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International Journal of River Basin Management

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Optimization of reservoir operation using linearprogram, case study of Riam Jerawi Reservoir,Indonesia

Bobby Minola Ginting, Dhemi Harlan, Ahmad Taufik & Herli Ginting

To cite this article: Bobby Minola Ginting, Dhemi Harlan, Ahmad Taufik & Herli Ginting(2017): Optimization of reservoir operation using linear program, case study of RiamJerawi Reservoir, Indonesia, International Journal of River Basin Management, DOI:10.1080/15715124.2017.1298604

To link to this article: http://dx.doi.org/10.1080/15715124.2017.1298604

Accepted author version posted online: 22Feb 2017.Published online: 10 Mar 2017.

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RESEARCH PAPER

Optimization of reservoir operation using linear program, case study of Riam JerawiReservoir, IndonesiaBobby Minola Gintinga, Dhemi Harlanb, Ahmad Taufikc and Herli Gintingd

aDepartment of Civil Engineering, Parahyangan Catholic University, Bandung, Indonesia; bDepartment of Civil Engineering, Bandung Institute ofTechnology, Bandung, Indonesia; cCenter for Research on New and Renewable Energy, Bandung Institute of Technology, Bandung, Indonesia;dDepartment of Physics, University of Sumatera Utara, Medan, Indonesia

ABSTRACTThe pattern of water use operation is important to ensure the continuity of water supply system of areservoir. This pattern, which is usually called power rule curve in a hydropower system, should beoptimally obtained, so that the yielded electrical power also becomes optimal while the continuityof reservoir storage can also be tenable. In this study, in order to satisfy the main objective of RiamJerawi Reservoir, where the energy of 6 MW should be provided every month, a linear optimizationmodel is applied for three objective functions such as maximizing total energy, maximizingminimum energy and minimizing energy shortage. In addition, a simulation model is alsopresented for comparison purpose particularly to emphasize some disadvantages of using thismodel. The results show that the optimum energy can be achieved by applying this optimizationmodel where the continuity of reservoir volume is also satisfied. Meanwhile, the simulation modelproduces the rule curve, which cannot satisfy the continuity criterion. This optimization model isexpected to be applied for other similar cases and give the optimum power rule curve for therelated stakeholders.

ARTICLE HISTORYReceived 1 December 2015Accepted 12 February 2017

KEYWORDSObjective functions;optimization model; powerrule curve; linear program

1 Introduction

Supply scarcity and expensive price of energy are potentialissues that still occur in Indonesia nowadays. Generally, vil-lage regions particularly on isolated areas have to face theseissues. Logistic problems and limitation of trade make deliv-ery cost and also expansion of electrical services very expens-ive. Meanwhile, energy supply is a social investment whichcannot be avoided to increase the economy growth and pros-perity. Of course, these issues then encourage a decentraliza-tion approach to rely on local resources in satisfying domesticneeds, for which the energy supply is an important require-ment to support economy and social growth. Also, low-costenergy for domestic needs can increase the quality of life.The uses of renewable energy and sustainable resources(e.g. water, sun, wind, biomass and geothermal) are thereforeexpected.

The continuity of electric supply in Indonesia is faced withcostive price due to high consumption of fuel whose costincreases rapidly. This fuel dependency is hard to avoid par-ticularly for isolated area where diesel is a main generator forelectricity. Therefore, it is prominent to find other energysources. Hydropower system is an alternative solution tosolve the energy crisis of electricity particularly in isolatedarea. Katingan is a regency located in Central Kalimantanprovince which connects the southern part of Central Kali-mantan province with the centre of growth in PangkalanBun and Sampit and also the northern part with the centreof growth in Palangkaraya, Pulang Pisau and Muarateweh.Kasongan as a capital city of Katingan regency is locatedexactly across the road of Central Kalimantan. The industrialdevelopment in Katingan regency is faced with energy crisiswhere until now the energy supply is limited to the existing

electrical generator. Katingan regency has a long river net-work located in Katingan catchment area, where the lengthof main river is approximately 650 km.

A reservoir uses its capacity (volume) and head of water toproduce the electrical power. One of many technical factorsthat must be concerned in designing a reservoir is the oper-ation pattern of water where in hydropower system’s pointof view it is usually called power rule curve. It is a very impor-tant issue because the continuity of reservoir capacity must betenable. If the reservoir rule curve is not designed properly,there will be lack of capacity during dry season or converselytoo much water spilling out of the reservoir. Therefore, a goodmanagement of a reservoir operation is required to produceoptimum results. In a good management of a reservoir oper-ation, an effective method should be applied which can opti-mize the use of river inflow for the capacity of a reservoir inorder to supply various purposes, minimize water losses andrisks such as flood problem and also reduce the environ-mental negative impacts. Achieving this effective method isa very complicated task since the roles of all stakeholdersare required. Nevertheless, with regard to the roles of engin-eers and practitioners, an effort can be undertaken such asperforming a study for optimizing a reservoir operation.Further, the results of the study could be used by other relatedstakeholders such as decision-makers.

There are two common techniques which havemostly beenused to describe the analysis of a reservoir operation. A simu-lation technique is a representation of a system under a givenset of conditions (Wurbs 1993). This technique will only pro-duce a pattern depicting a reservoir operation which is limitedto the user-specified set of variable values. Meanwhile, anoptimization technique can represent a reservoir operation

© 2017 International Association for Hydro-Environment Engineering and Research

CONTACT Bobby Minola Ginting bobbyminola.g@unpar.ac.id, bobbyminola.g@gmail.com

INTL. J. RIVER BASIN MANAGEMENT, 2017http://dx.doi.org/10.1080/15715124.2017.1298604

system by finding an optimum solution. One of the advantagesin using the optimization technique is that all optimizationmodels also ‘simulate’ the system (Wurbs 1993).While findingthe optimum solution for a set of constraints defined by users,an optimization technique will automatically perform thesimulation procedures. The advanced computer models forboth simulation and optimization techniques have been estab-lished since many years ago where Yeh (1985) and Wurbs(1996) are some of the pioneers.

Based on our experience, most practitioners in Indonesia,who are involved in hydropower projects, prefer performing asimulation technique to an optimization one, since it isadmittedly that a simulation technique is easier. They onlyneed to simulate the water balance based on the values theyhave specified. For example, they are given a task to providethe energy for a certain value, let us say ×MW. After obtain-ing the inflow discharge (or calculating with synthetic for-mula typically for 10 years due to the unavailability of theobserved inflow data) and some reservoir characteristicsdata such as topography and sedimentation volume whichare used as inputs, they compute the water balance withinthat period. In many cases, the maximum elevation of thereservoir is limited to a certain value due to several reasonseither technically or not. It means that one cannot obtainmore benefits from the capacity of the reservoir nor fromthe head elevation in order to yield more energy, even thoughit is possible theoretically. Therefore, if they get the negativewater balance for the targeted energy ×MW, they will justsay that the possible yielded energy should be less than ×MW. Even for a multi-purpose reservoir, for example, forhydropower and irrigation purposes, they would allocateless amount of water for the irrigation since they focusmore on the hydropower purpose, although actually theycould obtain better results. It is because, in simulation tech-nique, they can only adjust the targeted value in order toachieve the positive water balance. In general, it is like aniterative procedure. In the end, it could not be ensuredwhether the results have been optimum or not, even thoughthe positive water balance is achieved. In a few cases, afterthe reservoirs have been operating for 2 or 3 years, we oftenface some problems such as lack of water during dry seasonor too much water spilling out of the reservoirs. Our mainintention emerged from this problem. We would like toshow that an optimum result for a reservoir operationcould be achieved in a straightforward way by using anoptimization technique. One does not require performingan iterative procedure for a simulation technique anymorein order to obtain a rule curve of a reservoir. By performingan optimization technique, it can also be ensured in the endthat no other possibilities one can do to achieve better results,unless by changing the objective function.

There are three types of optimization technique which aremost commonly used such as linear, non-linear and dynamicprogramming. In this study, a linear program is used. Insteadof performing more complex technique like non-linear ordynamic programming, we would like to emphasize moreon linear programming step by step and reveal how the linearprogramming can reach an optimum solution. In this study,we write the codes in FORTRAN. We would like also todescribe some advantages of using an optimization modelin reservoir operation rather than a simulation model. Wetake the case study of Riam Jerawi Reservoir which is usedmainly for hydropower use.

In reality of course, we recognize that the optimization of areservoir operation cannot be separated from human judg-ment. However, in this study, we show that from another per-spective, this simple technique could be used as a usefulconsideration in order to obtain the optimum power rulecurve and electric power. The structure of this paper isdescribed as follows: In Section 2, we describe the technicaldata of Riam Jerawi Reservoir. In Sections 3 and 4,we show, respectively, the optimization and simulationtechniques as well as the mathematical formulation we usedin this study. In Section 4, we present that the use of simu-lation technique may cause some new problems particularlyin determining some unknown variables. We show our resultsin Section 5. In Section 6, we present our opinion to be dis-cussed about the recent optimization techniques whichwere proposed by Jordan et al. (2012) and Mower and Mir-anda (2013) and also we present our plans for future work.Finally, we give the summary and conclusions in Section 7.

2 Technical data of Riam Jerawi Reservoir

The technical data of Riam Jerawi Reservoir are obtainedbased on the design report (Laporan Akhir Proyek 2012).The analysis in this report was mainly focused on the feasi-bility study of Riam Jerawi Reservoir. The report was con-ducted by a consultant company. The field data, whichwere measured by the consultant, are topography aroundthe proposed location of the dam, the inflow discharge ofthe river for short period and some sediment samples bothsuspended and bed loads. Meanwhile, the others are second-ary data which are not directly measured by the consultant,such as the 90 m Digital Elevation Model, rainfall and clima-tology data for long period (10 years), land use maps, etc. Wedo not perform any computations, for example, inflow dis-charge or sediment calculations since we only focus on theoptimization technique for obtaining the optimum powerrule curve. Therefore, we just use these data in our compu-tation. However, in this section, we explain them brieflysuch as catchment area, inflow discharge, relationship curvebetween elevation-inundation area-storage capacity and sedi-mentation volume.

2.1 Catchment area

The Riam Jerawi Reservoir will be located at coordinate731,286, 9,912,314 with the catchment area approximatelyof 694.26 km2. The 90 m Digital Elevation Model is used astopographical data and the delineation of catchment area isperformed using Watershed Modeling System 8.1 as shownin Figure 1. The steps of delineating the catchment area arenot discussed here. Therefore, interested readers are referredto Aquaveo (2008).

2.2 Inflow discharge

Based on the design report (Laporan Akhir Proyek 2012), dueto the unavailability of the measured discharge data, theinflow discharge was computed using synthetic formula(rainfall-runoff procedure). The further explanation ofsome rainfall-runoff models could be read in Beven (2012).In the design report, NRECA model was used. NRECA is awater balance model which was developed by Norman Craw-ford and Steven Thurin in 1981 (Crawford and Thurin 1981).

2 B. M. GINTING ET AL.

It is a simple rainfall-runoff model based on monthly rainfalldata. This model can be divided into two parts such as directrunoff and base flow computations. The total values of bothparts are the inflow discharges. The scheme is depicted inFigure 2.

The main input parameters of this model are rainfall, cli-matology and land use coefficients. The rainfall and climatol-ogy data were collected from some gauging stations which arelocated inside the catchment area, for example, monthly rain-fall and climatology data such as temperature, sun radiation,humidity and average wind speed were collected from 1998 to2007. The land use maps show that almost 70% of the catch-ment areas are forests. Since the values of the monthly evapo-transpiration are required in the NRECA model, they werecomputed using the Penman Modified Method, which wasdeveloped by Howard Penman in 1948 (Penman 1948, Oliver2012). These values range from 5.14 to 6.38 mm/day. Afterobtaining the monthly evapotranspiration, the inflow dis-charges for every month during 10 years were obtained.Based on these values, the monthly inflow discharges forprobabilities of 20%, 50% and 80% could be obtained asshown in Table 1 and Figure 3. The values range from 7 to

33 m3/s. In this study, the optimization technique will befocused only on the discharge with probability of 50%.

2.3 Storage characteristics

As previously mentioned, the storage characteristics such asstorage capacity and inundation area were obtained from afield survey. These data are required to compute the evapor-ation losses on the reservoir which is the function of watersurface area. In the design report (Laporan Akhir Proyek2012), the evaporation was set to 4 mm/day. As shown inFigure 4, the lowest and highest contours range respectivelyfrom +105 m to +225 m with the maximum volume isapproximately 816 MCM. In reality, the bottom elevationof the reservoir will always change due to the sedimentationproblem. The sediment will be trapped at the toe of thedam and accumulate to a certain level during a certain period.In the report, the elevation of the crest of the spillway was setto +185 m which has the volume of 260 MCM. The dead sto-rage volume for 100 years was predicted to 1.27 MCM whichis located at the elevation of +108 m. The values of +185 mand +108 m are used respectively for the maximum and mini-mum boundary conditions of the elevation in both optimiz-ation and simulation models.

3 Optimization model

In order to ensure the continuity of the volume of a reservoir,an optimum power rule curve should be obtained by using anoptimization model. This has some boundary conditions,which has to be satisfied. In some cases, the boundary con-dition is usually called constraint. The continuity aspect ofa reservoir is based on the equilibrium of inflow and outflow.The inflow consists of river inflow and the outflow consists ofwater release, evaporation, seepage, infiltration and otherhydrologic processes which decrease the volume of water.

Figure 1. Catchment area of Riam Jerawi Reservoir (total area = 694.26 km2).

Figure 2. Scheme of NRECA model (after Laporan Akhir Proyek 2012).

INTERNATIONAL JOURNAL OF RIVER BASIN MANAGEMENT 3

An optimization technique is usually related to some math-ematical expressions which represent an objective function andsome constraints as a function of decision variables (Wurbs1996). The constraints are mass balance, storage characteristicsincluding the maximum and minimum volume, water releasebased on the objective, water losses due to evaporation, seepageand also other criteria such as maintenance flow, the maxi-mum and minimum capacity of turbines, etc. The objectivefunction is a mathematical formulation which defines a mainobjective. Based on Wurbs (1993) in general, some objectivefunctions with regard to the reservoir operation study can becategorized into three groups as follows:

a. Economic benefits and costsb. Water availability and reliabilityc. Hydroelectric power generation

and with regard to the hydroelectric power generation, thereare some common objective functions such as:

a. maximizing firm energyb. maximizing average annual energyc. minimizing energy shortagesd. maximizing the potential energy of water stored in the

system

In this study, since the Riam Jerawi Reservoir will be usedmainly for hydropower purpose, the objective function isthen taken with regard to the hydroelectric power generation.Therefore, we take three objective functions such as maximiz-ing total annual energy, maximizing the minimum energyand minimizing the energy shortage.

3.1 Mathematical formulation

As previously mentioned, in this section, we will explain theformulations of the three objective functions that we have cho-sen. The energy produced by a hydropower is computed as:

E = hN h1 h2 h3 Q r g Hnet, (1)

where E is energy (Watt), ηN, η1, η2 and η3 are, respectively,water to wire efficiency on energy output, adjustment of

efficiency when using annual flow duration curve, adjustmentof efficiency due to tailwater fluctuations and adjustment inconsidering the unscheduled down time, Q is inlet discharge(m3/s) and Hnet is net head (m). In this study, the average tail-water elevation and total head loss are taken based on LaporanAkhir Proyek (2012). The average tailwater elevation was pre-dicted approximately to +115 m, even though in fact it willalways vary. The head losses due to inlet, friction, expansionand outlet were taken into account based on the design layout.Also, in fact the total value of head loss will always vary since itdepends on the water surface elevation in the reservoir and thetailwater elevation. However, for simplification, the total headloss was estimated approximately to 5 m after considering allhead losses. The values of ηN, η1, η2 and η3 are set, respectively,to 0.80, 0.95, 0.97 and 0.97, ρ and g are set, respectively, to1000 kg/m3 and 9.81 m/s2.

The algorithm used in this optimization technique issketched in Figure 5. Subscript t in Figure 5 defines themonth where in this case t ranges from 1 to 12 (January toDecember). River inflow (It) is given in Table 1. Reservoir sto-rage (St) is the unknown variable that should optimally beobtained where the continuity criterion is only satisfied whenS12 ≥ S1. Reservoir release (rt) includes all amount of waterwhich are released out of the reservoir such as the waterreleased from the reservoir to the hydropower system (ht)and the water spilling out of the reservoir. In a hydropower sys-tem, the water is only used to rotate the turbine. Therefore, ifthe water is not used for other purposes or the evaporation isnot taken into account, it can be stated that theoretically thereis no loss of water, since all amount of water will flow againback into the river. The diversion release (dmt) usually includesthe irrigation and drinking water requirement, where in thiscase the value is zero. The maintenance flow (mft) is the mini-mum amount of water which is required for the river mainten-ance. It is obvious that now there are three unknown variableswhich must be optimized such as reservoir storage (St), reser-voir release (rt) and hydropower release (ht). In the Section3.1.1–3.1.3, the objective functions will be described.

3.1.1 Objective function: maximizing total energyThis objective function aims at maximizing the total annualenergy. It is written mathematically as:

f (Qt=1,...,12, Hnet, St=1,...,12) = max∑12t=1

Et

( ), (2)

where Et is the energy.

3.1.2 Objective function: maximizing minimum energyThis objective function aims at maximizing the minimumenergy which occurs in a year. The maximum energy couldbe obtained during the wet season and the minimum energymight occur during the dry season. Sometimes it would behard to predict exactly when the dry season occurs. Therefore,in this objective function, the minimum energy will besearched within the period and then optimized with an

Table 1. Inflow discharge in m3/s

Prob. (%) January February March April May June July August September October November December

20 29.62 31.53 25.74 26.85 17.43 21.21 14.59 12.31 11.96 16.39 33.47 18.9250 14.71 14.48 23.76 23.87 14.44 11.81 10.25 9.76 9.90 11.87 12.60 13.1680 11.63 8.90 12.97 14.63 10.88 10.03 8.91 8.30 8.07 7.67 7.31 7.75

Figure 3. Inflow discharge.

4 B. M. GINTING ET AL.

expectation that the maximum energy will also be optimal.This objective function is written as:

f (Qt=1,...,12, Hnet, St=1,...,12) = max[min(Et=1, . . . , E12)]. (3)

3.1.3 Objective function: minimizing energy shortageThis objective function aims at minimizing the differencebetween the yielded energy and the energy demand, whichis written mathematically as:

f (Qt=1,...,12, Hnet, St=1,...,12) = max[min(Edemand − Et=1 ,..., 12)],

(4)

where Edemand is the specified energy demand. Based onLaporan Akhir Proyek (2012), the Riam Jerawi Reservoir isexpected to produce the monthly energy in the range of 5–6 MW. Therefore, in this study, Edemand is set to 6 MW.

3.2 Linear program model

A linear program usually consists of an objective function andsome constrains. The standard form of a linear program iswritten as follows:

Max[c1x1 + c2x2 + c3x3 + · · · + cnxn]

Subject to

a11x1 + a12x2 + a13x3 + · · · + a1nxn = b1a21x1 + a22x2 + a23x3 + · · · + a2nxn = b2

· · ·am1x1 + am2x2 + am3x3 + · · · + amnxn = bm

x1 ≥ 0, x2 ≥ 0, · · · , xn ≥ 0

⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦.

(5)

It is shown that the objective function in Equation (5) is tomaximize the summation of some variables. All constraintsand variables are equalities and non-negative, respectively.If the objective function is to minimize the variable, Equation

(5) is slightly changed into Equation (6).

Min[c1x1 + c2x2 + c3x3 + · · · + cnxn]

= Max− [c1x1 + c2x2 + c3x3 + · · · + cnxn]. (6)

Based on Figure 5, now we formulate the constraints forour case as:

St − St−1 + rt = It − etSt ≤ Vmax

St ≥ Vmin

rt − ht ≥ 0rt + LIt ≥ dmt +mft

ht ≤ qmax

ht ≥ qmin

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

for t = 1, 2, 3, . . . , 12. (7)

The linear program shown in Equation (5) is only forequality constraints. Therefore, it cannot directly be used toarrange the constraints in Equation (7) since the non-equalityappears. Equation (7) must be converted by adding someslack variables zi. These variables can be either non-negativeor non-negative surplus variables which are shown inEquations (8) and (9), respectively.

ai1x1 + ai2x2 + ai3x3 + . . .+ ainxn ≤ biconvert intoai1x1 + ai2x2 + ai3x3 + . . .+ ainxn + zi = biwhere zi ≥ 0

⎡⎣

⎤⎦,(8)

ai1x1 + ai2x2 + ai3x3 + . . .+ ainxn ≥ biconvert intoai1x1 + ai2x2 + ai3x3 + . . .+ ainxn − zi = biwhere zi ≥ 0

⎡⎣

⎤⎦.(9)

Sometimes, it is usually found that some variables are notrestricted in sign or in other words the value can be eitherpositive or negative. For this condition, a different manneris applied into Equation (5) by replacing x1 with (x1′−x1′′).

Figure 4. Elevation versus storage capacity and inundation area.

INTERNATIONAL JOURNAL OF RIVER BASIN MANAGEMENT 5

This is shown in Equation (10).

Max [c1(x′1 − x′′1)+ c2x2 + c3x3 + . . .+ cnxn]

Subject to

a11(x′1 − x′′1)+ a12x2 + a13x3 + . . .+ a1nxn = b1a21(x′1 − x′′1)+ a22x2 + a23x3 + . . .+ a2nxn = b2

. . .

am1(x′1 − x′′1)+ am2x2 + am3x3 + . . .+ amnxn = bmx′1 ≥ 0, x1 ≥ 0, x2 ≥ 0, . . . , xn ≥ 0

⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦.

(10)

Applying equations (5), (6), (8) and (9) into Equation (7),the Equation (11) is yielded.

St − St−1 + rt = It − etSt + z1t = Vmax

St − z2t = Vmin

rt − ht − z3t = 0

rt + LIt − z4t = dmt +mftht + z5t = qmax

ht − z6t = qmin

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

for t = 1, 2, 3, . . . , 12

St ≥ 0, rt ≥ 0, ht ≥ 0, z1t ≥ 0, z2t ≥ 0,

z3t ≥ 0, z4t ≥ 0, z5t ≥ 0, z6t ≥ 0. (11)The last part of Equation (11) shows that all variables for

the optimization model must be non-negative. Of course it isnot required to apply Equation (10), since all variables such asstorage, hydropower release and reservoir release arerestricted in sign where the values cannot be negative. Forsimplification, the maintenance flow (mft) is set to 5% ofmonthly inflow. Therefore, Equation (11) is slightly changedinto Equation (12).

St − St−1 + rt = It − etSt + z1t = Vmax

St − z2t = Vmin

rt − ht − z3t = 0

rt + LIt − z4t = dmt + 0.05 Itht + z5t = qmax

ht − z6t = qmin

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

for t = 1, 2, 3, . . . , 12

St ≥ 0, rt ≥ 0, ht ≥ 0, z1t ≥ 0, z2t ≥ 0, z3t ≥ 0, z4t ≥ 0,

z5t ≥ 0, z6t ≥ 0. (12)

3.3 Solution of linear program model

In this study, the Simplex Method is used to solve the linearprogram. In this method, there are some criteria that must besatisfied in order to obtain a feasible and convergent solution.It is obvious that the well-known procedure such as theGauss–Jordan method is a very powerful technique in solvingsome sets of matrix, but since some unknown variables existin Equation (12), it is difficult to choose a pivot variable.Choosing the right pivot variable is really important inorder to perform an optimal computation. Therefore, inthis case, the Gauss–Jordan procedure cannot directly beapplied to solve Equation (12). With regard to the SimplexMethod, Equation (5) is changed into another form asshown in Equation (13).

Row 0 F − c1x1 − c2x2 − c3x3 − · · · − cnxn = 0

Row 1 a11x1 + a12x2 + a13x3 + · · · + a1nxn = b1Row 2 a21x1 + a22x2 + a23x3 + · · · + a2nxn = b2

· · ·Rowm am1x1 + am2x2 + am3x3 + · · · + amnxn = bmRow (m+ 1) x1 ≥ 0, x2 ≥ 0, . . . , xn ≥ 0.

(13)

The objective function is now changed into an equivalentform and it is called Row 0. With regard to Equations (8) and(9), all constraints should also be changed into other equival-ent forms. For some conditions, where all variables have thenon-negative coefficients, it could be stated that the currentbasic solution is optimal. Otherwise, a variable xi with a nega-tive coefficient in Row 0 should be chosen. Let us now con-sider that all coefficients c1, c2, c3,… , cn in Row 0 have thenegative values and choose the variable x1 as a basis. Thisvariable is then called entering variable. After choosing thevariable as a basis, a pivot Row should be determined. Inthis case, we choose Row 1 as the pivot. It should be notedthat all variables in Row 0 could freely be chosen as thebasis since they have negative values. Since now Row 1 is cho-sen as the pivot, Equation (13) changes into Equation (14).

Row 0 F + a12a11

− c2c1

( )x2 + a13

a11− c3

c1

( )x3

+ . . . + a1na11

− cnc1

( )xn = b1

a11

Row 1 x1 + a12a11

x2 + a13a11

x3 + . . .+ a1na11

xn = b1a11

Figure 5. Schematic of reservoir operation and the parameters.

6 B. M. GINTING ET AL.

Row 2a22a21

− a12a11

( )x2 + a23

a21− a13

a11

( )x3

+ . . .+ a2na21

− a1na11

( )xn = b2

a21− b1

a11

. . .

Rowmam2

am1− a12

a11

( )x2 + am3

am1− a13

a11

( )x3

+ . . .+ amn

am1− a1n

a11

( )xn = bm

am1− b1

a11

Row (m+ 1) x1 ≥ 0, x2 ≥ 0, . . . , xn ≥ 0.

(14)

Now, a new problem arises due to the difficulty in choos-ing the effective variable for pivot. The wrong choice may leadto an infeasible basic solution, for example, if Row 2 isselected as a pivot and the solution is infeasible, the valueof x1, x2, x3 or xn could be negative which does not satisfythe criteria in Row (m + 1). Therefore, we apply a simplemethod to determine the proper variable for pivot by com-paring the ratio of Right Hand Side (RHS) with Entering Vari-able Coefficient (EVC). The value should be a minimum one.Neither a non-minimum ratio nor a negative pivot elementwill produce a feasible solution (Oliver 2012). With regardto Equation (13) and choosing variable x1 as a basis, theratio of RHS/EVC for Row i = 1, … , m are b1/a11, b2/a21,… , bm/am1 respectively. After knowing the minimum ratio,the Gauss–Jordan procedure is performed. This will yield anew basic solution, where a looping procedure is performedin Equation (14) until the optimal solution is reached. Itshould also be noted, that some special conditions in a linearprogram such as alternate optimal solutions, degeneracy,unboudedness and infeasibility may occur. However, we donot discuss them further in this study. Interested readersare referred to Reeb and Leavengood (1998), Albright et al.(2011) and Source Material (2014).

Let us now combine the objective function in Equation (2)and Equation (12) into another form as Equation (15).

y −∑12i=1

Et = 0

St − St−1 + rt = It − etSt + z1t = Vmax

St − z2t = Vmin

rt − ht − z3t = 0rt + LIt − z4t = dmt + 0.05 It

ht + z5t = qmax

ht − z6t = qmin

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

for t

= 1, 2, 3, . . . , 12. (15)

It is shown that now Equation (15) has a form similarto Equation (13), where the coefficient of the objectivefunction in Row 0 has the negative value. With regardto the Simplex Method, the variable Et is called non-basic variable and the others are called basic variable(except for variable y in the objective function). By apply-ing Equation (1) into Equation (15), a new equation is

written as:

y − hN h1 h2 h3 r g∑12i=1

(htHnet t) = 0

St − St−1 + rt = It − etSt + z1t = Vmax

St − z2t = Vmin

rt − ht − z3t = 0rt + LIt − z4t = dmt + 0.05 It

ht + z5t = qmax

ht − z6t = qmin

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

for t

= 1, 2, 3, . . . , 12 (16)

Equation (16) shows that only hydropower release ht(not It) is used for rotating the turbine. The known valuesof the variables in Equation (16) are written now inTable 2.

The unknown variables qmax and qmin are the maxi-mum and minimum turbine capacity, respectively. Inthis study, qmax is set to have unrestricted value for notdetaining the turbine capacity to reach the maximumvalue. For the sake of simplicity, qmax is set to be a func-tion of qmin written in Equation (17), where qmin is set to14 m3/s.

qmax = 1.5 qmin. (17)

4 Simulation model

A simulation model has a relatively simpler procedure thanan optimization one, where the key point of performing asimulation model is to ensure the continuity of the reser-voir storage. As previously mentioned, in a simulationmodel, the reliability of a reservoir could be investigatedonly by observing the parameters such as storage level,reservoir release and the yielded energy at each time stepwhether satisfying the continuity criterion or not. In gen-eral, the mathematical formulation for a simulationmodel is similar to Equation (7), but the reservoir release(rt) in the first line of Equation (7) now becomes hydro-power release (ht), since the value of reservoir release (rt)cannot explicitly be determined in a simulation model.Therefore, the constraint in the fourth line of Equation(7) vanishes. The new equation for a simulation model is

Table 2. Value of variables

MonthInflow prob. of

50%Maintenance

flowLocalinflow

Diversionrelease

T It MFt Lit dmt

January 39.39 1.97 0.00 0.00February 35.03 1.75 0.00 0.00March 63.63 3.18 0.00 0.00April 61.88 3.09 0.00 0.00May 38.68 1.93 0.00 0.00June 30.60 1.53 0.00 0.00July 27.46 1.37 0.00 0.00August 26.15 1.31 0.00 0.00September 25.66 1.28 0.00 0.00October 31.80 1.59 0.00 0.00November 32.67 1.63 0.00 0.00December 35.25 1.76 0.00 0.00Vmax 260Vmin 1.27

Note: All units in MCM (million cubic metre).

INTERNATIONAL JOURNAL OF RIVER BASIN MANAGEMENT 7

written as:

St − St−1 + ht = It − et − sptSt ≤ Vmax

St ≥ Vmin

rt + LIt ≥ dmt + 0.05Itht ≤ qmax

ht ≥ qmin

⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

for t

= 1, 2, 3, . . . , 12. (18)

It is shown in Equation (18) that as the consequence, theremight be overflow water (spt) over the spillway if the amountof water is much higher than the hydropower release, wherethe summation of the hydropower release and the overflowwater is the reservoir release and written mathematically as:

if St−1 + It − (ht + et) ≥ Vmax thenspt = St−1 + It − (ht + et)− Vmax

elsespt = 0end if

where rt = ht + spt

. (19)

It might not be a problem for the downstream part of areservoir when the value of spt is high since there is no lossof water in order to satisfy the first and fourth line ofEquation (18). However, with regard to the efficiency of thehydropower system, a problem may arise. The higher valueof spt causes the higher inefficiency for the hydropower

system. This is a main disadvantage of a simulation model.It is shown that in order to solve Equation (18), the variableht should explicitly be determined by users. Afterwards, thereliability of this value is reviewed whether satisfying all con-strains in Equation (18) or not. Therefore, a simulation modelis simpler than an optimization one. However, in the end, wedo not know whether the value of ht, that we have deter-mined, has already been optimum for the energy or not.Sometimes it is hard to determine the value of ht, since itmay vary every month. Therefore, for the simulationmodel, we specify a minimum value of the energy whichshould be satisfied by the hydropower system, in this case,6 MW. Afterwards, the value of ht can be obtained fromEquation (1), where Q in Equation (1) equals ht. The valueof 6 MW could be increased or decreased as long as all con-straints in Equation (18) are satisfied. Also, the initial value ofSt should be determined. The flow chart of the simulationmodel is given in Figure 6.

5 Results and analysis

For both optimization and simulation models, the compu-tation is performed for 13 months starting from Januaryand ending in January for the next year. This aims at ensuringthe continuity aspect where volume of the reservoir in Janu-ary of the next year should be greater than or equal to thevolume in January of the previous year. Let us now define

Figure 6. Flow chart of simulation technique.

8 B. M. GINTING ET AL.

the unknown variables for the optimization model to be St, rtand ht. Since the computation is performed for 13 months,the unknown variables now become 39, consisting of 13

variables for each St, rt and ht. When the value of St for a cer-tain month has been determined, the value of water surfaceelevation and inundation area can be interpolated fromFigure 4. The value of water surface elevation is required tocompute both the value of Et and inundation area in orderto compute the value of et for the next month. Therefore, alooping procedure, which requires a correct pivot, is required.Otherwise, the feasible or the optimal objective function willnever be reached using the Simplex Method. In the simu-lation model, the unknown variable is only St, since ht canbe defined as we previously mentioned and rt is computedwith Equation (19).

Figure 7 and Table 3 show, respectively, the rule curvesand the summary of the optimization model for the threeobjective functions. It is shown that the lowest operatedwater surface elevation for the objective function of maximiz-ing minimum energy (OF-MME) is +177.52 m. This valuedoes not differ significantly with the objective function ofminimizing energy shortage (OF-MES) which is +177.53 m.Both of these values occur in February. Meanwhile, the lowestoperated water surface elevation for the objective function ofmaximizing total energy (OF-MTE) is +178.95 m whichoccurs in December. All the objective functions reach themaximum operated water surface elevation at +185 m inMay. The operated water surface elevations of both OF-MME and OF-MES show a similar characteristic, whereasthe operated water surface elevations of OF-MTE are differ-ent and never below the others.

With regard to hydropower release, from January to MarchOF-MME gives the higher values than the others with themaximum difference approximately of 2 MCM. Interestingly,from March to June OF-MTE gives the higher values withthe maximum difference approximately of 11 MCM, whereboth OF-MME and OF-MES keep producing constant values.From June to December, the constant hydropower releases aregiven by OF-MTE, whereas the values of the others keepincreasing. The maximum difference in this period is givenby OF-MES with the maximum difference approximately of6 MCM. A similar characteristic is shown for the yieldedenergy. From January to March, OF-MME produces the rela-tively constant and higher energy than the others with themaximum difference approximately of 0.43 MW (289 MW-hour). From March to June, OF-MTE produces the highest

Figure 7. Summary of the results of optimization model for discharge with prob-ability of 50% (a) the monthly operated water surface elevation at reservoir, (b)the monthly hydropower release (c) the monthly yielded energy.

Table 3. Summary of the results of optimization model for discharge with probability of 50%

Month

Storage (MCM)Water elevation at reservoir

(+m)Hydropower release

(MCM) Yielded energy (MW) Yielded energy (MW – Hour)

OF-MTE

OF-MME

OF-MES

OF-MTE

OF-MME

OF-MES

OF-MTE

OF-MME

OF-MES

OF-MTE

OF-MME

OF-MES

OF-MTE

OF-MME

OF-MES

January 219.08 209.28 206.81 179.37 178.04 177.71 36.29 38.53 38.95 5.66 5.87 5.90 4,208 4,367 4,389February 217.83 205.44 205.56 179.20 177.52 177.53 36.29 38.87 36.29 6.24 6.50 6.07 4,196 4,367 4,077March 245.17 232.44 232.44 182.92 181.19 181.19 36.29 36.63 36.75 5.99 5.88 5.90 4,459 4,377 4,391April 260.43 258.04 258.04 185.00 184.67 184.67 46.63 36.29 36.29 8.22 6.36 6.36 5,918 4,583 4,583May 260.43 260.43 260.43 185.00 185.00 185.00 38.68 36.29 36.29 6.60 6.19 6.19 4,909 4,606 4,606June 254.74 254.74 254.74 184.23 184.23 184.23 36.29 36.29 36.29 6.32 6.32 6.32 4,551 4,551 4,551July 245.90 245.90 245.90 183.02 183.02 183.02 36.29 36.29 36.29 6.00 6.00 6.00 4,466 4,466 4,466August 235.76 235.76 235.53 181.64 181.64 181.61 36.29 36.29 36.52 5.87 5.87 5.91 4,368 4,368 4,394September 225.13 224.19 224.90 180.20 180.07 180.17 36.29 37.23 36.29 5.92 6.06 5.92 4,266 4,367 4,264October 220.64 218.26 218.82 179.59 179.26 179.34 36.29 37.73 37.88 5.68 5.87 5.90 4,223 4,367 4,389November 217.02 212.10 212.84 179.09 178.43 178.53 36.29 38.83 38.64 5.82 6.15 6.13 4,188 4,430 4,417December 215.98 208.42 206.37 178.95 177.92 177.65 36.29 38.93 41.72 5.62 5.92 6.31 4,178 4,404 4,697January 219.08 209.28 206.81 179.37 178.04 177.71 36.29 38.53 38.95 5.66 5.87 5.90 4,208 4,367 4,389Min 215.98 205.44 205.56 178.95 177.52 177.53 36.29 36.29 36.29 5.62 5.87 5.90 4,178 4,367 4,077Max 260.43 260.43 260.43 185.00 185.00 185.00 46.63 38.93 41.72 8.22 6.50 6.36 5,918 4,606 4,697Total 53,927 53,251 53,223

Note: All units in MCM (million cubic metre).

INTERNATIONAL JOURNAL OF RIVER BASIN MANAGEMENT 9

energy of 8.22 MW (5918 MW-hour) among the others. Themaximum difference is 1.85 MW (1335 MW-hour). All objec-tive functions produce relatively similar energy from June toAugust. However, from August to December, the energy ofOF-MTE decreases. OF-MME produces the constant energy,while the energy of OF-MES increases. In this period, themaximum difference is 0.70 MW (519 MW-hour).

OF-MTE, OF-MME and OF-MES give the total energy ina year, respectively, of 53,927, 53,251 and 53,223 MW-hour.These values seem relatively similar, where for the totalenergy OF-MTE should be the best option. However, backto the main objective that the monthly energy demand of 6MW should be satisfied, now OF-MTE is not the best optionanymore. As shown in Figure 8, in January, OF-MTE cannotsatisfy the energy demand with the energy shortage of 0.34MW. Also, from August to December, OF-MTE producesthe energy lower than 6 MW, with the maximum shortageof 0.38 MW in December. Both OF-MME and OF-MES can-not satisfy the energy demand in January, where the energyshortages are, respectively, 0.13 and 0.10 MW, but thesevalues are lower than OF-MTE. From August to December,both OF-MME and OF-MES can relatively satisfy the energydemand, where OF-MES shows a better performance thanOF-MME. Therefore, in this case, OF-MES is the best optionin order to satisfy the main objective.

In optimization model, the unknown variables St, rt and htare determined automatically using the Simplex Method.However, as we previously mentioned, for simulation

model, the unknown variable St is computed based on thevalue of the previous step t-1. Therefore, the initial value ofSt should be known first. Actually, this initial value couldbe obtained iteratively or from the actual reservoir release.For the sake of simplicity and since the actual reservoir releaseis unavailable, we set the initial value of St for the simulationmodel to 219.08 MCM (at the water surface elevation of+179.37 m) which is similar to the largest storage value ofthe three objective functions of the optimization model inJanuary. Variable ht in simulation model is also unknown,but this value can be determined either constantly or itera-tively for the optimum result. It should be noted that itwould be hard to determine this value iteratively, sincethere will be too many possibilities. For the sake of simplicityand with regard to the main objective, variable ht is set to thevalue which always produces the energy equal to 6 MW.Another reason is that we can know whether the continuitycriterion will be satisfied or not for this value. We presentthe comparison between the result of simulation model andOF-MES in Figure 9.

It is shown that by setting the value of ht, which alwaysproduces the energy equal to 6 MW, the operated water sur-face elevation in January for the next year is lower than theelevation in January of the previous year, which means thatthe continuity criterion is not satisfied. If the computationis continued, one day the water in the reservoir would beempty, since the amount of water, which is released out ofthe reservoir, is higher than the inflow. This problem canbe anticipated either by reducing the value of ht or determin-ing this value iteratively, but the new problem might arise.The first problem is that by reducing the value of ht, themain objective of 6 MW yielded energy is not achieved.The second one, as we previously mentioned, is that thereare too many possibilities if the value of ht is determinediteratively. Actually, the second problem can be solvedusing optimization model. This is the essential part of optim-ization model, where when solving an optimization model,the simulation model is also being performed simultaneously.

6 Discussion

Nowadays, significant developments have been shown foroptimization model. For most readers, the optimization

Figure 8. The monthly yielded energy of the optimization model compared to the energy demand.

Figure 9. Comparison of monthly operated water surface elevation at reservoirbetween the simulation model and OF-MES.

10 B. M. GINTING ET AL.

model presented in this study might be quite old, since themore advanced optimization models such as non-linear anddynamic programming have intensively been used by otherresearchers. Some new techniques, which are not related tooptimization programming, are even used. We note thework of Mower and Miranda (2013), in which a new tech-nique was proposed to obtain the rule curve. Instead ofusing a complex model, they generated the rule curve byusing the historical water level data. They only required thelong-term data of water level for their model, which integratemany variables required by those complex models (Mowerand Miranda 2013). They processed the data with theEXPAND procedure in the Statistical Analysis System result-ing the 97.5 quantile model fit to the distribution of 60-dayssummed changes in water volume. They analysed the reser-voir whose main purpose was to control the flood. Thismethod is very promising and the idea behind is great,since the result can describe the existing rule curve, eventhough in fact there are always some differences between pre-dicted and existing rule curve. In our opinion, with regard toflood control objective, this method might be better thanother optimization models, since the procedure is relativelyeasier. Also, complex mathematical formulation is notrequired to be performed due to the use of real data, whichare sufficient to describe the real hydrologic phenomenon.

The model of Mower and Miranda (2013) has also severaldisadvantages. With regard to flood control objective, par-ticularly for large dams which are designed with the ProbableMaximum Flood (PMF), the absence of the PMF value in thecollected historical data would reduce the accuracy of result.The computed rule curve might underestimate the flood risk,where the existing rule curve has been designed beforehandwith the consideration of PMF. Another disadvantage,which was also stated in their paper, is the requirement oflong recorded data with keeping in mind that the good datashould always be ensured without manipulation. In ouropinion, with regard to the objective of maximizing energy,it would be hard to obtain a proper rule curve, due to thepresence of flood events. Even though the flood is an eventwith low probability of exceedance, its presence in a longdata measurement might cause a significant error for thecomputation. For example, let us assume that we have 10years historical data, where the first and fourth years arethe flood seasons with the return periods of 5 years and 100years, respectively. In the eighth year, the PMF occurs. Theuse of all data will produce overestimated results for theoptimization model with OF-MTE. Particularly in developingcountries such as Indonesia, it is also difficult to obtain thelong-term historical data.

We also note the works of Hydrologic Engineering Center(1991), Draper et al. (2003) and Jordan et al. (2012), that pre-sented the good explanation for combining the objectives ofeconomic and flood protection in an optimization model.

Both of these objectives were coupled implicitly leading toone objective function of minimizing the damage cost duringthe flood event. With regard to this method, we have a differ-ent opinion. We hypothesize that for multi-purposes reser-voir, for example, economic and flood protection, it wouldbe better to keep the main objective function of maximizingthe energy (for the economic purpose) and include theflood risk factor as the constraint. In this approach, the math-ematical formulation of the main objective function could beapplied similarly to OF-MTE, OF-MME or OF-MES, wherefor the constraints of flood risk, the formulation could be con-structed as a function of the damage costs. The minimumdamage cost is set to zero and the maximum one is specified.As shown in Figure 10, now the damage cost is defined as afunction of (Lit + rt – dmt +mft).

Our future study will be focused on an optimization modelfor a river from segment A to B (as shown in Figure 10). Aspecific location, whose risk factors are simulated, is specifiedwithin this segment. Therefore, in the future, the use of twomodels, such as hydrologic and hydraulic routing models,will be investigated. The total discharge of (Lit + rt – dmt +mft) will be treated as an input for both of these models.The simple hydrologic model such as Muskingum-Cungeor Kinematic Wave method could be used starting frompoint A to know the discharge at point B. The more advancedmodel such as 1D or 2D hydraulic model based on the shal-low water equations could also be used in order to simulatethe inundation area for determining the risk factor. The out-put from both of these models is water elevation, which isrelated to inundation area. Therefore, the flood risk con-straint can be determined based on the inundation area.

This proposed approach can be applied not only for opti-mizing energy, but also for checking the vulnerability of a rulecurve to flood problems at the downstream area of the dam.Let us now take an example of the Riam Jerawi’s rule curve,which has previously been obtained based on OF-MES. Asshown in Figure 7, there is a quite high difference of watersurface elevation of the rule curve in a year, which is approxi-mately 7.47 m. Since the time of a flood event cannot exactlybe predicted, the risk factors for the specified location (withinsegment AB) vary depending on the water elevation at thereservoir. If a flood occurs within January and February,the risk factors are lower than in May or June, since thereis a space of volume of more than 7 m in January to Februaryas a flood detention. Meanwhile, there is almost no space forflood detention in May or June, since the average water sur-face elevation at the reservoir in these months is +185 m. Byinvestigating this problem, some new scenarios for rule curvecould be obtained, as now the rule curve is also affected by therisk factor. We realize that an advanced formulation isrequired, since the computation of both hydrology andhydraulic models will be performed several times as muchas the number of simulation time. However, this is a

Figure 10. Concept of our future study to include risk factor in optimization model.

INTERNATIONAL JOURNAL OF RIVER BASIN MANAGEMENT 11

challenge, which is very interesting to be investigated in nextstudy. Therefore, a highly stable numerical model is required.With regard to the 2D hydraulic model, interested readers arereferred to Ginting (2011) and Ginting et al. (2011, 2012,2013).

7 Summary and conclusion

The optimization model for Riam Jerawi Reservoir has beenpresented. For comparison purpose, the simulation modelhas also been presented. It is shown that for achieving themonthly energy demand of 6 MW, some difficulties arise inthe simulation model particularly for determining the propervalue of hydropower release. As previously shown, the con-stant value of hydropower release for the energy of 6 MWproduces a rule curve which cannot satisfy the continuity cri-terion. Also, setting the value of hydropower release itera-tively causes difficulties due to many possibilities. We havealso explained that setting the value of hydropower releaseiteratively in simulation model is actually similar to perform-ing an optimization model but in a more effective way. In thisstudy, we only use the inflow discharge with the exceedanceprobability of 50%, since we only focus on the computationof optimization model. However, a similar way can also beapplied to the inflow discharge with the exceedance prob-ability of 20% or 80%.

In this study, the optimization model is a linear program,where the three unknown parameters, such as storage, reser-voir release and hydropower release are determined auto-matically. The storage is required to compute the othervariables such as water surface elevation and inundationarea based on Figure 4. Once the value of inundation areais obtained, the value of evaporation for the subsequenttime step can be computed. This requires a looping procedureuntil the objective function is solved and the feasibility is alsosatisfied. As also published in Source Material (2014), we con-clude that in order to achieve a feasible solution, the compu-tation of Simplex Method and Gauss–Jordan procedure cansimply be performed by following the two important steps.The first step is to choose a variable xi with a negative coeffi-cient in Row 0. The second one is to select the minimumvalue of the ratio of RHS with EVC.

The rule curve is selected based on OF-MES which givesthe total energy of 53,223 MW-hour in a year, even thoughthis value is lower than 53,927 MW-hour from OF-MTE.The reason is that OF-MES produces the lowest energy short-age among the others, which is suitable for the main objectiveof the 6 MW monthly energy demand. Although the optim-ization model presented in this study is a simple one, boththe advantage of using optimization model and disadvantageof using simulation model from practical point of view havebeen well explained. Therefore, the formulation of the optim-ization model presented in this study could be useful for somepractitioners and related stakeholders in designing a reser-voir. The more advanced optimization models, such asnon-linear and dynamic programming, are also interestingto be investigated in next study.

Acknowledgment

The authors appreciate all anonymous reviewers, editor andassociate editor for providing many constructive commentsand suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

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