Review of Gould–Dincer reservoir storage–yield–reliability estimates

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Advances in Water Resources 30 (2007) 1873–1882

Review of Gould–Dincer reservoir storage–yield–reliability estimates

Thomas A. McMahon a,*, Geoffrey G.S. Pegram b, Richard M. Vogel c, Murray C. Peel a

a Department of Civil and Environmental Engineering, The University of Melbourne, Vic., Australiab Civil Engineering Programme, University of KwaZulu-Natal, Durban, South Africa

c Department of Civil and Environmental Engineering, Tufts University, Massachusetts, USA

Received 9 November 2006; received in revised form 8 February 2007; accepted 8 February 2007Available online 21 February 2007

Abstract

The Gould–Dincer suite of techniques (normal, log-normal and Gamma), which is used to estimate the reservoir capacity–yield–reli-ability (S–Y–R) relationship, is the only known available procedure in the form of a simple formula, based on annual streamflow sta-tistics, that allows one to compute the S–Y–R relationship for a single storage capacity across the range of annual streamflowcharacteristics observed globally. Several other techniques are available but they are inadequate because of the restricted range of flowson which they were developed or because they are based on the Sequent Peak Algorithm or are not suitable to compute steady-statereliability values. This paper examines the theoretical basis of the Gould–Dincer approach and applies the three models to annualstreamflow data for 729 rivers distributed world-wide. The reservoir capacities estimated by the models are compared with equivalentestimates based on the Extended Deficit Analysis, Behaviour analysis and the Sequent Peak Algorithm. The results suggest that, inthe context of preliminary water resources planning, the Gould–Dincer Gamma model provides reliable estimates of the mean first pas-sage time from a full to empty condition for single reservoirs. Furthermore, the storage estimates are equivalent to deficits computedusing the Extended Deficit Analysis for values of drift between 0.4 and 1.0 and the values are consistent with those computed using aBehaviour simulation or a Sequent Peak Algorithm. Finally, a sensitivity analysis of the effect on storage of the four main streamflowstatistics confirms that the influential ones are mean and standard deviation, while effects of skew and serial correlation are orders ofmagnitude lower. This finding suggests that the simple reduced form of the Gould–Dincer equation may profitably be used for regionalstudies of reservoir reliability subject to climate change scenarios based on regional statistics, without having to perform calculationsbased on time series, which may not be easily obtained.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Reservoir theory; Storage–yield analysis; Dincer; Gould–Dincer

1. Introduction

During the preliminary design phase of a waterresources development or for a reconnaissance reviewof the yield from one or more reservoirs, it is useful tohave available a technique that one can use to examinethe reservoir capacity, yield and reliability (S–Y–R) rela-tionship for a single reservoir. Sometimes, particularly in

0309-1708/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2007.02.004

* Corresponding author. Fax: +61 3 8344 7731.E-mail addresses: [email protected] (T.A. McMa-

hon), [email protected] (G.G.S. Pegram), [email protected](R.M. Vogel), [email protected] (M.C. Peel).

countries with poor infrastructure, the historical flowrecord is short or even unavailable yet a preliminaryS–Y–R relationship is required. Furthermore, for waterresources assessments and to assess the impact of climatechange on hydrology at a continental scale, a simple pro-cedure using annual statistics is required in which changein precipitation and temperature can be related to reser-voir characteristics.

There are very few general S–Y–R procedures that areavailable for such preliminary analyses. Some are basedon limited empirical information and, therefore, are inade-quate for wide hydrologic application. Computer-basedmethods like Behaviour analysis are unsuitable where flow

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1874 T.A. McMahon et al. / Advances in Water Resources 30 (2007) 1873–1882

records are short or not available. For reconnaissance anal-ysis in countries with inadequate hydrologic records, reser-voir capacities can only be sized on the basis of an estimateof the mean flow and its variability. The authors are awareof only one technique – Gould–Dincer (G–D) suite ofequations – that potentially can be applied using annualstreamflows across the whole spectrum of global hydrol-ogy. It should be noted here that there are preliminarytechniques that can be used for a restricted range of hydrol-ogy (Buchberger and Maidment [1]) or are able to estimatethe reservoir capacity for no-failure yield (Vogel and Ste-dinger [24]; and Phien [17]) but cannot estimate capacity–steady state reliability relationships. The authors believethe G–D approach is the only suitable candidate to exam-ine such relationships and is, therefore, the subject of thispaper.

Although the G–D approach is widely recommended foruse as a preliminary S–Y–R technique (for example Gould[4], Teoh and McMahon [19], McMahon and Adeloye [7]),the techniques have not been fully reviewed. The firstauthor examined one aspect of the approach, but the anal-ysis was based on data limited to Australia and Malaysia(McMahon and Mein [8]; Teoh and McMahon [19]). Withthe much larger data set of annual streamflows consideredhere – 729 rivers globally with at least 25 years of continu-ous clean data – we believe it is opportune to carry out amore detailed review of the G–D techniques. This paperreviews the Gould–Dincer suite of equations that can beused to establish the relationship between reservoir capac-ity, yield and reliability over the most extensive range ofstreamflow records, and the widest range of climates, thathas ever been considered.

Following this introduction, we describe briefly in Sec-tion 2 the global annual streamflow data set used toassess the S–Y–R techniques. Next, we outline the back-ground theory of the three variant forms (normal, log-normal and Gamma) of the Gould–Dincer approach.In Section 4, we compare the mean first passage timefrom a full to empty reservoir (m0c) as defined theoreti-cally by Pegram [14] for a finite reservoir with an approx-imate equivalent recurrence interval calculated using theG–D normal model. In the next section, the G–Dapproach is applied to the data set and is compared tothree alternate S–Y–R techniques to assess the perfor-mance of the G–D methods. Parameter sensitivity in esti-mates of reservoir capacity based on the Gould–Dincermodel is explored in Section 6. There it is found thatthe effects of skew and serial correlation are orders ofmagnitude lower than those of changes of the meanand standard deviation. This finding suggests that thesimple reduced form of the Gould–Dincer equationmay profitably be used for regional water resourcesassessments and in studies of reservoir reliability subjectto climate change scenarios based on regional statistics,without having to perform calculations based on timeseries, which may not be easily obtained. Relevant con-clusions are drawn in Section 7.

2. Streamflow data

The streamflow data set used in this analysis consists ofhistorical streamflows for 729 rivers with 25 or more yearsof continuous annual data. The rivers are not regulated byreservoirs nor are they affected by diversions upstream ofthe gauging stations. The location of the 729 rivers suggeststhat the data are reasonably well distributed world-wide.Details of the data can be found in [10,12,13].

3. Gould–Dincer approach

The Gould–Dincer approach, which was offered to thefirst author by C.H. Hardison in 1966, is a modificationof a method for reservoir storage–yield analysis derivedby Professor T. Dincer, Middle East Technical University,Turkey. The Dincer method assumed the reservoir inflowswere normally distributed and serially uncorrelated. In1964, Gould [4] independently derived a similar reservoirstorage–yield relationship but incorporated inflows thatwere Gamma distributed. This method has become knownas the Gould Gamma method (McMahon and Adeloye[7]). To apply this method to skewed flows Gould [4] pro-vided a manual adjustment to modify normal flows toGamma distributed flows. Vogel and McMahon [23] pro-posed that the Wilson–Hilferty transformation [27] be usedinstead of the cumbersome Gould procedure to deal withskewed flows and derived an adjustment to storage as aresult of auto-correlation which produced a result identicalto that of Phatarfod [16], but used a completely indepen-dent approach. A further variation of the Gould–Dincerapproach that allows for lognormal inflows was offeredto the first author by G. Annanadale in 2004. To distin-guish among these three variations of the Gould–Dincerapproach we have labelled them: Gould–Dincer Normal(G–DN), Gould–Dincer Gamma (G–DG) and Gould–Dincer Lognormal (G–DLN). The following summarizesthe methodology.

3.1. Gould–Dincer Normal (G–DN)

The equation representing G–DN model is developed asfollows. Assuming normally distributed and independentannual flows (mean l and standard deviation r), consecu-tive n-year inflows (i.e., the sum of n consecutive annualflows) into a reservoir can be defined as:

n-year mean; ln ¼ nl ð1Þ

n-year standard deviation; rn ¼Xn

r2

� �0:5

¼ rffiffiffinp

ð2Þ

During a critical period (i.e., a period during which the res-ervoir contents decline from full to empty) of length n:

Sn;p ¼ Dn � X n;p ð3Þwhere Sn,p is the depletion (thought of as a positive quan-tity) of an initially full reservoir at the end of n years with-

T.A. McMahon et al. / Advances in Water Resources 30 (2007) 1873–1882 1875

out having spilled, Dn = anl is the target draft over n years,Xn,p is the n-year inflow with a probability of non-excee-dance of 100p% and a is a constant draft defined as a ratioof l. Assuming inflows are normally distributed

X n;p ¼ nlþ zprn ð4Þwhere zp is the standardised normal variate at 100p% prob-ability of non-exceedance (zp < 0 because we are looking atinflows below the mean). To obtain the maximum storagerequired to supply the draft, combine Eqs. (3) and (4)and differentiate with respect to n; this gives the requiredcapacity C to meet the target draft for 100p% probabilityof non-exceedance, i.e., for 100(1 � p)% reliability, andthe equivalent critical period ncrit in years as follows:

ncrit ¼z2

p

4ð1� aÞ2Cv2 ð5Þ

which, after back-substitution into Eq. (3), gives:

C ¼z2

p

4ð1� aÞCv2l ð6Þ

where Cv is the coefficient of variation of annual inflows tothe reservoir. ncrit is the period for the reservoir of capacityC to empty from an initially full condition.

By substituting 1�aCv¼ m in Eq. (6), we obtain the dimen-

sionless relationship:

K ¼z2

p

4mð7Þ

where K is the standardised storage (reservoir capacity di-vided by the standard deviation of annual flows) and m isknown as drift [14,21] or standardized net inflow [5], inother words, m is the inverse of the coefficient of variationof net inflow. Note that for a given reliability, K reduces asm increases.

To account for the auto-correlation effect on reservoircapacity one can adjust the reservoir capacity computedfrom Eq. (6) by (1 + q)/(1 � q) as follows [16,23]:

C ¼z2

p

4ð1� aÞCv2l1þ q1� q

ð8Þ

where q is the lag-one serial correlation coefficient.It is noted that probability p is the probability that

inflows into the reservoir will be just sufficient to allowthe reservoir to meet the targeted draft with reliability(1 � p). In terms of Pegram’s definitions of failure [14] dis-cussed in Section 4, 1/p is assumed to be a measure of themean first passage time from a full to an empty reservoir.

3.2. Gould–Dincer Gamma (G–DG)

If the inflows are assumed to be Gamma distributed, zp

in Eq. (8) is replaced by:

gp ¼2

c1þ c

6zp �

c6

� �n o3

� 1

� �ð9Þ

where gp is an approximate Gamma variate based on theWilson–Hilferty transformation [27] (see also Chowdhuryand Stedinger [3] for developments relating to the transfor-mation) and c is the coefficient of skewness of the annualinflows. If the flows are also auto-correlated, then c needsto be replaced by c 0 (Eq. (10)) in Eq. (9). Eq. (10) was firstproposed by Thomas and Burden [20] and published byLoucks et al. [6, p. 284]. This correction adjusts c and isseparate from the correction in Eq. (8) which deals withthe effect of auto-correlation on reservoir inflows and isindependent of the inflow distribution:

c0 ¼ c1� q3

1� q2ð Þ1:5

!ð10Þ

It should be pointed out that the Gamma transforma-tion (Eq. (9)) based on the Wilson and Hilferty transforma-tion breaks down for values of c > 4 [9]. This is equivalentto Cv = 1 in the lognormal domain, however, very few val-ues (only five out of the estimated values of c from theworld data-set) are >4.

3.3. Gould–Dincer Lognormal (G–DLN)

If the annual flows are considered to be lognormal, zp inEq. (8) can be replaced by Eq. (11) which is a rearrange-ment of Chow [2, Eq. (8-I-53)]

z0p ¼1

Cvezp

ffiffiffiffiffiffiffiffiffiffiffiffiffilnð1þCv2p

Þ�0:5 ln 1þCv2ð Þ � 1h i

ð11Þ

3.4. Attributes, limitations

A major advantage of the Gould–Dincer approach is thatit is based on a straight-forward and logical water balance ofsimultaneous inputs and outputs of a storage reservoir.Computationally, it is simple and, although the basic formu-lation assumes inflows are independent, the storage esti-mates can be adjusted to take the auto-correlation intoaccount as provided in Eq. (8). It has been shown elsewhere[19] that for carry-over or over-year storages the Gould–Dincer approach provides satisfactory estimates of reservoircapacity. This issue is examined further in Sections 4 and 5.

A limitation of G–D models relates to the definition ofprobability of failure (emptiness) (Pf). From Eq. (3), theprobability of failure is defined as the failure of the n-yearinflow to occur with a probability of non-exceedance of100p%. Given that the unit of time in a G–D analysis isa year, 1/p can be likened to and, for our analysis, has beenassumed equivalent to the mean first passage time from afull reservoir to an empty condition. The theory does notallow for failures beyond the first failure. The complementof Pf is an approximate measure of the reliability to meetthe target draft from a full condition.

The procedure is restricted to annual inflows and,hence, carryover storage. As a general rule reservoirs withm < 1 operate as over-year or carry-over storages and as


within-year systems if m P 1 [22,25]. Vogel et al. [26] sug-gested that m < Cv and m P Cv maybe a more appropriateguide. However, Montaseri and Adeloye [11] argue thatother variables in addition to m and Cv, such as reliability,may be needed to classify reservoirs as carry-over or within-year storages. In this paper, we have adopted the simple rulethat m < 1 or m P 1 and a further check was carried out byensuring that ncrit (Eq. (5)) was greater than 1 year.

Similar to some other S–Y procedures, net evaporationlosses cannot be explicitly taken into account, but proce-dures are available [7] to adjust the storage size once thecapacity has been determined using only inflows and drafts.

4. The adequacy of the Gould–Dincer approach

A specific reservoir size computed using the Gould–Din-cer approach is based on the definition of probability ofnon-exceedance of inflows (Eq. (3)). As noted earlier thishas been assumed equivalent to the mean first passage timefrom a full to an empty reservoir (m0c) as defined byPegram [14]. Employing numerical quadrature to solvethe integral storage equations and using the deferredapproach to the limit, Pegram [14] obtained a range ofresults in reservoir storage reliability with known errorbounds. Choosing finite reservoirs with a range of stan-dardized capacities, he selected inputs described by Normaland lognormal distributions with a range of values of drift,coefficient of skewness and lag-one serial correlation. Ofinterest here are the results for reservoirs fed with indepen-dent inputs, obtaining their reliability as measured by m00,the mean recurrence time of emptiness and m0c, the meanfirst passage time from full to empty; these were all com-puted to a precision of at least three significant figures.

In Fig. 1, the mean first passage time to empty fromfull, m0c, is plotted against the mean recurrence time of

1

10

100

1.000

10.000

1 10 1moo (unspecifi

moc

(uns

peci

fied

units

of t

ime)

Fig. 1. Plot of m0c (mean first passage time to empty from full storage) comparesults for finite reservoirs fed by normally distributed inflows.

emptiness, m00, i.e. the mean time between failures,based on results from Pegram [14]. The figure is basedon the standardised storage and drift with values rang-ing mainly from 0.5 to 8 (for K) and 0–1 (for m). Forpractical purposes and for annual flow data, values upto perhaps 1000 time units are relevant. It can be seenthat as drift, m, increases m0c approaches m00. This isreasonable given that as drift increases a reservoir tendsto spill more frequently, experiencing rarer emptyevents, so that the first passage time to empty from fullbecomes indistinguishable from the times between occur-rence of emptiness; it is noted in the figure that themeans of the two types of passage time are very closefor drifts above about 0.4.

To test the adequacy of the Gould–Dincer Normalprocedure (Eq. (7)), values of the mean first passage timefrom a full to empty condition were computed using Eq.(7) for appropriate 1/p values and were compared withPegram’s exact solution values of m0c [14] for the equiv-alent drift (m) and standardized storage values (K). Thecomparison is plotted in Fig. 2 for values of time unitsless than 1000. In the figure, m0c values for m = 0.3,0.5, 0.7 and 0.9 were interpolated from Table III inPegram [14]. The other values were read directly fromthe table. The slope of the regression without interceptis 1.00035. From this analysis it can be concluded thatG–DN for m < 1 is a satisfactory estimate of the meanfirst passage time from a full reservoir to an empty con-dition. We are unable, however, to independently test theadequacy of the other two forms of the G–D approachnamely G–D Gamma and G–D Lognormal, as the meanfirst passage times based on Gamma and Lognormalinflows are not available in Pegram [14]. The adequacyof these two latter variations along with G–DN areexamined in the following section.

00 1.000 10.000ed units of time)

m = 0m = 0.2m = 0.4m = 0.6m = 0.8m = 1.0

red with m00 (mean recurrence time of emptiness) based on Pegram’s [14]

1

10

100

1000

1 10 100 1000m0c (years)

Rec

urre

nce

inte

rval

base

d on

G-D

N (y

ears

)

Fig. 2. Comparison of m0c, mean first passage time from full reservoir to empty, for Gould–Dincer Normal procedure compared with Pegram’s exactsolution [14] using a range of drifts (0.2 6 m 6 0.9), standardised capacities in the range 0.25 6 C 6 8 and ncrit > 1. The 1:1 line is dashed.


5. Application of Gould–Dincer to world data

In this section, we compare the reservoir capacities esti-mated by the Gould–Dincer approach with those estimatedby the Extended Deficit Analysis (EDA), Behaviour analysisand the Sequent Peak Algorithm (SPA). The EDA tech-nique estimates for a river the deficit of providing a givendraft (from an initially full semi-infinite reservoir) at a givenlevel of reliability. It is based on estimating from the histor-ical flows a series of independent deficits that are assessed ina manner similar to a partial flood frequency analysis.Details of the technique are set out in Pegram [15] (alsosee [7] for an explanation). For Behaviour analysis, failureis measured as the percentage of time (annual time units inthis study) that the reservoir is empty. It is a technique thattracks the volumetric content of a finite reservoir by carryingout a water balance of inputs (inflows) and outputs (yields orreservoir releases and spills and other losses including netevaporation). In the analysis in this paper, losses are nottaken into account. In contrast to the other techniques,SPA estimates the firm yield which is the yield that can bemet over a particular planning period with a specified no-failure reliability. Based on historical streamflows it is anautomated version of the Rippl mass curve procedure [20]and has been used to estimate reservoir capacities world-wide. In a complementary (as yet unpublished) paper basedon the same 729 annual river flows adopted in this paper, weshow that these three techniques produce consistent esti-mates of reservoir capacity, with SPA and EDA procedurescomputing virtually the same capacities.

5.1. Gould–Dincer Gamma versus EDA

In order to apply the Gould–Dincer approach, we haveassumed in this application that the global annual flows areGamma distributed. Our analysis to be published elsewherewould suggest this is not an unreasonable assumption.

Although the G–D approach and EDA are based on dif-ferent probabilistic statements (G–D estimates are assumedto approximate the mean first passage time to empty froma full storage and EDA estimates the mean recurrence timeof emptiness), it is instructive to compare the storage esti-mates generated by both approaches. We are able to dothis for the range of drift 0.4 < m < 1 where the mean firstpassage time approximates the mean recurrence time ofemptiness (see Fig. 1 and Section 4). For this range of driftwe compared for the global rivers the G–D Gamma storageestimates using Eqs. (8)–(10) with the EDA values, bothbased on a = 0.75 (that is, a draft of 75% of the mean his-torical flow) and restricting the applications to the 305 riv-ers with c < 4.0 (see Section 3). In addition, we ensured thatncrit computed from Eq. (5) > 1 year. The G–DG estimatesare for 99% annual time reliability and EDA values are fora recurrence interval of 100 years.

In order to assess the similarity of the capacity estimatesbetween the two procedures, a weighted least squares(WLS) regression was applied, with weights proportionalto record lengths, and excluding the outlier indicated inFig. 3 with a cross. The overestimate by G–DG relativeto the EDA value for this (outlier) river record is mainlydue to an annual auto-correlation value of 0.74. This valueis in the upper 1% of q values in the data set and the stor-age estimate is 6.7 times larger (based on the adjustmentgiven in Eq. (8)) compared with a storage estimate forq = 0. The slope of the WLS regression is 0.99. From thisanalysis we can conclude that G–DG and EDA withinthe range adopted in Fig. 3 provide similar estimates of res-ervoir capacity for all practical purposes.

5.2. Gould–Dincer Gamma versus Behaviour analysis

Fig. 4 explores the relationship between Gould–DincerGamma and a Behaviour analysis for a = 0.75 and restrict-ing the applications to rivers with c < 4.0, m < 1.0 and

10

1.000

100.000

10 1.000 100.000Extended Deficit Analysis (106 m3)

Gou

ld-D

ince

r Gam

ma

(106

m3 )

0.10.1

Fig. 3. Comparison of Gould–Dincer Gamma procedure for a = 0.75, 99% annual time reliability with storage estimates using the Extended DeficitAnalysis for a = 0.75 and 100 years average recurrence interval. The analysis was restricted to 0.4 < m < 1.0, ncrit > 1 year and annual coefficient ofskewness of streamflow <4.0. The 1:1 line is dashed.


ncrit > 1 year. The analysis is for annual data and for 95%annual reliability. (We adopted 95% reliability to ensurethat with 25 or more years of data, at least one failureper simulation would be recorded during a Behavioursimulation.)

The figure shows a strong relationship between the twoestimates. The slope of the WLS regression, without anintercept term, is 0.958 which is statistically different fromunity at the 5% level of significance. A slope less than one isexpected because the definitions of failure adopted in thetwo procedures are different and result in the Behaviouranalysis estimates of reservoir capacity being larger than

10

1.000

100.000

10

Behaviour

Gou

ld-D

ince

r Gam

ma

(106

m3 )

0.10.1

Fig. 4. Comparison of Gould–Dincer Gamma reservoir estimates with Behavio0.4 < m < 1.0, ncrit > 1 year and annual coefficient of skewness of streamflow <

the G–DG estimates. In this paper the adopted failure cri-terion for the Behaviour analysis is the one used in waterengineering practice. It is the proportion of time units dur-ing simulation the reservoir fails to meet the targetdemand, whereas for G–DG the criterion is an approxi-mate estimate of the mean first passage time from full toempty which approximates the mean recurrence time ofemptiness for 0.4 < m < 1.0 (Section 4). Considering forthe moment the mean recurrence time of emptiness, ittherefore follows that for each G–DG failure (emptiness),there will be a corresponding period of failure observedduring the equivalent Behaviour analysis. However, for

1.000 100.000

analysis (106 m3)

ur estimates for a = 0.75 and 95% reliability. The analysis was restricted to4.0. The 1:1 line is dashed.


the latter each failure may last longer than one time unit.As a result, for the same reliability criterion, say 95% (or5% failure), there will be more time units of failure associ-ated with the Behaviour simulation than for G–DG. Thismeans the reservoir capacity estimated using G–DG willbe smaller than the Behaviour capacity for the same reli-ability or failure condition. For reservoir capacities wherem < 0.4, we observe from Fig. 1 that as m0c > m00 this effectwill be amplified.

Using the world data set the analysis confirms that theG–DG procedure provides reservoir capacity estimateswhich for the same numerical value of probability of failureare smaller (on average about 25%) than capacities com-puted using a Behaviour analysis. This result is consistentwith the definitions of failure adopted for the twoprocedures.

5.3. Gould–Dincer Gamma versus SPA

Figs. 2 and 3 confirmed that Gould–Dincer Gamma isable to reliably estimate the reservoir capacity for given tar-get draft and probability of failure (emptiness) defined asthe mean first passage time from a full to empty storage.Fig. 4 further showed that the G–DG reservoir capacityestimates were consistent with those estimated using aBehaviour analysis. The purpose of Fig. 5 is to observe,firstly, the difference in storage estimates using the threedistributions associated with the Gould–Dincer suite ofequations namely G–DN, G–DG and G–DLN and, sec-ondly, the relationship between estimates of reservoircapacity by the G–D procedures and SPA.

The data plotted in Fig. 5 are for the G–DN, G–DG andG–DLN and the SPA estimates based on a = 0.75, form < 1 and ncrit > 1 year. In order to provide an appropriatecomparison between the G–D and SPA estimates, the

10

1.000

100.000

10

Sequent Peak A

Gou

ld-D

ince

r(1

06m

3 )

0.10.1

Fig. 5. Comparison of reservoir capacities for a = 75% target draft based on theThe analysis was restricted to m < 1.0 and ncrit > 1 year. Probability of failure

probability of failure in the G–D analysis was defined as1/N where N is the number of years in the historical record.

Several observations follow from the figure. Firstly, theregression slope coefficients being 1.021, 1.006 and 1.013for G–DN, G–DG and G–DLN respectively are not signif-icantly different from unity at the 5% level. Secondly, theG–D estimates as a whole contain the 1:1 line with theG–DN, G–DG and G–DLN values, on average acrossthe range of capacities, being respectively 34% larger,32% smaller and 47% smaller than the equivalent SPAvalue. (The intercepts cause the averages to vary from eachother; the trend-lines through the respective sets have beenomitted as they are swamped by the data.) This resultemphasizes the importance of choosing the correct proba-bility distribution function for the reservoir inflows whenone is computing storage estimates using the Gould–Dincersuite of equations.

6. Parameter sensitivity in estimates of reservoir capacity

estimation

In this section, we explore the impact of parameter sen-sitivity in estimating reservoir capacity by considering theeffect of an equivalent change in the four parameters (l,r, c and q) on reservoir sizing. To do this we use theGould–Dincer Gamma procedure and determine theoreti-cally the effect of making a small change separately to eachof the four parameters in Eqs. (8)–(10). The resulting mar-ginal (akin to partial differential) error relationships aregiven as follows:

DCC¼ 1

a� 1

Dll

ð12Þ

DCC¼ 2

Drr

ð13Þ

1.000 100.000

lgorithm (106 m3)

G-DNG-DGG-DLN

three Gould–Dincer alternatives compared with Sequent Peak Algorithm.for G–D estimates based on 1/N. The 1:1 line is dashed.


DCC¼ c

zp � cp2

3

!1þ z cp

6� c2p2

36

n o2

1þ z cp6� c2p2

36

n o3

� 1

� �� 2

0BB@

1CCADc

c

ð14Þ

DCC¼ q 1þ gz

6� g2

36

2

z� g3

� � "

� 1þ gz6� g2

36

3

� 1

" #�1

� 2

g

!

�3cq 1� qð Þ 1� q2� ��2:5 þ 2

1� q2ð Þ

#Dqq

ð15Þ

where D represents a small change in the variable, z is thestandardised normal variate and g and p (both functionsof serial correlation) are defined in Appendix A. The deri-vation of these relationships is outlined in Appendix A.

Example results are shown in Table 1 for a +10%change in l, r, c and q, and adopting median values fromthe global data set of c = 0.55 and q = 0.1. We carried outthe analysis for a time reliability value of 95%(z = �1.645) and a target draft ratio a = 75% (i.e. 75%of the historical mean). Thus for these typical values, a+10% increase in each parameter, considered individu-ally, results in �40.0%, +20.0%, �2.52% and +1.95%change in estimated storage capacity respectively. For avalue of c = 1.0 (29% of values in the global data set aregreater than this value), the effect on storage of a 10%increase in c is a reduction in storage estimate of 5.50%.We also note from Eqs. (12)–(15) that changing the draftratio a only produces changes associated with the meanand not the other parameters.

These results are of particular interest to those wishingto investigate the impact of climate change on streamflowand the consequential effect on reservoir yield. This theo-retical approach is in contrast to using empirical modelslike those adopted by Vogel et al. [25] in the northeasternUnited States. As our understanding of the impact of cli-mate change on hydrology at the annual scale improves,through techniques like precipitation elasticity of stream-flow [18], information from exercises similar to the abovewill inform water managers to the likely impact of climatechange on reservoir yield. Furthermore, as our understand-ing of how climate change affects rainfall variability and

Table 1Change in storage estimate as a result of +10% change in Gould–DincerGamma parameters for reliability of 95% (z = �1.645), a = 75%, c = 0.55and q1 = 0.1

Parameter +10% change in parameter

l �40.0%r +20.0%c �2.52%q +1.95%

hence streamflow variability, such effects on reservoir yieldcan be explored.

7. Conclusions

Arising out of these analyses a number of conclusionsfollow.

1. The Gould–Dincer formulas which are based on a simplewater balance of storage content plus inflow less draft canaccommodate normal, Gamma and lognormal inflows.

2. Theoretically as drift m gets larger and approaches unity(effectively when m > 0.4), the mean first passage timefrom a full to an empty reservoir approaches the meanrecurrence time of emptiness for finite reservoirs fed byindependent normal inflows.

3. Mean first passage times from a full to an empty reser-voir calculated using the Gould–Dincer normal proce-dure were not significantly different to those calculatedby the Pegram [14] exact solution, for a range of driftsand standardized storage capacities.

4. For conditions in which the Gould–Dincer procedure isapplicable, reservoir capacities estimated by Gould–Dincer Gamma are, for all practical purposes, similarto those using Extended Deficit Analysis.

5. Gould–Dincer Gamma estimates were also comparedwith capacity estimates based on a Behaviour analysis.The analysis showed that for the same numerical valueof probability of failure the G–DG procedure providesreservoir capacity estimates that, on average, areabout 25% smaller than capacities computed using aBehaviour analysis. This observation is consistent withthe definitions of failure adopted for the twoprocedures.

6. Reservoir capacities determined by the Gould–Dincerformulae were, on average, 34% larger and 32% and47% smaller for the G–DN, G–DG and G–DLN modelsrespectively compared with capacities obtained from theSPA technique, in which the reciprocal of the length ofrecord was used as the probability of failure.

7. Using the Gould–Dincer Gamma model to represent thestorage–yield relationship, a sensitivity analysis for atypical set of values taken from the global data setand based on a 10% change in their value was carriedout. For a draft ratio a = 0.75, a +10% error in themean or standard deviation of flows resulted in storageestimates being underestimated by 40% or overestimatedby about 20% respectively. However, much smaller rela-tive changes in storage resulted for a 10% change intro-duced into the coefficient of skewness (�2.5%) or theauto-correlation (2.0%). Because these are an order ofmagnitude lower than the effects of mean and standarddeviation, they can be considered as second order effects.This result suggests that the reduced forms of the G–Dequations (Eqs. (6) and (7)) might profitably be usedfor assessing the effect of climate change scenarios onregional storage reliability.


Acknowledgements

We thank the Department of Civil and EnvironmentalEngineering, the University of Melbourne and the Austra-lian Research Council Grant DP0449685 for financiallysupporting this research. Our original streamflow data setwas enhanced by additional data from the Global RunoffData Centre (GRDC) in Koblenz, Germany. Streamflowdata for Taiwan and New Zealand were also provided byDr. Tom Piechota of the University of Nevada, Las Vegas.Professor Ernesto Brown of the Universidad de Chile, San-tiago kindly made available Chilean streamflows. Thanksfor South African reservoir data are also due to theDepartment of Water Affairs and Forestry with help fromconsultants WRP who extracted the details.

We are also grateful to Dr. Senlin Zhou of the Murray-Darling Basin Commission who completed early drafts ofthe computer programs used in the analysis.

Appendix A. Sensitivity of the Gould–Dincer Gamma

reservoir storage–yield relationships to inflow statistics

The Gould–Dincer Gamma storage equation (combin-ing Eqs. (8)–(10)) is

C¼ r2

l�Dð Þc2

1�q3

1�q2ð Þ1:5

!�2

� 1þ cz6

1�q3

1�q2ð Þ1:5

!� c2

36

1�q3

1�q2ð Þ1:5

!28<:

9=;

3

�1

264

375

2

1þq1�q

� �

ðA1Þ

where a in Eq. (2) is replaced by D/l where D is the targetdraft.

To examine the sensitivity of a storage estimate to themean l, we consider Eq. (A1) and let

k ¼ r2

c21þ cz

6

1� q3

1� q2ð Þ1:5

!� c2

36

1� q3

1� q2ð Þ1:5

!28<:

9=;

3

� 1

264

375

2

� 1þ q1� q

� �ðA2Þ

C ¼ kðl�DÞ�1 ðA3ÞoCol¼ �kðl� DÞ�2 ! oC

C¼ l

D� loll

ðA4Þ&ðA5Þ

oCC¼ 1

a� 1

oll

ðA6Þ

To examine the sensitivity of a storage estimate to the stan-

dard deviation r, we consider Eq. (A1)

C ¼ r2

ðl� DÞc02 1þ c0z

6� c02

36

3

� 1

" #21þ q1� q

� �

where c0 ¼ c1� q3

1� q2ð Þ1:5

!ðA7Þ

Let w ¼ 1

ðl� DÞc02 1þ c0z

6� c02

36

3

� 1

" #21þ q1� q

� �ðA8Þ

C ¼ wr2 ðA9ÞoCor¼ 2wr! oC

C¼ 2

orr

ðA10Þ&ðA11Þ

To examine the sensitivity of a storage estimate to thecoefficient of skewness c, we consider Eq. (A1) but first let

q ¼ r2 1þ q1� q

� �ðl� DÞ 1� q3

1� q2ð Þ1:5

!�2

ðA12Þ

C ¼ q1

c21þ cp

6z� c2p2

36

3

� 1

" #2

where p ¼ 1� q3

1� q2ð Þ1:5

!ðA13Þ

oCoc¼ q½ �2

o 1c2

ocþ q

1

c2

o½ �2

oc! oC

oc

¼ �2qc3½ �2 þ q

c2

o½ �oc

2

ðA14Þ&ðA15Þ

Consider only the derivative in second term of Eq. (A15)

o½ �2

oc¼

o 1þ z cp6� c2p2

36

n o3

� 1

� �oc

2

ðA16Þ

o½ �oc

2

¼ zp � cp2

3

� �½ �f g2 ðA17Þ

Substitute back into Eq. (A15) gives

oCoc¼ �2

qc3½ �2 þ q

c2zp � cp2

3

� �½ �f g2 ! oC

C

¼�2 q

c3 ½ �2 þ qc2 zp � cp2

3

� �½ �f g2

q 1c2 ½ �2

cocc

ðA18Þ&ðA19Þ

oCC¼ c zp � cp2

3

� �1þ z

cp6� c2p2

36

2

� 1þ zcp6� c2p2

36

3

� 1

" #�1

� 2

!occ

ðA20Þ

where p ¼ 1� q3

1� q2ð Þ1:5

!

To examine the sensitivity of a storage estimate to auto-correlation q, we consider Eq. (A1)

Let g ¼ c 1� q3ð Þ1� q2ð Þ1:5

and a ¼ r2

ðl� DÞ ðA21Þ&ðA22Þ

C ¼ a 1þ gz6� g2

36

3

� 1

" #21þ q1� q

� �1

g2ðA23Þ

Call C ¼ a � bðqÞ � cðqÞ � dðqÞ then ðA24ÞoCoq¼ a

oboq

cd þ abocoq

d þ abcodoq

ðA25Þ


Which, after rearrangement to obtain the relative changes,becomes

oCC¼ oq

q� q ob

og1

bþ od

og1

d

� �ogoqþ oc

oq1

c

� �ðA26Þ

Consider b, d and c in turn

First term bðqÞ ¼ 1þgz6� g2

36

3

�1

" #2

ðA27Þ

obog¼ 2 1þgz

6� g2

36

3

�1

" #�3 � 1þ gz

6� g2

36

2 z6�2g

36

� �!

ðA28Þ

obog

1

b¼ 1þgz

6� g2

36

2

z�g3

� �1þgz

6� g2

36

3

�1

" #�1

ðA29Þ

Second term dðqÞ ¼ 1

g2ðA30Þ

odog¼ � 2

g3! od

og1

d¼ � 2

gðA31Þ&ðA32Þ

Third term cðqÞ ¼ 1þ q1� q

ðA33Þ

ocoq¼ 2

1� qð Þ2! oc

oq1

c¼ 2

1� q2ð Þ ðA34Þ&ðA35Þ

We also needogoq

where g ¼ c 1� q3� �

1� q2� ��1:5

ðA36Þogoq¼ 3cq 1� qð Þ 1� q2

� ��2:5 ðA37Þ

Substituting Eqs. (A29), (A32), (A35) and (A37) into Eq.(A26) yields

oCC¼ q

" 1þ gz

6� g2

36

2

z� g3

� �1þ gz

6� g2

36

3

� 1

" #�1

� 2

g

!3cq 1� qð Þ 1� q2

� ��2:5 þ 2

1� q2ð Þ

#oqq

ðA38Þwhich, if q = 0, becomes

oCC¼ 2oq: ðA39Þ

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Documents

Review of Gould–Dincer reservoir storage–yield–reliability estimates