Scholarly article on topic 'A Graph-based Analytical Technique for the Improvement of Water Network Model Calibration'

A Graph-based Analytical Technique for the Improvement of Water Network Model Calibration Academic research paper on "Earth and related environmental sciences"

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Abstract of research paper on Earth and related environmental sciences, author of scientific article — Sophocles Sophocleous, Dragan Savic, Zoran Kapelan, Yibo Shen, Paul Sage

Abstract Correctly calibrated water distribution network models are valuable assets for water utilities. Among possible uses of hydraulic modelling, the detection and location of leakage hotspots are important operational considerations, with companies often spending large sums of money finding leaks, but many remaining undetected. For a more reliable modelling and calibration process, water utilities need to ensure that asset state and status is accurate. The paper considers a new graph-theory based technique, called pipe tree analysis, for clustering water distribution networks. The aim is to reduce the calibration problem size for leakage hotspot detection and to establish a foundation for improved model quality assurance. The pipe tree topological analysis is applied to divide the “Anytown” network from literature, into different pipe trees and combined with model pre-processing, to reduce the solution search space. A Genetic Algorithm is, then, used to solve the optimization problem of searching for calibration parameters values, while minimizing the differences between observations and model predictions. The new modelling method highlighted important calibration parameters and contributed to successful detection of model anomalies, such as unknown closed valves and leakage hotspots, providing additional benefits to optimisation-based calibration.

Academic research paper on topic "A Graph-based Analytical Technique for the Improvement of Water Network Model Calibration"

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Procedía Engineering 154 (2016) 27 - 35

Procedía Engineering

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12th International Conference on Hydroinformatics, HIC 2016

A graph-based analytical technique for the improvement of water

network model calibration

Sophocles Sophocleousa*, Dragan Savica, Zoran Kapelana, Yibo Shenb, Paul Sagec

aCentre for Water Systems, University of Exeter, North Park Road, Exeter, EX4 4QJ, UK bSevern Trent Water Ltd, Stratford Road, Longbridge, Warwick, CV34 6QW, UK cWITSConsult Ltd, Milton Rough, Acton Bridge, Northwich, CW8 2RF, UK

Abstract

Correctly calibrated water distribution network models are valuable assets for water utilities. Among possible uses of hydraulic modelling, the detection and location of leakage hotspots are important operational considerations, with companies often spending large sums of money finding leaks, but many remaining undetected. For a more reliable modelling and calibration process, water utilities need to ensure that asset state and status is accurate. The paper considers a new graph-theory based technique, called pipe tree analysis, for clustering water distribution networks. The aim is to reduce the calibration problem size for leakage hotspot detection and to establish a foundation for improved model quality assurance. The pipe tree topological analysis is applied to divide the "Anytown" network from literature, into different pipe trees and combined with model pre-processing, to reduce the solution search space. A Genetic Algorithm is, then, used to solve the optimization problem of searching for calibration parameters values, while minimizing the differences between observations and model predictions. The new modelling method highlighted important calibration parameters and contributed to successful detection of model anomalies, such as unknown closed valves and leakage hotspots, providing additional benefits to optimisation-based calibration.

CrownCopyright© 2016Publishedby ElsevierLtd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of HIC 2016

Keywords: Hydraulic modeling; Graph theory; calibration; topology

* Corresponding author. Tel.: +447927928661 E-mail address: ss694@exeter.ac.uk

1877-7058 Crown Copyright © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of HIC 2016

doi:10.1016/j.proeng.2016.07.415

1. Introduction

Quantification and localization of leaks within water distribution networks (WDNs) are of significant importance to a water utility both from operational and planning perspectives, as well as from reputation point of view. Continuous improvements on leakage detection and control are being applied based on the use of new available technologies. However, the whole leakage localization process may still have shortfalls in speed of detection (i.e., weeks, months) with a significant volume of water being lost before the leak is found [1]. To avoid these inconveniences, leakage detection based on mathematical models may be used by "comparing" and analysing the network monitoring data, with the network model simulated outputs [2]. Accurate determination of network parameters is usually associated with some system problems caused by incorrect pipe roughness, incorrect nodal demands and uncertain valve statuses [3]. Additional problems also arise when inconsistencies caused by pressure recoveries just downstream of pressure reducing valves are overlooked in the modelling process. Therefore, the simulated model results have some discrepancy with the pressure and flow measurement in networks. As a consequence, correctly calibrated water distribution network models are valuable assets for water utilities to perform reliable model simulations for maintaining and operating real-world WDNs [4].

Calibration consists of determining various model parameters, that, when input into a hydraulic simulation model, will yield a reasonable match between measured and predicted pressures and flows in the network [5]. Savic et al. [6] reviewed the WDN calibration problem and various solution approaches. Calibration approaches are classified into trial-and-error procedures, explicit methods and implicit models. Implicit methods are currently in widespread use through formulating an objective function that is expressed as the differences between observed and simulated head and flow parameters, and is aimed to be minimized. Any and all input data that have associated uncertainty in value are considered as candidates for adjustment during calibration. This is to obtain reasonable agreement between model-simulated hydraulics and actual field behaviour, which makes calibration a significantly complex task. Furthermore, the widely available and improved information of both the topological representation of the network (e.g., actual number and position of service connections) and related boundary conditions (e.g., demands) has considerably increased the size of the hydraulic model and, as a result, the complexity of network analysis. From the calibration perspective, the inverse problem is often ill-posed and, thus, the larger model complexity has a major impact on system observability (e.g., if system heads and flows can be estimated) and identifiability (e.g., if system parameters can be calibrated). Thus, in order to deal with the complex task of calibration a systematic approach to reduce the problem size is required.

The paper considers a new graph-theoretic methodology for the analysis and clustering of the pipe network topology. Combined with model pre-processing the ultimate aim is to present an integrated methodology for narrowing down the search for leakage hotspots in the network, providing additional benefits to the model calibration problem. Using pipe flow analysis as well as concepts from graph theory together with a Genetic Algorithm an optimization-based calibration problem is solved. By using this approach controllability and identification of calibration parameters is enhanced, while the solution search space is reduced. The paper is organized as follows: section 2 provides literature review on approaches towards network topology analysis and simplification, section 3 describes the graph-based pipe tree approach and the calibration methodology, section 4 presents the case study, section 5 compares and discusses the calibration results for the Anytown case study network with and without considering the pipe tree network analysis, followed by conclusions.

2. Background

It is currently a common practice in hydraulic modelling that the WDN is represented using graph theory, with different calculations and data analysis techniques undertaken for the entire network. However, the majority of research that makes use of this theory has been focusing on reliability, or skeletonization/simplification of models [7]. More recently different topological/connectivity analysis tools for WDN analysis have been developed. This arose from the need to select optimal segmentations of the network associated to a number of technical reasons for planning, management and operation of water networks such as: (1) WDN analysis and model calibration, (2) planning for sensor placement, (3) valve design and monitoring arrangements with respect to water quality purposes, pressure management and leakage control. Deuerlein [8] presented a graph-theoretic approach to decompose the network, by considering connectivity and classifying single network components, which provided insight in WDN

structure and identification of different supply areas. Perelman and Ostfeld [9] used Depth First Search and Breadth First Search to divide the system into clusters according to the flow directions in pipes. The resulting analysis divided the network into strongly and weakly connected clusters. Giustolisi and Ridolfi [10] proposed a community detection approach, based on modularity index, which was used as a metric for the strength of the network division into segments. An optimization-based approach was used to maximize modularity index for the purpose of locating measuring devices.

The topology of WDNs is naturally composed of several sub networks, which are hydraulically connected or disconnected. Connectivity properties vary in time as a result of changes in dynamic loading conditions. The developed methodology herein suggests a partition of a WDN as a function of its structural and connectivity properties (i.e., topology and hydraulics) to different pipe trees, which could be interpreted as cluster structures of the network. The notion of pipe trees was firstly introduced by [11] as a means of improving hydraulic model calibration. The developed clustering algorithm provides an improved understanding of the main structure of the system and the connections between its components. The model capabilities are demonstrated on the Anytown WDN from literature.

3. Methodology

3.1. The Pipe Tree approach

Each network can be represented by a family of pipe trees. Identification of pipe trees for network analysis is based on an established a hierarchy in the modelled network according to the simulated pipe flow direction. The WDN can be represented as a graph G (V, E) using graph-theory, where vertices V of the graph represent network nodes and the connecting links (i.e., pipes, pumping units and valves) as the edges E. The links are directed defining a positive direction of flow in each link established from the steady-state or extended period simulation of network hydraulics. When link flow is below an arbitrary low threshold value, defining a dead link, the edge is set as undirected. In EPS analysis used, the configuration of the network topology is time dependent. By taking simulated flow direction into consideration, each link element has a "from" and "to" node. From the frequency of occurrences as "from" and/or "to" nodes, all nodes are categorized into different classes. The resulting list classifies nodes as "source", "sink", "reticulation", "convergence", "conveyor" and a "dead end". A pipe tree begins at a "source" node and will end at a node where flows converge or at a downstream "sink" node. "Reticulation" nodes occur where flow passes through the node or splits at pipe branches. New pipe trees also begin from converging flow nodes obeying the same principle. Following nodal classification, the Breadth First Search (BFS) algorithm is used to explore the network and establish the topological order of how water reticulates from node to node, starting with the source node and up to the farthest network node. It begins at a real source node, such as reservoir or tank, and explores all the adjoining nodes and, then, for each of those nearest nodes it explores all their unexplored adjoining nodes, continuing until there are no more adjacent unvisited nodes. During BFS each node is assigned a number corresponding to number of topological steps done in exploring the network, or in other words the water reticulation distance away from the source. Then, Depth First Search (DFS) algorithm is used to explore the connectivity of the graph. The algorithm starts again at a real source node, traversing in the direction of the outgoing edges as far as possible before reaching a "convergence" node or "sink" node, where a pipe tree will end. At each step the procedure continues by choosing an unexplored link of the recently reached node. The DFS step completes when all edges and nodes of a pipe tree are explored. The DFS search finishes when all nodes of the graph have been visited. Individual pipe trees are constructed to represent a tree structure. They can comprise of at least a trunk pipe element, but more complex trees may also include more pipes. According to flow these can be described as boughs, branches, twigs and twiglets (and beyond). Trees are ranked, in order of source descending flows and flows departing from each converging node into the next downstream tree.

3.2. Illustrative example

The proposed approach is illustrated using the simple network example shown in Figure 1. This network (Figure 1a) comprises 10 directed edges (e1-e10), and one undirected edge (e11), ten junction nodes (n1-n10), and one fixed grade node (sl). The width of each edge represents flow in each link. Fixed grade node sl is identified as the real source node, and is the beginning point of pipe tree analysis. Nodes n1 and n3 involve reticulation nodes with at least two links exiting those nodes and only one node entering them. Nodes n2, n4, n6 and n8 are conveyor nodes as only one node exits and enters those nodes, while n5 and n9 involve sinks as no edges exit the nodes. Finally, n7 is a convergence node with more than one links entering the node, while n10 is a dead end as no flow is transmitted to that location. A pipe tree will normally consist of trunks, boughs, branches, twigs and twiglets. However, in the simple example the formed trees are smaller. By using BFS, the water reticulation steps from the source are defined, with dead end nodes not included in the search (Figure 1b). Two trees are constructed following DFS. This is achieved by exploring the network while taking sequence levels assigned at each node into consideration as well as their class (Figure 1c). Tree 1 begins at s1 and reticulates up to n7, comprising of edges el-e8 and nodes s1, n1-n6. Tree 2 begins at n7 and ends at n9. It consists of branches e9 and e10 and nodes n7-n9.

Fig 1. The pipe trees analysis (a) Node classification according to flow direction; (b) BFS resulting water reticulation topological steps; (c) DFS result for network connectivity and pipe tree assignment. Differently coloured dashed lines illustrate exploration paths at each DFS step.

3.3. Calibration problem formulation

A MATLAB optimization code was developed for model calibration and was linked to EPANET2 tool kit. The optimization process uses a non-dominated sorting genetic algorithm II (NSGA II) [12]. Valve status, pipe roughness and leakage coefficients were considered as decision variables. The calibration was defined as a nonlinear optimization problem with the single objective to minimize the weighted sum of squared differences between the field observed and simulated values of nodal heads and pipe flows. The calibration problem was subject to two sets of constraints: (1) the set of implicit type constraints considering mass and energy balance equations; and (2) the set of explicit constraints used as bounds for the algorithm solution search space for each decision variable. The optimization problem is formulated as follows:

Search for: X = (sk t, Kfjf) k= 1,......,NK\ i = 1,....., Nl\ j = 1,........, N]\

Minimize: F{i) = ZLiCffli Wvh (^^M)2 + Z»;=1 Wnf ^W^C^ (2)

Subject to: sk t e {0,1} (3)

0 < K? < K«ax (4)

If < ff < ff (5)

Where X represents a set of model calibration parameters, sk t is the status of a link k at time step t, belonging to a vector with values 0 and 1, Kf1 is the emitter coefficient for leakage node i in demand group n with 0 and Kjnax being the minimum and maximum values the emitter coefficient for group n can take, f? is the roughness coefficient for pipe j in pipe group g with fg and fg being the upper and lower limits a roughness coefficient, NK is the number of candidate links to calibrate, N1 is the number of the candidate leakage nodes in node group n, N] is the number of candidate roughness groups, F(Jc) is the objective function to be minimized, corresponding to weighted (Wnh, Wnf) goodness-of-fit between the field observed values and the model simulated values for nodal heads (Hsnh — Honh) and pipe flows (Qsnf — Qonf), respectively.

3.3 Artificial field data generation using fire flow hydrants and step testing

A hydraulic simulation analysis was carried out in EPANET2 by considering the true state of the network (Figure 2), i.e., the calibrated model. This created an artificial set of field (i.e., observed) pressure and flow measurements, without accounting for noise. The artificial data were assumed as collected by means of planned hydrant discharges during night fire flow field tests (NFFFT), opened to cause a controlled hydraulic stress to the system. Water discoloration risks were also taking into consideration. Furthermore, planned closures of pipes near the hydrants were introduced while the hydrants were open, in order to cause controlled redirection of flow in the network and variation in velocities of pipes adjacent to hydrants. A restricted number of nodes and pipes of the network were assumed to have pressure observed and flow metered. The NFFFT observations were used in the calibration process, which is still being further developed.

3.4 Model pre-processing for optimization solution search space reduction

A topological analysis of the WDN network was performed using pipe trees and was combined with a sensitivity analysis, which provided insight to the topological observability of the parts of the network where observations were unavailable. From this, it becomes possible to highlight subsets of pipe and node elements that can be used as surrogate valves and leakage hotspots, respectively. Potential candidates can include pipes immediately upstream and downstream of converging flow nodes. Similarly, those at the start of major boughs and branches can be included. Snapshots of pipe tree analyses were taken during the time steps where hydrants were opened, as well as during peak and minimum demand conditions. From these, the main network reticulation routes were identified, as well as areas where flows converge, during different hydraulic situations. Based on the topological and sensitivity analyses the model was pre-processed, in order to have as few as possible calibration model parameters and avoid unnecessary simulation of solutions that do not cause any impact on model fitness. The candidate closed pipes and leakage nodes were restricted to pipes and nodes with large sensitivity value. Moreover, pipes with flow less than 1% of the inlet during peak demand conditions were excluded from the optimization, together with pipes and nodes that are located downstream from the last pressure/flow measurement device. This is because pressure and/or flow at those locations are insensitive to any change in state or status making the network component unobservable.

4. Network analysis and Calibration of Anytown Network

To developed methodology was applied to an example pipeline network. The aim was to calibrate the hydraulic model to predict system state and status variables as accurately as possible. The results were compared to the case where calibration does not use pipe tree analysis. Two optimization problems were solved with respect to link status (allowing for valves with unknown statuses in real networks), leakage emitter coefficients (allowing to locate unknown leakage hotspots) and grouped pipe roughness coefficients (allowing for generally unknown pipe roughnesses in real networks). The first calibration problem size was only reduced using sensitivity

Fig 2. Anytown Network Layout and true state

Table 1. Anytown Network Pipe and Node Data

Pipe Length Diameter Status HW roughness coefficient HW Node Elevation Emitter Coefficient

analysis and model preprocessing. The second problem considered pipe tree and sensitivity analysis combined with model pre-processing, in order to reduce the solution search space. The EPANET network layout of the system is shown in Figure 2. The WDN contains 16 junction nodes, 32 pipes, one pump and has one source. One pipe (1034) was closed, while three leakage hotspots were introduced at nodes 40, 80 and 130, with emitter coefficients of 1, 0.6 and 0.8, respectively. Furthermore, all pipes were set to the same roughness coefficient values. This was considered as the true system state for artificial field data generation. Two planned field tests, included in the EPANET model as nodal demands, were operated at nodes 100 and 140. Generated field test data was obtained from three locations (50, 90, 170) recording pressures every 15 minutes, while pump flow and flows from two mains (1064 and 1046) supplying the hydrants were also obtained. A total of 96 data sets over 24 hrs, from midnight to midnight, have been used. The model that was considered for calibration assumed all pipes are open and no leaks exist in the network, while the model reported roughness values represent the true pipe roughness. The network configuration data for both "true" and "assumed true" system state are given in Table 1. For both optimization problems, links and nodes that were monitored for flow and pressure, respectively, were assumed to have their state and status known and thus, were removed from the solution space. Following model pre-processing the population of candidate calibration parameters was further reduced. Links that lead to isolation, or significantly restrict supply to demand nodes if closed were removed along with pipes carrying minimal flow relative to the inlet link, during peak demand. This reduced the first calibration problem from 32 to 25 candidate closed pipes. Candidate leakage nodes were reduced to 10 nodes after taking sensitivity analysis and location of monitoring devices into consideration. However, when pipe trees analysis was considered, candidate parameters were further reduced, as described in the next section. In both situations a total of 28 pipes were candidates for roughness calibration, clustered into 3 groups. In this instance, pipe tree analysis did not provide further insight for removing additional pipes from the solution search space. The following GA parameters were used for multiple optimisation runs: population size of 50, 500 generations, binary tournament selection operator, random-by-gene mutation with the probability of 0.25 and single-point crossover with the probability of 0.90.

ID (m) (m) True Assumed True Assumed group ID (m) Assumed True

'1002' 3657 406 1 1 100 90 1 '20' 6.24 0 0

'1004' 3657 406 1 1 100 90 1 '30' 15.24 0 0

'1006' 3657 406 1 1 100 90 1 '40' 15.24 0 1

'1008' 2743 305 1 1 100 90 1 '50' 15.24 0 0

'1010' 1830 305 1 1 100 90 1 '60' 15.24 0 0

'1012' 1830 254 1 1 100 90 1 '70' 15.24 0 0

'1014' 1830 305 1 1 100 90 1 '80' 15.24 0 0.6

'1016' 1830 254 1 1 100 90 1 '90' 15.24 0 0

'1018' 1830 305 1 1 100 90 1 '100' 15.24 0 0

'1020' 1830 254 1 1 100 90 1 '110' 15.24 0 0

'1022' 1830 254 1 1 100 90 1 '120' 36.60 0 0

'1024' 1830 254 1 1 100 90 1 '130' 36.60 0 0.8

'1026' 1830 305 1 1 100 100 2 '140' 24.40 0 0

'1028' 1830 254 1 1 100 100 2 '150' 36.60 0 0

'1030' 1830 254 1 1 100 100 2 '160' 36.60 0 0

'1032' 1830 254 1 1 100 100 2 '170' 36.60 0 0

'1034' 2743 254 0 1 100 100 2 '500' 100.00 - -

'1036' 1830 254 1 1 100 100 2

'1038' 1830 254 1 1 100 100 2

'1040' 1830 254 1 1 100 100 2

'1042' 1830 203 1 1 100 100 2

'1044' 1830 203 1 1 100 110 3

'1046' 1830 305 1 1 100 110 3

'1048' 1830 203 1 1 100 110 3

'1050' 1830 254 1 1 100 110 3

'1052' 1830 203 1 1 100 110 3

'1056' 1830 203 1 1 100 110 3

'1058' 1830 254 1 1 100 110 3

'1060' 1830 203 1 1 100 110 3

'1062' 1830 203 1 1 100 110 3

'1064' 3656 203 1 1 100 110 3

'1066' 3656 203 1 1 100 110 3

'1082' 3657 406 1 1 100 100 -

5. Results and Discussion

5.1. Pipe Tree Analysis

Figure 3 demonstrates the pipe tree analysis of the network during peak and low demand periods. The main flow routes supplying the network can be distinguished with link width being proportional to flow, while the size of each convergence node is proportional to the demand. Such a representation provides insight to which links are the hydraulically significant and, thus, components that do not affect the network can be removed from the solution search space for closed pipes. During peak demand the network is divided into total of ten

pipe trees (Fig. 3a). Eleven nodes are reported as convergence nodes, however, only eight nodes where demand exists are plotted. During minimum demand the network divides further to a total of eleven pipe trees (Fig. 3b), but interestingly the number of convergence nodes reduces to ten. This is a result of low flows in pipes, resulting in more frequent situations of flow reversal. The smaller number of convergence nodes mainly results from the formation of Tree 6, which constitutes of five pipes, whereas during the peak demand case those five pipes belonged to three different trees. Again, only the eight convergence nodes with demand are plotted, which provides the opportunity to eliminate some nodes from the population of candidate leakage nodes. During both the peak and minimum demand periods, the largest tree formed is Tree 1, originating from the source and reticulating within the network. In the first instance it consists of 13 pipes and six nodes. Tree 1 is formed of a trunk, three boughs and several branches. During the case of minimum flow in the network the tree components reduce to 12 due a flow reversal in one of the right hand side branches causing it to form a part of Tree 10, along with the upstream pipe that, in the first case, formed a part of Tree 8. It is of note that candidate subsets for closed pipes and leakage nodes, can include the major boughs and branches, (which includes the closed pipe) as well pipes upstream and downstream of convergence nodes, including nodes on them. This means any pipe being a part of those tree components can be closed. This tree configuration contributed in finding the unknown closed pipe on the branch of Tree 1. However, in this instance, including the three boughs of tree 1 as candidate closed pipes, would unnecessary increase the solution search space as the sensitivity and hydraulic disturbance as a result of a change in status of those pipes is considerably high. The reverse case occurs if there is small leakage on the pipes carrying major flows. By taking a close look into the sensitivity of components, the pipe flows and nodal demands, engineering judgement was used to choose tree components as candidate calibration parameters. The components of the tree considered for optimization search involved those that caused a hydraulic effect significant to be sensed at the measuring device location, but also did not largely disturb demand at nodes, or lead to pipe velocities able to cause discoloration. In this case several nodes and pipes were removed from the solution space. Similar approach was followed for the rest of the network components. This lead to a calibration

Fig. 3 Pipe Tree analysis of the Anytown network during (a) peak and (b) minimum demand. Each pipe is denoted by the assigned tree number.

problem with 21 pipes and 6 nodes for status and leakage emitter calibration, respectively, along with the 28 grouped pipes for roughness calibration.

5.2. Calibration results

Table 2. Selected optimal solutions comparison between the two calibration techniques

Best Run Objective error 1008 1010 1012 1014 1016 1018 1022 1024 1026 1028 1030 I 1032 ^ 1034 3 1036 1038 1042 1046 1048 1050 1052 1058 1060 1062 1064 1066

Without Pipe Trees

1 2 3 4.274 4.636 5.108

20 0 0 0 - - -

30 0.32 0 0 - - -

40 0.6 0.32 0 0.8 0.5 0.8

60 0 0 0 - - -

70 0 0 0.8 0 0.2 0

80 0 1 0.23 0.8 0.9 0.8

110 0 0.25 0 - - -

120 0 0 0 0 0 0

130 1 0.8 0.25 0.8 0.8 0.23

160 0.5 0 1 0 0 0

Ctoup 1 95 95 95 100 100 100

Ctoup 2 120 95 95 100 100 105

Ctoup 3 105 105 105 100 100 100

With Pipe Trees

1 2 3 0.021 0.153 0.359 1 1 1

The best-fit solutions from the optimization runs have been selected and are presented in Table 2 giving the values of each calibration parameter. Each solution consists of the identified status of candidate pipes, the identified nodes of positive emitter coefficients, as well as the roughness coefficient of each pipe group. The network components with no values shown in the right part of Table 2, indicate that those were removed by pipe-tree analysis from the calibration parameters. Optimized calibration without considering pipe tree analysis resulted in a best solution with an objective error of F=4.27. The correct closed pipe was reported, however, falsely reported closed pipes such as "1018" was also obtained in the final solution. Moreover, there were errors in the detection of leakage node locations and their emitter values, as well as the pipe roughness of each pipe group. In most cases the larger solution space causes a number of parameter combinations to lead to solutions that result in a reasonably small objective error. A reason for this might be that the algorithm was trapped in local optima. This is a frequent issue WDN model calibration, which is often underdetermined, with observations being less than the number calibration parameters. In this example the leakage emitters were reported at nodes upstream of the true leak locations. From this, hydraulic balance and a smaller difference between observed and simulated values was achieved by a compensation provided by larger roughness values. On the other hand, when pipe tree analysis was used, the solutions were significantly fitter with the best one achieving an objective error of F=0.02. The optimization algorithm successfully detected the closed link and the correct pipe roughness, although in some cases pipes adjacent to the true closed pipe with similar hydraulic effect were falsely reported. As a consequence, an impact was observed on the pipe roughness solution values as well. All best solutions that considered pipe tree analysis were pointing to the correct leakage locations, although the exact emitter values could not be detected.

6. Conclusions

A topological analysis approach has been presented that divides the pipe network into different trees by establishing a hierarchy in the network. The approach has been formulated and applied to analyze an example network, but the same arguments can be directly applied to any other network. A reduction in optimization solution space for the calibrating the hydraulic model and detecting leakage hotspots was achieved, with the approach appearing to provide additional benefits towards calibrations problem complexity reduction. The method identifies major and

minor flow routes in any network leveraging the usage of the hydraulic model and provides a systematic method to eliminate calibration parameters, thus reducing the solution space. The artificial case study has been successfully used to test for the detection of model anomalies, such as unknown valve statuses and leakage hotspots. Further development of the approach can potentially lead to additional modelling and calibration benefits, required to provide the new generation model tools suitable for general use.

7. Acknowledgements

This work is part of the first author's STREAM Engineering Doctorate (EngD) project, based at the University of Exeter and sponsored by the UK Engineering and Physical Science Research Council (grant EP/L015412/1), Severn Trent Water Ltd and WITSConsult Ltd.

8. References

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