A DSS for water resources management under uncertainty by scenario analysis

www.elsevier.com/locate/envsoft

Environmental Modelling & Software 20 (2005) 1031–1042

A DSS for water resources management under uncertaintyby scenario analysis

Stefano Pallottinoa, Giovanni M. Sechib, Paola Zuddasb,)

aDepartment of Computer Science, University of Pisa, Via F. Buonarroti, 56127 Pisa, ItalybDepartment of Land Engineering, University of Cagliari, Piazza d’Armi, 09123 Cagliari, Italy

Received 19 May 2004; received in revised form 19 August 2004; accepted 23 September 2004

Abstract

In this paper we present a scenario analysis approach for water system planning and management under conditions of climatic

and hydrological uncertainty. The scenario analysis approach examines a set of statistically independent hydrological scenarios, andexploits the inner structure of their temporal evolution in order to obtain a ‘‘robust’’ decision policy, so that the risk of wrongdecisions is minimised. In this approach uncertainty is modelled by a scenario-tree in a multistage environment, which includes

different possible configurations of inflows in a wide time-horizon. In this paper we propose a Decision Support System (DSS) thatperforms scenario analysis by identifying trends and essential features on which to base a robust decision policy. The DSS preventsobsolescence of optimiser codes, exploiting standard data format, and a graphical interface provides easy data-input and resultsanalysis for the user. Results show that scenario analysis could be an alternative approach to stochastic optimisation when no

probabilistic rules can be adopted and deterministic models are inadequate to represent uncertainty. Moreover, experimentation fora real water resources system in Sardinia, Italy, shows that practitioners and end-users can adopt the DSS with ease.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Water resources management; Multi-period dynamic network; Optimisation under uncertainty; Scenario analysis; Multi-scenario

aggregation

1. Introduction

Water Resources dynamic management (WR) prob-lems with a multi-period feature are associated withmathematical optimisation models that handle thou-sands of constraints and variables depending on thelevel of detail required to reach a significant represen-tation of the system (Loucks et al., 1981; Yeh, 1985).

One optimisation approach is to model the WRproblem as a dynamic multi-period network flowproblem, where all data are fixed and no level of

) Corresponding author. Tel.: C39 0706755320; fax: C39

0706755310.

E-mail addresses: [email protected] (S. Pallottino), [email protected]

(G.M. Sechi), [email protected] (P. Zuddas).

1364-8152/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.envsoft.2004.09.012

uncertainty is considered (Sechi and Zuddas, 1998;Kuczkera, 1992). Efficient optimisation algorithms havebeen used to solve this kind of problem (Sechi andZuddas, 2000).

But, WR problems are typically characterised bya level of uncertainty regarding, among other things, thevalue of hydrological exogenous inflows and demandpatterns. Assigning inaccurate values to them could wellinvalidate the results of the study. Consequently,deterministic models are inadequate for the representa-tion of these problems where the most crucial param-eters are either unknown or based on an uncertainfuture.

The traditional stochastic approach gives a probabi-listic description of the unknown parameters on thebasis of historical data. This is a very efficient approach

mailto:[email protected]



http://www.elsevier.com/locate/envsoft

1032 S. Pallottino et al. / Environmental Modelling & Software 20 (2005) 1031–1042

when a substantial statistical base is available andreliable probabilistic laws can adequately describeparameters’ uncertainty and their possible outcomes(Infanger, 1994; Kall and Wallace, 1994; Ruszczynski,1997). It is well known that stochastic optimisationapproaches cannot be used when there is insufficientstatistical information on data estimation to support themodel, when probabilistic rules are not available, and/orwhen it is necessary to take into account informationnot derived from historical data.

In these cases, the scenario analysis technique couldbe an alternative approach (Dembo, 1991; Rockafellarand Wets, 1991). Scenario analysis can model many realproblems where decisions are based on an uncertainfuture, whose uncertainty is described by means of a setof possible future outcomes, called ‘‘scenarios’’.

Some examples are given in Mulvey and Vladimirou(1989) for investment and production planning, inGlockner (1996) for air traffic management and inHoyland and Wallace (2001) for insurance policy andproduction planning.

The scenario analysis approach considers a set ofstatistically independent scenarios, and exploits theinner structure of their temporal evolution in order toobtain a ‘‘robust’’ decision policy, in the sense that therisk of wrong decisions is minimised. Rockafellar andWets (1991) proposed a scenario aggregation formula-tion that produces a tree structure of the scenarios, asexplained in detail in Section 3.

The aim of this paper is to apply the scenario analysisframework to WR problems and investigate its effec-tiveness with respect to traditional approaches.

A WRmodel is usually defined in a dynamic planninghorizon in which management decisions have to bemade sequentially or globally decided as a decisionstrategy referred to as a predefined scenario, wherea scenario represents a possible realisation of some setsof uncertain data in the time-horizon examined.

One common approach is to carry out a set ofexperiments on a number of generated series (parallelscenarios) followed by a simulation-testing phase ofeach scenario in order to validate the solutions underinvestigation. All the solutions (each one is a sequence ofdecisions) are completely independent one from theother because they are obtained from scenarios analysedseparately. As a consequence, the decisions adopted areclosely related to the scenario selected at the end of thesimulation and the study must start all over again ifa different scenario comes true.

To overcome the above difficulties, in this paper weanalyse the scenario approach for WR offering somegeneral rules for organising a predefined set of scenariosinto the scenario-tree and for identifying a complete setof decision variables relative to all the scenarios underinvestigation. The scenario-tree is obtained by aggregatecommon portions of scenarios; the aggregation condition

guarantees that the solution (that is, the decisions) inany given period is independent of the information notyet available, as detailed in Section 3.

Scenario analysis approach for WR was proposed in(Escudero, 2000); and in Wam-Me EU project, (Sechiand Zuddas, 2002), and tested on some real physicalsystems. Our proposal is to embed an evolution of theabove WR scenario analysis tool into a DSS that allowsin depth investigation of the robustness of the solutionand, if necessary, refinement of decisions. In fact, thedecision policy found in the scenario analysis can beconsidered ‘‘implementable’’ in the sense that thesequence of decisions is congruent, but it may notcorrespond to any solution obtainable by single-scenariooptimisation. In the proposed DSS, the availability ofan efficient computer graphical interface, that isdesigned to facilitate the use of models and database,helps end-users to evaluate with ease the best choice andreach a robust solution starting from the physicalsystem. The tool is a greatly improved version ofthe DSS WARGI (Sechi and Zuddas, 2000). Moreover,the proposed DSS allows ‘‘weight tuning phases’’ that theWR manager can use to refine the ‘‘relative importance’’that he should assign to each single scenario.

2. Water resources dynamic model

A successfully applied approach is to model theproblem by an optimisation network flow modelsupported by a multi-period dynamic graph where alldata are fixed and no level of uncertainty is considered(Kuczkera, 1992; Sechi and Zuddas, 1998). Networkflow models allow adopting highly efficient computa-tional algorithms even when thousands of variables andconstraints require management (Ahuja et al., 1993). InWR management problems, the authors explored thepossibility of maintaining network flow structure even ifnon-network constraints are present in the model(Manca et al., 2002) In Section 2.1 a detailed descriptionof a dynamic network flow model for water resourcesmanagement is provided. In this section we illustrate theway in which the associated multi-period dynamic graphcan be generated and describe the main components ofthe final mathematical deterministic model.

We formulate a WR management model in a de-terministic framework, i.e., having previous knowledgeof the time sequence of inflows and demand. We extendthe analysis to a sufficiently wide time-horizon andassume a time step (period), t. The scale and number oftime-steps considered must be adequate to reacha significant representation of the variability of hydro-logical inflows and water demands in the system.

Referring to a ‘‘static’’ or single-period situation, wecan represent the physical system by a direct network(basic graph), derived from the physical sketch. Fig. 1a

1033S. Pallottino et al. / Environmental Modelling & Software 20 (2005) 1031–1042

Fig. 1. (a) Water system. (b) Segment of a dynamic network.

shows a physical sketch of a simple water system. In thefigure, nodes maintain the shape of the commonhydraulic notation in order to recall the differentfunction of system components. Nodes could representsources, demands, reservoirs, groundwater, diversioncanal site, a hydropower station site, etc. Arcs representthe activity connections between them. Physical compo-nents corresponding to nodes and arcs can be in theproject stage (work planned) and/or operational (exist-ing works with a known dimension). Nodes correspond-ing to reservoirs represent the system memory since theycan store the resource in one period and transfer it ina successive period. A dynamic multi-period network isgenerated by replicating the basic graph for each periodt. We then connect the corresponding reservoir-nodesfor different consecutive periods by additional arcscarrying water stored at the end of each period. We callthese inter-period arcs. Fig. 1b shows a segment ofa dynamic network generated by the simple basic graphof Fig. 1a. Reservoir-nodes are supply-nodes that storeand supply the resource. Demand-nodes use andconsume the resource. Junction-nodes allow resourcepassage without consumption.

It may be convenient to add ‘‘dummy’’ nodes andarcs to represent not only physical components but alsoevents that may occur in the system. Fig. 2 shows thedynamic multi-period network, corresponding to that ofFig. 1b, including dummy nodes and arcs marked witha dot.

The basic graph is in the frame. The dummy node, U,represents a possible ‘‘external system’’ acting asa supposed source or demand of flow. In this way eacharc (i,U) represents a spillway from reservoir-nodes i,each arc (U,i) represents a supposed additional flow incase of shortage in order to meet request in the demand-nodes i and prevent solutions which are not feasible.Flow on arcs (U,i) highlights possible system deficits and

the need to modify the dimensions of the works or, alter-natively, to make recourse to external water resources.

The aim of this paper is not to identify thecomponents of the deterministic mathematical modelbut rather to provide formulation for a reduced modelthat can be adopted to formalize uncertainty in waterresources management. To illustrate our approach,we adopt deterministic Linear Programming (LP) asdescribed in the following section.

2.1. Definition of water resources optimisation modelcomponents

Though it is almost impossible to define a generalmathematical model for water resources planning andmanagement problems, our DSS makes it possible totake into account all possible general system compo-nents based on the most typical characterisation of thesetypes of models. Different components can be added ordeleted updating constraints and objectives. In thispaper, we describe only a few of them. A more detaileddescription of this approach can be found in Onnis et al.(1999) and Sechi and Zuddas (2000). Hereafter, we referto the dynamic network RZ (N,A) where N is the set ofnodes and A is the set of arcs. T represents the set oftime-steps t. Sets of nodes (subsets of N) can representreservoir-nodes, demand-nodes (such as civil, industrial,irrigation, etc.), hydroelectric nodes associated withhydroelectric plants, confluence nodes (such as riverconfluence, withdraw connections for demands satisfac-tion), etc. Sets of arcs (subsets of A) can representconveyance work arcs, artificial channels, transfer arcs,spilling arcs, etc.

Variables considered in the LP model can be dividedinto operation and project variables. Operation varia-bles can refer to different types of water transfer (flow onarcs) such as water transfer in space along an arc


Fig. 2. Dynamic multi-period network with dummy nodes and arcs.

connecting different nodes at the same time, watertransfer in an arc connecting similar nodes at differenttimes and so on. Project variables are associated to thedimension of future works: reservoir capacities, pipedimensions, irrigation areas, etc. Constraints in the LPmodel can represent mass balance equations, demandsfor the centres of water consumption, evaporation atreservoirs, relations between flow variables and projectworks, upper and lower bounds on decision variables.

The DSS allows the introduction of required datataking into account the different state of the hydrauliccomponents, that is if they are to be consideredoperational or in the project stage. To illustrate thisconcept, we provide hereunder, some details of possiblevariables, constraints and related data referred to asa reservoir-node and a demand-node.

For a reservoir-node, yt, represents the portion ofwater stored in the reservoir at the end of period t thatcan be used in subsequent periods. A correspondingconstraint, for each time-period t is:

rtminYmax%yt%rtmaxYmax

where Ymax represents the max storage volume for inter-period transfer arcs and rtmax

�rtmin

�represents the ratio

between max(min) stored volume in each period t andreservoir capacity. These constraints ensure that, in eachperiod, used volume yt of the reservoir is in theprescribed range. In an operational state Ymax is knownwhile in a project state it is a decision variable. In thelatter case it is bounded by:

m%Ymax%M

where M(m) represents the max(min) allowed capacity.For a demand-node, e.g. a civil demand, pt representsthe water demand at the civil demand centre in period t.A corresponding constraint, for each time-period t is:

ptZptdtP

where P represents the size of the population whosedemand can be fulfilled and pt the request program ineach period t. These constraints ensure the fulfilment ofthe demand in each period, no matter if it comes from

the system or from dummy resources. In an operationalstate, P is known while in a project state it is a decisionvariable. In the latter case it is bounded by:

Pmin%P%Pmax

where Pmax(Pmin) represents the max(min) estimatedpopulation.

Moreover, mass balance constraints are introduced,involving all flows that are going in or out of thereservoir or demand-node, including hydrological inputto the reservoir in each period.

The objective function considers costs, benefits, andpenalties associated with flow and project variables aswell as dummy costs or benefits associated with thedummy components of the multi-period dynamicnetwork.

2.2. Compact deterministic linear programming model

As is well known, a Linear Programming (LP)problem can be expressed in a compact standard formwhere all data are classified as:

- a ‘‘cost’’ vector c, in the objective function, whosecomponents cj can represent cost, benefit, penal-isation or a specific weight assigned by the managerto the variable xj;

- a RHS (Right Hand Side) vector b, in the constraintssystem, whose component bi can represent a supplyor a demand associated to a node i, i.e., to theactivity represented by node i as described in Section2.1;

- lower and upper bound vectors u and l, whosecomponents lj and uj represent lower and upperlimits (possible zero and infinity, respectively)imposed on the variable xj by physical, technolog-ical, environmental and/or political requirements;

- the matrix A represents the coefficient matrix of theconstraints system.

In the deterministic approach the hydrologicaldatabase is derived from available historical datasubmitted to statistical validation on the basis of


a forecast and adopted as reference scenario. As saidpreviously, in the deterministic optimisation model weassume that the manager has previous knowledge of thetime sequence of inflows and demands. As a conse-quence, the solution obtained is strictly connected to theadopted scenario. Given a set G of predefined scenarios,for a specific scenario g the corresponding LP model,(Pg) can be expressed as:

�Pg

�min cTgxg

s:t:AxgZbglg%xg%ug

The index g identifies vectors c, b, l and u of datarelated to the predefined scenario g. Moreover, xgrepresents the vector comprehensive of all operation andproject variables in scenario g and all constraints arerepresented as lower and upper bounds or in equationform. In this way, the compact LP model includes data,variables and constraints, such as those described inSection 2.1.

3. Water resources chance dynamic model

The deterministic models described in the previoussection are not adequate to describe the variability ofsome crucial parameters and, small differences in somedata in two different scenarios, can produce differentsolutions which differ significantly. In the deterministicapproach, a single scenario is taken as the one-timedatabase at the beginning of the optimisation processand the dynamic problem is converted into a staticmulti-period problem. Typically, most of the data inmodel (Pg) can be affected by uncertainty but a highlevel of uncertainty in WR problems is referred toexogenous inflows and demand patterns.

The last few years have shown just how unpredictablemeteorological events can be, especially in the Mediter-ranean area and, almost always, it is impossible torepresent these events by a probabilistic law. Thestochastic optimisation approach cannot be adoptedsince in WR it is unreliable to match a valid occurrenceprobability to each scenario. Moreover, some experi-mental tests using historical data, to simulate futureplanning horizons, have shown just how risky thestochastic decisions on the use of reservoir water inperiods of high variation in water availability have been.

The simulation approach studies a number of out-comes obtained by solving an optimisation problem (Pg)for each scenario g. During the optimisation process,different scenarios, corresponding to different dynamicmulti-period WR models, proceed independently of eachother obtaining a different water management policy for

each scenario. Simulation verifies the performance of allpolicies selecting one for future decisions. Usually, toreach a viable water management policy, a large numberof scenarios must be considered. The simulationapproach can prove very demanding from a computa-tional point of view, especially if continuously replicatedwhen the hydrological events occurring are verydifferent from those foreseen in the selected scenario.

The scenario analysis approach attempts to face theuncertainty factor by taking into account a set, G, ofdifferent supposed scenarios corresponding to thedifferent possible time evolution of some crucial data,such as inflows and demands, represented by vector bg inproblem (Pg), for each g˛G. Unlike simulation, thedifferent scenarios are considered together to obtaina global set of decision variables on the whole set ofscenarios. More precisely, two scenarios sharing a com-mon initial portion of data must be considered togetherand partially aggregated with the same decisionvariables for the aggregated part, in order to take intoaccount the two possible evolutions in the subsequentnon-common part. In this way, the set of parallelscenarios is aggregated by producing a tree structure,called scenario-tree. The aggregation rules guaranteethat the solution in any given period is independent ofthe information not yet available. In other words, modelevolution is only based on the information available atthe moment and, if necessary, scenario modification isallowed. We give detailed rules for the generation of thescenario-tree in Section 3.1.

The problem supported by the scenario-tree, isdescribed by a mathematical model that includes allsingle-scenario problems (Pg), c g˛G, plus some inter-scenario linking constraints representing the require-ment that if two scenarios g1 and g2 are identical up totime t on the basis of information available at that time,then the corresponding set of decision variables, x1 andx2, must be identical up to time t. This means that thesubsets of decision variables corresponding to theindistinguishable part of different scenarios must beequal among themselves. Moreover, a weight can beassigned to each scenario representing the ‘‘importance’’assigned by the manager to the running configuration.At times the weights can be viewed as the probability ofoccurrence of the examined scenario. More often theyare determined on the basis of background knowledgeabout the system.

The resulting mathematical model is named chancemodel to indicate that it is not stochastically based but,due to the impossibility of adopting probabilistic rulesand/or to the necessity of inserting information thatcannot be deduced from historical data, it attempts torepresent the set of possible performances of the system,as uncertain parameters vary. In Section 2.2 we describethe chance mathematical model produced by the scenarioanalysis approach.


3.1. Scenario-tree generation

In order to generalize the scenario approach for WR,we give some general rules to organise a predefined setof scenarios in a scenario-tree, and to manage the hugeamount of data needed to perform scenario analysis inthis field. These scenario aggregation rules are indepen-dent of the extension of the examined time-horizon, thenumber of time-periods and stages, the number ofpredefined scenarios and the adopted optimisationtechniques.

Fig. 3a shows a predefined set G of nine parallelscenarios before aggregation. Each scenario correspondsto a dynamic multi-period network, associated toa synthetic hydrological sequence, as shown in Fig. 2.In Fig. 3a each dynamic multi-period network isrepresented by a sequence of dots where a dot representsthe system in a time-period, i.e., a basic graph as inFig. 1a. Fig. 3b shows an example of the scenario-treederived from the parallel sequences.

To perform scenario aggregation, branching-timesand stages are defined. A branching-time t identifies thetime-period in which some scenarios, that are identicalup to that time-period, begin to differ. A stagecorresponds to the sequence of time-periods betweentwo branching-times. Stage 0 corresponds to the initialhydrological characterisation of the system up to thefirst branch time-period. This represents the root of thescenario-tree. In stage 1 a number b1 (3 in the figure) of

•-•-•-•--•-•-•-•-•-•-•-•-•-•-••-•-•-•--•-•-•-•-•-•-•-•-•-•-••-•-•-•--•-•-•-•-•-•-•-•-•-•-••-•-•-•--•-•-•-•-•-•-•-•-•-•-••-•-•-•--•-•-•-•-•-•-•-•-•-•-••-•-•-•--•-•-•-•-•-•-•-•-•-•-••-•-•-•--•-•-•-•-•-•-•-•-•-•-••-•-•-•--•-•-•-•-•-•-•-•-•-•-••-•-•-•--•-•-•-•-•-•-•-•-•-•-•

1 2 3 4 . . . . . . . . . . . . . . . . .8 .......time-periods

9 scenarios

1st 2nd branch-times

•-•-•-•-••-•-•-•-••-•-•-• •-•-•-•-••-•-•-•-••-•-•-• •-•-•-• •-•-•-•-••-•-•-•-•

•-•-•-• •-•-•-•-••-•-•-•-••-•-•-•-•0 stage 1st stage 2nd stage

a

b

1 bundle 3 bundles

Fig. 3. (a) Set G of nine parallel scenarios. (b) Scenario-tree

aggregation.

different possible hydrological configurations can occur,in stage 2 a number b1! b2 (9 in the figure) can occur,and so on and so forth.

Fig. 3b represents a tree with two branches: the firstbranching-time is the fourth time-period, the second isthe eighth period. In time-periods that precede the firstbranch, all scenarios are gathered in a single bundle andthree bundles are operated at second branch. Thisimplies that the zero bundle includes a unique group ofall scenarios; in the first stage 3 bundles are generatedidentifying the groups Gt:{(scen1,scen2,scen3); (scen4,-scen5,scen6); (scen7,scen8, scen9)} to include in eachbundle, while in the second stage the 9 scenarios rununtil they reach the end of the time-horizon.

Finally, the main rules adopted to organise the set ofscenarios are:

Branching to identify branching-times t as time-peri-ods at which to bundle parallel sequences,while identifying the stages at which todivide the scenario horizon.

Bundling to identify the number, bt, of bundles ateach branching-time.

Grouping to identify groups, Gt, of scenarios toinclude in each bundle.

The root of the scenario-tree corresponds to the timeat which decisions have been taken (common to allscenarios) and the leaves of the scenario-tree representthe performance of the system in the last stage. Eachback path from a leaf to the root identifies a possiblescenario.

3.2. The chance mathematical model

The chance model can be expressed as the collectionof one deterministic model for each scenario g˛G plusa set of congruity constraints representing the require-ment that the subsets of decision variables, correspond-ing to the indistinguishable part of different scenarios,must be equal among themselves. In this case, thechance mathematical model (PC) has the followingstructure:

ðPCÞ minPg

wgcgxg

s:t:AgxgZbg c g˛Glg%xg%ug c g˛Gx�˛S

where wg represents the weight assigned to a scenariog˛G; x* represents the vector of variables submitted tocongruity constraints; x*˛ S, the set of congruityconstraints.


Congruity constraints require that the flows (decisionvariables) in those scenarios that are indistinguishableup to branching-time t are the same up to t.Specifically, in these scenarios, the water stored ina reservoir at the end of the time t, to transfer in theperiod tC 1, must be the same. Of course, theseconstraints make the model redundant as, thanks toscenario aggregation, some scenarios are overlapped insome time-periods. As a consequence, the componentsof the model (variables and constraints), that areassociated to overlapped scenarios, can be reportedonly once. To illustrate this, Fig. 4 shows a branchcorresponding to a reservoir j for two scenarios, g1 andg2, identical up to the branching-time t. In particular,congruity constraint requires that the flow (storedvolume in a reservoir) from reservoir j at time t toreservoir j at time tC 1 in scenario g1 is equal to theflow from reservoir j at time t to reservoir j at timetC 1 in scenario g2.

Finally, they require that for each t, decisions takenup to that time must be the same for all scenarios thathave a common past and present in such a way that thechoices corresponding to different scenarios can beconsidered truly applied by the manager.

Regarding weight definitions, if the water managerwas able to evaluate the weight wg as the probability thatscenario g will occur, he could estimate it by somestochastic technique or statistical test. More often themanager has no, or few, possibilities to do this due tothe difficulty in deriving a probabilistic rule fromconceptual considerations. Instead, in scenario analysis,a weight wg assigned to a scenario g can be interpreted asthe ‘‘relative importance’’ of that scenario in theuncertain environment. In other words, in scenarioanalysis, weights are interpreted as subjective parame-ters assigned on the basis of the experience of the watermanagement board. Different weights can also beassigned to different stages. Then, the definitive weightin the objective function will be calculated considering

Fig. 4. Segment of branch at branching-time t referred to a reservoir j.

the contribution of scenarios and stages. A goodcompromise in weight settlement might be to assignscenario importance on the basis of subjective consid-erations, and assign weights to stages on the basis ofstatistical tests.

This kind of model can be solved by decompositionmethods such as Benders decomposition or Lagrangianrelaxation techniques, which exploit the special structureof constraints. When the problem is huge, it is possibleto resort to parallel computing (Glockner and Nem-hauser, 2000).

4. Using decision support system for water

resources management

The Decision Support System has been developed inorder to:

- guarantee simplicity of use in the input phase, inscenario setting and in processing output results;

- simplify modification of system configuration toperform sensitivity analysis and to process uncer-tainty;

- prevent obsolescence of the optimiser exploiting thestandard input format in optimisation codes.

A graphical interface allows scenario analysis, start-ing from a physical system following the main steps:

- time-period definition and scenario settlement.- system element characterisation.- connections topology and transfer constraints.- links to hydrological data and demand requirementfiles.

- planning and management rule definition.- benefits and costs attribution.- call to optimisers.- output processing.

The DSS has been developed and tested within a HP-Unix and PC-Linux environment. The various softwarecomponents have been coded in CCC and TCL-TKgraphic language. The graphical components are in-troduced on an empty canvas by means of a tool palette.A time range window provides data covering time rangeand time-steps. Once the graph has been completed, thenecessary data are inserted in correspondence to nodesand arcs in operation or project mode as required by thephysical system.

Different templates are associated with differentcomponents. Data can be provided in scalar, cyclic orvector modality. Hydrological input can be directlylinked to the appropriate files containing the generatedsynthetic series to be used by the DSS to generate thescenario-tree. Input data of scenario components are


Fig. 5. Scenario-tree for the sample system: two scenarios, two stages, one branching-time.

introduced with the appropriate tools defining numberof scenarios and setting branches and stages as inSection 2.1. Branch periods and multiplicity are thendefined.

Once the graph has been completed and all necessarydata have been inserted, it is possible to generate theMPS (Mathematical Programming System) file to feedto the solver. MPS is now the standard formatsupported by most efficient commercial and non-commercial mathematical programming computer co-des. Standard data format allows the use of any solver,possibly the most efficient or the most suitable or themost readily available in the work environment. More-over, it prevents obsolescence of the used solvers andensures portability of the DSS. The solver can beselected from a prefixed list (or configured if not in thelist) and launched.

A graphical post-processor manages the great quan-tity of information provided by the DSS. The user canselect what is of interest in the environment underexamination. Indeed, it is possible to show any flow inany scenarios in any period such as volume transferredto reservoir in a specific scenario, period by period, aswell as in different scenarios in the same graph.

4.1. Test case I: a sample system

To illustrate the scenario analysis approach wepresent, initially, a sample case derived from a simplewater system with a reservoir and a demand centre (e.g.civil demand). We assume that dimensions of thereservoirs and the demand centre are known, that isthe system is in an operational state. Variables of theoptimisation problem are referred to stored water ytg,flows transferred from reservoir to demand centre xtg,deficits utg in each period t and in each scenario g. Waterdemand ptg is assigned as population P is known (seeSection 2.1). We adopt, as historical data, a hydrologicalseries of 48 monthly time-periods (4 years) reproducingtypical behaviour of the Mediterranean system: a widerange of inflow variability between humid and drought

periods. Inflows average is adopted as civil demand bythe demand centre assuming that the water systemis balanced. Moreover, in order to facilitate resultinterpretation, we assume that the volume, Ymax, of thereservoir is large enough to prevent spillage and thatevaporation losses are negligible. We generate twoscenarios, g1 and g2, assuming that uncertain parame-ters correspond to hydrological inflows, inptg, i.e.,supplies in reservoir-node in period t in scenario g.Costs and bounds associated to variables are consideredwithout uncertainty and, as a consequence, are the samein both scenarios. We generate a scenario-tree with twostages and one branching-time t. Fig. 5 shows thescenario-tree for this simple example.

In the figure, variables up to branching-time t arereported without scenario subscribe because they are thesame in the two scenarios, as required by congruityconstraints. Dummy node U and dummy arcs, corre-sponding to deficit utg, are not reported in the figure. Thetwo scenarios are both identical to the historical data upto tZ 12th time-period. Scenario g2 follows thehistorical data from tC 1 to the last time-period, whilescenario g1 has the hydrological inflows reduced by 50%with respect to it. The scenario-tree is then generatedfollowing the aggregation rules in Section 3.1. Startingfrom a set of two parallel scenarios, stage 0 representsthe initial hydrological configuration from the 1st to the12th time-period. In stage 1, a number b1Z 2 of bundles(different possible hydrological configurations) canoccur so that GtZ {(scen1); (scen2)}. Then the twoscenarios run until they reach the end of the time-horizon. We assume that wg1Z wg2Z 1 and that ct andat represent costs per unit of flow xtg and utg, respectively,for gZ g1, g2.

The objective is to minimise operative total costs, thatis, transfer costs from reservoir to demand centre plusdeficit costs. The chance model, for this simple example,following notation in Section 2.1, can be written asfollows:

Xt

�ctxt

g1Cctxtg2Catutg1Catutg2

�objective function


ytg1Zytg2xtg1Zxt

g2

utg1Zutg2

9=; congruity constraints ðset S in the chance modelÞ tZ2;.;t

rtminYmax%ytg%rtmaxYmax bounds on stored volumes

yt�1g � ytg � xt

gZinptg mass balance in reservoir� node

xtgCutgZptg mass balance in demand� node

9=;tZ2;.;48; gZg1;g2

Hereunder we illustrate some results concerningstored volumes in reservoir, ytg, and transferred waterto demand centre, xtg, obtained by scenario analysis,solving the above optimisation chance model.

Fig. 6 shows stored volumes with tZ 12th time-period. Moreover, the figure shows stored volumesobtained by an optimisation deterministic model whenthe scarce scenario g1 is assumed as database. In thiscase we use g1 as an independent scenario and is calleds1 in the figure. The graph referred to deterministicresults with scenario s1 represents decisions, in watertransfer, in a deterministic optimisation process. Thezone (band) between the two graphics of the aggregatedscenarios, g1 and g2, represents the possible decisionsthat can be taken for stored volumes. Then we can saythat the part of s1 that does not stay between g1 and g2represents the error that the manager would have madeif he had adopted decision s1.

Fig. 7 shows the transferred water, xtg, from reservoirto the demand centre in aggregated scenarios, and in theindependent scenario. The behaviour of these flowsshows that in the scenario g2 demand is fulfilled while inscenario g1 deficits are present after branching-time.But, comparing with results in deterministic optimisa-tion under scenario s1, we can see that as regards thescarce resource conditions, scenario optimisation givesa smoother distribution, i.e., with a lower variance offlow distribution in scenario g1 even if the average isalmost the same as scenario s1. Then, the managementpolicy suggested by scenario analysis results lessdramatic and more implementable with respect to thatadopted in deterministic optimisation in the case of

0,00

50,00

100,00

150,00

200,00

250,00

300,00

350,00

0 5 10 15 20 25 30 35 40 45 50

time-periods

Mm

3

scenario g1

scenario g2

scenario s1

Fig. 6. Sample system: stored volumes in the reservoir.

scarce resources. These effects are even more evident ina real wider system as described in the following section.

4.2. Test case II: a real physical system

Scenario analysis was performed on the Flumendosa–Campidano–Cixerri (FCC) system in Sardinia, Italy.Fig. 8 shows a sketch of the system.

Since 1987, the Sardinia-Water-Plan has highlightedthe necessity of defining an optimal water worksassessment and the urgency of defining optimal man-agement rules for the water system. Correct evaluation ofsystem performances and requirements became increas-ingly urgent, as system managers were obliged to faceserious resource deficits caused by the drought events ofthe past decade accompanied by an almost totaluncertainty in hydrological inflows. The main infra-structures were built in the mid 1950s and supply most ofsouthern Sardinia. The main water supply source of thesystem is represented by the three reservoirs with a totalstorage capacity of 666.4 million cubic meters (Mm3).Gravity galleries connect the reservoirs. Total yearlyaverage distributed volume in the period examined is225 Mm3 for civil, industrial and agricultural demands.No significant aquifers are present in the system. Theprincipal works can be identified in 13 water supplysources (dams and weirs), 3 water diversion and commu-nication galleries, 5 main water supply and distributionnetworks, 2 hydroelectric power stations, 1 irrigationdistribution network, 11 pumping stations, and 2 drink-ing water plants.

A number of synthetic series is generated andadopted as the set of predefined scenarios. The basic

0,00

10,00

20,00

30,00

40,00

50,00

60,00

70,00

0 5 10 15 20 25 30 35 40 45 50

time-periods

Mm

3

scenario g1

scenario g2

scenario s1

Fig. 7. Water transfer from reservoir to the demand centre.


Fig. 8. Flumendosa–Campidano–Cixerri system.

hydrological data are derived from the report in RAS(2003) and different scenario generation techniques havebeen compared.

The graphical interface of the DSS makes it possibleto design the system following the network componentidentification like those described in Section 2.1 and the

simple rules reported in Section 3.1. Fig. 9 shows theinput window of the system examined.

Starting from a database with a time-horizon up to 75years, corresponding to 900 monthly time-periods, a setof 30 scenarios was submitted to statistical validation andselected. Scenario analysis was performed on a scenario-

Fig. 9. Input window of Flumendosa–Campidano–Cixerri system.


tree of 2 and 3 stages up to 30 leaves. Since each scenarioinvolves about 3000 variables, the chance model supportsseveral thousands of variables and constraints.

In this paper, we report some results selected amonga wide set of output information that the DSS provides.In particular, we report some results obtained adoptinga time-horizon of 48 time-periods and a branching-timein the 12th time-period as in the sample system.Illustrated results are referred to the ‘‘Nuraghe Arrubiu’’reservoir (the red arrow in Figs. 8 and 9) that isconsidered one of the main pivots of the system as itcan control water transfers to the principal demandcentres. Two scenarios are deduced from the last 4 yearsof hydrological inflows reported in RAS (2003). Weadopt these data as scenario g2 while scenario g1 isderived from it assuming that a reduction of 50% willoccur after the branching-time. Fig. 10 shows thebehaviour of stored volumes obtained by scenarioanalysis (aggregation of g1 and g2) and the behaviourobtained by deterministic optimisation using the reducedindependent scenario s1. The figure shows that decisionpolicy, corresponding to deterministic optimisation,induces an early empty reservoir with respect to thedecision policy given by scenario analysis in theequivalent scenario g1. This corresponds to the behav-iour of transferred water to demand centres as illustratedin Fig. 11. As in the sample system, decision policy g1,resulting from scenario analysis, exhibits smootherresources distribution and a lower variance with respectto deterministic policy s1.

5. Conclusion and perspectives

We presented a general-purpose scenario-modellingframework to solve water system optimisation problemswhen input data are uncertain, as an alternative to thetraditional stochastic approach, in order to achievea ‘‘robust’’ decision policy that should minimise the riskof wrong decisions. The proposed tool can performscenario analysis by generating a scenario-tree structure.It allows the exploitation of the state-of-the-art efficientcomputer codes for general-purpose Mathematical Pro-gramming supporting some thousands of variables andconstraints.

0,00

50,00

100,00

150,00

200,00

250,00

300,00

0 5 10 15 20 25 30 35 40 45

time-periods

Mm

3

scenario g1

scenario s1scenario g2

Fig. 10. Stored volumes in Nuraghe Arrubiu reservoir.

The aim of this paper is to contribute to themathematical optimisation of water resource systems,when the role of uncertainty is particularly important. Insuch a problem, which involves social, economical,political, and physical events, there is often no proba-bilistic description of the unknown elements available,either due to the lack of a substantial statistical base orbecause it is impossible to derive a probabilistic law fromconceptual considerations.

The high level of uncertainty regarding hydrologicalinflows has a dramatic impact on decisions adopted forassigning water resources to demand centres, makinglong-term management planning difficult. The unusuallysignificant reduction of resources in this last decade hascaused onerous deficits in demand fulfilment.

Experimentation has been performed on a real waterresource system in Sardinia, Italy, in collaboration withregional water managers showing that practitioners andend-users can adopt the DSS as a useful aid to decisionmaking. Moreover, the application of our approach toa real system demonstrates that it is simple and practicalto examine a number of different scenarios and toaggregate them in a scenario-tree, allowing accuratesensitivity analysis.

Another primary aim in planning this approach toWRanalysis was to create an easy-to-use tool for assistingwater managers in a DSS context, incorporating theimprovements made in the field of computer science andoperation research. Thanks to new methodologies anddevelopments inComputer Science, state-of-the-artMath-ematical Programming codes evolve continuously pro-ducing algorithms that improve computational efficiency,see CPLEX (2002). The standard input format, MPS,allows inserting the best state-of-the-art codes in the DSS.

Thus, the proposed DSS is a practicable instrumentwhich can be used by WR managers to defineprogrammed reduction of theoretical resource demand,in order to achieve optimal system management, both tofulfil consumers’ needs and minimise the cost deficit. Infact, as WR managers know well, the programming ofdeficits makes it possible to set up adequate preventivemeasures, which permit a notable reduction of manage-ment costs in the event of resource scarcity.

0,00

4,00

8,00

12,00

16,00

20,00

0 10 20 30 40 50 60

time-peirods

Mm

3

scenario g1

scenario g2

scenario s1

Fig. 11. Transfer water from Nuraghe Arrubiu reservoir to demand

centres.


References

Ahuja,R.K.,Magnanti, T.L.,Orlin, J.B., 1993.NetworkFlows:Theory,

Algorithms and Applications. Prentice Hall, Englewood Cliff, NJ.

CPLEX Optimisation Inc., 2002. Using the CPLEX Callable Library

and CPLEX Mixed Integer Library. CPLEX Optimisation, Inc.,

Incline Village, Nevada.

Dembo, R., 1991. Scenario optimisation. Annals of Operations

Research 30, 63–80.

Escudero, L.F., 2000. A robust approach for water resources planning

under uncertainty. Annals of Operations Research 95, 313–339.

Glockner, G.D., 1996. Effects of air traffic congestion delays under

several flow management policies. Transportation Research Re-

cord 1517, 29–36.

Glockner, G.D., Nemhauser, G.L., 2000. A dynamic network flow

problem with uncertain arc capacity: formulation and problems

structure. Operations Research 48 (2), 233–242.

Hoyland, K., Wallace, S.W., 2001. Generating scenario trees for

multistage decision problems. Management Science 47 (2), 295–307.

Infanger, G., 1994. Planning Under Uncertainty: Solving Large-scale

Stochastic Linear Programming. Boyd & Fraser Publishing

Company, Danvers, MA.

Kall, P., Wallace, S.W., 1994. Stochastic Programming. John Wiley

and Sons, New York.

Kuczkera, G., 1992. Network linear programming codes for water-

supply head-works modelling. Journal of Water Resources

Planning and Management 3, 412–417.

Loucks, D.P., Stedinger, J.R., Haith, D.A., 1981. Water Resource

Systems Planning andAnalysis. PrenticeHall, EnglewoodCliffs,NJ.

Manca, A., Podda, A., Sechi, G.M., Zuddas, P., 2002. The network

simplex equal flow algorithm in dynamic water resources manage-

ment. Proceedings of Sixth Congress of SIMAI, Chia Laguna, May

27–31, 2002.

Mulvey, J.M., Vladimirou, H., 1989. Stochastic network optimisation

models for investment planning. Annals of Operation Research 20,

187–217.

Onnis, L., Sechi, G.M., Zuddas, P., 1999. Optimisation Processes

Under Uncertainty. A.I.C.E., Milano, pp. 238–244.

RAS, 2003. Convention RAS-DIT-EAF, Regional Report. Piano

d’Ambito della Regione Sardegna, Cagliari.

Rockafellar, R.T., Wets, R.J.B., 1991. Scenarios and policy aggrega-

tion in optimisation under uncertainty. Mathematics of Operations

Research 16, 119–147.

Ruszczynski, A., 1997. Decomposition methods in stochastic pro-

gramming. Mathematical Programming 79, 333–353.

Sechi, G.M., Zuddas, P., 1998. Structure oriented approaches for

water system optimisation. Conference on Coping with Water

Scarcity, Hurgada, Egypt, September 1998.

Sechi, G.M., Zuddas, P., 2000. WARGI: water resources system

optimisation aided by graphical interface. In: Blain, W.R.,

Brebbia, C.A. (Eds.), Hydraulic Engineering Software. WIT-

PRESS, pp. 109–120.

Sechi, G.M., Zuddas, P., 2002. INCO-MED EU Project, Water

resources management under drought conditions (Wam-me). Work

Package 5.

Yeh, W.G., 1985. Reservoir management and operations models:

a state-of-the-art review. Water Resources Research 25 (12), 1797–

1818.

Documents

A DSS for water resources management under uncertainty by scenario analysis