PSZ 19: 16 (Pind. 1/97)
UNIVERSITI TEKNOLOGI MALAYSIA
DECLARATION OF THESIS/POSTGRADUATE THESIS PAPER AND
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Author’s Full Name
:
JOOMIZAN BINTI NOORDIN
Date of Birth
:
28TH AUGUST 1985
Title
:
RESERVOIR STORAGE SIMULATION AND FORECASTING
MODELS FOR MUDA IRRIGATION SCHEME, MALAYSIA
Academic Session
:
2009/2010/2
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Sir,
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RESERVOIR STORAGE SIMULATION AND FORECASTING MODELS FOR MUDA
IRRIGATION SCHEME, MALAYSIA
(JOOMIZAN BINTI NOORDIN)
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IRRIGATION SCHEME, MALAYSIA" is classified as RESTRICTED for a period
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(i)
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Thank you.
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Faculty of Civil Engineering
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RESERVOIR STORAGE SIMULATION AND FORECASTING MODELS
FOR MUDA IRRIGATION SCHEME, MALAYSIA
JOOMIZAN BINTI NOORDIN
A project report submitted in partial
fulfillment of the requirements for the award
of the Master of Engineering (Civil-Hydraulics and Hydrology)
Faculty of Civil Engineering
Universiti Teknologi Malaysia
APRIL 2010
ii
I declare that this thesis entitled “RESERVOIR STORAGE SIMULATION AND
FORECASTING MODELS FOR MUDA IRRIGATION SCHEME, MALAYSIA” is the
result of my own research except as cited in the references. The thesis has not been
accepted for any degree and is not concurrently submitted in candidature of any other
degree.
Signature :
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Name :
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Date :
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iii
Dedicated to the sweet memory of author’s family and Supervisor’s late
Grandmother
iv
ACKNOWLEDGEMENTS
All commendation and entreaty to Almighty Allah for his unbound graciousness
and unlimited kindness in all endeavors that made the author possible to complete the
Master of Engineering Degree.
The author would like to express her intense gratitude and indebtedness to her
respected supervisor, Associate Professor Dr. Md. Hazrat Ali, Faculty of Civil
Engineering, Universiti Teknologi Malaysia (UTM), for his persistent guidance,
invaluable suggestions, spontaneous support, and constant encouragement and
motivation in successful completion of this thesis. Special thanks to her co-supervisor,
Assoc. Prof. Dr. Sobri Harun, for guiding the author to select Associate Professor Dr.
Md. Hazrat Ali as the supervisor.
Sincere thanks are accredited to Assoc. Prof. Ir. Dr. Mohd Hanim Osman, Head
of the School of Postgraduate Studies and his staff, for making necessary arrangements
in finalising this study. The author is also grateful to her thesis panel members, for
providing suggestions towards completion of this study.
Appreciation is due to the staff of MADA, for their technical supports in
providing necessary observed data. The author expresses her profound appreciation to
the scholarship donor, Yayasan Sultan Iskandar Johor, for providing financial assistance
to support her study at Universiti Teknologi Malaysia. Special thanks are due to the
UTM library staff members, for their services during this study. Above all, the author
expresses indebtedness to her family members for their continuous inspiration and
selfless sacrifice throughout her life.
v
ABSTRACT
Reservoir operation policies aim at deriving maximum benefits from water that can
be stored in it and allocated to crops. Water shortage is the main constraint in
establishing stable irrigation water management in Muda Irrigation Scheme, Kedah,
Malaysia. Thus, the objective of this study is to a develop reservoir simulation model
and to consider stochastic models and Log-Pearson Type III distribution to generate
storage to compare with the observed storage, and to forecast future storage to
examine the performance of the reservoir with reliability under changing conditions.
The reservoir simulation model storage amounts were calculated for 1998-2008
using measured values of rainfall and evaporation (reservoir station no. 61), reservoir
inflow, release, seepage, spill, and Muda reservoir inflow. The developed reservoir
simulation model results simulated well with the mean monthly observed long-term
storage amounts (1998-2008), except for a few months where the model storages are
found relatively higher than the observed storage amounts. A stage-storage curve is
plotted using the monthly observed values of storage and water level from 19982008 to covert water level into storage and vice versa. The first order Markov model
with periodicity and Log-Pearson Type III distribution are considered to generate
storage amounts to compare with the mean monthly observed storages, and hence to
forecast future storage with reliability. The first order Markov model generated and
observed mean storage amounts were compared for each month. The comparison
results imply that the monthly statistical parameters of the historic record, except the
lag1 serial correlation between December and January months (i.e., over-year
monthly correlations), are preserved satisfactorily. The storage amounts are
forecasted for year 2009-2015 to be used in future reservoir operation, using first
order Markov model. The expected mean and minimum storage amounts for
different return periods are estimated, using Log-Pearson Type III distribution and
trendlines with equations and R2 values are shown, to help decision makers to
estimate future storage with corresponding return period under any changing weather
conditions and or demand.
vi
ABSTRAK
Polisi pengoperasian empangan yang mensasarkan keuntungan maksimum
dari air yang disimpan atau bagi tujuan penanaman. Masalah kekurangan air menjadi
masalah utama dalam memastikan pembangunan pertanian yang seimbang di Muda
Irrigation Scheme, Malaysia. Oleh yang demikian, tujuan kajian ini dijalankan adalah
untuk membina model simulasi yang menitikberatkan stokastik model dan juga
kaedah pengagihan Log-Pearson type III untuk mendapatkan jumplah simpanan
empangan dan membandingkan simpanan sediada dengan hasil daripada kiraan
model tersebut. Data dari tahun 1998-2008 digunakan untuk mengukur jumlah
curahan dan cairuapan (bagi empangan stesen 61), kadar alir masuk, pelepasan,
penyerapan, limpahan, dan Kadar alir dari empangan Muda. Model simulasi bagi
empangan yang diperolehi agak baik dengan purata bulanan jangka masa panjang
(1998-2008), kecuali untuk beberapa bulan di mana model simpanan didapati agak
tinggi daripada jumlah sebenar. Lengkungan simpanan-langkah telah diplotkan untuk
menukarkan paras air kepada simpanan. First Order Markov Model dan Log Pearson
Type III dipertimbangkan untuk mendapatkan jumlah simpanan dan dibandingkan
dengan purata bulanan simpanan sebenar, dan juga untuk meramalkan simpanan
dengan kemunginan tertentu. Setiap model yang diperolehi dibandingkan bagi setiap
bulan. Hasil perbandingan tersebut membayangkan statistical parameter bagi datadata yang direkodkan, kecuali bagi lag1 hubungan diantara bulan Januari dan
Disember adalah dijangkakan selamat. Jumlah simpanan yang diramalkan dari tahun
2009-2015 akan digunakan untuk operasi empangan pada masa akan datang. Purata
dan jumlah simpanan minimum yang dijangkakan bagi tempah kalaan yang berbeza
dijangkakan, menggunakan Log Pearson Type III dan didapati trendlines, R2
ditunjukkan untuk membantu membuat keputusan bagi pengurusan empangan pada
masa akan datang dengan sebarang perubahan cuaca.
vii
CONTENTS
CHAPTER
I
II
TITLE
PAGE
DECLARATION
i
DEDICATION
ii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
LIST OF TABLES
ix
LIST OF FIGURES
x
LIST OF ABBREVIATIONS
xi
INTRODUCTION
1.1
Background
1
1.2
Statement of the Problem
3
1.3
Justification of the Study
4
1.4
Objectives of the Study
5
1.5
Scope of the Study
5
LITERATURE REVIEW
2.1
General Remarks
6
2.2
Water Management in Reservoir
9
2.3
The integration of Reservoir Operation Policies
10
and Allocation of Irrigation Area
2.4
The Models
11
2.5
Forecasting
16
2.5.1
Markov First Order Method
17
2.5.2
Log Pearson Type III Distribution
18
2.6
III
Concluding Remarks
20
STUDY AREA AND DATA COLLECTION
3.1
Study Area
21
viii
3.2
Major River Systems and Climate Change in
23
Muda Irrigation Scheme
3.3
IV
Data Collection
25
MODEL DEVELOPMENT AND CONSIDERATIONS
4.1
General Remarks
27
4.2
Reservoir Simulation Model
28
4.3
Reservoir Forecasting Models
30
4.3.1
First Order Markov Model
31
4.3.2
First Order Markov Model with
33
Periodicity
4.3.3
4.4
V
VI
Log-Pearson Type III Distribution
Concluding Remarks
34
36
RESULTS AND DISCUSSION
5.1
Reservoir Simulation
37
5.2
Hydrologic Forecasting for Reservoir Storage
40
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
6.1
Summary and Conclusions
46
6.2
Recommendations
47
REFERENCES
48
APPENDICES
A
53
B
59
ix
LIST OF TABLES
TABLE NO.
TITLE
PAGE
Table 1
General features of MUDA Irrigation Scheme
23
Table 2
Observed Pedu storage with release after allowing for spill
26
(MCM)
Table 3
Step-by-step procedure of calculating model storage for Pedu
41
reservoir for March, using natural logarithm of observed
storage (MCM)
Table 4
Step-by-step
procedure
of
calculating
generated
and
43
forecasted storage for Pedu reservoir for March, using natural
logarithm of observed storage (MCM)
Table 5
Expected minimum storage in Pedu reservoir for different
return periods, using Log-Pearson Type III Distribution
45
x
LIST OF FIGURES
FIG. NO.
TITLE
PAGE
Figure 1
Location Map of Muda Irrigation Scheme
22
Figure 2
Major River Systems in Kedah
24
Figure 3
Pedu Water Level vs. Storage Curve using Observed Data
38
(1998-2008)
Figure 4
Mean, Minimum and Maximum Observed Monthly Pedu
38
Reservoir Storage with Release (MCM)
Figure 5
Comparison of Pedu Reservoir Simulation Model Results
39
with the Observed Mean Monthly Storages during 1998-2008
Figure 6
Pedu Reservoir Observed and Model Storages in Different
41
Months and Years
Figure 7
Pedu Reservoir Observed, Model and Forecasted Storages in
42
Different Months and Years
Figure 8
Reliability based Estimated Storage (MCM) in Different
44
Months, Considering Average Observed Flow and using LogPearson Type III Distribution
Figure 9
Reliability based Estimated Minimum Storage (MCM),
Considering Minimum Observed Storage for each Month
during
1998-2008
and
using
Distribution
Log-Pearson
Type
III
45
xi
LIST OF ABBREVIATIONS
A
Cross-sectional area
Ar (t )
Reservoir surface area
Aw (t )
Watershed area during period t
E (t )
Evaporation rate at the reservoir surface area
ET p
Potential evapotranspiration
ET p (t , t + Δt )
Evapotranspiration between t and t + Δt
IG (t )
Any other gain
GCM
General Circulation Model
GIS
Geographic Information System
I (t , t + Δ t )
Infiltration/percolation between t and t + Δt
IL (t )
Any other loss
LP
Linear Programming
MADA
Muda Agricultural Development Authority
P (t )
Precipitation falling on the reservoir surface area
P (t , t + Δ t )
Precipitation between t and t + Δt
Qi (t )
Inflow to the reservoir per unit watershed area during period t
Q j +1
Water delivered by conveyance system to the field (outflow)
Qs (t )
Uncontrolled releases downstream or spills from the reservoir
R (t )
Required reservoir release rate
R (t , t + Δt )
Runoff between t and t + Δt
SP
Seepage and percolation losses
Xi
Measured value of storage at time i
X i +1
Generated streamflow
xii
y
Average of the absolute values of deviations from the mean
Yi
Measured transformed value of storage at time i
Yi +1
Transformed generated streamflow
Y (t )
Reservoir yield
ε i +1
Random component with mean zero and variance σ ε2
μx
Mean of X
ρ x (1)
First order serial correlation
Δt
Time step
CHAPTER I
INTRODUCTION
1.1
Background
Allocation and management of water for agricultural purposes is a complex
issue affected by social, environmental and political factors. Reservoirs are very
useful choices for storage of irrigation water to use in drought periods. Optimal
operation of reservoir systems is important for effective and efficient management of
available water resources for maximum system net benefit (Consoli et al. 2008;
Nandalal and Sakthivadivel 2002; Raju and Kumar 1999; Suiadee and Tingsanchali,
2007; Shrestha et al., 1996).
In water resources development, reservoirs play a major role in modifying
uneven distribution of water both in space and time. To make the best use of the
available water, the optimal operation of reservoirs in a system is undoubtedly very
important though it is a very complicated task. During the last several decades many
attempts have been made towards solving this problem by various mathematical
means (Wurbs, 2005).
2
Irrigation projects which receive water from a reservoir can be challenging to
manage, since annual fluctuations in runoff from the reservoir's catchments area can
have considerable impact on the irrigation management strategy. (Shrestha et al.,
1996) mentioned that there is a general realization that many irrigation networks are
failing in their fundamental function of delivering water, where and when it is
needed, and in the right quantity. Irrigation departments, particularly in developing
countries, have been suffering financial setbacks and therefore implementing
improved techniques for operational management of these systems receives
inadequate attention.
Irrigation reservoir operation policies are aimed at deriving maximum
benefits from the water that can be stored in it and allocated to crops. Water releases
from reservoirs have to be conveyed through a hierarchical distribution system of
canals, branch canals, distributaries and field turnouts or outlets before they reach the
cropped fields. The operations are complex but substantial increases in benefits can
be derived even from relatively small increases in operating efficiency (Maidment
and Chow, 1981). Water management basically consists of determining when to
irrigate the amount to be applied during each state of plant growth, and the operation
and maintenance of the system. Water distribution systems and management
strategies that enable users to apply water uniformly and accurately require large
capital investments. All the three stages of the irrigation operation problems, namely,
determining the reservoir releases, transferring them to the field level, and allocating
the field supplies to crops are important components of operation of large irrigation
systems. Missing any one of these components can lead to low agricultural
productivities and operating efficiencies.
3
1.2
Statement of the Problem
Malaysia has mostly arid and semiarid climate, and spatial and temporal
distribution of rainfalls is irregular. Food demand is rapidly increasing with
increasing population. Water resources system of Malaysia has a complex structure
and financial opportunities to construct new dams are very restricted depends on the
purposes of construction, economical value and etc. Water resources managers and
decision makers paid a significant attention on optimum operation of reservoirs
during the last decade. Mathematical programming methods were the most widely
applied methods of optimization.
Water is main component for living beings on earth to continue their lives.
But, now-a-days the problem related to water such water shortage is very crucial.
This can be happened due to improper water management. The proper management
is very important in order to sustain the water resources with high water quality and
can reduce the problem on water shortage.
Reservoir is one of the sources of surface water. It can reserve the water and
supply water to the people. Reservoir will regulate inflows and provide outflows at
more regular rate, which is determined by water demand, temporarily storing the
surplus when inflows exceed outflows. These days reservoir has been facing a lot of
problems such as low water quality and water release not following an energyefficient schedule. It gives tendency of occurrence of water shortage.
According to previous studies, the water volume of Pedu and Muda dams
experience frequent deficit due to shortage of water supply from catchment. Thus,
the plan is needed to be modified periodically during real-time operation based on
4
current season data and climate change. The Muda irrigation scheme is highly
dependent on rainfall, fulfilling about 51% of the irrigation requirements. Two dams
(Pedu and Muda) contribute about 29%, while the uncontrolled river flow and
recycling supply contribute about 15 and 5%, respectively (Ali et al., 2000; MADA,
1987). In fact, the reservoirs were so depleted that irrigation for the 1978 dry season
crop was impossible, and again in 1983 and 1984, only half of the area could be
irrigated (Kitamura, 1990).
The shortage of reservoir water remains the most serious constraints on the
establishment of stable double cropping of rice. The efficient utilization of water
resources needs information, such as, annual effective rainfall, runoff, consumptive
use, and reservoir release, etc., thus, a reservoir simulation model often used to
predict the response of the system under a given set of conditions. On the other hand,
forecasting are used for warning of extreme events (e.g., floods and droughts), for
operation of water resources systems such as reservoir, hydropower generation
projects and etc. In addition, the models can be used to predict the future
performance of reservoirs.
1.3
Justification of the Study
Water management generally means the supply, conveyance, distribution, and
application of the right amount of water at the right time to the right place so that the
plants would thrive and produce good yield.
The shortage of reservoir water still remains the most serious constraints on
the establishment of stable double cropping of rice. Thus, a reservoir simulation
5
model needs to be developed to estimate the reservoir yield precisely. Long-range
water supply forecasting is an integral part of drought management and of water
supply management itself. Stochastic data generation aims to provide alternative
hydrologic data sequences that are likely to occur in future to assess the reliability of
alternative systems designs and policies, and to understand the variability in future
system performances. It is also very important to develop a stochastic hydrologic
model to generate the monthly streamflows and thus to estimate the future
streamflows with reliability.
1.4
Objectives of the Study
The main objectives to be carried out in this study are:
(i)
To develop a reservoir simulation model to simulate model storages with the
long-term observed storage amounts,
(ii)
To use stochastic models to generate storage and to compare with the
observed storage, and hence to forecast future storage with reliability.
1.5
Scope of the Study
The main scope of this study will be confined to the development of reservoir
simulation model, and utilization of stochastic models to generate and forecast
storage. The scopes of work that will be covered in this study are:
i.
Collection of various relevant historical data from MADA.
ii.
Development of various mandatory modules of reservoir systems.
CHAPTER II
LITERATURE REVIEW
2.1
General Remarks
In this study, a reservoir simulation model is to be derived and stochastic
models will be utilized. The literature reviews to be presented in this chapter are
concerned with the applications of different information available in the field of
water resources engineering and are divided into the categories, namely, (i)
Catchment hydrology, hydrological, and hydrodynamic modeling, (ii) Reservoir
operation, simulation, and optimization modeling, and (iii) Stochastic hydrologic
modeling for water resources forecasting.
Designing water resources systems are complex problems involving the
interaction of political and legal processes, governmental regulations, economics and
engineering aspects. Reservoir operators and managers have been faced with
difficulties in designing, operating, and managing multipurpose reservoir systems.
Many modeling techniques have been applied to solve problems in the areas of
planning, designing, and managing complex reservoir systems. (Tawfik, et. al., 1994)
mentioned that the application of modeling techniques in the planning area includes
determining the optimal reservoir size that satisfies downstream demands or finding
the best location to construct dam with minimum construction cost.
7
On the other hand, operation of a reservoir system requires a series of
decisions that determine how to accumulate and release water over time. Generally,
most reservoir systems are still managed based on fixed, predefined rules. These
rules are guiding the release of the reservoir system based on the current storage
level, the hydro-meteorological conditions, and the time of the year.
Application of optimization techniques to reservoir operation problems has
been a major focus of water resources planning and management. Various
mathematical programming techniques are described in several textbooks in
mathematics and water resources system analysis (e.g., Mays and Tung, 1992; Jain
and Singh, 2003). With the ever-growing use of computer technology in the
management process, mathematical models have been used for both simulation and
optimization (Belaineh et al., 1999; Hajilal et al., 1998; Maidment and Chow, 1981;
Ngo et al., 2007; Suiadee and Tingsanchali, 2007).
To achieve an optimal utilization of available resources, a scientific approach
should be adopted when planning and operating the existing reservoir. The need for a
proper management and optimum utilization of available water becomes crucial with
growing population, industrialization and rapid urbanization. An optimized operation
procedure is needed to accomplish the planning and management of a complex water
resources system (Jothiprakash and Shanthi, 2006).
Irrigation release decisions are periodic and sequential, and the consequences
of each decision can be evaluated only at the end of the season, after the crop yield is
known. This requires that the entire planning horizon be kept in view while making
irrigation decisions in a current interval (Rao et al., 1992). This can be realized on
the basis of such information as the storage in the reservoir, expected inflows, target
release requirements, and the impacts of intra-seasonal irrigation decisions on crop
yield (Hajilal et al., 1998). The operation rules are often evaluated using simulation
8
models. An efficient approach to define rules is by using optimization models in
combination with simulation models (Ngo et al., 2007).
Seong and Hyung, (2007) said that the factors affecting the amount of paddy
storage are rainfall, interception, evapotranspiration, deep percolation, paddy levee
height, and irrigation method. Among these factors, paddy levee height, irrigation
methods are chosen as the main control parameters to model water balance of paddy
fields. The intermittent irrigation method that carries out irrigation to paddy when the
ponding depth falls below a given threshold ponding depth, is adopted by this study.
Many researchers have described the hydrologic processes of surface and
subsurface flow on the soil water balance with hydrological modeling, but the
process of water balance under the water management of a rice paddy has not been
embedded in hydrological modeling. The model presented can be adapted to the
watershed-scale hydrological modeling to assess the quantitative effect of stream
discharge due to agricultural land use changes, especially for paddy field.
Irrigation managers therefore need to carefully plan the periodic reservoir
releases at head works, well in advance and for the entire season after assessing the
impacts of the decisions on crop yields. This can be done based on information of
storage in the reservoir, expected inflows, target release requirements, and crop yield
impacts of intra-seasonal irrigation decisions. However, current season inflows are
different from those used in deriving the irrigation release plan. Thus, the plan will
need to be modified periodically during real-time operation based on current season
data. This too must be done keeping the entire planning horizon in view. This two
phase approach to reservoir operation, that is, first planning the periodic releases in
advance for the entire season and then modifying the plan at the end of each period,
is referred to as real-time management. The management is real-time and adaptive
9
when the decisions are based not only on data of current system conditions but also
on future anticipated inflows (Labadie et. al., 1981).
2.2
Water Management in Reservoir
Water is drawn from the water supply, irrigation or even to run the
hydroelectric project. However, when we draw water directly from a stream, the
stream may unable to satisfy the water demands especially during low flow or
drought season (Chang Wei Chung, 2005). In order to solve such problem, we a need
reservoir to collect and stored the water for future use. Reservoir can be classified
into two types which are natural and manmade reservoir. Lakes and karst lakes is the
natural reservoir. While for manmade reservoirs are excavated reservoir, impounding
reservoir, tributary lateral reservoir etc.
According to Ladislav and Vojtech (1989), a water reservoir is an enclosed
area for the storage of water to be used at a later date; it can also serve to catch
floods to protect valleys downstream of it; to establish an aquatic environment; or to
change the properties of the water. A reservoir can be created by building a dam
across a valley, or by using natural or man-made depressions. The main parameters
of the reservoir are the volume, the area inundated and the range that the water level
can fluctuate.
Large dams are usually multipurpose structures. Besides providing water for
domestic, agriculture, and industrial uses (the main objectives of reservoir planning
and operation), hydropower electric production is another objective of development
of many river-reservoir systems. High efficiency, lower costs, and the specific
10
capabilities of hydropower plants for controlling the frequency of power networks
have made hydropower plants a necessary component of power systems. Flood
control and damage reduction is another objective for dam construction. A reservoir
reduces the peak flow of a flood hydrograph to an amount lower than the river
carrying capacity (Karamouz et al., 2003).
2.3
The Integration of Reservoir Operation Policies and Allocation of
Irrigation Areas
Reservoirs play an important role in water resources management. Mahdi et
al. (2007) stated that in Iran, as they have major storage utilities for regulating the
excess water for later deficit water periods or sometimes for drought years.
Appropriate reservoir operation and irrigation scheduling are needed for efficiently
utilizing water storage in reservoir-irrigation systems.
The system is characterized by two main components: the monthly reservoir
releases and the seasonal irrigation areas with the associated relationships defining
the interactions between them. The only input parameter of the reservoir is monthly
stream flow, while the output parameters are optimal irrigation areas supplied by
released water from the reservoir. The monthly released water consists of:
i.
reservoir releases when the reservoir is not completely full and the monthly
release is defined to be greater than monthly irrigation water demand by the
optimization model,
ii.
reservoir releases when the stored water exceeds the reservoir capacity.
11
In the first case, after supplying the demand by the irrigation system, the
excess water is conveyed through the downstream river of the irrigation intake of the
system. In other words, the spill is the volume of released water that occurs because
the monthly water demand for crops and fruits is less than what should be released
from the reservoir. It must be noted that there is another type of release policy when
the monthly inflow to the system is high enough so that the reservoir is completely
full and the excess water should be spilled from the crest of the spillway. In such a
case, after supplying the demand, the excess water will be spilled from the crest of
the spillway to bring the reservoir storage to the normal storage value. In other
words, when the total volume of monthly inflow and reservoir last-month storage is
more than the total capacity of the reservoir, water release from the crest of the dam
occurs.
2.4
The Models
The model described here predicts a crop yield through the use of crop yield
functions. In a sense it is similar to Standard Operating Rules, as it also uses as much
of the available water as needed in a season; however, SOP does not look ahead in
the season and neither does it consider the differences in the economic values of the
irrigations throughout the growing stages of the crops. Similarity is also observed in
the result. The models objective is to maximize the net benefits that can be obtained
from crop yields. Therefore, it maximizes the yield crops and minimizes the
shortage. Shortages throughout a season are distributed to minimize the amount of
yield that is lost for acre foot of shortage in release. Further explanation of the
models is discuss in the following part.
12
In recent years, optimization models have been successfully employed to
manage and operate reservoir systems. The choice of an optimization model is made
in respect with the characteristics of the system in consideration, the available data,
and pre-specified objectives and constraints. In many practical situations, operating
rules (also referred as operating policies) are established at the planning stage of the
proposed reservoir, to serve as guidelines for reservoir releases to meet the demands
planned (Tu et al., 2003)
Operators must evaluate tradeoffs among immediate and future uses of water
before the volume of the future supply becomes known. In the face of this
uncertainty, forecasts of future stream-flow can be helpful in determining efficient
operating decisions, and significant effort has been made to develop improved
forecasting methods (Faber and Stedinger 2001; Lettenmaier and Wood 1993).
Modeling techniques have been used in solving problems in reservoir
operation such as determining optimal releases or optimal storage volume to satisfy
various reservoir purpose; which may be providing irrigation water for agriculture,
generating hydroelectric power, protection of cities and towns against severe floods,
improving river navigation, fishing or recreation.
Sattari et al. (2009) investigated the efficiency of the Eleviyan irrigation dam
system (with a capacity of 60 hm3) which constructed to meet the irrigation and
municipal water needs of the Maraghan region (Northwestern Iran). They set up the
optimization model in three phases to maximize the water release for irrigation
purposes after the municipal water need were met. In the first phase, the inflows
measured in the 21 years prior to the construction of the reservoir, and in the second,
the inflows generated by the Monte Carlo simulation method, and in the third phase,
the inflows after the construction of the reservoir were used. They suggested that in
every future case, the changes in cropping pattern restrict the use of the optimization
13
results and more dynamic cropping pattern along with a more efficient operation and
water utilization may prepare the grounds for success. They concluded that Monte
Carlo could be used to generate data for the reservoir models to determine the
operational parameters and to create operational for proper plans and management.
Hydrological forecasting models have many similarities to the types of
simulation models developed for off-line studies for design, planning and other
applications. One of the first such approaches was the unit hydrograph (Shrestha et
al., 1996), in which a linear relationship is assumed between a unit depth of effective
rainfall falling in a given time, and the resulting runoff. The combined river flow
hydrograph is then estimated from the sum of these incremental contributions. This
approach is still widely used in flood estimation studies although, for real-time use,
other types of rainfall-runoff (or hydrologic) models are generally preferred, within
the general categories of physically-based, conceptual or data-driven models. Similar
types of model are also used in environmental forecasting applications for major
lakes and reservoirs, and in surge forecasting for coastal waters.
More empirical approaches, such as the Muskingum method and storage
routing approaches, are also widely used due to their relative computational
simplicity and modest data requirements, and are sometimes called hydrological flow
routing models. Indeed, some forms, such as the kinematic wave and MuskingumCunge approximations, are a simplified form of the equations of motion. Also, for
longer lead times, it can be reasonable to consider a mass or water balance alone, as
in the supply-demand modeling techniques which are widely used for water resource
applications.
Deepti and Maria (2009) presented a survey of simulation and optimization
modeling approaches used in reservoir systems operation problems. Optimization
methods have been proved of much importance when used with simulation modeling
14
and the two approaches when combined give the best results. They make a
conclusions that the reservoir management and operation practices must adapt
continuously to changes in water use priorities, physical and land use changes in the
river basin, technological developments, and changes in public policy expressed in
environment, safety, economic and technical regulations. Changing agricultural
practices can have a significant impact on water consumption. Optimizing the energy
generation through a hybrid of alternative sources and hydropower may provide
more overall benefit from reservoirs operation. Evolutionary algorithms have great
potential to deal with nonlinearity and multiobjective analysis; besides this, the other
feature that makes them attractive is that most of them can be directly linked with
simulation models.
Many studies have also sought to forecast seasonal and longer-term flows
using similar statistical approaches under meteorological conditions. Some possible
predictors include snow water equivalent, sea surface temperatures, and sea surface
pressure and indices linked to the El Niño-Southern Oscillation (ENSO) and North
Atlantic Oscillation (NAO) (Consoli et al., 2008)
Marcelo et al. (2001) considered six feature groups comprising of water
levels, rainfall, evaporation rate, discharges for rivers Malewa and Gilgil and one
pair of time harmonics were used to develop neural network models to forecast water
levels for Lake Naivasha in Kenya. The neural network models developed were able
to forecast effectively the reservoir levels for the lake for four consecutive months
after a given month and given data for six consecutive months prior to the month. It
was found that the more the number of feature groups used, the higher the ability of
neural networks to forecast accurately the reservoir levels. Data compression
generally reduced the size and computation time of the models. This can help in
water-use formulation and scheduling for domestic, municipal and agricultural uses.
Timely forecasting can also help in disaster monitoring, response and control in areas
prone to floods. For power generation, effective and timely reservoir level
15
forecasting can help in predicting power loads and management of power generation
for efficiency and optimisation. However, data compression introduces undesirable
qualities into the data that affects the forecasting ability. In addition, over
compression of data undermines the efficiency in forecasting reservoir levels.
El-Awar et al. (1998) presented a modified stochastic differential dynamic
programming algorithm for multireservoir system control. Hajilal et al. (1998)
applied dynamic programming method for reservoir optimization in Jayakwadi
Irrigation Project during both plan phase and real time operation phase in
Maharashtra, India. Duranyildiz et al. (1999) applied a change-constrained LP model
for optimization of the monthly operation of a real water supply system. Montaseri
and Adeloye (1999) applied Monte Carlo simulation method for investigating the
critical period of within-year and over-year reservoir systems and its relationship
with the currently used test for discriminating between two patterns of reservoir
behavior. Aksoy and Bayazit (2000) presented a model for the generation of daily
flows of an intermittent stream based on Markov Chains. Nandalal and Sakthivadivel
(2002) investigated the operational behavior of multi objective (irrigation and
hydropower) reservoir in the Walawe River, Sri Lanka by using stochastic dynamic
programming and simulation techniques. Sattari et al. (2006) developed a
deterministic nonlinear program (DNLP) to determine the optimum active capacity
of Keyserek reservoir active capacity in Iran.
Ngo et al. (2007) used a mathematical model to optimize the control
strategies for the Hoa Binh reservoir in Vietnam, by applying a combination of
simulation and optimization models. Ozturk et al. (2008) presented a linear stochastic
model methodology for modeling of suspended sediment data which fills dead
storage volume of reservoirs. Mathlouthi and Lebdi (2008) developed reservoir
operating rules for dry and wet periods, and their implementation in northern
Tunisia. Consoli et al. (2008) developed a nonlinear multi-objective allocation model
for Pozzillo irrigation reservoir in Italy. Sattari et al. (2009) compared classical
16
(Ripple diagram and sequent peak analysis) and modern methods (optimization
technique) for calculating dam capacity.
2.5
Forecasting
Forecasting is the process of estimation in unknown situations. Prediction is a
similar, but more general term. Both can refer to estimation of time series, crosssectional or longitudinal data. Usage can differ between areas of application: for
example in hydrology, the terms "forecast" and "forecasting" are sometimes reserved
for estimates of values at certain specific future times, while the term "prediction" is
used for more general estimates, such as the number of times floods will occur over a
long period. Risk and uncertainty are central to forecasting and prediction.
Forecasting is used in the practice of Customer Demand Planning in everyday
business forecasting for manufacturing companies. The discipline of demand
planning, also sometimes referred to as supply chain forecasting, embraces both
statistical forecasting and a consensus process.
In contrast to an earlier wave of optimism regarding the value of climate
forecasting in reducing the ravages of drought in Northeast Brazil (Galli, A. et al.,
1994), this paper argues that socio-economic, political, and cultural conditions can
compromise the use of seasonal forecasts by both farmers and policymakers. They
contend that seasonal forecasting faces a ‘new technology adoption’ problem in the
sense that the potential uses and limitations of the technology are not fully
understood and a process of learning must ensue in order to determine appropriate
use. At the same time, the presentation of the forecast and its mode of
communication to policymakers and farmers are critical to application success.
While much attention has been paid to the science of climate forecasting and its
17
application for drought mitigation, there is limited understanding of the sociopolitical environment through which climate forecasts are channeled and interpreted.
Once in the hands of policymakers, the science product loses – in a very critical
sense – its desired objectivity and becomes woven into a complex mesh of social,
economic, and cultural realities that influence how information is in fact used. It is
this interaction of policymaker, end-user, and scientist that is addressed here. They
also analyzed the policy process that underlies the application of seasonal climate
forecasting in the state of Ceará in Northeast Brazil specifically examining three
groups of variables: (1) the characteristics of the forecasts in terms of accuracy,
timing of release, data format, and mode of communication; (2) the policymaking
system at all relevant administrative levels; (3) and the relative social and economic
vulnerability of the population toward which the forecasts are directed. In addition,
this study explores still untapped opportunities for data use especially in long-term
drought-relief planning.
Forecasts are presented in the language of probabilities, but are often not
perceived as such. Probabilistic information is difficult to assimilate because people
do not think probabilistically nor do they interpret probabilities easily (Goddard et
al., 2003).
2.5.1 Markov First Order Methods
First order Markov models have been successfully applied to many problems.
Examples include modeling sequential data using Markov chains, and solving control
problems posed in the Markov decision processes (MDP) framework. If the Markov
model’s parameters are estimated from data, the standard maximum likelihood
estimates consider the first order (single-step) transitions only.
18
The purpose in the management of surface water resources is to find an
operating policy such that the maximum benefit or minimum cost is achieved in a
short or long period of time. This operating policy should map the periodic water
releases to system states (which are usually predefined discrete values) indicating the
different storage volumes at the beginning of each period of a cycle (e.g., year)
(Montoglou and Wilson, 1982). In optimization of some applications, when the
respective model is in the situation of non-stationary Markov decision process, the
preceding inflow in every time period can be considered as a new state variable. This
kind of state variable, along with other state variables for different reservoirs in the
system, forms a vector quantifying the system state.
First order Markov models have been successfully applied to many problems.
Examples include modeling sequential data using Markov chains, and solving control
problems posed in the Markov decision processes (MDP) framework. If the Markov
model’s parameters are estimated from data, the standard maximum likelihood
estimates consider the first order (single-step) transitions only. But for many
problems the first order conditional independence assumptions are not satisfied, as a
result of which the higher order transition probabilities can be poorly approximated
by the learned model.
2.5.2 Log Pearson Type III Distribution
The Pearson system was originally devised in an effort to model visibly
skewed observations. It was well known at the time how to adjust a theoretical model
to fit the first two cumulates or moments of observed data. Any probability
distribution can be extended straightforwardly to form a location-scale family.
Except in pathological cases, a location-scale family can be made to fit the observed
mean (first cumulates) and variance (second cumulates) arbitrarily well. However, it
was not known how to construct probability distributions in which the skewness
19
(standardized third cumulates) and kurtosis (standardized fourth cumulates) could be
adjusted equally freely. This need became apparent when trying to fit known
theoretical models to observed data that exhibited skewness. Pearson's examples
include survival data, which are usually asymmetric.
Pearson (1895) identified four types of distributions (numbered I through IV)
in addition to the normal distribution (which was originally known as type V). The
classification depended on whether the distributions were supported on a bounded
interval, on a half-line, or on the whole real line; and whether they were potentially
skewed or necessarily symmetric. Pearson (1901) fixed two omissions: it redefined
the type V distribution (originally just the normal distribution, but now the inversegamma distribution) and introduced the type VI distribution. Together the first two
papers cover the five main types of the Pearson system (I, III, VI, V, and IV). In a
third paper, Pearson (1916) introduced further special cases and subtypes (VII
through XII).
Rhind (1909) devised a simple way of visualizing the parameter space of the
Pearson system, which was subsequently adopted by Pearson (1916). Many of the
skewed and/or non-mesokurtic distributions familiar to us today were still unknown
in the early 1890s. What is now known as the beta distribution had been used in 1763
work on inverse probability. The Beta distribution gained prominence due to its
membership in Pearson's system and was known until the 1940s as the Pearson type I
distribution (I) distribution. The gamma distribution originated from Pearson's work
(Pearson 1893; Pearson, 1895) and was known as the Pearson type III distribution,
before acquiring its modern name in the 1930s and 1940s. Pearson's 1895 paper
introduced the type IV distribution, which contains Student's t-distribution as a
special case, predating William Sealy Gosset's subsequent use by several years. His
1901 paper introduced the inverse-gamma distribution (type V) and the beta prime
distribution (type VI).
20
The Pearson Type III Distribution was first applied in hydrology to describe
the distribution of annual maximum discharges. The Log Pearson Type III
Distribution is widely used in the U.S. to calculate flood recurrences because it has
been recommended by the U.S. Interagency Advisory Committee on Water Data. It
is the default distribution used by the U.S. Geological Survey for flood studies.
2.6
Concluding Remarks
Optimal reservoir operation is a complex multidimensional problem
involving various economic, social, and environmental issues and development
opportunities as well as the associated physical process of various types.
Multireservoir operating policies are usually defined by rules that specify either
individual reservoir desired (target) storage volumes or desired (target) releases
based on the time of year and the existing total storage volume in all reservoirs. A
considerable progress has been made in solving problems related to individual issue
or process, e.g., simulation of flow, simulation of water quality in river, reservoir
operation optimization, etc. Various techniques are well established for analysis of
individual problems. Long-range water supply forecasting is an integral part of
drought management and of water supply management itself. Stochastic data
generation aims to provide alternative hydrologic data sequences that are likely to
occur in future to assess the reliability of alternative systems designs and policies,
and to understand the variability in future system performances.
CHAPTER 3
STUDY AREA AND DATA COLLECTION
3.1
Study Area
The Muda Irrigation Project (Figure 1) covers a total gross area of 126,000ha,
out of which about 97,000 ha is under the double cultivation of paddy. It is the
largest double cropping area in Malaysia. The area is located at about 5°45' ~ 6°30' N
latitude and 100°10' ~ 100°30' E longitude in the vast flat alluvial Kedah–Perlis Plain
of about 20 km wide and 65 km long between the foothills of the Central Range and
the Straits of Malacca. The area is generally flat with slopes of 1 in 5,000 to 1 in
10,000 ranging from +4.5 m elevation in the inland fringe to +1.5 m elevation in the
coastal area (MADA 1977). The major part of Peninsular Malaysia is characterized
by a tropical rainforest climate. However, only the Muda area and its periphery
where there is a pronounced dry season are under the tropical monsoon climate as the
area is shielded by the rain-bearing winds of the Northeast monsoon and the
Southeast monsoon from the Central Range and Sumatra, respectively. There are
three large man-made lakes within the Ulu Muda area, namely, Muda, Ahning and
Pedu, formed by the construction of three correspondingly-named dams that regulate
water for domestic use and irrigation for most of Kedah, Penang and Perlis.
22
Figure 1: Location Map of Muda Irrigation Scheme
The water storage system covers three water storage reservoirs which include
Muda, Pedu and Ahning Reservoirs. The Muda and Pedu Dams were built in 1969
under the Muda Irrigation Scheme for the purpose of providing irrigation water to the
Muda area covering 96,000 hectares to enable double cropping of rice per year.
Ahning Dam was built by the Public Works Department for the main purpose of
supplying water for domestic and industrial uses.
In 1991, Ahning Dam was handed over to Muda for operations and
management work because the water from Pedu and Ahning Dams flows through the
same single river channel to arrive at the Pelubang Bifurcation where the water is
distributed to District I and II via the Northern Channel and to District III and IV via
the Central and Southern Channels respectively.
Since the Muda Reservoir has a large catchment area of (984 km2) but a low
storage capacity of 160 million m3, water from the Muda Reservoir is transferred to
the Pedu Reservoir, which has a higher storage capacity of 1073 million m3, via the
23
6.8 km long Saiong Tunnel for storage and release. Water for irrigation in the Muda
area is released through the Pedu and Ahning Dams. The general features of Muda
Irrigation Scheme are shown in Table 1.
Table 1: General Features of MUDA Irrigation Scheme
Muda Dam
Storage
Reservoir Area
Type
Max Height
Length
=
=
=
=
=
Pedu Dam
120 MCM
26 km2
Concrete
32 m
230 m
Storage
Reservoir Area
Type
Max Height
Length
Spillway Flood
Canal
=
860 MCM
65 km2
Rolled Rock Fill
60 m
200 m
280 m3/s
=
=
=
=
Drain
Total Length
=
1724 km
Density
=
17.7 m/Ha
Total Length
=
1420 km
Density
=
14.6 m/Ha
Saiong Tunnel
Total Length
3.2
= 6.6 km
Capacity
=
33-70 m3/s
Major River Systems and Climate Change in Muda Irrigation Scheme
The northern part of Peninsular Malaysia, where Muda Irrigation Scheme is
located, has two distinct seasons: a wet season between the months of May and
October coinciding with the south-west monsoon, and a dry season between
December and March during the northeast monsoon (which brings rain to the east
coast of Peninsular Malaysia). There is also a short dry season in the months of June
and July. The average annual rainfall is about 2,000 mm, with October usually the
wettest month, and another minor peak in April/May. The Ulu Muda forest forms the
24
headwaters of the Muda River which is the largest river system in Kedah (Figure 2).
The important tributaries of the Muda River, upstream of the Muda dam, are Sungai
Lasor, Sungai Teliang, Sungai Bohoi, Sungai Kawi and Sungai Kalir. The Pedu,
Ahning and Kedah are separate river systems from the Muda River but they also
originate from the Ulu Muda area.
Figure 2: Major River Systems in Kedah
25
The largest of the three lakes in the Ulu Muda area is the Pedu Lake which
covers an area of 15,500 ha. Although Muda Lake is smaller (5,200 ha) it has a larger
catchment area and there is a 6.6 km long tunnel that channels water from Muda
Lake to Pedu Lake. Kedah and Perlis are prone to seasonal drought and water stress,
and therefore the Ulu Muda forest plays an important role in regulating water flow to
the Muda River and its tributaries. The Ulu Muda forest provides upstream
protection of major rivers that supply water for domestic, industrial and agricultural
use to the people in the northern areas of Peninsular Malaysia.
Irrigation schemes that depend on the Ulu Muda catchment forest supply
water to the largest rice-growing state in the country. This has earned Kedah its
nickname of the “rice-bowl” of the Malaysia. The area under these irrigation
schemes, including the Muda irrigation scheme, is responsible for about 40% of the
country’s total rice production and directly benefits the livelihoods of 63,000
families. The electronics and heavy industries sector centred at Penang Island,
Seberang Perai and Kulim in southern Kedah are also highly dependent on the
continuous supply of clean water originating from the Ulu Muda forest. Penang has
one of the cheapest water rates in the country and this is one of the factors that make
it an attractive location for investments from multi-national companies.
3.3
Data Collection
The data from 1997-2008 was used in deriving both stochastic and
forecasting models. The computation work used the available historical data taken
from MADA office. For example, the processed observed Pedu storage including
release after allowing for reservoir spill is shown in Table 2. The models need the
26
following data to predict and simulate monthly yield of reservoir systems as shown
in Appendix A which includes this type of following data:
i.
Inflow
ii.
Dam releases
iii.
Storage
iv.
Spill Data
v.
Water level
vi.
Evaporation and Rainfall
Table 2: Observed Pedu storage with release after allowing for spill (MCM)
Month
1998
Jan 1080
Feb 1058
Mar 1267
Apr
932
May 663
June 475
July
472
Aug
522
Sep
620
Oct
710
Nov
811
Dec
961
1999
1002
1055
1184
1104
1185
1123
916
973
1092
1020
1083
1065
2000
1074
1084
1212
1115
1199
1119
1104
1131
1124
1122
1152
1150
Year
2001 2002 2003 2004 2005 2006 2007 2008
1118 968 682 713 608 679 908 1151
1110 936 716 679 599 761 923 1082
1175 988 780 761 617 882 1013 1175
1135 1033 775 656 685 843 982 1165
1184 864 676 575 491 803 948 1122
1034 673 499 412 435 844 862 1020
962 488 361 329 331 805 917 974
929 454 394 368 254 818 936 970
986 482 425 433 254 831 996 1089
918 587 510 564 275 819 1035 1063
1079 682 657 693 374 866 1091 1139
1078 813 789 692 536 984 1184 1216
CHAPTER 4
THEORETICAL CONSIDERATIONS AND MODEL DEVELOPMENT
4.1
General Remarks
One objective of reservoir management is to release or store water based on
its economic value. For agricultural purposes this implies evaluating the increased
crop yield gained by releasing water at a specific time/ storing it for later use. (Hanks
et. al., 1974) state that by developing the model with the objective of maximizing
crop yield it is subjected to physical constraints such as mass balance, storage limits
and target demands.
Irrigation reservoir operation policies are aimed at deriving maximum
benefits from the water that can be stored in it and allocated to crops. Water releases
from reservoirs have to be conveyed through a hierarchical distribution system of
canals, branch canals, distributaries and field turnouts or outlets before they reach the
cropped fields. The operations are complex but substantial increases in benefits can
be derived even from relatively small increases in operating efficiency (Maidment
and Chow, 1981).
The models either developed or used in order to carry out this study are of
different types in terms of their purposes, capabilities, interfaces, inputs, and outputs.
28
These mainly include water balance model, reservoir simulation, and stochastic
models. The Brief descriptions of the theories associated with each of the models are
presented in the following sections.
4.2
Reservoir Simulation Model
An individual reservoir simulation was conducted to check the performance
of simulated reservoir operations. The monthly observed inflow and initial storage
were input, and release and storage were simulated for reservoirs for which there was
observed operation data.
Model simulation is carried out to assess and to incorporate the effects of the
reservoir's performance into the operation of the system as a whole. The simulation is
carried out over the total historical record of inflow. The role of simulation in the
presented approach is two-fold:
i.
To determine the reservoir operation (reservoir releases) over a given time
period with known stream flows at input point to the reservoir. So that it can
operate reservoir to make sure best meet the flow demands.
ii.
In conjunction with release allocation algorithm, simulation provides
necessary information on the interaction among reservoirs (i.e. expected
levels of each individual demand fulfillment and deficit, additional flows
available to the reservoirs
situated downstream on the river course and
shortages in supply that a reservoir is going to encounter by following the
derived operating policy).
29
A reservoir firm yield analysis typically employs a numerical approximation
to the solution of the conservation of water mass equation. The reservoir simulation
model can be developed in the general form as shown below:
Where,
S(t)
=
initial storage volume at the beginning of the period t,
Qi(t)
=
the inflow to the reservoir per unit watershed area during
Aw(t)
=
the watershed area during period t,
P(t)
=
the precipitation falling on the reservoir surface area Ar(t)
E(t)
=
the evaporation rate at the reservoir surface of area Ar(t)
R(t)
=
the required reservoir release rate for the purpose of
period t,
maintaining
Qs(t)
in-stream inflow,
=
the uncontrolled release downstream or spills from the
reservoir,
IG(t) and IL(t) =
any other gains to or losses from the storage, respectively,
Y(t)
the reservoir yield,
=
Smax and Smin =
capacities,
the maximum and minimum available active storage
respectively.
The daily time-step firm yield Yday is determined using:
(2)
Where:
Sday(t+1)
=
the water in storage at the beginning of day (t+1),
Sday(t)
=
the water in storage at the beginning of the day (t),
30
Qi(t)
=
the average daily stream flow observed on the day t,
Qs day (t)
=
the spill from the reservoir
Moreover, the monthly time step firm yield Ymon is determined from:
(3)
For the jth month, which has nj,k day in year k (leap year to be considered also).
4.3
Reservoir Forecasting Models
Hydrological forecasts typically aim to translate meteorological observations
and forecasts into estimates of river flows. Techniques can include rainfallrunoff
(hydrologic) and hydrological and hydrodynamic flow routing models, and simpler
statistical and water-balance approaches. Additional components may also be
required for water quality applications, and for modeling specific features of a
catchment, such as reservoirs, and lakes, and the influence of snowmelt. Particularly
for short lead times, models may also need to be embedded in a forecasting system,
which controls the gathering of data, the model runs and the post-processing of
outputs.
The availability of real-time data also provides the option to update the model
states or parameters or to post-process the outputs to improve the accuracy of the
forecast; a process which is often called data assimilation or real-time updating. For
31
operational use, appropriate performance measures also need to be adopted for
forecast verification.
Stochastic data generation aims to provide alternative hydrologic data
sequences that are likely occur in future. These data sequences, particularly monthly
time series, are widely used in water resources planning and operation studies to
assess the reliability of alternative system design and policies, and to understand the
variability in future system performances (Loucks et. al., 1981). However, for valid
and realistic result, it is necessary that these synthetic monthly data sequences should
preserve both monthly and annual statistical parameters of historic data such as
mean, variance, correlations, etc. Two methods will be used in this study includes
first order Markov Process and Linear regression methods.
4.3.1 First Order Markov Model
In this study, a first order Markov process is used and is defined by the
equation:
(4)
Where is:
Xi
=
the value of the process at time I,
µx
=
the mean of x,
ρx(1) =
the first order serial correlation,
εi+1
the random component with E(ε)=0 and Var(ε)=σ2ε
=
Eq (4) also known as the first order autoregressive model since
is equal
to the regression coefficient β that could be obtained if the regression model was
used taking Y as
and X as
. This model states that the value of X in one time
32
period is dependent only on the value of X in the preceding time period plus a
random component. It is also assumed that
is dependent of
. The variance of
X is given by σ2ε and can be shown related to σ2ε by,
Or
(5)
If the distribution of X is N(
of
the distribution of ε is N(0,
can now be generated by selecting
distribution. If t is N(0,1) then tσε or
generating X’s that are N(
.
Random values
randomly from a N(0,
is
. Thus, a model for
and follow the first order Markov model is:
(6)
Eq(6) has been widely used for generating annual runoff from watershed
(Fiering and Jackson, 1971). Since t is N(0,1), it is possible to generate values of X
that are less than zero. If it occurs it is generally recommended that the negative X be
used to generate the next value of X and then discarded. This procedure will result in
slight bias. If the occurrence of negative X’s is common in the generation process, it
may indicate that X is not normally distributed. In this event, some other distribution
of ε must be used.
Eq(6) can be applied to the algorithms of data through the transformation
Yi=In(Xi). The generation model is given by:
(7)
33
Where is
,
is refer to mean, standard deviation and first order serial
correlation of the logarithm of the original data. Generation by Eq.(7) preserves the
mean, variance, coefficient of skew nets and first order serial correlation of the
original data but not of the data itself.
4.3.2 First Order Markov Method with Periodicity
The first order Markov model of the previous section assumes that the
process is stationary in its first three moments. It is possible to generalize the model
so that the periodicity in hydrologic data is accounted for the some extent. The main
application of this generalization has been in generating monthly stream flow where
pronounced seasonality in the monthly flow exists. The periodicity may affect not
only the mean, but also of the moments of the data as well as the first order serial
correlation.
Seasonal hydrologic time series, such as monthly flows, may be better
described by considering statistics on a seasonal basis. Let the seasonal time series,
Yv,r in which v = 1,2,…,n; and τ = 1,2,….ω, with n and ω denoting the number of
years of record and the number of season per year, respectively. The season to season
correlation coefficient (a measure of the strength of the linear relationship between
flows in successive months)
is determined by:
(8)
(9)
34
Hydrological forecasting models are subject to many sources of uncertainty,
including uncertainties in the input data, model parameters, initial conditions, and
boundary conditions. The model structure may also be inappropriate or less than
ideal for the situation under consideration. The extent to which these factors
influence the model performance should be evaluated during the model calibration,
and in subsequent monitoring of the operational performance, and ideally in realtime operation. Also, particularly where the lead time required exceeds the response
time of the hydrological system, meteorological forecasts may also be used as an
input to the models, which introduces another source of uncertainty, particularly for
longer lead times.
For instance, for monthly streamflow time series,
represents the
correlation between the flows of fifth month (May) with those fourth month (April).
With this notation, the multiseason, first order Markov model for normally
distributed flows becomes:
.(10)
4.3.3 Log-Pearson Type III Distribution
The Log Pearson Type III distribution is commonly used in hydraulic studies.
It is somehow similar to normal distribution, except instead of two parameters,
stanand deviation and mean, it also has skew. When the skew is small, Log Pearson
Type III distribution approximates normal.
Extreme values are selected maximum or minimum values of sets of data.
The widely used techniques for flood flow estimation are the Log-Pearson Type III,
the Gumbel extreme value distribution, and lognormal distribution (Al-Mashhadani,
35
E.H. and Beck, M.M. 1978, Pilon & Harvey, 1994). In this study, the Log-Pearson
Type III distribution is used, following the recommendation of the U.S. Water
Resources Council (1967, 1976, 1977, and 1981; Benson, 1968) method.
For Log-Pearson Type III Distribution, the first step is to take the logarithms
of the hydrologic data,
. The mean
, standard deviation s, and the
coefficient of skewness Cs are calculated for the logarithms of the data. The
frequency factor KT depends on the return period T and the coefficient of skewness
Cs. When Cs = 0, the KT is equal to the standard normal variable z. When
, KT
is approximated by Kite (1977) as
(11)
where
, and Cs is given by
(12)
and s is given by
(13)
The value of z corresponding to an exceedence probability of p (p = 1/T) can be
calculated by finding the value of an intermediate variable w:
(14)
Then calculating z using the approximation
36
(15)
When
is substituted for p in Eq. (14) and the value of z calculated by
Eq. (15) is given a negative sign. The error in this formula is less than 0.00045 in z
(Abramowitz & Stegun, 1965).
4.4
Concluding Remarks
In this study, a reservoir simulation model is developed that includes the
inflow, outflow, rainfall on the reservoir surface, evaporation at the reservoir surface,
the required reservoir release rate, seepage through the reservoir dam, uncontrolled
releases downstream or spills from the reservoir, any other gains to or losses from
storage, and reservoir yield. The Markov process with periodicity in hydrologic data
is considered to generate the monthly streamflows through the natural logarithms of
observed data transformation.
CHAPTER 5
RESULTS AND DISCUSSION
5.1
Reservoir Simulation
The relationship between reservoir storage and reservoir surface area is
defined by a site-specific storage-area curve. In a site-specific assessment, the
reservoir storage-surface-area relationship may be estimated using bathymetric
techniques or a plenimetry analysis of prereservoir construction topographic maps
and surveys. Here, instead of using storage-surface-area curve as it is not available,
storage-stage rating curve was plotted and used (Figure 3). The solution procedure is
two-step. Firstly, it needs to covert storage into water level (WL) to incorporate the
effects of rainfall (mm) and evaporation (mm) on the reservoir surface. Secondly, the
resulting water level from first step needs to be converted into storage again to get
the model storage.
In constructing the stage-storage curve, the monthly observed values of
storage and water level from 1998 through 2008 were used. The equation of the
curve was found to be WL=36.49S0.139 with coefficient of determination, R2 of 0.99,
where WL and S represent the water level (m MSL) and storage (Million Cubic
Meters, MCM), respectively. The observed mean, minimum and maximum monthly
Pedu reservoir storage with release (MCM) are calculated as shown in Figure 4.
38
Figure 3: Pedu Water Level vs. Storage Curve using Observed Data (1998-2008)
Figure 4: Mean, Minimum and Maximum Observed Monthly Pedu Reservoir
Storage with Release (MCM)
Typical simulation models associated with reservoir simulation include a
mass-balance computation of inflows, outflows, and changes in storage. The
reservoir simulation model is not able to generate an optimal solution to a reservoir
39
problem directly. However, when making numerous runs of a model with alternative
decision policies, it can detect at optimal or near-optimal solution. The model storage
amounts were calculated for 1998-2008 using measured values of rainfall and
evaporation (Pedu reservoir station no. 61), reservoir inflow, release, seepage, spill,
and Muda reservoir inflow. The additional inflow from the Muda reservoir was
channeled into Pedu reservoir applying Muda Agricultural Development Authority’s
used equation, Q = 285.14H1/2, where Q is in ft3/s and H is the difference in water
level between Pedu and Muda reservoirs (ft) and converted into cubic meter (m3) to
fulfill model requirements, and the model storage was considered at least equivalent
to the observed storage, as the inflows from the reservoir watershed area and from
Muda reservoir were not measured separately, i.e., nonavailable. Reservoir spill in
excess of 1050 MCM was also considered. The observed and model storage
capacities are compared as shown in Figure 4 and found to be simulated well, except
for few months where the model storage amounts are higher than the observed
storage (Figure 5).
Figure 5: Comparison of Pedu Reservoir Simulation Model Results with the
Observed Mean Monthly Storages during 1998-2008
40
5.2
Hydrologic Forecasting for Reservoir Storage
Long-range water supply forecasting is an integral part of drought
management and of water supply management itself. In this study, the multiseason,
first order Markov model with periodicity in normally distributed monthly
streamflows is used to generate monthly hydrologic data that preserve particularly
the over-year monthly correlations, in addition to other monthly parameters of
historic data. Stochastically generated hydrologic data have been used in the past by
water authorities worldwide for long-term planning of water resources development
projects. For valid and realistic results, it is necessary that the generated data
sequences preserve all statistical properties of historical data.
The student’s t-distribution quantiles for (n-1) degrees of freedom and 0.05
one-sided confidence limit are used in storage generation. The natural logarithm of
observed storage is used because of its higher accuracy of simulation (Hirsch, 1979).
The first order Markov model generated and observed mean storage amounts were
compared for each month. The results show that the monthly statistical parameters of
the historic record, except the lag1 serial correlation between December and January
months (i.e., over-year monthly correlations), are preserved satisfactorily (Figure 6).
The lag1 correlation coefficient for January is found to be poor, while the same for
February through December are satisfactorily found to be 0.976, 0.984, 0.891, 0.931,
0.962, 0.968, 0.978, 0.993, 0,983, 0.994, and 0.970, respectively. As an example, the
calculation procedure for Pedu reservoir storage generation (model storage) for
March is shown in Table 3, while the same for the other months can be found in
Appendix B.
41
Figure 6: Pedu Reservoir Observed and Model Storages in Different Months and
Years
Table 3: Step-by-step procedure of calculating model storage for Pedu reservoir for
March, using natural logarithm of observed storage (MCM)
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Month
March
LN of
Obs.
Storage
Std.
Dev.,
s
Lag1
Q
Mean
Q
Lag1
Mean
Q
Lag1
Cor.
Coeff.
Student’s
tdistribution
with (n-1)
degrees of
freedom
Lag1
Std.
Dev.,
s
Model
Storage,
Xi,j+1
7.144
7.077
7.100
7.069
6.896
6.659
6.635
6.425
6.782
6.921
7.069
0.081
0.059
0.067
0.057
0.002
0.073
0.080
0.147
0.034
0.010
0.057
6.964
6.961
6.988
7.012
6.842
6.574
6.521
6.396
6.634
6.828
6.987
6.889
6.889
6.889
6.889
6.889
6.889
6.889
6.889
6.889
6.889
6.889
6.792
6.792
6.792
6.792
6.792
6.792
6.792
6.792
6.792
6.792
6.792
0.984
0.984
0.984
0.984
0.984
0.984
0.984
0.984
0.984
0.984
0.984
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
0.054
0.054
0.062
0.070
0.016
0.069
0.086
0.125
0.050
0.012
0.062
7.231
7.105
7.124
7.088
6.896
6.688
6.666
6.481
6.795
6.924
7.084
42
The obtained lag1 correlation coefficients suggest that the model can
satisfactorily be applied for future storage prediction. The mean values of the
observed storages for each month (1998-2008) were considered as the anticipated
storage in the future, and the model was run to predict future storage amounts. In this
case, the lag1 correlation coefficient for January is also found to be poor, while the
same for February through December are satisfactorily found to be 0.976, 0.984,
0.892, 0.931, 0.962, 0.968, 0.979, 0.993, 0,983, 0.994, and 0.970, respectively. For
example, the step-by-step calculation procedure for forecasted Pedu reservoir storage
(forecasted model storage) for March is shown in Table 4. The results of the
forecasted storages for year 2009-2015 are shown in Figure 7, which could be used
in future reservoir operation and management.
Figure 7: Pedu Reservoir Observed, Model and Forecasted Storages in Different
Months and Years
43
Table 4: Step-by-step procedure of calculating generated and forecasted storage for
Pedu reservoir for March, using natural logarithm of observed storage
(MCM)
Year
Month
LN of
Obs.
Storage
Std.
Dev.,
s
Lag1
Q
Mean
Q
Lag1
Mean
Q
Lag1
Corr.
Coeff.
Student’s
tdistribution
with (n-1)
degrees of
freedom
Lag1
Std.
Dev.,
s
Model
Storage,
Xi,j+1
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
0.054
0.054
0.062
0.070
0.016
0.069
0.086
0.125
0.050
0.012
0.062
7.231
7.105
7.124
7.088
6.896
6.688
6.666
6.481
6.795
6.924
7.084
1.782
1.771
1.761
1.753
1.746
1.740
1.734
0.003
0.003
0.003
0.003
0.003
0.003
0.003
6.914
6.914
6.914
6.914
6.914
6.914
6.914
(a) Generated Storage
1998
1999
2000
2001
2002
2003 March
2004
2005
2006
2007
2008
7.144
7.077
7.100
7.069
6.896
6.659
6.635
6.425
6.782
6.921
7.069
0.081
0.059
0.067
0.057
0.002
0.073
0.080
0.147
0.034
0.010
0.057
6.964
6.961
6.988
7.012
6.842
6.574
6.521
6.396
6.634
6.828
6.987
(b)
2009
2010
2011
2012 March
2013
2014
2015
6.913
6.913
6.913
6.913
6.913
6.913
6.913
0.004
0.004
0.004
0.004
0.004
0.004
0.004
6.812
6.812
6.812
6.812
6.812
6.812
6.812
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
6.889 6.792 0.984
Forecasted Storage
6.898
6.898
6.898
6.898
6.898
6.898
6.898
6.800
6.800
6.800
6.800
6.800
6.800
6.800
0.984
0.984
0.984
0.984
0.984
0.984
0.984
For Log-Pearson Type III Distribution, the mean Pedu reservoir storage in
each month (January-December) is selected. The expected mean storage for different
return periods and probabilities of occurrences (i.e., 1/T) estimated using this
distribution are shown in Figure 8, and for clarity of Figure 8, only trendlines for a
few months are shown. Figure 8 would be helpful for the decision makers to estimate
future storage with corresponding return period.
44
Figure 8: Reliability based Estimated Storage (MCM) in Different Months,
Considering Average Observed Flow and using Log-Pearson Type III Distribution
Similarly, the expected minimum storage considering minimum observed
storage for each month during 1998-2008 is estimated, using this distribution (Table
5 and Figure 9). Figure 9 would also be helpful for the decision makers to estimate
future minimum storage for the changing weather conditions with corresponding
return period and or demand.
45
Table 5: Expected minimum storage in Pedu reservoir for different return periods,
using Log-Pearson Type III Distribution
Return Period,
Frequency
Forecasted Storage
T (year)
Factor, KT
(MCM)
2
0.053
433.41
Mean,
5
0.853
579.62
= 2.629
10
1.242
667.79
20
1.549
746.67
Standard Deviation,
25
1.636
770.67
s = 0.158
40
1.805
819.47
50
1.880
841.91
Skewness Coefficient,
60
1.938
859.95
Cs = -0.320
75
2.007
881.66
85
2.044
893.69
100
2.091
909.14
Statistical Parameters
Figure 9: Reliability based Estimated Minimum Storage (MCM), Considering
Minimum Observed Storage for each Month during 1998-2008 and using LogPearson Type III Distribution
CHAPTER 6
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
6.1
Summary and Conclusions
Reservoirs are the most important elements of complex water resource
systems. The management and operation simulation techniques are developed for
experimentation in order to analyze the performance of the reservoir under changing
conditions. A stage-storage curve for Pedu reservoir is constructed using the mean
monthly observed values of storage and water level from 1998 through 2008, and the
equation of the curve was found to be WL=36.49S0.139
with coefficient of
determination, R2 at 0.99.
The reservoir simulation model storages are compared with the long-term
observed mean storage amounts. The comparison results are found to be satisfactory,
except for few months where the model storage amounts are found to be higher than
the observed storage.
The first order Markov model generated and observed mean storage amounts
were compared for each month. The comparison results show that the monthly
statistical parameters of the historic record, except the lag1 serial correlation between
December and January months are preserved satisfactorily. The results of lag1
correlation coefficients suggest that the model can satisfactorily be applied for future
47
storage prediction. Thus, this model was run to predict future storage amounts for
2009-2015, considering the mean values of the observed storages for each month
(1998-2008).
The expected mean and minimum storage amounts for different return
periods are estimated, using Log-Pearson Type III distribution and trendlines with
equations and R2 values are shown, to help decision makers to estimate future storage
with corresponding return period under any changing weather conditions and or
demand.
6.2
Recommendations
The following recommendations are suggested for enhancement of the
models to improve their applicability in real situations:
1.
The reservoir model storage capacities were computed on the basis
that the model storage would be at least equivalent to the observed
storage, as the inflows from the reservoir watershed area and from
Muda reservoir were not measured separately. A research can be
taken to estimate the runoff from the watershed area in order to
achieve more realistic results,
2.
The stochastic generation of the reservoir storage can be performed by
several other rigorous stochastic models and a comparison can be
made with the results of this study, and
3.
Demand needs to be computed to achieve realistic application of the
study results.
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APPENDICES
54
APPENDIX A
Reservoir Simulation Model Results
Table 6: Reservoir Simulation Model Outputs for Pedu Reservoir
Year Month
Obs.
Observed Observed
Obs.
Obs.
Obs.
Obs.
Obs.
Measured
Water
Storage
Storage
Release
P(t)
E(t)
Seepage
Spill
(Model)
Level
MCM
gain from
MCM
mm
mm
MCM
MCM
Storage
m MSL
Muda
MCM
MCM
Jan
96.84 1042.48
50.36
37.27 26.00 205.60
0.10
0.00 1042.48
Feb
97.27 1066.25
28.26
7.70 78.00 205.60
0.13
0.00 1066.25
Mar
95.67 970.34
25.63 296.50 54.00 232.90
0.10
0.00 970.34
Apr
90.51 693.41
15.52 238.72 208.00 197.00
0.05
0.00 693.41
May
86.81 510.09
21.78 153.24 198.70 156.70
0.06
0.00 510.09
1998 June
85.22 436.04
21.74
38.48 157.40 125.10
0.07
0.00 436.04
July
85.54 450.87
34.69
21.30 325.70 119.80
0.00
0.00 450.87
Aug
87.05 521.21
104.20
0.83 447.50 113.00
0.00
0.00 521.21
Sept
89.06 619.92
74.57
0.00 276.80 117.10
0.00
0.00 619.92
Oct
90.44 689.52
105.25
20.27 363.40 99.70
0.07
0.00 689.52
Nov
92.84 811.17
217.44
0.00 405.20 86.80
0.09
0.00 811.17
Dec
95.01 929.44
136.25
31.79 142.00 107.20
0.10
0.00 929.44
Jan
96.22 1001.83
85.86
0.00 133.40 185.30
0.10
0.00 1001.83
Feb
97.32 1069.07
49.05
4.68 27.10 206.00
0.10
1.34 1069.07
Mar
97.45 1076.13
73.08 133.87 320.50 155.00
0.10 12.37 1076.13
Apr
96.77 1037.89
84.54
66.44 118.70 143.80
0.09
0.00 1037.89
May
96.57 1024.36
122.35 160.90 204.20 156.10
0.10
0.00 1024.36
55
1999 June
94.68 910.84
62.06 212.32 148.80 142.20
0.09
0.00 910.84
July
94.45 896.96
78.26
18.97 129.90 133.10
0.10
0.00 896.96
Aug
95.46 954.99
68.05
18.50 130.70 149.60
0.11
0.00 954.99
Sept
95.73 971.27
86.35 120.25 231.70 137.60
0.11
0.00 971.27
Oct
96.24 1003.76
176.35
15.75 348.50 102.00
0.12
0.00 1003.76
Nov
97.43 1075.21
131.98
33.12 227.50 111.20
0.14
7.82 1075.21
Dec
98.10 1113.66
237.86
15.19 301.70 126.30
0.14 75.58 1113.66
Jan
97.82 1096.74
91.83
23.72 86.20 186.70
0.13 34.55 1096.74
Feb
97.61 1083.88
50.88
33.65 77.50 187.00
0.11
5.91 1083.88
Mar
97.05 1052.56
66.93 161.60 282.30 158.00
0.09
0.48 1052.56
Apr
97.29 1066.99
174.99
64.52 420.00 135.50
0.09
3.56 1066.99
May
96.96 1048.88
117.23 149.74 219.40 148.70
0.11
0.00 1048.88
2000 June
96.85 1043.22
71.00
75.37 90.90 129.00
0.11
0.00 1043.22
July
97.04 1054.24
45.91
54.23 83.70 151.40
0.00
0.00 1054.24
Aug
97.18 1061.55
72.23
81.30 289.00 127.00
0.13
0.00 1061.55
Sept
96.27 1004.73
70.21 119.15 318.50 132.40
0.14
0.00 1004.73
Oct
97.09 1056.90
83.95
72.03 225.90 119.90
0.15
0.00 1056.90
Nov
96.91 1045.33
175.61 106.24 389.50 112.40
0.15
0.00 1045.33
Dec
97.10 1056.75
84.82 100.23 110.50 151.20
0.13
0.00 1056.75
Jan
97.43 1073.88
127.39
67.96 199.10 135.60
0.10
0.57 1073.88
Feb
97.36 1070.77
56.63
60.14 19.50 188.00
0.09
0.00 1070.77
Mar
97.39 1072.01
56.40 125.23 259.20 139.70
0.08 0.81 1072.01
Apr
96.74 1035.72
58.99
99.13 386.00 160.71
0.08
0.00 1035.72
May
96.40 1013.70
53.53 170.06 275.20 151.68
0.10
0.00 1013.70
2001 June
94.57 903.33
54.83 130.65 175.20 148.30
0.09
0.00 903.33
July
94.21 882.81
43.22
78.89 194.00 150.00
0.10
0.00 882.81
Aug
94.52 900.66
41.05
28.22 178.70 152.60
0.10
0.00 900.66
Sept
94.28 886.78
45.74
98.92 346.40 136.40
0.09
0.00 886.78
Oct
94.46 897.15
141.74
20.84 433.00 108.50
0.09
0.00 897.15
Nov
96.06 991.21
105.25
88.19 191.60 140.35
0.10
0.00 991.21
56
Dec
94.81 917.55
74.93 160.84 195.90 146.80
0.09
0.00 917.55
Jan
94.58 904.05
44.83
64.24 3.50 203.10
0.06
0.00 904.05
Feb
94.90 922.16
24.43
13.77 1.50 260.40
0.06
0.00 922.16
Mar
95.08 932.97
20.82
54.90 129.00 275.90
0.07
0.00 932.97
Apr
92.25 781.53
28.23 250.98 164.70 180.20
0.07
0.00 781.53
May
90.45 690.01
25.13 173.90 139.00 180.40
0.09
0.00 690.01
2002 June
86.97 517.56
24.39 155.23 131.00 146.90
0.09
0.00 517.56
July
85.48 447.93
23.23
39.58 201.90 135.60
0.11
0.00 447.93
Aug
85.62 454.44
24.67
0.00 174.70 124.10
0.11
0.00 454.44
Sept
86.21 481.94
44.64
0.00 243.50 137.20
0.12
0.00 481.94
Oct
88.05 569.30
145.34
18.08 498.70 127.70
0.13
0.00 569.30
Nov
89.83 658.42
89.62
23.21 219.10 113.90
0.13
0.00 658.42
Dec
90.15 674.62
68.49 138.09 125.10 161.70
0.12
0.00 674.62
Jan
90.29 682.15
40.54
0.00 5.00 203.00
0.10
0.00 682.15
Feb
90.82 708.77
19.01
7.63 1.50 231.50
0.09
0.00 708.77
Mar
90.44 689.72
28.08
90.03 263.40 231.80
0.09
0.00 689.72
Apr
88.82 607.39
28.20 168.04 115.50 183.50
0.08
0.00 607.39
May
86.67 503.26
41.25 172.50 237.00 139.20
0.10
0.00 503.26
2003 June
83.31 357.19
0.00 141.89 155.90 140.10
0.11
0.00 357.19
July
82.91 341.53
1.28
19.47 287.30 124.60
0.11
0.00 341.53
Aug
84.05 385.71
0.00
8.53 164.00 121.90
0.11
0.00 385.71
Sept
84.73 413.51
0.00
11.13 142.50 121.20
0.11
0.00 413.51
Oct
86.75 507.49
193.50
2.98 541.60 114.70
0.13
0.00 507.49
Nov
89.46 639.89
84.96
17.32 53.70 154.80
0.13
0.00 639.89
Dec
89.88 661.37
75.71 127.98 67.80 184.00
0.11
0.00 661.37
Jan
89.69 651.53
32.15
61.44 7.00 243.30
0.08
0.00 651.53
Feb
89.80 657.01
24.60
22.01 42.20 222.60
0.07
0.00 657.01
Mar
89.31 632.58
28.12 128.69 180.20 209.77
0.08
0.00 632.58
Apr
86.26 484.20
19.70 171.71 296.70 159.20
0.07
0.00 484.20
May
85.16 434.24
33.09 140.46 98.20 145.90
0.10
0.00 434.24
57
2004 June
82.35 320.40
0.00
91.62 142.00 124.00
0.10
0.00 320.40
July
82.40 322.52
3.29
5.99 212.90 141.00
0.11
0.00 322.52
Aug
83.60 367.55
0.31
0.00 233.50 128.70
0.11
0.00 367.55
Sept
85.15 433.41
86.19
0.00 373.50 108.00
0.11
0.00 433.41
Oct
87.62 548.11
130.28
15.51 209.20 119.70
0.12
0.00 548.11
Nov
88.91 611.90
77.85
80.65 58.20 173.70
0.12
0.00 611.90
Dec
88.00 566.13
91.33 125.90 112.60 198.10
0.13
0.00 566.13
Jan
88.10 571.33
33.38
37.16 0.60 228.10
0.13
0.00 571.33
Feb
88.29 580.71
23.76
18.70 73.00 231.10
0.12
0.00 580.71
Mar
88.16 574.36
22.99
42.56 90.30 240.30
0.14
0.00 574.36
Apr
86.64 501.85
34.57 182.95 192.10 206.50
0.13
0.00 501.85
May
83.25 354.45
1.30 136.64 199.00 148.80
0.14
0.00 354.45
2005 June
82.34 320.05
0.00 114.84 100.10 147.10
0.14
0.00 320.05
July
79.96 241.30
0.00
89.47 94.70 146.20
0.15
0.00 241.30
Aug
79.32 222.50
0.93
31.19 106.50 148.00
0.16
0.00 222.50
Sept
79.57 229.49
1.79
24.34 247.50 125.20
0.15
0.00 229.49
Oct
81.02 275.28
17.52
0.00 430.90 115.00
0.16
0.00 275.28
Nov
83.73 373.62
37.74
0.00 244.20 91.76
0.15
0.00 373.62
Dec
87.35 536.33
274.98
0.00 434.10 114.70
0.16
0.00 536.33
Jan
90.22 678.54
89.41
0.00 4.00 202.50
0.14
0.00 678.54
Feb
91.85 760.85
55.66
0.00 55.00 208.50
0.18
0.00 760.85
Mar
92.45 791.02
56.16
90.88 284.00 247.90
0.15
0.00 791.02
Apr
91.40 738.00
65.38 104.88 224.20 177.10
0.10
0.00 738.00
May
91.55 745.64
80.93
57.67 253.40 145.60
0.12
0.00 745.64
2006 June
91.39 737.51
73.69 106.22 186.80 126.50
0.12
0.00 737.51
July
91.68 752.14
53.54
52.98 167.00 133.20
0.12
0.00 752.14
Aug
92.26 781.42
34.96
36.24 160.50 142.80
0.12
0.00 781.42
Sept
91.99 767.96
50.86
63.43 240.70 132.20
0.12
0.00 767.96
Oct
92.49 793.26
119.96
25.91 275.50 142.10
0.15
0.00 793.26
Nov
93.86 863.45
79.82
2.21 98.20 157.70
0.18
0.00 863.45
58
Dec
93.88 864.37
55.74 119.15 89.50 162.70
0.21
0.00 864.37
Jan
94.09 876.01
92.01
32.19 152.00 186.00
0.24
0.00 876.01
Feb
94.80 916.53
40.98
6.85 157.50 210.00
0.24
0.00 916.53
Mar
95.23 941.30
31.90
72.11 72.00 198.00
0.13
0.00 941.30
Apr
93.80 860.39
46.44 121.64 316.90 189.40
0.11
0.00 860.39
May
93.34 836.29
68.48 111.59 251.20 159.90
0.15
0.00 836.29
2007 June
93.34 836.29
91.47
25.86 566.90 139.30
0.15
0.00 836.29
July
94.11 876.95
55.85
40.39 290.30 135.90
0.16
0.00 876.95
Aug
94.92 923.45
42.85
12.13 102.70 146.70
0.17
0.00 923.45
Sept
95.72 970.08
66.65
25.62 270.20 120.20
0.18
0.00 970.08
Oct
96.61 1026.41
85.35
8.59 308.00 129.00
0.20
0.00 1026.41
Nov
97.53 1079.49
116.13
40.93 169.70 126.30
0.19
4.82 1079.49
Dec
96.85 1041.47
187.50 142.13 294.70 135.60
0.20
0.00 1041.47
Jan
96.82 1040.98
70.43 110.51 24.00 189.50
0.17
0.00 1040.98
Feb
96.92 1047.45
61.12
34.78 158.00 199.90
0.15
0.00 1047.45
Mar
97.11 1056.87
55.96 124.74 187.00 185.00
0.14
0.00 1056.87
Apr
95.79 1056.87
70.17 115.08 437.00 184.30
0.12
0.00 1056.87
May
95.62 964.15
57.51 158.10 162.00 144.00
0.11
0.00 964.15
2008 June
94.82 917.71
68.31 102.73 244.50 129.80
0.15
0.00 917.71
July
94.61 905.49
68.08
68.66 156.70 138.20
0.17
0.00 905.49
Aug
95.54 959.60
56.06
10.57 325.50 125.00
0.18
0.00 959.60
Sept
96.83 1040.86
98.79
48.63 241.50 126.70
0.18
0.00 1040.86
Oct
97.30 1076.35
131.53
13.37 506.10 156.00
0.19 16.77 1076.35
Nov
96.34 1079.69
54.61
89.05 144.50 146.50
0.18 12.01 1079.69
Dec
95.74 1032.64
69.99 183.20 78.50 138.50
0.11
0.00 1032.64
59
APPENDIX B
Pedu Storage Generation Using Markov Model
Table 7: Step-by-step procedure of calculating model storage for Pedu reservoir for
March, using natural logarithm of observed storage (MCM)
Year
Month
1998
1999
2000
2001
Jan
2002
2003
2004
2005
2006
2007
2008
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Feb
LN of
Obs.
Storage
Std.
Dev.,
s
Lag1
Q
Mean
Q
Lag1
Mean
Q
Lag1
Cor.
Coeff.
Student’s
tdistribution
with (n-1)
degrees of
freedom
Lag1
Std.
Dev.,
s
Model
Storage,
Xi,j+1
6.984 0.063 7.002 6.787 6.831 0.000
6.314 0.012
7.182
6.910 0.039 7.002 6.787 6.831 0.000
2.920 0.044
6.900
6.979 0.061 7.002 6.787 6.831 0.000
2.353 0.069
6.930
7.019 0.074 7.002 6.787 6.831 0.000
2.132 0.048
6.943
6.876 0.028 7.002 6.787 6.831 0.000
2.015 0.041
6.843
6.525 0.083 7.002 6.787 6.831 0.000
1.943 0.050
6.947
6.569 0.069 7.002 6.787 6.831 0.000
1.895 0.092
6.917
6.411 0.119 7.002 6.787 6.831 0.000
1.860 0.173
7.008
6.520 0.084 7.002 6.787 6.831 0.000
1.833 0.019
6.941
6.811 0.008 7.002 6.787 6.831 0.000
1.812 0.078
6.801
7.049 0.083 7.002 6.787 6.831 0.000
1.796 0.086
6.936
6.964 0.054 6.984 6.792 6.787 0.976
6.314 0.063
7.034
6.961 0.054 6.910 6.792 6.787 0.976
2.920 0.039
6.991
6.988 0.062 6.979 6.792 6.787 0.976
2.353 0.061
7.015
7.012 0.070 7.019 6.792 6.787 0.976
2.132 0.074
7.039
6.842 0.016 6.876 6.792 6.787 0.976
2.015 0.028
6.847
6.574 0.069 6.525 6.792 6.787 0.976
1.943 0.083
6.608
6.521 0.086 6.569 6.792 6.787 0.976
1.895 0.069
6.562
6.396 0.125 6.411 6.792 6.787 0.976
1.860 0.119
6.456
6.634 0.050 6.520 6.792 6.787 0.976
1.833 0.084
6.658
6.828 0.012 6.811 6.792 6.787 0.976
1.812 0.008
6.832
60
2008
1998
1999
2000
2001
Mar
2002
2003
2004
2005
2006
2007
2008
1998
1999
2000
2001
Apr
2002
2003
2004
2005
2006
2007
2008
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
May
6.987 0.062 7.049 6.792 6.787 0.976
1.796 0.083
7.006
7.144 0.081 6.964 6.889 6.792 0.984
6.314 0.054
7.231
7.077 0.059 6.961 6.889 6.792 0.984
2.920 0.054
7.105
7.100 0.067 6.988 6.889 6.792 0.984
2.353 0.062
7.124
7.069 0.057 7.012 6.889 6.792 0.984
2.132 0.070
7.088
6.896 0.002 6.842 6.889 6.792 0.984
2.015 0.016
6.896
6.659 0.073 6.574 6.889 6.792 0.984
1.943 0.069
6.688
6.635 0.080 6.521 6.889 6.792 0.984
1.895 0.086
6.666
6.425 0.147 6.396 6.889 6.792 0.984
1.860 0.125
6.481
6.782 0.034 6.634 6.889 6.792 0.984
1.833 0.050
6.795
6.921 0.010 6.828 6.889 6.792 0.984
1.812 0.012
6.924
7.069 0.057 6.987 6.889 6.792 0.984
1.796 0.062
7.084
6.837 0.001 7.144 6.835 6.889 0.891
6.314 0.081
6.839
7.007 0.054 7.077 6.835 6.889 0.891
2.920 0.059
7.060
7.016 0.057 7.100 6.835 6.889 0.891
2.353 0.067
7.072
7.034 0.063 7.069 6.835 6.889 0.891
2.132 0.057
7.080
6.940 0.033 6.896 6.835 6.889 0.891
2.015 0.002
6.960
6.653 0.058 6.659 6.835 6.889 0.891
1.943 0.073
6.726
6.486 0.111 6.635 6.835 6.889 0.891
1.895 0.080
6.621
6.529 0.097 6.425 6.835 6.889 0.891
1.860 0.147
6.646
6.737 0.031 6.782 6.835 6.889 0.891
1.833 0.034
6.774
6.890 0.017 6.921 6.835 6.889 0.891
1.812 0.010
6.898
7.061 0.071 7.069 6.835 6.889 0.891
1.796 0.057
7.095
6.497 0.077 6.837 6.739 6.835 0.931
6.314 0.001
7.141
7.078 0.107 7.007 6.739 6.835 0.931
2.920 0.054
7.168
7.089 0.111 7.016 6.739 6.835 0.931
2.353 0.057
7.160
7.076 0.107 7.034 6.739 6.835 0.931
2.132 0.063
7.136
6.761 0.007 6.940 6.739 6.835 0.931
2.015 0.033
6.765
6.516 0.071 6.653 6.739 6.835 0.931
1.943 0.058
6.581
6.354 0.122 6.486 6.739 6.835 0.931
1.895 0.111
6.464
6.197 0.172 6.529 6.739 6.835 0.931
1.860 0.097
6.350
6.689 0.016 6.737 6.739 6.835 0.931
1.833 0.031
6.703
6.854 0.036 6.890 6.739 6.835 0.931
1.812 0.017
6.870
61
2008
1998
1999
2000
2001
June
2002
2003
2004
2005
2006
2007
2008
1998
1999
2000
2001
July
2002
2003
2004
2005
2006
2007
2008
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Aug
7.023 0.090 7.061 6.739 6.835 0.931
1.796 0.071
7.062
6.162 0.132 6.497 6.581 6.739 0.962
6.314 0.077
6.408
7.024 0.140 7.078 6.581 6.739 0.962
2.920 0.107
7.119
7.020 0.139 7.089 6.581 6.739 0.962
2.353 0.111
7.093
6.941 0.114 7.076 6.581 6.739 0.962
2.132 0.107
6.994
6.511 0.022 6.761 6.581 6.739 0.962
2.015 0.007
6.660
6.213 0.116 6.516 6.581 6.739 0.962
1.943 0.071
6.289
6.021 0.177 6.354 6.581 6.739 0.962
1.895 0.122
6.135
6.075 0.160 6.197 6.581 6.739 0.962
1.860 0.172
6.176
6.738 0.050 6.689 6.581 6.739 0.962
1.833 0.016
6.455
6.759 0.056 6.854 6.581 6.739 0.962
1.812 0.036
6.781
6.928 0.110 7.023 6.581 6.739 0.962
1.796 0.090
6.969
6.157 0.092 6.162 6.447 6.581 0.968
6.314 0.132
6.311
6.820 0.118 7.024 6.447 6.581 0.968
2.920 0.140
6.894
7.007 0.177 7.020 6.447 6.581 0.968
2.353 0.139
7.093
6.869 0.133 6.941 6.447 6.581 0.968
2.132 0.114
6.926
6.189 0.082 6.511 6.447 6.581 0.968
2.015 0.022
6.239
5.889 0.177 6.213 6.447 6.581 0.968
1.943 0.116
5.992
5.795 0.206 6.021 6.447 6.581 0.968
1.895 0.177
5.913
5.801 0.204 6.075 6.447 6.581 0.968
1.860 0.160
5.917
6.691 0.077 6.738 6.447 6.581 0.968
1.833 0.050
6.719
6.821 0.118 6.759 6.447 6.581 0.968
1.812 0.056
6.863
6.882 0.137 6.928 6.447 6.581 0.968
1.796 0.110
6.929
6.258 0.061 6.157 6.452 6.447 0.978
6.314 0.092
6.342
6.881 0.136 6.820 6.452 6.447 0.978
2.920 0.118
6.953
7.031 0.183 7.007 6.452 6.447 0.978
2.353 0.177
7.108
6.834 0.121 6.869 6.452 6.447 0.978
2.132 0.133
6.879
6.119 0.105 6.189 6.452 6.447 0.978
2.015 0.082
6.170
5.977 0.150 5.889 6.452 6.447 0.978
1.943 0.177
6.047
5.907 0.172 5.795 6.452 6.447 0.978
1.895 0.206
5.986
5.536 0.290 5.801 6.452 6.447 0.978
1.860 0.204
5.667
6.706 0.081 6.691 6.452 6.447 0.978
1.833 0.077
6.731
6.841 0.123 6.821 6.452 6.447 0.978
1.812 0.118
6.879
62
2008
1998
1999
2000
2001
Sept
2002
2003
2004
2005
2006
2007
2008
1998
1999
2000
2001
Oct
2002
2003
2004
2005
2006
2007
2008
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Nov
6.877 0.135 6.882 6.452 6.447 0.978
1.796 0.137
6.918
6.430 0.031 6.258 6.527 6.452 0.993
6.314 0.061
6.454
6.995 0.148 6.881 6.527 6.452 0.993
2.920 0.136
7.044
7.025 0.157 7.031 6.527 6.452 0.993
2.353 0.183
7.065
6.893 0.116 6.834 6.527 6.452 0.993
2.132 0.121
6.920
6.178 0.111 6.119 6.527 6.452 0.993
2.015 0.105
6.207
6.051 0.151 5.977 6.527 6.452 0.993
1.943 0.150
6.090
6.072 0.144 5.907 6.527 6.452 0.993
1.895 0.172
6.108
5.537 0.313 5.536 6.527 6.452 0.993
1.860 0.290
5.614
6.723 0.062 6.706 6.527 6.452 0.993
1.833 0.081
6.735
6.903 0.119 6.841 6.527 6.452 0.993
1.812 0.123
6.927
6.993 0.147 6.877 6.527 6.452 0.993
1.796 0.135
7.022
6.565 0.009 6.430 6.593 6.527 0.983
6.314 0.031
6.575
6.927 0.106 6.995 6.593 6.527 0.983
2.920 0.148
6.977
7.023 0.136 7.025 6.593 6.527 0.983
2.353 0.157
7.074
6.822 0.073 6.893 6.593 6.527 0.983
2.132 0.116
6.846
6.376 0.069 6.178 6.593 6.527 0.983
2.015 0.111
6.404
6.235 0.113 6.051 6.593 6.527 0.983
1.943 0.151
6.281
6.334 0.082 6.072 6.593 6.527 0.983
1.895 0.144
6.367
5.618 0.308 5.537 6.593 6.527 0.983
1.860 0.313
5.738
6.708 0.037 6.723 6.593 6.527 0.983
1.833 0.062
6.719
6.942 0.111 6.903 6.593 6.527 0.983
1.812 0.119
6.973
6.969 0.119 6.993 6.593 6.527 0.983
1.796 0.147
7.002
6.698 0.009 6.565 6.726 6.593 0.994
6.314 0.009
6.705
6.988 0.083 6.927 6.726 6.593 0.994
2.920 0.106
7.012
7.049 0.102 7.023 6.726 6.593 0.994
2.353 0.136
7.073
6.984 0.081 6.822 6.726 6.593 0.994
2.132 0.073
7.001
6.524 0.064 6.376 6.726 6.593 0.994
2.015 0.069
6.539
6.488 0.075 6.235 6.726 6.593 0.994
1.943 0.113
6.505
6.540 0.059 6.334 6.726 6.593 0.994
1.895 0.082
6.553
5.923 0.254 5.618 6.726 6.593 0.994
1.860 0.308
5.978
6.763 0.012 6.708 6.726 6.593 0.994
1.833 0.037
6.766
6.995 0.085 6.942 6.726 6.593 0.994
1.812 0.111
7.010
63
2008
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Dec
7.038 0.098 6.969 6.726 6.593 0.994
1.796 0.119
7.055
6.868 0.012 6.698 6.831 6.726 0.970
6.314 0.009
6.813
6.971 0.044 6.988 6.831 6.726 0.970
2.920 0.083
6.998
7.048 0.069 7.049 6.831 6.726 0.970
2.353 0.102
7.081
6.983 0.048 6.984 6.831 6.726 0.970
2.132 0.081
7.004
6.700 0.041 6.524 6.831 6.726 0.970
2.015 0.064
6.725
6.671 0.050 6.488 6.831 6.726 0.970
1.943 0.075
6.700
6.540 0.092 6.540 6.831 6.726 0.970
1.895 0.059
6.591
6.285 0.173 5.923 6.831 6.726 0.970
1.860 0.254
6.380
6.891 0.019 6.763 6.831 6.726 0.970
1.833 0.012
6.898
7.076 0.078 6.995 6.831 6.726 0.970
1.812 0.085
7.103
7.103 0.086 7.038 6.831 6.726 0.970
1.796 0.098
7.133