Application of Goal Programming

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Transcript of Application of Goal Programming

for Optimal Reservoir Operations
Virginia Polytechnic Institute and State University
in partial fulfillment ·of the requirements for the degree of
Master of Science
May, 1988
Blacksburg, Virginia
for Optimal Reservoir Operations
Civil Engineering
The optimal reservoir operations problem consists of obtaining releases, storages
of a reservoir and downstream reach routed flows such that benefits derived from
operating the reservoir are maximized. These are obtained on the basis of forecasted
inflows to the reservoir, and forecasted precipitation in the downstream reaches.
Five goal programming schemes, namely (i) preemptive goal programming (ii)
weighted goal programming (iii) minmax goal programming (iv) fuzzy goal program-
ming and (v) interval goal programming are considered .. The reservoir operation
problem is also formulated as a multiobjective linear program (MOLP). It is shown
that the optimal solutions of the goal programs are contained among the efficient
points of the MOLP. It is also shown that the min max and fuzzy goal programs can
yield inefficient points as optima; however, there exist alternate optima to these pro-
grams which are efficient. Therefore, it is suggested that one should solve an MOLP
for considering alternative efficient solutions. These techniques are applied to the
Green River basin system in Kentucky.
Acknowledgements
I would like to take this opportunity to express my sincere gratitude to my advisor,
Dr. G. V. Loganathan, without whose guidance this thesis could not have been a re-
ality. His constant encouragement kept me going when everything seemed bleak.
The training he has provided will stand me in good stead in my future pursuits.
I would also like to express my thanks to Dr. C. Y. Kuo and Dr. S. A. Ardekani for re-
viewing the manuscript. Their constructive criticisms and helpful suggestions have
gone a long way in improving the quality of this thesis.
My thanks are also due to all my friends at Virginia Tech who have helped me from
time to time. In particular, I would like to thank my roommates, during my stay in
Blacksburg, for keeping my spirits up during times of extreme distress.
Lastly, I would like to thank my parents whose moral support has been a continuing
source of encouragement in pursuing my goals throughout my academic career.
Acknowledgements iii
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Ru:n Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Mathematical Programming ........................................ 10
2. 7 Fuzzy Goal Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 Interval Goal Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Multiobjective linear programming (MOLP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.10 Application of Goal Programming for Reservoir Operations . . . . . . . . . . . . . . . . . . 39
Table of Contents iv
2.11 System Constraints ........................... ; . . . . . . . . . . . . . . . . . . . . 40
2.12 Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1 Routing Model of the Green River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.2 Storage Elevation Curves. . ........................................ 47
3.1.3 Penalty Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 47
3.3 Real Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Goal Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Summary ..................... -, ...... , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References .......• I •••• -· •••••• I •• I • I • I I I II •••••• I I •• - •••• I • I • I •••• I •• I e 7 4
The program AD BASE ... • .............. i •••• , ••••.•••••••••••••• :;; • • • • • • • • • 76
A.1.1.1 Execution control. parameters ............................. 77
A.1.1.2 1/0 Control Parameters ........•......................... 78
A.1.1.3 · ADBASE Input Format ................................... 79
A.1.1.4 Versatility of ADBASE ................................... 83
A.1.1.5 Sample Problem and Output .............................. 86
V·ita ....•......•.•....••••..••..••....•........•....................... 94
Figure 1. Development of a Rule Curve (Kuiper, 1965) ................... 9
Figure 2. A Single Reservoir System ................................ 16
Figure 3. Systems Model for the Green River Basin (Can et al., 1982) ........ 45
List of Illustrations vi
Table 1. Storage Penalties for Green and Nolin Reservoirs ................ 55
Table 2. Storage Penalties for Barren and Rough Reservoirs .............. 56
Table 3. Flow Penalties ........................................... 57
Table 4. Deviation Parameters for Fuzzy Goal Program .................. 59
Table 5; Input Parameters ......................................... 60
Table 6. Expected Rain ........................................... 61
Table 7. Reach Routing Equations(Can et al., 1982) ...................... 62
Table 8. Optimal Solutions ........................................ 64
Table 9. Targets ..........•..................................... 65
Table 11. Alternative Optimal Solutions for Min-Max formulation ............ 67
Table 12. Deviations for Min-Max GP using LINDO and ADBASE ............. 68
Table 13. Efficient Solutions ........................................ 69
Table 14. Parameters to be set in XTERNA and ADBASE data deck .......... 85
List of Tables vii
1.1 Introduction
The objective of constructing a water storage and diversion project is to augment
and/or control the natural flow of water in satisfying various purposes such as do-
mestic and industrial use, irrigation, navigation, hydropower production, low flow
augmentation and water quality. These projects generally have a number of stages
of development. Because of the complexity involved in design, construction, and
subsequently operation and maintenance, these systems are subject to detailed
analysis at every stage. Their environmental and economic impact is immense. A
typical storage and diversion project (Goodman, 1984; p.133)
• Disturbs natural state of an area. It changes land areas to water areas, also changes wild rivers to controlled rivers.
• Modifies stream flows downstream of reservoir; reduces flood peaks; reduces or raises minimum flows; may introduce abnormal and variableflows, such as the flows from penstocks during peak demand hours for power which can raise the water level by several feet downstream in a short period.
• Changes physical, chemical and bacteriological characteristics of water down- stream. Also, the groundwater level increases.
Reservoir Operations: An Overview 1
• Modifies types and quantities of fish and wildlife production.
• Alters erosion, sediment production, navigation pattern in the downstream channel and could lead to erosion of banks.
• Alters population and economic, social and political life of the area with associ- ated environmental impacts.
Like most water resources systems, the water storage systems involve the following
three subproblems: (1) Planning (2) Design and (3) Operations. Planning involves es-
timating engineering and economic objectives of a system which depend on the fol-
lowing data (Goodman, 1984; pp. 143-148) :
• Economic data showing the population distribution and the main occupation(s)
of the people in the region.
This factor determines whether a project will be useful to the people of a region
or not. Although this is the primary reason for building water storage facilities
other pertinent questions like environmental impact downstream of the storage
facility should be considered. The final decision should be made with regard to
the total economic and environmental impact of the project.
• Streamflow data showing all the relevant rainfall, runoff data pertaining to the
region.
This helps in determining the location and size of a water storage facility. The
rule curves which are plots of reservoir storage versus time are developed on the
basis of historically recorded low flows.
• Geographical maps of the basin showing streams, topography, soils and
geologic formations.
Reservoir Operations: An Overview 2
These maps are usually obtained from aerial photographs which are used for
agricultural soil classification and geologic interpretation. In addition satellite
photographs are also used to gain a perspective of the site. The dam site and the
material to build the dam are selected on the basis of the geology and climate
of a region. For example reinforced concrete arch dams are usually located in a
gorge, because of the best side support provided by hard rocks .
• Economic base projection determining the estimate of the future growth of the
population and the economy both in the near and long term future.
The forecasted population growth determines the increase in economic activity
for a particular region. In accordance with this economic base projection water
requirements for domestic and industrial use, irrigation, navigation, power, flood
control can be predicted.
Finally the above projections are combined into a development plan. Usually com-
parisons are made among the various alternative sites that a planner has in mind
(Goodman, 1984; p.130). An alternative must pass five feasibility tests namely engi-
neering, economic, financial, political and social. Engineering feasibility is ascer-
tained through design analysis which include structural stability, adequacy of
foundations, sources and amounts of construction material. Economic and other·
feasibilities are contingent on engineering feasibility because a project incapable of
producing the desired output is not going to produce the benefits needed for its jus-
tification from a purely economic standpoint. The alternatives are in general evalu-
ated in terms of benefit and cost which are defined in the sequel. Since all the
alternatives do not yield similar types of benefits, it is not always possible to evaluate
Reservoir Operations: An Overview 3
,-
these alternatives on an equitable basis. According to James and Lee (1971; pp.
165-168), these benefits and costs may be listed as
1. Primary Benefits are the benefits which accrue from the physical effects of the
project on the user.
• Direct benefits include, reduction in physical damage to items coming iri
contact with flood water, increase in farm income resulting from application
of i:"rigation water, the value consumers received from the use of electric
power, and savings in transportation costs for goods moved by navigation.
• Land enhancement benefits which result when a more productive land use
is made possible by a reliable water supply and could be distinguished from
direct benefits to the land use. For example, land enhancement benefits
equal the net crop income from the higher value crop with flood protection
less the net crop income from lower value crops with flood protection.
• Indirect benefits result as individuals realize the economic consequences of
technological external effects. In other words output intended for one pur-
pose ( low flow augmentation ) may achieve other beneficial effects ( navi-
gation ).
2. Secondary benefits denote the value added to activities influenced by the
project through economic rather than technological linkages. These include :
• Secondary benefits resulting from forward production linkages that increase
the net income of those who process project output. This is called the net
Reservoir Operations: An Overview 4
stemming-from benefit which is the income from processing project output
minus the sum of the income which would be obtained from processing out-
put displaced by the project and output which would have been obtained from
an an alternate investment by using the money spent on the project.
• Benefits which may result from increases in the net income of those who
provide goods and services to the project area. Cotton produced by an irri-
gation project will require the purchase of farm machinery, fertilizer which
will set up a chain reaction of economic activity. This is usually called the
induced by benefits.
• Employment benefits denote the economic value gained from the increased
employment opportunity from new jobs created to construct, maintain or op-
ernte the project.
• Public benefits are realized in achievement of goals other than economic
efficiency.The construction of water storage facilities leads to a dependable
water supply which in turn enables farmers to rotate crops and thereby re-
ducing the variability of annual income of farmers who are served by that
particular project. Economically backward regions are usually uplifted by
water supply projects which can enhance industrial as well as agricultural
production of a region. As for an example, the Rajasthan Canal in western
India has transformed a desert region into a region of vast agricultural land.
• Intangible benefits are the benefits which cannot be measured in terms of
monetary units. These include the effects on human beings physically,
through loss of health or life, which cannot be measured in terms of monetary
Reservoir Operations: An Overview 5
units. Similarly the loss of flora and fauna cannot be measured in terms of
monetary units. As for an example, the silent valley project in southern India
has .potentials of eradicating rare species of flora and fauna by construction
of a large dam.
The costs included in the calculation of benefit - cost ratio are :
• The project installation costs which include cost of construction, engineering
and administration and right of way costs and the cost of relocating facilities.
Cost of construction is the payment made to the contracting agency. Engineering
and administration costs is the amount paid to the consulting agency for doing
engineering analysis and inspecting construction sites. Right of way cost is the
opportunity cost of using the land required for project installation and mainte-
nance. The cost of relocating facilities is the amount required to move or modify
bridges, roads, railroads located on the project right of way.
• After installation, the project has continuing costs of operation, maintenance, and
replacement and other activities required to produce project output on a contin-
uing bc.isis. Operation includes the opening and closing of gates, overseeing hy-
droelectric plants, purchasing power for pumping. Maintenance includes
preventive maintenance to reduce anticipated breakdowns and repairs.
• Associated costs include the investment necessary to utilize project output. In-
duced costs are evaluated from an analysis of the nature and severity of partic-
ular adverse consequences. They should be evaluated as the smaller of the
value of the inflicted damage or the cost of damage prevention measures.
Reservoir Operations: An Overview 6
The traditional design of reservoir systems involve mainly the structural and
earthwork design. The usual storage structures are dams and reservoirs. The type
of dam selected for a project depends on site conditions, hydraulic conditions and
climatic effects. Stability and stress analyses are performed to ensure the safety of
dams (Morris and Wiggert, 1971; pp. 228-253). The operations procedure involves the
determination of storage levels and water releases from reservoirs based on the
given capacity, safe releases and hydraulic constraints. A long term release policy
is important in meeting the irrigation, domestic, industrial, hydropower, recreational,
navigational, instream flow and water quality demands. Such a release policy is im-
portant for a smooth operation, especially in view of the fluctuating natural
streamflows. Day to day real time operations of a reservoir involves short term
forecasts of precipitation and/or stream flows. Such an ability is extremely important
in the operation of spillway gates in flood situations. Short term forecasts of
streamflows also help in the conjunctive operation of navigational locks and diversion
of flows to penstocks for power production.
It is the operation of these water storage facilities of particular concern in this study.
The conventional operation of reservoirs primarily depends on a rule curve which is
a graph of reservoir storage versus time. A procedure for the development of a rule
curve is discussed in the section on regulation of reservoirs. Subsequently it is
shown how mathematical programming offers another avenue for development of
reservoir regulation procedures. This section also provides comments on some of
the limitations of rule curves.
Reservoir Operations: An Overview 7
1.2 Regulation of Reservoirs
1.2.1 Rule Curves
The conventional method of operating a reservoir on a day to day basis is with the
help of a rule curve. Rule curves are diagrams indicating storage requirements over
a time period. For a simple rule curve, consider the supply and demand curves
drawn on the same scale. Then the deficit at any time represents the amount of re-
lease from a reservoir and determines the amount of storage that should be main-
tained at that time. Figure 1 shows such a rule curve.
Referring to Figure 1, let the reservoir be used for water supply purposes. During
normal periods of river flow, the reservoir will be maintained at the normal pool level.
In general, the zone above the normal pool level is maintained for flood control. The
place between the normal and dead storage zones is called the active zone. Water
from the active zone is released based on the available volume and the expected in-
flows by using a rule curve. The maximum pool level may be maintained as long
as inflow does not exceed flood magnitude. Whenev_er inflow falls below the specified
yield, water is released to cover the deficiency. Referring to Figure 1, if the mass
curve of outlet release is shifted upwards, so that it becomes tangent to the mass
curve at a point where the reservoir is assumed to be full, then the difference be-
tween two curves yields the storage or deficit at that time depending upon whether
the outflow curve lies below or above the inflow curve, By plotting these differences
as shown in Figure 1-d, one obtains a rule curve.
This rule curve is designed to ensure that there will be adequate storage capacity
available to accommodate the design flood. This procedure can be applied for several
Reservoir Operations: An Overview 8
Maximum
(b)
(di
Reservoir Operations: An Overview
I I I
9
alternative demand curves, and the corresponding rule curves can be…