*Corresponding author. Tel.: #34-3-401-5785; fax: #34-3-401-5750.

E-mail address: cembrano@iri.upc.es (G. Cembrano).

Control Engineering Practice 8 (2000) 1177}1188

Optimal control of a water distribution network in asupervisory control system

G. Cembrano!,*, G. Wells!, J. Quevedo", R. PeH rez", R. Argelaguet"!Instituto de Robo& tica e Informa& tica Industrial, Universidad Polite&cnica de CatalunJ a, Consejo Superior de Investigaciones Cientn&xcas,

Gran Capita& n 2-4, 08034 Barcelona, Spain"Departament Enginyeria de Sistemes, Automa% tica i Informa% tica Industrial, Edixci d 'Automa% tica - TR11, Universidad Polite& cnica de CatalunJ a,

Campus de Terrassa, Rambla Sant Nebridi, s/n, 08222 Terrassa, Spain

Received 4 October 1999; accepted 31 March 2000

Abstract

This paper deals with the use of optimal control techniques in water distribution networks. An optimal control tool, developed inthe context of a European research project is described and the application to the city of Sintra (Portugal) is presented. ( 2000Elsevier Science Ltd. All rights reserved.

Keywords: Optimal control; Water networks; Telecontrol; Nonlinear optimization; Modelling; Simulation

1. Introduction

Water networks are generally composed of a large num-ber of interconnected pipes, reservoirs, pumps, valves andother hydraulic elements which carry water to demandnodes from the supply areas, with speci"c pressure levelsto provide a good service to consumers. The hydraulicelements in a network may be classi"ed into two catego-ries: active and passive. The active elements are thosewhich can be operated to control the #ow and/or thepressure of water in speci"c parts of the network, such aspumps, valves and turbines. The pipes and reservoirs arepassive elements, insofar as they receive the e!ects of theoperation of the active elements, in terms of pressure and#ow, but they cannot be directly acted upon.

A supervisory control system in a water network gen-erally includes a telemetry system which periodicallyupdates some information from a selected set of passiveelements and from most of (ideally all) the active ele-ments. This information is generally composed of pres-sure and #ow readings, as well as status of the activeelements, and it re#ects the instantaneous operating con-

dition of the water network. It also includes the mecha-nisms to actuate the active elements in the network andto control their performance. However, it is not alwaysobvious how to derive appropriate control strategies forthe active elements, in order to use the resources e$cient-ly and to meet the speci"c pressure and #ow needs in thewhole network, at all times.

Optimal control in water networks deals with theproblem of generating control strategies ahead of time, toguarantee a good service in the network, while achievingcertain performance goals, which may include one ormore of the following, according to the needs of a speci"cutility: minimization of supply and pumping costs, maxi-mization of water quality, pressure regulation for leakprevention, etc.

The implementation of optimization procedures mayprove an extremely e$cient measure to contribute to thecorrect management of water resources and to reduceoperational costs, as well as to improve the service toconsumers, as has been shown in several water utilitiesaround Europe. In the context of a European researchproject (WATERNET, ESPRIT IV, No. 22186), theauthors have developed an optimal control tool integ-rated in a supervisory control system which can be usedby a large class of water utilities. The whole system,including optimal control was demonstrated in a proto-type of the water network of the Sintra (Portugal) area. In

0967-0661/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 9 6 7 - 0 6 6 1 ( 0 0 ) 0 0 0 5 8 - 7

Fig. 1. Optimal control in water systems.

this paper, the concepts of optimal control in waternetworks are presented in Sections 1 and 2. Sections 3}5deal with the mathematical statement, the complexityand the methods of solving the optimal control problemrespectively and in Section 6, the application to thenetwork of Sintra (Portugal) is presented. Section 7 sum-marizes the conclusions of this work and presents futureresearch issues.

2. On-line optimal control

2.1. Inputs from the network

In a supervisory control system for a water network,the optimal control procedure interacts with the realnetwork through the SCADA (Supervisory Control andData Acquisition) system. It must receive informationabout the current state of the network, in terms of

f water volumes in reservoirs,f status of pumps and valves,f latest demand readings,f pressure and/or #ow readings at selected points.

In order to compute optimal control strategies aheadof time, the optimal control procedure must containa prediction model to generate forecasts of the nextfuture demands, based on a record of a number of pastvalues thereof. A number of methods for demand predic-tion may be used as described elsewhere (Quevedo, Cem-brano, GutieH rrez & Valls, 1988; Grin8 oH , 1991). In thiswork, the demand prediction model was obtained usingthe Fuzzy Inductive Reasoning Methodology, as de-scribed in LoH pez, Cembrano and Cellier (1996).

The demand information from the SCADA is used toupdate the past-demand record and a forecast of thedemand in the next period is performed. The demandforecasts are usually obtained with a horizon of the orderof one day. The hourly data is obtained by modulationwith appropriate demand patterns for each day in theweek.

2.2. Optimization execution

Additionally, the optimization module contains a hy-draulic model (see Section 3.2) of the network whichmakes it possible to test the e!ect produced by a controlaction (#ows through the active elements) on the net-work, in terms of

f water volumes in reservoirs,f pressure and/or #ow readings at selected points.

The optimal control procedure selects optimal strat-egies for the controllers of the active hydraulic elements,by searching in the space of possible controls and evalu-ating di!erent alternatives, as described later in this

document. In water networks where storage in reservoirsmust be planned ahead to meet the future demands withspeci"c pressure levels, the optimization involves thegeneration of controller strategies over a time period,called the optimization horizon, which may be of theorder of one day, at hourly intervals, for a common caseof water distribution utility.

2.3. Outputs to the remote units

The optimal control values are sent through theSCADA as setpoints to the local controllers in the re-mote stations that regulate the active elements.

Optimization updates must be performed regularlyduring the day. The general scheme of interaction withthe supervisory control system is shown in Fig. 1.

3. The optimal control problem

3.1. Network description

A convenient description of the dynamic model ofa water network is obtained by considering the setof #ows through the m active elements as the vector ofcontrol variables

u3Rm

and the set of nr

reservoir volumes monitored throughthe SCADA as a vector of observable state variables

r3Rnr .

Additionally, in order to complete the state description ofthe network, it may be necessary to de"ne a vector

q3Rnq

1178 G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188

containing the values of #ow at nq

speci"c passive ele-ments, which cannot be directly measured through theSCADA and a vector

p3Rnp

containing the values of pressures at np

speci"c nodes,not monitored through the SCADA.

Vectors p and q may be obtained from r and u throughexplicit or implicit, generally nonlinear, equations ex-pressing the hydraulic relationships which couple headand #ow. Well-known approximations of these hydraulicrelationships exist for di!erent types of hydraulic ele-ments. For pipes, for example, the Colebrook}White, theDarcy}Weisbach or the Hazen}Williams equations (seee.g. Isaacs & Mills, 1981; Jeppson, 1983; Quevedo, Cem-brano, Casanova & Ros, 1987) have been widely used innetwork #ow and pressure calculations. Similarly, a goodapproximation of the head-#ow relationship for pumpsmay be found in Anderson and Powell (1999) as well as inappropriate manufacturer's speci"cations. For valves,head-#ow equations, according to their characteristicsand their functionality are usually supplied by the manu-facturers. Some of the most relevant examples of theseequations may be found in Greco (1997). Hence,

p"h(r, u, d), (1)

q"g(r, u, d), (2)

where d3Rnd is a vector of stochastic disturbance con-taining the values of the demands at the n

dconsumer

nodes in the network.Since the model is used for predictive (optimal)

control, d will generally be a vector of demand forecasts,obtained through appropriate prediction models (seeSection 2.1).

An augmented state vector is de"ned as

x3Rn,(3)

n"nr#n

q#n

p,

where x contains the nrvalues of reservoir volume vector

r, the nq

values of the #ow vector q and the np

values ofthe pressure vector p.

3.2. Dynamic model

The dynamic model of the network may then bewritten, in discrete time, as

xk`1"f (xk, uk, dk), (4)

where f represents the mass and energy balance in thewater network and k denotes the instantaneous values atsampling time k.

3.3. Performance index

The control actions are generated by optimizing a per-formance index, i.e. a mathematical expression of thedesired exploitation goals.

In general, a performance index may contain one ormore of the following terms:

f The cost of water supply, treatment and elevation Cu,

which is a function of the #ows in the m active elementscontained in vector u.

f Arti"cial costs related to pressure regulation, ex-pressed in functions C

pof the pressures in n

pspeci"c

key nodes of the network, contained in vector p.f Arti"cial costs related to #ow regulation and/or water

quality, Cq, expressed as function of the #ows in

nq

speci"c pipes of the network, contained in vector q.

Then, a general performance index may be expressed asfollows: For an instantaneous evaluation,

Pk"m+i/1

cku(u

i)#

np+j/1

ckp(p

j)#

nq+l/1

ckq(q

l), (5)

where P is the instantaneous value of the performanceindex and k refers to the values of the instantaneousvalues at the kth time period.

For an evaluation over the entire optimization hor-izon, the performance index is

J"N~1+k/0

Pk, (6)

where N is the optimization horizon (in a number ofsampling periods).

3.4. Constraints

The performance optimization must take into accountthat a number of important constraints exists:

f On the control variables, such as the #ows across theactive hydraulic elements. Pumps and valves have spe-ci"c ranges of #ow for correct operation.

u3;, ;LRm

f On other variables, such as the levels in the reservoirs.These include capacity and operational limits on waterminimum and maximum allowed levels. Similar con-straints may be in order in the selected nq pipes and thenp pressure measurement nodes. Note that the optim-ization can only be performed in the space of controlswhile these constraints are expressed in terms of othervariables, which cannot be actuated directly.

r3S, SLRnr ,

q3Q, QLRnq ,

p3H, HLRnp .

G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188 1179

The optimization process consists of searching the spaceof admissible control actions to obtain the minimum (ormaximum) value of the performance index, while produ-cing only admissible states of the network.

It is important to remark that the performance indexcan only be minimized with respect to the control vari-ables (#ows across active hydraulic elements), since theseare the only variables which may be acted upon. However,the value of the index also depends on variables that arenot directly controlled, but must be evaluated for eachcontrol action. Similarly, the contraints apply both ondirectly controlled variables and on other variables relatedto passive elements, which cannot be operated directly.

In many practical cases a distinction can be madebetween `harda and `softa constraints, the former beingthose which must be ful"lled strictly and the latter, thosewhich may be violated to some moderate extent. Inparticular, the operational levels of reservoirs, the min-imum-pressure and the #ow-range constraints may usu-ally be de"ned so that only soft bounds are imposed. Inthis case, the optimization may proceed by augmentingthe performance index with penalty terms which imposethe approximate ful"lment of the soft constraints, usingLagrange multipliers (see e.g. Minoux, 1986).

4. Complexity of the optimal control problem inwater networks

The computational complexity of the optimal controlproblem in water networks is determined by a number ofimportant factors in the real network and in the model-ling assumptions. The main factors and their in#uence onthe optimization complexity are outlined below.

4.1. Size

The size of the optimization problem depends on thenumber of active hydraulic elements to be optimized, onthe number of passive elements which re#ect the state ofthe network, such as reservoirs and pressure- or #ow-evaluation nodes, but also, most importantly, on theoptimization horizon. Instantaneous optimization prob-lems pose few computational di$culties, but dynamicoptimization problems over 24 or more hourly intervals,which are common in many water utilities in Europe,impose a very high computational burden in non-trivialnetworks. This is so because of the combinatorial natureof the problem of searching the space of controls fora number of interacting active elements in a 24-step-ahead horizon.

4.2. Type of hydraulic elements involved

The type of hydraulic elements a!ects the design ofstrategy optimization programs insofar as it determines

the type of variables to be used to express #ows andpressures. This is important since very e$cient methodsexist for optimizing vast classes of problems involvingcontinuous variables and continuously di!erentiablefunctions, especially because it is possible to use gradi-ent-based searchs. Conversely, the methods of integerprogramming required to optimize problems with integeror binary variables are mainly of combinatorial natureand they cannot make use of the gradient concepts. Theyare, therefore, usually slow and not necessarily guaran-teed to converge. For this reason, it is often preferred towork with continuous approximations of the values ofthe discrete variables involved in the optimization, asfollows:

f Valves: They allow any value of the #ow in its opera-tional range. Therefore, continuous variables can beused to represent the value of the #ow through a valve.

f Fixed-speed pumps: For a speci"ed value of the down-stream pressure or the head increase, only one value ofthe #ow is possible according to the characteristicequation of the pump. Therefore, the instantaneousvalue of the #ow through a pump cannot be realisti-cally modelled using continuous variables. However,the volume of water through a pump in a speci"edperiod (e.g. hourly periods, as required for the dynamicoptimization) may attain any desired (feasible) valuethrough pulse-width modulation, i.e., by operating thepump only a portion of the period.

f Batteries of xxed-speed pumps: Utilizing a numberof "xed-speed pumps usually allows for a modulationin the #ows, which makes it easier to model the #owas a continuous variable. Additionally, continuous-variable optimization methods may be modi"ed toproduce sub-optimal solutions which are close to inte-ger numbers. An example of one such approach isgiven in Section 6.2.4.

4.3. Type of network description

The e!ect on the network of each candidate controlstrategy tested by the optimization must be evaluatedusing the network description and hydraulic couplingsstated in Eqs. (1)}(3). The complexity of these equationswill obviously in#uence the time required by the opti-mizer to evaluate each candidate control action.

f Flow-only models and, in particular, those modelswhere only the control variables are required to com-pute the change in the state of the network producedby a control action allow a very fast evaluation ofcontrol actions.

f Flow and pressure models, where a number of pres-sures and/or #ows in passive elements are required tocompute the change in the state of the network pro-duced by a control action, impose longer executiontimes for each test. The problem is easier when explicit

1180 G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188

Fig. 3. Tree-structure water network.

Fig. 2. Star-structure water network.

equations exist to compute the pressures and/or #owsin passive elements, in terms of the control variables. Ifthis is not the case, an iterative simulation proceduremust be run for each evaluation of a control action.This condition imposes considerably longer executiontimes.

4.4. Type of optimization equations

The availability and computational complexity of op-timization methods varies according to the form of theequations involved. In particular, for continuous-vari-able problems:

f Problems with linear performance indexes and linearconstraints have fast and e$cient solution methodsbased on the SIMPLEX method.

f Problems with quadratic performance indexes and lin-ear constraints have fast and e$cient methods basedon gradient techniques, which are guaranteed to con-verge in a number of iterations equal to the dimensionof the optimization space.

f Problems with other nonlinear convex performanceindexes and linear constraints may be solved using thesame methods as above, without the guarantee of thenumber of iterations for convergence.

f More general problems with convex performance in-dexes and nonlinear constraints de"ning a convexspace may be solved with gradient-based techniquese$ciently.

f Nonconvex problems pose serious di$culties, as thereis no guarantee that only one extremum (minimum ormaximum) exists. In these cases, random- and heuris-tic-search methods may be used.

f For integer or binary-variable systems, the optimiza-tion problems are of a combinatorial nature. Searchand pruning methods are available, such as thebranch-and-bound algorithms, which are e$cient insome speci"c cases with a moderate dimensionality.

4.5. Topology

The topology determines how an action in a certainelement of a water network a!ects the rest of it.For example, in some simple tree-like networks anaction at the end of a "nal branch may not a!ect the restof the network at all, while in a mesh-like network,a more global in#uence of the actuation of most of thehydraulic control elements is to be expected. This situ-ation is usually an important factor to be taken intoaccount for the selection of more or less de-centralizedschemes for the supervisory control system in generaland for the control strategy optimization in particular.Star-structure water networks generally contain onelarge source (usually with an associated reservoir) from

which water is supplied to a number of local small distri-bution areas, each of which may contain at most onereservoir, as shown in Fig. 2.

The special characteristics of this network allow theuse of spatial decomposition techniques for optimization,which make it possible to compute a solution to a largecomplex problem through the iterative solution ofa number of simple problems, see e.g. Coulbeck (1977).Note, in particular, that the sense of the #ows is "xed,from the central source to the demands.

Tree-structure water networks generally have onelarge source (and usually, an associated reservoir) fromwhich water is supplied to a small number of intermedi-ate locations, where it is stored and/or pumped to thedemand nodes, as shown in Fig. 3.

Note that here too, the sense of the #ows is de"ned,from the source to the demands.

A cascade water network is a special case of a tree-structure network where there is only one branch, asshown in Fig. 4.

A mesh-structure network usually containsseveral sources (e.g. with associated reservoirs) andis highly interconnected so that the demands can besupplied from more than one source and, in general,with di!erent operation patterns, so that the sense ofthe #ows is not "xed in some of the valves and pipes (seeFig. 5).

G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188 1181

Fig. 4. Cascade structure water network.

Fig. 5. Mesh-structure water network.

5. Solving the optimal control problem

5.1. State of the art

A number of large European utilities already utilizeoptimization programs to generate 24 h strategies aheadin time to minimize costs while maintaining a goodservice to consumers. Several types of optimization pro-cedures have been used, but most of them employ eitherdynamic programming or convex nonlinear program-ming techniques. In most cases, the size of the optimiza-tion problems involved makes it necessary to derivespecial methods, since the direct application of the tech-niques would not have produced acceptable computa-tion times.

In the 1970s, Fallside and Perry (1975) derived a hier-archical price-decomposition algorithm for the optimiza-tion of the Severn-Trent Water Authority network. Thesolution of the subproblems used a linearization aroundan operating point to reduce the mathematical complex-ity. The method was implemented in the Cambridge area,but it did not achieve completely satisfactory results inthe long run.

A more general scheme of hierarchical-decompositionoptimization in water networks was proposed byCoulbeck (1977) and tested in simulation for other UKnetworks.

In Barcelona, Solanas (1974) derived a successive ap-proximation dynamic programming algorithm for theBarcelona network, which was later improved and testedin simulation (Solanas & Montoliu, 1988).

In the US, Murray and Yakowitz (1979) developed thedi!erential dynamic programming techniques which al-lowed the use of the dynamic programming approach tothe optimization of a relatively small simulated waternetwork.

In the 1980s, one of the most relevant applications wasthe real implementation of the interaction predictionmethod developed by Carpentier (1983), in the networkof Paris Sud of the French Company Lyonnaise desEaux. This method was also based on the use of dynamicprogramming techniques and it made speci"c use of thestar structure of the distribution network.

Later, a nonlinear programming technique was de-veloped for the Barcelona network, which made use ofoptimal control concepts (Cembrano, Brdys, Quevedo,Coulbeck & Orr, 1988), and an improvement thereofwas implemented in the real network in 1990 (Grin8 oH &Cembrano, 1991). The Barcelona network is highly inter-connected and has a mesh structure, which makes itine$cient to use decomposition techniques.

In the 1990s, Brdys (1992) considered the problem ofoptimizing the con"guration of "xed-speed pumps in thewater networks of Leicestershire and Yorkshire. Hestudied a class of cascade hydraulic systems in two cases:with weak and with strong hydraulic interactions. Spatialand time decomposition are used in combination withbranch-and-bound mixed-integer programming tech-niques to solve the optimization problems of these twoareas. The method was implemented successfully by theYorkshire Water Authority (Brdys & Ulanicki, 1994).

Most recently, a number of European joint researchand development projects have attempted to solve moregeneral classes of optimization problems and to producetools applicable to a variety of water utilities throughoutEurope. For instance, the CLOCWiSe (INNOVATIONProgramme) project dealt with the general problem ofoptimized scheduling of water management and networkmaintenance, through the use of Constraint Logic Pro-gramming, producing a speci"c tool applicable to manywater control systems.

The WATERCIM project (ESPRIT Programme) paidspecial attention to the concept of integrating di!erentwater management tools, which had so far been usedindependently, with no information sharing.

The WATERNET project, which has produced thiswork, involved the development and dissemination ofadvanced techniques and tools for water managementand planning, including modelling, learning and know-ledge acquisition, simulation and state-estimation, optimalcontrol, quality analysis, and distributed-information-system design.

In the area of optimization, the project goal wasto demonstrate the bene"ts of introducing optimizationin a variety of water utilities. Additionally, theWATERNET optimal control procedure incorporatesaspects that had so far been avoided because of their

1182 G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188

Fig. 6. The network window. Fig. 8. The optimization results window showing a reservoir evolution.

Fig. 7. Edit constraints window.

computational complexity. This is the case of networkswith hydraulic models involving #ows and pressures forthe optimization and the interaction with the state es-timation and alarm management procedures to validateand correct optimization results.

5.2. The WATERNET optimal control module

5.2.1. StructureThe WATERNET optimization solver relies on

a powerful commercial optimization tool (CONOPT),which can e$ciently deal with linear or nonlinear costfunctions and constraints. It incorporates several optim-ization procedures and it can provide information on theoptimization process, as required by the user, with di!er-ent degrees of detail. The WATERNET optimization mod-ule also incorporates a commercial interface to theCONOPT program (GAMS, 1997) which makes it simplerfor a user who is knowledgeable in optimization to inputthe de"nition of the problem to the CONOPT program.

In addition, a higher-level user interface has been de-veloped and added, so that it is simple for a water facilityuser to introduce a water optimization problem, with noprevious knowledge in optimization and then to run theoptimization and analyze the results very easily. Thiscombination constitutes a powerful tool for the optimiza-tion of a very broad class of water systems.

5.2.2. The problem statement interfaceThe problem statement interface allows the user to

input the optimal control problem. In particular, it ispossible to create and visualize the network, through thenetwork window (Fig. 6), and to de"ne a performanceindex, as well as to impose constraints on the maximumand minimum values of the reservoir volumes, pump#ows, valve openings, pipe #ows and node pressures,through the Edit constraints window (Fig. 7).

5.2.3. The result presentation and analysis interfaceThis interface shows the results of the optimization

process (Figs. 8 and 9), in terms of strategies for thecontrol elements and evolution of the reservoir volumes.Additionally, it is possible to visualize details of theoptimization execution log.

5.2.4. Integration with other water management toolsThe WATERNET optimal control module is executed

in cooperation with other water management tools,corresponding to other modules in the WATERNETproject. In particular, it interacts with the Supervisionsystem, and with the Simulation, Learning and QualityControl subsystems.

At a "xed time everyday the optimization is startedaccording to the scheme shown in Fig. 10. First of all, itretrieves the current status of the network * reservoirvolumes, pump status, valve ratios * from the Super-vision Subsystem. From the Learning Subsystem, it getsthe demand-forecast results. Once the optimizationprocedure is run, it also interacts with the simulationmodule for the validation of a provisional set of control

G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188 1183

Fig. 9. The optimization results window, showing a valve strategy.

Fig. 10. Optimization strategy.

strategies in a more detailed network. It is not until theresults of the simulation in a detailed network appearreasonable to a human operator that the strategies areconsidered correct. Similarly, the provisional controlstrategies are analyzed by the Quality Control modulefor consistency with the quality requirements and re-optimizations must be run with new conditions, ex-pressed in constraints on the maximum or minimum#ows to be taken from the sources, until the qualitystandards are met. Only then are the optimization resultsconsidered "nal.

6. A case study: The Sintra network

6.1. Description of the network

SMAS-SINTRA is a public company responsible forthe drinking water network of the district of Sintra inPortugal as well as the wastewater service. The water

distribution network spreads in a wide area, forminga total of 900 km of main pipes covering almost all thewestern area of the Lisbon region. They serve a satellitezone of Lisbon comprising residential users who com-mute to Lisbon to work (approx. 100 000 inhabitants).

The network model includes 167 pipes, 204 nodes, 14pumps and 11 ON}OFF valves. A large number ofsmall-capacity reservoirs exists and the elevation variesgreatly within the network. The di!erence between theextreme reservoirs is 277 m. There are two main watersupplies, related to pumping stations UB1 and UB2 inFig. 6.

Because of its size, its mesh topology and the need fora #ow-and-pressure hydraulic calculation, this model isunmanageable for optimization purposes. In order toproduce a simpli"ed version of the original model, thefollowing criteria were used:

f Eliminating small-capacity reservoirs.f Eliminating small-diameter pipes, while preserving the

ontiguity of the network.f Keeping all the active elements and any other elements

supervised by the SCADA.

From this reduction, the operative model shown in thedisplay of Fig. 6 is generated. It contains: 7 reservoirs,2 pumping stations, 7 valves, 11 pipes and 6 consumptionnodes and has the agreement of SMAS experts. Thevalidity of the simpli"ed model was veri"ed by simula-tion. The pressures and #ows in all the elements of thesimpli"ed model were computed by simulation on both(original and simpli"ed models). It was con"rmed thatthe #ows and pressures computed with both models werenot signi"cantly di!erent. Details of these experimentsare reported in PeH rez, Cembrano and Quevedo (1998).

6.2. Statement of the optimal control problem

6.2.1. Optimization goals and performance indexThe optimization goal in the Sintra network is the

minimization of cost in acquisition and pumping opera-tions. For each pump i and for each time period k, this isexpressed as

Jk(i)"JkNL

(i)#JkL(i), (7)

where JNL

is the hourly pumping cost, expressed asa nonlinear function of the #ow through the pump andJL

is the acquisition cost, expressed as a linear function ofthe #ow:

JkNL

(i)"ak*tc*hk

iuki

gki

,

JkL(i)"jkuk

i*t, (8)

where i refers to the ith pump station, k refers to the kthtime period, a is the electricity tari! per energy unit, *tis the sampling period, c is the speci"c weight of water, *h

1184 G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188

Fig. 11. Setpoints for the pumping stations.

Table 1

Strategy Manual Optimal

Water source 1 1425 1573Water source 2 2068 1188Pumping Carenque 618 608Pumping Monte Abraao 39 19Total 4150 3388

is the pump head increase, u is the pump station #ow, g isthe e$ciency of the pump and j is a constant whichexpresses the cost of acquisition per unit volume. Then,the total cost is computed as a summation over theoptimization horizon and over all the pumps in thenetwork

J"N~1+k/0

+i

JkNL

(i)#JkL(i), (9)

where N is the optimization horizon expressed in numberof sampling periods. In this case, the horizon is equal to24 hourly intervals.

6.2.2. ConstraintsThe constraints are related to the mass balance in the

whole network, the head-#ow relationship in pipes andvalves, the operational limits on the reservoir volumes,and a pressure limitation at one junction node.

6.2.3. Initial and xnal conditionsThe initial condition of the network is read from the

SCADA system and is expressed in terms of the reservoirlevels at time 0.

In this case study, a "nal condition is imposed as well,stating that the "nal volumes in the reservoirs should bethe same as the initial ones. This condition makes sensewhen a 24 h periodic operation of the reservoirs issought.

6.2.4. Obtaining discrete values for pump setpointsAt each pumping station i of the network, there are

a number of identical pumps in parallel, with equalnominal #ow, q

i. Let s be an m-dimensional vector that

contains, in each component i, the number of pumpsswitched on at pumping station i. Then, at each samplinginterval k, the value of the total #ow through this station(corresponding to the ith component of vector u), may beexpressed as

u(i)k"s(i)kqi, (10)

so that optimizing s is equivalent to optimizing u. Integervalues of s correspond to setpoints indicating a certainnumber of pumps switched on during the correspondingtime interval. Non-integer values may be translated intosetpoints corresponding to switching on a number ofpumps for a fraction of the sampling interval.

Since it is important to avoid too frequent switchingoperations for the pumps, it is preferable to obtain valueswhich are close to integer numbers for the components ofvector s. In order to achieve this, a penalty function J

artis

added to the performance index. It penalises the #owvalues that do not correspond to an integer number ofpumps on:

Jkart

"

m+i/1

w(1!cos 2ps(i)k), (11)

where w is a penalty weight. This function takes min-imum (zero) value when the values of s are close tointegers, so that the set-points (expressed in number ofpumps on) can be approximated to the nearest integer forthe simulation and real implementation.

6.2.5. Optimization methodThe optimization method is a generalized reduced

gradient method, "rst suggested by Abadie and Carpen-tier (1969), implemented as a part of the CONOPT li-brary, which can cater for the nonlinear performanceindex and constraints. It starts by "nding a feasiblesolution; then, an iterative procedure follows, whichconsists of:

f "nding a search direction, through the use of the Jac-obian of the constraints, the selection of a set of basicvariables and the computation of the reduced gradient,

G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188 1185

Fig. 12. Reservoir levels, optimization vs. simulation.

f performing a search in this direction, througha pseudo-Newton process

until a convergence criterion is met. A description ofthe algorithm and its implementation may be foundin GAMS (1997). A more detailed version thereof isreported in Drud (1985, 1992).

6.3. Results

In this section, an example of the optimal controlresults is presented for one day of operation. Firstly, anoptimization is run; then, a simulation is carried out inorder to validate the strategy on a more detailed networkmodel and, "nally, the cost of the manual strategy is

1186 G. Cembrano et al. / Control Engineering Practice 8 (2000) 1177}1188

compared with the optimal one, computed through theoptimization procedure.

Fig. 11 shows the optimal control strategy for twopumping stations. As can be seen, the pumps are ONmostly (dotted line) when the electrical tari! is low (solidline). However, since some of the demands are very hardto meet, the pumps must occasionally work with anexpensive tari!.

Fig. 12 shows the evolution of the reservoirs, accordingto the simpli"ed and the detailed models. In both cases, itmay be observed that the evolution is satisfactory, sinceno extreme (over#owing or empty) conditions arereached while satisfying all the demands.

For illustration, the results of the optimal controlprocedure are compared to the actual manual strategywhich was used on one day (speci"cally December 15,1997).

In Table 1 the costs in ECU of the manual, andoptimized strategy are compared.

It is important to remark that the numeric values ofthis comparison must only be considered orientative,because the optimal control results are based on modelsthat are de"nitely a representative of the actual hydraulicsystem, but their parameters have not yet been thorough-ly calibrated, whereas the results of the manual strategyare retrieved from the historical data of the real network.In any event, it may be appreciated that, by implemen-ting the optimal strategy, a considerable saving mighthave been achieved. This includes, most importantly,a water cost reduction and an additional energetic costminimization, due to a good pump scheduling.

7. Conclusions

Optimal control in water networks is an e$cientmeans of scheduling water transfer operations to achievemanagement goals, such as cost minimization, qualityimprovement, pressure regulation, etc. So far, only veryfew water utilities have incorporated optimal controlprocedures in their supervision systems.

The optimal control module described in this paper isapplicable to a broad class of water networks and incor-porates a number of improvements over previous optim-ization applications. In particular, it is possible to solveoptimal control problems in networks where the hydrau-lic model includes the nonlinear coupling between #owsand pressures. Additionally, although the #ows aretreated as continuous variables, a solution that is close tothe discrete possible setpoints of the pumps is obtained.Furthermore, the interaction with other water manage-ment tools, such as the simulation and the quality controlprograms, guarantees that feasible and reliable solutionsare produced.

The case study presented in the paper shows thatsigni"cant savings may be achieved by using the optimal

control procedures to compute the strategies for pump-ing and water transfer operation, as compared to theactual strategies computed intuitively by an operator.This is the case in a variety of water distribution systems,where the structure, the size or the complexity of thehydraulic interactions make it hard for a person to pro-duce optimal control strategies.

Future research issues on this subject include the calib-ration of water network models, the improvement of themethods to cater for other water management needs,such as the minimization of switching operations inpumps and valves and the improvement of the techniquesfor obtaining integer strategies for the pumping opera-tions.

Acknowledgements

This work has been partly funded by the Commissionof the European Communities (ESPRIT-IV No. 22186)and by the Research Committee of the Generalitat deCatalunya (group SAC, ref. 1997AGR00098). Theauthors are members of CERCA (Association for theStudy and Research in Automatic control) and of LEA-SICA (Associated European Laboratory on IntelligentSystems and Advanced Control). The contribution ofMessrs. Carlos Martins Nunes and Jorge Vilala of SMAS(Servicios Municipalisados de Agua de Sintra, Portugal)in supplying data and validating the experiments isgratefully acknowledged.

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