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Evaluating Flexibility in Water Distribution System Design under Future
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Demand Uncertainty
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Innocent Basupi1 and Zoran Kapelan2
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Centre for Water Systems, College of Engineering, Mathematics and Physical Sciences,
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University of Exeter, North Park Road, Harrison Building, Exeter, EX4 4QF, UK (Email:
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[email protected]; [email protected])
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Telephone: +44(0)1392 723600; Fax: +44(0)1392 217965
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ABSTRACT
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Performance of Water Distribution Systems (WDSs) is highly dependent on consumer water
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demand that has to be met with adequate pressure. With uncertain future water demand due to
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the effect of rapid urbanisation and climate change, WDSs may underperform or end up being
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overdesigned due to the long-term unforeseen future conditions. Long-term planning of WDS
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therefore requires strategic, cost-effective and sustainable design intervention investment across
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the entire planning horizon that is uncertain in nature. However, making the most appropriate
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decisions on such intervention measures that keep up the performance of WDS under uncertainty
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is a challenge. This paper illustrates the importance of flexible WDS design under uncertain
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future water demand. The methodology has been tested on the New York Tunnels and the
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PhD Student, Centre for Water Systems, College of Engineering, Mathematics and Physical Sciences, University
of Exeter, North Park Road, Harrison Building, Exeter, EX4 4QF, UK. E-mail: [email protected]
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Professor, Centre for Water Systems, College of Engineering, Mathematics and Physical Sciences, University of
Exeter, North Park Road, Harrison Building, Exeter, EX4 4QF, UK. E-mail: [email protected]
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Anytown network interventions in the long-term planning. The approach is compared with the
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traditional deterministic intervention plans. The analysis results demonstrate that there is
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potential WDS performance value obtained when flexible design approach is adopted rather than
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the conventional deterministic practice.
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Keywords| Uncertainty; Flexible Design; Deterministic Design; Water Distribution Systems
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INTRODUCTION
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Water demand is one of the major factors that determine the hydraulic performance of a WDS.
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The effect of rapid urbanisation and climate change render water demand highly uncertain.
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Redesign of WDSs is necessary to keep up with the water service requirements. However, the
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extent of design has to be done now in order to supply adequate water to customers into the
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unknown future. Making appropriate and long-term decisions becomes a challenge that requires
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strategic methodologies that lessens the susceptibility of overdesigned or underperforming WDS
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as an attempt to better the traditional (deterministic) approach.
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The aim of this paper is to evaluate the flexibility inherent in the flexible WDS design approach
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as compared to the rigid and deterministic method that can lead to overdesigned or
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underperforming WDSs due to the uncertain nature of the long-term water demand.
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The paper is organised as follows: after this introduction, the background of flexible designs and
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application in engineering systems under uncertainty is briefly discussed and the WDS flexible
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design evaluation methodology under water demand uncertainty is explained. The methodology
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is then demonstrated on two literature case studies and the results are compared to both
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precautionary and staged deterministic approaches. Finally, conclusions are drawn.
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BACKGROUND
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Uncertainty in engineering system parameters requires a long-term planning, designing and
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management. Creaco et al. (2013) found out that long-term designs (i.e. sequencing of
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construction) perform better than the deterministic designs but uncertainty of design parameters
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was not considered. WDS design is traditionally based on the deterministic future water demand
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projections. The deterministic approach makes WDS susceptible to poor performance due to the
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actual demand most likely to differ from the projections. Building redundancy in WDS is one
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way of achieving robustness. Kapelan et al. (2005) and Babayan et al. (2005) worked on
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robustness based solutions to WDS design problems under water demand uncertainty. Other
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researchers formulated and solved similar robust WDS design problems under uncertainty
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(Lansey et al., 1989, Xu and Goulter (1999), Giustolisi et al., 2009). However, in all these
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approaches the future demand uncertainty has been addressed only passively by building in
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additional system redundancy via suitably sized interventions that are fixed over some pre-
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specified long-term planning horizon. Other researchers, Kang and Lansey (2012) considered
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scenario-based multi-stage construction of water supply infrastructure under demand uncertainty.
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In their method, scenarios (plausible futures with assumed probabilities of occurrence) are
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simultaneously considered to identify a solution that minimizes the expected costs (including
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overpayment and supplementary costs needed to meet the requirements) as a single objective.
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The method presented by Kang and Lansey (2012) adds robustness in system design by
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identifying solutions that allow for adaptive modifications with minimized overpayment and
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supplementary costs. However, optional paths for relevant levels of uncertain demands that are
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accounted for in the current study were not considered. The method presented here also
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simulates uncertain demands around the mean demand (without focus on different spatial
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distributions) at each stage as explained in the methodology section of this paper. Furthermore,
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the intervention plans are evaluated over a large number of possible futures compared to the
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scenario-based method.
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As pointed out by De Neufville (2004), alternative ways exist to more proactively manage future
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uncertainties by creating and maintaining flexibility in the engineering design. Huang et al.
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(2010) are the authors who also suggested using the concept of flexible design in the context of
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long-term planning of WDSs. However, the methodology presented here differs from the
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methodology shown in the above conference paper as follows: (a) Demand uncertainty is
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represented differently: in the methodology presented here, uncertain future demands are
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represented using the (Gaussian, can be any other) probability density functions with increasing
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mean (denoting increase in most likely future demands) and increasing variance (i.e. level of
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uncertainty) over the planning horizon whereas in the Huang et al. (2010) approach, uncertain
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future demands are represented using a scenario tree denoting a number of pre-defined demand
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scenarios with arbitrarily chosen probabilities for different paths on the tree. The approach
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adopted here seems more compatible with the way demand projections are normally made in the
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engineering practice; (b) Intervention plans are represented and evaluated differently: in the
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methodology presented here, flexible intervention plan is represented as a decision tree with
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threshold demand values defined at branching (i.e. decision) points. The associated intervention
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path probabilities are then estimated indirectly by using the Monte Carlo (MC) sampling method.
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This setup allows for decisions to be made according to how the uncertain water demands would
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actually evolve. In the Huang et al. (2010) approach, flexible intervention plan is represented as
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a set of time-staged interventions that are evaluated using the Enumeration method over a small
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number of pre-defined demand scenarios over the analyzed planning horizon (see above); (c)
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The designed WDS performance is evaluated differently: in the methodology presented here this
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is evaluated in terms of total cost and resilience as the average of all sampled demand profiles
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whereas in the approach by Huang et al. (2010), this is evaluated in terms of cost and pressure
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deficiency by summing up the products of individual WDS design costs and the assumed
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probability of each demand scenario analyzed.
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FLEXIBLE WDS DESIGN METHODOLOGY
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Uncertain Demands
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The future water demand is considered as the only source of uncertainty in this analysis i.e. to
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demonstrate the value of flexible WDS design approach. The nodal water demands at given
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points in time over some long-term planning horizon are assumed to follow a normal distribution
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function (Babayan et al., 2005, Kapelan et al., 2005) with mean values equal to the traditional,
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deterministic projection of future demands and the increasing standard deviation. The design
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methodology shown here is not limiting in this sense and any other probability density
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function(s) (PDF / PDFs) (or scenario based approach) can be used to describe uncertain future
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water demand.
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A single demand profile (or scenario) over the planning horizon is generated by randomly
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sampling system level demands (Dt1, Dt2, ..., Dtend shown in Figure 1) from respective, pre-
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specified demand PDFs at the end of each planning horizon stage (i.e. at time t1, t2 , ..., tend). Note
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that large number of demand profiles (or scenarios) is generated for the purpose of evaluation of
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flexible (and deterministic) WDS designs (i.e. 10,000 samples in the case studies shown here).
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Finally, the randomly generated system level demands are allocated to demand nodes
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proportionately, according to each node’s respective contribution to (i.e. percentage of) the
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system level (mean) demand. Note that this means that an absolute increase (i.e. in ‘l/s’) for a
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node with larger demand will be greater than the increase of the node with smaller demand.
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[Insert Figure 1]
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Representation of Flexible WDS Design
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The flexible design, i.e., intervention plan is represented using a decision tree (see Figure 1).
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Each branch on the tree represents a set of individual intervention measures (i.e. interventions)
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and each path on the tree (from the ‘root’ of the tree to its ‘leaves’) represents one possible (i.e.
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optional) intervention path across the analysed planning horizon (a total of 8 paths shown in
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Figure 1). The branching of the tree corresponds to the design stages of the planning horizon (t0-
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t1, t1-t2, etc.). Note that the decision tree used here is not a ‘full’ tree, i.e. it does not have
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combinations of tree nodes (opposite of binomial lattices). The decision tree explained here has
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optional intervention paths at the end of each time step of the planning horizon. These features of
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the decision tree enable it to deal with path-dependent, interdependent and irreversible flexible
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interventions. In this case, path-dependence means that the extent of future design interventions
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or the state of the system at any point in time depends on the previous intervention path
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undertaken (given changes in uncertain water demand). This feature is as opposed to the
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financial options (of which this methodology is analogous to) that have value which only
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depends on the price at any particular time. Interdependence refers to the interaction of
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interventions that influence each other’s performance, hence the overall performance of the
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whole system. The flexible designs are non-reversible because once they have been exercised,
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the interventions (e.g. structures) implemented are not going to be demolished.
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As shown in Figure 1, at each tree node a decision point exists. The decision to be made is which
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tree branch to follow (i.e. which set of optional interventions to implement) in the next planning
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horizon (i.e. design stage). This decision is taken based on the level of demand at that decision
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point relative to the pre-specified demand threshold (see intervention paths e.g. L-D, LH-D,
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LHH-D, etc. and the corresponding threshold demands TA, TB, TC shown in Figure 1). For
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example, if the generated system demand Dt1 at the end of first stage (i.e. at time t1) is larger than
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the corresponding threshold demand (TA) then path ‘H-D’ is followed. If, however, demand Dt1 is
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smaller than the threshold demand TA then the ‘L-D’ intervention path is followed. Furthermore,
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if Dt1 results in the selection of the ‘H-D’ path then Dt2 can only be used to select either ‘HL-D’
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or ‘HH-D’ path (depending on whether Dt2 is above or below TB-1), etc. The subscripts A, B and
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C represent the three design stages that are used here to demonstrate the evaluation of design
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plans. Therefore, as shown in Figure 1, for a given random demand profile, the demands
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generated at t1 t2 and tend are used to select (and eventually evaluate, see below) the WDS designs
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at stages 1, 2 and 3, respectively. Note that the demand threshold value is not linked directly to
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any pipe / tank / pump capacity. Also, when the flexible designs are evaluated, no further pipe
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and / or tank sizing, etc. are performed, i.e. the flexible design plans are not modified in any
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sense. Note that ‘do more / design’ and ‘do less / nothing’ interventions are represented with
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solid and broken lines, respectively. The threshold values essentially define indirectly the
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probabilities of optional paths on the intervention decision tree (given random future demands).
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Given a ‘certain system water demand level’ (say at t1) and assumed demand PDFs across the
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planning horizon, there is a probability associated with the likelihood of sampling system
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demand that is above or below that ‘certain water demand level’. This ‘certain water demand
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level’ is what is referred to as the threshold demand here.
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[Insert Figure 2]
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Evaluation of Flexible WDS Designs
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The WDS design performance at each stage of the planning horizon is evaluated by calculating
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the cumulative cost and the WDS resilience, as defined below. For this purpose, a number of
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random demand profiles (i.e. scenarios) are generated first, as described in the demand section
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above. Once this is done, for each demand profile generated, the corresponding intervention path
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on the decision tree (used to represent the evaluated flexible design) is determined, as explained
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in the previous section. Now, for each intervention path selected, the EPANET network
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hydraulic solver (Rossman, 2000) is run at the end of each design stage (i.e. at t1, t2,..., tend) with
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the aim to estimate the system resilience at these points in time (note that runs are not performed
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at t0 as initial resilience is known, i.e. needs to be estimated only once, prior to sampling
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demands). At the same time, the corresponding intervention (and if necessary operational) costs
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are evaluated and accumulated. Each time the hydraulic solver is run (e.g. at time t1) both the
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network configuration (and, if necessary, system operation) and system demands are updated to
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take into account the interventions ‘implemented’ during the analysed stage (t0-t1) and the
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modified end of stage demands (at t1). The above is repeated for all design stages resulting in two
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planning horizon profiles (resilience and cost) built for each analysed demand profile. At the
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end, the cumulative cost and system resilience profiles generated this way are averaged over a
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pre-specified, typically large number of Monte Carlo simulations (i.e. demand samples). The
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procedure for the evaluation of the flexible WDS design represented using a decision tree in
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Figure 1 is shown in Figure 3. Note that the approach presented here is generic in terms of the
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number of time-steps and the intervention paths considered.
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[Insert Figure 3]
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As mentioned above, in order to demonstrate the value embedded in the flexible intervention
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under the proposed methodology, the following two indicators were used here: (1) the total
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intervention cost incurred in the interventions that include duplicating existing or adding new
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pipes, cleaning and lining of existing pipes, addition of tanks and their locations, tank parameters
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(diameter, bottom elevation, minimum and maximum operating levels) and the pump schedule,
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i.e. the number of operating pumps at a given time. (2) the WDS resilience index (RI) , which
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serves as a measure of WDS’s intrinsic capability to ensure continuity of supply to users after
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sudden failure conditions.
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Total system intervention cost is estimated as follows:
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N
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Average Total Cost =
npu
1
(∑n Cj,cap +∑j=1 Cj,op )
(1+r)ti j=1
s ∑S
∑k=1
i=1
Ns
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(1)
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where Ns is the total number of samples (demand scenarios); r is the cost discount rate; ti is the
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time elapsed at the beginning of each design stage (design point); S is the total number of design
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stages; Cj,cap is the capital cost of the j-th rehabilitation intervention option in the i-th stage; n is
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the number of rehabilitation options implemented; Cj,op is the whole stage interval operating
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energy consumption cost due to the j-th pump and npu is the number of pumps in the system for a
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particular i-th stage. The total cost is cumulative across the entire planning horizon for every k-th
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sample.
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The WDS resilience was introduced by Todini (2000) as a measure of WDS’s intrinsic capability
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to ensure continuity of supply to users after sudden failure conditions. It is estimated as follows:
𝑛
∑𝑛𝑖=1
𝑞𝑖 (ℎ𝑎𝑣 − ℎ𝑟𝑒𝑞 )
𝑅𝑒𝑠𝑖𝑙𝑖𝑒𝑛𝑐𝑒 𝐼𝑛𝑑𝑒𝑥 =
𝑛𝑝𝑢 𝑃𝑗
𝑛𝑛
𝑟
∑𝑛𝑟=1
𝑄𝑟 𝐻𝑟 + ∑𝑗=1
( ⁄𝛾 ) − ∑𝑖=1
𝑞𝑖 ℎ𝑟𝑒𝑞
(2)
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Due to the multi-staged designs and a large number of demand scenarios considered in this
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methodology, the average resilience index (RIav) at each time-step is calculated as follows:
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𝑅𝐼𝑎𝑣
𝑠
∑𝑁
𝑘=1 𝑅𝐼𝑘
=
𝑁𝑠
(3)
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where qi corresponds to the flow of the i-th node; hav is the available piezometric head; hreq is the
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minimum required head; Qr is the reservoir flow; Hr is the reservoir head; Pj is the pump power;
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γ is the specific weight of water; nn is the number of demand nodes; nr is the number of
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reservoirs and npu is the number of pumps. Note that other resilience (and/or reliability/risk)
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measures could be used as well, if desired, e.g. the network resilience and the modified resilience
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index by Prasad and Park (2004) and Jayaram and Srinivasan (2008), respectively. For more
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information on the performance of these resilience indices, see Banos et al. (2011). The focus of
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this paper is on the flexible design concepts, not on the most appropriate resilience measure.
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Note that equation (1) uses the present value analysis for discounting future costs. System
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resilience improvement is used as a measure of benefit achieved by design interventions. Also
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note that this benefit is not expressed in monetary units which prevents the use of more
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conventional Net Present Value (NPV) analysis.
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The intervention plan analysis is subject to hydraulic model (i.e. mass and energy balance)
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equations:
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𝑖
∑𝑛𝑚=1
𝑞𝑚 −𝑞𝑑,𝑖 = 0
ℎ𝑖,𝑢 − ℎ𝑖,𝑑 − ∆ℎ𝑖 = 0
(𝑖 = 1, . . . , 𝑛𝑛 )
(4)
(𝑖 = 1, … , 𝑛𝑙 )
(5)
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where qm are the flows in all ni pipes; qd,i is the water demand at the i-th node; hi,u is the head at
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upstream node of the i-th pipe; hi,d is the head at downstream node of the i-th pipe; ∆hi is the
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difference between the i-th pipe’s total head loss and pumping head; nn is the number of network
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nodes; nl is the number of network links.
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Solutions whose performance is evaluated are kept within the practical constraints as shown in
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Figures 6, 7, 11 and 12. These constraints mean that a potential solution should have path-
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dependent, interdependent and irreversible interventions that can be implemented at most once in
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any particular uncertain water demand scenario across the planning horizon.
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Generation of Flexible WDS Designs
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The flexible and other WDS designs analysed in the case studies shown here have been derived
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from the corresponding optimised WDS designs reported previously in the literature. These
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designs were generated manually by using engineering judgment only. The relevant details of the
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designs generated this way can be found in the next section. Whilst this is not ideal (flexible
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designs could be optimised in the same way as the more conventional, deterministic ones) it is fit
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for the objective of this paper which is to compare the deterministic and flexible WDS designs
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and demonstrate that flexible WDS designs can outperform the more conventional (i.e.,
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deterministic) ones under conditions of uncertainty. Therefore, if the latter can be shown with
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manually generated flexible WDS designs (as it is done in the next section), then things can only
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be improved further by using some more formal methodology (e.g. optimisation based) to
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generate further improved flexible WDS designs.
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CASE STUDIES
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The flexibility valuation methodology presented in this paper has been illustrated on the New
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York Tunnels (Dandy et al., 1996) and the Anytown network (Walski et al., 1987) redesign
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problems. Figures 6-9 and Figures 11-14 show results for both networks that demonstrate the
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performance differences between deterministic and flexible design plans over the WDS planning
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horizon. The literature solutions for both networks were used in the current study as the basis for
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analysis since they were optimised for the deterministic water demand at the end of the planning
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horizon which is also used as the mean value in the current study. Two deterministic plans were
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considered, namely, the precautionary and the staged deterministic approaches. The
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precautionary implementation of the least-cost solution means all the interventions of the design
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are carried out at the beginning of the planning horizon. As for the staged approach,
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interventions were distributed across the planning horizon based on the critical demand nodes of
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the networks. Interventions that lead to the critical demand nodes were meant to be implemented
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at earlier stages.
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Data Analyses and Assumptions
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For both of the networks considered in this study (defined in the next section), a planning
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horizon of 60 years with three 20-year decision making intervals was used for analysis. The
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planning period is considered to be suitable for analysis of long-term WDS design and the
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associated climate change / urbanisation effects.
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Kapelan et al. (2005) assumed nodal demands as uncertain variables that follow a Gaussian PDF
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with a mean value equal to the deterministic value and a standard deviation equal to 10% of the
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mean value of the New York Tunnels network problem. Burovskiy et al. (2011) made similar
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assumption to Kapelan et al. (2005) but with 30% of the deterministic demands on the Anytown
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network problem. In a similar manner, Gaussian PDFs are assumed with standard deviations of
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5%, 10% and 20% of the mean water demand values (referred to as Case 2) at years 20, 40 and
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60, respectively, for both the New York Tunnels and the Anytown network solutions considered
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in the present study. These standard deviations reflect the fact that demands are more uncertain if
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projected for longer into the future. The initial water demand for each network was assumed to
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be known and it was fixed at a value which after a 30% increment results in the end of the
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planning horizon mean demand (the latter value being defined in the literature). The initial
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system water demands are 1552 ft3/s (43.9 m3/s) and 7538 gallons/minute (0.48 m3/s) for New
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York Tunnels and Anytown networks, respectively. The initial water demand is assumed to
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increase exponentially until the end of the planning horizon demand. Also, with regard to spatial
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distribution, all the nodal demands in the system are assumed to increase at the same rate
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(percentage). Due to the standard deviation assumptions made in this study, two more Cases
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(+25% and -25% of the Case 2 standard deviations across the planning horizon) were also
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investigated for sensitivity analysis. These two more cases are referred to as Cases 1 and 3,
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respectively (see Figure 4).
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[Insert Figure 4]
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As for the cost, time preference is accounted for by using present value analysis. Many water
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utilities adopt a discount rate close to the capital cost which is around 6-8% (Wu et al., 2010). In
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this study a discount rate for the cost of implementing interventions at different times as depicted
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by the solution plans across the planning horizon is 6%. Due to the assumption made here, other
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discount rates used for sensitivity analysis are 4.5% and 7.5%. The study considered the capital
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costs of pipes, tanks and pump operation only as they have a major contribution in the WDS
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design expenses. The additional costs components such as maintenance, labour, water loss, etc.,
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will unnecessarily obscure the intended purpose of the approach presented here. These additional
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aspects are common factors that apply to both staged deterministic and the flexible methods,
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which are more important in the comparison made in this study. A total of 10,000 Monte Carlo
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simulations of uncertain future water demand scenarios were used in the analysis of all the
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design plans presented in this study.
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The New York Tunnels Problem
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Description
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The network has a single source (i.e. reservoir), 19 demand nodes and 21 pipes (see Figure 5).
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There is a minimum head requirement of 255 feet (77.72 m) at all demand nodes except node 16
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and 17 that have head requirements of 260 feet (79.25 m) and 272.8 feet (83.15 m), respectively.
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The reservoir at node 1 has a fixed head of 300 feet (91.44 m). In the current study, the flexibility
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evaluation indicators are the total cost and the WDS resilience. The means of rehabilitation is to
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duplicate existing pipes with new ones at different design decision points as water demand
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evolves. An existing pipe can only be duplicated once across the entire planning horizon. In the
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present analysis, deterministic (precautionary and staged) and flexible design plans are evaluated
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separately under uncertain water demands.
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The precautionary intervention plan is based on the least-cost solution identified by Maier et al.
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(2003) by assuming that all interventions are implemented in the first stage of the planning
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horizon. The corresponding staged deterministic intervention plan was created by distributing the
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individual interventions of the aforementioned least-cost design across the planning horizon, in a
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way which is compatible with the forecasted demand increase and critical nodes. Finally, the
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flexible plan was created by adding or keeping the previous interventions on the decision tree
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paths that correspond to the possible higher and lower future demand forecasts. The flexible
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plan, which is intended to envelope uncertain demands in each stage, was created by including
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all and also adding more interventions (i.e. along higher demand paths) to those that were
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selected for the staged deterministic plan. It is important that the flexible designs created are
338
path-dependent and irreversible as explained in the methodology section. Along the lower
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demand paths, interventions from the previous stages are kept in the consecutive stages without
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any additional system reinforcement (i.e. do nothing). The threshold values were set manually in
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the example designs evaluated in this paper. For example, in Figure 6, in design stage 1, if we
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put a threshold demand of 1665 ft3/s (47.1 m3/s) this means that of the 10, 000 MC system
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demands simulated around an average deterministic value of 1723 ft3/s (48.8 m3/s), in
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approximately 75% cases, the demand value will be higher than 1665 ft3/s (47.1 m3/s). The
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percentage is determined by the relevant demand PDF defined at the end of stage 1. The higher
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demands are used to evaluate the more reinforced systems while those that are equal to or lower
347
than the threshold demand are used for less reinforced systems. The same evaluation procedure
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applies to the consecutive stages. The system performance at each design stage is averaged
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across all 10, 000 samples.
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The three intervention plans generated this way are shown in Figures 6 and 7. We acknowledge
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the fact that the staged and flexible intervention plans generated may not be optimal (hence
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potentially disadvantaged in that sense when compared to the precautionary plan) but are still
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comparable. The reason for this is that if staged and / or flexible plan show improvement over
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the precautionary one (in terms of average cost and / or resilience) this improvement can only be
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further increased if these two plans are optimised.
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The cost of pipe duplication intervention for the analysed plans in US dollars ($) is given by:
N
359
Average Total Cost =
np
1
∑
1.1Dj 1.24 Lj
(1+r)ti j=1
s ∑S
∑k=1
i=1
Ns
360
16
(6)
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where pipe diameter Dj and length Lj are in inches and feet, respectively. Note that this is an
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original cost model for the New York Tunnels network (see CWS, 2013a), which has been
363
extended to account for staged designs.
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[Insert Figure 5 here]
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Results and discussion
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The intervention plans’ evaluation results based on the previously explained methodology are
369
presented in Figures 6-9. The flexible, precautionary and staged deterministic design plans are
370
compared in terms of average cost and average resilience index.
371
372
The precautionary plan of the New York Tunnels problem in Figures 6 and 7 shows that all the
373
interventions (6 duplication pipes) are carried out in the first design stage, as it is normally done
374
in this type of approach. The duplication pipes selected are shifted towards improving available
375
head at high demand and far downstream nodes due to high headloss from source to critical
376
nodes (16, 18, 19 and 20) in the network. Also, it can be seen that the solution duplicates pipe 7
377
that has a relatively small (132 inches) existing diameter. The rest of the duplicated pipes (16,
378
17, 18, 19 and 21) have smaller diameter sizes of 72, 72, 60, 60 and 72 inches, respectively. For
379
the corresponding staged deterministic plan, the interventions were selected based on critical
380
nodes. For example, pipe 19 is duplicated with a 72 inches pipe in the first stage of the staged
381
deterministic plan. Pipe 19 conveys water to node 20 which has a high mean demand at the end
382
of the first design stage. It is also known that node 20 is one of the highest demand nodes in the
383
network with 170 ft3/s at the end of the planning horizon. The corresponding flexible design
17
384
interventions were selected as an attempt to add more reinforcement on the interventions that
385
were selected for the staged deterministic interventions. For example, in Figures 6 and 7 design
386
stage 2, pipes 16, 19 and 21 that have been selected in the staged deterministic plan are also part
387
of the alternative routes’ interventions in the flexible plan.
388
389
The solution evaluation results obtained indicate that there is value associated with flexibility in
390
the implementation of interventions across the WDS planning horizon. This is evidently
391
demonstrated in Figure 6 by the flexible intervention plan that shows a similar average total cost
392
(slightly lower) to the staged deterministic solution but has clearly higher end resilience. A
393
flexible plan which has an average total cost of $9.62 M results to an average resilience index of
394
0.441 while the deterministic has an end average resilience of 0.402 with costs of $38.64 M and
395
$9.67 M for precautionary and staged design, respectively.
396
397
Figure 7 shows a flexible plan that outperforms the resilience equivalent deterministic plan in
398
terms of average total cost. They all have similar resilience index of 0.402 but cost $7.79 M,
399
$38.64 M and $9.67 M for flexible, precautionary and staged deterministic plans, respectively.
400
The total cost or the end resilience difference between the deterministic and the flexible
401
intervention plans is the engineering value added by flexibility. For example, the $1.88 M
402
difference between the staged deterministic and the flexible approach reflects the cost reduction
403
that is introduced by the flexible plan. This reduction results due to the capability of the flexible
404
design plan to adapt to uncertain water demand. This capability means that the flexible approach
405
can effectively avoid implementing further interventions altogether if the future water demand
406
turns out not to increase (much) further. The deterministic intervention plans lack this attribute
18
407
which means considerable economic and WDS resilience values can be left unexploited. The
408
capability to adapt the WDS configuration when and where necessary allows postponing (or
409
avoiding) design intervention measures which, in turn, results in lower average present value
410
cost of the flexible design. Intervention measures implemented in future is an attribute of both
411
the staged deterministic and flexible plans but the latter allows for more interventions to be
412
possibly implemented in a certain route. This feature raises the average resilience but still
413
maintain a comparable average total cost.
414
415
[Insert Figure 6 here]
416
[Insert Figure 7 here]
417
418
Figures 8 and 9 show the profile of average resilience and the cumulative cost of the
419
precautionary, staged and the flexible intervention plans for Case 2. Table 1 provides additional
420
information of intervention plans and their respective stage cost and resilience performance. End
421
average resilience information resulting from all scenarios considered in this study is shown in
422
Table 2.
423
424
The least-cost design solution for New York Tunnels problem is $38.64 M (Maier et al., 2003).
425
The precautionary plan for this solution has a higher cost than both the staged deterministic and
426
the flexible plan that explains its higher initial average resilience (0.564). However, its average
427
resilience reduces because of the rising future demands until it intersects with the average
428
resilience (0.402) of the developmental deterministic plan at the end of the planning horizon.
429
This fact happens despite that the precautionary plan has a higher average total cost. The staged
19
430
deterministic design shows lower average total cost because of the discounted cost of future
431
interventions. The cost equivalent flexible plan’s average resilience starts off at a lower level
432
(0.243) with a lower initial cost ($2.40 M) than the staged deterministic plan ($3.18 M). This
433
happens due to the fact that the interventions are the same at that stage but the flexible plan
434
allows for doing nothing which explains the lower average total costs and average resilience
435
values.
436
437
In the second stage (at 40 years into the future) the cost equivalent flexible plan outperforms the
438
staged deterministic approach due to the possibility to take intervention paths with additional
439
system reinforcement if water demand is high. In this second stage, the cost equivalent flexible
440
solution has an average resilience of 0.167 and an average total cost of $4.08 M compared to the
441
average resilience of 0.149 and an average total cost of $4.42 M for the staged deterministic
442
approach.
443
444
More importantly, the cost equivalent flexible design plan outperforms both the precautionary
445
and the staged deterministic intervention plans in terms of the average resilience index at the end
446
of the planning horizon even though it has a similar or lower average total cost. The end
447
resilience equivalent solution also stresses that the same average resilience to the staged
448
deterministic approach can be achieved at lower average total cost. All these results reveal the
449
consequence of the built-in flexibility to adapt to changes in the future demand.
450
451
[Insert Figure 8 here]
452
[Insert Figure 9 here]
20
453
454
Table 2 provides the sensitivity analysis results of the performance of WDS in terms of average
455
resilience with varying level of demand uncertainty. The results indicate that the relative (i.e.
456
percentage) increase of average resilience from the deterministic to the cost equivalent flexible
457
plan increases with more water demand uncertainty. For example, an increase of an average
458
resilience at the end of the planning horizon for a comparable cost equivalent flexible
459
intervention to the deterministic plans is from 7.3% (Case 1) to 12.5% (Case 3). These average
460
resilience increment differences confirm that uncertainty does not always need to be avoided but
461
it can be an opportunity to be exploited (De Neufville, 2003).
462
463
The sensitivity analysis results for varying discount rates in Table 3 (cost equivalent flexible plan
464
only) show that reducing cost discount rate from 6% to 4.5% leads to 31.5 % increase of the
465
average total cost in the case of the staged deterministic solution. The increase is lower than
466
39.2% obtained from the flexible plan. Again, the flexible approach is more sensitive than the
467
staged deterministic plan when the discount rate is increased to 7.5%. The average total cost of
468
the flexible design is reduced by 24.4 % while in the case of staged deterministic plan is 20.4%.
469
This is a consequence of optional intervention paths that result in wider range of possible costs.
470
471
[Insert Table 1 here]
472
[Insert Table 2 here]
473
474
The Anytown Network Problem
475
Description
21
476
The Anytown WDS (Figure 10) was originally set up by Walski et al. (1987) as a realistic
477
example of a more challenging WDS rehabilitation problem even though it does not have all
478
features of real systems (e.g. multiple pressure zones, seasonal and local demand fluctuations,
479
fiscal constraints, uncertainty of future demands and pipe roughness, and complicated staging of
480
construction).
481
482
[Figure 10: The Anytown network]
483
484
The network consists of existing pipes in the central city (thick solid lines) that are difficult to
485
access making cleaning or pipe duplication more expensive. In the residential region (thin lines)
486
pipes are easier to access and therefore cheaper to clean or duplicate. For more details of
487
different costs involved in this problem see Walski et al. (1987). The dashed lines indicate the
488
new pipes for the planned extension to the north of the city which is also part of the network
489
rehabilitation. The network has two existing tanks. The treatment works is maintained at a fixed
490
level of 10 feet (3.05 m) and the two existing tanks operate with levels between 225 feet (68.58
491
m) and 250 feet (76.20 m). Water is pumped into the system from a nearby treatment plant by
492
three parallel identical pumps.
493
494
The objective of the problem is to determine the most economically effective solution to
495
reinforce the existing network to meet the future demands considering pumping (operational) and
496
capital costs. Rehabilitation options for each existing pipe include duplication, cleaning and
497
lining or do nothing. A pipe that has been cleaned and lined has a Hazen-Williams coefficient of
498
C = 125 compared to C = 130 for new pipes. New pipes can be chosen from a range of 10
22
499
possible diameters (6, 8, 10, 12, 14, 16, 18, 20, 24 and 30 inches). Any node (except node 1)
500
which is not already connected to the existing tank is considered as a potential location site of a
501
new tank. Each tank has an emergency volume and a normal operating volume. A maximum of
502
two new tanks each with its location, overflow elevation, normal day elevation, diameter and the
503
bottom elevation as decision variables are considered. The tank is connected to the demand
504
nodes by a riser pipe that also has to be sized. In addition to the rehabilitation of the network, the
505
operation schedule of the pumps for a typical day is to be selected. In order to optimise the
506
pumping schedule, design of new tanks that fill and empty over average daily flows and allow
507
for emergency flows make it difficult to choose between solutions since a number of solutions
508
can satisfy pressure requirements under average daily flows but the end-of-day tank levels may
509
differ from the start-of-day levels. Some of the solutions may satisfy the start and end-of-day
510
levels under average day flows but fail to satisfy the minimum required pressures under
511
instantaneous peak flows. The network solutions have to satisfy the minimum pressure and the
512
tanks level requirements to be feasible solutions.
513
514
To demonstrate the value of considering water demand uncertainty in the flexible design
515
approach, a 24 hour simulation (i.e. extended period simulation - EPS) with 1 hour hydraulic
516
time step for average day flows in each design stage considered is performed in the present
517
study. The WDS resilience is calculated based on the minimum pressure across the 24-hr
518
simulation. The total cost of an intervention plan is the cumulative capital cost of pipes, tanks
519
and the present value of pump operation over 60 years. The annual energy cost for pumping is
520
calculated by multiplying the energy used in a day (obtained from the EPANET 24-hr
521
simulation) by the unit cost of energy ($ 0.12 per kWh) and the number of days per year. These
23
522
annual costs are then discounted for each year of the design stage (i.e. time interval, e.g. 20
523
years) and these are then added up to estimate the total cost of energy at that stage. Under the
524
current study, the original Anytown network redesign problem described above has also been
525
considered for deterministic (precautionary and staged) and flexible designs across the planning
526
horizon. Furthermore, note that the Anytown network problem is normally solved for 5 operating
527
conditions (average day flow, instantaneous peak hour flow and three fire flows) (Walski et al.,
528
1987) but here, only the normal operation condition was analysed for simplicity in both
529
deterministic and the flexible design plans.
530
531
As it was done in the case of New York Tunnels, the precautionary intervention plan was created
532
by assuming that all interventions of the optimal solution identified by Farmani et al. (2005) are
533
occurring in the first stage. The staged deterministic intervention plan was generated by staging,
534
i.e. distributing aforementioned optimal interventions across the planning horizon and by leaving
535
out only pipe number 4 clean and line intervention since it is not necessary (the pipe has an
536
adequate roughness coefficient 130). However, cleaning and lining or doing nothing on pipe 4
537
has a negligible contribution in the solution objectives due to its short length. Also, three new
538
pipes (10, 14 and 16) were assumed to be already in place at the beginning of planning horizon
539
to avoid demand node isolation in the analysis. All three pumps were switched on for every hour
540
and a tank was set up at node 8. Finally, in each stage, the flexible plan was generated by
541
implementing all the staged interventions and adding more interventions in the higher demand
542
paths, all by using engineering judgment. Along the lower demand paths, interventions from the
543
previous stages are kept in the consecutive stages without any additional interventions. The
544
threshold values were also set manually in the flexible plans analyses by using engineering
24
545
judgment. The three intervention plans generated this way are shown in Figures 11 and 12. Even
546
though the design plans generated this way may not be optimal, the comparison still makes
547
sense. This is because of the fact that, if the flexible plan shows improvement over deterministic
548
plans in terms of average cost and / or resilience, there is only a potential to further enhance the
549
improvement if these two plans are optimised. Note also that the staged deterministic plan has
550
less possible solutions compared to the flexible plan which makes much better flexible solutions
551
difficult to achieve.
552
553
Results and Discussion
554
The design plan analysis results based on the new approach are presented in Figures 11-14 and
555
Tables 1-3. The results compare the two deterministic (precautionary and staged) and the flexible
556
approaches in terms of WDS average total cost and the average resilience index over the
557
planning horizon. Figures 11 and 12 show the precautionary, staged deterministic and the
558
flexible design interventions. In Figures 11 and 12, a cross through a pipe diameter (i.e. in the
559
first time-step) shows the diameter that is already in place (opposed to the original network
560
problem as explained in the network description). The shaded cells show design interventions
561
that are newly implemented in the time-step they are in. The precautionary approach means that
562
all the design interventions are implemented at the beginning of the planning horizon as in the
563
previous case study. The solution duplicates critical pipes. For example, pipes 1 and 2 convey
564
pumped water to the rest of the network. The selected developmental interventions (staged
565
deterministic and flexible) were meant to duplicate critical pipes in the earlier stages. For
566
example, pipes 1 and 2 are duplicated in the first design stage. The interventions in the staged
567
deterministic provide basis for the alternative routes in the flexible plan. This means that for any
25
568
given alternative design path, the interventions consist of at least the same intervention(s) as in
569
the staged deterministic approach. It can be observed that pipes 1, 2, 6, 26, 27, 29 and 30 have all
570
been selected in the first stage of both the staged deterministic and the flexible design plans.
571
572
Figure 11 confirms that a flexible intervention plan outperforms the deterministic design plans in
573
terms of end average resilience even though it has a similar or less cost than the latter. A flexible
574
plan, which has an average total cost of $8.42 M results in an end average resilience index of
575
0.164 while the two deterministic plans have an average end resilience of 0.117 with the costs of
576
$20.68 M and $18.43 M for the precautionary and staged designs, respectively. Figure 12 also
577
shows a flexible plan with less cost than the deterministic plans at equivalent end average
578
resilience. For an end average resilience of 0.117, a flexible plan has a cost of $17.79 M as
579
compared to the $20.68 and $18.43 M shown by the precautionary and the staged deterministic
580
plans, respectively. The differences mean there is value that has been derived from flexibility as
581
an opportunity presented by uncertainty. For example, the $0.64 M difference between the
582
staged deterministic and the flexible approach shows the potential cost reduction by the latter.
583
This reduction results due to the same reason to the previous case study that the flexible design
584
plan allows for postponement of design interventions up to such time that they would be needed.
585
In this case, the system avoids less desirable design for some water demands and captures the
586
more favourable ones. Flexible or conditional implementation of intervention measures in the
587
future can drastically increase WDS average resilience with an average total cost which is
588
similar or less than the deterministic intervention plans. In addition, design flexibility provides
589
for the possibility to implement additional interventions but still maintain the comparable
590
average total cost. This finding is attributable to the low initial cost and the future intervention
591
measures that have a lower present value cost.
26
592
593
[Insert Figure 11 here]
594
[Insert Figure 12 here]
595
596
Table 1 and Figures 13-14 display the average resilience and the cumulative average cost
597
performance profiles of the Anytown network design plans. A similar trend to the New York
598
network results indicates that in the initial stage, the deterministic approaches outperform the
599
flexible intervention plans in terms of average resilience which is explained by the fact that
600
flexible design approach has a ‘do nothing’ option which reduces the average system resilience
601
and cost. For example, in the initial stage, the staged deterministic has an average resilience of
602
0.147 and an average total cost of $13.59 M compared to the average resilience of 0.026 and the
603
average of 13.12 M for the cost equivalent flexible plan. With cumulative design interventions
604
that respond to uncertain demands as time passes on, the WDS average resilience at the end of
605
the planning horizon for a cost equivalent flexible intervention plan clearly outperforms both the
606
precautionary and the staged deterministic intervention plans in terms of average resilience.
607
Table 1 shows that the staged deterministic plan has an end average resilience of 0.117 whilst
608
the cost equivalent flexible approach has 0.164. The end resilience equivalent solution also
609
stresses the economic value that can be achieved by having the same resilience to the
610
deterministic approach in the flexible designs. Both deterministic plans (staged and
611
precautionary) show the potential of performing highly earlier on when water demand and
612
uncertainty is lower.
613
614
[Insert Figure 13 here]
27
615
[Insert Figure 14 here]
616
617
The sensitivity analysis in this case study confirms that the increase in uncertainty represented
618
by the standard deviation in water consumption (see Table 2) leads to higher percentage
619
increment of the average resilience index from the deterministic to the flexible plan. For
620
example, an increase of an average resilience at the end of the planning horizon for a comparable
621
flexible intervention to the staged deterministic plan is from 38.3% (Case 1) to 40.2% (Case 3).
622
These results also stress the point that there is value inherent in flexibility and the more uncertain
623
the future water demand is, the higher the value of flexible design. These also suggest that
624
uncertainty presents an opportunity that can be exploited.
625
626
In Table 3, the flexible design shows an increase of 23.5% whereas the staged deterministic
627
design has 23.3% when a lower cost discount rate of 4.5% is used. A similar trend is shown by a
628
higher discount rate of 7.5%, which shows a 15.8% and 15.4% reduction in the total cost of
629
flexible and staged deterministic designs, respectively. These cost increases’ differences mean
630
that the flexible design plan is more sensitive to cost discount rate than the staged deterministic
631
approach. This is a consequence of the built-in flexibility, i.e. optional intervention paths result
632
in wider range of possible costs. This finding implies that flexible WDS designs may not always
633
be the best choice, i.e. that discount rate for the long-term planning of WDS should be carefully
634
chosen.
635
636
The authors acknowledge that practicing engineers have been intuitively dealing with the issues
637
of staged and adaptive long-term planning of distribution systems based on multiple demand
638
scenarios. This study introduces formal concepts and techniques used in this context (e.g.
28
639
decision trees to represent flexible plans, MC simulation to evaluate flexible plans, etc.). The
640
reason why we are representing intervention plans using decision trees defined over the full
641
length of the planning horizon is to take the long term view, i.e. to make sure that what is
642
proposed for implementation in the near future (the first stage in this paper or next year for
643
practitioners) is compatible with different possible demand and other futures based on the current
644
best information available. This is consistent with Walski (2013) who also suggested that
645
decisions must be made in the short term but also fit into a long-term plan. Note that decision
646
trees can (and should) be updated as frequently as desired in the future, by using engineering
647
judgment or some other approach.
648
649
CONCLUSIONS
650
Methodologies that can identify design interventions that are adaptable to future climate and
651
urbanisation changes in the long-term planning of WDSs are essential. De Neufville (2004)
652
classified management of uncertainty in three ways as, controlling uncertainty by demand
653
management, protecting passively by building in robustness and lastly by protecting actively by
654
creating flexibility that managers can use to react to uncertainties. In this study, a methodology
655
that analyses the potential value of flexibility that is created in WDS design interventions under
656
uncertain water demand has been explored.
657
658
The new methodology was tested on two case studies based on the WDS redesign problem of
659
New York Tunnels and Anytown. Three alternative intervention plans, two deterministic (staged
660
and precautionary) and one flexible, were evaluated and compared against each other by
661
assuming uncertain future demands over some long-term planning horizon. The methodology
662
recognises uncertainty for flexibility purpose in the long- term planning of WDSs. The flexible
29
663
methodology gives managers opportunities to exploit the uncertain nature of water demand as a
664
design parameter. The approach allows the WDS to cope with the changes in water demand by
665
changing the actual design elements of the system when (and if) necessary.
666
667
The results obtained lead to the following conclusions:
668
1. It has been demonstrated that flexible WDS design (i.e. intervention plan) can have lower
669
economic cost and / or improved hydraulic performance (i.e. higher resilience) when
670
compared to the corresponding deterministic precautionary and staged designs (i.e.
671
intervention plans) under uncertain future water demand conditions. The value of
672
flexibility can be estimated as the difference in respective expected design costs (given
673
the same / similar design resiliencies) or as the difference in respective expected
674
resiliencies (given the same / similar design costs).
675
2. The value of flexibility comes from the ability of the flexible WDS design approach to
676
adapt the water distribution system to uncertain future water demands in a cost-effective,
677
resilient and timely manner. This is a consequence of the fact that flexible WDS design
678
approach allows both postponing (i.e. delaying in time) and implementing (or not
679
implementing) optional interventions that are compatible with future demands. The
680
staged deterministic WDS design allows only delaying interventions in time whilst the
681
precautionary approach does not allow either.
682
3. The flexible WDS design seems more sensitive to changes in the cost discount rate than
683
the staged deterministic plan. This is a consequence of the built-in flexibility, i.e. optional
684
intervention paths resulting in wider range of possible costs. This finding implies that
30
685
flexible WDS designs may not always be the best choice depending on the discount rate
686
used.
687
The above conclusions are based on the assumptions, data and cost models used in the two case
688
studies presented in this paper. Future work on larger, more complex WDSs is required to further
689
analyse and quantify the benefits of flexible WDS designs. Future work is also required to
690
identify the optimal staged and flexible design plans by formulating and solving the relevant
691
WDS optimisation problems.
692
693
Acknowledgements
694
This research work has been fully supported by a University of Exeter PhD scholarship, which is
695
gratefully acknowledged.
696
697
698
REFERENCES
699
700
Babayan, A., Kapelan, Z., Savic, D., and Walters, G. (2005). “Least-Cost Design of Water
701
Distribution Networks under Demand Uncertainty.” Journal of Water Resources
702
Planning and Management, 131(5) , 375-382.
703
Banos, R., Reca, J., Martínez, J., Gil, C., Márquez, A. L. (2011). “Resilience Indexes for Water
704
Distribution Network Design: A Performance Analysis Under Demand Uncertainty.”
705
Water Resources Management, 25, 2351-2366.
706
Burovskiy, P., Kapelan, Z., and Fu, G. (2011). “Stochastic Risk-Based Optimisation of Water
707
Distribution Systems with Unknown Water Demands.” Eleventh International
31
708
Conference on Computing and Control for the Water Industry proceedings, 83-88,
709
Exeter,UK: Centre for Water Systems, University of Exeter.
710
Centre for Water Systems (CWS), University of Exeter (2013a). (http://emps.exeter.ac.uk
711
/engineering/research/cws/resources/benchmarks/expansion/new-york-water.html)
712
(March 26, 2013).
713
Centre for Water Systems (CWS), University of Exeter (2013b). (http://emps.exeter.ac.uk
714
/engineering/research/cws/resources/benchmarks/expansion/anytown.html) (March 26,
715
2013).
716
Creaco E., Franchini M., Walski T. M. (2013). Accounting for phasing of construction within the
717
design of water distribution networks. Journal of Water Resources Planning and
718
Management (ISI), doi: 10.1061/(ASCE)WR.1943-5452.0000358, ASCE (ISSN: 1943-
719
5452).
720
721
722
723
724
725
Dandy, G. C., Simpson, A. R., and Murphy, L. J. (1996). “An improved genetic algorithm for
pipe network optimization.” Water Resources Research, 32(2), 449-458.
De Neufville, R. (2003). “Real Options: Dealing with Uncertainty in Systems Planning and
Design.” Integrated Assessment, 4(1) , 26-34.
De Neufville, R. (2004). “Uncertainty management for Engineering systems planning and
design.” Engineering Systems Symposium, MIT, Cambridge, MA.
726
Farmani, R., Walters, G. A., and Savic, D. A. (2005). “Trade-Off between Total Cost and
727
Reliability for Anytown Water Distribution Network.” Journal Of Water Resources
728
Planning and Management, 131(3), 161-171.
32
729
Giustolisi, O., Laucelli, D., and Colombo, A. F. (2009). “Deterministic versus Stochastic Design
730
of Water Distribution Networks.” Water Resources Planning and Management , 135(2),
731
117-127.
732
Huang, D., Vairavamoorthy, K., and Tsegaye, S. (2010). “Flexible Design of Urban Water
733
Distribution Networks.” World Environmental and Water Resources Congress 2010:
734
Challenges of Change, 4225-4236. ASCE.
735
Jayaram, N., and Srinivasan, K. (2008). “Performance-based optimal design and rehabilitation of
736
water distribution networks using life cycle costing.” Water Resources Research, 44,
737
W01417.
738
Kang, D., and Lansey, K. (2012). “Scenario-based Multistage Construction of Water Supply
739
Infrastructure.” World Environmental and Water Resources Congress 2012: Crossing
740
Boundaries, 3265 – 3274, ASCE.
741
742
Kapelan, Z., Savic, D., and Walters, G. (2005). “Multiobjective design of water distribution
systems under uncertainty.” Water Resources Research, 41, W11407.
743
Lansey, K. E., Duan, N., Mays, L., and Tung, Y. K. (1989). “Water Distribution System Design
744
under Uncertainties.” Journal of Water Resources Planning and Management, 115(5),
745
630-645.
746
Maier, H. R., Simpson, A. R., Zecchin, A. C., Foong, W. K., Phang, K. Y., Seah, H. Y., and Tan,
747
C. L. (2003). “Ant Colony Optimization for Design of Water Distribution Systems.”
748
Journal of Water Resources Planning and Management , 129(3), 200-209.
749
Prasad, T. D., and Park, N.-S. (2004). “Multiobjective Genetic Algorithms for Design of Water
750
Distribution Networks.” Journal of Water Resources Planning and Management , 130(1),
751
73-82.
33
752
Rossman, L. (2000). Epanet2 users manual. Washington, D.C.: USEPA.
753
Savic, D. A., and Walters, G. A. (1997). “Genetic Algorithms for Least-cost design of Water
754
Distribution Networks.” Journal of Water Resources Planning and Management, 123(2),
755
67-77.
756
757
758
759
Todini, E. (2000). “Looped water distribution networks design using a resilience index based
heuristic approach.” Urban Water, 2 , 115-122.
Walski, T. M. (2013). Long-term Water Distribution Design, World Environmental and Water
Resources Congress 2013: Showcasing the Future, 830 - 844, ASCE.
760
Walski, T. M., Brill, E.D., Jr., Gessler, J., Goulter, L.C., Jeppson, R.M., Lansey, K., Lee, H.-L.,
761
Liebman, J.C., Mays, L., Morgan, D.R., and Ormsbee, L., 1987. “Battle of the network
762
models: epilogue.” Water Resources Planning and Management, 113 (2), 191–203R.
763
Wang, T., and De Neufville, R. (2004). “Building Real Options into Physical Systems with
764
Stochastic Mixed-Integer Programming.” Real Options Annual International Conference,
765
17-19 June 2004, Montreal, Canada.
766
Wu, W., Maier, H. R., and Simpson, A. R. (2010). “Single-Objective versus Multiobjective
767
Optimization of Water Distribution Systems Accounting
768
Emissions by Carbon Pricing.” Water Resources Planning and Management, 136(5), 555
769
- 565.
770
771
for
Greenhouse
Gas
Xu, C., and Goulter, I. C. (1999). “Reliability-Based Optimal Design of Water Distribution
Networks.” Journal of Water Resources Planning and Management, 125(6), 352-362.
772
773
774
775
776
777
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778
779
780
List of Tables
781
Table 1: Design Plan end of Stage Average Costs and Average Total Costs (cumulative) for
782
783
784
785
786
New York Tunnels and Anytown networks (Case 2)
Table 2: Cost Equivalent Design Plan Analysis for varying Standard Deviations (New
York
Tunnels and Anytown networks)
Table 3: Cost Equivalent Design Plan Analysis for varying Cost Discount Rates (New York
Tunnels and Anytown networks) (Case 2)
35
Table 1: Design Plan end of Stage Average Costs and Average Total Costs (cumulative) for
New York Tunnels and Anytown networks (Case 2)
SOLUTION PLAN
DESIGN
TIME
NEW YORK TUNNELS
Average
Cost
ANYTOWN NETWORK
Average
Total Cost
(cumulative)
($M)
Average
Resilience
Index
(-)
0
0
0
0
0.564
38.64
16.23
0.164
16.23
0
0.468
38.64
3.35
0.172
19.57
tend
0
0.402
38.64
1.11
0.117
20.68
t0
0
0.310
0
0
0
0
t1
3.18
0.266
3.18
13.59
0.147
13.59
t2
4.42
0.149
7.60
3.53
0.169
17.12
tend
2.07
0.402
9.67
1.31
0.117
18.43
Flexible
t0
0
0.310
0
0
0
0
(cost equivalent)
t1
2.40
0.243
2.40
13.12
0.026
13.12
t2
4.08
0.167
6.47
4.24
0.169
17.36
tend
3.15
0.441
9.62
1.06
0.164
18.42
Flexible
t0
0
0.310
0
0
0
0
(resilience
equivalent)
t1
0.95
0.195
0.95
13.12
0.026
13.12
t2
3.96
0.137
4.91
3.55
0.035
16.67
tend
2.88
0.402
7.79
1.12
0.117
17.79
Precautionary
Staged Deterministic
Average
Total Cost
(cumulative)
($M)
Average
Resilience
Index
(-)
t0
0
0.310
t1
38.64
t2
Average
Cost
Note: Average Cost at any given time refers to the cost incurred before the corresponding design time
36
Table 2: Cost Equivalent Design Plan Analysis for varying Standard Deviations (New York
Tunnels and Anytown networks)
SOLUTION PLAN
STANDARD
DEVIATION
NEW YORK TUNNELS
Average
Total Cost
Precautionary
Staged Deterministic
Flexible
% Increase
from a Staged
Deterministic
to a Flexible
Approach
ANYTOWN NETWORK
(Case)
($M)
Average End
Resilience
Index
(-)
1
38.64
0.409
20.67
0.120
2
38.64
0.402
20.68
0.117
3
38.64
0.392
20.69
0.112
1
9.67
0.409
18.43
0.120
2
9.67
0.402
18.43
0.117
3
9.67
0.402
18.45
0.112
1
9.62
0.439
18.41
0.166
2
9.62
0.441
18.42
0.164
3
9.62
0.441
18.43
0.157
1
-
7.3
-
38.3
2
-
9.7
-
40.2
3
-
12.5
-
40.2
37
Average
Total Cost
($M)
Average End
Resilience
Index
(-)
Table 3: Cost Equivalent Design Plan Analysis for varying Cost Discount Rates (New York
Tunnels and Anytown networks) (Case 2)
SOLUTION PLAN
DISCOUNT
RATE
NEW YORK TUNNELS
Average
Total Cost
Precautionary
Staged Deterministic
Flexible
ANYTOWN NETWORK
(%)
($M)
Average End
Resilience
Index
(-)
Average
Total Cost
($M)
Average End
Resilience
Index
(-)
4.5
38.64
0.402
24.76
0.117
6
38.64
0.402
20.68
0.117
7.5
38.64
0.402
17.97
0.117
4.5
12.72
0.402
22.73
0.117
6
9.67
0.402
18.43
0.117
7.5
7.70
0.402
15.59
0.117
4.5
13.39
0.441
22.75
0.164
6
9.62
0.441
18.42
0.164
7.5
7.27
0.441
15.51
0.164
4.5
+31.5
-
+23.3
-
7.5
-20.4
-
-15.4
-
4.5
+39.2
-
+23.5
-
7.5
-24.4
-
-15.8
-
% Increase (+) / reduction (-)
Staged deterministic
Flexible approach
38