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Water Supply System Management Designand Optimization under Uncertainty
DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY WITH A MAJOR IN CIVIL ENGINEERING
In the Graduate College
THE UNIVERSITY OF ARIZONA
2007
2
THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation
prepared by GUNHUI CHUNG
entitled WATER SUPPLY SYSTEM MANAGEMENT DESIGN AND
OPTIMIZATION UNDER UNCERTAINTY
and recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of DOCTOR OF PHILOSOPHY
_______________________________________________________________________ Dr. Kevin Lansey Date: December 4, 2006 _______________________________________________________________________ Dr. Juan Valdes Date: December 4, 2006 _______________________________________________________________________ Dr. Larry W. Mays Date: December 4, 2006 _______________________________________________________________________ Dr. Donald R. Davis Date: December 4, 2006 _______________________________________________________________________ Dr. Guzin Bayraksan Date: December 4, 2006 _______________________________________________________________________ Dr. Bart Nijssen Date: December 4, 2006 Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ________________________________________________ Date: December 4, 2006 Dissertation Director: Dr. Kevin Lansey
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced
degree at The University of Arizona and is deposited in the University Library to be made
available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided
that accurate acknowledgment of source is made. Requests for permission for extended
quotation from or reproduction of this manuscript in whole or in part may be granted by the
copyright holder.
SIGNED: Gunhui Chung
4
ACKNOWLEDGEMENTS
I would like to express my gratitude to all those who help me to complete this dissertation.
I am deeply indebted to my advisor Dr. Lansey for all his support during this work. He was
the one who gave me the opportunity and his restless guides, suggestions and encouragement
made it possible for me to complete this dissertation.
I would like to thank Dr. Bayraksan for her invaluable advices to overcome whenever
seemingly unsolvable problems challenged me. Without her help, this study could not have
been completed.
I also want to thank my other committee, Dr. Valdes, Dr. Mays, Dr. Davis, and Dr. Nijssen
for their generosity and invaluable comments to my humble works. It was my honor to have
such respectful scholars as my committee members.
My special thank goes to Chinmaya for helping me out in C ++ coding. I am also grateful to
all friends of mine, including Jim, Amanda, Richard, Pasha, David, Doo Sun and Tae-
Woong, for their help and friendship. All of you listened to me sincerely, always took my
side and helped me to get over it when I was in trouble.
Especially, I would like to thank Derya, my best friend. She made my closed mind open with
her sincere friendship and helped me survive in unfamiliar environment. I am sure that she
will become a great scholar.
I also want to thank my family, father, mother and two brothers, Chul-Ho and Min-Suk, for
their endless support. Chul-Ho helped me a lot when I started working on computer
programming. Their warm heart is always with me wherever I go.
Finally, I would like to give my special thanks to my husband Inhong whose patient love
gave me strength to confront whatever I was up to.
5
TABLE OF CONTENTS ABSTRACT ........................................................................................................................ 9
U.S. Bureau of Reclamation (1987). Colorado River Simulation System: Overview,
Denver, Colorado.
44
Walski, T. M., Brill, E. D., Gessler, J., Goulter, I. C., Jeppson, R. M., Lansey, K., Lee,
H., Liebman, J. C., Mays, L., Morgan, D. R., and Ormsbee, L. (1987). “Battle
of the network models: epilogue.” Journal of Water Resources Planning and
Management, 113(2), 191-203.
Watkins, D. W. and McKinney, D. C. (1997). “Finding robust solutions to water
resources problems.” Journal of Water Resources Planning and Management,
123(1), 49-58.
Wilchfort, G. and Lund, J. R. (1997). “Shortage management modeling for urban water
supply systems.” Journal of Water Resources Planning and Management,
123(4), 250 - 258.
Wurbs, R. A. (1993). “Reservoir system simulation and optimization models.” Journal of
Water Resources Planning and Management, 119(4), 455-472.
Yang, S., Sun, Y., and Yeh, W. W-G. (2000). “Optimization of regional water
distribution system with blending requirements.” Journal of Water Resources
Planning and Management, 126(4), 229-235.
Yen, K. H., and Chen, C. Y. (2001). “Allocation strategy analysis of water resources in
South Taiwan.” Water Resources Management, 15, 283-297.
Zagona, E. A., Fulp, T. J., Shane, R., Magee, T., and Goranflo, H. M. (2001).
“Riverware: a generalized tool for complex reservoir system modeling.”
Journal of the American Water Resources Association, 37(4), 913-929.
45
APPENDICES
46
APPENDIX A: A GENERAL WATER RESOURECES
PLANNING MODEL USING DYNAMIC SIMULATION:
EVALUATION OF DECENTRALIZED TREATMENT
A. Graph
47
A General Water Resources Planning Model using Dynamic Simulation:
Evaluation of Decentralized Treatment
G. Chung1, K. Lansey2, P. Blowers3, P. Brooks4, W. Ela5, S. Stewart6 and P. Wilson7
ABSTRACT
Increasing population, diminishing supplies and variable climatic conditions can cause
difficulties in meeting water demands; especially in arid regions where water resources
are limited. Given the complexity of the system and the interactions among users and
supplies, a large-scale water supply management model can be useful for decision makers
to plan water management strategies to cope with future water demand changes. It can
also assist in deriving agreement between competing water needs and consensus and buy-
in among users of a proposed long-term water supply plans. The objective of this paper is
to present such a general water supply planning tool that is comprised of modular
components including water sources, users, recharge facilities, and water and wastewater
treatment plants. The model was developed in a dynamic simulation environment that
helps users easily understand the model structure.
1 Graduate Student, Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520-360-9554, E-mail: [email protected]) 2 Professor, Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520-621-2512, Fax: 1-520-621-2550, E-mail: [email protected]) 3 Assistant Professor, Department of Chemical and Environmental Engineering, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520- 626-5319, E-mail: [email protected]) 4 Assistant Professor, Department of Hydrology and Water Resources, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520- 621-3424, E-mail: [email protected]) 5 Associate Professor, Department of Chemical and Environmental Engineering, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520- 626-9323, E-mail: [email protected]) 6 Research scientist, Department of Hydrology and Water Resources, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520- 626-3892, E-mail: [email protected]) 7 Professor, Department of Agricultural and Resource Economics, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520- 621-6258, E-mail: [email protected])
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The model was applied to a realistic hypothetical system and simulated several
possible 20-year planning scenarios. In addition to water balances and water quality
analyses, construction and operation and maintenance of system components costs were
estimated for each scenario. One set of results demonstrates that construction of small-
cluster decentralized wastewater treatment system could be more economical than a
centralized plant when communities are spatially scattered or located at steep areas where
pumping costs may be prohibitive.
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1. INTRODUCTION
Increases in water demands have led to the need for innovative supply and demand
management to economically and efficiently operate a system within budget while
meeting user demands. A broad range of concerns resulting from modifying supplies and
demands must be considered in devising a water supply plan. The complexity of the
water supply system, however, makes it problematical to understand the interactions
between components; even for those intimately involved in the planning process. The
complicated system also causes difficulties in educating the public, improving existing
system operations, and finding low cost designs.
Thus, modeling tools that can represent a water supply system and demonstrate the
effects of management decisions can be extremely valuable. Several such tools have been
developed to simulate water supply systems. These models (Ocanas and Mays, 1981a and
1981b; Yen and Chen, 2001; Huang and Loucks, 2000; Cai et al., 2001 and 2003; Ejeta et
al., 2004; Yang et al., 2000; and Cohen et al.; 2004) tend to be system specific and are
generally inflexible in easily adapting to other systems and do not have user-friendly
interfaces.
This paper presents an integrated object-oriented dynamic simulation approach to
develop water supply system model that can be applied to design a long range plan. The
generality allows systems composed of multiple sources, users, and transportation and
treatment systems to be relatively easily organized for specific locations. The dynamic
simulation approach allows users and the general public to look inside the model and
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understand the relationships that comprise the model. This open architecture is
particularly useful in situations where several conflicting goals are to be addressed. In
addition to water balances, the model can also track water quality and costs over the
planning period duration.
2. LITERATURE REVIEW/BACKGROUND
Computer-based models together with their interactive interfaces are typically called
decision support systems (DSS) (Loucks, 1995). Despite software and hardware
limitation during the 1970s and 1980s, many site-specific river basin models were
developed and used by engineers in water management organizations for operational
planning of their basins (Zagona et al. 2001) such as the U.S. Bureau of Reclamation’s
(USBR) Colorado River Simulation System (CRSS), Tennessee Valley Authority’s
(TVA) Daily Scheduling Model, and the Potomac River Interactive Simulation Model
(PRISM). CRSS is representative of reservoir system models and captures a complicated
set of operating policies that balance end-of-water-year storage in Lakes Powell and
Mead (USBR 1987). PRISM was originally developed and implemented for a regional
water supply system for the Washington metropolitan area (Palmer et al. 1980).
To overcome the deficiencies of hard-wired models, several well-supported, general
river and reservoir modeling tools such as HEC-5 (Zagona et al. 2001) and HEC-3
(Wurbs 1993) have been developed that apply policy options that modelers can
parameterize and/or prioritize to represent the operations for a specific system. HEC
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ResSim (US Army 2003) and its predecessor, HEC-5, are two of the more widely used
and well documented reservoir-system simulation models for simulating the operation of
a system of reservoirs in a river network for flood control, water supply, hydropower, and
instream flow maintenance for water quality (Wurbs 1993 and Mays and Tung 1992).
The HEC-3 Reservoir System Analysis for Conservation program is much simpler than
HEC-5 but does not have the comprehensive flood-control capabilities of HEC-5 (Wurbs
1993).
Generalized mathematical water management models were also developed by
individual researchers. Ocanas and Mays (1981a), Huang and Loucks (2000), and Yang
et al. (2000) applied their reservoir/river management model to hypothetical river
networks. Other applications have a specific application network such as South Taiwan
(Yen and Chen 2001), the Aral Sea basin of central Asia (Cai et al. 2003), the Rio Grande
river from Elephant Butte, New Mexico to Fort Quitman, Texas (Ejeta et al. 2004), Syr
Darya River basin (Cai et al. 2001), and water supply system in southern Israel (Cohen et
al. 2004). However, these models were not generalized as they did not consider all
possible components and model generality and flexibility were insufficient to allow end-
users to easily modify the model. RiverWare (Biddle 2001, Magee and Goranflo 2002,
Gilmore et al. 2000, Fulp and Harkins 2001) is a general object-oriented model but is
limited to reservoir management.
All previous works mentioned above generally did not incorporate water quality
parameters in the models. The exceptions are Ocanas and Mays (1981a and 1981b) who
simulated biochemical oxygen demand (BOD) and total suspended solid (TSS). Cai et al.
52
(2001 and 2003) and Cohen et al. (2004) modeled salinity and Ejeta et al. (2004)
incorporated a total dissolved solid (TDS) component.
More recently, general system dynamics simulation object oriented models have been
developed. In one of the earliest applications, Palmer et al. (1993) tailored a dynamic
simulation software application to represent the Portland water supply system. Other
applications include river basin planning (Palmer et al. 2000), long-term water resource
planning and policy analysis (Simonovic et al. 1997; Simonovic and Fahmy 1999),
reservoir operation (Ahmad and Simonovic 2000), sustainability of a water resource
system, and water supply planning and management (Nandalal and Simonovic 2003).
System dynamics modeling was also used to model sea level rise in a coastal area by
Ruth and Pieper (1994).
Simonovic and Bender (1996) applied dynamic simulation in a collaborative
planning-support system to relate environmental issues, e.g., fish habitat, to hydroelectric
power generation. Stave (2003) prepared a system dynamics model of the Las Vegas
water supply system to increase public understanding of the value of water conservation.
Passell et al. (2002) presented a computerized dynamic simulation model of the
hydrology, ecology, demography and economy of the Middle Rio Grande Basin. Water
sustainability and groundwater storage in San Pedro River, Basin (AZ) was simulated by
Sumer et al. (2004).
In this paper, a set of modules are discussed that model various components of a
water supply system including water treatment and groundwater recharge. These modules
can be linked to represent the complete general water supply system and allow users to
53
evaluate alternative water management options. Total construction and operations costs
and water quality and availability are computed in the model. Given the current interest
in decentralized treatment, a hypothetical system is analyzed to evaluate the cost-
effectiveness of multiple treatment facilities within a community.
3. MODELING TOOLS
Various object-oriented dynamic simulation modeling tools are available including
Stella (http://www.isi.edu/isd/LOOM/Stella/), Dynamo (http://www.cs.auc.dk/
~normark/dynamo.html), Vensim (http://www.vensim.com/software.html), and Power-
sim (http://www.powersim.no). The power of object-oriented simulation is the ease of
constructing “what if” scenarios and tackling large, messy, real-world problems
(Nandalal and Simonovic, 2003). Powersim has flexibility in linking to other software
like Visual Basic, Visual C++, and Web program using Powersim SDK
(http://www.powersim.no). This feature makes Powersim Studio more powerful than
other object-oriented modeling languages. In this study, Powersim Studio 2003 was used
to develop water supply system modules and was linked to Visual Basic Studio for input
processing.
3.1 Modeling Objective
The goal of this effort is to develop a set of modules representing various water uses
and treatment options that can be easily combined to describe a general water supply
54
system. The modules can be tailored to the specific location by varying the module
parameters. Combining modules requires limited programming ability and promotes
rapid development of water resources management tools. Inclusion of a water quality
component is a unique feature of the tools and allows for examination of decentralized
treatment for specific waste streams or within a recycle/reuse system. In addition to the
modular structure, advantages of a dynamic simulation approach are the simple interfaces
and transparency in equations and relationships that comprise the model. This paper
describes the overall approach, the relationships comprising each module, and an
application to a hypothetical southwest US water supply system.
3.2 Generic System
Figure A.1 shows a general water supply system that includes all of the modeled
water supply, demand, and treatment components. Water supply components are
imported water, river and reservoir, subsurface, precipitation, and reclaimed water. Water
demand components are relationships describing the amount of water needed for various
purposes. Agricultural, domestic, industrial, large outdoor uses, such as parks, schools,
and golf courses, and environmental and riparian area are represented. The first four
sectors’ demands are computed by determining individual user or unit area demands and
aggregated over the sector. Environmental and recreational uses are based on estimated
in-stream water requirements. In Figure A.1, one of each supply/demand type is shown.
55
However, multiple components of the same type can be included in a model. For example,
each community within a watershed may have separate treatment facilities.
Water is conveyed to various users by pipes or canals that have defined capacities.
Mass balances within the system are computed accounting for appropriate connections
between uses and sources/sinks. A simple mass balance relationship is applied for most
systems:
∑∑ −==− −outflow
t,oinflow
t,itttt QQSSS ∆∆ (A.1)
where St is the storage in the system at time t and Qi,t and Qo,t are the inflows and
outflows during the time interval, ∆t, respectively.
Some interactions are described with more complex relationships. For example, flows
between surface and groundwater sources are based on hydraulic head differences.
Incidental discharge from the water distribution and sewer systems and planned aquifer
recharge from recharge basin are accounted for in mass balance relationships as
consumptive evaporative uses and flows into/out of the system. For planning purposes,
the dynamic simulation model performs calculations on a seasonal time step. Alternative
time steps can also be examined.
Simplified (lumped) and conventional water and wastewater treatment facilities are
included in the DSS. Lumped water and wastewater treatment facilities have constant
removal efficiencies and calculate effluent quality by a mass balance equation.
Conventional plant models are based upon current environmental engineering literature
and provide more detail on unit operation removal efficiencies. Rapid mixing and
flocculation, disinfection with chlorine, sedimentation, filtration, and sludge handling
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using drying beds comprise a conventional water treatment system and a conventional
wastewater treatment system is composed of primary settling, aeration tank, secondary
Table A.3. Sub-users and required model parameters for Large outdoor water user
User Required input School School density (number of school per population), average acreage per school,
water use per school acreage
Park Park density (number of parks per population), average acreage per park, water use per park acreage
Golf course Number of 9 and 18 hole courses per population, average water use per hole Private golf course Number of 9 and 18 hole private courses per population, average water use per hole
87
Table A.4. Parameters required for the domestic Area module
Name of model parameters Default value Unit Category
Number of households in 2000 (starting year) 11,784 houses
General
Market penetration rate 50 % Efficiency 90 % Incentive per home 1,000 USD/houses Incentive per toilet 100 USD/houses Incentive per washer 100 USD/houses Toilet flush frequency 5 flush/p/day
Toilet
Toilet water use built after 94 1.6 gal/flush Toilet water use built pre 80 6 gal/flush Toilet water use built between 80 and 94 3.5 gal/flush Number of households having toilet built pre 1994 10,314 houses Number of households having toilet built pre 1980 0 houses Showers per cap per day 0.9 shower/p/day
Shower Shower water use built before 94 5 gal/min Shower water use built after 94 2.5 gal/min Shower time 8 min/shower Faucet water use 2.5 gal/min
Faucet Faucets use per cap per day 4 min/p/day Aerator saving 2.94 gal/p Cooling season 2,500 hr/yr
Evaporative coolers
Coolers per house 1 1/houses Cooler water use with bleed off 8.1 gal/hr Cooler water use without bleed off 4 gal/hr Percent of cooler bleed off 20 % Percent of cooler without bleed off 80 % Percent of houses with evaporative cooler 90 % Dish cycles per day 0.2 1/p/da Dish-
Washer Water use per cycle of dish washer 10 gal Water use per bath 32.5 gal
Bathtub Number of bath per day 0.143 1/p/da Water use per front load clothes washer 42.3 gal
Clothes- washer Water use per top load clothes washer 18.5 gal
Number of cycle of clothes washer 0.3 1/p/da Fountain filling frequency 4 1/yr
Fountain Percent of houses having fountain 1 % Fountain storage 150 gal Number of fountain per house 1 1/houses Rainfall collection area 2,000 ft2/houses
Table A.4. Parameters required for the domestic Area module (Continued)
Name of model parameters Default value Unit Category
Evaporation from a pool having a cover 18.79 gal/day/pools
Pool
Evaporation from a pool without a cover 38.1 gal/day/pools Pool volume 16,830 gal/pools Drain frequency of a pool having a cover 0.1 1/yr Drain frequency of a pool without a cover 0.25 1/yr Percent of drained water of pool reaching aquifer 90 % Backwash amount of a pool 9.4 gal/day/pools Percent of houses with swimming pool 9.2 % Turf area per house 600 ft2/houses
Irrigation
Base drip area per house 1,200 ft2/houses Water use of drip irrigation system 0.91 af/acre/yr Water use of turf area 3.65 af/acre/yr Percent of recharging of outdoor irrigation 0.5 % Percent of houses with permanent irrigation system 75 % Irrigation system efficiency 70 %
*gal – gallon, p – person, min – minute, hr – hour, af – acre-ft, USD – US Dollars
89
Table A.5. Parameters required for riparian area module Input Parameter Value Unit
Total open channel & sand area 750,000 m2 Total extra-riparian area 20,000,000 m2
Percent of each tree Riparian area Tamarix 20 %
Cottonwood 80 %
Extra-riparian area Mesquite 15 % Grassland 85 %
90
Table A.6. Lumped water quality module parameters
Components Water
loss (kafy)
Removal efficiency (%)
BOD TSS Hardness Giardia
Water treatment system 1 1.5 90 90 90 3.0log Water treatment system 2 1.5 90 90 90 3.0log Wastewater treatment system 1 1.5 90 90 90 2.0log Wastewater treatment system 2 1.5 90 90 90 2.0log Wastewater treatment system 3 4.0 90 90 90 2.0log Advanced water treatment system 1 1.5 95 95 95 3.5log Advanced water treatment system 2 1.5 95 95 95 3.5log Advanced wastewater treatment system 1 4.0 95 95 95 2.5log Advanced wastewater treatment system 2 1.5 95 95 95 2.5log Advanced wastewater treatment system 3 1.5 95 95 95 2.5log Recharge facility - 30 30 30 2.0log
Components BOD (mg/l)
TSS (mg/l)
Hardness (mg/l as CaCo3)
Giardia (#/ml)
Initial water quality of precipitation 5 5 2 0 Initial water quality of imported water 30 30 150 100 Initial water quality of groundwater 30 30 250 0 Initial water quality of river 30 30 200 100 Initial water quality of reservoir 30 30 200 20
Components BOD (mg/l)
TSS (mg/l)
Hardness (mg/l as CaCo3)
Giardia (#/ml)
Waste quality deterioration of Agricultural area 1 150 130 200 25 Waste quality deterioration of Agricultural area 2 150 130 200 25 Waste quality deterioration of Agricultural area 3 150 130 200 25 Waste quality deterioration of Agricultural area 4 150 130 200 25 Waste quality deterioration of Domestic area 1 200 180 10 50 Waste quality deterioration of Domestic area 2 200 180 10 50 Waste quality deterioration of Domestic area 3 200 180 10 50 Waste quality deterioration of Domestic area 4 200 180 10 50 Waste quality deterioration of Industrial area 250 200 100 10 Waste quality deterioration of Large outdoor area 1 150 130 120 10 Waste quality deterioration of Large outdoor area 2 150 130 120 10
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Table A.7. Parameters required for conventional water treatment plant module
Input Value Units Alum 10 mg/l Rotation speed for turbine impeller 100/60 rps Impeller diameter 40 % of basin width ft Water density 62.4 lb/ft3 Detention time of rapid mixing (DT) 30 sec Velocity gradient in flocculation 50 1/sec Detention time of flocculation (FDT) 30 min Influent temperature 25 C Dosage of chlorine 100 mg/l Total organic carbon 20 mg/l Detention time of sedimentation (SDT) 5 hrs Depth of sedimentation basin (DEPTH) 12 ft Percentage of solids in the sludge 75 % Effective sand size in filtration 0.5 mm Uniformity coefficient of sand 1.2 Depth of sand in filtration 30/12 ft Backwash time in filtration 30 min Time between backwash in filtration 24 hr Maximum allowable average filtration rate (QAVE) 50 m/day Influent aluminum hydroxide 10 mg/l Influent turbidity 10 ntu Price of dry alum 5 USD/lb Liquid chlorine cost 20 USD/tonBackwash material cost for filtration 5,265 USD/yr Backwash labor cost for filtration 93.7 USD/yr Pumping efficiency of backwash pumps 90 %
92
Table A.8. Parameters required for conventional wastewater treatment plant module
Parameter Value Units Primary sedimentation: Constant in Voshel-Sak Model 0.139 - Constant in Voshel-Sak Model 0.27 - Constant in Voshel-Sak Model 0.22 - Sludge settling characteristics: Thickening constant 24.2 - Thickening constant 198.7 - Thickening constant 2.5 - Thickening constant 2.375 - Thickening constant 2.803 - Activated sludge kinetics: Growth yield coefficient 0.4 g cell/g BOD5 Half-velocity constant 60 g BOD5/m3 Maximum specific utilization coefficient 5 day-1 Endogenous decay coefficient 0.04 day-1 Fraction of cells degradable 0.77 - Conversion 1.42 g BODL/g VSS Conversion 1.5 g BODL/g BOD5 Secondary sedimentation: Constant in Chapman Model 5.69 - Constant in Chapman Model 0.00403 - Constant in Chapman Model 11.91 - Aeration: Alpha factor in aeration 0.8 - Beta factor in aeration 0.95 - DO concentration in aeration tank 1.5 g/m3 DO saturation concentration 9.17 g/m3 Temperature mixed liquor 20 oC Oxygen transfer efficiency 0.08 - Density of air 1.2 Kg/m3 Weight fraction of oxygen in air 0.232 - Temperature correction constant 1.024 - Mixing Requirement 28.8 m3 air/m3/d Gravity thickening: TSS of thickener supernatant 200 g/m3 Anaerobic digestion: Primary digestion reaction rate constant 0.632 - Primary digestion reaction rate constant 3.003 - Temperature of digester influent 20 oC Methane Production 0.35 m3/kg BODL
Parameter Value Units Average ambient temperature 10 oC Efficiency of heat exchanger 0.85 - Heat conduction coefficient 1 W/m2-oC Outside surface area and volume ration for digester 0.4 - Worth of digester gas 2.5 $/106 kl Soluble BOD5 in digester supernatant 500 g/m3 Factor accounting for the effect of rising gas on thickening in SD 0.25 - Thickening constant for digested sludge 292.6 - Thickening constant for digested sludge 2.9 - TSS of digester supernatant 4,000 g/m3 Height of digester 10 m Vacuum filtration: Form time per cycle time 0.33 - Pressure applied on vacuum filter 83,300 Nt/m2 Viscosity of filtrate 0.00089 Nt-sec/m2 Cycle time 6 min Specific resistance of sludge 1.00E+12 m/kg TSS of filtrate 2,000 g/m3
94
Table A.9. Parameters required for population modules Initial population
(person) Initial households (houses)
Initial growth rate (%)
Projected growth rate (%)
Domestic area 1 100,000 36,030 2.5 1.5 Domestic area 2 110,000 41,030 2.5 1.5 Domestic area 3 120,000 51,030 2.5 1.5 Domestic area 4 130,000 66,030 2.5 1.5 Initial number of
business Water use rate (afy/business)
Initial growth rate (%)
Projected growth rate (%)
Industrial area 931 1.844 2.5 1.5
95
Table A.10. Network geometry data of hypothetical water supply system
From imported water (2,800 ft) Water treatment plant 1 2,000 3 216 Canal Water treatment plant 2 2,100 3 216 Canal Agricultural area 1 2,000 10 240 Canal Agricultural area 2 2,000 3 240 Canal Agricultural area 3 2,400 4 240 Canal Agricultural area 4 2,400 3 240 Canal Recharge facility 0 5 158 Canal
From river (2,100 ft) Water treatment plant 1 2,000 3 12 Canal Water treatment plant 2 2,100 3 60 Canal Agricultural area 1 2,000 7 12 Canal Agricultural area 2 2,000 10 24 Canal Agricultural area 3 2,400 10 12 Canal Agricultural area 4 2,400 10 12 Canal
From reservoir (5 ft) Water treatment plant 2 2,100 3 12 Canal Agricultural area 1 2,000 7 12 Canal Agricultural area 2 2,000 7 24 Canal Agricultural area 3 2,400 7 24 Canal Agricultural area 4 2,400 7 24 Canal
From groundwater (300 ft) Water treatment plant 1 2,000 0.02 60 Canal Water treatment plant 2 2,100 0.02 60 Canal Agricultural area 1 2,000 Pumping Agricultural area 2 2,000 Pumping Agricultural area 3 2,400 Pumping Agricultural area 4 2,400 Pumping Large outdoor area 1 1,700 0.02 60 Canal Large outdoor area 2 1,700 0.02 60 Canal Domestic area 1 1,800 0.02 60 Canal Domestic area 2 2,600 0.02 60 Canal Domestic area 3 2,200 0.02 60 Canal Domestic area 4 2,600 0.02 60 Canal Industrial 1,800 0.02 60 Canal
From water treatment plant 1 (2,000 ft) Agricultural area 1 2,000 5 60 Pipe Agricultural area 2 2,000 20 Decision Alternative flow Agricultural area 3 2,400 20 Decision Alternative flow Agricultural area 4 2,400 20 Decision Alternative flow Large outdoor area 1 1,700 5 60 Pipe Large outdoor area 2 1,700 20 Decision Alternative flow Domestic area 1 1,800 20 Decision Alternative flow Domestic area 2 2,600 20 Decision Alternative flow Domestic area 3 2,200 20 Decision Alternative flow Domestic area 4 2,600 20 Decision Alternative flow Industrial 1,800 4 60 Pipe
96
Table A.10. Network geometry data of hypothetical water supply system (Continued)
From wastewater treatment plant 1 (2,100 ft) Riparian area 0 9 12 Canal Agricultural area 1 2,000 5 60 Pipe Agricultural area 2 2,000 5 60 Pipe Agricultural area 3 2,400 20 Decision Alternative flow Agricultural area 4 2,400 20 Decision Alternative flow Large outdoor area 1 1,700 5 60 Pipe Large outdoor area 2 1,700 20 Decision Alternative flow Recharge facility 0 8 12 Pipe
From wastewater treatment plant 2 (1,700 ft) Riparian area 0 9 12 Canal Agricultural area 1 2,000 20 Decision Alternative flow Agricultural area 2 2,000 5 60 Pipe Agricultural area 3 2,400 20 Decision Alternative flow Agricultural area 4 2,400 20 Decision Alternative flow Large outdoor area 1 1,700 20 Decision Alternative flow Large outdoor area 2 1,700 5 60 Pipe Recharge facility 0 8 12 Pipe
From wastewater treatment plant 3 (2,500 ft) Riparian area 0 9 12 Canal Agricultural area 1 2,000 5 60 Pipe Agricultural area 2 2,000 5 60 Pipe Agricultural area 3 2,400 5 60 Pipe Agricultural area 4 2,400 5 60 Pipe Large outdoor area 1 1,700 20 Decision Alternative flow Large outdoor area 2 1,700 20 Decision Alternative flow Recharge facility 0 8 12 Pipe
98
Table A.11. Conservation measures for domestic/industrial use Users Programs Objective systems Domestic indoor
Incentive programs
Front load washing machine for existing and new houses Faucet, Shower and Toilet for houses built before 1994
Domestic outdoor
Incentive programs
Evaporation cooler and fountain for existing and new houses Incentives to purchase pool covers for existing and new houses Water irrigation efficiency increasing for existing and new houses Grey water reuse system for new houses
Ordinances
Reduced swimming pool use – Public education Restrict future swimming pool development Eliminate existing swimming pools Discharge pool water for eventual reuse for existing and new swimming pools Outdoor water use restriction for existing and new houses Landscaping standards and regulation for existing houses Landscaping standards and regulation for new houses Rainwater harvesting for new houses Rainwater harvesting for existing houses Water loss due to violation
Industrial
Incentive program Incentives to purchase pool covers for existing and new houses
Ordinances
Toilet for existing and new businesses Reduced swimming pool use – Public education Restrict future swimming pool development Eliminate existing swimming pools Discharge pool water for eventual reuse for existing and new swimming pools Outdoor water use restriction for existing and new houses Water irrigation efficiency increasing for existing and new houses Large water user audits for existing and new businesses
Population Growth restriction
99
Table A.12. Parameter values before and after a conservation measure implementation for domestic/industrial uses
Grey water reuse system Reuse system cost = $500/houses
Large water user audits 0 af/yr 22 af/yr
Water loss due to violation – Domestic outdoor
21,400 gal/violations
26031 violations/house/yr 0 violation
Violations in industrial area 21,400 gal/violations
1444 violations/business/yr 0 violation
Car wash
<<Commercial area>> Water use for commercial car = 10 gal/car Number of washes of commercial car = 0 /day
<<Domestic area>>water use for a car = 15 gal/car Number of washes of commercial car = 60 /day
100
Table A.13. Government subsidy for conservation incentives
Alternatives Government investment ($/yr)
Cost ($/house)
Front load washing machine for existing and new houses 100,000 70 Shower, toilet, and faucet for houses built before 1994 100,000 100 Grey water reuse system for new houses - 200 Evaporation cooler and fountain for existing and new houses – domestic and industrial area 100,000 70
Incentives to purchase pool covers 100,000 70
Table A.14. Cost and water savings next 8 years (2012-2020) resulting when individual conservation measures are implemented in year 2012 in domestic area 4
Alternatives
Fresh water use in
domestic area
(km3/yr)
Operation cost in
domestic area
($/yr x 105)
Operation cost
reduction (%)
Fresh water use reduction
(%)
Base condition 0.054 18.93 Front load clothes washer for existing houses 0.052 18.32 3.25 3.59 Shower, toilet, and faucet for existing houses 0.054 18.79 0.76 1.10 Evaporative cooler and fountain for existing houses 0.052 18.27 3.48 3.82 Grey water reuse system for new houses 0.046 17.24 8.95 15.07 Reduced swimming pool use for existing and new houses 0.054 18.87 0.33 0.33
Reduced incentives to purchase pool covers and splash recovery system for existing and new houses 0.054 18.83 0.55 0.55
Reduced restrict future swimming pool development for existing and new houses 0.054 18.81 0.64 0.64
Reduced eliminate exiting swimming pools 0.054 18.91 0.09 0.09 Reduced discharge pool water for eventual reuse 0.054 18.93 0.00 0.00 Water drip irrigation efficiency for existing and new houses 0.054 18.92 0.06 0.41
Landscaping standards and regulations for existing houses 0.049 17.18 9.23 9.23
Landscaping standards and regulations for new users - It is turned on if the same regulation for existing houses is on.
0.053 18.34 3.11 3.11
Outdoor water use restriction 0.054 18.93 0.00 0.00 Water loss due to violations 0.054 18.91 0.11 0.11 Rainwater harvesting for new houses 0.054 18.77 0.86 0.86 Rainwater harvesting for existing houses 0.053 18.45 2.55 2.55
101
Table A.15. Cost and water savings next 8 years (2012-2020) resulting when individual conservation measures are implemented in year 2012 in industrial area
Conservation measure
Operation cost in
industrial area
($/yr x 10 5)
Fresh water use
in industrial
area (kafy)
Operation cost
Savings (%)
Fresh water
use Savings
(%)
Base condition 1.99 4.63 Toilet for existing and new houses 1.98 4.62 0.32 0.32 Reduced swimming pool use for existing and new houses 1.99 4.63 0.09 0.09
Reduced incentives to purchase pool covers and splash recovery system for existing and new houses
1.99 4.63 0.15 0.15
Reduced restrict future swimming pool development for existing and new houses 1.99 4.63 0.00 0.00
Reduced eliminate exiting swimming pools 1.98 4.62 0.38 0.38 Reduced discharge pool water for eventual reuse 1.99 4.63 0.00 0.00 Water drip irrigation efficiency for existing and new houses 1.98 4.61 0.49 0.49
Outdoor water use restriction 1.98 4.61 0.49 0.49 Large water user audits 1.98 4.61 0.37 0.48
Table A.16. Consumptive use in the domestic areas and groundwater storage Domestic area/ groundwater storage
No conservation measures
All conservation measures implementing Difference (%)
Domestic area 1 (km3/yr) 0.013 0.002 -84.03 Domestic area 2 (km3/yr) 0.016 0.003 -80.81 Domestic area 3 (km3/yr) 0.018 0.004 -77.05 Domestic area 4 (km3/yr) 0.024 0.007 -70.76 Groundwater storage (km3) 27.44 27.69 0.89
102
Table A.17. Groundwater storage after 20-year simulation resulting under alternative scenarios of water availability
Alternatives Storage after 20-year (kaf)
Difference (%)
Base condition 6,299 No imported water 5,823 7.56 No reclaimed water 5,773 8.35 No imported water and reclaimed water 5,122. 18.68
Table A.18. Water and wastewater constructed under different conditions
Condition Water plant built Wastewater plant built Number of treatment plant
Table A.22. Annual construction, expansion, and O&M cost for each conveyance and treatment system (× 106$/yr) (Continued) Lumped system Conventional treatment system
alternatives Wastewater treatment system 1 Wastewater treatment system 1 Construction Expansion O&M Construction Expansion O&M
Figure A.1. Schematic of a generic water supply system. Solid lines indicate supply flows and dashed lines indicate effluent flows. All components (uses and supplies) have a consumptive use loss.
PRECIPITATION
IMPORTED WATER
Environmental/ Recreational area
Domestic area Large outdoor area
Water treatment plant
Advanced water treatment plant Wastewater
treatment plant
Advancedwastewater
treatment plant
Recharge Facility
GROUNDWATER
RIVER
RESERVOIR
On-site treatment plant
Consumptive Use
Agricultural area
Industrial area
109
Figure A.2. Water balance for four domestic areas
Figure A.3. Powersim representation of a conventional water treatment system
110
Figure A.4. Powersim representation of a conventional wastewater treatment system
Figure A.5 Monthly rainfall for Coolidge, AZ (Jan. 1987 – Dec. 2004) taken from the Arizona Meteorological Network (AZMET) (http://ag.arizona.edu/azmet/.html)
Figure A.7. Salt River, AZ monthly inflows and outflows at Roosevelt Dam in the site of USGS 09497500 and USGS 09502000 (Jan. 1987-Sep. 2004) from NWISWeb (http://waterdata.usgs.gov/nwis)
Figure A.8. Water use in domestic Area 1 with no conservation measures, with implementation of a grey water reuse and a fixture replacement incentive programs
DO1_baseDO2_baseDO3_baseDO4_baseDO1_on all programsDO2_on all programsDO3_on all programsDO4_on all programs
Figure A.9. Comparison of water use in four domestic areas for no conservation measures (base) and after implementing all conservation measures (DO – Domestic Area) in 2012. The variability in use after implementation is related to the climatic conditions.
Figure A.10. Cost for constructing and operating entire system over time for no conservation measures (base) and after implementing all conservation measures in year 2012
Figure A.11. Inflows from various sources to agricultural area 1 during growing season for meeting consumptive use with reclaimed water and imported water
base conditionCWT1 onlyCWT2 onlyCWWT1 onlyCWWT2 onlyCWWT3 onlyCWWT1 and CWWT2CWWT2 and CWWT3CWWT1 and CWWT3
(b)
Figure A.17. Cost of water supply system over time for different treatment plant configurations ((a) lumped representation (b) conventional water treatment system)
TSS in agricultural area TSS in large outdoor area TSS in domestic area TSS in industrial areaBOD in agricultural area BOD in large outdoor area BOD in domestic area BOD in industrial area
Figure A.20. Water quality (BOD and TSS) over time supplied to various users
The developed model was applied to two hypothetical water communities. The
capacities for the system components including water transport and treatment facilities
are model decision variables. An explicit representation of energy consumption cost for
the transporting water in the model assists in determining the efficacy of satellite
wastewater treatment facilities.
1 Graduate Student, Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520-360-9554, E-mail: [email protected]) 2 Professor, Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520-621-2512, Fax: 1-520-621-2550, E-mail: [email protected])
128
Although the water supply systems studied contained highly nonlinear terms in the
formulation as well as several hundred decisions variables, the stochastic search
algorithm, SFLA, successfully found solutions that satisfied all the constraints for the
studied networks.
129
1. INTRODUCTION AND BACKGROUND
A water supply system is a collection of water transport structures, pumping stations,
and water treatment and storage facilities that are managed to supply the desired amount
of water with the desired quality to consumers. With increasing water demand from
domestic and industrial areas, sustainable water supply becomes more important and the
development of a long-term water supply plan is challenging because of complexity of
the system and uncertainties in the future.
In the southwest Unites States and many arid and semi-arid regions, groundwater is a
major water source but, oftentimes, it has been mined to meet the increased water
demands. As a result, ground water table level have fallen requiring better management
plans or identifying alternative water sources. Reclaimed and imported water are
potential alternatives that may replace groundwater for agricultural and other purposes. In
the future as new supplies become more limited, water reclaimed, after a high level of
treatment, may be necessary to meet potable demands.
Water supply planning requires considering current demand, future growth, and
available supplies. Supply costs include capital for construction, operation and
maintenance. As reclaimed water becomes a more integral part of the water supply
system, the cost of transporting water from a treatment facility to users becomes more
critical in decision making. This introduces another complexity to the planning process in
that the cost of distribution may exceed the gains in economies of scale that are obtained
130
if wastewater treatment is centralized. Hence, smaller distributed wastewater treatment
facilities may be a cost effective alternative over a single central plant.
Little research has been conducted on water supply system planning optimization.
Ocanas and Mays (1981a and b) formulated and solved a water reuse planning
optimization model using non-linear programming under steady and dynamic conditions.
The steady state model consisted of a nonlinear objective function, linear and nonlinear
constraints for a single period. A large-scale generalized reduced gradient technique was
used to solve this optimization problem (Ocanas and Mays 1981a). In the follow-up
paper, the same technique with successive linear programming methods was applied to a
dynamic water reuse planning model with single and multiple periods (Ocanas and Mays
1981b). Water quality was considered in both papers. In the dynamic model, the capacity
expansion of treatment facilities was considered at the beginning of the period and
operation costs were included in the objective function. This model provided a basis of
the optimization structure for a water supply management system. Conveyance systems
were considered as lumped units without detailed representations of energy loss and
capacity. Later, Ejeta et al. (2004) applied a general approach to the studies in Rio
Grande in New Mexico and Texas including a total suspended solids (TSS) constraint.
The objective of this study was to maximize total net benefit.
In this paper, the water reuse planning system formulated by Ocanas and Mays
(1981a and b) is extended to consider component hydraulic capacities and improve
scalability. Decision variables include the capacities of water transport facilities - such as
pipes, pumps, and canals - as well as the capacity of treatment facilities. This step extends
131
previous work in this area by optimizing overall system costs with an explicit
representation of energy consumption costs and evaluating the tradeoff of multiple
satellite wastewater treatment facilities. The problem considered in this study is highly
nonlinear and deals with a large-scale water supply system that involves several hundred
decisions. A stochastic search algorithm is successfully applied to determine the optimal
water supply plans for two hypothetical communities.
2. PROBLEM DESCRIPTION
The overall planning goal is to minimize the total costs of construction, operation and
maintenance of the water supply system. The problem considered in this paper extends
earlier work by considering operations costs as a function of flow rates and selected
component sizes. The supply system may have multiple sources and users, and contain
one or more water and wastewater treatment facilities. Unlike previous models,
conveyance system hydraulics are directly embedded in the model to more realistically
estimate operation expenses. One or more planning periods can be represented to allow
delaying expansion investments or to take advantage of economies of scale by
constructing excess capacity in early decision periods.
Figures B.1 and B.2 show system schematics for two communities. Potential flow
paths are shown between sources, sinks and users that are denoted as nodes. Water uses
are agricultural, domestic, industrial and large outdoor irrigation (golf courses, school,
and parks). Available water supply sources can include surface reservoirs or groundwater
132
aquifers that have storage capabilities and rivers that cannot store water over time. Any
flow provided to domestic and industrial users from surface water sources must be treated
at a water treatment facility (WT) while surface water provided to agricultural and
outdoor uses does not need to be treated. Wastewater return flows from domestic and
industrial uses must be treated at a wastewater treatment plant (WW). After wastewater
treatment, reclaimed water may be supplied to agricultural or large irrigation areas,
discharged to the river, or recharged to the aquifer through infiltration basins.
Groundwater banking can also be achieved by recharging imported waters or surface
supplies.
The primary system constraints are to satisfy conservation of mass at all locations and
components in the system. In addition, each user has defined water quality and demand
requirements, while all supplies are limited by flow capacity or storage volume.
Conveyance system conditions related to canal capacity and pipeline/pump sizes are
formulated to ensure proper sizing and energy consumption. The mathematical form of
the problem is given in the next section.
3. PROBLEM FORMULATION
A general system can be represented by a set of N nodes and A arcs. Arcs denote
water transmission systems while nodes are locations where water is collected from or
split between a set of arcs. The set of nodes is comprised of several subsets representing
sources (IS) and users (IR). Sources are further divided into storage sources (ISS) that
133
have storage carried over in time (groundwater aquifers and surface reservoirs) and non-
storage sources (INS) that cannot store water over time (rivers and imported waters (IIW)).
Note that a river is represented as upstream (IRU) and downstream nodes (IRD) that are
connected by a river arc. Water is withdrawn by users from upstream nodes and returned
from treatment plants to the downstream river. Water user nodes represent domestic,
industrial, and irrigational purposes and water and wastewater treatment plants (IWT, and
IWWT, respectively).
An arc transmits flow from a node i to a node j. Arcs represent the sets of canals (IC),
pipe connections (IP), pump connections (IU). Pipe connections may have pump stations
at their upstream source depending on the elevation difference and energy losses between
the two connected nodes. Recharge basin arcs (IB) can be used to transmit imported or
treated water to the aquifer to bank or recharge water in the aquifer. In addition, rivers
and users may recharge the aquifer through seepage or infiltration that is represented by
infiltration arcs (II).
The capacities of the water transmission structures such as pipes, pumps, and canals
and treatment plants are described as structural variables. Flow allocations over the water
supply network are operational variables.
The objective function consists of the function of construction and expansion, and
operations and maintenance (O&M) costs for all components (pipes, canals, pumps, and
treatment facilities) or:
654321* ),()(),()()(Minimize fwqfwfHfffZ t
itij
ti
tij
tij
tij
tij +++++= χκκ (B.1)
Each term in Eq. B.1 is non-linear relationship with respect to the decision variables.
134
Pipe construction and expansion costs are given by (Clark et al. 2002):
ij
tij
tij
tij
tij
ti tjtij
tij
tij
tij
tij
tij
L
x
f
71.093.08.173.0
, ,
83.19.154.1
1
0022.023.002.0062.0
0062.00018.062.035.0198.57
)(
κκκκ
κκκκ
κ
+++−
++++= ∑ ∑=∈ =∈TI TIP P
(B.1a)
Canal construction and expansion costs are (US Army Corp of Engineers 1980):
∑ ∑=∈ =∈⎭⎬⎫
⎩⎨⎧
+=TI TIC Cti tj
ttijij
ttij
tij
CITYENRLCITYENR
f
, ,
2
2
287730.55
287739.0
)(
κκ
κ (B.1b)
Pump construction and expansion costs are given by (Walski et al. 1987):
( )∑ ∑=∈ =∈=
TII TIIUP UPti tjtij
tij
tij
tij
H
Hf
, ,
4.07.0
3
500
),(
U Uχ
χ (B.1c)
Water and wastewater treatment facility construction and expansion are approximated by
(Tang et al. 1987):
( ) ( )∑∑ =∈=∈+=
TITI WWTWT titi
titi
ti
ti
ti
tij
w.yw.y
wqf
,,
4
5454228 + 921081135987 + 132897
),(
(B.1d)
Operation and maintenance of pipes, canals, pumps, and treatment facilities are given by
(Clark et al. 2002; US Army Corp of Engineers 1980; Walski et al. 1987; Tang et al.
1987):
135
( ) [
( ) ( ) ] ]∑∑
∑ ∑
∑ ∑
∑ ∑ ∑
≤∈≤∈
≤∈ ≤∈
≤∈ ≤∈
∈ ≤∈ ≤∈
+++⎭⎬⎫
⎩⎨⎧
++∆+
⎟⎟⎠
⎞⎜⎜⎝
⎛+++
⎢⎣
⎡+
+
ottiti
tiotti
ti
ti
otti ottj
ooij
oij
oij
oij
tij
otti ottj
o
ijoij
oijij
o otti ottj ijoij
tijo
ti
tij
w.ywy
CITYENRqqq
ENRLqqL
LqxI
wqf
|,|,
|, |,
935.058.0
|, |,
572.0
|, |,
5
54542 + 12108 36097.28
2877320456047.79
1850)0135.0078.0(0254.0
)3.07.27(1
1
),(
WWTWT
UP UP
C C
P P
II
II II
I I
O I I
U Uµ
(B.1e)
where IP, IU, IC, IWT, and IWWT are the set of pipe, pump, and canal arcs and water and
wastewater treatment plant nodes, respectively. The superscript t indicates the
construction and expansion time while o denotes the operation period used for evaluating
O&M costs. T and O represent the time sets for construction and expansion, and
operation and maintenance, respectively.
In terms of construction decisions in the above equations, a binary decision variable x
for a pipe identifies whether or not pipe i in the set of links, IP, will be installed with
diameter κ over its length of L. Similarly, µ is a binary decision variable representing
whether or not a pump will be installed with design pump discharge and head, χ and H,
respectively. Note that pumps can be installed at the beginning of defined pipelines or
stand alone as pump connections. The binary decision variable, yi, indicates whether a
water or wastewater treatment plant, will be built with design capacity, w. If only a single
plant capacity is to be determined no binary variable is needed. No discrete variables are
needed for the other components as their decision variables are permitted to go to zero.
136
Canal variables are represented by the continuous canal depth κ for the given canal length
L.
Operation and maintenance costs (Eq. B.1e) are calculated for each component and
summed over the planning period, O. O&M costs are functions of the component
capacity and flow, q, that is also a model decision variable. The double summations for
the pipe, canal and pump are used to indicate the corresponding connection from a node, i,
to a node, j.
The objective also includes several system defined parameters. ∆ in the pump terms
corresponds to the elevation difference between two nodes, i and j. CITY is a
construction cost factor that varies by location. The ENR cost factor at year t is used to
consider inflation rate in the estimation of construction costs that is given by:
For example, for m = 3, rank 1 goes to memeplex 1, rank 2 goes to memeplex 2, rank 3
goes to memeplex 3, rank 4 goes to memeplex 1 again, and so on.
Step 4: Memetic evolution within each memeplex: Evolve each memeplex Yk, k = 1, …,
m according to the frog leaping algorithm (FLA) outlined below.
147
Step 5: Shuffle memeplexes: After a defined number of memetic evolutionary steps
within each memeplex, replace Y1, …, Ym into X, such that X = Yk, k = 1, …, m. Sort X
in order of decreasing performance value. Update the population best frog’s position, PX.
Step 6: Check convergence: If the convergence criteria are satisfied, stop. Otherwise,
return to Step 3. Typically, the decision on when to stop is made by a pre-specified
number of consecutive time loops when at least one frog carries the “best memetic
pattern” without change. Alternatively, a maximum total number of function evaluations
can be defined.
4.2 Local Exploration: Frog Leaping Algorithm (FLA)
Step 0: Set im (iteration count) and iN (shuffle count) equal to zero. The number of
iterations and shuffles are limited to user-defined values ic and is, respectively. Form an
initial random set of frogs and evaluate each frogs objective function value.
Step 1: Set im = im + 1.
Step 2: Set iN = iN + 1.
Step 3: Construct a sub-memeplex: The frogs’ goal is to move towards the optimal ideas
by improving their memes. An individual frog is updated using the presently available
information. The new frog is returned to the memeplex and another frog is updated. This
strategy is consistent with evolution of ideas since the best information available is used
unlike a genetic algorithm in which the entire population is updated prior to using any
information gained. To complete this process, a subset of the memeplex, (Yk, k = 1, …,
148
m), called a sub-memeplex, (Ziq, iq = 1, …, q) is considered. Figure B.3 presents the
structure of population, memeplex, and sub-memeplex. The sub-memeplex selection
strategy from a memeplex (Yk) having n frogs is to give higher weights to frogs that have
higher performance values and less weight to those with lower performance values. The
weights are assigned with a triangular probability distribution, i.e.,
pj = 2(n+1-j)/(n(n+1)) , j = 1, ... , n
such that, within a memeplex, (Yk, k = 1, …, m), the frog with best performance has the
highest probability of being selected for the sub-memeplex, p1 = 2/(n+1) and the frog
with worst performance has the lowest probability, pn = 2/(n(n+1)).
Here, q distinct frogs are selected randomly from n frogs in each memeplex (Yk, k =
1, …, m) to form the sub-memeplex array (Ziq, iq = 1, …, q). The sub-memeplex is sorted
so that frogs are arranged in order of decreasing performance. Record the best (iq = 1; iq
= 1, ..., q) and worst (iq = q; iq = 1,…, q) frog’s position in the sub-memeplex as vectors
PB and PW, respectively.
Step 4: Improve the worst frog’s position: When improving a frog’s position, it can adapt
their ideas from the best frog within the memeplex (group), PB, or from the global
(population) best, PX. The direction, step size and new position are first computed for the
frog with worst performance in the sub-memeplex (Ziq, iq = 1, …, q). The computation
includes identifying the direction of improvement (gradient) and the magnitude of change
(step length) in that direction. The direction of change (positive or negative) is defined by
the movement toward the sub-memeplex best or (PB - PW) where P represents the location
vector. This change involves both in magnitude as well as direction of the decisions.
149
The magnitude of the step size is randomly selected as a proportion of change
direction. It is limited by the maximum step size, Smax. The present version of the
algorithm considers a pre-specified fraction of the bound of the variable value as Smax.
Mathematically, the step size is defined as:
Step size, |s| = MIN[2(rand (PB - PW)), Smax ]
where rand is a random number in the range [0,1]. The new position is then computed by:
Z(iq = q) = PW + s (B.35)
Note that the multiplier of 2 in the step size calculation allows the frog’s new position to
be between the two frogs or move beyond the better frog’s location depending on rand.
Then, if the new position is beyond feasible boundary for any decision variable, it is
forced to be the boundary value. This modification improves upon convergence seen in
Eusuff et al (2006).
Compute the new performance value f(iq = q). If the new f(iq = q) is better than the old
f(iq = q), i.e., if the move produces a benefit, then replace the old Z(iq = q) with the new one
and go to Step 7. Otherwise go to Step 5.
Step 5: If Step 4 cannot produce a better result then the step and new position are
computed for that frog by:
Step size, |s| = MIN[2(rand(PX - PW)), Smax ]
and the new position is computed by Eq. B.35.
Compute the new performance value f(iq = q) for point Z(iq = q). If the new f(iq = q) is better
than the old f(iq = q), i.e., if the evolution produces a benefit, then replace the old Z(iq = q)
with new one and go to Step 7. Otherwise go to Step 6.
150
Step 6: Censorship: If the new position is either infeasible or worse than the old position,
the spread of the defective meme is stopped by randomly generating a new frog ‘r’ at a
feasible location to replace the frog whose new position was not favorable to progress.
Compute f(r) and set Z(iq = q) = r and f(iq = q) = f(r).
Step 7: Upgrade the memeplex: After the memetic change for the worst frog in the sub-
memeplex, replace Ziq in their original locations in Yk. Sort Yk in order of decreasing
performance value.
Step 8: If iN < is , go to Step 2.
Step 9: If im < ic , go to Step 1. Otherwise return to global search to shuffle
memeplexes.
Steps 4 and 5 of the FLA are similar in philosophy to particle swarm optimization. A
descent direction is identified for a particular frog and the frog is moved in that direction.
Here, however, since the global search is also introduced in the shuffle operation, only
the local minimum is used rather than the complete population best (Step 4) unless no
improvement is made (Step 5). Since a descent direction is implicitly applied, it may be
fruitful to perform a line search rather than a random step but the simpler approach is
taken here.
151
5. APPLICATIONS
The optimization problem (Eqs. B.1 - B.32) has been formulated for the water supply
systems shown in Figures B.1 and B.2 for 20-year planning periods. New structural
component construction is permitted at the outset (year 1) and new components or
existing component expansion may be added after 10 years. Biochemical Oxygen
Demand (BOD) is used as the representative water quality parameter.
5.1 Single Wastewater Treatment Plant System
The first system to be optimized (Figure B.1) consists single water and wastewater
plants, multiple sources (imported water, groundwater aquifer, and surface water) and
two demands centers (domestic and agricultural). Three types of water transport
structures are used depending upon the connection: canal, pipe and/or pump. All canal
flows for the conveyance of imported and raw water sources are driven by gravity.
Agricultural areas and water treatment plant may directly pump groundwater from
aquifers available near their location so do not require a pipe link. Other flows are
transported through pipes that may require a pump station to supply the energy necessary
to pass flow through the pipeline and satisfy the minimum pressure head requirement at
the outlet (14.0 m of water = 137.9 kPa = 20 psi). Groundwater replenishment through
recharge basins is assumed to occur at a constant rate of 9.1 m/yr (30 ft/yr). Seepage
losses from users to the aquifer are assumed as 0.1% of total user demand.
152
Input parameters for the single wastewater treatment plant system are shown in
Tables B.1 and B.2. The system contains the total of eighteen arcs (Figure B.1). Origin,
destination nodes and arc lengths are listed in Table B.3. As seen in Figure B.1, the
network consists of six canal depth construction decisions (6) and seven pump/pipeline
arcs with their three design decisions (pipe diameter and pump flow capacity and head)
for a total of 21 decisions. The network also includes two pump links for which the pump
design flow and head (four total design decisions) must be selected and two treatment
facilities with plant capacity as decision variables (total of 2 decisions). Thus, 33 design
decision variables are to be determined for each of the two planning periods or a total of
66 design decisions.
Ten of the 18 arc flows in each period are independent control decision variables that
are also selected by the optimization model. The remaining 8 are dependent variables that
are computed from the mass balance constraints defined in Eqs. B.15 and B.16.
Therefore, the final optimization problem contains a total 86 of decision variables for the
two design periods (66 design and 20 control decision variables).
The following SFLA parameters were selected from experience and preliminary
testing: the total number of population (F = 3000), memeplex (m = 10), frogs in each
memeplex (N = 300), evolutionary steps (iN = 300), and frogs in a sub-memeplex (iq =
300) are established and applied in the single wastewater treatment plant system. The
problem was run on a Dell Inspiron with a Centrino Duo T2300 1.6GHz and 1GB of
RAM and was solved in 5.5 min. after 131 thousand function evaluations. Figure B.4
shows the progress of solution with respect to the number of function evaluations. The
153
penalty term that accounts for constraint violations fell dramatically in early iterations
after which the total system cost gradually decreased. Total construction and operation
cost for the single treatment plant system for the 20-year period is $ 771 million (present
value for year 0) or an annual cost of $47 million.
Table B.4 lists the optimal component designs and the optimal network solution is
depicted graphically in Figure B.5. Water and wastewater treatment plant capacities are
0.05 and 0.05 km3/yr during the first design period, respectively. The water and
wastewater treatment plants are not expanded at year 10 suggesting that economies of
scale made oversizing in the first period to be more desirable than future expansion.
Increased transport capacity was required to/from the domestic area in year 10 to
convey the increased demands. Although the flow allocation from the aquifer to the farms
remains constant over time, the groundwater pump capacity was expanded to overcome
the required lift to overcome the drop in the aquifer water level (Table B.5).
The system has abundant amount of water in downstream river and subsurface (Table
B.5) to preserve sustainability. Therefore, the domestic area is supplied mostly from river
through water treatment plant, and pumped water from the aquifer is the main source for
agricultural area.
BOD concentrations in the aquifer and the downstream river remain steady and below
their 30 mg/l water quality requirements (Table B.5). Influents to domestic and
agricultural area also have better quality than the required, 5 mg/l and 30 mg/l,
respectively.
154
As shown in Table B.7, pipe construction is the dominant cost for this system and
economies of scale compel some pipe installations to be constructed in the initial period.
Pipe and pump connections require significant operation costs as compared with the
operation of canal and treatment plants.
5.2 Multiple Wastewater Treatment Plant System
As shown in Figure B.2, multiple water users and wastewater treatment plants are
included in the multiple wastewater treatment plant system in order to investigate a more
generalized system. This network is consists of six users - three domestic areas, one
industrial, one agricultural, and one large outdoor area – and three wastewater treatment
plants. In general, input parameters used for the multiple wastewater treatment plant
system are the same as for the single wastewater treatment plant system except for the
initial population at the domestic areas (Table B.8). Table B.9 summarizes the multiple
wastewater treatment plant system nodal parameters. This network has 44 arc
connections and their lengths are given in Table B.10. The structural design include 6
canals (6 parameters for canal depth), 29 pipes of which pump station could be built
depending on energy relationship (29 parameters for each pipe diameter, pump design
capacity, and pump head), 2 pumps (2 parameters for each pump design capacity and
head), and 4 treatment plants (4 parameters for capacities). The structures can be built or
expanded in year 1 and 10. Total structural design variables are 101 (= 6 + 3 × 29 + 2 × 2
+ 4) for each design period and a total 202 for whole operational period.
155
Flow allocations through twenty-three arcs out of forty four are defined as operation
decision variables while the remaining 21 arcs are dependent variables that are computed
by mass balance equations. In total, the final problem consists of 248 decision variables
(2 design periods × 124 decision variables (101 and 23 for design and operation
variables, respectively)).
The problem was solved using the computer system cited in the previous section in
about 70 minutes and nearly 582 thousands function evaluations. The optimal cost for the
system was $837 million as the present value in the starting year of the planning period
and the estimated annual cost was $51 million. Final optimal solution is depicted in
Figure B. 6. Although the optimal solution found may not be the global optimal solution
due to high discrete nonconvexity associated with the study system, the optimization
process demonstrates the improvement in overall system cost and reduction in the penalty
term (Figure B.7).
Since the system has sufficient local water to meet user demands, imported water is
not purchased. Domestic areas 1 and 2 are supplied from the aquifer which has good
enough water quality to supply domestic demands, and domestic area 3 and industrial
area are supplied from upstream river through water treatment plant.
Pipe, pump, and canal capacities are given in Table B.11. Table B.12 summarizes the
capacities for water and wastewater treatment plants. Canals carry water from the
upstream river to agricultural and large outdoor areas. Large outdoor turf use is partially
supplied with treated wastewater. As for the single treatment plant case, most pipe
construction occurred at the outset due to economies of scale. Pipe construction cost
156
dominates the total system cost (Table B.13). Water quality parameters for influent and
effluent are satisfactory and listed in Table B.14. Water elevation, source discharge,
water demand and population are summarized in Tables B.15 and B.16, respectively.
157
6. CONCLUSIONS
As water demands grow, water supply system capacities must be increased to provide
the desired water. A poorly designed system can waste money and energy. Optimization
of the system can assist decision makers make good decisions to respond to long term
changes. In this study, a large-scale general water supply system optimization model is
developed using deterministic nonlinear programming. The approach was applied to two
moderate and larger hypothetical water networks.
Difficulties in solving the problem using gradient based NLP methods led to the
application of a heuristic stochastic search algorithm, the Shuffled Frog Leaping
Algorithm (SFLA). As the example water communities, the single (7 nodes) and multiple
wastewater treatment plant system (13 nodes) have been optimized in terms of the total
system cost. Total number of decision variables in the single and multiple wastewater
treatment plant system application are 86 and 248, respectively. The resulting annual
minimum costs were $47 million and $51 million, respectively.
The developed single and multiple wastewater treatment plant applications have
sufficient local water to supply user demand, so no external water is purchased.
Economical scale suggests construction of enough treatment and transportation facilities
at the outset in many connections. Pipe construction cost and pipe and pump operation
cost dominates total operation cost. Supply water to user has better quality than the
required and water source keep enough and better quality water to preserve environment.
Further research efforts are needed to develop more detail water supply system, for
158
example, by introducing discrete pipe diameter or uncertainties in the parameters.
Different kind of random search technique such as Shuffled Complex Evolution (SCE)
can be applied and compare the results to find a proper algorithm for the water supply
system.
159
7. NOMENCLATURE
Indices and Sets
N a set of nodes in a network (sources, users, and treatment plants)
A a set of arcs (i, j) from a node i to a node j in a network, N∈∀ ji,
W a set of pollutants
T a set of design period t, 6,1=∈∀ Tt
O a set of operation period t, 6,1=∈∀ Oo
Subsets,
CI a set of canal connections, AIC ⊆
PI a set of pipe connections, AIP ⊆
UI a set of pump connections, AIU ⊆
BI a set of connections through a recharge basin to an aquifer, AIB ⊆
II a set of seepage from users to an aquifer and riverbed infiltration, AII ⊆
RI a set of users, NIR ⊆
SI a set of sources, NIS ⊆
SSI a set of storage sources, NII SSS ⊆⊆
NSI a set of non-storage sources, NII SNS ⊆⊆
IWI imported water, NIII SNSIW ⊆⊆⊆
RUI river upstream node, NIII SNSRU ⊆⊆⊆
160
RDI river downstream node, NIII SNSRD ⊆⊆⊆
WTI a set of water treatment plants, NIWT ⊆
WWTI a set of wastewater treatment plants, NIWWT ⊆
Data
f Darcy Weisbach coefficient
ijn Manning coefficient of pipes and canals from i to j CP II U∀
ijz Channel side slope from i to j CI∀
COND hydraulic conductivity
I interest rate
CITY city multiplier
IWA imported water available
oiPOP population at an operation year o at i RI∈i , O∈o
At year 0, 0iPOP initial population at a node i
ooi POPGRPOPPOP )1(0 +=
oPP precipitation at an operation year o O∈o
iAREA area of a node i if a node i is a storage source SI∈i
iCU consumptive use at a node i N∀
iLOSS seepage loss to an aquifer at a node i N∀
iEL elevation at a node i N∀
ijL length for an arcs (i, j) A∀
161
iREL required water elevation at a node i if a node i is a storage source SI∈i
iRQ required discharge at a node i if a node i is a non-storage source SI∈i
iWQR water quality requirement at a node i SR II U∀
iWQ∆ water quality increasing at a node i RI∀
iWQRE water quality removal efficiency at a node i TS II U∀
0ikc water quality of pollutant k in a node i in year 0 if a node i is a storage source
SSI∈i
BOT river bottom width
LRI river length
VRI average velocity in river
Hmin,j minimum pressure requirement at the end of pipe and pump connections
Areabasin groundwater basin area
V recharge rate
Decision variables
tijx takes value 1 if an arcs (i, j) is built at a design period t
and 0 otherwise CP II U∀ , T∈t
tijµ takes value 1 if a pump in an arcs (i, j) is built at a design period t
and 0 otherwise UP II U∀ , T∈t
tiy takes value 1 if a water and wastewater treatment plant (i) is built at a design period t
and 0 otherwise WWTWT II U∀ , T∈t
162
tijκ pipe diameter [L] or canal depth [L] for an arcs (i, j) at a design period t
CP II U∀ , T∈t
tijχ capacity of pump for an arcs (i, j) [L3/T] at a design period t
UP II U∀ , T∈t
tijH Design head of pump for an arcs (i, j) [L] at a design period t
UP II U∀ , T∈t
oijq flow allocation for an arcs (i, j) at an operation year o [L3/T] A∀ ,
O∈o
tiw capacity at a node i at a design period t [L3] WWTWT II U∀ , T∈t
oijk
c water quality concentration of pollutant k for an arcs (i, j) in an operation year o
oiWEL water table elevation at a node i in an operation year o SI∀ , O∈o
oij∆ elevation differences for the pump installation at an arcs (i, j) in an operation year o
⎪⎩
⎪⎨⎧ ∈
−⎪⎩
⎪⎨⎧ ∈
=otherwiseEL
iifWEL
otherwiseEL
jifWEL
i
oi
j
oj SS II
oijS Channel bottom slope for an arcs (i, j) in an operation year o ⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆=
ij
oij
L CI∀ ,
O∈o
oi
D demand at a node i in an operation year o RI∀ , O∈o
Indices and sets related to SFLA
U(i) frogs in entire population, i = 1, …, F
163
Yi frogs in a memeplex, i = 1, …, m
Zi frogs in a sub-memeplex, i = 1, …, iq
Data related to SFLA
F total population
M number of memeplexes
N number of frogs in each memeplex
d number of decision variables
im iteration count in frog leaping algorithm
iN shuffle count
Smax maximum step size
ic maximum number of iterations
is maximum number of shuffles
Variables related to SFLA
F(i) performance value of a frog U(i)
PX best frog in entire population F
PB best frog in a sub-memeplex
PW worst frog in a sub-memeplex
s step size
164
8. REFERENCES
Clark, R. M., Sivaganesan, M., Selvakumar, A., and Sethi, V. (2002). “Cost models for
water supply distribution systems.” Journal of Water Resources Planning and
Management, 128(5), 312-321.
Ejeta, M. Z., McGuckin , J. T., and Mays, L. W. (2004). “Market exchange impact on
water supply planning with water quality.” Journal of Water Resources
Planning and Management, 130(6), 439–449.
Eusuff, M., Lansey, K., and Pasha, F. (2006). “Shuffled frog-leaping algorithm: a
memetic meta-heuristic for discrete optimization.” Engineering Optimization,
38(2), 129-154.
Ocanas, G. and Mays, L. W. (1981a). “A model for water reuse planning.” Water
Resources Research, 17(1), 25–32.
Ocanas, G. and Mays, L. W. (1981b). “Water reuse planning models: extensions and
applications.” Water Resources Research, 17(5), 1311-1327.
Tang, C. C., Brill, E. D., and Pfeffer, J. T. (1987). “Optimization techniques for
secondary wastewater treatment system.” Journal of Environmental
Engineering, 113(5), 935-951.
US. Army Corps of Engineers, (1980). Methodology for areawide planning studies.
Engineer Technical. Letter No. 1110-2-502, Washington, D.C.
United States Environmental Protection Agency, (2000). EPANET2 Users Manual.
EPA/600/R-00/057, Cincinnati, OH.
165
Walski, T. M., Brill, E. D., Gessler, J., Goulter, I. C., Jeppson, R. M., Lansey, K., Lee,
H., Liebman, J. C., Mays, L., Morgan, D. R., and Ormsbee, L. (1987). “Battle
of the network models: epilogue.” Journal of Water Resources Planning and
Management, 113(2), 191-203.
166
9. TABLES
Table B.1. Input parameters for the single wastewater treatment plant system application
Parameter Value Unit Darcy-Weisbach coefficient, f 0.02 Manning's coefficient, n 0.014 Canal side slope, S 2 Hydraulic conductivity, COND 9.144 m/yr Imported water, IWA 0.062 km3/yr Initial population, POP0 300,000 Population growth rate, POPGR 2.7 %/yr Interest rate, I 2.0 %/yr City multiplier, CITY 1 Annual precipitation, PP 69.8 mm/yr Basin area, AREA 132,771 km2 Required water elevation of groundwater, REL 397 m Required discharge of downstream river, RQ 2.83 m3/s Initial water quality of groundwater, 0
ikc , i = GW 0.04 mg/l
Water quality of imported water, 0ikc , i = IW 30.0 mg/l
Precipitation water quality, 0ikc , i = PP 1.0 mg/l
Upstream river bottom width, BOT 3.05 m Upstream river length, LRI 4,023 m Average velocity of upstream river, VRI 1.52 m/s Agricultural consumptive use, CU 0.26 km3/yr
Operation at year 1 (× 106 $/yr) 21.22 5.39 0.07 0.06
Expansion (× 106 $) 5.86 2.77 0.00 0.15 Operation at year 10
(× 106 $/yr) 22.72 6.22 0.13 0.06
172
Table B.8. Additional input parameters for the multiple wastewater treatment plant system application
Parameter Value Initial population for domestic area 1 300,000 Initial population for domestic area 2 400,000 Initial population for domestic area 3 600,000
Table B.9. Nodal input parameters for the multiple wastewater treatment plant system application
Table B.15. Source storage, river discharge and water quality from wastewater treatment plants in the multiple wastewater treatment plant system application
Figure B.1. Single wastewater treatment plant supply system schematic. Bold arcs represent the 10 decision variables and thin arcs are dependent flows that are computed from mass balance constraints (Note the number in arcs are correspond those in Table B.3).
181
Figure B.2. Multi-wastewater plant system schematic (WT – water treatment plant, DO1 , DO2, DO3 – the first, second and third domestic areas, respectively, ID – industrial area, WW1, WW2, WW3 – the first, second and third potential wastewater treatment plants, respectively, AG – agricultural area, LO – large outdoor area) Note the number in arcs are correspond those in Table B.10.
182
Population
Memeplex 1
Memeplex 2
Memeplex 3
Sub-memeplexSub-memeplex
Sub-memeplex
Sub-memeplexSub-memeplex
Sub-memeplex
Sub-memeplexSub-memeplex
Sub-memeplex
Figure B.3. Representation of population, memeplexes, and sub-memeplexes in SFLA
183
Figure B.4. Best solution and penalty term change with the number of function evaluations in the single WWT plant system
184
800
Imported water
River Upstream
Groundwater Domestic Area
Agricultural Area
Wastewater Treatment Plant (0.1 km3/yr)
11
14
1500
Pipe/PumpD = 600 / 1500 mmQ = 0.70 / 2.88 cms
Pipe/PumpD = 900 mm
Qp = 0.74 / 1.03 cmsHp = 107 m
Q = 1.11 / 1.42 cms
Water Treatment Plant (0.05 / 0.1 km3/yr)3
CanalW = 1 m
Q = 0.34 / 2.07 cms
5
10
Groundwater
PumpQp = 5.13 / 8.77 cms
Hp = 87 mQ = 7.67 cms
Pipe/PumpD = 1200 mm
Qp = 0.74 / 0.93 cmsHp = 152 / 183 m
Q = 1.08 / 1.40 cms
CanalW = 1 m
Q = 0.39 cms
18Infiltration
0.1 kafy
Rec
harg
e
River Downstream
4Canal
W = 1 mQ = 0.59 cms
7
PumpD = 600 / 900 mm
Qp = 0.68 / 0.69 cmsHp = 61 / 74 mQ = 1.03 cms
PumpD = 200 / 200 mm
Q = 0.02 cms
9PumpQp = 0.24 / 0.55 cms
Hp = 145 mQ = 0.36 / 0.80 cms
Figure B.5. Optimal solution of single wastewater plant system application showing the flow allocations and arc capacities/design variables – results from first design period / second design period, respectively. (Note the single value indicates no expansion in the capacity or flow allocations)
Figure B.6. Optimal solution of multiple wastewater plant system application showing the flow allocations and arc capacities/design variables – results from first design period / second design period, respectively. (Note the single value indicates no expansion in the capacity or flow allocations)
186
Figure B.7. Optimal system cost and penalty term changes with the number of function evaluations for the multiple WWT plant system.
187
APPENDIX C: RELIABLE WATER SUPPLY SYSTEM
DESIGN UNDER UNCERTAINTY
C. Graph
188
Reliable Water Supply System Design under Uncertainty
G. Chung1, K. Lansey2, and G. Bayraksan3
ABSTRACT
Long term reliability is the most important design factor for water supply systems. Water
supply systems are particularly impacted by uncertain future conditions. Many research
efforts have attempted to account for data uncertainty while simultaneously improving
economical feasibility and attaining system reliability. However, the large problem size
and the correlated uncertainties make solving the problem difficult to solve. To consider
correlated uncertainties in water demand and supply, this study applies the robust
optimization approach of Bertsimas and Sim to a water supply system design problem.
Robust optimization aims to find a solution that remains feasible under data
uncertainty. For instance, a water supply system will be “robust” so that it can meet
demand under extreme drought conditions. However, such a system can be too
conservative and costly. It is possible to vary the degree of conservatism to allow for a
decision maker to understand the trade-off between system reliability and economical
feasibility/cost.
In this study, the uncertainty factors are controlled by the degree of conservatism such
that the system stability is guaranteed under uncertain conditions. The degree of
1Graduate Student, Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520-360-9554, E-mail: [email protected]) 2 Professor, Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520-621-2512, Fax: 1-520-621-2550, E-mail: [email protected]) 3 Assistant Professor, Department of Systems Industrial Engineering, The University of Arizona, Tucson, AZ 85721, USA (Tel: 1-520-621-2605, E-mail: [email protected] )
189
conservatism is presented as a form of the probability bound of constraint violation. As a
result, the total cost increases as the degree of conservatism is increased, i.e., the
probability bound of constraint violation is decreased. A trade-off appears to exist
between the level of conservatism and imported water purchase, i.e., cost increase. It was
found that the robust optimization approach can be a useful tool to find a solution that
prevents system failure at a certain level of risk within the available budget.
190
1. INTRODUCTION
A municipal water supply system is defined as the physical infrastructure to treat,
deliver water to and collect water from users. The capacities of alternative components
are based upon predictions of future population and climatic conditions. Uncertainty in
predicting these conditions is inherent in all water supply systems. Thus, to reduce the
risk of failure during future operations, it is desirable to consider these uncertainties
during the planning process. A decision made with a deterministic model may result in
two consequences; lower net benefits than expected (i.e., it is more costly to provide the
desired water) or some probability of system failure, where failure is defined as not
meeting a given demand or other system constraint (Watkins and McKinney 1997).
These consequences may be rectified in real-time operations at some cost but flexibility
must be built into the system during the design process to allow for those adjustments.
Deterministic optimization that is based on satisfying demand/supply conditions without
consideration of uncertainty removes this flexibility. Thus, a reliability-based design tool
is needed that can assist decision makers plan a long-term water supply scheme to cope
with the future changes in water demands and supplies.
The complexity of the system and the correlated uncertainties make incorporating
uncertainty a challenging exercise. A number of stochastic optimization approaches have
been applied to water supply system design and operation. Most works have adopted
multi-stage linear or nonlinear optimization techniques. The main objectives of these
studies were to minimize expected total cost for water transfer to spot-markets (Lund and
191
Israel 1995); to develop long- as well as short-term water supply management strategies
(Wilchfort and Lund 1997); to manage water supply capacity under water shortage
conditions (Jenkins and Lund 2000); and to design and operate a water supply system
(Elshorbagy et al. 1997).
Some water supply optimization studies have considered the aspect of system failure
risk. For example, Fiering and Matalas (1990) investigated the robustness of water supply
planning with respect to global climate change for regions where water storage capacity
is limited. Watkins and McKinney (1997) considered uncertainty factors by introducing
the standard deviation of the objective function as a constraint into a two-stage stochastic
model by Lund and Israel (1995). This is embedded in the robust optimization framework
of Mulvey et al. (1995).
Chance constraint modeling intends to limit decisions more directly by considering
uncertainty in model input. For instance, chance constraint model may explicitly limit the
probability of not being able to meet a constraint. Chance constraint models, while
intuitively easy to model, are usually non-convex causing difficulties in optimization and
the approach requires numerical integration of the probability distribution or, if an
invertible probability distribution is assumed to hold, has difficulty considering parameter
correlations.
In this paper, the robust optimization framework of Bertsimas and Sim (2004) is used
to develop a reliable water supply system design. A robust solution can be defined as one
that remains feasible under uncertainty. This type of robust optimization was first
introduced by Soyster (1973) to solve linear programming problems. Soyster’s model,
192
which is linear, significantly constrains the objective function to assure robustness; thus
conservative solutions are found that may be practically unrealistic. Ben-Tal and
Nemirovski (1999 and 2000), El-Ghaoui and Lebret (1997), and El-Ghaoui et al. (1998)
extended the Soyster model. These extensions, however, introduced a higher degree of
non-linearity. Since real systems themselves are likely to be nonlinear, these approaches
make the problem more complicated and difficult to find a solution. The approach of
Bertsimas and Sim controls the degree of conservatism for the system reliability without
increasing the difficulty in solving the original problem.
2. ROBUST OPTIMIZATION FRAMEWORK
The classical assumption in deterministic mathematical programming is that all
parameters (input data) are known precisely and can be represented by some nominal
values. This is rarely the case in real applications since many parameters contain
uncertainties such as in measurement and/or uncertainties due to future. One way to deal
with uncertainty is to design a system that is “robust” to changes in the parameters. That
is, the system remains feasible and operates in a near-optimal fashion for a variety of
values that the uncertain parameters can take. Soyster (1973) formulated the following
deterministic linear programming model to find a solution that is feasible for all uncertain
data belonging to a convex set:
maximize cx
subject to ∑=
≤n
jijij bxa
1
~ , jij Ka ∈~ , nj ,...,1= , i∀ (C.1)
193
0≥jx , j∀
where jK is a nonempty convex set and it considers “columnwise” uncertainty,
Sahinidis, N. and Tawarmalani, M. (2005). GAMS/BARON Solver Manual
Soyster, A. L. (1973). “Convex programming with set-inclusive constraints and
applications to inexact linear programming.” Operations Research, 21, 1154-1157.
Tang, C. C., Brill, E. D., and Pfeffer, J. T. (1987). “Optimization techniques for
secondary wastewater treatment system.” Journal of Environmental Engineering,
113(5), 935-951.
US. Amy Corps of Engineers, (1980). Methodology for areawide planning studies.
Engineer Technical. Letter No. 1110-2-502, Washington, D.C.
228
Watkins, D. W. and McKinney, D. C. (1997). “Finding robust solutions to water
resources problems.” Journal of Water Resources Planning and Management, 123(1),
49 – 58.
Wilchfort, G. and Lund, J. R. (1997). “Shortage management modeling for urban water
supply systems.” Journal of Water Resources Planning and Management, 123(4),
250-258.
Walski, T. M., Brill, E. D., Gessler, J., Goulter, I. C., Jeppson, R. M., Lansey, K., Lee,
H., Liebman, J. C., Mays, L., Morgan, D. R., and Ormsbee, L. (1987). “Battle of the
network models: epilogue.” Journal of Water Resources Planning and Management,
113(2), 191-203.
229
8. TABLES
Table C.1. Gammas for robust formulation
Variables Description Range Γ1 Flow from precipitation to groundwater in operation period 1 [0-1] Γ2 Flow from precipitation to groundwater in operation period 2 [0-2] Γ3 Flow from precipitation to groundwater in operation period 3 [0-3] Γ4 Flow from precipitation to groundwater in operation period 4 [0-4] Γ5 Flow from precipitation to groundwater in operation period 5 [0-5] Γ6 Flow from precipitation to groundwater in operation period 6 [0-6] Γ7 Flow from precipitation to groundwater in operation period 7 [0-7] Γ8 Flow from precipitation to groundwater in operation period 8 [0-8] Γ9 Flow from precipitation to groundwater in operation period 9 [0-9] Γ10 Flow from precipitation to groundwater in operation period 10 [0-10] Γ11 Flow from precipitation to river [0-1] Γ12 Domestic area demand satisfaction [0-2] Γ13 Agricultural area demand satisfaction [0-2] Γ14 Imported water availability [0-2]
230
Table C.2. Choice of iΓ as a function of the maximum probability of constraint violation
Table C.3. Input parameters used for the hypothetical water supply system
Parameter Value Unit Darcy-Weisbach coefficient, f 0.02 Manning's coefficient, n 0.014 Canal side slope, z 2 Hydraulic conductivity, COND 9.14 m/yr Imported water availability, WI 19.6 m3/s Initial population, POP0 1,200,000 Population growth rate, POPGR 2.7 %/yr Interest rate, I 2.0 %/yr City multiplier, CITY 1 Annual precipitation, P 533.4 mm/yr Basin area, Ab 12,645 km2 Basin area contributing to imported water, '
bA 13,909 km2 Required groundwater storage, RS2 9.93 km3 Required downstream river flow, RQ3 11.4 m3/s Unit cost of purchasing imported water, CIW 0.81 $/m3 Agricultural consumptive use (1 – 5 periods), o
AGD 12.5 m3/s
Agricultural consumptive use (6 – 10 periods), oAGD 11.3 m3/s
Table C.4. Node characteristics for hypothetical water supply system
Nodes Area (km2) Loss (m3/s) Agricultural area 1,214 0 Domestic area 2,974 0.0002 Imported water 0 0 Water treatment plant 0 0.0002 Wastewater treatment plant 0 0.0002
232
Table C.5. Arc lengths, type and elevation differences of arcs in the hypothetical water supply system
Links Origin Destination Length (m) Elevation difference (m) Connection type 1 RIU
* WT 4,506 -30 Canal 2 RIU AG 45,062 -91 Canal 3 IW WT 16,093 -61 Canal 4 IW GW 77,249 -152 Canal 5 IW AG 61,155 -122 Canal 6 GW DO 0 244 Pump 7 GW AG 0 152 Pump 8 WT DO 4,506 -30 Pipe/Pump 9 DO WW 16,093 -15 Pipe/Pump 10 WW AG 16,093 -3 Pipe/Pump 11 WW RID 45,062 -46 Pipe/Pump
* IW - Imported water, GW - Groundwater, RIU – Upstream river, RID – Downstream river, WT - Water treatment plant, DO - Domestic area, AG - Agricultural area, WW - Wastewater treatment plant
* FRIUTWT - flow allocation from upstream river to water treatment plant, FRIUTAG – flow allocation from upstream river to agricultural area, FIWTWT - flow allocation from imported water to water treatment plant, FIWTGW - flow allocation from imported water to groundwater, FIWTAG - flow allocation from imported water to agricultural area, FGWTDO - flow allocation from groundwater to domestic area, FGWTAG - flow allocation from groundwater to agricultural area, FWTTDO - flow allocation from water treatment plant to domestic area, FDOTWW - flow allocation from domestic area to wastewater treatment plant, FWWTAG - flow allocation from wastewater treatment plant to agricultural area, FWWTRID - flow allocation from wastewater treatment plant to downstream river
Table C.8. Flow allocations along operational periods when probability of violation is 0.1 Operational
Figure C.1. Water supply system network schematic. Bold arcs represent 14 conveyance structures to be sized and thin arcs represent infiltration from users and sources to the aquifer.
Figure C.3. Infrastructure in water supply system network before optimization. schematic. (q – flowrate; d = canal depth; K = pipe diameter; X = pump capacity; H = pump head)
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Figure C.4. Optimal water supply system operation of nominal problem at year 1 (precipitation – arcs 12, 13, 14, and 15; infiltration – arcs 16, 17, and 18; q – flowrate; d = canal depth; K = pipe diameter; X = pump capacity; H = pump head)
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Figure C.5. Optimal water supply system operation at year 1 when probability violation is 0.1 (precipitation – arcs 12, 13, 14, and 15; infiltration – arcs 16, 17, and 18; q – flowrate; d = canal depth; K = pipe diameter; X = pump capacity; H = pump head)
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Figure C.6. Optimal total cost of the water supply system as a function of the probability bound of constraint violation
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Figure C.7. Total amount of imported water purchased as a function of the probability bound of constraint violation