Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=trbm20
Download by: [Institut Teknologi Bandung] Date: 13 March 2017, At: 02:20
International Journal of River Basin Management
ISSN: 1571-5124 (Print) 1814-2060 (Online) Journal homepage: http://www.tandfonline.com/loi/trbm20
Optimization of reservoir operation using linear program, 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 Riam Jerawi 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: 22 Feb 2017.
Published online: 10 Mar 2017.
Submit your article to this journal
Article views: 6
View related articles
View Crossmark data
RESEARCH PAPER
Optimization of reservoir operation using linear program, case study of Riam Jerawi Reservoir, Indonesia
Bobby Minola Gintinga, Dhemi Harlanb, Ahmad Taufikcand Herli Gintingd
aDepartment of Civil Engineering, Parahyangan Catholic University, Bandung, Indonesia;bDepartment of Civil Engineering, Bandung Institute of Technology, 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
ABSTRACT
The pattern of water use operation is important to ensure the continuity of water supply system of a reservoir. This pattern, which is usually called power rule curve in a hydropower system, should be optimally obtained, so that the yielded electrical power also becomes optimal while the continuity of reservoir storage can also be tenable. In this study, in order to satisfy the main objective of Riam Jerawi Reservoir, where the energy of 6 MW should be provided every month, a linear optimization model is applied for three objective functions such as maximizing total energy, maximizing minimum energy and minimizing energy shortage. In addition, a simulation model is also presented for comparison purpose particularly to emphasize some disadvantages of using this model. The results show that the optimum energy can be achieved by applying this optimization model where the continuity of reservoir volume is also satisfied. Meanwhile, the simulation model produces the rule curve, which cannot satisfy the continuity criterion. This optimization model is expected to be applied for other similar cases and give the optimum power rule curve for the related stakeholders.
ARTICLE HISTORY Received 1 December 2015 Accepted 12 February 2017 KEYWORDS
Objective functions;
optimization model; power rule curve; linear program
1 Introduction
Supply scarcity and expensive price of energy are potential issues that still occur in Indonesia nowadays. Generally, vil- lage regions particularly on isolated areas have to face these issues. 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 which cannot 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 domestic needs, for which the energy supply is an important require- ment to support economy and social growth. Also, low-cost energy 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 therefore expected.
The continuity of electric supply in Indonesia is faced with costive price due to high consumption of fuel whose cost increases rapidly. This fuel dependency is hard to avoid par- ticularly for isolated area where diesel is a main generator for electricity. Therefore, it is prominent to find other energy sources. Hydropower system is an alternative solution to solve the energy crisis of electricity particularly in isolated area. Katingan is a regency located in Central Kalimantan province which connects the southern part of Central Kali- mantan province with the centre of growth in Pangkalan Bun and Sampit and also the northern part with the centre of growth in Palangkaraya, Pulang Pisau and Muarateweh.
Kasongan as a capital city of Katingan regency is located exactly across the road of Central Kalimantan. The industrial development in Katingan regency is faced with energy crisis where 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 length of main river is approximately 650 km.
A reservoir uses its capacity (volume) and head of water to produce the electrical power. One of many technical factors that must be concerned in designing a reservoir is the oper- ation pattern of water where in hydropower system’s point of view it is usually called power rule curve. It is a very impor- tant issue because the continuity of reservoir capacity must be tenable. If the reservoir rule curve is not designed properly, there will be lack of capacity during dry season or conversely too much water spilling out of the reservoir. Therefore, a good management of a reservoir operation is required to produce optimum 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 in order to supply various purposes, minimize water losses and risks such as flood problem and also reduce the environ- mental negative impacts. Achieving this effective method is a very complicated task since the roles of all stakeholders are required. Nevertheless, with regard to the roles of engin- eers and practitioners, an effort can be undertaken such as performing a study for optimizing a reservoir operation.
Further, the results of the study could be used by other related stakeholders such as decision-makers.
There are two common techniques which have mostly been used to describe the analysis of a reservoir operation. A simu- lation technique is a representation of a system under a given set of conditions (Wurbs1993). This technique will only pro- duce a pattern depicting a reservoir operation which is limited to the user-specified set of variable values. Meanwhile, an optimization technique can represent a reservoir operation
© 2017 International Association for Hydro-Environment Engineering and Research
CONTACT Bobby Minola Ginting [email protected], [email protected] http://dx.doi.org/10.1080/15715124.2017.1298604
system by finding an optimum solution. One of the advantages in using the optimization technique is that all optimization models also‘simulate’the system (Wurbs1993). While finding the optimum solution for a set of constraints defined by users, an optimization technique will automatically perform the simulation procedures. The advanced computer models for both 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 a simulation technique to an optimization one, since it is admittedly that a simulation technique is easier. They only need to simulate the water balance based on the values they have specified. For example, they are given a task to provide the 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 the observed inflow data) and some reservoir characteristics data such as topography and sedimentation volume which are used as inputs, they compute the water balance within that period. In many cases, the maximum elevation of the reservoir is limited to a certain value due to several reasons either technically or not. It means that one cannot obtain more benefits from the capacity of the reservoir nor from the head elevation in order to yield more energy, even though it is possible theoretically. Therefore, if they get the negative water balance for the targeted energy × MW, they will just say that the possible yielded energy should be less than × MW. Even for a multi-purpose reservoir, for example, for hydropower and irrigation purposes, they would allocate less amount of water for the irrigation since they focus more on the hydropower purpose, although actually they could obtain better results. It is because, in simulation tech- nique, they can only adjust the targeted value in order to achieve the positive water balance. In general, it is like an iterative procedure. In the end, it could not be ensured whether the results have been optimum or not, even though the positive water balance is achieved. In a few cases, after the reservoirs have been operating for 2 or 3 years, we often face some problems such as lack of water during dry season or too much water spilling out of the reservoirs. Our main intention emerged from this problem. We would like to show that an optimum result for a reservoir operation could be achieved in a straightforward way by using an optimization technique. One does not require performing an iterative procedure for a simulation technique anymore in order to obtain a rule curve of a reservoir. By performing an optimization technique, it can also be ensured in the end that no other possibilities one can do to achieve better results, unless by changing the objective function.
There are three types of optimization technique which are most commonly used such as linear, non-linear and dynamic programming. In this study, a linear program is used. Instead of performing more complex technique like non-linear or dynamic programming, we would like to emphasize more on linear programming step by step and reveal how the linear programming can reach an optimum solution. In this study, we write the codes in FORTRAN. We would like also to describe some advantages of using an optimization model in reservoir operation rather than a simulation model. We take the case study of Riam Jerawi Reservoir which is used mainly for hydropower use.
In reality of course, we recognize that the optimization of a reservoir 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 useful consideration in order to obtain the optimum power rule curve and electric power. The structure of this paper is described as follows: In Section 2, we describe the technical data of Riam Jerawi Reservoir. In Sections 3 and 4, we show, respectively, the optimization and simulation techniques as well as the mathematical formulation we used in this study. In Section 4, we present that the use of simu- lation technique may cause some new problems particularly in determining some unknown variables. We show our results in Section 5. In Section 6, we present our opinion to be dis- cussed about the recent optimization techniques which were proposed by Jordanet 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 obtained based 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, which were measured by the consultant, are topography around the proposed location of the dam, the inflow discharge of the river for short period and some sediment samples both suspended 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. We do not perform any computations, for example, inflow dis- charge or sediment calculations since we only focus on the optimization technique for obtaining the optimum power rule curve. Therefore, we just use these data in our compu- tation. However, in this section, we explain them briefly such as catchment area, inflow discharge, relationship curve between elevation-inundation area-storage capacity and sedi- mentation volume.
2.1 Catchment area
The Riam Jerawi Reservoir will be located at coordinate 731,286, 9,912,314 with the catchment area approximately of 694.26 km2. The 90 m Digital Elevation Model is used as topographical data and the delineation of catchment area is performed using Watershed Modeling System 8.1 as shown inFigure 1. The steps of delineating the catchment area are not discussed here. Therefore, interested readers are referred to Aquaveo (2008).
2.2 Inflow discharge
Based on the design report (Laporan Akhir Proyek2012), due to the unavailability of the measured discharge data, the inflow discharge was computed using synthetic formula (rainfall-runoff procedure). The further explanation of some rainfall-runoff models could be read in Beven (2012).
In the design report, NRECA model was used. NRECA is a water balance model which was developed by Norman Craw- ford and Steven Thurin in 1981 (Crawford and Thurin1981).
It is a simple rainfall-runoff model based on monthly rainfall data. This model can be divided into two parts such as direct runoff and base flow computations. The total values of both parts are the inflow discharges. The scheme is depicted in Figure 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 are located 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 to 2007. 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 were computed using the Penman Modified Method, which was developed by Howard Penman in 1948 (Penman1948, Oliver 2012). These values range from 5.14 to 6.38 mm/day. After obtaining the monthly evapotranspiration, the inflow dis- charges for every month during 10 years were obtained.
Based on these values, the monthly inflow discharges for probabilities of 20%, 50% and 80% could be obtained as shown inTable 1andFigure 3. The values range from 7 to
33 m3/s. In this study, the optimization technique will be focused only on the discharge with probability of 50%.
2.3 Storage characteristics
As previously mentioned, the storage characteristics such as storage capacity and inundation area were obtained from a field survey. These data are required to compute the evapor- ation losses on the reservoir which is the function of water surface area. In the design report (Laporan Akhir Proyek 2012), the evaporation was set to 4 mm/day. As shown in Figure 4, the lowest and highest contours range respectively from +105 m to +225 m with the maximum volume is approximately 816 MCM. In reality, the bottom elevation of the reservoir will always change due to the sedimentation problem. The sediment will be trapped at the toe of the dam and accumulate to a certain level during a certain period.
In the report, the elevation of the crest of the spillway was set to +185 m which has the volume of 260 MCM. The dead sto- rage volume for 100 years was predicted to 1.27 MCM which is located at the elevation of +108 m. The values of +185 m and +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 an optimization 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 of a reservoir is based on the equilibrium of inflow and outflow.
The inflow consists of river inflow and the outflow consists of water release, evaporation, seepage, infiltration and other hydrologic 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 Proyek2012).
An optimization technique is usually related to some math- ematical expressions which represent an objective function and some constraints as a function of decision variables (Wurbs 1996). The constraints are mass balance, storage characteristics including the maximum and minimum volume, water release based on the objective, water losses due to evaporation, seepage and also other criteria such as maintenance flow, the maxi- mum and minimum capacity of turbines, etc. The objective function is a mathematical formulation which defines a main objective. Based on Wurbs (1993) in general, some objective functions with regard to the reservoir operation study can be categorized into three groups as follows:
a. Economic benefits and costs b. Water availability and reliability c. Hydroelectric power generation
and with regard to the hydroelectric power generation, there are some common objective functions such as:
a. maximizing firm energy
b. maximizing average annual energy c. minimizing energy shortages
d. maximizing the potential energy of water stored in the system
In this study, since the Riam Jerawi Reservoir will be used mainly for hydropower purpose, the objective function is then taken with regard to the hydroelectric power generation.
Therefore, we take three objective functions such as maximiz- ing total annual energy, maximizing the minimum energy and minimizing the energy shortage.
3.1 Mathematical formulation
As previously mentioned, in this section, we will explain the formulations of the three objective functions that we have cho- sen. The energy produced by a hydropower is computed as:
E=hNh1h2h3Qrg Hnet, (1) whereEis energy (Watt),ηN,η1,η2 andη3 are, respectively, water to wire efficiency on energy output, adjustment of
efficiency when using annual flow duration curve, adjustment of efficiency due to tailwater fluctuations and adjustment in considering the unscheduled down time,Qis inlet discharge (m3/s) andHnetis net head (m). In this study, the average tail- water elevation and total head loss are taken based on Laporan Akhir Proyek (2012). The average tailwater elevation was pre- dicted approximately to +115 m, even though in fact it will always vary. The head losses due to inlet, friction, expansion and outlet were taken into account based on the design layout.
Also, in fact the total value of head loss will always vary since it depends on the water surface elevation in the reservoir and the tailwater elevation. However, for simplification, the total head loss was estimated approximately to 5 m after considering all head losses. The values ofηN,η1,η2andη3are set, respectively, to 0.80, 0.95, 0.97 and 0.97, ρand g are set, respectively, to 1000 kg/m3and 9.81 m/s2.
The algorithm used in this optimization technique is sketched in Figure 5. Subscript t in Figure 5 defines the month where in this case tranges from 1 to 12 (January to December). River inflow (It) is given inTable 1. Reservoir sto- rage (St) is the unknown variable that should optimally be obtained where the continuity criterion is only satisfied when S12≥S1. Reservoir release (rt) includes all amount of water which are released out of the reservoir such as the water released 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, if the water is not used for other purposes or the evaporation is not taken into account, it can be stated that theoretically there is no loss of water, since all amount of water will flow again back into the river. The diversion release (dmt) usually includes the irrigation and drinking water requirement, where in this case 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 variables which must be optimized such as reservoir storage (St), reser- voir release (rt) and hydropower release (ht). In the Section 3.1.1–3.1.3, the objective functions will be described.
3.1.1 Objective function: maximizing total energy
This objective function aims at maximizing the total annual energy. It is written mathematically as:
f(Qt=1,...,12,Hnet,St=1,...,12)=max 12
t=1
Et
, (2)
whereEtis the energy.
3.1.2 Objective function: maximizing minimum energy This objective function aims at maximizing the minimum energy which occurs in a year. The maximum energy could be obtained during the wet season and the minimum energy might occur during the dry season. Sometimes it would be hard to predict exactly when the dry season occurs. Therefore, in this objective function, the minimum energy will be searched 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 18.92
50 14.71 14.48 23.76 23.87 14.44 11.81 10.25 9.76 9.90 11.87 12.60 13.16
80 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.
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 shortage This objective function aims at minimizing the difference between the yielded energy and the energy demand, which is 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 on Laporan Akhir Proyek (2012), the Riam Jerawi Reservoir is expected to produce the monthly energy in the range of 5–
6 MW. Therefore, in this study,Edemandis set to 6 MW.
3.2 Linear program model
A linear program usually consists of an objective function and some constrains. The standard form of a linear program is written as follows:
Max[c1x1+c2x2+c3x3+ · · · +cnxn]
Subject to
a11x1+a12x2+a13x3+ · · · +a1nxn=b1
a21x1+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 to maximize the summation of some variables. All constraints and 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 onFigure 5, now we formulate the constraints for our case as:
St−St−1+rt=It−et
St≤Vmax
St ≥Vmin
rt−ht ≥0 rt+LIt ≥dmt+mft
ht≤qmax
ht ≥qmin
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
fort=1, 2, 3, . . ., 12. (7)
The linear program shown in Equation (5) is only for equality constraints. Therefore, it cannot directly be used to arrange the constraints in Equation (7) since the non-equality appears. Equation (7) must be converted by adding some slack variableszi. These variables can be either non-negative or non-negative surplus variables which are shown in Equations (8) and (9), respectively.
ai1x1+ai2x2+ai3x3+. . .+ainxn≤bi
convert into
ai1x1+ai2x2+ai3x3+. . .+ainxn+zi=biwhere zi≥0
⎡
⎣
⎤
⎦,
(8) ai1x1+ai2x2+ai3x3+. ..+ainxn≥bi
convert into
ai1x1+ai2x2+ai3x3+. ..+ainxn−zi=biwhere zi≥0
⎡
⎣
⎤
⎦. (9) Sometimes, it is usually found that some variables are not restricted in sign or in other words the value can be either positive or negative. For this condition, a different manner is applied into Equation (5) by replacing x1 with (x1′−x1′′).
Figure 4.Elevation versus storage capacity and inundation area.
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=b1
a21(x′1−x′′1)+a22x2+a23x3+. . .+a2nxn=b2
. . .
am1(x′1−x′′1)+am2x2+am3x3+. . .+amnxn=bm
x′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−et
St+z1t =Vmax
St−z2t=Vmin
rt−ht−z3t =0 rt+LIt−z4t=dmt+mft
ht+z5t =qmax
ht−z6t=qmin
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎣
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎦
fort=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 is not required to apply Equation (10), since all variables such as storage, hydropower release and reservoir release are restricted in sign where the values cannot be negative. For simplification, the maintenance flow (mft) is set to 5% of monthly inflow. Therefore, Equation (11) is slightly changed into Equation (12).
St−St−1+rt=It−et
St+z1t=Vmax
St−z2t =Vmin
rt−ht−z3t =0 rt+LIt−z4t=dmt+0.05It
ht+z5t=qmax
ht−z6t =qmin
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎣
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎦
fort=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 linear program. In this method, there are some criteria that must be satisfied in order to obtain a feasible and convergent solution.
It is obvious that the well-known procedure such as the Gauss–Jordan method is a very powerful technique in solving some sets of matrix, but since some unknown variables exist in Equation (12), it is difficult to choose a pivot variable.
Choosing the right pivot variable is really important in order to perform an optimal computation. Therefore, in this case, the Gauss–Jordan procedure cannot directly be applied to solve Equation (12). With regard to the Simplex Method, Equation (5) is changed into another form as shown in Equation (13).
Row 0 F−c1x1 − c2x2−c3x3 − · · · −cnxn=0 Row 1 a11x1+a12x2+a13x3+ · · · +a1nxn=b1
Row 2 a21x1+a22x2+a23x3+ · · · +a2nxn=b2
· · ·
Rowm am1x1+am2x2+am3x3+ · · · +amnxn=bm
Row (m+1) x1≥0,x2≥0, . . .,xn≥0.
(13)
The objective function is now changed into an equivalent form 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 the non-negative coefficients, it could be stated that the current basic solution is optimal. Otherwise, a variablexiwith a nega- tive coefficient in Row 0 should be chosen. Let us now con- sider that all coefficients c1,c2,c3,…,cnin Row 0 have the negative values and choose the variable x1 as a basis. This variable is then called entering variable. After choosing the variable as a basis, a pivot Row should be determined. In this case, we choose Row 1 as the pivot. It should be noted that all variables in Row 0 could freely be chosen as the basis since they have negative values. Since now Row 1 is cho- sen as the pivot, Equation (13) changes into Equation (14).
Row 0 F + a12
a11−c2
c1
x2+ a13
a11−c3
c1
x3
+. . . + a1n
a11−cn
c1
xn= b1
a11
Row 1 x1+a12
a11
x2+a13
a11
x3+. . .+a1n
a11
xn= b1
a11 Figure 5.Schematic of reservoir operation and the parameters.
Row 2 a22
a21−a12
a11
x2+ a23
a21−a13
a11
x3
+. . .+ a2n
a21−a1n
a11
xn= b2
a21−b1
a11
. . . Rowm am2
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 lead to an infeasible basic solution, for example, if Row 2 is selected as a pivot and the solution is infeasible, the value ofx1, x2, x3 orxncould be negative which does not satisfy the criteria in Row (m+ 1). Therefore, we apply a simple method to determine the proper variable for pivot by com- paring the ratio ofRight Hand Side(RHS) withEntering Vari- able Coefficient(EVC). The value should be a minimum one.
Neither a non-minimum ratio nor a negative pivot element will produce a feasible solution (Oliver 2012). With regard to Equation (13) and choosing variable x1 as a basis, the ratio ofRHS/EVCfor Rowi= 1, …,mareb1/a11,b2/a21,
…,bm/am1 respectively. After knowing the minimum ratio, the Gauss–Jordan procedure is performed. This will yield a new basic solution, where a looping procedure is performed in Equation (14) until the optimal solution is reached. It should also be noted, that some special conditions in a linear program such as alternate optimal solutions, degeneracy, unboudedness and infeasibility may occur. However, we do not discuss them further in this study. Interested readers are 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−12
i=1Et =0 St−St−1+rt =It−et
St+z1t=Vmax
St−z2t =Vmin
rt−ht−z3t=0 rt+LIt−z4t =dmt+0.05It
ht+z5t =qmax
ht−z6t=qmin
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦ fort
=1, 2, 3, . . ., 12. (15)
It is shown that now Equation (15) has a form similar to Equation (13), where the coefficient of the objective function in Row 0 has the negative value. With regard to the Simplex Method, the variable Et is called non- basic variable and the others are called basic variable (except for variable yin the objective function). By apply- ing Equation (1) into Equation (15), a new equation is
written as:
y−hNh1h2h3rg12
i=1
(htHnett) =0 St−St−1+rt=It−et
St+z1t=Vmax
St−z2t =Vmin
rt−ht−z3t=0 rt+LIt−z4t =dmt+0.05It
ht+z5t=qmax
ht−z6t =qmin
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦ fort
=1, 2, 3, . . ., 12 (16)
Equation (16) shows that only hydropower release ht
(not It) is used for rotating the turbine. The known values of the variables in Equation (16) are written now in Table 2.
The unknown variables qmax and qmin are the maxi- mum and minimum turbine capacity, respectively. In this study, qmax is set to have unrestricted value for not detaining the turbine capacity to reach the maximum value. For the sake of simplicity, qmax is set to be a func- tion of qmin written in Equation (17), where qmin is set to 14 m3/s.
qmax=1.5qmin. (17)
4 Simulation model
A simulation model has a relatively simpler procedure than an optimization one, where the key point of performing a simulation model is to ensure the continuity of the reser- voir storage. As previously mentioned, in a simulation model, the reliability of a reservoir could be investigated only by observing the parameters such as storage level, reservoir release and the yielded energy at each time step whether satisfying the continuity criterion or not. In gen- eral, the mathematical formulation for a simulation model 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 Month
Inflow prob. of 50%
Maintenance flow
Local inflow
Diversion release
T It MFt Lit dmt
January 39.39 1.97 0.00 0.00
February 35.03 1.75 0.00 0.00
March 63.63 3.18 0.00 0.00
April 61.88 3.09 0.00 0.00
May 38.68 1.93 0.00 0.00
June 30.60 1.53 0.00 0.00
July 27.46 1.37 0.00 0.00
August 26.15 1.31 0.00 0.00
September 25.66 1.28 0.00 0.00
October 31.80 1.59 0.00 0.00
November 32.67 1.63 0.00 0.00
December 35.25 1.76 0.00 0.00
Vmax 260
Vmin 1.27
Note: All units in MCM (million cubic metre).
written as:
St−St−1+ht =It−et−spt
St≤Vmax
St ≥Vmin
rt+LIt≥dmt+0.05It
ht≤qmax
ht ≥qmin
⎡
⎢⎢
⎢⎢
⎢⎢
⎣
⎤
⎥⎥
⎥⎥
⎥⎥
⎦ fort
=1, 2, 3, . . ., 12. (18)
It is shown in Equation (18) that as the consequence, there might be overflow water (spt) over the spillway if the amount of water is much higher than the hydropower release, where the summation of the hydropower release and the overflow water is the reservoir release and written mathematically as:
if St−1+It−(ht+et)≥ Vmaxthen spt =St−1+It−(ht+et)−Vmax
else spt =0
end if wherert =ht+spt
. (19)
It might not be a problem for the downstream part of a reservoir when the value ofsptis high since there is no loss of water in order to satisfy the first and fourth line of Equation (18). However, with regard to the efficiency of the hydropower system, a problem may arise. The higher value of 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 variable ht should explicitly be determined by users. Afterwards, the reliability of this value is reviewed whether satisfying all con- strains in Equation (18) or not. Therefore, a simulation model is simpler than an optimization one. However, in the end, we do 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 it may vary every month. Therefore, for the simulation model, we specify a minimum value of the energy which should be satisfied by the hydropower system, in this case, 6 MW. Afterwards, the value of ht can be obtained from Equation (1), where Qin Equation (1) equalsht. The value of 6 MW could be increased or decreased as long as all con- straints in Equation (18) are satisfied. Also, the initial value of St should be determined. The flow chart of the simulation model is given inFigure 6.
5 Results and analysis
For both optimization and simulation models, the compu- tation is performed for 13 months starting from January and ending in January for the next year. This aims at ensuring the continuity aspect where volume of the reservoir in Janu- ary of the next year should be greater than or equal to the volume in January of the previous year. Let us now define
Figure 6.Flow chart of simulation technique.
the unknown variables for the optimization model to beSt,rt
and ht. Since the computation is performed for 13 months, the unknown variables now become 39, consisting of 13
variables for eachSt,rtandht. When the value ofStfor a cer- tain month has been determined, the value of water surface elevation and inundation area can be interpolated from Figure 4. The value of water surface elevation is required to compute both the value of Et and inundation area in order to compute the value of etfor the next month. Therefore, a looping procedure, which requires a correct pivot, is required.
Otherwise, the feasible or the optimal objective function will never be reached using the Simplex Method. In the simu- lation model, the unknown variable is only St, sinceht can be defined as we previously mentioned and rt is computed with Equation (19).
Figure 7and Table 3 show, respectively, the rule curves and the summary of the optimization model for the three objective functions. It is shown that the lowest operated water surface elevation for the objective function of maximiz- ing minimum energy (OF-MME) is +177.52 m. This value does not differ significantly with the objective function of minimizing energy shortage (OF-MES) which is +177.53 m.
Both of these values occur in February. Meanwhile, the lowest operated water surface elevation for the objective function of maximizing total energy (OF-MTE) is +178.95 m which occurs in December. All the objective functions reach the maximum operated water surface elevation at +185 m in May. The operated water surface elevations of both OF- MME and OF-MES show a similar characteristic, whereas the operated water surface elevations of OF-MTE are differ- ent and never below the others.
With regard to hydropower release, from January to March OF-MME gives the higher values than the others with the maximum difference approximately of 2 MCM. Interestingly, from March to June OF-MTE gives the higher values with the maximum difference approximately of 11 MCM, where both OF-MME and OF-MES keep producing constant values.
From June to December, the constant hydropower releases are given by OF-MTE, whereas the values of the others keep increasing. The maximum difference in this period is given by OF-MES with the maximum difference approximately of 6 MCM. A similar characteristic is shown for the yielded energy. From January to March, OF-MME produces the rela- tively constant and higher energy than the others with the maximum 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 4,389 February 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 4,077 March 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 4,391 April 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 4,583
May 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 4,606
June 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 4,551 July 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 4,466 August 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 4,394 September 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 4,264 October 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 4,389 November 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 4,417 December 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 4,697 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 4,389
Min 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 4,077
Max 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 4,697
Total 53,927 53,251 53,223
Note: All units in MCM (million cubic metre).
energy of 8.22 MW (5918 MW-hour) among the others. The maximum difference is 1.85 MW (1335 MW-hour). All objec- tive functions produce relatively similar energy from June to August. However, from August to December, the energy of OF-MTE decreases. OF-MME produces the constant energy, while the energy of OF-MES increases. In this period, the maximum difference is 0.70 MW (519 MW-hour).
OF-MTE, OF-MME and OF-MES give the total energy in a year, respectively, of 53,927, 53,251 and 53,223 MW-hour.
These values seem relatively similar, where for the total energy OF-MTE should be the best option. However, back to the main objective that the monthly energy demand of 6 MW should be satisfied, now OF-MTE is not the best option anymore. As shown inFigure 8, in January, OF-MTE cannot satisfy the energy demand with the energy shortage of 0.34 MW. Also, from August to December, OF-MTE produces the energy lower than 6 MW, with the maximum shortage of 0.38 MW in December. Both OF-MME and OF-MES can- not satisfy the energy demand in January, where the energy shortages are, respectively, 0.13 and 0.10 MW, but these values are lower than OF-MTE. From August to December, both OF-MME and OF-MES can relatively satisfy the energy demand, where OF-MES shows a better performance than OF-MME. Therefore, in this case, OF-MES is the best option in order to satisfy the main objective.
In optimization model, the unknown variablesSt,rtandht
are determined automatically using the Simplex Method.
However, as we previously mentioned, for simulation
model, the unknown variable St is computed based on the value of the previous stept-1. Therefore, the initial value of St should be known first. Actually, this initial value could be obtained iteratively or from the actual reservoir release.
For the sake of simplicity and since the actual reservoir release is unavailable, we set the initial value ofStfor the simulation model to 219.08 MCM (at the water surface elevation of +179.37 m) which is similar to the largest storage value of the three objective functions of the optimization model in January. 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 it would be hard to determine this value iteratively, since there will be too many possibilities. For the sake of simplicity and with regard to the main objective, variablehtis set to the value which always produces the energy equal to 6 MW.
Another reason is that we can know whether the continuity criterion will be satisfied or not for this value. We present the comparison between the result of simulation model and OF-MES inFigure 9.
It is shown that by setting the value of ht, which always produces the energy equal to 6 MW, the operated water sur- face elevation in January for the next year is lower than the elevation in January of the previous year, which means that the continuity criterion is not satisfied. If the computation is continued, one day the water in the reservoir would be empty, since the amount of water, which is released out of the reservoir, is higher than the inflow. This problem can be anticipated either by reducing the value ofhtor determin- ing this value iteratively, but the new problem might arise.
The first problem is that by reducing the value of ht, the main objective of 6 MW yielded energy is not achieved.
The second one, as we previously mentioned, is that there are too many possibilities if the value of ht is determined iteratively. Actually, the second problem can be solved using 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 for optimization 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 reservoir between the simulation model and OF-MES.
model presented in this study might be quite old, since the more advanced optimization models such as non-linear and dynamic programming have intensively been used by other researchers. Some new techniques, which are not related to optimization programming, are even used. We note the work of Mower and Miranda (2013), in which a new tech- nique was proposed to obtain the rule curve. Instead of using a complex model, they generated the rule curve by using the historical water level data. They only required the long-term data of water level for their model, which integrate many variables required by those complex models (Mower and Miranda 2013). They processed the data with the EXPAND procedure in the Statistical Analysis System result- ing the 97.5 quantile model fit to the distribution of 60-days summed changes in water volume. They analysed the reser- voir whose main purpose was to control the flood. This method is very promising and the idea behind is great, since the result can describe the existing rule curve, even though in fact there are always some differences between pre- dicted and existing rule curve. In our opinion, with regard to flood control objective, this method might be better than other optimization models, since the procedure is relatively easier. Also, complex mathematical formulation is not required to be performed due to the use of real data, which are sufficient to describe the real hydrologic phenomenon.
The model of Mower and Miranda (2013) has also several disadvantages. With regard to flood control objective, par- ticularly for large dams which are designed with the Probable Maximum Flood (PMF), the absence of the PMF value in the collected 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 beforehand with the consideration of PMF. Another disadvantage, which was also stated in their paper, is the requirement of long recorded data with keeping in mind that the good data should always be ensured without manipulation. In our opinion, with regard to the objective of maximizing energy, it would be hard to obtain a proper rule curve, due to the presence of flood events. Even though the flood is an event with low probability of exceedance, its presence in a long data measurement might cause a significant error for the computation. For example, let us assume that we have 10 years historical data, where the first and fourth years are the flood seasons with the return periods of 5 years and 100 years, respectively. In the eighth year, the PMF occurs. The use of all data will produce overestimated results for the optimization model with OF-MTE. Particularly in developing countries such as Indonesia, it is also difficult to obtain the long-term historical data.
We also note the works of Hydrologic Engineering Center (1991), Draperet al. (2003) and Jordanet al. (2012), that pre- sented the good explanation for combining the objectives of economic and flood protection in an optimization model.
Both of these objectives were coupled implicitly leading to one objective function of minimizing the damage cost during the 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 would be better to keep the main objective function of maximizing the energy (for the economic purpose) and include the flood risk factor as the constraint. In this approach, the math- ematical formulation of the main objective function could be applied similarly to OF-MTE, OF-MME or OF-MES, where for the constraints of flood risk, the formulation could be con- structed as a function of the damage costs. The minimum damage cost is set to zero and the maximum one is specified.
As shown inFigure 10, now the damage cost is defined as a function of (Lit+rt–dmt+mft).
Our future study will be focused on an optimization model for a river from segment A to B (as shown in Figure 10). A specific location, whose risk factors are simulated, is specified within this segment. Therefore, in the future, the use of two models, 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-Cunge or Kinematic Wave method could be used starting from point A to know the discharge at point B. The more advanced model such as 1D or 2D hydraulic model based on the shal- low water equations could also be used in order to simulate the inundation area for determining the risk factor. The out- put from both of these models is water elevation, which is related 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 rule curve 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. As shown inFigure 7, there is a quite high difference of water surface elevation of the rule curve in a year, which is approxi- mately 7.47 m. Since the time of a flood event cannot exactly be predicted, the risk factors for the specified location (within segment AB) vary depending on the water elevation at the reservoir. If a flood occurs within January and February, the risk factors are lower than in May or June, since there is a space of volume of more than 7 m in January to February as a flood detention. Meanwhile, there is almost no space for flood detention in May or June, since the average water sur- face elevation at the reservoir in these months is +185 m. By investigating this problem, some new scenarios for rule curve could be obtained, as now the rule curve is also affected by the risk factor. We realize that an advanced formulation is required, since the computation of both hydrology and hydraulic models will be performed several times as much as the number of simulation time. However, this is a
Figure 10.Concept of our future study to include risk factor in optimization model.
