optimization of hedging rules for hydropower...

Scientia Iranica A (2017) 24(5), 2242{2252

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Optimization of hedging rules for hydropower reservoiroperation

K. Sasirekaa;� and T.R. Neelakantanb,1

a. School of Civil Engineering, SASTRA University, Thanjavur 613401, India.b. Centre for Advanced Research in Environment, School of Civil Engineering, SASTRA University, Thanjavur, 613401, India.

Received 22 September 2015; received in revised form 30 December 2015; accepted 30 May 2016

KEYWORDSHedging rule;Reservoir operation;Hydropower;Optimization;Genetic algorithm;Arti�cial bee colonyalgorithm;Imperialisticcompetitivealgorithm.

Abstract. Reservoir operation plays an important role in economic development ofa region. Hedging operations were used for municipal, industrial, and irrigation watersupplies from reservoirs in the past. However, hedging operation for hydropower reservoiroperation is very rare. A practically simple and useful new form of Standard OperationPolicy and a new form of hedging rules for hydropower production are introduced in thispaper and demonstrated with a case study for hydropower reservoir operation of Indirasagarreservoir system in India. The performance of optimal hedging rules is compared with thatof a new standard operation policies and the superiority (reliability increases by about 10%)of the hedging rules is demonstrated. When the number of decision variables is increasedfrom 5 to 15, energy production increases by 0.7%, the spill is reduced by 16.8%, andreliability slightly decreases by 2.1%. A bi-level simulation-optimization algorithm is usedfor optimizing the hedging rules. For optimization, Genetic Algorithm, arti�cial bee colonyalgorithm, and imperialistic competitive algorithms are utilized. The results indicate thatall the three algorithms are competitive and arti�cial bee colony algorithm is marginallybetter than the other two.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

In a de�cit prone system, availability of water fromnatural river ow is not reliable as the ow is variable.Hence, reservoirs are used to store river water for lateruse. The objective of reservoir operation is to improvereliability of water supply by e�cient operation duringnormal and drought situations. Reservoir operation isbased on certain rules, which attempt to achieve ob-

1. Present address: Department of Civil Engineering,Kalasalingam Academy of Research and Education,Kalasalingam University, Krishnankoil 626126, India.

*. Corresponding author. Tel. +91 9842634831;Fax: +91 4362 264120E-mail addresses: [email protected] (K. Sasireka);[email protected] (T.R. Neelakantan)

doi: 10.24200/sci.2017.4153

jectives, and �nding optimal reservoir operation rulesis an important research area. Monitory bene�ts canbe observed in measuring the performance of operationrules. However, often converting the intangible bene�tslike equivity or environmental protection into monitoryterms is di�cult. Hence, release based statistics areused to measure the performance. Hedging rule forreservoir operation is inspired from hedging applicationin �nancial management. It considers preservation ofsome water to meet future demands. It increases wateravailability in the reservoir by accepting small currentde�cits and avoids unacceptable large de�cits in future.It distributes the de�cit magnitude across time tominimize the impact of larger shortage. Thus, hedgingprovides insurance for high-value water uses whereverreservoirs have low re�ll potentials or experience highlyunpredictable in ows [1].

Some of the di�erent forms of hedging rules pro-

K. Sasireka and T.R. Neelakantan/Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 2242{2252 2243

posed for reservoir operation include two-point linearhedging rule [2], one-point hedging rule [3], three-point linear hedging rule [4,5], discrete phased hedgingrules [3,6,7], multi-period ahead hedging rule [8], andtype II two-point hedging rule [9]. The historicaldevelopment of hedging rules for reservoir operationis discussed in the next section.

1.1. Hedging rules for reservoir operationBower et al. [10] introduced the concept of hedgingrule and, after a gap of two decades, Hashimoto etal. [11] explained that the hedging rules were moresuitable when the loss function was non-linear. Moyet al. [12] demonstrated that a single-period severeshortage caused more damage than a few smallershortages spread in time that amounted to the sametotal shortage when added. E�ects of hedging with anexplicit demand-management policy based operationof reservoir using the hedging rule were �rst studiedby Bayazit and Unal [2] through stochastic simulationanalysis and were further extended by Srinivasan andPhilipose [4,5]. Though the hedging rule concept wasknown earlier, its �rst application to a �eld problemwas reported in 1994. Shih and ReVelle [3] proposeda continuous linear hedging rule and formulated anon-linear non-separable mixed-integer programmingmodel for a water supply system having 36-monthcritical period to reduce the maximum shortage andthey demonstrated the discrete phased hedging rules.They further modi�ed the formulation to maximizethe number of months during which no allocationwas required [13]. A longer data sequence of 25years of monthly data was used by Neelakantan andPundarikanthan [6,7] for minimizing the sum of thesquared de�cits of a multi-reservoir water supply sys-tem through a discrete phased hedging policy.

By modifying a linear model into a quadraticmodel, Tu et al. [14] showed improved results throughdiscrete phased hedging rules for a multi-purpose multi-reservoir system. Several hedging rules, with variouslead-time and reduction percentages, were developedto investigate the e�ects on shortage characteristics.Reis et al. [15] developed a reservoir operation modeland solved the problem by hybrid model of GA andlinear programming to determine operational decisions.From their results, it was understood that the modelwas well suited for generating operating policy in theform of hedging rules without a prior imposition of theirform. Shiau [16] analyzed a reservoir that served forboth urban and agricultural demands. The de�nitionfor the reservoir index, which may be well stated asthe probability that the summation of reservoir storageand in ow is su�cient to meet the demand, was usedto trigger the hedging as well as to �nd a hedgingfactor [16]. Several hedging rules, with various lead-time and reduction percentages, were developed to

investigate the e�ects on shortage characteristics. Sub-sequently, Shiau and Lee [17] simultaneously minimizedthe maximum monthly de�cit and `de�cit to demandratio' over the analysis horizon to derive optimal 2-point hedging rules by employing a compromise pro-gramming based method.

Karamouz and Araghinejad [18] adopted a dis-crete phased hedging rule for a reservoir system thatsupplied water for municipal, industrial, and irrigationdemands. Shiau [8] introduced the multi-period aheadhedging rule and applied the same for a reservoirsystem serving domestic and agricultural demands.The e�ect of spill and evaporation while applying thehedging rules was studied by Celeste and Billib [19].They found that preserving water before droughtshould be judged carefully, especially in the regionswhere the evaporation rate was high. Karamouz etal. [20] applied hedging based optimization model toa reservoir, which served domestic, industrial, andirrigation demands. They obtained hedging rules basedon precipitation, temperature, and generated stream ow. Recent attempts were made by Zeng et al. [21]for applying hedging rules for hydropower reservoiroperation. They used `energy available' on the x-axisand `current generation' on the y-axis.

2. Hedging rules for hydropower reservoiroperation

From the above listed literature, it is evident thathedging rules have been applied for domestic, indus-trial, and irrigation water supplies. The bene�t ofthese cases is a function of water owrate. However,in the case of hydropower generation, the bene�t isa function of product of head of water and owrate.Hence, the hedging rules used for other purposes cannotbe directly used for hydropower reservoir operation.

The power generation (P ) from hydropower reser-voir is directly proportional to both owrate (Q) andavailable head (H) at the turbine (P _ QH). Therelationship between head and storage available in thereservoir is non-linear. If the water availability in areservoir is more, the head availability is also more.Hence, for a given quantity of power generation, whenthe water availability in a reservoir is less, more dis-charge is required. Thus, for a given power generation,water demand is a function of available storage. How-ever, water demand in the case of domestic, industrial,or irrigation water supply is not dependent on availablewater in the reservoir. According to the hedging rulesdeveloped for the operation of domestic, industrial, orirrigation water supply reservoir, water release dependson available water alone (since the water demand istaken as a constant for a given period) and, hence,in the graphical form, `available water' is taken on x-axis and `release' is taken on y-axis. In other words,

2244 K. Sasireka and T.R. Neelakantan/Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 2242{2252

Figure 1. Operation rules for hydropower production in two di�erent forms.

water demand to domestic, industrial, or irrigationwater supply reservoir is a �xed value in a givenperiod. However, in the case of hydropower reservoir,water demand is not constant; but, power demand isconstant. Hence, the rules with `storage available' onx-axis and `power production' on y-axis are similarto the operation rules applied for municipal watersupply (Figure 1). Hence, the operation rules may bepresented in the graphical form with `power generationpossible' (based on releasing all the available water orthe turbine capacity whichever is less) on x-axis and`power generation' (suggested as per the rule) on y-axis.However, providing the rule with `storage available' onx-axis and `release of water' on y-axis is more readable.

Hydropower systems generally have more thanone turbine. When water availability is less andpower generation is to be partial, either the turbinesmay be partially loaded or few of the turbines maybe fully loaded while the others are not operated.Operating a limited number of turbines may be e�cientin many practical situations. In this research work,one such system is demonstrated. Assume there arethree similar turbines connected to a reservoir and ifthe available water is equal to S1, one of the turbinescan be operated with its full production capacity; ifthe available water is equal to S2, two of the turbinescan be operated with their full production capacity.However, if water availability is more than S1 but lessthan S2, only one turbine will be operated with its fullproduction capacity and the remaining water will beused for future power production. When the availablewater is S1, releasing all the water will produce 33% ofthe power generation capacity. Similarly, S2 and S3 attheir corresponding availability will produce 67% and100% of the power generation capacity.

In Figure 1, the straight line connecting the pointsP0, P1, P2, and P3 makes an angle of 45� with thehorizontal line. As per the rule:

If 0 � available water < S1; number of

turbines to be operated is 0;

If S1 � available water < S2; number of

turbines to be operated is 1;

If S2 � available water < S3; number of

turbines to be operated is 2; and

If S3 � available water; number of

turbines to be operated is 3: (1)

Figure 1 shows that the release is to be decreasedas the available water increases, which is indicatedby the falling curves between the points (P1, P1'),(P2, P2'), and (P3, P3'). When the available-waterreaches the full reservoir capacity, the head availabilitycannot increase further. Thus, at this level, the releasebecomes constant irrespective of available water. Thus,in the release rule at K in the x-axis, the rule curvebecomes horizontal.

So far, the values S1, S2, and S3 are describedas the minimum available storage levels at which 1,2, and 3 turbines can be used respectively; however,the trigger values can be set more than these min-imum values. When the reservoir is operated usingthe minimum trigger values, the operation is namedStandard Operation Policy-Power (SOPP ). As ex-plained in the earlier paragraphs, the SOPP is di�erentfrom SOP used for operation of reservoirs of otherpurposes. According to SOPP , if su�cient water isavailable to operate n number of turbines (at theirfull loading condition), they will be operated (at theirfull loading condition). Shifting trigger to a highervalue implies preserving some water for future use,which is nothing but hedging. Considering hedgingpolicy, �nding the trigger values to maximize thebene�ts is an optimization problem, which is attemptedin this study. The objective of this study is todemonstrate the optimization of the above-explainedhedging rule for maximizing the total power productionwith constraints on reliability through a case study of


Indirasagar Reservoir in India. Further, three di�erentoptimization algorithms, which are popular and seemto be more promising, are utilized and the comparisonof their e�ciencies is presented.

3. Optimization algorithms

Optimization algorithms that mimic the natural hap-penings are gaining importance as they prove to bee�cient. Evolution mimicry and swarm intelligencemimicry are very popular now. Evolution algorithmslike Genetic Algorithm (GA) imitate the natural adap-tation process. The favorable traits in the chromo-somes are carried to the successive generations bycombination and mutual transfer of chromosomal ma-terial during breeding and random mutation. Swarmalgorithms imitate the behavior of social animals (ant,�sh, bee, frog, and bird) in sharing the tasks like for-aging. It has been observed that the knowledge gainedby interactions among the members of the swarmalong with experience of individual members optimizesthe operations. In the recent years, plenty of newalgorithms have been proposed and many have beenapplied in the civil engineering and water management�elds [22-26]. Many variations and hybrids of thesealgorithms have also been described in the literature.Though the scope of this paper is not to evaluate themall, an attempt is made to compare the e�ectiveness ofa few popularly used algorithms for getting a guidelineon the direction of future research. Particle SwarmAlgorithm, Ant Colony Algorithm, and Arti�cial BeeColony (ABC) algorithm are few important swarmintelligence algorithms. Among them, ABC algorithmis relatively newer, which has been claimed to be oneof the e�cient algorithms [27,28]. Further, ImperialistCompetitive Algorithm (ICA) is also claimed to bea promising algorithm. Hence, the basic GA, ABC,and ICA algorithm are used for optimizing the hedgingrules and the comparisons of optimization algorithmsare presented in this paper.

In this study, a bi-level simulation-optimizationapproach is used. The simulation model is a monthlymass-balance model that uses the hedging parametersand �nds the performance by operating through 384months. The parameters of hedging rules are takenas decision variables. The simulation model is used asa sub-model of an optimization model for optimizingthe hedging parameters and is called repeatedly bythe optimization algorithm. The optimization modelgenerates values for hedging rule parameters or decisionvariables (individual in GA, location of honey in ABC,location of a country in ICA) and passes them to thesimulation model for (�tness in GA, quality of sourcein ABC, cost in ICA) evaluation. The performanceis passed back to the optimization model to adjustthe values of hedging parameters for improving the

performance. This process is repeated until satisfyingthe stopping criteria.

3.1. Genetic algorithmA simple GA is used in this study. Descriptions of GAcan be found in Deb [29]. GA is a population basedoptimization algorithm in which each individual rep-resents a solution. The initial population is generatedrandomly. GA involves coding of decision variables,�tness evaluation of solutions, and genetic operationsfor �nding new solutions. Binary coding is used in thisstudy for representing decision variables. Fitness ofsolutions is evolved using the objective function andviolation of constraints. When a constraint is violated,a big penalty is added with objective function value soas to make it less �t. With the known �tness values,using the GA operators crossover and mutation, a newpopulation is generated. GA has been used for reservoiroperation by many studies, while few have applied it foroptimizing operations of hydropower reservoir [30,31].

3.2. Arti�cial Bee Colony (ABC) algorithmABC algorithm has been used since 2005 and is aswarm intelligence based optimization algorithm. Thisalgorithm is brie y explained below and detailed de-scriptions and applications can be found elsewhere [32-34]. In real bee colony, the foraging is performedby bees that are classi�ed as scout bees, employedbees, and onlooker bees. First, the scout bees ex-plore the honey source by random search. After aperiod, scout bees return to the hive with the honeycollected and perform a dance called waggle dance.The bees communicate the information (amount ofhoney, direction, and distance of source) about thehoney source they have identi�ed through the waggledance to the onlooker bees waiting in the dancearea. Some of the onlooker bees select to explorethe neighborhood of a good honey source indicatedin the waggle dance based on its quality. By waggledance, the bees that �nd good source attract onlookersand go back to the neighborhood of the same sourceaccompanied by the attracted onlookers. The numberof attracted onlookers depends on the quality of thesource identi�ed, which is indicated by the waggledance. A bee, which goes to the neighborhood of thesource already identi�ed by it, is called an employedbee. A bee returns and stays in the hive as a scoutbee after exhausting the search in an area. Some of thescout bees decide to perform random search. Theseprocesses continue repeatedly.

The location of honey represents a solution (val-ues for decision variables) to the optimization problem.The quality of the location (the amount of honey)represents the �tness of the solution. The �tnessis calculated using objective function equation. Thepresent study is aimed to maximize the total energy


generation as explained in Eq. (6). The more thetotal energy generation, the higher is the �tness. Thelocation (decision vector value) considered by the itharti�cial bee is denoted by zi. Each location ordecision vector has D number of values representing theoptimization parameters and each component of zi isrepresented by zi;j . The arti�cial onlooker bee choosesa honey source depending on the relative �tness, whichis calculated as follows:

ri =fiPNn=1 fn

; (2)

where N is the number of employed arti�cial bees ornumber of honey sources and fi is the �tness of alocation or solution. The new search location for anarti�cial bee based on its previous memory is calculatedas follows:znewi;j = zold

i;j + Randi;j�zoldi;j � zold

k;j�; (3)

where i 2 (1; 2; :::; N), k 2 (1; 2; :::; N), j 2 (1; 2; :::; D),k 6= i, and Randi;j are random numbers between {1and 1. In this study, k is selected randomly. A moredetailed explanation of the algorithm can be foundelsewhere [32-34].

3.3. Imperialist Competitive Algorithm (ICA)ICA was �rst reported in 2007 and is a socio-politicallymotivated optimization algorithm. This algorithmis brie y explained below and detailed descriptionsand applications can be found in [35-39]. In thisalgorithm, the number of countries is considered toform a population and the optimization process startswith a random initial population. The countries arecategorized into either colony-country or imperialist-state. An imperialist-state and its colonies togetherform an empire. The population is initially assumedto have few empires. ICA works based on imperialisticcompetition among these empires. In the process ofcompetition, weak empires deteriorate and strong onesincrease the number of colonies attached to them.

As in the ABC, the location of a country decidesthe cost; location is a set of values for decision variables;however, the lower the cost, the more powerful is thecountry. At the beginning, location of each country isset randomly. The number of countries to be consid-ered (population size) and the number of empires arearbitrarily decided. The most powerful countries areselected as imperialistic-states and other countries areconsidered as colonies. The number of colony countriesto be attached to an imperialistic-state is decided basedon the relative power of the imperialistic-state. Oncethe number of colony countries is decided, they arerandomly chosen and attached to the imperialistic-states.

In the iterations for optimization, each colonycountry moves toward the imperialist-state by a ran-dom value. If the movement is too small, too many

explorations are necessary. If the movement takes thecolony very close to imperialistic-state, it reduces thesearch ability. If the movement takes the colony too farfrom the imperialistic-state, divergence may occur. Inthe new position, if a colony is more powerful than itsimperialistic-state, the colony becomes imperialistic-state and imperialistic-state becomes a colony.

All empires try to bring colonies under theircontrol from other empires. This competition reducesthe power of few empires and, at the same time,increases the power of other empires. The weakestcolony of the weakest empire is taken for transfer ofcontrol. Calculation of empire's total power is givenbelow:

TPj = CIj +R1

Pncji=1 Cj;incj

; (4)

where TPj is total power of the empire j, CIj is the costof imperialistic-state in the empire j, R1 is a randomnumber (taken as 0.1), ncj is number of colonies in theempire j, and Cj;i is cost of colony i in the jth empire.Probability of taking the weakest colony by an empireis evaluated based on the relative total power amongthe empires. The random number generated and theprobability a�ect the control transfer. In the processof imperialistic competition and control transfer, anempire subsides when it drops o� all its colonies andthe imperialistic-state becomes a colony. Revolution,similar to mutation process in GA, is also appliedto increase the exploration and prevent convergenceon local minima. The above-described process ofICA, the collapsed empires, and colonies move towardthe imperialistic-state and, �nally, there will be onlyone location. In this situation, all the colonies andimperialistic-state have the same power as they are allin the same location.

4. Indirasagar reservoir system

Indirasagar reservoir is located at Narmada River inIndia at 22�1700000 N, 76�2800000 E. The reservoir isbuilt for the main purpose of hydropower generation.The reservoir has a storage capacity of 12,212 mil-lion m3 at a full reservoir level of 262.13 m and itscapacity at minimum draw down level of 237.70 m is1357 million m3. The powerhouse level is 196.6 m andthere are 8 turbines with a power generation capacityof 125 MW each. The overall e�ciency of the turbinesis 85%.

Thirty-two years of in ow into the reservoir isused in this study. Monthly period is used and, hence,for the 32 years, the total number of periods is 384.The evaporation loss is calculated by multiplying waterspread area and rate of evaporation. In a monthlystep, the average storage ((beginning storage + endstorage)/2) is worked and the corresponding water


spread area is found using the storage-water spreadarea relationship shown below:

WSA = (�2� 10�6 � S2) + (0:0972� S); (5)

where S is storage in 106 m3 and WSA is water spreadarea in 106 m2.

4.1. Objective function and constraintsThe objective is to maximize the total energy genera-tion, that is:

Max TE (6)

where TE =P384

mon=1Emon =P384

mon=1 c�Rmon�Hmon,Emon is the energy produced in month mon; c isenergy production coe�cient, which includes overallturbine e�ciency of 85%; Rmon is the release made forpower generation alone; Spillmon is the variable; andHmon is the average head of water available for powerproduction. The ow continuity constraint, and theminimum and maximum storage constraints are usedas follows:

Smon+1 =Smon + Imon � Evpmon �Rmon

� Spillmon; 8mon (7)

Smin � Smon � Smax; 8mon: (8)

8mon is used as a notation for all months.There are 8 turbines in the system and each has

a capacity of 125 MW. In this study, the turbinesare considered for either full loading or no-loadingconditions. Hence, power production for a month(Pmon) can be any one from the set of (0, 125, 250,375, 500, 625, 750, 875 and 1000 MW):

Pmon 2 (0; 125; 250; 375; 500; 625; 750; 875;

and 1000 MW); 8mon: (9)

Reliability can be de�ned as the probability of a systemto be in success state. A minimum power productionPmin is expected at all times. If Pmon � Pmin, thenthe month is considered successful and otherwise, themonth is considered to be a `failure' month. Theoptimization process is set to allow a certain number of`failures' through a reliability constraint. A reliabilityindex is de�ned as follows:

Reliability index = RI

=�

1�Number of failuresNumber months

�� 100:

(10)

Hence, a constraint is adopted as given below:

RI � RImin: (11)

In case of RI < RImin, a big penalty ({3841000) isadded with objective function value so that the solutionviolating this constraint is forced to be ineligible foroptimum solution.

The economic impact due to continuous failureperiods is more critical than the intermittent failures.Hence, with the total number of failure periods remain-ing �xed, continuous failures are undesirable comparedto intermittent failures [40]. The characteristic ofrecovery from failure to success is generally providedby a special reliability function, called resilience. Theability of the system to recover from failure state tosuccess state will be de�ned as resilience [40]. In thisstudy, both reliability and resilience constraints areused for �nding the optimal operation rules. Oncethe system performance enters a failure mode, untilit recovers to a success mode, the whole duration isconsidered as a failure event.

To provide a good measure of resilience, thefollowing two resilience indicators are used. Within theoperation horizon (384 months), Maximum Numberof Consecutive Failure months (MNCF) is taken asa resilience indicator. However, using this resilienceindicator alone may have some drawbacks [40]. Hence,another resilience indicator is also considered based onthe mean recovery time from a failure event or the meanduration of a failure event or Mean Down Time (MDT).The MNCF and MDT are constrained by user speci�edMNCFmax and MDTmax. Thus, the constraints arewritten as:

MNCF � MNCFmax; (12)

MDT � MDTmax: (13)

In this study, if MNCF > MNCFmax or MDT >MDTmax or MNCF > MNCFmax and MDT >MDTmax, a big penalty ({3841000) is added withobjective function value so that the solution is forcedto be ineligible for optimum solution. Apart from theabove constraints, the set of hedging rules similar toEq. (1) is used. Application of hedging rule is analyzedthrough three di�erent cases.

4.2. Hedging rulesIn the �rst case, a single set of hedging rules isapplied for all the 12 months of a year. Six di�erentpower production options (0, 125, 250, 500, 750, and1000 MW) are considered. To describe the optimumoperation rules, �ve storage levels or decision variables(S1 to S5) are to be determined. Similar to Eq. (1), �vedi�erent storage levels and six (including zero) di�erentpower production options are considered.

In the second case, a year is divided into two sea-sons of six months each (July to December and Januaryto June). For each season, a set of hedging rules isused. In each season, six di�erent power production


options are considered. The power production optionsconsidered for the �rst and second seasons are (0, 125,500, 625, 750, and 1000 MW) and (0, 125, 250, 375,500, and 625 MW), respectively. In the �rst season,the in ows are generally more and, hence, higher powerproduction options are considered. In this case, thereare 10 decision variables (S1 to S5 for season 1; andS6 to S10 for season 2), which are to be found byoptimization.

In the third case, a year is divided into three sea-sons based on the in ow (July to Septemeber, Octoberto December, and January to June). Generally, the�rst season gets more in ows (mean = 6793 millionm3/mon) and the options considered are (0, 125, 500,625, 750, and 1000 MW). The second season gets mod-erate in ows (mean = 1739 million m3/mon) and thepower production options considered are (0, 125, 250,500, 750, and 1000 MW). The third season gets lessin ows (mean = 928 million m3/mon) and the powerproduction options considered are (0, 125, 250, 375,500, and 625 MW), respectively. Hence, consideringall the three seasons, the number of decision variablesis 15 (S1 to S5 for season 1; S6 to S10 for season 2;and S11 to S15 for season 3), which are to be found byoptimization.

5. Results and discussions

Though the reservoir has eight turbines, the studyreveals that the reliability index cannot be 100% evenfor a single turbine operation (125 MW) and, hence,

Pmin is taken as 125 MW. The minimum availablestorage values required for operating the turbines arelisted in Table 1. The minimum available storagevalues for each power production level are determinedbased on the relationship, P / QH. For example,for a production of 125 MW, the `minimum availablestorage' (= 1022 million m3) is an available storagethat is to be fully released such that this release witha mean head (of initial and �nal heads) could justgenerate 125 MW. However, for an available storageof 5000 million m3, a release of 751 million m3 issu�cient to produce 125 MW as the mean head is more.Based on the Indirasagar reservoir hydraulic data,relationships between `available storage' and releaseare evaluated for various power production levels usingcurve �tting, which are presented in Table 1. The`minimum available storage values required' and the`release equations' given in Table 1 form the SOPP .The results of SOPP , when applied for Cases 1, 2, and3, are presented in Table 2.

When the limiting values in the Constraints (12)-(14) are modi�ed, di�erent optimal results are ob-tained. The optimum results obtained through hedgingare presented in Table 3 for all the three cases. Amongthe three cases, Case 1 produces the lowest total energywith the highest RI. Case 3 produces maximum totalenergy while not compromising much on RI. Further,the total spill is also low in Case 3. MNCF does notplay signi�cant role to di�erentiate among the threecases. Among the three cases, Case 3 is expectedto perform best as the number of rules used for the

Table 1. Minimum requirement of the available storage for operating turbines.

Number ofturbines

Powergeneration

(MW)

Min. availablestoragerequired(106 m3)

Release equation (R) (106 m3)

1 125 1022 �307:8S3 � 10�12 + 1035:6S2 � 10�8 � 1197:6S � 10�4 + 11282 250 1955 �684:6S3 � 10�12 + 2320:8S2 � 10�8 � 2712:0S � 10�4 + 2395:83 375 2827 �1083:6S3 � 10�12 + 3763:8S2 � 10�8 � 4506:6S � 10�4 + 3815:44 500 3657 �1488:0S3 � 10�12 + 5353:2S2 � 10�8 � 6631:86S � 10�4 + 54335 625 4456 �2250:0S3 � 10�12 + 7780:8S2 � 10�8 � 9411:0S � 10�4 + 7250:46 750 5234 �3117:0S3 � 10�12 + 11160:0S2 � 10�8 � 13797:6S � 10�4 + 9837:67 875 5997 �4306:8S3 � 10�12 + 15496:2S2 � 10�8 � 19249:8S � 10�4 + 12889:28 1000 6751 �5584:2S3 � 10�12 + 20494:8S2 � 10�8 � 25917:0S � 10�4 + 16620:0

Table 2. Performance of SOPP for all cases.

Case RI (%)No. of timeszero poweris produced

MNCF MDT Energy production(TW-h�)

Spill(106 m3)

1 84.89 58 2 1.18 112.07 569652 85.93 54 3 1.17 112.25 611643 85.42 56 3 1.22 111.96 62773

(�1 TW-h = 1012 Watt-hour)


Table 3. Optimal reservoir operation policies and their performances.

Case Rule RI (%)

No. oftimes zeropower isproduced

MNCF MDTEnergy

production(TW-h)

Spill(106 m3)

Parameter Parameter Parameter1 S1 = 1600 { { 98.96 4 3 2.00 116.45 95066

S2 = 4400 { {S3 = 8700 { {S4 = 9200 { {S5 = 9500 { {

2 S1 = 1600 S6 = 2050 { 97.40 10 5 2.00 116.90 84287S2 = 5900 S7 = 5050 {S3 = 6000 S8 = 9450 {S4 = 9000 S9 = 9750 {S5 = 9100 S10 = 11350 {

3 S1 = 2000 S6 = 3000 S11 = 1250 96.88 12 3 1.50 117.27 79070S2 = 4500 S7 = 4500 S12 = 4500S3 = 5500 S8 = 10000 S13 = 5000S4 = 8000 S9 = 10500 S14 = 8500S5 = 8500 S10 = 11000 S15 = 9000

operation is higher. However, only marginal di�erencescould be seen when analyzing the results. It could beseen from Tables 2 and 3 that the percentage of powerfailure over the operating horizon is higher in SOPPin all three cases compared to Hedging. The averagepower failure is 15% in SOPP , whereas in hedging rulethe power failure is only 2%. Total energy productionis also higher in the case of hedging. Reliability ofthe operation is presented in this study through RI.In the case of SOPP , the RI is about 85%, whereasthe RI is above 98.96%, 97.40%, and 96.88% in Cases1, 2, and 3, respectively. When the total number offailures is very low, a very small higher value of MDTand/or MNCF does not really matter. Decisions basedon MDT and MNCF are essential when the RI is notsigni�cantly di�erent between di�erent sets of rules,which is not so in the present study. Comparison ofSOPP and hedging rules clearly indicates that: (1)Hedging rules provide better results than SOPP forboth total energy production and RI, and (2) the totalspill is higher when hedging is adopted. More spills areexpected when hedging is adopted due to the highertrigger values used for hedging rules.

The hedging policy for Case 1 is presented in thegraphical form in Figures 2 and 3. While Figure 2presents the rules in the form of available-storageversus power production; Figure 3 presents the rules

Figure 2. Optimal hedging policy for Case 1 in the formof available-storage versus power production.

Figure 3. Optimal hedging policy for Case 1 in the formof available-storage versus release.


Table 4. Comparison of accuracy and number of function-evaluations required by the optimization algorithms.

Method Number ofdecision variables

Mean accuracy (%)Mean number of

evaluations ofthe simulation

GA5

96.3 34216ABC 98.1 36018ICA 98.4 31563

GA10

91.7 254016ABC 94.5 216117ICA 91.8 235521

GA15

89.1 953921ABC 92.1 864270ICA 90.3 901543

in the form of available-storage versus release. ThoughFigure 2 looks simple, Figure 3 provides additionaldetails on the amount of water to be released and,hence, it is more preferable. The equations for thecurves in Figure 3 for various power generation optionsare given in Table 1. In Figure 3, along with hedgingrules, SOPP is presented. The deviation of heding rulesfrom SOPP can be observed in Figure 3. The hedgingrules adopt a high trigger value of available-storagefor all power production levels and the deviations arelarger except for 125 MW. Optimization of the hedgingrules results in storage targets of 1600, 4400, 8700,9200, and 9500 million m3 to produce 125, 250, 500, 750and 1000 MW, respectively. It can be observed that theincremental storage values for producing higher powergradually decrease, which may be attributed to therelationship between elevations and storage capacity ofthe reservoir.

5.1. Comparison of optimization algorithmsIn this study, three di�erent algorithms are used foroptimization. In the GA, crossover probability of 0.8,mutation probability of 0.05, single point crossover,population size of 100, and a maximum of 10000generations are used. For the ABC algorithm, 50number of employed bees or honey sources, 45 numberof onlooker bees, 5 scout bees, and a maximum of10000 cycles are considered. In the case of ICA, aninitial population of 100 countries, of which the best9 are imperialists and the remaining 91 are colonies,a maximum of 10000 iterations, and a revolutionprobability of 0.05 are considered. All these valuesfor di�erent parameters are decided based on fewinitial trials. It is found that the performance of thealgorithms varies with di�erent trials. The initial pop-ulation and probabilities in uence the performance toa great extent. It is found di�cult to make generalized

comments on the performance based on the experiencegained through this study. A more elaborate statisticalstudy with benchmark optimization problems may bemore suitable for comparing the algorithms. However,based on this study, it can be concluded that all thethree algorithms can be judged as close competitors.The problem size (number of decision variables) couldhelp a little to decide on the performance. All threealgorithms have almost the same performance for Case1 with 5 decision variables. However, when thenumber of decision variables increases to 10 or 15, ABCalgorithm performs slightly better in terms of accuracyand convergence speed, while the GA and ICA performalmost similarly. Out of 30 di�erent trials made with15 decision variables, ABC algorithm is better in 11trials, GA is better in 8 trials, ICA is better in 9trials, and in 2 trails all the three algorithms performalmost similarly. However, in all the three optimizationprocedures, the �nal optimum solutions are obtainedby �ne tuning through reducing the zone of search insteps. The detailed comparative results of accuracyand mean number of function evaluations of di�erentalgorithms are shown in Table 4.

6. Conclusions

In this study, hedging was introduced for the opera-tion of a reservoir, which was meant for hydropowergeneration. Di�erent forms of hedging rules werediscussed and a useful form was identi�ed theoreticallyand demonstrated with case study results. Further,a new form of SOP was also proposed in this study,especially for hydropower operation of reservoir. Theresults proved the superiority of hedging rules overSOPP . When the number of decision variables in-creased, though the reliability was did not decreasemuch, the produced total energy increased and, fur-


ther, the spill was reduced signi�cantly. In contrast tohedging rule performance, while operating with SOPP ,spill increased (10.20%) with the increase in decisionvariables. These indicate the advantages of the hedgingrules. The results reported in this paper are speci�c tothe case study. Though similar results may be expectedin other cases, for generalization of the conclusions,more number of similar studies need to be performed.In this work, the planning model was optimized with 5,10, and 15 numbers of decision variables. The numbersof decision variables were decided based on the in owvariability. However, for real-time operation modelor more detailed planning model, the optimizationwill become more complex and pose a challenge formany optimization algorithms. The hedging rules wereoptimized using GA, ABC, and ICA algorithms. Theexperience could not allow to conclude superiority ofany one algorithm. ABC algorithm performed slightlybetter for larger optimization problems, but it has to beanalyzed and justi�ed further with di�erent problems.However, it can be concluded that a more detailedstatistical analysis may be required to identify theperformance of these algorithms.

References

1. Draper, A.J. and Lund, J.R. \Optimal hedging andcarryover storage value", J. Water Resour. Plan.Manag., 130(1), pp. 83-87 (2004).

2. Bayazit, M. and Unal, E. \E�ects of hedging onreservoir performance", Water Resour. Res., 26(4),pp. 713-719 (1990).

3. Shih, J.S. and ReVelle, C. \Water-supply operationsduring drought: Continuous hedging rule", J. WaterResour. Plan. Manag., 120(5), pp. 613-629 (1994).

4. Srinivasan, K. and Philipose, M.C. \Evaluation andselection of hedging policies using stochastic reservoirsimulation", Water Resour. Manag., 10(3), pp. 163-188 (1996).

5. Srinivasan, K. and Philipose, M.C. \E�ect of hedgingon over-year reservoir performance", Water Resour.Manag., 12(2), pp. 95-120 (1998).

6. Neelakantan, T.R. and Pundarikanthan, N.V. \Hedg-ing rule optimization for water supply reservoirs sys-tem", Water Resour. Manag., 13, pp. 409-426 (1999).

7. Neelakantan, T.R. and Pundarikanthan, N.V. \Neu-ral network-based simulation-optimization model forreservoir operation", J. Water Resour. Plan. Manag.,126(2), pp. 57-64 (2000).

8. Shiau, J.T. \Optimization of reservoir hedging rulesusing multiobjective GA", J. Water Resour. Plan.Manag., 135(5), pp. 355-363 (2009).

9. Shiau, J.T. \Analytical optimal hedging with explicitincorporation of reservoir release and carryover storagetargets", Water Resour. Res., 47(1), W01515 (2011).

10. Bower, B.T., Hufschmidt, M.M. and Reedy, W.W.\Operating procedures: Their role in the design ofwater-resources systems by simulation analyses, indesign of water-resource systems" , Edited by A. Maasset al., pp. 443-458, Harvard Univ. Press, Cambridge,Mass (1962).

11. Hashimoto, T., Stedinger, J.R. and Loucks, D.P.\Reliability, resiliency and vulnerability criteria for wa-ter resources system performance evaluation", WaterResour. Res., 18(1), pp. 14-20 (1982)

12. Moy, W.S., Cohon, J.L. and ReVelle, C.S. \A pro-gramming model for analysis of reliability, resilience,and vulnerability of a water supply reservoir", WaterResources Research, 22(4), pp. 489-498 (1986).

13. Shih, J.S and ReVelle, C. \Water supply operationsduring drought: A discrete hedging rule", EuropeanJournal of Operational Research, 82, pp. 163-175(1995).

14. Tu, M.-Y., Hsu, N.-S., Tsai, F.T.-C. and Yeh, W.W.-G. \Optimization of hedging rules for reservoir oper-ations", J. Water Resour. Plan. Manag., 134(1), pp.3-13 (2008).

15. Reis, L.F.R., Bessler, F.T., Walters, G.A. and Savic,D. \Water supply reservoir operation by combinedGA-linear programming approach", Water Resour.Manag., 20(2), pp. 227-255 (2006).

16. Shiau, J.T. \Water release policy e�ects on the short-age characteristics for the Shihmen reservoir system",Water Resour. Manag., 17(6), pp. 463-480 (2003).

17. Shiau, J.T. and Lee, H.C. \Derivation of optimalhedging rules for a water-supply reservoir throughcompromise programming", Water Resour. Manag.,19(2), pp. 111-132 (2005).

18. Karamouz, M. and Araghinejad, S. \Drought mitiga-tion through long-term operation of reservoirs: Casestudy", Journal of Irrigation and Drainage Engineer-ing, 134(4), pp. 471-478 (2008).

19. Celeste, A.B. and Billib, M. \The role of spill andevaporation in reservoir optimization models", WaterResour. Manag., 24(4), pp. 617-628 (2010).

20. Karamouz, M., Imen, S. and Nazif, S. \Developmentof a demand driven hydro-climatic model for droughtplanning", Water Resour. Manag., 26(2), pp. 329-357(2012).

21. Zeng, Y., Wu, X., Cheng, C. and Wang, Y. \Chance-constrained optimal hedging rule for cascade hy-dropower reservoirs", J. Water Resour. Plan. Manag.,140(7), 04014010 (2014).

22. Tahershamsi, A., Kaveh, A., Sheikholeslami, R. andTalatahari, S. \Big bang-big crunch algorithm forleast-cost design of water distribution systems", Inter-national Journal of Optimization in Civil Engineering,2(1), pp. 70-79 (2012).

23. Kaveh, A., Ahmadi, B., Shokohi, F. and Bohlooli,N. \Simultaneous analysis, design and optimization ofwater distribution systems using supervised chargedsystem search", International Journal of Optimizationin Civil Engineering, 3(1), pp. 37-55 (2013).


24. Sheikholeslami, R., Kaveh, A., Tahershamsi, A. andTalatahari, S. \Application of charged system searchalgorithm to water distribution networks optimiza-tion", International Journal of Optimization in CivilEngineering, 4(1), pp. 41-58 (2014).

25. Tahershamsi, A., Kaveh, A., Sheikholeslami, R. andKazemzadeh Azad, S. \An improved �re y algorithmwith harmony search scheme for optimization of waterdistribution systems", Scientia Iranica, 21(5), pp.1591-1607 (2014).

26. Kaveh, A., Shokouhi, F. and Ahmadi, B. \Analysisand design of water distribution systems via collid-ing bodies optimization", International Journal ofOptimization in Civil Engineering, 4(2), pp. 165-185(2014).

27. Karaboga, D. and Basturk, B. \ A comparative studyof arti�cial bee colony algorithm", Applied Mathemat-ics and Computation, 214, pp. 108-132 (2009).

28. Singh, A. \An Arti�cial Bee Colony algorithm forthe leaf-constrained minimum spanning tree problem",Applied Soft Computing, 9(2), pp. 625-631 (2009).

29. Deb, K., Multi-Objective Optimization Using Evolu-tionary Algorithm, John Wiley and Sons, Ltd (2001)

30. Cheng, C.T., Wang, W.C., Xu, D.M. and Chau, K.W.\Optimizing hydropower reservoir operation using hy-brid GA and chaos", Water Resour. Manag., 22(7),pp. 895-909 (2008).

31. Jothiprakash, V. and Arunkumar, R. \Optimizationof hydropower reservoir using evolutionary algorithmscoupled with chaos", Water Resour. Manag., 27(7) pp.1963-1979 (2013).

32. Ozkan, C., Kisi, O. and Akay, B. \Neural net-works with arti�cial bee colony algorithm for mod-eling daily reference evapotranspiration", IrrigationScience, 29(6), pp. 431-441 (2011).

33. Kisi, O., Ozkan, C. and Akay, B. \Modeling discharge-sediment relationship using neural networks with ar-ti�cial bee colony algorithm", Journal of Hydrology,428-429, pp. 94-103 (2012).

34. Liao, X., Zhou, J., Ouyang, S., Zhang, R. and Zhang,Y. \Multi-objective arti�cial bee colony algorithm forlong-term scheduling of hydropower system: A casestudy of China", Water Utility Journal, 7, pp. 13-23(2014).

35. Kaveh, A. and Talatahari, S. \Optimum design ofskeletal structures using imperialist competitive al-gorithm", Computers and Structures, 88(21-22), pp.1220-1229 (2010).

36. Niknama, T. Fard, E.T., Pourjafarian, N. and Rousta,A. \An e�cient hybrid algorithm based on modi�edimperialist competitive algorithm and K-means fordata clustering", Engineering Applications of Arti�cialIntelligence, 24(2), pp. 306-317 (2011).

37. Bagher, M., Zandieh, M. and Farsijani, H. \Balancingof stochastic U-type assembly lines: An imperialistcompetitive algorithm", International Journal of Ad-vanced Manufacturing Technology, 54(1-4), pp. 271-285 (2011).

38. Talatahari, S., Kaveh, A. and Sheikholeslami, R.\Chaotic imperialist competitive algorithm for opti-mum design of truss structures", Structural and Mul-tidisciplinary Optimization, 46(3), pp. 355-367 (2012).

39. Nazari-shirkouhi, S., Eivazy, H., Ghodsi, R., Rezaie,K. and Atashpaz-Gargari, E. \Solving the integratedproduct mix-outsourcing problem using the imperialistcompetitive algorithm", Expert Systems with Applica-tions, 37(12), pp. 7615-7626 (2010).

40. Srinivasan, K., Neelakantan, T.R., Narayan, P.S.and Nagarajukumar, C. \Mixed-integer programmingmodel for reservoir performance optimization", J. Wa-ter Resour. Plan. Manag., 125(5) pp. 298-301 (1999).

Biographies

Kandasamy Sasireka obtained her BE degree inCivil Engineering from Bharathidasan University andMTech from National Institute of Technology, Tiruchi-rappalli, in 1998 and 2003, respectively. She wasawarded PhD in 2017 from SASTRA University.

Thurvas Renganathan Neelakantan is a SeniorProfessor in the Department of Civil Engineering atKalasalingam University. Earlier he served as facultymember at SASTRA University in India and at Uni-versity of Kentucky in USA. He received his BE in CivilEngineering from Madurai Kamaraj University, MEin Hydrology and Water Resources Engineering fromAnna University, Chennai, and PhD from Anna Uni-versity, Chennai, in 1991, 1993, and 1998, respectively.His research interests include water resources engi-neering, management, optimization, neural networks,and environmental modeling. He has published andpresented various papers in journals and at conferencesin these �elds.

optimization of hedging rules for hydropower...

Documents