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Stochastic simulation for the operation of a flood control reservoir. Heyland, Stuart Dalton 1973

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STOCHASTIC SIMULATION FOR THE OPERATION OF A FLOOD CONTROL RESERVOIR by STUART DALTON HEYLAND B.Sc. ( C i v i l Engineering), University of Alberta, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of C i v i l Engineering We accept this thesis as conforming t the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 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 requirements 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 study. 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 copying 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 g r a n t e d by the Head of my Department or 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 or 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 allowed without my w r i t t e n p e r m i s s i o n . Department o f C i v i l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date Marr.h 30, A B S T R A C T Most flood control reservoirs are operated on the basis of " f i x e d r u l e s . " The discharge from the reservoir at any-time i s related to the reservoir l e v e l at that time. The rules are usually derived from the most severe floods which have oc-curred. However, i f a method of forecasting future flows is avai l a b l e , as i s the case with snowmelt floods, t h i s informa-t i o n should be u t i l i z e d i n the operation of the reservoir. A stochastic simulation technique is proposed for generating equally l i k e l y series of future flows over the complete flood season. Two possible operating techniques have been considered in conjunction with the flow simulation procedure. One i n -volves minimizing the expected downstream damages based on a damage function curve. The other operating routine obtains the average solution based on the assumption of perfect hind-sight with each of the several possible flow sequences. Methods for using this information to determine appropriate operating procedures are given. i i TABLE OF CONTENTS Chapter Page I. INTRODUCTION ~. . 1 II. STOCHASTIC HYDROLOGY 5 III . THE SIMULATION MODEL 9 I I I . l Main Model -- Watershed Storage Component. 9 III . 2 Main Model -- Snowmelt Component 11 111.3 Main Model -- P r e c i p i t a t i o n Component. . . 15 111.4 Main Model -- Random Variable Component. . 15 111.5 Sub-Model -- Temperature Component . . . . 18 111.6 Sub-Model -- P r e c i p i t a t i o n Component . . . 21 IV. OPERATING ROUTINES 28 IV. 1 Flood Damage Function Approach 28 IV. 2 Running Average Approach 30 V. SUMMARY AND CONCLUSIONS 31 V. l Flood Damage Function Approach 31 V.2 Running Average Approach 35 V.3 Total Volume of Runoff 40 V.4 Conclusions 49 V.4.1 The Generating Scheme 49 V.4.2 The Operating Routines 49 REFERENCES 51 APPENDIX I Computer Program f o r Operating Routine Using A Damage Function Curve 54 APPENDIX II Computer Program for Operating Routine Using A Running Average Method 57 i i i LIST OF TABLES Table Page 1. Data for Years Used i n Regression Analysis. . . . 10 2. S t a t i s t i c a l Data for the Main Model 16 3. S t a t i s t i c a l Data for the Temperature Sub-Model. . 21 4. S t a t i s t i c a l Data for the P r e c i p i t a t i o n Sub-Model. 23 5. A p r i l to September Volumes of Runoff Past Hope. . 32 6. Reservoir Operation for 1964 Using Damage Function Routine (Total volume equals forecast volume) 36 7. Reservoir Operation f o r 1964 Using Damage Function Routine (Total volume equals actual volume) 37 8. Reservoir Operation for 1964 Using Running Average Routine (Total volume equals forecast volume) 38 9. Reservoir Operation f o r 1964 Using Running Average Routine (Total volume equals actual volume). 39 10. Results Using the Damage Function Operating Routine 43 11. Results Using the Running Average Operating Routine 47 iv LIST OF FIGURES Figure Page 1. Hypothetical Hydrograph 13 2. Residuals for Main Model (1967 Hope Flows Only). 17 3. Long Term Mean Daily Temperatures 19 4. Residuals for Average Daily Temperature Sub-Model (1967 Values only) 20 5. Actual and Generated Hydrographs )u for 1964 Hope Flows 24 6. Actual and Generated Hydrographs for 1965 Hope Flows 25 7. Actual and Generated Hydrographs for 1966 Hope Flows 26 8. Actual and Generated Hydrographs for 1967 Hope Flows 27 9. Damage Function Curve 29 10. Actual and Generated Hydrographs for 1964 Hope Flows . . . 33 11. Cost-Regulated Flow Curve for May 5, 1964 . . . . 34 12. E f f e c t of Total Volume on Regulated Flow (1964) (Damage Function Routine) 41 13. Maximum Discharge vs. Regulated Flow (1964) . . . 42 14. Eff e c t of Error i n Forecast Volume on Cost (Damage Function Routine) 45 15. Effect of Total Volume on Regulated Flow (1964) (Running Average Routine) 46 16. E f f e c t of Error i n Forecast Volume on Cost (Running Average Routine) 48 v LIST OF APPENDICES Appendix Page I. Computer Program f o r Operating Routine Using A Damage Function Curve 54 II. Computer Program for Operating Routine Using A Running Average Method 5 7 v i ACKNOWLEDGEMENT The author wishes to thank Professor S. 0. Russell, Department of C i v i l Engineering, U.B.C, for his assistance i n the preparation of thi s thesis. Gratitude i s also expressed to the University of B r i t i s h Columbia for f i n a n c i a l support in the form of a research assistantship and the provision of computer funds. v i i CHAPTER I INTRODUCTION With hindsight, the operation of a flood control res e r v o i r would be easy. Using the known streamflows i t would not be d i f f i c u l t to calculate how to make the best use of the available storage to minimize flood damage. In an actual oper-ation, hindsight is unavailable and there are problems of un-certa i n t y ; uncertainty about what the succeeding flows w i l l be and uncertainty about how the operator w i l l respond to the s i t u a t i o n i n the l i g h t of the information available to him at any given time. This information includes the flows that have occurred and an estimate of future flows. At the beginning of a flood there i s considerable f l e x i b i l i t y available to the operator. The reservoir i s usually empty and the operator can control the outflow to almost any value he chooses by storing the excess inflow in the rese r v o i r . But i t is d i f f i c u l t to know what i s best to do. He may be using storage which would be best preserved f o r use l a t e r in the season or he may be reserving storage unnecessarily, thus not making optimal use of the available storage and releasing excess flows. As time passes, the information w i l l improve but the f l e x i b i l i t y of operation decreases as the reservoir f i l l s . This f l e x i b i l i t y ranges between two extremes; maximum at the beginning of the 2 flood when information i s at a minimum and zero when the reser-v o i r is f u l l . The conservative approach to the fl o o d control problem has been " f i x e d r u l e " operation such as that used by the Tennessee Vall e y Authority (1). "This method i s based on the premise that the space reserved for flood storage i n a reservoir should be completely f i l l e d only when a flood approaching the maximum probable occurs or when the r a i n f a l l pattern causing a flood of major proportions has f u l l y developed and has been la r g e l y d i s s i -pated. Of course, this applies only to reservoirs lacking s u f f i c i e n t capa-c i t y to store the maximum flood. F i l l -ing then involves r e l a t i n g the discharge from the dam at any time to the r i s i n g reservoir l e v e l at that time." This approach is conservative i n that storage is wasted i n a l l but the very worst floods. With r a i n floods the " f i x e d r u l e " curve is probably the best s o l u t i o n since i n most cases there is not enough time to re-evaluate the s i t u a t i o n as the flood develops. Snowmelt floods are not so rapid so there is more time to plan operat-ing procedures for any p a r t i c u l a r flood. Also, there i s usu-a l l y much more warning. Consequently a more f l e x i b l e approach should be possible. The problem investigated was flood control for snow-melt floods. The Fraser River flows at Hope, B r i t i s h Columbia were chosen for study purposes. These were assumed to be the 3 natural flows. Hope flows were selected because of the duration and q u a l i t y of the recorded measurements (2), and because long-range forecasts of the t o t a l volume of runoff past Hope ( A p r i l 1 to September 30) are available (3). These volume forecasts are subject to error and the actual fl o o d hydrograph shape de-pends very much on the weather pattern during the melt season. For s i m p l i c i t y i t was assumed that a reservoir was constructed at Hope and that i t i s close to the flood damage area. The r e s e r v o i r s i z e was a r b i t r a r i l y set at 2,000,000 cfs-days and the dam was assumed to have an unlimited discharge capacity. The problems of flood trav e l time and involuntary storage are avoided i n this way. The approach used was to generate a number of equally possible hydrographs and then select the course of action which would minimize the expected damage. Since the outcome depends not only on the hydrographs but also on the action taken, two d i f f e r e n t operating rules were t r i e d . The operation of the reservoir was updated as the fl o o d season progressed. Most of the study was concerned with developing a computer model fo r generating the hydrographs. Quick and Pipes (8) have developed a simulation model which requires meteorolo-g i c a l data. However, temperature and p r e c i p i t a t i o n forecasts are not a v a i l a b l e . The generating model had to be simple to keep computing costs within reasonable bounds yet p h y s i c a l l y r e a l i s t i c . The model used i s not a f u l l hydrologic forecasting 4 model. In practice the method could be t r i e d and gradually "tuned" with experience, p a r t i c u l a r l y the operation rules. Other approaches such as "Stochastic Dynamic Programming" (4) have been t r i e d but at t h i s point i n time appear too compli-cated and cost l y . Chapter II describes Stochastic Hydrology as back-ground to the model development. The flow simulation system used i n this study is outlined i n Chapter III and the t r i a l s using the system are given i n Chapter IV. A summary and con-clusions are included i n Chapter V. CHAPTER II STOCHASTIC HYDROLOGY A natural hydrologic phenomena i s stochastic i n na-ture; that i s , i t changes with time i n accordance with the laws of p r o b a b i l i t y as well as with the sequential r e l a t i o n -ship between i t s occurrences. Streamflow simulation or stoch-a s t i c hydrology refers to the process of obtaining information from the h i s t o r i c a l record and using this information and a randomization routine to produce possible future sequences of streamflows. The occurrence of each of these sequences of streamflows i s assumed equally l i k e l y . The information obtained from the h i s t o r i c a l records!. is used to b u i l d a mathematical model of the stochastic hydro-l o g i c system. One such model of time series that has been used in hydrologic studies i s the l i n e a r autoregression model of the nth order which has the general form of: Q- = a ^ - ^ + a-Q-.- + . . . + a nQ T. n + e -where Qj i s the flow attributed to the Ith time i n t e r v a l , usually taken as a monthly period; a l ' a2' * * '» a n a r e t^ i e r e 8 r e s s i ° n c o ~ e f f i c i e n t s ; a n d e T i s a random v a r i a b l e . For n = 1, the above equation becomes the f i r s t order Markov process: 5 6 Qi = a ( V i + e i where a is the Markov process c o e f f i c i e n t . In a study of monthly r i v e r flows, F i e r i n g and Thomas (9) used this model. Their model also included the ef f e c t of seasonal variations in the flow. This approach i s used for planning purposes. Russell and Caselton (5) have developed a stochas-t i c simulation model for snowmelt runoff into Okanagan Lake in E r i t i s h Columbia. It is used for generating a series of monthly inflows during the high flow season, A p r i l to July, based on a forecast of the t o t a l inflow volume for a s p e c i f i c year. It has the general form of: JMAY " IMAY + bMAY (I , - 1 ^ ) + i o ^ l r 2 I3* etc MAY ) G Z C ' in which and MAY MAY = synthesized inflow for May; = average inflow for May from record; M^AY a n < * rMAY = r e 8 r e s s i ° n a n <* c o r r e l a t i o n co-e f f i c i e n t s for estimating May inflow from t o t a l May - July in-flow; MAY = standard deviation of recorded May inflows. in which I t = I f ± io I t = synthetic inflow A p r i l to July i n c l u s i v e ; 1^ = forecast inflow A p r i l to July; a = standard error of forecast; 7 and i = random normal deviate with mean 0 and variance 1. This model i s used as an aid in regulating the lake, both for the control of floods and storage of water for i r r i g a -t i o n . Chow and K a r e l i o t i s (6) have treated a watershed as a stochastic hydrologic system whose components of p r e c i p i t a t i o n , runoff, storage and evapotranspiration are simulated as stochas-t i c processes by time series models. The hydrologic system model was formulated on the basis of the p r i n c i p l e of conser-vation of mass and made up of component stochastic processes. This model can be used for planning purposes. Some of the concepts outlined above have been used i n this study. In p a r t i c u l a r the ideas of Chow and K a r e l i o t i s (6) about hydrologic models using component processes and those of Russell and Caselton (5) about simulation for a s p e c i f i c year i n the l i g h t of the available forecast information have been used. The Fraser watershed has been treated as a hydrologic system where the runoff i s due to the combined ef f e c t s of watershed storage, snowmelt, and p r e c i p i t a t i o n . Chow and K a r e l i o t i s have stated that "the autoregres-sion model may be used as a model representing hydrologic se-quences whose nonrandomness i s due to storage i n the hydrologic system." (6) In this study the watershed storage i n the Fraser system has been represented by a f i r s t order Markov process. 8 Snowmelt has been represented as the resu l t of heat energy applied to a receding snowpack. The volume of the remaining snowpack is based on a forecast of the t o t a l volume of run-off for a given year. The p r e c i p i t a t i o n input i s simulated from an appropriate p r o b a b i l i t y d i s t r i b u t i o n . Multiple c o r r e l a t i o n analysis was used i n the development of the long range d a i l y r i v e r flow simulation model as presented i n the following chapter. CHAPTER III THE SIMULATION MODEL In order to determine an optimum operation p o l i c y for a fl o o d control reservoir which would minimize the downstream damages a forecast system for daily'flows over the complete floo d season i s required. The period A p r i l 1 to September 30 inc l u s i v e i s considered the flood season. F i f t e e n years of record approximating the range of fl o o d volumes and peaks were selected for the regression analysis. These values have been tabulated i n Table 1. The generating system i s comprised of a main model and two sub-models. The main model simulates the i n t e r a c t i o n be-tween the runoff, watershed storage, snowmelt and p r e c i p i t a t i o n components of the system. One sub-model generates temperature values and the other p r e c i p i t a t i o n values for the flood season. Both these sub-models provide inputs to the main model. I I I . l MAIN MODEL -- WATERSHED STORAGE COMPONENT If there are no inputs to the runoff process other than from watershed storage then this can be represented by a f i r s t order Markov model of the form: Qj. = A * Q j . , + e j 9 TABLE 1 DATA FOR.YEARS USED IN REGRESSION ANALYSIS TOTAL APR.-SEPT. PEAK FLOW VOLUME OF RUNOFF YEAR (1000 cfs) (1000 cfs-days) 1963 272 26,810 1969 276 25,912 1966 279 29,162 1959 299 30,010 1965 303 27,628 1970 306 21,123 1968 312 31,237 1960 330 29,732 1961 336 27,197 1958 345 27,612 1957 369 29,169 1967 382 31,535 1955 400 29,426 1964 408 34,390 1948 536 33,905 10 11 where Q T = the flow on day I; Qj_^ = the flow on day 1-1; A = the Markov process c o e f f i c i e n t ; £j = a random vari a b l e . With Fraser River flows i t was found that the error term increased with the magnitude of the flow; therefore, in order to make the errors normally d i s t r i b u t e d the equation was expressed i n the form: 1 = A + e o n 1 where and A = a constant; £j = a normally d i s t r i b u t e d random variable, III.2 MAIN MODEL -- SNOWMELT COMPONENT The major part of the runoff i n the Fraser River during the period of A p r i l to September i s comprised of snowmelt. This has been represented as being a function of the heat energy applied to the remaining snowpack. The remaining snowpack on a given day was represented by the t o t a l A p r i l to September runoff, less the cumulative runoff up to that day, less the watershed storage. The watershed stor-12 age i s represented by the area under the exponential decay curve. This curve represents the flows that would take place i f there were no further inputs to the system (see Figure 1). Day number 1 = March 31; Day number 184 = September 30; Qj = flow i n cfs on day I; V o l j = cumulative runoff i n cfs-days on day I; Vo^ = 0 Vol.. R 4 = cumulative A p r i l to September runoff > obtained from the forecast (3) On any day I, the remaining snowpack is represented by shaded area C i n Figure 1. It i s equal to: V o l l g 4 - A - B N = 1, 1-1 K = recession constant K < 1 This expression is an approximation of the volume B as shown i n Figure 1 for i t i s only true at time t = i n f i n i t y . A more exact expression for B would be: B = Q I - 1 * Z I ( M M = 1> 1 8 5 " 1 However, the s i m p l i f i e d expression appeared adequate for this study. where A = ZQ N  = V o l I - l arid B = T^t* Q I - 1 I I 184 DAY NUMBER FIGURE I * HYPOTHETICAL H Y D R O G R A P H 13 14 The remaining snowpack on day I then equals: V o l i s 4 " V o l i - i " A = V o l l g 4 - ( V o l j . , + j * , Q l _ l } The remaining snowpack i s greater than or equal to zero. The heat energy was indexed by using the average of the recorded maximum and minimum a i r temperatures at B a r k e r v i l l e , B r i t i s h Columbia. The best f i t equation for any given day was obtained by using the average temperature over the previous four days less a base temperature. A base temperature serves to d i f -f e rentiate between freezing and melting conditions. If the heat index i s represented by T, then: T = ATEM I_ 1 + ATEMj _ 2 .+ A.TEMI_3 .+. ATEMj_^ _ fi T 1 1 - r m ~ where Tj = heat index on day I, °F; Tj i s greater than or equal to zero; ATEMT , = average of recorded maximum and minimum a i r [ j , temperatures on day 1-1, °F; B.T. = base temperature,°F; a base temperature of 26°F gave the best f i t equation. Therefore, the snowmelt component (SM) is equal to the remaining snowpack m u l t i p l i e d by the heat index, or: K SMj = B * Tj * { V o l 1 8 4 " ^ o l I - l + TTX QI-1 )} + e I SM j is greater than or equal to zero; 15 where B = a constant; e T = a random v a r i a b l e . 111.3 MAIN MODEL - PRECIPITATION COMPONENT The d a i l y p r e c i p i t a t i o n values recorded at Barker-v i l l e were used to index the p r e c i p i t a t i o n that occurred i n the basin. The best f i t equation was obtained by lagging the recorded value by four days. The p r e c i p i t a t i o n component on a given day I, written as Rp equals: R = r * P + e I L *I-4 I where C = a constant; Pj_4 = recorded p r e c i p i t a t i o n value on day 1-4; e-j- = a random v a r i a b l e . 111.4 MAIN MODEL - RANDOM VARIABLE COMPONENT Combining the watershed storage, snowmelt, and p r e c i p i -t a t i o n terms and f i t t i n g with least squares resulted in the follow-ing equation: ^1 A . n + T> * r . . « , . K !I-1 = A + B * T T * { V o l 1 8 4 - C V O I L ! + TTirQl.l)> + C * PI-4 + e I 16 where A, B and C are constants; £j i s a normally d i s t r i b u t e d random variable; K was obtained by t r i a l and error. The error term, e^, was expressed as: S * t j where S = standard error of estimate; t j = a random, normally d i s t r i b u t e d deviate for day I, with mean 0 and variance = 1. The assumption that the errors are normally d i s t r i b u t e d i s supported by Figure 2. The residuals for the main model (1967 flows at Hope only) have been plotted on normal p r o b a b i l i t y paper and thi s plot indicates that the assumption i s not i n - • v a l i d . A residual i s the difference between the actual QT/QT 1 r a t i o and the forecast value. The various s t a t i s t i c a l data have been l i s t e d i n Table 2. TABLE 2 STATISTICAL DATA FOR THE MAIN MODEL Regression constants: A = 0.980976E-00 B = 0.275145D-09 C = 0.498420D-01 K * 0.9823 Standard Error of Estimate = 0.388248E-01 Mu l t i p l e C o r r e l a t i o n C o e f f i c i e n t = 0.58844 R Squared - 0.34626 0.08i 0.06 0.04 0.02 < I ooo! LU or - 0 . 0 2 - 0 . 0 4 - 0 . 0 6 \ r i — r i I I i r r 1 — T - 0.08" J L J I : I L 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.8 PERCENTAGE LESS THAN FIGURE 2 : R E S I D U A L S FOR MAIN MODEL ( 1967 HOPE FLOWS O N L Y ) 18 III.5 SUB-MODEL - TEMPERATURE COMPONENT A model for generating temperature values was obtained in a manner si m i l a r to that used for obtaining the main model. The p a r t i c u l a r s considered s i g n i f i c a n t were the time of year and the persistence e f f e c t . The time of year was represented by the long term mean for a given day and the persistence e f f e c t by the previous day's temperature. The long term mean d a i l y temp-erature was obtained by drawing a smooth curve through the long term mean monthly temperatures (7). (See Figure 3). The aver-age temperature on day I can be written as: where ATEMj = A + B * LTMT + C * (ATEMj^ - LTMj^) + e T A, B, and C are the regression constants; LTMj = long term mean d a i l y temperature on day I; e T = a normally d i s t r i b u t e d random variable = S * t T ; S = standard error of estimate; t T = a random normally d i s t r i b u t e d deviate for day I, with mean = 0 and variance = 1. The assumption that the errors are normally d i s t r i b u t e d is sup-ported by Figure 4 where the residuals for the temperature model (1967 temperatures at B a r k e r v i l l e only) have been p l o t t e d on nor-mal p r o b a b i l i t y paper. The various s t a t i s t i c a l data have been l i s t e d i n Table 3. A P R I L M A Y J U N E J U L Y A U G U S T S E P T E M B E R FIGURE 3 : LONG TERM M E A N DAILY T E M P E R A T U R E S . 15. O r - i 1 1 r o P E R C E N T A G E L E S S T H A N FIGURE 4 ' R E S I D U A L S FOR A V E R A G E DAILY T E M P E R A T U R E S U B - M O D E L (1967 VALUES ONLY). 21 TABLE 3 STATISTICAL DATA FOR THE TEMPERATURE SUB-MODEL Regression Constants: A = -1.63573 B = 1.03304 C = 0.76121 Standard Error of Estimate = 4.09683 Multiple Correlation C o e f f i c i e n t = 0.90716 R Squared = 0.82294 III.6 SUB-MODEL - PRECIPITATION COMPONENT This input to the main model was expressed i n terms of a uniform p r o b a b i l i t y d i s t r i b u t i o n . A d i s t r i b u t i o n was c a l -culated for each of the six months the model covers. The precip-i t a t i o n on day I can be expressed as: P T = A + B * LOG X where A and B are constants; X = percentage of the time the,:-precipitation i s greater than a given value. The constants, A and B, were determined by the least squares f i t t i n g technique. An error term was not considered 22 necessary because of the high values obtained for the c o e f f i -cients of determination. X is a random, uniformly d i s t r i b u t e d variable i n the range 0 < X <_ C, where C is a constant. The various s t a t i s t i c a l data for the six months involved have been tabulated i n Table 4. The meteorological data was gathered from reference 10. The main model combined with the two sub-models i s used to generate hydrographs of equal p r o b a b i l i t y of occurrence. Using the correct volumes of runoff for the years 1964 to 1967, t y p i c a l generated hydrographs and the actual hydrographs have been plotted in Figures 5 to 8 for these years. The operating rules that were used i n conjunction with the generated hydro-graphs to minimize the flood damages are described in Chapter IV. TABLE 4 STATISTICAL DATA FOR THE PRECIPITATION SUB-MODEL MONTH PRECIPITATION LONG TERM MEAN (ins.) PRECIPITATION MEAN FOR 15 YRS ANALYSIS (ins.) A B SEE R R2 RANGE APRIL 2.42 2.96 0.817672 -0.489779 0.0356173 0.98494 0.97011 0<X<46.57 MAY 2.56 2.68 0.764929 -0.469772 0.0252893 0.99162 0.98330 0<X<42.49 JUNE 4.36 3.28 0.851696 -0.499811 0.0116023 0.99843 0.99687 0<X<50.59 JULY 3.78 3.42 0.899729 -0.534941 0.0302875 0.99074 0.98156 0<X<48.08 AUGUST 4.27 5.12 1.291160 -0.760168 0.0289341 0.99578 0.99158 0<X<49.95 SEPTEMBER 3.33 3.97 0.982526 -0.564413 0.0170083 0.99737 0.99474 0<X<55.05 SEE = Standard Error of Estimate R = Multiple Correlation Coefficient Long term mean monthly precipitation based on the period 1931-1960. 0 4 450.0 375.0 300.0 o o o £ 225.0 o 150.0 75.0 0.0 ACTUAL HYDROGRAPH - GENERATED HYDROGRAPH Total Volume Used Equals *1 Actual Volume * 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 TIME IN DAYS ( DAY NUMBER I - MARCH 31, 1964 ) FIGURE 5 ' ACTUAL AND GENERATED HYDROGRAPHS FOR 1964 HOPE F L O W S . 450.0 ACTUAL HYDROGRAPH 375.0 - GENERATED HYDROGRAPH f Total Volume Used \ «• Equals Actual Volume •» 300.0 o o o CO 5 225.01 o 150.0 75.0 O.O1 _L 0.0 20.0 FIGURE 6 40.0 60.0 80.0 100.0 120.0 140.0 160.0 TIME IN DAYS ( DAY NUMBER 1= MARCH 31,1965 ) ACTUAL AND GENERATED HYDROGRAPHS FOR 1965 HOPE FLOWS 180.0 450.0 ACTUAL HYDROGRAPH 375.0 300.0 O o o CO I i . u o _ l u. 225.0 150.0 - GENERATED HYDROGRAPH rTotal Volume Used rTotal olu e sed "» «• Equals Actual Volume * 75.0 0.0' _L J_ 0.0 20.0 FIGURE 7 40.0 60 .0 80.0 100.0 120.0 140.0 160.0 T I M E IN D A Y S ( D A Y N U M B E R I = M A R C H 3 1 , 1 9 6 6 ) ACTUAL AND GENERATED HYDROGRAPHS FOR 1966 HOPE F L O W S . 180.0 450.0 375.0 300.0 o o o CO LL. u o 225.0 150.0 75.0 O.O1 ACTUAL HYDROGRAPH -GENERATED HYDROGRAPH rTotal Volume Used \ I Equals Actual Volume •> 1 0.0 20.0 FIGURE 8 40.0 60.0 80.0 100.0 120.0 140.0 160.0 TIME IN DAYS ( DAY NUMBER I = MARCH 31,1967 ) ACTUAL AND GENERATED HYDROGRAPHS FOR 1967 HOPE FLOWS 180.0 CHAPTER IV OPERATING ROUTINES IV.1 FLOOD DAMAGE FUNCTION APPROACH In order to simplify the problem a damage function curve was assumed as shown in Figure 9 where the damage is re-lated to the maximum flow at Hope. In practice a rea l damage function curve of any shape could be used. For ease of compu-tat i o n the curve was a r b i t r a r i l y defined by the parabolic equa-tio n : C = 5 * Q 2 - 20.0 where C = flood damage index cost; Q = the maximum discharge i n hundreds of thousands of c f s . It was assumed that the lower l i m i t for damage to begin to occur was equal to 200,000 c f s . The amount of damage was assumed to depend only on the maximum discharge and not on the duration o flooding. A limited amount of reservoir storage was assumed. This was set equal to 2,000,000 cfs-days. A range of regulated flow values was selected, and as each hydrograph was generated, the re su i t i n g maximum discharge for each regulated flow was calculated. The damage incurred at each of these levels was costed on the basis of the above equation. A running average of each level's 28 F IGURE 9 : D A M A G E F U N C T I O N C U R V E 29a cost was maintained over a l l the generated sequences of flow. F i f t y such runs were assumed adequate. On the basis of aver-age minimum cost a regulated flow value was selected. The t o t a l volume of runoff used in the generating model had the greatest e f f e c t on the optimum cost obtained. As time progresses, the operator has two options or a combination of the two available to him. He can revise the t o t a l volume e s t i -mate and thereby the optimum regulated flow and/or he can take corrective action i n response to the inflows that have occurred and the storage used. As a complex hydrologic forecasting model or f i e l d measurements would be required to revise the volume term, the l a t t e r approach was t r i e d . Knowing the flows, temp-eratures and p r e c i p i t a t i o n s that had occurred up to any given time, the model was run again i n combination with the remain-ing storage and a new estimate of the optimum regulated flow value was obtained. This method of r e v i s i n g the reservoir oper-ation was done p e r i o d i c a l l y throughout the stori n g process. Throughout the flood season the reservoir was operated on the basis of the following rules: 1. If the inflow was less than the estimated optimal regulated flow the outflow was set equal to the inflow. 2. If the inflow was greater than the e s t i -mated optimal regulated flow the outflow was set equal to the estimated optimal regulated flow. 3. Because the duration of the flooding was not considered important, the outflow was never set less than the maximum value used. 30 IV.2 RUNNING AVERAGE APPROACH Once again, the amount of storage available was set equal to 2,000,000 cfs-days. For each hydrograph generated the " i d e a l " regulated flow was calculated. This " i d e a l " regulated flow was such that the storage used was exactly equal to the amount avai l a b l e . Each one of the generated hydrographs i s operated at maximum e f f i c i e n c y . This method assumes a l i n e a r damage function curve. A running average of the " i d e a l " regulated flow was maintained over the flow sequen-ces generated. F i f t y such runs were considered adequate. On the basis of the average " i d e a l " regulated flow a regulated flow value was selected. As i n the previous method, the information gathered during the flood season was used to p e r i o d i c a l l y re-vise the reservoir operation. Operating rules, numbered 1 to 3, have been applied here also. The t o t a l A p r i l to September volumes of runoff have been forecast since 1964 (3). The years 1964 through 1969 were used i n te s t i n g the two procedures outlined above. The results are given i n Chapter V. CHAPTER V SUMMARY AND CONCLUSIONS To i l l u s t r a t e the procedure used and the results ob-tained, the floo d control operation for the 1964 flows at Hope are described herewith. The 1964 t o t a l A p r i l to September v o l -ume of runoff past Hope was forecast (3) as 55 m i l l i o n acre-feet or 27.8 m i l l i o n cfs-days. The actual volume of runoff was 34.4 m i l l i o n cfs-days. The forecast and the actual volumes for the years 1964 to 1970 have been l i s t e d i n Table 5. V. 1 FLOOD DAMAGE FUNCTION APPROACH On day 6 (Ap r i l 5, 1964) using the forecast t o t a l volume, and knowing the flows, p r e c i p i t a t i o n and temperatures for days 1 to 5, the model was used to generate 50 hydrographs. A t y p i -c a l generated hydrograph and the actual hydrograph for 1964 are shown i n Figure 10. An optimum regulated flow value of 250,000 cfs was obtained on the basis of minimum expected damage. The cost-regulated flow curve has been plo t t e d in Figure 11. The computer program developed for this optimization i s given i n Appendix I. As the flows approached the lower bounding l i m i t f or damage to begin the model was run again. At the end of day 60 (May 29, 1964) knowing the flows, temperatures and p r e c i p i t a t i o n 31 TABLE 5 APRIL TO SEPTEMBER VOLUMES OF RUNOFF PAST HOPE FORECAST VOLUME ACTUAL YEAR 1000 Ac.Ft. 1000 cfs-dayg VOLUME 1000 cfs-days 1964 55,000 27,800 34,390 1965 57,000 28,800 27,628 1966 54,000 27,300 29,162 1967 68,000 34,400 31,535 1968 57,000 28,800 31,237 1969 53,400 27,000 25,912 1970 45,600 23,100 21,123 32 450.0 375.0 ACTUAL HYDROGRAPH GENERATED HYDROGRAPH /Total Volume Used Equals \ Forecast Volume o o o o 300.0 225.0 o 150.0 75.0 0.0 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 TIME IN DAYS ( DAY NUMBER I - MARCH 31,1964 ) FIGURE 10 • A C T U A L AND GENERATED HYDROGRAPHS FOR 1964 HOPE FLOWS. 180.0 180 200 220 240 260 280 300 REGULATED FLOW ( 1 , 0 0 0 CFS) NOTE : TOTAL V O L U M E OF RUNOFF USED EQUALS FORECAST (27,800,000 CFS-DAYS ). FIGURE II • C O S T - R E G U L A T E D FLOW CURVE FOR A P R . 5 , 1964 . 35 that had occurred over days 1 to 60 an optimal regulated flow value of 230,000 cfs was obtained. The flow on day 60 was equal to 193,000 c f s . This regulated flow value was used un-t i l approximately 25 per cent of the storage had been used. At t h i s point the optimal regulated flow value was again re-vised. A value of 260,000 cfs was obtained this time. The procedure was repeated again when approximately 50 and 75 per cent of the storage had been used. The reservoir operation has been tabulated in Table 6. The maximum outflow using t h i s procedure equalled 360,000 c f s . This procedure was also run using the actual t o t a l volume of runoff. The res u l t s are given i n Table 7. The max-imum discharge in this case equalled 330,000 c f s . V. 2 RUNNING AVERAGE APPROACH On day 6, th i s method produced a regulated flow value of 230,200 cfs using the forecast t o t a l volume. The computer program developed f o r this procedure i s given i n Appendix I I . On day 60 a regulated flow value of 218,700 cfs was ob-tained. As in the damage function approach, this value was re-vised when approximately 25, 50 and 75 per cent of the storage had been used. The rese r v o i r operation has been l i s t e d i n Table 8. The maximum outflow equalled 347,800 c f s . The procedure was also run using the actual t o t a l volume of runoff. The re s u l t s are given i n Table 9. The maxi-mum discharge i n this case equalled 328,700 c f s . TABLE 6 RESERVOIR OPERATION FOR 1964 USING DAMAGE FUNCTION ROUTINE (Total volume equals forecast volume) INFLOW EST. REG. FLOW OUTFLOW AS STORAGE REMAINING DAY 1000 c f s . 1000 c f s . 1000 c f s . 1000 cfs-days 1000 cfs 60 193 230.0 193 0 2000 61 207 207 0 2000 62 231 230 -1 1999 63 255 230 -25 1974 64 284 230 -54 1920 65 302 230 -72 1848 66 316 230 -86 1762 67 337 230 -107 1655 68 350 230 -120 1535 69 359 260.0 260 -99 1436 70 373 260 -113 1323 71 391 260 -131 1192 72 387 260 -127 1065 73 393 290.0 290 -103 962 74 400 290 -110 852 75 402 290 -112 740 76 400 290 -no 630 77 403 290 -113 517 78 407 360.0 360 -47 470 79 402 360 -42 428 80 402 360 -42 386 81 405 360 -45 341 82 405 360 -45 296 83 408 360 -48 248 84 400 360 -40 208 85 389 360 -29 179 86 369 360 -9 170 87 343 360 + 17 187 88 330 360 +30 217 89 321 360 + 39 256 90 304 360 + 56 312 36 TABLE 7 RESERVOIR OPERATION FOR 1964 USING DAMAGE FUNCTION ROUTINE (Total volume equals actual volume) EST. REG. STORAGE INFLOW FLOW OUTFLOW AS REMAINI1 DAY 1000 c f s . 1000 c f s . 1000 c f s . 1000 cfs-days 1000 c f s - i 60 193 310 193 0 2000 61 207 207 62 231 231 63 255 255 64 284 284 65 302 302 0 2000 66 316 310 -6 1994 67 337 310 -27 1967 68 350 310 -40 1927 69 359 310 -49 1878 70 373 310 -63 1815 71 391 310 -81 1734 72 387 310 -77 1657 73 393 310 -83 1574 74 400 330 330 -70 1504 75 402 330 -72 1432 76 400 330 -70 1362 77 403 330 -73 1289 78 407 330 -77 1212 79 402 330 -72 1140 80 402 330 -72 1068 81 405 330 330 -75 993 82 405 330 -75 918 83 408 330 -78 840 84 400 330 -70 770 85 389 330 -59 711 86 369 330 -39 672 87 343 330 -13 659 88 330 330 0 659 89 321 330 +9 668 90 304 330 + 26 694 37 TABLE 8 RESERVOIR OPERATION FOR 1964 USING RUNNING AVERAGE ROUTINE (Total volume equals forecast volume) EST. REG. STORAGE INFLOW FLOW OUTFLOW AS REMAINING DAY 1000 c f s . 1000 c f s . 1000 c f s . 1000 cfs-days 1000 cfs-days 60 193 218.7 193.0 0 2000.0 61 207 207.0 0 2000.0 62 231 218.7 -12.3 1987.7 63 255 218. 7 -36.3 1951.4 64 284 218 .7 -65.3 1886.1 65 302 218.7 -83.3 1802.8 66 316 218. 7 -97.3 1705.5 67 337 218.7 -118.3 1587.2 68 350 255.2 255.2 -94.8 1492.4 69 359 255. 2 -103.8 1388.6 70 373 255.2 -117.8 1270.8 71 391 255.2 -135.8 1135.0 72 387 255. 2 -131.8 1003.2 73 393 281.1 281.1 -111.9 891.3 74 400 281.1 -118.9 772.4 75 402 281.1 -120.9 651.5 76 400 281.1 -118.9 532.6 77 403 347.8 347.8 -55.2 477.4 78 407 347.8 -59.2 418. 2 79 402 347.8 -54.2 364.0 80 402 347.8 -54.2 309.8 81 405 347.8 -57.2 252.6 82 405 347.8 -57.2 195.4 83 408 347.8 -58.2 137.2 84 400 347.8 -52.2 85.0 85 389 347 .8 -41.2 43.8 86 369 347.8 -21.2 22.6 87 343 347.8 + 4.8 27.4 88 330 347.8 + 17.8 45.2 89 321 347.8 + 26.8 72.0 90 304 347.8 + 43.8 115.8 38 TABLE 9 RESERVOIR OPERATION FOR 1964 USING RUNNING AVERAGE ROUTINE (Total volume equals actual volume) EST. REG. STORAGE INFLOW FLOW OUTFLOW AS REMAINING DAY 1000 c f s . 1000 c f s . 1000 c f s . 1000 cfs.-days 1000 cfs-days 60 193 298.4 193.0 0 2000.0 61 207 207.0 ••: 0 2000.0 62 231 231.0 0 2000.0 63 255 255.0 0 2000.0 64 284 284.0 0 2000.0 65 302 298.4 -3.6 1996.4-66 316 298.4 -17.6 1978.8 67 337 298.4 -38.6 1940.2 68 350 298.4 -51.6 1888.6 69 359 298.4 -60.6 1828.0 70 373 298.4 -74.6 1753.4 71 391 298.4 -92.6 1660.8 72 387 298.4 -88.6 1572.2 73 393 299 .8 299. 8 -93.2 1479 .0 74 400 299. 8 -100.2 1378.8 75 402 299.8 -102.2 1276.6 76 400 299.8 -100.2 1176.4 77 403 299. 8 -103.2 1073.2 78 407 328.7 328. 7 - 78.3 994.9 79 402 328. 7 - 73.3 921.6 80 402 328.7 - 73.3 848.3 81 405 328.7 - 76.3 772.0 82 405 328.7 - 76.3 695.7 83 408 328. 7 - 79.3 616.4 84 400 328.7 - 71.3 545.1 85 389 322 .4 328.7 - 60.3 484.8 86 369 328.7 - 40.3 444.5 87 343 328.7 - 14.3 430.2 88 330 328.7 - 1.3 428.9 89 321 328.7 + 7.7 436.6 90 304 328.7 + 24.7 461.3 39 40 V.3 TOTAL VOLUME OF RUNOFF The forecast t o t a l volume of runoff (April to September) was greatly in error for 1964. The actual volume of runoff was 34.4 m i l l i o n cfs-days as compared to the forecast value of 27.8 m i l l i o n cfs-days. If no corrective action had been taken by the operator, the maximum discharge would have equalled 408,000 cfs or the years peak flow for either of the two pro-cedures outlined. In order to i l l u s t r a t e the effects of the t o t a l volume of runoff term a l l further results are given i n terms of no corrective action during the flood season. Using the damage function procedure with various t o t a l volumes of runoff for 1964 resulted i n a range of estimated regulated flow values. These have been plotted in Figure 12. The corresponding r e s u l t i n g maximum discharges are shown i n Figure 13. Figure 12 indicates that a conservative regulated flow value would have been obtained i f the correct t o t a l volume of runoff had been used. The "perfect" s o l u t i o n for 1964 would have been to operate the reservoir at a regulated flow value of 299,800 c f s . The "perfect" sol u t i o n is defined as that regulated flow value which would r e s u l t i n having the storage used equal the storage available (2,000,000 cfs-days) and the r e s u l t i n g maximum discharge equal the regulated flow. Table 10 l i s t s the res u l t s obtained for the years 1964 to 1969. The item e n t i t l e d "extra cost" i s the difference be-tween the cost (based upon the damage function curve used) of ~ 350 co o o o _ 300 o S 2 5 0 3 O U l oc 200 FORECAST TOTAL VOLUME = 27,800,000 C F S - D A Y S PERFECT SOLUTION = 299 ,800CFS o oo o ooo _^ 0- • ~~ L. o o A C T U A L TOTAL VOLUME = 34,400,000 CFS-DAYS -1 I 27 28 29 30 /3I 32 33 34 TOTAL VOLUME OF RUNOFF (APRIL TO SEPTEMBER ) ( 1,000,000 CFS-DAYS) 35 FIGURE 12 * E F F E C T OF TOTAL VOLUME ON R E G U L A T E D FLOW (1964) . (DAMAGE FUNCTION ROUTINE) . 4^ 42 FIGURE 13: MAXIMUM DISCHARGE V S . REGULATED F L O W . (1964) TABLE 10 RESULTS USING THE DAMAGE FUNCTION OPERATING ROUTINE YEAR PERFECT SOLUTION FORECAST TOTAL VOLUME FORECAST VOLUME ERROR ESTIMATED REGULATED FLOW RESULTING MAXIMUM ' DISCHARGE EXTRA COST (cfs) (cfs-days) (cfs-days) (cfs) (cfs) ($) 1964 299,800 27,800,000 -6,590,000 250,000 408,000 38.29 1965 191,600 28,800,000 +1,172,000 260,000 260,000 13.80 1966 203,000 27,300,000 -1,862,000 220,000 220,000 3.60 1967 290,700 34,400,000 +2,865,000 340,000 340,000 15.55 1968 247,700 28,800,000 -2,437,000 250,000 250,000 0.57 1969 181,300 27,000,000 +1,088,000 230,000 230,000 6.45 43 44 the maximum discharge and the cost of the "perfect s o l u t i o n . " This "extra cost" item has been plotted against the error i n the forecasted t o t a l volume of runoff for the six years con-sidered. The error i s equal to the forecast volume minus the actual volume. (See Figure 14). The importance of a reason-able estimate f o r the forecast t o t a l volume i s emphasized by t h i s curve. Using the running average approach with various t o t a l volumes of runoff for 1964 also resulted in a range of e s t i -mated regulated flow values. These have been p l o t t e d i n Figure 15. Using the actual t o t a l volume of runoff would r e s u l t i n a solution less conservative than that obtained using the damage function approach. Table 11 l i s t s the r e s u l t s using the running average approach for the years 1964 to 1969 and Figure 16 shows the extra cost incurred as a r e s u l t of an error i n the forecast t o t a l volume. Again, the importance of a reasonable estimate for the forecast t o t a l volume i s i l l u s t r a t e d by t h i s curve. On the average, this method produced results comparable to those of the previous method. $ 60.0 £ 40.0 o o < rr x 20.0 UJ 0.0 - 8 1 1 1 1 1 1 - \ 1 \ V 1 \ \ 1 :> ^ ^ 0 0 ->-1 1 0 -6.0 -4.0 -2 .0 " 0 . 0 / 2.0 4.0 ERROR IN FORECAST TOTAL VOLUME (I,OOO.OOOCFS-DAYS) FIGURE 14 : EFFECT OF ERROR IN FORECAST VOLUME ON C O S T . (DAMAGE FUNCTION ROUTINE) . Ln 200 27 28 29 30 31 32 33 34 TOTAL VOLUME OF RUNOFF (APRIL TO SEPTEMBER) (1,000,000CFS-DAYS) 35 FIGURE 15= E F F E C T OF TOTAL VOLUME ON R E G U L A T E D FLOW (1964) . £ (RUNNING AVERAGE ROUTINE) . TABLE 11 RESULTS USING THE RUNNING AVERAGE OPERATING ROUTINE YEAR PERFECT SOLUTION FORECAST TOTAL VOLUME FORECAST VOLUME ERROR ESTIMATED REGULATED FLOW RESULTING MAXIMUM DISCHARGE EXTRA COST (cfs) (cfs-days) (cfs-days) (cfs) (cfs) ($) 1964 299,800 27,800,000 -6,590,000 230,200 408,000 38.29 1965 191,600 28,800,000 +1,172,000 241,400 241,400 9.14 1966 203,000 27,300,000 -1,862,000 216,600 216,600 2.85 1967 290,700 34,400,000 +2,865,000 315,600 315,600 7.55 1968 247,700 28,800,000 -2,437,000 233,100 292,000 11.95 1969 181,300 27,000,000 +1,088,000 225,100 225,100 5.34 47 $ 60.0 <o 40.0 o o < or \— x ui 20.0 0.0 1 A \ i i i 1 i \ \ v -i \ N i 0 1 o — t r - ^ 1 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 ERROR IN FORECAST TOTAL VOLUME (I.OOO.OOOCFS-DAYS ) FIGURE 16= E F F E C T OF ERROR IN FORECAST VOLUME ON C O S T . « (RUNNING A V E R A G E ROUTINE ). 49 V.4 CONCLUSIONS V.4.1 The Generating Scheme. This study has attempted to show that a simple generating model i n combination with a set of operating rules may be used to minimize the damage due to a snow-melt flood. The generating scheme appears to be f a i r l y good; how-ever, i t may be improved by incorporating the following r e f i n e -ments: 1. Separate the snowmelt volume of runoff from that due to r a i n f a l l . The volume of runoff due to r a i n f a l l can be accounted for by examining, the recession limb of the hydrograph. Here the flows are usually dependent upon the Markov chain plus a r a i n input. 2. Expand the data base for the main model and the two sub-models. The p r e c i p i t a t i o n model i s now biased on the high side, e s p e c i a l l y during the l a s t two months where any input to the system is usually dependent upon r a i n alone. 3. Correlate the temperature and p r e c i p i t a -t i o n sub-models. This would help to prevent any excessive increases i n the rate of flow. For ex-ample, i t i s quite u n l i k e l y that a record high temperature and a record high amount of p r e c i p i -t a t i o n would occur on the same day. 4. D i f f e r e n t i a t e between p r e c i p i t a t i o n i n the form of snow and p r e c i p i t a t i o n as r a i n f a l l . Not only i s the quantity of p r e c i p i t a t i o n r e l a t e d to the temperature but also i t s q u a l i t y , i . e . , i t s phase. 5. Properly account for the volume of runoff volume under the exponential decay curve as explained i n section III.2. 6. Check the data for consistency over the years of record used i n the analysis, e.g., double mass curve analysis. 7. Use the natural flows at Hope, i . e . , e l i m i -nate the effects of the Nechako diversion. V.4.2 The Operating Routines. Two aspects of either of 50 of the operating routines used require further "tuning." One i s the t o t a l volume of runoff and the other is the corrective action to be employed by the operator. As indicated e a r l i e r , the t o t a l volume of runoff used i n the operating routines has the greatest e f f e c t on the regulated flow value obtained. An e f f e c t i v e method of r e v i s i n g the course of action the operator should take would be to improve the forecast t o t a l volume,and consequently the estimated regulated flow value. This could possibly be done through the use of the model developed by Quick and Pipes (8) or f i e l d measurements. An alternative to these two suggestions would be to synthesize the t o t a l volume of runoff as has been done by Russell and Caselton (5). Due to the li m i t e d number of forecast years for the t o t a l volume of runoff past Hope thi s i s not presently possible. The sum of the forecast errors i s not equal to zero and the errors are not normally d i s t r i b u t e d . In combination with an improved t o t a l volume term, a refinement of the rules of operation i n response to the combined inflows and decreasing available stor-age would aid in the operation of the re s e r v o i r . The best solu-t i o n may be a combination of the two routines previously out-lin e d . The development of these rules would require experience in operating the reservoir. R E F E R E N C E S 1. Tennessee Valley Authority, "Floods and Flood Control," T e c h n i c a l Report No. 26, Knoxville, Tennessee, 1961, pp. 134-137. 2. Surface Water Data, B r i t i s h Columbia, Water Survey of Canada, Inland Waters Branch, Department of Energy, Mines and Resources, Ottawa, Canada, 1948-1970. / 3. B r i t i s h Columbia Snow Survey B u l l e t i n , Water Investigations Branch, Water Resources Service, Department of Lands, Forests and Water Resources, V i c t o r i a , B r i t i s h Columbia, A p r i l 1, 1964-1970. 4. Tsou, C. A. Determination of Reservoir Daily Operation P o l i c i e s by Stochastic Dynamic Programming. Thesis sub-mitted to the University of B r i t i s h Columbia, i n August 19 70, i n p a r t i a l f u l f i l m e n t of the requirements for the degree of Master of Applied Science. 5. Russell, S. 0. and William F. Caselton. "Reservoir Oper-ation with Imperfect Flow Forecasts," Journal of the H y d r a u l i c s D i v i s i o n , ASCE, Vol. 97, No. HY2, February, 1971, pp. 323-331. 6. Chow, Ven Te and S. J . K a r e l i o t i s . "Analysis of Stochastic Hydrologic Systems," Water Resources Research, V o l . 6, No. 6, December, 1970, pp. 1569-1582. 7. Province of B r i t i s h Columbia, C l i m a t e of B r i t i s h Columbia, Department of Agr i c u l t u r e , Report for 1966. 8. Quick, Michael C., and Anthony Pipes. U n i v e r s i t y of B r i t i s h Columbia Watershed Budget Model. 9. Thomas, H. A., and M. B. F i e r i n g . "The Mathematical Synthe-s i s of Streamflow Sequences for the Analysis of River Basins by Simulation," Design of Water Resources Systems, Maass et a l . (Eds.), Harvard University Press, Cambridge, Mass., 1962, pp. 459-538. 51 52 Monthly Record, Meteorological Observations in Canada, Department of Transport, Meteorological Branch, Government of Canada, Ottawa, Ontario. A P P E N D I C E S APPENDIX I $ L I S T * S O U R C E * 6 C COMPUTER PROGRAM FOR OPERATING ROUTINE 7 0 USING A DAMAGE FUNCTION CURVE 8 C 9 C FLOW = THE ACTUAL A P R I L TO SEPTEMBER FLOWS 10 C 0 = THE GENERATED APRIL TO SEPTEMBER FLOWS 11 C VOL = CUMULATIVE VOLUME OF RUNOFF IN C F S - D A Y S 12 C LTM = THE LONG TERM MEAN DAILY TEMPERATURE 13 C TMAX = THE MAXIMUM DAILY TEMPERATURE 14 C TMIN = THE MINIMUM DAILY TEMPERATURE 15 C ATEM * THE AVERAGE DAILY TEMPERATURE 16 C RAIN = THE DAILY P R E C I P I T A T I O N IN INCHES 17 REAL LTM*184} 18 DIMENSION FLOW! 184) t T M A X U 8 4 ) •TMIN I184 ) , V O L 1184 ) , 19 IRA I N U 8 4 ) , Q ( 1 8 4 9 , A T E M ( 1 8 4 ) 20 R E A D « 5 , 1 0 1 , E N D ± 1 0 4 ) N , I F L O W t J ) , J = l , 1 8 4 ) , I T M A X * J ) , J = 21 1 1 , 1 8 4 ) , ( T H I N I J U J - l f 184) 22 101 F 0 R M A T ( I 4 / 1 6 ( 1 1 F 7 . G / ) , 8 F 7 . 0 / 9 ( 2 0 F 4 . 0 / ) . 4 F 4 . 0 / 9 23 1 ( 2 0 F 4 . 0 / ) , 4 F 4 . 0 ) 24 104 R E A D { 4 , 1 0 0 , E N D = 1 0 5 ) I L T M ( J ) , J = 2 , 1 8 4 ) 25 100 F 0 R M A T I 9 ( 2 X , 2 0 F 5 . 1 / ) , 3 F 5 . 1 ) 26 105 R E A D ( 3 , 1 0 6 , E N D = 1 0 7 > i M, (RAINt J ) , J = 1 , 1 8 4 ) 27 106 F 0 K M A T U 4 / 9 * 2 X , 2 0 F 5 . 2 / ) , 2 X , 4 F 5 . 2 ) 28 107 V O L U ) = 0 . 0 29 DO 7 1=2 ,5 30 VOLf I ) = V O L U - l ) f F L G W U ) 31 Q ( l ) = F L O W U ) 32 A T E M { I ) = < T M A X ( I ) + T M I N ( I ) ) / 2 . 0 33 7 CONTINUE 34 A = 0 . 9 8 2 3 / ( 1 . 0 - 0 . 9 8 2 3 ) 35 B l = 0 . 9 8 0 9 7 6 E 00 36 8 2 = 0 . 2 7 5 1 4 5 0 - 0 9 37 8 3 = 0 . 4 9 8 4 2 0 0 - 0 1 38 W R I T E ( 6 , 2 0 0 ) 39 200 F O R M A T U H 0 , 2 X , " L E V E L * , 4 X , * H Y D R 0 G R A P H « , 3 X , * C 0 S T » , 5 X 40 1 » • A V E R A G E * » 4 X » S M A X I M U M * , 5 X » ' A V G . M A X . * , 3 X , 41 2 ' R E G U L A T E D * » 3 X » * U N U S E S T O R A G E * , 3 X , • A V G . UNUSE S T O R . * 42 3 / 3 X * * N U M B E R * ,4X- I *NUMBER' , 1 6 X , ' C O S T * | 6 X , « FLOW(CFS )» 43 4 , 3 X , . * F L 0 W ( C F S ) • #3X, * F L O W ( C F S ) • , 4 X , • I C F S - D A Y S ) * , 6 X , 44 5 ' ( C F S - D A Y S ) • / ) 45 DO 571 K=2 ,50 46 C V O L ( 1 8 4 ) = THE TOTAL A P R I L TO SEPTEMBER VOLUME OF RUNOFF 47 V O L f 1 8 4 ) = 2 7 8 0 0 0 0 0 . 0 48 S = S C L O C K ( 0 . ) 49 Z=RANDN(S) 50 XZJ=RAND*S) 51 C C A L C U L A T E ^VALUES FOR TEMPERATURE SUB-MODEL 52 DO 2 1=6 ,184 53 ZN=FRANDN(0 . ) 54 A T E M ! I ) = — 1 . 6 3 5 7 3 + 1 . 0 3 3 0 4 * L T M C I ) + 0 . 7 6 1 2 1 * I A T E M < I - i ) 55 i - L T M C 1 - 1 ) ) + 4 . 0 9 6 8 3 * Z N 56 2 CONTINUE 57 C C A L C U L A T E VALUES FOR P R E C I P I T A T I O N SUB-MODEL 58 DO 5 1=6,184 59 XU=FRANDIO. ) 60 I F J X U . E Q . O . O ) GO TO 11 54 61 I F ( I . L E . 3 1 ) GO TO 12 r r IFi< I . L E . 6 2 . A N D . I . G T . 3 1 ) GO TO 13 62 63 IF( I * L E . 9 2 . A N D . I . G T . 6 2 ) GO TO 14 64 I F < I . L E . 1 2 3 . A N D . I . G T . 9 2 ) GO TO 15 65 IF( I . L E . 1 5 4 . A N D . I . G T . 1 2 3 ) GO TO 16 66 IF( I . L E . 1 8 4 . A N 0 . I . G T . 1 5 4 ) GO TO 17 67 12 R A I N U ) = 0 . 8 1 7 6 7 2 - 0 . 4 8 9 7 7 9 * A L O G 1 0 ( X U * 1 0 0 . 0 ) 68 GO TO 18 69 13 RAINI I ) = 0 . 7 6 4 9 2 9 - 0 . 4 6 9 7 7 2 * A L 0 G 1 0 < X U * 1 0 0 . 0 ) 70 GO TO 18 71 L4 R A L N l I ) = 0 . 8 5 1 6 9 6 - 0 . 4 9 9 S 1 1 * A L O G 1 0 < X U * 1 C O . O ) 72 GO TO 18 73 15 R A I N ( I ) = 0 . 8 9 9 7 2 9 - 0 . 5 3 4 9 4 1 * A L 0 G 1 0 I X U * 1 C 0 . 0 ) 74 GO TO 18 75 16 R A I N ( I ) = 1 . 2 9 1 1 6 0 - 0 . 7 6 0 1 6 8 * A L O G 1 0 < X U * 1 0 0 . 0 ) 76 GO TO 18 77 17 RAIN( I ) = 0 . 9 8 2 5 2 6 - 0 . 5 6 4 4 1 3 * A L O G 1 0 f X U * 1 0 0 . 0 ) 78 GO TO 18 79 11 1=1-1 80 GO TO 5 81 18 IF<<RAIN!I i . L T . O . O ) GO TO 19 82 GO TO 5 83 19 RAIN I I ) = 0 . 0 84 5 CONTINUE 85 C GENERATE APRIL TO SEPTEMBER FLOWS 86 DO 3 J = 6 , 1 8 4 87 YN=FRANDN(0 . ) 88 TEMP=(ATEM( J - U * A T E M t J - 2 ) + A T E M U - 3 ) * A T E M I J - 4 ) ) / 4 . 0 89 1 - 2 6 . 0 90 I F C T E M P . L T . O . O ) GO TO 20 91 GO TO 30 92 20 TEMP=0.0 93 30 C = B 2 * T E M P * ( VOL 1 1 8 4 ) - ( V O L * J - i ) + A * « ( J - l 1 ) » 94 I F t C . L T . O . O ) GO TO 50 95 GO TO 59 96 50 C = 0 . 0 97 59 D=B1+C+B3*RAIN(J -4 ) > 0 . 0 3 8 8 2 4 8 * Y N 98 Q < J ) = D * Q { J - 1 ) 99 V O L ( J ) = V O H J - l ) + Q ( J ) 100 3 CONTINUE 101 C C A L L THE FLOOD DAMAGE COST SUBROUTINE 102 C A L L C O S T ( K , Q I 103 71 CONTINUE 104 END 105 C 106 C SUBROUTINE FOR COSTING THE FLOOD DAMAGES 107 C 108 SUBROUTINE C O S T ( K . Q ) 109 C QREG = THE REGULATED FLOW VALUE 110 C QMAX = THE MAXIMUM RESERVOIR OUTFLOW 111 INTEGER Y I 1 0 0 ) 112 DIMENSION C O S ( l l ) , A C O S l l l , 5 0 ) , S T O R i 1 8 4 1 , Q M A X U 1 ) , 113 1 Q ( 1 8 4 3 , A F S T ( 1 1 , 5 0 ) , A V G Q I 1 1 , 5 0 ) 114 C S E L E C T A RANGE OF REGULATED FLOW VALUES 115 QREG=2C00OO.O 116 D E L = 1 0 C 0 0 . O 117 DO 6 L = l , l l 118 A C O S ( L , 1 ) = 0 . 0 119 A F S T ( L , 1 ) = 0 . 0 120 A V G Q < L , 1 ) = 0 . 0 121 1=0 122 QMAX(L)=Q(5) 123 C C A L C U L A T E THE DAYS OVER WHICH STORAGE IS REQUIRED 124 DO A J = 6 , 1 8 4 125 IF (QJ J ) . G T . Q R E G ) GO TO 60 126 I F < Q ( J ) . L E . Q R E G . A N D . Q ( J ) . G T . Q M A X I L ) } Q M A X ( L ) = Q t J ) 127 GO TO 4 128 60 1=1 + 1 129 Y M ) = J 130 4 CONTINUE 131 I F U . E Q . O ) GO TO 64 132 M=Y(1) 133 KK^M-1 134 N=YII } 135 C THE I N I T I A L AMOUNT OF STORAGE A V A I L A B L E EQUALS TWO 136 C MILL ION C F S - D A Y S 137 S T Q R < K K ) = 2 0 0 0 0 0 0 . 0 ' 138 0 C A L C U L A T E THE UNUSED STORAGE AND THE MAXIMUM DISCHARGE 139 DO 8 J=M,N 140 STOR ( J )=STOR( J - D - ( Q ( J J -QREGI 141 I F I S T O R i J ) . L T . 0 i 0 ) GO TO 61 142 I F ( S T O R ( J ) . G T . 2 0 0 0 0 0 0 . 0 ) S T O R ( J ) = 2 0 0 0 0 0 0 . 0 143 I F ( Q R E G . G T . Q M A X ( L ) ) QMAX(L)=QREG 144 GO TO 8 145 61 S T G R I J ) = 0 . 0 146 B = Q ( J ) - i S T O R ( J - U 147 IF i (B .GT .QMAX<L)1 GO TO 62 148 GO TO 8 149 62 QMAXIL)=B 150 8 CONTINUE 151 GO TG 65 152 64 STORCN)=200C000;0 153 C C A L C U L A T E THE FLOOD DAMAGE COST 154 65 C O S ( L ) = 5 . 0 * 1 Q M A X < L I t 1 0 0 0 0 0 . 0 ) * * 2 . 0 - 2 0 . 0 155 I F I C O S ( L ) . L T . O . O ) C O S ( L ) = 0 . 0 156 IF (QMAX(L J . G T . Q R E G ) S T O R ( N ) = 0 . 0 157 M=K-1 158 RM*M 159 C KEEP A RUNNING AVERAGE OF THE FLOOD DAMAGE C O S T , THE 160 C MAXIMUM D I S C H A R G E , AND THE UNUSED STORAGE 161 A C O S ( L , K ) = ( A C O S ( L , M } * ( R M - l . O ) + C O S t L ) J / R M 162 A V G Q ( L , K ) = ( A V G Q l L , M ) * ( R M - 1 . 0 ) + Q M A X I L ) ) / R M 163 A F S T ( L , K ) = ( A F S T ( L , M ) * ( R M - 1 . 0 ) + S T 0 R 1 N ) 1 / R M 164 IF ;<K.EQ.50) GO TO 63 165 GO TC 66 166 63 W R 1 T E ( 6 , 1 0 2 J L , K , C O S ( L ) i A C O S i L , K ) , Q M A X C L ) • 167 1 A V G Q ( L , K ) , Q R E G , S T O R ( N ) » A F S T ( L , K ) 168 102 F 0 R W A T ( 2 X , I 3 , 9 X « 1 I 3 * 2 X , 2 F 1 0 . 2 , 2 X , 4 F 1 2 . 0 , 5 X , F 1 2 . 0 ) 169 66 QREG=QREG+DEL 170 6 CONTINUE 171 RETURN 172 END 56 END OF F I L E $S IG APPENDIX II $ L I S T * S O U R C E * 6 C COMPUTER PROGRAM FOR OPERATING ROUTINE 7 C USING A RUNNING AVERAGE METHOD 8 C 9 C FLOW = THE ACTUAL APRIL TO SEPTEMBER FLOWS 10 C Q = THE GENERATED APRIL TO SEPTEMBER FLOWS 11 C VOL = CUMULATIVE VOLUME OF RUNOFF IN C F S - D A Y S 12 C GREG = THE " I D E A L " REGULATED FLOW VALUE 13 C AQREG = THE AVERAGE " I D E A L " REGULATED FLOW VALUE 14 0 LTM = THE LONG TERM MEAN DAILY TEMPERATURE 15 C TMAX = THE MAXIMUM DAILY TEMPERATURE 16 C TMIN = THE MINIMUM DAILY TEMPERATURE 17 C ATEM = THE AVERAGE DAILY TEMPERATURE 18 C RAIN = THE DAILY P R E C I P I T A T I O N IN INCHES 19 REAL LTM(184 ) 20 INTEGER Y U G O ) 21 DIMENSION F L O « ( 1 8 4 ) , T M A X ( 1 8 4 ) i T M I N I 1 8 4 3 , V O L ( 1 8 4 ) , 22 I Q ( 1 8 4 ) , Q R E G { 1 0 0 < ) , A Q R E G U 0 0 ) , R A I N ( 1 8 4 ) , A T E M < 1 8 4 ) 23 R E A D ! 5 , 1 0 1 , E N O * 1 0 4 ) N , ( F L O W ( J ) , J = i , 1 8 4 ) , C T M A X t J ) , J = 24 1 1 , 1 8 4 ) , 1 T M I N < J ) , J = 1 , 1 8 4 ) 25 101 F 0 R M A T t I 4 / 1 6 < l l F 7 . 0 / ) , 8 F 7 . O / 9 ( 2 0 F 4 . 0 / ) , 4 F 4 . 0 / 9 26 1 ( 2 0 F 4 . 0 / ) , 4 F 4 . 0 ) 27 104 R E A D ( 4 , 1 0 0 , E N D = 1 0 5 ) ( L T M ( J ) , J = 2 , 1 8 4 ) 28 100 F 0 R M A T ( 9 t 2 X , 2 G F 5 . 1 / ) , 3 F 5 . 1 ) 29 105 R E A D 1 3 , 1 0 6 , E N D * 1 0 7 ) M,<RAIN I J ) , J = l , 1 8 4 ) 30 106 F 0 R M A T ( I 4 / 9 ( 2 X , 2 0 F 5 . 2 / ) , 2 X , 4 F 5 . 2 ) 31 107 V O L ( 1 ) = 0 . 0 32 DO 7 1=2,5 33 V 0 L U ) = V 0 L ( I - 1 ) * F L 0 W ( I ) 34 Q( I )=FLOW<I i 35 ATEM (I )= (TMAX{ I I + T M I N d ) ) / 2 . 0 36 7 CONTINUE 37 Q R B G ( 1 ) = 2 0 0 C 0 0 . 0 38 A Q R E G U ) = 0 . 0 39 A = Q . 9 8 2 3 / ( 1 . 0 - 0 * 9 8 2 3 ) 40 B 1 * 0 . 9 8 0 9 7 6 E 00 41 B2f=0 .275145D-09 42 8 3 = 0 . 4 9 8 4 2 0 0 - 0 1 43 DO 71 K=2 ,50 44 C V 0 L ( 1 8 4 ) = THE TOTAL APRIL TO SEPTEMBER VOLUME OF RUNOFF 45 V O L t 1 8 4 ) = 2 7 8 0 0 0 0 0 . 0 46 S = S C L O C K ( 0 . ) 47 Z=RANDNtS) 48 XZ=RAND(S) 49 C C A L C U L A T E VALUES FOR TEMPERATURE SUB-MOOEL 50 DO 2 1=6,184 51 Z N £ F R A N D N ( 0 . ) 52 A T E M C I ) = - 1 . 6 3 5 7 3 * - 1 . 0 3 3 0 4 * L T M I I ) + 0 . 7 6 1 2 1 * 4 A T E M ( 1 - 1 ) 53 1 - L T M U - l ) ) * 4 . 0 9 6 8 3 * Z N 54 2 CONTINUE 55 C C A L C U L A T E VALUES FOR P R E C I P I T A T I O N SUB-MODEL 56 DO 5 1=6 ,184 57 XU=FRAND(0 . ) 58 I F ( X U . E Q . O . O ) GO TO 11 59 I F . U . L E . 3 1 ) GO TO 12 60 IFs< I . L E . 6 2 . A N D i I . G T . 3 l ) GO TO 13 57 61 IF( I . L E . 9 2 . A N D . I . G T . 6 2 ) GO TO 14 62 I F ( I . L E . 1 2 3 . A N D i I . G T . 9 2 ) GO TO 15 63 I F l I . L E . 1 5 4 . A N D . I . G T . 1 2 3 ) GO TO 16 64 IF( I . L E . 1 8 4 . A N D . I . G T . 1 5 4 ) GO TO 17 65 12 R A I N i I ) = 0 . 8 1 7 6 7 2 - 0 . 4 8 9 7 7 9 * A L O G 1 0 I X U * I G O . O ) 66 GO TO 18 67 13 R A I N * I ) = 0 . 7 6 4 9 2 9 - 0 . 4 6 9 7 7 2 * A L O G 1 0 < X U * 1 0 0 . 0 ) 68 GO TO 18 69 14 R A I N < I ) = 0 . 8 5 1 6 9 6 - 0 . 4 9 9 8 1 1 * A L 0 G 1 0 < X U * 1 0 0 . 0 ) 70 GO TO 18 71 15 R A I N ( I ) = 0 . 8 9 9 7 2 9 - 0 . 5 3 4 9 4 1 * A L 0 G 1 0 ( X U * 1 C 0 . 0 ) 72 GO TO 18 73 16 R A I N t I ) = 1 . 2 9 1 1 6 0 - 0 . 7 6 0 1 6 8 * A L O G 1 0 f X U * 1 0 0 . 0 ) 74 GO TO 18 75 17 R A I N ( I 1 = 0 . 9 8 2 5 2 6 - 0 . 5 6 4 4 1 3 * A L 0 G 1 0 I X U * 1 0 0 . O l 76 GO TO 18 77 11 1=1-1 78 GO TO 5 79 18 I F ( R A I N U ) . L T . O . O ) GO TO 19 80 GO TO 5 81 19 RAIN (I )=0 .0 82 5 CONTINUE 83 0 i GENERATE APRIL TO SEPTEMBER FLOWS 84 DO 3 J = 6 , 1 8 4 85 YN=FRANDN(0 . ) 86 TEMP={ATEMi J - l ) + ATEMC J - 2 ) + A T E M ( J - 3 ) • A T E M I J 87 1 - 2 6 . 0 88 I F I T E M P . L T . 0 . 0 ) GO TO 20 89 GO TO 30 90 20 TEMP=0.0 91 30 C = B 2 * T E M P * ( V 0 H 1 8 4 ) - { V O L < J - 1 ) + A * Q { J - 1 ) 5 ) 92 I F I C . L T . O . O ) GO TO 50 93 GO TO 59 94 50 C = 0 . 0 95 59 D=81+C+B3*RAIN(J -4 ) * - 0 .0388248*YN 96 Q I J ) = D * 0 ( J - 1 ) 97 V O L * J ) = V O H J - 1 ) * Q ( J ) 98 3 CONTINUE 99 0 C A L C U L A T E THE " I D E A L " REGULATED FLOW VALUE 100 DO 6 L = l , 2 0 101 1 = 0 102 C THE I N I T I A L AMOUNT OF STORAGE A V A I L A B L E EQUALS 103 C MILLION C F S - D A Y S 104 STOR=2G0000O.0 105 DO 4 J = 6 , 1 8 4 106 I F ( Q ( J ) . G T . Q R E G I L ) ) GO TO 60 107 GO TO 4 108 60 I=H)1 109 Y< I )=J 110 4 CONTINUE 111 M=Y(1) 112 N=Y( I ) 113 C0UNT=N-M+1.0 114 D O J = M , N 115 S T O R = S T O R - 1 Q 1 J ) - Q R E G I L ) ) 116 8 CONTINUE 117 QREGR+l )=QREG< L ) - S T 0 R / C O U N T 118 LL=IQREGIL + 1 ) + 5 0 ) / 1 0 0 119 Q R e G ( L + l ) = 1 0 0 . 0 * L L 120 B=ABS (QREG(L-t-l)—QREGiL ) ) 121 I F t B . E Q - O . O ) GO TO 102 122 6 CONTINUE 123 102 RK?PK 124 C KEEP A RUNNING AVERAGE OF THE " I D E A L " REGULATED 125 C FLOW VALUE 126 A Q R E G t K J = < A G R E G ( K - l ) * { R K - 2 . 0 ) + Q R E G t L + l ) ) / { R K - 1 . 0 ) 127 LL=<AQREG<K)+501/100 128 A Q R B G ( K ) = 1 0 0 . 0 * L L 129 71 CONTINUE 130 W R I T E ( 6 , 2 0 6 ) A Q R E G * K ) , Q R E G ( L + 1 ) 131 206 F 0 R M A T I 2 X , * R E G U L A T E D FLOW<K) = * , F 8 . 0 , I X , f C F S * • 4 X , 132 1 • R E G U L A T E D F L G W I L + 1 ) = * , F 8 „ 0 , I X , « C F S • / ) 133 END END OF F I L E $SIG 

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