optimal operation of ground‐water supply distribution systems€¦ · for such cases, it is more...

O P T I M A L O P E R A T I O N O F G R O U N D - W A T E R S U P P L Y

D I S T R I B U T I O N S Y S T E M S

By S. Pezeshk, ~ Associate Member, ASCE, O. J. Helweg, 2 Fellow, ASCE, and K. E. Oliver, ~ Associate Member, ASCE

ABSTRACT: Pumping costs are the major operating expense of ground-water supply systems. This paper presents a nonlinear optimization model to minimize pumping costs for both a well field and a main water-supply distribution system. Con- siderations are given to individual well losses, pump efficiencies, and the hydraulic losses in the pipe network. In addition, the transient drawdown at each well is included in a well-field model. When the demand served is less than the total capacity, there is a potential for reducing costs in the selection of pumps to meet that demand. A simulation model in conjunction with an optimization algorithm is assembled and optimized using the general nonlinear optimization program MI- NOS. For a given demand, the optimization procedure provides the best combination of pumps to meet that demand. Two example problems are given to evaluate the validity of the underlying assumptions and to demonstrate some of the characteristics of the proposed procedure.

INTRODUCTION

Ground water supplies a grea ter p ropor t ion of municipal and industr ial (M&I) water than most people realize. Of noncommuni ty water users ( i .e . , users of water not suppl ied by a utility) 97% depend on ground water for their potable water supply, and 80% of smal l -communi ty public-water-supply systems use ground water . These small utilities serve 70,000,000 people in the Uni ted States. Approx ima te ly 40% of the large utility water suppliers tap ground water ( "Na t iona l " 1986). Normal ly , larger water suppliers use well fields connected to a pipe ne twork to deliver water to a central pumping station.

For example, Memphis Light , Gas, and W a t e r ( M L G W ) , the water- supply utility for the met ropol i tan area in, and around, the city of Memphis , Tennessee, delivers water to a populat ion of about 1,000,000 people. M L G W operates 10 pumping stations, each connected to a well field. Each well field has between 10 and 25 wells for a total of 161 wells. The pumping costs to deliver water exceeds $4,000,000 annually, comprising the ma jo r par t of the operat ion and maintenance ( O & M ) costs. Obviously M L G W and other utilities a t tempt to minimize these costs. Because of t rea tment requirements , the wells do not pump water direct ly into the water mains, but del iver the water to a reservoir from which the service pumps draw. Thus, the optimization of well-field opera t ion is separable from the opera t ion of the pumping stations.

Though related, the del ivery costs for the main distr ibution system and a well field can be further separa ted into the cost to opera te an individual well (pump) and the cost to collect the water from each pump into the pipe network. The present paper shows an approach to opt imize the del ivery

1Assoc. Prof., Dept. of Civ. Engrg., Memphis State Univ., Memphis, TN 38152. 2prof., Dept. of Civ. Engrg., Memphis State Univ., Memphis, TN. 3Grad. Res. Asst., Dept. of Civ. Engrg., Memphis State Univ., Memphis, TN. Note. Discussion open until March 1, 1995. To extend the closing date one month,

a written request must be filed with the A~CE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 18, 1993. This paper is part of the Journal of Water Resources Planning and Management, Vol. 120, No. 5, September/October, 1994. �9 ISSN 0733-9496/94/0005-0573/$2.00 + $.25 per page. Paper No. 5375.

573

J. Water Resour. Plann. Manage. 1994.120:573-586.

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costs for both a well field and the main distribution system, considering the individual well losses, pump efficiencies, and hydraulic losses in the pipe network.

MLGW, the nation's largest three-service utility, obtains water from the Memphis Sands, one of the most prolific aquifers in the southern United States. Each of the 10 well fields delivers water to the reservoirs of the pumping stations attached to that particular well field. The service pumps (usually three or four in each pumping station) pump the water throughout the water mains, maintaining the required pressure.

When the full capacity of a well field or of pumping Stations is not required, there is an opportunity to minimize pumping costs. There are several methods utilities use to determine which pumps to operate: (1) Turning on pumps in order of decreasing capacity; (2) turning on pumps in order of decreasing efficiency; (3) turning on pumps to equalize on-line time; (4) turning on pumps arbitrarily at the operator decision; and (5) turning on pumps in order of increasing cost of water pumped. This last criterion, which we believe is the most logical starting point, is deceptively simple because of the water-distribution complexity, large number Of pumps, and interactions that cause the cost of water from each pump to change with each combination of pumps used. The contribution of the present paper is to consider all well, pump, and pipe losses and interactions in developing a nonlinear optimization procedure to minimize pumping costs.

The overall objective is to deliver the required water at minimum cost. Pump, well, and pipe-network data are input into simulation and optimization models that comprise the overall system operation strategy, which is then summarized in pumping schedules.

A simple ground-water supply system is shown in Fig. 1. The wells of a given field deliver water to a pumping-station reservoir. The reservoir serves as an interface between the well field and the main distribution system. Service pumps are located in each pumping station to deliver water through the main distribution system. If the relatively minor changes in the water elevations of the interfacing reservoirs are negligible, the well pumping costs and the high-service pumping costs are then separable and can be optimized separately. First, the least-cost pumping algorithm is developed for each well field, then the optimal pumping schedule for the main distribution system is constructed.

BACKGROUND

Quite frequently, studies reveal that a linearized approach is used beyond its limits of applicability to solve the nonlinear optimal operation of ground- water distribution systems. Many simplifying assumptions are made to reduce the dimensionality and complexity of the optimal pump scheduling problem. However, in real engineering situations where the qualitative nature of the behavior is completely unknown, a linearized approach does not provide adequate representation of the nonlinear nature of the problem. For such cases, it is more appropriate to optimize a ground-water distribution system on the basis of a nonlinear optimization problem when the system has an inherent tendency to possess nonlinear behavior.

A number of investigators have addressed the problem of optimal pump operation. Some have linearized the system, and others have made simplifying assumptions to reduce the dimensionality and complexity of the problem. Lansey and May (1989) and Brion and Mays (1991) developed an optimal operation of pumping stations by interfacing KYPIPE, a simulation

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Well fwld

J 1 l k

I

Main distribution system WeU.Oeld

�9 - well pump, service pump

J r - reservoir

~ t - external demand

FIG. 1. Ground-Water Supply System

program (Wood 1980) and an optimization model (GRG2). Brion and Mays (1991) applied their methodology to the Austin, Texas, water-distribution system and were able to achieve a reduction in operating costs. Most re- cently, Jowitt and Germanopoulos (1992) presented a procedure based on linear programming. Their method relies on a set of assumptions decoupling the operation of pumps from the nonlinear hydraulic characteristic of the network.

One class of optimal operation that is based on simplifying assumptions to reduce the dimensionality and complexity of problem is dynamic programming. Dynamic-programming methodology has been approached by many researchers (Sterling and Coulbeck 1975; Joalland and Cohen 1980; Coulbeck and Orr 1983; Ormsbee et al. 1987). Sabet and Helweg (1985) described a dynamic-programming model that considered an algorithm for pump operation in a small pipe-network system serving a population of 45,000 or less. The primary intent of their model was minimization of costs relative to system hydraulics and varying power unit costs.

The formulation of the optimization problem must be such that sufficient representation of the network hydraulic characteristics and pumping costs is included, without the resulting solution being too complex for computer implementation. The need for an efficient solution is increased for real-time operation. The rapidly varying nature of consumer demands requires pump schedules to be obtained in real time for the full benefits of optimal control to be achieved.

As stated, our aim is to provide efficient computer-based-decision support to optimally control a complex water-supply distribution system. There are two separable tasks necessary for optimizing the water-distribution sys-

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85

75 ~''~ ~ - 'I "~'--[~-] ..._ _.c<

" - : 55

45 0 10000 20000 30000 40000 50000 60000 70000

Q (cmd)

FIG. 2. Head-Discharge Curves

0.90

0.85-

.o

0.80

0.75- o 20000 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 60000 10000 80000

O (cmd)

FIG. 3. Efficiency Curves

70O00

tem: (1) A least-cost strategy to operate each well field; and (2) a least-cost strategy to operate high-service pumps. Both tasks require information about each pump and the interaction between pumps. The pump characteristic curves can be obtained by an on-line pump test. These curves are the head- discharge curve, efficiency curve, and cost-discharge curve. The pump char-

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320000,

270000

220OOO

17001)0

120000

�9 �9 ~ [ - ~ - D ]

70000 10000 20000 30000 40000 50000 60000 70000 80000

Q (crnd)

FIG. 4. Cost-Discharge Curves

acteristic curves used in example 2 are shown in Figs. 2, 3, and 4 where cmd = cubic meters per day (m3/d). In general, knowing the pumping cost associated with each pump is a good approximation of least-cost operation. However, this procedure might not provide optimal scheduling because it neglects consideration for interaction among pumps and energy losses in the distribution system.

The following sections formulate a nonlinear optimization procedure for least-cost operation of a well field and a water-distribution system. The method is illustrated with two example problems.

FORMULATION OF OBJECTIVE

The equations used in the optimization procedure can be modeled by classic nonlinear algorithms. The objective function is formulated as the least amount of energy to produce a minimum specified flow:

Minimize ~ ~TDHiQi (1) i= 1 ei

where I = total number of pumps in the distribution system; Qi = flow at pump i; ei = efficiency for pump i; y = specific weight of water; and TDHi = total dynamic head developed at pump i.

The relationship of flow Qi versus TDH,. is referred to as the pump- characteristic curve. This characteristic curve can be approximated by a quadratic equation

TDHi = A t dhtQ2., ~ , q_ --,Rtclh()~, q_ --,(?(dh (2)

in which A ~ah, --,171 t.dh, and _zc 'dh = constants for pump i. Similarly, the efficiency ei is represented in terms of the flow Q~ in a quadratic form

e, = A e Q 2 + B~Qz + C7 (3)

where the coefficients A~, B~, and C e can be determined by a curve fitting

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of the efficiency curve for pump i as shown in Fig. 3. Substitution of (2) and (3) into (1) yields an objective expressed in terms of I variables

tdh ~'~ 3 R tdh 0 2 M i n i m i z e ~t(Ai ~Zi + + C ~ a h a i )

e 2 e ~=1 (A~Qi + B~Q~ + C e) (4)

The objective function is modified in the next section by the addition of penalty variables. The penalty variables are used to hasten the convergence of certain constraints.

The objective function is not continuous; it consists of discrete values, one value for each possible combination of operating pumps. Neither is the objective function necessarily convex; there may be several local optima that depend on the decision path followed. An initial starting point is carefully chosen to increase the possibility of achieving a global optimum. Since the unconstrained pump flows from the on-line pump tests are known, a good initial point is to proportion each pump flow so that the total flow from the pumps equals the specified demand. Using these values, a simulation routine such as the KYPIPE (Wood 1980) model can produce the initial estimates for each flow and head in the system.

FORMULATION OF CONSTRAINTS

In modeling a water-distribution system, two basic equations are required to achieve a feasible solution: flow continuity and conservation of energy.

For each junction node j, a continuity relationship equating the flow into the junction node to the external demand is written as

Q~ = Dj; j = 1 , . . . , J (5) k = l

where Kj = number of pipes that connect to junction node j; Qk = flow in pipe k into junction node j; and Dj = external demand at junction node j. As in the classical pipe-network algorithms, each pipe must have an assumed direction of flow.

For each closed loop l, energy conservation is written as

Kt

FkQ~, = 0; l = 1 . . . . , L (6) k = l

where K~ = number of pipes in closed loop l; Fg = a head-loss coefficient for pipe k (determined by the physical characteristics of the pipe and including valve losses); Qk = flow in pipe k for closed loop l; and n = a coefficient for head loss (n = 1.852 for the Hazen-Williams equation).

The constraint that requires the system to supply a minimum flow is

l J

Z Qi-> Z Dj (7) i = l / = 1

where J = total number of junction nodes; and Dj = external demand at junction node j. If Dj = actual demand, then this constraint forces the system to not tap the reservoirs. A larger value of Dj can require the system to replenish the reservoirs in preparation for the next high-demand period.

For the main distribution system, each junction node j may be restricted to a maximum pressure p~,x and a minimum pressure, p~i,

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p? , • > p j > p,~,n; j = 1 . . . . , J (8)

The energy conservation must also be written between two fixed-grade nodes (from the water surface in a well or pump E,- to the water surface in a reservoir or elevated tank E,) a total of M times such that

M = I + R - 1 (9)

where M = number of pseudoloops; and R = number of reservoirs or elevated tanks. Each pump i and each reservoir r must be included in at least one pseudoloop equation

grn

E r = E i "Jr T D H i - ~ F~Q~ (10) k = l

where Er = water-surface elevation in reservoir r; E~ = water-surface elevation at the intake structure of pump or well i; and Km = number of pipes in pseudoloop m.

For a well, the term E~ in (10) is not constant. The water levels in a well field are not static because of regional and local changes induced by pumping wells within the field (Claborn and Rainwater 1991). The water level in the well at any time can be estimated using an appropriate unsteady drawdown equation such as the Theis equation for confined aquifers or the Boulton equation for unconfined aquifers. These equations provide estimates of the drawdown at a point defined as the effective radius of the well as a function of time, pumping rate, and aquifer properties. The water level in the well can be defined by

Ei = Zi - S W L i -- Si (11)

where Z~ = pump block elevation for well i; S W L i = static water level at well i before pumping; and sg = drawdown at well i computed from the appropriate equation as a function of time since pumping began and the pumping rate. The drawdown at well i can be represented in terms of Qg in an exponential form such as

s, = a,Q~i + [3~Q, (12)

where a~, [3~, and 8~ are coefficients determined for the drawdown curve of well i.

For a pump, combining (2) and (10) yields

gm

Er = Ei q- --iAtdh()2~-~i + --,R'dhO~, + --iC'dh -- ~ FkQ~ (13) k = l

For a well, a combination of (2), (10), (11), and (12) becomes

Er = Zi - - S W L i - a~Q~, - [~iQs + --iAtdhO2~z.i

g m + --,R ~"hO.~, + v,(:~dh _ ~ FkQ~

k = l (14)

Eqs. (13) and (14) are not applicable when the pump is off (Qi = 0). We are faced with a binary problem that cannot be handled when some of the pumps are off. To handle this problem, multiply (13) and (14) by Qi

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Qi A:ahO ~ , -t- _,R ~ahr -t- v,('Tt.dh "t- E i - E, - ~ F~QT, = 0 (15) k = l

and

Qi (A~dhO ~.~, -F R t.ahO.-t ~:., q- Ct'dhv, q- Zi - Er - SWZi

) - ot, e ~ , - ~3iQi- ~ FkQT, = 0 (16) k = l

If pump i is off, then Q~ is zero; otherwise, (13) or (14) are zero, and Q; is the discharge from the pump. The optimization program determines a feasible solution with the least cost. In addition, to make sure that during an optimization problem the right-hand side of (15) and (16) becomes zero, introduce a penalty variable T~ for each closed and pseudoloop equation. The penalty variable T~ is included in the objective function multiplied by a large number Tp. This approach hastens convergence.

The new objective function is

Minimize ~ (.3'TDHiQ~+ T~ Tp) (17) i = 1 \ ei

where the value of the penalty parameter Tp is chosen to be

- - 1,000 (18) \ e~ / max

After introduction of the penalty variable T~ for a well field, (15) takes the form

Q, A 'dhi Qi2 +_,R~.dhO,~, + v,C~ ah + E l - E r - ~ FkQT, = Ti (19) k = l

and for a distribution system, (16) takes the form

Q~ (A~dhQ~ + B~dhQ~ + C~ ah + Zi - E, - SWL~ N

) - a i Q ~ ' - ~ e l - ~'~ F~Q~ = T~ (20) k = l

EXAMPLE APPLICATION

In the following example problems a skeletal system that only includes the major mains are used in presenting the distribution system. A schematic model of the computer model is developed for each example problem and presented. Once a skeletal model of the distribution system has been developed, a database must be established. The required database includes information on each pipe and node included in the model as well as information on all pumps, tanks, and flow-control valves. In addition, the demand associated with the various nodes must be determined.

Pipes in the skeletal model are selected based on size and importance. Lengths must be determined based on the actual dimensions and from maps. Roughness coefficients must be determined by field test, if possible, or by

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3 - p ipe 3

[ - ~ - n o d e 3

�9 - we l l p u m p

- we l l f i e ld r e s e r v o i r

~ pump 2

pump6 6 [ " ~ 4 [-~] 3 [ 2 1 -

pump 7

FIG. 5. Example 1

TABLE 1. Optimum Combinations of Pumps for Example 1

a Case no. (gpm)* Combination of pumps

(1) (2) (3)

1 1,142 7 2 1,371 6 3 2,383 6, 7 4 2,508 2, 6 5 3,446 2, 6, 7

"1 gpm = 5.44 emd.

calibration analysis. Calibration can be done systematically using optimization procedures or by a trial-and-error method using a simulation routine such as the KYPIPE (Wood 1980) program. For the example problems presented here, we assume that the skeletal model has already been calibrated and is ready to be used. Information for junction nodes includes demand and elevation.

As part of the optimal operation study, the pumps must be field tested to determine actual operating characteristics such as efficiency and cost of operation. Each pump may be tested individually or in parallel combination by the step-drawdown method to obtain data over a wide variety of operating conditions. Each pump is started against a closed valve in the discharge line to obtain a shutoff head. After the pump reaches full speed, a pressure reading is taken from a gauge located in the discharge line between the pump and the valve. The power consumption of the pump is also recorded. Once shutoff head and corresponding power are obtained, the valve is opened to the next "s tep." For the remaining steps, the pressure, discharge, drawdown, static water level, and power consumption values are recorded.

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01

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O

f~3

12

-

F~]

-

p-

pip

e 12

nod

e 12

serv

ice

pu

mp

stor

age

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~ t

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26

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�9

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-Z-1

11

12

191

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18

19

22

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11

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27

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]

32 m

5

6

9 ~

10

~

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13

Illl

20

23

1151

31

i

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]

7

3O

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21

28

A

A

[]ff

] pu

mp

D

) B

pu

mp

C

FIG

. 6.

E

xam

ple

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255000-

250000

245000

240000

235000

230000 300000

1 }

,j4

310000 320000

10 / w" 5

I I

330000 340000 350000

Q (cmd)

FIG. 7. Optimum Combinations of Pumps

TABLE 2. Optimum Combinations of Pumps for Example 2

a Case no. (gpm)* Combination of pumps

(1) (2) (3)

1 2 3 4 5 6 7 8 9

10

56,918 58,185 59,559 60,022 60,808 61,449 62,259 62,443 63,246 63,550

25A 26B 27C 30A 31B 32C 25A 26B 27C 31B 32C 33D 25A 27C 30A 31B 32C 33D 25A 26B 27C 30A 32C 33D 25A 27C 28D 30A 32C 33D 25A 26B 27C 28D 30A 32C 25A 27C 28D 31B 32C 33D 26B 27C 28D 30A 32C 33D 25A 26B 28D 30A 32C 33D 25A 27C 28D 30A 31B 33D

"1 gpm = 5.44 cmd.

This process is repeated, increasing the valve opening each time until the valve is completely open.

Example 1 A well f ie ld is s h o w n in Fig. 5. I t cons i s t s of e igh t p ipes , f o u r j u n c t i o n

nodes , t h r e e wel l p u m p s , a n d t h e wel l - f ie ld r e se rvo i r . T h e r e a re a t o t a l o f n ine c o n s t r a i n t s in t h e M I N O S p r o g r a m : f o u r f r o m (5), o n e f r o m (6), o n e

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from (7), and three from (20). The program is able to choose the optimal combination of pumps shown in Table 1 by using a starting point that divides the desired minimum flow among the three wells in proportion to the unconstrained flow from the pump tests.

Example 2 An example distribution system is shown in Fig. 6. It consists of 34 pipes,

16 junction nodes, eight pumps, and two reservoirs (elevated tanks). There are a total of 51 constraints in the example: 16 from (5), nine from (6), one from (7), 16 from (8), and nine from (19). There are two of each pump A, B, C, and D. These pumps are roughly similar with regards to the head- discharge curves, efficiency curves, and cost-discharge curves.

The specified demand is from 305,000 to 345,000 cmd (m3/d) (56,000 to 63,500 gal./min). Six of the pumps are needed to supply this demand. There are 28 possible combinations of six pumps. Ten combinations are found to be optimal for a range of flows, and these 10 combinations are shown in Fig. 7 and listed in Table 2. The optimization algorithm can select the 10 points, but the initial starting conditions need to be specified more carefully than using the proportion of unregulated flow from each pump as an initial starting point.

APPLICATION

These procedures are being developed for MLGW, whose average daily water demand is 757,000 m3/d. The MLGW distribution system has 35 service pumps located at 10 pumping stations. MLGW serves three high- pressure districts north, south, and east of Memphis. There are 13 booster stations and 15 elevated tanks to regulate the pressure in the system. The distribution system is somewhat overtaxed in the eastern part of the system, where the newest station, constructed in the late 1970s, needs help to meet continuing rapid growth in the suburbs.

MLGW presently uses a Harris M9000 SCADA system to control the wells and high-service pumps. The SCADA system is the most popular method for controlling medium and large water-supply systems. The least- cost algorithm displays which well to bring on-line as demand increases in each of the 10 well fields. In the main distribution system, pressure readings from each fire station are also monitored. This information, together with the observed water-use patterns, may improve the initial feasible starting point (which is beyond the scope of this paper). Presently, the algorithm to optimally operate the high-service pumps is being studied to improve convergence and robustness.

CONCLUSIONS

All optimization models require accurate simulation of system behavior and response. In the case of optimal control of a water-distribution system, the response of the water-distribution systems is described in the form of the pressure and flow variation within the network, the changes and characteristics of static and dynamic water level, and values for pump head and pump discharge. The modeled system must represent the actual response in order for the optimization algorithm to work accurately to provide mean- ingful results. Therefore, the simulation model must be calibrated for the system to be optimized.

Significant savings may be obtained from just knowing the cost of water

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from each pump in a system. With good initial starting points, an optimization model can choose a combination of pumps to minimize energy requirements for a specified demand. Present algorithms are sensitive to starting conditions and need to be robust. Work is proceeding to achieve these objectives.

ACKNOWLEDGMENT

The support of Memphis Light, Gas, and Water is gratefully appreciated.

APPENDIX I. REFERENCES

Brion, L. M., and Mays, L. W. (1991). "Methodology for optimal operation of pumping stations in water distribution systems." J. Hydr. Engrg., ASCE, 117(11), 1551-1569.

Claborn, B. J., and Rainwater, K. A. (1991). "Well-field management for energy efficiency." J. Hydr. Engrg., ASCE, 117(10), 1290-1303.

Coulbeck, B., and Orr, C. H. (1983). "Computer control of water supply; optimized pumping in water supply systems--2." Res. Rep. 33, Leicester Polytechnic, Leices- ter, England.

Joalland, G., and Cohen, G. (1980). "Optimal control of a water distribution network by two multi-level methods." Automatica, Oxford, England, 16(1), 83-88.

Jowitt, P. W., and Germanopoulos, G. (1992). "Optimal pump scheduling in water- supply networks." J. Water Resour. Plng. and Mgmt., 118(4), 406-422.

Lansey, K. E., and Mays, L. W. (1989). "Optimization model for water distribution system design." J. Hydr. Engrg., 115(10), 1401-1419.

"National water summary 1986--hydrological events and ground-water quality." (1986). Water-supply paper 2325, U.S. Geological Survey, Washington, D.C., 3- 8.

Ormsbee, L. E., Walski, T. M., Chase, D. V., and Sharp, W. W. (1987). "Techniques for improving energy efficiency at water supply pumping stations." Tech. Rep. EL-87-16, Environmental Laboratory, U.S. Army Waterways Experiment Station, Vicksburg, Miss.

Sabet, M. H., and Helweg, O. J. (1985). "Cost effective operation of urban water supply system using dynamic programming." Water Resour. Bull., 21(1), 75-81.

Sterling, M. J. H. and Coulbeck, B. (1975). "A dynamic programming solution to the optimisation of pumping costs." Proc., Instn. Civ. Engrs., 59, 789-797.

Wood, D. J. (1980). User's manual--computer analysis of flow in pipe networks including extended period simulations. University of Kentucky, Lexington, Ky.

APPENDIX II. NOTATION

The following symbols are used in this paper:

AT, BT, and C7 = quadratic coefficients to calculate ei for pump i; A,dh l:ttdh and Qdh = quadratic to calculate TDHi for pump i; i ~ - - i ~

D~ - external demand at junction node j; Dr = external flow into reservoir or elevated tank r; Ei = water level at beginning node of pseudoloop (well,

pump); E, = water level at final node of pseudoloop (reservoir,

elevated tank); ei = efficiency for pump i;

Fk -- loss coefficient for pipe friction in pipe k; Hk = head loss in pipe k;

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I

J K

K~

L M n

ej p~ax p~nin

Q~ R

S W L i

si TDHi

Zi ai, [~i, and ~i

= total number of pumps for distribution system, total number of wells for well field;

= total number of junction nodes; = total number of pipes; = number of pipes for junction node j; = number of pipes for closed loop l; = number of pipes for pseudoloop m; = total number of closed loops; = total number of pseudoloops; = exponent in pipe friction head-loss equation; = head pressure for junction node j; = maximum head pressure for junction node ]; = minimum head pressure for junction node j; = flow in pipe k; = total number of elevated tanks for distribution sys-

tem, total number of reservoirs for well field; = static water level at well i before pumping; = drawdown at well i; = total dynamic head developed by pump i; = penalty variable for pump i; = penalty parameter; = pump block elevation for well i; and = drawdown coefficients for well i.

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optimal operation of ground‐water supply distribution systems€¦ · for such cases, it is more...

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