 1 , N I 3 6 3 E L R ( I ( = £ L R ( I l / B E T A 3 6 4 0 0 1 2 9 J = 1 , N I 3 6 5 1 2 9 4 ( I , J ) = - P ( I , J I 3 6 6 1 2 8 A I I , I t = A I I . l l - l . / B E T A 3 6 7 C A L L F S L E ( N I , 2 7 , A , 1 , 2 6 , E L R , V , I P E R M , 2 7 , A . D E T , J E X P I 3 6 8 P W E R N S = 0 . 3 6 9 E X P A N R = 0 . 3 7 0 DO 1 3 0 1 = 1 , N I 3 7 1 I K = N I + 1 - I 3 7 2 P I ( I R l = A B S I P I ( I R » I 3 7 3 E L R ( I R t = E L R ( I R t * B E T A 3 7 4 P W E R N S = P W E R N S * V ( I R ) * P I ( I R J 3 7 5 E X P A N R = E X P A N R - E L R ( I R l * P I ( I R » 3 7 6 1 3 J W R I T E ( 6 , 6 3 3 ) I R , P I ( I R » , E L R ( I R ) . V ( I R ) 3 7 7 6 3 3 F O R M A T ! I 5 , 2 F 1 1 . 3 , F 1 4 . l l 3 7 8 P W E R N S = P W E R N S / 1 0 0 0 . 3 7 9 W R I T E I 6 . 6 3 4 ) P r f E R N S , E X P A N R 3 8 0 6 3 4 F O R M A T ! • P R E S E N T V A L U E OF E X P E C T E D S U B S Y S T E M R E T U R N = ' , F 7 . 3 , ' B I L L 3 8 1 * I C N D O L L A R S ' / ' V A L U E O F E X P E C T E D A N N U A L S U B S Y S T E M R E T U R N = ' , F 7 . 1 , 3 8 2 * ' M I L L I O N D O L L A R S ' ) 3 8 3 P V S Y S T = P V S Y S T + P H E R N S 3 8 4 C D E T E R M I N E N U M B E R O F T R A J E C T O R I E S A N D S T O R E V A L U E S 0 / P 3 8 5 C F R O M E A C H T R A J E C T O R Y I N F I R S T P E R I O D O F Y E A R 3 8 6 2 1 3 D O 1 3 1 N 0 = 1 , N V 3 8 7 DO 1 3 1 M R = 1 , N P D S 3 8 8 M R N Q = M R * ( N O - 1 I * N P D S 3 8 9 D V D S ( M R N Q ) = 0 . 3 9 0 G E N N S ! M R , N O , N S I = 0 . 3 9 1 G N M C N S 1 M R , N O , N S I = 0 . 3 9 2 Q N S ( M R , N Q , N S I = 0 . 3 9 3 1 3 1 P R B I ( M R , N O , N S » = 0 . 3 9 4 N T R J = 0 79. 3 9 5 DO 1 3 2 1 = 1 , N I 3 9 6 I F ( P K I ) . E O . O . ) G O T O 1 3 2 3 5 7 N T R J = . N T R J + 1 3 9 3 P I ( N T R J l = P I( I ) 3 9 9 DO 1 3 3 N O = l , N V 4 0 0 N Q P = N P D S * N C 4 0 1 P R B I ( N P D S . N O . N S » = P R B I ( N P D S , N O , N S I + P I ( I » * F L O A T < I » 4 0 2 G E N N S t N P D S , N O , N S l = G E N N S < N P D S , N O , N S » - P I U I * G E N I I N OP , I ) 4 0 3 G N M C N S ( N P D S , N O , N S ) = G N M ; N S ( N P D S , N O , N S I • P I ( I I * 3 E N M C I ( N Q P . I I 4 0 4 O N S I N P D S . N Q . N S I = O N S ( N ° D S . N G » N S » * P I t It * 3 1 ( N O P , I » 4 0 5 I F < I . E O . l ) G 0 T O 1 4 1 4 0 6 C N O D I S C O N T I N U I T Y I N D V D S F O R S T A T E S N E A R F U L L P O O L . 4 0 7 D V D S I N O P » = D V D SI N O P > «-P I I NT R J ) * ( V D S 1 1 + 1 , N O P ) — V D S ( I - 1 , N Q P ) ) / ( 2 . * S I ) 4 0 3 GO T O 1 3 3 4 C 9 1 4 1 0 V D S ( N Q P l = D V D S l N O P I - P I I N T R J t * ( V D S ( 2 . N O P J - V D S t 1 , N O P M / S I 4 1 0 1 3 3 U L T J < N O P , N T R J > = U L T J I N O P , I ) 4 1 1 1 3 2 C O N T I N U E 4 1 2 C I N T E R P O L A T E A N D M E A N V A L U E S 0 / P F R O M E A C H 4 1 3 C P E R I O D D U R I N G Y E A R F O R E A C H T R A J E C T O R Y 4 1 4 DO 1 3 4 N 0 = 1 , N V 4 1 5 N Q P = N P D S * N O 4 1 6 D O 1 3 4 M = 2 , N P D S 4 1 7 M R N Q = N Q P - M + 1 4 1 8 M P = N P 0 S - M + 1 4 1 9 DO 1 3 4 1 = 1 , N T R J 4 2 0 E N D J = U L T J ( N O P , I ) 4 2 1 J 1 = I F I X < E N O J ) 4 2 2 D J = E N D J - F L O A T ( J i ) 4 2 3 J 2 = J 1 - 1 4 2 4 G E N N S ( M R , N Q , N S ) = G E N N S ( M R , N O , N S M - P I I I I * ( G E N 1 ( M R N Q , J I ) + D J * ( G E N 1 ( M R N Q 4 2 5 * , J 2 l - G E N l ( M R N O , J l I ) ) 4 2 6 G N M C N S ( M R , N O , N S ) = 3 N M C N S ( M R , N O , N S ) + P I ( I ) * ( G E N M C 1 ( M R N O , J 1 ) • O J * ( G E N M C 4 2 7 * 1 ( M R N S , J 2 ) - G E N M C 1 ( M R N O , J l ) ) I 4 2 8 O N S I M R . N O . N S t = O N S ( M R , N O , N S ) * P I ( I I * ( 0 1 ( M R N O , J l ) + D J * ( G l ( M R N O . J ? ) - 0 1 ( 4 2 9 * M R N C , J 1 ) I ) 4 3 0 P R B I I M R , N O , N S l = P R B I I M R , N O , N S ) - P I ( I ) * E N D J 4 3 1 I F ( E N O J . E O . R N I ) G 0 T O 1 5 1 4 3 2 VD S I N I X , M R N O I = 2 . * V D S [ N I , M R N O ) - V D S I N I — I , M R N O ) 4 3 3 D S 0 M R 2 = ( V D S I J 2 + 1 , M R N O ) - V D S ( J 1 , M R N O ) ) / 2 . 4 3 4 I F ( J l . E Q . 1 ) G 0 T O 1 5 2 4 3 5 D S Q M R 1 = ( V D S ( J 2 • M R N O I - V D S ( J 1 - 1 , M RN Q) » I Z . 4 3 6 G O T O 1 5 3 4 3 7 1 5 2 O S O M R l = V D S ( 2 , M R N 0 l - V D S ( 1 . M R N Q ) 4 3 8 1 5 3 D V D S ! M R N O ) = D V D S ( M S N Q M - P I I I ) * ( D S Q M R 1 *0 J * ( D S 3 M * 2 - D S Q M R 1 1 ) / S I 4 3 9 G O T O 1 3 4 4 4 0 1 5 1 D V D S t M R N O l = D V D S ( M R N O ) + P I ( I ) * ( V D S ( N I X , M R N 0 I - V 3 S ( N I , M R N O ) I / S I 4 4 1 1 3 4 U L T J I N O P . I ) = U L T J ( M R N C . J l I - D J * I U L T J I M R N Q , J 2 l - J L T J ( M R N O . J 1 t I 4 4 2 R E T U R N 4 4 3 E N D E N D O F F I L E 80. 444 4 4 5 C 4 4 6 C H O W A R D S ME TH 0 0 F O R R E S E R V O I R S NOT IN A S E R I E S W I T H ANY O T H E R S 4 4 7 C 4 4 8 S U B R O U T I N E I N DSUB I / O E L T A/ , / NI / , / G E N MAX / , / DV DI / , H H , / C 1 / , P 3 R TN , 4 4 9 * A N N V 0 L , G E N N S , Q N S , G N M C N S , F E A L 0 C , T H A L 0 C , F E D M R f P S 3 N , / P W E R N S / l 4 5 0 COMMON NS , B E T A , N P D S . N S U B S . NV , PT H , MHL MT , P V O L ( 5 ) , O R B I t 1 2 , 5 , 3) 4 5 1 R E A L F E D M R ( N P D S ) . F E A L O C ( N P D S , N S U B S I . T H A L O C l N P D S , N S U B S ) , 4 52 * A N N V O L ( N V . N S U B S ) , P O R T N ( N P D S , N S U BS ) , HH ( 2 6 , N SU3 S ) , V J ( 27 , 2 ) . D V J 3 3 ( 2 7 ) 4 5 3 * « U L T J ( 6 1 , 2 7 » , G E N 1 ( 6 1 , 2 6 ) , 0 1 ( 6 1 , 2 6 1 , G E N M C 1 ( 6 1 , 2 6 ) , H ( 2 7 ) , M R V O L . 4 54 * V ( 2 7 ) , P I ( 2 7 ) , E R ( 2 7 ) , P ( 2 7 , 2 7 ) , A ( 2 7 , 2 7 ) , P S G N ( N 3 D S ) , 4 5 5 * G N M C N S ( N P D S . N V , N S U B S ) , G E N N S ( N P D S , N V , N S U B S ) , O N S < N P D S , N V , N S J 3 S ) 4 5 6 I N T E G E R I P E R M ( 5 4 ) , I A / 1 / , I B / 2 / 4 5 7 R N I = F L O A T ( M t - l . E - 6 4 5 8 N I X = N I + 1 4 5 9 DO 1 0 2 1 = 1 , N I 4 6 0 1 0 2 H ( I ) = H H ( I , N S ) 4 6 1 H ( N I X ) = H ( N I ) 4 6 2 C O M I T OMN C O N S T R A I N T D U R I N G I N I T I A L MODEL D E V E L O P M E N T 4 6 3 Q M N = 0 . 0 4 6 4 C C A L C E N D - O P - Y E A R S T A T E V A L U E E S T I M A T E S 4 6 5 DO 1 0 3 1 = 1 , N I X 4 6 6 1 0 3 V( I ) = F L O A T 1 I ) * D V D I 4 6 7 I C O U N T = 0 4 6 8 2 1 2 I C C U N T = I C 0 U N T + 1 4 6 9 DO 1 0 4 1 = 1 , N I X 4 7 C E R I I ) = 0 . 4 7 1 DO 1 0 5 J = l , N I X 4 7 2 1 0 5 P ( I , J 1 = 0 . 4 7 3 1 0 4 C O N T I N U E 4 7 4 C l . N V P R O B A B L E A N N U A L I N F L O W S 4 7 5 DO 2 0 1 N 0 = 1 , N V 4 7 6 C E N D - O F - Y E A S S T A T E V A L U E S 4 7 7 DO 1 0 6 1 = 1 , N I X 4 7 3 1 0 6 VJ( I , I A ) = V ( I J * B E T A 4 7 9 C l . N P D S P E R I O D S 4 8 0 DO 2 0 2 M R = 1 , N P D S 4 8 1 P S G = P S G N ( M R ) 4 8 2 M R N O = M R - ( N O - I ) * N P D S 4 8 3 0 0 1 0 9 K = 2 . N I 4 8 4 1 0 9 0 V J D Q ( K I = V J l K - l , 1 A ( - V J ( K , I A l 4 8 5 D V J D O U l = D V J D 0 ( 2 ) 4 8 6 D V J D O l N I X ) = D V J D Q ( N I » 4 8 7 F E M X = F E A L O C ( M R , N S ) 4 3 8 F E M N = F h M X - T H A L O C ( M R , N S ) 4 8 9 I F { F E M N . G T . G E N M A X ) F E M N = G E N M A X 4 9 0 C I N F L O W T H A T W I L L OCCUR I N P E R I O D MR G I V E N A N N V O L ( N O ) 4 9 1 MR V O L = A N N V O L 1 N C , N S ) * P O R T N I M R . N S ) 4 9 2 S T O R E J = R N I 4 9 3 O P T J = R N I 494 C I , N I X R E S E R V O I R S T A T E S 4 9 5 DO 2 0 3 I R = 1 , N I X 4 9 6 I = N I X - I R + 1 4 9 7 H I = h ( I » 4 9 8 R J M X = F L O A T ( I l - M R V O L 4 9 9 R J O F = R J M X - C M N 5 0 0 I F C C O P T J . E Q . l . l . O R . C R J O F . L T . l . O U G O TO 1 1 0 5 0 1 K T 0 P = N I X - I F I X ( 0 P T J * . 5 ) 5 0 2 DO 2 0 4 K R = K T O P , N I 5 0 3 K 1 = N I X - K R - 1 81. 5 0 4 O P T J = F L O A T ( K l J - . 5 5 0 5 R L S = R J M X - 0 P T J 5 0 6 H J = ( H ( K l - 1 ) * H ( K I ) ) / 2 . 5 0 7 D G E N D Q = C 1 » ( H I * H J > 5 0 8 G E N = R L S * D G F N D 0 5 0 9 D V D 0 = 0 V J D 0 ( K 1 J + G E N M R ( P T H , G E N , F E M X , P S G , G E N M A X , F E M N , D G E S D Q ) 5 1 0 I F I O V O O . L E . O . ) G O TO 1 1 2 5 1 1 2 0 4 C O N T I N U E 5 1 2 GO TO 1 1 3 5 1 3 1 1 0 0 P T J = 1 . 5 1 4 GO TO 1 1 4 5 1 5 1 1 2 I F ( K R . E Q . K T O P ) G 0 TO 1 1 3 5 1 6 R J 1 = 0 P T J 5 1 7 R J 2 = R J 1 + 1 . 5 1 8 R K 1 = R J 1 5 1 9 K 2 = K 1 - 1 5 2 0 1 1 6 O P T J = ( R J 1 + R J 2 1 / 2 . 5 2 1 I F (R J 2 - R J l . L T . O . 10 I G O TO H i 5 2 2 J l = I F I X ( O P T J ) 5 2 3 J 2 = J l + l 5 2 4 O J = O P T J - F L O A T ( J l l 5 2 5 H J = H ( J l l + ( H ( J 2 J - H ( J l ) l * D J 5 2 6 R L S = R J M X - C P T J 5 2 7 D G E N D 0 = C 1 * ( H I + H J ) 5 2 3 G E N = R L S * O G E N D C 5 2 9 D V J D Q J = D V J D Q ( K l > + ( D V J D O ( K 2 I - D V J D Q I K 1 ) I * ( O P T J - R K l I 5 3 0 D V D Q = D V J D O J + G E N M R ( P T H , G E N , F E M X , P S G , G E N M A X , F E M N , O G E N D O ) 5 3 1 I F I O V O O . L E . O . I R J 1 = 0 P T J 5 3 2 I F ( O V D O . L E . 0 . J G O T O 1 1 6 5 3 3 R J 2 = 0 P T J 5 3 4 GO TO 1 1 6 5 3 5 1 1 3 I F ( 0 P T J . L T . 1 . I 0 P T J = 1 . 5 3 6 I F ( O P T J . G T . R J O F ) O P T J = R J C F 5 3 7 I F ( O P T J . G T . S T O R E J ) O P T J = S T O R E J 5 3 3 1 1 4 J 1 = I F I X ( 0 P T J ) 5 3 9 J 2 = J 1 + 1 5 4 0 R L S = R J M X - O P T J 5 4 1 0 J = 0 P T J - F L Q A T ( J 1 I 5 4 2 H J = H ( J l ) + ( H l J 2 ) - H ( J l I I * D J 5 4 3 D G E N D 0 = C 1 * < H I + H J ) 5 4 4 G E N = R L S * D G E N D Q 5 4 5 I F ( G E N . G T . G E N M A X I G E N = G E N M A X 5 4 6 V J E N D = V J ( J l , I A ) + ( V J ( J 2 , I A l - V J ( J 1 , 1 A ) I * D J 5 4 7 I F ( G E N . G T . F E M X » G E N B E N = F E M X * P T H + 1 G E N - F E M X I*PSG 5 4 3 I F ( G E N . L E . F E M X ) G E N 6 E N = G E N * P T H 5 4 9 C H I G H P R I C E TO BE P A I D I F E N E R G Y TO M E E T F E M N MUST B E 5 5 0 C O B T A I N E D O U T S I D E S Y S T E M OR NOT AT A L L 5 5 1 I F I G E N . L T . F E M N ) G E N B E N = G E N B E N - ( F E M N - G E N J * 2 . * PT H 5 5 2 V J ( I , I B ) = G E N 8 E N * V J E N D 5 5 3 I F ( I . E O . N I X ) G O TO 2 0 3 5 5 4 G E N K M R N O , H = G E N 5 5 5 0 1 ( M R N Q , I I = R L S 5 5 6 K 1 = I F I X ( 0 P T J - . 5 I 5 5 7 D V J D O J = D V J D O ( K l )«•< D V J D O (K 1+ 1 ) - D V J D 0 ( K 11 ) * ( D P T J + . 5 - F L 0 A T( K l ) ) 5 5 8 C OMR S H O U L D B E V A L U E OF L A S T U N I T OF WATER R E L E A S E ( * M I L L / S I VOL I 5 5 9 G E N M C K M R N O , I ) = - D V J D O J / D G E N D O 5 6 0 2 0 3 U L T J ( M R N Q , I » = 0 P T J 5 6 1 S T O R E J = 0 P T J 5 6 2 C R E V E R S E V A L U E S OF F I R S T S U B S C R I P T S OF V J S E T S 5 6 3 I C = I A 82. 5 6 4 I A = I B 5 6 5 2 0 2 I B = I C 5 6 6 C E X P E C T E D A N N U A L R E T U R N A N D S T A T E T R A N S I T I O N P R O B A B I L I T I E S F R O M E A C H 5 6 7 C S T A T E I A F T E R O N E P R O B A B L E A N N V O L 5 6 8 N O N P = N O * N P D S 5 6 9 0 0 2 0 1 1 = 1 , N I X 5 7 0 E R < I > = E R ( I l * V J ! I , 1 A ) * P V O L ! N C ) 5 7 1 J l = I 5 7 2 D J = 0 . 5 7 3 J 2 = J 1 5 7 4 D O 1 2 1 M = l , N P D S 5 7 5 M R N O = N O N P + 1 - M 5 7 6 E N C J = U L T J ( M R N Q , J l > + D J * ( U L T J ( M R N O , J 2 I - U L T J { M R M 0 , J l l I 5 7 7 J l = 1 F I X ( E N D J ) 5 7 8 D J = E N D J - F L O A T ( J l ( 5 7 9 1 2 1 J ? = J l + 1 5 8 0 P J 2 = P V 0 L ( N 0 ) * D J 5 8 1 P U , J 2 ) = P ( I . J 2 I + P J 2 5 8 2 2 0 1 P I I , J 1 ) = P ( I , J 1 I - P V 0 L ( N 0 I - P J 2 5 8 3 C N E W V E S T I M A T E S 5 8 4 D O 1 2 3 1 = 1 , N I X 5 8 5 ER I I l = E R t I J / B E T A 5 8 6 D O 1 2 4 J = l , N I X 5 8 7 P J = P ( I , J ( 5 8 8 E R { I >=ER ! I l - P J * V ( J ) 5 8 9 1 2 4 A 1 I • J l = - P J 5 9 0 1 2 3 A ( I , I ) = A ( I , I 1 + 1 . / B E T A 5 9 1 C S O L V E S Y S T E M C p L I N E A R E Q U A T I O N S 5 9 2 C A L L F S L E I N I X , 2 7 , A , l , 2 7 , E R , P 3 , I P E R K , 2 7 , A , D E T , J E X P ) 5 9 3 K V = 0 5 9 4 DO 1 2 5 1 = 1 , N I X 5 9 5 E R ( I ) = E R ( I ) * B E T A 5 9 6 C C H E C K V A L U E I M P R O V E M E N T 5 9 7 I F ( A B S I P I ( I ) - V ( I I I . G T . D E L T A * ° I ( I ) ) K V = 1 5 9 3 1 2 5 V I I I = P I I I I 5 9 9 I F < I C O U N T . G T - M H L M T l G O T O 2 1 4 6 0 0 I F I K V . G T . O l G O T O 2 1 2 6 0 1 2 1 4 W R I T E 1 6 . 6 3 2 ) N S , I C O U N T 6 0 2 6 3 2 F O R M A T ! ' - S U B S Y S T E M ' , 1 2 , ' V A L U E D E T E R M I N A T I O N : • / • N O . O F C Y C L E S = * . 6 0 3 * I 2 / ' S T A T E t l ) P H I I E R ( t M I L L / Y R ) V A L U E ( 1 MI L L ) ' ) 6 0 4 D O 1 2 6 1 = 1 , N I 6 0 5 D O 1 2 7 J = 1 , N I 6 0 6 1 2 7 A ( I , J ) = P ( J , I I 6 0 7 A l I , 1 l = A ( I , 1 ( - 1 . 6 0 8 A t 1 , I ) = A ( 1 , I ) + 1 . 0 6 0 9 1 2 6 P I ( I I = 0 . 6 1 0 P I ( 1 1 = 1 . 6 1 1 C A L L F S L E C N I , 2 7 , A , 1 , 2 7 , P I , P I , I P E R M , 2 7 , A , D E T , J E X P I 6 1 2 P W E R N S = 0 . 6 1 3 E X P A N R = 0 . 6 1 4 D O 1 3 0 1 = 1 , N I 6 1 5 I R = N I * 1 - I 6 1 6 P I ( I R J = A B S < P I ( I R 1 1 6 1 7 P W E R N S = P W E R N S » V ( I R ) * P I I I R ) 6 1 8 E X P A N R = E X P A N « « - E R ( I R ) * P I ( I R » 6 1 9 1 3 0 W R I T E I 6 , 6 3 3 1 I R , P I ( I R I , E R I IR» ,V( I RI 6 2 3 6 3 3 F 0 R M A T ( I 5 , 2 F 1 1 . 3 . F 1 4 . 1 ) 6 2 1 P W E R N S = P W E R N S / 1 0 0 0 . 6 2 2 W R I T E ( 6 , 6 3 4 ) P W E R N S , E X P A N R 6 2 3 6 3 4 F O R M A T ! ' P R E S E N T V A L U E O F E X P E C T E D S U B S Y S T E M R E T U R N = ' , F 7 . 3 , ' B I L L 83. 6 2 4 * I O N D O L L A R S ' / ' V A L U E O F E X P E C T E D A N N U A L S U B S Y S T E M R E T U R N = ' , F 7 . l , 6 2 5 * ' M I L L I O N D O L L A R S ' ) 6 2 6 C D E T E R M I N E N U M B E R O F T R A J E C T O R I E S A N D S T O R E V A L U E S 0 / P 6 2 7 C F R O M E A C H T R A J E C T O R Y I N F I R S T P E R I O D O F Y E A R 6 2 8 D O 1 3 1 N 0 = 1 , N V 6 2 9 C O 1 3 1 M R = 1 , N P D S 6 3 0 M R N O = M R - ( N C - l l * N P D S 6 3 1 G E N N S ( M R , N O « N S ) = 0 . 6 3 2 G N * C N S ( M R , N O , N S ) = 0 . 6 3 3 O N S ( M R , N Q , N S ) ^ = 0 . 6 3 4 1 3 1 P R B I ( M R , N Q , N S ) = 0 . 6 3 5 N T R J = 0 6 3 6 D O 1 3 2 1 = 1 , N I 6 3 7 I F ( P I ( I I . E O . 0 . ) G O T O 1 3 2 6 3 8 NT R J = NT R J + 1 6 3 9 P I ( N T R J ) = P I ( I I 6 4 0 DO 1 3 3 N 0 = 1 , N V 6 4 1 N 0 P = N P D S * N C 6 4 2 P R B I ( N P D S , N O , N S » = P R B l ( N P D S . N Q , N S ) «• P I ( I ) * F L O A T ( I ) 6 4 3 GE N N S < N P D S , N Q , N S I = G E N N S ( N P D S , N Q . N S ) * P I ( I ) * G E N 1 ( N O P , I ) 6 4 4 G N M C N S I N P D S . N 0 , N S I = G N M C N S 1 N P D S , N O , N S ) * P I ( I I * G E N M C 1 ( N 0 P , I ) 6 4 5 Q N S t N P D S . N Q , N S ) = Q N S ( N ° D S , N O , N S ) * P I ( 1 1 * 3 11 N O P , I ) 6 4 6 1 3 3 U L T J l N Q P , N T R J ! = U L T J ( N Q P , I l 6 4 7 1 3 2 C O N T I N U E 6 4 8 C I N T E R P O L A T E A N D M E A N V A L U E S 3 / P F R O M E A C H 6 4 9 C P E R I O D D U R I N G Y E A R F O R E A C H T R A J E C T O R Y 6 5 0 D O 1 3 4 N C = l , N V 6 5 1 N Q P = N P D S * N Q 6 5 2 D O 1 3 4 M = 2 , N P D S 6 5 3 M R N O = N O P - M - l 6 5 4 M R = N P D S - M + 1 6 5 5 D O 1 3 4 1 = 1 , N T R J 6 5 6 E N D J = U L T J ( N 0 P , I ) 6 5 7 J 1 = I F I X ( E N C J ) 6 5 8 D J = E N D J - F L C A T ( J 1 1 6 5 9 J 2 = J 1 + 1 6 6 0 G E N N S ( M R . N O . N S I = G E N N S I M R . N O . N S ) * P I ( I ( * [ G E N 1 ( M R N O , J 1 > - D J * ( G E N I I M R N Q 6 6 1 * , J 2 ) - G E N 1 ( M R N 0 , J l ) ) ) 6 6 2 G N M C N S ( M R , N Q , N S ) = G N M C N S ( M R , N O , N S ) + P I ( I I * ( G E N M C I < M R N O , J 1 ) + D J * < G E N M C 6 6 3 * 1 ( M R N O . J 2 I - G E N M C 1 ( M R N O . J l ) ) ) 6 6 4 O N S ( M R . N O . N S ) = O N S I M R , N O . N S I * P I ( I I * ( 0 1 ( M R N O . J I I + D J * ( O K M R N 3 , J 2 I - 0 1 ( 6 6 5 * M R N 0 , J 1 I ( ) 6 6 6 P R B I ( M R . N O . N S l = P R B I ( M R , N O . N S I + P K I 1 * E N D J 6 6 7 1 3 4 U L T J ( N O P , I t = U L T J ( M R N C , J 1 ) - - D J * ( U L T J ( M R N O , J 2 ) - U L T J ( M R N O , J 1 I I 6 6 3 I F ( K C H K O . E O . l ) W R I T E ( 6 , 6 3 5 l ( ( P R B I ( M R , N O , N S I , M R = 1 , N P D S I , N 0 = 1 , N V ) 6 6 9 6 3 5 F O R M A T l 5 1 1 2 F 7 . 2 / I ) ' 6 7 0 R E T U R N 6 7 1 E N D E N D O F F I L E 84. 6 7 2 6 7 3 C 6 7 4 C SUBPROG TO C A L C U L A T E THE M A R G I N A L RETURN FROM THE LAST UN IT OF R E L E A S E 6 7 5 C 6 76 F U N C T I O N GENMR I / P T H / , / G E N / , / F E M X / , / P S G / , / G E N * A X / , / F E M N / , / O G E N 0 3 / I 6 7 7 GENMR=PTH*OGENOO 6 7 3 I F ( G E N . G T . F E M X ) G E N M R = P S G * D G E N D O 6 7 9 I F ( G E N . G T . G E N M A X I G E N M R = 0 . 6 8 0 C H I G H P R I C E TO BE P A I D I F ENERGY TO MEET FEMN MUST BE 6 8 1 C O B T A I N E D O U T S I D E S Y S T E M OR NOT AT ALL 6 82 I F ( G E N . L T . F E M N I G E N M R = 3 . * P T H * D G E N D 0 6 8 3 R E T U R N 6 8 4 END END OF F I L E APPENDIX B SAMPLE OUTPUT SHOWING THE OPTIMAL ALLOCATION OF FIRM ENERGY DEMAND This computer simulation used the data described in Section 4.1 for the system specified in Section 4.2. In the following output, Subsystem 1 refers to the Mica subsystem and Subsystem 2 refers to the Williston Lake. LEAR 86 OMITTED IN PAGE NUMBERING, 86a. SYSTEM DATA: R=0.010 NV= 5 NSUBS= 2 FED= 32000.GWH THRMX= 1000.MW PTH = 25 .0MILLS/KWH MHLMT= 6 ITLMT= 4 DAMP=0.20 SUBSYSTEM D A T A : DELTA DOLMT NI S I ( M K C F T I SUBSYSTEM I : 0 . 0 1 0 0 0 . 5 0 0 0 26 1 5 . 0 SUBSYSTEM 2: 0 . 0 1 0 0 0 . 3 0 0 0 26 4 0 . 0 CAPMAX(MW) NRJS 2 0 0 0 . 0 240 5. 0 SUBSYSTEM 1 VALUE DETERMINATION • • NO. OF CYCLES = 4 STAT E l I) P H ! ) ER($MILL/YR» VALUE( $MI LL) 26 0 . 0 4 3 5 . 1 1 0 3 8 6 4 5 . 0 25 0 . 0 4 2 8 . 949 3 8 6 3 8 . 7 24 0 . 0 4 2 2 . 5 8 3 3 8 6 3 2 . 3 2 i 0 . 0 4 1 5 . 8 4 9 3 8 6 2 5 . 6 22 0 . 0 4 0 8 . 9 3 4 3 8 6 1 8 . 7 21 0 . 0 4 0 1 . 9 7 7 3 8 6 1 1 . 8 20 0 . 0 3 9 4 . 7 2 1 3 8 6 0 4 . 6 19 0 . 3 2 9 3 8 7 . 2 1 4 3 8 5 9 7 . 1 18 0 . 665 3 7 9 . 6 2 6 3 8 5 8 9 . 5 17 0 . 0 0 6 3 7 2 . 0 1 0 3 8 5 8 1 . 8 16 0 . 0 0 0 3 6 4 . 4 3 4 3 8 5 7 4 . 0 15 0 . 0 0 0 3 5 6 . 8 4 2 3 8 5 6 6 . 4 14 0 . 0 0 0 3 4 8 . 9 5 2 3 8 5 5 8 . 5 13 0 . 0 3 4 1 . 5 5 7 3 8 5 5 0 . 4 12 0 . 0 3 3 3 . 7 1 4 3 8 5 4 2 . 2 11 0 . 0 3 2 5 . 8 6 6 3 8 5 3 3 . 9 10 0 . 0 3 1 8 . 9 1 7 3 8 5 2 4 . 9 9 0 . 0 3 0 5 . 7 0 9 3 8 5 1 0 . 5 8 0 . 0 2 8 5 . 0 6 0 3 8 4 8 9 . 9 7 0 . 0 2 6 2 . 7 3 2 3 8 4 6 7 . 5 6 0 . 0 2 4 0 . 4 4 7 3 8 4 4 5 . 2 5 0 . 0 2 1 8 . 1 2 0 3 8 4 2 2 . 9 4 0 . 0 1 9 5 . 9 0 4 3 8 4 0 0 . 7 3 0 . 0 1 7 3 . 9 3 3 3 8 3 7 8 . 7 2 0 . 0 1 5 2 . 1 0 0 3 8 3 5 6 . 9 1 0 . 0 0 0 1 3 0 . 1 9 8 3 8 3 3 5 . 0 PRESENT VALUE OF EXPECTED SUBSYSTEM RETURN VALUE OF EXPECTED ANNUAL SUBSYSTEM RETURN = 3 8 . 5 9 2 B I L L I O N DOLLARS 3 8 2 . 1 MILLION DOLLARS 86b. SUBSYSTEM 2 VALUE DETERMINATION * NO. OF CYCLES = 5 STATE(I» PI (I ) E R ( $ M I L L / Y R J V A L U E ! $ M I L L 1 26 0. 0 4 7 3 . 5 4 9 4 1 0 2 3 . 1 25 0.0 4 6 5 . 6 5 2 4 1 0 1 5 . 1 24 0 . 0 4 5 6 . 9 8 0 4 1 0 0 6 . 0 23 0 . 0 4 4 7 . 5 2 4 4 0 9 9 5 . 9 22 0 . 0 0 4 4 3 7 . 8 9 4 4 0 9 8 4 . 8 21 0 . 0 4 7 4 2 7 . 1 8 1 4 0 9 7 2 . 4 20 0 . 135 4 1 4 . 9 9 0 4 0 9 5 9 . 6 L9 0 . 6 9 1 4 0 4 . 2 2 9 4 0 9 4 6 . 1 18 0 . 1 2 1 3 9 2 . 4 3 9 4 0 9 3 1 . 9 17 0 . 0 0 2 3 8 1 . 0 7 8 4 0 9 1 7 . 5 16 0 . 0 0 0 3 6 7 . 7 8 4 4 0 9 0 2 . 2 15 0 . 0 0 0 3 5 7 . 4 4 7 4 0 8 8 7 . 1 14 0 . 0 3 0 3 4 4 . 8 3 3 4 0 8 7 1 . 3 13 0 . 0 0 0 3 3 7 . 5 1 5 4 0 8 5 6 . 4 12 0 . 0 0 0 3 3 0 . 9 7 9 4 0 8 4 0 . 5 11 0 . 0 0 0 3 2 5 . 7 5 6 4 0 8 2 4 . 0 10 0 . 0 0 0 3 2 1 . 3 6 7 4 0 8 0 7 . 8 9 0 . 0 0 0 3 1 6 . 7 3 6 4 0 7 8 9 . 9 8 0 . 000 3 0 7 . 7 3 6 4 0 7 6 7 . 5 7 0 . 0 0 0 2 7 0 . 7 4 6 4 0 7 2 9 . 7 6 0 . 0 2 3 0 . 5 0 3 4 0 6 8 9 . 5 5 0 . 0 1 9 0 . 2 0 2 4 0 6 4 9 . 2 4 0 . 0 1 4 9 . 9 6 8 4 0 6 0 8 . 9 3 0 . 0 1 0 9 . 8 1 9 4 0 5 6 8 . 8 2 0 . 0 6 9 . 9 5 9 4 0 5 2 8 . 9 1 0 . 000 3 0 . 1 6 6 4 0 4 8 9 . 1 PRESENT VALUE OF EXPECTED SUBSYSTEM RETURN VALUE OF EXPECTED ANNUAL SUBSYSTEM RETJRN 4 0 . 9 4 7 B I L L I O N DOLLARS 4 0 5 . 4 MILL ION DDLLARS LEAF 87 OMITTED IN PAGE NUMBERING. 87a. SUBSYSTEM PRODUCTION: DEC NOV PSG = 2 4 . 0 1 8 . 0 SUBSYSTEM 1: FEALOC = 1234, FEMN N0=1 PRBI GEN GENMC N0=2 PRBI GEN GENMC NQ=3 PRBI GEN GENMC N0=4 PRBI GEN GENMC NQ=5 PRBI GEN GENMC MEAN GENMC 86 9 . 2 1 . 06 1 2 3 6 . 2 4 . 3 2 1 . 7 7 1 3 5 8 . 2 4 . 1 2 2 . 1 8 1 4 7 6 . 2 4 . 0 2 2 . 63 1 6 5 4 . 2 4 . 0 2 3. 69 2 0 5 6 . 24 . 0 2 4 . 0 1 1 6 1 . 7 9 6 . 2 3 . 6 5 1 1 6 9 . 2 4 . 3 2 4 . 2 3 1165. 2 4 . 0 2 4 . 5 2 1 1 6 3 . 2 4 . 0 2 4 . 8 5 1161 . 2 3 . 9 2 5 . 7 2 1 1 5 4 . 2 3 . 9 2 4 . 0 OCT 1 5 . 0 9 7 6 . 6 1 0 . 2 4 . 81 9 8 1 . 2 4 . 5 2 5 . 13 968 . 2 4 . 3 2 5 . 2 8 97 8 . 2 4 . 1 ' 2 5 . 38 96 8 . 2 4 . 1 2 6 . 0 0 9 7 6 . 2 4 . 1 2 4 . 2 SEP 1 2 . 0 9 4 2 . 5 7 7 . 2 4 . 2 8 9 3 7 . 2 4 . 6 2 4 . 2 3 9 3 6 . 2 4 . 6 2 4 . 0 8 9 4 1 . 2 4 . 4 2 3 . 8 5 9 5 0 . 2 4 . 4 2 6 . 0 0 1597 . 2 4 . 3 2 4 . 5 AUG 12 . 0 2 2 6 4 . 1 6 7 9 . 2 4 . 7 4 2 2 7 0 . 2 4 . 6 2 3 . 8 3 2 2 3 3 . 2 4 . 8 2 3 . 24 2 2 6 2 . 2 2 . 6 2 2 . 3 2 2 2 5 8 . 2 2 . 3 2 4 . 5 7 2 5 9 8 . 1 2 . 1 2 2 . 8 J J L 6 . 0 1 1 3 7 . 822 . 1 8 . 6 0 1 099 . 2 4 . 4 16. 84 1 138 . 2 4 . 4 1 5 . 4 9 1 18 1 . 2 3 . 1 1 3 . 6 0 1 135 . 22 .3 1 4 . 4 7 1 1 7 9 . 1 1 . 8 2 2 . 8 MEAN MARGINAL COST OF SUBSYSTEM GENERATION = 2 3 . 8 MILLS/KWH MEAN ANNUAL SUBSYSTEM GENERATION = 1 5 3 6 3 . GWH ALLOCATIONS: FIRM E = 0 . 4 2 0 0 . 4 2 0 0 . 3 7 0 0 . 3 8 0 0 . 9 0 0 0 . 4 9 0 THERML = 0 . 5 0 0 0 . 5 0 0 0 . 5 0 0 0 . 5 0 0 0 . 8 0 0 0 . 5 0 0 87b. JUN • MAY APR 6 . 0 6 . 0 1 2 . 0 1169. 1336. 1 3 6 1 . 8 0 4 . 9 7 1 . 9 9 6 . 12 .08 1 0 . 2 7 1 2 . 3 1 9 6 1 . 9 8 5 . 9 9 4 . 2 5 . L 2 5 . 5 2 5 . 9 9 . 86 8 . 4 1 1 1 . 3 1 1 1 3 3 . 1 2 4 8 . 1 2 8 8 . 25 . 0 2 4 . 3 2 4 . 8 7. 59 5 .84 8 . 8 2 1114 . 1284 . 1 3 3 4 . 2 3 . 7 2 4 . 8 2 4 . 2 4 . 3 3 2 . 6 0 5 . 5 9 1 1 4 6 . 1285 . 1 3 5 7 . 2 3 . 1 2 3 . 5 2 3 . 4 4 . 4 4 1 .68 4 . 2 7 1 1 7 5 . 1 3 3 5 . 1 2 9 4 . 1 2 . 3 1 2 . 7 1 3 . I 2 3 . 4 2 3 . 9 2 3 . 7 0 . 4 8 0 0 . 5 2 0 0 . 5 2 0 0 . 5 0 0 0 . 5 0 0 0 . 5 0 0 MAR FEB JAN 1 8 . 0 1 8 . 0 2 4 . 0 1 1 5 5 . 1 0 8 0 . 1 2 4 4 . 7 8 9 . 7 1 5 . 8 7 9 . 1 4 . 2 8 1 6 . 1 0 18 .32 7 8 4 . 7 1 9 . 8 7 6 . 2 6 . 5 2 6 . 9 2 6 . 6 1 3 . 9 9 1 6 . 0 0 1 8 . 3 2 1 0 2 0 . 7 9 1 . 9 2 9 . 2 5 . 0 2 5 . 1 2 5 . 2 1 1 . 8 5 1 4 . 7 5 1 8 . 3 2 1 1 3 1 . 1 0 7 4 . 1 3 2 9 . 2 4 . 1 2 4 . 1 2 4 . 2 8 . 6 9 1 1 . 5 9 1 8 . 3 2 1 1 5 4 . 1 0 7 7 . 2 3 1 2 . 2 3 . 6 2 3 . 5 2 4 . 1 7 . 6 1 1 0 . 6 0 1 8 . 3 2 1 2 4 7 . 1 1 2 0 . 2 6 3 6 . 2 0 . 4 1 9 . 3 1 8 . 9 2 4 . 2 2 4 . 1 2 4 . 3 0 . 4 0 0 0 . 4 0 0 0 . 4 1 0 0 . 5 3 0 0 . 5 0 0 0 . 5 0 3 MILLS/KWH GWH GWH MI LLS/KWH GWH MILLS/KrfH GWH MILLS/KWH GWH MILLS/KWH GWH MILLS/KWH MILLS/KWH LEAF 88 OMITTED IN PAGE NUMBERING. 88a. PSG DEC 2 4 . 0 SUBSYSTEM 2 : FEALOC = 1 7 0 4 . FEMN N0=1 PR8I GEN GENMC N0=2 PRBI GEN GENMC NQ=3 PRBT GEN GENMC N0=4 PRB I GEN GENMC NQ=5 PRB I GEN GENMC MEAN GENMC 1 3 3 9 . 2 0 . 5 5 1 6 9 1 . 2 4 . 8 2 1 . 03 1 7 3 3 . 2 4 . 5 2 1 . 6 8 1988 . 2 4 . 2 2 2 . 3 3 2 1 7 5 . 2 3 . 5 2 3 . 86 2 1 8 1 . 2 1 . 7 2 4 . 0 NOV 1 8 . 0 1604 . 1 2 3 8 . 2 2 . 0 3 1425. 2 4 . 9 2 2 . 6 9 1 6 1 3 . 2 4 . 5 2 3 . 21 1 6 0 8 . 2 4 . 1 2 3 . 7 4 1598 . 2 3 . 3 2 5 . 0 2 1594 . 21 . 9 2 3 . 9 OCT 1 5 . 0 1 6 6 1 . 1296 . 2 2 . 9 3 1364. 2 5 . 1 2 3 . 9 1 1 6 6 8 . 2 4 . 6 2 4 . 24 1 6 5 5 . 2 4 . 1 2 4 . 5 8 1670. 2 3 . 5 2 5 . 5 5 1 6 5 4 . 2 2 . 0 2 4 . 0 SEP 1 2 . 0 1538, 1172, 2 3 . 6 0 1 2 0 8 . 2 5 . 5 2 4 . 9 4 1 5 3 6 . 2 4 . 6 2 5 . 14 1 5 5 1 . 2 4 . 3 2 5 . 2 4 I 53 0. 2 3 . 6 2 5 . 9 5 1 5 4 4 . 2 2 . 3 2 4 . 1 AUG 1 2 . 0 2 5 2 . 1 0 5 . 2 1 . 7 7 1 1 8 . 2 6 . 1 2 3 . 02 2 4 3 . 2 4 . 8 2 3 . 0 5 2 7 1 . 2 4 . 6 2 2 . 8 8 2 8 3 . 2 3 . 9 2 3 . 9 8 7 2 7 . 22 . 8 2 4 . 4 JUL 6 . 0 123.5. 87 0 . 19 . 5 7 874 . 2 6 . 2 2 0. 7 7 1 1 4 9 . 2 4 . 9 2 0 . 5 2 1 2 2 7 . 2 4 . 0 19. 8 5 1 2 4 4 . 2 1 . 7 2 0 . 24 1 2 4 4 . 1 3 . 9 2 3 . 2 MEAN MARGINAL COST OF SUBSYSTEM GENERATION = 2 4 . 0 MI LLS/KWH MEAN ANNUAL SUBSYSTEM GENERATION = 1 6 3 2 0 . GWH ALLOCATIONS: FIRM E = 0 . 5 8 0 0 . 5 8 0 0 . 6 3 0 0 . 6 2 0 0 . 1 0 0 0 . 5 1 0 THERML = 0 . 5 0 0 0 . 5 0 0 0 . 5 0 0 0 . 5 0 0 0 . 2 0 0 0 . 5 0 0 MEAN ANNUAL SYSTEM GENERATION = 3 1 6 8 3 . GWH PRESENT VALUE OF EXPECTED SYSTEM RETURN = 7 9 . 5 4 0 $BILL ION VALUE OF EXPECTED ANNUAL SYSTEM RETURN = 7 8 7 . 5 SMILLION 8 8 b . JUN MAY APR 6 . 0 6 . 0 1 2 . 0 1 2 b 6 . 1 2 3 3 . 1 2 5 6 . 9 0 1 . 8 6 8 . 3 9 1 . 1 4 . 9 1 1 2 . 3 2 1 3 . 1 0 8 9 4 . 8 6 2 . 8 9 6 . 2 6 . 9 2 7 . 7 2 8 . 2 1 5 . 6 2 1 2 . 5 4 1 3 . 2 1 1 1 2 4 . 9 2 1 . 9 0 1 . 2 4 . 7 2 5 . 4 2 5 . 7 1 4 . 8 0 1 1 . 5 6 1 2 . 4 1 1186 . 1081 . 1 0 4 7 . 2 4 . 2 2 4 . 1 2 4 . 9 1 3 . 3 7 9 . 3 0 1 0 . 8 6 1 2 5 5 . 1 2 2 1 . 1 2 2 4 . 2 2 . 6 2 2 . 6 2 2 . 3 12 . 62 8 . 3 2 9 . 32 1 2 6 4 . 1 2 4 0 . 1 2 7 1 . 1 4 . 8 1 5 . 4 1 5 . 9 2 3 . 5 2 3 . 8 2 4 . 1 0 . 5 2 0 0 . 4 8 0 0 . 4 8 0 0 . 5 0 0 0 . 5 0 0 0 . 5 0 0 MAR FES JAN 1 8 . 0 1 8 . 0 2 4 . 0 1 7 3 2 . 1 6 2 0 . 1 7 9 0 . 1 3 6 7 . 1 2 5 5 . 1 4 2 5 . 1 5 . 1 8 1 7 . 0 2 19 .11 1 3 7 5 . 1 2 4 7 . 1 4 3 3 . 2 8 . 2 2 7 . 7 2 3 . 2 1 5 . 2 7 1 7 . 0 8 1 9 . 1 1 1395 . 1 2 6 0 . 1 4 3 7 . 2 5 . 8 2 5 . 8 2 6 . 1 1 4 . 7 3 1 6 . 8 6 1 9 . 1 1 1 5 6 0 . 1 4 6 4 . 1 5 8 7 . 2 5 . 0 2 5 . 0 2 5 . 0 1 3 . 4 8 1 5 . 8 7 1 9 . 1 1 1 7 4 7 . 1 6 3 1 . 2 1 7 7 . 2 2 . 4 2 2 . 2 2 2 . 2 1 2 . 6 1 1 5 . 9 1 1 9 . 1 1 2 1 4 6 . 2 1 7 3 . 2 2 0 1 . 1 5 . 9 1 5 . 8 1 5 . 8 2 4 . 2 2 4 . 1 2 4 . 2 0 . 6 0 0 0 . 6 0 0 0 . 5 9 0 0 . 5 0 0 0 . 5 0 0 0 . 5 0 0 MILLS/KWH GWH GrfH GWH M I L L S / M H GWH MILLS/KWH GWH MILLS/KWH GWH MILLS/KdH GWH MILL S/KWH MILLS/KWH 

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