Reservoir capacity and yield

RESERVOIR CAPACITY AND YIElD

DEVELOPMENTS IN WATER SCIENCE, 9

advisory editor

VEN TE CHOW Professor of Hydraulic Engineering Hydrosystems Laboratory University of fllinois Urbana, fll., U.S.A.

FUR THER TITLES IN THIS SERIES

1 G. BUGLIARELLO AND F. GUNTER COMPUTER SYSTEMS AND WATER RESOURCES

2 H L. GOLTERMAN PHYSIOLOGICAL LIMNOLOGY

3 Y. Y. HAIMES, W. A. HALL AND H. T. FREEDMAN MULTI OBJECTIVE OPTIMIZATION IN WATER RESOURCES SYSTEMS: THE SURROGATE WORTH TRADE-OFF METHOD

4 J. J. FRIED GROUNDWATER POLLUTION

5 N. RAJARATNAM TURBULENT JETS

6 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS

7 V. HALEK AND J. SVEC GROUNDWATER HYDRAULICS

8 J. BALEK HYDROLOGY AND WATER RESOURCES IN TROPICAL AFRICA

RESERVOIR CAPACITY AND YIElD THOMAS A. McMAHON & RUSSEL G. MEIN Department of Civil Engineering, Monash University, Clayton, Vic., Australia

ELSEVIER SCIENTIFIC PUBLISHING COMPANY 1978

Amsterdam - Oxford - New York

ELSEVIER SCIENTIFIC PUBLISHING COMPANY 335 Jan van Galenstraat P.O. Box 211, Amsterdam, The Netherlands

Distributors for the United States and Canada: ELSEVIER NORTH-HOLLAND INC. 52, Vanderbilt Avenue New York, N.Y. 10017

Lihr:uy of ('()ngn's~ Cataloging in Publication nata

HCHahon, Thomas ]\quinas. Reservoir capacity and yield. (Developments in water science; v. 9) Bibliography: p. Includes index. 1. Reservoirs. I. Mein, Russell G., joint author. II. Title. III. Series. TD395.H24 1978 628.1'3 77-18704 ISBN 0-444-41670-6

ISBN 0-444-41670-6 (Vol. 9) ISBN 0-444-41669-2 (Series) © Elsevier Scientific Publishing Company, 1978 All rights reserved. No part of this pUblication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Scientific Publishing Company, P.O. Box 330, Amsterdam, The Netherlands

Printed in The Netherlands

PREFACE

The text for this book has evolved from the notes written for a workshop held at Monash University in May 1975.

The format of the workshop, the

first of a series on specific topics in water engineering, was about one half lectures supported by printed notes, and one half exercises involving both manual and computer applications ot the theory. For this text, the printed notes have been revised and expanded, and the exercises have been replaced by worked examples.

Most of the latter have

been worked using the streamflow data of one river, the Mitta Mitta (Appendix E), chosen because of its median value of variability with respect to other Australian streams;

compared to North American and European data

it would be classed in the high range of variability. The aim of the text is to provide a comprehensive review and classification of most of the currently used storage-estimation procedures.

The

essential features of each method are presented and limitations inherent in the assumptions are discussed.

Recommendations based on the results of

considerable research effort in this department over a period of several years are made.

The book is written for the practising engineer involved

in storage estimation and for graduate level study in this field. The authors are indebted to the contributions of several postgraduate students who have worked, or are presently working in the Department under the supervision of the senior author.

These are Dr. C. Joy,

Dr. G. Codner, Mr. G. Philips, Mr. C. Teoh, Mr. S. Fletcher and Mr. R. Srikanthan.

Discussions with Dr. R. Phatarfod of the Mathematics

Department at Monash University have also provided a strong source of stimulation.

The authors' colleague, Professor E. M. Laurenson, the third

of the original workshop instructors, has given unfailing support and assistance in this and many other areas.

Finally, for the production of the

manuscript itself, many people have contributed;

in particular, Mrs. J. Helm

typed the final draft most ably, Mr. R. Alexander drafted the diagrams, and Mr. D. Holmes completed the photographic reproduction.

We are very

appreciative of their efforts. T. A. McMahon, R. G. Mein, Department of Civil Engineering, Monash University. September, 1977.

This Page Intentionally Left Blank

CONTENTS

Chapter 1

INTRODUCTION

1.1

THE DESIGN PROCESS

2

1.2

CLASSIFICATION OF RESERVOIR CAPACITY-YIELD PROCEDURES

3

1.3

PROCEDURES IN CURRENT USE

5

DEFINITION OF TERMS

6

2.1

TIME INTERVAL

6

2.2

INFLOW DATA

6

Chapter 2

2.3

2.2.1

Measures of Central Tendency

7

2.2.2

Measures of Variability

9

2.2.3

Measures of Skewness

9

2.2.4

Measure of Persistence

10

2.2.5

Typical Parameter Values

11

2.2.6

Standard Errors of Parameters

11

STORAGE TERMS

14

2.3.1

Active Storage

14

2.3.2

Within-year Storage

14

2.3.3

Carryover Storage

14

2.3.4

Conceptual Storages

14

2.4

RELEASE

15

2.5

RELEASE RULE OR OPERATING RULE

16

2.6

PROBABILITY OF FAILURE AND RELIABILITY

16

2.7

NOTATION

18

CRITICAL PERIOD TECHNIQUES

19

3.1

CRITICAL PERIOD

19

3.2

METHODS WHICH INDICATE RESERVOIR FULLNESS WITH TIME

20

Chapter 3

3.2.1

Mass Curve Method (Rippl Diagram)

20

3.2.2

Residual Mass Curve Method

22

3.2.3

Behaviour (or Simulation) Analysis

24

3.2.4

Semi-Infinite Reservoir

27

3.3

METHODS BASED ON RANGE Hurst's Procedure

3.3.1

3.4

30 31

3.3.2

Fathy and Shukry

33

3.3.3

Sequent Peak Algorithm

33

METHODS BASED ON LOW FLOW SEQUENCES

35

3.4.1

Minimum Flow Approach

35

3.4.2

Alexander's Method

38

3.4.3

Dincer's Method

46

3.4.4

Gould's Gamma Method

49

3.4.5

Carryover Frequency Mass Curve Analysis

52

3.4.5.1

Overlapping Sequence Approach

52

3.4.5.2

Independent Series Approach

55

3.4.5.3

Independent versus Overlapping Series

57

3.4.6

Wi thin-year Frequency Mass Curve Analysis

59

3.4.7

Regional Within-year Storage Estimates

63

3.4.8

Bias in Mass Curve Frequency Analysis

64

3.4.9

Combining Carryover and Seasonal Storages - Hardison's Approach

65

3.5

OTHER CRITICAL PERIOD METHODS

67

3.6

SUMMARY

67

3.7

NOTATION

68

PROBABILITY MATRIX METHODS

71

4.1

GENERAL CLASSIFICATION OF MORAN DERIVED METHODS

71

4.2

A SIMPLE MUTUALLY EXCLUSIVE MODEL

73

Chapter 4

4.2.1

The Discrete Equations for the Mutually Exclusive Model - General case

76

4.3

A SIMPLE SIMULTANEOUS MODEL

78

4.4

COMPUTATION OF STEADY STATE CONDITION

79

4.5

DISCUSSION - MORAN TYPE MODELS

81

4.5.1

82

Further Modifications

4.6

4.7

4.8

GOULD'S PROBABILITY MATRIX METHOD

83

4.6.1

Procedure

83

4.6.2

Practical Considerations

90

RELATED PROBABILITY MATRIX METHODS

93

4.7.1

McMahon's Empirical Equations

93

4.7.2

Probability Routing

96

4.7.3

Hardison's Generali zed Method

97

OTHER MODELS

101

4.8.1

Me1entijevich

4.8.2

K1emes

102

4.8.3

Phatarfod

102

101

4.9

SUMMARY

IDS

4.10

NOTATION

IDS

USE OF STOCHASTICALLY GENERATED DATA

107

5.1

TIME-SERIES COMPONENTS

108

5.2

HISTORICAL DEVELOPMENTS TO 1960

110

5.3

ANNUAL MARKOV MODEL

111

Chapter 5

5.3.1

Practical Considerations

112

5.4

THOMAS AND FIERING SEASONAL MODEL

ll4

5.5

MODIFICATIONS FOR NON-NORMAL STREAMFLOWS

115

5.5.1

Modifying ti

115

5.5.2

Moment Transformation Equations

ll7

5.5.3

Normalizing Flows

121

5.6

TWO TIER MODEL

124

5.7

OTHER CONSIDERATIONS

125

5.8

MODEL VERIFICATION AND PERFORMANCE

126

5.8.1

Unrepresentative Streamflow Data

135

5.9

SIMULATION

When and How to Use Generated Data

5.9.1 5.10

Chapter 6

6.1

136

GENERALIZED RESERVOIR CAPACITY-YIELD RELIABILITY RELATIONS

140

5.10.1

Gould's Synthetic Data Procedure

140

5.10.2

Gug1ij's and Svanidze's Synthetic Data Procedures

5.11

135

NOTATION

144 144

QUANTITATIVE ASSESSMENT OF CAPACITY-YIELD TECHNIQUES FOR SINGLE RESERVOIRS

147

CRITICAL PERIOD AND PROBABILITY MATRIX METHODS

147

6.1.1

Mass Curves and Minimum Flow (Waitt)

147

6.1.2

Alexander's Method

149

6.1.3

Overlapping Series Frequency Mass Curve Method (Thompson)

6.1. 4

150

Independent Series Frequency Mass Curve Method (Stall)

151

6.1.5

Gould's Probability Matrix Method

152

6.1.6

Further Comparison of Gould and Behaviour Methods

153

6.1. 7

Summary

154

6.2

CAPACITIES BASED ON STOCHASTIC DATA GENERATION

155

6.3

RAPID RESERVOIR CAPACITY-YIELD PROCEDURES

159

6.4

SAMPLING ERROR OF STORAGE AND DRAFT ESTIMATES

164

6.5

RECOMMENDATIONS

169

6.6

NOTATION

170

MULTI-RESERVOIR SYSTEMS

171

A TYPICAL PROBLEM

171

Chapter 7 7.1

7.1.1

Traditional Solution

172

7.1. 2

Recycled Historical Sequences

174

7.2

STOCHASTICALLY GENERATED FLOWS

175

7.2.1

Key Station Approach

175

7.2.2

Principal Component Approach

176

7.2.3

Regression Method

176

7.2.4

Residual Approach

176

7.2.5

Multi-site Model Performance

177

7.2.6

Use of Generated Data

177

7.2.7

Application to Multi-storage Systems

178

7.3

TRANSITION MATRIX APPROACH

179

7.4

OTHER ALTERNATIVES

180

7.5

NOTATION

181

REFERENCES APPENDIX A·

181 Procedure to adjust Storage Estimate for Net Evaporation Loss

APPENDIX B

Adjustment for Assumption of Independence of Annual Flows

APPENDIX C

APPENDIX E

193

Theoretical Justification of a Non-Seasonal Markov Model

APPENDIX D

190

199

Newton-Raphson Method for solving an Inexplicit Variable

202

Flow Tables for Mitta Mitta River

203

AUTHOR INDEX

205

SUBJECT INDEX

207

This Page Intentionally Left Blank

CHAPTER 1

INTRODUCTION The storage required on a river to meet a specific demand depends on three factors;

the variability of the river flows, the size of the demand,

and the degree of reliability of this demand being met.

As this and sub-

sequent chapters will show a large number of procedures have been proposed to estimate storage requirements.

This text is concerned with examining

and classifying these procedures with the aim of recommending the ones most suitable for particular requirements. In its simplest form the problem being tackled is shown in Fig. 1.1. It is required to divert water from the stream with flow sequence Q(t) to meet the demand of perhaps an urban area or of a rural irrigation scheme. Alternatively it may be necessary to augment the low flow periods of the river.

In any event, the question being posed is:

"How large does the

reservoir capacity CC) need to be to provide for a given controlled release or draft DCt) with an acceptable level of reliability?"

Other variations

of this question are possible (such as determining release for a given capacity) but the basic problem remains unaltered;

the relationship between

inflow characteristics, reservoir capacity, controlled release, and reliability must be found. Following definition of terms in the next chapter, Chapters 3-5 examine in detail all the common and some relatively unknown procedures for

Stream flow sequence

QW . . Demand area

Controlled release sequence OW Reservoir with active storage capacity C

FIG. 1.1

SPil~ An idealized view of the reservoir capacity-yield problem.

2

solution of the single reservoir problem.

The performance of several of

the methods is assessed in Chapter 6, where recommendations for use of particular procedures are made. The use of more than one reservoir storage to satisfy the demand adds a significant degree of complication to the problem.

The reservoirs may be

on the same stream, different streams, or not on any stream (e.g. pumped storage).

Additional complexity may result from topographical or other

constraints which restrict flow between reservoirs and thus reduce system flexibility. 1.1

The multi-reservoir problem is discussed in Chapter 7.

THE DESIGN PROCESS In the early analysis of a water supply development, a number of

alternative darn sites would be investigated, not only for the construction requirements but also from the hydrologic point of view.

For such studies

and for hydrologic reconnaissance or regional reviews, quick and relatively simple techniques for estimating the reservoir capacity-yield relationship are required. The methods which can be used for rapid assessment are designated as preZiminary design techniques. Simplifying assumptions are often made; for example, releases may be assumed to be constant, evaporation and sedimentation losses ignored, the probability of failure may not be considered, and the seasonal characteristics of the river flows may not be taken into account.

For these preliminary methods, accuracy is reduced for ease of

application. After using preliminary design techniques to eliminate unsuitable reservoir sites from consideration, the remaining few should be evaluated using a finaZ design technique.

These techniques are often more compli-

cated because they take into account most, or all, of the factors which influence storage.

Thus, properties of the river inflOWS, variation of

releases with season, the possibility of water restrictions, the effect of evaporation, and the probability of not being able to meet the demand must be realistically treated. In the text, recommended methods are designated as being suitable as preliminary or as final design techniques.

3

1.2

CLASSIFICATION OF RESERVOIR CAPACITY-YIELD PROCEDURES Reservoir capacity-yield procedures can be classified into three main

groups although the distinction between groups is not always clear-cut. The first group (critical period techniques) includes methods in which a sequence (or sequences) of flows for which demand exceeds inflows is used to determine the storage size.

Those methods related to Moran Darn Theory

or similar procedure are included in the second group, part of which is grouped under a general umbrella of probability matrix methods.

The third

group consists of those procedures which are based on generated data.

A

detailed classification along with author references is given in Fig. 1.2. The methods are discussed in detail under these groupings in Chapters 3, 4 and 5, respectively. Briefly, critical period methods are those in which the required reservoir capacity is equated to the difference between the water released from an initially full reservoir and the inflows, for periods of low flow. For the procedures designated as mass curve, minimum flow, or range, the storage is normally associated with the severest drought sequence in the historical record.

If historical data is used with these procedures, an

estimate of the risk of being unable to meet the design releases (probability of failure) cannot be made.

In contrast, other critical period

methods enable the reliability of the reservoir to meet the demand to be estimated. The second group of procedures is considered to be a development of Moran's Theory of Storage (1954, 1955, 1959).

In essence Moran derived an

integral equation relating inflow to reservoir capacity and releases such that the probable state of the reservoir contents at any time could be defined.

However, except for idealized conditions the solution was

intractable.

Subsequently Morml considered time and flow to be discon-

tinuous variables ffild showed how reservoir capacity, release and inflow could be related to each other by a system of simultaneous equations, but the method has several shortcomings.

Gould (1961) modified Moran's approach

to a general procedure of direct practical use to the water engineer.

In

this context it is worth noting that a Russian, Savarenskiy, published in 1938 similar ideas to those presented later by Moran and Gould but it is only recently that his contributions have become known in English technical literature. Although procedures for estimating reservoir capacity-yield relationships using streamflow data generated by stochastic methods were first used

RESERVOIR CAPACITY - YIELD ANALYSIS

PROCEDURES BASED ON DATA GENERATION

GENERAL WITH PROBABILITY

OVER FREQUENCY

YEAR FREQUENCY

Hazen

(1914)

Sudler

(1927)

Barnes (1954)

Rippl

(1883)

King

(1920)

Waitt (1945)

Hurst 0951.56.57.6S} Alexander (1962) Thompson l1 950) U.S.G.S. (c1960) Stall (1962) Hardison ('965)

Fathy & Shukry C1956} Dincer

Thomas (1963) (Sequent peak)

Gould

Wilson (1949)

(1965)

Hurst

(1953.55)

Law

(1962)

Guglij

(1969)

Svandze (1964)

(1964)

MORAN THEORY

I CONTINUOUS TIME Moran

(1956)

Gani

(1965)

Gant & Prabhu (1958.59) Gani & Pyke (1960.52)

Melentijevich (1966) (1967)

DISCONTINUOUS TIME

(1976)

I CONTINUOUS

DISCONTINUOUS

~

VOLUMES

Gani &. Prabhu (1957) (1958)

Prabhu

Ghosal

Langbein

- - - - - - - - - - - - - - - - - - - Hardison (r965)

I

I

NON-SEASONAL

SEASONAL

------r=

I

(1959.60>

INDEPENDENT Moran (1954)

CORRELATED Lloyd

(1963)

INDEPENDENT Moran

(1955)

Lloyd &. Odoom (1964)

FIG. 1.2

A classification of reservoir capacity-yield procedures

(19611

Gould Maass

Gould £1960 McMahon (1916)

I CORRELATED

Dearlove & Harris (1965) Venetis (1969)

5

more than sixty years ago, it was not until the advent of high-speed digital computers in the sixties that such procedures became established in engineering hydrology.

Stochastic data generation is the basis of the third

group of storage-yield procedures. It should be noted that many of the methods shown in Fig. 1.2 are included for only their historical importance in the development of a particular technique or groups of techniques;

they are often impractical

or use unacceptable assumptions in their derivation. 1.3

PROCEDURES IN CURRENT USE Very little published information is available on techniques currently

in use by water authorities around the world.

A questionnaire survey of

Australian water authorities by the senior author in 1974 showed that, in general, storage capacity designs were based on mass curve or simulation analyses using historical streamflows.

These two methods were used both

for preliminary and final design calculations.

In about one half of the

cases, the probability of the reservoir not being able to meet the demand was computed, although it was never used as the sole design criterion. Data generation techniques have been used by about one half of the Australian water authorities although more than that indicated their belief in the potential of the method. There is no reason to assume that the methods in current use in Australia are any different to those in current use overseas.

6

CHAPTER 2

DEFINITION

OF TERMS

This chapter is concerned with defining and explaining several of the terms used in reservoir capacity-yield analyses.

The meaning of some of

these terms sometimes differs from one author to another;

it is therefore

important that the reader be clear as to which interpretation is used in this text. The definitions given include those for several statistical measures. These are necessary to specify the characteristics of the river inflows to the reservoir because these characteristics have a major influence on storage requirements.

Other important terms discussed in this chapter

include several storage terms, release, release rule, and definition of probability of failure. 2.1

TIME INTERVAL The time interval required for the inflow data depends on the size of

the storage and on the degree of accuracy required.

For small storages

designed to provide water in excess of the river flow for only a month or two in the year, daily flow data are required.

For larger storages, monthly

data are usually adequate to define the variations of streamflow with season (seasonality), although annual data can often provide sufficiently accurate results for preliminary design estimates. As a general rule, monthly data are used for most studies.

With this

time interval the data processing time is not excessive, variations in streamflow and releases throughout the year are adequately accounted for, and records are readily available.

A minor drawback in dealing with monthly

flow volumes is that the calendar months are not equal in length;

the effect

of this on storage is small, however, and is usually ignored. 2.2

INFLOW DATA It is not possible to predict the future sequence of flows of a

natural stream.

All methods therefore use historical flow data or parameters

derived from it, and thus implicitly assume that these data are representative of the true streamflow characteristics.

Hence, any value of storage

(or draft) estimated using historical data has a sampling error inherent in it (see Sec. 2.2.6).

7

In the analytical procedures that follow it is assumed that inflows into the reservoir occur as daily, monthly, or annual discrete events. These will be available as measured data at the site in question, or will have been estimated by either regression analysis (see, for example, Searcy, 1966, or Brown, 1961) or with a deterministic process model such as the Stanford Watershed Model (Crawford and Linsley, 1966). It is also assumed that the data have been checked for homogeneity

and consistency.

In this context homoger:eity requires that identical flow

events in a time series are equally likely to occur at all times.

Consis-

tency requires that there has not been any physical change at the stream gauging station that might affect the recorded flows.

Searcy and Hardison

(1960) discuss this aspect in detail. The inflow data can be represented as a frequency distribution of flows, such as those plotted for several rivers in Fig. Z.1.

Often these

distributions can be approximated by standard theoretical distributions such as the Normal, log-Normal, Gamma, Weibull, Extreme Value Type I, and logPearson Type III.

These distributions are defined by parameters of the

flows, for example, mean, standard deviation,and coefficient of skewness. Another important flow parameter is the lag-one serial correlation which describes flow persistence. If the flow volumes in successive time intervals are designated as Xl' x z' ... ' Xi' ... , xn ' the parameters are defined as follows; 2.2.1

Measures of Central Tendency * arithmetic mean n

L x.

x

1

1

(2.1)

n

* median The median is the middle value or the variate which divides the flow frequency distribution into two equal portions. The armithmetic mean is more commonly used because of its computational simplicity.

In extremely skewed distributions, however, the

median will provide a better indication of central tendency.

8

27 24 21

DIAMANTINA RIVER

~ t: V

WARRAGAMBA RIVER

~18

6 5 4

~

15

g-:J 12 L.t 9

5-3

~ 2 u. 1

6 3

O+------r--~~-L~--r_--~

o

1

2

3

O~--r_-r--~~~~~~~

4

8 7 >- 6 5 ~ 4 g- 3 L.t 2

4

6

Annual flow (xl0 9 m 3 )

MITTA MITTA RIVER

g

2

0

Annual flow (xl09 m3 )

8

MEKONG RIVER

7 ~ 6

~ 5 5- 4

~ 3

u. 2 1 04-~~---r--~----~~'_--~

1

2

o

3

100

Annual flow (xl0 9 m 3 ) 18 16 14

YARRA RIVER

140

160

180

BATANG PADANG RIVER

7 ~ 6 ~ 5

~12

~10

4 v 3 L.t 2 1+----'

5- 8 ~

120

Annual flow (xl0 9 m 3 )

:J

C'

6

u. 4 2 O+---r_----._----~~~~~

o

0.2

0.1

0.3

0.4

O+---r_--~--~~~~--~

o

0.5

0.6

0.7

0.8

Annual flow (x 10 9 m 3 )

8 7 >- 6

g5 ~ 4 g- 3 L.t 2 1 O~----._--~----._--~~~

0.6

0.8

1.0

FIG. 2.1

1.2

1.4

Frequency distributions of annual 'flows of selected rivers.

0.9

200

9

2.2.2

Measures of Variability * standard deviation 1 s

n

[-1 n-

L (x.1

1 n \ [-1 ( L

n-

2

x.1 - n

(2.2) 1

X-2)]2

(2.3)

For computational convenience Eq. 2.3 is preferred.

The standard deviation

is the basic measure of variability.

* variance is the square of the standard deviation. * coefficient of variation

cv

(2.4)

The coefficient of variation is a dimensionless measure of variability and is widely used in hydrology. * index of variaJ;iUty 1

[-1 n-

n

L (log 10 x.1

(2.5)

The index of variability is the standard deviation of logarithms of flows. 2.2.3

Measures of Skewness The lack of symmetry of a distribution is called skewness. * coefficient of skewness a

Cs where

(2.6)

53 a

n (n-l) (n-2)

n

L ex.1 - x) 3

(n-l~ (n-2) [L

x 3 _ 3x

(2.7)

L x2

+

2nx 3]

(2.8)

This dimensionless measure relates to the third moment of the data and is one measure defining the shape of the distribution.

Data with

positive skewness are skewed to the right (Fig. 2.2). Another measure of skewness used in hydrology is given by: * Pearson second coefficient of skewness, given by 3 (mean - median) standard deviation

(2.9)

10

Median I Mean:

>-

+'

(/)

s::: Q)

"'C

I

>-

:!: ..0 ~

..0

o

~

a.. Magnitude

Magnitude (b)

(a) FIG. 2.2

Skewed distributions, (a) Positively skewed; (b) Negatively skewed.

Typically, flow distributions have a positive skewness as shown in Fig. 2.1.

The degree of skewness generally decreases as the time interval

of the data increases.

Thus, the distribution for annual flows will

normally be less skewed than the distribution for monthly flow of the same river. 2.2.4

Measure of Persistenct Persistence is the non-random characteristic of a hydrologic time-

series.

For example, a month with high streamflow will tend to be followed

by another of high flow rather than by one of low flow.

This feature, which

is important in storage-yield studies, but which is not a parameter that can be included in theoretical distributions, is quantitatively characterized by the serial correlation coefficient.

It indicates how strongly one event

is affected by a previous event. * serial correlation 1

---k n-

~

_1_

n-k

where

n-k

I

x2 i

n-k 1 n-k n-k \ x.l x.l + k - (----k)2 \ x.l L\ x.l + k L nL

(2.10)

_

lag k serial correlation coefficient, and lag between flow events.

Except for some procedures using stochastic data generation, lag one serial correlation (k chapters.

= 1 in Eq. 2.10) is the only lag considered in later

11

It should be noted that in addition to Eq. 2.10 there are several other procedures for calculating serial correlation of time-series data. The characteristics of each are discussed by Wallis and O'Connell (1972). For reservoir capacity-yield analyses the differences among the procedures are of little importance and Eq. 2.10 is recommended. Serial correlation is usually significant for monthly flow data.

For

annual flow data the majority of streams do not have a serial correlation coefficient significantly different from zero;

however, there are still a

large number of streams with significant coefficients. 2.2.5

Typical Parameter Values In order to illustrate this discussion dealing with parameter values,

monthly and annual flow parameters for five Australian and two South-east Asian streams are tabulated in Table 2.1.

As well, Fig. 2.3 shows, for

156 Australian streams, frequency histograms of coefficient of variation, coefficient of skewness, and serial correlation coefficient of annual flows. Various continental values are superimposed for comparison along with estimates of continental mean annual runoff.

(The latter values are taken

from Australian Dept. of National Resources, 1976.) 2.2.6

Standard Errors of Parameters It must be emphasized that the parameter values defined in the previous

section are no more than estimates of the population values.

An

indication

of the magnitude of the error of the estimate is given by the standard error of the parameter.

These are defined as follows:

standard error of mean standard error of standard deviation

(2.11) 1

(2.12)

s/(2n)"

standard error of coefficient of variation

(2.13)

standard error of coefficient of skewness

(2.14 ) 1

standard error of serial correlation coefficient where

(n - k - 1)" n - k

s

standard deviation of flow volumes,

n

number of items of data, and

k

lag between flow events.

(2.15)

TABLE 2.1

Annual, monthly seasonal and non-seasonal parameters for selected Australian and South-east Asian streams.

River (Country) (Area km 2 ) Diamantina (Australia) (115 000) Warragamba (Aus t Tali a) (8750) Mi tta Mitta (Aus t ra1 i a) (4710) Mekong (Thailand/ Laos) (299 000) Yarra (Australia) (334) Bat.ang Padang (Malaysia) (378)

I

King (Australia) (451)

.

Parameter

x

C V C s r

x CV C s r

x

C V C s r

x C V C s T

x C CV s r

x C V C rs

x

C V Cs r

Annual

.

All Months

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

-

4.5+ 1.82 2.63 0.33

10.1 0.97 0.69 0.62

28.7 1.74 2.51 0.59

'8.1 1.62 1.67 0.91

11.9 1.62 1.86 0.18

6.0 2.78 3.83 0.97

5.4 3.21 3.97 0.71

2.5 2.36 2.56 0.66

0.3 1.84 1.47 0.01

0.6 1.96 3.06 0.40

0.9 1.98 2.76 0.27

1.0 2.07 2.10 -0.16

-

5.2 1.60 2.76 0.20

8.5 2.34 5.76 0.76

10.3 2.62 4.43 0.58

6.8 2.28 5.58 0.20

10.7 1. 81 2.83 0.39

15.3 1. 83 2.69 0.53

14.0 1. 58 2.74 0.53

9.7 1.85 4.59 0.42

5.7 1.35 2.93 0.30

6.8 1. 76 3.85 0.57

3.2 1.55 2.77 0.27

3.8 1.72 3.39 0.32

-

3.1 0.66 1.79 0.58

2.1 0.58 1.12 0.60

2.4 1.03 3.27 0.54

3.7 1. 74 4.88 0.86

4.9 1.13 4.11 0.80

8.1 1.16 3.17 0.63

12.0 0.87 1.34 0.59

15.6 0.76 1.30 0.65

15.6 0.52 0.88 0.73

16.2 0.56 0.58 0.65

10.6 0.62 0.62 0.79

5.7 0.69 1.72 0.61

-

3.2 0.19 -0.14 0.89

2.3 0.17 -0.18 0.91

2.2 0.16 -0.00 0.88

2.1 0.19 0.20 0.86

3.0 0.30 0.96 0.69

6.4 0.32 0.71 0.56

13.0 0.25 0.49 0.62

23.1 0.22 -0.03 0.49

20.7 0.22 0.53 0.52

12.7 0.28 0.47 0.50

7.0 0.30 1.26 0.85

4.4 0.22 -0.00 0.95

-

3.3 0.89 5.15 0.53

1.9 0.72 3.89 0.35

2.0 0.74 2.29 0.36

3.1 1.24 3.31 0.51

5.4 1.05 2.23 0.38

9.0 1.03 3.52 0.59

13.3 0.58 0.91 0.49

15.8 0.51 0.67 0.46

15.6 0.48 0.83 0.44

14.3 0.61 1.48 0.52

9.9 0.75 1.53 0.61

6.4 1.11 4.11 0.36

-

9.6 0.32 1.63 0.74

6.9 0.27 0.70 0.62

6.9 0.28 0.66 0.75

8.3 0.27 0.30 0.66

9.3 0.29 0.32 0.53

7.1 0.23 0.31 0.47

5.9 0.22 0.54 0.56

5.6 0.21 0.35 0.25

6.9 0.27 1.98 0.58

9.43 0.31 0.59 0.59

12.4 0.34 0.77 0.48

11.7 0.35 0.79 0.71

4.4 0.74 1.19 0.16

4.2 0.70 0.84 0.24

4.6 0.69 1. 74 -0.35

8.2 0.39 0.32 0.03

9.6 0.54 1.03 0.18

10.7 0.58 1.12 0.01

11.6 0.35 0.24 0.44

12.0 0.42 0.46 0.35

11.0 0.39 0.92 0.36

9.3 0.45 0.55 0.41

7.8 0.49 0.90 -0.05

6.6 1.11 5.11 -0.11

7 1.19 1. 85 0.11

2.80 4.71 0.53

122 1.11 2.67 0.30

2.14 4.73 0.45

270 0.57 1.50 0.06

1.06 1.91 0.71

480 0.17 -0.08 0.45

0.90 1.27 0.74

538 0.40 0.77 0.12

0.99 1. 78 0.62

1700 0.18 0.60 0.47

0.40 1.46 0.60

2340 0.19 -0.31 -0.11

0.63 1.64 0.32

-

Mean is expressed as depth of runoff in mm.

tMonth1y flows are expressed as percentage of mean.

x Cs

is mean;

C

v

is coefficient of variation;

is coefficient of skewness;

r

is serial correlation.

13

EUROPE & ASIA NORTH AMERICA

25 AUSTRALIA

>~ 20

,"",500

E E

:::J

::t: o

u.

Q)

.... 400

a

~ 15

§ 300 ~

10

5 OL..LJL..LJL..f-I-U.L....&.+...L.L.L..L+L~.&..,

2.0 1.0 0.5 1.5 o Annual coefficient of variation ~OTHER

CONTINENTS "AUSTRAlIA

~AUSTRAlIA

25

>-

1./ NORTHERN

c:

I' HEMISPHERE

u

~ 20

>c:

CT

20

u

~

Q)

U.

g.

15

15

Q) ~

u. 10

-

5

10

-

5

o -[ 2 3 Annual coefficient of skewness

FIG. 2.3

n-n,

-0.2 0 +0.2 +0.4 +0.6 Annual serial correlation

Annual flow parameters for Australian streams compared with other continental values. (Lines indicate average values for the continents indicated.)

14 As a general rule, the interpretation of these errors may be likened to the interpretation of the standard deviation of a variable.

If we

assume the parameters values are normally distributed (this is a reasonable approximation in most circumstances) two-thirds of the values will lie within ± one standard error. 2.3

STORAGE TERMS

2.3.1

Active Storage The active storage of a reservoir is the water stored above the level

of the lowest offtake.

It is thus equal to the total volume of water stored

less the volume of "dead" storage (the volume below the level of the offThroughout this text the terms storage and active storage are used

take).

synonymously. 2.3.2

Within-year storage Many small reservoirs fill up and spill on the average several times

a year.

These reservoirs are constructed to provide water over a short

drawdown period of only a month or two of low flows.

The estimation of the

storage required in this case is termed a within-year storage analysis. 2.3.3

Carryover storage Where the reservoir fills up and spills only every few years on the

average, the water stored at the end of one year is carried over to the next.

This is called carryover storage.

On the other hand seasonal storage

results from the fluctuations of inflows and outflows during the year.

In

procedures that utilize only annual data, the seasonal effects are not taken into account. cedures;

In this text such procedures are known as carryover pro-

those concerned only with seasonal storage are known as within-

year procedures.

Figure 2.4 illustrates the difference between these two

components. 2.3.4

Conceptual Storages A finite storage is a conventional storage which can spill and run dry.

Not all reservoir storage-yield procedures assume finite storages.

infinite storage is one that can spill but never run dry.

A semi-

It is a

conceptual tool and the consequences of using it are discussed in Chapter 3. Another conceptual storage is the infinite storage which can empty but never spill.

15

FULL

....C (JI

....4lC

Carryover storage

0

(J

... ...>4l

'0 (JI

4l

a:

EMPTY

n+12

n+6

n

Time (months) FIG. 2.4

2.4

Illustration of carryover and within-year storages showing the increase of storage necessary to cater for seasonal fluctuations.

RELEASE

Release is the volume of controlled water released from a reservoir during a given time interval.

The term release is used synonymously in

this text with the terms yield, draft, outflow and regulation and describes regulated flow from the reservoir.

Spill is regarded as uncontrolled flow

from the reservoir and will take place only when the water stored in the reservoir is above full supply level. Release is often expressed as a percentage of mean flow having values generally around 50 - 70%, and because of net evaporation losses rarely exceeds 90%. Data from Australian water authorities indicate that the median regulation of Australian reservoirs is 65%. markedly across a continent.

Potential regulation varies

Hardison (1972) has shown that for mainland

United States potential regulation varies from 57% in Lower Colorado to 95% in Tennessee.

In Australia, potential values vary from 70% in the arid

zone to 95% in Tasmania (McMahon, 1977). To estimate the design capacity of a reservoir it is necessary first to estimate the demands which will be placed upon the storage at some time (or times) in the future.

This is a difficult and uncertain task which is

beyond the scope of this text.

A general discussion of ways of tackling

this problem is given in Linsley and Franzini (1974).

16

2.5

RELEASE RULE OR OPERATING RULE Usually the volume of water released from a reservoir is equal to the

volume of water required (or demanded) by the consumers.

However, there

may be periods when either the reservoir level is so low that the water required cannot be supplied, or that prudence dictates that only part of the water demanded is released from storage (for example, water restrictions for an urban centre).

Another factor in the decision may be the time of the

year and the expected inflows for the subsequent period.

The way in which

releases are controlled is called the release or operating rule. The simplest release rule is to supply all of the water demanded [Fig. 2.5(a)].

In this situation, the draft is independent of reservoir

content and season.

If there is insufficient water in the reservoir to meet

the required draft, the storage empties.

Release

~ 100

~ 100

"0

"0

c: ('iJ

c:

('iJ

E Q) Cl

E

Q)

0

Cl

C

0

0

FIG. 2.5

C

0

Water stored (al

Water stored (b)

Example of two operating rules: (a) Simple operating rule; (b) Operating rule with restrictions.

The more complicated release rule shown in Fig. 2.5(b) is typical of that used by a metropolitan water supply authority.

As the volume of water

stored in the headwater reservoirs decreases, restrictions are placed on users so that demand falls and releases are lowered.

It will be noted in

later chapters that few procedures can accommodate such an operating rule. In the majority of reservoir capacity-yield techniques, constant draft is assumed, that is, seasonal fluctuations in demand are not considered. 2.6

PROBABILITY OF FAILURE AND RELIABILITY A number of definitions of probability of failure of a reservoir are

given in the technical literature.

Probably the most common

one defines

probability of failure as the proportion of time units during which the

17 reservoir is empty to the total number of time units used in the analysis. Hence, P

~

e

where

p

(2.16) the number of time units during which the storage is empty, and

N

the total number of time units in the streamflow sequence.

The corresponding definition of reZiabiZity is defined as: R

e

1 - P

(2.17)

e

These definitions of probability of failure and of reliability are not very realistic for most situations.

A city water supply reservoir, for

instance, would never be permitted to empty; apply long beforehand. to Eq. 2.16 but where

restrictions on releases would

An alternative definition sometimes used is similar p

is taken as the number of months during which

restrictions are necessary, that is, months during which the reservoir cannot meet the demand under the adopted operating rule. Another definition of reliability, voZumetric reZiabiZity, is equivalent to Fiering's (1967) performance index;

it relates the volume of

water supplied to the volume of water demanded for the study period as follows: R

v

actual supply demand

(2.18)

This definition has merit for overall reservoir performance, but can mask the severity of any restrictions imposed. The definition of probability of failure used for the remainder of this text is Eq. 2.16 unless otherwise indicated.

Although it may be

somewhat unrealistic in practice, it enables comparisons to be made between different methods.

The reader can, of course, use an alternative definition

to suit his purpose for most of the methods recommended.

18

2.7

NOTATION

(n-l~ (n-2)

a

(xi-x) 3

l:

(Eq. 2.7)

coefficient of variation

cs

coefficient of skewness

I

index of variability (standard deviation of logarithms of flows)

v

k

lag between flow events under analysis

n

number of items of data

N

number of time units

p

number of months reservoir is empty

p

e

probability of failure (emptiness) serial correlation coefficient

r

lag k serial correlation coefficient

1 - Pe

R

reliabilitv

R v

volumetric reliability defined as water supplied divided by water demanded

s

standard deviation

e

X.

1

~

=

' ·1 th perlo . d fl ow volumes d urlng

flow volumes in successive time intervals x

mean flow

19

CHAPTER 3

CRITICAL PERIOD TECHNIQUES The methods presented in this chapter typify two general approaches to the reservoir capacity-yield problem.

The first group of methods all use

the historical inflows and projected demand to simulate the volumetric behaviour of the reservoir, that is, the state of fullness versus time.

The

second group of methods has in common that only the periods of low flow (droughts) in the record are used in the analysis. Some of the methods of each group provide a reservoir size that will not fail for the historical inflow sequence;

the remaining methods allow

the user to determine the storage size for a given probability of failure. However, all methods base the estimate of required storage capacity on sequences of low flows and hence can be placed under the general heading of critical period techniques.

3.1

CRITICAL PERIOD A criticaZ period is defined as a period during which a reservoir goes

from a full condition to an empty condition without spilling in the intervening period.

The start of a critical period is a full reservoir;

of the critical period is when the reservoir first empties. I

FULL

~

a critical

Thus, only one

I

period ~

-----------c--I 1

I

I 1

I

·1 I 1

I I I

I I

I I

I I

I I I EMPTY~-+-1-9-40-4r-1-9-41~~19-4-2-r-1-94-3~--1-94-4-+~19~4-5~~1~94-6~ Years

FIG. 3.1

the end

Behaviour diagram showing critical periods. {Mitta Mitta River; draft = 75%)

20

faiZure can occur during a critical period. where there are two critical periods.

Figure 3.1 gives an example

Note that the remaining failures

(empty condition) of the reservoir in years 1945 and 1946 are not included in a critical period. This definition is not universally accepted.

For example, the

U.S. Army Corps of Engineers (1975) define the critical period from the full condition through emptiness to the full condition again and use the term

criticaZ draw down period to apply from fullness to emptiness. 3.2

METHODS WHICH INDICATE RESERVOIR FULLNESS WITH TIME

3.2.1

Mass Curve Method (Rippl Diagram) The mass curve technique (following Ripp1, 1883) would appear to be

the first known rational method for estimating the size of storage required to meet a given draft (see classification, Fig. 1.2).

15000 M

E

'"o
10000

5: o

;;::

., (I)

.>

-3

A

5000

E

"

<.)

I

1957 58

59 1960 61

62

63

I

64 1965 66

I

I

67

I

68 1969

Year

FIG. 3.2

Reservoir capacity-yield analysis by mass curve (Mitta Mitta River; draft

75%) .

The steps in the procedure, as shown in Fig. 3.2, are as follows: (i)

For the proposed dam site in question, construct a mass or cumulative curve of the historical streamflows. (As a general rule this curve should be constructed using monthly flows.)

(ii)

Superimpose on the mass curve the cumulative draft line for the reservoir such that it is tangential to each hump of the mass inflow curve.

(iii)

Measure the largest intercept between the mass inflow curve and the cumulative draft line.

21 In the example shown in Fig. 3.2 the intercept C2 is greater than intercept C and so the design capacity would be taken as C . (No allowance l 2 has been made for evaporation.) Thus, a reservoir of capacity C , full at time zero (Jan. 1957), and 2 experiencing the historical inflows and assumed drafts, will be full at A, begin to empty from

A to

B, fill again from

E, just empty when it reaches

F

B to

D, spill from

and begin refilling again.

Period

D to EF

the critical period in this example.

Assumptions: (i)

The reservoir is full at time zero and consequently at the beginning of the critical period.

(ii)

In using the historical flow data it is implicitly assumed that future flow sequences will not contain a more severe drought than the historical flow sequence.

Limitations: (i)

The draft is usually taken to be constant.

Seasonality in

demand could be included but restrictions in releases (as a function of storage content, for example) are difficult to handle. (ii)

Storage capacities estimated by mass curve procedures increase with increasing length of record.

It is therefore

difficult to relate storage size to economic life. (iii)

It is not possible to compute a storage size for a given probability of failure.

On the other hand the rank 1

drought in a record of length of failure of P = 1/(N (iv)

+

N has an implied probability

1) as shown in Sec. 3.4.5.

The method of analysis does not take into account net evaporation losses.

If required, an additional amount of

storage has to be added to the mass curve estimate to cover this loss.

Details of a procedure to do this are

given in Appendix A. In this context it is of interest to note that King (1920) modified Rippl's method to include evaporation losses and incident rainfall in the mass balance of water over the critical period.

Attributes: (i)

The procedure is simple and it is widely understood.

is

22

(ii)

It takes into account seasonality, serial correlation

and other flow parameters insofar as they are included in the historical flows used in the analysis.

EXAMPLE 3.1 Using the mass curve method, find the storage required on the Mitta Mitta River (Appendix E) to meet a draft of 75% of the mean flow.

*

*

*

*

*

Because the scale of the graph required to plot the entire 34 years of data for the Mitta Mitta River is too large for this page size, only a portion of the plot (the last 13 years) is shown in Fig. 3.2. The slope of the draft line is 79.6 x 106m3jmonth determined as 75% of the mean monthly flow (106.1 x 10 6m3).

After drawing the draft line

tangential to major peaks, the storage is determine from the maximum intercept between the draft line and the mass curve. From Fig. 3.2 the required storage is 1110 x 10 6m3 , corresponding to a critical period of 3 years and 4 months.

(Note that the storage has not reached a full condition at the

end of the period of record.)

3.2.2

Residual Mass Curve Method The residual mass curve is a slightly more complicated version of the

mass curve. (i)

The steps in the procedure (see Fig. 3.3) are: Subtract the mean flow from each flow value of the record. (If monthly data are used, subtract the monthly mean; for annual data use the annual mean.)

The resulting flows

are called residual values. (ii)

Plot the residual mass curve (cumulative values) and superimpose the cumulative draft line (expressed as a residual) such that the draft line is tangential to each hump of the residual mass curve.

(iii)

Measure the largest intercept between the mass inflow curve and the draft lines.

23

3000

'"E

~ 2000

~ 1000

:5 "'"'

1

Draftmean flow

= 26.5 x 106 m 3 Imonth

III

E

_ -1000 III :::

-0

-~ -2000

0::

-3000 1935

1940

FIG. 3.3

1945

1950 1955 Year

1960

1965

1969

Reservoir capacity-yield analysis by residual mass curve (Mitta Mitta River; draft = 75%).

Overall the assumptions, limitations and advantages of the residual mass curve approach are similar to the mass curve procedure.

However, it

does have an added disadvantage in that the method is not as easily understood as the straightforward mass curve procedure.

On the other hand, it

permits the use of a cumulative inflow scale many times larger than that used for a mass curve of the same data and thus improves graphical accuracy.

EXAMPLE 3.2 Using the residual mass curve method, determine the storage required on the Mitta Mitta River (Appendix E) to meet a draft of 79.6 m3/month (75% of the mean flow).

*

*

*

*

*

The mean monthly flow (106.1 x 10 6m3) is subtracted from the monthly flows to determine the "residual" flows.

The cumulative residual flows

(residual mass curve) are plotted in Fig. 3.3. To determine the slope of the draft line, the mean flow rate is subtracted from the draft rate. (79.6 - 106.1) x 106m3/month.

Thus the slope of the draft line is Lines of this slope are drawn tangent to the

humps on the curve and the maximum deficit determined. The answer is 1110 x 10 6m3 as shown in Fig. 3.3. Note that the scale of the residual mass curve allows more accuracy than a mass curve drawn on the same scale.

24

3.2.3

Behaviour (or Simulation) Analysis In behaviour or simulation analysis, the changes in storage content

of a finite reservoir are calculated using a mass storage equation thus: (3.1) subject to where

Zt+l

0

~

Zt+l

~

storage at en d

C 0

. . d f t h e t t htlme perlo

(= storage at beginning of t+lth period),

storage at beginning of tth time period, . fl ow d ' . . d, ln urlng t t htlme perlo · t h . . d, re 1ease d urlng t tlme perlo net evaporation loss from reservoir during tth time period, L t C

other losses, and acti ve storage capacity.

(I f the active storage

is reduced by sedimentation during the life of the reservoir this needs to be taken into account.) The usual time period is one month, but other periods can be used. The net evaporation loss is the difference between the evaporation from the proposed reservoir and the evapotranspiration from the proposed reservoir site and depends on the surface area of water in the reservoir.

Other

losses are comparatively small and are usually neglected. The steps in constructing a behaviour diagram are as follows: (i)

Arbitrarily choose a reservoir of active capacity assume that it is initially full, that is, Zo

(ii)

C, and

= C.

Apply Eq. 3.1 month by month for the whole historical record.

D can either be assumed constant or to vary t seasonally. It could also vary as a function of Zt' (~Et

(iii) (iv)

may be included if necessary.)

Plot Zt+l against time on a monthly time scale (see Fig. 3.4). Compute the probability of failure (Eq. 2.16) by dividing the number of time periods for which the reservoir is empty by the total number of time periods.

(v)

If the probability of failure is unacceptable, choose a new value of

C and repeat the steps.

(The process is

thus an iterative one to determine the storage size for a particular design probability of failure.)

2S

100

~

~~ u ~

r

~

~

~

u

60

0

~

40

~

~ ~ ~

0

m20 0 1935

\ ~

1940

1945

1950

1955

1960

1965

LL

1969

Years

FIG. 3.4

Behaviour diagram for Mitta Mitta River (draft = 75%) with a storage such that the number of months empty is approximately 5% of the total number of months.

It is possible to set a storage size for which the reservoir just empties once for the period of historical data.

This storage is the same as

that found using the mass curve and residual mass curve methods.

Assumptions: (i) (ii)

The reservoir is initially full. The historical data sequence is representative of future river flows.

Limitations: (i)

The reservoir is initially full.

The significance of this

on storage size can be checked by examining a behaviour diagram for various starting conditions.

Analysis based on

generated data suggests that at least 100 years of streamflow are required for some rivers before the effect of the initially full assumption can be ignored (McMahon et aZ.,1972). (ii)

The analysis is based on the historical record.

The

sequencing of flows may not be representative of the population of flows. (iii)

Non-continuous records cannot be easily handled because of difficulties of assigning the initial reservoir condition after a break in the streamflow data.

26

(iv)

Demands (and hence releases) that are related to growth rates in time (for example, increased urban water demands through population increases) are not easily taken into account because of the difficulty of relating the demand in a future year to a specific year of the historical flow record.

Some guidance in dealing with this problem is

given in Chapter 7.

Attributes: (i)

The historical behaviour analysis is a simple procedure and displays clearly the behaviour of the stored water. The behaviour diagram can be readily understood by non-technical people.

(ii)

The procedure takes into account serial correlation, seasonality and other flow parameters insofar as they are included in the historical flows used in the analysis.

(iii)

The procedure can be applied to data based on any time interval.

(iv)

Not only can seasonal drafts be easily taken into account, but also complicated operating policies can be modelled. For example, there are real difficulties in including release as a function of demand (itself a function of some climatic variable) and the contents of the reservoir.

EXAMPLE 3.3 Use the behaviour analysis method to compute the storage required on the Mitta Mitta River (Appendix E) to supply a uniform draft of 75% of the mean flow with a probabi Ii ty of failure of 5%.

* Mean flow Draft

*

*

*

*

106.1 x 106m3/month (Appendix E). 79.6 x 106m3/month.

The method is an iterative one.

A storage capacity is chosen, the

inflows are routed through the storage, and the probability of failure (the number of months that the reservoir is empty divided by the total number of months) is calculated using Eq. 2.16.

27

The progressive results are tabulated below. Storage Estimate (l06 m3)

Trial 1

Note:

Corresponding probability of failure

(%)

1000

0.73

2

800

4.17

3

700

6.37

4

750

5.39

5

760

4.90

6

755

5.39

A probability of failure of exactly 5% cannot be obtained because

it is calculated on a monthly basis;

5% of 34 years is 20.4 months.

Thus, for a steady draft and storage initially full, a capacity of 760 x 10 6m3 is required.

A plot of the behaviour diagram is given in

Fig. 3.4.' (One can note that if the same analysis is followed for an initial condition of empty, the storage required is found to be 825 x 10 6m3 .)

3.2.4

Semi-Infinite Reservoir The mass curve method does not enable a probability of storage failure

to be calculated.

Perhaps the earliest method proposed to calculate failure

probabilities was to run the streamflow sequence through a semi-infinite storage (Hazen, 1914).

(As defined in Sec. 2.3.4., a semi-infinite storage

can spill but never run dry.)

Unfortunately, this method leads to error in

calculation of the probability of failure of a finite reservoir as shown below. The depletion diagram for a semi-infinite reservoir is a behaviour diagram without the lower boundary (empty) condition. full line in Fig. 3.5.

It is shown as the

Drawdown or depletion of the reservoir content ABC

occurs because inflows are less than outflow.

At

C, inflows begin to

dominate and the reservoir depletion decreases until spill occurs at

E.

A depletion or behaviour diagram for the finite reservoir is shown in the same figure as a broken line.

For this situation, a reservoir of some

finite capacity has been assumed and the lower boundary condition taken into account.

The effect of this is shown as BC'E'.

The probability of failure

using the semi-infinite depletion diagram is given as:

28

II> )(


2

·c ;;::: .!!1 )( c:
E ~c:

II> (/)

u::

Full

.:: c

o

0 > '';:;

~ II> 11>II> 0Il> .,

0:0

FIG. 3.5

p

where

Comparison of behaviour diagrams using finite and semi-finite storages.

.

(3.2)

seml i.

number of months of emptiness, and

N

total in months of the record used in the analysis.

1

From the behaviour diagram of the finite reservoir the probability of failure is found as: (3.3)

where

mi

number of months of emptiness based on the behaviour diagram for the finite reservoir.

From Fig. 3.5, it is seen that Eii > Lm . Thus the assumption of i semi-infinite storage always results in the overestimation of the probability of failure, and therefore the overestimation of the required behaviour (finite) storage for a given probability of failure.

Barnes (1954) argued

that this overestimation provided a safety factor which was necessary for urban water supply design.

For six locations on Australian rivers, storage

estimates for two conditions of draft and one probability of failure are compared in Table 3.1 using finite and semi-infinite behaviour analysis. The table shows that this overestimation is about 20%.

29 TABLE 3.1

Percentage increase in required reservoir capacity of semi-infinite storage over finite storage for 5% probability of failure using monthly data.

Rivers (Australian gauging station no.)

50% draft

90% draft

Yarra (229102)

14

26

Murrumbidgee (410008)

14

12

Lachlan ( 412010)

17

14

Warragamb.a (212240)

41

21

Namoi ( 419007)

14

27

Burdekin (120090)

37

Mean

6

--

--

23

18

EXAMPLE 3.4 Use the semi-infinite storage procedure to calculate the storage capacity required to supply 75% with no failures on the Mitta Mitta River (Appendix E).

*

*

*

*

*

The procedure is the same as a behaviour analysis but with an arbitrarily high storage capacity.

The behaviour diagram for the semi-

infinite case is shown in Fig. 3.6. From the figure it can be seen that for the period of record a storage of capacity 1110 x 10 6m3 or more will not fail. To calculate the storage size for a given, say 5%, probability of failure with this method, a horizontal line is drawn such that for 5% of the time period the reservoir contents are below it.

The line is shown in

Fig. 3.6, giving the storage corresponding to 5% probability of failure as 900 x 10 6m3 . (Note that the storage determined in this way is an overestimate as shown in Sec. 3.2.4.)

30

1000

E 800 "'o ., 600

. a Cl

ci5

400 This line gives 5%

proba bility of failure

200 o~~~

1935

__~=-~~__~~~~__~~

1940

1945

1950

1955

1960

1965

1969

Year

FIG. 3.6

3.3

Behaviour diagram for semi-infinite storage on Mitta Mitta River for 75% Draft (Example 3.4).

METHODS BASED ON RANGE In Sec. 3.2.2 we discussed how a residual mass curve is used to

determine storage size. Fig. 3.7.

Consider the four residual mass curve diagrams in

In Figs. 3. 7(a), 3.7(b) and 3. 7(c), it is seen that the range (R)

FIG. 3.7

( a)

(b)

(e)

(d)

Definition of range (R). For (a), R = R1; (b) R = Rl; (c) R = Rl + R2 ; (d) R = Rl + R2 ·

31

of the cumulative sums of departures from the mean streamflow will be equal to the critical period storage estimate assuming the draft to be equal to the mean annual streamflow.

However, in Fig. 3.7(d) the range R

Rl +

but the required storage will be equal to the larger of Rl or R2

R2,

(This

characteristic will be examined further in our discussion of the sequent peak procedure.)

The range defined as

R in Fig. 3.7 is the basis of a

number of important reservoir capacity-yield procedures.

These are now

discussed. 3.3.1

Hurst's Procedure Hurst was concerned with storages on the River Nile.

It was an

unusual problem in that he was dealing with large equatorial lakes where a big increase in volume could be obtained by small structures at outlets.

lake

Consequently, storage sizes capable of providing very high regu-

lation (approaching 100%) were being examined.

Hurst's work (1951, 1956,

1965) is concerned more with mathemati cal experiment rather than theory, but from it, he was able to formulate a general solution to the reservoir capacity-yield problem. From an analysis of some 700 natural time-series including streamflows, rainfalls, temperatures, pressures, tree rings and lake varves, Hurst found that the range could be related to the length of record as follows: R/s where

R

(N/2) K

(3.4)

range defined as the sum of the cumulative departures from the mean,

s

standard deviation of the time-series data,

N

length of the time-series, and

K

exponent, and was found by Hurst to have a mean value of 0.72 and a standard deviation of 0.09.

As an aside it should be noted that for a purely random time-series process

K tends towards 0.5 (Feller, 1951).

series yield on the average values of Hurst phenomenon.

The fact that natural time-

K greater than 0.5 is known as the

This aspect is considered again in Sec. 5.7.

In developing his general solution to the storage-yield problem, Hurst computed than mean

K and the storage size to "guarantee" a uniform draft x

Bless

using a mass curve analysis for records of natural phenomena.

The generalized storage relations (for all observations) are (Hurst, 1951):

32

(i)

for draft

mean (x) R

C for draft

(ii)

mean

<

- O.OS

log) 0 (C/R) or where

(3.5)

-

,

0.94 - 0.96 [(x_B)/S)2

C/R

C

required reservoir capacity,

R

range given by Eq. 3.4,

B

(3.6)

1. 05 (x-B)/s

s

draft parameter which is defined x-B s standard deviation of flow, and

x

mean flow.

(3.7)

as

draft

(3. S)

Examining these relationships using only Hurst's river flow data, Joy (1970) found that the generalized curves were not lines of best fit and that the results exhibited large scatter as shown in Fig. 3.S.

Hurst

admi tted the generali zed nature of his results in replying to discussions by Chow (Hurst, 1951, p. SOO).

A further limitation of the procedure is

that only the rank 1 values of storage are considered and no other estimates of risk of storage failure can be estimated.

Nevertheless, Hurst's work is

of monumental significance because of the importance of the value of

K in

stochastic data generation models (Sec. 5.7). CIA = 0.94 - 0.96 [G- B}/s]

'/ 2

log1Q(C/RJ

-0.08 -1.05 Cx-B}/s

=

1.0

1.0

0.8 0.8

a:: ~

---

(J

0.6

(J

0.6 0.4

o o

0.4

o

00

00

8",8 6'

o

0.2

0 0

o

o

0.2'

0.4

0.6

80

'0

o

o

0.8

1.0

o

o

o

o

o

0

~x-BJ/sJ'/2 FIG. 3.8

o o

0.2

o o

o 0.6

0.8

(x-BJ/s

Comparisons of Hurst's generalized equations with data for river flows used in their derivation.

1.0

33

3.3.2

Fathy and Shukry Fathy and Shukry (1956) agreed with Hurst that for 100% regulation

the equivalent mass curve reservoir capacity was given by Hurst's range

R.

But for lower drafts they disagreed entirely with Hurst and developed their own method.

However, as shown by Joy (1970) their technique is no more than

a mathematical representation of the minimum flow approach of Waitt' (1945) given in Sec. 3.4.1. 3.3.3

Sequent Peak Algorithm Thomas (Thomas and Burden, 1963) proposed the sequent peak algorithm

as a method to circumvent the need to locate the correct starting storage as is required in the mass curve procedure.

In essence, the storage size

calculated using the sequent peak procedure is equal to the range of the net cumulative inflow (inflow less draft) estimated for the historical record concatenated with itself. The sequent peak procedure is described by Fiering (1967) as follows. Given an

N year record of streamflow at the site of a proposed dam and the

desired drafts, it is required to find a reservoir of minimum capacity such that the design draft can always be satisfied if the flows and drafts are repeated in a cyclic progression of cycles of requires only two cycles.

N periods each.

The solution

The reason for the second period can be seen in

Fig. 3.7(d) where the storage by mass curve is max (R I , R2) but the range This anomaly arises because the two periods in which there

equals RI + R2'

are net outflows are broken by a net inflow period so that no continuous depletion of the reservoir occurs as it does in the three other examples in Fig. 3.7. The steps in the sequent peak procedure are: (i)

Calculate X. - D, (that is, flow - draft) for all l

i

l

= 1, 2, ... ,2N and calculate the net cumulative t

inflow (ii)

L

i=l

(X.- D.) for t l

= 1, 2, ... ,2N. (Column 4 in Table 3.2).

l

Locate the first peak (local maximum), PI, in the column of cumulative net inflows (column 4).

(iii)

(iv)

Locate the sequent peak, P2 , which is the next peak of greater magnitude than the first. Between this pair of peaks find the lowest trough (local minimum), T I , and calculate PI - T I .

34 (v)

Starting with P2' find the next sequent peak, P 3 , that has magnitude greater than P2 ·

(vi)

(vii) (viii)

Find the lowest trough, T2 , between P2 and P3 and calculate P2 - T2' Starting with P3' find P4 and T3, calculate P3 - T3' Continue for all sequent peaks in the series for 2N

(ix)

periods.

The required reservoir capacity is: (3.9)

C

TABLE 3.2

Sequent peak procedure for Mitta Mitta River (75% draft = 79.6 x 106m3/month).

Flow Month

Flow

( 1)

10 6m3 (2)

-79.6 10 6m3 (3)

1

56

-23.6

2

32

-47.6

3

32

-47.6

4

38

-41. 6

5

31

-4S.6

6

113

33.4

7

lS9

109.4

8

529

449.4

9

217

137.4

10

152

72.4

11

SO

0.4

12

S4

4.4

13

53

-26.6

14

27

l:

(Flow - 79.6) 10 6m3 (4)

0

Remarks (5) PI First peak

-23.6 -71. 2 -llS. S -160.4 -209.0

Tl Lowest point between peaks

-175.6 -66.2 383.2 520.6 593.0 593.4 597.S

P 2 Second peak higher

than first

571. 2 -52.6 5lS.6

35

Limitations and Attributes The sequent peak algorithm was designed for use with replicates of generated data rather than with a single historical record and so several of the limitations attributed to it here are a consequence of using a single historical record. Unlike the mass curve procedure it can handle variable drafts so long as they can be specified without reference to reservoir content (for example, seasonal drafts).

But like the mass curve, the required storage is a

function of record length and we are unable to determine capacities other than those that satisfy the rank 1 low flow sequence. The use of generated flows allows a probability of failure to be indirectly obtained.

For a single sequence the method calculates the

storage required to meet the worst drought in the period of record. for a record length of P

=

N years the implied probability of failure is

l/(N+l) as discussed in Sec. 3.4.5.

the desired value of

Thus,

P.

N can therefore be chosen to give

For replicated sequences, the required storage is

the average of the individual storage estimates. Because the storage size is estimated from two cycles of a record of length

N years, the computed value is equal to or greater than the equi-

valent mass curve value. of Fig. 3.7 occurs.

But this difference will only occur when case (d)

An analysis by Srikanthan (private communication, 1977)

of 16 Australian streams (discussed in Sec. 5.8) showed that there were differences between the sequent peak and mass curve values in only two streams.

The differences were respectively 68% and 83%.

Computationally the algorithm is fast and in contrast to behaviour analysis a trial and error solution is not required.

Like the behaviour

analysis, it uses the historical data directly and consequently the effects of seasonality, serial correlation and other flow parameters are taken directly into account. (1967) and

~10ss

(197l).

A description of its use may be found in Fiering

36 EXAMPLE 3.5 Use the sequent peak method to determine the storage capacity required on the Mitta Mitta River (Appendix E) to provide a controlled draft of 75% without failures. *

*

*

*

*

The mean monthly flow for the Mitta Mitta is 101.1 x 10 6m3; 75% draft is 79.6

x

therefore

106m3/month.

The first few months of the calculation are shown in Table 3.2.

The

draft has been subtracted from each monthly flow and the residuals accumulated.

The first summed residual value is itself a peak (designated Pi)'

because the succeeding values decrease.

The next peak is shown as P2 .

The storage required to cover this flow sequence is Pi

Tl where Tl is the

lowest cumulative residual value between the peaks, that is, (-209,0) = 209 x 10 6m3 .

o-

After traversing through the historical data twice, the successive P-T values are found to be:

209, 957, 960, 242, 1032, 227, 287, 104, 204, 220, 194, 197, 157, 16, 414, 323, 283, 682, 1107, 957, 960, 242, 1032, 227, 287, 104, 204, 220, 194, 197, 157, 16, 414, 323, 283, 682 (x 10 6m3 ).

lllO

x

The required storage capacity is the largest value, that is, 10 6m3 . This example shows the importance of running the historical flows

through twice.

If only one sequence had been used, the maximum storage

capacity would not have been found.

3.4 3.4.1

METHODS BASED ON LOW FLOW SEQUENCES Minimum Flow Approach This approach was proposed by Waitt in 1945.

From the historical

streamflow record, the lowest sub-sequences of flows of various durations are selected and plotted as mass flow against corresponding duration. Constant drafts are plotted as straight lines on this 50-called drought curve graph.

The critical storage is given by the ordinate at the origin

when the draft line is-tangential to the drought curve. are related to Fig. 3.9 are as follows:

The steps which

37

Drought curve based on lowest recorded flows

12

0

FIG. 3.9

(i)

60

12

Time (months)

CDESIGN

Reservoir capacity-yield analysis by minimum flow approach.

From the historical sequence, select the lowest (rank 1) flows for various durations - these are usually taken as 12, 24, 36, 48, etc. consecutive months.

Intermediate

values could also be selected which would allow for seasonal inflow variations. (ii)

Plot only the rank 1 flow volumes against the corresponding durations on arithmetic graph paper.

(iii)

Superimpose the draft line such that it passes through the origin.

(iv)

Measure the largest intercept between the inflow curve and draft line.

This gives the storage estimate

denoted by CCRIT' Waitt recommended that in practice one full year's supply be added to the storage size as a safety fac,tor.

This is illustrated in Fig. 3.9,

and the reservoir would be designed for a capacity equal to COESIGN' In effect the procedure without the added safety factor gives the same answer as the mass curve procedure.

38 EXAMPLE 3.6 Using the minimum flow procedure find the storage capacity required to supply 75% draft (no failures) on the Mitta Mitta River (Appendix E).

*

*

*

*

*

A search of the flow records for the Mitta Mitta produces the following: Duration (months)

Lowest total flow for that period (l06 m3)

5

26

10

238

20

618

30

1356

40

2077

50

3341

60

4275

70

4925

80

5730

90

6615

100

7704

These are plotted on Fig. 3.10 with the draft line (75% draft 'is 79.6

x

106m3/month).

storage requi red.

The largest intercept between the two curves is the

From Fig. 3.10 this occurs for a duration of 40 months

(= critical period);

the storage can be measured from the graph or

calcylated thus: 40

x

79.6 - 2077 say

Note:

1107 1110 (x 10 6m3).

Waitt (1945) proposed a "safety factor" of 12 months draft. This + 12 x 79.6 = 2062 (x 10 6m3).

would give a storage capacity of 1107

3.4.2

Alexander's Method Neither the mass curve nor the minimum flow approach nor the range

provides an estimate of the probability of storage failure.

Alexander (1962)

extended these earlier approaches by developing a series of drought curves for different probabilities of occurrence and from these derived generalized storage-regulation-probability curves.

39

8000 7000

;;

6000

E <0

5000

0 ~

0 ;:

]

4000 3000

0 I-

2000 1000 0

10

FIG. 3.10

20

30 40 50 60 Months duration

70

80

90

100

Storage capacity for the Mitta Mitta River (75% Draft) determined using the minimum flow procedure.

In effect Alexander's approach may be considered as a development of Waitt's drought curve procedure.

Alexander assumed that annual streamflow

could be represented by a Gamma distribution from which he specified the mass inflows as shown in Fig. 3.11 for various recurrence intervals. Recurrence interval is the reciprocal of probability thus: T

r

recurrence interval (years)

=

b'l't pro b a l l y of occurrence

'"'" <1l

III

>III

> .;:; :J <.> III

'"c:

0 <.>

/ /

c:

/ /

.E'"

/

"",e

~ 0

<;)<~"'//////

;:

.S

''""

Historical flows rank 1

.,,/

<1l

~

0

2

4

5

6

Time (years)

FIG. 3.11

Introduction of recurrence interval (and hence probability) to the minimum flow approach (Alexander, 1962).

(3. 10)

40

Draft lines were also superimposed on the drought curves as Waitt had done and storage capacity estimated as the difference between cumulative inflows and outflows. It is necessary to understand a little about the Gamma distribution and its properties.

Two Parameter Gamma Distribution The probability function of the two parameter Gammadistribution is defined as follows: f(x) where

=

T

a-I e- x/S

(3.11)

(a)

x

the flow value,

a

the shape parameter, the scale parameter, and

S T

x

Sa

the Gamma function

(a)

0

I

x

a-l -x e dx

To estimate the probability of a flow of magnitude some value

x

being less than

X given that the flows are Gamma distributed, one calculates: Prob (x

~

X)

I

X

f(x) dx

o a

S

1

IX x a-l

e

-xiS

(3.12)

dx

T(a).O

As shvwn diagrammatically in Fig. 3.12, this equation estimates the area of the shaded portion of the probability distribution.

x

FIG. 3.12

Probability of

x

being less than

X.

41 To calculate the parameters a and B of the Gamma distribution, Thorn's (1958) maximum likelihood method is used, thus:

1+(1+4A)~ 3

&

where

(3.13)

4A ~

A

e

(that is, log of mean - mean of logs.).

There is a small correction that should be applied to paper) but for Alexander's application it is not significant.

a (see Thorn's The scale

parameter is calculated as follows:

B= where

x

X

(3.14)

a

= mean annual streamflow.

Detai ls of A lexa:nder IS Method

In developing his method, Alexander used a property of the Gamma distribution which states that the sum of

n

independent Gamma variables

with parameters a and B is a Gamma variable with parameters: an and as

x n

(3.15)

nx

Sn where

na

xn an

nx na

(3.16 )

S

an

shape parameter of

n

consecutive years flow, and

Bn

scale parameter of

n

consecutive years flow.

Thus assuming annual streamflow to be Gamma distributed and knowing the shape parameter of the annual flows for a given stream, Eq. 3.15 allowed the shape parameter to be calculated for all flow.

n

consecutive years

As illustrated in Fig. 3.13, Alexander could then calculate the

drought curves for a'given probability using Eq. 3.12. Using this concept he developed a series of curves relating constant regulation or release to reservoir capacity as a ratio of a mean annual flow and recurrence interval for a shape parameter of unity (Fig. 3.14). Storage capacities for streams with shape parameters that are not unity are obtained by dividing the estimated capacity for ~ parameter.

=1

by the correct shape

42 ~

"'"» "> :g <.J

" VI

C

0

<.J

C

~ ~ 0

~ VI VI

::2'" 1

0

2

3

4

Time (consecutive years)

ii

2ii

3ii

4&

Shape parameter of Gamma distribution

FIG. 3.13

Development of Alexander's drought curve. (Shaded area indi cates probabi l i ty of n consecutive years flow being less than value indicated on ordinate.)

The computational steps are as follows: (i)

Decide on the percentage regulation with respect to the mean flow (D).

(ii)

This must be constant.

Compute the shape parameter ~ of the Gamma distribution for annual flows at the site in question using Thorn's procedure Eq. 3.13.

(iii)

Decide on the probability of failure or recurrence interval CTr) of the critical design period.

(iv)

Use Fig. 3.14 (Alexander, 1962, Figure 6) to estimate '] and CPt knowing Tr and D where ,] as ratio of mean annual flow for ~ period (years) for a

(v)

=

reservoir capacity 1, and CPj

= critical

= 1.

Compute required reservoir capacity (in volume units) thus, reservoir capacity

C

'1

=~

x

(3.17)

a

(vi)

The corresponding critical period (in years) is CP 1 critical period CP = a

(3.18)

43

0

~O

c o

i

70

..

V , j

I

/\

u ~

II

/

V

j

v, Vj VI V/ 'i J V I V ~ v/VI / I / / 1/ ~ V 1/T V~ 1/(h

V

~

I

Regulation 0 with capacity 1'1 giving I probability of failure'of =1 r for

V /I f/ / ~ ) ~ ~ / ~ ~I Y

r,

Il

50

J

/

/ / /.1 V/ / //-~ / / ~~ J I J

I

/

/

Q)

30

I vi.s> 0-"-

,0

...

V

60

40

:'1(

"

')..

c

cf

",

pj~Dr '?-JG:rK ~,1 1\ / K\V .I~ k '~OOI ,#_ c-~l"> :(\ \V\ /K ~t?v~# ;1 \ V\ ) /V)~0~v

en Q) a: en cu

~O ~~

~",v ~KI"\Y/Kl><~o ~o (~

:::J

Q)

1

~'v -

0«'

Q

1

\'~ I,<;, '1-2;..-X, f"-.

g0

,... 80

JI

I\.G~\)

0.5

1

2

Reservoir Capacity

FIG. 3.14

3

4

a.

5

(1'1)

Alexander's reservoir capacity-regulationprobability curves (Alexander, 1962).

6 7 8 g

44

Assumptions: (i) (ii) (iii)

Annual flows are independent. Annual flows are Gamma distributed. The reservoir is full at the beginning of the critical period.

Limitations: (i)

Regarding assumption (i), approximately 25% of Australian streams exhibit lag one annual serial correlations that are significantly different from zero at 5% level of significance. Since the median value of Australian and Northern Hemisphere serial correlations are similar (Fig. 2.3), it would be expected that the same proportion of serial correlation values would be significant in the Northern Hemisphere.

If a

procedure has been used that is based on historical data (for example, behaviour analysis) serial correJation effects would have been implicitly taken into account.

Therefore,

for consistency with procedures using the low flow periods directly, an adjustment to the storage estimate should be made to account for the assumption of zero serial correlation. Unfortunately, no comprehensive study of the effect of the independence assumption of annual flows on the reservoir capacity-yield-probability relationship is available. However, some guidance is given in Appendix B. (ii)

Annual flows for some streams may not be Gamma distributed although studies of annual flow data for Australian streams suggests that this distribution is generally suitable (McMahon, 1977).

As the shape parameter increases, the Gamma

distribution approaches the normal distribution.

Thus for

the more normally distributed Northern Hemisphere streams, the Gamma distribution should be satisfactory. (iii)

Because annual data are used, seasonal variations are ignored and storage estimates will tend therefore to underestimate storages based on monthly data, particularly for short duration critical periods.

(iv)

Releases must be assumed constant.

45

(v)

The reservoir must be assumed to be initially full and no variation from this condition is possible.

(vi)

The method of analysis does not take into account net evaporation losses.

If required, an additional amount

of storage has to be added to the computed value to cover this loss (see Appendix A) .

Attributes: (i) (ii)

The procedure is rapid and is simple to perform. It is sufficiently accurate to be used as a preliminary design or reconnaissance technique, but not as a final design tool.

(iii)

Through estimating the recurrence interval of minimum flows, failure probabilities are obtained.

EXAMPLE 3.7 Determine the storage required for 75% draft and 5% annual probability of failure for the Mitta

River (Appendix E) using Alexander's

~1itta

procedure.

*

*

*

*

*

From the data (Appendix E) and working in units of 10 6m3 , mean flow x loge x

1274 7.1496

Sum of logs (base e) of annual flows From Eq. 3.13

= 238.042

A

7.1496

(238.042)/34

0.1484 4A ~

1 + (1 + 3) 4A

3.53.

From Fig. 3.14 using recurrence interval failure) and percentage draft D

Tr

= 75%, we find

20 years (5% probability of

46

2.1, and

Tl

Storage capacity

C

[Eq. 3.17]

This capacity has to be adjusted to compensate for the annual serial correlation of 0.06 (Appendix E).

The adjustment factor, from Fig. B.l,

is approximately 1.06. Required storage

758 x 1. 06 800 (x 10 6 m3 ).

Length of critical period CP

3.4.3

=~ 3.53 =

3 . 4 years.

Dincer's Method The derivation which follows is essentially due to Professor T. Dincer,

Middle East Technical University, Ankara (C.H. Hardison, personal communication, 1966). Consider a sequence of annual flows, which are assumed independent, wi th a mean

x

and standard devi ation

s.

n-yearly mean

x

n-yearly standard deviation

s

n

nx

(3.19)

n

(3.20)

As a consequence of the Central Limit theorem, the distribution of n

consecutive annual flows approaches normality as

n

increases.

Therefore, the lower p-percentile flow is given by: nx - z

p

,

n ~s

(3.21)

n-year flow with a probability of occurrence of p%,

where

that is, for p% of time n-year flow z

p

standardized normal variate at p%.

~

qn,p' and

47

During a critical period, required storage

(3.22)

n,p

D n

C

depletion of an initially full storage at end of

C

where

outflow - inflow

n,p

an n-year period during which time the n-year flow (q Dn where

D

n,p

) has a probability of occurrence of p%, and

constant draft from the reservoir over Dnx

n

years (3.23)

constant draft as ratio of mean annual flow.

substituting Eqs. 3.21 and 3.23 in Eq. 3.22, C

n,p

Dnx

nx + z

nx (D-l) + z

p

(3.24)

p

To obtain the length of the critical period and the maximum required storage differentiate Eq. 3.24 with respect to z 2 ~ C 2 CP 4(l-D)2 v z 2 C -E- C 2 and hence T 4(l-D) v x where

CP C

n

and equate to zero giving (3.25) (3.26)

length of the cri ti cal drawdown period in years, annual coefficient of variation,

v C

maximum required storage in volume units, and

T

= maximum required storage expressed as a ratio of mean annual flow.

Assumptions, Limitations and Attributes In addition to the basic critical period assumptions (an initially full reservoir and only one failure during the critical period), the above derivation is based on a number of other assumptions, namely: (i)

that the critical period is large enough so that the n-year flows will tend towards a Normal distribution;

Cii) (iii)

that annual flows are independent;

and

that the draft rate is uniform.

For non-normally distributed annual flows the procedure will tend to overestimate storage need, but this is offset by a tendency for underestimation due to initially full conditions and the condition of no repeated

48

failures without spill.

In addition, the procedure is based on annual data

and seasonal need is not taken into account.

An adjustment to correct for

the independence assumption is given in Appendix B.

Notwithstanding these

limitations, it has been found that the procedure provides reasonably reliable storage estimates at high drafts (greater than 50%).

EXAMPLE 3.8 Use Dincer's method and the annual flow data for the Mitta Mitta River (Appendix E) to estimate the storage capacity required for a controlled release of 75% with a probability of failure of 5%.

*

*

*

*

*

c

From Eq. 3.26,

x

x

We have z

p

1274 (x 10 6 m3 ) (Appendix E) 1.65 from tables of the Normal distribution

for a one-tail (p D

cv

=

5%)

0.75 -

x

731 1274

(Appendix E)

0.57

Therefore, storage capacity is:

c

(1274) (1.65)2(0 57)2 4(1 - 0.75) . 1127 (x 10 6 m3 ).

The adjustment factor for annual serial correlation of 0.06 (Appendix E) is obtained from Fig. B.l as 1.06. or 1200 (x 10 6m3).

Therefore required storage

is 1130 x 1.06

The length of critical drawdown period is given by Eq. 3.25:

CP

z 2 --L-

4(1-0)2

cv 2

(1.65)2 2 4(0.25)2 (0.57) 3.5 years.

49 3.4.4

Gould's Gamma Method This method (Gould, 1964) can be thought of as a modification of

Dincer's method (Sec. 3.4.3) although it was derived independently.

It uses

the fact that, while parameters for the Normal distribution are easy to calculate and probability tables for it are readily available, the Gamma distribution usually is a better approximation to the distribution of annual flow data.

The procedure is therefore to use the Normal dis!ribution for

the calculations, and then to apply a correction to approximate the Gamma distribution. The mean and variance of a one parameter Gamma distribution, G(c), are equal and equivalent to the shape parameter, say

X,

convert the mean,

c.

It is possible to

and standard deviation, s, of a Normal distribution to

Gamma units by dividing them both by

s2/x.

The resultant Normal distri-

bution will have the same mean and variance, that is Gamma units. For a Normal distribution Dincer (Sec. 3.4.3, Eq. 3.26) showed that z 2

-1'..- c

c where

4(l-D)

2

X

C

required storage in volume units,

D

draft as ratio of mean annual flow,

C v x

mean annual flow, and

z

standardized Normal variate at p%.

coefficient of variation,

p

Substituting for

z

cy where

v

x

and

C 2 v

with Gamma units gives

2

-p4(l-D)

(3.27)

storage volume in' Gamma units.

C

Y

Gould argued that the difference flow of a Gamma

c

and variance both equal to of

p

d

between the lower

c

percentile

is approximately constant for a given value

over a large range of shape parameter

in Table 3.3.

p

distribution and that of a Normal distribution with mean

Because of the constancy of

c. d

Values of

for a given

d

are given

p, according to

Gould the critical period for a Gamma distributed inflow is the same as that for normally distributed inflow with the same mean and coefficient of variation, and that the storage required for an inflow that has a Gamma distribution is

d

Gamma units less than that required for a Normal

50 TABLE 3.3 Lower

distribution.

Values of z

p

p percentile value

and

d. d

z

p

0.5

3.30

d not constant

1.0

2.33

1.5

2.0

2.05

1.1

3.0

1. 88

0.9

4.0

1. 75

0.8

5.0

1.64

7.5

1. 44

10.0

1. 28

0.6 } Gould's Gamma method 0.4 not recommended for 0.3 use in this range.

In other words, as shown diagrammatically in Fig. 3.15,

inflows for. a Gamma distribution will be greater than for the Normal case and the storage should be decreased by

d

Gamma units.

Hence,

z

2

C

~

T

~

4(l-D)

Y

z

where

4c(l-D)

Y

T

2

- d

(3.28) d c

(3.2fl)

required storage divided by mean annual flow

y

in Gamma units, that is, storage / c. The required storage can be converted from Gamma units to units of volume as a ratio of mean annual flow by multiplying the right-hand side of • s2 1 Eq. 3.28 by (~) -=-.

x

x

Magnitude of n consecutive year flow.

W

d is approximately constant as c varies

FIG. 3.15

Diagrammatic illustration of the difference between inflows for the Normal and Gamma distributions.

S1

z 2

P

.4(T=D)

where

C 2 _ dC 2 v

(3.30)

v

required storage divided by mean annual flow

T

in volume units per year. Thus Eq. 3.30 is the same as Eq. 3.26 except for the correction term dC 2. v

Assumptions, Limitations and Attributes By assuming flows to be Gamma distributed rather than normally distributed, Gould overcomes one of Dincer's major assumptions, but in doing so, introduces a small approximation relating to the difference between the Normal and Gamma distributions.

Other points concerning Dincer's

procedure also apply to Gould's Gamma method.

It will be noted in

Chapter 6 that this procedure provides very good estimates of storage over the whole range of practical interest.

EXAMPLE 3.9 Use Gould's Gamma procedure and the annual flows for the Mitta Mitta River (Appendix E) to estimate the storage required on this river to meet a draft of 75% with a probability of failure of 5%.

r4(6 *

z

Equation 3.30 We have

z

p

*

C

L )

*

*

*

2

v

2

1.64 from tables of the Normal distribution

for p

=

5%,

D

0.75,

d

0.6 from Table 3.3 corresponding to z

P

:>...

=

x T

731 1274

"

0.57

(Appendix E).

1. 64 2 [ 4(0.25)-

0.68.

Therefore storage capacity

0.68 x (0.68

x

1274)

866 (x 10 6m3).

=

1.64,

52

Adjustment for annual serial correlation of 0.06 (Appendix E) is obtained from Fig. B.l as 1.06. 866 x 1. 06

Required storage

920 (x 10 6m3).

3.4.5

Carryover Frequency Mass Curve Analysis Carryover frequency analysis is generally similar to the previous

techniques.

Instead of using theoretical distributions like the Gamma dis-

tribution the inflow probability is defined by empirical procedures. of these are now discussed.

Two

In both of them, the strearnflows are ranked;

in the first, overlapping sequences are adopted while independent sequences are used in the other approach. 3.4.5.1

Overlapping Sequence Approach

In a flow sequence of N' months, there are (N' - n sub-sequences of

n

consecutive months' duration.

+

1) overlapping

The absolute severity

(in a low flow sense) of any sub-sequence is measured by its rank.

The

severest sequence of a particular duration, as measured in terms of minimum flow, is ranked 1, the next worst sequence is ranked 2, etc. concepts are illustrated in Fig. 3.16.

These

To obtain a relative measure of

severity, commonly rankings are translated into recurrence intervals (return periods) or probabilities, using Eq. 3.31 .

.,

"C

2

""E'"

Cl

~

o

;;::

Tim e (months)

~

~

~ ~ ~

.

FIG. 3.16

~

Ranked 4 monthly sub - sequences from total sequence of 24 months

Rank and sub-sequence concepts (for a 4-monthly sub-sequence, the lowest consecutive four months of flow are ranked as no. 1).

53

p

where

probabi l i ty of occurrence m N'

m

(3.31)

N'+l

rank of the sub-sequence, and length of record in months.

By considering a range of durations one can relate duration and associated low flow values to probability of occurrence, and hence construct drought curves (cumulative inflow curves) for given levels 9f severity (probability of occurrence) as we have done in the previous procedures.

A

draft line is then superimposed on the inflow curve and storage is computed as in the minimum flow approach.

This method was proposed by Thompson

(1950) .

Assumptions and Limitations Being a critical period technique, two assumptions - an initially full storage and no repeated failures - are implicitly made.

These result in the

procedure tending to underestimate the number of behaviour failures (that is, failures that would result if the same flows were run through a behaviour analysis for a storage of similar capacity) and the reservoir capacity is underestimated. This underestimation is compensated to various degrees by the "clustering" effect inherent in ranking overlapping sequences.

Due to

overlap, any sequence tends to contain the last (n - 1) months of the previous sequence.

Thus, radically different values of mass flows cannot

be expected, especially for sequences of long duration.

As a result,

sequences of similar rank tend to cluster around the same portion of the streamflow record and define the same historical values.

As a consequence,

the procedure tends to overestimate the number of behaviour failures in certain portions of the streamflow-record and the reservoir capacity is overes timated. Thus there are two compensating discrepancies in this technique.

It

underestimates the number of behaviour failures due to the neglect of repeated failures.

However,

due to clustering it overestimates the

number of behaviour failures in certain portions of the streamflow record. For short critical periods, clustering is minimal; consequently there is little or no compensation for repeated failures, and the technique will tend to underestimate the required behaviour storage.

54 On the other hand, for long critical periods, clustering is severe. If the critical period is sufficient long, overestimation of the required storage may be so severe that there is little difference between the rank and higher ranked values.

EXAMPLE 3.10 Find the storage required on the Mitta Mitta River (Appendix E) to provide for a 75% draft and a probability of failure of 5% using the overlapping sequence procedure.

*

*

*

*

*

The procedure requires searches through the data file (Appendix E) to find the lowest, second lowest etc. cumulative flows corresponding to a number of durations. The iowest (that is, rank 1) sequence has a recurrence interval (To

liP).

From Eq. 3.31,

T

o

NI + 1 m

409

-1-

409 months 34 years.

of 409 2

The second lowest (that is, rank 2) sequence has a recurrence interval 204.5 months 17 years. Thus, to get a recurrence interval of 20 years, it is necessary to

plot these two values on log-probab.ili ty paper and interpolate. The results of the searches for different durations are as follows: Duration Months

Lowest Total Flow (10 6m3)

Second Lowes t Total Flow (106m~

10

238

244

20

618

689

30

1356

1395

40

2077

2125

50

3341

3346

60

4275

4287

70

4925

4927

80

5730

5898

90

6615

6667

100

7704

7733

55 After interpolating between the two figures to get the flow corresponding to a 20 year recurrence interval for each duration, the results are plotted in Fig. 3.17. The greatest intercept between the draft line (79.6 x 106m3/mont~ and the 1 in 20 year flow line is the required storage, 1070 x 10 6m3 , and corresponds to a critical period of 40 months.

In practice, the storage estimate

needs to be increased by approximately 20% to account for bias due to crossnesting of flows (Sec. 3.4.8).

7000

6000 ME 5000

'"o ';; 4000

~ 3000

Storage required = 1070

u::

1 in 20

2000 1000

o

10

I

!

I

I

!

I

I

30

40

50

60

70

80

90

100

Duration of sequence (months)

Fig. 3.17

3.4.5.2

Storage estimate for Mitta Mitta River (75% Draft) using the overlapping series approach (Example 3.10).

Independent Series Approach

Based on the work of Hudson and Roberts (1955), Stall and Neill (1961) presented another method for estimating the frequency distribution of the flow series.

From the (N' - n + 1) overlapping series of

duration, the lowest flow is selected and ranked as 1.

n

months

However, unlike the

overlapping series approach, the next step is to neglect all other sequences which overlap the rank 1 sequence. from the remaining flow record.

The rank 2 sequence is then selected

Higher rankings are determined in a similar

manner with none of the flows for any particular rank overlapping the sequence of flows for any other rank. Stall (1962) extended this technique to storage-yield analysis in a similar manner to the previous techniques.

follow and can be related to Fig. 3.18.

The steps in his procedure

56 8000

7000

;:;

6000

E ~ 5000 )(

;

4000 0

u::

3000 Storage required = 980 (x 106 m 3 )

2000 1000

0

70

10

80

90

100

Sequence duration in months

FIG. 3.18

(i)

Reservoir capacity-yield analysis by Stall's procedure (Independent Series Approach) for Mitta Mitta River (Draft and probability of failure = 5%).

Convert the rank of each

n

75%

month sub-sequence to

recurrence interval by Eq. 3.32. N + 1

T. where

(3.32)

m

l

recurrence interval (in years) for an

T. l

independent series, m

the rank of the sub-sequence, and

N

the length of the streamflow record in years.

The interpretation of this equation must be understood in the light of the duration of the event under review (n duration).

event of rank

m is defined as l2N + n m

where

months in

The average recurrence interval of an n-month

(3.33)

total possible number of independent events

l2N/m

in the sequence. The recurrence interval

Tn

has units of

n

months and is

converted to years thus:

T. l

n

12 N +

Tn

n/12 m

N + 1 m

(3.34)

57 An alternative to Eq. 3.32 is N + 1 - 2c

T.

where

c

(3.35)

m- c

1

is a function of the distribution of the data

under analysis, and in general is taken as c = 0.3 giving T.

1

N + 0.4

(3.36)

m - 0.3

This latter equation is recommended for use by United States Corps of Engineers, Hydrologic Engineering Center (see United States Army Corps of Engineers, 1975 and Beard, 1962). (ii)

Inflow volumes (mass values) are plotted against duration for design recurrence intervals.

(iii)

The appropriate constant draft line is superimposed on the inflow curves.

(iv)

The required capacity is estimated as the largest interval between the inflow curve and the demand line.

In Fig. 3.18 a storage of capacity of 980

x

10 6m3 will maintain the

constant 75% draft for inflows with recurrence intervals less than one in 20 years.

In practice, the storage capacity would need to be increased to

cover evaporation losses and a bias resulting in part from cross-nesting of the mass curves as explained in Section 3.4.7.

Storages should be increased

by 20% to take this bias into account. 3.4.5.3

Independent versus Overlapping Series

From Eq. 3.31 the equivalent recurrence interval in years for a rank m event by the overlapping procedure is given as follows: T

where

NI + 1

---:i.2Iil

o

T

o

year s

(3.37)

recurrence interval (in years) for an overlapping series.

From Eqs. 3.32 and 3.37 it is seen that for a particular flow sequence to have the same recurrence interval in both procedures, a value of given rank in the independent series must be approximately 1/12 of the rank in the overlapping series. Recall that in the overlapping series clustering is more severe for long duration sequences than for short ones.

Thus, for long duration

sequences the effect of removing overlapping flows in the independent

58

series is to remove from further analysis a large number of flows of rank similar to that of the ranked flow.

In other words, for the independent

series and for long duration sequences, we are ranking from a decreasing pool of flows, whose actual flow values tend to become rapidly high.

This

effect is reduced for short duration sequences. It is not possible to predict when the rank of independent series (multiplied by 12) will underestimate the overlapping series rank.

Because

of this factor the independent series will tend to have lower flows than the overlapping series for the corresponding rank.

On the other hand, the

effect of clustering results in the overlapping series having the lower flows particularly for sub-sequences of long duration.

Somewhere between

these two opposite effects, the results will be equivalent. Thus for critical periods of short duration, the independent approach will tend to overestimate the equivalent storage of the overlapping series. On the other hand, for long duration critical periods the independent approach underestimates the overlapping series storage estimates. It is difficult theoretically to relate the independent approach to a behaviour analysis.

Empirically, it has been found that the independent

series overestimates the behaviour storage at low drafts and underestimates it at high drafts.

This conclusion fits the general observations made with

respect to the overlapping series. From the above discussion it may be concluded that, at least from a theoretical point of view, the carryover frequency mass curve techniques described above are inadequate for several reasons.

Unfortunately the mag-

nitudes of these inadequacies are not easily quantified and this fact needs to be kept in mind if the procedures are used in design.

The independent

series approach, slightly modified, is recommended by the United States Army Corps of Engineers (1975).

EXAMPLE 3.11 Use the independent series method to determine the storage required on the Mitta Mitta River (Appendix E) to meet a 75% draft with a failure rate of once every 20 years on the average. *

*

*

*

*

The procedure is to search through the data record to determine first the lowest total flow for a given duration.

Then, excluding those months

59 from the record and not allowing "straddling" of them by a sequence, the remaining flow record is searched for the second lowest total flow for that duration.

This is repeated for several durations, as shown in the following

tab Ie. Duration (Months)

Lowest Flow (l06 m35

Second Lowest Flow (10 6m3 )

10

238

244

20

618

837

30

1356

1480

40

2077

2367

50

3341

3763

60

4275

4301

70

4925

5386

80

5730

6030

90

6615

6827

100

7704

7798

From Eq. 3.32 the lowest flow for a given duration has a recurrence interval of 34 + 1 --1-

35 years.

The second lowest flow has a recurrence interval of 34 +

--2-

= 17.5 years.

Thus, to get a 20-year recurrence interval, interpolation (on logprobability paper) is necessary. The resultant values are plotted on Fig. 3.18 and the largest intercept between the draft line (79.6 x 10 6m3) and the inflow curve gives the storage, 980 x 10 6m3 corresponding to a critical period of 30 months. In practice, the storage should be increased by 20% to account for bias due to cross-nesting of flows (Sec. 3.4.8).

3.4.6

Within-year Frequency Mass Curve Analysis Annual low flow frequency curves are the basis of the United States

Geological Survey approach to within-year storage requirements (see, for example, Hardison and Martin, 1963). follows:

The steps in this procedure are as

60 (i)

For duations 1, 7, 15, 30, 60, 120 and 183 consecutive days, the mi.nimum flows for each year of the historical record are determined.

(it)

These flows are ranked (separately for each duration) in order of magnitude, rank 1 being given to the lowest flow.

(iii)

For each flow a recurrence interval (or probabili ty of occurrence) is estimated by the formula:

P

r

where

(iv)

N

1

T

+

1

(3.38)

m

recurrence interval (years), P

probability, and

m

rank of low flOl" for given duration.

Flows for each duration are plotted against recurrence interval on Extreme Value probability paper, and smooth eye-fitted curves are drahn (Fig. 3.19).

(The scales are

chosen because they tend to lineari se the data and, as well, Gumbel's (1954) low flow theoretical curves can be superimposed on the diagram.)

sfI II>

"'.§

5~

4

"i:' '<.>"

.t:.

!I>

'5

"'" a;'"

«>

1

'f I

183

~

120

:B'"

>-

c

2

SO

0

~

30 Cl 7

1.1

15

2

3

5

10

20

50

Recurrence interval (years)

FIG. 3.19

Annual low flow frequency curves for Brandywine Creek at Chadds Ford, Pa., USA. (U.S.G.S. 7600)

61

(v)

Flows for a given recurrence interval are read from the low flow frequency curves and replotted as inflows against duration (like the drought curves in earlier approaches) on arithmetic graph paper (Fig. 3.20).

50

M-

./

E 40

./

'"0

./

x Q)

./ ./

Required reservoir capacity

30

./

./ ./

E

./

:l

"0 > ~ 0

;;::

20

.E

10

O~~~

____

o

~

____- L____-L____

40

~

____

80 Duration

FIG. 3.20

(vi) (vii)

~

____L -_ _

120

~L-

__

~

____

160

(days)

Mass inflow-duration curve for within-year storage analysis (Data are for Brandywine Creek at Chadds Ford, Pa., U.S.G.S. 7600).

Constant draft lines are superimposed on the diagram. The largest intercept between the draft line and inflow curves is taken as the reservoir capacity required to meet the draft at the design level of reliability (or probability of failure). For this situation the probability expresses the chance that the reservoir, if operated under the design conditions, will fail (empty) at least once within any year.

Asswrrptions: (i)

The reservoir is assumed to be initially full.

Unlike

carryover storage situations this assumption will probably be met in most situations, particularly as the levels of regulation are usually very low. (ii)

Failures that occur after the end of the critical period are neglected.

~

200

62

Limitations: (il (ii)

Variable draft conditions cannot be easily treated. The use of frequency curves introduces a bias in the computed storage estimates.

This bias is due in part

to cross-nesting of the mass curves as explained in Sec. 3.4.8.

Computed estimates of capacity should be

increased by 10% to take this bias into account. (iii)

This method further underestimates the required storage due to the method of establishing the low flows.

As the

ranked values are necessarily from different years, there is no allowance for two or more independent events in the one year that are more severe than the next rank event (from a different year).

This effect would be small

except for low recurrence interval events (say less than 10 years). (iv)

The frequency analysis is based on daily flows.

This

considerably increases the computational requirements where these have not already been processed. (v)

The method of analysis does not take into account net evaporation losses.

If required, an additional amount

of storage has to be added to the computed value to cover this loss (Appendix A).

Attributes: (i)

If low flow frequency curves are readily available for the site in question, the method is quick and simple.

(ii)

Notwithstanding the bias in the storage

estimate~,

the

within-year reservoir capacity determined using the annual low flow frequency procedure gives satisfactory estimates if compared with those determined using

annual mass curves (Hardison, 1965).

63 EXAMPLE 3.12 Using the annual low flow frequency curves for Brandywine Creek at Chadds Ford, Pa., Fig. 3.19, determine the storage required to provide a 30 96 draft with a 5% annual probabi Ii ty of fai lure. Brandywine Creek = 1046 m3/s).

*

*

*

(l>lean flow rate for

*

*

The 5% annual probability of failure corresponds to a recurrence interval of 20 years.

From Fig. 3.19 the flow rates for durations of 7, 30,

60, 120 and 183 days corresponding to a 20 year recurrence interval can be

read off as follows: Duration ( days)

Flow Rate (m 3s- 1)

Flow Volume (x 10 6m3)

7

1. 62

0.97

30

1. 89

4.67

60

2.29

11.4

120

2.70

28.0

183

3.25

50.6

These points are plotted on the graph (Fig. 3.20) and the storage determined from the maximum intercept between the demand and mass inflow curves, that is, 5.1 x 10 6m3 . In practice, the storage estimate needs to be increased by 10% to account for the bias due to cross-nesting.

3.4.7

Regional Within-year Storage Estimates Hardison (1965) also provides a technique for obtaining an approximate

estimate of seasonal storage requirements by using the median annual 7-day flow as an index.

Based on 72 streams in eastern United States, he related

seasonal storage need to the median annual 7-day low flow as shown in Fig. 3.21.

Storages taken from these curves are subject to the bias inherent

in using low flow frequency curves to compute storage-draft relations and the storage estimates should be increased by 10% (see next section).

64

60

~

.2 Cii ::l

c: c:

'"

c:

'" Q)

E

'0 'EQ) ~ Q)

.3....

0'"

~ .l)

'"~

10

.2

« 0

0

0.3

0.4

0.5

Median Annual 7-day Low Flow. (ratio to mean annual flow)

FIG. 3.21

3.4.8

Areal draft-reservoir capacity relationship for 5% probability of failure as a function of median annual fow flow. (Parameter is storage capacity in percent of mean annual flow.) (Hardison, 1965.)

Bias in Mass Curve Frequency Analysis A further procedural error is associated with the use of frequency

curves to estimate reservoir capacity.

This is evident in both the inde-

pendent and overlapping series, and the within-year frequen;y curves.

This

effect, which results in under-estimation of the equivalent behaviour capacity, has been attributed principally to the cross-nesting of the mass curves of each record (Hardison, 1965) and can be illustrated as follows. Consider the example given in Table 3.4 which is similar to that used by Hardison to illustrate the bias.

If the two years of low flow data are

time-wise ordered the rank 1 and rank 2 deficiences (or storage sizes) are 3000 and 2500 m3/s day respectively. Yet if the flows are ordered by rank (which is the procedure in low flow frequency analysis) the rank 1 and 2 deficiences are 3000 and 2000 m3/s day. The difference in rank 2 estimates results from the cross-nesting effects.

65 TABLE 3.4

Example of calculations to show bias in frequency-mass curve storage procedure. Flow in m3 js

Ordering

By year By rMk

period

10 day period

200

300

2000

3000 1

100

500

2500 2

1000

100

300

2500

3000 1

500

2000 2

1000

5d~

1929 1930 1 2

Deficiency in m3 js days for a draft of 600 m3 js 10 day 5d~ period period

200

Hardison (unpublished paper, 1965) has examined this bias for the independent series Md suggests that the under-estimation of storage is approximately 20% for streams with an Mnual coefficient of variation less than 0.4.

For within-year analyses the degree of under-estimation is

about 10%. 3.4.9

Combining Carryover and Seasonal Storages - Hardison's Approach In order to correct his carryover storage estimates based on annual

data for seasonal (or within-year) need, Hardison (1965) utilized the Mnual low flow frequency mass-curve method (Sec. 3.4.6).

He proposed two solutions

of which the one described below assumes that the probability distribution of seasonal storage is independent of carryover storage.

The alternative

but quicker procedure assumes that the average seasonal storage requirement for 100% regulation is 0.4 times the mean annual flow. Hardison's steps for combining seasonal and carryover storage are as follows (Fig. 3.22): (i)

Divide the seasonal storage-probability curve (calculated using the within-year frequency mass curve analysis) into about eight segments Md compute the mean storage Md corresponding incremental probability for each segment.

(ii)

For a selected amount of total storage, the required carryover storage for each segment of the seasonal curve is computed by subtracting the seasonal amount from the total amount.

66

1.2

\

1.0

\ \

Q)

Cl tV

....

£!II

".:;....

Q)

\

:t: 0

c: ::l 0.8 ....

\

carryover

a:

•\ ,

,

iii ::l

\

c: 0.6 c:

\

tV

cr- c: tV

Q)

, \

"\

Q)

~ 0.4

"\ '.~ combined

.....

"',

0.2

95

.0.1

0

98 99

Probability of failure ('Yo)

FIG. 3.22

(iii)

Combination of seasonal and carryover storage probabilities.

For each segment, the probability of the required carryover storage taken from the carryover storageprobabili ty curve is multiplied by the incremental probability of the selected amount of total storage in (ii).

(If there are segments of the seasonal

storage curve which correspond to storage values equal to or greater than the total storage, then the probability for the corresponding carryover storage for those segments is taken to be unity.) (iv)

This is repeated for other points of total storage to give the total storage versus probability curve as shown in Fig. 3.22.

In the above analysis, the carryover storage-probability curve must be based on annual data.

If monthly data were used the computed carryover

storage would automatically have taken the seasonal need into account.

67

3.5

OTHER CRITICAL PERIOD METHODS Because of distinctive seasonal variations (four wet and eight dry

months each year) Wilson (1940) was able to define clearly the starting and finishing months of critical periods and hence compute the storage size necessary to offset specific inflow conditions.

He evaluated the proba-

bility of failure associated with each initial period and summed these to give the total probability of failure.

The procedure is not of general

applicability. Law (1953, 1955) developed massed curves of rainfall expressed as a percentage of average rainfall for various durations and coefficients of variation and for given probabilities of occurrence.

Through regression

analysis, the rainfalls were converted to streamflow for the same durations. Assuming an initally full reservoir, Law used a cumulative depletion diagram to calculate the required reservoir capacity.

The generalized procedure is

appli cab Ie to the Bri tish Isles for which the empirical massed rainfall curves were derived.

Outside this region, a complete analysis would need

to be carried out before the procedure could be applied.

Other procedures

are less complex. 3.6

SUMMARY Critical period procedures for estimating reservoir capacity-yield

relationships were reviewed under three main headings - methods which indicate reservoir fullness with time, methods based on the range of flows and methods based on low flow sequences.

Another classification that could

have been used is based on whether the procedure allows storage to be related to probability of failure.

Those methods that do not consider

probability of failure - mass and residual mass curves, Hurst and sequent peak, and Waitt's minimum flow approach - are considered to be inadequate. However, the sequent peak algorithm, although reviewed as a technique to be used with only an historical record, was developed as an efficient approach to be used with generated sequences;

in that context it is possible to

calculate the storage-probability relationship. Examination of the assumptions and theoretical basis of the cri tical period procedures not included above, showed that all are deficient in one way or another.

For example, all assume that the reservoir is initially full.

Alexander, Dincer and Gould Gamma procedures also assume that annual serial correlation is zero.

In addition, Dincer's procedure is based on the

68

assumption that

n

consecutive year flows are normally distributed whereas

Alexander and Gould assume that flows are Gamma distributed.

Because of

cross-nesting of mass curves (Sec. 3.4.8) carryover and within-year frequency analysis underestimates storage need.

In addition the overlapping

carryover frequency procedure is inadequate because of effects of dependence. From this review it is concluded at this stage that the Alexander and Gould Gamma approaches appear to be suitable preliminary design procedures, and that behaviour analysis of finite reservoirs is a useful technique to display clearly the behaviour of the reservoir contents.

3.7

NOTATION

A

loge x - loge x

B

draft parameter in Hurst's equations (Eqs. 3.6, 3.7)

c

variable in Eq. 3.35.

C

reservoir capacity

C1 ' C2

various reservoir capacity estimates

CCRIT

reservoir capacity for critical drawdown using the Minimum Flow approach (Sec. 3.4.1)

CDESIGN

design reservoir capacity including safety factor

CP

critical period

CP I

critical period for a

C

coefficient of variation

C n,p

storage capacity for n year inflow with p% probability of occurrence

C

reservoir capacity in gamma units

d

difference between the lower p percentile flow of Gamma distribution Gec) and a Normal distribution N(c,c)

D

draft as ratio of mean flow

D.

draft

D n

constant draft over

D

draft during tth period

f(x)

probability density function

G( c)

Gamma dis tribution with mean and variance equal to

K

Hurs t exponen t

V

y

1

t

(Eq. 3.13) i.e. log of mean - mean of logs.

=

n

1

year period

c

69

number of months of emptiness for semi-infinite storage

,1',. 1

number of months of emptiness for semi-infinite storage (Sec. 3.2.4) water losses from the reservoir other than evaporation during time t m

rank of event

m.

number of months of emptiness for a finite storage

1

number of months of emptiness for a fini te storage n

1 ength of a sub-sequence of monthly flows

n

number of items of data

N

number of flow events

N

number of years of data

N'

number of months of data

N( c, C)

normal distribution with mean and variance equal to

p

percentage ch"ance of occurrence

p

probability of failure or probability of occurrence

c

probability of failure using behaviour diagram p

.

semI

probabili ty of failure using semi-infinite depletion diagram sequent peaks n

year flow with a probability of occurrence of p%

. fl ow d ' . d In urlng t th perla range of flows range of flows standard deviation

s s

n

standard deviation of

n

year flows

time

t

T.

recurrence interval for independent partial duration series

T

recurrence interval of an n-month event

T a

recurrence interval for overlapping partial duration series

T

recurrence interval (years)

1

n

r

sequent troughs x

flow volume

70

mean flow

x x

mean of

n

n

year flows

X

value of flow

x.l

flow

z

a'

z p

standardized normal variate

Zt,Zt+l

reservoir storage contents at the beginning and the end of tth time interval

a

shape parameter in Gamma distribution

a

n

n year flow Gamma shape parameter

a

estimate of

S

scale parameter in Gamma distribution

Sn

n year flow Gamma scale parameter

B

estimate of

~Et

net evaporation loss during time

T

reservoir capacity divided by mean annual flow

Tj

reservoir capacity divided by mean annual flow for

T

reservoir capacity divided by mean annual flow in Gamma units

Y

TCa)

a

S

Gamma function

CEq. 3.13)

CEq. 3.13) t

=

1

71

CHAPTER 4

PROBABILITY MATRIX METHODS The second group in the classification of reservoir capacity-yield procedures (Fig. l. 2) is headed "Moran Related and Other Techniques". Virtually all of the methods shown are based on the theory presented by Moran in his book Theory of Storage (1959). In terms of practical usefulness, the most important methods in this group are those described as probability matrix methods.

The other

techniques are mainly of theoretical interest and are important only because of their r6le in the development of the procedures which use historical data. This chapter includes some of the theoretical development, but is mainly devoted to details of selected probability matrix procedures.

4.1

GENERAL CLASSIFICATION OF MORAN DERIVED METHODS In Fig. 1.2 we see that the Moran approach can be subdivided into

three groups: (i)

those in which both time and volume are considered as

continuous variables.

The most common continuous time

model is the 'random buckets in the bath' model. Moran's (1959, p. 79) description of the model is of a man pouring buckets, at random instants of time, into a bath which has no plug".

Generally, the continuous

time model is the most complex and least realistic of the various classes of techniques due to Moran [see, for example, Gani (1955), Gani and Prabhu (1958, 1959) and Gani and Pyke (1960, 1962)].

The assumed form of the inflow distribution

(often Poisson), the darn size (possibly infinite) and the 'buckets in the bath' approach all contribute to the unrealistic nature of the solution.

Thus, continuous

time solutions are of theoretical interest only; (ii)

those in which time is discontinuous but water volumes aPe

continuous.

Moran (1955) derived the following integral

equations describing a mutually exclusive (see below) situation.

72

For x ( C - D D

g(x)

=

f(x)!

D+x

g(x) dx +

!

o

f(x+D- t) get) dt

(4.1)

D

For x > C - D C

D

g(x)

=

f(x) !

o

+ f(x

+

g(x) dx + ! f(x

+

D - t) get) dt

D

D - C)

!

(4.2)

get) dt

C

where

x

inflows,

C

reservoir capacity,

D

constant release during unit period,

f(x) g(x)

inflow probability function, and probability function of storage content plus inflow during unit period.

Solutions for particular inflow distributions and release rules have been obtained by Gani and Prabhu (1957), Prabhu (1958a) and Ghosal (1959, 1960).

Of these the most

potentially useful solution is that due to Prabhu (1958a) in which the inflows were assumed Gamma distributed and releases were assumed constant.

However, evaluation of

this solution is very complex; (iii)

those in which time and water volumes are both discrete

variables.

This approach by Moran (given in his 1954

paper) and followed by others (for example Ghosal, 1962 and Prabhu, 1958b) is the basis of the practical applications of his work. Basically it involved sub-dividing the reservoir volume into a number of parts, thus creating a system of equations which approximate the integral equations (Eqs. 4.1 and 4.2). This approximation primarily affects the results at the storage boundaries (that is, full and empty) but is satisfactory if the sub-division of the storage volume is fine enough. Two main assumptions can be made about the inflows and outflows, which occur at discrete time intervals.

The first, given by Moran (1954),

assumes that the inflows and outflows do not occur at the same time. this model, termed the "mutually exclusive" model, the unit period is

In

73

sub-divided into a wet season (all inflows and no outflows) followed by a dry season (all releases but no inflows). The alternative assumption, not given by Moran but which is only a simple further development, is that inflows and outflows occur simultaneously - the "simultaneous" model. Both of these models are discussed in detail in the following sections. 4.2

A SIMPLE MUTUALLY EXCLUSIVE MODEL It is convenient to choose the inflows, draft, and storage capacity as

integer multiples of some arbitrary volume unit.

Consider the following

example: Reservoir capacity:

2 units

Constant draft: Inflows:

unit per time period

discrete and independent and distributed as in Fig. 4.1.

Note that the sum of the probabilities

equals unity.

Relative Frequency (Probability)

1/5

o

1

2

3

Units of flow FIG. 4.1

Distribution of reservoir inflows.

For the mutually exclusive model we have: Z t+l

o

~

M

if M < Z + X < K t t

Zt+l

(4.3) (4.4)

Zt

(4.5) Zt + Xt ·· 'd store d water at t h e b eglnnlng 0 f t h e t th perlo,

Zt+l

stored water at the end of the tth period or at

Zt+l where

if Z + Xt t

ifK

~

the beginning of the (t+l)th period, K

X t M

capacity of reservoir, . fl ow d ' . d an d In urlng t th perlo, constant volume released at the end of the unit period.

74

Gi ven this information about capacity, draft and inflows, the firs t step is to set up the "transition matrix" of the storage contents. A transition matrix shows the probability of the storage finishing in any particular state at the end of a time period for each possible initial state at the beginning of that period.

The transition matrix for the above

example is a (2 x 2) matrix representing an empty condition and a half full condition as follows: Initial State Zt Empty Finishing

Full 1

°

Empty

State

°

Zt+l

1

Full

I

1 + -2 5 5

1 5

-

-1 + 1

5

5

-

2

(4.6)

2 + -1 + 1 5 5 5

-

2 =

1

(always check)

1

Each element of the transition matrix is found by applying Eqs. 4.3 to 4.5 to determine the inflows (and hence probability) of the storage beginning and ending in the state corresponding to that element.

In the

computations the boundary conditions (empty and full) must be considered and, for the mutually exclusive model, the inflows must be considered separately and prior to the outflows. Consider the element (0,0) in Eq. 4.6 which represents a reservoir starting empty and finishing empty.

This can happen if there are no

inflows for the period (probability 1/5) or if there is one unit of inflow (probability 2/5).

In the latter case the release of one unit reduces the

reservoir contents back to zero.

Hence, if the reservoir starts empty

there is a probability of 0.6 that it will still be empty at the end of the time period. Consider now the element (1,0) which represents a reservoir starting empty and finishing half full.

If there are two units of inflow

(probability 1/5) followed by one unit of release the reservoir will finish half full.

If there are three units of inflow (probability also 1/5) the

reservoir will spill because its capacity is only 2 units, then after 1 unit of release, it will again finish half full. from empty to half full is 2/5.

Thus the probability of going

75

Note that the reservoir can never finish (and hence start) in the full condition because of the mutually exclusive assumption about inflows and outflows.

Note also that the reservoir must finish in some condition

thus the sum of the probabilities in any column must be unity. Let us now assume that the time unit is equal to one year and that the reservoir of capacity 2 units is empty at the beginning of the year one, that is, the initial probability distribution of storage contents is: Storage

o (4.7)

1

State

2

L=l Since the transition matrix expresses the conditional probability of final storage contents given the various values of initial contents, the probability distribution of final contents can be found by the matrix product of the transition matrix and the probability distribution of initial contents. Therefore, at the end of year one (or at the beginning of the year two) the probability of storage content will be: 0.6 [ 0.4

0.2J

OJ

0.6 x 1 + 0.2 x [ 0.4 x 1 + 0.8 x 0

0.8

transition matrix

state of storage at beginning of year one

=

state of storage at end of year one

[0.6 ] 0.4 (4.8)

L 1.0

The quantitative process in Eq. 4.8 may be described as follows.

The

transition matrix shows the probability of the reservoir finishing in a specific state, given an initial state.

If the initial state is known in

terms of probability, then the joint probability will indicate the likelihood of the storage ending in a specific state.

In Eq. 4.8 the transition

matrix shows the probability of going from state 0

+

state 0 as 0.6, and

the probability of being in state 0 at the beginning of year one is 1, thus the probability of ending in state 0 is 0.6 x 1

=

0.6.

But also it is

possible to arrive at state 0 from state 1 which from the transition matrix has a probability of 0.2.

The likelihood of being in state 1 at the

beginning of the year one is 0, thus the probability of ending in state 0 but beginning in state 1 is 0.2 x 0

=

O.

Hence the combined probability of

ending in state 0 at the end of the first year is 0.6 + 0 argument holds for state 1.

= 0.6.

A similar

76

The process can now be repeated, using the state vector as the new starting condition.

Therefore, at the end of the second year, the proba-

bility of storage content will be: [ 0.6 0.4

0.2J 0.8

transi tion matrix

[0.6 x 0.6 + 0.2 x 0.4 J 0.4 x 0.6 + 0.8 x 0.4

[0.6 ] 0.4 state of storage at end of year one or beginning of year two

state of storage at end of year two

[0.44 ] 0.56

(4.9)

L=1.00

At the end of the third year, the probability of storage content will be: [0.6 0.4

0.2J 0.8

[0.44] 0.56

= [0.6 x 0.44 0.4 x 0.44

+ +

0.2 x 0.56] = [0.38 ] 0.62 0.8 x 0.56

(4.10)

L=1.00

At the end of the fourth year, the probability of storage content will be: 0.6 [ 0.4

0.2J 0.8

0.38J [ 0.62

=

[0.6 x 0.38 + 0.2 x 0.62] = [0.35] 0.4 x 0.38 + 0.8 x 0.62 0.65

(4.11)

L=1. 00 At the end of the eighth year the probability of the storage content will be: 0.33J [ 0.67

(4.12)

At the end of the ninth period it will be: 0.33J [ 0.67 It will be noticed that as successive years are

(4.13)

consi~ered,

the

probability vector of storage content becomes less affected by the initial starting conditions(in this example, the reservoir was assumed empty) and approaches a constant or steady state situation, which is independent of the initial conditions.

From the steady state vector (Eq. 4.13) it is seen

that there is a 1/3 chance that the reservoir will be empty at the end of any year. 4.2.1

of

The Discrete Equations for the Mutually Exclusive Model General Case.

Consider a reservoir with discrete inflows, X , and a constant draft t M during unit time period t. Zt is the stored content at the

77

beginning of time

t.

All volumes are multiples of a constant water volume.

The reservoir is divided into K-M+I discrete zones 0, 1, 2, ... ,K-M where

o

is the empty zone. From continuity it follows that

o

Zt

+ X

:: ~1

t

(4.14)

M < Zt + Xt <

K :: Zt in which

+

K

(4.15) (4.16)

Xt

M is released at the end of the wet season.

Let the probability

of the reservoir being in its empty zone at the end of a period be: pI

o

prob {Zt+l = o}

(4.17)

prob {Zt + X :: M}. t

(4. 18)

Assuming that the X values are serially independent, then t prob {Zt = olx t ' M}* + prob {Zt = llx t ' M - I}

Pb

+

+ prob {Z M

pI

Po

o

where

L

i=O

t

= Mix

M-l qi + PI

L

q.+

i=O

l

t

= o}

...

(4.19)

+ PM qo

(4.20)

P.

probability that the content of the reservoir

qi

probabili ty of receiving an inflow of

Pi

probability that the content of the reservoir

l

is in zone

i

before the inflows arrive, i

units

during the wet season, and is in zone

i

at end of period.

In a similar manner equations can be derived for each of the K-M+l zones that the reservoir contents can occupy. follow.

M

I

pI

0

M-l q.

i=O

pI

1

qM+l

pI

= qM+2

2

l

L

qo

0

0

0

P

qM

ql

qo

0

0

PI

qM+l

q2

ql

qo

0

P2

i=O

I

i=K-l * prob {Zt

olx

t

...

The resulting set of K-M+l equations

qi

0

q. l

:: M} means probability of Zt = 0 given that Xt :: M.

78

or in condensed form, [T] [P]

[P '] 4.3

(4.21)

A SIMPLE S lMUL TANEOUS MODEL Consider the previous example but now with inflows and outflows

occurring simultaneously. Reservoir capacity:

2 units

Constant draft:

unit per time period

Inflows:

discrete and independent, and distributed as in Fig. 4.1.

For the simultaneous model it follows that: 0,

(4.22)

o K

that is, X and t

M < K

(4.23)

M ;: K

(4.24)

M occur simultaneously.

Again the transition matrix is computed.

In contrast to the mutually

exclusive case it is now possible for the reservoir to finish in a full condition at the end of a period;

the resulting matrix is:

Initial State Empty

Full

a Finishing

Empty 0

I

-

5

State

I

2

I

5

5

I

2

5

5

2

5

=

I

1

2

I

I

Zt+l Full

+

Zt

I

-

5

+

1

0

I

5 I

5

(4.25)

2 1 1 + 5 + 5 5

-

1

the elements being determined in the same way as before.

For example, to

start full and end full (with one unit of release) is possible for any inflow except zero inflow.

Hence the probability is the sum of the

probability of inflows of one, two, and three units of inflow, that is, O.S.

79

As before, if one examines the probability of reservoir contents in successive years, one sees that the effect of initial conditions soon dies out.

For example, consider the storage initially empty.

.6 .4.2 .2 OJ [1]° [0.60J 0.20, [0.40J 0.24 , [0.29J 0.25 , ~0'18J 0.25 ,.. [0.13 0.25l [.2.2 .4.8 ° 0.20 0.36 0.46 0.56 0.62J End of End of End of End of 1st yr. 2nd yr. 3rd yr. 5th yr.

(4.26)

steady state

Now consider the storage initially full.

6 .2 0] [OJ° [O'0.20,0.24,0.25,0.25, OOJ [0.04] [0.07J [o.l1J .. [0.13] .2.4.2 [.2.4.8 1 0.80 0.72 0.68 0.64 0.25 0.62 End of End of End of End of 1st yr. 2nd yr. 3rd yr. 5th yr.

(4.27)

steady state

It is observed that steady state conditions are independent of either initial condition.

4.4

COMPUTATION OF STEADY STATE CONDITION Given the transition matrix of the storage content for a mutually

exclusive or simultaneous model, three methods are available to compute the steady state condition. (i)

Use the method adopted in the previous numerical examples in which the probability distribution of storage contents was computed year by year using [Pl + = [Tl [Pl . At steady state t t l [Pl + = [Pl . This approach also allows us to determine year t l t by year the probable change in storage content, for example, one is able to determine the probable rate of filling of a newly constructed dam.

(ii)

A second way to find the steady state condition of stored contents is to "power-up" the transition matrix thus:

(4.28) where [T] m

transition matrix, and positive integer large enough for the resulting matrix to be equivalent to steady state conditions.

80 For example, consider the transition matrix used previously.

[6.2

.2

.2

.4

.4

O~'

.8

.20

[:

.2

.2

.4

0:]

.4

.8

x

['

.2

.4

.Z

.4

[40

.28

OJ' [22

.16

.24

.24

.25

.25

.25

.36

.52

.72

.53

.59

.66

[22

.16 .Z5

[14

.13

.25

o~' .25

.25

.25

012J .25

.53

.59

.66

.61

.62

.63

[14 .25

.25

.25

[13 .25

.25

.25

.61

.62

.63

.62

.62

.62

.13

012J

.2

:] = [40

.8

.20

.24

.28

OO:J .24

.36

.52

.72

09J

.13

13J (4.29)

As a general rule squaring the transition matrix four or five times is sufficient to determine the steady state condition.

This condition is reached when each column

contains the same set of elements. (iii)

The third approach is based on the fact that at steady state [T 1 [P 1t

=

[P 1t + 1

In the numerical example of the simultaneous model above (Eq. 4.25), the equations become: 0.6 Po + 0.2 PI + 0 P z

Po

(4.30)

0.2 Po + 0.4 P + 0.2 P 2 1

PI

(4.31)

0.2 Po + 0.4 PI + 0.8 P 2

P 2

(4.32)

I t wi 11 be found that these three equations are not

independent.

To get a solution one of them (anyone) must

be replaced by the equation (4.33) Solving for PO' PI' P2 as a system of simultaneous equations gives

81

Po

0.125

PI

0.250

P

0.625

2

These values are the same as those found by method (i) (Eq. 4.27) and by method (ii) (Eq. 4.29). 4.5

DISCUSSION - MORAN TYPE MODELS The mutually exclusive model (Sec. 4.2) overestimates both the

probability of failure and the probability of spill.

This is because of

the assumption that inflows always precede outflows;

thus the reservoir

can never be full at the end of a time period. The simultaneous model, on the other hand, is more realistic in that it is more representative of reservoir inflow and outflow conditions.

Limi tations: For both models there are a number of limitations. (i)

Inflows are assumed to be independent.

For a monthly time

period this assumption is not valid, and the required storage capacity would be seriously underestimated.

Annual flows are

independent for many streams, but this period is normally too long for final design computations.

If annual flows are not

independent (50% of Australian streams have serial correlation coefficients greater than 0.1) the storage is underestimated. This is discussed further in Appendix B. (ii)

As a general rule, seasonality cannot be taken into account unless seasonal data is 'used and serial correlation assumed to be zero.

Note that if this were done, the definition of

probability of a given state would change.

For rivers

exhibiting a distinctive wet and dry season, the mutually exclusive approach could be adopted for annual data. (iii)

As described, the techniques assume constant draft, but varying drafts can be accommodated so long as they are stored-water dependent and not time dependent.

(iv)

A computer is required for solution of realistic problems. (The above examples were grossly simplified for illustrative purposes. )

82

(v)

An iterative solution is necessary if the storage capacity or release for a given probability of failure is to be determined. In the analysis the capacity and draft are assumed to be known and probability of emptiness is computed.

If capacity or draft

is unknown, a value is assumed and the computed probability is compared with the design probability.

Attributes: (i)

The procedures sample all years of stream flow records without reference to the historical sequencing (but serial correlation is assumed zero).

(ii)

Computed storage estimates are independent of the starting conditions.

This is in contrast to critical period techniques

and is one of the prime advantages of matrix methods. (iii)

N.on-continuous records can be handled.

As annual flows are

assumed to be independent, the sequencing of flows becomes unimportant and so records with missing data can be used as effectively as continuous records. (iv)

Estimates of probability of the state of the reservoir can be computed either at steady state or as a time dependent function of starting conditions.

4.5.1

Further Modifications An exampie of Moran's

procedur~

is given by Jarvis (1964) who applied

the technique to the Ord River storage in Western Australia.

His unit time

period was a year and he assumed that the annual flows were independent. His release rule was more complex (and more realistic) than the constant release adopted by Moran. In 1955 Moran modified the discrete model to deal with seasonal flows. Transi tion matrices were prepared for each season and were multiplied together to yield an annual transition matrix.

However, as before, the

seasonal flows must be assumed to be independent.

Lloyd and Odoom (1964)

adopted a somewhat similar approach. Harris (1965) gave a worked example of Moran's seasonal method applied to a British catchment.

He found the flows to be seasonal and independent,

and prepared wet and dry season transition matrices which were multiplied together to give the

ann~al

transition matrix.

83

Lloyd (1963) partly overcame the independence assumption in the Moran approach by assuming that the inflows were represented by a bivariate distribution rather than a simple histogram. number of equations to be solved.

In effect, this squares the

Dearlove and Harris (1965) made the

technique more applicable by combining Lloyd's approach with Moran's seasonal method, but computationally the problem is large and therefore limited.

However, Doran's recent work (Doran, 1975) on the divided interval

technique for solving the transition matrix may overcome tlris limitation. Venetis (1969) developed monthly bivariate transition matrices from generated flows using Roesner and Yevjevich's model (1966).

Following Moran,

and Dearlove and Harris, he multiplied the matrices together to obtain an annual transition matrix. 4.6

GOULD'S PROBABILITY

~MTRIX

METHOD

Gould (1961) modified the simultaneous Moran-type model to account for both the seasonality and serial correlation of inflows.

He did this by

using the transition matrix with a yearly time period, but accounting for wi thin-year flows by using behaviour analysis.

Thus monthly flow variations,

monthly serial correlation and draft variations can be included. In the approach storage capacity and draft are given, and probability of failure of the reservoir is determined.

If draft or storage size is to

be determined then a trial and error method needs to be utilised. 4.6.1

Procedure The procedure to determine storage capacity for given draft and

probability of failure criteria is as follows: (i)

Decide on the releases lD ) required from the reservoir in t volume units per month and how they are to vary from month to month throughout the year.

It is usual that releases

are seasonal, although the method requires that annual releases are not time variant.

Operating policy may require

the releases to be reduced if the stored contents fall below some predetermined values; be defined at this stage.

thus the release rule needs to In addition, net evaporation loss

from the reservoir which is a function of surface area (hence stored content) needs to be defined.

84

(ii)

Decide on the design probability of failure (P ) which e is defined as the probability of the reservoir running dry in any month.

(iii)

Assume a first trial, reservoir capacity e1 (volume units).

(iv)

Set up a tally sheet (Fig. 4.2) to construct the transition matrix.

Figure 4.2 is shown here for descriptive purposes.

Wi th the computations being done on a computer, the "tally sheet" is actually a computer matrix. Zt

0

Zt+l

(End of period)

FIG. 4.2

0 1 2 3 4

1

(Beginning of period)

2

3

4

19 l

Hal Hbl

16 17 18 19

·HcJ

f,

Example of tally sheet for transition matrix in Gould's procedure.

Divide the trial reservoir capacity

e

into

K zones.

As a general rule twenty is sufficient (see Sec. 4.6.2). The volume of each zone including the top and bottom ones is given as follows: (4.34)

Volume (W) 18,

1,2,

0,

19

~ --------------~V ~------------/

t

t

.....

empty zone

18 zones of

full zone (e1)

of zero volume

of equal volume

of zero volume

(W) (v)

Apply the continuity equation (same as for behaviour analysis) on a monthly basis taking one year of data at a time as described below. (4.35)

85

where Zt' Zt+l X t D t 6E t

ilie storage content U ilie beginning th Md end of the t month, the inflow during the tth month, ' ilie release d urlng t h e t ili month, Md the net evaporation loss during the tth month.

The net evaporation loss is the difference between the evaporation from the proposed reservoir and the evapotranspiration from the proposed reservoir site and will be a function of water surface area which in turn is a function of volume of water in active storage plus dead storage capacity. For each year of data (there are a total of

N years),

Eq. 4.35 is applied month by month to determine the zone in which the reservoir finishes at the end of the year. This is done for each possible starting zone, and the element corresponding to the starting and finishing zone in the tally sheet (Fig. 4.2) is incremented by one.

Any

failures (reservoir emptying) that occur during the year are also noted. In applying the continuity equation, the seasonal variability of D should be taken into account in cont junction with any restrictions imposed on the release as a result of the present state (Zt) of the storage Md also the 6E

t

losses.

To illustrate the procedure, suppose the historical inflow data for a reservoir site begins in 1929. The steps are: (a)

Apply the continuity equation to 1929 (the first

year), assuming the reservoir contents are initially in state O. Route the flows month by month observing the operating rules associated with D , the losses 6E , and t t the boundary conditions of emptiness and spill. In the example illustrated in Fig. 4.3, no restrictions are applied to D and evaporation losses are zero. At the t beginning of the year Zt = 0 and the reservoir content ends the year in zone 3, that is, Zt+l = 3, and the corresponding element on the tally sheet is incremented. It is noted as (a) in this case in Fig. 4.2.

86 19

19

18

19 18 17

17

A

FIG. 4.3 (b)

1929

1929

1929

(a)

(b)

(c)

Monthly routing example in Gould's Procedure. ~ext

apply the continuity equation to 1929

beginning with the reservoir in state 1.

Route

flows as before. Hence Zt

= 1, Zt+l = 3.

See (b) on the tally sheet.

Other zones are completed in a similar manner. (c)

Finally, begin in zone 19 (that is, the full zone)

and then apply continuity equation. Hence Zt ~~en

= 19, Zt+ 1 = 17 and record.

It is shown as (c).

the routing is completed, the sum of the elements of

the tally sheet should be N x K (N years of flow x K zones) with subtotals of (vi)

N in each colurrrn of the matrix.

Divide each column by

N to convert the element contents

to probability, thus forming the transition matrix which expresses the state of the reservoir contents at the end of a year as a probability relationship of the state of the reservoir contents at the beginning of that year. (vi i)

In addition to the transi bon matrix, it is necessary to estimate the probability of failure (that is reservoir emptiness) as a function of the starting zone.

For

example, consider Fig. 4.3: in case (a) - reservoir started in zone 0 and failed (emptied) in two months (July and August); in case (b) - reservoir started in zone 1 and failed in one month (August); in case (c) - reservoir started in zone 19, and no failures were recorded.

87 From this,another tally sheet (or vector in the computer) is built up in which the probability of failure is related to the contents of the reservoir at the beginning of the year.

The second tally sheet is shown in Fig. 4.4.

Zt

------

19

No. of months of failure

FIG. 4.4

Example of tally sheet for monthly failures in Gould's Procedure.

As there is a total of 12 N months routed through each zone, the values in Fig. 4.4 may be converted to probability by dividing each entry by 12 N.

This step is different to

that proposed by Gould in that he defined failure on an annual basis, that is, one or more monthly failures within

a year constituted one annual failure.

In effect, Gould's

measure overestimates the monthly behavioural capacities by about 20%.

(Based on an analysis of six Australian rivers

and two draft conditions, percentage increases are given in Table 4.1.)

For this reason the procedure recommended above,

that is defining monthly failures from monthly data, is preferred to Gould's approach. TABLE 4.1

Percentage increase in required capacity using Gould's definition of failure over a behaviour analysis using monthly data.

Ri vers (Australian gauging station no. )

50% draft

90% draft

Yarra (229102)

29

17

Murrumbidgee (410008)

32

16

Lachlan (412010)

15

15

Warragamba (212240)

41

7

Namoi ( 419007)

21

23

Burdekin (120090)

20

26

Mean

26

17

88

(viii) Compute

the steady state matrix by one of the three

procedures given in Sec. 4.4. for this operation.)

(A computer is required

That is: m

(a)

Power up the transition matrix [T]

(b)

set up simultaneous equations and solve.

or Remember

to replace one equation by Po + PI + .•• + PK- l

=1

l4.36)

(see Eqs. 4.30 - 4.32 as an example); (c)

or

compute the stored content year by year starting

initially with the reservoir in a known or assumed state. (ix) Multiply the steady state probability of the reservoir being in a particular zone [from step (viii)] with the probability of failure for that zone computed in step (vii); the resulting sum of the products is the probability of failure of the reservoir.

An example is shown in Table 4.2

where the probability of failure is computed to be 0.02. TABLE 4.2

Zone

(1)

Example of combining steady state probability with the conditional failure probability in Gould's procedure.

Probability of starting in a particular zone (steady state) (2)

Conditional prob abi 1 i ty of failure in any month within any year for that zone (3)

Product of probabili ties

(2)

x

(3)

0

0.016

0.385

0.0062

1

0.017

0.154

0.0026

2

0.064

0.051

0.0033

3

0.094

0.026

0.0024

4

0.202

0.026

0.0053

5

0.216

0.000

0.0000

19

0.101

0.000

0.000

L= 0.0200 (x)

If the computed probability of failure does not

e~ual

the

design probability, choose a new capacity and proceed from step (iv) again.

89

Assu;nptions:

(i)

Annual flows are assumed independent, that is, serial correlation is zero.

Limitations:

(i)

The analysis requires a computer for the computations.

(i i)

The explici t result of the Gould analys is (like other probability matrix methods) is probability of failure for an assumed storage size.

If storage size or draft

is required to be calculated, a trial and error approach must be adopted.

Usually three or four iterations are

sufficient. (iii)

Annual serial correlation is assumed to be zero.

For

many streams this assumption is satisfactory but for many others it is not. Nevertheless, as recommended for other procedures in which annual flows are assumed to be independent, storage estimates should be corrected for the assumption of independence irrespective of whether or note the serial correlation value is statistically significant (see Appendix B for method). Gould (1964) does provide a correction for storage estimates on streams with a significant serial correlation coefficient, but it is of limited value because it does not cover the required range of design criteria. AUributes:

(i)

The procedure samples all years of data without reference to the historical sequencing of annual flows.

This is in

contrast to many critical period procedures which sample only periods of critical duration. (ii)

Computed storage estimates are independent of the initial reservoir conditions.

(iii)

Non-continuous records can be handled.

As annual flows are

assumed to be independent, the sequencing of flows is unimportant and so records with missing data can be used as effectively as continuous records. (iv)

No assumptions need to be made about the distribution of the historical flows.

Relatively speaking this is not a significant

advantage as many other storage procedures also do not require

90 an assumption about a theoretical distribution of flow for their implementation. (v)

Monthly parameters and monthly serial correlations, except that between the last month of year of year

i+l,

i

and the first month

are automatically taken into account as yearly

flow sequences are routed on a monthly basis through the reservoir. (vi)

Probability of failure is computed either at steady state or as a time dependent function of the starting conditions. The latter attribute is a most important characteristic of probability matrix methods.

(vii)

Varying drafts and complicated release rules can be handled easily.

(viii)

Net evaporation losses from the reservoir can be related through the stored water content to the surface water area during the analysis.

4.6.2

Practical Considerations (i)

To assist in determining a given reservoir capacity it has been found useful to plot probability of failure (on a Normal probability scale) against storage capacity (on a logarithmic scale) as this tends to linearise the relationship and reduce the number of iterations required to determine the capaci ty for the given design probabi li ty.

(il)

For more variable streams difficulties have been experienced at high drafts (> 70%) with the limited accuracy (round-off errors) associated with mini-computers in solving for steady state using (20 x 20) or larger matrices.

In these cases,

as a check to the solution of simultaneous

equ~tions

it is

recommended that the transition matrix be powered up and a check made to ensure that the sum of each column equals unity. (iii)

Probability matrix solutions are affected by zone size and hence the number of zones.

Joy (1970, Appendix V) examined

this question using Moran's mutually exclusive model and found that for streams with C < 0.5, 20 zones were required v to adequately define the storage size; for 0.5 < C < 0.85, v 30 zones were required and for C ~ 0.85, 40-50 zones were v required. Teoh (personal

communication, 1977) analysed ten streams with

Cv varying between 0.19 and 1.79 using Gould's procedure.

From

91 his results it is concluded that as a general rule: for C < 0.5 usc 10 zones, v 1.0 usc 20 zones, for 0.5 ~ C < v for 1.0 ~ C < 1.5 use 30 zones, and v for C ;: 1.5 use 40 zones. v The differences between these and Joy's results (which were based on Moran's rather than Gould's model) arc consistent with Doran's divided interval approach (Doran, 1975) and Klemes' (1977) recent analysis. If an insufficient number of zones is used, sometimes a hunting effect in the storage-probability relabon becomes evident.

An example is shown in Fig. 4.5 for the Nogoa River

in Queensland.

Results arc plotted for 10, 20, 30 and 40 zones. ~

25

+----+ ./0.

10 zones 20 zones 30 zones 40 zones

20

~ ~

"

15

'0 .~

10

~

:c

"'0

.0

ct

5

O+-----~----_r-----+----~~~

o

2 Reservoir capacity

llG. ->'5

3

4

(xl0 6 m3 )

Effect of llllmber of zones on reservoir capaci ty probabili ty of failllre relation (Nogoa River Australian gauging station no. 13020J).

(iv) At this state little guidance can be given regarding the effect of beginning a Gould analysis in different months because insufficient research information is available.

It appears that, for

at least some rivers, the derived storage using a GOllld analysis docs depend on the starting month if the draft is high.

It is,

therefore recommendecl that before a fina 1 des i gn capaci ty is chosen, four separate Gould analyses be carried out, each heginning three months apart, to check the significance of the starting month.

92

EXAMPLE 4.1 For the Mitta Mitta River (Appendix E) use Gould's probability matrix procedure to determine the storage required to meet a draft of 75% of the mean flow with a 5% probability of failure.

*

*

*

*

*

The Gould procedure requires a computer for efficient solution; for each estimate of storage capacity the method requires each year of flow to be routed through the storage for each possible starting condition.

In

Sec. 4.6.2 it is shown that the number of zones required depends on the coefficient of variation of annual flows, C ' For the Mitta Mitta River, v 15 zones should therefore suffice.

C is equal to 0.57; v

The procedure is an iterative one, the probability of failure being calculated for the input draft and the storage capacity estimate. For the Mitta Mitta at 75% draft and a storage capacity of 910 x 10 6 m3 the following (15 x 15) transition matrix is obtained (terms are espressed as probability): Starting Zone

o o

3

4

6

Zt 7

8

9

10

11

12

13

14

.147 .147 .147 .147 .147 .118 .118 .088 .088 .000 .000 .000 .000 .000 .000 .118 .118 .118 .118 .118 .088 .029 .059 .000 .088 .000 .000 .000 .000 .000 .029 .029 .029 .029 .029 .088 .088 .000 .059 .000 .088 .000 .000 .000 .000

3

.029 .029 .029 .029 .029 .029 .059 .088 .000 .059 .000 .088 .000 .000 .000

N~ 4

.029 .029 .029 .029 .029 .000 .029 .059 .088 .000 .059 .000 .088 .000 .000

.118 .118 .118 .088 .088 .059 .000 .029 .059 .088 .000 .059 .000 .088 .029

~

+

~

5

~

6

.059 .059 .059 .088 .059 .059 .059 .000 .029 .059 .088 .000 .059 .000 .059

....~ ....~

7

.029 .029 .029 .029 .059 .059 .059 .059 .000 .029 .059 .088 .000 .059 .029

~

8

.029 .029 .029 .029 .029 .088 .059 .059 .059 .000 .029 .059 .088 .000 .029

9

.029 .029 .029 .029 .029 .029 .088 .059 .059 .059 .000 .029 .059 .088 .029

.aoo

10

.000 .000 .000 .000 .000 .000 .029 .088 .059 .059 .059

11

.029 .029 .029 .029 .029 .000 .000 .029 .088 .059 .059 .059 .000 .029 .029

.020 .059 .088

12

.059 .059 .059 .059 .029 .029 .000 .000 .029 .088 .059 .059 .059 .000 .029

13

.147 .147 .147 .147 .176 .llS .147 .147 .147 .176 .265 .265 .294 .353 .353

14

.147 .147 .147 .147 .147 .235 .235 .235 .235 .235 .235 .294 .324 .324 .324

To compute the steady state (or long term) probabilities of the reservoir being in any particular zone the transition matrix can be powered up or solved as a system of simultaneous equations (Sec. 4.4).

The latter

option involves less computation and the result is tabulated below with the respective probability of failure for each starting zone:

93

Zone

Steady State Probabi l i ty of being in Zone

Probabi l i ty of Fai lure from Starting in Zone

(2)

(1)

Contribution to Overall Probability of Failure

(3)

(2) x (3)

0

0.034

0.502

0.0171

1

0.026

0.480

0.0125 0.0065

2

0.018

0.360

3

0.017

0.260

0.0044

4

0.015

0.157

0.0024

5

0.057

0.083

0.0047

6

0.039

0.044

0.0017 0.0007

7

0.044

0.017

8

0.029

0.007

0.0002

9

0.051

0

0

10

0.055

0

0

11

0.031

0

0

12

0.028

0

0

13

0.276

0

0

14

0.279

0

0 I:

=

0.0502

that is, the probability of failure of a 910 x 10 6m3 storage is 5.02%. This estimate needs adjustment for the effects of the annual serial correlation of 0.06.

From Fig. B.l the adjustment factor is 1.06.

Therefore, the final answer by Gould method is 910 x 1.06 6 (x 10 m3).

4.7 4.7.1

=

960

RELATED PROBABILITY MATRIX METHODS McMahon's Empirical Equations For 156 Australian streams, McMahon (1976) used Gould's modified

procedure to estimate the theoretical storage capacities for four draft conditions (90%, 70%, 50% and 30%) and three probability of failure values (2±%, 5% and 10%).

These capacities were related by least squares analysis

to the appropriate coefficient of variation of annual flows by the following simple relationship:

L)~

where

aC b

C/x

T

(4.37)

v

C

storage capacity in volume units,

x

mean annual flow in volume units,

T

reservoir capacity divided by mean annual flow,

C v a, b

coefficient of variation of annual flows, and empirically derived constants tabulated in Table 4.3.

In addition to

a

and

b

in Table 4.3, standard errors of estimate

in percent and coefficients of determination are shown.

The constants were

not based on regional analysis and are considered to apply to the whole of Australia. TABLE 4.3

Draft

(%)

90

Reservoir capacity-yield equation coefficients, standard errors and coefficients of determination for Mc,lahon I s empi rical method. (e = standard error of estimate, r2 = coefficient of determination) Probabili ty of failure (% ) Parameter

2.5

5

a

7.50

5.07

b

l. 86

l. 81

e

+ 18, -15

r2

97

a

2.51

b 70

50

30

+21, -17

r2

96

a

0.98

b

+12, -11

1. 83

e

+58, -36

r2

81

3.08 1. 82

+43, -30

98

87

1. 81

1. 21

1. 79

+25, -20

1. 91

e

10

1. 74

+29, -23

94

92

0.75

0.51

1. 93

+63, -39 79

1. 83

+61, -38 79

a

0.28

0.22

0.15

b

1. 53

1. 49

1. 79

e

+44, -31

r2

82

+61, -38 72

+64, -39 77

Assumptions, Limitations and Attributes: As the storage values used in the regression analysis are Gould estimates, a major assumption relates to the neglect of annual serial correlations.

Corrections given in Appendix B should be made.

Another

95

assumption is that capacity for given conditions is related only to the coefficient of variation.

The proportion of variance accounted for shown

in the table suggests that this is a reasonable assumption. Because of the errors noted above and limitations due to constant oraft, this procedure is regarded only as a preliminary procedure.

However, as a

preliminary procedure it is based on monthly flol"s and on a large number of well-distributed Australian streams and therefore should provide reasonable estimates of storage at least within the Australian environment.

EXAMPLE 4.2 Compute the storage required on the Mitta Mitta River (Appendix E) to meet a draft of 75% with a 5% probabi li ty of fai lure using McMahon's Empirical equations.

*

*

*

*

*

From Eq. 4.37 storage,

C

(aC

b

v

-

) x

From Appendix E for the Mitta Mitta C

0.57

v

Table 4.3 gives values of ability of failure.

a

and

b

for various drafts and prob-

Since a draft of 75% is not mentioned specifically, it

is necessary to interpolate on a

log-linear plot of draft versus storage

as follows:

ae b v

C (l06 m3)

1. 81

1. 83

2331

1. 79

0.66

841

0.75

1. 93

0.25

319

0.22

1. 49

0.10

127

Draft (%)

a

b

90

5.07

70

1. 81

50 30

Interpolation for 75% draft on the

log-linear plot of storage versus

draft (Fig. 4.G) gives a storage estimate of 1090 (x 10 6 m3).

96 Adjust for annual serial correlation.

From Fig. B.l for 75% draft

and annual serial correlation of 0.06 correction factor is 1.06 approximately.

Thus the estimate of storage requirement by McMahon's procedure is: 1090

x

1.06

X

2000

1000 M

X

800

E ~

600 500

0 ~

400

~

CD ~

X

300

~

0 ~

00

200

X

100

FIG. 4.b

4.7.2

0

Interpolation on log-linear plot for McMahon's Empirical procedure (Example 4.2).

Probability Routing Langbein's Probability Routing method (Langbein, 1958) is very similar

to Moran's (1954) probability matrix procedure except that Langbein modified his technique to deal with correlated annual inflows.

Both the stream-

flow regime and reservoir storage were divided into low, medium and high sub-regimes.

By classifying each flow into the same streamflow regime as

its predecessor, three separate streamflow histograms were obtained.

Thus

in setting up his system of equations describing the cumulative probability

97 of reservoir contents, Langbein used the inflow distribution appropriate to the state of the reservoir.

In this way he was able to take annual serial

correlation into account in an approximate way. 4.7.3

Hardison's Generalized Method Hardison (1965) generalized Langbein's probability routing procedure

using theoretical distributions of annual inflow and assuming serial correlation to be zero.

This is equivalent to Moran's (1954) model except

that Hardison used a simultaneous model rather than the mutually exclusive type adopted by Moran.

The annual storage estimates are shown graphically

in Figs. 4.7, 4.8 and 4.9 for log-normal, Normal and Weibull t distributions of annual flows.

The percentage chance of deficiency shown in the figures

is defined by Hardison as the percentage of years that the indicated storage capacity would be insufficient to supply the design draft. In addition to the carryover storage based on annual data, Hardison presented a procedure for determining the combined carryover plus seasonal storage requirement.

The latter procedure is discussed in Sec. 3.4.9.

Procedure: (i)

Compute the mean, standard deviation and coefficient of skewness of both the annual flows and their common logari thms.

(ii)

The appropriate distribution depends on the parameters computed in (i) as follows: (a)

Adopt a log-normal distribution if the coefficient

of skewness of the logarithms of flows is algebraically greater than -0.2. (b)

Adopt a Normal distribution if the coefficient of

skewness of the absolute flows is algebraically less than +0.2 or if the coefficient of variation of the flows is less than 0.25. tThe probability density function of the two parameter Weibu1l distribution is: f(x) where

11

11 (~) 8

8

11-1

exp [- (~e)

11

1

shape parameter, and characteristic drought when Prob (x

e)

lie.

98

........,.... 0

10· PERCENT CHANCE OF DEFICIENCY

5 PERCENT CHANCE OF DEFICIENCY

2 . PERCENT CHANCE

. PERCENT CHANCE

c:

::t \..

(ij

::t

c: c:

<'G

c:

<'G

Q)

E

..... 0 0

:;:: <'G \..

<'G 1/1 <'G

.....>.

3

Co

2

'v<'G <'G

() \..

0

... >

Q)

1/1 Q)

0:

Variability index FIG.

Variability index

4.

Hardison's relationship of reservoir capaci ty-yieldpercentage chance of defi ciency for log-normal distributions of inflow (Hardison, 1965).

eel

Adopt a Weibull distribution if neither a log-normal

nor a Normal distribution is selected except when the

thms of

flows have a negative skew coefficient

greater than 1.5. (iii)

Decide upon the reliabili ty (that is, Hardison'

5

percentage

chance of deficiency) and the required draft. (iv)

Based on (ii), enter appropriate Fig. 4.7,4,8 or 4.9

and

estimate reservoir capaci ty as a ratio of mean annual flow. If the log-normal distribution is adopted use as the variability

index CEq, 2.5) as the measure of variability;

otherwise

use the coefficient of variation, (v)

Compute the serial correlation of annual flows and adjust appropriately the required storage capacity (Appendix B).

99

:;;

10 - PERCENT CHANCE 4 .--_......--O_F---,D_E_F_IC_I,.E_N_C_Y-.-_-.

5 - PERCENT CHANCE OF DEFICIENCY

o

c

2

3

"'iii :J C C

2

('II

C

('II

Q)

E

o o

2 - PERCENT CHANCE

1 - PERCENT CHANCE

o

~

Q)

(/)

Q)

a:

0.1

0.2

0.3 0.4 0.5 Coefficient of variation FIG. 4.8

Coefficient of variation

Hardison's relationships of reservoir capacity-yieldpercentage chance of deficiency for Normal distribution of inflows (Hardison, 1965).

Assumptions, Limitations and Attributes: Like most probability matrix procedures the serial correlation of annual flows is assumed independent.

Adjustments are given in Appendix B.

As analysis is based on annual flows, seasonal variations are not accounted for;

Hardison provides a generalized procedure to correct for this, but it

is valid only for streams like those in the eastern United States where variability is low (Sec. 3.4.7). As a preliminary procedure for estimating carryover storage the procedure is reasonably quick, theoretically sound (except for neglecting annual serial correlation) and covers all but the most variable streams (with C v

>

1.66) that one is likely to encounter in practice.

100

:::0 c

4

~ :l C

3

...

:l

c

1'0

10-PERCENT CHANCE OF DEFICIENCY

5- PERCENT CHANCE OF DEFICIENCY

2

C 1'0 /I)

E

....0

0

0

...

1'0 11/ 1'0

1 - PERCENT' CHANCE

2 - PERCENT CHANCE OF DEFICIENCY

:p

4

OF DEFICIENCY

>- 3


·u 1'0

Co

2

1'0 (.)

...

·0 ...>

/I)

11/

00

/I)

0.20.4 Coefficient of variation

0 Coefficient of variation

a:

FIG.

4.9

Hardison's relationship of reservoir capacity-yieldpercentage chance of deficiency for Weibull distributions of inflows (Hardison, 1965).

EXAMPLE 4.3 Compute the storage required on the Mitta Mitta River (Appendix E) to provide 75% draft with a 5% probability of failure using Hardison's Generalized method.

*

*

*

*

*

As described in Sec. 4.7.3, it is necessary to compute the mean, standard deviation, and skewness of both the annual flows and the common logarithms of annual flows. Appendix E.

Parameters for the flows are given in

These calculated annual parameters are:

101

~m

Stmdard Deviation

Skew

Flows

1274

731

1.50

Log 1O (flow)

3.04

0.24

-O.OOS

Using these parameters the choice of distribution indicated under procedure step (ii) in Sec. 4.7.3 is the log-normal.

Thus the curves in

Fig. 4.7 are used. From the curve for 5% chmce of deficiency md a variability index equal to the standard deviation of the logs (that is, 0.24) the reservoir capacity for 75% draft is approximately 90% of the mean annual flow, that is, 1150 x 10 6m3 • After applying the correction for correlated annual flows (from Fig. B.l the correction factor is 1.06 for a serial correlation of 0.06) the estimated storage by Hardison's method is 1220 x 10 6m3 .

4.S 4.S.l

OTHER MODELS Melentijevich Assuming an infinite storage md independent normal inflows,

Melentijevich (1966) obtained expressions for both time dependent and steady state distributions of reservoir content.

In addition, he gave results for

the mean, varimce and skewness of the surplus, the deficit md the range of storage.

However, he made no mention of how they could be applied in a

design situation. In considering finite reservoirs, Melentijevich used a random walk model md a behaviour analysis of 100,000 rmdom normally distributed numbers.

From the analysis, he obtained an expression for the density

function of the stationary distribution of storage contents.

The solution

is complex and limited in use because of the assumptions of normality, independence md neglect of seasonality. retical interest.

The results are only of theo-

102

4.8.2

Klemes By considering a reservoir to have only two states - wet and dry -

Klemes (1967) was able to reduce the probability of failure within a limited period to the classical occupancy problem (that is, the determination of the number of arrangements of placing a number of identical balls into a number of identical cells).

His technique is restricted to a

uniform release or a randomly varying one.

The major limitation relates to

the unit period being one year and, as a consequence, it is not possible to discern a failure within a year.

This could be overcome by working with

monthly rather than annual flows but the number of possible arrangements becomes very large. 4.8.3

Phatarfod Phatarfod's procedure (1976) is based on random walk theory and is

concerned"with finding the probability of the contents of a finite reservoir being equal to or less than some value capacity of the reservoir.

~C

where

~C

> 0

and

C is the

The physical process of dam fluctuations can be

likened to a random walk with impenetrable barriers at full supply and empty conditions.

Phatarfod used Wald's identity, an approximate technique,

to solve a problem with absorbing barriers and a relation connecting the two kinds of random walks. Steps in his method, which assumes the draft is the unit of measurement, are: (i) (ii)

Assume a constant draft' D as a ratio of mean annual flow. Calculate 4

h

Y

a

Q::L

e

).l -

o

(4.39)

2

where ).l

20

(4.40)

y

mean flow in draft units

liD,

= standard deviation of annual flow in draft units, = ).lC ' v

C v y

(iii)

(4.38)

2

Decide on

coefficient of variation of annual flow, and coefficient of skewness of Annual flow. P

and

~

where

P

is the probability of the

reservoir contents being less than £C total capacity.

and

C is the

103 (iv)

Solve for

y,

the unique positive solution

(other than unity) of Py v-I + Py v-2 + ... + Py - (l-P) where

For example, if t

(v)

= 1/3,

H- 1 + [1 +

=

(4.41)

= v.

l/t

Y

o

4(1~P)l\

(4.42)

Use the Newton-Raphson iteration to solve for e, which is the unique positive solution of:

(vi)

e (1- e)

=

where

r

+r) + ea(l-r) + [{il+r)+ eaCl-r)}2-4rl~J ( 4 . 4 3) h loge [Cl---"----~~--"--'~~'----'-~-'-"--"-~ annual serial correlation coefficient.

Calculate required capacity of the reservoir as C where

=

v log y

e

Dx

(4.44)

C

capacity in volume units, and

x

mean annual flow in volume units/year.

This model assumes that annual flows are Gamma distributed and is based on a fixed draft.

It is considered to be a preliminary design

procedure, although the solution of Eqs. 4.41 and 4.43 can be quite timeconsuming.

The procedure is a useful preliminary way to determine the

likelihood of the reservoir falling below some level and the possibility of restrictions in releases. is limited to

v

Because of the approximation made, the procedure

being less than or equal to about 5.

(Phatarfod, personal

communication, 1977.)

EXAMPLE 4.4 Find the storage required on the Mitta Mitta River (Appendix E) to provide for 75% draft with a probability of the reservoir being less than one third full using Phatarfod's procedure. *

*

*

*

*

For the Mitta Mitta River (Appendix E) annual flow parameters are: coefficient of variation

0.57

coefficient of skewness

1.50.

104 Draft

D

0.75

In draft units, mean flow

lJ

liD

and standard deviation

(J

11

1. 33,

=

Cv

(1.33) (0.57) 0.76 From Eq. 4.38,

h

From Eq. 4.39,

a

From Eq. 4.40,

e

4

1. 78

2y

=

(0.76) (1.50)

0.57

2 2a

Y 1.33 _ 2 (0. 76) lJ -

--r:so

0.32. Choose t

= 1/3

P of 0.05 gives the chance

so that the probability

of the reservoir being less than or equal to 1/3 full. From Eq. 4.42,

y

H- 1

+ [1 + 4(l-P)] \

H- 1

+ [1

P

!

+ 4(1 - 0.05)] } 0.05

3.89. Solve Eq. 4.43 for

e:

eel-e)

= h loge [(l+r)+ 8a(1-r)+

where

r

[{(~+r)+

annual serial correlation

8a(1-r)}2 - 4r]!]

=

0.06

8(1-0.32)= 1.78 loge [1.06+ 8(0.57) (0.94)+ [{1;06+ 8(0.57(0.94)}2 - 0.24]!] Using the Newton-Raphson iteration procedure (Appendix D), the required value of

8

From Eq. 4.44:

is found to be 1.824. Storage

v log Y D 8

x

3 loge 3.89 1. 824

Dx

2.23 (0.75)(1274) 2130 (x 10 6m3) [Note that this is the reservoir size for which there is a probability of 5% Thus the figure of 2130 x 10 6m3 is

of being only one third (or less) full.

105 not directly comparable with reservoir sizes based on 5% probability of failure.

However, one can run the Gould procedure and compute the

probability of being in the lower one-third of the storage from 'the steady state matrix. For 2130 x 10 6m3 storage the answer is 3.8%.]

4.9

SUMMARY From a theoretical point of view the Gould procedure as described in

Sec. 4.6 stands out as the most acceptable reservoir capacity-yield technique. Essentially the procedure involves only one major assumption and overcomes most of the disadvantages of other probability matrix procedures and critical period approaches.

Based on these reasons and the satisfactory

results of several extensive testing programs

using Australian streamflow

data (reported in Chapter 6) the Gould procedure, modified as outlined in Sec. 4.6.1 and corrected for annual serial correlation, is recommended as a final design tool for establishing the single reservoir capacity-yieldprobability of failure relationship. In this chapter it was also noted from a theoretical point of view that there are two preliminary procedures which are suitable for storage analysis - Hardison's (1965) Generalized carryover procedure and McMahon's (1976) Empirical Equations.

Both are based on results generalized from

applying probability matrix methods, but the empirical constants in the latter procedure are based only on Australian streamflow data. 4.10 a

NOTATION variable in Phatarfod's procedure (Sec. 4.8.3.)

a

variable in McMahon's Empirical procedure (Sec. 4.7.1)

b

variable in McMahon's Empirical procedure (Sec. 4.7.1)

C

reservoir capacity

C1, C2

various reservoir capacity estimates

C v

coefficient of variation

D D

draft as ratio of mean flow . dra f turIng d · t th perlod

e

variable in Phatarfod's procedure (Sec. 4.8.3)

f(x)

probability density function

t

106 g(x)

probability function of storage content plus inflow during unit period

h

variable in Phatarfod's procedure (Eq. 4.38)

K

reservoir capacity as a multiple of constant water volume in Moran analysis (Secs. 4.1-4.3)

K

number of zones in Gould analysis (Sec. 4.6)

!C

proportion of reservoir capacity being 1/2, 1/3, 1/4 or 1/5 (Eq. 4.41)

m

positive integer exponent large enough so that resulting matrix is equivalent to steady state (Secs. 4.4 and 4.6)

M

release as a multiple of constant water volume in Moran analysis (Secs. 4.1-4.3)

N

number of years of data

P

prohability of reservoir contents being less than some amount (Sec. 4.8.3)

P.

probability that the content of the reservoir is in zone i at the heginning of the period

P!

probability that the content of the reservoir is in zone i at the end of the period

1

1

rPj

probahi Ii ty vector

qi r

probability of receiving an inflow of

t

time

[T]

transition matrix of reservoir contents

units

annual serial correlation coefficient

v

lit in Phatarfod's procedure (Sec. 4.8.3)

W

zone volume in Gould's probability matrix procedure (Sec.4.6.1)

x

flow volume

x

mean flow

X t y

inflow during tth period variahle in Phatarfod's procedure (Sec. 4.8.3)

Zt,Zt+l

reservoir storage contents at the beginning and the end of tth time interval

y

coefficient of skewness of annual flows

LlE

net evaporation loss during time

t

t

~

shape parameter in Weibull distribution

e e

variable in Phatarfod's procedure (Sec. 4.8.3)

characteristic drought in Weibull distribution

jJ

mean annual flow in draft units

a

standard deviaition of annual flow in draft units

T

reservoir capacity divided by mean annual flow

107 CHAPTER 5

USE OF STOCHASTICALLY GENERATED

DATA

The third grouping of storage estimation methods is based on the use of generated or synthetic data. same as described previously; are changed.

In essence, however, the methods are the the difference is that the input streamflows

The technique involves using a stochastic generation model to

produce "streamflow" sequences with the same statistical properties as the historical record.

It is then possible to determine the storage capacity

(using some standard method) corresponding to each sequence, thus providing a designer with a distribution of values.

This in turn gives him an idea of

the confidence which can be placed on the adopted design value.

"Synthetic

flows (or stochastic data) do not improve poor records but merely improve the quality of designs made with whatever records are available."

(Fiering

and Jackson, 1971, p.24.) In this chapter we restrict our examination of data generation processes to operational aspects of Markovian models that are used for generating annual and monthly streamflows.

Readers requiring more detail

are referred to the many excellent texts, reports and papers devoted to stochastic processes.

A selection of these is included in the references.

It should also be noted that data generation procedures will be dealt with

here not in terms of the physical mechanics underlying the streamflow process but rather from an operational point of view.

In this regard we

commend readers to Klemes' (1974) paper for some thought-provoking comments on the relationships between physically based and operational models. This chapter is divided into several distinct parts.

The first

examines the time-series components making up the streamflow process.

This

is followed by a review of historical developments in data generation procedures up to 1960.

Next, the methodology and performance of Markovian

data generation procedures are discussed in detail.

Simulation and the use

of generated data are then considered and finally several procedures based on generated data for making preliminary estimates of reservoir capacity are reviewed.

108 5.1

TIME-SERIES COMPONENTS From a stochastic point of view, streamflow data can be regarded as

consisting of four components (Kottegoda, 1970);

trend (T ), periodic or t seasonal (St)' correlation (K ), and random (E ) components which can be t t combined simply as: (5.1) These components are represented pictorially in Fig. 5.1.

Time

FIG. 5.1

Time

Time-series components of the streamflow process.

To obtain representative stochastic data, it is necessary to identify and measure the strength of each component.

A fifth component not included

in Eq. 5.1 relates to catastrophic events.

This aspect is beyond the scope

of this

text

and relates to the so-called 'Noah and Joseph' effects and

the 'Hurst' phenomenon.

Data generation models accounting for these effects

are still at the research stage (Sec. 5.7). A sequence of values arranged in order of their occurrence is called a time-series.

A time-series is considered to be stationary if the

statistical properties characterising i t are time invariant.

In this

discussion it is assumed that the data are stationary or can be made so by a simple transformation.

For example, to partially eliminate the non-

stationary effect of seasonality, monthly data can be standardised by the following equation:

x't

(5.2)

109 where

monthly flows,

x t x, t x.

standardised monthly flows, mean monthly flow for the j th month, and

s.

standard deviation of monthly flows for the J.th month.

J

J

One characteristic of a time-series is persistence which relates to the sequencing of the data.

In Chapters 3 and 4 it was noted that this

property is very important in storage-yield analysis.

In streamflow, per-

sistence arises from natural catchment storage effects which tend to delay the runoff;

over a short time period high flows in one interval will

tend to be followed by high flows in the following interval.

The longer the

time period the lesser the effect and for many streams it is negligible for annual flows. The usual quantitative measure of persistence is serial correlation .. Serial correlation coefficients may be calculated for the correlation between the flow in any given time period (for example, month or year) and the flow in lag.

k

time periods earlier where

In many studies only the lag

k (= 1, 2, ... ) is called the

one serial correlation is considered,

that is, the persistence between an event and the immediately preceding event.

Lag one models have been shown to be operationally satisfactory in

several studies (for example, Kottegoda, 1970;

Philips, 1972;

Wright,1975).

The algorithm to compute serial correlation is given as Eq. 2.10. For a sample of finite size, computed values of serial correlation (r , where

is the lag) may differ from zero because of sampling errors.

k

k

Thus it is necessary to test the values to determine if they are significantly different from zero. purpose.

Yevjevich (1972b) outlines a test for this

The confidence limits (eL) for a computed value of

r

k

are given

by: -l±z

Cl

IN-k-l (5.3)

N- k where

z

Cl

the standardized normal deviate corresponding to the

N

Cl

level of significance, and

number of flow events.

falls outside the confidence limits,

If

significantly different from zero at the

Cl

is considered to be level of significance.

Equation 5.3 may be used to test the statistical significance of k

>

1 if

k

is small relative to

N.

r

k

for

110 5.2

HISTORICAL DEVELOPMENTS TO 1960 Hazen (1914) is considered to be the first to recognise the

desirability of extending hydrolgic data.

He combined standardised annual

flows for fourteen streams in the northwest of U.S.A. to produce a synthesized record of 300 years.

His procedure has a number of limitations.

The streams were geographically close, and more than half the records were based on the period 1900-1910, so that records tended to be repeated.

The

technique of combining the flows forces the residual massed curve to pass through zero at least fifteen times thus restricting the range of the combined data. requirement.

This would result in an underestimation of the storage But in Hazen's procedure this effect was compensated for

because his storage was determined on the basis of a semi-infinite reservoir. Hazen's curves are still used today to assess preliminary storage sizes in the eastern United States. Sudler (1927) utilized historical and representative annual flows which were entered on fifty cards.

These cards were shuffled and dealt

without replacement to produce a sequence of 50 years. twenty times, producing in all 1000 years of data.

This was repeated

The procedure is

limited by the process of non-replacement so that each 50 years is specified by the same parameter set.

The method assumes serial correlation to be zero.

Nevertheless, this was probably the first truly stochastic streamflow generation model. By dealing the cards without replacement, Sudler's mass curve passed through zero at the end of every 50 years. the range is curtailed.

Thus, as for Hazen's mass curve,

Because he dealt with finite storages, Sudler does

not have a compensating error as a result of his type of storage. Consequently, Sudler's technique would tend to underestimate the required, storage capacity. In estimating the capacity of the Upper Yarra Dam in Victoria (Australia), Barnes (1954) found that the annual flows were normally distributed and independent, and used a Monte Carlo approach to generate 1000 years of data.

Historically, this approach contrasts with the earlier

procedures in that they were distribution free methods.

Barnes adopted a

design criterion of probability of failure of 1 in 40 but used a semiinfinite storage approach as an added safety factor.

III

From 1936 to the 1960's, Hurst studied the river Nile, and developed various card sampling techniques to generate annual flows which were used in simulated operational studies of the Aswan High Dam.

Details can be

found in Hurst's text (1965), pp. 41-42. 5.3

ANNUAL MARKOV MODEL The Russian mathematician Markov (1856-1922) introduced the concept of

a process in which the probabi I i ty distribution of the outcome of any trial depends only on the outcome of the directly preceding trial and is independent of the previous his tory of the process.

In this case the "trial"

is the passage of one year and its "outcome" is the streamflow for that year.

If the probability distribution of annual streamflow is either

independent of previous streamflows or correlated only with the previous years flow, we have a "simple" or "lag one" Markov process.

The concept

has been extended to include cases of lag greater than one.

The Markov

process was the basis of the developments at the Universities of Colorado and Harvard in stochastic streamflow generation procedures during the early 1960's (Julian, 1961;

Yevdjevich, 1961;

Brittan, 1961;

Maass et aL., 1962).

Brittan (1961) proposed the following Markov model to represent actual streamflows: (5.4) where xi+l' xi x s

annual runoffs for (i+l)th and ith years, mean annual historical flow, standard deviation of annual flows, annual lag one serial correlation coefficient, and normal random variate with a mean of zero and a variance of unity.

This equation was adopted in order that the expected vaZues of the mean, standard deviation and serial correlation of the computed xi+l's would be equal to the respective values of those parameters derived from the historical record and used in the right-hand side of the equation. Moreover, if the that the

xi+l

xi

values are normally distributed, then it follows

values will also be normally distributed.

(Appendix C

shows theoretically that this algorithm does preserve the mean, standard deviation and serial correlation of the flows.)

112

5.3.1

Practical Considerations (i)

The model CEq. 5.4) consists of two components: a deterministic or correlation component and a random component

[x

+

rl(x

- x)]

i

[tis (1 - rf)11.

If r = 0, the model is purely random. This sometimes l occurs with annual data, for example, as found by Barnes (1954) in generating 1000 years of inflows for the Upper Yarra Darn in Victoria.

For the model as proposed,

r 1 cannot exceed unity and for annual data is generally less than 0.4. (ii)

To use the model to generate annual flows, we need to compute the mean, standard deviation and serial correlation of the historical annual flows, and to assume that the flows are normally distributed.

(iii)

The normal random variate, ti' is generated by an appropriate routine which is available for all computers.

One method is

to generate pseudo-random numbers which are usually uniformly distributed with a mean of 1 and variance of 1/12.

If we

add 12 of these numbers together and subtract 6 the resulting variate may be regarded as a normal random variate with a mean of zero and a variance of unity - designated as N(O,I). (iv)

To initialize the model operation, Xl is set equal to

X.

Consequently the first ten or so generated flows should be discarded as they will be dependent on this initialisation procedure.

A similar initialisation procedure is used for

other variations of the Markov model. (v)

This and some other models can generate negative flows. When this occurs the negative value is to calculate the next flO\oJ, after which it is set to zero.

Such a procedure

is acceptable so long as the proportion of negative flows is not too high (say no more than 5%).

In addition, one

should check the difference in mean flow of the generated sequence with the negative values included and with them set to zero.

If the difference is greater than say one

percent, the model is probably unsatisfactory for that stream. (vi)

Sample calculations of annual generated flows for a stream with the following historical parameters are given in

113 Table 5.1 for the Yarra River at Doctor's Creek (229103) in Victoria, using: N

= 77 years, x = 180 10 m , s = 72 10 m , r l = 0.12

and a sequence of random numbers, ti' which are normally distributed with a mean of zero and a variance of one.

TABLE 5.1

i

1

x.

1

Sample calculation of annual Markov streamflow model.

x

X.1

r l (xi-

x)

x+rl (xi- x) deterministic component

t.

-0.52

180

0

0

180

1

\ s (l-rir± random component

xi + 1

-37

143

2

143

-37

-4

176

0.61

44

220

3

220

40

5

185

-0.36

-26

159

4

159

-21

-3

177

-0.39

-28

149

5

149

-31

-4

176

0.08

6

182

6

182

2

0

180

-0.93

-66

114

7

114

-66

-8

172

-0.03

-2

170

8

170

-10

-1

179

0.80

57

236

9

236

56

7

187

1.67

119

306

It illustrates very clearly the relative importance in the model of the deterministic and random components.

Even

though the serial correlation is about average, the fluctuation in the deterministic component is small relative to the fluctuations caused by the random component.

Even

if the serial correlation were 0.5 (an approximate upper limit for annual flows) the random component would still contribute 75% of the variance in the generated flows. (vii)

In the above example using annual data, ti is defined as a random normal variate, N(O,l). this assumption is not acceptable.

However, for many streams (For Australia, it is

valid for only about 20% of streams.)

In order to provide

for this non-Gaussian situation, the model can be modified in several ways which are outlined in Sec. 5.5.

114

In the annual Markov model as outlined above, only two of the four components assumed to make up the streamflow process, as defined in Eq. 5.1, are accounted for explicitly.

Trend and periodicity are not considered.

How then do we treat trend?

Unless there is an a priori reason for

knowing the type of trend, a non-parametric test such as Kendall's rank correlation procedure (Kendall and Stuart, 1968) should be used to measure its strength.

Trends can be modelled by fitting either polynomials or

moving averages although there are difficulties with both approaches (Tintner, 1968). The most common form of periodicity relates to seasonality, particularly with respect to monthly data generation.

Here the most

appropriate practical model is the one proposed by Thomas and Fiering (1962). 5.4

THOMAS AND FIERING SEASONAL MODEL The algorithm for the Thomas and Fiering seasonal model is as follows: x '+ 1 + b. (x. - x.) + t. s. 1 (l-r~) ~ J 1 J 1 J+ J J

(5.5)

generated flows during the (i+l)th, ith seasons reckoned from the start of the synthesized sequences, mean flows during (j+l)th, jth seasons within

b. J

a repetitive annual cycle

of seasons (if months

are being used, then 1

~

~

12),

least squares regression coefficient for estimating (j+1)th flow from the jth flow b. ]

t.

1

(5.6)

normal random variate with mean of zero and variance of unity, standard deviations of flows during the . l)th , J.th seasons, an d ( J+

r.

J

correlation coefficient between flows in ]. th an d (.J+l ) th seasons.

To use the model to generate monthly flows at a site, 36 parameters monthly means, standard deviations and lag one serial correlations - are required.

These are obtained from analysis of monthly historical flows.

115

To run the model, set xl where

ti

' and compute successively x , x , ... JAN 2 3 is the only unknown and for each step it is calculated as a

pseudo-random normal variate.

x

Thus,

xJAN ) xFEB )

2

xFEB

+

bFEB/JAN(x l -

x3

x MAR

+

bMAR/FEB(x2-

= xJAN

+

bJAN/OEC(X12-XOEC)

X

x

13

+

tl sFEB (1 -

+

t2 sMAR (1

-

+

t

-

s (l 12 JAN

r2FEB/JAN)~ r2MAR/FEB)~

(S.7a) (5.7b)

r2JAN/DEC)~

(S.7c)

As defined above this model is restricted to normally distributed flows, that is,

ti

is considered to be a Normal random variate.

In order

to cater for non-normal streams the model can be modified as shown in the next Section. MODIFICATIONS FOR NON-NORMAL STREAMFLOWS

5.5

To cater for non-normal annual and monthly streamflows, three alternatives are available: (i)

(ii)

modify

t.

1

by an appropriate transformation;

modify the streamflow parameters and the model algorithms such that the final generated data are distributed like the historical flows upon which they are based;

(iii)

and

generate normally distributed flows and apply inverse normalizing equations.

5.5.1

Modifying tj In dealing with the problem of skewed data, Thomas and Burden (1963)

transformed the Normal variate, t., to a skewed variate, t , with an Y

l

approximate Gamma distribution (designated as 'like Gamma' in the following text), using the Wilson and Hilferty (1931) transformation thus: t

2 Y t,j

Y

where Yt,j

[

1

+

Y

t,~

t

i

y2.~

t,J - 36

(5.8)

coefficient of skewness of the like Gamma variate,

ti

N (0,1)

t

like G(O,l'Y t .), and ,J

Y

3

repetitive annual cycles of seasons usually 1

~

j

~

12.

116

In order to maintain the historical skewness in the generated flows, the historical skewness is increased to account for the effect of serial correlation.

Using expectation theory Thomas and Burden derived the

following algorithm to do this. (5.9)

seasonal coefficient of skewness for (j+1) th and jth seasons. To apply this method, called the like Gamma transformation, Eq. 5.5 is replaced by

t

Y

from Eq. 5.8,

y

.

t,]

t.

1

in

being calculated with

Eq. 5.9. The Wilson and Hilferty transformation is an approximation to the Gamma distribution but it breaks down for large skews and serial calculations (McMahon and Miller, 1971;

Phatarfod, 1976);

the limits are given

in Fig. 5.2. In general these limits do not affect annual models but, for monthly models restrict the use of this method to flow sequences with low to medium variability. 0.6

////

Wilson and Hilferty/ transformation / unacceptable 0.4

0.2

0+-----r-----r----.,----1~~~ 3 2 C s

·0.2

FIG 5.2

Limits of applicability of the Wilson and Hilferty approximation.

Kirby (1972) provided an alternative transformation to Eq. 5.8 which remains theoretically satisfactory over the whole range of hydrologic interest. (5.10)

where A, Band G are a function of skewness and given in Kirby's paper, and

117

EXAMPLE 5.1 Over the period of record the January flows for the Mitta Mitta River (Appendix E) exhibit a skewness of 1.8;

for February, it is 1.1.

The serial correlation coefficient of monthly flows between January and February is 0.58.

[N CO, 111

Show how a random number from a Normal distribution

can be transformed to a like Gamma skewed variable [G(O,l,1

*

*

*

*

.)].

t, J

*

A random number taken from the Normal distribution [N(O,l)J is - 0.4305.

- r~ I. J ]

From Eq. 5.9:

1.1 - (0.58)3(1.8) (1 - 0.58 2 )3/2

= From Eq. 5.8:

t

1. 385

-2 -

Y

Yt,j 1.

=-

~05

[1 [

+

1 +

'J

It,j t.1

It ,]. ~

6

(1. 385) (- 0.4305)

6

2

Yt, j _ (1.385)2] 3 36

2 1.385

0.5655

That is, the corresponding number to the normally distributed - 0.4305 is skewed gamma distributed - 0.5655.

Random numbers transformed in this way

can be used directly in the Thomas and Fiering seasonal model (Eq. 5.7).

5.5.2

Moment Transformation Equations Matalas (1967) presented moment transformation equations which

theoretically preserve the moments and the lag one serial correlation coefficients.

This method assumes that the Zogarithms of the flows are

normally distributed.

Thus the procedure is first to calculate a series of

logarithms using a Normal model, and then obtain absolute flows by exponentiation.

The generating algorithms and the parameter estimation

equations for the three parameter log-normal model are as follows:

118

Generating Algorithm: R2.)! - j+l + Bj (X i - X-].) + t i Sj+l (1 - ] X

(5.12)

(5.13)

where

N(O,l),

t. l

Xi+l

generated flow logarithms, and

xi+l

generated flow in absolute units.

Other symbols are defined below.

Parameter Estimation: x.

Aj

s~

exp

J

J

+

sj

exp (0.5 [2(S~

+

Xj )

+ it.)] - exp

J

J

(5.14) (S~

+ 2X.)

J

(5.15)

J

exp [3S~] - 3 exp [S~] + 2 J

g. J

J

(5.16 )

{exp [S2] _ l} 3/2 J

exp [So S. 1 R.] ]

r.

J+

J

(5.17)

J

where Historical data mean

x.

standard deviation

S.

coefficient of skewness

gj r.

J

lag one serial correlation and

Log-transformed value

J

J

R. Sj+lJ S J [ j

B.

J

(5.18)

To solve for A., X., S., R. begin with Eq. 5.16 and solve for S .. J

J

J

J

J

This is not explicit in S.and an iterative solution is required. J

One fast

converging technique is the Newton-Raphson method providing a reasonable initial guess is used.

The procedure is given in Appendix D.

been determined, then use Eqs. 5.14, 5.15 and 5.17 to obtain

Once S. has

xJ.,

J

A. and R.. J

J

119 EXAMPLE 5.2 Use moment transformation equations to transform the monthly parameters for the Mitta Mitta River (Appendix E) so as to preserve these characteristics in a generation model using normally distributed random numbers.

*

*

*

*

*

For January the historical parameters are (from Appendix E): 3 39.9 (x 106)m 3 26.4 (x 106)m

mean, x standard deviation, s serial correlation, r

0.58

skewness, g

1.8

From Eq. 5.16 : exp 3S~

-

J

gj

{exp

3 exp S2 + 2 J

S~

J

- 1}3/2

This cannot be solved explicitly for Sj; required.

a trial and error procedure is

If a reasonable first trial value is obtainable, the Newton-

Raphson procedure (Appendix 0) gives rapid convergence to the solution. Using a starting value of 0.5 for Sl; f(Sp

[exp (3Sf)

that is,

3 exp (Sf) + 2] - gl (exp Sf

-

si 1)

= 0.25 3/2

exp (0.75) - 3 exp (0.25) + 2 - 1.8 [exp (0.25) - 1]

(Eq. 05) 3/2

0.0075. 3 exp 3Sf - 3 exp Sf - 1.5 gl(exp Sf - 1)0.5 exp S1

(Eq. D6)

3 exp (0.75) - 3 exp (0.25) - (1.5)(1.8)[exp (0.25)-1]0.5exp (0.25) 0.6513. Therefore, second estimate

0.25 -

- 0.0075 0.6513

0.2615. Similarly, after repeating the above, the third estimate is found to be 0.260752 and the fourth, 0.260747, (that is, convergence). Thus Sf

0.26075 and

Sl

0.5106.

120 Equation 5.15 can be rearranged to give:

0.5 log [26.4 2 /(exp (2 x 0.2607) - exp (0.2607)] 3.748. Equation 5.14 can be rearranged to give:

39.9 - exp (0.5 x 0.2607 + 3.748) 8.469. Rl cannot be obtained from Eq. 5.17 until S2 is found from Eq. 5.16 (S2

0.3419) . Equation 5.17 can be rearranged thus:

log [0.58,j{exp (0.5106)2 - l}{exp (0.3419 2 )- 1}+ 1]/(0.5106)(0.3419) 0.6054. Similarly the computations can be done for the other 11 months; results are given in Table 5.2. TABLE 5.2

Log-transformed parameters for the Mitta Mitta River. Log-Transformed Parameters

Month

1 2 3 4 5 6 7 8 9 10 11

12

Mean

Standard Deviation

3.748 3.720 3.323 3.856 3.879 4.657 5.697 5.833 5.842 6.390 5.990 4.445

0.5106 0.3419 0.7546 0.9124 0.8424 0.7420 0.3945 0.3944 0.2829 0.1893 0.2018 0.4891

A. ]

- 8.469 - 16.83 5.774 - 24.88 - 6.860 - 36.25 -169.3 -170.2 -159.7 -399.9 -273.0 - 23.39

Serial Correlation 0.605 0.658 0.626 0.892 0.854 0.684 0.609 0.665 0.737 0.654 0.805 0.639

the

121

If a two parameter log-normal model is used, gj will be assumed zero

= 0 and Eq. 5.13 becomes

in which case A.

J

(5.19) and the model parameters for input into Eq. 5.12 can be determined explicitly from Eqs. 5.14, 5.15 and 5.17.

The model modified in this way

is based on a two parameter log-normal distribution rather than the three parameter one. For annual data generation, Eqs. 5.12 to 5.17, which are set down above with monthly subscripts, are modified appropriately. An important limitation arises in applying two parameter models to streams with high coefficients of variation.

For this distribution, the

coefficient of skewness (C ) is related to the coefficient of variation (C ) s v in the following manner (Chow, 1964, p. 8-17):

= 3C v

C

+

v

(5.20)

3

In practice, the implied values of skewness are modified among other things by the effect of serial correlation and so the generated value is always less than the value given by Eq. 5.20. 5.5.3

Normalizing Flows This procedure was proposed by Beard and has been adopted by the

United States Army Corps of Engineers (Beard,1972).

The following equations

are for annual flows: (i)

Compute logarithms of all flows after a small increment has been added to each in order to eliminate zero values.

(ii)

Compute mean, standard deviation and coefficient of skewness of log flows.

(iii)

Standardize the log values

L-=-.Z

v

where

s

s

v

standardized log flow,

y

loge (x + E)

y

mean of loge (x + E) flows,

y

(5.21)

y

standard deviation of loge (x + E) flows,

X

historical flows, and

E

small increment.

122

(iv)

Normalize the standardized values,

v,

to eliminate

skewness using the inverse Wilson and Hilferty transformation thus: v =

where v

(vi)

g

+

2

1)1/3_ I} + ~

(5.22)

6

normalized values, and coefficient of skewness of the log values.

g

(v)

~ {(~ v

Compute serial correlation of normalized values. Generate standardized variates by the Normal Markov process (5.23)

generated standardized variates N(O,l), lag one serial correlation of normalized variates, and ti (vii)

(viii)

=

Normal random deviate N(O,l).

Apply inverse transformations as follows: v

=

x

= exp (y

{[! (v

~) + 1]3 - l} -2

(5.24)

vs ).

(5.25)

g

6

+

Y

Subtract the small increment

(E)

added in step (i) .

If negative

flows result, set them to zero. Because of the initial decrease in skewness as a result of taking logarithms, the procedure does not suffer from the Wilson-Hilferty limitation (Fig. 5.2) but it has been found that the serial correlations of the absolute flows are poorly :,reserved.

These and other aspects are covered

more fully in Sec. 5.8 which deals with the performance of these models. For monthly flows, the mean, standard deviation and toefficient of skewness need to be modified.

Details are given in Beard's (1972) report.

EXAMPLE 5.3 Apply the normalizing flow procedure to the annual flows for the Mi tta Mi tta River (Appendix E) to demonstrate this method of flow generation.

*

*

*

*

*

123 Following the steps listed in Sec. 5.5.3 and referring to Table 5.3: (i)

= 0.01) to all of the annual flows

Add 0.01 (that is, s

in column (2) of the table, and enter the natural logarithm of each in column (3). (ii)

Standardize the flows using Eq. 5.21. mean of Column (3)

7.00121

standard deviation of Column (3)

0.559156

Column (3) - 7.0012 0.5592

then Column (4) (iii)

Use Eq. 5.22 to normalize the flows.

then Column (5)

First, calculate

= - 0.0825

skewness of Column (3)

= _ 0.0825 6 [(-0.0 825 (Column (4)) 2 +

(iv)

First, calculate:

+

1) 1/3 -

1]

- 0.0825 6

Use Eq. 5.23 to generate standardized variates.

=-

calculate serial correlation of Column (5) Next, assume an initial value of vi;

First, 0.008.

zero is appropriate.

Hence, for a random normal variate (say

tl

1.0752)

=

0.008(0) + 1.0752 (1 - 0.008 2)t 1.0752;

for t

z=

-

0.0064, say, 0.008 (1.075Z) - 0.0064 (1 -

0.0082)~

0.0064. Note: (v)

The very low serial correlation means that the vi+1

Apply the inverse transformation Eq. 5.Z4. V

z

= {[-

0.~825

(v

2

- -

0.~825)

+

1r -

- 1.1053. From Eq. 5.25, x

exp

(y

+ v

2

s) y

exp (7.00121 - 1.1053 x 0.5592) 591.76

1} _

0.~825

t .. l

124

Hence the first generated flow in the sequence is 591.76 - 0.01 = 591.75 ~ 592. Subsequent flows are calculated in the same way.

(vi)

In normal application the first few generated flows would be discarded to remove the effect of the starting condition applied in (iv). TABLE 5.3

(1)

Flow x 10 6m3 (2)

Year

5.6

Derivation of normalized annual flows for the Mitta Mitta River. Loge(Flow + 0.01) (3)

Standardized Flows (4)

Normalized Flows (5)

1936

1553

7.3480

0.6202

0.6120

1937

650

6.4770

- 0.9374

- 0.9392

1969

1010

6.9177

- 0.1493

- 0.1639

TWO TIER MODEL Monthly models do not necessarily preserve the annual flow charac-

teristics.

To overcome this deficiency, Harms and Campbell (1967) extended

the Thomas and Fiering model to constrain the annual and monthly flows separately, and also to preserve the annual serial correlation. flows were generated by an annual Normal Markov model.

Annual

A Thomas and

Fiering log-normal model was used for monthly generation, but the values were adjusted to sum to the appropriate annual values by the following algori thm:

x ..

1J

x! .

-12-I x ..

1]

j =1

where

Qi

1J

x'.

adjusted monthly generated flow volumes,

X

unadjusted monthly generated flow volumes,

Qi

annual generated flow volumes,

i

year, and

1J

ij

month.

(5.26)

125

Results presented in the Harms and Campbell paper suggest that the model works well.

In cases where the annual flows are not normally dis-

tributed, a skewed distribution could be used in place of the normality assumption.

One minor drawback with this approach is that the method of

adjusting monthly data does not allow the monthly serial correlation coefficient from the end of one year to the beginning of the next to be preserved. 5.7

OTHER CONSIDERATIONS Many other considerations are involved in stochastic generation of

streamflow other than those aspects treated in this chapter, for example, correlograms, partial correlation functions, spectral analysis, and daily models.

Nevertheless, the annual and monthly models outlined above do

provide a basis for the practical application of data generation techniques to single site situations.

In this text it would be out of place to

consider in detail multi-site models but some background material is given in Chapter 7. This discussion would be incomplete without a brief comment on the use of so-called long memory models.

Three data generation models, the

ARIMA, Broken Line and Fractional Gaussian Noise models fall into this category.

They have been proposed as replacements for Markovian schemes in

order that the Hurst phenomenon (see Sec. 3.3.1) is preserved in the streamflow sequence.

In Markovian models the exponent

K in Eq. 3.4 tends to

0.5, yet in reality its observed mean value is about 0.72. But are these three models of importance to the practising water engineer?

High

K values in Eq. 3.4 imply long-term persistence and result

in longer and more extreme events in the flow sequence.

From Eq. 3.4, high

K values result in larger ranges than would otherwise occur, and consequently relatively larger storage sizes. important are these effects.

What is not clear is how

For example, Wallis and Matalas (1972)

observed that differences in storage estimates between the Markovian and fast Fractional Gaussian Noise models occurred only for drafts greater than 80%.

Yet Kottegoda (1970) found for British rivers that the Fractional

Gaussian Noise model gave unrealistically high estimates of storage if compared with those estimated from historical or Markovian sequences.

While

there are these and other differences in detail, the consensus of opinion in the literature suggests that high reliabilities.

K should be preserved for high drafts and

126

5.S

MODEL VERIFICATION AND PERFORMANCE Before a data generation model is used in storage-yield analysis it

is necessary to check not only that it satisfactorily reproduces the main statistical characteristics defining the streamflow process, but also that critical periods are being satisfactorily generated.

A validation procedure

for an annual or a monthly model might include the following tests: (i)

comparison of the mean and variability of various statistics (annual and monthly means, standard deviations, coefficients of skewness and serial correlations) computed from many sets of generated data (each of length equal to that of the historical record) with the actual values of those statistics computed from the historical record;

(ii)

comparison of flow duration and frequency curves based on the generated data with the corresponding curves based on the historical record;

(iii)

comparison of correlograms based on monthly generated data with that derived from the historical record;

(iv)

and

comparison of the mean and variability of reservoir storage estimates based on replicated generated data compared with estimates using historical data.

The number of replicates of generated data that are required will vary with respect to streamflow variability.

Twenty-five are generally

sufficient. In the above tests, it should be noted that the historical values from (i) and the flow distribution (used in the flow duration comparison) in (ii) are part of the model structure, and, therefore, are more likely to be satisfactorily modelled than the other factors listed. To illustrate the level of performance achieved with the Thomas and Fiering model, published results (McMahon et al., 1972a, 1972b, 1973) have been included here as Tables 5.4, 5.5 and 5.6,

In addition, results from

a recently completed evaluation of a number of annual Markov models are available (R. Srikanthan, personal communication, 1977). In Tables 5.4 to 5.6 historical monthly flows and storage-yield results are compared with results from generated data using the Thomas and Fiering monthly model incorporating three of the distributions discussed

TABLE 5.4

Comparison of historical and generated monthly and annual streamflow parameters

River Australian Stream Gauging Station No. (Years of Record)

O'Shannassy 229103 (59)

M:JNTHLY Distribution

Historical LGLT

LN-3 Historical LGLT LN -2 LN-3 Torrens 504501 (72)

Historical LGLT LN-2 LN-3

Warragamba 212240 (72)

(Mm3)

ANNUAL

Standard Deviation

Skew

Seri al Correlation

(Mm3)

LN-2

Gordon 308007 (36)

Mean

Historical LGLT LN-2 LN-3

8.8

5.7

1.2

0.76

8.8 (0.4) 8.9 (0.5) 9.1 (0.5)

5.8 (0.6) 5.9 (0.5) 7.3 (0.7)

1.6 (0.7) 1.S (0.3) 2.3 (0.5)

0.78 (0.03) 0.78 (0.02) 0.74 (0.02)

1.4

0.38

1.6 (0.5) 1.9 (0.8) 1.3 (0.3)

0.38 (0.06) 0.39 (0.07) 0.37 (0.05)

150

109

153 (5.4) 152 (5.8) 152 (3.8)

113 (8.3)

III (12.2)

III (4.6)

Standard Deviation (Mm 3 ) 31

Skew

Serial Correlation

0.7

0.04

32 (6.4) 32

1.1 (0.9)

(4.2)

(0.4) 0.9 (0. S)

0.13 (0.13) 0.06 (0.15) 0.08 (0.15)

420

-0.1

0.06

390 (66) 390 (79) 390

(60)

0.4 (0.5) 0.7 (0.6) 0.4 (0.6)

0.00 (0.18) -0.07 (0.14 ) -0.01 (0.19)

39

1.5

0.02

2.5 (0.9) 2.4 (1. 1) 1.5 (0.8)

-0.02 (0.07) -0.02 (0.10) 0.00 (0.12)

42 (6.0)

1.0

4.2

7.8

3.1

0.57

4.3 (0.4) 4.4 (0.4) 4.7 (0.4)

9.9 (1. 5) 9.0 (1.5) 9.3 (0.9)

5.9 (2.0) 4.9 (1. 8) 3.6 (0.7)

0.56 (0.07) 0.61 (0.05) 0.60 (0.04)

4.6

0.45

1200

2.7

0.29

15.0 (8.0) 6.2 (1. 7) 4.2 (1.3)

0.34 (0.16) 0.47 (0.07) 0.43 (0.06)

2500 (2500) 920 (230) 970 (190)

4.4 (2.0) 2.3 (1.2) 1.7 (0.7)

0.04 (0.15) -0.03 (0.08) 0.00 (0.08)

93

200

110 560 (35) (690) 91 170 (9.0) (35) 110 190 (10) (32)

Generated parameters are means of 20 replicates of length shown. of generated parameters.

49 (9.6) 43 (10) 43 (5.6)

Values in parentheses are standard deviations

TABLE 5.5 Statistic

Comparison of historical and generated monthly parameters for O'Shannassy River in Victoria Model Historical

Mean (Mm3)

Note:

A

M

J

J

5.4

3.8

3.7

4.2

6.5

8.8

12.1

A 14.9

S

0

N

D

14.3

13.0

10.6

8.3 8.1

5.4

3.8

3.7

4.2

6.7

8.8

12.1

14.9

14.3

13.0

10.5

5.4

3.9

3.8

4.2

6.4

8.5

12 .5

15.3

14.7

13.1

10.6

8.0

LN-3

5.7

4.4

3.8

4.2

6.4

8.5

13.3

16.9

14.3

13.3

10.7

7.5

2.0

1.1

1.2

1.9

3.7

5.1

5.2

5.7

5.2

4.9

4.7

4.8

LGLT

2.0

1.0

1.2

1.9

4.3

5.3

5.6

6.0

5.3

4.8

4.6

4.2

LN-2

2.3

1.5

1.4

1.7

3.3

4.6

6.3

6.8

6.2

5.1

4.3

3.9

LN-3

2.7

3.3

1.7

1.6

3.6

4.6

5.3

13.4

5.3

6.0

4.6

2.7

1.2

0.5

0.8

1.6

1.7

2.0

0.9

0.4

1.0

0.7

0.9

3.4

LGLT

1.3

0.5

0.8

1.4

2.4

1.9

1.3

1.0

1.1

0.5

l.0

1.9

LN-2

l.2

l.0

0.8

1.1

1.3

1.4

1.4

1.2

l.2

0.9

1.0

1.3

LN-3

1.3

1.2

0.9

1.0

1.3

1.4

1.3

1.1

0.9

0.7

0.8

l.4

0.86

0.64

0.56

0.44

0.64

0.82

0.68

0.73

0.61

0.76

0.66

0.70

LGLT

0.86

0.69

0.59

0.39

0.77

0.87

0.67

0.72

0.77

0.77

0.73

0.90

LN-2

(j.92

0.68

0.51

0.33

0.62

0.87

0.79

0.80

0.66

0.75

0.50

0.80

LN-3

0.98

0.81

0.42

0.40

0.61

0.94

0.94

0.73

0.75

0.74

0.33

0.85

Historical Serial Correlation

M

LGLT

Historical Skew

F

LN-2

Historical Standard Deviation CMm3)

J

Generated parameters are mean values of fifty replicates.

129 TABLE 5.6

Reservoir capacity estimates based on generated flows compared with historical values.

River Australian Stream Gauging Station No. (Years of Record)

Annual coefficient of variation

O'Shannassy 229103

Model LGLT

a

LN-2

LN-3

a

a

0.29

94

0.23

90 ( 7)

74 (7)

92 (10)

0.44

86 (10)

85 (8)

116 (12)

0.77

94 (19)

59 (13)

89 (13)

1. 07

107 (24 )

53 (20)

60 (12)

97

l57

(SO)

Gordon 308007 (36 ) Yarra 229103 (50) Torrens 504501 (72)

Warragamba 212240 (72)

aMedian value based on ten replicates. Note:

Storage estimates for conditions of 1% probability of failure and 50% draft are expressed as percentages of the long-term historical Gould values. Generated storage estimates are means of 20 replicates of length shown. Values in parentheses are standard deviations expressed as percentages of appropriate mean value.

earlier - two and three parameter log-normal (denoted by LN-2 and LN-3) and the like Gamma distribution (LGLT).

In using the latter distribution a

logarithmic transformation was initially applied to the data.

These tables

highlight a number of points: (i)

In Tables 5.4 and 5.5, except for LGLT for Warragamba and the LN-3 August standard deviation, the annual and monthly means and standard deviations compare well with the historical estimates.

(ii)

Coefficients of skewness are generally too high but are considered satisfactory except for the LGLT monthly estimate for Warragamba.

Seasonal historical skews are erratic and

the variations are poorly modelled (Table 5.5).

Some workers

recommend smoothing seasonal coefficients of skewness (and other moments) prior to analysis (Beard, 1965).

130 (iii)

Monthly serial correlations are well modelled (Tables 5.4 and 5.5) yet high annual serial correlations are poorly simulated, for example, Warragamba in Table 5.4.

This

inadequacy is typical of monthly Markov models and can be overcome by using the two tier approach outlined in Sec. 5.6. (iv)

In Table 5.6 reservoir capacity estimates using Gould's procedure (Sec. 4.6) and based on generated flows are compared with historical values. discrepancies.

The results show large

Overall,the storage values from the LGLT

model compare most favourably with the historical estimates. At least two of the storage estimates for the LN-3 model deviate considerably from their historical values.

The

LN-2 model shows even greater discrepancies and is considered satisfactory for only one river. From this analysis, it can be seen that using the historical values as a basis of comparison, the LGLT model shows least variations yet exhibi ts some unsatisfactory parameter estimates.

On the other hand, the LN-2 and

LN-3 models reproduce the parameter values, but deviate markedly from the historical storage estimates.

Thus no model is wholly satisfactory.

Tables 5.7, 5.8 and 5.9 deal with a more detailed evaluation of Markov models than that discussed above, although the evaluation was restricted to the annual lag one type (R. Srikanthan, personal communication, 1977).

In all, 16 rivers * which represent the range of streamflow varia-

bility. encountered across the Australian continent were examined.

Up to

seven variations of model distributions were considered and for each case 5000 years of data were generated.

In addition to the parameters and

characteristics listed at the beginning of this section results were examined for the range, Hurst's

H and

K exponents, run lengths, extreme

events, spectral values and distribution types.

* In Table 5.7, 5.8 and 5.9, the names and national stream gauging numbers refer to the following Australian rivers as follows:

(1) King (309001),

(2) Wilmot (315003), (3) South Johnstone (112101), (4) Yarra (229103), (5) Murray (401201), (6) South Esk (318001), (7) Wungong (615071), (8) Serpentine (615074), (9) Loddon (407203), (10) Torrens (504501), (11) Ord, (809302), (12) Peel (419004), (13) Warragamba (212240), (14) Burnett (136001), (15) Wide Bay Creek (138002) and (16) Goulburn (21006) .

131

TABLE 5.7

Parameter and

Di5tr1-

Comparison of historical parameters with generated parameters based on annual flows generated using an annual Markov model.

f-_____________Ri_v_e_'_'_e_fe_'_en_'_e_n_o_._ _ _ _ _ _ _ _ _ _ _ _ _ _~

button

Hist. ::nm)

2230

1620

1710

2350

1610

1720

2320

1620

1710

540

)60

238

214

122

370

236

210

119

530 540

61

10

11

12

13

14

15

16

144

94

209

120

44

126

27

205

121

43

124

26

132

96

210

123

44

127

26

119

43

124

26

61 61

LN-2

540

)70

239

216

123

61

145

96

209

123

45

127

LN-3

540

)70

239

217

123

62

145

97

213

124

46

132

540

370

240

216

122

63

0.40

0.47

0.47

0.50

0.57

0.78

0.40

0.46

0.47

0.49

0.56

LN-2

0.40

0.46

0.47

0.49

LN-J

0.40

0,46

0.47

0.49

0.40

0.46

0.48

0.46

0.8

1.3

0.9

0.6

1.0

1.2

0.9

0.6

0.8

Hist.

0.18

0.22

0.38

0.18

0.22

0.37

0.18

0.22

0.38

146

97

214

124

46

131

28

0.78

0.80

0.82

1.11

1.14

1.26

1.79

0.81

1.11

loll

1, 15

0.74

0.78

0.79

0.79

1.09

1.11

1. 22

1. 71

1.07

1.13

1.23

1.64

0.56

D,75

0.76

0.76

0.77

1.02

0.56

0.74

0.76

0.76

0.78

1.05

1.07

1.158

0.79

0.86 0 1 . 1 1

1.24

1.33

1.80

2.4

2.5

3.6

0.39

Hist.

-0.2

0.3

0.2

-0.2

0.4

0.2

0.77

D.578

0.7 0.7

-9

LN-2 LN-3

Hist.

-0.09 -0.06 -0.10

0.9

1.5

1.2

1.4

1.2

0.9

1.1

2.8

88~

::: ~~~:: ~ 2.32.0

GC08C0GG@ A @

0.7

1.2

0.8

1.]

1.2

0.12

0.11

0.01

0.8

0.9

0.9

~ee 0.14

8

C":\r,:-...-0

0.9

1.]

27

0

1.3 1.5

0.02

1.1

~

2.6 1.9

2.5

1.9

2.9

2.9

2.8

'd

0.24

0.30

0.20

0.28

0.18

0.25

0.31

0.1'.1

0.29

0.16

0,30

0,20

0,27

0.18

1.1

@8 0.07

2.6

@)~

2.7

0.21

0.21

0.13

0,10 8 0 , 2 1

0,22

0,16

0.01

0.30

0.22

0,27

0.16

0.11

0.09

0.00

0.20

0,19

0.13

0.01

0.06

0.22

0,]0

0.20

0,27

0.19

0,11

0,09

0,00

0.20

0,19

0.13

0.01

0.06

0.22

0.29

0.19

0.26

0,16

-0.10 -0.06 -0.10

,

0.13

G

-0.088-0,12

LN-2 LN-3

__

-

0.14

80,22

~~ ~_-=--_~- _98 o,o~~~~_~~~ S88888 wh~r~

,0[ .. normal distributlon K .. Kirby's transformation B .. 8eard's method

W - Weibull LN-2 ,. T.... o parameter log normal

G .. like Gamma distribution LN-3 = Three parameter log nonnal

1.)Z

TABLE 5.8

Parameter and error

% negative flows

8

Distribution*

0

Ri ver reference no. 9

10

11

lZ

13

14

15

16

N

-

-

-

-

-

-

-

-

W

@])

G

K

LN-2

% increase in x

Percentage of negative flows in generated sequences from annual Markov model and their effect on mean flow estimate.

e

-

1.6

-

0

LN-3

g

B

0

-

g 0

3.3 0

80

N

-

-

W

0.9

-

-

G

C[D

0.1

0.7

-

K

LN-Z LN-3 B

0

C[D 0

-

-

-

0

Ci]) CQ)~@ C8) 0.9 6.5 ~

-

-

0

0

0.4

0.8

0

0

0

o 0

~ 16. 0

0

®

~ 3l. 0

QJ) cQ)§@ @ 0

0

0

-

-

9 CO>

0.6

-

0

cQ) 0

0

-

g

0.1

0.4

0

0.6

0

0

0.9 0

0

-

-

~~ 1.6

5.8

Z.O

3.8

0

0

cQ) CQ) ~ 0

0

0

* For definition of symbols, see Table 5.7. The characteristics compared in Tables 5.7 to 5.9 include the mean flow (x), the annual coefficients of. variation (C ) and skewness (C )' the v s annual serial correlation coefficient (r), storage capacities for 50% and 90% draft rates (Sso and S90) using the Sequent Peak method, the rank one two-year and ten-year consecutive low flows (1:2 and 1:10), the percentage of generated negative flows and the percentage increase in mean in setting them to zero.

In all, seven distributions or variations are examined - Normal

eN), two and three parameter log-normal (LN-Z and LN-3), like Gamma (G); Kirby's modification (K), Beard's normalizing procedure (B) and a Weibull distribution (W).

The values given in Tables 5.7 to 5.9 are averages based

on replicates in length equal to the historical records and equivalent to 5000 years of flow.

To assist in evaluating the performance of the various

distributions, generated values outside ± 5%, ± 10% or ± 25% from the appropriate historical values are circled in three tables. indicate an arbitrary level of performance.

These limits

133

TABLE 5.9

Parameter and

River reference no.

Distribution·

Hist.

10 0.00 0, 01

8

5"

G

Comparison of historical low flow and storage characteristics with those based on annual flows generated using an annual Markov model (in units of mean annual flow) .

0.00

0,23

88 0.01

0. 23

LN-2

0.20

0.23

0.50

0,50

-

0.48

0.94

II

12

13

14

15

16

l.2

1.2

1.5

1.5

1.1

2.4

-

-

-

-

-

1.~ 8~® l.~ ~:~ ~: 8~ @ 0.56@8 e@ 9880.58

8°.21 GO.57 o •

0.21

85 1.4

8

1.7

1.3

1.00

1.1

G

LN·3

His!.

1.4

1.3

1.1

1.5

1.3

0.9

1.4

J.4

0.9

"'

0.24

0.50

0,58

1.2

0.86

1.0

l.0

1.6

1.6

1.7

2.2

3.3

5.2

3.5

5.9

5.9

13.7

8.8

11.1

eG@

11.6

15,1

2.6

5.1

6.9

12.5

9.8

11.0

12.4

2.4

6.1

6.5

11.7

9.7

10.4

13.4

1.10

0.94

0.32

0.28

0.23

0.11

0.30

'.7

2.9

0.77

0.63

0,58

-

8

O,8S D,79

0,86

0. 90

0.56

8

0 . 72

7.9

7.2

7.7

7.7

6.3

7.3

7.2

7.2

LN-2

7.7

7.4

7.6

7.1

6.8

LN·,

7.6

7.4

7.5

@

6.5

7.7

7.'

7.6

7.1

0.8

8.9

8.2

For definition of symbols, see Table 5.7.

0.33

-

-

-

0.21

-

0.26

0.01

.

~~(§B@5

0.02

0.11 0.10

0.27

0.11

0.23

0,19

0.11

0.23

0.36

0.36

0.32

0.26

'.6

6.0

5.4

4.8

4.0

0.03

4.'

2.7

7.4

G

-

0.08 .

0.46

9.0

0.23

8· 8 · 8 '~0.02 0.54

'10

17.1

2.2

0.54

8.2

IS.1

8

12,0

0.60

8.9

12.7

10.0

0.75

9.1

10.1

10.7

12.2

0.86

8.9

14 . 2 12.8

10.2

6.2

13.78

0.82

9.2

8

6.8

0.86

'.9

4.9

4.7

0.84

Hist.

8

2.5

0.88

LN·'

6.2 6.3

2.1

13.9

2.0

0.97

LN-Z

e

2.2

0.88

G

2.6

1.4

1.4

0.22

2.0

LN-2

5.1

1.8

0 : 9 0 . ; 7 8 8 5 2.5

~ 4.5

0.20

(!.43~5"

3.1

..

0.20

0.69

0.43

0.67

1.1

0.18

0.14

LN-3

Hist.

0.18

8

6.3

134 Overall the models preserved the mean and coefficient of variation satisfactorily but the coefficient of skewness was poorly modelled for the two parameter log-normal distribution

and Beard's normalizing procedure.

The overestimation of LN-2 is expected because the implied skewness given in Eq. 5.20 is always considerably greater than the observed skewness for C greater than unity (McMahon, 1975, p. 384). The fourth input parameter v to the models - serial correlation - is well preserved except for Beard's model which underestimates all but one historical value.

A further relevant

factor relates to the proportion of negative generated flows and their effect on the mean.

This is shown in Table 5.8 for the eight streams with

a coefficient of variation greater than 0.7.

Those with a coefficient of

variation less than 0.7 generate less than one percent negative flows, and were not tabulated.

The table shows that except for the two parameter log-

normal distribution and Beard's model, all others generate too many negative flows and are regarded as being unsatisfactory.

From Tables 5.7 and 5.8 we

conclude that for annual data generation streams with an annual coefficient of skewness of less than 0.3, the most suitable model is like Gamma;

for

other streams with an annual coefficient of variation of less than 0.7, the three parameter log-normal or like Gamma are suitable;

for streams with

larger variability, the two parameter log-normal model is recommended. Unlike the generated parameter values which should approach the historical values input into the model, the correct generated values for other characteristics like low flow values and storage capacities are .unknown.

Nevertheless, if one takes several sets of flows drawn from a

range of geographic, climatic and hydrologic regions as in Table 5.9, estimates of generated characteristics based on historical input parameters should be a reasonable approximation overall of the historically observed characteristics.

Accepting this approach and examining the two storage

values (S50 and S90) and the 2 year and 10 year low flow sums

(~2and ~10)

in Table 5.9, it is concluded that Beard's procedure performed the most satisfactorily, followed by the two and three parameter log-normal distributions.

Generally all the models overestimated the historical storage

values but Beard's estimates are on the average no more than 10% different to the historical estimates.

This is considered to be very satisfactory.

For streams with low skewness (say Cs < 0.3) the Gamma model is recommended.

135 This detailed review confirms our observations regarding the results in Tables 5.4, 5.5 and 5.6, namely that no model 'is wholly satisfactory for all purposes.

Consequently, if one is using a data generation model in

practice, one needs to understand clearly the objectives of the study which should influence model choice. 5.8.1

Unrepresentative Streamflow Data Input parameters to generating models are based on historical data.

If the parameters are not representative of the flow population, generating more data will not improve the relevant information.

For this reason it is

important to use the "best" estimates of the parameters. be wary of bias in historical data.

Modellers should

If suspected, it may be necessary to

employ regional techniques to estimate the parameters (Benson and Matalas, 1967). 5.9

A similar approach is recommended for ungauged catchments.

SIMULATION Simulation analysis is defined as "

a process which duplicates the

essence of a sys tern or acti vi ty wi thout actually attaining reality i tsel f" (Hufschmidt and Fiering, 1967).

This involves developing an algebraic model

of all the inherent characteristics and probable responses of the system to an operating rule.

Simul ation analysis has been described as a "brute

force" fitting technique (Fiering, 1961);

however, planners are often

forced to use the method to effectively deal with large and complex systems that become intractable with analytical techniques. Digital simulation does have several limitations which need to be recognised. (i)

Where a large number of variables has to be optimised, it may become infeasible to examine all possible combinations (Dorfman, 1965).

(ii)

It is a trial and error approach that does not necessarily lead to an optimal solution (Maass et al., 1962).

Notwithstanding these limitations simulation analysis using stochastically generated data as input is a very powerful tool particularly for a large complex system; are dealt with in Chapter 7.

aspects relating to multi-reservoir systems

136 5.9.1

When and How to use Generated Data In reservoir capacity-yield analysis data generation procedures should

be used to provide alternative yet equally likely flow sequences to the historical one.

In a large number of generated sequences, some will contain

less severe droughts than the historical record and some more severe.

When

the sequences are used in simulation or behaviour studies with a range of assumed storages and demand values a quantitative picture of the probability of failure-storage-yield relation can be built up. The point is best illustrated by an example.

If a behaviour analysis

(see Sec. 3.2.3) is carried out on the Mitta Mitta River (Appendix E) for various combinations of draft and probability of failure a series of curves will be obtained (Fig. 5.3).

From such a diagram the storage required for

any given draft and probability of failure can be obtained;

for example,

for 75% draft and 5% probability of failure, the storage required is 760 x 10 6m3 . 100~----------------------------------

____________________,

Probability of failure

10%

90

2%

5%

80

70

~

60

.:::
£5

50

40

30

20

10~----'-----'-----r----Lr-----r-----r---~r---~----~----~~

o

200

400

600

800

1000

1200

1400

1600

Storage (106m 3)

FIG. 5.3

Relationship of draft and probability of failure to reservoir capacity for the Mitta Mitta River.

1800

2000

137

This single estimate of the storage required (that is, 760 x 10 6m3) is based on the historical record which itself is only a sample of the total flow record for the river.

It is probable that other flow sequences, with

the same mean, standard deviation, and serial correlations, and equally

likely to occur could result in different storage estimates. Table 5.10 shows the storage estimated from flow sequences generated using a three parameter log-normal distribution.

The historical parameters

were converted to the equivalent log-normal parameters using moment transformation equations (Sec. 5.5.2) and each sequence generated was of the same length as the historical to avoid the effect of storage estimate increasing with record length (Sec. 3.2.1).

To estimate the storage a

fixed release (75%) of the historical mean flow was used. It will be noted from the table that the mean flows of the generated sequences vary, but the overall average (1282) is consistent with the historical mean flow (1274).

The storage estimates vary widely, however,

being quite sensitive to the mean flow (and other parameters) of the corresponding sequence. The values of storage should be Extreme Value distributed (Burges and Linsley, 1971) and are shown plotted on Extreme Value paper in Fig. 5.4. 1400 1300 1200

x

x x

1100 1000

;;

x

900

E <0

0 Gl

x

600 700

'"~ 600

x

0

iii

500 400

x

300 200 099

0.95 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.15

O. I

0.07

0.05

Risk that storage estimate is too small

FIG. 5.4

Risk that indicated storage is too small for design conditions on Mitta Mitta River.

0.03

0.02

138 TABLE 5.10

Storage estimated for generated flow sequences for the Mitta Mitta River (Appendix E). (Draft = 79.6 x 1Q6 m3/ mon th, probability of failure = 5%, sequence length = 34 years.)

Mean annual flow

Storage estimate

(l06 m3 )

(l06 m3)

1

1205

575

Sequence

Probability of estimate being too low (= rank)

25

0.68

2

1186

620

0.56

3

1404

891

0.24

4

1213

745

0.40

5

1253

666

0.52

6

1244

783

0.36

7

1093

1181

0.08

8

1214

801

0.28

9

1508

401

0.96

10

1261

1109

0.12

11

1460

501

0.80

12

1031

1229

0.04

13

1296

439

0.88

14

1152

724

0.44

15

1242

702

0.48

16

1291

617

0.60

17

1404

454

0.84

18

1409

608

0.64

19

1311

902

~.

20

1230

516

0.76

21

1219

1001

0.16

22

1518

547

0.72

23

1225

801

0.28

24

1426

411

0.92

1282

718

-

125

238

Mean Standard deviation

20

139 From such a graph the risk of the storage estimate being too low is clearly shown. For example, using historical data a storage of 760 x 10 6m3 is required, but there is a 20% chance that a storage greater than 930 x 10 6m3 is necessary to meet the design requirements, that is 75% draft, 5% probability of failure. Thus the use of generated data gives the designer a clearer picture of the variability of his estimate and allows him to choose a value dependent on the risk acceptable to him. Stochastic data generation processes cannot be used to provide estimates of historical flows.

Data estimation or extension should be

based on deterministic methods.

In addition, if the objectives of the

study can be achieved using analytical procedures with historical flows, there is no need for analysis to be based on generated data.

However,

where the problem cannot be solved analytically, simulation based on generated data is appropriate.

Beard (1972) considers the problem in the

following manner. "The design of water resource projects is commonly based on assumed recurrence of past hydrologic events.

By generating

a number of hydrologic sequences, each of a specified desired length, it is possible to create a much broader base for hydrologic design.

While i t is not possible to create information

that is not already in the record, it is possible to use the information more systematically and more effectively." In storage-yield analysis, for cases in which a data generation approach is adopted, simulation of the system is carried out either by behaviour type analysis or sequent peak algorithm for each generated sequence or replicate.

The behaviour approach is preferred to the sequent

peak algorithm which is computationally more efficient but cannot be used for problems in which draft is related to water in storage and where evaporation losses need to be directly taken into account. Twenty-five replicates are sufficient to indicate the variability in the objective being examined.

In this context, our studies suggest that

if it is necessary to negate the initial starting conditions of the storages, for highly variable streams and high drafts at least 100 years of records should be used in each replicate.

On the other hand, if starting

conditions are known the length of each replicate should be related to the objective of the analysis.

140 5.10

GENERALIZED RESERVOIR CAPACITY-YIELD-RELIABILITY RELATIONS Gould (1964), Guglij (c. 1959) * and Svanidze (1964) provide curves

relating carryover storage to draft, flow variability, flow distribution shape, persistence and probability of failure.

To establish these

relationships, the authors stochastically generated annual flow data and analysed these for storage need. 5.10.1

Gould's Synthetic Data Procedure Using a three parameter log-normal model, Gould (1964) generated

240 sets of data, each equivalent to 10,000 years of independent flow events. The 240 sets were analysed for different combinations of eight draft rates, five storage sizes and six values of skewness.

The results are presented in

Figs. 5.5, 5.6 and 5.7 where mean flow - draft standard deviation

and

(5.27)

reservoir capacity standard deviation'

(5.28)

Lines for three probability of failure values are also shown on the figures.

Gould also provides a correction factor (Eq. 5.29 below) for

assuming annual serial correlation is zero.

This equation was based on

analysis of additional synthesized data.

Procedure: The procedure for estimating the required carryover storage for given values of constant draft and probability is as follows: (i)

Compute the mean, standard deviation, coefficient of skewness and serial correlation of annual flows.

(ii)

To allow for annual serial correlation, modify the design probability by the fOllowing equation: (P' - l2r)/(1.7r + 1)

P

where

(5.29)

P

equivalent probability of failure after

P'

design probability of failure, and

r

annual serial correlation coefficient.

adjusting for serial correlation,

* A specific reference to Guglij's analysis is unavailable.

Curves developed

by him were provided by Dr. S. Selvalingam, Asian Institute of Technology, Bangkok.

141

6

4

2

O+-~~-'------.------r------r------

5

-2~--------------------------------~

FIG. 5.5

Storage and draft ratios based on Gould's synthetic data for coefficient of skewness equalling zero (Gould, 1964). 10.----------------------,~------r_--_,

8

6

4

O+---~~------._----_r----_,------~

2

3

4

5

k1 +O.15 -2

-4~---------------------------------

FIG. 5.6

Storage and draft ratios based on Gould's synthetic data for coefficient of skewness equalling one (Gould, 1964).

142 10~--------------------------~------~----

8

4

5

-2

-41-______________________________________

FIG. 5.7

(iii) (iv)

~

Storage and draft ratios based on Gould's synthetic data for coefficient of skewness equalling two (Gould, 1964).

Compute kl from Eq. 5.27. According to the coefficient of skewness enter Fig. 5.5, 5.6 or 5.7 and estimate k2 for the appropriate probability of failure.

(Interpolate between lines or between figures

if necessary.) (v)

Compute carryover reservoir storage from Eq. 5.28.

Comments: The procedure utilizes the results of an analysis of synthesized flows from a data generation model using a three parameter log-normal distribution.

It was noted in Sec. 5.8 that this distribution does not

model the time-series parameters nor the low flow values particularly well. From Fig. 5.7 it will be seen that for kl drafts, k2 becomes negative. skewnesses as well.

>

0.85, which can occur at low

This unrealistic result can occur for other

As will be noted in Sec. 6.3, the procedure tends to

underestimate capacity for the smaller reservoirs.

143 EXA~lPLE

5.4

Use Gould's Synthetic Data procedure to estimate the storage required to meet 75% draft with a 5% probability of failure for the Mitta Mitta River (Appendix E). *

*

*

*

*

Following the steps given in Sec. 5.10.1: (i)

From Table E3 for annual flows: 1274 (x 10 6 m3 )

!vIe an

Standard Deviation

(ii)

731 (x 10 6 m3 )

Coeff. of Skewness

1.5

Serial correlation

0.06

Modify probability: From Eq. 5.29:

P

(P' - 12r)/(1.7r (5 - 12

x

+

1)

0.06)/(1.7

x

0.06 + 1)

4.28/1.1 3.9% (iii)

From Eq. 5.27:

kl

mean flow - draft standard deviation 1274 - 0.75 731

x

1274

0.44 (i v)

From Fig. 5.6:

k2 (P

3.9%, skew

1. 0)

2.0

From Fig. 5.7:

k2 (P

3.9%, skew

2.0)

1.3

Interpol ating:

k2 (P

3.9%, skew

1. 5)

2.0 + 1.3 2 1. 65

(v)

From Eq. 5.28:

Reservoir capacity

standard deviation x k2 731 x 10 6 x 1.65

144

Guglij's and Svanidze's Synthetic Data Procedures

5.10.2

Guglij and Svanidze used Monte Carlo models and a modification of the Pearson Type III distribution known as the Kritskii-Menkel curve (Kartvelishvili, 1969, p. 35) to generate long sequences of annual flows (1000 years or more) for various combinations of coefficients of variation,

skewness and serial correlation.

These data sequences were used to deter-

mine finite storage capacities (as ratios of mean annual flow) for given values of draft and reliability.

Guglij results are summarized in 66 graphs

(provided by S. Selvalingam, personal communication, 1976).

Svanidze

includes more details in his 120 graphs (Kartvelishvili, 1969, Appendix VII). But both sets of results are of use only for streams with low variability less than 0.4 or 0.8 for Svanidze's curves and 1.4 for Guglij's relations. Moreover, their usefulness for high drafts is limited because the computed storage sizes are limited to values less than 2.8.

These relationships

include annual serial correlation as a parameter which is not available in Gould's Synthetic Data curves. It wi 11 be noted in Sec. 6.3 that reservoi r capacity estimates from

these procedures are not as satisfactory as other estimates;

consequently

other procedures are recommended before these.

5.11

NOTATION

a.

number of days in month

A

variahle in Kirby's transformation (Eq. 5.10)

A.

location parameter in three parameter log-normal distribution

J B

variable in Kirby's transformation CEq. 5.10)

B

Beard's procedure in Tables 5.7, 5.8 and 5.9

1

J b.

B.

J C v

regression coefficient for estimating Cj+l)th flow

log transformed regression coefficient coefficient of variation

C

coefficient of skewness

CL

confidence limits

g

s

i

gj

coefficient of skewness of standardized log flows (Sec. 5.5.3) ·· coe ff lClent a f s k ewness of f lows during J. th season

G

like-Gamma distribution in Tables 5.7, 5.8 and 5.9

G

variable in Kirby's transformation CEq. 5.10)

GCO,I,y)

Gamma distribution with zero mean, unite variance and skew = y

H

variable in Kirby's transformation (Eq. 5.11)

k

lag between flow events under analysis

145 draft ratio defined as mean less draft, divided by standard deviation of flow storage ratio defined as reservoir capacity divided by standard deviation of flow K

Kirby's procedure in Tables 5.7, 5.8 and 5.9

K

correlation component of the streamflow process

LGLT

like-Gamma Logarithmic Transformation model (Sec. 5.5.1)

t

LN-2

2 parameter Log-normal model

LN- 3

3 parameter Log-normal model

N

Normal distribution in Tables 5.7, 5.8 and 5.9

N

number of flow events

N (0,1)

normal distribution with mean zero and variance equal to one

P

probability of failure or probability of occurrence

P'

design probability of failure in Gould's synthetic data procedure (Sec. 5.10.1) annual generated flow volumes annual serial correlation coefficient lag one serial correlation of normalized variates (Sec. 5.5.3)

r

serial correlation between (j+l)th and jth months lag

k

serial correlation coefficient

lag one serial correlation coefficient log transformed serial correlation coefficient s s.

J

standard deviation . . stan d ar d d eV1at1on of mont h ly flows f or·J th month

s

standard deviation of loge (x + €) flows . t·lon of flows for J.th season 1og t rans f orme d s t an d ar d d eVla

St

seasonal component of the streamflow process

SSO ,S90

storage capacities for 50 and 90% draft rate us.ing the sequent peak procedure

Y S. J

t

time

t.

normal random variate

1

G(O,l,y

.) like-Gamma variate t,J

Kirby's modified like Gamma variate CEq. 5.10) trend component of the streamflow process standardized log flow values (Sec. 5.5.3) W

Weibull distribution in Tables 5.7, 5.8 and 5.9

x .. 1J x.

unadjusted monthly generated flow volumes CEq. 5.26) . .th perlo . d f low volumes durlng 1

x

monthly flow

1

t

146 x! .

adjusted monthly generated flmv volumes CEq. 5.26)

Xl

standardized monthly flow

lJ

t X

Xj ,X j +l

\

mean flow . J' th and CJ' +1) th season mean mont hI y fl ow durlng . th perla . d f low ln t

X.

generated flow logarithms for (i+l)th and ith seasons .th 1og t rans f orme d mean flow for J season

Y

mean of loge

\+1'\ J Y

loge

ex

+ E) flow

ex

+ E)

flows

za' zp

standardized normal variate

Yj 'Y j +l

seasonal coefficients of skewness for jth and

Yt,j

coefficient of skewness of the like -Gamma variate

U+l) th seasons small increment

E E

t

random component in streamflow process

v

normalized flow values

1: 7 ,L)0

sum of the rank one 2 and 10 year consecutive low flows respectively

147 CHAPTER 6

QUANTITATIVE ASSESSMENT OF TECHNIQUES FOR

CAPACITY· YIELD

SINGLE RESERVOIRS

Over the past several years, the authors have been involved in four comparative studies using Australian streams in which reservoir capacityyield techniques have been assessed. been published (Joy, 1970; Codner and McMahon, 1973;

Results of three of the studies have

Joy and McMahon, 1972;

~jcMahon

McMahon, Codner and Joy, 1973);

and Codner, 1973; the fourth will

be published shortly (C.H. Teoh, personal communication, 1977). In all of the above studies the storage estimates provided by the behaviour analysis method (with the reservoir initially full) were taken as the benchmark to which other methods could be compared.

It was selected

for this purpose because of its current widespread use by water authorities as a final design technique, and because it suffers from only one major drawback - that of dependence of storage es timates on the initial condi tion chosen. 6. I

CRITICAL PERIOD AND PROBABILITY MATRIX METHODS The first study was concerned with illustrating quantitatively the

inadequacies in some of the techniques discussed in Chapters 3 and 4 by applying them to six Australian rivers on which major dams already existed or were proposed.

The rivers and their flow characteristics are given in

Table 6.1. In Figs. 6.1 and 6.2 storage estimates at 50% and 90% drafts respectively for the six rivers are compared among the critical period techniques - minimum flow (Waitt), Alexander's method corrected for the assumption of zero serial correlation, overlapping series mass curve frequency method (Thompson), independent series mass curve frequency method (Stall) and behaviour estimates for 5% probability of failure.

(Note that

to obtain a suitable scale, storages have been standardised by dividing by the standard deviation of monthly flows.

Reference numbers refer to the

rivers listed in Table 6.1.) 6 .1.1

~lass

Curves and Minimum Flow (Waitt)

From Figs. 6.1 and 6.2 it is seen that the Waitt technique (excluding the additional one year's supply) which is equivalent to a mass curve

TABLE 6.1

No.

Rivers investigated in evaluation of critical period and probability matrix methods.

River (Australian Streamgauging Station Reference Number)

1

Yarra (229103)

2

Murrumbidgee ( 410008)

3

Annual Area (sq. km)

Period of Record

Mean

Monthly

(mm)

Coeff. of Var.

Coeff. of Skew

Serial Corre1.

Coeff. of Var.

Coeff. of Skew

Serial Corre1.

334

1892-1968

540

0.40

0.77

0.12

l.00

1.77

0.62*

13000

1927-1960

128

0.85

l. 99

0.08

l.39

3.66

0.60*

Lachlan ( 412010)

8290

1931-1967

110

l.09

l. 89

0.14

l. 79

3.42

0.61*

4

Warragamba (212240)

8750

1881-1959

122

1.11

2.67

0.30*

2.14

4.71

0.45*

5

Namoi ( 419007)

5700

1924-1960

76

1. 06

l. 96

0.23

l.97

4.22

0.42*

6

Burdekin (120090)

114000

1915-1950

57

l.00

l.72

0.07

2.86

5.05

0.41*

* Values significantly different from zero at the 5% level.

149 0 10

Ii

(4)

(3)

0 6

• •

8

Waitt Thompson Stall Alexander

(5)

0 6

SI

(f

"

(6) 0 (2) 4

•

0

(1)

I

•

•

0

•

• • ,,/ " • "•

2

"

O~---------.----------r----------'~

o

FIG. 6.1

2

Sbl (f

4

6

Comparison of critical period storage estimates (ordinate) with historical behaviour estimates (abscissa) at 50% draft and 5% probability of failure.

analysis, overestimates the behaviour storage other critical period techniques.

values computed using all

This is to be expected as the approach

allows only one failure whereas in the other techniques we have adopted a 5% probability of failure criterion. 6.1.2

Alexander's Method It was suggested earlier (Sec. 3.4.2) that Alexander tends to under-

estimate the behaviour storage for short critical periods (neglect of seasonality) and to approximate it for long critical periods.

From Fig. 6.1

it is seen that the Alexander method underestimates the behaviour storage

for rivers 1, 3, 4, 5 and 6, but overestimates it for river 2, possibly because the flows for 2 are not adequately described by a two-parameter Gamma distribution.

For long critical periods, as expressed in Table 6.2,

the behaviour storages are closely approximated except for river 4.

150

80

o (4)

o

Waitt

6. Thompson • Stall

•

60

Alexander

(3)

0

•

(2)

40 (5)

0

•

e (1)

(6)

0

0

•

20

• o FIG. 6.2

6.1.3

• ••

•

• 20

40

60

Comparison of critical period storage estimates (ordinate) with historical behaviour estimates (abscissa) at 90% draft and 5% probability of failure.

Overlapping Series Frequency Mass Curve Method (Thompson) Earlier it was shown (Sec. 3.4.5) that the relationship of the

Thomp~on

storage estimate to the behaviour estimate depends on the

clustering effect of sequences of the same duration, which leads to overestimation of the behaviour storage results. critical periods.

This is significant for long

At short critical periods, due to the neglect of repeated

failures, it can be shown theoretically that Thompson's method tends to underestimate the required behaviour storage. These views are confirmed in practice by Fig. 6.1 which shows that the Thompson procedure underestimates the behaviour storage for rivers 1, 2, 5 and 6 and overestimates it for rivers 3 and 4.

From Table 6.2 it is

seen that rivers 1, 2, 5 and 6 have short critical periods whereas rivers 3 and 4 have long ones. For 90% draft, the duration of Thompson's critical period approaches that of Waitt for rivers 2, 3, 4, 5 and 6.

In this investigation, the

151 TABLE 6.2

Duration of critical periods for storages designed for 5% probability of failure (years).

Method

1

2

Mass curve and Waitt

1.5

2.0

flow with probabi li ty (Alexander)

0.5

2.2

Overlapping series (Thompson)

1.5

Independent series (Stall)

0.8

River* 3 4

5

6

8.3

5.0

6.7

3.1

3.2

3.0

2.7

1.5

8.3

5.0

2.0

1.5

2.0

3.0

3.0

0.5

1.5

50?, Draft ~linimum

10

90% Draft

Mass curve and Waitt

7.5

Minimum flow with probability (Alexander)

10

Ove r 1 app ing series (Thompson) Independent series (Stall)

1.5

20

18

15

15

15

+

greater than 50

->

+

greater than 12

->

5.0

I

6.7

I 8.3 I

6.7

I

8.3

* As listed in Table 6.1. maximum duration of critical periods examined for Thompson's procedure was 12 years. ri vers .

This was insufficient to define Thompson's storage for these Hotvever, for river 1 repeated fai lure compens ates the cl us tering

effect and so the Thompson method approximates the behaviour storage. In view of these inadequacies, we recommend that this procedure should not be used. 6.1.4

Independent Series Frequency Mass Curve Method (Stall) It was shown earlier that Stall's method tends to overestimate the

required behaviour storage for short critical periods and to underestimate it for long cri tical periods. of Figs. 6.1 and 6.2.

This is confi rmed by the empirical results

Table 6.2 contains the durations for Stall's critical

periods. From Fig. 6.1 it is seen that Stall's procedure overestimates the required storage for rivers 1 and 5, and underestimates it for rivers 2, 3, 4 and 6.

From Table 6.1, rivers 1 and 5 have short critical periods while

rivers 2, 3, 4 and 6 have longer critical periods.

152

At 90% draft, Stall's procedure underestimates all the behaviour storages.

From Table 6.1 we see that all rivers have long critical periods,

except for river I in which Stall and behaviour estimates are very similar. Because of these limitations we recommend that Stall's procedure should not be used. 6.l.5

Gould's Probability

~latrix

Method

This is the only method in the second group of storage-yield techniques for which there are comparative results.

The limitations of Moran's

discrete probability matrix models were discussed in detail in Chapter 4. In Figs. 6.3 and 6.4 results from Gould's probability matrix method (as proposed by Gould and also as modified to cater for monthly failures rather than annual failures) are compared with behaviour estimates for conditions of 50% and 90% drafts and 5% probability of failure. In both Figs. 6.3 and 6.4 the original Gould method overestimates the behaviour storage.

On the other hand, the modified Gould method fits the

behaviour relationship satisfactorily. 8 6

Gould

•

Modified Gould (4)

6.

6

• (6)

(5)

6. 6.

4 (2)

•

•

•

6.

(ll

./ • 6.

2

O~---------.----------.----------'r---------~

o

FIG. 6.3

2

4

Sb/

6

8

(f

Comparison of Gould's probability matrix storage estimates (ordinate) with historical behaviour estimates (abscissa) at 50% draft and 5% probability of failure.

153

(4) to. (3)

to.

60

(5) (2)

•

to. to.

•

•

40

(6)

•

to.

s/ () 20

(ll

•

6

•

6.

Gould

•

Modified Gould

O~-----------'~-----------r------------~----~

o

FIG. 6.4

6.1.6

20

40

60

Comparison of Gould's probability matrix storage estimates (ordinate) with historical behaviour estimates (abscissa) at 90% draft and 5% probability of failure.

Further Comparison of Gould and Behaviour Methods A second study, which was aimed at establishing empirical equations

relating storage to other flow parameters, involved 156 Australian streams (Sec. 4.7.1, McMahon, 1977).

In the analysis for each stream, storage

sizes required to meet conditions of 5% probability of failure and 50% and 90% drafts were computed using Gould's modified approach (20 zones) and two

behaviour approaches, one assuming the reservoir to be initially full and the other initially empty.

Storage estimates expressed as ratios of the

mean annual flow are plotted in Figs. 6.S and 6.6 which represent respectively 50% and 90% drafts. Overall, the fit at 50% draft is satisfactory with the behaviour empty case tending to overestimate the steady state storage requirement for the more variable streams.

In other words, the initial emptiness state is

significant for the conditions examined.

At 90% draft, there is a tendency

for steady state storage need to be underestimated for the behaviour full case as a result of the initially full assumption.

154 2.5

/

,!,V/ 1.0

'i

£

"cc'"

0.5

'"c

.,'" E --.,

0.25

X

0>

:::: 0

§

0.1

.&til E

:;:;

~

0.05

:;

o .;;

a-Behaviour Method:

Initially empty

til

6-Behaviour Method:

Initially full

~ 0.025 III

0.01 / / /

o;y /

0.005~/~__~~______~~~__~~____~~______~~____~~____-L________~

0.005

0.01

0.025

0.05

Gould Estimate

FIG. 6.5

0.1

0.25

0.5

1.0

2.5

(Storage/mean annual flowl

Comparison of behaviour results (initially full and empty) with modified Gould estimates for 156 Australian streams at 50% draft and 5% probability of failure.

It is believed that in both cases the effect of ignoring annual serial correlation in Gould's procedure is masked by the various inadequacies in the behaviour approaches.

The results confirm the modified Gould pro-

cedure as being analytically superior to the behaviour approaches. 6.1.7

Summary From these two studies we conclude that the modified Gould technique

is a suitable analytical storage-yield procedure for final design.

In

addition, it is noted that the behaviour analysis, which was used as a basis for comparison, gave results consistent with theoretically acceptable procedures so long as the effect of initial conditions is recognized.

155 25

I

I

I

I

I

I

/

/

/

/ /

/ /

lOt-

/

!

/

/

-

/

~! ~~{~
5.0 t-

fA!

B

:&

2.5 -

1.0

/

!

d~

r

0.5 t-

/

0.25 t-

~

/

-

1,

!

g1

!

~lr

/

:or

! !

-

0-

Behaviour Method:

Initially em pty

6-

Behaviour Method:

Initially full

-

/

/

O.l~____~~I__~~I~__~IL-____~~l

0.1

0.25

0.5

1.0

Gould Estimate

FIG. 6.6

5.2

____~I~____L-I______~

2.5

5.0

10

{Storage/mean annual flowl

Comparison of behaviour results (initially full and empty) with modified Gould estimates for 156 Australian streams at 90% draft and 5% probability of failure.

CAPACITIES BASED ON STOCHASTIC DATA GENERATION By generating several streamflow sequences, all with the same

statistical characteristics and all assumed to be equally likely to occur, the designer is able to determine something about the precision of his storage estimate.

This result contrasts strongly with methods like the

)ehaviour analysis using historical data in which only one value is )btained;

nothing can be deduced about the distribution of this single

,stimate. A study of three rivers (numbers 1,4 and 5 in Table 6.1) involved :omparing historical behaviour and modified Gould estimates with those

25

f-'

Uo

1.0

5.1

1.14

~

80% bond 0.8

/ /

Range~

",

\

\

/ /

M

\

/

\

/

\

\

M

\

E

.... ....

"",,,,,,,""

I

E 3

I

.... ....

0.6

.... ....

01

'-

•


01

'" '0

ci) 0.4

I

,,

,

0.2

,,

---

"

""

,

/

-.-• I

....

/

,- "

,- /

.- .-

/

,"

I

2

(j)

I

I I

•

Q)

/

---

Historical Behaviour Historical Gould

x

Generated Behaviour

0

Generated Gould

o

... ...

I

80% band'\.

....

'" .8

... ...

I

X

X

... ...

I

'"0

\

/

'"0

'"

4 Range~

"-

.....

.... ...

•

Historical Behaviour

X

Generated Behaviour

0

Generated Gould

I

- i-Historical Gould

0 50

100

Generated Period

200

500

(years)

FIG. 6.7 Behaviour and modified Gould storage estimates based on historical and generated data for conditions of 50% draft and 1% probability of failure [Namoi River at Keepit No. 2 (419007) New South Wales].

20

50

Generated Period

100

200

(years)

FIG. 6.8 Behaviour and modified Gould storage estimates based on historical and generated data for conditions of 90% draft and 1% probability of failure [Namoi River at Keepit No.2 (419007) New South Wales].

TABLE 6.3

River t

Reservoir capacltles estimated by behaviour and * Gould techniques using historical and generated data. Behaviour

Historical

50 years

Gould 200 years

Historical

50 years

200 years

50% draft, 1% ~_robabi1ity of failure Yarra Warragamba Namoi

0.22

0.18 (+41%, -10%)

0.19 ( +4%, -6%)

0.21

1.0 (+25%, -26%)

1.2 (+22%, -9%)

1.2

1.0 (+83% , -19%)

1.2 (+26% , -14%)

1.1

0.94 (+35%, -43%)

0.89 (+26% , -31%)

0.94

4 (+110%, -30%)

8 (+93% , -34%)

9

2.5 (+70%, -15%)

5.1 (+22% , -38%)

5.3

1.5 0.9

0.19

0.17 ( +10% , -10%)

(+7% , -3%)

0.9 ( +43%, -34%)

(+23%,

l.2

1.2 -6%)

(+28% , -28%)

1.0 (+8% , -18%)

0.94 (+47%, -23%)

0.89 (+22% , -15%)

:----

90% draft, 5% probabi1i ty of failure Yarra Warragamba Namoi

0.89 11

4.6

* Storage estimates are expressed as ratios of mean annual flow.

5 (n. a.) 4.2 (+34% , -52%)

7

(n. a. , -22%) 5.1 (+33%, -23%)

Synthetic estimates are the median values of ten replicates. Values in brackets represent the variation of storage sizes within which eight of the ten replicates lie.

tPor details see Table 6.1.

158 obtained using generated data of length 20-500 years.

Results for one

river are shown in Figs. 6.7 and 6.8 and for all three rivers in Table 6.3. A Thomas and Fiering monthly model using the like Gamma distribution and an initial logarithmic transformation (LGLT) was adopted as the data generation procedure.

Assuming that the monthly generated flows are representative of

alternative historical sequences, a number of observations follow: (i)

It is observed that for the behaviour analysis a wide range of storage values results from the different flow sequences. This variation is indicative of the sampling error associated wi th a hydrologic sequence.

Sampling error of the storage

estimate is discussed in Sec. 6.4. (ii)

By increasing the record length, sampling variability is lowered.

Generally, too, the storage variations reSUlting

from the modified Gould analysis were smaller than those found using the behaviour technique. (iii)

For the behaviour technique as typified in Figs. 6.7 and 6.8, average storage estimates increase to a maximum value

with increasing record length.

This feature is a consequence

of critical period approaches in which the storage is assumed initially full so that the failure in the sequence is dependent on the initial storage condition, whereas additional failures can result without the storage refilling.

Thus proportionally

less failures will occur during the earlier part of the sequence as compared with those occurring in the later part. Results suggest that for highly variable streams and high drafts at least 100 years of streamflows are required before the effect of the initially full assumption can be ignored. Because the Gould procedure is not dependent on the initial conditions of the storage, this limitation does not occur with it. (iv)

The results also confirm the consistency of the Gould procedure.

For each river the Gould results show less

variability between the historical and generated flow estimates and within the generated flow estimates than do the corresponding behaviour values.

This mainly arises because

of the problem of the initial starting conditions.

159 The results shown in Table 5.9 are also pertinent to this evaluation. There it was found that the storage estimates based on annual generated data using Beard's technique overestimated the historical values by about 10%. This result is not inconsistent with the results in Table 6.3 where the estimates based on replicates of 200 years of generated data are within 10% of the historical Gould value. 6.3

RAPID RESERVOIR CAPACITY-YIELD PROCEDURES Rapid storage-yield procedures suitable for preliminary reservoir

capacity-yield estimation were evaluated in the fourth study (C.H. Teoh, personal communication, 1977).

Rapid procedures are considered to include

any method that would allow a storage-yield estimation to be made quickly (say within an hour) without the aid of a digital computer. cedures were evaluated:

Eight pro-

three critical period approaches - Alexander's,

Dincer's and Gould's Gamma;

two probabi Ii ty matrix methods - Hardison's

Probabi lity Routing and McMahon's Empiri cal; data - Gould's, Guglij's and Svanidze's.

and three based on generated

In the evaluation, four combi-

nations of draft and reliability were adopted (30% draft, 95% reliability; 50/90;

70/98 and 90/95) and reservoir capacities were computed for

36 Australian and 12 Malaysian streams with annual coefficients of variation ranging from 0.12 to 1.8 and annual coefficients of skewness ranging from -0.8 to 3.7.

Storage estimates using Gould's transition matrix procedure

and the behaviour method were used to determine datum values against which the estimates from the other procedures could be tested.

Comparisons are

shown in Figs. 6.9 to 6.15. From these figures a number of points should be noted. (i)

(a)

With regard to Alexander's procedure (Fig. 3.14),

for drafts less than 40%, recurrence intervals are restricting; (b)

no values could be estimated for 30% draft.

There is a tendency as seen in Fig. 6.9 for

Alexander's storage estimates based on the same draft values to be generally overestimating or underestimating what are expected to be the correct estimates. (c)

For drafts of 85% and above, Alexander's curves

(Fig. 3.14) require extrapolation for use.

160

20

,0

" "Qj

I

""C

c:

'" ~

<'" 30% 50% 70% 90%

draft. draft. draft, <:iraft.

95% 90% 98% 95%

reliability reliability reliability reliability

05

05

•

2

5

1

2

5

10

20

Gould's probability matrix estimate (reservoir capacity / mean annual flow)

FIG. 6.9

Comparison of Alexander's estimates with Gould's probability matrix values.

20

++

.*

'0

'"

0;

.,E

2

'""

" ~E

5

c:

o

x o c,

+

30% 50% 70% 90%

draft. draft. draft, draft.

95% 90% 98% 95%

5

10

reliability reliability reliability reliability

05

05

2

5

1

2

20

Gould's probability matrix estimate ( reservoir capacity I mean annual flow)

FIG. 6.10

Comparison of Dincer's estimates with Gould's probability matrix values.

161

20

+

+

+ •

+ it •

..E .

j: .0Y'J·

2

E

;(~<&

""

'0 "S o

:t4'l • . / 0 . ;:;''1 x

.5

o

XX

x

Xx.,p>.0

~~..


/ .0.

05 •

x

00'"

x o

30% 50% n 70% + 90%

+

draft, draft. draft. draft.

95% 90% 98% 95%

reliability reJiabiIJty reliability reliability

00

.)(

0

'"

05

2

5

1

2

5

10

20

Gould's probability matrix estimate (reservoir capacity / mean annual flow)

FIG. 6.11

Comparison of Gould's Gamma estimates with Gould's probability matrix values.

//

/

~

.~ '"'"

/

20

r::+: 0.0;;/.

10

, "

~'" :0

o~

2

..

1

ii

,~,

.0

o

~

"'~~~

5

~ /~.:x~o

x

2

:x:

A

b.-¥<.~

1

05

'"

IAli

05

+

30% 50% 70% 90%

draft, draft, draft. draft,

95% 90% 98% 95%

reliability reliability reliability reIJability

4)(

2

5

1

2

5

10

20

Gould's probability matrix estimate ( reservoir capacity / mean annual flow)

FIG. 6.12

Comparison of Hardison's probability routing estimates with Gould's probability matrix values.

162

20

10

~

'" E

5

:;:;

"'"

'"c: o .c

'"

::;

2

u

x

::;

30% 50% 70% 90%

o l:!.

+

05

2

5

1

draft, draft, draft, draft.

95% 90% 98% 95%

5

10

2

reliability reliability reliability reliability

20

Gould's probability matrix estimate (reservoir capacity / mean annual flow)

FIG. 6.13

Comparison of McMahon's empirical estimates with Gould's probability matrix values.

20

!

10

'" E

~

"

~

~

2

u

~ -;;

,

1

c:

>..

'"

5

*

'"

+

"0

:; o

+f'

h +-t ,.. + +

"

•

2

Cl +

30% 50% 70% 90%

draft, draft, draft. draft,

95% 90% 98% 95%

reliability reliability reliability reliability

05

05

1

2

5

1

2

5

10

20

Gould's probability matrix estimate (reservoir capacity / mean annual flow)

FIG. 6.14

Comparison of Gould's synthetic data estimates with Gould's probability matrix values.

163

20

10

x o

30% draft, 95% reliiibility 50% draft. 90% reliability

6

70% draft. 98"10 reliability

+ 90% draft, 95% reliability 05

05

1

5

1

10

20

Gould's probability matrix estimate ( reservoir capacity / mean annual flow)

FIG. 6.15

(ii)

Comparison of Guglij's synthetic data estimates with Gould's probability matrix values.

Overall, Dincer's values overestimate the Gould storage estimates (Fig. 6.10).

The magnitude of overestimate is

large and unacceptable for 30% drafts, but for storage sizes greater than unity the overestimation is less than 10%. (iii)

For Gould's Gamma procedure, the results are satisfactory although a slight bias is evident for some drafts (Fig. 6.11). The increase in variability for storages less than unity is expected, as reservoir capacities based on annual flows are compared with results determined from monthly data.

(iv)

(a)

Hardison's procedure tends to slightly overestimate

storages greater than unity and severely underestimate very small ones (Fig. 6.12). (b)

Some values are missing in the upper range because

the curves (Fig. 4.8 to 4.10) at high drafts and high variability are restricting. (v)

As expected, results using McMahon's procedure fi t well because 36 of the 48 streams shown in Fig. 6.13 were used to develop the empirical relations.

Nevertheless, for the

Malaysian streams which were not l:sed to develop the empirical equations no anomalies are evident although some of the streams have coefficients of variation outside the range of data used to develop the relationships.

164 (vi)

Gould's Synthetic Data procedure performs poorly for smaller storages but satisfacorily for storages greater than unity (Fig. 6.14).

(vii)

Only Malaysian data were used to test Guglij's method mainly because most of the Australian streams were outside the range of parameters given in the original paper (Sec. 5.10.2).

Guglij was used in preference to Svanidze

because he covers a wider range of condi tions.

The results

in Fig. 6.15 are very similar to the smaller storage estimates from Gould's Synthetic Data procedure (Fig. 6.14) where for storages less than unity the estimates are unacceptable. From this evaluation three procedures appear to be satisfactory. Gould's Gamma and

Mc~1ahon'

s Empiri cal methods are s lightly superior to

Alexander, but it must be noted that the regression equations for McMahon's method were based on Australian data and until it is thoroughly tested using data from other continents, its use should be restricted to Australian streams. 6.4

SAMPLING ERROR OF STORAGE AND DRAFT ESTIMATES Few papers provide an estimate of the sampling error of storage size.

Some procedures are given here: (i)

Gould (1964) assumed that the draft is near-normally distributed and used the results of his Synthetic Data procedure (Sec. 5.10.1) in the estimation of sampling error.

Based on the results of that procedure, it is

recommended that the following error analysis be applied to storage sizes (in terms of mean annual flow) greater than unity.

Gould showed that there is an ct% chance that

the draft which would result in a p% probability of failure is less than D', where 1

D' where

D

D -

t

a,n-l

C v

[lIn

+

0.5 (EL +

FL)J"

(6.1 )

design draft (as percentage of mean annual flow), single-tailed Student's n-l degrees of freedom,

t

for a% and

165 years of recorded flow,

n

coeffi cient of variation of annual flows, c(2k + b) 2 1 (6.2) [ QlS (k2 + b)2

C v E

J

rn

_ ck2 2 n(k + b)3

F

(6.3)

reservoir capacity/ standard deviation

k2

Variables

b

and

TABLE 6.4

Parameter

c

are tabulated in Table 6.4.

Values of b and c in Eqs. 6.2 and 6.3 for different values of probability of failure and coefficients of skewness.

Probabi Ii ty of Failure (p)

Coeffi cient of Skewness 0

1

1.5

2.8

(%) 0.5 b

c

(ii)

2

3

4

5.5

2

1

2.1

3.1

4

5

0.5

1.7

2.5

3

0.5

3.3

3.8

4

4.2

2

2.4

2.7

2.9

3

5

1.7

2

2.1

2.2

Alternatively, one could determine the magnitude of sampling error from one of the theoretical methods - the most suitable being Gould's Gamma approach (Sec. 3.4.4). Equation 3.30 gives

[,(1 'P'D)- d] C/

T

where

T

(6.4)

required storage divided by mean annual flow in volume units per year.

Differentiating with respect to z 2 IH

i'lT T

P

2 [ 4 (1 - D) i'lC 2

v

CV

d]

Cv

C i'lC V v

leads to (6.5) (6.6)

166

where the standard error of the estimate of C ' v denoted by ~Cv' is given by Eq. 2.13. Equation 6.6 gives the sampling error with respect to the storage

estimate. Again using Eq. 6.4 as the basis of the storage. draft relation, the sampling error with respect to the draft, may also be derived by differentiating with respect to D, leading to:

D'

D -

(6.7)

Since (from Eq. 6.4) 2

D

1 - ~

1

4

(;- +

Cv

dj

2

D'

-

~

4(~2

+

d)

[1

2T +

v

~C

C3(~ 2 V C v

v +

dJJ

(6.8)

which is an alternative equation to Eq. 6.1, but with a much simpler derivation.

EXAMPLE 6.1 Previous examples have used various procedures to estimate the storage requi rement on the Mi tta Mi tta River (Appendix E) to meet a 75% draft with a 5% probability of failure.

However, due to sampling errors,

it is possible that the realizable draft may be lower (or higher) than the design value.

Use each of the alternative Eqs.

6.1 and 6.8 to determine

the magnitude of realizable draft which has a 90% chance of being equalled or exceeded. * (i)

*

*

*

*

For the variables in Eq. 6.1:

o

design draft

n

number of years of record

C(

0.10 from Student- t

0.75

distribution

34 1. 3.

167 coefficient of variation of annual flows

C

v

0.57

(Appendix E) b

2.1

c

2.05

k2

1. 65

} from Table 6.4 (for skewness

1.5, P

5%)

(from Example 5.4)

Equation 6.2 gives: E

_1_ [0.15

!34

2.05 (2 x 1.65 + 2.1) ] (1.65 + 2.1)2

- 0.11 Equation 6.3 gives: 2.05 (1.65)2

F

34(1.65 + 2.1)

3

0.003 Substitute in Eq. 6.1 to obtain: D'

0.75 - (1.31) (0.57) "'1/1/34 + 0.5 (0.1l2

+

0.003 2 )

0.61 That is, Eq. 6.1 predicts that, for a probability of failure of 5%, there is a 90% chance that the realizable draft from the storage obtained using the design draft will be 61% or more. (ii)

For the variables in Eq. 6.8 and results from Example 3.9, z

1.64 (from Normal distribution, p = 5%)

T

0.68

C v d

0.57

P

0.6

Equation 2.13 gives one standard error of

C v

1 + 2C

( (

2n

V)

1 + 2 x 0.57, t 2 x 34 )

0.18 Now, one standard error is equivalent to 67% confidence(Normal distribution).

6e v

For 90% confidence multiply 0.18 by 1.28. 0.23

168 Substitute in Eq. 6.8 to get: 2 l.64 D' 0.68 4 - - + 0.6 2 0.57

~

2(0.68) (0.23)

•

0.57 3

]

0.68 + 0.6 0.57 2

0.59 That is, Eq. 6.S predicts that the realizable draft which one can be 90% sure of being met is 59% or more.

(iii)

An alternative approach to sample error analysis is an empirical examination of the effect of changes in parameter values on storage size using stochastic data generation models. Results of such an analysis are shown in Table 6.5.

A monthly

log-normal Markov model was used to generate sequences of flows for four rivers in which random errors were superimposed on. the input his torical parameters of the model.

Storage

estimates for each replicate were calculated using Gould's procedure. TABLE 6.5 River (Annual Coeff. of Skewness)

Effect of Flow Parameters on Reservoir Size * Mean

l.0

0.9

Coeff. of Skewness 0.8

LN2 Model Gordon (0.24)

100 (17)

130

Yarra (0.40)

100 (10)

127

Torrens (0.78) Warragamba (1.11)

Ignore Skew LN2

1.0

2.0

Seri al Correlation 1.0

LN3 Model

1.1

1.2

LN2 Model

167 (19)

73 (17)

100 (17)

SO (15)

100

(17)

lOS (8)

105 (15)

(13)

187 (20)

76 (10)

100 (26)

76 (12)

100 (10)

106 (14)

114 (16)

100 (23)

147 (31)

219 (34)

66 (23)

100 (25)

86 (38)

100 (23)

110 (22)

122 (22)

100 (29)

140 (30)

288 (51)

Not avail ab Ie

100 (29)

123 (38)

125 (36)

(21)

* Storage estimates are mean values of 20 replicates and are expressed as percentages of the long term Gould values for the appropriate parameter.

Values in brackets are standard deviations expressed as

percentages of corresponding mean value.

169 In Table 6.5 it is seen that with a decrease in mean flow, storages increase markedly to compensate for the more severe flow conditions.

A 10% decrease in mean results in a 30-40%

increase in storage requirement.

A further decrease in the

mean results in a relatively larger increase in storage. As the variability of the streamflow increases, this effect becomes more significant (for example, the storage increase is larger for the Warragamba river than for the Gordon river). For the case of serial correlation the magnitude of errors in storage is considerably less than for errors in the mean. Increasing the coefficients of skewness in the three parameter log-normal model results in an increase in the means and hence lower storage requirements.

This effect is even more

marked for the two parameter model where the implied skewness (Eq. 5.20) is large, resulting in less storage than required by the three parameter model. 6.5

RECOMMENDATIONS Based on the theoretical considerations presented in earlier chapters

and on the assessment of the methods when applied to a wide range of rivers, certain recommendations can be made.

These are:

Within-year Storage (Critical period less than say 6 months): If low flow frequency curves are available, use the within-year frequency-massed curve procedure (Sec. 3.4.6) and increase the estimated storage capacity by 10%.

If frequency curves are not available, use a

historical behaviour analysis.

Preliminary Design of Large Storage (Critical period say equal to or greater than 6 months): Use Gould's Gamma procedure (Sec. 3.4.4) or McMahon's Empirical equations (Sec. 4.7.1) if Australian streams are being considered.

Adjust

the storage estimate to account for annual serial correlation (Appendix B). Alexander's (Sec. 3.4.2) and Dincer's (Sec. 3.4.3) procedures can be used to estimate the length of the critical period.

Final Design of Large Storage: Use Gould's Probability Matrix procedure modified as recommended to define monthly failures (Sec. 4.6) and adjust the storage estimates to account for annual serial correlation (Appendix B).

170

Alternatively, a behaviour (simulation) analysis using stochastic data could be adopted (Sec. 5.9). In either situation as a check, an historical behaviour analysis with empty and full initial conditions should be carried out (Sec. 3.2.3).

6.6

NOTATION

b

variable in Eqs. 6.2 and 6.3

c

variable in Eqs. 6.2 and 6.3

C

coefficient of variation

d

difference hetween the lower p percentile flow of Gamma distribution G(c) and a Normal distribution N(c,c)

v

D

draft as ratio of mean flow

0'

minimum draft for a% chance based on sampling error analysis (Sec. 6.4)

E

variable in Gould's sampling error analysis (Eq. 6.1)

F

variable in Gould's sampling error analysis CEq. 6.1)

k2

storage ratio defined as reservoir capacity divided by standard deviation of flow

n

number of items of data

p

percentage chance of occurrence

S

storage estimates

Sb

storage estimate based on behaviour single tailed student's t for a% and n-l degrees of freedom standardized normal variate percentage change of occurrence

lIC LIT

V

incremental change in C v incremental change in T

a

standard deviation of monthly flows (Figs. 6.1 to 6.4)

T

reservoir capacity divided by mean annual flow

171 CHAPTER 7

MULTI - RESERVOIR SYSTEMS So far this text has been concerned wi th the storage capaci ty-draftprobability relationship for a single reservoir.

Many water supply systems

however consist of more than one reservoir and for these cases the single reservoir technique is only a very approximate guide to the capacity-yieldprobability relationship of either an independent reservoir or the whole system.

In most cases of multi-reservoir systems, the extent of the

reservoir interconnection either among the storages themselves or at the point of demand

determines whether a single or a multi-reservoir analysis

is most appropriate. 7.1

A TYPICAL PROBLEM The general problem can best be illustrated by reference to a real

situation.

Consider, for example, the Canberra water supply system in

Australia. Canberra, wi th a population of 161 000 in 1971 and with an annual

growth rate in population of 9.5 percent, was served by three headwater storages located in series along the nearby Cotter River.

The locations of

the three reservoirs, the major supply lines and stream gauging stations in relation to the city are shown in Fig. 7.1.

Water from the Cotter reservoir,

the most downstream storage, can only be supplied to the city by pumping. Releases from Corin reservoir supplement the Bendora reservoir, water from •

Cotter Dam

Gl

CANBERRA

,

4.7 Mm 3 480 km 2

9

\ " G4<~;~ \

Proposed

"I

~

Googong Dam\ '; 118 Mm 3 ~ Bendora Dam 10.7 Mm 3 290 km 2

'!. Stream gauging • Station

Carin Dam 75Mm 3 200 km 2

FIG. 7.1

Canberra water supply system.

172 which feeds by gravity through a conduit to the city service basins. Investigation reveals that annual per capita consumption is constant, but seasonal variations from 370 t/capita day during winter to about 1150 t/ capita day during summer can be expected. Relative to many water resources engineering studies, adequate hydrologic data are available. continuous monthly data;

At gauging station Gl, there are 57 years of seven and eight years of data for concurrent

periods are available at sites G2 and G3 respectively.

Monthly rainfall and

pan evaporation data for the urban area are also available for the 57 year period.

On the adjacent Queanbeyan river, 57 years of monthly streamflow

data are also available (site G4 in Fig. 7.1). A preliminary hydrologic and topographic survey indicated that a possible dam site existed at Googong (site G4 in Fig. 7.1).

The designers

then require answers to questions such as the following: (i)

For the projected demand increase with time how long would the system with the added reservoir operate at a given level of reliability?

(ii)

Alternatively to (i), what extra releases can be safely made with the new reservoir in the system?

(iii)

Given

the~esent

system and demands, at what time will its

reliability fall below an accepted design level?

In other

words, when does the new reservoir need to be ready to add to the system? (iv)

How large a reservoir needs to be added to the system to ensure reliability in a given period?

Like any water resources project, an important initial step is to define clearly the objectives and the design criteria.

In this example, the

objectives are engineering ones, namely the design capacity of the reservoir and construction commencement date.

Often the design criteria will be

related to economic factors modified by social and environmental aspects, but for our purposes we assume the design criteria are related to the reliability of supplying the predicted demand given a specific operating policy for the system. 7.1.1

Traditional Solution Traditionally, this problem would have been tackled by mathematical

simulation of the whole system. following components:

The model used would have included the

173

(i)

the streamflow inputs to each reservoir including the proposed one.

If historical data for particular sites

are inadequate then some estimation procedure (for example, regression of streamflows) would be necessary; (ii)

the transfer of water between the reservoirs, and the supply to the point of the demand.

This component would

include as constraints the conduit capacities of the transfer links and pump capacities where appropriate; (iii)

the operating rule for the system (Sec. 2.5) which governs the proportion of the demand to be supplied;

(iv)

a constant annual demand (probably taken as an extrapolated population estimate times a per capita demand, plus an allowance for evaporation losses) broken down into seasonal components.

All these components would be used in the model to simulate the entire system on a monthly basis;

that is, for each reservoir the operation would

be equivalent to a behaviour analysis. The objectives of the simulation study could have been subject to some constraints.

For example, for a given maximum capacity of the proposed

reservoir determined by the site, the system could be modelled to determine the level of demand which could be supplied without failure occurring.

This

in turn would lead to an estimate of the design life of the system from the population growth predictions. If storage capacity was not a limitation, the capacity of the proposed reservoir would have been adjusted until the system could just meet the design demand without failure,that is, the same criterion as used in a mass curve analysis. The commencement date of construction of the proposed reservoir would be estimated from an examination of the growth trend in demand and the potential yield of the original reservoir system, taking due cognizance of the possibility of some severe future drought, the time taken for the reservoir to fill, and of the estimated construction time.

Probability of failure. The description above has given the historical approach to analysis of a reservoir system.

In recent years many water authorities have

accepted the concept of probability of failure as a more useful design standard.

The definition used for failure is not normally that of the

174

system storage being empty (because it is unrealistic), but is more likely to correspond to Eq. 2.18 which is a volumetric reliability (ratio of water supplied to water demanded).

Another definition in use is the proportion

of time that water restrictions are necessary, weighted if desired by the severity of restriction imposed. To simulate the system and compute the probability of failure is similar to the analysis described above.

The only extra computational step

is the keeping of a record of the number of failures (according to the definition adopted) during the simulation. For a demand that is growing with time it is of particular interest to compute the probability of failure versus time relationship.

This would

indicate the point in time where extra storage is going to be needed to maintain the required design reliability of the system.

Since the simulation

or behaviour analysis requires a constant annual demand, it is necessary to repeat the entire simulation for different demand levels and to calculate the probability of failure corresponding to each level.

Then a plot of the

expected system reliability (probability of failure) versus time can be made. 7.1.2

Recycled Historical Sequences The traditional approach does not give the water planner or decision

maker any idea of the level of risk t of the system being unable to meet the design demand during a future drought.

Nor do they get any idea of the

likelihood of the new reservoir being required before the proposed commissioning date. One approach that has been used in recent years to answer these questions is to carry out the

N simulation analyses as described above using

N year historical sequence concatenated with itself and beginning each

separate analysis in a new year of the historical record and continuing for N successive years.

In this way the

historical record provide

N analyses based on

N estimates of storage size.

N years of

In addition, if

demand is superimposed in the form of an annual trend rather than being considered constant at an extrapolated value,

N estimates of the time

tIt is important that the reader not confuse risk with probability of

failure.

Risk is related to the error in the estimate of the parameter

concerned, be it storage capacity for a given draft, or draft for a given storage capacity.

Probability of failure can be defined in several ways,

but in all of them it is a measure of the proportion of system failures which can be expected over an extended period.

175 period from year one to the commissioning date of the proposed dam could be made.

The analysis is completed by ranking the

N values of storage

capacity or time to commissioning, plotting them on probability paper and reading off the value of storage or time to commissioning equivalent to some probability of occurrence.

A major limitation with this type of

approach is that the storage estimates and time values are not independent and the probability associated with any estimate could be in (serious) error. This difficulty is similar to that observed in the overlapping sequence approach as discussed in Sec. 3.4.5.1. Where annual flows are independent, a valid variation of this procedure is to choose at random yearly flows (but at the same time maintaining for each reservoir the same flows in the one year) from the

N years of

data to make up synthetic flow sequences \'ihich are then used as described above.

Such a procedure eliminates dependence but suffers from other limi-

tations;

for example, the synthesized flows are restricted to those

observed in the historical record, and the inflows to each reservoir (and the climatic variable upon which a variable demand would be based) are totally dependent. 7.2

STOCHASTI CALL Y GENERATED FLOWS The inadequacies of the above methods may be overcome by using sto-

chastically generated data for both the inflows and the climatic variables in a manner to preserve the time-series structure of all variables and their cross-correlations. 7.2.1

At least four approaches are available.

Key Station Approach In a key station approach (Thomas and Fiering, 1962), flows are

generated at a key station (chosen because of its record length, reliability or relative significance of a reservoir located on it) by a single site procedure (Chapter 5).

Using this synthetic record as an independent

variable, regression equations are used to estimate flows at all other sites. The regression algorithms include a random term which accounts for the variability between the key and secondary stations.

A basic limitation with

this approach is that it ignores the serial correlation at the secondary station and introduces a spurious serial correlation of value r r2 x y,x where r is the serial correlation at the key station and r is the

x

y,x

cross-correlation between the key station (y) and secondary site (x) (Fiering 1964).

Relative to other limitations in the data generation

process, for example the selection of distribution types (Sec. 5.5), this inadequacy may not necessarily be serious.

176 7.2.2

Principal Component Approach A principal component approach (Fie:ring, 1964) allows gauging station

flows to be linearly combined to represent fictitious gauging station variables.

These variables are orthogonal and consequently the generation

of data at the fictitious sites may proceed independently using single site methods.

After generation the synthetic data are recombined through an

inverse transformation to form generated flows at the original gauging stations.

Matalas (1967) noted two related errors in this technique but

Fiering states that they are only of minor significance (Codner, 1974). 7.2.3

Regression Method The Hydrologic Engineering Center, U.S. Army Corps of Engineers,

recommends a regression method (Beard, 1972).

A separate regression

equation is used to generate streamflows for each calendar month at each gauging station.

Each equation incorporates either a lag zero or a lag one

component (but not both) between the site in question and all other sites. In addition, a random term dependent on the amount of variance not accounted for in the deterministic component is included.

Thus, the generated data

for some stations are mainly dependent on current flows, while those for other stations are primarily related to the previous month's flow.

Other

aspects of the procedure are identical to Beard's single site method discussed in Sec. 5.5.3. 7.2.4

Residual Approach In his residual approach, Matalas (1967) adopted the following

algorithm to generate flows at

m multi-sites: (7.1)

(m x 1) matrices whose elements are standardized flow residuals for (i+l)th and ith consecutive periods, (m x 1) matrix whose elements are random normal deviates, and [AJ, [BJ

(m x m) matrices whose elements consist of a combination of the lag zero and lag one cross-correlations between sites.

Each flow value is related to the flow at all other sites in both the present and preceding time intervals.

The elements of [AJ and [BJ must be

177

defined in such a way that the generated multi-variate sequences resemble the historical multi-variate sequences in terms of mean, variance, skew, serial correlation and lag zero and lag one cross-correlations for all sites.

The derivation of [A] and [B] is given by Matalas.

The solution of

[A) is straightforward but [B] poses difficulties sometimes (Fiering, 1964; Young and Pisano, 1968;

Crosby and Maddock, 1970) .

To apply the Matalas residual method to a seasonal (monthly) problem, two methods are available.

Young and Pisano (1968) first transformed all

flows to a Normal distribution by either taking logarithms or square roots, and removing the seasonal effect by standardizing the transformed flows with respect to the monthly means and standard deviations. then used to calculate [A) and [B).

The data are

Next, flows are generated by Eq. 7.1

and finally, inverse transformations applied.

Theoretically, the method

preserves only the transformed flows but this is not serious. An alternative, theoretically more correct, approach utilizes Matalas moment transformation equations.

The steps in this procedure, which is

more complex than the method of Young and Pisano, are given in McMahon, Codner and Philips (1973). 7.2.5

Multi-site Model Performance ~lul ti-si te

(Beard, 1965;

model performances as presented in the literature

Codner, 1974;

and Fiering, 1962;

Doran et al., 1973;

McMahon, 1971;

Fiering, 1964;

McMahon et al., 1973;

Thomas

Young and

Pisano, 1968) suggest that models generate flows which are of the same order of variability as that found for single-site models.

Generally too, as for

the single site, the theoretically correct models preserve the parameters slightly better than the less rigorous procedures.

However, no detailed

studies have been made comparing and evaluating the various approaches. 7.2.6

Use of Generated Data By replacing the climatic and streamflow variables in the

Recycled Historical Sequence approach (outlined in Sec. 7.1.2) with stochastic data generated by one of the procedures described above, one has available a powerful technique for simulating a water resources system. Wi th each set of data the required information is found (for example, required capacity of added reservoir, time before new reservoir is required) by simulating the multi-reservoir system, accounting for all flow constraints between reservoirs and for seasonal variations in demand.

l7R

At the end of this process there are thus twenty to thirty estimates of the variable in question, enough for a designer to plot a distribution of them and to calculate the confidence limits which surround the chosen value (be it the mean value, or any other). following way:

The answer may be expressed in the

for example, a new reservoir must be added to the existing

system within 2 years ± 3 months to maintain the design reliability of supply.

Contrast this answer wi th a behaviour analysis using historical

data which can only give a single estimate of each variable. It is also worth noting that the problem of initial conditions inherent in the behaviour analysis technique is less important in multireservoir systems.

This is because most applications involve the addition

of a reservoir to an existing system and the actual initial conditions of the system can be specified. 7.2.7

Application to Multi-storage Systems Consider the problem of deciding the construction commencement date

of a headwater storage as outlined at the beginning of this chapter. effect, a multi-site monthly data generation model is required;

In

four stream-

gauging sites (for which flow volume needs to he adjusted to he equivalent to reservoir inflows) and one climatic variable (monthly rainfall), the latter being considered as a fifth gauging station for the purposes of the data generation procedure.

In the original study upon which this discussion is

based (HcMahon, 1971), the key station concept was adopted because at that time the data at two sites were considered to be too short to establish stable seasonal parameters. The system outlined in Sec. 7.1 was simulated using a behaviour approach for each reservoir; monthly inflows and design drafts were generated, and apportionment of flows between reservoirs and drafts to the city were all based on specific operating rules.

Evaporation losses in the

model were taken into account by an equivalent population release.

Given

initial water contents of the three original reservoirs, and some criterion for introduction of another headwater storage (in the original study three criteria were used:

no restriction policy, reliability approach and a

minimum cost function) the time from year zero to when the proposed darn was required could be estimated from the simulation.

In the original analysis

10 and 18 replicates were adopted, but 25 is considered to be a more desirable number.

The results of one analysis are shown in Fig. 7.2.

This

brief illustration emphasizes the usefulness of simulation analysis using stochastically generated data in providing useful design information not possible by traditional procedures.

179 1.0

"~

'5 0-

08

v

'-

C>~ ,S ca v V '..c 0.6 !/)

E

.-

"'" '0

."'= v

>.

Gi

5

0.4

)(

~.D

:0

'"

..c ~

0.2

0..

2 4 6 8 10 Time to when Googong Dam is ready for use (years)

FIG. 7.2

7.3

Simulation result using stochastically generated data. (Curve gives designer a picture of the risks associated with any given completion date).

TRANSITION MATRIX APPROACH An alternative concept to the multi-reservoir system approaches noted

above is based on Gould's probability matrix method (Sec. 4.6).

For a

single reservoir system, it is possible to construct the transition matrix of stored contents by considering mass continuity on a monthly basis.

From

the transition matrix the time variant and steady state conditions of the reservoir can be determined. This concept can be extended to a multi-reservoir system in which the transition matrix is developed for the system as a whole by setting up each element of the transition matrix conditional on the state of all the other reservoirs in the system.

This approach is still being investigated

(Fletcher, personal communication, 1977) but it appears to offer a viable alternative for multi-reservoir analysis to procedures involving data generation.

180 7.4

OTHER ALTERNATIVES Mathematical programming and primarily dynamic programming are

alternatives to simulation, but for complex systems it is necessary to introduce simplifying assumptions often negating the validity of the results for final design, but give useful preliminary estimates.

Readers

interested in this field are referred to Hall and Dracup (1970), Hall and Howell (1970), Asfur and Yeh (1971) and Manoel and Schonfeldt (1977). 7.5

NOTATION

[AJ

(m x m) matrix whose elements consist of a combination of lag-zero and lag-one cross correlations between m sites

[BJ

(m x m) matrix whose elements consist of a combination of lag-zero and lag-one cross correlations between m sites (m x 1) matrices whose elements are standardized flow . . reS1. duals f or (.1+1 ) th an d'1 th consecut1ve per10ds (m x 1) matrix whose elements are random normal deviates

181

REFERENCES Alexander, G.N., 1962.

The use of the Gamma distribution in estimating

regulated output from storages.

Civil Engineering Transactions, The

Institution of Engineers, Australia, CE4(1) : 29-34. Alexander, G.N., 1964. storage capacity.

Effect of variability of streamflow on the optimum In Water Resources Use and Management.

Melbourne

University Press, Melbourne. Anderson, R.L., 1941.

Distribution of the serial correlation coefficient.

Annals of Mathematical Statistics, 8 : 1-13. Asfur, H. and Yeh, W. W-G., 1971. of water resources systems.

An annotated bibliography on the design

University of California Water Resources

Center, No. 134. Australia, Department of National Resources, 1976.

Water Resources, 1975.

Review of Australia's

Australian Government Printing Service,

Canberra. Barnes, F.B., 1954.

Storage required for a city water supply.

Journal of

the Institution of Engineers, Australia, 26 : 198. Beard, L.R., 1965.

Use of interrelated records to simulate streamflow.

Journal of the Hydraulics Division, ASCE, 91 (HY5) : 13-22. Beard, L.R., 1962.

Statistical methods in hydrology.

U.S. Army Engineer

District, Corps of Engineers, Sacramento California. Beard, L.R., 1972.

Hydrologic data management.

In Hydrologic Engineering

Methods for Water Resources Development, Vol. 2, Corps of Engineers, U.S. Army. Benson, M.A., and Matalas, N.C., 1967. regional statistical parameters. Bibra, E.E. and Riggs, H.C.W., 1971.

Synthetic hydrology based on

Water Resources Research, 3 : 931-935. Victorian River Gaugings to 1969.

State Rivers and Water Supply Commission. Brittan, M.R., 1961.

Probability analysis to the development of a synthetic

hydrology for the Colorado River.

In Part IV of Past and Probable

Future Variations in Streamflow in the Upper Colorado River. University of Colorado.

182 Brown, J.A.E., 1961.

Streamflow correlation in the Snowy Mountains area.

,Journal of the Institution of Engineers, Australia, 33 (3) : 85-95. Burges, S.J., 1972.

So~e

problems with log-normal Markov runoff models.

Journal of the HydI'aulics Division, ASCE, 98 (HY9): 1487-1496. Burges, S.3. and Linsley, R.K., 1971.

Some factors influencing required Journal of the HydI'aulics Division, ASCE,

reservoir storage. 97 (7) : 977-991. Chow, V.T. (ed.), 1964.

Handbook of Applied HydI'ology. McGraw Hill,

New York. Codner, G.P., 1974.

Simulation of Melbourne water supply scheme on a

digital computer.

Thesis (Ph.D.), Monash University.

codner, G.P. and McMahon, T.A., 1973.

Log-normal streamflow generation

Journal of the HydI'aulics Division, ASCE.

models re-examined. 99 (HY9): 1421-1431.

Crawford, N.H. and Linsley, R.K., 1966. Stanford Watershed Model IV.

Digital simulation in hydrology:

Department of Civil Engineering,

Stanford University, Technical Report No. 39. Crosby, D.S. and Maddock, T., 1970.

Estimating coefficients of a flow

generator for monotone samples of data.

Water Resources Research,

6 (4) : 1079-1086. Dear10ve, R.E. and Harris, R.A., 1965.

Probability of emptiness III.

Proc. Reservoir Yield Symposium.

(Water Research Association).

Doran, D.G., 1975.

In

An efficient transition definition for discrete state

reservoir analysis: The divided interval technique.

Water Resources

Research, 11 (6) : 867-873. Doran, D.G., Lindner, M.A., and Wright, G.L., 1973. correlated streamflow synthesis.

Spatially and serially

Hydrology Symposium 19?3, The

Institution of Engineers, Australia. Dorfman, R., 1965.

Formal models in the design of water resource systems.

Water Resources Research, 1 (3) : 329-336. Fathy, A. and Shukry, A.S., 1956. long-term storage,

The problem of reservoir capacity for

Journal of the Hydraulics Division, ASCE.

82 (HY5) , Paper 1082. Feller, W., 1951.

The asymptotic distribution of the range of sums of

independent random variables. 22 : 427-432.

Annals of Mathematical Statistics,

183 Fiering, M.B., 1961.

Queueing theory and simulation in reservoir design.

Journal of the Hydraulics Division, ASCE. 87 (HY6): 39-69. Fiering, M.B., 1964.

Multivariate technique for synthetic hydrology.

Journal of the Hydraulics Division, ASCE, 90 (HY5): 43-60. Fiering, M.B., 1967.

Streamflow Synthesis, Macmillan, London.

Fiering, M.B., and Jackson, B.B., 1971.

Synthetic Streamflows, American

Geophysical Union, Water Resources Monograph 1. Gani, J., 1955.

Some problems in the theory of provisioning and of dams.

Biometrica, 42 : 179. Gani, J. and Prabhu, N.U., 1957.

Stationary distributions of the negative

Journal of the Royal

exponential type for the infinite dam.

Statistical Society, Series B, 19 Gani, J. and Prabhu, N.U., 1958.

342.

Remarks on a dam with Poisson type inputs.

Australian JournaZ of AppZied Science, 10 : 113. Gani, J. and Prabhu, N.U., 1959.

The time dependent solutions for a

storage problem with Poisson inputs.

Journal of Mathematics and

Mechanics, 8 : 653. Gani, J. and Pyke, R., 1960.

The content of a dam as the supremum of an

infinitely divisible process.

Journal of Mathematics and Mechanics,

9 : 639. Gani, J. and Pyke, R., 1962.

Inequalities for first emptiness

probabilities of a dam with ordered inputs.

Journal of the Royal

Statistical Society, Series B, 24 : 102. Ghosal, A., 1959.

On the continuous analogue of Holdaway's problem for

the finite dam. Ghosal, A., 1960.

Australian Journal of Applied Science, 10 : 365.

Emptiness in the finite dam.

Annals of Mathematical

Statistics, 31 : 803. Ghosal, A., 1962.

Finite dam with negative binomial inputs.

Australian

Journal of Applied Science, 13 : 71. Gould, B.W., 1961. of dams.

Statistical methods for estimating the design capacity

Journal of the Institution of Engineers, Australia,

33 (12) : 405-416. Gould, B.W., 1964.

Statistical methods for reservoir yield estimation.

Water Research Foundation of Australia, Report No.8.

184 Gould, B.W., 1964.

In Water

Discussion of paper Alexander, 1964.

Resources Use and Management, Melbourne University Press, Melbourne. Gumbel, E.J., 1954.

Statistical theory of droughts.

Proceedings ASCE,

80 : 439.

Water Resources Systems Engineering,

Hall, W.A. and Dracup, J.A., 1970. McGraw-Hill, N.Y. Hall, W.A. and Howell, D.T., 1970. supply.

Optimal allocation of stochastic water

Journal of Irrigation and Drainage Division, ASCE,

96 : 395-402. Hardison, C.H., 1965.

Storage to augment low flows.

Reservoir Yield Symposium. Hardison, C.H., 1972.

In Proceedings of

(Water Research Association).

Potential United States water-supply development.

Journal of the Irrigation and Drainage Division, ASCE, IR3 : 479-492. Hardison, C.H., and Martin, R.O.R., 1963.

Water-supply characteristics of

streams in the Delaware River basin and in southern New Jersey.

Geological Survey, Water Supply Paper 1669-N. Harms, A.G., and Campbell, T.H., 1967.

An extension to the Thomas-Fiering

model for the sequential generation of streamflow.

Water Resources

Research, 3 : 653. Harris, R.A., 1965.

Probability of reservoir yield failure using Moran's

steady state probability method and Gould's probability routing method. Hazen,

~.,

Journal of the Institution of Water Engineers, 19 : 302.

1914.

Storage to be provided in impounding reservoirs for

municipal water supply.

Transactions, ASCE, 77 : 1539.

Hudson, H.E., and Roberts, W.J., 1955.

1952-1955 Illinois drought with

special reference to impounding reservoir design.

Illinois State

Water Survey Bulletin, No. 43. Hufschmidt, M.M., and Fiering, M.B., 1967.

Design of Water-Resource Systems. Hurst, H.E., 1951.

Simulation Techniques for the

Macmillan, London.

Long term storage capacity of reservoirs.

Transactions, ASCE, 116 : 770-799. Hurst, H.E., 1956.

Methods of using long term storage in reservoirs.

Proceedings of the Institution of Civil Engineers, Paper 6059, 5

519.

185 Hurst, H.E., Black, R.P. and Simaika, Y.M., 1965.

Long Tepm Stopage.

Constable, London. Jarvis, C.L., 1964.

Application of Moran's theory of dams to the Ord

JouPnal of HydI'ology, 2 : 232.

River project. Joy, C.S., 1970.

On the relationship between reservoir capacity and yield.

Thesis, (M. Eng. Sc.), Monash University. Joy, C.S. and McMahon, T.A., 1972.

Reservoir-yield estimation procedures.

Civil Engineeping Tpansactions, The Institution of EngineeI's, Australia, CE14 (1) : 28-36. Julian, P.R., 1961.

A study of statistical predictability of stream runoff

in the Upper Colorado River basin.

In Part II of Past and ppobable

Futupe VaY'iations in Stpeamflow in the Uppep Colopado RiVeI'. University of Colorado. Kartvelishvili, N.A., 1969.

Theory of stochastic processes.

In HydI'ology

and RiVer Runoff Regulation, Israel Prog. for Scientific Translations.

The Advanced Theory of Statistics,

Kendall, M.G. and Stuart, A., 1968. Vol. 3, Charles Griffin and Co. King, C.W., 1920.

Supply of water for towns in New South Wales.

Transactions, The Institution of Engineers, Australia, 1 : 262. Kirby, W., 1972.

Computer-orientated Wilson-Hilferty transformation that

preserves the first three moments and the lower-bound of the Pearson Ty e 3 distribution. Klemes, V., 1967.

Watep Resources ReseaI'ch, 8 (5) : 1251.

Reliability of water supply performed by means of a

storage reservoir within a limited period of time.

JouPnal of

Hydrology, 5 : 70. Klemes, V., 1974.

The Hurst phenomenon

a puzzle?

Watep Resources

ReseaI'ch, 10 (4) : 675-688. Klemes, V., 1977.

Discrete representation of storage for stochastic

reservoir optimization. Kottegoda, K.T., 1970.

Mss.

Statistical methods of river flow synthesis for

water resources assessment.

Institution of Civil EngineeI's,

Supplement 18, Paper 7339S. Langbein, W.E., 1958.

Queueing theory and water storage.

ppoceedings,

JouPnal of the Hydraulics Division, ASCE, 84 : Paper 1811.

186 Law, F., 1953.

The estimation of the reliable yield of a catchment by

correlation of rainfall and runoff.

Journal of the Institution of

Water Engineers, 7 (3) : 273. Law, F., 1955.

Estimation of the yield of reservoired catchments.

Journal of the Institution of Water Engineers, 9 Lins ley, R. K. and Franzini, J. B., 1972.

467.

Water Resources

McGraw-Hi 11. Lloyd, E.H., 1963.

A probability theory of reservoirs with serially

correlated inputs.

Journal of Hydrology, 1 (2) : 99-128.

Lloyd, E.H. and Odoom, S., 1964.

Probability theory of reservoirs with

Journal of Hydrology, 2 : 1.

seasonal input. Maass, A., Hufschmidt,

M.~l.,

and Fair, G.M., 1962.

Dorfman, R., Thomas, H.A., Marglin, S.A.,

Design of Water Resources Systems.

MacMillan and Co., 620 pp. McMahon, T.A., 1971. generated data.

Simulation Analysis of a water supply system using

Proceedings, Fourth Australasian Conference on

Hydraulics and Fluid Mechanics, 188-195. McMahon, T.A., 1975.

Predictive use of catchment outputs.

In

Prediction in Catchment Hydrology, (T.G. Chapman and F.X. Dunin, Eds.) Australian Academy of Science, 371-425. McMahon, T .A., 1976.

Preliminary estimation of reservoir storage for

Australian streams.

Civil Engineering Transactions, The Institution

of Engineers, Australia, CE18 (2) : 55-59. McManon, T.A., 1977.

Some statistical characteristics of annual stream-

flows of Northern Australia.

Hydrology Symposium, The Institution

of Engineers, Australia, National Conference Publication No. 77/5, 131-135. McMahon, T.A., 1977.

Australia's surface water resources I - Potential

development based on hydrologic factors. McMahon, T.A. and Codner, G.P., 1973. reservoir capacity.

Mss.

Inadequate hydrologic data and

In Decisions with Inadequate Hydrologic Data

(D.A. Woolhiser, Ed.) Proceedings, Second International Hydrology

symposium, Fort Collins. McMahon, T.A. and Miller, A.J., 1971.

Application of the Thomas and

Fiering model to skewed hydrologic data. 7 (5) : 1338.

Water Resources Research,

187 ~IcMahon,

LA., Codner, G.P. and Joy, C.S., 1972.

Reservoirstorage-yield

estimates based on historical and generated streamflows.

Civil

Engineering Transactions, The Institution of Engineers, Australia, CE14 (2) : 147-152. McMahon, T.A., Codner, G.P. and Philips, C., 1972.

Single and multisite

Nordic Hydrology, 3 : 214-238.

operational hydrology.

McMahon, T.A., Codner, G.P. and Philips, C., 1973.

Manoe1, P.J. and Schonfe1dt, C.B., 1977. expanding water supply.

A note on single and

Nordic Hydrology, 4 : 54-55.

mu1tisite operational hydrology.

Economic optimization of an

HydroZogy Symposium, The Institution of

Engineers, Australia, National Conference Publication No. 77/5, 88-92. Mata1as, N.C., 1967.

Mathematical assessment of synthetic hydrology. Water Resources Research, 3 (4) : 937.

Matalas, N.C. and Benson, M.A., 1968. coefficient of skewness. Melentijevich, M.J., 1966.

Note on the standard error of the

Water Resources Research, 4 (1) : 204-205.

Storage equations for linear flow regulations,

Journal of Hydrohogy, 4 : 201-223. Moran, P.A.P., 1954.

A probability theory for dams and storage systems.

Australian Journal of Applied Science, 6 : 116. Moran, P.A.P., 1955.

A probability theory of dams and storage systems:

modifications of the release rules.

Australian Journal of Applied

Science, 6 : 117. Moran, P.A.P., 1959. Moss, M.E., 1971.

The Theory of Storage.

Methuen, London.

Stochastic methods in urban water systems.

In

Treatise on Urban Water Systems (Albertson, M.L., Tucker, L.S. and Taylor, D.C., Eds.) Colorado State University, Fort Collins, 384-401. Perrins, S.J. and Howell, D.T., 1971.

Neglect of streamflow serial

correlation as a source of error in reservoir yield estimation.

The Institution of Engineers, Australia, Hydrology Papers 1971, 66-70. Phatarfod, R. M.

1976.

Some aspects of stochastic reservoir theory.

Journal of HydroZogy, 30 : 199-217.

188 Philips, C., 1972.

Stochastic analyses and generation of streamflow data.

Thesis (M. Eng. Sc.), Monash University. Prabhu, N.U., 1958 (a).

On the integral equations for the finite dam.

Quarterly Journal of Mathematics, Series 2, 9 : 183. Prabhu, N.U., 1958 (b).

Some exact results for the finite dam.

Annals

of Mathematical Statistics, 29 : 1234. Rippl, W., 1883.

Capacity of storage reservoirs for water supply.

Minutes of Proc., Institution of civil Engineers, 71 : 270-278. Roesner, L.A. and Yevjevich, V.M., 1966.

Mathematical models for time

series of monthly precipitation and monthly runoff.

Hydrology

Paper 15, Colorado State University, Fort Collins, Colorado. Savarenskiy, A. D., 1940.

Metod Rascheta Regulirovaniya Stoka,

Gidrotekhnicheskoe Stroitelstvo 2 : 24-28. Searcy, J.K., 1960.

Graphical correlation of gauging - station records.

Geological Survey, Water Supply Paper l54l-C. Stall, J.B., 1962.

Reservoir mass analysis by a low-flow series.

Journal of the Sanitary Engineering Division, ASCE, 88 (SA5): 21-40. Stall, J.B. and Neill, J.C., 1961. flow analysis.

A partial duration series for low

Journal of Geophysical Research, 66 (12) : 4119-4125.

Sudler, C.H., 1927.

Storage required for regulation of streamflow. Transactions, ASCE, 61 622.

Svanidze, G. G., 1964.

Osnovy rascr.eta regulirovaniya rechnogo stoka

metodom Monte-Karlo (Elements of river runoff regulation computation by Monte Carlo method), Tbilisi, Izdatel'stro Metsniereba. Thorn, H. S. C., 1958.

A note on the gamma distribution.

Monthly Weather

Review, 86 (4) : 117-122. Thomas, H.A. and Burden, R.P., 1963.

Management.

Operations Research in Water Quality

Harvard Water Resources Group.

Thomas, H.A. and Fiering, M. B., 1962.

Mathematical synthesis of stream-

flow sequences for the analysis of river basins by simulations. In Design of Water Resources Systems (A.Maass et al., Eds.) Harvard University. Thompson, R. W. S., 1950.

The application of statistical methods in the

determination of the yield of a catchment from runoff data.

Journal of the Institution of Watel' Engineers, 4 : 394-428.

189

Tintner, G., 1968.

Econometrics.

John Wiley

United States Corps of Engineers, 1975.

&Sons.

Hydrologic Engineering Methods

for Water Resources Development, Vol. 8 Reservoir Yield.

Hydrologic

Engineering Center, Davis, California. Venetis, C., 1969.

A stochastic model of monthly storage.

Water Resources

Research, 5 (3) : 729. Waitt, F.W.F., 1945.

Studies of droughts in the Sydney catchment areas.

Journal of the Institution of Engineers, Australia, 17 (4-5) : 90-97. Wallis, J.R. and O'Connell, P.E.,1972.

Water Resources Research, Wilson, H. McL., 1940.

e

Small sample estimation of p.

(3) : 707-712.

Safe yield from Mundaring reservoir.

Journal

of the Institution of Eng1:neers, Australia, 12 (2) : 39. Wilson, E.B. and Hi1ferty, M.M., 1931.

Distribution of Chi-square.

Proceedings National Academy of Science, 17 : 684-688. Wright, G.L., 1975.

Mu1ti1ag Markov models for eastern Australian streams.

Hydrology Symposium 1975, The Institution of Engineers, Australia. National Conference Publication 75/3. Yevdjevich, V.M., 1961.

Some general aspects of fluctuations of annual

runoff in the upper Colorado River basin.

In Part II of Past and

Probable Future Variations in Streamflow in the Upper Colorado River, University of Colorado. Yevjevich, V.M., 1972 (a).

Probability and Statistics in Hydrology.

Water Resources Publications, Fort Collins. Yevjevich, V.M., 1972 (b).

Stochastic Processes in Hydrology,

Water Resources Publications, Fort Collins. Young, G.K. and Pisano, W.C., 1968. residuals. 909-923.

Operational hydrology using

Journal of the Hydraulics Division, ASCE, 94 (HY4)

190

APPENDIX

A

PROCEDURE TO ADJUST STORAGE ESTIMATE FOR NET EVAPORATION LOSS Prior to dam construction, the long term rainfall-runoff relation for the area which will be flooded by the proposed reservoir can be expressed as follows: P

RB where

B

(Al)

- ETB

RB

mean annual runoff before reservoir inundation,

P

mean annual rainfall, and

B

mean annual evapotranspiration.

ETB

After the reservoir is filled, the relation can be expressed as:

where

(A2)

PA - EO A

RA RA

mean annual runoff after reservoir is filled,

PA

mean annual rainfall, and

EO

A

mean annual evaporation from the water surface.

Assuming that the mean annual rainfall before and after construction remains approximately equal, that is, P

B

= PA, then (A3)

(A4)

noting that nE

~

0, where

nE

= mean annual net evaporation loss.

Lake or open surface water evaporation can be estimated by one of the recognized theoretical or empirical procedures (see Chow, 1964, Section 11) or by applying an annual pan coefficient (p) to tank evaporation data (Ep) , thus: P E

P

(AS)

Pre-dam evapotranspiration estimates are difficult to determine. One apporach is through Eq. Al, thus: (AI)

191

Estimates of mean annual rainfall (P ) are readily available. Mean annual B runoff (R ) can be calculated using data for the catchment or estimated B from regional runoff maps. Another factor that is required is the length of the critical drawdown period.

It can be found from Fig. Al in which the critical period in

years is related to the draft and to the annual coefficient of variation.

100~----~-------+--~~~------~----~

50~----~----~4-------~-----+-

70

0.5~~~+4~----4-------~-----+----~

0.1 L-_ _ _ _L-._ _--'_ _ _-L_ _ _--L_ _....J

o

2

Coefficient of Variation

FIG. Al

Critical period in years versus annual coefficient of variation and draft (Curves are based on Eq. 3.25 and points on Alexander's method - Sec. 3.4.2)

192

This figure was derived from Eq. 3.25 and computations were carried out for a probability of failure of 5%.

As an approximate check on the relationship,

values for 50% and 70% drafts are superimposed using Alexander's procedure (Sec. 3.4.2).

The fit is satisfactory.

It should be noted that estimates

of critical periods of one year or less will not be very reliable. Thus the final adjustment factor for net reservoir evaporation loss is given by combining mean annual net evaporation loss, mean surface reservoir area and drawdown period as follows: 0.7 A LIE CP where LISE A

(A6)

amount that computed storage needs to be increased to account for the net reservoir evaporation loss, (m 3) surface area of the reservoir at full supply level, (m 2)

LIE

is determined from Eq. A4, (m/year)

CP

is estimated from Fig. AI. (years).

The factor 0.7 relates the mean reservoir surface area exposed to evaporation during a critical drawdown period to the total area at full supply level.

This factor is based on an analysis of the storage capacity-surface

area curves for six Australian dam sites:

Bendora, Corin and Cotter dams

on the Cotter River (N.S.W.), proposed Sites 10 and 15 on the Mitchell River (Vic.), and the Talbot dam site on the Thomson River (Vic.). NOTATION A

surface area of reservoi.r at full supply level

CP

critical period

E

tank evaporation data

EO

mean annual evaporation from reservoir water surface

P

pan coefficient

P

A ETB

mean annual evapotranspiration

PA' PB mean annual precipitation equivalent mean annual runoff from area inundated by reservoir RA mean annual runoff from area of proposed reservoir RB fiE

mean annual evaporation loss

LISE

increase

to account for the net reservoir evaporation loss

193

APPENDIX

B

ADJUSTMENT FOR ASSUMPTION OF INDEPENDENCE OF ANNUAL FLOWS No comprehensive study of the effect on the reservoir capacity-yield relationship of the assumption of independence of annual flows is generally avai lab Ie.

However, several incomp lete studies (in the sense of covering

the needs of users of this text) are available and the results of these are summarized below under two headings - the effect of neglecting serial correlation on the required storage capacity and the effect on yield. Increase in storage required to compensate for the neglect of serial correlation The results of five studies (Thomas and Burden 1963, Gould 1964, Joy 1970, Perrins and Howell 1971 and McMahon and Codner 1973) which examine the quantitative increase in storage required to compensate for the independence assumption are summarized in Tables Bl, B2 and B3.

Results are

for three levels of serial correlation 0.1, 0.2 and 0.3 and for a range of models.

For the studies of Thomas and Burden, Gould, and Perris and Howell

results have been generalized. In looking over the tables, it can be seen that there are discrepancies betl"een studies (and wi thin studies).

Al though these are probably

attributable to the various distributions, parameters, and definition of probability (or reliability) used in the analyses, it is difficult for the user to isolate the adjustment factor for a particular river. A recommended alternative to Tables Bl to B3 is Fig. B.l which shows the increase in storage relative to the increase in serial correlation. The lines in this figure are based on the average reservoir capacity computed for 156 Australian streams using an historical behaviour analysis assuming the reservoir to be alternatively initially empty and initially full.

This average value was divided by the equivalent Gould estimate

modified for monthly failures and the resulting ratio, which is the required correction factor, was plotted against the annual serial correlation. Separate analyses were carried out for 50% and 90% drafts but only one probability of failure condition of 5% was examined (McMahon, 1976).

194 TABLE Bl.

Storage adjustment factors (percentage increase required) to account for assumption of independence in reservoir capacity-yield analysis for serial correlation of 0.1.

Probabili ty of failure

Streamflow characteris ti cs

Draft (%) 100

90

80

60

70

50

Thomas and Burden (1963) - theoretical simul ation study 1/25 years 1/50 years 1/25 years 1/50 years Mc~lahon

Normal distribution Cv = 0.25

}

Gamma distribution Cs = 1

}

8

0*

0*

5

13*

0*

8

0*

0*

11

17*

0*

and Codner (1973) ** - simulation of four Australian rivers

5% probability (monthly) LN -2 distribution

Gould (1964)

-

0.7

8

Cv = 1.0 Cv = 1.9 Cv = 2.1

6

['"

10 23

theoreti cal simulation study (results generali zed with respect to skewness)

5% probability (annual) LN-3 distribution

r

I

0.5

<= 1 Cv = 2

I

20

I

I

* The results marked by an asterisk are based on storage estimates reported only to one significant figure.

In these cases the

adjustment factor could be in error by up to 25%. **

This study was based on a seasonal monthly model and adjustments were made to the monthly serial correlations rather than the annual one.

195 TABLE B2.

Storage adjustment factors (percentage increase required) to account for assumption of independence in reservoir capacity-yield analysis for serial correlation of 0.2.

Probabi l i ty of of failure

Streamflow characteristics

Draft (%) 100

90

80

70

60

50

Thomas and Burden (1963) - theoretical simulation study 1/25 years 1/50 years 1/25 years 1/50 years

Normal distribution = C 0.25 v

}

Gamma distribution C 1 s =

}

17

17*

0*

10

25*

0*

25

0*

0*

16

33

0*

McMahon and Codner (1973)** - simul ation of four Australian rivers 5% probabi Ii ty (monthly)

LN -2 . .

dls~n-

butlon

Gould ---

(1964)

=

5% probability (annual)

Joy (1970)

-

r

0.7 v C = 1.0 v C = 1.9 v C = 2.1 v

r

5 14 22 25

theoretical simulation study (results generalized with respect to skewness) LN-3 distribution

v ' 0.5 C'v = I C = 2 v

I

45

I

I

I

theoretical simulation study using Lloyd's model (1963 )

S% probabi l i ty

Normal distribution C = 0.4 v

22

* The results marked by an asterisk are based on storage estimates reported only to one significant figure.

In these cases the

adjustment factor could be in error by up to 25%. **

This study was based on a seasonal monthly model and adjustments were made to the monthly serial correlations rather than the annual one.

196

TABLE B3.

Storage adjustment factors (percentage increase required) to account for assumption of independence in reservoir capacity-yield analysis for serial correlation of 0.3. Draft (%)

Streamflow characteristics

Reliability

100

90

80

60

70

50

Pcrrins and Howell (1971) - theoretical simulation study 80%

Normal

95%

distribution

99%

C

v

=

33 58

57

0.5

48

lbe fit between the results for the five studies and the lines in Fig. B.l is poor.

This is because the individual studies are for specific

cond; tions whereas the lines arc generalized results from a range of streams and conditions.

,.....

Draft 90% 50%

(,)

...- _-Appendix B

*

*

0

181

A

• 0

Gould 1964 Thomas & Burden 1963 McMahon & Codner 1973 Joy 1970 Perrins & Howell 1971

1.8

...

0

+'

90%

(,)

ell

u..

1.6- c: 0

'';:; (,)

Q)

1.4-: 0

()

1.2

* 0.1

0.2

0.3

0.4

0.5

--11-+---+---; Serial Correlation

0.8

FIG. B.l

I

Reservoir-capacity correction factor for annual serial correlation.

197

Decrease in yield required to compensate for the neglect in serial correlation Hardison (1965) carried out a simulation analysis using a twoparameter log-normal distribution to assess the effect of neglecting serial correlation on yield.

Figure B.2 shows for a 2% chance of deficiency the

adjustment that should be subtracted from the allowable draft computed with no serial correlation between years. Perrins and Howell (1971) provide some data but it is limited because the simulation runs are based on a normally distributed Markov model with C

v

0.5.

I

~j I

,,

~~ 0.6 Index of Variability

o

023

0.49

0.78

1.16

1.66

Coefficient of Variation

FIG. B. 2

Effect of serial correlation on draft at 2% chance of deficiency for log-normal distributions of annual discharge for indicated serial correlation (Hardison, 1965). (The parameter shown is the length of the critical period in years. The adjustment should be subtracted from the allowable draft computed with no serial correlation between years. )

198

NOTATION C

coeffjcient of variation

C

coefficient of skewness

LN-2

2 parameter Log-normal model

LN-3

3 parameter Log-normal model

v s

19!

APPENDIX C

THEORETICAL JUSTIFICATION 01' A NON-SEASONAL MA.RKOV MODEL

Expectation theory is used to justify the form of the algorithm of the Markov model. (i)

Consider a model of the form: X+b(X.-X)

(Cl)

1

where

generated flows,

\+1' Xi X

mean flow, and

b

regression coefficient between ith and (i+l)th flows.

The expected value or mean of \+1' E(\+l)' is given by: E[X + b(\- X)] E(X) + b E(X ) i

b E (X)

X+bX-bX

X

(C2)

Therefore, the mean is preserved by this type of generation model. Consider now the variance. E(X?l+ 1) - [E(X.1+ 1)]2 E[ (X + b (X. - X)) 2] 1

(C3)

F

X) + b 2 (X.

E(F + 2Xb (X.

1

1

-

X) 2)

X2

E(X2 + 2bXX. - 2bX 2 + b 2X?- 2b 2 X.X + b 2X2) 1 1

E (F) + 2bXE(\) + b 2E (F)

-

1

2bE (X2) + b 2 E (Xf)

-

F

2b 2X E (Xi)

X2

X2 + 2bX2

(C4)

but

b

si+l r -s-.1

and si+l

s.

1

=

5

200

standard deviation of (i+l)th and ith flows, and r

correlation coefficient between ith and (i+l)th flows.

=

(e5)

Therefore, the model preserves variance when r r f ± 1.

±

1, but not when

Consequently, the model is unsatisfactory.

(ii)

Consider now a model of the form:

x + b(X i where

t

- X) +

S

tiel

(C6)

N(O,l) and

i

standard deviation of flows.

s

This is Brittan's model as given in Sec. 5.3.

The expected value of

Xi+l is

,

E[X + b(\- X) + s ti (1

E (Xi +1)

r2) 2]

,

(C7)

b E eX) + s (1 - r 2)2 E(t.)

E (X) + b E(X ) i

1

X

(C8)

Therefore, mean is again preserved. The variance is again computed as: E(X?l+ 1) - [E(x.l+ 1)]2 Let

(C9)

c

X) + c t 2 E[X2+ b [X. - X)2 + E[X

+ b(\1

i

X2

]2 -

c 2t 2 + 2bX(X.- X) 1

1

+ 2c X t. + 2bc (X.- X) t.] - x2 1

1

1

X2 + b 2E (X?) - 2b 2 X E (X.) + b 2E(X)2 1

1

+

c 2 E(t?) 1

+ 2bXE[X - X) + 2cX E(t ) + 2bc E(t ) E(X ) i i i i - 2bc E(t ) E(X) - X2 i b 2E (X?) - b 2X2 + s2 [1 1

b 2 [E(x2) - X2] + s2[l

(ClO)

1

[Recall that s 2

[[(X2) - X2) and that b 1 r 2s2 + s2 _ s2r2 52

Therefore, the variance is preserved.

= r]

(ell)

(C12)

201 Now check that dependence is preserved as lag one serial correlation (r). r

By definition, COY (Xi+l \)

(VAR X.1+ 1) ~ (VAR Xi) COY (X i + Xi) l 2 s

2

where COY (X i +l Xi) = covariance of s2r

r

(C13)

~

(C14) Xi and Xi+l

COY (X i +l Xi)

(CIS)

X) ] E[(\+l- X) (X.1 X E (Xi) + X2 E(X i +l Xi) - X E(X i +l ) E[X (X + b(X.- X) + ct.)] - x2 i 1 1 E(X X. + bX.1 - bX X. + c X t i ) - x2 1 1 x2 + b E(X?) - b X2 - x2 1 b [E eX?) - x2] 1 b s2

(C16)

(C17)

b

(C18)

Therefore, dependence is preserved. Thus the model preserves the three parameters:

mean, standard

deviation and serial correlation and is therefore in theory a satisfactory generation model. NOTATION regression 1 s(l - r2)2

b

c

coeffici~nt

E (x)

covariance between X th and x.th flows i+l 1 expected value of x

N(O,l)

normal distribution with mean zero and variance equal to one

r

serial correlation coefficient

COV(\+l'\)

S.

standard deviation of flows in ith time period

t.

normal random variate

VAR(X ) i X

variance of flow sequence

1 1

mean flow generated flows

X.

1

202

APPENDIX

D

NEWTON-RAPHSON METHOD FOR SOLVING FOR AN INEXPLICIT VARIABLE Given f(x)

=

0, to solve for

Assume first trial solution is

x. x .. 1

Application of the Newton-Raphson method gives an estimate of a better value of

xi'

Call it

x i +l '

f(xl f(Xjl I------~

x

FIG. D.I

Illustration of Newton-Raphson Solution. d ~;x) = slope, of tangent at x

X.

1

(Fig. D.1)

(DI) (D2)

(D3) Thus

xi+l

=

Xi -

f(x i )

(D4)

~ 1

If

xi+l - Xi

exceeds the allowable error, then use

xi+1

to find

another estimate xi+2' Equation 5.16 can be treated as a function of S

thus:

f(S2) = exp (3S 2) - 3 exp (S2) + 2 - g[exp (S2)_ 1]3/2 and

(DS)

20:

APPENDIX

TABLE E1.

Year

E

Monthly flows for ~1itta Mitta River at Ta11andoon (Station 4012(4), Victoria, Australia, for period 1936-1939 inclusive. Units are 10 6m3 .

Jan. Feb. Mar. Apr. May

June July Aug. Sep. Oct. Nov. Dec.

Total

32 27 16 44

32 26 15 179

38 20 20 130

31 27 26 94

113 28 44 183

189 32 47 179

529 54 58 395

217 171 91 318

152 125 52 363

80 56 19 276

84 31 9 99

1553 650 413 2266

19

14 46 12 15 12

33 27 12 76 17

44 20 112 51 64

44 32 149 52 39

42 101 347 110 64

60 63 215 139 43

93 100 316 201 39

58 136 232 241 46

31 52 149 113 26

28 23 64 52 17

509 710 1634 1107 401

21

7 69 35 22 33

14 47 30 23 28

16 44 36 67 32

56 91 80 69 52

42 444 253 59 95

154 302 237 81 117

146 164 276 126 174

101 162 300 158 236

89 109 185 252 238

30 58 94 65 86

685 1548 1578 1012 1151

35 41 28 84 59

49 25 16 48 65

67 19 22 28 28

130 36 49 26 30

44 100 120 44 46

49 159 534 58 64

79 297 312 158 68

113 321 207 253 149

164 250 472 297 122

220 253 260 338 80

167 126 349 195 249

73 63 241 84 153

1190 1690 2610 1613 1113

1955 1956 1957 1958 1959

53 139 54 43 32

56 64 36 22 26

53 88 38 17 36

31 481 32 21 44

48 414 42 89 23

121 548 65 105 39

180 513 117 191 42

638 456 69 471 96

417 402 69 165 245

449 382 132 426 211

241 231 60 154 96

123 116 43 72 46

2410 3834 757 1776 936

1960 1961 1962 1963 1964

23 37 32 44 17

15 17 19 28 14

11

20 33 16 15 22

148 32 41 49 26

112 41 139 46 80

217 78 86 60 451

279 111 144 141 271

223 139 127 163 305

218 95 169 137 421

132 58 90 105 178

75 54 53 46 91

1473 717 928 850 1888

1965 1966 1967 1968 1969

32 15 58 7 35

15 14 22 2 20

16 12 15 6 42

19 25 15 80 43

22 44

28 68 20 51 178

67 136 35 222 132

152 212 52 155 197

78 242 91 342 115

62 152 20 163 75

48 204 10 73 62

553 1139 369 1230 1010

1936 1937 1938 1939

56 53 16 6

1940 1941 1942 1943 1944

43 88 14 37 22

1945 1946 1947 1948 1949

14 14 30 57 39

16 44 22 33

1950 1951 1952 1953 1954

22

12 20 12

22

12 16 12 14 15 16 1 27

15

128 84

Extracted from Bibra, E.E. and Riggs, H.C.W. (1971): Victorian River Gaugings to 1969.

State Rivers and Water Supply Commission.

204 TABLE E2.

Month

Monthly flow parameters for Mitta Mitta River at Tallandoon (1936-69).

Mean Flow (lOD m3)

Standard Deviation (l06 m3)

Coefficient of Variation

26.4

0.66

Coeffi cien t of Skewness

Serial Correlation

January

39.9

February

26.9

15.4

0.57

1.1

0.60

March

31.1

32.3

1.03

3.3

0.54

0.58

1.8

April

46.8

81. 7

1. 74

4.9

0.85

May

62.1

70.1

1.13

4.1

0.81

June

102.5

118.9

1.16

3.2

0.63

July

152.9

132.2

0.86

1.3

0.59

August

198.8

151. 4

0.76

1.3

0.65

September

198.8

103.5

0.52

0.89

0.73

October

206.5

115.8

0.56

0.58

0.65

November

134.6

83.1

0.62

0.62

0.78

December

72.6

49.9

0.69

1.7

0.61

TABLE E3.

Annual flow parameters for Mitta Mitta River at Tal1andoon (1936-69).

Mean Flow (l06 m3)

Standard Deviation (l06 m3)

Coe f fi ci en t of Variation

Coefficient of Skewness

Serial Correlation

1274

731

0.57

1. 50

0.06

205

AUTHOR INDEX

Alexander, G.N. 39-45, 67, 147, 149, 151, 159-60, 169, 181, 191 Anderson, R.L. 118 Asfur, H. 180, 181 Australia, Department of National Resources. 11, lSI

Franzini, J.B. 15, 186 Gani, J. 71,72,183 Ghosa1, A. 72, IS3 Gould, B.W. 3, 49-50, 67, 83-91, 105, 140-2, 152-169, 179, lS3-4, 193-6

Barnes, F.B. 28, 110, 112, 181

Guglij, G.

Beard, L.R. 57, 121-2, 129, 132-4,

Gumbel, E.J. 60, lS4

139, 159, 176-7, 181 Benson, M.A. 135, 181, 187 Bibra, E.E. 181, 203 Black, R.P. 185 Brittan, M.R. Ill, 181 Brown, J.A.E. 7, lS2 Burden, R.P. 33, 115-6, 188, 193-5 Burges, S.J. 137, 182 Campbell, T.H. 124-5, 184 Chow, V.T. 121, 182, 190 Codner, G.P. 147, 176-7, 182, 186,

140, 144, 159, 163-4

Hall, W.A. 180, IS4 Hardison, G.H. 7, 15, 46, 59, 62-6, 97-100, 105, 159, 161, 163, 184, 197 Harms, A.G. 124-5, 184 Harris, R.A. 82, 83, 182, 184 Hazen, A. 27, 110, 184 Hilferty, M.M. 115, 122 Howell, D.T. ISO, 184, 187, 192, 196- 7 Hudson, H.E. 55, 184

187, 192-5

Hufschmidt, M.M. 135, 184, 186

Crawford, N.H. 7, 182

Hurst, H.E. 31-32, Ill, 184-5

Crosby, D.S. 177, 182

Jackson, B.B. 107, 183

Dear1ove, R.E. 83, 182

Jarvis, C.L. 82, 185

Dincer, T. 46-7, 49, 51, 67, 159-60,

Joy, C.S. 32, 33,90, 91, 147, 185,

163, 169 Doran, D.G. 83, 91, 177, 182 Dorfman, R. 135, IS2, 186 Dracup, J .A. ISO, lS4

lS7, 193-6 Julian, P.R. Ill, ISS Kartvelishvili, N.A. 144, ISS Kendall, M.G. 114, ISS

Fair, G.M. 186

King, C.W. 21, 185

Fathy, A. 33, IS2

Kirby, W. 116, 132, 185

Feller, W. 31, lS2

K1emes, V. 91, 102, 107, 185

Fletcher, S.J. 179

Kottegoda, K. T. 108, 109, 125, 185

Fiering, M.B. 17, 33, 35, 107, 114-5, 124, 126, 135, 158, 176, 176-7, 183, 184, 188

Langbein, W.B. 96-97, 185 Law, F. 67, 186 Lindner, M.A. 182

206 Linsley, R.K. 7, 15, 137, 182, 186

Stall, J.B. 55, 151-2, 188

Lloyd, E.H. 82, 83, 186

Stuart, A. 114, 185

Maas, A. Ill, 135, 186 McMahon, T.A. 15, 25, 44, 93-5, 105,

Sudler, C.H. 110, 188 Svanidze, G.G. 140, 144, 159, 188

116, 126, 134, 147, 153, 159,

Teoh, C.T. 90, 147, 159

162-4, 169, 177, 178, 182, 185,

Thorn, II.S.C. 41, 188

186, 187, 193-6

Thomas, H.A. 33, 114-6, 124, 126,

Maddock, T. 177, 182

158, 175, 177, 186, 188, 193-5

Manoe1, P.J. 180, 187

Thompson, R.W.S. 53, 147, 150-1, 188

Marglin, S.A. 186

Tintner, G. 114, 189

Martin, R.O.R. 59, 184 Matalas, N.C. 117-8, 125, 135, 176, 181, 187 Me1entijcvich, M.J. 101, 187

United States Corps of Engineers, 20, 57, 58, 121, 176, 189 United States Geological Survey, 60, 61

Miller, A.J. 116, 186 Moran, P.A.P. 3, 71-3, 81-3, 90, 96, 187

Venetis, C. 83, 189 Waitt, F.W.F. 33, 36-8,147,150-1,

Moss, M.E. 35, 187 Neill, J .C. 55, 188

189 Wallis, J .R. 11, 125, 189 Wilson, E.B. 115-6, 122, 189

O'Connell, P.E. 11, 189 Odoom, S. 82, 186 Pcrrins, S.J. 187, 193, 196-: Pisano, W.C. 177, 189 Phatarfod, R.M. 102-3, 116, 187 Philips, C. 109, 177, 187, 188 Prabhu, N.U. 71, 72, 183, 188 Pyke, R. 71, 183 Rippl, W. 20, 188 Riggs, H.C.W. 181, 203 Roberts, W.J. 55, 184 Roesner, L.A. 83, 188 Savarenskiy, A.D. 3, 188 Schonfeldt, C.B. 180, 187 Searcy, J.K. 7, 188 Se1va1ingam,

S. 140, 144

Shukry, A.S. 33, 182 Simaika, Y.M. 185 Srikanthan, R. 35, 126, 130

Wilson, H. McL. 67, 189 Wright, G.L. 109, 182, 189 Yeh, W. W-G. 180, 181 Yevjevich, V.M. 83, 109, 111, 188, 189 Young, G.K. 177, 189

207

SUBJECT INDEX

Active storage 14 Alexander's method 38, 67, 169 evaluation of 149, 159 Annual flows parameters 13 generating model for III ARIMA model 125 Beard's normalizing procedure 121, 132 Behaviour analysis 24, 169 comparison with Gould method 153 for multi-reservoir systems 172, 174, 178 use with generated data 155, 170 Broken line model 125 Carryover frequency mass curve analysis 52 Carryover storage 14 and seasonal storage 65 Classification of reservoir capacity-yield procedures 3 Moran derived methods 71 Coefficient of skewness 9 Coefficient of variation 9 effect on zone size (Gould) 91 Conceptual storages 14 Critical drawdown period 20 Critical period 19 and evaporation loss 191 Critical period techniques 3, 19 evaluation of 147 Cross-nesting of mass curves 64, 68 Design preliminary 2, 169 procedures in current use 5, 173 final 2, 169 Design process 2 Demand 15 allowance for increase in 174

208 Dincer's method 46, 67, 169 evaluation of 159 Distribution of flows 7, 44 Gamma 40, 49, 72, 103 Gumbel (Extreme-value) 60, 137 like Gamma 115,129,132 Log-normal 97, 117, 129, 132, 142, 168 Normal 46, 49, 97, 110, 132 Weibu11 97, 132 Draft 15 sampling error of 164 Drought curves 36, 42 Dynamic programming 180 Evaporation loss, adjustment for 190 Extreme-value distribution 60, 137 Failure, probability of (see Probability of failure) Final des ign 2 recommended procedures 169 Finite storage 14 Flow parameters (see Inflows, Distribution of flows) Fractional Gaussian Noise model 117 Frequency distributions of parameters 7 Gamma distribution 40, 49, 72, 103 shape parameter for 41, 49 Gamma units 49 Generated data (see also Stochastic data generation) comparison with historical data 126 negative flows 112 when and how to use single reservoir 136 multi-reservoir systems 177 Gould's Gamma method 49, 67, 169 evaluation of 159 use for determination of sampling error 165 Gould's Probability Matrix method 83, lOS, 169 effect of starting month 91 effect of zone size 90 evaluation of 152 failure vector for 87

209 for multi-reservoir systems 179 transition matrix for 84 use with generated data 155 Gould's Synthetic Data procedure 140 evaluation of 159 use for determination of sampling error 164 Gug1ij 's method 144 evaluation of 159 Gumbel distribution 60, 137 Hardison's method for combination of carryover and seasonal storage 65 for probability routing 97, 105 evaluation of 159 Hurst phenomenon 31, 108 importance of 125 Hurst's storage procedure 31, 67 Independent series 55 evaluation of 151 vs. overlapping series 57 Index of variability 9, 98 Infinite storage 14 Inflows 6 and outflows, occurrence of 72, 78 distribution of 7 (see also Distribution of flows) estimated 7 generated 107, 111, 114 sampling error of 6, 11 statistical parameters of 7, 11, 108 Initial conditions effect of full assumption 25 independence of effect of 82 Kirby's transformation 116, 132 Langbein's Probability Routing method 96 Like Gamma distribution 115, 132 Log-normal distribution 97, 117, 129, 132, 142, 160 Low flow frequency curves 59, 65, 169 bias in 62, 64

210

Low flow sequences independent series 55 methods based on 36 minimum flow approach 36 overlapping series 52 ~!C~lahon

I

s empirical equations 93, 105, 169

evaluation of 159 Markov model Ill, 130, 199 ~lass

curve method 20, 67 evaluation of 147 frequency analysis 52, 59 bias 64 cross-nesting 64, 68 res idual 22, 67

Matalas transformation equations 117 multi-site 176 Mean of flows 7 ~ledian

of flows 7

Moment transformation equations 117 Monte Carlo model, use of 110, 144 Moran probabi Ii ty matrix procedures 3 general classification of 71 mutually exclusive model 73 equations for general case 76 transition matrix for 74 simultaneous model 78 Mul ti-reservoir probl em 2, 171 flow generation for 175 transition matrix approach 179 system reliability 172 Mutually exclusive model (Moran) 73 Negative generated flows Newton-Raphson method 117, 201 Noah and Joseph effects 108 Normal distribution 46, 49, 97, 110, 132 Normalizing flows 121, 132 Operating rule 16 for multi-reservoir systems 178 Outflows 15

211

Overlapping series 52, 68 clustering 53 evaluation of ISO vs. independent series 57 Performance index (Fiering's) 17 Persistence 109 measure of 10, 109 Phatarfod's procedure 102 Preliminary design 2 recommended procedures for 169 Principal component method 176 Probabi 1 i ty mat rix methods 71 evaluation of 147 Probability of failure and risk 174 annual vs. monthly 87 definition of 16 for multi-reservoir systems 173 Probabili ty routing 96 (see llardison' s method) Random walk 101, 102 Range, definitions of 30 Recycled historical sequences 174 Regional within-year storage estimates 63 Regression method for multi-site generation 176 Regulation 15 Release 15 rule 16 Reliability 17, 172 volumetric 17 Residual mass curve 22 Restrictions 16 Risk definition of 174 of storage estimate being low 137 Samp Ii ng error of draft estimate 164 of flows 6, 11 of storage estimate 164 Seasonal storage 14, 65

212 Semi-infinite storage 14, 27 Sequent peak algorithm 33 use of 67, 132 Serial correlation adjustment for effect of 169, 193 confidence limits for 109 definition of 10 preservation by generating models 130 significance 44, 109 Sequences low flow 36 independent 55, 151 overlapping 52, ISO recycled historical 174 Shape parameter, Gamma distribution 41, 49 Simulation 135 (see Behaviour analysis) Simultaneous model (Moran) 78 Skewness measures of 9 modifications to cater for 115, 117, 121 preservation by generating models 129 Spill IS Standard error of parameters 11 Stochastic data generation multi-site models 175 key station method 175 performance of 177 principal component method 176 regression method 176 residual approach 176 single site models annual flows 110 evaluation of storage estimates using ISS generalized results using 140 modifications for non-normal flows 115, 117, 121 monthly 114 number of sequences required 139 theoretical justification of 199 two tier model 124 verification and performance 126

213

Steady state probability 79 Storage, definitions of 14 Storage estimates effect of record length 158 effect of serial correlation 169, 193 rapid procedures for 159 sampling error of 164 Svanidze's method 144 evaluation of 159 Thomas and Fiering mode 1 114, 126 theoretical justification of 199 Time interval 6 Time series components of flows 108 Transition matrix Gould's method 84 Moran model 74, 78 steady state 76, 79 multi-reservoir systems 179 Two tier model 124 Variability, measures of 9 Waitt's procedure 36, 67 evaluation of 147 Weibull distribution 97, 132 Wilson and Hilferty transformation 115 Within-year frequency mass curve analysis 59 Within-year storage 14 regional estimates of 63 Yield 15 adjustments for serial correlation 197

This Page Intentionally Left Blank