From the SelectedWorks of Ahmad ShahiriParsa
2015
Introduction to Linear Programming as a Popular
Tool in Optimal Reservoir Operation, a Review
Ahmad ShahiriParsa
Mohammad Noori
Mohammad Heydari, Dr
Faridah Othman
Kourosh Qaderi
Available at: https://works.bepress.com/ahmad-shahiriparsa/5/
Advances in Environmental Biology, 9(3) February 2015, Pages: 906-917
AENSI Journals
Advances in Environmental Biology
ISSN-1995-0756
EISSN-1998-1066
Journal home page: http://www.aensiweb.com/AEB/
Introduction to Linear Programming as a Popular Tool in Optimal Reservoir
Operation, a Review
1Mohammad
Heydari,
2Faridah
Othman,
3Kourosh
Qaderi,
4Mohammad
Noori and
5Ahmad
ShahiriParsa
1
PhD candidate, Faculty of engineering, University of Malaya, Kuala Lumpur
Associated professor, Department of civil Engineering, Faculty of Engineering, Kuala Lumpur, Malaysia
Assistant Professor, Department of Water Engineering, Shahid Bahonar University of Kerman, Iran
4
PhD candidate, Faculty of Engineering, Ferdowsi University, Mashhad, Iran
5
Graduated student, Faculty of Engineering, University of UNITEN, Kuala Lumpur, Malaysia
2
3
ARTICLE INFO
Article history:
Received 21 November 2014
Received in revised form 4 December
2014
Accepted 3 January 2015
Available online 28 January 2015
Keywords:
Linear programming, LP, optimal
operation,
multiple
reservoirs,
optimization, mathematical modeling.
ABSTRACT
Water is a rare and vital natural resource for all biological phenomena and human
activities that continuously is needed to be known at any time and place. Taking into
the burgeoning growth of population and consequently increasing human needs, the
limitation of water resources is a considerable challenge for human. Moreover,
asymmetrical distribution of rain time and location in most countries has caused that the
water resources management and programming be considered. In order to resolve this
problem, researchers are trying to use some techniques in relation with programming
and management for a long time. Most of practical and applied problems can be
modeled as a linear programming problem regarding all intrinsic complexities. The
mentioned reason and also presence of different solving software of linear
programming problems have caused that linear programming be used as one of the most
practical methods in the field of dam operation for years. In this research, we introduce
optimal operation problems of reservoirs by using linear programming techniques and
discuss about them. Also, objective and multi objective models were introduced by
using some questions. Finally, some popular methods in the field of modeling such
problems are introduced.
© 2015 AENSI Publisher All rights reserved.
To Cite This Article: Mohammad Heydari, Faridah Othman, Kourosh Qaderi, Mohammad Noori, Ahmad ShahiriParsa. Introduction to
linear programming as a popular tool in optimal reservoir operation, a Review. Adv. Environ. Biol., 9(3), 906-917, 2015
INTRODUCTION
Water is the most important requires for all living creatures after oxygen. Life and health of all beings
containing human, plants and animals, depends on water. Therefore, nowadays, water is known as human
treasure. Although 75 percent of planet earth is composed of water, but only one percent of the fresh water is
usable. In spite of the fact that the amount of usable water (drinking water) on earth is limited, but this
insignificant amount is not spread on the earth uniformly. This limitation is one of the most important and
essential challenges in countries with arid and semi-arid regions.
On one hand, limited access to water resources and on the other hand, human need for water, necessitate the
proper management strategies. Taking into aims like providing water, controlling floods, hydro power
production, tourism, etc. dams are designed and constructed in order to resolve such problems. Providing water
for municipal, agricultural and industrial consumption is one of the main purposes for reservoir operation and
planning. In most countries, agricultural purposes has the highest water level consumption. So, optimal
operation and management of water resources, among giving proper response to the needs of this part, leads to
reduced waste water and increasing the level of yield of production and gaining sustaining development in
agriculture [1]. Flood control and decreasing its resulted damage are another aims of constructing dams. Besides
dams as main sources in providing water, these also have high capacity in the field of growing and developing
tourism industry. In most countries in the world, dams and their reservoirs are counted as the most important
tourist absorption and absorb numerous tourists annually. One of the aims of constructing dams is hydroelectric
Corresponding Author: Faridah Othman, Associated professor, Department of civil Engineering, Faculty of Engineering,
Kuala Lumpur, Malaysia. Ph: 0379674584; E-mail: Faridahothman@um.edu.my
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Mohammad Heydari et al, 2015
Advances in Environmental Biology, 9(3) February 2015, Pages: 906-917
energy. Nowadays, the hydropower and thermal energies have the highest share in producing the world`s
electricity. Although, problems and limitations of producing electricity in thermal power sources and due to
technical issues, the imperatives of environmental criteria, resources constrains have caused that, by the time the
general trend in the world of power generation, hydroelectric plants will be more attentive. The potential energy
of water behind a dam, provide hydroelectric energy. In this case the energy of the water depends on stored
water of dam and height difference between the water source and the withdrawal of water from the dam. Power
generation in hydroelectric plants has a lot of advantages. Perhaps these benefits have caused that this
production method has comparative advantages and are considered in the world, especially in countries where
water resources are relatively substantial. By constructing the dam in areas of water in the river and the regions
where rainfall is high and installing turbine, the gravitational potential energy of water behind the dam can be
used to generate hydroelectric power.
The next issue is the optimal operation of the reservoir, considering the objectives like drinking water
needs, industrial, agricultural, hydroelectric purposes, flood control, tourism and etc. For this purpose, efficient
approaches and appropriate solutions must be considered for operating reservoirs as one of the most important
components of water resources. Application of such approaches leads to create balance between available
limited resources and high consumption, optimization of water use in agriculture, municipal and industry and
finally sustainable development in water resources management. Nowadays, the water management and water
protection are high importance in developing and developed countries. In order to system enhancement and
equitable management of water resources, using the principles and technical planning is necessary. Using
planning technique practical and practicable in order to optimize water resources, due to its simplicity and
applicability has special status.
Charles Revelle in 1969 decided to act for design and reservoir management by linear programming and
using a Linear Decision Rule (LDR). In this linear decision making method, reservoir outflow in whole
operation period calculated as the difference between the storage of the reservoir at the beginning of the period
and decision parameter by solving linear programming [2]. In 1970, Loucks applied the linear model with its
probable limitation and its deterministic equivalent for solving the system of reservoirs.
Cai and his collaborators in 2001, used genetic algorithm with linear programming in complex problems of
water reservoir. The gained results have been reported very satisfactory [3]. in 2005, Reis et al. used
combination of Genetic Algorithm (GA) and Linear Programming (LP) method designed and solved planning
and decision-making for reservoirs of water systems during the probabilistic [4]. in 2006, Reise and his
associates in performed a combination method using genetic algorithm (GA) and linear programming (LP) in
order to achieve operational decisions for a system reservoir that is applied during optimization term. This
method identifies a part of decision variables named Cost Reducing Factors (CRFs) by Genetic Algorithm (GA)
and operational variables by Linear Programming (LP) [5].
Optimization Process:
The optimization process of this study is presented in Figure 1, and it consists of seven vital steps.
1) A detailed view
2) Problem definition
3) Developing mathematical model
4) Finding solution for the model
5) Sensitive analysis phase
6) Validation
7) Performing the solution
Fig. 1: Schematic representation of optimization process.
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1. The first step in this process is expanding a clear understanding of the problem with a detailed view of the
real world. To this end, a number of primitive solutions to achieve objectives must be defined that consider
different aspects of the problem. Also, some doubts should be in the minds of decision makers as to which
opinion to achieve the goal is best.
2. Problem definition phase is a phase in which a precise and clear statement of the problem taken from
observations must be made and be gained in identification phase and transparency step of the problem. The
mentioned problem definition should define purpose, the influence of the initial solutions, assumptions, barriers,
limitations and possible available information on sources and markers involved in the problem. Experience
shows that erroneous definition of the problem leads to analysis failure.
3. When we define the problem, the next step is developing a mathematical model. The mathematical model is
mathematical performance of the system or real problem and is able to perform different aspects of the problem
in interpretable form. At first it may be that qualitative model structure itself, including unofficial descriptive
approach. In this unofficial qualitative model, an official model may develop. (Part 3 contains some applied
models in order to optimize reservoir operations of dams.)
4. After formulating and developing the model, it is turn to find a solution for the model. Usually the optimal
solution for the model with evaluation outcome sequence is found. This sequence of operation starts from a
primitive solution that is as input of the model and the generation of the developed solution as output that is
known as repeat is resulted. The developed output is resubmitted as a new input and the process is repeated
under the certain circumstances.
5. Another important phase of this study is sensitive analysis phase. Performing the sensitivity analysis allows
us to determine the necessary accuracy on input data and understanding the decision variables that have the
highest influence on the solution. The sensitivity analysis allows the analyst to see how sensitive the preferred
option of changing assumptions and data is. By performing the sensitivity analysis, we will be able to recognize
how strong is a preferred opinion and input data what need to change to become an optimal choice. In sensitivity
analysis, analyst modifies the suppositions or data to enhance the considered option and convert it into optimal
choice. Amount of default modification is measuring is determining the power of the model.
6. A solution should be tested. Often the solution is tested in a short or long term. The proposed solution must
be validated against the actual performance observation while the test is being made, and also it should be
independent of how an optimal solution is obtained.
7. The final phase is performing the solution. This is the step of using optimal outputs for decision making
process. Usually, analyst converts his mathematical findings as a series of understandable and applicable
decisions. It may be necessary to train decision-makers to help them apply the findings to attain the required
changes from the current situation to the desired situation. Also, they need to be supported until they learn the
mechanism of maintenance and upgrading the solution.
Fig. 2: Allocating the capacity of reservoir to different volumes (adopted from [6]).
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Reservoirs Operation Modelling:
The main objective of optimization is finding the best acceptable solution. There may be different solutions
for a problem that in order to compare them and choose the optimal solution, the objective function is defined.
The choice of this function depends on the nature of the problem. An Optimization problem with one objective
is called (single objective problems), and When several objectives and criteria are considered for minimizing or
maximizing the optimization problem, then it is called (multi-objective optimization problems). Some
conditions of optimization model structures for the operation of the reservoir are given in below.
Single objective models:
Objective function by minimizing the total flood damage in its simplest form is defined as a function of
flow rate in vulnerable areas as follows:
Min Z =
(t Є T)
(1)
Subject to:
f(It,Rt) = 0
(2)
Rt ≥ Rmin
(3)
Rt ≤ Rmax
(4)
St+1 = St + It - Rt
(5)
Smin ≤ St+1 ≤ Smax
(6)
│Rt+1 - Rt│≤ ЄR
(7)
Where:
Z: value of objective function
Qt: flow rate at damage point
t: time index
T: time horizon of operation
It: flow rate
Rt: reservoir outflow that their relationship is expressed by the function as a power constraint.
St: reservoir storage
Rmin and Rmax: minimum and maximum release values.
ЄR: limiting the maximum difference between release values at two successive time steps
It: inflow to the reservoir
In other models as seen in the relations 8 to 11, the released volume in studied periods is considered as a
function of the river basin or reservoir storage. Among these, the function that estimated best value of the
objective function is selected as optimal function. This function performs some constant factors for any period
during the statistical term. According to it, the release amount can take a percentage of the river basin, save
volume or a combination of both. The following is with agricultural purpose and for a reservoir:
Min Def =
2
(8)
Subject to:
St+1 =St - Qt - Rt – Lt (St, St+1)- Spt
(9)
Smin ≤ St ≤ Smax
(10)
Rmin ≤ Rt ≤ Rmax
(11)
Where:
Def: the monthly shortage
Rt: released volume of reservoir
Dt: amount of the required monthly
Dmax: maximum required during a statistical period
St: storage volume at the beginning of month
Qt: inflow to river during the period t
Lt (St, St+1) is calculations of evaporation losses in period t that creates a set of implicit nonlinear equations.
Spt: is overflow from the reservoir during the period.
Smax and Smin are in order the maximum and minimum volume of reservoir storage, in order to providing the
dead storage of the reservoir and renewable and purposes of flood control volume.
Rmin and Rmax are in order the minimum and maximum outflow from the reservoir.
If a single-objective optimization problem has multiple optimal solutions, it does not matter which one is
selected, as they give the same objective function value.
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Mohammad Heydari et al, 2015
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Yield Model:
Discharge model is a linear optimization model. Discharge refers to the flow of future periods with
relatively high credit (or with probability equal to or exceeds above) can be supplied. In this model, the
relationship between the two series is creating volumetric balance in storage volume during the year.
Inter year relations:
Sy+ Qy – YFIRM – α P,Y Yp– Ey – Ry = S y+1
αP,Y=
Sy≤ka0
Ey= E0 + [ Sr +∑ ( St + St+1 / 2 ) ᵞ1].E
In these formulas the various parameters are as follows:
Sy: The storage at the beginning of the period
Qy: Annual input
YFIRM: Annual primary requires
Yp: Annual secondary require
Ey: Evaporation
Ry: Additional output
ka0: Storage volume of out year
St: Storage volume at the start of the period
E: Amount of annual evaporation
E0: The fixed amount of annual evaporation
ᵞ1: The relative rate of evaporation of the month
(12)
(13)
(14)
(15)
Extra year relations:
St + Qt– YFIRM,t –Ypt – et –Rt = St+1
(16)
KD + ka0+ St+Kft≤ K
(17)
et= ᵞt .E0+ ( St + St+1 ) ᵞt .Et
(18)
In this relation we have:
Qt: Average of monthly period
et : Monthly evaporation
KD: Dead storage
Kft: Flooding control volume
Because the critical condition determines the reservoir volume, sometimes the relation 15 can be written as
follows.
St + βt
–et – rt =St+1
(19)
In this case the model is called the approximate discharge model:
βt: The relative coefficient of flow in the driest month of the year
To fit the size of the reservoir, this model depends on value and river discharge. In some projects, due to the
large fluctuations in discharge and a mismatch between the needs, the model requires reservoir storage volume
greater than the existing one. Advantage of discharge model is in performing the results and the simplicity of
application in simulation. Moreover, the simulation results clearly show the deficit and reduce the severity of the
shortage.
Multi-objective:
Multi-objective optimization problems (MOPs) are common. They can be either defined explicitly as
separate optimization criteria or formulated as constraints. Formally, this can be defined as follows.
Definition of Multi-objective Optimization Problem: A general MOP includes a set of n parameters
(decision variables), a set of k objective functions, and a set of m constraints. Objective functions and
constraints are functions of the decision variables. The optimization goal is to:
Maximize y= f (x) = (f1 (x), f2 (x)… fk (x))
(20)
Subject to:
e(x) = (e1(x),e2(x),…, em(x)) ≤ 0
(21)
Where
x = (x1, x2… xn) X
(22)
y =(y1, y2… yn) Y
(23)
And x is the decision vector, y is the objective vector, X is denoted as the decision space, and Y is called
the objective space.
The constraints e (x) ≤ 0 determine the set of feasible solutions.
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Feasible Set: The feasible set X f is defined as the set of decision vectors x that satisfy the constraints e.x:
Xf = {x X | e (x) ≤ 0}
(24)
The image of Xf , i.e., the feasible region in the objective space, is denoted as
Yf = f (Xf) = Ux Xf {f(x)}.
(25)
Without loss of generality, a maximization problem is assumed here. For minimization or mixed
maximization/minimization problems the definitions presented in this section are similar.
In single-objective optimization, the feasible set is completely (totally) ordered according to the objective
function f(x).
The structure of a basic model that is the base of many optimization modes of reservoirs’ operation is as
follows:
Minimize Z=
(26)
Subject to:
St+1=St+it-Rt-Et-Lt
(t=1,2,3,…n)
(27)
Smin≤St≤cap
(t=1,2,3,…n)
(28)
0≤Rt≤Rmax,t
(t=1,2,3,…n)
(29)
St.Et.Lt.Rt≥0
(t=1,2,3,…n)
(30)
In which, the variables are defined as follows:
Z: objective function
Loss: operation cost in month t that is function of output and required storage and volume of the reservoir in
month t.
Rt: released volume of reservoir
Dt: amount of monthly need
St: volume of storage reservoir in month t
n: length of planning period
Smax and Smin are in order the maximum and minimum storage of water in reservoir.
Rmax is the maximum outflow from the reservoir during period t.
Cap: total volume of water stored in the reservoir.
Et: amount of evaporation of the reservoir in month t.
Lt: volume of water leaks in the reservoir in month t.
It: volume of inflow...

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