INTRODUCTION
Generally, rule curves of a reservoir are basic monthly guides for
long run of reservoir operation. The rule curves have a lower bound that
is set to store water for reducing the risk of water shortage in the
future. The rule curves also have an upper bound set to maintain water
level for controlling flood volume. They are to be created when
initially implementing the reservoirs and generally modified after being
used for a certain period of time since total water requirements (e.g.,
water supply, industrial demand and irrigation requirement) supported by
the systems usually increase with time.
A simulation model is applied to find the suitable rule
curves[1,2]. The model is straightforward and applicable for both simple
and complex systems. Generally, this approach assesses the effectiveness
(i.e., objective function) of the systems based on several sets of trial
rule curves that are adjusted from the preceding ones. However,
depending on the result of the adjustment, it does not guarantee to
yield the optimal rule curves. Often, a frequency of water deficit was
used as the objective function for searching of this approach. However,
an extreme maximum magnitude of deficit water possibly occurs because of
regard the frequency only.
A dynamic programming (DP) is another optimization technique
applied to search the non-linear problems of water resource[3-6].
Unfortunately, the application of DP to multi-reservoir system is
limited due to a curse of dimensionality. Chleeraktrakoon and
Kangrang[7] applied the DP with a principle progressive optimality to
determine the optimal rule curves using a magnitude of water shortage
and excess release as the objective function. However, this method does
not guarantee it as the proper objective function for searching rule
curves.
Recently, genetic algorithms (GAs) embedded the simulation model
(HEC-5) have proposed to search the rule curves of the reservoir
system[8-11]. The best part of GAs is that they can handle any type of
objective function. A shortage index (SI) was used as the objective
function for searching the curves without any constraint. This objective
function considers only the deficit water, so it may not cover the
situation of excess water. In order to derive the optimal rule curves, a
suitable objective function is required. Often, the obtained rule curves
are not feasible for reservoir operation because of the large variations
of the intervals between the upper and lower rule curves. Therefore, a
smoothing function constraint is required to include into the model for
fitting the rule curves.
This paper thus proposes the smoothing-function constraint for
fitting rule curves and presents the suitable objective function for
determining the optimal rule curves using the genetic algorithm (GAs)
with the simulation model. The proposed approach is applied to the
Bhumibol and Sirikit Reservoirs (the Chao Phraya River Basin, Thailand).
Simulation models: Simulation models (i.e., HEC-3, HEC-5) are
generally used to study the efficiency of the reservoir operation. This
study conducted the simulation model based on those concepts, because it
is easily connected with an optimization (GAs) model. The developed
simulation model can be used to determine both reservoir storage
requirements and operational strategies for flood control or
conservation. Generally, the multi-reservoir operating policies are
based on the rule curves of individual reservoirs and the principles of
water balance concept. The reservoir system is operated along the
standard operating policy expressed in Eq.(1) and Fig. 1.
[FIGURE 1 OMITTED]
[R.sub.[upsilon],[tau]] = {[D.sub.[tau]] + [W.sub.[upsilon],[tau]]
- [y.sub.[tau]], for [W.sub.[upsilon],[tau]] [greater than or equal]
[Y.sub.[tau]] + [D.sub.[tau]] [D.sub.[tau]], for [x.sub.[tau]] [less
than or equal to] [W.sub.[upsilon],[tau]] <[y.sub.[tau]] +
[D.sub.[tau]] [D.sub.[tau]] + [W.sub.[upsilon],[tau]] - [x.sub.[tau]],
for [x.sub.[tau]] - [D.sub.[tau]] [less than or equal to]
[W.sub.[upsilon],[tau]] < [x.sub.[tau]] 0, otherwise (1)
in which R[upsilon],[tau] is the release discharges form the
reservoir during year [epsilon] and period [tau] ([tau] = 1 to 12
representing month, January to December). [D.sub.[tau]] is the water
requirement of month [tau], [x.sub.[tau]] is lower rule curve of month
[tau],[y.sub.[tau]] is upper rule curve of month [tau] and
[W.sub.[epsilon],[tau]] is the available water calculated using simple
water balance described in Eq.(2) as
[W.sub.[upsion],[tau]+1] = [S.sub.[upsilon],[tau]] +
[Q.sub.[upsilon],[tau]] - [R.sub.[upsilon],[tau]] - [E.sub.[tau]] - DS
(2)
where [S.sub.[epsilon] [tau]]t is the stored water at the end of
month [tau], [Q.sub.[epsilon] [tau]] is monthly reservoir inflow,
[E.sub.[tau]] is average value of evaporation loss and DS is the minimum
reservoir storage capacity (the capacity of dead storage). In the
mention figure and equation, if available water is in a range of the
upper and lover rule level, then demands are satisfied in full. If
available water over tops the upper rule level, then the water is
spilled from the reservoir in downstream river in order to maintain
water level at upper rule level and if available water is below the
lower rule level, reduce supply is made. The policy usually reserves the
available water [W.sub.[epsilon],[tau]] for reducing the risk of water
shortage in future, when 0 [less then or equal to]
[W.sub.[epsilon],[tau]] < [x.sub.[tau]]-[d.sub.[tau]].
At the end of simulation program, the situation of water shortage
and excess release water (e.g., the number of failure year, the number
of excess release water, the average annual shortage) will be recorded.
Integration of the GAs and simulation model: The algorithms of
connection the developed simulation model into the GAs are described as
follows. GAs requires encoding schemes that transform the decision
variables into chromosome. Then, the genetic operations (reproduction,
crossover and mutation) are performed. These genetic operations will
generate new sets of chromosomes. The most common encoding schemes use
binary strings as indicated in Fig. 2. Each bit of the binary string is
called a gene. The chromosome in Fig. 2 contains five decision
variables, each represented by six bits. In this study, each decision
variable represents a monthly level of the rule curves of reservoirs.
[FIGURE 2 OMITTED]
After the chromosomes (rule curves) of the initial population have
been determined, the release of the system in every period is calculated
by the developed simulation model corresponding to each chromosome. The
release of the system for each chromosome is retuned to the GAs to
evaluate its fitness. The situation of water shortage of the system is
defined as fitness function in this study. Next, the reproduction
including selection, crossover and mutation is performed for creating a
new rule curve parameters in next generation. This procedure is repeated
until the criterion is satisfied as described in Fig. 3. Each parameter
of the fitness functions is applied into the model to find the suitable
objective function. The objective function of each search is to minimize
the parameter of the fitness functions. There are 48 parameters (rule
curve levels) of two reservoirs which are represented by the
chromosomes. This study used population size = 80, crossover probability
= 0.9, mutation probability = 0.01.
[FIGURE 3 OMITTED]
There are six objective functions which chosen for searching the
optimal rule curve. First, the shortage index (SI) which proposed by the
US Army crops of Engineers[12] and can be summarized as
SI = 100/N [N.summation over (i=1)]([[[Sh.sub.i]/[D.sub.i]].sup.2])
(3)
in which N is the total number of periods, [Sh.sub.i] is water
deficit during the period i, [D.sub.i] is target demand during the
period i. A month is taken as the period of reservoir operation.
The others are the average water shortage (Aver-MCM/year), the
maximum magnitude of water shortage (Max--MCM/year), Frequency of water
shortage (Fre, times/year), Total square deficit (RMS--MCM2) and sum of
above mention (SUM) which described as follows:
Aver = 1/n [n.summation over ([upsilon]=1)] [Sh.sub.[upsilon]] (4)
Max = Maximun([Sh.sub.[upsilon]]), for [upsilon] = 1,...,n (5)
Fre = [P.sub.i]/n (6)
RMS = [n.summation over ([upsilon]=1)] [([[D.sub.i] -
[R.sub.2]]).sup.2] ,[for all][R.sub.i] < [D.sub.i] (7)
SUM = 1/4[Aver/full(Aver) + Max/full(Max) + Fre + RMs/full(RMS)]
(8)
where n is the total number of considered year. [Sh.sub.[upsilon]]
is water deficit during year [tau] [p.sub.i] be total number of annual
failure (year that release does not met 100% of target demand),
[R.sub.i] is supply water during the period i.
To reduce the fluctuate of rule curve in order to obtain the
optimal rule curves which are suitable in the practice, the moving
average is chosen as a base of the smoothing function constraint for
fitting the rule curves, for each curve can be present as
[absolute value of [[[x.sub.[tau]-2] + [x.sub.[tau]-1] +
[x.sub.[tau]]]/3 - [x.sub.[tau]]] [less than or equal to] 0.1T for [tau]
= 3,...12 (9)
[absolute value of [[[x.sub.12] + [x.sub.[tau]-1] +
[x.sub.[tau]]]/3 - [x.sub.[tau]]] [less than or equal to] 0.1T for [tau]
= 2 (10)
[absolute value of [[[x.sub.12] + [x.sub.12-[tau]] +
[x.sub.[tau]]]/3 - [x.sub.[tau]]] [less than or equal to] 0.1T for [tau]
= 1 (11)
where x is rule curve level and T is active storage of each
reservoir. These smoothing functions are integrated into the fitness
function in the procedure of searching.
ILLUSTRATIVE APPLICATION
The proposed approach was applied to search the optimal rule curve
of a system of major multi-purpose storages (the Bhumibol and Sirikit
Reservoirs) locating in the watershed area of the Chao Phraya River
(Thailand). Figure 4 and 5 present the location of the Chao Phraya River
and the schematic diagram of water resource systems within the drainage
basin. The solid lines represent the systems where they are considered
in the application. They include the discharges of the two reservoirs
and the side flows of River Wang and River Yom. The dashed lines stand
for the systems in which they are ignored. For example, these are the
discharges of River Sakae Krang and River Tha Chin, the releases of the
Pasak Reservoir and the return flows of irrigation projects.
[FIGURE 4 OMITTED]
Two sequences of 21-year (1975-1995) monthlyflow records of
stations P.12 and SK covering several dry and flooding years were
commonly used for searching the optimal upper and lower rule curves. The
other average hydrological data for each month included series of
evaporation losses and precipitation of the reservoirs and those of side
flows of stations W.4A (River Wang) and Y.5 (River Yom). The report of
the Electrical Generating Authority of Thailand[2] was used to provide
for the considered waterrequirement information of the applied basin.
Results of the illustrative application are presented as follows.
[FIGURE 5 OMITTED]
Suitable objective function: Figure 6 and 7 respectively present
the optimal rule curves of the Bhumibol and Sirikit Reservoirs using all
objective functions for searching. The figures appear that the patterns
of the rule curves between the two reservoirs generally agree with each
other due to the seasonality effects on reservoir inflows and considered
water demands. However, there are large variations of the intervals
between the upper and lower rule curves.
[FIGURE 6 OMITTED]
The rule curves of each objective function were then assessed to
examine the situations of water shortage and excess release by
considering related characteristics (e.g., frequency, magnitude and
duration). A Monte Carlo simulation study against 500 samples of
generated monthly flows of stations P.12 and SK[13] was used to compute
the interval (mean [+ or -] standard deviation) of the referred
statistics for the assessment. In the following, the obtained assessment
results of the considered water-deficit and excessrelease properties for
each objective function are presented.
[FIGURE 7 OMITTED]
Table 1 and 2 respectively show the assessment intervals of water
shortage and excess release characteristics for all objective functions.
They indicate that the rule curves of using average water shortage
(Aver) as the objective function gives the magnitude bounds of water
deficit that are generally less than using the others, while the other
bounds are not different. In addition, the maximum magnitude bounds of
excess release of using the mention objective function are less than
using the others. Therefore, the average water shortage (Aver) is the
most suitable for using as an objective function of searching rule
curve.
Smoothing function: The average water shortage (Aver) was then used
to search the rule curve with the smoothing function constraint. The
developed model was applied to determine the optimal rule curves of the
Bhumibol and Sirikit Reservoirs (the Chao Phraya River Basin, Thailand).
Figure 8 and 9 present the optimal rule curves of GAs connected
simulation with the smoothing function constraint and without it as well
as the existing curves of the HEC-3[14] simulation approach[2] for the
Bhumibol and Sirikit Reservoirs respectively.
[FIGURE 8 OMITTED]
They demonstrate that the optimal rule curves of the search using
smoothing function constraint are smoothly than their search without the
constraint. Moreover, the pattern of the obtained curves which using
smoothing constraint is similar to the existing ones. Thus smoothing
function constraint can reduce the variation of the upper and lower rule
curves.
[FIGURE 9 OMITTED]
The rule curves of them were then assessed to examine the
situations of water shortage and excess release by comparing related
characteristics (e.g., frequency, magnitude and duration) of the
referred circumstances with those of the optimal curves. The Monte Carlo
simulation study against 500 samples of generated monthly flows of
stations P.12 and SK[13] was used to compute the interval (mean [+ or -]
standard deviation) of the referred statistics for the assessment. In
the following, the obtained assessment results of the considered
water-deficit and excess-release properties for the three cases are
presented.
[FIGURE 10 OMITTED]
Figure 10 and 11 show the assessment intervals of water-shortage
characteristics for the GAs connected simulation with the smoothing
function constraint and without it as well as the HEC-3 simulation
approach. They appear that the proposed technique gives the water
deficit characteristics (e.g., the durations and magnitudes of water
deficits) which are smaller than the existing one does. It can be these
concluded that the GAs connected simulation with the smoothing function
constraint is able to reduce the effect of the correlations on the water
shortage situation. Moreover, the figures also demonstrate the water
deficit characteristics of both using smoothing constraint and without
smoothing constraint are not much different. Figure 12 and 13 present
the referred statistics of excess releases for the three rules. It is
evident that the both of using GAs connected simulation techniques do
not yield greater excess releases, as compared with the existing one.
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
CONCLUSION
Rule curves are fundamental guidelines for operating a reservoir
system. The objective of this paper is to find the suitable objective
function and to propose a smoothing function constraint for searching
the optimal rule curves by using GAs connected simulation technique. The
curves of all objective functions are compared and assessed on the
properties (frequency, magnitudes and consecutive duration) of water
deficit and excess release using the Monte Carlo simulation. The results
show that the average water shortage is the optimal objective function
for searching the optimal rule curves. It can represent the situations
of water deficit and excess release.
[FIGURE 13 OMITTED]
To reduce the fluctuation of rule curves, the moving average is
applied to be the constraint of the searching rule curve. Results
indicate that the smoothing function constraint can reduce the variation
of the upper and lower rule curves. The optimal rule curves of the
developed model which using the average water shortage and smoothing
function are used for evaluating the existing rule curves. Results
demonstrate that the optimal rule curves of the GAs technique are more
mitigate the situation of water deficit and excess release than the
existing rule curves. They are also concluded that the GAs connected
simulation with smoothing constraint is more effective than the model
without constraint.
ACKNOWLEDGEMENTS
The authors would like to acknowledge the support by the Faculty of
Engineering, Mahasarakham University. Thanks are also due to Mr.
Pitthaya Sangsom for helpful development the GAs technique.
REFERENCES
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11. Chang, J.F., L. Chen and C.L. Chang, 2005. Optimizing reservoir
operating rule curves by genetic algorithms. Hydrological Processes, 19:
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12. Hydrologic Engineering Center (HEC), 1975. Hydrologic
Engineering Methods for Water Resources Development, Vol. 8, Reservoir
Yield. US Army Corp of Engineers:Davis, California, U.S.A.
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AnongritKangrang and ChavalitChaleeraktrakoon
(1)Faculty of Engineering, Mahasarakham University, Khamriang
Campus Kantharawichai, Mahasarakham, 44150, Thailand (2)Department of
Civil Engineering, Faculty of Engineering, Thammasat University Klong
Luang, Pathumthani, 12120, Thailand
Corresponding Author: Anongrit Kangrang, Faculty of Engineering,
Mahasarakham University, Khamriang Campus,Kantharawichai, Mahasarakham,
44150, Thailand, E-mail: anongrit@hotmail.com
Table 1: Frequency, magnitude and successive period of water shortage
for all objective functions
Frequency Magnitude (MCM/year) Duration (year)
Objective
functions (times/ Average Maximum Average Maximum
year)
SI [mu] 0.142 25 391 2.0 3.1
[sigma] 0.080 21 243 0.9 1.7
Aver [mu] 0.208 29 207 2.1 3.8
[sigma] 0.082 14 102 0.7 1.8
Max [mu] 0.807 139 437 8.5 18.1
[sigma] 0.092 30 98 5.4 7.6
Fre [mu] 0.123 45 707 1.9 2.8
[sigma] 0.070 35 457 0.8 1.5
RMS [mu] 0.196 31 333 2.1 3.7
[sigma] 0.087 19 177 0.8 1.7
SUM [mu] 0.156 28 353 2.0 3.3
[sigma] 0.078 19 218 0.8 1.7
Note: [micro] = mean, s = standard deviation
Table 2: Frequency, magnitude and successive period of excess release for all objective functions
Objective Frequency Magnitude (MCM/year) Duration(year)
functions (times Average Maximum Average Maximum
/year)
SI [mu] 0.848 1,038 4,757 9.0 18.2
[sigma] 0.072 205 2,002 5.2 7.1
Aver [mu] 0.843 1,188 4,446 8.8 17.2
[sigma] 0.073 194 1,825 5.6 7.0
Max [mu] 0.856 1,160 5,403 8.3 17.1
[sigma] 0.065 208 2,031 4.2 6.5
[mu] 0.814 1,036 4,882 7.1 15.4
[sigma] 0.078 205 1,932 3.2 5.9
RMS [mu] 0.836 1,289 5,578 7.9 16.3
[sigma] 0.071 231 2,014 4.2 6.3
SUM [mu] 0.848 1,082 4,602 8.7 17.7
[sigma] 0.071 196 1,922 4.4 6.7