River Basin Modeling 11 · 2017-08-26 · River Basin Modeling 11 Multipurpose river basin...

11River Basin Modeling

Multipurpose river basin development typicallyinvolves the identification and use of bothstructural and nonstructural measures designed toincrease the reliability and decrease the cost ofmunicipal, industrial, and agriculture water sup-plies, to protect against droughts and floods, toimprove quality, to provide for commercialnavigation and recreation, to enhance aquaticecosystems, and to produce hydropower, asappropriate for the particular river basin. Struc-tural measures may include diversion canals,reservoirs, hydropower plants, levees, floodproofing, irrigation delivery and drainage sys-tems, navigation locks, recreational facilities,groundwater wells, and water and treatmenttreatment plants along with their distribution andcollection systems. Nonstructural measures mayinclude land-use controls and zoning, floodwarning and evacuation measures, and economicincentives that affect human behavior with regardto water and watershed use. Planning the devel-opment and management of water resource sys-tems involves identifying just what and when andwhere structural or nonstructural measures areneeded, the extent to which they are needed, andtheir combined economic, environmental, eco-logical, and social impacts. This chapter intro-duces some modeling approaches for doing this.Having just reviewed some water quality mod-eling approaches in the previous chapter, thischapter focuses on quantity management.

11.1 Introduction

Various types of models can be used to assistthose responsible for planning and managingvarious components of river systems. Thesecomponents include streams, rivers, lakes, reser-voirs, and wetlands, and diversions to demandsites that could be within or outside the basinboundaries. Each of these components can beimpacted by water management policies andpractices. The management of any single com-ponent can impact the performance of othercomponents in the basin. Hence, for the overallmanagement of the water in river basin systems, asystems view is usually taken. This systems viewrequires the modeling of multiple interacting andinterdependent components. These multicompo-nent models are useful for analyzing alternativedesigns and management policies for improvingthe performance of integrated river basin systems.

The discussion in this chapter is limited towater quantity management. Clearly the regimesof flows, velocities, volumes, and other propertiesof water quantity will impact the quality of thatwater as well. However, unless water allocationsallocations and uses are based on requirements forwater quality, such as for the dilution of pollution,water quality does not normally affect waterquantity. For this reason among others, it iscommon to separate discussions of water quantity

© The Author(s) 2017D.P. Loucks and E. van Beek, Water Resource Systems Planning and Management,DOI 10.1007/978-3-319-44234-1_11

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management from water quality management(Chap. 10). However, when attempting to predictthe impacts of any management policy on bothwater quantity and quality, both water quantityand quality models are needed.

This chapter begins with a discussion ofselecting appropriate model time periods that willdepend on the issues being addressed as well ason the variability of the water supplies anddemands. Discussed next are methods for esti-mating streamflows at various sites of interestthroughout a basin based on gage (measured)flows at other sites. Following these discussionsseveral methods are reviewed and compared forestimating reservoir storage requirements forwater supplies. Model components are definedfor withdrawals and diversions, and for reservoirstorage. Reservoir storage can serve the needsover time for water supply, flood control, recre-ation, and hydroelectric power generation. Nextflood control structures, such as levees andchannel flow capacity improvements at potentialflood damage in a river basin are introduced.These components are then combined into amultiple purpose multi-objective planning modelfor a hypothetical river basin. The chapter con-cludes with an introduction to some dynamicmodels for assisting in the scheduling and timesequencing of multiple projects within a riverbasin.

11.2 Model Time Periods

When analyzing and evaluating various watermanagement plans designed to distribute thenatural unregulated flows over time and space, itis usually sufficient to consider average condi-tions within discrete time periods. In optimizationmodels, weekly, monthly, or seasonal flows arecommonly used as opposed to daily flows. Theshortest time period duration usually consideredin optimization models developed for identifyingand evaluating alternative water managementplans and operating policies is one that is no lessthan the time water takes to flow from the upper

end of the applicable river basin to the lower endof the basin. In this case stream and river flowscan be defined by simple mass balance or conti-nuity equations. For shorter duration time periodsflow routing may be required.

The actual length or duration of eachwithin-year period defined in a model may varyfrom period to period. Modeled time period timeperiods need not be equal. Generally what isimportant is to capture in the model the neededcapacities of infrastructure that are determined inlarge measure by the variation in supplies anddemands. These variations should be captured inthe model by appropriately selecting the numberand duration of time periods. If say over athree-month period there is little variation in bothwater supplies and demands or for the purposeswater serves, such as flood control, hydropower,or recreation, there is no need to divide thatthree-month period into multiple time periods.

Another important factor to consider in mak-ing a decision regarding the number and durationof time periods to include in any model is thepurpose for which the model is to be used. Someanalyses are concerned only with identifyingdesigns designs and operating policies of variousengineering projects for managing water resour-ces at some fixed time (say a typical year) in thefuture. Multiple years of hydrological records areused, usually in simulation models, to obtain anestimate of just how well a system might per-form, at least in a statistical sense, in that futuretime period. The within-year period durationscan have an impact on those performance indi-cator values as well as on the estimate ofover-year as well as within-year storage that maybe needed to meet various goals. These staticanalyses are not concerned with investmentproject scheduling or sequencing.

Dynamic planning models are used to esti-mate the impacts of changing conditions overtime. These changes could include hydrologicalinputs, economic, environmental and otherobjectives, water demands, and design andoperating parameters. As a result, dynamicmodels generally span many more years than do

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static models, but they may have fewerwithin-year periods.

11.3 Streamflow Estimation

Water resource managers need estimates ofstreamflows at each site, where managementdecisions are being considered. These streamflowestimates can be based on the results ofrainfall-runoff models or on measured historicalflows at gage sites. For modeling alternativemanagement policies, these gage-based flows atthe sites of interest should be those that wouldhave occurred under natural conditions. These arecalled naturalized flows that have been derivedfrom measured flows or rainfall–runoff modelsand then adjusted to take into account anyupstream regulation and diversions. Many gageflow values reflect actions such as diversions andreservoir releases that occurred upstream thataltered the downstream flows, unless suchupstream water management and use policies areto continue, these measured gage flows should beconverted to unregulated or natural flows prior totheir use in management models.

Assuming that unregulated streamflow dataare available at gage sites, these data can be usedto estimate the unregulated flows at sites wherethey are needed. These sites would include anyplace where diversions might occur or wherereservoirs for regulating flows might be built.

Consider, for example, the simple river basinillustrated in Fig. 11.1. Assume streamflowshave been recorded over a number of years at

gage sites 1 and 9. Knowledge of the flows Qst in

each period t at gage sites 1 and 9 permits theestimation of flows at any other site in the basinas well as the incremental flows between thosesites in each period t.

The method used to estimate flows at ungagedsites will depend on the characteristics of thewatershed or river basin. In humid regions wherestreamflows increase in the downstream directiondue to rainfall runoff, and the spatial distributionof average monthly or seasonal rainfall is more orless the same from one part of the river basin toanother, the runoff per unit land area is typicallyassumed constant. In these situations, estimatedflows, qst , at any site s can be based on thewatershed areas, As, contributing flow to thosesites, and the corresponding streamflows andwatershed areas above the nearest or most rep-resentative gage sites.

For each gage site, the runoff per unit land areacan be calculated by dividing the gage flow Qs

t bythe upstream drainage area, As. This can be donefor each gage site in the basin. Thus for any gagesite g, the runoff per unit drainage area in monthor season t is Qg

t divided by Ag. This runoff perunit land area times the drainage area upstream ofany site s of interest will be the estimatedstreamflow in that period at that site s. If there aremultiple gage sites, such as illustrated inFig. 11.1, the estimated streamflow at someungaged site s can be a weighted combination ofthose unit area runoffs times the area contributingto the flow at site s. The nonnegative weights, wg,that sum to 1, reflect the relative significance ofeach gage site with respect to site s. Their values

Fig. 11.1 River basingage sites wherestreamflows are measuredand recorded

11.2 Model Time Periods 471

will be based on the judgments of those who arefamiliar with the basin’s hydrology.

Qst ¼

Xg

wgQgt =A

g

( )As ð11:1Þ

In all the models developed and discussedbelow, the variable Qs

t will refer to the meannatural (unregulated) flow (L3/T) at a site s in aperiod t.

The difference between the natural stream-flows at any two sites is called the incrementalflow. Using Eq. 11.1 to estimate streamflows willresult in positive incremental flows. The down-stream flow will be greater than the upstreamflow. In arid regions runoff is not constant overthe region. Incremental flows may not exist andhence due to losses, the flows may be decreasingin the downstream direction. In these cases thereis a net loss in flow in the downstream direction.This might be the case when a stream originatesin a wet area and flows into a region that receivesless rainfall. In such arid areas the runoff is oftenless than the evapotranspiration and infiltrationinto the ground along the stream channel.

For stream channels where there exist rela-tively uniform conditions affecting loss loss,where there are no known sites where the streamabruptly enters or exits the ground, as can occurin karst conditions, the average streamflow for aparticular period t at site s can be based on thenearest or most representative gage flow, Qg

t , anda loss rate per unit length of the stream or river,Lgs between gage site g and an ungaged site s. Ifthere are at least two gage sites along the portionof the stream or river that is in the dry region, onecan compute the loss of flow per unit streamlength, and apply this loss rate to various sitesalong the stream or river. This loss rate per unitlength may not be constant over the entire lengthbetween the gage stations, or even for all flowrates, however. Losses will likely increase withincreasing flows simply because more watersurface is exposed to evaporation and seepage. Inthese cases one can define a loss rate per unit

length of stream or river as a function of themagnitude of flow.

In watersheds characterized by significantelevation changes and consequently varyingrainfall and runoff runoff, other methods may berequired for estimating average streamflows atungaged sites. The selection of the most appro-priate method to use, as well as the mostappropriate gage sites to use for estimating thestreamflow, Qs

t , at a particular site s can be amatter of judgment. The best gage site need notnecessarily be the nearest gage to site s, butrather the site most similar with respect toimportant hydrologic variables.

The natural incremental flow between any twosites is simply the difference between theirrespective natural flows.

11.4 Streamflow Routing

If the duration of a within-year period is less thanthe time of flow throughout the stream or riversystem being modeled, and the flows vary withinthe system, some type of streamflow routingmust be used to keep track of where the varyingamounts of water are in each time period. Thereare many proposed routing methods (as descri-bed in any hydrology text or handbook, e.g.,Maidment 1993). Many of these more traditionalmethods can be approximated with sufficientaccuracy using relatively simple methods. Twosuch methods are described in the followingparagraphs.

The outflow, Ot, from a reach of stream orriver during a time period t is a function of theamount of water in that reach, i.e., its initialstorage, St, and its inflow, It. Because of bankstorage, that outflow is often dependent on whe-ther the quantity of water in the reach is increas-ing or decreasing. If bank inflows and outflowsare explicitly modeled, or if bank storage isnot that significant, the outflow from a reachin any period t can be expressed as a simpletwo-parameter power function of the forma(St + It)

b. Mass balance equations, that may take


into account losses, update the initial storagevolumes in each succeeding time period. Thereach–dependent parameters a and b can bedetermined through calibration procedures suchas genetic algorithms (Chap. 5) using a time seriesof reach inflows and outflows. The resultingoutflow function is typically concave (theparameter b will be less than 1), and thus theminimum value of St + Itmust be at least 1. If dueto evaporation or other losses the reach volumedrops below this or any preselected higheramount, the outflow can be assumed to be 0.

Alternatively one can adopt a three- orfour-parameter routing approach that fits a widerrange of conditions. Each stream or river reachcan be divided into a number of segments. Thatnumber n is one of the parameters to be deter-mined. Each segment s can be modeled as astorage unit, having an initial storage volume, Sst,and an inflow, Ist. The three-parameter approachassumes the outflow, Ost, is a linear function ofthe initial storage volume and inflow:

Ost ¼ a Sst þ bIst ð11:2Þ

Equation 11.2 applies for all time periodst and for all reach segments s in a particularreach. Different reaches will likely have differentvalues of the parameters n, α, and β. The cali-brated values of α and β are nonnegative and nogreater than 1. Again a mass balance equationupdates each segment’s initial storage volume inthe following time period. The outflow from eachreach segment is the inflow into the succeedingreach segment.

The four-parameter approach assumes that theoutflow, Ost, is a nonlinear function of the initialstorage volume and inflow

Ost ¼ ða Sst þ bIstÞc ð11:3Þ

The parameter γ is greater than 0 and nogreater than 1. In practice γ is very close to 1.Again the values of these parameters, includingthe number of reaches n, can be found usingnonlinear optimization methods, such as geneticalgorithms, together with a time series ofobserved reach inflows and outflows.

Note the flexibility available when using thethree- or four-parameter routing approach. Evenblocks of flow can be routed a specified distancedownstream over a specified time, regardless ofthe actual flow. This can be done by setting α andγ to 1, β to 0, and the number of segments n tothe number of time periods it takes to travel thatdistance. This may not be very realistic, but thereexist some river basin reaches where managersbelieve this particular routing applies.

11.5 Lakes and Reservoirs

Lakes and reservoirs are sites in a basin wheresurface water storage needs to be modeled. Thus,variables defining the water volumes at thosesites must be defined. Let Sst be the initial storagevolume of a lake or reservoir at site s in periodt. Omitting the site index s for the moment, thefinal storage volume in period t, St+1, (which isthe same as the initial storage in the followingperiod t + 1) will equal the initial volume, St,plus the net surface and groundwater inflows, Qt,less the release or discharge, Rt, and evaporationand seepage losses, Lt. All models of lakes andreservoirs include this mass balance equation foreach period t being modeled.

St þQt�Rt�Lt ¼ Stþ 1 ð11:4Þ

The release from a natural lake is a function ofits surrounding topography and its water surfaceelevation. It is determined by nature, and unlessit is made into a reservoir its discharge or releaseis not controlled or managed. The release from areservoir is controllable, and, as discussed inChaps. 4 and 8, is usually a function of thereservoir storage volume and time of year.Reservoirs also have fixed storage capacities,K. In each period t, reservoir storage volumes, St,cannot exceed their storage capacities, K.

St �K for each period t: ð11:5Þ

Equations 11.4 and 11.5 are the two funda-mental equations required when modeling watersupply reservoirs. They apply for each period t.

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The primary purpose of all reservoirs is toprovide a means of regulating downstream sur-face water flows over time and space. Otherpurposes may include storage volume manage-ment for recreation and flood control, and storageand release management for hydropower pro-duction. Reservoirs are built to alter the naturalspatial and temporal distribution of the stream-flows. The capacity of a reservoir together withits release (or operating) policy determine theextent to which surface water flows can be storedfor later release.

The use of reservoirs for temporarily storingstreamflows often results in a net loss of totalstreamflow due to increased evaporation andseepage. Reservoirs also bring with them chan-ges in the ecology of a watershed and riversystem. They may also displace humans andhuman settlements. When considering newreservoirs, any benefits derived from regulationof water supplies, from flood damage reduction,from hydroelectric power, and from any navi-gational and recreational activities should becompared to any ecological and social losses andcosts. The benefits of reservoirs can be substan-tial, but so may the costs. Such comparisons ofbenefits and costs are always challengingbecause of the difficulty of expressing all suchbenefits and costs in a common metric (Chap. 9).

Reservoir storage capacity can be dividedamong the three major uses: (1) the active stor-age used for downstream flow regulation and for

water supply, recreational development orhydropower production; (2) the the dead storageused for sediment collection; and (3) the floodstorage capacity reserved to reduce potentialdownstream flood damage during flood events.These separate storage capacities are illustratedin Fig. 11.2. The distribution of active and floodcontrol storage capacities may change over theyear. For example there is no need for floodcontrol storage in seasons that are not likely toexperience floods.

The next several sections of this chapteraddress how these capacities may be determined.

11.5.1 Estimating Active StorageCapacity

11.5.1.1 Mass diagram AnalysesPerhaps one of the earliest methods used to cal-culate the active storage capacity required tomeet a specified reservoir release, Rt, in asequence of periods t, was developed by Rippl(1883). His mass diagram analysis is still usedtoday by many planners. It involves finding themaximum positive cumulative differencebetween a sequence of prespecified (desired)reservoir releases Rt and known inflows Qt. Onecan visualize this as starting with a full reservoir,and going through a sequence of simulations inwhich the inflows and releases are added andsubtracted from that initial storage volume value.

Fig. 11.2 Total reservoirstorage volume capacityconsisting of the sum ofdead storage, activestorage, and flood controlstorage capacities


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Doing this over two cycles of the record ofinflows will identify the maximum deficit vol-ume associated with that release. This is therequired reservoir storage. Having this initialstorage volume, the reservoir would always haveenough water to meet the desired releases.However, this only works if the sum of all thedesired releases does not exceed the sum of allthe inflows over the same sequence of timeperiods. Reservoirs cannot make water.

Equation 11.6 represents this process. Theactive storage capacity, Ka, will equal the maxi-mum accumulated storage deficit one can find

over some interval of time within two successiverecord periods, T.

Ka ¼ maximumXj

t¼i

Rt�Qtð Þ" #

; ð11:6Þ

where 1 ≤ i ≤ j ≤ 2T.Equation 11.6 is the analytical equivalent of

graphical procedures proposed by Rippl forfinding the active storage requirements. Two ofthese graphical procedures are illustrated inFigs. 11.3 and 11.4 for a 9-period inflow recordof 1, 3, 3, 5, 8, 6, 7, 2, and 1. Rippl’s original

Fig. 11.3 The Rippl ormass diagram method foridentifying reservoir activestorage capacityrequirements. The releasesRt are assumed constant foreach period t

Fig. 11.4 Alternative plotfor identifying reservoiractive storage capacityrequirements

11.5 Lakes and Reservoirs 475

approach, shown in Fig. 11.3, involves plottingthe cumulative inflow

Ptt¼1 Qs versus time

t. Assuming a constant reservoir release, R, ineach period t, a line with slope R is placed so thatit is tangent to the cumulative inflow curve. Tothe right of these points of tangency the releaseR exceeds the inflow Qt. The maximum verticaldistance between the cumulative inflow curveand the release line of slope R equals the maxi-mum water deficit, and hence the required activestorage capacity. Clearly, if the average releaseR is greater than the mean inflow, a reservoir willnot be able to meet the demand no matter what itsactive storage capacity.

An alternative way to identify the requiredreservoir storage capacity is to plot the cumula-tive nonnegative deviations,

Pts ðRs � QsÞ, and

note the biggest total deviation, as shown inFig. 11.4.

These graphical approaches do not accountfor losses. Furthermore, the method shown inFig. 11.3 is awkward if the desired releases ineach period t are not the same. The equivalentmethod shown in Fig. 11.4 is called the sequentpeak method. If the sum of the desired releasesdoes not exceed the sum of the inflows, calcu-lations over at most two successive hydrologicrecords of flows are needed to identify the largestcumulative deficit inflow. After that the proce-dure will produce repetitive results. It is mucheasier to consider changing release values whendetermining the maximum deficit by the sequentpeak method.

11.5.1.2 Sequent peak analysesThe sequent peak procedure is illustrated inTable 11.1. Let Kt be the maximum total storagerequirement needed for periods 1 up throughperiod t. As before, let Rt be the required releasein period t, and Qt be the inflow in that period.Setting K0 equal to 0, the procedure involvescalculating Kt using Eq. 11.7 consecutively forup to twice the total length of record. Thisassumes that the record repeats itself to take careof the case when the critical sequence of flowsoccurs at the end of the streamflow record, as

indeed it does in the example 9-period record of1, 3, 3, 5, 8, 6, 7, 2, and 1.

Kt ¼ Rt�Qt þKt�1 if positive;

¼ 0 otherwiseð11:7Þ

The maximum of all Kt is the required storagecapacity for the specified releases Rt and inflows,Qt. Table 11.1 illustrates this sequent peak pro-cedure for computing the active capacity Ka, i.e.,the maximum of all Kt, required to achieve arelease Rt = 3.5 in each period given the series of9 streamflows. Note this method does not requireall releases to be the same.

11.5.2 Reservoir Storage-YieldFunctions

Reservoir storage-yield functions define theminimum active active storage capacity requiredto insure a given constant release rate for aspecified sequence of reservoir inflows. Massdiagrams, sequent peak analyses, and linearoptimization (Chap. 4) are three methods that canbe used to define these functions. Given the samesequence of known inflows and specified relea-ses, each method will provide identical results.Using optimization models, it is possible toobtain such functions from multiple reservoirsand to account for losses based on storage vol-ume surface areas, as will be discussed later.

There are two ways of defining a linear opti-mization (linear programming) model to estimatethe active storage capacity requirements. Oneapproach is to minimize the active storagecapacity, Ka, subject to minimum required con-stant releases, Y, the yield. This minimum activestorage capacity is the maximum storage volume,St, required given the sequence of known inflowsQt, and the specified yield, Y, in each periodt. The problem is to find the storage volumes, St,and releases, Rt that

Minimize Ka ð11:8Þ


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subject tomass balance constraints

St þQt�Rt ¼ Stþ 1 t ¼ 1; 2; . . .; T;T þ 1 ¼ 1

ð11:9Þ

capacity constraints

St �Ka t ¼ 1:2; . . .; T ð11:10Þ

minimum release constraints

Rt � Y t ¼ 1:2; . . .; T ð11:11Þ

for various values of the yield, Y.

Alternatively one can maximize the constantrelease yield, Y, for various values of activestorage capacity, Ka, subject to the same con-straint Eqs. 11.9–11.11.

Maximize Y ð11:12Þ

Constraints 11.9 and 11.11 can be combinedto reduce the model size by T constraints.

St þQt�Y � Stþ 1 t ¼ 1; 2; . . .; T ;T þ 1 ¼ 1

ð11:13Þ

The solutions of these two linear programmingmodels, using the 9-period flow sequence referred

Table 11.1 Illustration of the sequent peak procedure for computing active storage requirements


to above and solved for various values of yield orcapacity, respectively, are plotted in Fig. 11.5.The results are the same as could be found usingthe mass diagram or sequent peak methods.

There is a probability that the storage-yieldfunction just defined will fail. A record of only 9flows, for example, is not very long and hencewill not give one much confidence that they willdefine the critical low-flow period of the future.One rough way to estimate the reliability of astorage-yield function is to rearrange and rankthe inflows in order of their magnitudes. If thereare n ranked inflows there will be n + 1 intervalsseparating them. Assuming there is an equalprobability that any future flow could occur inany interval between these ranked flows, there isa probability of 1/(n + 1) that a future flow willbe less than the lowest recorded flow. If thatrecord low flow occurs during a critical low-flowperiod, more storage may be required than indi-cated in the function.

Hence for a record of only 9 flows that areconsidered representative of the future, one canbe only about 90% confident that the resultingstorage-yield function will apply in the future.One can be only 90% sure of the predicted yieldY associated with any storage volume K. A muchmore confident estimate of the reliability of anyderived sstorage-yield function can be obtainedby synthetic flows to supplement any measuredstreamflow record, taking parameter uncertainty

into account (as discussed in Chaps. 6 and 8).This will provide alternative sequences as well asmore intervals between ranked flows.

While the mass diagram and sequent peakprocedures are relatively simple, they are notreadily adaptable to reservoirs where evaporationlosses and/or lake level regulation are importantconsiderations, or to problems involving morethan one reservoir. Mathematical programming(optimization) methods provide this capability.These optimization methods are based on massbalance equations for routing flows through eachreservoir. The mass balance or continuity equa-tions explicitly define storage volumes (andhence storage areas from which evaporationoccurs) at the beginning of each period t.

11.5.3 Evaporation Losses

Evaporation losses, Lt, from lakes and reservoirs,if any, take place on their surface areas. Hence tocompute these losses their surface areas must beestimated in each period t. Storage surface areasare functions of the storage volumes, St. Thesefunctions are typically concave, as shown inFig. 11.6.

In addition to the storage area-volume func-tion, seasonal surface water evaporation lossdepths, Et

max, must be assumed, perhaps based onmeasured evaporation losses over time.

Fig. 11.5 Storage-yieldfor the sequence of flows 1,3, 3, 5, 8, 6, 7, 2, and 1


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Multiplying the average surface area, At, basedon the initial and final storage volumes, St and St+1, by the loss depth, Et

max, yields the volume ofevaporation loss, Lt, in the period t. The linearapproximation of that loss is

Lt ¼ ao þ a St þ Stþ 1ð Þ=2½ �Emaxt ð11:14Þ

Letting

at ¼ 0:5 a Emaxt ð11:15Þ

the mass balance equation for storage volumesthat include evaporation losses in each periodt can be approximated as

1� atð ÞSt þQt�Rt�aoEmaxt ¼ 1þ atð ÞStþ 1

ð11:16Þ

If Eq. 11.16 are used in optimization modelsfor identifying preliminary designsof a proposedreservoir and if the active storage storage capacityturns out to be essentially zero, or just that requiredto provide for the fixed evaporation loss, ao Et

max,then clearly any reservoir at the site is not justified.These mass balance equations together with anyreservoir storage capacity constraints should beremoved from the model before resolving it again.This procedure is simpler than introducing 0,1integer variables that will remove the terms aoEtmax in Eq. 11.16 if the active storage volume is 0

(using methods discussed in Chap. 4).

An alternative way to estimate evaporationloss that does not require a surface area—storagevolume relationship, such as shown in Fig. 11.6,is to define the storage elevation-storage volumefunction. Subtracting the evaporation loss depthfrom the initial surface elevation associated withthe initial storage volume will result in anadjusted storage elevation which in turn definesthe initial storage volume after evaporation losseshave been deducted. This adjusted initial volumecan be used in continuity Eqs. 11.9 or 11.13.This procedure assumes that evaporation is onlya function of the initial storage volume in eachtime period t. For relatively large volumes andshort time periods such an assumption is usuallysatisfactory.

11.5.4 Over- and Within-YearReservoir Storageand Yields

An alternative approach to modeling reservoirs isto separate out over-year storage and within-yearstorage, and to focus not on total reservoirreleases, but on parts of the total releases that canbe assigned specific reliabilities. These releasecomponents we call yields. To define these yieldsand the corresponding reservoir rules, we dividethis section into four parts. The first outlines amethod for estimating the reliabilities of variousconstant annual minimum flows or yields. The

Fig. 11.6 Storage surfacearea as a function ofreservoir storage volumealong with its linearapproximation. The slopeparameter a is the assumedincrease in surface areaassociated with a unitincrease in the storagevolume


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second discusses a modeling approach for esti-mating over-year and within-year active storagerequirements to deliver a specified annual andwithin-year period yields having a specified re-liability. The third and fourth parts expand thismodeling approach to include multiple yieldshaving different reliabilities, evaporation losses,and the construction of reservoir operation rulecurves using these flow release yields.

It will be convenient to illustrate the yieldmodels and their solutions using a simpleexample consisting of a single reservoir and twowithin-year periods per year. This example willbe sufficient to illustrate the method that can beapplied to models having more within-yearperiods. Table 11.2 lists the nine years of

available streamflow data for each within-yearseason at a potential reservoir site. Thesestreamflows are used to solve and compare thesolutions of various yield models as well as toillustrate the concept of yield reliability.

11.5.4.1 Reliability of Annual yieldsThe maximum flow that can be made available ata specific site by the regulation of the historicstreamflows from a reservoir of a given size isoften referred to as the “firm yield” or “safeyield.” These terms imply that the firm or safeyield is that yield which the reservoir will alwaysbe able to provide. Of course, this may not betrue. If historical flows are used to determine thisyield, then the resulting yield may be better

Table 11.2 Recorded unregulated historical streamflows at a reservoir site


called an ‘‘historical yield.’’ Historical and firmyield are often used synonymously.

A minimum flow yield is 100% reliable onlyif the sequence of flows in future years will neversum to a smaller amount than those that haveoccurred in the historic record. Usually onecannot guarantee this condition. Hence associ-ated with any historic yield is the uncertainty,i.e., a probability, that it might not always beavailable in the future. There are some ways ofestimating this probability.

Referring to the nine-year streamflow recordlisted in Table 11.2, if no reservoir is built toincrease the yields downstream of the reservoirsite, the historic firm yield is the lowest flow onrecord, namely 1.0 that occurred in year 5. Thereliability of this annual yield is the probabilitythat the streamflow in any year is greater than orequal to this value. In other words, it is theprobability that this flow will be equaled orexceeded. The expected value of the exceedanceprobability of the lowest flow in an n-year recordis approximately n/(n + 1), which for then = 9 year flow record is 9/(9 + 1), or 0.90. Thisis based on the assumption that any future flowhas an equal probability of being in any of theintervals formed by ordering the record of 9flows from the lowest to the highest value, andthat the lowest value has a rank of 9.

Ranking the n flows of record from the highestto the lowest and assigning the rank m of 1 to thehighest flow, and n to the lowest flow, theexpected probability p that any flow of rankm willbe equaled or exceeded in any year is approxi-mately m/(n + 1). An annual yield having aprobability p of exceedance will be denoted as Yp.

For independent events, the expected numberof years until a flow of rank m is equaled orexceeded is the reciprocal of its probability ofexceedance p, namely 1/p = (n + 1)/m. The re-currence time or expected time until a failure (aflow less than that of rank m) is the reciprocal ofthe probability of failure in any year. Thus, theexpected recurrence time Tp associated with aflow having an expected probability p of excee-dance is 1/(1 − p).

11.5.4.2 Estimation of ActiveReservoir StorageCapacitiesfor Specified Yields

A reservoir with active over-year storage capac-ity provides a means of increasing the magnitudeand/or the reliabilities of various annual yields.For example, the sequent peak algorithm definedby Eq. 11.7 provides a means of estimating thereservoir storage volume capacity required tomeet various “firm” yields Y0.9, associated withthe nine annual flows presented in Table 11.1.The same yields can be obtained from a linearoptimization model that minimizes activeover-year storage capacity, Ka

o,

Minimize Koa ð11:17Þ

This active over-year storage capacity mustsatisfy the following storage continuity andcapacity constraint equations involving onlyannual storage volumes, Sy, inflows, Qy, yields,Yp, and excess releases, Ry. For each year y:

Sy þQy � Yp � Ry ¼ Syþ 1 ð11:18Þ

Sy �Koa ð11:19Þ

Once again, if the year index y = n, the lastyear of record, then year y + 1 is assumed toequal 1. For annual yieldsof 3 and 4, theover-year storage requirements are 3 and 8,respectively, as can be determined just byexamining the right-hand column of annual flowsin Table 11.2.

The over-year model, Eqs. 11.17–11.19,identifies only annual or over-year storagerequirements based on specified (known) annualflows, Qy, and specified constant annual yields,Yp. Within-year periods t requiring constantyields ypt that sum to the annual yield Yp mayalso be considered in the estimation of therequired over-year and within-year or total activestorage capacity, Ka. Any distribution of theover-year yield within the year that differs fromthe distribution of the within-year inflows may


require additional active reservoir storagecapacity. This additional capacity is called thewithin-year storage capacity.

The sequent peak method, Eq. 11.7, can beused to obtain the total over-year and within-yearactive storage capacity requirements for specifiedwithin-year period yields, ypt. Alternatively alinear programming model can be developed toobtain the same information along with associ-ated reservoir storage volumes. The objective isto find the minimum total active storage capacity,Ka, subject to storage volume continuity andcapacity constraints for every within-year periodof every year. This model is defined as

minimize Ka ð11:20Þ

subject to

Sty þQty � ypt � Rty ¼ Stþ 1;y 8t; y ð11:21Þ

Sty �Ka 8t; y ð11:22Þ

In Eq. 11.21, if t is the final period T in year y,the next period T + 1 = 1 in year y + 1, or year 1if y is the last year of record, n.

The within-year storage requirement, Kaw, is

the difference in the active capacities resultingfrom these two models, Eqs. 11.17–11.19, andEqs. 11.20–11.22.

Table 11.3 shows some results from solvingboth of the above models. The over-year storagecapacity requirements, Ka

o, are obtained fromEqs. 11.17–11.19. The combined over-year andwithin-year capacities, Ka, are obtained fromsolving Eqs. 11.20–11.22. The difference betweenthe over-year storage capacity, Ka

o, required tomeet only the annual yields and the total capacity,Ka, required to meet each specified within-yearyield distribution of those annual yields is thewithin-year active storage capacity Ka

w.Clearly, the number of continuity and reser-

voir capacity constraints in the combinedover-year and within-year model (Eqs. 11.20–11.22) can become very large when the numberof years n and within-year periods T are large.Each reservoir site in the river system will

require 2nT continuity and capacity constraints.Not all these constraints are necessary, however.It is only a subset of the sequence of flows withinthe total record of flows that generally determinesthe required active storage capacity Ka of areservoir. This is called the critical period. Thiscritical period is often used in engineering studiesto estimate the historical yield of any particularreservoir or system of reservoirs.

Even though the severity of future droughts isunknown, many planners accept the traditionalpractice of using the historical critical droughtperiod for reservoir design and operation studieson the assumption that having observed such anevent in the past, it is certainly possible toexperience similar conditions in the future. Insome parts of the world, notably those countriesin the lower portions of the southern hemisphere,historical records are continually proven to beunreliable indicators of future hydrological con-ditions. In these regions especially, syntheticallygenerated flows based on statistical methods(Chap. 6) are more acceptable as a basis for yieldestimation.

Over and within-year storage CapacityTo begin the development of a smaller, but moreapproximate, model, consider each combinedover-year and within-year storage reservoir toconsist of two separate reservoirs in series(Fig. 11.7). The upper reservoir is the over-yearstorage reservoir, whose capacity required for anannual yield is determined by an over-yearmodel, e.g., Eqs. 11.17–11.19. The purpose ofthe ‘‘downstream’’ within-year reservoir is todistribute as desired in each within-year periodt a portion of the annual yield produced by the‘‘upstream’’ over-year reservoir. Within-yearstorage capacity would not be needed if thedistribution of the average inflows into the upperover-year reservoir exactly coincided with thedesired distribution of within-year yields.Otherwise within-year storage may be required.The two separate reservoir capacities summedtogether will be an approximation of the totalactive reservoir storage requirement needed toprovide those desired within-year period yields.


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Assume the annual yield produced andreleased by the over-year reservoir is distributedin each of the within-year periods in the sameratio as the average within-year inflows dividedby the total average annual inflow. Let the ratioof the average period t inflow divided by the totalannual inflow be βt. The general within-yearmodel is to find the minimum within-year storagecapacity, Ka

w, subject to within-year storage vol-ume continuity and capacity constraints.

Minimize Kwa ð11:23Þ

subject to

st þ btYp � ypt ¼ stþ 1 8t T þ 1 ¼ 1

ð11:24Þ

Table 11.3 Active Storagerequirements for various within-year yields

Fig. 11.7 Approximating a combined over-year andwithin-year reservoir as two separate reservoirs, one forcreating annual yields, the other for distributing them asdesired in the within-year periods


st �Kwa 8t ð11:25Þ

Since the sum of βt over all within-year peri-ods t is 1, the model guarantees that the sum ofthe unknown within-year yields, ypt, equals theannual yield, Yp.

The over-year model, Eqs. 11.17–11.19, andwithin-year model, Eqs. 11.23–11.25, can becombined into a single model for an n-yearsequence of flows

Minimize Ka ð11:26Þ

subject to

Sy þQy � Yp � Ry ¼ Syþ 1 8yif y ¼ n; yþ 1 ¼ 1

ð11:27Þ

Sy �Koa 8y ð11:28Þ

st þ btYp � ypt ¼ stþ 1 8tif t ¼ T ; T þ 1 ¼ 1

ð11:29Þ

st �Kwa 8t ð11:30Þ

Xt

ypt ¼ Yp ð11:31Þ

Ka �Koa þKw

a ð11:32Þ

Constraint 11.31 is not required due toEq. 11.29, but is included here to make it clearthat the sum of within-year yields will equal theover-year yield. Such a constraint will berequired for each yield of reliability p if multipleyields of different reliabilities are included in themodel. In addition, constraint Eq. 11.30 can becombined with Eq. 11.32, saving a constraint. Ifthis is done, the combined model contains2n + 2T + 1 constraints, compared to the moreaccurate model, Eqs. 11.20–11.22, that contains2nT constraints.

If the fractions βt are based on the ratios of theaverage within-year inflow divided by average

annual inflow in the two within-year periodsshown in Table 11.2, 0.25 of the total annualyield flows into the fictitious within-year reser-voir in period t = 1, and 0.75 of the total annualyield flows into the reservoir in period t = 2.Suppose the two desired within-year yields are tobe 3 and 0 for periods 1 and 2, respectively. Thetotal annual yield, Y0.9, is 3. Assuming the nat-ural distribution of this annual yield of 3 inperiod 1 is 0.25 Y0.9 = 0.75, and in period 2 it is0.75 Y0.9 = 2.25, the within-year storage requiredto redistribute these yields of 0.75 and 2.25 tobecome 3 and 0, respectively, is Ka

w = 2.25.From Tables 11.2 or 11.3 we can see that anannual yield of 3 requires an over-year storagecapacity of 3. Thus, the estimated total storagecapacity required to provide yields of 3 and 0 inperiods 1 and 2 is the over-year capacity of 3plus the within-year capacity of 2.25 equaling5.25. This compares with 3 plus 2.5 of actualwithin-year capacity required for a total of 5.50,as indicated in Table 11.3.

There are ways to reduce the number ofover-year constraints without changing thesolution of the over-year model. Sequences ofyears whose annual inflow values equal orexceed the desired annual yield can be combinedinto one constraint. If the yield is an unknownvariable then the mean annual inflow can be usedas the desired annual yield since it is the upperlimit of the annual yield. For example inTable 11.2 note that the last three years and thefirst year have flows equal or greater than 4, themean annual inflow. Thus, these four successiveyears can be combined into a single continuityequation

S7 þQ7 þQ8 þQ9 þQ1 � 4Yp � R7 ¼ S2

ð11:33Þ

This saves a total of 3 over-year continuityconstraints and 3 over-year capacity constraints.Note that the excess release, R7, represents theexcess release in all four periods. Furthermore,not all reservoir capacity constraint Eq. 11.28 areneeded, since the initial storage volumes in theyears following low flows will probably be lessthan the over-year capacity.


There are many ways to modify and extendthis yield model to include other objectives, fixedratios of the unknown annual yield for eachwithin-year period, and even multiple yieldshaving different exceedance probabilities p.

The number of over-year periods being mod-eled compared to the number of years of flowrecords determines the highest exceedanceprobability or reliability a yield can have; e.g.,9/10 or 0.9 in the 9-year example used here. Ifyields having lower reliabilities are desired, suchas a yield with a reliability of 0.80, then the yieldvariable YP can be omitted from Eq. 11.27 in thatcritical year that determines the requiredover-year capacity for a 0.90 reliable yield.(Since some outflow might be expected, even if itis less than the 0.90 reliable yield, the outflowcould be forced to equal the inflow for that year.)If a 0.70 reliable yield is desired, then the yieldvariables in the two most critical years can beomitted from Eq. 11.27, and so on.

The number of years of yield failure deter-mines the estimated reliability of each yield. Anannual yield that fails in f years has an estimatedprobability (n − f)/(n + 1) of being equaled orexceeded in any future year. Once the desiredreliability of a yield is known, the problem is toselect the appropriate failure years and to specifythe permissible extent of failure in those f failureyears.

To consider different yield reliabilities p letthe parameter αy

p be a specified value between 0and 1 that indicates the extent of a failure in yeary associated with an annual yield having a reli-ability of p. When αy

p is 1 there is no failure, andwhen it is less than 1 there is a failure, but aproportion of the yield Yp equal to αy

p is released.Its value is in part dependent on the conse-quences of failure and on the ability to forecastwhen a failure may occur and to adjust thereservoir operating policy accordingly.

Theover-year storage continuity constraintsfor n years can now be written in a form appro-priate for identifying any single annual yield Yphaving an exceedance probability p.

Sy þQy � apyYp � Ry ¼ Syþ 1 8y ify ¼ n; yþ 1 ¼ 1

ð11:34Þ

When writing Eq. 11.34, the failure year oryears should be selected from among those inwhich permitting a failure decreases the requiredreservoir capacity Ka. If a failure year is selectedthat has an excess release, no reduction in therequired active storage capacity will result, andthe reliability of the yield may be higher thanintended.

The critical year or years that determine therequired active storage volume capacity may bedependent on the yield itself. Consider, forexample, the 7-year sequence of annual flows (4,3, 3, 2, 8, 1, 7) whose mean is 4. If a yield of 2 isdesired in each of the 7 years, the critical yearrequiring reservoir capacity is year 6. If a yield of4 is desired (again assuming no losses), thecritical years are years 2–4. The streamflows andyields in these critical years determine therequired over-year storage capacity. The failureyears, if any, must be selected from within thecritical low-flow periods for the desired yield.

When the magnitudes of the yields areunknown, some trial and error solutions may benecessary to ensure that any failure years arewithin the critical period of years for the associ-ated yields. To ensure a wider range of applicableyield magnitudes, the year having the lowest flowwithin the critical period should be selected as thefailure year if only one failure year is selected.Even though the actual failure year may followthat low-flow year, the resulting required reser-voir storage volume capacity will be the same.

Multiple Yields and Evaporation LossesThe yield models developed so far define onlysingle annual and within-year yields. Incrementalsecondary yields having lower reliabilities canalso be included in the model. Referring to the9-year streamflow record in Table 11.3, assumethat two yields are desired, one 90% reliable andthe other 70% reliable. Let Y0.9 and Y0.7 representthose annual yields having reliabilities of 0.9 and


0.7, respectively. The incremental secondaryyield Y0.7 represents the amount in addition toY0.9 that is only 70% reliable. Assume that theproblem is one of estimating the appropriatevalues of Y0.9 and Y0.7, their respectivewithin-year allocations ypt and the total activereservoir capacity Ka that maximizes somefunction of these yield and capacity variables.

In this case the over-year andwithin-yearcontinuity constraints can be written

Sy þQy � Y0:9 � a0:7y Y0:7 � Ry ¼ Syþ 1 8yif y ¼ n; yþ 1 ¼ 1

ð11:35Þ

st þ btðY0:9 þ Y0:7Þ � y0:9;t � y0:7;t ¼ stþ 1 8tif t ¼ T ; T þ 1 ¼ 1

ð11:36Þ

Now an additional constraint is needed toinsure that each within-year yield of a reliabilityp adds up to the annual yield of the same relia-bility. Selecting the 90% reliable yield,

Xt

y0:9;t ¼ Y0:9 ð11:37Þ

Evaporation losses must be based on anexpected storage volume in each period and yearsince the actual storage volumes are not identi-fied using these yield models. The approximatestorage volume in any period t in year y canbe defined as the initial over-year volume Sy,plus the estimated average within-year volume(st + st+1)/2. Substituting this storage volumeinto Eq. 11.14 (see also Fig. 11.6) results in anestimated evaporation loss Lyt.

Lyt ¼ ao þ aðSy þðst þ stþ 1Þ=2Þ� �

Emaxt

ð11:38Þ

Summing Lyt over all within-year periods tdefines the estimated annual evaporation loss, Ey.

Ey ¼Xt

ao þ aðSy þðst þ stþ 1Þ=2Þ� �

Emaxt

ð11:39Þ

This annual evaporation loss applies, ofcourse, only when there is a nonzero activestorage capacity requirement. These annualevaporation losses can be included in theover-year continuity constraints, such asEq. 11.35. If they are, the assumption is beingmade that their within-year distribution will bedefined by the fractions βt. This may not berealistic. If it is not, an alternative would be toinclude the average within-year period losses, Lt,in the within-year constraints.

The average within-year period loss, Lt, canbe defined as the sum of each loss Lyt defined byEq. 11.38 over all years y divided by the totalnumber of years, n.

Lt ¼Xny

ao þ aðSy þðst þ stþ 1Þ=2Þ� �

Emaxt =n

ð11:40Þ

This average within-year period loss, Lt, canbe added to the within-year’s highest reliabilityyield, ypt, forcing greater total annual yields of allreliabilities to meet corresponding totalwithin-year yield values. Hence, combiningEq. 11.37 and 11.38, for p equal to 0.9 in theexample,

Yp ¼Xt

ypt þXny

½ao þ aðSy þðstþ 1Þ=2Þ�Emaxt =n

( )

ð11:41Þ

Since actual reservoir storage volumes in eachperiod t of each year y are not identified in thismodel, system performance measures that arefunctions of those storage volumes, such ashydroelectric energy or reservoir recreation, areonly approximate. Thus, as with any of thesescreening models, any set of solutions should be


evaluated and further improved using more pre-cise simulation methods.

Simulation methods require reservoir operat-ing rules. The information provided by thesolution of the yield model can aid in defining areservoir operating policy for such simulationstudies.

Reservoir Operation RulesReservoir operation rules are guides for thoseresponsible for reservoir operation. They apply toreservoirs being operated in a steady-state con-dition (i.e., not filling up immediately after con-struction or being operated to meet a set of newand temporary objectives). There are severaltypes of rules but each indicates the desired orrequired reservoir release or storage volumes atany particular time of year. Some rules identifystorage volume targets (rule curves) that theoperator is to maintain, if possible, and othersidentify storage zones, each associated with aparticular release policy. This latter type of rulecan be developed from the solution of the yieldmodel.

To construct an operation rule that identifiesstorage zones, each having a specific releasepolicy, the values of the dead and flood storagecapacities, KD and Kf are needed together withthe over-year storage capacity, Ka

o, and within -year storage volumes st in each period t. Sinceboth Ka

o and all st derived from the yield model

are for all yields, Yp, being considered, it isnecessary to determine the over-year capacitiesand within-year storage volumes required toprovide each separate within-year yield ypt.Plotting the curves defined by the respectiveover-year capacity plus the within-year storagevolume (Ka

o + st) in each within-year period t willdefine a zone of storage whose yield releasesypt from that zone should have a reliability of atleast p.

For example, assume again a 9-year flowrecord and 10 within-year periods. Of interest arethe within-year yields, y0.9,t and y0.7,t, havingreliabilities of 0.9 and 0.7. The first step is tocompute the over-year storage capacity require-ment, Ka

o, and the within-year storage volumes,st, for just the yields y0.9,t. The sum of thesevalues, Ka

o + st, in each period t can be plotted asillustrated in Fig. 11.8.

The sum of the over-year capacity andwithin-year volume Ka

o + st in each period t de-fines the zone of active storage volumes for eachperiod t required to supply the within-year yieldsy0.9,t. If the storage volume is in this shaded zoneshown in Fig. 11.8, only the yields y0.9,t shouldbe released. The reliability of these yields, whensimulated, should be about 0.9. If at any timet the actual reservoir storage volume is withinthis zone, then reservoir releases should notexceed those required to meet the yield y0.9,t ifthe reliability of this yield is to be maintained.

Fig. 11.8 Reservoirrelease rule showing theidentification of the mostreliable release zoneassociated with thewithin-year yields y0.9,t


The next step is to solve the yield model forboth yields Y0.9 and Y0.7. The resulting sum ofover-year storage capacity and within-year stor-age volumes can be plotted over the first zone, asshown in Fig. 11.9.

If at any time t the actual storage volume is inthe second lighter shaded zone in Fig. 11.9, boththe release should be the sum of the most reliableyield, y0.9,t and the incremental secondary yieldy0.7,t. If only these releases are made, the prob-ability of being in that zone, when simulated,should be about 0.7. If the actual storage volumeis greater than the total required over-year stor-age capacity Ka

o plus the within-year volume st,the non-shaded zone in Fig. 11.23, then a release

can be made to satisfy any downstream demand.Converting storage volume to elevation, thisrelease policy is summarized in Fig. 11.10.

These yield models focus only on the activestorage capacity requirements. They can be a partof a model that includes flood storage require-ments as well (as previously discussed in thischapter). If the actual storage volume is withinthe flood control zone in the flood season,releases should be made to reduce the actualstorage to a volume no greater than the totalcapacity less the flood storage capacity.

Once again, reservoir rules developed fromsimplified models such as this yield model areonly guides, and once developed these rules

Fig. 11.9 Reservoirrelease rule showing theidentification of the secondmost reliable release zoneassociated with the totalwithin-year yields y0.9,t + y0.7,t

Fig. 11.10 Reservoirrelease rule defined by theyield model


should be simulated, evaluated, and refined priorto their actual adoption.

11.6 Drought and Flood RiskReduction

11.6.1 Drought Planningand Management

Droughts are natural hazards that unlike floods,tornadoes, and hurricanes, occur slowly andgradually over a period of time. The absence of aprecise drought threshold introduces someuncertainty about whether a drought exists and, ifit does, its degree of severity. The impacts ofdrought are nonstructural and typically spreadover a larger geographical area than are damagesresulting from other natural hazards. All of thesedrought characteristics have impacted the devel-opment of effective drought preparedness plans.

Droughts result from a deficiency of precipi-tation compared to normal (long-term average)amounts that, when extended over a season orespecially over a longer period of time, isinsufficient to meet the demands of humanactivities and the environment. All types ofdrought results in water shortages for one ormore water-using activities.

Droughts differ from one another in threeessential characteristics: intensity, duration, andspatial coverage. Moreover, many disciplinaryperspectives of drought exist. Because of thesenumerous and diverse disciplinary views, con-fusion often exists over exactly what constitutesa drought. Regardless of such disparate views,the overriding feature of drought is its negativeimpacts on people and the environment.

11.6.1.1 Drought TypesDroughts are normally distinguished by type:meteorological, hydrological, agricultural, andsocioeconomic. Meteorological drought isexpressed solely on the basis of the degree ofdryness in comparison to some normal or aver-age amount and the duration of the dry period.Drought intensity and duration are the keydescriptors of this type of drought. Agricultural

drought links various characteristics of meteo-rological drought to agricultural impacts, focus-ing on precipitation shortages, differencesbetween actual and potential evapotranspiration,and soil water deficits.

Hydrological droughts are described based onthe effects of low precipitation on surface orsubsurface water supply (e.g., streamflow,reservoir storage, lake levels, and groundwater )rather than with precipitation shortfalls. Hydro-logical droughts usually lag the occurrence ofmeteorological and agricultural droughts becausemore time elapses before precipitation deficien-cies are detected in rivers, reservoirs, ground-water aquifers, and other components of thehydrologic system. As a result, hydrologicaldroughts are typically detected later than otherdrought types. Water uses affected by droughtcan include multiple purposes such as powergeneration, flood control, irrigation, domesticdrinking water, industry, recreation, and ecosys-tem preservation.

Socioeconomic droughts are linked directly tothe supply of some economic good. Increases inpopulation can alter substantially the demand forthese economic goods over time. The incidenceof socioeconomic drought can increase becauseof a change in the frequency of meteorologicaldrought, a change in societal vulnerability towater shortages, or both. For example, poor landuse practices such as overgrazing can decreaseanimal carrying capacity and increase soil ero-sion, which exacerbates the impacts of, andvulnerability to, future droughts.

11.6.1.2 Drought ImpactsThe impacts of drought are often widespreadthrough the economy. They can be direct andindirect. Restrictions in water use resulting fromdrought is a direct or first-order impact ofdrought. However, the consequences of suchrestrictions could result in loss of income, farmand business foreclosures, and government reliefprograms) are possible indirect second- orthird-order impacts.

The impacts of drought appear to be increas-ing in both developing and developed countries,which in many cases reflects the persistence of


non-sustainable development and populationgrowth. Lessening the impacts of future droughtevents typically requires the development ofdrought risk policies that emphasize a wide rangeof water conservation and early warning mea-sures. Drought management techniques are oftenconditional on the severity of the drought. Iden-tifying the actions to take and the thresholdsindicating when to take them are best accom-plished prior to a drought, as agreements amongstakeholders are easier to obtain when individu-als are not having to deal with the impacts of anongoing drought.

Drought impacts can be economic, environ-mental, and social.

Economic impacts can include direct losses toagricultural and industrial users, losses in recre-ation, transportation, and energy sectors. Otherindirect economic impacts can include resultingunemployment and loss of tax revenue to local,state, and federal governments.

Environmental losses include damages toplant and animal species in natural habitats, andreduced air and water quality; an increase inforest and range fires; the degradation of land-scape quality; and possible soil erosion. Theselosses are difficult to quantify, but growingpublic awareness and concern for environmentalquality has forced public officials to focus greaterattention on them.

Social impacts can involve public safety,health, conflicts among water users, and inequi-ties in the distribution of impacts and disasterrelief programs. As with all natural hazards, theeconomic impacts of drought are highly variablewithin and among economic sectors and geo-graphic regions, producing a complex assortmentof winners and losers with the occurrence of eachdisaster.

11.6.1.3 Drought Preparednessand Mitigation

Droughts happen, and it makes no sense to waituntil realizing a drought is happening beforepreparing plans and policies to mitigate theadverse impacts from a drought. As evidenced bythe ongoing drought (at this writing) in Califor-nia, and the even more severe drought those in

southeastern Australia recently witnessed,drought management has to involve the institu-tions that not only manage water supply systems,but all those who use water, and all those whomake land-use decisions that impact water run-off. It can involve hydrologic modeling methodsdiscussed in Chap. 6, and reservoir modeling asdiscussed in Chaps. 4 and 8. Appendix C of thisbook (contained on a disk or downloadable fromthe web) discusses drought management model-ing methods and options in more detail.

11.6.2 Flood Protectionand DamageReduction

Next consider the other extreme—floods. Twotypes of structural alternatives are often used forflood risk reduction. One is the provision of floodstorage capacity in reservoirs designed to reducedownstream peak flood flows. The other is channelenhancement and/or flood-proofing structures thatare designed to contain peak flood flows andreduce damage. This section introduces methodsof modeling both of these alternatives for inclusionin either benefit–cost or cost-effectiveness analy-ses. The latter analyses apply to situations in whicha significant portion of the flood control benefitscannot be expressed in monetary terms and the aimis to provide a specified level offlood protection atminimum cost.

The discussion will first focus on the estima-tion of flood storage capacity in a single reservoirupstream of a potential flood damage site. Thisanalysis will then be expanded to includedownstream channel capacity improvements.Each of the modeling methods discussed will beappropriate for inclusion in multipurpose riverbasin planning (optimization) models havinglonger time step durations than those required topredict flood peak flows.

11.6.2.1 Reservoir Flood StorageCapacity

In addition to the active storage capacities in areservoir, some capacity may be allocated for thetemporary storage of flood flows during certain


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periods in the flood season of the year, as shownin Fig. 11.2. Flood flows usually occur over timeintervals lasting from a few hours up to a fewdays or weeks. Computational limitations make itimpractical to include such short time durationsin many of our multipurpose planning modelsthat typically include time periods of a week, or10 days, or months or seasons spanning severalmonths. If we modeled these short daily orhourly durations, flood routing equations wouldhave to be included in the model; a simple massbalance would not be sufficient. Neverthelessthere are ways of including unknown floodstorage variables within longer period optimiza-tion models .

Consider a potential flood damage site alonga river. A flood control reservoir can be builtupstream of that potential damage site. Thequestion is how much flood storage capacity, ifany, should the reservoir contain. For variousassumed capacities and operating policies, sim-ulation models can be used to predict the impacton the downstream flood peaks. These hydraulicsimulation models must include flood routingprocedures from the reservoir to the downstreampotential damage site and the flood control op-erating policy at the reservoir. For variousdownstream flood peaks, water elevations andassociated economic flood damages on thefloodplain can be estimated. To calculate theexpected annual damages associated with anyupstream reservoir capacity , the probability ofvarious damage levels being exceeded in anyyear needs to be calculated.

The likelihood of a flood peak of a givenmagnitude or greater is often described by itsexpected return period. How many years wouldone expect to wait, on average, to observeanother flood of equal or greater than a flood ofsome specified magnitude? This is the reciprocalof the probability of observing such a flood orgreater in any given year. A T-year flood has aprobability of being equaled or exceeded in anyyear of 1/T. This is the probability that could becalculated by adding up the number of years anannual flood of a given or greater magnitudeis observed, say in 1000 or 10,000 years,

divided by 1000 or 10,000, respectively.A one-hundred-year flood or greater has aprobability of 1/100 or 0.01 of occurring in anygiven year. Assuming annual floods are inde-pendent, if a 100-year flood occurs this year, theprobability that a flood of that magnitude orgreater occurring next year remains 1/100 or0.01.

If PQ is the random annual peak flood flowand PQT is a particular peak flood flow having areturn period of T years, then by definition theprobability of an actual flood of PQ equaling orexceeding PQT is 1/T.

Pr½PQ�PQT ¼ 1=T � ð11:42Þ

The higher the return period, i.e., the moresevere the flood, the lower the probability that aflood of that magnitude or greater will occur.Equation 11.42 is plotted in Fig. 11.11.

The exceedance probability distributionshown in Fig. 11.11 is simply 1 minus thecumulative distribution function FPQ(�) of annualpeak flood flows. The area under the function isthe mean annual peak flood flow, E[PQ].

The expected annual flood damage at apotential flood damage site can be estimatedfrom an exceedance probability distribution ofpeak flood flows at that potential damage sitetogether with a flow or stage damage function.The peak flow exceedance distribution at anypotential damage site will be a function of the

E020

121u

Pr [

PQ >

PQ

]

T

peak flood flow

00

1

1/T

PQT

Fig. 11.11 Probability of annual peak flood flows beingexceeded

11.6 Drought and Flood Risk Reduction 491

upstream reservoir flood storage capacity Kf

and the reservoir operating policy.The probability that flood damage of FDT

associated with a flood of return period T will beexceeded is precisely the same as the probabilitythat the peak flow PQT that causes the damagewill be exceeded. Letting FD be a random flooddamage variable, its probability of exceedance is

Pr½FD� FDT � ¼ 1=T ð11:43Þ

The area under this exceedance probabilitydistribution is the expected annual flood damage, E[FD].

E½FD� ¼Z10

Pr½FD� FDT �dFDT ð11:44Þ

This computational process is illustratedgraphically in Fig. 11.12. The analysis requiresthree input functions that are shown in quadrants(a), (b), and (c). The dashed-line rectanglesdefine point values on the three input functions inquadrants (a), (b), and (c) and the correspondingprobabilities of exceeding a given level of dam-ages in the lower right quadrant (d). The distri-bution in quadrant (d) is defined by theintersections of these dashed-line rectangles. Thisdistribution defines the probability of equaling orexceeding a specified damage in any given year.The (shaded) area under the derived function isthe annual expected damage, E[FD].

The relationships between flood stage anddamage, and flood stage and peak flow, definedin quadrants (a) and (b) of Fig. 11.12, must beknown. These do not depend on the flood storage

Fig. 11.12 Calculation of the expected annual flooddamage shown as the shaded area in quadrant (d) derivedfrom the expected stage damage function (a), the expected

stage-flow relation (b), and the expected probability ofexceeding an annual peak flow (c)


capacity in an upstream reservoir. The infor-mation in quadrant (c) is similar to that shown inFig. 11.11 defining the exceedance probabilitiesof each peak flow. Unlike the other three func-tions, this distribution depends on the upstreamflood storage capacity and flood flow releasepolicy. This peak flow probability of exceedancedistribution is determined by simulating theannual floods entering the upstream reservoir inthe years of record.

The difference between the expected annualflood damage without any upstream flood storagecapacity and the expected annual flood damageassociated with a flood storage capacity of Kf isthe expected annual flood damage reduction.This is illustrated in Fig. 11.13. Knowing theexpected annual damage reduction associatedwith various flood storage capacities, Kf, permitsthe definition of a flood damage reductionfunction, Bf(Kf).

If the reservoir is a single purpose floodcontrol reservoir , the eventual tradeoff isbetween the expected flood reduction benefits,Bf(Kf), and the annual costs, C(Kf), of thatupstream reservoir capacity. The particularreservoir flood storage capacity that maximizesthe net benefits, Bf(Kf) − C(Kf), may be appro-priate from a national economic efficiency per-spective but it may not be best from a localperspective. Those occupying the potentialdamage site may prefer a specified level of pro-tection from that reservoir storage capacity,rather than the protection that maximizesexpected annual net benefits, Bf(Kf) – C(Kf).

If the upstream reservoir is to serve multiplepurposes, say for water supply, hydropower, andrecreation, as well as for flood control, theexpected flood reduction benefit function justderived could be a component in the overallobjective function for that reservoir.

(a)(b)

(c) (d)

Fig. 11.13 Calculation of expected annual flood damage reduction benefits, shown as the darkened portion ofquadrant (d), associated with a specified reservoir flood storagecapacity


Total reservoir capacity K will equal the sumof dead storage capacity Kd, active storagecapacity Ka, and flood storage capacity Kf,assuming they are the same in each period t. Insome cases they may vary over the year. If therequired active storage capacity can occupy theflood storage zone when flood protection is notneeded, the total reservoir capacity K will be thedead storage, Kd, plus the maximum of either(1) the actual storage volume and flood storagecapacity in the flood season or (2) the actualstorage volume in non-flood season.

K�Kd þ St þKf for all periods t in flood season

plus the following period that

represents the end of

the flood season

ð11:45Þ

K �Kd þ St for all remaining periods t

ð11:46Þ

In the above equations the dead storagecapacity, Kd, is assumed known. It is included inthe capacity Eqs. 11.45 and 11.46 assuming thatthe active storage capacity is greater than zero.Clearly, if the active storage capacity were zero,there would be no need for dead storage.

11.6.2.2 Channel CapacityThe unregulated natural peak flow of a particulardesign flood at a potential flood damage site can bereduced by upstream reservoir flood storagecapacity or it can be contained within the channelat the potential damage site by levees and otherchannel-capacity improvements. In this section,the possibility of levees or dikes and other channelcapacity or flood-proofing improvements at adownstream potential damage site will be consid-ered. The approach used will provide a means ofestimating combinations of flood control storagecapacity in upstream reservoirs and downstreamchannel capacity improvements that together willprovide a prespecified level of flood protection atthe downstream potential damage site.

Let QNT be the unregulated natural peak flowin the flood season having a return period ofT years. Assume that this peak flood flow is thedesign flood for which protection is desired. Toprotect from this design peak flow, a portionQS ofthe peak flow may be reduced by upstream floodstorage capacity. The remainder of the peak flowQRmust be contained within the channel. Hence ifthe potential damage site s is to be protected froma peak flow of QNT, the peak flow reductions dueto upstream storage, QS, and channel improve-ments, QR, must at least equal to that peak flow.

QNT �QSþQR ð11:47Þ

The extent to which a specified upstreamreservoir flood storage capacity reduces the de-sign peak flow at the downstream potentialdamage site can be obtained by routing thedesign flood through the reservoir and thechannel between the reservoir and the down-stream site. Doing this for a number of reservoirflood storage capacities permits the definition ofa peak flow reduction function, fT(Kf).

QS ¼ fTðKf Þ ð11:48Þ

This function is dependent on the relativelocations of the reservoir and the downstreampotential damage site, on the characteristics andlength of the channel between the reservoir anddownstream site, on the reservoir flood controloperating policy, and on the magnitude of thepeak flood flow.

An objective function for evaluating these twostructural flood control measures should includethe cost of reservoir flood storage capacity,CostK(Kf), and the cost of channel capacityimprovements, CostR(QR), required to contain aflood flow of QR. For a single purpose, singledamage site, single reservoir flood control prob-lem, the minimum total cost required to protectthe potential damage site from a design flood peakof QNT, may be obtained by solving the model:

minimize CostK Kf

� �þCostRðQRÞ ð11:49Þ


subject to

QNT � fTðKf ÞþQR ð11:50Þ

Equations 11.49 and 11.50 assume that adecision will be made to provide protection froma design flood QNT of return period T; it is only aquestion of how to provide the required protec-tion, i.e., how much flood storage capacity andhow much levee protection.

Solving Eqs. 11.47 and 11.48 for peak flowsQNT of various return periods T will identify therisk-cost tradeoff. This tradeoff function mightlook like what is shown in Fig. 11.14.

11.6.2.3 Estimating Risk of LeveeFailures

Levees are built to reduce the likelihood offlooding on the flood plain. Flood flows pre-vented from flowing over a floodplain due to alevee will have relatively little effect on users ofthe flood plain, unless of course the levee fails tocontain the flow. Levee failure can result fromflood events that exceed (overtop) the designcapacity of the levee. Failure can also result fromvarious types of geostructural weaknesses. If anyof the flow in the stream or river channel passesover, through or under the levee and onto theflood plain, the levee is said to fail. The proba-bility of levee failure along a river reach is in parta function of the levee height, the probabilitydistribution of flood flows in the stream or riverchannel, and the probability of geo-structural

failure. The latter depends in part on how wellthe levee and its foundation is constructed. Somelevees are purposely designed to ‘‘fail’’ at certainsites at certain flood stages to reduce the likeli-hood of more substantial failures and flooddamages further downstream.

The probability of levee failure given the floodstage (height) in the stream or river channel isoften modeled using two flood stages. The USArmy Corps of Engineers calls the lower stage theprobable non-failure point, PNP, and the higherstage is called the probable failure point, PFP(USACE 1991). At the PNP, the probability offailure is assumed to be 15%. Similarly, theprobability of failure at the PFP is assumed to be85%. A straight-line distribution between thesetwo points is also assumed, as shown inFig. 11.15. Of course these points and distribu-tions are at best only guesses, as not many, if any,data will exist to base them on at any given site.

To estimate the risk of a flood in the flood-plain protected by a levee due to overtopping orgeo-structural levee failure, the relationshipsbetween flood flows and flood stages in thechannel and on the floodplain must be defined,just as it had to be to carry out the analysesshown in Figs. 11.12 and 11.13.

Assuming no geo-structural levee failure, theflood stage in the floodplain protected by a leveeis a function of the flow in the stream or riverchannel, the cross sectional area of the channelbetween the levees on either side, the channelslope and roughness, and the levee height. Iffloodwaters enter the floodplain, the resultingwater level or stage in the floodplain will dependon the topological characteristics of the floodplain. Figure 11.16 illustrates the relationshipbetween the flood stage in the channel and theflood stage in the flood plain, assuming nogeo-structural failure of the levee. Obviouslyonce the flood begins overtopping the levee, theflood stage in the flood plain begins to increase.Once the flood flow is of sufficient magnitudethat its stage without the levee is the same as thatwith the levee, the existence of a levee has only anegligible impact on the flood stage.

Figure 11.17 illustrates the relationshipbetween flood flow and flood stage in a

Fig. 11.14 Tradeoff between minimum cost of floodprotection and flood risk, as expressed by the expectedreturn period


floodplain with and without flood levees, againassuming no geo-structural levee failure.

Combining Figs. 11.16 and 11.17 defines therelationship between reach flow and channelstage. This is illustrated in the upper left quadrantof Fig. 11.18.

Combining the relationship between floodflow and flood stage in the channel (upper leftquadrant of Fig. 11.18) with the probability dis-tribution of levee failure (Fig. 11.15) and theprobability distribution of annual peak flowsbeing equaled or exceeded (Fig. 11.11), provides

Fig. 11.15 Assumed cumulative probability distribution of levee failure expressed as function of flood stage in riverchannel

Fig. 11.16 Influence of a levee on the flood stage infloodplain compared to stream or river flood stage. Thechannel flood stage where the curve is vertical is the stage

at which the levee fails due to overtopping or fromgeo-structural causes


an estimate of the expected probability of leveefailure. Figure 11.19 illustrates this process offinding, in the lower right quadrant, the shadedarea that equals the expected annual probabilityof levee failure from overtopping and/orgeo-structural failure.

The channel flood-stage function, S(q), ofpeak flow q shown in the upper left quadrant ofFig. 11.19 is obtained from the upper left

quadrant of Fig. 11.18. The probability of leveefailure, PLF(S), a function of flood stage, S(q),shown in the upper right quadrant is the same asin Fig. 11.15. The annual peak flow exceedanceprobability distribution, FQ(q), (or its inverseQ(p)) in the lower left quadrant is the same asFig. 11.12 or that in the lower left quadrant (c) ofFig. 11.13. The exceedance probability functionin the lower right quadrant of Fig. 11.19 is

Fig. 11.17 Relationshipbetween flood flow andflood stage in a floodplainwith and without floodlevees, again assuming nogeo-structural levee failure

Fig. 11.18 Deriving therelation (shown in theupper left quadrant)between flood flow andflood stage in the channel


derived from each of the other three functions, asindicated by the arrows, in the same manner asdescribed in Fig. 11.12.

In mathematical terms, the annual expectedprobability of levee failure, E[PLF], found in thelower right quadrant of Fig. 11.19, equals

E½PLF� ¼Z10

PLF½SðqÞ�f ðqÞdq

¼Z1

0

PLF½SðQðpÞÞ�dp ¼Z1

0

PLF0ðpÞdp;

ð11:51Þ

where PLF′(p) is the probability of levee failureassociated with a flood stage of S(q) having anexceedance probability of p.

Note that if the failure of the levee was onlydue to channel flood stages exceeding the leveeheight (i.e., if the probability of geo-structuralfailure were zero) the expected probability oflevee failure would be simply the probability of

exceeding a channel flow whose stage equals thelevee height, as defined in the lower left quadrantof Fig. 11.19. This is shown in Fig. 11.20.

Referring to Fig. 11.20, if the levee height isincreased, the horizontal part of the curve in theupper right quadrant would rise, as would thehorizontal part of the curve in the upper leftquadrant as it shifts to the left. Hence given thesame probability distribution as defined in thelower left quadrant, the expected probability ofexceeding an increased levee capacity woulddecrease, as it should.

11.6.2.4 Annual Expected Damagefrom Levee Failure

A similar analysis can provide an estimate of theexpected annual flood plain damage for a streamor river reach. Consider, for example, a parcel ofland on a flood plain at some location i. If aneconomic efficiency objective were to guide thedevelopment and use of this parcel, the ownerwould want to maximize the net annual eco-nomic benefits derived from its use, Bi, less the

Fig. 11.19 Derivation ofthe probability ofexceeding a givenprobability of levee failure,shown in lower rightquadrant. The shaded areaenclosed by this probabilitydistribution is the annualexpected probability oflevee failure


annual (non-flood damage) costs, Ci, and theexpected annual flood damages, EADi. The issueof concern here is the estimation of theseexpected annual flood damages.

Damages at location i resulting from a floodwill depend in part on the depth of flooding atthat location and a host of other factors (floodduration, velocity of and debris in flood flow,time of year, etc.). Assume that the flood damageat location i is a function of primarily the floodstage, S, at that location. Denote this potentialdamage function as Di(S). Such a function isillustrated in Fig. 11.21.

Integrating the product of the annual excee-dance probability of flood stage, Fs(S), and thepotential damages, Di(S), over all stages S willyield the annual expected damages, E[Di], forland parcel i.

E½Di� ¼Z

DiðSÞFsðSÞdS ð11:52Þ

The sum of these expected damage estimatesover all the parcels of land i on the floodplain isthe total expected damage that one can expecteach year, on average, on the floodplain.

EAD ¼Xi

E½Di� ð11:53Þ

Fig. 11.20 Calculation ofannual probability ofequaling or exceeding anyspecified probability oflevee overtopping, shownin the lower right quadrant.The shaded area in thisquadrant is the expectedprobability of leveeovertopping, assumingthere is no geo-structuralfailure

Fig. 11.21 An example function defining the damagesthat will occur given any flood stage S to a parcel of land i


Alternatively the annual expected flood dam-age could be based on a calculated probability ofexceeding a specified flood damage, as shown inFig. 11.12. For this method the potential flooddamages, Di(S), are determined for various stagesS and then summed over all land parcels i foreach of those stage values S to obtain the totalpotential damage function, D(S), for the entirefloodplain, defined as a function of flood stage S.

DðSÞ ¼Xi

DiðSÞ ð11:54Þ

This is the function shown in quadrant (a) inFig. 11.12.

Levee failure probabilities, PLF′(p), based onthe exceedance probability p of peak flows, orstages, as defined in Fig. 11.31 and Eq. 11.49can be included in calculations of expectedannual damages. Expressing the damage func-tion, D(S), as a function, D′(p), of the stageexceedance probability p and multiplying thisflood damage function D′(p) times the probabil-ity of levee failure, PLF′(p) defines the jointexceedance probability of expected annual dam-ages. Integrating over all values of p yields theexpected annual flood damage, EAD.

EAD ¼Z1

0

D0ðpÞ PLF0ðpÞ dp ð11:55Þ

Note that the flood plain damages and prob-ability of levee failure functions in Eq. 11.55both increase with increasing flows or stages, butas peak flows or stages increase, their exceedanceprobabilities decrease. Hence with increasingp the damage and levee failure probabilityfunctions decrease. The effect of levees on theexpected annual flood damage, EAD, is shown inFig. 11.22. The “without levee” function in thelower right quadrant of Fig. 11.18, is D′(p). The“with levee” function is the product of D′(p) andPLF′(p). If the probability of levee failure, PLF′(p) function were as shown in Fig. 11.18, i.e., ifit were 1.0 for values of p below some overtop-ping stage associated with an exceedance prob-ability p*, and 0 for values of p greater than p*,

then the function would appear as shown “withlevee—overtop only” in Fig. 11.22.

11.6.2.5 Risk-Based AnalysesRisk-based analyses attempt to identify theuncertainty associated with each of the inputsused to define the appropriate capacities of var-ious flood risk reduction measures. There arenumerous sources of uncertainty associated witheach of the functions shown in quadrants (a), (b),and (c) in Fig. 11.25. This uncertainty translatesto uncertainty associated with estimates of floodrisk probabilities and expected annual flooddamage reductions obtained from reservoir floodstorage capacities and channel improvements.

Going to the substantial effort and cost ofquantifying these uncertainties, which them-selves will be surely be uncertain, does howeverprovide additional information. The design ofany flood protection plan can be adjusted toreflect attitudes of stakeholders toward theuncertainty associated with specified flood peakreturn periods or equivalently their probabilitiesof occurring in any given year.

For example, assume a probability distribu-tion capturing the uncertainty about the expectedprobability of exceedance of the peak flows at thepotential damage site (as shown in Fig. 11.12) isdefined from a risk-based-analysis. Figure 11.23shows that exceedance function together with its90% confidence bands near the higher flood peakreturn periods. To be, say, 90% sure that pro-tection is provided for the T-year return periodflow, PQT, one may have to for an equivalentexpected T + Δ year return period flow, PQT+Δ,i.e., the flow having a (1/T) − δ expected prob-ability of being exceeded. Conversely, protectionfrom the expected T + Δ year peak flood flowwill provide 90% assurance of protection fromflows that will occur less than once in T years onaverage.

If society wanted to eliminate flood damage itcould do it, but at a high cost. This would requireeither costly flood control structures or elimi-nating economic activities on lands subject topossible flooding. Both reduce expected eco-nomic returns from the floodplain. Hence suchactions are not likely to be taken. There will


always be a risk of flood damage. Analyses suchas those just presented help identify these risks.Risks can be reduced and managed but not

eliminated. Finding the best levels of flood pro-tection and flood risk, together with risk insur-ance or subsidies (illustrated in Fig. 11.24) is thechallenge for public and private agencies alike.Floodplain management is as much concernedwith good things not happening on them as withbad things—like floods—happening on them.

Fig. 11.22 Calculation ofexpected annual flooddamage taking into accountprobability of levee failure

E02 0

1 30

h

P r [

PQ >

PQ

]

T

peak flood flow

00

1

1/T

PQT PQT+

1/T-δ

90% confidenceestimates

Fig. 11.23 Portion of peak flow probability of excee-dance function showing contours containing 90% of theuncertainty associated with this distribution. To be 90%certain of protection from a peak flow of PQt, protectionis needed from the higher peak flow, PQt+Δ expected onceevery T + Δ years, i.e., with an annual probability of 1/(T + Δ) or (1/T) − δ of being equaled or exceeded

Fig. 11.24 Relationship between expected economicreturn from flood plain use and risk of flooding. Thelowest flood risk does not always mean the best risk, andwhat risk is acceptable may depend on the amount ofinsurance or subsidy provided when flood damage occurs


11.7 Hydroelectric PowerProduction

Hydropower plants, Fig. 11.25, convert theenergy from the flowing water to mechanical andthen electrical energy. These plants containingturbines and generators are typically locatedeither in or adjacent to dams. Pipelines (pen-stocks) carry water under pressure from thereservoir to the powerhouse. Power transmissionsystems transport the produced electrical energyfrom the powerhouse to where it is needed.

The principal advantages of using hydro-power are the absence of polluting emissionsduring operation, its capability to respond rela-tively quickly to changing utility load demands,and its relatively low operating costs. Disad-vantages can include high initial capital cost andpotential site-specific and cumulative

environmental impacts. Potential environmentalimpacts of hydropower projects include alteredflow regimes below storage reservoirs or withindiverted stream reaches, water quality degrada-tion, mortality of fish that pass through turbines,blockage of fish migration, and flooding of ter-restrial ecosystems by impoundments. However,in many cases, proper design and operation ofhydropower projects can mitigate some of theseimpacts. Hydroelectric projects can also provideadditional benefits such as from recreation inreservoirs or in tailwaters below dams.

Hydropower plants can be either conventionalor pumped storage. Conventional hydropowerplants use the available water from a river, stream,canal system, or reservoir to produce electricalenergy. In conventional multipurpose reservoirsand run-of-river systems, hydropower productionis just one of many competing purposes for which

Fig. 11.25 Hydropower system components


water resources may be used. Competing wateruses may include irrigation, flood control, navi-gation, downstream flow dilution for qualityimprovement, and municipal and industrial watersupply. Pumped storage plants pump the water,usually through a reversible turbine that acts as apump, from a lower supply source to an upperreservoir. While pumped storage facilities are netenergy consumers, they are income producers.They are valued by a utility because they can bebrought online rapidly to operate in a peak powerproduction mode when energy prices are thehighest. The pumping to replenish the upperreservoir is performed during off-peak hourswhen electricity costs are low. Then they arereleased through the power plant when the elec-tricity prices are higher. The system makesmoney even though it consumes more energy.This process benefits the utility by increasing theload factor and reducing the cycling of its baseload units. In most cases, pumped storage plantsrun a full cycle every 24 h (DOE 2002).

Run-of-river projects use the natural flow ofthe river and produce relatively little change inthe stream channel and stream flow. A peakingproject impounds and releases water when theenergy is needed. A storage project extensivelyimpounds and stores water during high-flowperiods to augment the water available duringlow-flow periods, allowing the flow releases andpower production to be more constant. Manyprojects combine the modes.

The power capacity of a hydropower plant isprimarily the function of the flow rate throughthe turbines and the hydraulic head. Thehydraulic head is the elevation difference thewater falls (drops) in passing through the plant orto the tailwater, which ever elevation differenceis less. Project design may concentrate on eitherof these flow and head variables or both, and onthe hydropower plant installed designed capacity.

The production of hydroelectric energy duringany period at any particular reservoir site isdependent on the installed plant capacity; the flowthrough the turbines; the average effective pro-ductive storage head; the duration of the period; theplant factor (the fraction of time energy is pro-duced); and a constant for converting the product of

flow, head, and plant efficiency to electrical energy.The kilowatt-hours of energy, KWHt, produced inperiod t is proportional to the product of the plantefficiency, e, the productive storage head Ht, andthe flow qt through the turbines.

A cubic meter of water, weighing l03 kg,falling a distance of 1 m, acquires 9.81 × 103 J(nm) of kinetic energy. The energy generated inone second equals the watts (joules per second)of power produced. Hence an average flow of qtcubic meters per second falling a height of Ht

meters in period t yields 9.81 × 103qtHt watts or9.81 qtHt kilowatts of power. Multiplying by thenumber of hours in period t yields thekilowatt-hours of energy produced given a headof Ht and an average flow rate of qt. The totalkilowatt-hours of energy, KWHt, produced inperiod t assuming 100% efficiency in conversionof potential to electrical energy is

KWHt ¼ 9:81 qtHtðseconds in period tÞ=ðseconds per hourÞ

¼ 9:81 qtHtðseconds in period tÞ=3600ð11:56Þ

Since the total flow, QtT through the turbines

in period t, equals the average flow rate qt timesthe number of seconds in the period, the totalkilowatt-hours of energy produced in periodt given a plant efficiency (fraction) of e equals

KWHt ¼ 9:81 QTt Ht e=3600

¼ 0:002725 QTt Ht e ð11:57Þ

The energy required for pumped storage,where instead of producing energy the turbinesare used to pump water up to a higher level, is

KWHt ¼ 0:002725 QTt Ht=e ð11:58Þ

For Eqs. 11.57 and 11.58, QtT is expressed in

cubic meters and Ht is in meters. The storagehead, Ht, is the vertical distance between thewater surface elevation in the lake or reservoirthat is the source of the flow through the turbinesand the maximum of either the turbine elevationor the downstream discharge elevation. In vari-able head reservoirs, storage heads are functions

11.7 Hydroelectric Power Production 503

of storage volumes (and possibly the reservoirrelease if the tailwater elevation affects the head).In optimization models for capacity planning,these heads and the turbine flows are among theunknown variables. The energy produced isproportional to the product of these twounknown variables. This results in non-separablefunctions in model equations that must be writtenat each hydroelectric site for each time period t.

A number of ways have been developed toconvert these non-separable energy productionfunctions to separable ones for use in linearoptimization models for estimating design andoperating policy variable values. These methodsinevitably increase the number of model variablesand constraints. For a preliminary screening ofhydropower capacities prior to a more detailedanalysis (e.g., using simulation or other nonlinearor discrete dynamic programming methods) onecan (1) solve the model using both optimistic andpessimistic assumed fixed head values, (2) com-pare the actual derived heads with the assumedones and adjust the assumed heads, (3) resolve themodel, and (4) compare the capacity values. Fromthis iterative process, one should be able toidentify the range of hydropower capacities thatcan then be further refined using simulation.

Alternatively average heads, Hto, and flows,

Qto, can be used in a linear approximation of the

non-separable product terms, QtTHt.

QTt Ht ¼ Ho

t QTt þQo

t Ht � Qot H

ot ð11:59Þ

Again, the model may need to be solvedseveral times in order to identify reasonablyaccurate average flow and head estimates, Qt

o andHto, in each period t.The amount of electrical energy produced is

limited by the installedkilowatts of plant capacityP as well as on the plant factor pt. The plantfactor is a measure of hydroelectric power plantuse in each time period. Its value depends on thecharacteristics of the power system and thedemand pattern for hydroelectric energy. Theplant factor is defined as the average power loadon the plant for the period divided by theinstalled plant capacity. The plant factor accountsfor the variability in the demand for hydropower

during each period t. This factor is usuallyspecified by those responsible for energy pro-duction and distribution. It may or may not varyfor different time periods.

The total energy produced cannot exceed theproduct of the plant factor pt, the number ofhours, ht, in the period, and the plant capacity P,measured in kilowatts.

KWHt �P htpt ð11:60Þ

11.8 Withdrawals and Diversions

Major demands for the withdrawal of waterinclude those for domestic or municipal uses,industrial uses (including cooling water), andagricultural uses including iirrigation. These usesgenerally require the withdrawals of water from ariver system, from reservoirs, or from other sur-face or groundwater bodies. The water withdrawnmay be only partially consumed, and that which isnot consumed may be returned, perhaps at a dif-ferent site, at a later time period, and containingdifferent concentrations of constituents.

Water can also be allocated to instream usesthat alter the distribution of flows in time andspace. Such uses include (1) reservoir storage,possibly for recreational use as well as for watersupply; (2) for flow augmentation, possibly forwater quality control or for navigation or forecological benefits; and (3) for hydroelectricpower production. The instream uses may com-plement or compete with each other or withvarious off-stream municipal, industrial, andagricultural demands. One purpose of developingmanagement models of river basin systems is tohelp derive policies that will best serve thesemultiple uses, or at least identify the tradeoffsamong the multiple purposes and objectives.

The allocated flow qts to a particular use at site

s in period t must be no greater than the total flowavailable, Qt

s, that site and in that period.

qst �Qst ð11:61Þ

The quantity of water that any particular userexpects to receive in each particular period is


termed the target allocation. Given amulti-period (e.g., annual) known or unknowntarget allocation Ts at site s, some (usuallyknown) fraction, ft

s, of that target allocation willbe expected in period t. If the actual allocation,qts, is less than the target allocation, ft

sTs, therewill be a deficit, Dt

s. If the allocation is greaterthan the target allocation, there will be an excess,Ets. Hence, to define those unknown variables the

following constraint equation can be written foreach applicable period t.

qst ¼ f st Ts�Ds

t þEst ð11:62Þ

Even though allowed, one would not expect asolution to contain nonzero values for both Dt

s

and Ets.

Whether or not any deficit or excess allocationshould be allowed at any demand site s dependson the quantity of water available and the lossesor penalties associated with deficit or excess al-locations to that site. At sites where the benefitsderived in each period are independent of theallocations in other periods, the losses associatedwith deficits and the losses or benefits associatedwith excesses can be defined in each periodt (Chap. 9). For example, the benefits derivedfrom the allocation of water for hydropowerproduction in period t in some cases will beessentially independent of previous allocations.

For any use in which the benefits are depen-dent on a sequence of allocations, such as atirrigation sites, the benefits may be based on theannual (or growing season) target water alloca-tions Ts and their within season distributions,ftsTs. In these cases one can define the benefitsfrom those water uses as functions of theunknown season or annual targets, Ts, where theallocated flows qt

s must be no less than thespecified fraction of that unknown target.

qst � f st Ts for all relevant t ð11:63Þ

If, for any reason, an allocation variable valueqts must be low, or even zero, due to other more

beneficial uses, then clearly from Eq. 11.63 the

annual or growing season target allocation Ts

would be low (or zero) and presumably so wouldbe the benefits associated with that target value.

Water stored in reservoirs can often be used toaugment downstream flows for instream usessuch as recreation, navigation, and water qualitycontrol. During natural low-flow periods in thedry season, it is not only the increased volumebut also the lower temperature of the augmentedflows that may provide the only means ofmaintaining certain species of fish and otheraquatic life. Dilution of wastewater or runofffrom non-point sources may be another potentialbenefit from flow augmentation. These and otherfactors related to water quality management arediscussed in greater detail in Chap. 10.

The benefits derived from navigation on apotentially navigable portion of a river systemcan usually be expressed as a function of thestage or depth of water in various periods.Assuming known stream or river flow-stagerelationships at various sites in the river, a pos-sible constraint might require at least a minimumacceptable depth, and hence flow, for those sites.

11.9 Lake-Based Recreation

Recreation benefits derived from natural lakes aswell as reservoirs are usually dependent on theirstorage levels. Where recreational facilities havebeen built, recreational benefits will also bedependent on recreational target lake levels aswell. If docks, boathouses, shelters, and otherrecreational facilities were installed based onsome assumed (target) lake level, and the lakelevels deviate from the target value, there can bereduced recreational benefits. These storage tar-gets and any deviations can be modeled similar toEq. 11.62. The actual storage volume at thebeginning of a recreation period t equals the targetstorage volume less any deficit or plus any excess.

Sst ¼ Ts � Dst þEs

t ð11:64Þ

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The recreational benefits in any recreationalperiod t can be defined based on what they wouldbe if the target were met less the average of anylosses that may occur from initial and finalstorage volume deviations, Dt

s or Ets, from the

target storage volume in each period of therecreation season (Chap. 9).

11.10 Model Synthesis

Each of the model components discussed abovecan be combined, as applicable, into a model of ariver system. One such river system together withsome of its interested stakeholders is shown inFig. 11.26.

One of the first tasks in modeling this basin isto identify the actual and potential system com-ponents and their interdependencies. This is

facilitated by drawing a schematic of the systemat the level of detail that will address the issuesbeing discussed and of concern to these stake-holders. This schematic can be drawn over thebasin as in Fig. 11.27. The schematic without thebasin is illustrated in Fig. 11.28.

A site number must be assigned at each pointof interest. These sites are usually where somedecision must be made. Mass balance and otherconstraints will need to be defined at each ofthose sites.

As shown in the schematic in Fig. 11.28, thisriver has one streamflow gage site, site 1, tworeservoirs, sites 3 and 5, two diversions, sites 2and 3, one hydropower plant, site 5, and a leveedesired at site 4 to help protect against floods inthe urban area. The reservoir at site 5 is apumped storage facility. The upstream reservoirat site 3 is used for recreation, water supply, and

Fig. 11.26 A multipurpose river system whose management is of concern to numerous stakeholders


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Fig. 11.27 A schematic representation of the basin components and their interdependencies drawn over the mapimage of the basin

Fig. 11.28 Schematic ofriver system showingcomponents of interest atdesignated sites

11.10 Model Synthesis 507

flood control. The downstream reservoir isstrictly for hydropower production.

Before developing a model of this river sys-tem, the number of time periods t to include inthe model and the length of each within-yeartime period should be determined. If a riversystem’s reservoirs are to contain storage for thedistribution of water among years, calledover-year storage, then a number of periodsencompassing multiple years of operation mustbe included in the model. This will allow anevaluation of the possible benefits of storingexcess water in wet years for release in dry years.

Many reservoir systems completely fill almostevery year, and in such cases one is concernedonly with the within-year operation of the sys-tem. This is the problem addressed here. Tomodel the within-year operation of the system, ayear is divided into a number of within-yearperiods. The number of the periods and theduration of each period will depend on thevariation in the hydrology, the demands, and onthe particular objectives, as previously discussed.

Once the number and duration of the timesteps to be modeled have been identified, thevariables and functions used at each site must benamed. It is convenient to use notation that canbe remembered when examining the modelsolutions. The notation made up for this exampleis shown in Table 11.4.

For this example assume we are interested inmaximizing a weighted combination of all thenet economic benefits derived from all the des-ignated uses of water. There could, and no doubtshould, be other objective components defined aswell, as discussed in Chap. 9. Nevertheless theseeconomic objective components serve the pur-pose here of illustrating how a model of this riversystem can be constructed:

The overall objective might be a weightedcombination of all net benefits, NBs, obtained ateach site s in the basin:

MaximizeXs

wsNBs ð11:65Þ

This objective function does not identify howmuch each stakeholder group would benefit andhow much they would pay. Who benefits andwho pays, and by how much, may matter. If it isknown how much of each of the net benefitsderived from each site are to be allocated to eachstakeholder group i, then these allocated fraction,fi, of the total net benefits, NBs, can be includedin the overall objective:

MaximizeXi

wi

Xs

fiNBs ð11:66Þ

Using methods discussed in Chap. 9, solvingthe model for various assumed values of theseweights can help identify the tradeoffs betweendifferent conflicting objectives, Eq. 11.65, orconflicting stakeholder interests, Eq. 11.66.

The next step in model development is todefine the constraints applicable at each site. It isconvenient to begin at the most upstream sitesand work downstream. As additional variables orfunctions are needed, invent notation for them.These constraints tie the decision variablestogether and identify the interdependenciesamong system components. In this example thesite index is shown as a superscript.

At site 1:

• No constraints are needed. It is the gage site.

At site 2:

• the diverted water, X2(t), cannot exceed thestreamflow, Q2(t), at that site.

Q2ðtÞ�X2ðtÞ 8t in the irrigation season

ð11:67Þ

• the diversion flow, X2(t), cannot exceed thediversion channel capacity, X2.

X2 �X2ðtÞ 8t in the irrigation season

ð11:68Þ


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Table 11.4 Names associated with required variables and functions at each site in Fig. 11.28

The units of these variables and parameters, however defined, must be consistent


• the diversion flow, X2(t), must meet the irri-gation target, δt

2T2

X2ðtÞ� d2t T2 8t in the irrigation season

ð11:69Þ

• NB2 = benefit function associated with theannual target irrigation allocation, T2, less theannual cost function associated with thediversion channel capacity, X2.

At site 3:

• storage volume mass balances (continuity ofstorage), assuming no losses.

S3ðtþ 1Þ ¼ S3ðtÞþQ3ðtÞ�X3ðtÞ�R3ðtÞ8t; T þ 1 ¼ 1

ð11:70Þ

• define storage deficits, D3(t), and excesses,E3(t), relative to recreation target, T3.

S3ðtÞ ¼ T3 � D3ðtÞþE3ðtÞ8t in recreation season plus

following period:

ð11:71Þ

• diverted water, X3(t), cannot exceed diversionchannel capacity, X3.

X3ðtÞ�X3 8t ð11:72Þ

• reservoir storage capacity constraints involv-ing dead storage, KD

3 , and flood storage, KF3,

capacities.

S3ðtÞ�K3 � K3D � K3

F

8t in flood season plus following period:

S3ðtÞ�K3 � K3D

for all other periods t:

ð11:73Þ

• NB3 = sum of annual benefit functions for T3

and KF3 less annual costs of K3 and X3 less

annual recreation losses associated with allD3(t) and E3(t).

At site 4:

• define deficit diversion, D4(t), from site 3,associated with target, δt

4T4, if any.

X3ðtÞ ¼ d4t T4 � D4ðtÞ 8t ð11:74Þ

• channel capacity, Q4, must equal peak floodflow, PQt

4, associated with selected returnperiod, T.

Q4 ¼ PQ4T ð11:75Þ

• NB4 = sum of annual benefit functions for T4

less annual cost of Q4 less annual lossesassociated with all D4(t).

At site 5:

• continuity of pumped storage volumes,involving inflows, QI5(t), and outflows,QO5(t), and assuming no losses.

S5ðtþ 1Þ ¼ S5ðtÞþQI5ðtÞ � QO5ðtÞ 8tð11:76Þ

• active storage capacity involving dead stor-age, KD

5 .

S5ðtÞ�K5 � K5D 8t ð11:77Þ

• pumped inflows, QI5(t), cannot exceed theamounts of water available at the intake. Thisincludes the release from the upstream reser-voir, R3(t), and the incremental flow,Q5(t) − Q3(t).

QI5ðtÞ�Q5ðtÞ � Q3ðtÞþR3ðtÞ 8tð11:78Þ

• define the energy produced, EP5(t), given theaverage storage head, H(t), flow through theturbines, QO5(t), and efficiency, e.

EP5ðtÞ ¼ ðconst:ÞðHðtÞÞðQO5ðtÞÞe 8tð11:79Þ


• define the energy consumed, EC5(t), frompumping given the amount pumped, QI5(t).

EC5ðtÞ ¼ ðconst:ÞðHðtÞÞðQI5ðtÞÞ=e 8tð11:80Þ

• Energy production, EP5(t), and consumption,EC5(t), constraints given power plant capac-ity, P5.

EP5ðtÞ�P5 ðhours of energy production in tÞ8t

ð11:81Þ

EC5ðtÞ�P5ðhours of pumping in tÞ 8tð11:82Þ

• Define the average storage head, H(t), basedon storage head functions, h(S5(t)).

HðtÞ ¼ hðS5ðtþ 1ÞÞþ hðS5ðtÞÞ� �=2 8tð11:83Þ

• NB5 = Sum of benefits for the energy pro-duced, EP5(t), less the costs of the energyconsumed, EC5(t), less the annual costs ofcapacities K5 and P5.

Equations 11.67–11.83 together with objec-tive Eq. 11.65 or 11.66 define the generalstructure of this river system model. Before themodel can be solved, the actual functions mustbe defined. Then they may have to be madepiecewise linear if linear programming is to bethe optimization procedure used to solve themodel. The process of defining functions mayadd variables and constraints to the model, asdiscussed in Chaps. 4 and 9.

For T within-year periods t, this static modelof a single year includes between 14T + 8 and16T + 5 constraints, depending on the number ofperiods in the irrigation and recreation seasons.This number does not include the additionalconstraints that surely will be needed to define

the functions in the objective function compo-nents and constraints. Models of this size andcomplexity, even though this is a rather simpleriver system, are usually solved using linearprogramming algorithms simply because othernonlinear or dynamic programming (optimiza-tion) methods are more difficult to use.

The model just developed is for a typicalsingle year. In some cases it may be moreappropriate to incorporate over-year as well aswithin-year mass balance constraints, and yieldswith their respective reliabilities, within thismodeling framework. This can be done as out-lined in Sect. 5.4 of this chapter.

The information derived from optimizationmodels of river systems such as this one shouldnot be considered as a final answer. Rather it isan indication of the range of system design andoperation policies that should be further analyzedusing more detailed analyses. Optimizationmodels of the type just developed serve as waysto eliminate inferior alternatives from furtherconsideration more than as ways of finding asolution all stakeholders will accept as the best.

11.11 Project Scheduling

The river basin models discussed thus far in thischapter deal with static planning situations in thatsystem components and their capacities oncedetermined are not assumed to change over time.Project capacities, targets, and operating policiestake on fixed values and one examines “snapshotsteady-state” solutions for a particular time in thefuture. These “snapshots” only allow for fluctua-tions caused by the variability of supplies anddemands. The non-hydrologic world is seldomstatic, however. Targets and goals and policieschange in response to population growth, invest-ment in agriculture and industry, and shiftingpriorities for water use. In addition, financialresources available for water resources investmentare limited and may vary from year to year.

Dynamic planning models can aid thoseresponsible for the long-run development and


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expansion of water resources systems. Althoughstatic models can identify target values and sys-tem configuration designs for a particular periodin the future, they are not well adapted tolong-run capacity expansion planning over a 10-,20-, or 30-year period. But static models mayidentify projects for implementation in earlyyears which in later year simulations do notappear in the solutions (Chap. 4 contains anexample of this).

This is the common problem in capacityexpansion, where each project has a fixed con-struction and implementation cost as well asvariable operating, repair, and maintenance costcomponent. If there are two mutually exclusivecompeting projects, one may be preferred at asite when the demand at that site is low, but theother may be preferred if the demand is, as it islater projected to be, much higher. Which of thetwo projects should be selected now when thedemand is low, given the uncertainty of theprojected increase over time, especially assumingit makes no economic sense to destroy andreplace a project already built?

Whereas static models consider how a waterresources system operates under a single set offixed conditions, dynamic expansion modelsmust consider the sequence of changing condi-tions that might occur over the planning period.For this reason, dynamic expansion models arepotentially more complex and larger than aretheir static counterparts. However, to keep thesize and cost of dynamic models within thelimitations of most studies, these models aregenerally restricted to very simple descriptions ofthe economic and hydrologic variables of con-cern. Most models use deterministic hydrologyand are constrained either to stay within prede-termined investment budget constraints or tomeet predetermined future demand estimates.

Dynamic expansion models can be viewed asnetwork models for solution by linear or dynamicprogramming methods. The challenge in riversystem capacity expansion or projectschedulingmodels is that each component’s performance, orbenefits, may depend on the design and operatingcharacteristics of other components in the sys-tem. River basin project impacts tend to be

dependent on what else is happening in the basin,i.e., what other projects are present and how theyare designed and operated.

Consider a situation in which n fixed-scalediscrete projects may be built during the planningperiod. The scheduling problem is to determinewhich of the projects to build and in what order.The solution of this problem generally requires aresolution of the timing problem. When shouldeach project be built, if at all?

For example, assume there are n = 3 discreteprojects that might be beneficial to implementsometime over the next 20 years. Let this 20-yearperiod consists of four 5-year construction peri-ods y. The actual benefits derived from any newproject may depend on the projects that alreadyexist. Let S be the set of projects existing at thebeginning of any construction period. Finally letNay(S) be the maximum present value of the totalnet benefits derived in construction period y as-sociated with the projects in the set S. Here‘‘benefits’’ refer to any composite of systemperformance measures.

These benefit values for various combinationsdiscrete projects could be obtained from staticriver system models, solved for all combinationsof discrete projects for conditions existing at theend of each of these four 5-year periods. It mightbe possible to just do this for one or two of thesefour periods and apply applicable discount ratesfor the other periods. These static models can besimilar to those discussed in the previous sectionof this chapter. Now the challenge is to find thesequencing of these three projects over the peri-ods y that meet budget constraints and thatmaximize the total present value of benefits.

This problem can be visualized as a network.As shown in Fig. 11.29, the nodes of this net-work represent the sets S of projects that exist atthe beginning of the construction period. Forthese sets S we have the present value of theirbenefits, NBy(S), in the next 5-years. The linksrepresent the project or projects implemented inthat construction period. Any set of new projectsthat exceeds the construction funds available forthat period is not shown on the network. Thoselinks are infeasible. For the purposes of thisexample, assume it is not financially feasible to


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add more than one project in any single con-struction period. Let Cky be the present value ofthe cost of implementing project k in construc-tion period y.

The optimal is to find the best (maximumbenefits less costs) paths through the network.Each link represents a net benefit, NBy(S)over the next 5-years obtained from the set of

Fig. 11.29 Project scheduling options. Numbers in nodes represent existing projects. Links represent new projects, thedifference between the existing projects at both connecting nodes

11.11 Project Scheduling 513

projects, S, that exist less the cost of adding anew project k.

Using linear programming, one can define acontinuous nonnegative unknown decision vari-able Xij for each link between node i and node j.It will be an indicator of whether a link is on theoptimum path or not. If after solving the modelits value is 1, the link connecting nodes i andj represents the decision to make in that con-struction period. Otherwise its value is 0 indi-cating the link is not on the optimal path. Thesequence of links having their Xij values equal to1 will indicate the most beneficial sequence ofproject implementations.

Let the net benefits associated with node i bedesignated NBi (that equals the appropriateNBy(S) value), and the cost, Cky, of the newproject k associated with that link. The objectiveis to maximize the present value of net bene-fits less project implementation costs over allperiods y.

MaximizeXi

Xj

ðNBi � CijÞXij ð11:84Þ

Subject toContinuity at each node

Xh

Xhi ¼Xj

Xij

for each node i in the network:ð11:85Þ

Sum of all decision variable values on thelinks in any one period y must be 1. For examplein period 1

X00 þX01 þX02 þX03 ¼ 1 ð11:86Þ

The sums in Eq. 11.86 are over nodes h hav-ing links to node i and over nodes j having linksfrom node i.

The optimal path through this network canalso be solved using dynamic programming.(Refer to the capacity expansion problem illus-trated in Chap. 4). For a backward movingsolution procedure, let

s = subset of projects k not contained in theset S (s 62 S).

$y = the maximum project implementationfunds available in period y.

Fy(S) = the present value of the total benefitsover the remaining periods, y, y + 1, …, 4.

FY+1(S) = 0 for all sets of projects S followingthe end of the last period.

The recursive equations for each constructionperiod, beginning with the last period, can bewritten

FyðSÞ ¼ maximum fNByðSÞ �Xk2s

Cky þFyþ 1ðSþ sÞg

for all S s 62 SXk2s

Cky � $y

ð11:87Þ

Defining Fy′(S) as the present value of thetotal benefits of all new projects in the set S im-plemented in all periods up to and includingperiod y, and the subset s of projects k beingconsidered in period y now belonging to the setS of projects existing at the end of the period, therecursive equations for a forward moving solu-tion procedure beginning with the first period,can be written

F0yðSÞ ¼ maximumf NByðS� sÞ �

Xk2s

Ck;y þF0y�1ðS�sÞg

for all S s 2 SXk2s

Cky � $y

ð11:88Þ

where F0′(0) = 0.Like the linear programming model, the

solutions of these dynamic programming modelsidentify the sequencing of projects recognizingtheir interdependencies. Of interest, again, iswhat to do in this first constriction period. Theonly reason for looking into the future is to makesure, as best as one can that the first period’sdecisions are not myopic. Models like these canbe developed and solved again with more upda-ted estimates of future conditions when nextneeded.

Additional constraints and variables might beadded to these scheduling models to enforcerequirements that some projects precede others orthat if one project is built another is infeasible.


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These additional restrictions usually reduce thesize of a network of feasible nodes and links, asshown in Fig. 11.29.

Another issue that these dynamic models canaddress is the sizing or capacity expansionproblem. Frequently, the scale or capacity of areservoir, pipeline, pumping station, or irrigationis variable and needs to be determined concur-rently with the solution of the scheduling andtiming problems. To solve the sizing problem,the costs and capacities in the scheduling modelbecome variables.

This project scheduling problem by its verynature must deal with uncertainty. A relativelyrecent contribution to this literature is the work ofHaasnoot et al. (2013), Walker et al. (2013).

11.12 Conclusions

This chapter on river basin planning modelsintroduces some ways of modeling river basincomponents, separately and together within anintegrated model. Ignored during the develop-ment of these different model types were theuncertainties associated with the results of thesemodels. As discussed in Chaps. 7 and 8, theseuncertainties may have a substantial effect onmodel solution and the decision taken.

Most of this chapter has been focused on thedevelopment of simplified screening models,using simulation as well as optimization methods,for identifying what and where and when infras-tructure projects should be implemented, and ofwhat capacity. The solution of these screeningmodels, and any associated sensitivity and uncer-tainty analyses, can be of value prior to committingto more costly design modeling exercises.

Preliminary screening of river basin systems,especially given multiple objectives, is a chal-lenge to accomplish in an efficient and effectivemanner. The modeling methods and approachesdiscussed in this chapter serve as an introduc-tion to that art.

References

Haasnoot, M., Kwakkel, J. H., Walker, W. E., & ter Maat,J. (2013). Dynamic adaptive policy pathways: Amethod for crafting robust decisions for a deeplyuncertain world. Global environmental Change, 23,485–498.

Maidment, D. R. (1993). Handbook of hydrology. NY:McGraw-Hill Inc.

Rippl, W. (1883). The capacity of storage reservoirs forwater supply. In Proceedings of the Institute of CivilEngineers (Brit.) (Vol. 71).

U.S. DOE. (2002, April). Hydropower program. http://hydropower.inel.gov or http://hydropower.id.doe.gov/

USACE. (1991). Hydrology and hydraulics workshop onRiverine Levee Freeboard, Monticello, Minnesota,Report SP-24, Hydrologic Engineering Center, USArmy Corps of Engineers, 272 pp.

Walker, W. E., Haasnoot, M., & Kwakkel, J. H. (2013).Adapt or Perish: A review of planning approaches foradaptation under deep uncertainty. Sustainability, 5,955–979.

Additional References(Further Reading)

Becker, L., & Yeh, W. W. G. (1974). Timing and sizingof complex water resource systems. Journal of theHydraulics Division, ASCE , 100(HYIO), 1457–1470.

Becker, L., & Yeh, W. W.-G. (1974a). Optimal timing,sequencing, and sizing of multiple reservoir surfacewater supply facilities. Water Resources Research, 10(1), 57–62.

Bedient, P. B., & Huber, W. C. (1992). Hydrology andfloodplain analysis (2nd ed.). Reading: AddisonWestley.

Basson, M. S., Allen, R. B., Pegram, G. G. S., & vanRooyen, J. A. (1994). Probabilistic management ofwater resource and hydropower systems. HighlandsRanch: Water Resources Publications.

Chin, D. A. (2000). Water-resources engineering. UpperSaddle River: Prentice Hall.

Erlenkotter, D. (1973a). Sequencing of interdependenthydroelectric projects. Water Resources Research, 9(1), 21–27.

Erlenkotter, D. (1973b). Sequencing expansion projects.Operations Research, 21(2), 542–553.

Erlenkotter, D. (1976). Coordinating scale and sequencingdecisions forwater resources projects.EconomicModel-ing for Water Policy Evaluation, NorthHolland/TIMSStudies in the Management Sciences, 3, 97–112.

11.11 Project Scheduling 515

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http://dx.doi.org/10.1007/978-3-319-44234-1_8

http://hydropower.inel.gov

http://hydropower.inel.gov

http://hydropower.id.doe.gov/

Erlenkotter, D,. & Rogers, J. S. (1977). Sequencingcompetitive expansion projects. Operations Research,25(6), 937–951.

Hall, W. A., Tauxe, G. W., & Yeh, W. W.-G. (1969). Analternative procedure for optimization of operationsfor planning with multiple river, multiple purposesystems. Water Resources Research, 5(6), 1367–1372.

Jacoby, H. D., & Loucks, D. P. (1972). Combined use ofoptimization and simulation models in river basinplanning.Water Resources Research, 8(6), 1401–1414.

James, L. D., & Lee, R. R. (1971). Economics of waterresources planning. New York: McGraw-Hill BookCompany.

Linsley, R. K., Franzini, J. B., Freyberg, D. L., &Tchobanoglous, G. (1992). Water-resources engineer-ing. NY: McGraw-Hill Inc.

Loucks, D. P. (1976). Surface water quantity manage-ment, Chap. 5. In A. K. Biswas (Ed.), Systemsapproach to water management. New York:McGraw-Hill Book Company.

Loucks, D. P., Stedinger, J. R., & Haith, D. A. (1981).Water resource systems planning and analysis. Engle-wood Cliffs: Prentice Hall Inc.

Major, D. C., & Lenton, R. L. (1979). Applied waterresources systems planning. Englewood Cliffs: Pren-tice Hall Inc.

Maass,A.,Hufschmidt,M.M.,Dorfman,R., Thomas,H.A.,Jr.,Marglin, S. A., & Fair, G.M. (1962).Design of waterresource systems. Cambridge: Harvard University Press.

Mays, L. W. (2001). Water resources engineering. NY:Wiley.

Mays, L. W., & Tung, Y.-K. (1992). Hydrosystemsengineering and management. NY: McGraw-Hill Inc.

Morin, T. L. (1973). Optimal Sequencing of CapacityExpansion Projects. Journal of the Hydraulics Divi-sion, ASCE, 99(HY9), 1605–1622.

Morin, T. L., & Esogbue, A.M.O. (1974). A useful theoremin the dynamic programming solution of sequencing andscheduling problems occurring in capital expenditureplanning.Water Resources Research, 10(1), 49–50.

National Research Council. (1999). New strategies forAmerica’s Watersheds. Washington, DC: NationalAcademy Press.

National Research Council. (2000). Risk analysis anduncertainty in flood damage reduction studies. Wash-ington, DC: National Academy Press.

O’Laoghaire, D. T., & Himmelblau, D. M. (1974).Optimal expansion of a water resources system.New York: Academic Press Inc.

ReVelle, C. (1999). Optimizing reservoir resources. NY:Wiley.

Thomas, H. A., Jr., & Burden, R. P. (1963). Operationsresearch in water quality management. Cambridge:Harvard Water Resources Group.

USACE. (1991a). Benefit determination involving exist-ing levees. Policy Guidance Letter No. 26, Headquar-ters, U.S. Army Corps of Engineers, Washington, DC.

USACE. (1999).Risk analysis ingeotechnical engineering forsupport of planning studies, ETL 1110-2-556, Headquar-ters, U.S. Army Corps of Engineers, Washington, DC.

Viessman, W., Jr., &Welty, C. (1985).Water managementtechnology and institutions. NY: Harper & Row, Pubs.

Wurbs, R. A., & James, W. P. (2002). Water resourcesengineering. Upper Saddle River: Prentice Hall.

Exercises

11:1 Using the following information pertain-ing to the drainage area and discharge inthe Han River in South Korea, develop anequation for predicting the natural unreg-ulated flow at any site in the river, byplotting average flow as a function ofcatchment area. What does the slope of thefunction equal?

Gage point Catchmentarea (km2)

Average flow(106 m3/year)

First bridge ofthe Han River

25,047 17,860

Pal Dang dam 23,713 16,916

So Yang dam 2703 1856

Chung Ju dam 6648 4428

Yo Ju dam 10,319 7300

Hong Chun dam 1473 1094

Dal Chun dam 1348 1058

Kan Yun dam 1180 926

Im Jae dam 461 316

11:2 In watersheds characterized by significantelevation changes, one can often developreasonable predictive equations for averageannual runoff per hectare as a function ofelevation. Describe how onewould use sucha function to estimate the natural averageannual flow at any gage in a watershedwhich is marked by large elevation changesand little loss of water from stream channelsdue to evaporation or seepage.

11:3 Compute the storage-yield function for asingle reservoir system by the mass dia-gram and modified sequent peak methodsgiven the following sequences of annualflows: (7, 3, 5, 1, 2, 5, 6, 3, 4). Nextassume that each year has two distinct


hydrologic seasons, one wet and the otherdry, and that 80% of the annual inflowoccurs in season t = 1 and 80% of theyield is desired in season t = 2. Using themodified sequent peak method, show theincrease in storage capacity required forthe same annual yield resulting fromwithin-year redistribution requirements.

11:4 Write two different linear programmingmodels for estimating the maximum con-stant reservoir release or yield Y given afixed reservoir capacity K, and for esti-mating the minimum reservoir capacityK required for a fixed yield Y. Assume thatthere are T time periods of historical flowsavailable. How could these models beused to define a storage capacity-yieldfunction indicating the yield Y availablefrom a given capacity K?

11:5(a) Construct an optimization model for esti-mating the least-cost combination of activestorage capacities, K1 and K2, of tworeservoirs located on a single stream, usedto produce a reliable constant annual flowor yield (or greater) downstream of the tworeservoirs. Assume that the cost functionsCs(Ks) at each reservoir site s are knownand there is no dead storage and no evap-oration. (Do not linearize the cost func-tions; leave them in their functional form.)Assume that 10 years of monthly unregu-lated flows are available at each site s.

(b) Describe the two-reservoir operating pol-icy that you would incorporate into amodel model to check the solutionobtained from the optimization model.

11:6 Given the information in the accompany-ing tables, compute the reservoir capacitythat maximizes the net expected flooddamage reduction benefits less the annualcost of reservoir capacity.

Reservoircapacity

Floodstage forflood ofreturnperiod T

Annualcapacitycost

T = 1 T = 2 T = 5 T = 10 T = 100

0 30 105 150 165 180 10a

5 30 80 110 120 130 25

10 30 55 70 75 80 30

15 30 40 45 48 50 40

20 30 35 38 39 40 70

a10 is fixed cost if capacity > 0; otherwise, it is 0

Flood stage Cost of flood damage

30 0

50 10

70 20

90 30

110 40

130 50

150 90

180 150

11:7 Develop a deterministic, static, within-yearmodel for evaluating the developmentalternatives in the river basin shown in theaccompanying figures. Assume that thereare t = 1, 2, 3,…, n within-year periodsand that the objective is to maximize thetotal annual net benefits in the basin. Thesolution of the model should define thereservoir capacities (active + flood storagecapacity), the annual allocation targets, thelevee capacity required to protect site 4from a T-year flood, and the within-year

period allocations of water to the uses atsites 3 and 7. Clearly define all variablesand functions used, and indicate how themodel would be solved to obtain themaximum net benefit solution.

1 2

Capacities K1 K2

Exercises 517

Site Fraction ofgage flow

1 0 Potential reservoir for watersupply

2 0.3 Potential reservoir for watersupply, flood control

3 0.15 Diversion to a use, 60% ofallocation returned to river

4 – Existing development,possible flood protection fromlevee

5 0.6 Potential reservoir for watersupply, recreation

6 – Hydropower; plantfactor = 0.30

7 0.9 Potential diversion to anirrigation district

8 1 Gage site

For simplicity, assume no evaporation lossesor dead storage requirements. Omitting theappropriate subscripts t for time periods and s forsite, let T, K, D, E, and P be the target, reservoircapacity, deficit, excess, and power plant capac-ity variables, respectively. Let Qt and Rt be thenatural streamflows and reservoir releases, and Stbe the initial reservoir storage volumes in periodt. Kf will denote the flood storage capacity at site2 that will contain a peak flow of QS and QR isthe downstream channel flood flow capacity. Therelationship between QS and Kf is defined by thefunction κ(QS). The unregulated design floodpeak flow for which protection is required is QN.KWH will be the kilowatt-hours of energy,H will be net storage head, ht the hours in aperiod t. The variable q will be the water supplyallocation. Benefit functions will be B(), L() will

denote loss functions and C() will denote the costfunctions.

11:8 List the potential difficulties involvedwhen attempting to structure models fordefining

(a) Water allocation policies for irrigationduring the growing season.

(b) Energy production and capacity ofhydroelectric plants.

(c) Dead storage volume requirements inreservoirs.

(d) Active storage volume requirements inreservoir.

(e) Flood storage capacities in reservoirs.(f) Channel improvements for damage

reduction.(g) Evaporation and seepage losses from

reservoirs.(h) Water flow or storage targets using

long-run benefits and short- run lossfunctions.

11:9 Assume that demand for water supplycapacity is expected to grow as t(60 − t),for t in years. Determine the minimumpresent value of construction cost of somesubset of water supply options describedbelow so as to always have sufficientcapacity to meet demand over the next30 years. Assume that the water supplynetwork currently has no excess capacityso that some project must be built imme-diately. In this problem, assume that pro-ject capacities are independent and thuscan be summed. Use a discount factor

Levee Potential flooddamage site Irrigation

area

Gagesite

1

2

3

45

6

7

8


equal to exp(−0.07 t). Before you start,what is your best guess at the optimalsolution?

Project number Construction cost Capacity

1 100 200

2 115 250

3 190 450

4 270 700

11:10 (a) Construct a flow diagram for a simu-lation model designed to define astorage-yield function for a singlereservoir given known inflows in eachmonth t for n years. Indicate how youwould obtain a steady-state solutionnot influenced by an arbitrary initialstorage volume in the reservoir at thebeginning of the first period. Assumethat evaporation rates (mm per month)and the storage volume/surface areafunctions are known.

(b) Write a flow diagram for a simulationmodel to be used to estimate the prob-ability that any specific reservoir ca-pacity,K, will satisfy a series of knownrelease demands, rt, downstream givenunknown future inflows, it. You neednot discuss how to generate possiblefuture sequences of streamflows, onlyhow to use them to solve this problem.

11:11 (a) Develop an optimization model forfinding the cost-effective combinationof flood storage capacity at anupstream reservoir and channelimprovements at a downstreampotential damage site that will protectthe downstream site from a prespeci-fied design flood of return periodT. Define all variables and functionsused in the model

(b) How could this model be modified toconsider a number of design floodsT and the benefits from protecting the

potential damage site from thosedesign floods? Let BFT be the annualexpected flood protection benefits atthe damage site for a flood havingreturn periods of T.

(c) How could this model be furthermodified to include water supplyrequirements of At to be withdrawnfrom the reservoir in each month t?Assume known natural flows Qs

t ateach site s in the basin in each month t.

(d) How could the model be enlarged toinclude recreation benefits or losses atthe reservoir site? Let Ts be theunknown storage volume target andDs

t be the difference between thestorage volume Sst and the target Ts ifSst – Ts > 0, and Es

t be the difference ifTs – Sst > 0. Assume that the annualrecreation benefits B(Ts) are a functionof the target storage volume Ts and thelosses LDðDs

t Þ and LEðEst Þ are associ-

ated with the deficit Dst and excess Es

t

storage volumes.11:12 Given the hydrologic and economic

data listed below, develop and solve alinear programming model for esti-mating the reservoir capacity K, theflood storage capacity Kf, and therecreation storage volume target T thatmaximize the annual expected floodcontrol benefits, Bf(Kf), plus theannual recreation benefits, B(T), lessall losses LDðDtÞ and LEðEtÞ associ-ated with deficits Dt or excesses Et inthe periods of the recreation season,minus the annual cost C(K) of storagereservoir capacity K. Assume that thereservoir must also provide a constantrelease or yield of Y = 30 in eachperiod t. The flood season begins atthe beginning of period 3 and laststhrough period 6. The recreation sea-son begins at the beginning of period4 and lasts though period 7.

Exercises 519

Period 1 2 3 4 5 6 7 8 9

Inflowstoreservoir

50 30 20 80 60 20 40 10 70

Bf ðKf Þ ¼12Kf if Kf � 560þ 8ðKf � 5Þ if 5�Kf � 15140þ 4ðKf � 15Þ if Kf � 15

8<:

CðKf Þ ¼45þ 10K if 0�Kf � 595þ 6ðKf � 5Þ if 5�Kf � 20185þ 10ðKf � 20Þ if 20�Kf � 40385þ 15ðK � 40Þ if Kf � 40

8>><>>:

B(T) = 9T, where T is a particularunknown value of -reservoirstorage

LDðDtÞ = 4Dt, where Dt is T − St if T ≥ StLEðEtÞ = 2 Et, where Et is St − T if St ≥ T

11:13 The optimal operation of multiple reser-voir systems for hydropower productionpresents a very nonlinear and often dif-ficult problem.

Use dynamic programming to determine theoperating policy that maximizes the total annualhydropower production of a two-reservoir sys-tem, one downstream of the other. The releasesR1t from the upstream reservoir plus the unreg-ulated incremental flow (Q2t − Q1t) constitutethe inflow to the downstream dam. The flows Q1t

into the upstream dam in each of the four seasonsalong with the incremental flows (Q2t − Q1t) andconstraints on reservoir releases are given in theaccompanying two tables:

Upstream dam (flow in 106 m3/period)

Seasont

InflowQ1t

Minimumrelease

Maximumrelease throughturbines

1 60 20 90

2 40 30 90

(continued)

Qt = inflowRt = release

(excess)

St = initial storageKf = flood storageK = total capacity

Y = 30

-15

45

5 20 40

12

10

15

K

C(K)6Bf (Kf)

Kf

5 150

12

84


Upstream dam (flow in 106 m3/period)

Seasont

InflowQ1t

Minimumrelease

Maximumrelease throughturbines

3 80 20 90

4 120 20 90

Downstream dam (flow in 106 m3/period)

Seasont

Incrementalflow,(Q2t − Q1t)

Minimumrelease

Maximumreleasethroughturbines

1 50 30 140

2 30 40 140

3 60 30 140

4 90 30 140

Note that there is a limit on the quantity of waterthat can be released through the turbines for energygeneration in any season due to the limited capacityof the power plant and the desire to producehydropower during periods of peak demand.

Additional information that affects the opera-tion of the two reservoirs are the limitations onthe fluctuations in the pool levels (head) and thestorage-head relationships:

Data Upstreamdam

Downstreamdam

Maximum head,Hmax (m)

70 90

Minimum head,Hmin (m)

30 60

Maximum storagevolume, Smax (m3)

150 × 106 400 × 106

Storage-net headrelationship

H = Hmax(S/Smax)0.64

H = Hmax(S/Smax)0.62

In solving the problem, discretize the storagelevels in units of 10 × 106 m3. Do a preliminaryanalysis to determine how large a variation in

storage might occur at each reservoir. Assumethat the conversion of potential energy equals tothe product RiHi to electric energy is 70% effi-cient independent of Ri and Hi. In calculating theenergy produced in any season t at reservoir i,use the average head during the season

�Hi ¼ 12HiðtÞþHiðtþ 1Þ½ �

Report your operating policy and the amountof energy generated per year. Find another fea-sible policy and show that it generates lessenergy than the optimal policy.

Show how you could use linear programmingto solve for the optimal operating policy byapproximating the product term Ri �Hi by a linearexpression.

11:14 You are responsible for planning a pro-ject that may involve the building of areservoir to provide water supply benefitsto a municipality, recreation benefitsassociated with the water level in the lakebehind the dam, and flood damagereduction benefits. First you need todetermine some design variable values,and after doing that you need to deter-mine the reservoir operating policy.

The design variables you need to determineinclude

• the total reservoir storage capacity (K),• the flood storage capacity (Kf) in the first

season that is the flood season,• the particular storage level where recreation

facilities will be built, called the storage target(ST) that will apply in seasons 3, 4, and 5—the recreation seasons and finally,

• the dependable water supply or yield (Y) forthe municipality.

Exercises 521

Assume you can determine these designvariable values based on average flows at thereservoir site in six seasons of a year. Theseaverage flows are 35, 42, 15, 3, 15, and 22 in theseasons 1–6, respectively.

The objective is to design the system to maxi-mize the total annual net benefits derived from

• flood control in season 1,• recreation in seasons 3–5, and• water supply in all seasons,

less the annual cost of the

• reservoir and• any losses resulting from not meeting the

recreation storage targets in the recreationseasons.

The flood benefits are estimated to be 2 Kf0.7.

The recreation benefits for the entire recre-ation season are estimated to be 8 ST.

The water supply benefits for the entire yearare estimated to be 20 Y.

The annual reservoir cost is estimated to be3 K1.2.

The recreation loss in each recreation seasondepends on whether the actual storage volume islower or higher than the storage target. If it islower the losses are 12 per unit average deficit inthe season, and if they are higher the losses are 4per unit average excess in the season. It is pos-sible that a season could begin with a deficit andend with an excess, or vice versa.

Develop and solve a nonlinear optimizationmodel for finding the values of each of the designvariables: K, Kf, ST, and Y and the maximumannual net benefits. (There will be other variablesas well. Just define what you need and put it alltogether in a model.)

Does the solution give you sufficient infor-mation that would allow you to simulate thesystem using a sequence of inflows to thereservoir that are different than the ones used toget the design variable values? If not how wouldyou define a reservoir operating policy? Afterdetermining the system’s design variable valuesusing optimization, and then determining thereservoir operating policy, you would then sim-ulate this system over many years to get a betteridea of how it might perform.

11:15 Suppose you have 19 years of monthlyflow data at a site where a reservoir couldbe located. How could you construct amodel to estimate what the requiredover-year and within year storage neededto produce a specified annual yield Y thatis allocated to each month t by someknown fraction δt. What would be themaximum reliability of those yields? Ifyou wanted to add to that an additionalsecondary yield having only 80% relia-bility, how would the model change?Make up 19 annual flows and assume thatthe average monthly flows are specifiedfractions of those annual flows. Just usingthese annual flows and the averagemonthly fractions, solve your model.

ST

Kf


11:16

(a) Develop an optimization model forestimating the least-cost combinationof active storage K1 and K2 capacitiesat two reservoir sites on a singlestream that are used to produce areliable flow or yield downstream ofthe downstream reservoir. Assume10 years of monthly flow data at eachreservoir site. Identify what otherdata are needed.

(b) Describe the two-reservoir operatingpolicy that could be incorporated intoa simulation model to check thesolution obtained from the optimiza-tion model.

Define

CsðKsÞ cost of active storage capacity at site s;where s = 1, 2

Ksd dead storage capacity of reservoir at

site s; Ksd = 0

Sst storage volume at beginning of periodt at site s

Ls loss of water due to evaporation at sites; Ls = 0

R12t release from reservoir at site 1 to site 2

in period tYt yield to downstream in period tQs

t 10 years of monthly natural flowsavailable at each site s

aos area associated with dead storage

volume at site sas area per unit storage volume at site sets evaporation depth in period t at site s

11:17 Given inflows to an effluent storagelagoon that can be described by a simplefirst-order Markov chain in each of

T periods t, and an operating policy thatdefines the lagoon discharge as a func-tion of the initial volume and inflow,indicate how you would estimate theprobability distribution of lagoon stor-age volumes.

11:18 (a) Using the inflow data in the tablebelow, develop and solve a yield

model for estimating the storage capac-ity of a single reservoir required toproduce a yield of 1.5 that is 90% reli-able in both of the two within-year pe-riods t, and an additional yield of 1.0that is 70% reliable in period t = 2.

(b) Construct a reservoir-operating rule thatdefines reservoir release zones for theseyields.

(c) Using the operating rule, simulate the 18periods of inflow data to evaluate theadequacy of the reservoir capacity andstorage zones for delivering the requiredyields and their reliabilities. (Note thatin this simulation of the historical recordthe 90% reliable yield should be satis-fied in all the 18 periods, and theincremental 70% reliable yield shouldfail only two times in the 9 years.)

(d) Compare the estimated reservoir capac-ity with that which is needed using thesequent peak procedure.

Year Period Inflow

1 1 1.0

2 3.0

2 1 0.5

2 2.5

3 1 1.0

2 2.0

(continued)

1 2

Capacities K1 K2

Exercises 523

Year Period Inflow

4 1 0.5

2 1.5

5 1 0.5

2 0.5

6 1 0.5

2 2.5

7 1 1.0

2 5.0

8 1 2.5

2 5.5

9 1 1.5

2 4.5

11:19 One possible modification of the yieldmodel of would permit the solutionalgorithm to determine the appropriatefailure years associated with any desiredreliability instead of having to choosethese years prior to model solution. Thismodification can provide an estimate ofthe extent of yield failure in each failureyear and include the economic conse-quences of failures in the objectivefunction. It can also serve as a means ofestimating the optimal reliability withrespect to economic benefits and losses.Letting Fy be the unknown yield reduc-tion in a possible failure year y, then inplace of αpyYp in the over-year continuityconstraint, the term (Yp − Fy) can beused. What additional constraints areneeded to ensure (1) that the averageshortage does not exceed (1 − αpy)Yp or(2) that at most there are f failure yearsand none of the shortages exceed(1 − αpy)Yp.

11:20 In Indonesia there exists a wet seasonfollowed by a dry season each year. Inone area of Indonesia all farmers withinan irrigation district plant and grow rice

during the wet season. This crop bringsthe farmer the largest income per hectare;thus they would all prefer to continuegrowing rice during the dry season.However, there is insufficient water dur-ing the dry season for irrigating all5000 ha of available irrigable land forrice production. Assume an availableirrigation water supply of 32 × 106 m3 atthe beginning of each dry season, and aminimum requirement of 7000 m3/ha forrice and 1800 m3/ha for the second crop.

(a) What proportion of the 5000 hashould the irrigation district managerallocate for rice during the dry seasoneach year, provided that all availablehectares must be given sufficientwater for rice or the second crop?

(b) Suppose that crop production func-tions are available for the two crops,indicating the increase in yield perhectare per m3 of additional water,up to 10,000 m3/ha for the secondcrop. Develop a model in which thewater allocation per hectare, as wellas the hectares allocated to each crop,is to be determined, assuming aspecified price or return per unit ofyield of each crop. Under what con-ditions would the solution of thismodel be the same as in part (a)?

11:21 Along the Nile River in Egypt, irrigationfarming is practiced for the production ofcotton, maize, rice, sorghum, full andshort berseem for animal production,wheat, barley, horsebeans, and winterand summer tomatoes. Cattle and buffaloare also produced, and together with thecrops that require labor, water. Fertilizer,and land area (feddans). Farm types ormanagement practices are fairly uniform,and hence in any analysis of irrigation


policies in this region this distinctionneed not be made. Given the accompa-nying data develop a model for deter-mining the tons of crops and numbers ofanimals to be grown that will maximize(a) net economic benefits based onEgyptian prices, and (b) net economicbenefits based on international prices.Identify all variables used in the model.

Known parameters

Ci miscellaneous cost of land preparation perfeddan

PEi Egyptian price per 1000 tons of crop i

PIi international price per 1000 tons of crop i

v value of meat and dairy production peranimal

g annual labor cost per workerfP cost of P fertilizer per tonfN cost of N fertilizer per tonYi yield of crop i, tons/feddanα feddans serviced per animalβ tons straw equivalent per ton of berseem

carryover from winter to summerrw berseem requirements per animal in winterswh straw yield from wheat, tons per feddansba straw yield from barley, tons per feddanrs straw requirements per animal in summerlNi N fertilizer required per feddan of crop ilPi P fertilizer required per feddan of crop ilim labor requirements per feddan in month m,

man-dayswim water requirements per feddan in

month m, 1000 m3

him land requirements per month, fraction(1 = full month)

Required Constraints. (Assume knownresource limitations for labor, water, and land)

(a) Summer and winter fodder (berseem)requirements for the animals.

(b) Monthly labor limitations.(c) Monthly water limitations.(d) Land availability each month.

(e) Minimum number of animals required forcultivation.

(f) Upper bounds on summer and winter toma-toes (assume these are known).

(g) Lower bounds on cotton areas (assume this isknown).

Other possible constraints

(a) Crop balances.(b) Fertilizer balances.(c) Labor balance.(d) Land balance.

11:22 In Algeria there are two distinct croppingintensities, depending upon the avail-ability of water. Consider a single cropthat can be grown under intensive rota-tion or extensive rotation on a total ofA hectares. Assume that the annual waterrequirements for the intensive rotationpolicy are 16,00 m3 per ha, and for theextensive rotation policy they are4000 m3 per ha. The annual net produc-tion returns are 4000 and 2000 dinars,respectively. If the total water available is320,000 m3, show that as the availableland area A increases, the rotation policythat maximizes total net income changesfrom one that is totally intensive to onethat is increasingly extensive.

Would the same conclusion hold if instead offixed net incomes of 4000 and 2000 dinars perhectares of intensive and extensive rotation, thenet income depended on the quantity of cropproduced? Assuming that intensive rotationproduces twice as much produced by extensiverotation, and that the net income per unit of cropY is defined by the simple linear function5 − 0.05Y, develop and solve a linear program-ming model to determine the optimal rotationpolicies if A equals 20, 50, and 80. Need this netincome or price function be linear to be includedin a linear programming model?

Exercises 525

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River Basin Modeling 11 · 2017-08-26 · River Basin Modeling 11 Multipurpose river basin...

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