on a dynamic programming approach [Murray and Weiss- beck, 1980]. But the .... Given the values of the inflows Qt, releases R t, and fixed losses CU t (and ...
WATER
RESOURCES
RESEARCH,
VOL.
23, NO.
10, PAGES
1819-1823, OCTOBER
1987
Improved Algorithms for Reservoir Capacity Calculation Incorporating Storage-Dependent Losses and Reliability Norm SHARAD M. LELE x Departmentof ComputerScienceand Automation,Indian Institute of Science,Ban•Talore,India Two algorithms that improve upon the sequent-peakprocedurefor reservoircapacity calculation are presented.The first incorporatesstorage-dependent losses(like evaporation losses)exactly as the standard linear programmingformulation does.The secondextendsthe first so as to enable designingwith lessthan maximum reliability even when allowable shortfallin any failure year is also specified.Together, the algorithmsprovide a more accurate,flexible and yet fast method of calculatingthe storagecapacity requirementin preliminary screeningand optimization models.
1.
INTRODUCTION
The sequent-peakprocedure is a well-known simplistic algorithm for determining the storage capacity requirement for a reservoir [Thomas and Fiering, 1963]. It has also been extended to cascadedmultireservoir systemsin analyses based on a dynamic programming approach [Murray and Weissbeck, 1980]. But the procedure suffers from serious limitations:
1. Storage-dependentlossesor releasescannot be included in the calculations.
In this paper we present two simple algorithms that overcome limitations (1) and (3) completely. In part I we describe an algorithm that is based on the sequent-peakprocedure but takes into considerationstorage-dependentlossesas "exactly" as they are included in the standard LP formulation. (Although they are called "evaporation losses" hereinafter, it should be noted that they could be any kind of storagedependent lossesor withdrawals.) In part II we show how this improved algorithm may be used in a procedure that would enable one to calculate the active storage capacity requirement when the reliability norm specifiesnot only the maximum number of failure years but also the allowable shortfall in any failure year. Finally, we indicate how the procedure may be able to overcome limitations (2) and (4) to some
2. It cannot be usedfor complex multireservoir systems. 3. It cannot be used when less than maximum reliability (percentageof periods in which target demand is met) is desired; specificationof the shortfall that may be allowed in any extent. failure year is, therefore,out of the question. 4. It cannot be usedwhen the reservoiroperating policy is not the standard one. (A standard operating policy (SOP) is 2. PART I: AN ALGORITHM FOR INCORPORATING one under which the seasonal release is equal to the target STORAGE-DEPENDENT LOSSES INTO releaseor all the available water, whicheveris less.) THE SEQUENT-PEAK PROCEDURE On the other hand, the procedure has the significantadvanWe begin by defining the notation used. The original tage of not being limited by the number of years and seasons sequent-peak procedure is outlined in section 2.1, and the alof inflows used in the analysis,a factor that is crucial in detergorithm is then derivedfrom it in the subsequentsections. mining practical solvability in analysesbased on the commonly used linear programming (LP) formulation [Loucks et al., a slope of the area-capacity curve for the reservoir, L-1. 1981, pp. 236-238]. Further (at least in the case of single or cascaded multireservoir systems),it enables the analyst to b reservoirwater spreadat deadstoragelevel,/2; tackle problems with significantly nonlinear objective funcAt reservoirwater spreadat the beginningof time tions with the help of direct-searchoptimization methods. periodt, Le; Some attempts have been made to overcome the shortCU t fixed lossfrom the reservoirin time period t comings mentioned above. Modifications have been easily (suchas localconsumptive use),/2; made to enable calculation of capacity for different levels of ½t averagerate of evaporationfrom reservoirsurface reliability [Loucks, 1976]. Tejada-Guibert [1978] has presentin period t, L; ed an iterative algorithm for calculating storage capacity at E Vt volume of water lost owing to evaporationin maximum reliability in which evaporative losses are accuperiodt, L3' rately included and also a half-interval search procedure for ICRIT last period in critical sequenceof inflows; determiningthe storagerequired for a desiredreliability. None IOVF last period in which the reservoir spilled over of these algorithms, however, is able to provide for specifibefore encounteringthe critical sequenceof inflows' cation of the extent of shortfall that may be allowed in any K a approximateactivestoragecapacityof reservoir failure or "short" year. calculated from sequent-peak procedure ignoring •Now at Energy and ResourcesGroup, University of California, Berkeley.
storage-dependent losses,/2' K,•* exactactivestoragecapacity,L3' K t sequentialcumulativedeficitat the end of period t, L3;
Copyright 1987 by the American GeophysicalUnion.
Qt inflowintoreservoir duringperiodt,/•' R• release fromreservoir duringperiodt,/•;
Paper number 7W0617. 0043-1397/87/007W-0617505.00 1819
1820
LELE' IMPROVED ALGORITHMS FOR RESERVOIR CAPACITY CALCULATION
St active storagein the reservoir at the beginning
Now, the original storage-continuity relationship in the standard LP model [Loucks et al., 1981, pp. 343-348] is
of period t, L3' Y S
total number of years in the inflow sequence' number of within-year seasonsthat each year is divided
T
St + Qt- Rt- CUt-
into'
t = 1,...,
T
which, using (3), can now be rewritten as
total number of periods in the inflow sequence (= YS).
St(1-- eta/2) + Qt - Rt - CUt - etb> St+•(1 + eta/2)
Note that it is assumed as usual that if t = T then t + 1 = 1,
i.e., the given set of inflows repeats itself. In other words, the sequence of inflows may be looked upon as not a "straight line" but a "closedcircle." Therefore a phrase such as "backward from t = t• to t = t2" means(when t2 > t•) "from t = t•
down to t= 1 and then t= T down to t=t2" Similarly, "forward from t = t• to t = t2" means (when t2 < t•) "up to t = T and then from t = 1 up to t = t2." 2.1.
EVt > St+l
or
St+• < [St(1-eta/2)+Qt-Rt-CUt-etb]/(1
+eta/2 ) (4)
The other set of constraints in the LP model that goes along with (4) is the set of constraints specifyingthe minimum active storage capacity:
K•* > St
t= 1,'",
T
Sequent-Peak Procedure
(The symbol K•* is used instead of K, to indicate that this Given the values of the inflows Qt, releasesRt, and fixed would be the exact value of capacity required and not the lossesCU t (and assumingEVt = 0) for t = 1, ---, T, the mini- approximate one given by the sequent-peakprocedure.) This mum active storagerequirementK a can be calculatedusing constraint, when combined with (4), gives a rule for determining St+ •, given St and Ka*, which is the sequent-peakprocedureas follows. Stepl. SetK 0=0. Step2. Fort= 1 to T, calculateKt= max {O,(Kt_• + Rt St+• = min {K•*,
+ CUt- Qt)}.
- [(St(1--eta/2)+Qt-Rt--CUt--etb)/(1 +eta/2)]}
Step 3. If K r - K o, then go to step 4' elseif this is the first iteration, then set K 0 -K r and go to step 2' else STOP' sequent-peak analysis failed becausegross utilization is great-
(5)
Note that if St+• is equal to K•*, it indicatesthat the difference between the secondterm on the right-hand side and K•* er than the average inflow. is the volume of water that has spilled over. Step4. Ka = maximum{K,} overt = 1,.--, T. The questionof courseis "how can the valuesof St and K,* Note that Rt and CU t values are actually given/specified be known (when Ka* itself is the basic unknown to be deteronly for t = 1 to S; since they repeat from year to year, R t = mined)?". We shall now outline the algorithm that showshow Rt_ s for t > S, etc. this can be done systematically. Evaporation loss in any period t is actually proportional to The starting point of the algorithm is the fact that if one is the average reservoir water spread area during that period, sure that during a certain period (or sequenceof periods) there which in turn dependson the storage levels in the reservoir at has been no spillover, then for only that period (or sequenceof the beginning and the end of that period, i.e., St and St+•. periods) Since it is not possibleto determine St without knowing K•, the above procedure obviously cannot include any such St+•=[St(1--eta/2)+Qt-Rt--CUt--etb]/(1 +eta/2 ) (6)
storage-dependent losses. 2.2.
Inclusion of Evaporation Losses
The manner in which evaporation lossesare incorporated in the standard LP formulation provides a pointer to the modifications required in the sequent-peak procedure. In the LP formulation, the area-capacityrelationship for the reservoiris approximated as
At = aSt + b
(1)
(which temporarily obviates the need to know K•*). Now, there does exist such a sequenceof periods within which one can be sure that there have been no spillovers, namely, the critical sequenceof inflows. Further, one knows that (1) the active storage at the end of the sequencewill be zero (the reservoir will be at its maximum draw-down level) and (2) it is possible to identify the beginning and end of this sequence from the calculations in the sequent-peak procedure. The identification of the critical sequenceis a simple matter. In step 4 of the sequent-peakprocedure,all the values of the
(Note that this linearization of the area-storagerelationship is an approximation that may lead to distortions in some cases, sequentialcumulativedeficit K t have to be examined.Clearly, but it is neverthelessused very commonly. Our algorithm uses the value of t correspondingto the largestdeficit K t (which is the same approximation and so is "exact" only to the extent then taken as K•) is nothing but the last critical period, say that the LP formulation is.) The evaporation loss in any period t is taken to be the
product of the average reservoir water spread during period t and the averageevaporation rate for that period. Hence
EV, = et[(At + At+ ,)/2]
(2)
Using (1) and (2), one obtains an expressionfor the evaporation loss as a function of the active storage volumes'
EV, = et[a(St + St+•)/2 + b] = St(aet/2)+ St+•(aet/2) + etb (3)
ICRIT.
Hence from remark 1 above,
SICRIT + 1 --- 0
Moreover, the beginning of the critical sequenceis the last time the reservoir spilled over or "overflowed" before this lowinflow sequencewas encountered, or going backward from
t = ICRIT, it would be the first period for which K t = 0. Let this period be IOVF. Note that the storagein the reservoirat the end of this period as indicated by the sequent-peakpro-
cedure(say,SiOVF+ •') would be equal to K•. One could then
LELE:IMPROVED ALGORITHMS FORRESERVOIR CAPACITYCALCULATION
rewrite (6) as
St=[St+ 1(1+eta/2)+Rt+CUt+etb-Qt]/(1-eta/2) and calculate
backward
from t -
ICRIT
to t -
(7)
IOVF
1821
reliability when the allowable shortfall in any failure year is also specified.
3.1. Further Notation and Definitions
q- 1.
In addition to the symbols defined in section 2, we define Note that SlovF+l is the storagevolume that should have been there in the reservoir at the beginning of the critical the following: sequenceif the reservoir were to provide for all the releases, f number of failure years allowed; consumptive losses and exact evaporation lossesin that sed reliability (expressedas a fraction); quenceof periods.Hence in general,SiovF+ • would be greater •z fraction of normal yield desired in a failure than SiOVF+ 1' (= Ka), and in fact, it is the desiredexact active year (= 1 - maximum allowable fractional shortfall); storage,i.e.,
Rt* targetreleasein periodt l-L3].
Ka* = SIOVF+1
(8)
This claim, however, needsto be qualified. Since the critical sequencecould be different for different levelsof grossutilization, the inclusion of the evaporation lossesmay lead to a shift in the critical sequenceitself. Such a shift would manifest itself in a negativevalue of St (i.e., a deficit) in some period subsequent to ICRIT. To take care of this possibility,the remaining valuesof St need to be calculatedusingrule (5) with Ka* from (8) above and checkedfor negativity.If any St is found to be negative,the sequent-peakprocedurewould have to be repeated with the new (nonzero) values of evaporative lossesEVtt (obtained using the current values of Sts in (3)); ICRIT and IOVF would have to be recalculated and the procedure repeated all over again. 2.3.
Al•lorithm I
Thus the algorithm may be statedsuccinctlyas follows. Step 1: Initialization. 1. Initialization of Qt, R, e, CU t, a, b.
d = (Y --f)/(Y
Step 2: Sequent-PeakProcedure. 1. K o -0.
2. CalculateK t = max {0, (Kt_ • + Rt + CUt + EVt -- Qt)} T.
+ 1)
Therefore
f=rnd
{Y-(Y+
1)d}
where rnd means rounded off. So, the actual set of releases
should be Rt = •zRt*for all the within-year periodsin a failure yearand Rt = Rt* for all otherperiods. In the LP
formulation
that
allows
for less than
maximum
reliability, the identification of the actual failure years has to be done by trial and error, since the critical years are not known beforehand [Loucks et al., 1981, pp. 348-349]. Now, however, using the algorithm from part I, we can identify the critical yearsdynamically,i.e., during the calculationitself, and modify or "reset"the releasesfor those years to lower values. 3.2.
2. E V•= 0 for all t = 1,..., T.
for all t - 1,---,
Given the desired degree of reliability d, the allowable number of failure years can be calculated using the definition of reliability:
Procedure
II
Step l. Set Rt = Rt* for all t = 1,---,T. Step 2. Execute "modified" algorithm (of part I of this paper) to determineKa* and ICRIT for current set of releases Step .3. If this is the (f + 1)th iteration (i.e., if releaseshave
3. If K r = K o, then go to number4; elseif this is the first been "reset" forf years), then STOP; else go on to step 4. iteration in step 2, then put K o = Kr and go to number 2; else Step 4. From ICRIT, determine the current critical year, STOP: sequent-peakanalysisfailed becausegrossutilization is greater than the averageinflow.
4. Find the period for which K t is the maximum of all Kt, t - 1, .-., T; call this ICRIT.
5. Search backward from t- ICRIT until K t -0 is encounteredfor the first time. Assignto IOVF this value of t. Step 3: Recalculation. 1. SlCRIT+ I = 0. 2.
For t = ICRIT
+ 1 backward to t = IOVF
+ 1, calcu-
late St = [St+i(1 + eta/2) + Rt + CU t + etb-Qt]/(1 -eta/2 ). 3. Ka* = SIOVF+1. Step 4: Checking. 1. For t = ICRIT
+ 1 forward to t = IOVF-
1, calculate
St+1=min{Ka*,[St(1-eta/2)+Qt-Rt-CUt-etb-]/(1 +eta/2)}. 2. If any St is negative,then go to number 3; elseSTOP; Ka* is the requiredexactactivestorage. 3. Calculate EVt- et•a(St + St+•)/2 + b] for all t = 1, ß--, T, and go back to step 2. 3.
say ICY. If ICY is the same as for the last iteration, then reset Rt = •zRt* for t correspondingto periodsin year previousto ICY; elseresetRt = art* for t correspondingto periodsin the ICYth year. Go back to step 2. Sometimes the critical period happens to be the first season of the "wet" year immediately following a drought sequence. In such a case it is customary to choose the last year of the drought sequenceas the critical year rather than the wet year to which ICRIT belongs. Thus it is possible to generate a complete and accurate "family" of storage-yield curves for a set of inflows of any length. Clearly, the above procedure could be modified easily so as to reset the releases for one season at a time rather than
for all the seasonsin a year at once. It can also be easily altered to suit the concept of providing secondary yields of lessthan maximum reliability (in addition to a firm yield with maximum reliability) as outlined by Loucks [1976, pp. 168169] or Loucks et al. [1981, pp. 350-351].
PART II: PROCEDURE TO CALCULATE THE ACTIVE
STORAGECAPACITY REQUIREMENTFOR DIFFERENT LEVELS OF RELIABILITY WITH A SPECIFIC DEGREE OF SHORTFALL IN ANY FAILURE YEAR
In this part a procedureis outlined which enablesone to calculate the active storage capacity for less than maximum
4.
CONCLUDING
REMARKS
The algorithm for incorporating storage-dependentlosses was tested on a real 50-years and 12-months inflow sequence and was found to give the same resultsas those obtained by using the LP formulation to the last significantdigit. The
1822
LELE.' IMPROVED ALGORITHMS FOR RESERVOIR CAPACITY CALCULATION
second procedure was also implemented as a computer program; it provided a very quick and flexible way of accurately determining the effect of changesin the reliability norm, including in the percentageshortfall allowed, on the storage capacity requirement. Some remarks may be pertinent here as to the usefulness and scope of the algorithms. First, the inclusion of evaporation losses is itself of significance, because although the averageannual evaporationlossmay be only about 6-10% of the storagecapacity of the reservoir,the differencein the storage capacity requirement as calculated before and after the inclusion of evaporation lossesis usually much higher, with the magnitude of the increasedependingon the length of the critical sequence.(For example, this increasewas 30% in the case of the same inflow data mentioned above.) Consequently, a reasonably accurate estimation of the evaporative lossesis necessaryfor the accurate estimation of the active storage capacity requirement, especially at sites with high rates of
with the storagecapacity being calculatedat each point in the searchusing one of our algorithms. For example,for a singlereservoir 100-year and 12-month model, if the objectivefunction is nonlinear (say, becauseof a nonlinearly varying dam cost), the LP approach requires the use of a piecewiselinear approximation and integer programming; the problem would then be of more
than
1200 variables
and more
than
2400
constraints.Instead, one could use algorithm I and pose the problem as one in which only the monthly releasesare the
decisionvariables.This would reducethe problemto a 12variable (and maybe 12 bounds) problem; existing computerized algorithms can solve nonlinear constrained optimization problems of this size quite rapidly. (See Lele [1986, chap. 3] for such an application.) Since the algorithm calculatesthe valuesof S, for all t, it could be usedevenin the case of a hydropowergenerationmodel wherein the objectivefunction is nonlinear becausethe generation head is a function of the storage level.
When specifyingthe minimum reliability norm for water resourcesprojects,decision-makingbodies often also specify the extent of shortfall that may be allowed in any failure year. For instance,the Planning Commissionin India specifiesthat ating policies,too. For instance,a very general S-type linear the reliability of hydropower and irrigation projects may not be less than 90% and 75%, respectively,and the shortfall in decision rule (LDR) of the kind any failure year may not be more than 15-20% (Y. D. Pendse, R, = csS , + ds Central Water Commission, New Delhi, personal communication, 1985). The LP formulation uses chance-constrained (subscripts denotesthat the decisionparametersare different modelsfor suchproblems,which require the estimationof the only from seasonto season,not year to year) couldbe incor- cumulative distribution functions for the seasonal inflows;
evaporation. Second,although we have developedthe algorithms for the case when the reservoir operating policy is the SOP, they could be used (with minor modifications) under other oper-
porated easily by substituting(c + e,a/2) for (e,a/2) and they also usually do not specifythe extent of shortfall. Rather (d + be,)for (be,)in the statementof the algorithmin section than go through these complex calculations,planners often 2.3. Similarly, it should be possibleto incorporate an SQ-type prefer to designapproximately.In doing so, they err on the LDR. Since LDRs have not been found to be very appropriate side of overdesigning,i.e., for excessivelyhigh reliability and
for reservoircapacityscreeningmodels[Stedinger,1984], this may not be a very useful avenuefor further exploration; nevertheless, it servesto demonstratethe inherentflexibility of the algorithms. A more useful extension would be one whereby a "less than 100% minimum storage reliability" couldbe incorporated;resettingR, to (R,* -- Smin) rather than
low shortfall.
Calculations
show that
the difference
between
the storagecapacity requirementsat, say, 99% and 90% reliability could be very significant; for the inflow sequencementioned above, it was 30%. Similarly, exact consideration of shortfall could also affect the storage capacity calculation significantly. Overdesigningcould turn out to be quite costly, 0•R,*in step4 of procedure II for ff numberof periods(where especiallywhen one considersthe environmental impacts of Smi n is the minimumactivestoragerequirementthat may be land submergence.Procedure II provides a simple, fast, and violated /• fraction of the time; if= rnd {YS/•}) should accurate method for recalculating the capacity and thus for achieve that. examining the tradeoffs between reliability and economic
Any algorithm that calculatesK,* directly obviatesthe and/or environmental costs.
need to include the 2T constraints (T constraints for storage
continuityand T for minimum storage)and the T variables Acknowledgments.The author wishes to thank S. Vedula of the (corresponding to the storagein eachtimeperiod)in the reser- Indian Institute of Science,Bangalore, and Daniel P. Loucks of Corvoir capacityoptimizationproblem.It is this large numberof nell University, Ithaca, for their encouragement. The prompt and constraintsand variablesthat imposessignificantlimits on the valuable comments of the anonymous referee are also gratefully acknowledged. size of the LP problem that can be solved practically (and forces analysts to look for alternativesto the "complete" REFERENCES model like the "critical sequenceof years"model or the "ap- Lele, S. M., An optimization model for macro-hyddprojects incor-
proximate within-year and over-year separatedsequences" porating energy/economiccosts of submergence,M.S. thesis, 160 pp., Indian Inst. of Sci.,Bangalore, 1986. model [Loucks et al., 1981, pp. 343-348]). Conversely,our algorithms,with their very smallcomputationalrequirements Loucks, D. P., Surface water quantity management models, in Sysand their "self-contained" form, could be used for solving
tems Approach to Water Management,edited by A. K. Biswas,pp. 167-170, McGraw-Hill,
New York, 1976.
modelswith a very largenumberof periodsand evenin multi- Loucks, D. P., J. R. Stedinger, and D. A. Haith, Water Resources reservoir
models.
SystemsPlanningand Analysis,chaps. 5 and 7, Prentice-Hall, Englewood Cliffs, N.J., 1981. Murray, M., and K. Weissbeck,Computer program for river hydro development,Intl. Water Power Dam Constr.,30(3), 30-40, 1980. year releases,the active storagecapacity,and related con- Stedinger,J. R., The performanceof LDR models for preliminary straints), it is possible to solve nonlinear problems using design and reservoir operation, Water Resour.Res., 20(2), 215-224,
Further, since the remaining constraints and variables are much fewer in number (typically involving only the within-
direct-search methods for constrained nonlinear optimization,
1984.
LELE: IMPROVED ALGORITHMS FOR RESERVOIR CAPACITY CALCULATION
Tejada-Guibert,J. A., Algoritmospara la determinacionde capacida-
1823
S. M. Lele,EnergyandResources Group,Building T-4, University des de embalse(Algorithms for the determination of reservoir capacities),paper presentedat the Eighth Latin-American Congress of California,Berkeley,CA 94720. on Hydraulics, Quito, Ecuador, i978. Thomas, H. A., Jr., and M. B. Fiering, OperationsResearchin Water (ReceivedMay 7, 1987; Quality Management,pp. 1-17, Harvard Water ResourcesGroup, revisedJuly 13, 1987; Cambridge, Mass,, 1963. acceptedJuly 16, 1987.)
