lansey-1994,from google

OPTIMAL PUMP OPERATIONS CONSIDERING PUMP SWITCHES

By Kevin E. Lansey,' Associate Member, ASCE, and Kofi Awumah 2

AaSTRACT: A methodology for determining optimal pump operation schedules for water-distribution systems is presented. In addition to minimizing the energy- consumption cost, the model includes a constraint to limit the number of pumps that are switched on during the planning period. A two-level approach is adopted whereby the system hydraulics are analyzed in an off-line mode to generate simplified hydraulic and cost functions for an on:line model. These functions developed for each pump combination allow for rapid evaluation within a dynamic programming optimization algorithm. Constraints can also be included for tank levels, rate of change of tank level, pump switches during each period, and maximum energy consumption. Solution times for small to medium-sized systems show that the model can be used in an on-line mode to determine pump operation schedules; however, the applicability of the model is limited by the number of pumps in the system. An extensive sensitivity that highlights operational problems and the utility of the approach is presented.

INTRODUCTION

Interest by practitioners and researchers in pump scheduling for municipal water-supply systems has grown recently. Researchers have focused upon the objective of minimizing pumping costs. Operating cost, however, is not the only criterion used by operators to judge the quality of a pump schedule. Pump switching, in many cases, is equally, if not more, important.

For example, the prime stated objective in one city's operator's manual is to minimize the number of pump switches while maintaining adequate pressure. This paper presents an approach for including pump switches within a least-cost optimization model that is applicable to a class of systems.

Previous efforts have resulted in a range of tools for developing optimal pump schedules that only consider the cost of pumping (Ormsbee and Lan- sey, in press, 1993). One of the earliest works was presented by DeMoyer and Horowitz (1975), who applied dynamic programming (DP) using a simple hydraulic relationship. This publication was followed by a series of DP approaches (Sterling and Coulbeck 1975; Jolland and Cohen 1980; Zes- sler and Shamir 1989). Ormsbee et al. (1989) formulated and solved a DP model for a single-tank system after developing a set of system curves. The curves relate the minimum cost of pumping and the change of tank level with the demand level and present tank level, which represent the network hydraulics.

Other solution techniques have also been applied to the least-cost-operation problem. Linear programming was applied by Jowitt (1989) and others by linearizing or simplifying the network hydraulics. To fully account for the nonlinear network relationships, several authors have applied non-

~Asst. Prof., Dept. of Civ. Engrg. and Engrg. Mech., Univ. of Arizona, Tucson, AZ 85721.

aRes. Assoc., Dept. of Civ. Engrg. and Engrg. Mech., Univ. of Arizona, Tucson, AZ.

Note. Discussion open until July 1, 1994. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 26, 1991. This paper is part of the Journal of Water Resources Planning and Management, Vol. 120, No. 1, January/February, 1994. 9 ISSN 0733-9496/94/0001-0017/ $1.00 + $.15 per page. Paper No. 3213.

17

J. Water Resour. Plann. Manage. 1994.120:17-35.

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linear programming. Chase and Ormsbee (1989), Lansey and Zhong (1990), and Brion and Mays (1991) linked network-simulation models with nonlinear optimization algorithms to determine optimal pump operations. None of these algorithms considers the number of pump switches as a criterion for making operation decisions.

Little and McCrodden (1989) solved a mixed-integer linear-programming problem for a pipeline water-supply system. Since flow is pumped to a free outlet, a linear program can be formulated. The integer portion incorporates the discrete operations and a maximum power-usage constraint but does not add pump-switching constraints.

The approach in the present paper is similar to that of Ormsbee et al. (1989), in which the network hydraulics are simplified to nonlinear relationships. Unlike Ormsbee et al., the curves are developed for each pump combination. Independent pump combinations are then examined in a dynamic-programming problem with a constraint on the number of pump switches. This methodology can be used to solve for optimal operations of networks with one or two tanks and a reasonable number of pumps.

The next section of this paper is a detailed description of the model and solution method. A numerical example follows, with a discussion about the use of this type of model.

PROBLEM FORMULATION

The objective of the problem is to provide a pump-operation schedule, for the planning horizon, that has the lowest energy cost, while limiting the number of pump switches. Energy-cost savings can be obtained because the unit cost of energy may vary over the planning period and the proper use of the pumps in conjunction with the flow of water into and out of the elevated storage tanks can yield lower operational cost. In addition, the efficiency at which the pumps operate depends on the head and the discharge at which they operate.

Another important cost issue that deserves consideration is pump maintenance. An operation schedule in which pumps are turned on and off many times may reduce energy consumption. However, this schedule may increase the wear on the pumps and the resulting pump-maintenance costs. This cost has not been quantified, but it can be assumed that it increases as the number of pump switches increases. Hence the number of pump switches is used as a surrogate measure for the intangible wear-and-tear cost.

Since switching is not a quantifiable cost, the operation problem has two objectives that do not have common units. Therefore, the switching objective is introduced into the model as a constraint. The operator can then evaluate the trade-off of increasing cost to reduce the number of switches by reviewing the model results. The minimum cost-pump switch problem can be stated mathematically as

NLOAD NPUMF ( HPi, ie ~ minimize total energy = i=1~ ip=l~ Xijp, CtQP~jp, 550epijp! . . . . . (1)

Subject to

q, = QEXT for all nodes and loads . . . . . . . . . . . . . . . . . . . . . (2) iI~NLN

HL,,, + ~ HPUMP,p = AE for all loops and loads . . . (31 ill~ NLL ip ~ NPL

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Q Ti,~ At Ti= Ti_, + - ATk

/)max ~ P >- Pmin

Zmax ~ Z~- Znain

TNLOA D = TFINA L

PSWma x ~ Paw i

NLOAD

NSWmax >~ E i~1

Xi,jp = 0 or 1

for each tank and all times . . . . . . . . . . . . . . (4)

for all nodes and loads . . . . . . . . . . . . . . . . . . . . . . (5)

for all tanks and loads . . . . . . . . . . . . . . . . . . . . . . (6)

for each tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

for each load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

PSW~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

for all pumps and loads . . . . . . . . . . . . . . . . . . . . . . (10)

(To is given) where NLOAD = number of periods in the planning horizon; NPUMP = number of pumps in all pump stations; Ci = unit cost of energy in period i; QPijp, HPijp, and epijp = pump discharge, pressure head, and wire-to-water e~ficiency for pump jp during period i, respectively; X~,jp = integer on-off variable for pump jp in period i; and X~jp defines the pump- operation schedule and is the operation decision considered in this model.

Eq. (2) requires that mass be conserved at each node; Qexr are the nodal demands; and q~l is the flow in pipe il in the set of NLN links that are connected to the node. Constraint 3 represents the conservation of energy for all loops in the system, where HLilt is head loss along pipe link ill, HPUMPIp is the head supplied by pump ip, and zXE is the difference between fixed grade nodes or zero for a closed loop. NLL and NLP define the sets of links and pumps in each loop. The tank levels are updated in constraint 4, where QT~,,~ is the flow into tank k during period i, ATk is the area of tank k, and At is the length of time period. This paper considers only a single tank system (k = 1). Given the dimension of the tank-transition functions developed later and the optimization scheme, k is limited to 1 or 2. Constraints 2, 3, and 4 are sets of nonlinear equations written for each demand condition through a day; P, the vector of nodal pressure heads for all loads and nodes, are bounded in (5) by Pmax and Pmin, the maximum and minimum allowable pressure nodal heads, respectively; Tm~x and Tmirl are the maximum and minimum allowable tank-water-surface elevations, respectively; and T is a vector of the tank-water-surface elevations; To, the initial water surface elevation is known by observation and the final tank elevation; TNLOAO, is predefined as TF1NAL; PSWmax is the maximum number of pump switches allowed for each period; NSWma x is the maximum total number of pump switches allowed over the planning horizon; and PSWi the number of pump switches during period i.

A pump switch is defined as turning on a pump that was not operating in the previous period. The number of switches for period i is

NPUMP

PSW~ = ~ max(0, X~.jp - Xi-ljp) . . . . . . . . . . . . . . . . . . . . . . . . . (11) jp=l

SOLUTION METHOD

The purpose of this model is to determine the optimal pump operations in real time (i.e., on-line) for a system with known hydraulic characteristics and forecasted demands. This goal implies that a short decision period of

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a few hours is used, which accounts for the variation in water demand and energy cost. It also implies that the model should have a short execution time. However, for most networks for every new scenario to be analyzed (choice of pumps operations or changes in nodal demand), a complete hydraulic analysis of the system must be performed. Except for the smallest of systems, this hydraulic simulation is too time-consuming to be done on- line.

A two-level approach is therefore adopted for solving the aforementioned model. The first level is termed "preoptimization work," and involves the generation of hydraulic data for the second level, which is the actual optimization step. A dynamic-programming approach is used to solve the optimization problem. Rather than solve a system of nonlinear equations to determine the state transition (tank-level change) during a given period, polynomial functions are evaluated that approximate these transitions and are developed in the first level.

The index jp in the objective function [(1)] refers to individual pumps in the system. However, since the pump discharge and head are related to the set of operating pumps, the operating pump combination is considered as the decision in the optimization model. Eq. (1) is therefore modified so that it refers to pump combinations. If the system contains N pumps with different characteristics, the total number of pump combinations, NPC, equals 2 N, which includes the case of no operating pumps. The objective function in (1) is then replaced by NLOAD NPC

E c,xci,j,,cec,.j, o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12) i= l jpc= l

where ECi,/e c = energy required, in kilowatt hours (kW.h), to operate pump combmation jpc in demand period i; XCi, jr, c = 1 if pump combination jpc is operating in period i, and 0 if it is not; and Ci = unit cost of energy [$/(kW. h)] in period i. This cost coefficient can account for both variable energy rates and energy surcharges as a result of exceeding limits set on energy use.

PREOPTIMIZATION WORK

To avoid performing hydraulic simulations during an on-line optimization, an approach similar to that of Ormsbee et al. (1989) is applied. Simple functions that describe the network hydraulics and energy consumption are developed. Since pump switches are of concern, these curves are generated for each pump combination. Data required for this preoptimization model are: (1) Pump-characteristic curves; (2) pump-efficiency curves (wire to water); (3) allowable maximum and minimum tank-water-surface elevations; (4) nodal demand; (5) allowable maximum and minimum allowable nodal pressure heads; and (6) other physical characteristics of the network such as pipe sizes, lengths, friction coefficients, and types and location of valves.

The data needed by the on-line model are the rate of change of tank- water-surface levels and the rate of pump-energy consumption. The rate of water-surface-level change depends on the operating pumps, the initial water levels, and the network demand. The energy consumption depends on the same factors and the pump efficiencies,

These data, if given in the form of simple nonlinear functions (curves), will reduce the problem dimensions, required computer memory, and the

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optimization computation time. The energy-consumption rate is used to determine the pump-operation costs. The water-level-change relationships are used as the transition function in the DP optimization and to check the tank-level bounds.

The procedure to generate the functions is as follows. For a given demand load, a pump combination is selected. The allowable range of tank-water depths is discretized into intervals that represent the water-surface elevation at the beginning of a period. A hydraulic simulator, such as KYPIPE (Wood 1980), is executed for the current pump combination at each starting water- surface elevation and the energy consumption and the rate of change of tank water-surface levels are determined. The process is repeated for all discrete water levels.

Using least-squares regression, curves are fit for the energy demand versus starting water surface elevation (r) and the rate of change of water-surface elevation (t) in a tank versus starting water-surface elevation. Typical curves are shown in Figs. 1 and 2. The next pump combination is selected and the process is repeated. After all pump combinations have been considered for the current demand, similar curves are developed for each demand level.

In fitting the regression curves, a general polynomial least-squares method was used in which the model determines the best degree of the polynomial. Preliminary evaluation showed that a quadratic function acceptably fits the rate of change of tank-water-surface elevation (although in some instances a linear function was also obtained). In the case of energy demand by the pumps, a cubic function was adequate for all cases. Thus, three regression parameters were estimated for the former and four parameters were estimated for the latter.

To consider the nodal-pressure-head constraints, the maximum and rain-

n" "1- "~ 1.5-

LU

z 1

UJ ~9 Z

< 0.5 I o LL O LU 0-

n- )1( )K >1( )1( ~[< ))C 3LC )1( )K

-0.5 341 3~,2 ' 3~,3 344 ' 345

STARTING WATER SURFACE ELEVATION (M)

>< Pump J6 -~- Pump J7 ~ Pumps J6 and J7

FIG. 1. Rate of Water-Surface-Elevation Change versus Starting Water-Surface Elevation for Pump J6, Pump J7, and Pumps J6 and J7

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450

~ 400- 09

~; 350-

n >. 300- rn

250- r r

O 200- LIJ n"

>- 150- (9 r r LIJ z 100" LIJ

50 341

.k

. . . . . . . . . X )K >K )K )]d ' )K ~.r(

3a,2 ' 343 ' 3a,4 ' STARTING WATER SURFACE ELEVATION (M)

345

~ Pump J6 ~ Pump J7 ~ Pumps J6 and J7

FIG. 2. Pump-Energy Requirements versus Starting Water-Surface Elevation for Pump J6, Pump J7, and Pumps J6 and J7

imum allowable pressure heads are computed by the hydraulic-simulation model. For every demand and pump combination, the starting tank-water surface elevations below which the minimum allowable nodal pressure head will be violated is noted. Similarly, the maximum starting water-surface elevation above which the maximum pressure head violation occurs is also recorded.

This preoptimization work is completed off-line and generates all the data required by the on-line model to describe the system hydraulics. The only network variable that will vary in the short term is the demand. Typically, demand levels are assumed to vary as incremental proportions of the average demands. This implies that the demands will increase uniformly throughout the system, which may not always be correct. Although any number of demand conditions that typically occur on the system can be evaluated in the preoptimization, the time involved for the preoptimization and data- storage requirements may be limiting. Therefore, a base water demand will commonly be used and data generated for percentages of the base demand. Through time as other system parameters change, a revised run of the preoptimization work can be done to incorporate changes in system characteristics such as pump efficiencies or pipe-friction coefficients.

The hydraulic model evaluates the system for each pump combination and demand level. Therefore, it is not necessary that all pumps be located within one pump station. Pumps at any number of locations can be considered.

ON-LINE DYNAMIC PROGRAMMING MODEL

Dynamic programming has been applied to the pump-operation problem without considering pump switches for general systems with up to three

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tanks and for systems with a special structure with larger dimensions. The least cost-pump switch model in this paper is directed at small systems with up to about eight pumps and one or two tanks. The application in this paper is a single-tank system. Adaptation is necessary for larger systems either in the programming approach or by system decomposition. Oftentimes, large systems consist of a number of small subsystems, with each subsystem rep- resenting a pressure zone that can be considered hydraulically independent of the others. Therefore, each subsystem can be analyzed or operated in- dependently.

As in other DP formulations for this problem, the problem states cor- respond to tank water levels and time is broken into stages. To consider the multiperiod pump switching constraint [(10)], the standard DP procedure must be modified by expanding the state space. The state space for this problem is shown in Fig. 3. It is noted that within each tank level state (Ti), there exists a set of states corresponding to the pump combinations (XCi) and cumulative number of switches from time 1 to i (NSWi).

The recursive equation for this problem is

MIN

F,(XCi, T,, USWg) = XC, [r~(XC~, Ti-1) + F~_I(XC~_I, Ti_l, NSW~_I)]

i = 1 . . . . . NLOAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

where F~(XCi, T~, NSW~) = minimum total cost of operating the system from time 1 to i and ending in the state defined by XC~, T~, NSWg; and r~

Tank Level, T

7

J

2 f

f

J

3

Cumulative Number of Switches, NSW 6

5 4 j

3 / XC=l

/ / 4

Pump Combination XC

FIG. 3.

Stage i

State-Space Grid for Pump Operation with Switching Constraint

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= transition cost for operating the system in period i starting at tank level Ti_ 1, using pump combination XG and ending in tank level T~.

The DP proceeds forward in time from the known initial tank-water- surface level and pump combination. At each stage i, the procedure evaluates all initial tank level states Ti_ 1 for a selected pump combination. By choosing the pump combination, the switch states can also be determined. The evaluation for each pump combination, beginning tank level, and number of switches is described in detail in the following. After one pump combination is examined for each beginning state, the next pump combination is considered. When all pump combinations for all states have been examined, the minimum cost for each pump combination for each ending state is retained. The model then proceeds to the next stage. The model then continues until all stages have been examined including the last stage, which requires that the final target tank level be satisfied. The minimum overall solution is determined by minimizing (14) for each pump combination at the last period. Although the least cost will be associated with the maximum allowable pump switches, the solufion algorithm will supply the least cost for each allowable number of switches. An operator can then judge the trade-off between cost and number of switches.

The evaluation for each pump combination and tank level state is as follows. Initially, the pump combination is screened for feasibility with respect to minimum and maximum pressure heads. This screening is completed by comparing the current starting water-surface elevation with the minimum and maximum water-surface elevations, which ensures that the minimum and maximum pressure head constraints, respectively, are satisfied. As noted earlier, these minimum and maximum elevations are determined in the preoptimization work. Additional screenings can be performed to ensure the rate of tank-level change does not exceed a defined value or to account for energy surcharge fees that the energy consumption for a given period does not exceed a threshold value.

After the initial screening, the ending-period water-surface elevation is found by evaluating the tank-transition function (rate of change in water surface elevation) for a given pump combination

Ti.jpc = tjpc(T,_l, XCi, D,) 'At + T~_I . . ! . . . . . . . . . . . . . . . . . . . . . . . (14)

where tipc = rate of water-surface-elevation change for pump combination jpc; and D~ = the demand level for period i. This step determines the tank and pump combination state for the next stage.

To determine the switch state, the switch transition function is evaluated to determine the number of switches for a feasible pump combination, XG_I , in the beginning state for the last stage, i.e.

NSW, = PSW,(XC,_a, XC~) + NSWi_I . . . . . . . . . . . . . . . . . . . . . . . . (15)

PSWj is then checked to verify that it does not exceed the allowable switches in a period [(9)].

The total operation cost for each number of pump switches is computed by summing the present stage's operation cost (return function, r~) with the cumulative cost up to the last stage for pump combination, XG-1 . This cost is compared to others for the same operating pump combination (XG) and the same total number of switches (NSW~); and the lowest-cost value is retained for each pump-combination-number-of-switches state. This effort requires one tank-level transition-function evaluation tip c for each pump

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combination and beginning tank level, and comparison over all pump combinations that could feasibly occur at the beginning state.

Since it is possible to move to a pump combination-switch state from several pump combinations in stage i - 1, two pieces of data are needed for each state to define the optimal state in the previous stage (XCi and XCi_ 1). It is easier, and requires less computer storage, to retain the optimal operation history from stage 0 to stage i for each tank level-pump combination-number of switches state as the algorithm proceeds. Since the operations are stored, only the last-stage cumulative costs must be retained. Also, since the entire operation history is retained, it is available at the last stage. The optimal pump operation history is then used to obtain the optimal tank trajectory using the simplified curves.

Since the water-surface elevation is a continuous variable but is considered at discrete levels it is rounded to the nearest DP state. The optimal solution will be more accurate with higher levels of discretization; however, the computational effort also increases. The trade-off between solution accuracy and level of discretization is presented next.

APPLICATION OF MODEL TO EXISTING SYSTEM

Description of System The optimization approach was applied to an existing network to deter-

mine the optimal pump-operation schedule and analyze some important factors in pump scheduling. The Austin, Tex., water-distribution system consists of eight independent pressure zones. The model was applied to the northwest B pressure zone (NWB), which spans an area of over 12,000 acres (4,900 ha) and supplies a population of about 30,000. NWB comprises the Jollyville pumping station that pumps water from Jollyville reservoir into the network, which includes one elevated storage tank (the Pond Springs Reservoir). A schematic of northwest B pressure zone is shown in Fig. 4, modeled with 126 links and 98 nodes.

The Jollyville pump station contains five pumps (pumps J6, J7, J8, J9, and J10). Although pump J8 has been permanently retired, it has been retained in the analysis portion of this study for demonstration purposes. Also, the pumping capacities of pumps J9 and J10 have been modified to be more similar to the other pumps for this analysis. The modification was made because their actual capacities are quite high [around 22,740 L/min (6,000 gpm)] relative to the average demand for the day under consideration [18,200 L/rain (4,795 gpm)]. Leaving the pumps with their actual capacities will essentially result in the system being reduced to a three-pump system (the other two large pumps rarely being considered by the model). Table 1 lists three head-discharge points for each pump, which can be used to fit quadratic pump curves. These pumps are installed in parallel, and pump curves for all pump combinations can be developed from these data.

The Pond Springs Reservoir has a capacity of 1.14 107 L (3,000,000 gal.) and a tank diameter of 34.5 m (113 ft). The active storage heights range between the overflow at 344.4 m (1,130 ft) above mean sea level (MSL) and allowable minimum water level at 341.4 m (1,120 ft) above MSL which gives an operating tank volume of 2.96 106 L (780,000 gal.). This operational range is intended to give a large contingency storage for emei-- gency use (such as fire fighting) and to maintain adequate pressure head in the network.

The source storage reservoir is the Jollyville Reservoir, which has an 11,000,000 gal. capacity and an overflow at 308.5 m (1,012 ft) above MSL.

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= Pump Station 9 Node o Reservoir

and Pump Station

FIG. 4. Schematic of Water-Distribution System for City of Austin NWB Pressure Zone

Pump (1) J6 J7 J8 J9 J10

TA

Cutoff head (m) (2)

52.8 53.1 47.3 52.8 53.1

iLE 1. Pump Characteristics

Head (m) (5) 35.1 30.5 38.8 35.1 30.5

Design Flow

Head Flow (m) (L/min) (3) (4)

47.3 7,580 37.9 15,160 45.8 3,790 47.3 7,580 37.9 15,160

High Flow

Flow (L/min)

(6) 11,578 18,192 4,684

11,578 18,192

For the day considered in the analysis, it is modeled as a fixed grade node set at 306.4 m (1,005.3 ft) above MSL. Fixing the elevation is a reasonable assumption since the tank has a very large diameter.

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Preoptimization Work To obtain the required hydraulic data for the network, the operating

range of water-surface elevations for Pond Springs Reservoir was discretized into six levels. The hydraulic-simulation model KYPIPE (Wood 1980) was used to evaluate the rate of change of the tank-water-surface elevation and energy requirements for a given spatial demand pattern and the demand factors listed in Table 2. The demand factors apply to all nodes. A quadratic function fit the data points for the rate of change in water-surface elevation, and a cubic function was used for the energy requirements. Typical plots of rate of change of water-surface elevation (t) versus starting water-surface elevation and energy required (r) versus starting water-surface elevation are shown in Figs. 1 and 2. Also, if pressure violations occurred, the minimum allowable was set at 138 kPa (20 psi) and the maximum set at 551 kPa (80 psi), the starting tank elevation at which it occurred was saved. As noted, this elevation is checked during the optimization analysis to verify if a pump combination can supply pressure head. This procedure was completed for all pump combinations, including the no-pump case.

For the application that has a one-day planning horizon and using 2-hr decision intervals, 12 demand levels are needed. A total of 2,304 KYPIPE model runs (32 pump combinations 6 tank elevations 12 demand periods) and the regression analysis were performed in 1.3 hr on an IBM- compatible personal computer (PC) (80386) with a 33-MHz processing speed using the 1986 version of KYPIPE. The total data generated for the on-line model were quite small (48 kilobytes of storage memory).

This preoptimization computational time, if included in the on-line model, would clearly be unacceptable. However, it is not prohibitive even for a midsized system. As an example, for a system with one tank and eight pumps, it is possible to generate these curves over a weekend using an off- line personal computer. One would expect to develop curves for a range of demands varying from a low of about 30% of the average to an instantaneous

TABLE 2. Global Demand and Energy Cost Factors for NWB System in Austin, Tex.

Time of day (hr) Demand factor Energy factor a (1) (2) (3)

0-2 2-4 4-6 6-8 8-10 10-12 12-14 14-16 16-18 18-20 20-22 22-24

0.8792 0.7893 0.8892 1.2290 1.4290 1.1890 0.8393 0.8393 0.7394 1.1391 1.0890 0.9392

1.00 1.00 1.00 1.60 1.60 1.80 1.80 1.60 1.60 1.60 1.00 1.00

~Energy factors were only used during some of the sensitivity analysis runs. The energy factors for the historical event and the number of pump switches sensitivity analysis were constant (1.0) for the entire day.

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peak of near 1.8 times the average. Using a 3% increment, a total of 50 demand levels would be evaluated. Continuing to use six discrete tank levels (more or less may be used depending upon the range of the tank) a total of 76,800 simulations are needed, or about 45 hr of PC time, for a network of this size. The computation time increase is linear in the number of demand and tank levels, and exponential with the number of pumps.

In this example, curves are generated for all pump combinations. It may be possible to reduce computation requirements by eliminating pump combinations using heuristics, or not simulating conditions that are infeasible due to pressure requirements or tank levels. It may also be possible to reduce simulation times by starting the simulation model with a good starting point from a previous pump combination. Since the solution space is discretized in the optimization, it may be possible to linearly interpolate between demand levels to determine the rate of water-surface elevation change at intermediate demands for which curves are not generated. This approach was taken by Ormsbee et al. (1989) in their minimum-cost tank-levePchange curves. In the optimization, the final tank level will be rounded to the nearest state so a small error in its estimate may be tolerable. In this case, curves would be generated for fewer demand levels.

Optimal Operations for Historical Event The optimal operation of the Austin NWB system was determined using

historical data for September 29, 1988. The demand pattern is based on an average demand of 18,200 L/rain (4,795 gpm) multiplied by the demand factors given in Table 2. It was assumed that pumps are switched at the beginning of each 2-hr time interval. When a pump is switched on it must remain on for the entire time interval. The tank water levels were disctetized at 3-in. intervals. A constant unit cost for energy of $0.07/(kW. h) was used. Following the actual operations, the starting water-surface elevation in the Pond Springs tank was 343.1 m (1,125.8 ft), and the target ending elevation was 343.1 m (1,125.5 ft). On this date the operators only used pumps J6, J7, and J8. Therefore, for this analysis only these three pumps are included in the optimization model; and it was assumed that no pumps were operating at the beginning of the first period.

The actual pump schedule followed and the optimal schedule computed by the optimization model are given in Table 3. The tank water-level trajectories followed on the actual date and the optimal solution are shown in Fig. 5.

There was a significant difference in the two schedule's costs. The actual operation cost was $231.0, the optimal solution was $211.63, or a 9% energy- cost savings. The computational time for this model run was 17 sec on an IBM-compatible personal computer (80386 with a 20 MHz processing speed). In addition to finding a better cost schedule, the optimal solution was achieved with fewer pump switches. It must be recalled, however, that the optimization analysis is a postevent analysis. The operators may have predicted different demands than actually occurred or may have attempted to maintain a high reliability level by keeping the tank full.

Substantial cost savings were obtained because the optimal solution used two pumps in only three periods compared to the nine periods during the actual Operations. The aCtual operations kept the tank water level high (by using two pumps most of the time) while the model took advantage of the storage by "emptying" the tank for a long period (see Fig. 5). The model

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TABLE 3. Actual and Optimal Pump Schedules for September 29, 1988

Time of day (hr) Actual pumps Optimal pumps (1) (2) (3)

0-2 2-4 4-6 6-8 8-10 10-i2 12-14 14-16 16-18 18-20 20-22 22-24

[Total cost (dollars)] [Number of pump switches]

J7 J7

J6, J7 J6, J7 J6, J7 J6, J7

J7 J6, J8 J6, J8 J6, J8 J6, J8 J6, J8 231.00

4

J7 J7

J7, J8 J7, J8

J7 J7 J7 J7 J7

J6, J7 J6, J7

J6 211.63

3

~ 345[-

344.5-t . . . I - -% 344

H / z ~ 341.5-

341 0 ~i 1'0 1'5 2'0 25

TIME OF THE DAY (HOURS)

Actual Trajectory ~ Optimal Trajectory

FIG. 5. Actual and Optimal Tank Trajectories for September 29, 1988

also ensured that the pumps being used were performing at near peak efficiency.

If the allowable operating range for the Pond Springs Reservoir for that day was increased or the final tank-elevation requirement relaxed, a further cost reduction may be available. To meet the defined boundary condition, the best operations for this date are to empty the tank then fill it at the end of the day. Thus for most of the day the system was virtually reduced to a "pump through" system (or a network without a tank).

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Sensitivity Analysis To examine the impact of decisions that must be made outside of the

optimization model, an analysis of the system was performed considering the following factors: (1) Allowable number of pump switches; (2) starting and ending tank-water levels; (3) long-term planning; (4) length of constant demands intervals; and (5) tank-level discretization. The first three factors are operational decisions; the others influence the accuracy and computational effort of the optimization model. All runs in this analysis used the September 29, 1988 temporal demand distribution and assumed five pumps to be available, including the two with modified characteristic curves. In addition, all pumps were assumed to be off at the beginning of the cycle. To speed their execution, many of these runs were made on a Convex minicomputer at the University of Arizona.

The effect of the number of pump switches allowed for the planning period on the optimal cost was investigated since the allowable number of pump switches is not known until the system is analyzed with a model such as the one presented in this paper. For example, four pump switches were made by the NWB operators on September 29, 1988. As discussed in the previous section, the model found a lower-cost solution, which needed only three pump switches. If the cost increases when using fewer switches, a decision regarding the trade-off between turning on a pump and spending more on energy must be made.

In general, there will be a minimum number of switches necessary to operate a system for a given set of demands. The optimization model can provide the least-cost solution for different numbers of pump switches. Table 4 shows results for two model runs with different starting and ending tank- water elevations. In the first run, the starting water-surface elevation (at the beginning of the day) was 344.4 m (1,130 ft) and the final target water- surface elevation (at the end of the day) was 342.9 m (1,125.0 ft). A constant unit cost of energy was used in this run. For this condition, allowing only one pump switch was not feasible. As the allowable number of pump switches is increased the optimal cost decreases since more pump combinations are feasible (i.e., the problem is less constrained). When the required number of switches was set to a high value, however, the cost increased since it forced many switches, and the problem becomes more constrained. Increas- ing cost, however, occurred well above acceptable values for the number of switches.

TABLE 4. C

Number of pump switches (1)

aStarting bStarting

)timal Cost'and Number of Pump Switches

Optimal Cost (Dollars)

Boundary condition A ~ (2)

Infeasible 260.44 196.08 177.06 174.75 174.97 174.75

Boundary condition B b (3)

Infeasible 346.42 312.92 230.08 212.68 209.09 207.73

WSE = 344.4 m (1,130 ft); ending WSE = 342.9 m (1,125 ft). WSE = 342 m (1,120 ft); ending WSE = 345 m (1,130 ft).

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In the second run, the starting and ending water-surface elevations (WSE) were 341.4 and 344.4 m (1,120 and 1,130 ft), respectively. The unit energy cost varied through the day by the factors listed in Table 2. Again in this case, the one-pump-switch condition was infeasible. The same trend of decreasing cost with increasing allowable number of pump switches is ob- served. In both solutions, there is a steep drop in cost from the fewest feasible number of pump switches solution to the next highest allowable. The first run decreased from $260.44 to $196.00, or a 25% cost savings; the cost in second run dropped from $312.78 to $230.08, or 26.5% cost savings. Thereafter, the drop in cost with increasing number of pump switches be- come smaller (e.g., increasing the allowable pump switches in the first run from four to seven resulted in a 1% decrease in cost). A minor cost increase occurs from five to six switches. Since discrete operations are required, this small variation is expected. A limit of four pump switches for the first conditions appears to be reasonable because there is little to gain in terms of energy-cost savings beyond that number of pump switches. A limit of five pump switches would be appropriate for the second condition for the same reason.

As is expected, the optimal solution depends on the starting and final target tank-water levels. Obviously, the lowest-cost operations are for days when the tank is initially full and left empty at the end of the day. In this case, the volume of water pumped is the smallest and the PUmps will "push" against the lowest head. However, these conditions will not be permitted since the cost to fill the tanks on the prior day and to operate the system with initially empty tanks on the following day would be prohibitive. Fig. 6 shows the optimal cost versus the ending tank-water level for three different starting conditions and required ending condition of 342.9 m (1,125 ft). As discussed, the cost increases with lower initial tank levels. Operators

220

210. I-'-

O 200J o (..9 Z

~ 190.

__1 < 180- F- (l .

0 170.

34i .5 3~,2 342.5 34-3 34~3.5 344 344.5 ENDING WATER SURFACE ELEVATION (M)

160 341 345

Starting WSE=341.4 + Starting WSE=342.9 ~ Starting WSE=344.4

FIG. 6. Optimal Costs for Different Boundary Conditions and Five Pump Switches

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should review their system to ensure that appropriate boundary conditions are used. Since pump switches are also of concern, the pumps operating at the end of the optimization period should also be considered in the decision process. The analysis of the ending pump status is well beyond the scope of this paper.

To overcome the problem of defining boundary conditions, it may be useful to plan operations for more than one day. Planning operations for multiple days can result in lower costs by allowing the model to decide the ending water levels for intermediate days. For systems with limited operational storage, the effect of the final tank level on operations early in the period will be small after a long period. A long period will be as short as two or three days for many systems. This long-term impact can be tested by examining planning periods of different duration.

This long-range planning is also useful for systems with tanks that are large enough to hold significant storage volumes or when energy rates vary from day to day (e.g., weekend and weekdays). In these situations, water can be stored during lower-energy-cost days, and the tanks can be emptied during higher-energy-cost periods. In these systems, however, the final tank condition will have more impact on the overall operations.

For long-term planning for the Austin system, results were obtained for one-, two-, and three-day planning periods, which allow five pump switches per day. The tank level at the end of the planning period was 342.9 m (1,125 ft). The unit cost of energy was not considered to vary from day to day but varied during each day by the factors in Table 2. The demand pattern for each day was identical to the September 29 demands. The energy cost was $186.01 per day using a single-day planning horizon, $183.34 per day when a two-day horizon was used, and $182.90 per day for a three-day planning horizon.

280- t -

O 260- O L9 Z

~ 240-

. . I < 220-

O 200-

180 0 1 ~ ~ ~, ~ 6

STAGE DURATION (HRS)

300

I --k- 5 Pump Switches -=- 6 Pump Switches } FIG. 7. Effect of Stage Duration on Optimal Pumping Cost

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188.

I ' - ~9 O o ...I <

I-- 0- O

186

184-

182"

180"

178

176"

174 0 1'0 2o ab 4b 5o go 70

TANK LEVEL INTERVAL (CM)

FIG. 8. Effect of State Space Discretization on Optimal Cost

For this system, lower tank levels are maintained for most of the periods. However, since energy cost is low during the night the tank is filled to a level near 343 m each day at midnight. By analyzing the system in this way, it appears that the 342.9-m ending level is reasonable and cost-effective. It is also interesting to note that the operation cost only increases to $185 per day when 10 total switches are allowed in the three-day period. It thus appears that pump-switch strategies may also be improved by considering longer planning horizons.

This system demonstrates the need to look beyond one day when making decisions. Beyond this simple analysis, more research is needed to assist operators in properly defining their boundary conditions. In addition, efforts for making more accurate demand forecasts, which is a second major dif- ficulty in long-term planning should continue.

The effect of using longer time intervals was examined to see if coarse demand estimates would be useful. The time-interval duration also impacts the computer-memory requirements and the solution time for the analysis of even a single day's operation. Time intervals of 0.5-6 hr were used in the model for a single-day run. Fig. 7 shows the results obtained for five and six pump switches. The boundary conditions were the beginning and ending state equaling 342.9 m (1,125 ft). It is noted that six pump switches was infeasible for a 6-hr stage interval. The results show that, in general, as the time intervals decrease, the optimal solution improves. Using shorter time periods allows more flexibility in making decisions since pumps must operate during the entire period. The breakpoint for an acceptable interval that reduces the problem dimensionality (by using reasonably longer time intervals) and still obtains reasonable results must be selected on an individual system basis.

The effect of the level of discretization of the state variable also affects the solution time and accuracy. As the tank level grid is made finer the

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optimal solution approaches the true optimal solution. Fig. 8 shows results of the solution cost and the level discretization of state variable for this application with beginning and ending tank levels of 343.2 m (1,126 ft). As the grid is made finer the cost increases, since the significance of rounding within each state transformation is reduced. Given the uncertainty in demand prediction and their assumed spatial distribution, an extremely fine grid is probably not justified. In addition, as a day progresses the optimization model can be reexecuted if the projected trajectory diverges from the actual tank levels.

SUMMARY AND CONCLUSION

To date, all pump-scheduling models consider only the objective of minimizing the energy-consumption cost. A methodology considering an equally important goal of avoiding many pump switches was presented. In this method, the hydraulic analysis of the system for each scenario of demand pattern and pump combinations is completed in an off-line mode to provide regression equations for an on-line model. The optimization can then determine optimal solutions in an on-line mode on a personal computer using a constrained dynamic-programming algorithm.

Due to the problem of the dimension of the variables, the model is only applicable to small to midsize systems. The size of the system that can be analyzed depends on the number of pumps and tanks in it and the allowable number of pump switches. It does not depend on the size of the pipe network. It also depends on the level of accuracy of the desired optimal solution. The model can, however, be applied to large systems (with a large number of pumps) if the number of pumps that can be used during a day is limited so that only a few candidate pumps are considered.

A sensitivity analysis was done to demonstrate how external operating decisions and optimization model criteria affect the optimal solutions. It showed that acceptable solutions can be obtained even if the state and stage spaces are not made extremely fine. As such, the method can be used for planning on daily basis as well as longer planning periods. Additional work must be done to improve (and apply) the method for two tank systems and to more quickly find a more exact solution.

ACKNOWLEDGMENTS

The writers appreciate the detailed comments of the anonymous review- ers, which greatly improved the presentation and clarity of this paper. Kwame Agyare and Keshaw MaUick are acknowledged for their assistance in performing model runs and in programming. This research was supported by the U.S. Geological Survey (USGS), Department of the Interior, under USGS award number 14-08-0001-Gt893.

APPENDIX. REFERENCES

Brion, L. M., and Mays, L. W. (1991). "Methodology for optimal operation of pumping stations in water distribution systems." J. ofHydr. Engrg., ASCE, 117(11) 1-19.

Chase, D. V., and Ormsbee, L. E. (1989). "Optimal pump operation of water distribution system with multiple storage tanks." Proc., Conf. on Water Resour. Plnng. and Mgmt., ASCE, New York, N.Y., 733-736.

DeMoyer, R., and Horowitz, L. (1975). A systems approach to water distribution modeling and control. Lexington Books, Lexington, Mass.

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Jolland, G., and Cohen, G. (1980). "Optimal control of water distribution network by two multilevel methods." Automatica, Vol. 16, 83-88.

Jowitt, P., Garrett, R., Cook, S., and Germanopoulis, G. (1988). "Real-time fore- casting and control for water distribution." Computer applications in water supply; Vol. 2, B. Coulbeck and C. Orr, eds., John Wiley & Sons, Inc., New York, N.Y., 329-355.

Lansey, K., and Zhong, Q. (1990). "A methodology for optimal control of pump stations." Proc., 1990 ASCE Water Resour. Plnng. and Mgmt. Specialty Conf., ASCE, New York, N..Y.

Little, K., and McCrodden, B. (1989). "Minimization of raw water pumping costs using MILP." L Water Resour. Plnng. and Mgmt., ASCE, 115(4), 511-522.

Ormsbee, L., Walski, T., Chase, D., and Sharp, W. (1989). "Methodology for improving pump operation efficiency." J. Water Resour. Plnng. and Mgmt. , ASCE, 115(2), 148-164.

Sterling, M., and Coulbeck, B. (1975). "A dynamic programming solution to optimization of pumping costs." Proe., Institute of Civil Engineers, London, England, 59(Part 2), 813-818.

Wood, D. J. (1980). Computer analysis of flow in pipe networks including extended period simulation; user's manual. Office of Engineering Continuing Education and Extension, University of Kentucky, Lexington, Ky.

Zessler, U., and Shamir, U. (1989). "Optimal operation of water distribution systems." J. Water Resour. Plnng. and Mgmt., ASCE, 115(6), 735-752.

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