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Operational decision making for a multi-purpose reservoir with total seasonal inflow forecast Caselton, William F. 1970

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OPERATIONAL DECISION MAKING FOR A MULTI-PURPOSE RESERVOIR WITH TOTAL SEASONAL INFLOW FORECAST by WILLIAM F. CASELTON B.Sc. (Mechanical Engineering) Uni v e r s i t y of Leeds, England, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 a g r e e 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 p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment 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 u n d e r s t o o d 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 . Depar tment o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada A B S T R A C T This study investigates the operational decision process f o r Okanagan Lake, a natural lake regulated by a dam at the outlet f o r flood c o n t r o l , i r r i g a t i o n and water supply purposes. In addition, the Lake supports a s u b s t a n t i a l t o u r i s t industry. The Lake i s p r i n c i p a l l y supplied by snowmelt and a forecast of t o t a l inflow volume during the c r i t i c a l runoff season i s a v a i l a b l e to a s s i s t the operator. The operational decision process was found to d i f f e r from the sequential decision basis of many Operations Research techniques and the absence of information on costs and benefits precluded the use of conven-t i o n a l optimization procedures. The importance of making the best use of the inflow forecast to achieve the operational goals was recognized and was used as the basis of the decision analysis developed. The method developed assesses possible immediate operational de-cis i o n s by evaluating the effectiveness of future discharges to correct for past decision errors. The evaluation i s made i n terms of the pr o b a b i l -i t i e s of exceeding Operational constraints and of achieving operational goals. The method involves simulation of sets of monthly inflows f o r the remainder of the runoff season given an inflow volume forecast and know-ledge of the probable accuracy of the forecast; computation of water l e v e l s which would occur with various operating procedures; frequency analysis of the r e s u l t i n g l e v e l s ; i n t e r p r e t a t i o n of the frequencies as p r o b a b i l i t i e s ; and presentation of the r e s u l t i n g information describing the operational i i i i i s i t u a t i o n i n r e a d i l y assimilable form. TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES v i Chapter I. INTRODUCTION 1 II . GENERAL DESCRIPTION OF OKANAGAN LAKE AND ITS OPERATION . . 4 I I I . OKANAGAN LAKE INFLOW FORECASTING 9 IV. THE OPERATIONAL DECISION PROCESS: ANALYSIS AND ASSESS-MENT OF A DECISION 11 V. GENERATION AND SIMULATION METHODS 22 VI. COMPUTATIONAL PROCEDURE FOR DECISION ASSESSMENT 33 VII. RESULTS AND INTERPRETATION 36 VIII. SUMMARY 51 BIBLIOGRAPHY 53 APPENDIX I 54 APPENDIX II 56 i v LIST OF TABLES Table Page II.1 OKANAGAN LAKE WATER DEMANDS 7 V I I . l HISTORIC MONTHLY INFLOW ANALYSIS 36a VII. 2 TYPICAL RESULTS 50 v LIST OF FIGURES Figure Page 4.1 ASSESSMENTS OF A SINGLE DECISION 20 4.2 COMPARISON OF TWO DECISION ALTERNATIVES 21 7.1 CUMULATIVE DISTRIBUTION OF HISTORIC INFLOWS FOR APRIL . 38 7.2 CUMULATIVE DISTRIBUTION OF HISTORIC INFLOWS FOR MAY . . 39 7.3 CUMULATIVE DISTRIBUTION OF HISTORIC INFLOWS FOR JUNE . . 40 7.4 CUMULATIVE DISTRIBUTION OF HISTORIC INFLOWS FOR JULY . . 41 7.5 CUMULATIVE DISTRIBUTION OF HISTORIC TOTAL INFLOWS FROM APRIL TO JULY 42 7.6 CUMULATIVE DISTRIBUTIONS OF SYNTHETIC MAY INFLOWS FOR VARIOUS FORECASTED TOTAL INFLOWS FROM APRIL TO JULY . 43 7.7 DECISION ASSESSMENT FOR FEBRUARY 47 7.8 DECISION ASSESSMENT FOR APRIL 48 7.9 DECISION ASSESSMENT FOR APRIL WITH IMPROVED FORECAST ACCURACY 49 v i A C K N O W L E D G E M E N T The author wishes to express his gratitude to his supervisor, Mr. S. 0; Russell f or his valuable guidance and encouragement during the research, development and preparation of t h i s t h e s i s . The author i s also g r a t e f u l to Mr. T. A. J . Leach, Assistant Chief Engineer, and Mr. H. I. Hunter, Chief Hydrologist, both of the Water Investigations Branch, B.C. Water Resources Service, Department of Lands, Forests and Water Resources, V i c t o r i a , B.C., for t h e i r a s s i s t -ance i n providing information on the operation and forecast procedures f o r Okanagan Lake. y i i CHAPTER I INTRODUCTION Operational decision p o l i c i e s for water resources projects have been obtained by a number of methods, emphasis currently being placed on simulation and optimization techniques. Generally a decision p o l i c y i s obtained by considering a time period covering a large number of succes-sive decisions and t r e a t i n g a l l aspects i n a deterministic fashion. The analysis and r e s u l t i n g p o l i c y are r e l a t e d to t h i s t h e o r e t i c a l context but do not consider the predicament faced by the operator at the time he makes an i n d i v i d u a l decision. Unless a " r i g i d r u l e " philosophy i s being imposed on the operator, the decision sequence which w i l l unfold i n prac-t i c e w i l l consist of a s e r i e s of short term commitments, each one repre-senting the operator's best e f f o r t to resolve the predicament facing him at the time of making the commitment. Thus i t follows that, i n order to achieve the best decision sequence i n actual p r a c t i c e , each i n d i v i d u a l d e c i s i o n must be the best under the p r e v a i l i n g circumstances. "Best" i n t h i s context need not necessarily imply the maximum economic return, but, because of the operator's hazy view of the future, a decision which w i l l permit the greatest future c o r r e c t i v e action may be regarded as the most desira b l e . This study s p e c i f i c a l l y considers the p o s i t i o n of the operator when making an i n d i v i d u a l decision. The operational s i t u a t i o n of an e x i s t i n g multi-purpose natural r e s e r v o i r , Okanagan Lake, B r i t i s h Columbia, was 2 chosen as the b a s i s of the study. A major f a c t o r i n the operation of t h i s Lake i s that i t i s fed p r i n c i p a l l y from snowmelt, w i t h the r e s u l t that the i n f l o w volume can be p r e d i c t e d by d i r e c t measurement of the snowpack. This p r e d i c t i o n i s not p r e c i s e as many other f a c t o r s i n f l u -ence the amount of water which w i l l u l t i m a t e l y reach the Lake. I t does, however, provide v a l u a b l e i n f o r m a t i o n on the probable f u t u r e i n f l o w volume during the run-off season. The importance of using t h i s i n f o r -mation to i t s best e f f e c t was recognized and was a major f a c t o r i n prompting t h i s study. The o p e r a t i o n a l requirements of Okanagan Lake i n v o l v e p r o v i d i n g adequate storage capacity p r i o r to peak inflo w s to prevent f l o o d i n g and ensuring that the Lake i s f u l l at the end of the run-off season to meet consumptive demands throughout the summer. With present consumptive de-mands the l a t t e r w i l l a l s o provide over-year storage and hence some pro-t e c t i o n against very low i n f l o w s i n the f o l l o w i n g year. The extent to which these two requirements c o n f l i c t i s dependent upon many f a c t o r s , one of which i s the accuracy w i t h which one can p r e d i c t not only f u t u r e t o t a l i n f l o w volume but a l s o the rates at which t h i s i n f l o w w i l l occur. The a v a i l a b l e volume f o r e c a s t provides no i n f o r m a t i o n on i n f l o w r a t e s and a major p o r t i o n of t h i s study i s concerned w i t h overcoming t h i s d e f i c i e n c y . I t was a n t i c i p a t e d that an e x i s t i n g Operations Research technique would form the b a s i s of the d e c i s i o n e v a l u a t i o n . However, i t was found that most of the a v a i l a b l e techniques r e q u i r e d accurate and comprehensive data on the economic value of consumed water and costs associated w i t h extreme Lake l e v e l s . This i n f o r m a t i o n was not a v a i l a b l e f o r Okanagan 3 Lake. Furthermore } because of the intangible nature of many of the bene-f i t s Cand costs) associated with the operation of the Lake, i t was appar-ent that t h i s information was e s s e n t i a l l y unobtainable with any degree of accuracy. This precluded the a p p l i c a t i o n of many optimization/decision making methods and led to the development of the method described i n this t h e s i s . The method proposed does not attempt to determine the optimal decision but combines h i s t o r i c inflow data, Lake operating constraints, and the forecast information, together with an estimate of the probable forecast accuracy, to provide an i n d i c a t i o n of the operational s i t u a t i o n . This s i t u a t i o n i s described i n terms of the probable consequences of var-ious possible courses of action on Lake l e v e l with respect to l i m i t s and storage goals. The actual decision would then have to be made by the operator i n the l i g h t of this s i t u a t i o n information and his own experience and judgement. The method i s described i n the context of the Okanagan Lake problem, but would require l i t t l e modification for many t y p i c a l multi-purpose reser-v o i r s i t u a t i o n s . Chapter VIII discusses some possible extensions of the method. CHAPTER II GENERAL DESCRIPTION OF OKANAGAN LAKE AND ITS OPERATION Descri p t i o n Okanagan Lake i s located i n a semi-arid region i n the i n t e r i o r of the Province of B r i t i s h . Columbia. It i s fed p r i n c i p a l l y by snowmelt from the surrounding h i l l s with most runoff occurring during the l a t e spring months. The Lake has a surface area of 84,200 acres (131.5 sq. mis.) and i t i s c o n t r o l l e d at i t s outlet at Penticton over a normal operating range of 4.0 f t . This provides a l i v e storage volume of 337,000 acre f t . Dis-charge flow from the Lake i s l i m i t e d by the capacity of the channel down-stream of the Penticton control structure. On occasions the discharge capacity must be further c u r t a i l e d to avoid flooding at a downstream junction with the Similkameen River. The maximum discharge capacity was taken as 1800 c . f . s . which i s equivalent to 108,000 acre f t . per month. Variations i n a v a i l a b l e discharge capacity at c e r t a i n times or under cer-t a i n conditions were not considered i n this study but could be r e a d i l y accommodated by the method developed (see Chapter V I I I ) . Estimates of net monthly inflows into the Okanagan Lake have been determined from the net inflows to the Lake (computed from Lake elevation changes and outflows) plus an allowance for the volume of water intercepted and consumed before entry into the Lake. This i n f o r -4 5 mation f l j was provided f o r a period of 48 years by the B r i t i s h Columbia Water Resources S e r v i c e , the B r i t i s h Columbia Department of Lands, Forests and Water Resources, V i c t o r i a . These computed net i n f l o w s were used as b a s i c data i n t h i s study and i n t e r c e p t e d volumes ( i n upstream storage r e -s e r v o i r s ) were accounted f o r by i n c l u d i n g them i n the demand volumes. The t o t a l annual net i n f l o w v a r i e s between 96,000 acre f t . and 796,000 acre f t . w i t h an average of 401,000 acre f t . The l i v e storage i s thus about 84 per cent of the average annual net i n f l o w and 42 per cent of the maximum recorded i n f l o w . Approximately 90 per cent of the annual net i n f l o w occurs during the A p r i l to J u l y p e r i o d w i t h the highest i n f l o w u s u a l l y o c c u r r i n g i n May. The average annual peak monthly i n f l o w i s 198,500 acre f t . and the highest recorded i n f l o w f o r one month i s 402,000 acre f t . The computed net monthly i n f l o w s f o r 48 years are shown i n Appendix 1 and monthly averages and standard d e v i a t i o n s are given i n Table V I I . 1 . Evaporation l o s s e s from the Lake are l a r g e and may be as high as 50,000 acre f t . per month during the summer months. As net i n f l o w s are computed from Lake e l e v a t i o n changes, evaporation losses are a u t o m a t i c a l l y i n c o r -porated so that when evaporation l o s s e s exceed i n f l o w i n a p a r t i c u l a r month, then a negative i n f l o w w i l l be recorded f o r that month. Okanagan Lake serves three major purposes: (.1) Flood c o n t r o l ; (2) Storage f o r i r r i g a t i o n requirements; (3) Recreation and tourism. B e n e f i t s from the Lake are secondary or i n d i r e c t and a l s o predominantly of 6 an i n t a n g i b l e nature. Typically- f or a natural r e s e r v o i r i n an area which has been developing over a long period of time, there i s l i t t l e a v a i l a b l e q u a n t i t a t i v e data r e l a t i n g to these benefits and, consequently, methods for conversion of inta n g i b l e benefits to monetary units could not be applied. R e a l i s t i c a l l y , i t could be assumed that information of t h i s type i s unobtainable unless inordinate e f f e c t and funds were directed towards i t s c o l l e c t i o n . This fa c t had a s i g n i f i c a n t bearing on the effectiveness of the Operations Research techniques which were consid-ered and on the method which was subsequently developed. The t o t a l consumptive demand on the Okanagan Lake Basin, based upon 1966 figures [2] i s 216,000 acre f t . , a f i g u r e which allows.for return flows to the Lake. This quantity meets the requirements of domestic water supply, i r r i g a t i o n , and minimum flows i n the Okanagan River which c a r r i e s the discharge from"the Lake. Table II.1 gives a breakdown of the annual t o t a l water demand and estimated demands for the months A p r i l , May, 'June and July. These monthly demands (and to some extent the annual demand) w i l l vary from year to year depending on c l i m a t i c conditions, etc., but i t i s assumed that a conservative e s t i -mate of immediate future seasonal demands w i l l always be possible with reasonable accuracy. TABLE II.1 OKANAGAN LAKE WATER DEMANDS USE . ANNUAL TOTAL DEMAND MONTHLY DEMANDS APRIL MAY JUNE JULY I r r i g a t i o n 100,000 Neglig. 10,000 10,000 25,000 Domestic and Waterworks 8,000 1,000 1,000 1,000 1,000 Minimum Flow Okanagan River 108,000 1,000 8,000 8,000 8,000 TOTAL 216,000 9,000 19,000 34,000 34,000 Lake Operation The Lake l e v e l i s co n t r o l l e d by operating the gates i n the con-t r o l dam at Penticton i n response to discharge decisions' made at i n t e r -vals varying from one week to one month. The discharge decisions are based upon current inflow forecasts, inflows to date, downstream con-d i t i o n s on the Okanagan River, and past operational experience. The major operational constraints are the maximum and minimum Lake l e v e l s , set'at elevations 102.5 f t . and 98.5 f t . r e s p e c t i v e l y , and 8 the maximum discharge capacity. Flood control requirements during the runoff period and storage requirements to meet the i r r i g a t i o n needs must be met by operation within these l i m i t s . Recreational and tourism require-ments are met by maintaining the Lake between the same l i m i t s . Operation of the Lake i s made d i f f i c u l t by the l i m i t e d accuracy of the t o t a l volume forecast for the runoff season and the d i f f i c u l t y i s compounded by the i n a b i l i t y to predict the timing and rate of runoff. CHAPTER III OKANAGAN LAKE INFLOW FORECASTING Most of the inflow to Okanagan Lake originates from snowmelt and thus measurement of the snowpack within the Okanagan Lake Basin, which completely melts each year, provides a basis for forecasting the t o t a l inflow volume. A v a r i e t y of other factors influence the t o t a l inflow and major discrepancies can occur between estimates based s o l e l y on snowpack measure-ments and the inflow which subsequently occurs. Retention of runoff i n the s o i l , groundwater recharge, evaporation, and p r e c i p i t a t i o n during the runoff season are some of the factors involved. Investigations into the runoff process of the Okanagan Lake Basin are expected to provide better estimates of runoff contributing to groundwater recharge but factors such as p r e c i p i t a t i o n and evaporation during the snowmelt season depend upon short term meteorological e f f e c t s and are not predictable more than a few hours i n advance. Forecasting of the inflow rates during weekly or monthly periods, while highly, desirable from an operational standpoint, i s not attempted because these are also dependent on short term meteorological e f f e c t s . Volume forecasts are based on p r e d i c t i o n equations which are obtained by applying conventional multiple regression methods to h i s t o r i c records of snowpack measurements, antecedent p r e c i p i t a t i o n , and subsequent Lake inflows. 9 10 Commencing with receipt of the f i r s t snowpack data i n l a t e Feb-ruary, a forecast of the t o t a l inflow which w i l l occur by the end of July i s made. As time elapses the forecast i s revised f o r the A p r i l to July, May to July, and June to July periods. These forecasts, as would be expected, becoming progressively more accurate as the season advances. Although forecasts could be made commencing November or December of the previous year since some of the antecedent factors are known at that time, they would be of very low accuracy. The s i g n i f i c a n c e of early forecasts i s considered i n conjunction with the method developed. A value of the Standard Error of Estimate f o r the forecast i s given by the multiple regression analysis and i s used as an i n d i c a t i o n of forecast accuracy i n this study. The assumptions involved are d i s -cussed i n Chapter V-of th i s t h e s i s . CHAPTER IV THE OPERATIONAL DECISION PROCESS; ANALYSIS AND ASSESSMENT OF A DECISION The Decision Process Available decision analysis techniques i n Operations Research concentrate on determining an optimal sequence of decisions i n order to maximize e x p l i c i t l y defined returns while not v i o l a t i n g any of the sys-tem constraints. Stochastic optimization techniques, which recognize the unpre-d i c t a b i l i t y of future events, o f f e r the most r e a l i s t i c simulation of operational decision s i t u a t i o n s but become highly involved when applied to r e a l l i f e s i t u a t i o n s . The return or objective function must s t i l l be e x p l i c i t l y stated but i n this instance i t i s the expected return that i s maximized ( i . e . , the sum of a l l possible returns, each m u l t i p l i e d by i t s p r o b a b i l i t y of occurrence); thus introducing a further degree of a r t i f i -c i a l i t y to the analysis. Even a f t e r considerable s i m p l i f i c a t i o n stochastic optimization techniques place extreme demands on computing f a c i l i t i e s and the process of s i m p l i f i c a t i o n may have to be c a r r i e d out beyond the point where the r e s u l t s remain meaningful. Deterministic optimization techniques r e l y upon the maximization of the return function over a long simulation run and reduce the computa-t i o n a l load to a more manageable l e v e l . This approach introduces the per-spective of a long range planner to operational decision making which may 11 12 not be appropriate. An optimal long run operating p o l i c y may conceivably allow occasional minor flooding i n order to r e a l i z e more than compensating benefits i n future months or years. However, i n r e a l i t y , i t i s d i f f i c u l t to envisage an operator, with dubious knowledge of the future, a c t u a l l y permitting flooding to any degree i f i t i s within his power to prevent i t . As stated previously, the lack of s u i t a b l e information on economic returns from operation of the Lake precludes developing an objective func-t i o n and further d i s q u a l i f i e s the above techniques from a p p l i c a t i o n to t h i s p a r t i c u l a r problem. Since standard techniques appeared to be i n a p p l i c a b l e , an examina-t i o n of the Okanagan Lake operational decision s i t u a t i o n was made i n order to define the actual decision process involved and seek a basis for i t s analysis. Although operational decisions may be made at i n t e r v a l s from one week to one month i t was found desirable for study purposes to base the analysis on a monthly decision period. This coincides with the r e -corded monthly inflow data and the monthly updating of the inflow forecast for the Lake. At the time of making a discharge decision for the forthcoming month, the operator has exact information on the present Lake l e v e l and i t s p o s i t i o n with respect to the upper and lower l e v e l l i m i t s . The maximum discharge capacity at h i s disposal w i l l also be known together with the storage goal at the end of July. His view of the future i s confined to the current updated t o t a l inflow forecast. In the absence of forecasts or foresight extending a number of years into the future, the operator cannot determine a strategy which w i l l r e s u l t i n an optimal return over a period 13 beyond the immediate summer ahead. Achievement of f u l l storage at the end of the current period, provided t h i s i s accomplished without flooding, w i l l however be synonymous with, a long term optimal operation. On t h i s basis i t was concluded that an operational decision method which enabled the Lake to be brought to a f u l l storage condition i n July each year, or at l e a s t maximized the p r o b a b i l i t y of achieving this goal, would meet the long term optimization objective. (It i s not suggested that t h i s w i l l n e c e s s a r i l y be the general case i n a l l r e s e r v o i r s , but i t s p e c i f i c a l l y applies to Okanagan Lake). A perfect forecast of the t o t a l inflow and inflow rates w i l l always enable the operator to achieve the desired storage l e v e l without flooding provided the inflow i n any month do es not exceed the a v a i l a b l e storage volume plus maximum discharge i n one month. In the p r a c t i c a l s i t u a t i o n , with imperfect volume forecast and no inflow rate forecast, the a b i l i t y of the operator to achieve these goals diminishes and c o n f l i c t s a r i s e i n meeting the operational requirements. I t i s apparent, though, that the best possible use of the a v a i l a b l e forecast information i s e s s e n t i a l to determining the best operating decision. S i g n i f i c a n t l y , l i t t l e work has been ca r r i e d out on decision methods incorporating forecast i n f o r -mation except perhaps i n Game Theory which confines i t s e l f mainly to s i n g l e decisions and simply stated returns. The factors which determine i f the operational goals can be met during the runoff season, and at the same time determine a "correct" operational decision are: (a) past decisions; 14 (b) actual inflows to the present time; (c) lake l e v e l at the present time; (d) future monthly inflows; (e) future decisions and decision errors. Both (a) and (b) are embodied i n the Lake l e v e l at the time of making the new decision and these factors can be reduced to (c) alone without loss of relevant information. The only information a v a i l a b l e on future inflows i s contained i n the current t o t a l inflow forecast, but t h i s does not provide a d i r e c t estimate of inflow by months. This prevents estimation of what future decisions might be and consequently provides no basis for the simulation of a future decision sequence. The remaining "dynamic" factors which are known and upon which an operational decision must be based are now: (a) present lake l e v e l ; (b) current t o t a l inflow forecast; together with the constraint and goal f a c t o r s : (a) upper and lower l i m i t s of Lake l e v e l ; (b) maximum discharge capacity; (c) desired terminal l e v e l at the end of July. The analysis of the operational decision process to t h i s point reveals that i t consists of a sequence of v i r t u a l l y self-contained single decision problems. This contrasts with the strongly interdependent decision sequence usually found i n dynamic decision models. Furthermore, the c r i t e r i a for the "best" decision must now be defined i n a context which i s somewhat d i f f e r e n t 15 from the conventional decision sequence a n a l y s i s . As the runoff season advances the operator acquires more and more accurate information on the future inflow s i t u a t i o n and this enables him to make improved corrections to the Lake l e v e l . At-.the same time h i s capacity to make these corrections i s progressively reduced by the reduction i n time a v a i l a b l e f o r the correc-t i v e discharges. The operator must therefore attempt to make current de-ci s i o n s which w i l l ensure that he w i l l be able to achieve the operational requirements without r e q u i r i n g corrective discharges i n excess of those at h i s disposal at any time during the runoff season. In other words, the "best" current decision w i l l maximize the effectiveness of future correc-t i v e capacity. This i s the basis of assessment adopted i n th i s study, but before describing the method developed, the terms cowecti-ve action and oovTective capacity must be defined. Cowective act-ion would be a low or minimum discharge following an error of too large a discharge, and a high or maximum discharge following an error of too small a discharge. Corrective capacity for a period extending one month beyond the current decision month would be the range of volumes, from minimum to maximum, which could be discharged during the second month. For a period extending two months beyond the current decision month, the cor r e c t i v e capacity becomes greater. Maximum and minimum discharges can now be sustained for two months following the current month. I t 16 should be noted that the c o r r e c t i o n i s being a p p l i e d s o l e l y to the p o t e n t i a l e r r o r i n the current d e c i s i o n , t h i s being the only commit-ment under c o n s i d e r a t i o n . As the pe r i o d i s lengthened the c o r r e c t i v e capacity increases but at the same time the i n f l o w volume f o r the p e r i o d w i l l a l s o i n c r e a s e and w i l l , to some extent, o f f s e t the c o r r e c t i v e c a p a c i t y . While the above i s a convenient way to develop the concept of c o r r e c t i v e c a p a c i t y , i n r e a l i t y , the p e r i o d , which ends at a f i x e d p o i n t i n time ( i n the case of t h i s study, 31 J u l y ) , w i l l p r o g r e s s i v e l y shorten and the c o r r e c t i v e c a p a c i t y reduces. For purposes of comparison of v a r -ious d e c i s i o n s i t i s necessary to o b t a i n some q u a n t i t a t i v e measure of the e f f e c t i v e n e s s of the c o r r e c t i v e c a p a c i t y which may be subsequently a p p l i e d . This i s accomplished by s i m u l a t i n g the Lake l e v e l response w i t h a mathematical model and s u b j e c t i n g t h i s model to probable seasonal i n f l o w s i t u a t i o n s , a current discharge d e c i s i o n , and maximum appropriate c o r r e c t i v e discharges. The r e s u l t i n g Lake l e v e l s (maximum, minimum, e t c . , from each simul a t i o n ) can then be represented i n the form of a histogram, a s i n g l e p o i n t on the histogram i n d i c a t i n g the p r o b a b i l i t y of a c e r t a i n Lake l e v e l being exceeded. The histogram provides the re q u i r e d q u a n t i -t a t i v e measure of e f f e c t i v e n e s s of the c o r r e c t i v e ^ c a p a c i t y . D e c i s i o n A n a l y s i s The various steps i n the method of d e c i s i o n a n a l y s i s are o u t l i n e d below. 17 1. A decision at a p a r t i c u l a r point i n time i s considered and i s considered to be f i r m for a period of one month. 2. The f u l l range of possible current discharge decisions i s assessed i n conjunction with only extreme (and therefore known) cor r e c t i v e discharges i n the future. Corrective action w i l l not however be the same for avoiding v i o l a t i o n of the upper and lower l e v e l constraints and i t w i l l be necessary to separate-l y analyze each decision with respect to maximum and minimum Lake l e v e l s , and terminal l e v e l s . - maximum cor r e c t i v e action associated with the peak or max-imum l e v e l c r i t e r i o n w i l l be a f u l l discharge during the remaining months i n the runoff period (following the de-c i s i o n month); - the maximum corrective action associated with the minimum l e v e l c r i t e r i o n w i l l be a minimum ( i . e . , zero voluntary) discharge during the remaining months; - co r r e c t i v e action for the terminal l e v e l c r i t e r i o n w i l l be the same as for minimum l e v e l . 3. The current value of the t o t a l future inflow forecast and an estimate of i t s accuracy are used as a basis for generating a synthetic population of " a c t u a l " future inflows which might occur following this forecast; i n turn these " a c t u a l " inflows are used as a basis for the generation of sets of synthetic monthly inflows, each set representing one runoff season. The generation procedure adopted i s described i n the following chapter. 1 8 4. Maximum, minimum and terminal l e v e l s are computed from the Lake l e v e l response model for each set of monthly inflows, a range of current discharge.decisions, and the appropriate cor-r e c t i v e actions. A large number of simulations are run and a frequency analysis made of the r e s u l t i n g maximum, minimum and terminal l e v e l s f o r each current discharge decision. In prac-t i c e i t was found only necessary to test the largest and smallest possible current decisions and one or two intermediate decisions f o r c a l i b r a t i o n purposes. 5. The frequency r e s u l t s obtained are interpreted as p r o b a b i l i t i e s of exceeding, or f a l l i n g below, various l e v e l s and are presented i n the form of cumulative p r o b a b i l i t y curves f o r each current decision analyzed. When plotted on normal p r o b a b i l -i t y graph paper, the cumulative p r o b a b i l i t y data c l o s e l y approxi-mates s t r a i g h t l i n e s . Figure 4.1 shows a t y p i c a l set of pro b a b i l -i t y l i n e s f o r a sing l e possible current discharge d e c i s i o n D^. C r i t i c a l Lake l e v e l s are superimposed to aid i n the i n t e r p r e t a t i o n . I f the discharge decision were taken i n the current month, and maximum appropriate c o r r e c t i v e action was taken i n sub-sequent months, the p r o b a b i l i t i e s that the upper and lower Lake l e v e l l i m i t s would be exceeded are indicated by points (a) and (b) i n Figure 4.1. S i m i l a r l y , point (c) indicates the p r o b a b i l i t y that the desired terminal l e v e l w i l l be achieved. 19 The comparison of the two decision a l t e r n a t i v e s i s shown i n Figure 4.2. In this case decision i s the larger discharge decision and w i l l r e s u l t i n reduced p r o b a b i l i t y of exceeding the upper l e v e l l i m i t but w i l l also reduce the prob-a b i l i t y of achieving the desired terminal l e v e l . At the same time the p r o b a b i l i t y of f a l l i n g below the lower l i m i t i s r e -duced. If and T)^ represented the minimum and maximum d i s -charge de c i s i o n p o s s i b i l i t i e s then a l l intermediate decision l i n e s w i l l f a l l between these pairs of p r o b a b i l i t y l i n e s . The r e s e r v o i r operator may now assess the extent to which a present decision w i l l r e s t r i c t the influence of future corrective action on achieving the primary operational requirements and goals i n p r o b a b i l i s -t i c terms. The decision p r o b a b i l i t y l i n e s also provide a d d i t i o n a l i n f o r -mation. The slope of the l i n e s i s an i n d i c a t i o n of the variance of the possible outcomes. As the accuracy of the forecast improves t h i s v a r i -ance i s reduced and the slope of the l i n e s i s reduced. A h o r i z o n t a l l i n e represents a forecast with zero error. The h o r i z o n t a l separation of a p a i r of dec i s i o n l i n e s i s an i n d i c a t i o n of the e f f e c t which the v a r i a t i o n i n discharge f o r the current decision month can have on future consequences. Lo ke LeveI Ft. above Datum Datum 00 Ft. below Datum Terminal Decision ' D| followed by mox. corrective action (Zero Disch.) to end of period. Upper Le 99 99 Lower Level Limit Decision Di followed at max. corrective action ( Max. Disch.) to end of period. Cumu loti ve H — Probability 00. % Decision Di followed at max. co rre c ti v e actionlZero Disch.) to end of period. Figure 4 • I ASSESSMENT OF A SINGLE DECISION O Lake LeveI F i g u r e 4 2 C O M P A R I S O N OF DECISION ALTERNATIVES. CHAPTER V GENERATION AND SIMULATION METHODS There are no u n i v e r s a l generating techniques a v a i l a b l e for produc-ing appropriate synthetic hydrologic events following a forecast. The out-comes following a forecast and t h e i r p r o b a b i l i t i e s are dependent upon the nature of the quantity being forecasted, the type of p r e d i c t i v e model, measurement errors i n the model v a r i a b l e s , and numerous other f a c t o r s . The generation method developed for the purposes of this study i s con-sidered s a t i s f a c t o r y . It could provide the basis for a more precise method but a more rigorous approach was beyond the scope of this study and i n any event not warranted by the accuracy of the o r i g i n a l data. Monthly Inflow Generation In order to assess an operational decision i t i s subjected to a large number of seasonal sets of monthly inflows which might occur follow-ing a forecast of given accuracy. These are generated from the forecast information i n two stagesi Generation of a synthetic " a c t u a l " t o t a l inflow. The forecasting method for Okanagan Lake i s currently based on a p r e d i c t i v e equation ob-tained by multiple l i n e a r regression [3J. This y i e l d s a t o t a l inflow pre-d i c t i o n and a value of the Standard Error of Estimate. It i s assumed at t h i s point that the p r e d i c t i v e model i s of the correct form and that the forecast error i s normally d i s t r i b u t e d with mean zero and standard devia-22 23 t i o n equal to the standard error of estimate { 6 J . These assumptions are d i f f i c u l t to j u s t i f y i n the absence of a long h i s t o r i c record of the forecast performance but are the best which can be made with a v a i l a b l e information. Since there w i l l undoubtedly be s i g n i f i c a n t measurement error and interdependency i n the "independent" variables of the multiple l i n e a r regression, there i s no r e a d i l y a v a i l a b l e technique to e s t a b l i s h the true nature and probable magnitude of the forecast error. On the basis of these assumptions a series of synthetic " a c t u a l " inflows which might follow a forecast of given value and standard error of estimate are generated from the following equation: T i = T F + S F " fci ( 5 < 1 ) where T. = an "a c t u a l " t o t a l inflow and i = 1, . . . n, where n i s the t o t a l number of "actual" inflows to be generated. T = forecast t o t a l inflow. F s = standard error of estimate of the forecast, r t. = random normal deviate, mean zero, variance 1.0. l Both the forecast t o t a l inflow and the synthetic " a c t u a l " t o t a l inflows are f o r the period commencing i n the current decision month and termina-t i n g at the end of the runoff period. Generation of a Set of Monthly Inflows From a Synthetic "Actual" T o t a l Inflow. The generation procedure adopted i s s i m i l a r to that o r i g i -24 n a l l y proposed by Thomas and F i e r i n g [4], but the simple s e r i a l c o r r e l a t i o n component i s replaced by a simple l i n e a r regression component r e l a t i n g the inflow of a p a r t i c u l a r month to the t o t a l inflow from the beginning of that month through, to the end of the runoff period. The Thomas and F i e r i n g generation equation can be stated i n general terms as: y s = Y.+ b(x-X) + s.tCl-R2) (5.3) together with the simple l i n e a r regression r e l a t i o n s h i p : y = Y + b(x-X) (5.4) based upon h i s t o r i c values of the dependent va r i a b l e y and independent v a r i a b l e x, where Ys = a synthetic value of y Y = mean of the recorded y b = regression c o e f f i c i e n t R = c o r r e l a t i o n c o e f f i c i e n t from the simple l i n e a r regression X = mean of the recorded x values s = standard deviation of the recorded y values. t = a unit random normal deviate Regardless of the quantities represented by x and y, i t can be shown that the above generation equation w i l l produce a series of synthetic y values with the same mean and variance as the recorded y. 25 From Equation C5.3) E(y s) = ECY) + E[bCx-X)J + E [ s . t C l - R 2 ) i 2 J where E( ) i s the expected value. But Y, b, s, and R are a l l constants; also: E(x-X) = 0 E(t) = 0 therefore E ( y s ) = Y Equation (5.3) i s a l i n e a r combination of the two independent random var-iables x and t [cov(x, t) =0 ], hence: v ( y ) = b 2 v ( x - x ) + s 2 ( i - R 2 ) V ( t ) where V( ) i s the variance. But t i s a unit random normal deviate; therefore V(t) = 1.0 V(y s) = b2V(.x-X) + s 2 ( l - R 2 ) C5.5) Let SS^ = corrected sum of squares of recorded x values. Then: SS V(x-X) = — n 26 where n = number of recorded values. Let SS = corrected sum of squares of recorded y values. Then: y 2 y n From simple l i n e a r regression theory: 9 b 2SS R ~ SS y Therefore, from Equation (5.5), SS SS SS V(y s) = b 2 ^ + ^ (1 - b 2 ^ ) SS y z n 2 = s i . e . , the variance of the recorded y and the standard deviation of y = s. J s Thus the mean and standard deviation of the h i s t o r i c y values w i l l be preserved i n a generation of the form of Equation (5.3) [5] provided that the standard deviation of the independent (x) v a r i a b l e i n the generation i s the same as the standard deviation of the h i s t o r i c values of x used i n obtaining the regression and c o r r e l a t i o n c o e f f i c i e n t s . Thomas and F i e r i n g used Equation (5.3) to produce a series of consecutive synthetic monthly values. The independent v a r i a b l e of the regression r e l a t i o n s h i p Equation (5.4) was the previous month's h i s t o r i c inflow and i n the generation was the previous month's synthetic inflow. The c y l i c a l nature of the continuous generation provides the r e q u i s i t e 27 h i s t o r i c standard deviation i n the independent v a r i a b l e of the generation equation and hence reproduces the h i s t o r i c standard deviation i n the de-pendent or synthesized v a r i a b l e . This study required r e a l i s t i c sets of synthetic monthly inflows with the sum of these inflows s p e c i f i e d by an 'actual' t o t a l . The form of the monthly inflow generation equation based upon t o t a l " a c t u a l " inflow i s : i ! = I. +b.. [ CT - ^ E i ' ) - T ] +s.- t (l-R 2. ) ^ (5.7) 3 3 j k a n=l n J J J Subscript i indicates the current decision month; k indicates the l a s t month of the. runoff period; j = i , . . . , k I = synthetic inflow f o r the j t h month I. = mean h i s t o r i c inflow for the i t h month J b = regression c o e f f i c i e n t j t h month to t o t a l inflow ^ from j t h to kth month R , = c o r r e l a t i o n c o e f f i c i e n t i t h month to t o t a l inflow i k J from j t h to kth month T = t o t a l " a c t u a l " inflow from i t h to kth month a T., = mean h i s t o r i c t o t a l inflow from i t h to kth month jk s = h i s t o r i c standard deviation of i t h month 3 t = unit random normal deviate A value f o r T i s obtained from Equation (5.1). The constraint a that the sum of the generated monthly inflows should be equal to T i s 2 8 met as the r e g r e s s i o n and c o r r e l a t i o n c o e f f i c i e n t s f o r the l a s t (kth) month, a n d a r e D O t a 1-0. Thus f o r the l a s t month the generation equation becomes: I, = I + (T - E I ) - T.. k k a n kk n=i But the mean h i s t o r i c t o t a l i n f l o w from the k t h to the k t h month T, , i s kk equal to the mean h i s t o r i c i n f l o w f o r the k t h month I ; t h e r e f o r e : K. k - i , I = T - E I k a n n=i Hence: k - 1 , T = E I + I, a n k n=l k , = E I i . e . , the sum of the monthly i n f l o w s . n=i A requirement of the generation based on t o t a l i n f l o w i s that the monthly i n f l o w s generated over a s p e c i f i e d p e r i o d from the complete set of h i s t o r i c i n f l o w s f o r the same period should have the same mean and standard d e v i a t i o n as the h i s t o r i c monthly values. Equation (5.5) can be used to demonstrate that t h i s i s always the case f o r the f i r s t month of the p e r i o d being generated: 29 V ( y . ) = b 2 V(x-X) + s 2 (1-R 2) s i 1 1 1 can be rewritten: V ( y s l ) = b 2 V(x) + (1-R*) VCY L) but from simple l i n e a r regression theory: „ b 2SS b 2V(X) R = X SS y V(Y) therefore: vCysl) = VCY 1) + b2 Lvcx) - vcx)j When V(x) = V(X), i . e . , when the variance of the independent v a r i a b l e of the generation i s equal to i t s h i s t o r i c variance, then the second term goes to zero and: VCy s l) = VCY X) hence, the standard deviations of y .. and Y, are the same. s i 1 The expression for the second month's variance reveals that i n t e r -dependence ex i s t s between successive monthly generations. 30 v(y s 2) = V C Y 2 ) + b 2 [yCx-y s l) - V ( X - Y 1 ) J and V ( y s 3 ) = V C Y 3 ) + b2 { V ( x - y s 2 ) - V ( X - Y 1 - Y 2 ) J , etc. As a r e s u l t of the dependency of the x and Ys-^> ^ s 2' " ' ' e t c ' e x P a n s i ° n of the variance terms w i l l introduce covariances. While i t i s possible to express these covariances i n terms of the various c o e f f i c i e n t s and s t a t i s t i c a l parameters for the generated variables these would have to be equated with values of covariance f o r the h i s t o r i c variables X , Y ^ , etc. As an a l t e r n a t i v e to this involved procedure a series of generation runs on the computer for various periods based upon the h i s t o r i c period t o t a l s confirmed that the h i s t o r i c mean and standard deviation were being maintained i n the generated monthly values. In the proposed a p p l i c a t i o n of the generation equation, the mean and standard deviation of the t o t a l inflows would be determined by the forecast rather than the h i s t o r i c data. The e f f e c t of th i s change on the mean and variance of the generated values w i l l be: E(y ) = Y + b .• E(x - X ) s = Y + b ( x - X ) and VCy ) = V ( Y ) + b 2 [ V(x) - V C X ) J 31 Thus the mean w i l l be s h i f t e d by an amount which i s dependent upon the deviation of the forecast t o t a l inflow value from the h i s t o r i c mean and the l i n e a r dependency of the monthly inflow on the t o t a l inflow. Sim-i l a r l y , the variance w i l l be reduced by an amount dependent on the reduc-t i o n i n variance of the t o t a l inflows following the forecast from the h i s t o r i c variance and the l i n e a r dependency of the monthly inflow on t o t a l inflow. Simulation of Lake Level Response A simple continuity model was used to determine the e f f e c t of i n -flows, compulsory demands, and discharge decisions, on the Lake l e v e l . 3 1 3 ° A 3 J 3 where: E . , = Lake elevation at end of i t h month E^Q = Lake elevation at beginning of j t h month A = Surface area of Lake i I = Synthetic inflow for jth. month 0 = Compulsory demand for j t h month D = Discharge decision during j t h month. This model assumes that the Lake area remains constant over the range of 32 le v e l s considered. This i s v i r t u a l l y true for Okanagan Lake over the r e l a t i v e l y narrow operating range of 4 f t . CHAPTER VI COMPUTATIONAL PROCEDURE FOR DECISION ASSESSMENT Data Analysis for Synthetic Inflow Generation H i s t o r i c records of net monthly inflows f o r Okanagan Lake extend-ing over a period of 48 years were analyzed to obtain: Ca) Mean and standard deviation f o r each month's inflow; Cb) T o t a l inflows for periods commencing i n each month and terminating at the end of the runoff season, i . e . , July 31; (c) Simple l i n e a r regression and c o r r e l a t i o n c o e f f i c i e n t s for each month's inflow r e l a t e d to the t o t a l inflow values obtained i n (b). The above, and following computational steps are i d e n t i f i e d i n the computer program shown i n Appendix I I . Generation of Sets of Monthly Inflows from the Forecast The monthly inflow sets generated commence i n the decision month and terminate i n the l a s t month of the runoff season. (a) Given values of the inflow forecast and standard error of estimate are inserted i n Equation (5.1) to obtain a synthetic " a c t u a l " inflow t o t a l . A value of the random normal deviate i s obtained from a standard computer sub-program. Cb) The synthetic " a c t u a l " inflow t o t a l , appropriate s t a t i s t i c a l parameters, and c o e f f i c i e n t s are used i n Equation (5.7) to obtain a synthetic value f o r the inflow during the current decision month. This value i s stored. 33 34 (c) The above generated inflow i s subtracted from the t o t a l " a c t u a l " inflow to obtain a revised t o t a l i n -flow which i s then used to generate an inflow f o r the second month, again using Equation (5.7) and the appropriate parameters and c o e f f i c i e n t s f o r the second month. (d) The above procedure (c) i s repeated u n t i l monthly i n -flow values have been generated for each month i n the runoff period. Procedure (a), (b),(c) above w i l l produce one set of synthetic monthly i n -flows. When th i s i s repeated (a) w i l l produce a d i f f e r e n t value of syn-t h e t i c " a c t u a l " t o t a l inflow and hence a new set of monthly inflow values. Decision Evaluation The Lake elevation at the beginning of the decision month i s taken as datum l e v e l (E.„ = 0 . 0 ) . JO a) Maximum l e v e l evaluation. The values of the synthetic inflow, demand and a discharge decision f o r the f i r s t month of the period are inserted i n Equation (5.8) to obtain the Lake l e v e l at the end of the f i r s t month. The l e v e l at the end of the f i r s t month becomes the l e v e l at the beginning of the second month of the period and the process i s repeated but with the decision discharge equal to maximum discharge ( i . e . , maximum cor r e c t i v e discharge with respect to maximum le v e l s ) for a l l months remaining to the end of the period. The maximum l e v e l of the period i s stored. b) Minimum l e v e l evaluation. The computational procedure i s s i m i l a r to (a) with the exception that a minimum (zero) discharge decision i s applied to the months following the decision month. The minimum l e v e l 35 occurring over the period i s stored. c) Terminal l e v e l evaluation. The Lake l e v e l at the end of the runoff period obtained i n (b) above i s stored as terminal l e v e l . This completes the analysis of one set of monthly inflows. The program returns to step (a) [under the Generation of Sets of Monthly In-flows from the Forecast section] and the procedure i s repeated for a number of seasonal periods with the same current month discharge decision. Output The maximum,- minimum, and terminal l e v e l values obtained from each runoff period simulation are each analyzed to produce cumulative frequency values at i n t e r v a l s of 0.5 f t . over the range of Lake l e v e l s 5 f t . above and below the datum l e v e l . This data i s then plotted i n the form of cumulative d i s t r i b u t i o n curves f o r maximum, minimum and terminal l e v e l s . These r e s u l t s being associated with a sing l e discharge decision. Evaluation of a l l ' Current Discharge Decision P o s s i b i l i t i e s The decision evaluation described i n the preceding sections i s repeated with other possible discharge decisions i n the current month. In p r a c t i c e i t i s only necessary to perform the decision evaluation for the maximum, minimum, and one or two intermediate discharge decisions to provide adequate information on the f u l l range of operational p o s s i -b i l i t i e s . In the computer program shown in.Appendix I I , two current d i s -charge decisions (one of which i s a zero discharge) and the three l e v e l values are analyzed concurrently. CHAPTER VII RESULTS AND INTERPRETATION Analysis of Data A c e r t a i n amount of data analysis was necessary to e s t a b l i s h the basic s t a t i s t i c a l parameters and regression c o e f f i c i e n t s required i n the generation. A d d i t i o n a l information on simple s e r i a l c o r r e l a t i o n (one month lag) was obtained f o r comparative purposes. Table VII.1 gives: - means and standard deviations of h i s t o r i c monthly inflows; - means of h i s t o r i c inflow t o t a l s f o r each month through to the end of July; - regression and c o r r e l a t i o n c o e f f i c i e n t s f o r the simple l i n e a r regression r e l a t i n g the inflow of each month to the inflow t o t a l to the end of July; - c o r r e l a t i o n c o e f f i c i e n t s f o r the simple s e r i a l c o r r e l a t i o n with one month lag. Figures 7.1, 7.2, 7.3 and 7.4 show: - cumulative d i s t r i b u t i o n s of A p r i l , May, June and July h i s t o r i c inflows with normal cumulative d i s t r i b u t i o n of the same mean and standard deviation superimposed. Figure 7.5 shows: - cumulative d i s t r i b u t i o n of h i s t o r i c inflow t o t a l s f o r the period A p r i l to July with a normal cumulative d i s -t r i b u t i o n of the same mean and standard deviation super-imposed . 36 TABLE VII..1 HISTORIC MONTHLY INFLOW ANALYSIS AUG SEPT OCT NOV DEC JAN FEB MAR APR MAY JUNE JULY Average Inflow -4.4 -10.2 -1.3 4.3 7.5 6.4 7.3 14.7 56.3 193.5 113.4 13.6 Standard Deviation 17.2 16.1 14.8 11.8 10.8 10.3 8.4 8.5 34.8 81.1 56.8 23.4 Average Inflow to July 405.2 398.8 391.5 376.8 320.5 127.0 13.6 Simple Regression C o e f f i c i e n t .01 .02 .02 .09 .54 75 1.00 R Correlation C o e f f i c i e n t ,21 ,40 .32 .34 .87 .96 1.00 R Simple S e r i a l Correlation C o e f f i c i e n t (One month lag) .04 -.19 .06 .42 .29 ,43 .57 00 ON 37 Figure 7.6 shows: - cumulative d i s t r i b u t i o n s of synthetic May inflows generated from t y p i c a l low, medium and high fore-cast inflows for the period A p r i l to July . The cumulative d i s t r i b u t i o n for the h i s t o r i c May i n -flows i s superimposed. Figures 7.1, 7.2, 7.3, 7.4 and 7.5 show that the h i s t o r i c monthly and t o t a l inflows over the c r i t i c a l runoff period have d i s t r i -butions which may be considered to approximate to normal d i s t r i b u t i o n s with the same mean and standard deviation. Thus the generation methods used, which produce normally d i s t r i b u t e d values, w i l l produce synthetic values with cumulative d i s t r i b u t i o n s s i m i l a r to the h i s t o r i c records provided the means and standard deviations are maintained. Figure 7.6 shows that synthetic inflows generated for a p a r t i c u -l a r month, i n t h i s case May, on the basis of a range of forecast t o t a l inflows with t y p i c a l standard error of estimates, does not produce monthly inflow values which are s i g n i f i c a n t l y outside the h i s t o r i c range. 1 0 0 2 5 0 In f low in thousand acre ft Figure 7 • I C U M U L A T I V E D I S T R I B U T I O N O F H I S T O R I C M O N T H L Y I N F L O W S FOR APRIL. Figure 7-2 C U M U L A T I V E D I S T R I B U T I O N O F H I S T O R I C M O N T H L Y I N F L O W S FOR MAY. Figure 7-3 o CUMULATIVE DISTRIBUTION OF HISTORIC MONTHLY INFLOWS FOR JUNE. 1 0 0 8 0 6 0 « 4 0 > a E U 2 0 0 / // / / ^ — N o r n i a l ( 1 3 6 , 2 3 -// // Mean Standa 1 3 - 6 thousa rd Deviat ion nd acre ft. 2 3 - 4 thous and acre f t . II // - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Inf low in thousand acre ft. F i g u r e 7 4 C U M U L A T I V E D I S T R I B U T I O N O F H I S T O R I C M O N T H L Y I N F L O W S FOR J U L Y . Figure 7 - 5 C U M U L A T I V E D I S T R I B U T I O N O F H I S T O R I C T O T A L I N F L O W S F R O M A P R I L TO J U L Y . F S 4 orecast - 2 0 0 tandard Error 0 , 0 0 0 acre OOOacre ft. -of Forecast Ft. Forec Stand Forec as t = 4 0 0 , O C ard Error of ast 8 0 , 0 0 0 a f S \ui<-tnr i c s / 1 / 1 / Va lue s . t • M / / ' /Foreca Standa 120 ,00 st = 6 0 0 , 0 0 0 rd Error of 0 acre ft. acre ft . Fore cast 9 / • / / / 4 / / -/ / - 5 0 5 0 100 150 2 0 0 2 5 0 3 0 0 Inflow in thousand acre ft. 3 5 0 4 0 0 4 5 0 Figure 7 - 6 C U M U L A T I V E D I S T R I B U T I O N O F S Y N T H E T I C M A Y I N F L O W S F O R V A R I O U S F O R E C A S T E D T O T A L I N F L O W S F R O M A P R I L T O J U L Y . 44 Decision Assessment The decision assessment method was applied to a number of hypo-t h e t i c a l operational s i t u a t i o n s and subsequently to the actual conditions experienced during the 1970 runoff season. The res u l t s obtained from three of the hypothetical s i t u a t i o n s are used to demonstrate the e f f e c t of time and forecast accuracy on the decision making s i t u a t i o n . The basic data described i n Chapter II was used and i t was assumed that the Lake l e v e l was 2 f t . below the upper l i m i t at the beginning of the decision month. Figure 7.7 shows the s i t u a t i o n for the decision month of February with a forecast of 400,000 acre f t . and a forecast standard error of estimate of 160,000 acre f t . This low forecast accuracy i s chosen to r e f l e c t the d i f f i c u l t y i n obtaining a good forecast i n mid-winter before accurate snowpack data i s a v a i l a b l e . Decision A represents the minimum discharge decision i n February and Decision B the maximum. The upper and lower l e v e l l i m i t s and terminal l e v e l goal have been super-imposed to f a c i l i t a t e i n t e r p r e t a t i o n . Low p r o b a b i l i t i e s a, a', b, b' are indicated for both maximum and minimum decisions and the commitment r i s k associated with either of these extreme decisions i s low with respect to the upper and lower l i m i t s . The p r o b a b i l i t y of achieving the desired terminal l e v e l i s shown to be s i g -n i f i c a n t l y reduced from 81 per cent to 61 per cent i f a maximum discharge d e c i s i o n i s made i n February. This p r o b a b i l i t y exists i n sp i t e of taking f u l l c o r r e c t i v e action i n subsequent months. Under these circumstances there i s l i t t l e or no c o n f l i c t between the l e v e l l i m i t and terminal goal c r i t e r i a and a minimum discharge decision i s appropriate. 45 Figure 7.8 indicates the s i t u a t i o n i n A p r i l with no change i n the forecast inflow; t h i s w i l l often occur i n p r a c t i c e when no s i g n i f i c a n t runoff occurs i n February or March, but with the standard error of e s t i -mate reduced to 80,000 acre f t . r e f l e c t i n g the improved forecasting accuracy as the season progresses. The p r o b a b i l i t y of f a l l i n g below the lower Lake l e v e l i s n e g l i g i b l e for both extreme decisions (and there-fore a l l intermediate decisions) and attention may be concentrated on the maximum and terminal l e v e l c r i t e r i a . Decision A i s the most desirable f o r achieving the terminal l e v e l goal, but Figure 7.8 shows that i t would r e s u l t i n a r e l a t i v e l y high (30 per cent) p r o b a b i l i t y that the upper l e v e l l i m i t would be exceeded, t h i s again i n s p i t e of f u l l c o r r e c t i v e maximum discharge i n a l l subsequent months. An intermediate decision must now be considered. While increas-ing discharge decisions i n A p r i l w i l l r e s u l t i n a reduction i n p r o b a b i l i t y of exceeding the upper l e v e l l i m i t i t w i l l at the same time reduce the p r o b a b i l i t y of achieving the desired terminal l e v e l . Assessment of i n t e r -mediate decisions i s f a c i l i t a t e d by c a l i b r a t i o n between the extreme d e c i -sion l i n e s ; however, t h i s was found to be e s s e n t i a l l y l i n e a r u n t i l the de-c i s i o n month of June. The decision make must now make his f i n a l decision on the basis of a compromise and may introduce other factors such as past operational experience, information not included i n th i s a n a l y s i s , and so on. Figure 7.9 shows the e f f e c t of a further improvement i n forecast accuracy i n A p r i l . The increased h o r i z o n t a l separation of the pair s of decision l i n e s f o r maximum and minimum A p r i l discharges indicates that the 46 range of decisions has a greater apparent e f f e c t on the outcome p r o b a b i l i -t i e s and the reduced slope r e f l e c t s the reduced range of possible Lake l e v e l outcomes. Hie more accurate forecast e s s e n t i a l l y increases the constraint on possible future inflows and there i s consequently l e s s uncertainty about the consequences of the various possible discharges. The above r e s u l t s are shown i n tabular form i n Table VII.2. 4 0 r 3 0 2 0 0 0 - 1 0 •20 - 3 0 4 0 Upper Limit of Lake Level Desired Terminal Le ve I Lake Level Ist February 9999 999 99 Lower Limit of Lake Level J/o Pro ba bility Decision month •  FEBRUARY Period evaluated 1 l s ,February to 31 s t Ju I y Forecast : Season Inflow 4 0 0 , 0 0 0 acre ft. Standard Error of Estimate : l60,000acre ft 1 X • Figure 7 7 D E C I S I O N A S S E S S M E N T O F F E B R U A R Y 001 3 0 -Decision month : APRIL Period evaluated = Ist APRIL to 3 l s t JULY Forecast = Season Inflow 400,000acre ft. Standard Error of Estimate : 80 ,000acre ft. F i g u r e 7 - 8 4 0 L D E C I S I O N A S S E S S M E N T F O R A P R I L . 4 0 s . E £!f 'ON "a" Decision month : APRIL Period evaluated : Ist APRIL to 3 l s f JULY Forecast •  Season inflow 4 0 0 , 0 0 0 acre ft. Standard Error of Estimate 4 0 , 0 0 0 a c r e ft. i l l I i I I I F i g u r e 7 - 9 D E C I S I O N A S S E S S M E N T F O R A P R I L W I T H I M R O V E D F O R E C A S T A C C U R A C Y 50 TABLE VII.2 TYPICAL RESULTS DISCHARGE DECISION MAXIMUM MINIMUM 0.6 % 2.5 % 61.0 % 3.0 % 0.4 % 81.0 % Fi g . 7.7 Decision Month February Forecast 400,000 acre f t . Std. Error 160,000 acre f t . P r o b a b i l i t y of: exceeding upper l i m i t exceeding lower l i m i t achieving terminal l e v e l F i g . 7.8 Decision Month A p r i l Forecast 400,000 acre f t . Std. Error 160,000 acre f t . P r o b a b i l i t y of: exceeding upper l i m i t exceeding lower l i m i t achieving terminal l e v e l 1.0 % 30.0 % .09 % < .01 % 67.0 % 96.0 % Fi g . 7.9 Decision Month A p r i l Forecast 400,000 acre f t . Std. Error 40,000 acre f t . P r o b a b i l i t y of: exceeding exceeding achieving upper l i m i t lower l i m i t terminal l e v e l .05 % < .01 % 97.0 % 20.0 •% < .01 % >99.99 % CHAPTER VIII SUMMARY The problem of operating a multi-purpose res e r v o i r with the assistance of a t o t a l future inflow forecast has been considered. Stan-dard Operations Research techniques, and p a r t i c u l a r l y optimization methods, were not found to be applicable to the operational problem on an e x i s t i n g r e s e r v o i r due to the lack of suitable data. Investigation of the decision making process indicated that i t consisted of a se r i e s of e s s e n t i a l l y self-contained i n d i v i d u a l decisions which i d e a l l y gave the operator the greatest a b i l i t y to e f f e c t future c o r r e c t i v e action and hence achieve future goals. A method of decision assessment which overcame the data deficiency and rel a t e d c l o s e l y to the actual decision process was developed. The assessment i s presented i n the form of p r o b a b i l i t i e s of exceeding Lake l e v e l l i m i t s and achieving Lake storage goals at the end of the runoff season f o r a l l possible current decisions. The analysis considers forecast inflow and accuracy and correc-t i v e actions which may be taken i n the future to overcome forecast and de-c i s i o n e rrors. The assessment does not y i e l d an e x p l i c i t optimal decision but provides information on the current operational s i t u a t i o n . The opera-tor can then proceed to make h i s decision i n the l i g h t of th i s information, hi s past experience, and any other considerations he may f e e l applicable. The structure of the proposed method i s f l e x i b l e and could be adapted to many operational s i t u a t i o n s where some kind of forecast i s involved. The 51 52 incorporation of forecast information into the simulation and analysis of operational s i t u a t i o n s also makes i t possible to demonstrate the value of the forecast and i t s accuracy. B I B L I O G R A P H Y Present, Future and Ultimate Water Requirements in the South Thompson Watershed and their Effect in Combination With the Shuswap River-Okanagan Lake Water Supply Canal Division [Scheme 3). V i c t o r i a , B.C.: Water Resources Service, Department of Lands, Forests and Water Resources, July 1968. Present (.1966) and Future Water Requirements in Okanagan and North Okanagan. V i c t o r i a , B.C.: Water Resources Service, Department of Lands, Forests and Water Resources. Private correspondence with Mr. H. I. Hunter, Chief Hydrologist B.C. Water Resources Service, Department of Lands, Forests and Water Resources, V i c t o r i a , B.C. Mass, Arthur, Maynard M. Hufschmidt, Robert Dorfman, Harold A. Thomas, J r . , Stephen A. Marglin, Gordon Maskew F a i r . Design of Water Resources Systems. Cambridge, Mass.: Harvard University Press, 1962. F i e r i n g , M. B. Streamflow Synthesis. London: Macmillan, 1967 Draper, N. R. and H. Smith. Applied Regression Analysis. New York: John Wiley and Sons, 1966. 53 APPENDIX I TABLE OF MONTHLY VIRGIN INFLOWS IN THOUSAND ACRE FEET OKANAGAN LAKE BASIN PERIOD: CLIMATIC YEARS 1921 - 1968 (48 YEARS) CL. YEAR APRIL MAY JUNE JULY AUG SEPT OCT NOV DEC JAN FEB MAR TOTAL 1921 31.8 226.8 150.5 -12.0 -22.9 - 7.4 1.3 3.9 8.9 - 5.4 13.4 396.0 1922 29.9 165.1 128.7 -10.0 - 8.6 - 3.3 r 0.4 4.4 4.1 6.6 - 1.0 25.3 332.8 1923 54.4 187.9 181.8 19.5 - 6.1 -23.9 - 5.9 - 5.4 11.4 -19.7 14.8 15.9 424.4 1924 16.1 137.2 31.2 -20.3 -16.3 -28.3 - 2.9 5.3 3.6 11.2 10.3 8.0 155.2 1925 63.0 190.3 65.3 -17.2 -26.8 -30.8 - 8.5 -10.2 10.0 21.0 - 3.4 11.1 263.8 1926 57.2 97.3 5.7 -15.4 -22.2 -22.3 - 1.1 1.9 - 2.3 10.3 - 7.3 - 1.2 100.7 1927 24.1 124.6 132.9 3.4 - 5.4 :.41.2 59.4 24.7 27.2 12.9 8.7 25.0 478.7 1928 83.2 402.2 140.9 56.6 -17.0 -21.2 - 8.9 31.3 -21.7 0.2 - 7.4 14.1 652.4 1929 16.3 52.8 62.8 -16.8 7.5 22.1 -10.6 -12.4 - 9.0 - 0.6 3.6 7.9 123.6 1930 51.6 55.3 52.7 - 9.2 - 6.9 -15.0 -19.8 • 0.2 -17.4 3.6 - 4.8 7.9 98.2 1931 23.1 84.1 26.6 1.9 -27.8 -18.3 -14.8 1.1 3.6 - 7.3 - 6.5 30.7 96.2 1932 68.9 209.2 113.9 - 0.7 1.3 -14.8 -10.0 21.9 0.0 4.2 4.2 14.0 412.2 1933 48.9 201.3. 180.8 27.3 - 9.4 3.6 20.2 14.3 33.7 25.2 9.2 36.6 591.6 1934 223.4 148.1 29.6 3.0 - 0.9 -12.5 - 6.9 22.7 16.0 13.5 14.3 29.9 480.0 1935 34.9 216.9 138.6 92.4 22.1 - 5.9 - 7.8 2.4 6.6 26.0 - 5.2 13.0 534.0 1936 93.0 208.5 100.0 12.5 -10.7 -16.1 -11.2 -22.0 15.9 - 6.0 15.0 13.8 392.8 1937 37.9 174.3 162.2 11.2 -14.6 -12.8 - 3.0 21.7 8.2 6.5 9.0 18.0 418.5 1938 66.5 204.7 67.1 - 3.0 -14.4 - 7.8 -11.4 -12.4 6.9 11.1 - 6.6 13.2 313.9 1939 50.5 136.6 63.7 0.4 -26.0 -18.6 -15.7 - 1.1 11.5 - 0.3 1.4 24.5 226.7 1940 55.5 132.3 30.7 -13.9 -27.7 -12.3 -11.9 - 6.8 1.6 11.7 - 0.1 12.8 171.9 1941 82.0 98.7 69.4 10.7 - 5.0 16.0 30.2 17.6 23.5 - 4.3 11.7 2.9 353.5 1942 84.0 235.3 167.8 49.8 -11.5 -23.3 -17.6 - 3.8 15.6 - 2.1 11.4 2.7 508.5 1943 58.8 128.8 104.0 7.1 -14.4 -28.4 2.5 - 8.9 - 9.7. 2.1 14.7 -4.4 252.1 1944 33.0 117.8 107.1 3.0 - 8.3 - 6.2 4.5 12.7 3.5 12.7 13.6 6.4 300.1 1945 27.1 269.0 139.7 0.4 -13.8 -18.5 15.2 2.1 13.1 15.6 6.3 16.1 472.2 1946 91.9 346.6 163.4 9.6 - 2.7 -22.7 -15.2 7.2 - 4.6 0.4 8.8 18.6 601.2 1947 50.7 104.3 58.2 - 1.9 - 9.8 -10.3 6.1 8.9 - 0.4 5.5 3.0 3.3 217.6 1948 56.8 368.0 208.2 38.4 66.4 6.7 15.1 - 5.5 6.3 - 4.6 17.3 23.7 796.5 Ln APPENDIX I (continued) CL. YEAR APRIL MAY JUNE JULY AUG SEPT OCT NOV DEC JAN FEB MAR TOTAL 1949 101.4 272.1 61.0 18.3 9.1 -16.9 -11.0 3.7 27.3 -13.1 8.7 21.1 481.9 1950 39.9 208.9 216.8 17.4 11.7 -25.3 5.0 8.7 16.3 - 4.7 30.6 22.0 547.5 1951 99.7 346.2 114.0 28.4 -15.3 - 5.1 10.0 3.7 14.6 - 3.9 18.8 14.4 625.7 1952 106.4 279.7 90.5 25.0 -10.2 -16.9 -11.0 -10.6 1.2 23.1 10.4 4.3 492.1 1953 31.2 163.5 139.5 18.0 16.5 -25.1 - 0.8 11.4 11.4 6.3 11.4 11.9 395.2 1954 20.5 245.1 162.9 67.1 26.0 19.3 5.0 29.8 10.4 8.7 7.9 14.4 617.3 1955 33.2 130.6 216.8 68.8 - 4.3 - 5.9 1.6 0.3 20.5 17.2 1.2 26.2 506.4 1956 93.0 296.5 145.2 20.8 2.4 -16.0 - 3.5 - 1.4 10.4 1.2 12.1 15.2 576.1 1957 42.4 304.1 71.1 3.9 20.1 - 4.3 - 4.3 2.0 6.2 20.5 21.4 21.1 504.4 1958 54.2 241.7 68.5 1.3 -22.0 -15.2 6.6 4.6 17.3 18.1 7.2 20.3 402.3 1959 52.7 287.0 204.5 20.0 -10.9 28.5 23.9 18.8 17.4 13.2 19.4 10.5 685.0 1960 71.2 178.4 107.3 - 1.1 -14.2 - 7.9 -10.3 5.4 - 2.2 5.0 9.9 12.6 353.8 1961 34.4 227.9 99.2 6.1 -13.6 -43.1 -12.9 -12.0 13.5 3.9 12.7 6.2 322.3 1962 54.8 135.4 99.5 12.8 0.3 -22.5 10.3 2.2 - 1.6 0.6 3.5 18.4 313.6 1963 34.8 118.0 52.2 22.5 2.4 - 5.1 -17.1 6.1 8.0 8.8 0.9 12.6 244.0 1964 25.6 158.5 220.5 41.4 20.8 10.7 18.3 19.7 - 5.2 29.8 10.0 9.7 559.8 1965 84.0 205.9 120.2 8.9 12.6 - 9.2 - 1.2 3.7 1.4 0.4 3.5 9.1 439.3 1966 36. 7 120.6 58.2 16.7 -12.4 -10.7 -19.2 4.3 11.4 8.8 4.3 5.9 224.7 1967 24.1 166.5 171.6 - 8.1 -27.0 -23.1 -11.5 - 8.7 8.7 - 6.9 14.8 23.5 324.1 1968 21.4 210.0 177.7 12.4 12.0 - 2.3 - 7.3 7.1 10.0 8.0 1.4 20.9 471.3 U l APPENDIX II COMPUTER PROGRAM FOR OPERATIONAL DECISION ASSESSMENT (See Chapter VI) $COMPILE INTEGER M, Y, MM1, MMMl, YY, YYY INTEGER YX, • MX DIMENSION QO-2,26), QAVE(12), BC12), R(12), S(12) DIMENSION EL(4), EMAX(4), EMIN(4), ETER(4), SMAX(4,20) DIMENSION SMIN(4,20), STERC4,20), DEMC12) DIMENSION QQ(12,501), QSAV(12) DIMENSION SSMAX(4,20), SSMTN(4,20), SSTER(4,20) WRITE (6,4) 4 FORMAT (4OX,'OKANAGAN LAKE - SITUATION 1st MAY, 1970') C SPECIFY DECISION MAKING MONTH . MONTH = 5 WRITE (6,10) MONTH 10 FORMAT (25X, 'DECISION MONTH', IX, 13) C SPECIFY FORECAST TOTAL INFLOW TO JULY 31st (TE) TF = 297.0 WRITE (6,11) TF 11 FORMAT (15X, 'FORECAST TOTAL INFLOW TO JULY', F10.1) C SPECIFY FORECAST STD ERROR AS A FRACTION OF TF FSERR = 0.2*TF WRITE (6,12) FSERR 12 FORMAT ('FORECAST STD. ERROR', F10.1, 'THOU. A.FT.') C SPECIFY MAXIMUM DISCHARGE DECISION DURING DECISION MONTH DECIS =45.0 WRITE (6,13) DECIS-13 FORMAT ('MAXIMUM DISCH. DECISION DURING DECISION MONTH' F10.1) C SPECIFY MAXIMUM DISCHARGE CAPACITY IN THOU. A.FT./MONTH SPILMAX =90.0 WRITE (6,14) SPILMAX 14 FORMAT ('MAXIMUM DISCHARGE CAPACITY', F10.1) C SPECIFY COMPULSORY DEMANDS FOR EACH MONTH DEM(3) =9.48 DEM(4) =9.60 DEM(5) =15.00 DEM(6) =18.9 DEM(7) =18.96 AREA = 84.2 (See Chapter VI, Section: Data Analysis for Synthetic Inflow 5 READ (5,6) ((Q(M,Y), M =1,12), Y = 1,26) Generation) 6 FORMAT (12F9.1) D = 25.0 56 5 7 A P P E N D I X I I (Continued) 2 0 DO 2 5 M = 1 , 1 2 Q A V E CM) = 0 . 0 DO 2 4 Y = 1 , 2 5 Q A V E CM) = A Q V E ( M ) + Q ( M , Y ) / D 2 4 C O N T I N U E 2 5 C O N T I N U E 3 0 DO 4 9 M = 1 , 1 2 B l = 0 . 0 B 3 = 0 . 0 B 5 = 0 . 0 B 6 = 0 . 0 B 7 = 0 . 0 3 1 DO 3 5 Y = 1 , 2 5 Y Y = Y QS = 0 . 0 Q S A V E = 0 . 0 MM = M C = F L O A T ( 8 - M O IF ( M . G T . 7 ) C = F L O A T ( 2 0 - M ) I F ( M M . G T . 1 2 ) Y Y = Y + 1 I F ( M M . G T . 1 2 ) M M = 1 QS = QS + Q ( M M , Y Y ) / C Q S A V E = Q S A V E + Q A V E ( M M ) / C MM = MM + 1 I F ( M M . N . E . * ) GO to 3 3 B l = B l + Q C M , Y ) * Q S B 2 = D * A S A V E * Q A V E ( M ) B 3 = B 3 + Q S - Q S B 4 = D * Q S A V E * Q S A V E B 5 = B 5 + Q ( M , Y ) * Q ( M , Y ) B 6 = D * Q A V E ( M ) * Q A V E ( M ) B 7 = B 7 + ( Q ( M , Y ) - Q A V E ( M ) ) * ( Q ( M , U ) - Q A V E ( M ) ) 3 5 C O N T I N U E B ( M ) = ( B 1 - B 2 ) / ( B 3 - B 4 ) R ( M ) = ( B 1 - B 2 ) / ( S Q R T ( ( B 3 - B 4 ) * ( B 5 - B 6 ) ) ) S C M ) = S Q R T ( B 7 / D ) Q S A V ( M ) + Q S A V E 4 9 C O N T I N U E (See Chapter VI3 Section: Generation of Sets of Monthly Inflows From the Forecast) G E N E R A T E MONTHLY I N P U T S F O R A G I V E N F O R E C A S T T = R A N D N ( 1 0 . ) DO 2 1 5 N = 1 , 4 DO 2 1 4 1 = 1 , 2 0 58 SMAX(N,L) = 0.0 SMIN(N,1) = 0.0 STERCN.l) =0.0 214 CONTINUE 215 CONTINUE DO 279 YY = 1,500 YX = YY C GENERATE AN ACTUAL TOTAL INPUT TO JULY 31st T = TF + RANDNC0.)*FSERR GENERATE MONTHLY INPUTS MX = MONTH 120 IF(MX.GT.12)YY = YX = 1 IF(MX.GT.12)MX = 1 C = FLOAT(8-MX) IF(MX.GT.7) C + FLOAT(20 - MX) QQ(MX,YX) = QAVE(MX) + B(MX)*(T/C - QSAV(MX))+ 1 RANDN(0.)*S(MX)*SQRT(1-R(MX)*R(MX)) T = T - QQ(MX,YX) MX = MX + 1 IF(MX.NE.8)GO to 120 (See Chapter VI3 Section: Decision Evaluation) Y = YY DO 222 N = 1,4 EL(N) = 0.0 EMAX(N) =0.0 EMIN(N) =0.0 ETER(N) =0.0 222 CONTINUE K = MONTH M = MONTH 225 IF(M.GT.12) Y = Y +1 IF(M.GT.12) M = 1 SPIL = SPILMAX IF(M.EQ.K) SPIL = DECIS EL(1) = EL(1) + (QQ(M,Y) = DEM(M))/AREA IF(M.EQ.K)EL(2) = EL(1) IF(M.NE.K)EL(2) = EL(2) + (QQ(M,Y) - DEM(M) - SPIL)/AREA EL(4) = EL(4) + (QQ(M,Y) - DEM(M) - SPIL)/AREA IF(M.EQ.K)EL(3) = EL (.4) IF(M.NE.K)EL(3) = EL(3) + (QQ(M,Y) - DEM(M))/AREA DO 227 N = 1,4 IF(EL(N).GT.EMAX(N))EMAX(N) = EL(N) IF(EL(N).LT.EMIN(N))EMIN CN) = EL CN) IF(M.EQ.7)ETER(N) = EL(N) 227 CONTINUE M = M + 1 IF(M.NE.8) GO to 225 59 (See Chapter VI, Section: Output) C SORT 250 DO 259 N = 1,4 251 DO 258 I = 1,20 J = 1-10 XL = FLOAT(J)/2.0 IF(EMAX(N) .GT.XL)SMAX(N,1) = SMAXCN,1) + 1.0 IF(EMINCN) .GT.XL)SMIN(N,1) = SMIN(N,1) + 1.0 IF(ETER(N) .GT.XL) STERCN, 1) == STER(N,1) + 1.0 258 CONTINUE 259 CONTINUE IF(YY.NE.500) GO to 279 WRITE C6,325) YY 325 FORMAT('GENERATION PERIOD',3X,14,3X,'YEARS') WRITE (6,330) 330 FORMAT (' -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0') DO 350 N = 1,4 DO 351 1 = 1,20 SSMAX(N,1) = SMAX(N,1)*100.0/FLOAT(YY) SSMIN(N,1) = SMIN(N,1)*100.0/FLOAT(YY) SSTER(N,1) = STER(N,1)*100.0/FL0AT(YY) 351 CONTINUE 350 CONTINUE 280 DO 300 N = 1,4 WRITE (6,285) N 285 F0RMAT(//'DECISION', 3X, 12, /) 290 WRITE (6,291)(SSMAX(N,1), 1 = 1,20) 291 FORMAT (20F5.1) 292 WRITE (6,293)(SSMIN(N,1), 1 = 1,20) 293 FORMAT (20F5.1) 294 WRITE (6,295)(SSTER(N,1),1 = 1,20) 295 FORMAT (20F5.1) 300 CONTINUE 279 CONTINUE STOP END 

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