Reservoir Capacity and Yield 0444416706, 9780444416704, 9780080870007

Table of contents :
Content:
Edited by
Page iii

Copyright page
Page iv

Preface
Page v
T.A. McMahon, R.G. Mein

Chapter 1 Introduction
Pages 1-5

Chapter 2 Definition of Terms
Pages 6-18

Chapter 3 Critical Period Techniques
Pages 19-70

Chapter 4 Probability Matrix Methods
Pages 71-106

Chapter 5 Use of Stochastically Generated Data
Pages 107-146

Chapter 6 Quantitative Assessment of Capacity - Yield Techniques for Single Reservoirs
Pages 147-156,156a,158-170

Chapter 7 Multi - Reservoir Systems
Pages 171-180

References
Pages 181-189

Appendix A
Pages 190-192

Appendix B
Pages 193-198

Appendix C
Pages 199-201

Appendix D
Page 202

Appendix E
Pages 203-204

Author Index
Pages 205-206

Subject Index
Pages 207-213

Citation preview

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

-3

A

5000

E

"

.;:; :J

III

'"c:

0

/ /

c:

/ /

.E'"

/

"",e

~ 0

:(\ \V\ /K ~t?v~# ;1 \ V\ ) /V)~0~v

en Q) a: en cu

~O ~~

~",v ~KI"\Y/Kl>

"'.§

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
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:

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

.:::.

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

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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