Optimal Operation of GroundâWater Supply ... - Semantic Scholar
OPTIMAL OPERATION
OF GROUND-WATER
DISTRIBUTION
SUPPLY
SYSTEMS
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By S. Pezeshk, ~ Associate Member, ASCE, O. J. Helweg, 2 Fellow, ASCE, and K. E. Oliver, ~ Associate Member, ASCE ABSTRACT: Pumpingcosts are the major operating expense of ground-water sup-
ply systems. This paper presents a nonlinear optimization model to minimizepumping costs for both a well field and a main water-supply distribution system. Considerations 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 MINOS. 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 g r e a t e r p r o p o r t i o n of municipal and industrial (M&I) water than most p e o p l e realize. O f n o n c o m m u n i t y water users (i.e., users of water not supplied by a utility) 97% d e p e n d on ground water for their potable water supply, and 80% o f small-community public-water-supply systems use ground water. T h e s e small utilities serve 70,000,000 p e o p l e in the United States. A p p r o x i m a t e l y 40% of the large utility water suppliers tap ground water ( " N a t i o n a l " 1986). Normally, larger water suppliers use well fields connected to a pipe n e t w o r k to deliver w a t e r to a central p u m p i n g station. For example, M e m p h i s Light, Gas, and W a t e r ( M L G W ) , the watersupply utility for the m e t r o p o l i t a n a r e a in, and around, the city of Memphis, Tennessee, delivers water to a population of about 1,000,000 people. M L G W operates 10 pumping stations, each connected to a well field. E a c h well field has between 10 and 25 wells for a total of 161 wells. The p u m p i n g costs to deliver water exceeds $4,000,000 annually, comprising the m a j o r part of the operation and maintenance ( O & M ) costs. Obviously M L G W and other utilities attempt to minimize these costs. Because of t r e a t m e n t requirements, the wells do n o t p u m p water directly into the w a t e r mains, but deliver the water to a reservoir from which the service pumps draw. Thus, the optimization of well-field o p e r a t i o n is s e p a r a b l e from the o p e r a t i o n of the p u m p ing stations. Though related, the delivery costs for the main distribution system and a well field can b e further s e p a r a t e d into the cost to o p e r a t e an individual well (pump) and the cost to collect the water from each p u m p into the pipe network. The present p a p e r shows an a p p r o a c h to optimize the delivery 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 groundwater 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 574
J. Water Resour. Plann. Manage. 1994.120:573-586.
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1
lk
J
I Main distribution system
Wellfwld 9 Jr ~t FIG. 1.
-
well pump, service pump
-
reservoir
-
external demand
WeU.Oeld
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 recently, 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 sys575
J. Water Resour. Plann. Manage. 1994.120:573-586.
85
..._ _.c< -
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75~''~ ~
'I"~'--[~-]
"-: 55
45 10000
0
20000
30000
40000
50000
60000
70000
Q (cmd)
FIG. 2. Head-Discharge Curves
0.90
0.85.o
0.80
0.75o
10000
20000
30000
40000
50000
60000
70O00
80000
O (cmd)
FIG. 3. Efficiency Curves
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 headdischarge curve, efficiency curve, and cost-discharge curve. The pump char576
J. Water Resour. Plann. Manage. 1994.120:573-586.
320000,
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270000 9
9
~[-~-D]
220OOO
17001)0
120000
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 i= 1
(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 pumpcharacteristic 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 577
J. Water Resour. Plann. Manage. 1994.120:573-586.
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
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Minimize
~=1
tdh ~'~ 3 ~Zi + e 2 (A~Qi
~t(Ai
R tdh 0 2 +
C~ahai)
e + 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
i=l
/=1
(7)
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, 578
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p ? , • > p j > p,~,n;
j = 1....
(8)
,J
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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
Er
=
E i "Jr
TDHi -
~
(10)
F~Q~
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 = Z i -
SWLi
(11)
-- Si
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 -- ~
(13)
FkQ~
k=l
For a well, a combination of Er = Zi
--
SWLi -
a~Q~, -
(2), (10), (11), and (12) becomes
[~iQs +
--iAtdhO2~z.i gm
+ --,R~"hO.~, + v,(:~dh _ ~
FkQ~
(14)
k=l
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
579 J. Water Resour. Plann. Manage. 1994.120:573-586.
Qi A:ahO ~, -t- _,R~ahr -t- v,('Tt.dh"t- E i - E, - ~
F~QT,
= 0
(15)
= 0
(16)
k=l
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and
Qi (A~dhO ~.~, -F R t.ahO.-t~:., q- Ct'dhv, q- Zi - Er - SWZi - ot, e ~ , -
~3iQi- ~
FkQT,
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) ei
(17)
i=1 \
where the value of the penalty parameter Tp is chosen to be --
1,000 \
e~
/
(18) 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 -
Er-
~
FkQT,
= Ti
(19)
= T~
(20)
k=l
and for a distribution system, (16) takes the form
Q~ (A~dhQ~ + B~dhQ~ + C~ah + Zi - E, - SWL~ N
- aiQ~'-
~el-
~'~ F~Q~ 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 580 J. Water Resour. Plann. Manage. 1994.120:573-586.
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3
-
pipe 3
[-~
-
node 3
9
-
well pump
-
w e l l field 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 (gpm)* (2)
Combination of pumps (3)
1
1,142
7
2 3 4 5
1,371 2,383 2,508 3,446
6 6, 7 2, 6 2, 6, 7
Case no. (1)
"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 "step." For the remaining steps, the pressure, discharge, drawdown, static water level, and power consumption values are recorded. 581 J. Water Resour. Plann. Manage. 1994.120:573-586.
J. Water Resour. Plann. Manage. 1994.120:573-586.
01 GO f~3
-
F~]
p-
-
12
t
-
-
25 _~
pump A
1131
FIG. 6.
27
Example 2
pump C
1141
19
12
9
23
- I-Z-1
5
18
pum ) B
26
'~
11ol 22
1
~
32
33
191
11
storage t a n k
external demand
I-rl
4
service p u m p
n o d e 12
pipe 12
IS]
9
]
1151
20
Illl
13
m
6
m
pump C
|
pump D
10 ~ -
[]ff]
21
14
['~ 7
28
29
i
A A
pump,
9
pump ]
pump D
3O
31
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255000-
/
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250000
245000
10
w"
,j4
240000
1
235000
230000 300000
5
}
I
310000
320000
I
330000
340000
350000
Q (cmd)
FIG. 7. Optimum Combinations of Pumps TABLE 2. Optimum Combinations of Pumps for Example 2
a Case no. (1)
(gpm)* (2)
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
Combination of pumps (3) 25A 25A 25A 25A 25A 25A 25A 26B 25A 25A
26B 26B 27C 26B 27C 26B 27C 27C 26B 27C
27C 27C 30A 27C 28D 27C 28D 28D 28D 28D
30A 31B 31B 32C 31B 32C 30A 32C 30A 32C 28D 30A 31B 32C 30A 32C 30A 32C 30A 31B
32C 33D 33D 33D 33D 32C 33D 33D 33D 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 field is s h o w n in Fig. 5. I t c o n s i s t s o f e i g h t p i p e s , f o u r j u n c t i o n n o d e s , t h r e e well p u m p s , a n d t h e w e l l - f i e l d r e s e r v o i r . T h e r e a r e a t o t a l o f n i n e c o n s t r a i n t s i n 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
583 J. Water Resour. Plann. Manage. 1994.120:573-586.
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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 headdischarge 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 highpressure 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 leastcost 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 meaningful 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 584 J. Water Resour. Plann. Manage. 1994.120:573-586.
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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, Leicester, 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 watersupply 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., 38. 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 A,dh i ~ -l:ttdh -i ~ and Qdh D~ Dr Ei
= = = =
E, = ei = Fk -Hk =
quadratic coefficients to calculate ei for pump i; quadratic to calculate TDHi for pump i; external demand at junction node j; external flow into reservoir or elevated tank r; water level at beginning node of pseudoloop (well, pump); water level at final node of pseudoloop (reservoir, elevated tank); efficiency for pump i; loss coefficient for pipe friction in pipe k; head loss in pipe k; 585
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I = total number of pumps for distribution system, total number of wells for well field; J = total number of junction nodes; K = total number of pipes; = number of pipes for junction node j; K~ = number of pipes for closed loop l; = number of pipes for pseudoloop m; L = total number of closed loops; M = total number of pseudoloops; n = exponent in pipe friction head-loss equation; ej = head pressure for junction node j; p~ax = maximum head pressure for junction node ]; p~nin = minimum head pressure for junction node j; Q~ = flow in pipe k; R = total number of elevated tanks for distribution system, total number of reservoirs for well field; S W L i = static water level at well i before pumping; si = drawdown at well i; TDHi = total dynamic head developed by pump i; = penalty variable for pump i; = penalty parameter; Zi = p u m p block elevation for well i; and ai, [~i, and ~i = drawdown coefficients for well i.
586
J. Water Resour. Plann. Manage. 1994.120:573-586.
OF GROUND-WATER
DISTRIBUTION
SUPPLY
SYSTEMS
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By S. Pezeshk, ~ Associate Member, ASCE, O. J. Helweg, 2 Fellow, ASCE, and K. E. Oliver, ~ Associate Member, ASCE ABSTRACT: Pumpingcosts are the major operating expense of ground-water sup-
ply systems. This paper presents a nonlinear optimization model to minimizepumping costs for both a well field and a main water-supply distribution system. Considerations 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 MINOS. 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 g r e a t e r p r o p o r t i o n of municipal and industrial (M&I) water than most p e o p l e realize. O f n o n c o m m u n i t y water users (i.e., users of water not supplied by a utility) 97% d e p e n d on ground water for their potable water supply, and 80% o f small-community public-water-supply systems use ground water. T h e s e small utilities serve 70,000,000 p e o p l e in the United States. A p p r o x i m a t e l y 40% of the large utility water suppliers tap ground water ( " N a t i o n a l " 1986). Normally, larger water suppliers use well fields connected to a pipe n e t w o r k to deliver w a t e r to a central p u m p i n g station. For example, M e m p h i s Light, Gas, and W a t e r ( M L G W ) , the watersupply utility for the m e t r o p o l i t a n a r e a in, and around, the city of Memphis, Tennessee, delivers water to a population of about 1,000,000 people. M L G W operates 10 pumping stations, each connected to a well field. E a c h well field has between 10 and 25 wells for a total of 161 wells. The p u m p i n g costs to deliver water exceeds $4,000,000 annually, comprising the m a j o r part of the operation and maintenance ( O & M ) costs. Obviously M L G W and other utilities attempt to minimize these costs. Because of t r e a t m e n t requirements, the wells do n o t p u m p water directly into the w a t e r mains, but deliver the water to a reservoir from which the service pumps draw. Thus, the optimization of well-field o p e r a t i o n is s e p a r a b l e from the o p e r a t i o n of the p u m p ing stations. Though related, the delivery costs for the main distribution system and a well field can b e further s e p a r a t e d into the cost to o p e r a t e an individual well (pump) and the cost to collect the water from each p u m p into the pipe network. The present p a p e r shows an a p p r o a c h to optimize the delivery 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
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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 groundwater 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 574
J. Water Resour. Plann. Manage. 1994.120:573-586.
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1
lk
J
I Main distribution system
Wellfwld 9 Jr ~t FIG. 1.
-
well pump, service pump
-
reservoir
-
external demand
WeU.Oeld
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 recently, 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 sys575
J. Water Resour. Plann. Manage. 1994.120:573-586.
85
..._ _.c< -
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75~''~ ~
'I"~'--[~-]
"-: 55
45 10000
0
20000
30000
40000
50000
60000
70000
Q (cmd)
FIG. 2. Head-Discharge Curves
0.90
0.85.o
0.80
0.75o
10000
20000
30000
40000
50000
60000
70O00
80000
O (cmd)
FIG. 3. Efficiency Curves
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 headdischarge curve, efficiency curve, and cost-discharge curve. The pump char576
J. Water Resour. Plann. Manage. 1994.120:573-586.
320000,
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270000 9
9
~[-~-D]
220OOO
17001)0
120000
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 i= 1
(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 pumpcharacteristic 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 577
J. Water Resour. Plann. Manage. 1994.120:573-586.
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
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Minimize
~=1
tdh ~'~ 3 ~Zi + e 2 (A~Qi
~t(Ai
R tdh 0 2 +
C~ahai)
e + 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
i=l
/=1
(7)
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, 578
J. Water Resour. Plann. Manage. 1994.120:573-586.
p ? , • > p j > p,~,n;
j = 1....
(8)
,J
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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
Er
=
E i "Jr
TDHi -
~
(10)
F~Q~
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 = Z i -
SWLi
(11)
-- Si
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 -- ~
(13)
FkQ~
k=l
For a well, a combination of Er = Zi
--
SWLi -
a~Q~, -
(2), (10), (11), and (12) becomes
[~iQs +
--iAtdhO2~z.i gm
+ --,R~"hO.~, + v,(:~dh _ ~
FkQ~
(14)
k=l
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
579 J. Water Resour. Plann. Manage. 1994.120:573-586.
Qi A:ahO ~, -t- _,R~ahr -t- v,('Tt.dh"t- E i - E, - ~
F~QT,
= 0
(15)
= 0
(16)
k=l
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and
Qi (A~dhO ~.~, -F R t.ahO.-t~:., q- Ct'dhv, q- Zi - Er - SWZi - ot, e ~ , -
~3iQi- ~
FkQT,
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) ei
(17)
i=1 \
where the value of the penalty parameter Tp is chosen to be --
1,000 \
e~
/
(18) 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 -
Er-
~
FkQT,
= Ti
(19)
= T~
(20)
k=l
and for a distribution system, (16) takes the form
Q~ (A~dhQ~ + B~dhQ~ + C~ah + Zi - E, - SWL~ N
- aiQ~'-
~el-
~'~ F~Q~ 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 580 J. Water Resour. Plann. Manage. 1994.120:573-586.
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3
-
pipe 3
[-~
-
node 3
9
-
well pump
-
w e l l field 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 (gpm)* (2)
Combination of pumps (3)
1
1,142
7
2 3 4 5
1,371 2,383 2,508 3,446
6 6, 7 2, 6 2, 6, 7
Case no. (1)
"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 "step." For the remaining steps, the pressure, discharge, drawdown, static water level, and power consumption values are recorded. 581 J. Water Resour. Plann. Manage. 1994.120:573-586.
J. Water Resour. Plann. Manage. 1994.120:573-586.
01 GO f~3
-
F~]
p-
-
12
t
-
-
25 _~
pump A
1131
FIG. 6.
27
Example 2
pump C
1141
19
12
9
23
- I-Z-1
5
18
pum ) B
26
'~
11ol 22
1
~
32
33
191
11
storage t a n k
external demand
I-rl
4
service p u m p
n o d e 12
pipe 12
IS]
9
]
1151
20
Illl
13
m
6
m
pump C
|
pump D
10 ~ -
[]ff]
21
14
['~ 7
28
29
i
A A
pump,
9
pump ]
pump D
3O
31
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255000-
/
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250000
245000
10
w"
,j4
240000
1
235000
230000 300000
5
}
I
310000
320000
I
330000
340000
350000
Q (cmd)
FIG. 7. Optimum Combinations of Pumps TABLE 2. Optimum Combinations of Pumps for Example 2
a Case no. (1)
(gpm)* (2)
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
Combination of pumps (3) 25A 25A 25A 25A 25A 25A 25A 26B 25A 25A
26B 26B 27C 26B 27C 26B 27C 27C 26B 27C
27C 27C 30A 27C 28D 27C 28D 28D 28D 28D
30A 31B 31B 32C 31B 32C 30A 32C 30A 32C 28D 30A 31B 32C 30A 32C 30A 32C 30A 31B
32C 33D 33D 33D 33D 32C 33D 33D 33D 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 field is s h o w n in Fig. 5. I t c o n s i s t s o f e i g h t p i p e s , f o u r j u n c t i o n n o d e s , t h r e e well p u m p s , a n d t h e w e l l - f i e l d r e s e r v o i r . T h e r e a r e a t o t a l o f n i n e c o n s t r a i n t s i n 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
583 J. Water Resour. Plann. Manage. 1994.120:573-586.
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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 headdischarge 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 highpressure 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 leastcost 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 meaningful 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 584 J. Water Resour. Plann. Manage. 1994.120:573-586.
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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, Leicester, 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 watersupply 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., 38. 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 A,dh i ~ -l:ttdh -i ~ and Qdh D~ Dr Ei
= = = =
E, = ei = Fk -Hk =
quadratic coefficients to calculate ei for pump i; quadratic to calculate TDHi for pump i; external demand at junction node j; external flow into reservoir or elevated tank r; water level at beginning node of pseudoloop (well, pump); water level at final node of pseudoloop (reservoir, elevated tank); efficiency for pump i; loss coefficient for pipe friction in pipe k; head loss in pipe k; 585
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I = total number of pumps for distribution system, total number of wells for well field; J = total number of junction nodes; K = total number of pipes; = number of pipes for junction node j; K~ = number of pipes for closed loop l; = number of pipes for pseudoloop m; L = total number of closed loops; M = total number of pseudoloops; n = exponent in pipe friction head-loss equation; ej = head pressure for junction node j; p~ax = maximum head pressure for junction node ]; p~nin = minimum head pressure for junction node j; Q~ = flow in pipe k; R = total number of elevated tanks for distribution system, total number of reservoirs for well field; S W L i = static water level at well i before pumping; si = drawdown at well i; TDHi = total dynamic head developed by pump i; = penalty variable for pump i; = penalty parameter; Zi = p u m p block elevation for well i; and ai, [~i, and ~i = drawdown coefficients for well i.
586
J. Water Resour. Plann. Manage. 1994.120:573-586.
